LMFDB - Embedded newform 1020.3.c.a.749.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1020,3,Mod(749,1020)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1020, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1020.749");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1020 = 2^{2} \cdot 3 \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1020.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$27.7929869648$
Analytic rank:	$0$
Dimension:	$64$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			749.9
Character	$\chi$	$=$	1020.749
Dual form			1020.3.c.a.749.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.75746 - 1.18170i) q^{3} +(4.98117 + 0.433573i) q^{5} -13.3287i q^{7} +(6.20715 + 6.51700i) q^{9} +O(q^{10})$	\(q+(-2.75746 - 1.18170i) q^{3} +(4.98117 + 0.433573i) q^{5} -13.3287i q^{7} +(6.20715 + 6.51700i) q^{9} +7.28217i q^{11} +14.2252i q^{13} +(-13.2230 - 7.08182i) q^{15} +4.12311 q^{17} +30.5400 q^{19} +(-15.7505 + 36.7532i) q^{21} +29.9932 q^{23} +(24.6240 + 4.31940i) q^{25} +(-9.41481 - 25.3054i) q^{27} +48.5070i q^{29} -48.6377 q^{31} +(8.60536 - 20.0803i) q^{33} +(5.77895 - 66.3923i) q^{35} -13.4646i q^{37} +(16.8100 - 39.2254i) q^{39} -18.6706i q^{41} +70.3336i q^{43} +(28.0933 + 35.1535i) q^{45} +39.8435 q^{47} -128.653 q^{49} +(-11.3693 - 4.87229i) q^{51} +40.7085 q^{53} +(-3.15735 + 36.2737i) q^{55} +(-84.2129 - 36.0893i) q^{57} +9.13866i q^{59} +28.3187 q^{61} +(86.8629 - 82.7331i) q^{63} +(-6.16767 + 70.8581i) q^{65} -46.3846i q^{67} +(-82.7049 - 35.4430i) q^{69} -95.5663i q^{71} -73.9456i q^{73} +(-62.7955 - 41.0089i) q^{75} +97.0616 q^{77} +46.8555 q^{79} +(-3.94248 + 80.9040i) q^{81} +132.752 q^{83} +(20.5379 + 1.78767i) q^{85} +(57.3208 - 133.756i) q^{87} +18.7147i q^{89} +189.603 q^{91} +(134.116 + 57.4754i) q^{93} +(152.125 + 13.2413i) q^{95} -43.0964i q^{97} +(-47.4578 + 45.2015i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 64 q - 4 q^{9}+O(q^{10}) $ 64 * q - 4 * q^9	$ 64 q - 4 q^{9} - 46 q^{15} + 28 q^{19} + 40 q^{21} + 10 q^{25} - 256 q^{31} - 260 q^{39} - 264 q^{45} - 552 q^{49} + 174 q^{55} - 296 q^{61} - 28 q^{69} - 184 q^{75} + 496 q^{79} + 596 q^{81} - 34 q^{85} - 56 q^{91} - 340 q^{99}+O(q^{100}) $ 64 * q - 4 * q^9 - 46 * q^15 + 28 * q^19 + 40 * q^21 + 10 * q^25 - 256 * q^31 - 260 * q^39 - 264 * q^45 - 552 * q^49 + 174 * q^55 - 296 * q^61 - 28 * q^69 - 184 * q^75 + 496 * q^79 + 596 * q^81 - 34 * q^85 - 56 * q^91 - 340 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1020\mathbb{Z}\right)^\times$.

$n$	$241$	$341$	$511$	$817$
$\chi(n)$	$1$	$-1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−2.75746	−	1.18170i	−0.919153	−	0.393901i
$3$	−2.75746	−	1.18170i	−0.919153	−	0.393901i
$4$		0			0
$5$	4.98117	+	0.433573i	0.996233	+	0.0867146i
$5$	4.98117	+	0.433573i	0.996233	+	0.0867146i
$6$		0			0
$7$		−	13.3287i		−	1.90410i	−0.305950	−	0.952048i	$-0.598974\pi$
$7$		−	13.3287i		−	1.90410i	0.305950	−	0.952048i	$-0.401026\pi$
$8$		0			0
$9$	6.20715	+	6.51700i	0.689684	+	0.724111i
$10$		0			0
$11$			7.28217i			0.662015i	0.943628	+	0.331008i	$0.107389\pi$
$11$			7.28217i			0.662015i	−0.943628	+	0.331008i	$0.892611\pi$
$12$		0			0
$13$			14.2252i			1.09425i	0.837052	+	0.547123i	$0.184277\pi$
$13$			14.2252i			1.09425i	−0.837052	+	0.547123i	$0.815723\pi$
$14$		0			0
$15$	−13.2230	−	7.08182i	−0.881534	−	0.472121i
$16$		0			0
$17$	4.12311			0.242536
$17$	4.12311			0.242536
$18$		0			0
$19$	30.5400			1.60737			0.803685	−	0.595055i	$-0.202870\pi$
$19$	30.5400			1.60737			0.803685	+	0.595055i	$0.202870\pi$
$20$		0			0
$21$	−15.7505	+	36.7532i	−0.750025	+	1.75015i
$22$		0			0
$23$	29.9932			1.30405			0.652025	−	0.758197i	$-0.273920\pi$
$23$	29.9932			1.30405			0.652025	+	0.758197i	$0.273920\pi$
$24$		0			0
$25$	24.6240	+	4.31940i	0.984961	+	0.172776i
$26$		0			0
$27$	−9.41481	−	25.3054i	−0.348697	−	0.937236i
$28$		0			0
$29$			48.5070i			1.67265i	0.548231	+	0.836327i	$0.315301\pi$
$29$			48.5070i			1.67265i	−0.548231	+	0.836327i	$0.684699\pi$
$30$		0			0
$31$	−48.6377			−1.56896			−0.784479	−	0.620155i	$-0.787070\pi$
$31$	−48.6377			−1.56896			−0.784479	+	0.620155i	$0.787070\pi$
$32$		0			0
$33$	8.60536	−	20.0803i	0.260768	−	0.608493i
$34$		0			0
$35$	5.77895	−	66.3923i	0.165113	−	1.89692i
$36$		0			0
$37$		−	13.4646i		−	0.363908i	−0.983307	−	0.181954i	$-0.941758\pi$
$37$		−	13.4646i		−	0.363908i	0.983307	−	0.181954i	$-0.0582423\pi$
$38$		0			0
$39$	16.8100	−	39.2254i	0.431025	−	1.00578i
$40$		0			0
$41$		−	18.6706i		−	0.455380i	−0.973734	−	0.227690i	$-0.926883\pi$
$41$		−	18.6706i		−	0.455380i	0.973734	−	0.227690i	$-0.0731173\pi$
$42$		0			0
$43$			70.3336i			1.63567i	0.575455	+	0.817833i	$0.304825\pi$
$43$			70.3336i			1.63567i	−0.575455	+	0.817833i	$0.695175\pi$
$44$		0			0
$45$	28.0933	+	35.1535i	0.624295	+	0.781189i
$46$		0			0
$47$	39.8435			0.847733			0.423867	−	0.905725i	$-0.360673\pi$
$47$	39.8435			0.847733			0.423867	+	0.905725i	$0.360673\pi$
$48$		0			0
$49$	−128.653			−2.62558
$50$		0			0
$51$	−11.3693	−	4.87229i	−0.222927	−	0.0955351i
$52$		0			0
$53$	40.7085			0.768086			0.384043	−	0.923315i	$-0.374531\pi$
$53$	40.7085			0.768086			0.384043	+	0.923315i	$0.374531\pi$
$54$		0			0
$55$	−3.15735	+	36.2737i	−0.0574064	+	0.659521i
$56$		0			0
$57$	−84.2129	−	36.0893i	−1.47742	−	0.633145i
$58$		0			0
$59$			9.13866i			0.154892i	0.996997	+	0.0774462i	$0.0246766\pi$
$59$			9.13866i			0.154892i	−0.996997	+	0.0774462i	$0.975323\pi$
$60$		0			0
$61$	28.3187			0.464240			0.232120	−	0.972687i	$-0.425434\pi$
$61$	28.3187			0.464240			0.232120	+	0.972687i	$0.425434\pi$
$62$		0			0
$63$	86.8629	−	82.7331i	1.37878	−	1.31322i
$64$		0			0
$65$	−6.16767	+	70.8581i	−0.0948872	+	1.09012i
$66$		0			0
$67$		−	46.3846i		−	0.692308i	−0.938178	−	0.346154i	$-0.887487\pi$
$67$		−	46.3846i		−	0.692308i	0.938178	−	0.346154i	$-0.112513\pi$
$68$		0			0
$69$	−82.7049	−	35.4430i	−1.19862	−	0.513667i
$70$		0			0
$71$		−	95.5663i		−	1.34600i	−0.739640	−	0.673002i	$-0.765004\pi$
$71$		−	95.5663i		−	1.34600i	0.739640	−	0.673002i	$-0.234996\pi$
$72$		0			0
$73$		−	73.9456i		−	1.01295i	−0.862254	−	0.506477i	$-0.830948\pi$
$73$		−	73.9456i		−	1.01295i	0.862254	−	0.506477i	$-0.169052\pi$
$74$		0			0
$75$	−62.7955	−	41.0089i	−0.837273	−	0.546785i
$76$		0			0
$77$	97.0616			1.26054
$78$		0			0
$79$	46.8555			0.593108			0.296554	−	0.955016i	$-0.404163\pi$
$79$	46.8555			0.593108			0.296554	+	0.955016i	$0.404163\pi$
$80$		0			0
$81$	−3.94248	+	80.9040i	−0.0486725	+	0.998815i
$82$		0			0
$83$	132.752			1.59943			0.799714	−	0.600382i	$-0.204985\pi$
$83$	132.752			1.59943			0.799714	+	0.600382i	$0.204985\pi$
$84$		0			0
$85$	20.5379	+	1.78767i	0.241622	+	0.0210314i
$86$		0			0
$87$	57.3208	−	133.756i	0.658860	−	1.53742i
$88$		0			0
$89$			18.7147i			0.210277i	0.994458	+	0.105139i	$0.0335287\pi$
$89$			18.7147i			0.210277i	−0.994458	+	0.105139i	$0.966471\pi$
$90$		0			0
$91$	189.603			2.08355
$92$		0			0
$93$	134.116	+	57.4754i	1.44211	+	0.618015i
$94$		0			0
$95$	152.125	+	13.2413i	1.60132	+	0.139383i
$96$		0			0
$97$		−	43.0964i		−	0.444293i	−0.975013	−	0.222146i	$-0.928694\pi$
$97$		−	43.0964i		−	0.444293i	0.975013	−	0.222146i	$-0.0713063\pi$
$98$		0			0
$99$	−47.4578	+	45.2015i	−0.479372	+	0.456581i
$100$		0			0
$101$			72.2619i			0.715465i	0.933824	+	0.357732i	$0.116450\pi$
$101$			72.2619i			0.715465i	−0.933824	+	0.357732i	$0.883550\pi$
$102$		0			0
$103$		−	89.3407i		−	0.867385i	−0.901061	−	0.433693i	$-0.857210\pi$
$103$		−	89.3407i		−	0.867385i	0.901061	−	0.433693i	$-0.142790\pi$
$104$		0			0
$105$	−94.3912	+	176.245i	−0.898964	+	1.67852i
$106$		0			0
$107$	−93.5164			−0.873985			−0.436993	−	0.899465i	$-0.643956\pi$
$107$	−93.5164			−0.873985			−0.436993	+	0.899465i	$0.643956\pi$
$108$		0			0
$109$	77.5459			0.711431			0.355715	−	0.934594i	$-0.384237\pi$
$109$	77.5459			0.711431			0.355715	+	0.934594i	$0.384237\pi$
$110$		0			0
$111$	−15.9112	+	37.1281i	−0.143344	+	0.334487i
$112$		0			0
$113$	−15.8816			−0.140545			−0.0702724	−	0.997528i	$-0.522387\pi$
$113$	−15.8816			−0.140545			−0.0702724	+	0.997528i	$0.522387\pi$
$114$		0			0
$115$	149.401	+	13.0042i	1.29914	+	0.113080i
$116$		0			0
$117$	−92.7056	+	88.2980i	−0.792356	+	0.754684i
$118$		0			0
$119$		−	54.9555i		−	0.461811i
$120$		0			0
$121$	67.9701			0.561736
$122$		0			0
$123$	−22.0631	+	51.4834i	−0.179375	+	0.418564i
$124$		0			0
$125$	120.784	+	32.1920i	0.966269	+	0.257536i
$126$		0			0
$127$			92.8562i			0.731151i	0.930782	+	0.365575i	$0.119128\pi$
$127$			92.8562i			0.731151i	−0.930782	+	0.365575i	$0.880872\pi$
$128$		0			0
$129$	83.1135	−	193.942i	0.644291	−	1.50343i
$130$		0			0
$131$		−	90.2128i		−	0.688647i	−0.938851	−	0.344323i	$-0.888108\pi$
$131$		−	90.2128i		−	0.688647i	0.938851	−	0.344323i	$-0.111892\pi$
$132$		0			0
$133$		−	407.058i		−	3.06059i
$134$		0			0
$135$	−35.9250	−	130.132i	−0.266111	−	0.963942i
$136$		0			0
$137$	121.516			0.886977			0.443488	−	0.896280i	$-0.353741\pi$
$137$	121.516			0.886977			0.443488	+	0.896280i	$0.353741\pi$
$138$		0			0
$139$	31.9966			0.230192			0.115096	−	0.993354i	$-0.463282\pi$
$139$	31.9966			0.230192			0.115096	+	0.993354i	$0.463282\pi$
$140$		0			0
$141$	−109.867	−	47.0831i	−0.779196	−	0.333923i
$142$		0			0
$143$	−103.590			−0.724408
$144$		0			0
$145$	−21.0313	+	241.621i	−0.145044	+	1.66635i
$146$		0			0
$147$	354.756	+	152.030i	2.41331	+	1.03422i
$148$		0			0
$149$			78.2024i			0.524848i	0.964953	+	0.262424i	$0.0845220\pi$
$149$			78.2024i			0.524848i	−0.964953	+	0.262424i	$0.915478\pi$
$150$		0			0
$151$	149.668			0.991178			0.495589	−	0.868557i	$-0.334952\pi$
$151$	149.668			0.991178			0.495589	+	0.868557i	$0.334952\pi$
$152$		0			0
$153$	25.5928	+	26.8703i	0.167273	+	0.175623i
$154$		0			0
$155$	−242.273	−	21.0880i	−1.56305	−	0.136052i
$156$		0			0
$157$		−	201.290i		−	1.28210i	−0.767498	−	0.641052i	$-0.778498\pi$
$157$		−	201.290i		−	1.28210i	0.767498	−	0.641052i	$-0.221502\pi$
$158$		0			0
$159$	−112.252	−	48.1054i	−0.705988	−	0.302550i
$160$		0			0
$161$		−	399.769i		−	2.48304i
$162$		0			0
$163$			16.8171i			0.103172i	0.998669	+	0.0515861i	$0.0164277\pi$
$163$			16.8171i			0.103172i	−0.998669	+	0.0515861i	$0.983572\pi$
$164$		0			0
$165$	51.5710	−	96.2921i	0.312551	−	0.583588i
$166$		0			0
$167$	−170.328			−1.01993			−0.509963	−	0.860196i	$-0.670341\pi$
$167$	−170.328			−1.01993			−0.509963	+	0.860196i	$0.670341\pi$
$168$		0			0
$169$	−33.3565			−0.197375
$170$		0			0
$171$	189.567	+	199.029i	1.10858	+	1.16391i
$172$		0			0
$173$	−14.8370			−0.0857630			−0.0428815	−	0.999080i	$-0.513654\pi$
$173$	−14.8370			−0.0857630			−0.0428815	+	0.999080i	$0.513654\pi$
$174$		0			0
$175$	57.5718	−	328.205i	0.328982	−	1.87546i
$176$		0			0
$177$	10.7992	−	25.1995i	0.0610123	−	0.142370i
$178$		0			0
$179$			155.033i			0.866106i	0.901368	+	0.433053i	$0.142564\pi$
$179$			155.033i			0.866106i	−0.901368	+	0.433053i	$0.857436\pi$
$180$		0			0
$181$	−211.969			−1.17110			−0.585550	−	0.810636i	$-0.699122\pi$
$181$	−211.969			−1.17110			−0.585550	+	0.810636i	$0.699122\pi$
$182$		0			0
$183$	−78.0876	−	33.4643i	−0.426708	−	0.182865i
$184$		0			0
$185$	5.83789	−	67.0695i	0.0315562	−	0.362538i
$186$		0			0
$187$			30.0251i			0.160562i
$188$		0			0
$189$	−337.287	+	125.487i	−1.78459	+	0.663952i
$190$		0			0
$191$		−	298.521i		−	1.56293i	−0.623946	−	0.781467i	$-0.714472\pi$
$191$		−	298.521i		−	1.56293i	0.623946	−	0.781467i	$-0.285528\pi$
$192$		0			0
$193$		−	262.057i		−	1.35781i	−0.734228	−	0.678903i	$-0.762456\pi$
$193$		−	262.057i		−	1.35781i	0.734228	−	0.678903i	$-0.237544\pi$
$194$		0			0
$195$	100.740	−	188.100i	0.516617	−	0.964615i
$196$		0			0
$197$	−271.215			−1.37673			−0.688364	−	0.725365i	$-0.741671\pi$
$197$	−271.215			−1.37673			−0.688364	+	0.725365i	$0.741671\pi$
$198$		0			0
$199$	−175.753			−0.883179			−0.441590	−	0.897217i	$-0.645585\pi$
$199$	−175.753			−0.883179			−0.441590	+	0.897217i	$0.645585\pi$
$200$		0			0
$201$	−54.8129	+	127.904i	−0.272701	+	0.636337i
$202$		0			0
$203$	646.533			3.18489
$204$		0			0
$205$	8.09507	−	93.0013i	0.0394881	−	0.453665i
$206$		0			0
$207$	186.172	+	195.465i	0.899382	+	0.944277i
$208$		0			0
$209$			222.398i			1.06410i
$210$		0			0
$211$	−198.030			−0.938533			−0.469267	−	0.883057i	$-0.655482\pi$
$211$	−198.030			−0.938533			−0.469267	+	0.883057i	$0.655482\pi$
$212$		0			0
$213$	−112.931	+	263.520i	−0.530193	+	1.23718i
$214$		0			0
$215$	−30.4948	+	350.344i	−0.141836	+	1.62951i
$216$		0			0
$217$			648.276i			2.98745i
$218$		0			0
$219$	−87.3817	+	203.902i	−0.399003	+	0.931059i
$220$		0			0
$221$			58.6520i			0.265394i
$222$		0			0
$223$			6.87693i			0.0308382i	0.999881	+	0.0154191i	$0.00490825\pi$
$223$			6.87693i			0.0308382i	−0.999881	+	0.0154191i	$0.995092\pi$
$224$		0			0
$225$	124.696	+	187.286i	0.554203	+	0.832382i
$226$		0			0
$227$	−91.3375			−0.402368			−0.201184	−	0.979553i	$-0.564479\pi$
$227$	−91.3375			−0.402368			−0.201184	+	0.979553i	$0.564479\pi$
$228$		0			0
$229$	155.566			0.679329			0.339664	−	0.940547i	$-0.389687\pi$
$229$	155.566			0.679329			0.339664	+	0.940547i	$0.389687\pi$
$230$		0			0
$231$	−267.643	−	114.698i	−1.15863	−	0.496528i
$232$		0			0
$233$	−161.438			−0.692866			−0.346433	−	0.938075i	$-0.612607\pi$
$233$	−161.438			−0.692866			−0.346433	+	0.938075i	$0.612607\pi$
$234$		0			0
$235$	198.467	+	17.2751i	0.844540	+	0.0735109i
$236$		0			0
$237$	−129.202	−	55.3693i	−0.545157	−	0.233626i
$238$		0			0
$239$			212.830i			0.890501i	0.895406	+	0.445250i	$0.146885\pi$
$239$			212.830i			0.890501i	−0.895406	+	0.445250i	$0.853115\pi$
$240$		0			0
$241$	445.835			1.84994			0.924969	−	0.380043i	$-0.124091\pi$
$241$	445.835			1.84994			0.924969	+	0.380043i	$0.124091\pi$
$242$		0			0
$243$	106.476	−	218.431i	0.438172	−	0.898891i
$244$		0			0
$245$	−640.844	−	55.7806i	−2.61569	−	0.227676i
$246$		0			0
$247$			434.438i			1.75886i
$248$		0			0
$249$	−366.059	−	156.874i	−1.47012	−	0.630016i
$250$		0			0
$251$			110.922i			0.441922i	0.975283	+	0.220961i	$0.0709193\pi$
$251$			110.922i			0.441922i	−0.975283	+	0.220961i	$0.929081\pi$
$252$		0			0
$253$			218.415i			0.863301i
$254$		0			0
$255$	−54.5198	−	29.1991i	−0.213803	−	0.114506i
$256$		0			0
$257$	−56.1145			−0.218344			−0.109172	−	0.994023i	$-0.534820\pi$
$257$	−56.1145			−0.218344			−0.109172	+	0.994023i	$0.534820\pi$
$258$		0			0
$259$	−179.465			−0.692916
$260$		0			0
$261$	−316.120	+	301.090i	−1.21119	+	1.15360i
$262$		0			0
$263$	171.027			0.650293			0.325146	−	0.945664i	$-0.394586\pi$
$263$	171.027			0.650293			0.325146	+	0.945664i	$0.394586\pi$
$264$		0			0
$265$	202.776	+	17.6501i	0.765192	+	0.0666043i
$266$		0			0
$267$	22.1152	−	51.6050i	0.0828285	−	0.193277i
$268$		0			0
$269$			200.140i			0.744016i	0.928230	+	0.372008i	$0.121331\pi$
$269$			200.140i			0.744016i	−0.928230	+	0.372008i	$0.878669\pi$
$270$		0			0
$271$	−9.71558			−0.0358508			−0.0179254	−	0.999839i	$-0.505706\pi$
$271$	−9.71558			−0.0358508			−0.0179254	+	0.999839i	$0.505706\pi$
$272$		0			0
$273$	−522.822	−	224.055i	−1.91510	−	0.820713i
$274$		0			0
$275$	−31.4546	+	179.316i	−0.114380	+	0.652059i
$276$		0			0
$277$		−	15.2582i		−	0.0550839i	−0.999621	−	0.0275420i	$-0.991232\pi$
$277$		−	15.2582i		−	0.0550839i	0.999621	−	0.0275420i	$-0.00876799\pi$
$278$		0			0
$279$	−301.902	−	316.972i	−1.08209	−	1.13610i
$280$		0			0
$281$			140.166i			0.498813i	0.968399	+	0.249406i	$0.0802355\pi$
$281$			140.166i			0.498813i	−0.968399	+	0.249406i	$0.919764\pi$
$282$		0			0
$283$			36.7624i			0.129903i	0.997888	+	0.0649513i	$0.0206892\pi$
$283$			36.7624i			0.129903i	−0.997888	+	0.0649513i	$0.979311\pi$
$284$		0			0
$285$	−403.831	−	216.279i	−1.41695	−	0.758874i
$286$		0			0
$287$	−248.854			−0.867088
$288$		0			0
$289$	17.0000			0.0588235
$290$		0			0
$291$	−50.9271	+	118.836i	−0.175007	+	0.408373i
$292$		0			0
$293$	−440.326			−1.50282			−0.751410	−	0.659836i	$-0.770626\pi$
$293$	−440.326			−1.50282			−0.751410	+	0.659836i	$0.770626\pi$
$294$		0			0
$295$	−3.96228	+	45.5212i	−0.0134314	+	0.154309i
$296$		0			0
$297$	184.278	−	68.5602i	0.620464	−	0.230843i
$298$		0			0
$299$			426.659i			1.42695i
$300$		0			0
$301$	937.454			3.11446
$302$		0			0
$303$	85.3922	−	199.259i	0.281822	−	0.657621i
$304$		0			0
$305$	141.060	+	12.2782i	0.462492	+	0.0402564i
$306$		0			0
$307$		−	18.3930i		−	0.0599120i	−0.999551	−	0.0299560i	$-0.990463\pi$
$307$		−	18.3930i		−	0.0599120i	0.999551	−	0.0299560i	$-0.00953672\pi$
$308$		0			0
$309$	−105.574	+	246.353i	−0.341664	+	0.797260i
$310$		0			0
$311$			121.314i			0.390076i	0.980796	+	0.195038i	$0.0624830\pi$
$311$			121.314i			0.390076i	−0.980796	+	0.195038i	$0.937517\pi$
$312$		0			0
$313$			241.333i			0.771033i	0.922701	+	0.385517i	$0.125977\pi$
$313$			241.333i			0.771033i	−0.922701	+	0.385517i	$0.874023\pi$
$314$		0			0
$315$	468.549	−	374.446i	1.48746	−	1.18872i
$316$		0			0
$317$	487.588			1.53813			0.769067	−	0.639169i	$-0.220721\pi$
$317$	487.588			1.53813			0.769067	+	0.639169i	$0.220721\pi$
$318$		0			0
$319$	−353.236			−1.10732
$320$		0			0
$321$	257.868	+	110.509i	0.803326	+	0.344264i
$322$		0			0
$323$	125.920			0.389845
$324$		0			0
$325$	−61.4443	+	350.282i	−0.189059	+	1.07779i
$326$		0			0
$327$	−213.830	−	91.6363i	−0.653913	−	0.280233i
$328$		0			0
$329$		−	531.060i		−	1.61416i
$330$		0			0
$331$	−263.720			−0.796736			−0.398368	−	0.917226i	$-0.630423\pi$
$331$	−263.720			−0.796736			−0.398368	+	0.917226i	$0.630423\pi$
$332$		0			0
$333$	87.7488	−	83.5769i	0.263510	−	0.250982i
$334$		0			0
$335$	20.1111	−	231.050i	0.0600332	−	0.689700i
$336$		0			0
$337$			213.013i			0.632087i	0.948745	+	0.316044i	$0.102355\pi$
$337$			213.013i			0.632087i	−0.948745	+	0.316044i	$0.897645\pi$
$338$		0			0
$339$	43.7927	+	18.7673i	0.129182	+	0.0553607i
$340$		0			0
$341$		−	354.188i		−	1.03867i
$342$		0			0
$343$			1061.67i			3.09526i
$344$		0			0
$345$	−396.600	−	212.406i	−1.14956	−	0.615670i
$346$		0			0
$347$	−384.201			−1.10721			−0.553603	−	0.832781i	$-0.686748\pi$
$347$	−384.201			−1.10721			−0.553603	+	0.832781i	$0.686748\pi$
$348$		0			0
$349$	175.928			0.504090			0.252045	−	0.967715i	$-0.418897\pi$
$349$	175.928			0.504090			0.252045	+	0.967715i	$0.418897\pi$
$350$		0			0
$351$	359.974	−	133.928i	1.02557	−	0.381560i
$352$		0			0
$353$	−504.077			−1.42798			−0.713990	−	0.700156i	$-0.753114\pi$
$353$	−504.077			−1.42798			−0.713990	+	0.700156i	$0.753114\pi$
$354$		0			0
$355$	41.4350	−	476.032i	0.116718	−	1.34093i
$356$		0			0
$357$	−64.9411	+	151.537i	−0.181908	+	0.424475i
$358$		0			0
$359$		−	9.62957i		−	0.0268233i	−0.999910	−	0.0134117i	$-0.995731\pi$
$359$		−	9.62957i		−	0.0268233i	0.999910	−	0.0134117i	$-0.00426919\pi$
$360$		0			0
$361$	571.694			1.58364
$362$		0			0
$363$	−187.425	−	80.3205i	−0.516321	−	0.221268i
$364$		0			0
$365$	32.0608	−	368.335i	0.0878378	−	1.00914i
$366$		0			0
$367$		−	648.885i		−	1.76808i	−0.467411	−	0.884040i	$-0.654813\pi$
$367$		−	648.885i		−	1.76808i	0.467411	−	0.884040i	$-0.345187\pi$
$368$		0			0
$369$	121.676	−	115.891i	0.329746	−	0.314068i
$370$		0			0
$371$		−	542.591i		−	1.46251i
$372$		0			0
$373$		−	96.4552i		−	0.258593i	−0.991606	−	0.129296i	$-0.958728\pi$
$373$		−	96.4552i		−	0.258593i	0.991606	−	0.129296i	$-0.0412719\pi$
$374$		0			0
$375$	−295.014	−	231.498i	−0.786705	−	0.617329i
$376$		0			0
$377$	−690.021			−1.83030
$378$		0			0
$379$	24.1078			0.0636091			0.0318045	−	0.999494i	$-0.489875\pi$
$379$	24.1078			0.0636091			0.0318045	+	0.999494i	$0.489875\pi$
$380$		0			0
$381$	109.728	−	256.047i	0.288001	−	0.672039i
$382$		0			0
$383$	−535.837			−1.39905			−0.699526	−	0.714607i	$-0.746606\pi$
$383$	−535.837			−1.39905			−0.699526	+	0.714607i	$0.746606\pi$
$384$		0			0
$385$	483.480	+	42.0833i	1.25579	+	0.109307i
$386$		0			0
$387$	−458.364	+	436.572i	−1.18440	+	1.12809i
$388$		0			0
$389$		−	718.536i		−	1.84714i	−0.383435	−	0.923568i	$-0.625259\pi$
$389$		−	718.536i		−	1.84714i	0.383435	−	0.923568i	$-0.374741\pi$
$390$		0			0
$391$	123.665			0.316279
$392$		0			0
$393$	−106.605	+	248.758i	−0.271259	+	0.632972i
$394$		0			0
$395$	233.395	+	20.3153i	0.590873	+	0.0514311i
$396$		0			0
$397$		−	287.466i		−	0.724096i	−0.932160	−	0.362048i	$-0.882078\pi$
$397$		−	287.466i		−	0.724096i	0.932160	−	0.362048i	$-0.117922\pi$
$398$		0			0
$399$	−481.022	+	1122.45i	−1.20557	+	2.81315i
$400$		0			0
$401$			622.455i			1.55226i	0.630575	+	0.776128i	$0.282819\pi$
$401$			622.455i			1.55226i	−0.630575	+	0.776128i	$0.717181\pi$
$402$		0			0
$403$		−	691.881i		−	1.71683i
$404$		0			0
$405$	−54.7159	+	401.287i	−0.135101	+	0.990832i
$406$		0			0
$407$	98.0515			0.240913
$408$		0			0
$409$	−352.901			−0.862840			−0.431420	−	0.902151i	$-0.641987\pi$
$409$	−352.901			−0.862840			−0.431420	+	0.902151i	$0.641987\pi$
$410$		0			0
$411$	−335.075	−	143.596i	−0.815267	−	0.349381i
$412$		0			0
$413$	121.806			0.294930
$414$		0			0
$415$	661.262	+	57.5579i	1.59340	+	0.138694i
$416$		0			0
$417$	−88.2294	−	37.8105i	−0.211581	−	0.0906728i
$418$		0			0
$419$		−	622.927i		−	1.48670i	−0.668903	−	0.743350i	$-0.733236\pi$
$419$		−	622.927i		−	1.48670i	0.668903	−	0.743350i	$-0.266764\pi$
$420$		0			0
$421$	−242.097			−0.575052			−0.287526	−	0.957773i	$-0.592833\pi$
$421$	−242.097			−0.575052			−0.287526	+	0.957773i	$0.592833\pi$
$422$		0			0
$423$	247.314	+	259.660i	0.584668	+	0.613853i
$424$		0			0
$425$	101.527	+	17.8093i	0.238888	+	0.0419043i
$426$		0			0
$427$		−	377.450i		−	0.883958i
$428$		0			0
$429$	285.646	+	122.413i	0.665841	+	0.285345i
$430$		0			0
$431$		−	454.978i		−	1.05563i	−0.849359	−	0.527816i	$-0.823011\pi$
$431$		−	454.978i		−	1.05563i	0.849359	−	0.527816i	$-0.176989\pi$
$432$		0			0
$433$			382.897i			0.884288i	0.896944	+	0.442144i	$0.145782\pi$
$433$			382.897i			0.884288i	−0.896944	+	0.442144i	$0.854218\pi$
$434$		0			0
$435$	343.518	−	641.408i	0.789695	−	1.47450i
$436$		0			0
$437$	915.992			2.09609
$438$		0			0
$439$	126.812			0.288865			0.144433	−	0.989515i	$-0.453864\pi$
$439$	126.812			0.288865			0.144433	+	0.989515i	$0.453864\pi$
$440$		0			0
$441$	−798.571	−	838.433i	−1.81082	−	1.90121i
$442$		0			0
$443$	254.887			0.575365			0.287682	−	0.957726i	$-0.407115\pi$
$443$	254.887			0.575365			0.287682	+	0.957726i	$0.407115\pi$
$444$		0			0
$445$	−8.11419	+	93.2210i	−0.0182341	+	0.209485i
$446$		0			0
$447$	92.4120	−	215.640i	0.206738	−	0.482416i
$448$		0			0
$449$			863.869i			1.92398i	0.273076	+	0.961992i	$0.411959\pi$
$449$			863.869i			1.92398i	−0.273076	+	0.961992i	$0.588041\pi$
$450$		0			0
$451$	135.962			0.301469
$452$		0			0
$453$	−412.703	−	176.863i	−0.911044	−	0.390426i
$454$		0			0
$455$	944.444	+	82.2068i	2.07570	+	0.180674i
$456$		0			0
$457$		−	222.183i		−	0.486178i	−0.970004	−	0.243089i	$-0.921839\pi$
$457$		−	222.183i		−	0.486178i	0.970004	−	0.243089i	$-0.0781607\pi$
$458$		0			0
$459$	−38.8183	−	104.337i	−0.0845714	−	0.227313i
$460$		0			0
$461$			427.601i			0.927551i	0.885953	+	0.463775i	$0.153506\pi$
$461$			427.601i			0.927551i	−0.885953	+	0.463775i	$0.846494\pi$
$462$		0			0
$463$			437.238i			0.944358i	0.881503	+	0.472179i	$0.156532\pi$
$463$			437.238i			0.944358i	−0.881503	+	0.472179i	$0.843468\pi$
$464$		0			0
$465$	643.137	+	344.444i	1.38309	+	0.740739i
$466$		0			0
$467$	407.465			0.872516			0.436258	−	0.899822i	$-0.356304\pi$
$467$	407.465			0.872516			0.436258	+	0.899822i	$0.356304\pi$
$468$		0			0
$469$	−618.245			−1.31822
$470$		0			0
$471$	−237.865	+	555.049i	−0.505022	+	1.17845i
$472$		0			0
$473$	−512.181			−1.08284
$474$		0			0
$475$	752.019	+	131.915i	1.58320	+	0.277715i
$476$		0			0
$477$	252.684	+	265.297i	0.529736	+	0.556179i
$478$		0			0
$479$			510.262i			1.06527i	0.846347	+	0.532633i	$0.178797\pi$
$479$			510.262i			1.06527i	−0.846347	+	0.532633i	$0.821203\pi$
$480$		0			0
$481$	191.537			0.398206
$482$		0			0
$483$	−472.408	+	1102.35i	−0.978071	+	2.28229i
$484$		0			0
$485$	18.6854	−	214.670i	0.0385267	−	0.442619i
$486$		0			0
$487$		−	918.558i		−	1.88616i	−0.332572	−	0.943078i	$-0.607916\pi$
$487$		−	918.558i		−	1.88616i	0.332572	−	0.943078i	$-0.392084\pi$
$488$		0			0
$489$	19.8728	−	46.3724i	0.0406396	−	0.0948310i
$490$		0			0
$491$			293.290i			0.597332i	0.954358	+	0.298666i	$0.0965417\pi$
$491$			293.290i			0.597332i	−0.954358	+	0.298666i	$0.903458\pi$
$492$		0			0
$493$			199.999i			0.405678i
$494$		0			0
$495$	−255.994	+	204.580i	−0.517159	+	0.413293i
$496$		0			0
$497$	−1273.77			−2.56292
$498$		0			0
$499$	47.2362			0.0946618			0.0473309	−	0.998879i	$-0.484928\pi$
$499$	47.2362			0.0946618			0.0473309	+	0.998879i	$0.484928\pi$
$500$		0			0
$501$	469.671	+	201.277i	0.937468	+	0.401750i
$502$		0			0
$503$	−45.4136			−0.0902855			−0.0451428	−	0.998981i	$-0.514374\pi$
$503$	−45.4136			−0.0902855			−0.0451428	+	0.998981i	$0.514374\pi$
$504$		0			0
$505$	−31.3308	+	359.949i	−0.0620412	+	0.712770i
$506$		0			0
$507$	91.9790	+	39.4174i	0.181418	+	0.0777464i
$508$		0			0
$509$			864.768i			1.69896i	0.527625	+	0.849478i	$0.323083\pi$
$509$			864.768i			1.69896i	−0.527625	+	0.849478i	$0.676917\pi$
$510$		0			0
$511$	−985.596			−1.92876
$512$		0			0
$513$	−287.529	−	772.827i	−0.560485	−	1.50648i
$514$		0			0
$515$	38.7357	−	445.021i	0.0752150	−	0.864118i
$516$		0			0
$517$			290.147i			0.561212i
$518$		0			0
$519$	40.9124	+	17.5329i	0.0788293	+	0.0337821i
$520$		0			0
$521$			474.781i			0.911287i	0.890162	+	0.455644i	$0.150591\pi$
$521$			474.781i			0.911287i	−0.890162	+	0.455644i	$0.849409\pi$
$522$		0			0
$523$		−	956.640i		−	1.82914i	−0.404427	−	0.914570i	$-0.632529\pi$
$523$		−	956.640i		−	1.82914i	0.404427	−	0.914570i	$-0.367471\pi$
$524$		0			0
$525$	−546.593	+	836.980i	−1.04113	+	1.59425i
$526$		0			0
$527$	−200.538			−0.380528
$528$		0			0
$529$	370.589			0.700547
$530$		0			0
$531$	−59.5566	+	56.7250i	−0.112159	+	0.106827i
$532$		0			0
$533$	265.593			0.498298
$534$		0			0
$535$	−465.821	−	40.5462i	−0.870693	−	0.0757873i
$536$		0			0
$537$	183.203	−	427.497i	0.341160	−	0.796084i
$538$		0			0
$539$		−	936.875i		−	1.73817i
$540$		0			0
$541$	220.979			0.408464			0.204232	−	0.978923i	$-0.434530\pi$
$541$	220.979			0.408464			0.204232	+	0.978923i	$0.434530\pi$
$542$		0			0
$543$	584.496	+	250.485i	1.07642	+	0.461298i
$544$		0			0
$545$	386.269	+	33.6218i	0.708751	+	0.0616914i
$546$		0			0
$547$			314.582i			0.575104i	0.957765	+	0.287552i	$0.0928413\pi$
$547$			314.582i			0.575104i	−0.957765	+	0.287552i	$0.907159\pi$
$548$		0			0
$549$	175.778	+	184.553i	0.320179	+	0.336161i
$550$		0			0
$551$			1481.40i			2.68857i
$552$		0			0
$553$		−	624.521i		−	1.12933i
$554$		0			0
$555$	−95.3540	+	178.043i	−0.171809	+	0.320797i
$556$		0			0
$557$	−1096.11			−1.96788			−0.983940	−	0.178497i	$-0.942877\pi$
$557$	−1096.11			−1.96788			−0.983940	+	0.178497i	$0.942877\pi$
$558$		0			0
$559$	−1000.51			−1.78982
$560$		0			0
$561$	35.4808	−	82.7931i	0.0632456	−	0.147581i
$562$		0			0
$563$	386.925			0.687257			0.343628	−	0.939106i	$-0.388344\pi$
$563$	386.925			0.687257			0.343628	+	0.939106i	$0.388344\pi$
$564$		0			0
$565$	−79.1086	−	6.88581i	−0.140015	−	0.0121873i
$566$		0			0
$567$	1078.34	+	52.5479i	1.90184	+	0.0926771i
$568$		0			0
$569$			113.274i			0.199076i	0.995034	+	0.0995381i	$0.0317365\pi$
$569$			113.274i			0.199076i	−0.995034	+	0.0995381i	$0.968263\pi$
$570$		0			0
$571$	434.868			0.761589			0.380795	−	0.924660i	$-0.375650\pi$
$571$	434.868			0.761589			0.380795	+	0.924660i	$0.375650\pi$
$572$		0			0
$573$	−352.763	+	823.158i	−0.615642	+	1.43658i
$574$		0			0
$575$	738.552	+	129.552i	1.28444	+	0.225309i
$576$		0			0
$577$		−	283.194i		−	0.490805i	−0.969421	−	0.245402i	$-0.921080\pi$
$577$		−	283.194i		−	0.490805i	0.969421	−	0.245402i	$-0.0789200\pi$
$578$		0			0
$579$	−309.673	+	722.610i	−0.534842	+	1.24803i
$580$		0			0
$581$		−	1769.41i		−	3.04546i
$582$		0			0
$583$			296.446i			0.508484i
$584$		0			0
$585$	−500.066	+	399.633i	−0.854813	+	0.683133i
$586$		0			0
$587$	80.5141			0.137162			0.0685810	−	0.997646i	$-0.478153\pi$
$587$	80.5141			0.137162			0.0685810	+	0.997646i	$0.478153\pi$
$588$		0			0
$589$	−1485.40			−2.52190
$590$		0			0
$591$	747.865	+	320.496i	1.26542	+	0.542295i
$592$		0			0
$593$	583.731			0.984370			0.492185	−	0.870491i	$-0.336198\pi$
$593$	583.731			0.984370			0.492185	+	0.870491i	$0.336198\pi$
$594$		0			0
$595$	23.8272	−	273.742i	0.0400458	−	0.460071i
$596$		0			0
$597$	484.631	+	207.688i	0.811777	+	0.347885i
$598$		0			0
$599$		−	1080.54i		−	1.80391i	−0.431834	−	0.901953i	$-0.642133\pi$
$599$		−	1080.54i		−	1.80391i	0.431834	−	0.901953i	$-0.357867\pi$
$600$		0			0
$601$	689.594			1.14741			0.573706	−	0.819062i	$-0.305505\pi$
$601$	689.594			1.14741			0.573706	+	0.819062i	$0.305505\pi$
$602$		0			0
$603$	302.289	−	287.917i	0.501308	−	0.477474i
$604$		0			0
$605$	338.570	+	29.4700i	0.559620	+	0.0487107i
$606$		0			0
$607$		−	466.053i		−	0.767798i	−0.923375	−	0.383899i	$-0.874581\pi$
$607$		−	466.053i		−	0.767798i	0.923375	−	0.383899i	$-0.125419\pi$
$608$		0			0
$609$	−1782.79	−	764.010i	−2.92740	−	1.25453i
$610$		0			0
$611$			566.781i			0.927629i
$612$		0			0
$613$			754.913i			1.23151i	0.787939	+	0.615753i	$0.211148\pi$
$613$			754.913i			1.23151i	−0.787939	+	0.615753i	$0.788852\pi$
$614$		0			0
$615$	−132.222	+	246.881i	−0.214995	+	0.401433i
$616$		0			0
$617$	685.857			1.11160			0.555800	−	0.831316i	$-0.312412\pi$
$617$	685.857			1.11160			0.555800	+	0.831316i	$0.312412\pi$
$618$		0			0
$619$	−633.092			−1.02277			−0.511383	−	0.859353i	$-0.670867\pi$
$619$	−633.092			−1.02277			−0.511383	+	0.859353i	$0.670867\pi$
$620$		0			0
$621$	−282.380	−	758.988i	−0.454718	−	1.22220i
$622$		0			0
$623$	249.442			0.400388
$624$		0			0
$625$	587.686	+	212.722i	0.940297	+	0.340355i
$626$		0			0
$627$	262.808	−	613.252i	0.419152	−	0.978074i
$628$		0			0
$629$		−	55.5160i		−	0.0882608i
$630$		0			0
$631$	374.848			0.594054			0.297027	−	0.954869i	$-0.404005\pi$
$631$	374.848			0.594054			0.297027	+	0.954869i	$0.404005\pi$
$632$		0			0
$633$	546.061	+	234.013i	0.862655	+	0.369689i
$634$		0			0
$635$	−40.2599	+	462.532i	−0.0634015	+	0.728397i
$636$		0			0
$637$		−	1830.12i		−	2.87303i
$638$		0			0
$639$	622.805	−	593.195i	0.974656	−	0.928317i
$640$		0			0
$641$		−	36.2900i		−	0.0566147i	−0.999599	−	0.0283073i	$-0.990988\pi$
$641$		−	36.2900i		−	0.0566147i	0.999599	−	0.0283073i	$-0.00901171\pi$
$642$		0			0
$643$			421.273i			0.655168i	0.944822	+	0.327584i	$0.106234\pi$
$643$			421.273i			0.655168i	−0.944822	+	0.327584i	$0.893766\pi$
$644$		0			0
$645$	498.090	−	930.022i	0.772233	−	1.44189i
$646$		0			0
$647$	−999.955			−1.54553			−0.772763	−	0.634695i	$-0.781126\pi$
$647$	−999.955			−1.54553			−0.772763	+	0.634695i	$0.781126\pi$
$648$		0			0
$649$	−66.5492			−0.102541
$650$		0			0
$651$	766.070	−	1787.59i	1.17676	−	2.74592i
$652$		0			0
$653$	783.570			1.19995			0.599977	−	0.800017i	$-0.295176\pi$
$653$	783.570			1.19995			0.599977	+	0.800017i	$0.295176\pi$
$654$		0			0
$655$	39.1138	−	449.365i	0.0597158	−	0.686053i
$656$		0			0
$657$	481.903	−	458.992i	0.733490	−	0.698617i
$658$		0			0
$659$			123.974i			0.188124i	0.995566	+	0.0940620i	$0.0299852\pi$
$659$			123.974i			0.188124i	−0.995566	+	0.0940620i	$0.970015\pi$
$660$		0			0
$661$	117.706			0.178072			0.0890360	−	0.996028i	$-0.471621\pi$
$661$	117.706			0.178072			0.0890360	+	0.996028i	$0.471621\pi$
$662$		0			0
$663$	69.3093	−	161.731i	0.104539	−	0.243937i
$664$		0			0
$665$	176.489	−	2027.62i	0.265398	−	3.04906i
$666$		0			0
$667$			1454.88i			2.18122i
$668$		0			0
$669$	8.12649	−	18.9628i	0.0121472	−	0.0283451i
$670$		0			0
$671$			206.221i			0.307334i
$672$		0			0
$673$			539.523i			0.801669i	0.916150	+	0.400835i	$0.131280\pi$
$673$			539.523i			0.801669i	−0.916150	+	0.400835i	$0.868720\pi$
$674$		0			0
$675$	−122.527	−	663.786i	−0.181521	−	0.983387i
$676$		0			0
$677$	645.528			0.953512			0.476756	−	0.879036i	$-0.341812\pi$
$677$	645.528			0.953512			0.476756	+	0.879036i	$0.341812\pi$
$678$		0			0
$679$	−574.417			−0.845975
$680$		0			0
$681$	251.859	+	107.934i	0.369837	+	0.158493i
$682$		0			0
$683$	−796.825			−1.16665			−0.583327	−	0.812237i	$-0.698249\pi$
$683$	−796.825			−1.16665			−0.583327	+	0.812237i	$0.698249\pi$
$684$		0			0
$685$	605.290	+	52.6860i	0.883636	+	0.0769139i
$686$		0			0
$687$	−428.968	−	183.833i	−0.624407	−	0.267588i
$688$		0			0
$689$			579.087i			0.840475i
$690$		0			0
$691$	−1302.35			−1.88474			−0.942369	−	0.334575i	$-0.891407\pi$
$691$	−1302.35			−1.88474			−0.942369	+	0.334575i	$0.891407\pi$
$692$		0			0
$693$	602.476	+	632.550i	0.869374	+	0.912770i
$694$		0			0
$695$	159.381	+	13.8729i	0.229325	+	0.0199610i
$696$		0			0
$697$		−	76.9808i		−	0.110446i
$698$		0			0
$699$	445.158	+	190.772i	0.636850	+	0.272921i
$700$		0			0
$701$		−	260.882i		−	0.372157i	−0.982535	−	0.186078i	$-0.940422\pi$
$701$		−	260.882i		−	0.372157i	0.982535	−	0.186078i	$-0.0595778\pi$
$702$		0			0
$703$		−	411.210i		−	0.584936i
$704$		0			0
$705$	−526.850	−	282.164i	−0.747305	−	0.400233i
$706$		0			0
$707$	963.155			1.36231
$708$		0			0
$709$	440.863			0.621809			0.310905	−	0.950441i	$-0.399368\pi$
$709$	440.863			0.621809			0.310905	+	0.950441i	$0.399368\pi$
$710$		0			0
$711$	290.839	+	305.357i	0.409057	+	0.429476i
$712$		0			0
$713$	−1458.80			−2.04600
$714$		0			0
$715$	−516.000	−	44.9140i	−0.721679	−	0.0628167i
$716$		0			0
$717$	251.502	−	586.869i	0.350769	−	0.818506i
$718$		0			0
$719$			1359.12i			1.89029i	0.326648	+	0.945146i	$0.394081\pi$
$719$			1359.12i			1.89029i	−0.326648	+	0.945146i	$0.605919\pi$
$720$		0			0
$721$	−1190.79			−1.65158
$722$		0			0
$723$	−1229.37	−	526.845i	−1.70038	−	0.728692i
$724$		0			0
$725$	−209.521	+	1194.44i	−0.288994	+	1.64750i
$726$		0			0
$727$			67.9032i			0.0934019i	0.998909	+	0.0467009i	$0.0148708\pi$
$727$			67.9032i			0.0934019i	−0.998909	+	0.0467009i	$0.985129\pi$
$728$		0			0
$729$	−551.723	+	476.490i	−0.756821	+	0.653622i
$730$		0			0
$731$			289.993i			0.396707i
$732$		0			0
$733$			728.945i			0.994468i	0.867617	+	0.497234i	$0.165651\pi$
$733$			728.945i			0.994468i	−0.867617	+	0.497234i	$0.834349\pi$
$734$		0			0
$735$	1701.18	+	911.100i	2.31454	+	1.23959i
$736$		0			0
$737$	337.781			0.458318
$738$		0			0
$739$	1008.11			1.36416			0.682080	−	0.731278i	$-0.261076\pi$
$739$	1008.11			1.36416			0.682080	+	0.731278i	$0.261076\pi$
$740$		0			0
$741$	513.377	−	1197.95i	0.692817	−	1.61666i
$742$		0			0
$743$	696.093			0.936869			0.468434	−	0.883498i	$-0.344818\pi$
$743$	696.093			0.936869			0.468434	+	0.883498i	$0.344818\pi$
$744$		0			0
$745$	−33.9065	+	389.539i	−0.0455120	+	0.522871i
$746$		0			0
$747$	824.015	+	865.147i	1.10310	+	1.15816i
$748$		0			0
$749$			1246.45i			1.66415i
$750$		0			0
$751$	783.678			1.04351			0.521756	−	0.853095i	$-0.325277\pi$
$751$	783.678			1.04351			0.521756	+	0.853095i	$0.325277\pi$
$752$		0			0
$753$	131.077	−	305.864i	0.174073	−	0.406193i
$754$		0			0
$755$	745.520	+	64.8919i	0.987444	+	0.0859496i
$756$		0			0
$757$		−	511.207i		−	0.675307i	−0.941270	−	0.337654i	$-0.890367\pi$
$757$		−	511.207i		−	0.675307i	0.941270	−	0.337654i	$-0.109633\pi$
$758$		0			0
$759$	258.102	−	602.271i	0.340055	−	0.793505i
$760$		0			0
$761$		−	371.165i		−	0.487734i	−0.969809	−	0.243867i	$-0.921584\pi$
$761$		−	371.165i		−	0.487734i	0.969809	−	0.243867i	$-0.0784160\pi$
$762$		0			0
$763$		−	1033.58i		−	1.35463i
$764$		0			0
$765$	115.832	+	144.942i	0.151414	+	0.189466i
$766$		0			0
$767$	−129.999			−0.169491
$768$		0			0
$769$	−61.6158			−0.0801245			−0.0400623	−	0.999197i	$-0.512756\pi$
$769$	−61.6158			−0.0801245			−0.0400623	+	0.999197i	$0.512756\pi$
$770$		0			0
$771$	154.733	+	66.3107i	0.200692	+	0.0860061i
$772$		0			0
$773$	−87.8197			−0.113609			−0.0568045	−	0.998385i	$-0.518091\pi$
$773$	−87.8197			−0.113609			−0.0568045	+	0.998385i	$0.518091\pi$
$774$		0			0
$775$	−1197.66	−	210.086i	−1.54536	−	0.271078i
$776$		0			0
$777$	494.868	+	212.075i	0.636896	+	0.272940i
$778$		0			0
$779$		−	570.201i		−	0.731965i
$780$		0			0
$781$	695.930			0.891075
$782$		0			0
$783$	1227.49	−	456.684i	1.56767	−	0.583249i
$784$		0			0
$785$	87.2740	−	1002.66i	0.111177	−	1.27727i
$786$		0			0
$787$			1333.59i			1.69453i	0.531171	+	0.847264i	$0.321752\pi$
$787$			1333.59i			1.69453i	−0.531171	+	0.847264i	$0.678248\pi$
$788$		0			0
$789$	−471.600	−	202.103i	−0.597719	−	0.256151i
$790$		0			0
$791$			211.680i			0.267610i
$792$		0			0
$793$			402.839i			0.507994i
$794$		0			0
$795$	−538.289	−	288.291i	−0.677093	−	0.362630i
$796$		0			0
$797$	309.090			0.387817			0.193908	−	0.981020i	$-0.437884\pi$
$797$	309.090			0.387817			0.193908	+	0.981020i	$0.437884\pi$
$798$		0			0
$799$	164.279			0.205605
$800$		0			0
$801$	−121.964	+	116.165i	−0.152264	+	0.145025i
$802$		0			0
$803$	538.484			0.670590
$804$		0			0
$805$	173.329	−	1991.31i	0.215315	−	2.47368i
$806$		0			0
$807$	236.506	−	551.879i	0.293069	−	0.683864i
$808$		0			0
$809$			1189.17i			1.46992i	0.678110	+	0.734960i	$0.262799\pi$
$809$			1189.17i			1.46992i	−0.678110	+	0.734960i	$0.737201\pi$
$810$		0			0
$811$	960.641			1.18451			0.592257	−	0.805749i	$-0.298237\pi$
$811$	960.641			1.18451			0.592257	+	0.805749i	$0.298237\pi$
$812$		0			0
$813$	26.7903	+	11.4809i	0.0329524	+	0.0141217i
$814$		0			0
$815$	−7.29143	+	83.7686i	−0.00894654	+	0.102784i
$816$		0			0
$817$			2147.99i			2.62912i
$818$		0			0
$819$	1176.90	+	1235.64i	1.43699	+	1.50872i
$820$		0			0
$821$		−	1071.24i		−	1.30480i	−0.757877	−	0.652398i	$-0.773763\pi$
$821$		−	1071.24i		−	1.30480i	0.757877	−	0.652398i	$-0.226237\pi$
$822$		0			0
$823$			915.823i			1.11279i	0.830919	+	0.556393i	$0.187815\pi$
$823$			915.823i			1.11279i	−0.830919	+	0.556393i	$0.812185\pi$
$824$		0			0
$825$	298.633	−	457.287i	0.361980	−	0.554287i
$826$		0			0
$827$	−505.257			−0.610952			−0.305476	−	0.952200i	$-0.598816\pi$
$827$	−505.257			−0.610952			−0.305476	+	0.952200i	$0.598816\pi$
$828$		0			0
$829$	−386.448			−0.466161			−0.233081	−	0.972457i	$-0.574881\pi$
$829$	−386.448			−0.466161			−0.233081	+	0.972457i	$0.574881\pi$
$830$		0			0
$831$	−18.0307	+	42.0740i	−0.0216976	+	0.0506305i
$832$		0			0
$833$	−530.451			−0.636796
$834$		0			0
$835$	−848.430	−	73.8495i	−1.01608	−	0.0884425i
$836$		0			0
$837$	457.915	+	1230.79i	0.547091	+	1.47048i
$838$		0			0
$839$		−	610.689i		−	0.727877i	−0.931423	−	0.363939i	$-0.881432\pi$
$839$		−	610.689i		−	0.727877i	0.931423	−	0.363939i	$-0.118568\pi$
$840$		0			0
$841$	−1511.92			−1.79777
$842$		0			0
$843$	165.635	−	386.503i	0.196483	−	0.458485i
$844$		0			0
$845$	−166.154	−	14.4625i	−0.196632	−	0.0171153i
$846$		0			0
$847$		−	905.950i		−	1.06960i
$848$		0			0
$849$	43.4423	−	101.371i	0.0511688	−	0.119400i
$850$		0			0
$851$		−	403.846i		−	0.474555i
$852$		0			0
$853$		−	1500.11i		−	1.75863i	−0.476243	−	0.879314i	$-0.658002\pi$
$853$		−	1500.11i		−	1.75863i	0.476243	−	0.879314i	$-0.341998\pi$
$854$		0			0
$855$	857.970	+	1073.59i	1.00347	+	1.25566i
$856$		0			0
$857$	127.645			0.148944			0.0744721	−	0.997223i	$-0.476273\pi$
$857$	127.645			0.148944			0.0744721	+	0.997223i	$0.476273\pi$
$858$		0			0
$859$	46.8391			0.0545274			0.0272637	−	0.999628i	$-0.491321\pi$
$859$	46.8391			0.0545274			0.0272637	+	0.999628i	$0.491321\pi$
$860$		0			0
$861$	686.205	+	294.072i	0.796986	+	0.341547i
$862$		0			0
$863$	−554.819			−0.642895			−0.321448	−	0.946927i	$-0.604169\pi$
$863$	−554.819			−0.642895			−0.321448	+	0.946927i	$0.604169\pi$
$864$		0			0
$865$	−73.9056	−	6.43292i	−0.0854400	−	0.00743691i
$866$		0			0
$867$	−46.8768	−	20.0890i	−0.0540678	−	0.0231707i
$868$		0			0
$869$			341.210i			0.392646i
$870$		0			0
$871$	659.831			0.757556
$872$		0			0
$873$	280.859	−	267.506i	0.321717	−	0.306421i
$874$		0			0
$875$	429.076	−	1609.88i	0.490372	−	1.83987i
$876$		0			0
$877$			775.419i			0.884172i	0.896973	+	0.442086i	$0.145761\pi$
$877$			775.419i			0.884172i	−0.896973	+	0.442086i	$0.854239\pi$
$878$		0			0
$879$	1214.18	+	520.335i	1.38132	+	0.591962i
$880$		0			0
$881$			826.534i			0.938177i	0.883151	+	0.469089i	$0.155418\pi$
$881$			826.534i			0.938177i	−0.883151	+	0.469089i	$0.844582\pi$
$882$		0			0
$883$		−	1422.79i		−	1.61131i	−0.592383	−	0.805657i	$-0.701813\pi$
$883$		−	1422.79i		−	1.61131i	0.592383	−	0.805657i	$-0.298187\pi$
$884$		0			0
$885$	64.7183	−	120.840i	0.0731280	−	0.136543i
$886$		0			0
$887$	503.369			0.567496			0.283748	−	0.958899i	$-0.408422\pi$
$887$	503.369			0.567496			0.283748	+	0.958899i	$0.408422\pi$
$888$		0			0
$889$	1237.65			1.39218
$890$		0			0
$891$	−589.156	−	28.7098i	−0.661230	−	0.0322220i
$892$		0			0
$893$	1216.82			1.36262
$894$		0			0
$895$	−67.2182	+	772.245i	−0.0751041	+	0.862844i
$896$		0			0
$897$	504.184	−	1176.49i	0.562078	−	1.31159i
$898$		0			0
$899$		−	2359.27i		−	2.62432i
$900$		0			0
$901$	167.846			0.186288
$902$		0			0
$903$	−2584.99	−	1107.79i	−2.86267	−	1.22679i
$904$		0			0
$905$	−1055.85	−	91.9041i	−1.16669	−	0.101552i
$906$		0			0
$907$			1552.85i			1.71208i	0.516911	+	0.856039i	$0.327082\pi$
$907$			1552.85i			1.71208i	−0.516911	+	0.856039i	$0.672918\pi$
$908$		0			0
$909$	−470.931	+	448.541i	−0.518076	+	0.493444i
$910$		0			0
$911$			1017.10i			1.11646i	0.829686	+	0.558230i	$0.188519\pi$
$911$			1017.10i			1.11646i	−0.829686	+	0.558230i	$0.811481\pi$
$912$		0			0
$913$			966.725i			1.05884i
$914$		0			0
$915$	−374.458	−	200.548i	−0.409244	−	0.219178i
$916$		0			0
$917$	−1202.42			−1.31125
$918$		0			0
$919$	−1102.70			−1.19989			−0.599946	−	0.800040i	$-0.704811\pi$
$919$	−1102.70			−1.19989			−0.599946	+	0.800040i	$0.704811\pi$
$920$		0			0
$921$	−21.7351	+	50.7179i	−0.0235994	+	0.0550683i
$922$		0			0
$923$	1359.45			1.47286
$924$		0			0
$925$	58.1590	−	331.553i	0.0628746	−	0.358436i
$926$		0			0
$927$	582.233	−	554.552i	0.628083	−	0.598222i
$928$		0			0
$929$		−	560.608i		−	0.603454i	−0.953394	−	0.301727i	$-0.902437\pi$
$929$		−	560.608i		−	0.603454i	0.953394	−	0.301727i	$-0.0975631\pi$
$930$		0			0
$931$	−3929.08			−4.22028
$932$		0			0
$933$	143.357	−	334.517i	0.153651	−	0.358539i
$934$		0			0
$935$	−13.0181	+	149.560i	−0.0139231	+	0.159957i
$936$		0			0
$937$			673.304i			0.718574i	0.933227	+	0.359287i	$0.116980\pi$
$937$			673.304i			0.718574i	−0.933227	+	0.359287i	$0.883020\pi$
$938$		0			0
$939$	285.184	−	665.467i	0.303711	−	0.708697i
$940$		0			0
$941$			438.910i			0.466429i	0.972425	+	0.233214i	$0.0749244\pi$
$941$			438.910i			0.466429i	−0.972425	+	0.233214i	$0.925076\pi$
$942$		0			0
$943$		−	559.990i		−	0.593839i
$944$		0			0
$945$	−1734.49	+	478.833i	−1.83544	+	0.506701i
$946$		0			0
$947$	−1842.75			−1.94588			−0.972939	−	0.231063i	$-0.925780\pi$
$947$	−1842.75			−1.94588			−0.972939	+	0.231063i	$0.925780\pi$
$948$		0			0
$949$	1051.89			1.10842
$950$		0			0
$951$	−1344.50	−	576.185i	−1.41378	−	0.605872i
$952$		0			0
$953$	198.284			0.208063			0.104031	−	0.994574i	$-0.466826\pi$
$953$	198.284			0.208063			0.104031	+	0.994574i	$0.466826\pi$
$954$		0			0
$955$	129.430	−	1486.98i	0.135529	−	1.55705i
$956$		0			0
$957$	974.033	+	417.420i	1.01780	+	0.436175i
$958$		0			0
$959$		−	1619.64i		−	1.68889i
$960$		0			0
$961$	1404.63			1.46163
$962$		0			0
$963$	−580.471	−	609.446i	−0.602773	−	0.632862i
$964$		0			0
$965$	113.621	−	1305.35i	0.117742	−	1.35269i
$966$		0			0
$967$			889.888i			0.920256i	0.887853	+	0.460128i	$0.152197\pi$
$967$			889.888i			0.920256i	−0.887853	+	0.460128i	$0.847803\pi$
$968$		0			0
$969$	−347.219	−	148.800i	−0.358327	−	0.153560i
$970$		0			0
$971$			553.795i			0.570335i	0.958478	+	0.285167i	$0.0920492\pi$
$971$			553.795i			0.570335i	−0.958478	+	0.285167i	$0.907951\pi$
$972$		0			0
$973$		−	426.473i		−	0.438307i
$974$		0			0
$975$	583.359	−	893.279i	0.598317	−	0.916183i
$976$		0			0
$977$	−1047.10			−1.07175			−0.535877	−	0.844296i	$-0.680019\pi$
$977$	−1047.10			−1.07175			−0.535877	+	0.844296i	$0.680019\pi$
$978$		0			0
$979$	−136.283			−0.139207
$980$		0			0
$981$	481.340	+	505.367i	0.490662	+	0.515155i
$982$		0			0
$983$	257.621			0.262076			0.131038	−	0.991377i	$-0.458169\pi$
$983$	257.621			0.262076			0.131038	+	0.991377i	$0.458169\pi$
$984$		0			0
$985$	−1350.97	−	117.592i	−1.37154	−	0.119382i
$986$		0			0
$987$	−627.556	+	1464.38i	−0.635821	+	1.48366i
$988$		0			0
$989$			2109.53i			2.13299i
$990$		0			0
$991$	−1574.78			−1.58909			−0.794543	−	0.607208i	$-0.792289\pi$
$991$	−1574.78			−1.58909			−0.794543	+	0.607208i	$0.792289\pi$
$992$		0			0
$993$	727.196	+	311.638i	0.732322	+	0.313835i
$994$		0			0
$995$	−875.453	−	76.2016i	−0.879853	−	0.0765846i
$996$		0			0
$997$		−	1435.38i		−	1.43970i	−0.694129	−	0.719850i	$-0.744210\pi$
$997$		−	1435.38i		−	1.43970i	0.694129	−	0.719850i	$-0.255790\pi$
$998$		0			0
$999$	−340.727	+	126.767i	−0.341068	+	0.126894i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1020.3.c.a.749.9	✓	64
3.2	odd	2	inner	1020.3.c.a.749.55	yes	64
5.4	even	2	inner	1020.3.c.a.749.56	yes	64
15.14	odd	2	inner	1020.3.c.a.749.10	yes	64

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1020.3.c.a.749.9	✓	64	1.1	even	1	trivial
1020.3.c.a.749.10	yes	64	15.14	odd	2	inner
1020.3.c.a.749.55	yes	64	3.2	odd	2	inner
1020.3.c.a.749.56	yes	64	5.4	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1020 = 2^{2} \cdot 3 \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1020.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.75746 - 1.18170i) q^{3} +(4.98117 + 0.433573i) q^{5} -13.3287i q^{7} +(6.20715 + 6.51700i) q^{9} +O(q^{10})\)	\(q+(-2.75746 - 1.18170i) q^{3} +(4.98117 + 0.433573i) q^{5} -13.3287i q^{7} +(6.20715 + 6.51700i) q^{9} +7.28217i q^{11} +14.2252i q^{13} +(-13.2230 - 7.08182i) q^{15} +4.12311 q^{17} +30.5400 q^{19} +(-15.7505 + 36.7532i) q^{21} +29.9932 q^{23} +(24.6240 + 4.31940i) q^{25} +(-9.41481 - 25.3054i) q^{27} +48.5070i q^{29} -48.6377 q^{31} +(8.60536 - 20.0803i) q^{33} +(5.77895 - 66.3923i) q^{35} -13.4646i q^{37} +(16.8100 - 39.2254i) q^{39} -18.6706i q^{41} +70.3336i q^{43} +(28.0933 + 35.1535i) q^{45} +39.8435 q^{47} -128.653 q^{49} +(-11.3693 - 4.87229i) q^{51} +40.7085 q^{53} +(-3.15735 + 36.2737i) q^{55} +(-84.2129 - 36.0893i) q^{57} +9.13866i q^{59} +28.3187 q^{61} +(86.8629 - 82.7331i) q^{63} +(-6.16767 + 70.8581i) q^{65} -46.3846i q^{67} +(-82.7049 - 35.4430i) q^{69} -95.5663i q^{71} -73.9456i q^{73} +(-62.7955 - 41.0089i) q^{75} +97.0616 q^{77} +46.8555 q^{79} +(-3.94248 + 80.9040i) q^{81} +132.752 q^{83} +(20.5379 + 1.78767i) q^{85} +(57.3208 - 133.756i) q^{87} +18.7147i q^{89} +189.603 q^{91} +(134.116 + 57.4754i) q^{93} +(152.125 + 13.2413i) q^{95} -43.0964i q^{97} +(-47.4578 + 45.2015i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 4 q^{9}+O(q^{10}) \) 64 * q - 4 * q^9	\( 64 q - 4 q^{9} - 46 q^{15} + 28 q^{19} + 40 q^{21} + 10 q^{25} - 256 q^{31} - 260 q^{39} - 264 q^{45} - 552 q^{49} + 174 q^{55} - 296 q^{61} - 28 q^{69} - 184 q^{75} + 496 q^{79} + 596 q^{81} - 34 q^{85} - 56 q^{91} - 340 q^{99}+O(q^{100}) \) 64 * q - 4 * q^9 - 46 * q^15 + 28 * q^19 + 40 * q^21 + 10 * q^25 - 256 * q^31 - 260 * q^39 - 264 * q^45 - 552 * q^49 + 174 * q^55 - 296 * q^61 - 28 * q^69 - 184 * q^75 + 496 * q^79 + 596 * q^81 - 34 * q^85 - 56 * q^91 - 340 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more