LMFDB - Embedded newform 1980.2.y.c.1297.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1980,2,Mod(1297,1980)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1980, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 1, 2]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1980.1297");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1980 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1980.y (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:	$15.8103796002$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 660)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1297.12
Character	$\chi$	$=$	1980.1297
Dual form			1980.2.y.c.1693.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.20524 + 0.369998i) q^{5} +(1.41964 - 1.41964i) q^{7} +O(q^{10})$	$q+(2.20524 + 0.369998i) q^{5} +(1.41964 - 1.41964i) q^{7} +(-0.476743 - 3.28218i) q^{11} +(-0.145122 - 0.145122i) q^{13} +(3.21300 - 3.21300i) q^{17} -5.69223 q^{19} +(0.957657 - 0.957657i) q^{23} +(4.72620 + 1.63187i) q^{25} +9.39080 q^{29} -2.39625 q^{31} +(3.65591 - 2.60538i) q^{35} +(-7.69393 - 7.69393i) q^{37} -11.5294i q^{41} +(-5.75826 - 5.75826i) q^{43} +(-2.21349 - 2.21349i) q^{47} +2.96927i q^{49} +(-2.25736 + 2.25736i) q^{53} +(0.163067 - 7.41441i) q^{55} +13.4763i q^{59} +6.44414i q^{61} +(-0.266335 - 0.373724i) q^{65} +(6.59960 + 6.59960i) q^{67} +6.84581 q^{71} +(8.61931 + 8.61931i) q^{73} +(-5.33630 - 3.98270i) q^{77} -6.14021 q^{79} +(4.69214 + 4.69214i) q^{83} +(8.27424 - 5.89664i) q^{85} -12.6882i q^{89} -0.412041 q^{91} +(-12.5528 - 2.10612i) q^{95} +(2.74044 + 2.74044i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 8 q^{5}+O(q^{10}) $ 24 * q - 8 * q^5	$ 24 q - 8 q^{5} - 8 q^{11} - 16 q^{23} + 24 q^{25} + 16 q^{31} - 40 q^{37} - 64 q^{47} - 24 q^{53} + 24 q^{55} + 48 q^{67} - 24 q^{77} - 48 q^{91} - 72 q^{97}+O(q^{100}) $ 24 * q - 8 * q^5 - 8 * q^11 - 16 * q^23 + 24 * q^25 + 16 * q^31 - 40 * q^37 - 64 * q^47 - 24 * q^53 + 24 * q^55 + 48 * q^67 - 24 * q^77 - 48 * q^91 - 72 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1980\mathbb{Z}\right)^\times$.

$n$	$397$	$541$	$991$	$1541$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	2.20524	+	0.369998i	0.986215	+	0.165468i
$5$	2.20524	+	0.369998i	0.986215	+	0.165468i
$6$		0			0
$7$	1.41964	−	1.41964i	0.536572	−	0.536572i	−0.385948	−	0.922520i	$-0.626126\pi$
$7$	1.41964	−	1.41964i	0.536572	−	0.536572i	0.922520	+	0.385948i	$0.126126\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.476743	−	3.28218i	−0.143743	−	0.989615i
$11$	−0.476743	−	3.28218i	−0.143743	−	0.989615i
$12$		0			0
$13$	−0.145122	−	0.145122i	−0.0402496	−	0.0402496i	0.686696	−	0.726945i	$-0.259061\pi$
$13$	−0.145122	−	0.145122i	−0.0402496	−	0.0402496i	−0.726945	+	0.686696i	$0.759061\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.21300	−	3.21300i	0.779266	−	0.779266i	−0.200440	−	0.979706i	$-0.564237\pi$
$17$	3.21300	−	3.21300i	0.779266	−	0.779266i	0.979706	+	0.200440i	$0.0642372\pi$
$18$		0			0
$19$	−5.69223			−1.30589			−0.652944	−	0.757406i	$-0.726466\pi$
$19$	−5.69223			−1.30589			−0.652944	+	0.757406i	$0.726466\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.957657	−	0.957657i	0.199685	−	0.199685i	−0.600180	−	0.799865i	$-0.704904\pi$
$23$	0.957657	−	0.957657i	0.199685	−	0.199685i	0.799865	+	0.600180i	$0.204904\pi$
$24$		0			0
$25$	4.72620	+	1.63187i	0.945240	+	0.326375i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	9.39080			1.74383			0.871914	−	0.489659i	$-0.162879\pi$
$29$	9.39080			1.74383			0.871914	+	0.489659i	$0.162879\pi$
$30$		0			0
$31$	−2.39625			−0.430379			−0.215189	−	0.976572i	$-0.569037\pi$
$31$	−2.39625			−0.430379			−0.215189	+	0.976572i	$0.569037\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	3.65591	−	2.60538i	0.617961	−	0.440390i
$36$		0			0
$37$	−7.69393	−	7.69393i	−1.26487	−	1.26487i	−0.948702	−	0.316172i	$-0.897602\pi$
$37$	−7.69393	−	7.69393i	−1.26487	−	1.26487i	−0.316172	−	0.948702i	$-0.602398\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	11.5294i		−	1.80058i	−0.435287	−	0.900292i	$-0.643353\pi$
$41$		−	11.5294i		−	1.80058i	0.435287	−	0.900292i	$-0.356647\pi$
$42$		0			0
$43$	−5.75826	−	5.75826i	−0.878127	−	0.878127i	0.115214	−	0.993341i	$-0.463245\pi$
$43$	−5.75826	−	5.75826i	−0.878127	−	0.878127i	−0.993341	+	0.115214i	$0.963245\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−2.21349	−	2.21349i	−0.322871	−	0.322871i	0.526997	−	0.849867i	$-0.323318\pi$
$47$	−2.21349	−	2.21349i	−0.322871	−	0.322871i	−0.849867	+	0.526997i	$0.823318\pi$
$48$		0			0
$49$			2.96927i			0.424181i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−2.25736	+	2.25736i	−0.310072	+	0.310072i	−0.844937	−	0.534866i	$-0.820362\pi$
$53$	−2.25736	+	2.25736i	−0.310072	+	0.310072i	0.534866	+	0.844937i	$0.320362\pi$
$54$		0			0
$55$	0.163067	−	7.41441i	0.0219880	−	0.999758i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			13.4763i			1.75446i	0.480067	+	0.877232i	$0.340612\pi$
$59$			13.4763i			1.75446i	−0.480067	+	0.877232i	$0.659388\pi$
$60$		0			0
$61$			6.44414i			0.825088i	0.910938	+	0.412544i	$0.135360\pi$
$61$			6.44414i			0.825088i	−0.910938	+	0.412544i	$0.864640\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−0.266335	−	0.373724i	−0.0330347	−	0.0463548i
$66$		0			0
$67$	6.59960	+	6.59960i	0.806269	+	0.806269i	0.984067	−	0.177798i	$-0.0568972\pi$
$67$	6.59960	+	6.59960i	0.806269	+	0.806269i	−0.177798	+	0.984067i	$0.556897\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	6.84581			0.812448			0.406224	−	0.913774i	$-0.366845\pi$
$71$	6.84581			0.812448			0.406224	+	0.913774i	$0.366845\pi$
$72$		0			0
$73$	8.61931	+	8.61931i	1.00881	+	1.00881i	0.999961	+	0.00885306i	$0.00281805\pi$
$73$	8.61931	+	8.61931i	1.00881	+	1.00881i	0.00885306	+	0.999961i	$0.497182\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−5.33630	−	3.98270i	−0.608128	−	0.453871i
$78$		0			0
$79$	−6.14021			−0.690827			−0.345414	−	0.938451i	$-0.612261\pi$
$79$	−6.14021			−0.690827			−0.345414	+	0.938451i	$0.612261\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	4.69214	+	4.69214i	0.515029	+	0.515029i	0.916063	−	0.401034i	$-0.131349\pi$
$83$	4.69214	+	4.69214i	0.515029	+	0.515029i	−0.401034	+	0.916063i	$0.631349\pi$
$84$		0			0
$85$	8.27424	−	5.89664i	0.897468	−	0.639580i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	12.6882i		−	1.34495i	−0.740120	−	0.672475i	$-0.765231\pi$
$89$		−	12.6882i		−	1.34495i	0.740120	−	0.672475i	$-0.234769\pi$
$90$		0			0
$91$	−0.412041			−0.0431936
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−12.5528	−	2.10612i	−1.28789	−	0.216083i
$96$		0			0
$97$	2.74044	+	2.74044i	0.278250	+	0.278250i	0.832410	−	0.554160i	$-0.186961\pi$
$97$	2.74044	+	2.74044i	0.278250	+	0.278250i	−0.554160	+	0.832410i	$0.686961\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	13.4635i		−	1.33966i	−0.742513	−	0.669832i	$-0.766366\pi$
$101$		−	13.4635i		−	1.33966i	0.742513	−	0.669832i	$-0.233634\pi$
$102$		0			0
$103$	7.22358	−	7.22358i	0.711760	−	0.711760i	−0.255143	−	0.966903i	$-0.582123\pi$
$103$	7.22358	−	7.22358i	0.711760	−	0.711760i	0.966903	+	0.255143i	$0.0821226\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−0.266335	+	0.266335i	−0.0257475	+	0.0257475i	−0.719863	−	0.694116i	$-0.755796\pi$
$107$	−0.266335	+	0.266335i	−0.0257475	+	0.0257475i	0.694116	+	0.719863i	$0.255796\pi$
$108$		0			0
$109$	5.07239			0.485847			0.242923	−	0.970045i	$-0.421894\pi$
$109$	5.07239			0.485847			0.242923	+	0.970045i	$0.421894\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−4.00794	+	4.00794i	−0.377035	+	0.377035i	−0.870031	−	0.492996i	$-0.835902\pi$
$113$	−4.00794	+	4.00794i	−0.377035	+	0.377035i	0.492996	+	0.870031i	$0.335902\pi$
$114$		0			0
$115$	2.46620	−	1.75754i	0.229974	−	0.163891i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	9.12257i		−	0.836264i
$120$		0			0
$121$	−10.5454	+	3.12952i	−0.958676	+	0.284501i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	9.81864	+	5.34737i	0.878206	+	0.478283i
$126$		0			0
$127$	11.4505	−	11.4505i	1.01607	−	1.01607i	0.0161984	−	0.999869i	$-0.494844\pi$
$127$	11.4505	−	11.4505i	1.01607	−	1.01607i	0.999869	−	0.0161984i	$-0.00515634\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	10.8893i		−	0.951403i	−0.879607	−	0.475701i	$-0.842194\pi$
$131$		−	10.8893i		−	0.951403i	0.879607	−	0.475701i	$-0.157806\pi$
$132$		0			0
$133$	−8.08089	+	8.08089i	−0.700702	+	0.700702i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	13.0742	+	13.0742i	1.11700	+	1.11700i	0.992179	+	0.124822i	$0.0398359\pi$
$137$	13.0742	+	13.0742i	1.11700	+	1.11700i	0.124822	+	0.992179i	$0.460164\pi$
$138$		0			0
$139$	−11.3316			−0.961134			−0.480567	−	0.876958i	$-0.659569\pi$
$139$	−11.3316			−0.961134			−0.480567	+	0.876958i	$0.659569\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−0.407131	+	0.545503i	−0.0340460	+	0.0456172i
$144$		0			0
$145$	20.7090	+	3.47458i	1.71979	+	0.288548i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−5.06586			−0.415012			−0.207506	−	0.978234i	$-0.566535\pi$
$149$	−5.06586			−0.415012			−0.207506	+	0.978234i	$0.566535\pi$
$150$		0			0
$151$		−	14.6099i		−	1.18894i	−0.804118	−	0.594470i	$-0.797362\pi$
$151$		−	14.6099i		−	1.18894i	0.804118	−	0.594470i	$-0.202638\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−5.28431	−	0.886608i	−0.424446	−	0.0712141i
$156$		0			0
$157$	8.77316	+	8.77316i	0.700174	+	0.700174i	0.964448	−	0.264273i	$-0.0851321\pi$
$157$	8.77316	+	8.77316i	0.700174	+	0.700174i	−0.264273	+	0.964448i	$0.585132\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	2.71905i		−	0.214291i
$162$		0			0
$163$	−2.73430	+	2.73430i	−0.214167	+	0.214167i	−0.806035	−	0.591868i	$-0.798391\pi$
$163$	−2.73430	+	2.73430i	−0.214167	+	0.214167i	0.591868	+	0.806035i	$0.298391\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−13.6725	+	13.6725i	−1.05801	+	1.05801i	−0.0597971	+	0.998211i	$0.519045\pi$
$167$	−13.6725	+	13.6725i	−1.05801	+	1.05801i	−0.998211	+	0.0597971i	$0.980955\pi$
$168$		0			0
$169$		−	12.9579i		−	0.996760i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	4.71150	+	4.71150i	0.358208	+	0.358208i	0.863152	−	0.504944i	$-0.168487\pi$
$173$	4.71150	+	4.71150i	0.358208	+	0.358208i	−0.504944	+	0.863152i	$0.668487\pi$
$174$		0			0
$175$	9.02615	−	4.39282i	0.682313	−	0.332066i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	3.38697i		−	0.253154i	−0.991957	−	0.126577i	$-0.959601\pi$
$179$		−	3.38697i		−	0.253154i	0.991957	−	0.126577i	$-0.0403990\pi$
$180$		0			0
$181$	18.6393			1.38545			0.692723	−	0.721204i	$-0.256411\pi$
$181$	18.6393			1.38545			0.692723	+	0.721204i	$0.256411\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−14.1202	−	19.8137i	−1.03814	−	1.45673i
$186$		0			0
$187$	−12.0774	−	9.01386i	−0.883188	−	0.659159i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−4.01094			−0.290222			−0.145111	−	0.989415i	$-0.546354\pi$
$191$	−4.01094			−0.290222			−0.145111	+	0.989415i	$0.546354\pi$
$192$		0			0
$193$	−0.897030	−	0.897030i	−0.0645697	−	0.0645697i	0.674085	−	0.738654i	$-0.264538\pi$
$193$	−0.897030	−	0.897030i	−0.0645697	−	0.0645697i	−0.738654	+	0.674085i	$0.764538\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	3.10561	−	3.10561i	0.221265	−	0.221265i	−0.587766	−	0.809031i	$-0.699992\pi$
$197$	3.10561	−	3.10561i	0.221265	−	0.221265i	0.809031	+	0.587766i	$0.199992\pi$
$198$		0			0
$199$			3.98233i			0.282300i	0.989988	+	0.141150i	$0.0450799\pi$
$199$			3.98233i			0.282300i	−0.989988	+	0.141150i	$0.954920\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	13.3315	−	13.3315i	0.935689	−	0.935689i
$204$		0			0
$205$	4.26585	−	25.4251i	0.297940	−	1.77576i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	2.71373	+	18.6829i	0.187713	+	1.29233i
$210$		0			0
$211$		−	5.21688i		−	0.359145i	−0.983745	−	0.179572i	$-0.942529\pi$
$211$		−	5.21688i		−	0.359145i	0.983745	−	0.179572i	$-0.0574714\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−10.5678	−	14.8289i	−0.720720	−	1.01132i
$216$		0			0
$217$	−3.40180	+	3.40180i	−0.230929	+	0.230929i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−0.932553			−0.0627303
$222$		0			0
$223$	3.45760	−	3.45760i	0.231538	−	0.231538i	−0.581796	−	0.813335i	$-0.697650\pi$
$223$	3.45760	−	3.45760i	0.231538	−	0.231538i	0.813335	+	0.581796i	$0.197650\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	1.54979	−	1.54979i	0.102863	−	0.102863i	−0.653802	−	0.756665i	$-0.726827\pi$
$227$	1.54979	−	1.54979i	0.102863	−	0.102863i	0.756665	+	0.653802i	$0.226827\pi$
$228$		0			0
$229$			12.0816i			0.798377i	0.916869	+	0.399189i	$0.130708\pi$
$229$			12.0816i			0.798377i	−0.916869	+	0.399189i	$0.869292\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	13.0324	+	13.0324i	0.853782	+	0.853782i	0.990597	−	0.136815i	$-0.0436865\pi$
$233$	13.0324	+	13.0324i	0.853782	+	0.853782i	−0.136815	+	0.990597i	$0.543687\pi$
$234$		0			0
$235$	−4.06230	−	5.70027i	−0.264995	−	0.371845i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−2.82112			−0.182483			−0.0912416	−	0.995829i	$-0.529084\pi$
$239$	−2.82112			−0.182483			−0.0912416	+	0.995829i	$0.529084\pi$
$240$		0			0
$241$			1.09501i			0.0705357i	0.999378	+	0.0352678i	$0.0112284\pi$
$241$			1.09501i			0.0705357i	−0.999378	+	0.0352678i	$0.988772\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.09863	+	6.54796i	−0.0701886	+	0.418334i
$246$		0			0
$247$	0.826068	+	0.826068i	0.0525615	+	0.0525615i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−11.2736			−0.711586			−0.355793	−	0.934565i	$-0.615789\pi$
$251$	−11.2736			−0.711586			−0.355793	+	0.934565i	$0.615789\pi$
$252$		0			0
$253$	−3.59976	−	2.68665i	−0.226315	−	0.168908i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	11.6860	+	11.6860i	0.728953	+	0.728953i	0.970411	−	0.241458i	$-0.0776256\pi$
$257$	11.6860	+	11.6860i	0.728953	+	0.728953i	−0.241458	+	0.970411i	$0.577626\pi$
$258$		0			0
$259$	−21.8451			−1.35739
$260$		0			0
$261$		0			0
$262$		0			0
$263$	9.66997	+	9.66997i	0.596276	+	0.596276i	0.939319	−	0.343044i	$-0.111458\pi$
$263$	9.66997	+	9.66997i	0.596276	+	0.596276i	−0.343044	+	0.939319i	$0.611458\pi$
$264$		0			0
$265$	−5.81324	+	4.14280i	−0.357104	+	0.254490i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			14.0667i			0.857662i	0.903385	+	0.428831i	$0.141074\pi$
$269$			14.0667i			0.857662i	−0.903385	+	0.428831i	$0.858926\pi$
$270$		0			0
$271$			18.9922i			1.15369i	0.816852	+	0.576847i	$0.195717\pi$
$271$			18.9922i			1.15369i	−0.816852	+	0.576847i	$0.804283\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	3.10292	−	16.2902i	0.187113	−	0.982338i
$276$		0			0
$277$	−17.6125	+	17.6125i	−1.05823	+	1.05823i	−0.0600349	+	0.998196i	$0.519121\pi$
$277$	−17.6125	+	17.6125i	−1.05823	+	1.05823i	−0.998196	+	0.0600349i	$0.980879\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			3.46118i			0.206477i	0.994657	+	0.103238i	$0.0329205\pi$
$281$			3.46118i			0.206477i	−0.994657	+	0.103238i	$0.967080\pi$
$282$		0			0
$283$	11.7091	+	11.7091i	0.696033	+	0.696033i	0.963553	−	0.267519i	$-0.0862039\pi$
$283$	11.7091	+	11.7091i	0.696033	+	0.696033i	−0.267519	+	0.963553i	$0.586204\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−16.3675	−	16.3675i	−0.966143	−	0.966143i
$288$		0			0
$289$		−	3.64668i		−	0.214511i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	5.64180	+	5.64180i	0.329597	+	0.329597i	0.852433	−	0.522836i	$-0.175126\pi$
$293$	5.64180	+	5.64180i	0.329597	+	0.329597i	−0.522836	+	0.852433i	$0.675126\pi$
$294$		0			0
$295$	−4.98621	+	29.7185i	−0.290308	+	1.73028i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−0.277954			−0.0160745
$300$		0			0
$301$	−16.3493			−0.942356
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−2.38432	+	14.2109i	−0.136526	+	0.813714i
$306$		0			0
$307$	−13.4805	+	13.4805i	−0.769375	+	0.769375i	−0.977997	−	0.208621i	$-0.933102\pi$
$307$	−13.4805	+	13.4805i	−0.769375	+	0.769375i	0.208621	+	0.977997i	$0.433102\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	24.9510			1.41484			0.707421	−	0.706792i	$-0.249858\pi$
$311$	24.9510			1.41484			0.707421	+	0.706792i	$0.249858\pi$
$312$		0			0
$313$	5.09323	−	5.09323i	0.287886	−	0.287886i	−0.548358	−	0.836244i	$-0.684747\pi$
$313$	5.09323	−	5.09323i	0.287886	−	0.287886i	0.836244	+	0.548358i	$0.184747\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−16.9245	−	16.9245i	−0.950575	−	0.950575i	0.0482601	−	0.998835i	$-0.484632\pi$
$317$	−16.9245	−	16.9245i	−0.950575	−	0.950575i	−0.998835	+	0.0482601i	$0.984632\pi$
$318$		0			0
$319$	−4.47700	−	30.8223i	−0.250664	−	1.72572i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−18.2891	+	18.2891i	−1.01763	+	1.01763i
$324$		0			0
$325$	−0.449055	−	0.922697i	−0.0249091	−	0.0511820i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−6.28470			−0.346487
$330$		0			0
$331$	9.45326			0.519598			0.259799	−	0.965663i	$-0.416344\pi$
$331$	9.45326			0.519598			0.259799	+	0.965663i	$0.416344\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	12.1119	+	16.9956i	0.661743	+	0.928567i
$336$		0			0
$337$	−18.9524	+	18.9524i	−1.03240	+	1.03240i	−0.0329454	+	0.999457i	$0.510489\pi$
$337$	−18.9524	+	18.9524i	−1.03240	+	1.03240i	−0.999457	+	0.0329454i	$0.989511\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	1.14239	+	7.86492i	0.0618642	+	0.425909i
$342$		0			0
$343$	14.1527	+	14.1527i	0.764176	+	0.764176i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−1.52241	+	1.52241i	−0.0817273	+	0.0817273i	−0.746789	−	0.665061i	$-0.768405\pi$
$347$	−1.52241	+	1.52241i	−0.0817273	+	0.0817273i	0.665061	+	0.746789i	$0.268405\pi$
$348$		0			0
$349$	−30.5386			−1.63469			−0.817346	−	0.576147i	$-0.804556\pi$
$349$	−30.5386			−1.63469			−0.817346	+	0.576147i	$0.804556\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	11.8541	−	11.8541i	0.630930	−	0.630930i	−0.317371	−	0.948301i	$-0.602800\pi$
$353$	11.8541	−	11.8541i	0.630930	−	0.630930i	0.948301	+	0.317371i	$0.102800\pi$
$354$		0			0
$355$	15.0967	+	2.53294i	0.801248	+	0.134434i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−8.44992			−0.445970			−0.222985	−	0.974822i	$-0.571580\pi$
$359$	−8.44992			−0.445970			−0.222985	+	0.974822i	$0.571580\pi$
$360$		0			0
$361$	13.4015			0.705342
$362$		0			0
$363$		0			0
$364$		0			0
$365$	15.8185	+	22.1968i	0.827981	+	1.16183i
$366$		0			0
$367$	−13.8240	−	13.8240i	−0.721608	−	0.721608i	0.247324	−	0.968933i	$-0.420449\pi$
$367$	−13.8240	−	13.8240i	−0.721608	−	0.721608i	−0.968933	+	0.247324i	$0.920449\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			6.40924i			0.332751i
$372$		0			0
$373$	18.0741	+	18.0741i	0.935844	+	0.935844i	0.998063	−	0.0622190i	$-0.0198177\pi$
$373$	18.0741	+	18.0741i	0.935844	+	0.935844i	−0.0622190	+	0.998063i	$0.519818\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−1.36281	−	1.36281i	−0.0701884	−	0.0701884i
$378$		0			0
$379$			12.9668i			0.666057i	0.942917	+	0.333029i	$0.108071\pi$
$379$			12.9668i			0.666057i	−0.942917	+	0.333029i	$0.891929\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−6.34581	+	6.34581i	−0.324256	+	0.324256i	−0.850397	−	0.526142i	$-0.823638\pi$
$383$	−6.34581	+	6.34581i	−0.324256	+	0.324256i	0.526142	+	0.850397i	$0.323638\pi$
$384$		0			0
$385$	−10.2943	−	10.7573i	−0.524644	−	0.548240i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	24.1883i		−	1.22639i	−0.789930	−	0.613197i	$-0.789883\pi$
$389$		−	24.1883i		−	1.22639i	0.789930	−	0.613197i	$-0.210117\pi$
$390$		0			0
$391$		−	6.15390i		−	0.311216i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−13.5407	−	2.27187i	−0.681304	−	0.114310i
$396$		0			0
$397$	−9.80413	−	9.80413i	−0.492055	−	0.492055i	0.416898	−	0.908953i	$-0.363117\pi$
$397$	−9.80413	−	9.80413i	−0.492055	−	0.492055i	−0.908953	+	0.416898i	$0.863117\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−7.66805			−0.382924			−0.191462	−	0.981500i	$-0.561323\pi$
$401$	−7.66805			−0.382924			−0.191462	+	0.981500i	$0.561323\pi$
$402$		0			0
$403$	0.347748	+	0.347748i	0.0173226	+	0.0173226i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−21.5848	+	28.9209i	−1.06992	+	1.43356i
$408$		0			0
$409$	4.63333			0.229104			0.114552	−	0.993417i	$-0.463457\pi$
$409$	4.63333			0.229104			0.114552	+	0.993417i	$0.463457\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	19.1314	+	19.1314i	0.941396	+	0.941396i
$414$		0			0
$415$	8.61122	+	12.0834i	0.422708	+	0.593150i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	25.0447i		−	1.22352i	−0.791045	−	0.611758i	$-0.790463\pi$
$419$		−	25.0447i		−	1.22352i	0.791045	−	0.611758i	$-0.209537\pi$
$420$		0			0
$421$	13.6388			0.664715			0.332357	−	0.943153i	$-0.392156\pi$
$421$	13.6388			0.664715			0.332357	+	0.943153i	$0.392156\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	20.4285	−	9.94206i	0.990926	−	0.482261i
$426$		0			0
$427$	9.14833	+	9.14833i	0.442719	+	0.442719i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	5.00629i		−	0.241145i	−0.992705	−	0.120572i	$-0.961527\pi$
$431$		−	5.00629i		−	0.241145i	0.992705	−	0.120572i	$-0.0384730\pi$
$432$		0			0
$433$	−22.4842	+	22.4842i	−1.08052	+	1.08052i	−0.0840597	+	0.996461i	$0.526789\pi$
$433$	−22.4842	+	22.4842i	−1.08052	+	1.08052i	−0.996461	+	0.0840597i	$0.973211\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−5.45121	+	5.45121i	−0.260767	+	0.260767i
$438$		0			0
$439$	−33.1102			−1.58026			−0.790132	−	0.612937i	$-0.789988\pi$
$439$	−33.1102			−1.58026			−0.790132	+	0.612937i	$0.789988\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	1.72992	−	1.72992i	0.0821909	−	0.0821909i	−0.664816	−	0.747007i	$-0.731490\pi$
$443$	1.72992	−	1.72992i	0.0821909	−	0.0821909i	0.747007	+	0.664816i	$0.231490\pi$
$444$		0			0
$445$	4.69463	−	27.9807i	0.222547	−	1.32641i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	6.22627i		−	0.293836i	−0.989149	−	0.146918i	$-0.953065\pi$
$449$		−	6.22627i		−	0.293836i	0.989149	−	0.146918i	$-0.0469353\pi$
$450$		0			0
$451$	−37.8415	+	5.49654i	−1.78188	+	0.258822i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−0.908650	−	0.152454i	−0.0425982	−	0.00714717i
$456$		0			0
$457$	−15.7357	+	15.7357i	−0.736085	+	0.736085i	−0.971818	−	0.235733i	$-0.924251\pi$
$457$	−15.7357	+	15.7357i	−0.736085	+	0.736085i	0.235733	+	0.971818i	$0.424251\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			17.9297i			0.835069i	0.908661	+	0.417534i	$0.137106\pi$
$461$			17.9297i			0.835069i	−0.908661	+	0.417534i	$0.862894\pi$
$462$		0			0
$463$	−24.8279	+	24.8279i	−1.15385	+	1.15385i	−0.168078	+	0.985774i	$0.553756\pi$
$463$	−24.8279	+	24.8279i	−1.15385	+	1.15385i	−0.985774	+	0.168078i	$0.946244\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	19.9139	+	19.9139i	0.921507	+	0.921507i	0.997136	−	0.0756289i	$-0.0240964\pi$
$467$	19.9139	+	19.9139i	0.921507	+	0.921507i	−0.0756289	+	0.997136i	$0.524096\pi$
$468$		0			0
$469$	18.7381			0.865243
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−16.1545	+	21.6449i	−0.742783	+	0.995233i
$474$		0			0
$475$	−26.9026	−	9.28900i	−1.23438	−	0.426209i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−19.8344			−0.906256			−0.453128	−	0.891445i	$-0.649692\pi$
$479$	−19.8344			−0.906256			−0.453128	+	0.891445i	$0.649692\pi$
$480$		0			0
$481$			2.23312i			0.101821i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	5.02938	+	7.05730i	0.228373	+	0.320456i
$486$		0			0
$487$	26.4609	+	26.4609i	1.19906	+	1.19906i	0.974448	+	0.224611i	$0.0721112\pi$
$487$	26.4609	+	26.4609i	1.19906	+	1.19906i	0.224611	+	0.974448i	$0.427889\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			39.2359i			1.77069i	0.464933	+	0.885346i	$0.346078\pi$
$491$			39.2359i			1.77069i	−0.464933	+	0.885346i	$0.653922\pi$
$492$		0			0
$493$	30.1726	−	30.1726i	1.35891	−	1.35891i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	9.71855	−	9.71855i	0.435937	−	0.435937i
$498$		0			0
$499$		−	32.5595i		−	1.45756i	−0.684747	−	0.728781i	$-0.740087\pi$
$499$		−	32.5595i		−	1.45756i	0.684747	−	0.728781i	$-0.259913\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−11.0832	−	11.0832i	−0.494177	−	0.494177i	0.415443	−	0.909619i	$-0.363627\pi$
$503$	−11.0832	−	11.0832i	−0.494177	−	0.494177i	−0.909619	+	0.415443i	$0.863627\pi$
$504$		0			0
$505$	4.98146	−	29.6902i	0.221672	−	1.32120i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			13.4112i			0.594440i	0.954809	+	0.297220i	$0.0960595\pi$
$509$			13.4112i			0.594440i	−0.954809	+	0.297220i	$0.903941\pi$
$510$		0			0
$511$	24.4726			1.08260
$512$		0			0
$513$		0			0
$514$		0			0
$515$	18.6025	−	13.2570i	0.819722	−	0.584175i
$516$		0			0
$517$	−6.20981	+	8.32034i	−0.273107	+	0.365928i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	10.0696			0.441155			0.220578	−	0.975369i	$-0.429206\pi$
$521$	10.0696			0.441155			0.220578	+	0.975369i	$0.429206\pi$
$522$		0			0
$523$	32.2051	+	32.2051i	1.40823	+	1.40823i	0.769090	+	0.639141i	$0.220710\pi$
$523$	32.2051	+	32.2051i	1.40823	+	1.40823i	0.639141	+	0.769090i	$0.279290\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−7.69914	+	7.69914i	−0.335380	+	0.335380i
$528$		0			0
$529$			21.1658i			0.920252i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−1.67316	+	1.67316i	−0.0724728	+	0.0724728i
$534$		0			0
$535$	−0.685876	+	0.488789i	−0.0296530	+	0.0211322i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	9.74568	−	1.41558i	0.419776	−	0.0609733i
$540$		0			0
$541$		−	17.7894i		−	0.764828i	−0.923991	−	0.382414i	$-0.875093\pi$
$541$		−	17.7894i		−	0.764828i	0.923991	−	0.382414i	$-0.124907\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	11.1859	+	1.87678i	0.479149	+	0.0803922i
$546$		0			0
$547$	2.25426	−	2.25426i	0.0963852	−	0.0963852i	−0.657270	−	0.753655i	$-0.728289\pi$
$547$	2.25426	−	2.25426i	0.0963852	−	0.0963852i	0.753655	+	0.657270i	$0.228289\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−53.4546			−2.27724
$552$		0			0
$553$	−8.71686	+	8.71686i	−0.370678	+	0.370678i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−25.8844	+	25.8844i	−1.09676	+	1.09676i	−0.101970	+	0.994788i	$0.532514\pi$
$557$	−25.8844	+	25.8844i	−1.09676	+	1.09676i	−0.994788	+	0.101970i	$0.967486\pi$
$558$		0			0
$559$			1.67130i			0.0706885i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−20.9843	−	20.9843i	−0.884382	−	0.884382i	0.109594	−	0.993976i	$-0.465045\pi$
$563$	−20.9843	−	20.9843i	−0.884382	−	0.884382i	−0.993976	+	0.109594i	$0.965045\pi$
$564$		0			0
$565$	−10.3214	+	7.35555i	−0.434225	+	0.309450i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−6.26435			−0.262615			−0.131308	−	0.991342i	$-0.541918\pi$
$569$	−6.26435			−0.262615			−0.131308	+	0.991342i	$0.541918\pi$
$570$		0			0
$571$		−	20.9444i		−	0.876497i	−0.898854	−	0.438249i	$-0.855599\pi$
$571$		−	20.9444i		−	0.876497i	0.898854	−	0.438249i	$-0.144401\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	6.08886	−	2.96331i	0.253923	−	0.123578i
$576$		0			0
$577$	4.53489	+	4.53489i	0.188790	+	0.188790i	0.795173	−	0.606383i	$-0.207380\pi$
$577$	4.53489	+	4.53489i	0.188790	+	0.188790i	−0.606383	+	0.795173i	$0.707380\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	13.3223			0.552700
$582$		0			0
$583$	8.48523	+	6.33287i	0.351422	+	0.262281i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	1.98863	+	1.98863i	0.0820795	+	0.0820795i	0.746955	−	0.664875i	$-0.231515\pi$
$587$	1.98863	+	1.98863i	0.0820795	+	0.0820795i	−0.664875	+	0.746955i	$0.731515\pi$
$588$		0			0
$589$	13.6400			0.562026
$590$		0			0
$591$		0			0
$592$		0			0
$593$	5.13480	+	5.13480i	0.210861	+	0.210861i	0.804633	−	0.593772i	$-0.202362\pi$
$593$	5.13480	+	5.13480i	0.210861	+	0.210861i	−0.593772	+	0.804633i	$0.702362\pi$
$594$		0			0
$595$	3.37534	−	20.1175i	0.138375	−	0.824736i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	9.94504i		−	0.406343i	−0.979143	−	0.203172i	$-0.934875\pi$
$599$		−	9.94504i		−	0.406343i	0.979143	−	0.203172i	$-0.0651249\pi$
$600$		0			0
$601$		−	39.3997i		−	1.60715i	−0.595206	−	0.803573i	$-0.702930\pi$
$601$		−	39.3997i		−	1.60715i	0.595206	−	0.803573i	$-0.297070\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−24.4132	+	2.99955i	−0.992536	+	0.121949i
$606$		0			0
$607$	19.6389	−	19.6389i	0.797118	−	0.797118i	−0.185522	−	0.982640i	$-0.559398\pi$
$607$	19.6389	−	19.6389i	0.797118	−	0.797118i	0.982640	+	0.185522i	$0.0593975\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			0.642452i			0.0259908i
$612$		0			0
$613$	20.4336	+	20.4336i	0.825305	+	0.825305i	0.986863	−	0.161558i	$-0.0516519\pi$
$613$	20.4336	+	20.4336i	0.825305	+	0.825305i	−0.161558	+	0.986863i	$0.551652\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−28.8165	−	28.8165i	−1.16011	−	1.16011i	−0.984451	−	0.175659i	$-0.943794\pi$
$617$	−28.8165	−	28.8165i	−1.16011	−	1.16011i	−0.175659	−	0.984451i	$-0.556206\pi$
$618$		0			0
$619$			14.4468i			0.580665i	0.956926	+	0.290333i	$0.0937660\pi$
$619$			14.4468i			0.580665i	−0.956926	+	0.290333i	$0.906234\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−18.0127	−	18.0127i	−0.721662	−	0.721662i
$624$		0			0
$625$	19.6740	+	15.4251i	0.786959	+	0.617005i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−49.4411			−1.97135
$630$		0			0
$631$	17.2681			0.687434			0.343717	−	0.939073i	$-0.388314\pi$
$631$	17.2681			0.687434			0.343717	+	0.939073i	$0.388314\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	29.4878	−	21.0145i	1.17019	−	0.833934i
$636$		0			0
$637$	0.430906	−	0.430906i	0.0170731	−	0.0170731i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	32.4657			1.28232			0.641160	−	0.767407i	$-0.278454\pi$
$641$	32.4657			1.28232			0.641160	+	0.767407i	$0.278454\pi$
$642$		0			0
$643$	−1.61548	+	1.61548i	−0.0637083	+	0.0637083i	−0.738243	−	0.674535i	$-0.764344\pi$
$643$	−1.61548	+	1.61548i	−0.0637083	+	0.0637083i	0.674535	+	0.738243i	$0.264344\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	16.0536	+	16.0536i	0.631132	+	0.631132i	0.948352	−	0.317220i	$-0.102749\pi$
$647$	16.0536	+	16.0536i	0.631132	+	0.631132i	−0.317220	+	0.948352i	$0.602749\pi$
$648$		0			0
$649$	44.2316	−	6.42473i	1.73624	−	0.252193i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−1.62087	+	1.62087i	−0.0634294	+	0.0634294i	−0.738110	−	0.674681i	$-0.764281\pi$
$653$	−1.62087	+	1.62087i	−0.0634294	+	0.0634294i	0.674681	+	0.738110i	$0.264281\pi$
$654$		0			0
$655$	4.02903	−	24.0136i	0.157427	−	0.938288i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−39.4221			−1.53567			−0.767833	−	0.640649i	$-0.778665\pi$
$659$	−39.4221			−1.53567			−0.767833	+	0.640649i	$0.778665\pi$
$660$		0			0
$661$	−34.4378			−1.33947			−0.669737	−	0.742598i	$-0.733593\pi$
$661$	−34.4378			−1.33947			−0.669737	+	0.742598i	$0.733593\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−20.8103	+	14.8304i	−0.806987	+	0.575099i
$666$		0			0
$667$	8.99317	−	8.99317i	0.348217	−	0.348217i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	21.1508	−	3.07220i	0.816519	−	0.118601i
$672$		0			0
$673$	−18.8034	−	18.8034i	−0.724819	−	0.724819i	0.244763	−	0.969583i	$-0.421290\pi$
$673$	−18.8034	−	18.8034i	−0.724819	−	0.724819i	−0.969583	+	0.244763i	$0.921290\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−19.7637	+	19.7637i	−0.759582	+	0.759582i	−0.976246	−	0.216664i	$-0.930482\pi$
$677$	−19.7637	+	19.7637i	−0.759582	+	0.759582i	0.216664	+	0.976246i	$0.430482\pi$
$678$		0			0
$679$	7.78086			0.298602
$680$		0			0
$681$		0			0
$682$		0			0
$683$	7.11481	−	7.11481i	0.272240	−	0.272240i	−0.557761	−	0.830002i	$-0.688339\pi$
$683$	7.11481	−	7.11481i	0.272240	−	0.272240i	0.830002	+	0.557761i	$0.188339\pi$
$684$		0			0
$685$	23.9943	+	33.6691i	0.916775	+	1.28643i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	0.655184			0.0249605
$690$		0			0
$691$	−32.5558			−1.23848			−0.619240	−	0.785202i	$-0.712559\pi$
$691$	−32.5558			−1.23848			−0.619240	+	0.785202i	$0.712559\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−24.9890	−	4.19268i	−0.947885	−	0.159037i
$696$		0			0
$697$	−37.0438	−	37.0438i	−1.40313	−	1.40313i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			28.0136i			1.05806i	0.848604	+	0.529029i	$0.177444\pi$
$701$			28.0136i			1.05806i	−0.848604	+	0.529029i	$0.822556\pi$
$702$		0			0
$703$	43.7956	+	43.7956i	1.65178	+	1.65178i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−19.1132	−	19.1132i	−0.718826	−	0.718826i
$708$		0			0
$709$			6.43629i			0.241720i	0.992670	+	0.120860i	$0.0385652\pi$
$709$			6.43629i			0.241720i	−0.992670	+	0.120860i	$0.961435\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−2.29478	+	2.29478i	−0.0859403	+	0.0859403i
$714$		0			0
$715$	−1.09966	+	1.05233i	−0.0411249	+	0.0393549i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	7.33610i		−	0.273590i	−0.990599	−	0.136795i	$-0.956320\pi$
$719$		−	7.33610i		−	0.273590i	0.990599	−	0.136795i	$-0.0436802\pi$
$720$		0			0
$721$		−	20.5097i		−	0.763821i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	44.3828	+	15.3246i	1.64834	+	0.569141i
$726$		0			0
$727$	−32.8750	−	32.8750i	−1.21927	−	1.21927i	−0.967890	−	0.251375i	$-0.919117\pi$
$727$	−32.8750	−	32.8750i	−1.21927	−	1.21927i	−0.251375	−	0.967890i	$-0.580883\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−37.0026			−1.36859
$732$		0			0
$733$	9.39670	+	9.39670i	0.347075	+	0.347075i	0.859019	−	0.511944i	$-0.171074\pi$
$733$	9.39670	+	9.39670i	0.347075	+	0.347075i	−0.511944	+	0.859019i	$0.671074\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	18.5148	−	24.8074i	0.682000	−	0.913792i
$738$		0			0
$739$	−30.5118			−1.12240			−0.561198	−	0.827682i	$-0.689659\pi$
$739$	−30.5118			−1.12240			−0.561198	+	0.827682i	$0.689659\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−2.01757	−	2.01757i	−0.0740175	−	0.0740175i	0.669129	−	0.743146i	$-0.266667\pi$
$743$	−2.01757	−	2.01757i	−0.0740175	−	0.0740175i	−0.743146	+	0.669129i	$0.766667\pi$
$744$		0			0
$745$	−11.1715	−	1.87436i	−0.409291	−	0.0686713i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			0.756196i			0.0276308i
$750$		0			0
$751$	37.7271			1.37668			0.688340	−	0.725388i	$-0.258340\pi$
$751$	37.7271			1.37668			0.688340	+	0.725388i	$0.258340\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	5.40565	−	32.2185i	0.196732	−	1.17255i
$756$		0			0
$757$	−9.84866	−	9.84866i	−0.357956	−	0.357956i	0.505103	−	0.863059i	$-0.331454\pi$
$757$	−9.84866	−	9.84866i	−0.357956	−	0.357956i	−0.863059	+	0.505103i	$0.831454\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			5.64635i			0.204680i	0.994749	+	0.102340i	$0.0326330\pi$
$761$			5.64635i			0.204680i	−0.994749	+	0.102340i	$0.967367\pi$
$762$		0			0
$763$	7.20094	−	7.20094i	0.260692	−	0.260692i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1.95571	−	1.95571i	0.0706165	−	0.0706165i
$768$		0			0
$769$	−5.11155			−0.184327			−0.0921637	−	0.995744i	$-0.529378\pi$
$769$	−5.11155			−0.184327			−0.0921637	+	0.995744i	$0.529378\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	16.1849	−	16.1849i	0.582131	−	0.582131i	−0.353357	−	0.935489i	$-0.614960\pi$
$773$	16.1849	−	16.1849i	0.582131	−	0.582131i	0.935489	+	0.353357i	$0.114960\pi$
$774$		0			0
$775$	−11.3252	−	3.91037i	−0.406812	−	0.140465i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			65.6278i			2.35136i
$780$		0			0
$781$	−3.26369	−	22.4692i	−0.116784	−	0.804011i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	16.1009	+	22.5930i	0.574666	+	0.806379i
$786$		0			0
$787$	−26.4921	+	26.4921i	−0.944342	+	0.944342i	−0.998531	−	0.0541891i	$-0.982743\pi$
$787$	−26.4921	+	26.4921i	−0.944342	+	0.944342i	0.0541891	+	0.998531i	$0.482743\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			11.3796i			0.404613i
$792$		0			0
$793$	0.935187	−	0.935187i	0.0332095	−	0.0332095i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	15.3289	+	15.3289i	0.542979	+	0.542979i	0.924401	−	0.381422i	$-0.124566\pi$
$797$	15.3289	+	15.3289i	0.542979	+	0.542979i	−0.381422	+	0.924401i	$0.624566\pi$
$798$		0			0
$799$	−14.2239			−0.503204
$800$		0			0
$801$		0			0
$802$		0			0
$803$	24.1809	−	32.3993i	0.853327	−	1.14335i
$804$		0			0
$805$	1.00604	−	5.99616i	0.0354584	−	0.211337i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	24.2292			0.851853			0.425927	−	0.904758i	$-0.359948\pi$
$809$	24.2292			0.851853			0.425927	+	0.904758i	$0.359948\pi$
$810$		0			0
$811$		−	2.59463i		−	0.0911099i	−0.998962	−	0.0455549i	$-0.985494\pi$
$811$		−	2.59463i		−	0.0911099i	0.998962	−	0.0455549i	$-0.0145056\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−7.04149	+	5.01812i	−0.246653	+	0.175777i
$816$		0			0
$817$	32.7774	+	32.7774i	1.14674	+	1.14674i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	41.9633i		−	1.46453i	−0.681021	−	0.732264i	$-0.738464\pi$
$821$		−	41.9633i		−	1.46453i	0.681021	−	0.732264i	$-0.261536\pi$
$822$		0			0
$823$	−9.78175	+	9.78175i	−0.340970	+	0.340970i	−0.856732	−	0.515762i	$-0.827509\pi$
$823$	−9.78175	+	9.78175i	−0.340970	+	0.340970i	0.515762	+	0.856732i	$0.327509\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	30.2752	−	30.2752i	1.05277	−	1.05277i	0.0542446	−	0.998528i	$-0.482725\pi$
$827$	30.2752	−	30.2752i	1.05277	−	1.05277i	0.998528	−	0.0542446i	$-0.0172751\pi$
$828$		0			0
$829$			45.9706i			1.59662i	0.602245	+	0.798312i	$0.294273\pi$
$829$			45.9706i			1.59662i	−0.602245	+	0.798312i	$0.705727\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	9.54025	+	9.54025i	0.330550	+	0.330550i
$834$		0			0
$835$	−35.2099	+	25.0923i	−1.21849	+	0.868356i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			1.15601i			0.0399100i	0.999801	+	0.0199550i	$0.00635229\pi$
$839$			1.15601i			0.0399100i	−0.999801	+	0.0199550i	$0.993648\pi$
$840$		0			0
$841$	59.1871			2.04094
$842$		0			0
$843$		0			0
$844$		0			0
$845$	4.79440	−	28.5753i	0.164932	−	0.983020i
$846$		0			0
$847$	−10.5279	+	19.4134i	−0.361743	+	0.667054i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−14.7363			−0.505154
$852$		0			0
$853$	−5.35709	−	5.35709i	−0.183423	−	0.183423i	0.609422	−	0.792846i	$-0.291401\pi$
$853$	−5.35709	−	5.35709i	−0.183423	−	0.183423i	−0.792846	+	0.609422i	$0.791401\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	30.8816	−	30.8816i	1.05489	−	1.05489i	0.0564918	−	0.998403i	$-0.482009\pi$
$857$	30.8816	−	30.8816i	1.05489	−	1.05489i	0.998403	−	0.0564918i	$-0.0179915\pi$
$858$		0			0
$859$			22.3650i			0.763083i	0.924352	+	0.381541i	$0.124607\pi$
$859$			22.3650i			0.763083i	−0.924352	+	0.381541i	$0.875393\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	35.4659	−	35.4659i	1.20727	−	1.20727i	0.235365	−	0.971907i	$-0.424371\pi$
$863$	35.4659	−	35.4659i	1.20727	−	1.20727i	0.971907	−	0.235365i	$-0.0756287\pi$
$864$		0			0
$865$	8.64675	+	12.1332i	0.293998	+	0.412543i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	2.92730	+	20.1533i	0.0993019	+	0.683653i
$870$		0			0
$871$		−	1.91549i		−	0.0649041i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	21.5302	−	6.34758i	0.727854	−	0.214587i
$876$		0			0
$877$	−15.6784	+	15.6784i	−0.529421	+	0.529421i	−0.920400	−	0.390978i	$-0.872137\pi$
$877$	−15.6784	+	15.6784i	−0.529421	+	0.529421i	0.390978	+	0.920400i	$0.372137\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−45.1013			−1.51950			−0.759752	−	0.650214i	$-0.774679\pi$
$881$	−45.1013			−1.51950			−0.759752	+	0.650214i	$0.774679\pi$
$882$		0			0
$883$	31.2495	−	31.2495i	1.05163	−	1.05163i	0.0530373	−	0.998593i	$-0.483110\pi$
$883$	31.2495	−	31.2495i	1.05163	−	1.05163i	0.998593	−	0.0530373i	$-0.0168902\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	13.5535	−	13.5535i	0.455081	−	0.455081i	−0.441956	−	0.897037i	$-0.645715\pi$
$887$	13.5535	−	13.5535i	0.455081	−	0.455081i	0.897037	+	0.441956i	$0.145715\pi$
$888$		0			0
$889$		−	32.5111i		−	1.09039i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	12.5997	+	12.5997i	0.421633	+	0.421633i
$894$		0			0
$895$	1.25317	−	7.46909i	0.0418889	−	0.249664i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−22.5027			−0.750507
$900$		0			0
$901$			14.5057i			0.483256i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	41.1041	+	6.89650i	1.36635	+	0.229247i
$906$		0			0
$907$	0.756930	+	0.756930i	0.0251334	+	0.0251334i	0.719562	−	0.694428i	$-0.244343\pi$
$907$	0.756930	+	0.756930i	0.0251334	+	0.0251334i	−0.694428	+	0.719562i	$0.744343\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	44.1348			1.46225			0.731125	−	0.682244i	$-0.238996\pi$
$911$	44.1348			1.46225			0.731125	+	0.682244i	$0.238996\pi$
$912$		0			0
$913$	13.1635	−	17.6374i	0.435648	−	0.583712i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−15.4588	−	15.4588i	−0.510496	−	0.510496i
$918$		0			0
$919$	−1.00299			−0.0330855			−0.0165427	−	0.999863i	$-0.505266\pi$
$919$	−1.00299			−0.0330855			−0.0165427	+	0.999863i	$0.505266\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−0.993478	−	0.993478i	−0.0327007	−	0.0327007i
$924$		0			0
$925$	−23.8075	−	48.9186i	−0.782787	−	1.60843i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	6.85691i		−	0.224968i	−0.993654	−	0.112484i	$-0.964119\pi$
$929$		−	6.85691i		−	0.224968i	0.993654	−	0.112484i	$-0.0358807\pi$
$930$		0			0
$931$		−	16.9018i		−	0.553933i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−23.2985	−	24.3464i	−0.761943	−	0.796212i
$936$		0			0
$937$	6.67134	−	6.67134i	0.217943	−	0.217943i	−0.589688	−	0.807631i	$-0.700749\pi$
$937$	6.67134	−	6.67134i	0.217943	−	0.217943i	0.807631	+	0.589688i	$0.200749\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	11.6576i		−	0.380027i	−0.981781	−	0.190014i	$-0.939147\pi$
$941$		−	11.6576i		−	0.380027i	0.981781	−	0.190014i	$-0.0608532\pi$
$942$		0			0
$943$	−11.0412	−	11.0412i	−0.359550	−	0.359550i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	9.57867	+	9.57867i	0.311265	+	0.311265i	0.845399	−	0.534135i	$-0.179362\pi$
$947$	9.57867	+	9.57867i	0.311265	+	0.311265i	−0.534135	+	0.845399i	$0.679362\pi$
$948$		0			0
$949$		−	2.50170i		−	0.0812087i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	14.0583	+	14.0583i	0.455392	+	0.455392i	0.897139	−	0.441748i	$-0.145641\pi$
$953$	14.0583	+	14.0583i	0.455392	+	0.455392i	−0.441748	+	0.897139i	$0.645641\pi$
$954$		0			0
$955$	−8.84511	−	1.48404i	−0.286221	−	0.0480225i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	37.1211			1.19870
$960$		0			0
$961$	−25.2580			−0.814774
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−1.64627	−	2.31007i	−0.0529953	−	0.0743638i
$966$		0			0
$967$	27.3654	−	27.3654i	0.880014	−	0.880014i	−0.113522	−	0.993535i	$-0.536213\pi$
$967$	27.3654	−	27.3654i	0.880014	−	0.880014i	0.993535	+	0.113522i	$0.0362132\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	30.5076			0.979034			0.489517	−	0.871994i	$-0.337173\pi$
$971$	30.5076			0.979034			0.489517	+	0.871994i	$0.337173\pi$
$972$		0			0
$973$	−16.0868	+	16.0868i	−0.515718	+	0.515718i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	22.2577	+	22.2577i	0.712088	+	0.712088i	0.966972	−	0.254884i	$-0.0820373\pi$
$977$	22.2577	+	22.2577i	0.712088	+	0.712088i	−0.254884	+	0.966972i	$0.582037\pi$
$978$		0			0
$979$	−41.6451	+	6.04903i	−1.33098	+	0.193328i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−35.3554	+	35.3554i	−1.12766	+	1.12766i	−0.137106	+	0.990556i	$0.543780\pi$
$983$	−35.3554	+	35.3554i	−1.12766	+	1.12766i	−0.990556	+	0.137106i	$0.956220\pi$
$984$		0			0
$985$	7.99769	−	5.69955i	0.254828	−	0.181603i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−11.0289			−0.350698
$990$		0			0
$991$	−34.0251			−1.08084			−0.540421	−	0.841394i	$-0.681735\pi$
$991$	−34.0251			−1.08084			−0.540421	+	0.841394i	$0.681735\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−1.47345	+	8.78200i	−0.0467117	+	0.278408i
$996$		0			0
$997$	−40.0954	+	40.0954i	−1.26984	+	1.26984i	−0.323662	+	0.946173i	$0.604914\pi$
$997$	−40.0954	+	40.0954i	−1.26984	+	1.26984i	−0.946173	+	0.323662i	$0.895086\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1980.2.y.c.1297.12		24
3.2	odd	2		660.2.x.a.637.2	yes	24
5.3	odd	4	inner	1980.2.y.c.1693.11		24
11.10	odd	2	inner	1980.2.y.c.1297.11		24
15.2	even	4		3300.2.x.c.1693.12		24
15.8	even	4		660.2.x.a.373.1	✓	24
15.14	odd	2		3300.2.x.c.1957.11		24
33.32	even	2		660.2.x.a.637.1	yes	24
55.43	even	4	inner	1980.2.y.c.1693.12		24
165.32	odd	4		3300.2.x.c.1693.11		24
165.98	odd	4		660.2.x.a.373.2	yes	24
165.164	even	2		3300.2.x.c.1957.12		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
660.2.x.a.373.1	✓	24	15.8	even	4
660.2.x.a.373.2	yes	24	165.98	odd	4
660.2.x.a.637.1	yes	24	33.32	even	2
660.2.x.a.637.2	yes	24	3.2	odd	2
1980.2.y.c.1297.11		24	11.10	odd	2	inner
1980.2.y.c.1297.12		24	1.1	even	1	trivial
1980.2.y.c.1693.11		24	5.3	odd	4	inner
1980.2.y.c.1693.12		24	55.43	even	4	inner
3300.2.x.c.1693.11		24	165.32	odd	4
3300.2.x.c.1693.12		24	15.2	even	4
3300.2.x.c.1957.11		24	15.14	odd	2
3300.2.x.c.1957.12		24	165.164	even	2

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:	\( N \)	\(=\)	\( 1980 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1980.y (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(2.20524 + 0.369998i) q^{5} +(1.41964 - 1.41964i) q^{7} +O(q^{10})\)	\(q+(2.20524 + 0.369998i) q^{5} +(1.41964 - 1.41964i) q^{7} +(-0.476743 - 3.28218i) q^{11} +(-0.145122 - 0.145122i) q^{13} +(3.21300 - 3.21300i) q^{17} -5.69223 q^{19} +(0.957657 - 0.957657i) q^{23} +(4.72620 + 1.63187i) q^{25} +9.39080 q^{29} -2.39625 q^{31} +(3.65591 - 2.60538i) q^{35} +(-7.69393 - 7.69393i) q^{37} -11.5294i q^{41} +(-5.75826 - 5.75826i) q^{43} +(-2.21349 - 2.21349i) q^{47} +2.96927i q^{49} +(-2.25736 + 2.25736i) q^{53} +(0.163067 - 7.41441i) q^{55} +13.4763i q^{59} +6.44414i q^{61} +(-0.266335 - 0.373724i) q^{65} +(6.59960 + 6.59960i) q^{67} +6.84581 q^{71} +(8.61931 + 8.61931i) q^{73} +(-5.33630 - 3.98270i) q^{77} -6.14021 q^{79} +(4.69214 + 4.69214i) q^{83} +(8.27424 - 5.89664i) q^{85} -12.6882i q^{89} -0.412041 q^{91} +(-12.5528 - 2.10612i) q^{95} +(2.74044 + 2.74044i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 8 q^{5}+O(q^{10}) \) 24 * q - 8 * q^5	\( 24 q - 8 q^{5} - 8 q^{11} - 16 q^{23} + 24 q^{25} + 16 q^{31} - 40 q^{37} - 64 q^{47} - 24 q^{53} + 24 q^{55} + 48 q^{67} - 24 q^{77} - 48 q^{91} - 72 q^{97}+O(q^{100}) \) 24 * q - 8 * q^5 - 8 * q^11 - 16 * q^23 + 24 * q^25 + 16 * q^31 - 40 * q^37 - 64 * q^47 - 24 * q^53 + 24 * q^55 + 48 * q^67 - 24 * q^77 - 48 * q^91 - 72 * q^97

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