LMFDB - Embedded newform 156.3.d.a.53.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [156,3,Mod(53,156)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(156, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("156.53");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 156 = 2^{2} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	156.d (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:	$4.25069212402$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 15x^{6} + 10x^{5} + 97x^{4} + 252x^{3} + 700x^{2} + 1696x + 3792 $ x^8 - 2x^7 + 15x^6 + 10x^5 + 97x^4 + 252x^3 + 700x^2 + 1696*x + 3792
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{5}\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			53.8
Root			$-1.83932 - 0.968627i$ of defining polynomial
Character	$\chi$	$=$	156.53
Dual form			156.3.d.a.53.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.83932 + 0.968627i) q^{3} -5.58779i q^{5} +3.90626 q^{7} +(7.12352 + 5.50049i) q^{9} +O(q^{10})$	\(q+(2.83932 + 0.968627i) q^{3} -5.58779i q^{5} +3.90626 q^{7} +(7.12352 + 5.50049i) q^{9} -14.3353i q^{11} +3.60555 q^{13} +(5.41249 - 15.8656i) q^{15} +26.6835i q^{17} -2.16283 q^{19} +(11.0911 + 3.78371i) q^{21} +17.4950i q^{23} -6.22343 q^{25} +(14.8981 + 22.5177i) q^{27} -43.2229i q^{29} -24.0555 q^{31} +(13.8856 - 40.7026i) q^{33} -21.8274i q^{35} +3.55594 q^{37} +(10.2373 + 3.49243i) q^{39} +48.3100i q^{41} -84.6256 q^{43} +(30.7356 - 39.8048i) q^{45} +33.8105i q^{47} -33.7411 q^{49} +(-25.8464 + 75.7631i) q^{51} +9.24821i q^{53} -80.1028 q^{55} +(-6.14098 - 2.09498i) q^{57} +36.2385i q^{59} +23.8927 q^{61} +(27.8263 + 21.4863i) q^{63} -20.1471i q^{65} -112.566 q^{67} +(-16.9462 + 49.6741i) q^{69} -93.9449i q^{71} +24.0719 q^{73} +(-17.6703 - 6.02818i) q^{75} -55.9974i q^{77} +44.4386 q^{79} +(20.4892 + 78.3658i) q^{81} +55.1829i q^{83} +149.102 q^{85} +(41.8669 - 122.724i) q^{87} -77.4146i q^{89} +14.0842 q^{91} +(-68.3014 - 23.3008i) q^{93} +12.0855i q^{95} +143.143 q^{97} +(78.8513 - 102.118i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 6 q^{3} - 8 q^{7} - 22 q^{9}+O(q^{10}) $ 8 * q + 6 * q^3 - 8 * q^7 - 22 * q^9	$ 8 q + 6 q^{3} - 8 q^{7} - 22 q^{9} + 4 q^{15} - 24 q^{19} + 16 q^{21} - 28 q^{25} + 36 q^{27} + 96 q^{31} + 4 q^{33} - 96 q^{37} + 76 q^{43} - 136 q^{45} + 204 q^{49} - 62 q^{51} - 80 q^{55} + 184 q^{57} - 104 q^{61} + 36 q^{63} - 384 q^{67} - 8 q^{73} - 100 q^{75} + 328 q^{79} + 50 q^{81} - 64 q^{85} - 204 q^{87} - 52 q^{91} + 72 q^{93} + 416 q^{97} + 284 q^{99}+O(q^{100}) $ 8 * q + 6 * q^3 - 8 * q^7 - 22 * q^9 + 4 * q^15 - 24 * q^19 + 16 * q^21 - 28 * q^25 + 36 * q^27 + 96 * q^31 + 4 * q^33 - 96 * q^37 + 76 * q^43 - 136 * q^45 + 204 * q^49 - 62 * q^51 - 80 * q^55 + 184 * q^57 - 104 * q^61 + 36 * q^63 - 384 * q^67 - 8 * q^73 - 100 * q^75 + 328 * q^79 + 50 * q^81 - 64 * q^85 - 204 * q^87 - 52 * q^91 + 72 * q^93 + 416 * q^97 + 284 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/156\mathbb{Z}\right)^\times$.

$n$	$53$	$79$	$145$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	2.83932	+	0.968627i	0.946441	+	0.322876i
$3$	2.83932	+	0.968627i	0.946441	+	0.322876i
$4$		0			0
$5$		−	5.58779i		−	1.11756i	−0.829316	−	0.558779i	$-0.811270\pi$
$5$		−	5.58779i		−	1.11756i	0.829316	−	0.558779i	$-0.188730\pi$
$6$		0			0
$7$	3.90626			0.558037			0.279018	−	0.960286i	$-0.409991\pi$
$7$	3.90626			0.558037			0.279018	+	0.960286i	$0.409991\pi$
$8$		0			0
$9$	7.12352	+	5.50049i	0.791503	+	0.611166i
$10$		0			0
$11$		−	14.3353i		−	1.30321i	−0.758558	−	0.651605i	$-0.774096\pi$
$11$		−	14.3353i		−	1.30321i	0.758558	−	0.651605i	$-0.225904\pi$
$12$		0			0
$13$	3.60555			0.277350
$13$	3.60555			0.277350
$14$		0			0
$15$	5.41249	−	15.8656i	0.360832	−	1.05770i
$16$		0			0
$17$			26.6835i			1.56962i	0.619738	+	0.784809i	$0.287239\pi$
$17$			26.6835i			1.56962i	−0.619738	+	0.784809i	$0.712761\pi$
$18$		0			0
$19$	−2.16283			−0.113833			−0.0569166	−	0.998379i	$-0.518127\pi$
$19$	−2.16283			−0.113833			−0.0569166	+	0.998379i	$0.518127\pi$
$20$		0			0
$21$	11.0911	+	3.78371i	0.528149	+	0.180177i
$22$		0			0
$23$			17.4950i			0.760654i	0.924852	+	0.380327i	$0.124189\pi$
$23$			17.4950i			0.760654i	−0.924852	+	0.380327i	$0.875811\pi$
$24$		0			0
$25$	−6.22343			−0.248937
$26$		0			0
$27$	14.8981	+	22.5177i	0.551780	+	0.833989i
$28$		0			0
$29$		−	43.2229i		−	1.49045i	−0.666816	−	0.745223i	$-0.732343\pi$
$29$		−	43.2229i		−	1.49045i	0.666816	−	0.745223i	$-0.267657\pi$
$30$		0			0
$31$	−24.0555			−0.775984			−0.387992	−	0.921663i	$-0.626831\pi$
$31$	−24.0555			−0.775984			−0.387992	+	0.921663i	$0.626831\pi$
$32$		0			0
$33$	13.8856	−	40.7026i	0.420775	−	1.23341i
$34$		0			0
$35$		−	21.8274i		−	0.623639i
$36$		0			0
$37$	3.55594			0.0961066			0.0480533	−	0.998845i	$-0.484698\pi$
$37$	3.55594			0.0961066			0.0480533	+	0.998845i	$0.484698\pi$
$38$		0			0
$39$	10.2373	+	3.49243i	0.262496	+	0.0895496i
$40$		0			0
$41$			48.3100i			1.17829i	0.808026	+	0.589146i	$0.200536\pi$
$41$			48.3100i			1.17829i	−0.808026	+	0.589146i	$0.799464\pi$
$42$		0			0
$43$	−84.6256			−1.96804			−0.984018	−	0.178069i	$-0.943015\pi$
$43$	−84.6256			−1.96804			−0.984018	+	0.178069i	$0.943015\pi$
$44$		0			0
$45$	30.7356	−	39.8048i	0.683013	−	0.884551i
$46$		0			0
$47$			33.8105i			0.719372i	0.933073	+	0.359686i	$0.117116\pi$
$47$			33.8105i			0.719372i	−0.933073	+	0.359686i	$0.882884\pi$
$48$		0			0
$49$	−33.7411			−0.688595
$50$		0			0
$51$	−25.8464	+	75.7631i	−0.506791	+	1.48555i
$52$		0			0
$53$			9.24821i			0.174495i	0.996187	+	0.0872473i	$0.0278070\pi$
$53$			9.24821i			0.174495i	−0.996187	+	0.0872473i	$0.972193\pi$
$54$		0			0
$55$	−80.1028			−1.45641
$56$		0			0
$57$	−6.14098	−	2.09498i	−0.107737	−	0.0367540i
$58$		0			0
$59$			36.2385i			0.614213i	0.951675	+	0.307106i	$0.0993607\pi$
$59$			36.2385i			0.614213i	−0.951675	+	0.307106i	$0.900639\pi$
$60$		0			0
$61$	23.8927			0.391683			0.195842	−	0.980636i	$-0.437256\pi$
$61$	23.8927			0.391683			0.195842	+	0.980636i	$0.437256\pi$
$62$		0			0
$63$	27.8263	+	21.4863i	0.441688	+	0.341053i
$64$		0			0
$65$		−	20.1471i		−	0.309955i
$66$		0			0
$67$	−112.566			−1.68009			−0.840045	−	0.542517i	$-0.817471\pi$
$67$	−112.566			−1.68009			−0.840045	+	0.542517i	$0.817471\pi$
$68$		0			0
$69$	−16.9462	+	49.6741i	−0.245597	+	0.719914i
$70$		0			0
$71$		−	93.9449i		−	1.32317i	−0.749871	−	0.661584i	$-0.769885\pi$
$71$		−	93.9449i		−	1.32317i	0.749871	−	0.661584i	$-0.230115\pi$
$72$		0			0
$73$	24.0719			0.329752			0.164876	−	0.986314i	$-0.447278\pi$
$73$	24.0719			0.329752			0.164876	+	0.986314i	$0.447278\pi$
$74$		0			0
$75$	−17.6703	−	6.02818i	−0.235604	−	0.0803757i
$76$		0			0
$77$		−	55.9974i		−	0.727239i
$78$		0			0
$79$	44.4386			0.562514			0.281257	−	0.959633i	$-0.409249\pi$
$79$	44.4386			0.562514			0.281257	+	0.959633i	$0.409249\pi$
$80$		0			0
$81$	20.4892	+	78.3658i	0.252953	+	0.967479i
$82$		0			0
$83$			55.1829i			0.664854i	0.943129	+	0.332427i	$0.107868\pi$
$83$			55.1829i			0.664854i	−0.943129	+	0.332427i	$0.892132\pi$
$84$		0			0
$85$	149.102			1.75414
$86$		0			0
$87$	41.8669	−	122.724i	0.481228	−	1.41062i
$88$		0			0
$89$		−	77.4146i		−	0.869827i	−0.900472	−	0.434913i	$-0.856779\pi$
$89$		−	77.4146i		−	0.869827i	0.900472	−	0.434913i	$-0.143221\pi$
$90$		0			0
$91$	14.0842			0.154772
$92$		0			0
$93$	−68.3014	−	23.3008i	−0.734424	−	0.250546i
$94$		0			0
$95$			12.0855i			0.127215i
$96$		0			0
$97$	143.143			1.47570			0.737850	−	0.674965i	$-0.235841\pi$
$97$	143.143			1.47570			0.737850	+	0.674965i	$0.235841\pi$
$98$		0			0
$99$	78.8513	−	102.118i	0.796477	−	1.03149i
$100$		0			0
$101$		−	3.99037i		−	0.0395086i	−0.999805	−	0.0197543i	$-0.993712\pi$
$101$		−	3.99037i		−	0.0395086i	0.999805	−	0.0197543i	$-0.00628840\pi$
$102$		0			0
$103$	−67.2250			−0.652670			−0.326335	−	0.945254i	$-0.605814\pi$
$103$	−67.2250			−0.652670			−0.326335	+	0.945254i	$0.605814\pi$
$104$		0			0
$105$	21.1426	−	61.9750i	0.201358	−	0.590238i
$106$		0			0
$107$			83.3513i			0.778984i	0.921030	+	0.389492i	$0.127349\pi$
$107$			83.3513i			0.778984i	−0.921030	+	0.389492i	$0.872651\pi$
$108$		0			0
$109$	152.682			1.40076			0.700378	−	0.713773i	$-0.253015\pi$
$109$	152.682			1.40076			0.700378	+	0.713773i	$0.253015\pi$
$110$		0			0
$111$	10.0965	+	3.44438i	0.0909593	+	0.0310305i
$112$		0			0
$113$			176.746i			1.56413i	0.623199	+	0.782064i	$0.285833\pi$
$113$			176.746i			1.56413i	−0.623199	+	0.782064i	$0.714167\pi$
$114$		0			0
$115$	97.7586			0.850075
$116$		0			0
$117$	25.6842	+	19.8323i	0.219523	+	0.169507i
$118$		0			0
$119$			104.233i			0.875905i
$120$		0			0
$121$	−84.5012			−0.698357
$122$		0			0
$123$	−46.7944	+	137.168i	−0.380442	+	1.11519i
$124$		0			0
$125$		−	104.920i		−	0.839357i
$126$		0			0
$127$	113.451			0.893318			0.446659	−	0.894704i	$-0.352614\pi$
$127$	113.451			0.893318			0.446659	+	0.894704i	$0.352614\pi$
$128$		0			0
$129$	−240.279	−	81.9706i	−1.86263	−	0.635431i
$130$		0			0
$131$		−	183.454i		−	1.40041i	−0.713941	−	0.700206i	$-0.753091\pi$
$131$		−	183.454i		−	1.40041i	0.713941	−	0.700206i	$-0.246909\pi$
$132$		0			0
$133$	−8.44858			−0.0635232
$134$		0			0
$135$	125.824	−	83.2473i	0.932032	−	0.616647i
$136$		0			0
$137$		−	242.400i		−	1.76934i	−0.466217	−	0.884670i	$-0.654383\pi$
$137$		−	242.400i		−	1.76934i	0.466217	−	0.884670i	$-0.345617\pi$
$138$		0			0
$139$	−8.43247			−0.0606652			−0.0303326	−	0.999540i	$-0.509657\pi$
$139$	−8.43247			−0.0606652			−0.0303326	+	0.999540i	$0.509657\pi$
$140$		0			0
$141$	−32.7497	+	95.9989i	−0.232268	+	0.680843i
$142$		0			0
$143$		−	51.6867i		−	0.361445i
$144$		0			0
$145$	−241.521			−1.66566
$146$		0			0
$147$	−95.8020	−	32.6826i	−0.651715	−	0.222330i
$148$		0			0
$149$			240.727i			1.61562i	0.589443	+	0.807810i	$0.299347\pi$
$149$			240.727i			1.61562i	−0.589443	+	0.807810i	$0.700653\pi$
$150$		0			0
$151$	132.525			0.877650			0.438825	−	0.898573i	$-0.355395\pi$
$151$	132.525			0.877650			0.438825	+	0.898573i	$0.355395\pi$
$152$		0			0
$153$	−146.772	+	190.081i	−0.959296	+	1.24236i
$154$		0			0
$155$			134.417i			0.867208i
$156$		0			0
$157$	279.914			1.78289			0.891445	−	0.453130i	$-0.149693\pi$
$157$	279.914			1.78289			0.891445	+	0.453130i	$0.149693\pi$
$158$		0			0
$159$	−8.95807	+	26.2587i	−0.0563400	+	0.165149i
$160$		0			0
$161$			68.3401i			0.424473i
$162$		0			0
$163$	−26.2876			−0.161273			−0.0806367	−	0.996744i	$-0.525695\pi$
$163$	−26.2876			−0.161273			−0.0806367	+	0.996744i	$0.525695\pi$
$164$		0			0
$165$	−227.438	−	77.5897i	−1.37841	−	0.470240i
$166$		0			0
$167$			5.93886i			0.0355621i	0.999842	+	0.0177810i	$0.00566018\pi$
$167$			5.93886i			0.0355621i	−0.999842	+	0.0177810i	$0.994340\pi$
$168$		0			0
$169$	13.0000			0.0769231
$170$		0			0
$171$	−15.4070	−	11.8966i	−0.0900993	−	0.0695710i
$172$		0			0
$173$		−	64.2120i		−	0.371168i	−0.982628	−	0.185584i	$-0.940582\pi$
$173$		−	64.2120i		−	0.371168i	0.982628	−	0.185584i	$-0.0594177\pi$
$174$		0			0
$175$	−24.3103			−0.138916
$176$		0			0
$177$	−35.1016	+	102.893i	−0.198314	+	0.581316i
$178$		0			0
$179$		−	250.877i		−	1.40155i	−0.713384	−	0.700774i	$-0.752838\pi$
$179$		−	250.877i		−	1.40155i	0.713384	−	0.700774i	$-0.247162\pi$
$180$		0			0
$181$	−189.445			−1.04666			−0.523328	−	0.852132i	$-0.675310\pi$
$181$	−189.445			−1.04666			−0.523328	+	0.852132i	$0.675310\pi$
$182$		0			0
$183$	67.8391	+	23.1431i	0.370705	+	0.126465i
$184$		0			0
$185$		−	19.8699i		−	0.107405i
$186$		0			0
$187$	382.516			2.04554
$188$		0			0
$189$	58.1957	+	87.9600i	0.307914	+	0.465397i
$190$		0			0
$191$			82.1711i			0.430215i	0.976590	+	0.215108i	$0.0690102\pi$
$191$			82.1711i			0.430215i	−0.976590	+	0.215108i	$0.930990\pi$
$192$		0			0
$193$	−295.084			−1.52893			−0.764467	−	0.644663i	$-0.776998\pi$
$193$	−295.084			−1.52893			−0.764467	+	0.644663i	$0.776998\pi$
$194$		0			0
$195$	19.5150	−	57.2041i	0.100077	−	0.293354i
$196$		0			0
$197$			70.0389i			0.355527i	0.984073	+	0.177764i	$0.0568862\pi$
$197$			70.0389i			0.355527i	−0.984073	+	0.177764i	$0.943114\pi$
$198$		0			0
$199$	−168.999			−0.849239			−0.424619	−	0.905372i	$-0.639592\pi$
$199$	−168.999			−0.849239			−0.424619	+	0.905372i	$0.639592\pi$
$200$		0			0
$201$	−319.611	−	109.034i	−1.59011	−	0.542460i
$202$		0			0
$203$		−	168.840i		−	0.831723i
$204$		0			0
$205$	269.946			1.31681
$206$		0			0
$207$	−96.2313	+	124.626i	−0.464886	+	0.602060i
$208$		0			0
$209$			31.0049i			0.148349i
$210$		0			0
$211$	−215.277			−1.02027			−0.510134	−	0.860095i	$-0.670404\pi$
$211$	−215.277			−1.02027			−0.510134	+	0.860095i	$0.670404\pi$
$212$		0			0
$213$	90.9975	−	266.740i	0.427218	−	1.25230i
$214$		0			0
$215$			472.870i			2.19940i
$216$		0			0
$217$	−93.9671			−0.433028
$218$		0			0
$219$	68.3479	+	23.3167i	0.312091	+	0.106469i
$220$		0			0
$221$			96.2087i			0.435334i
$222$		0			0
$223$	−403.897			−1.81120			−0.905598	−	0.424137i	$-0.860578\pi$
$223$	−403.897			−1.81120			−0.905598	+	0.424137i	$0.860578\pi$
$224$		0			0
$225$	−44.3327	−	34.2319i	−0.197034	−	0.152142i
$226$		0			0
$227$		−	180.191i		−	0.793794i	−0.917863	−	0.396897i	$-0.870087\pi$
$227$		−	180.191i		−	0.793794i	0.917863	−	0.396897i	$-0.129913\pi$
$228$		0			0
$229$	−148.025			−0.646399			−0.323200	−	0.946331i	$-0.604759\pi$
$229$	−148.025			−0.646399			−0.323200	+	0.946331i	$0.604759\pi$
$230$		0			0
$231$	54.2406	−	158.995i	0.234808	−	0.688290i
$232$		0			0
$233$			248.041i			1.06455i	0.846571	+	0.532276i	$0.178663\pi$
$233$			248.041i			1.06455i	−0.846571	+	0.532276i	$0.821337\pi$
$234$		0			0
$235$	188.926			0.803940
$236$		0			0
$237$	126.176	+	43.0444i	0.532386	+	0.181622i
$238$		0			0
$239$		−	361.855i		−	1.51404i	−0.653392	−	0.757020i	$-0.726655\pi$
$239$		−	361.855i		−	1.51404i	0.653392	−	0.757020i	$-0.273345\pi$
$240$		0			0
$241$	274.070			1.13722			0.568611	−	0.822607i	$-0.307481\pi$
$241$	274.070			1.13722			0.568611	+	0.822607i	$0.307481\pi$
$242$		0			0
$243$	−17.7317	+	242.352i	−0.0729702	+	0.997334i
$244$		0			0
$245$			188.539i			0.769545i
$246$		0			0
$247$	−7.79820			−0.0315717
$248$		0			0
$249$	−53.4516	+	156.682i	−0.214665	+	0.629246i
$250$		0			0
$251$			156.485i			0.623447i	0.950173	+	0.311724i	$0.100906\pi$
$251$			156.485i			0.623447i	−0.950173	+	0.311724i	$0.899094\pi$
$252$		0			0
$253$	250.797			0.991292
$254$		0			0
$255$	423.348	+	144.424i	1.66019	+	0.566369i
$256$		0			0
$257$		−	108.439i		−	0.421941i	−0.977492	−	0.210971i	$-0.932338\pi$
$257$		−	108.439i		−	0.421941i	0.977492	−	0.210971i	$-0.0676624\pi$
$258$		0			0
$259$	13.8904			0.0536310
$260$		0			0
$261$	237.747	−	307.899i	0.910909	−	1.17969i
$262$		0			0
$263$		−	316.993i		−	1.20530i	−0.798007	−	0.602649i	$-0.794112\pi$
$263$		−	316.993i		−	1.20530i	0.798007	−	0.602649i	$-0.205888\pi$
$264$		0			0
$265$	51.6771			0.195008
$266$		0			0
$267$	74.9858	−	219.805i	0.280846	−	0.823240i
$268$		0			0
$269$		−	160.970i		−	0.598401i	−0.954190	−	0.299200i	$-0.903280\pi$
$269$		−	160.970i		−	0.598401i	0.954190	−	0.299200i	$-0.0967199\pi$
$270$		0			0
$271$	−183.482			−0.677057			−0.338528	−	0.940956i	$-0.609929\pi$
$271$	−183.482			−0.677057			−0.338528	+	0.940956i	$0.609929\pi$
$272$		0			0
$273$	39.9897	+	13.6424i	0.146482	+	0.0499720i
$274$		0			0
$275$			89.2148i			0.324417i
$276$		0			0
$277$	372.697			1.34547			0.672737	−	0.739881i	$-0.265118\pi$
$277$	372.697			1.34547			0.672737	+	0.739881i	$0.265118\pi$
$278$		0			0
$279$	−171.360	−	132.317i	−0.614194	−	0.474255i
$280$		0			0
$281$			249.004i			0.886137i	0.896488	+	0.443068i	$0.146110\pi$
$281$			249.004i			0.886137i	−0.896488	+	0.443068i	$0.853890\pi$
$282$		0			0
$283$	371.492			1.31269			0.656346	−	0.754460i	$-0.272101\pi$
$283$	371.492			1.31269			0.656346	+	0.754460i	$0.272101\pi$
$284$		0			0
$285$	−11.7063	+	34.3145i	−0.0410747	+	0.120402i
$286$		0			0
$287$			188.711i			0.657531i
$288$		0			0
$289$	−423.009			−1.46370
$290$		0			0
$291$	406.429	+	138.652i	1.39666	+	0.476468i
$292$		0			0
$293$		−	529.503i		−	1.80718i	−0.428400	−	0.903589i	$-0.640923\pi$
$293$		−	529.503i		−	1.80718i	0.428400	−	0.903589i	$-0.359077\pi$
$294$		0			0
$295$	202.493			0.686419
$296$		0			0
$297$	322.798	−	213.568i	1.08686	−	0.719086i
$298$		0			0
$299$			63.0793i			0.210967i
$300$		0			0
$301$	−330.569			−1.09824
$302$		0			0
$303$	3.86518	−	11.3300i	0.0127564	−	0.0373926i
$304$		0			0
$305$		−	133.507i		−	0.437729i
$306$		0			0
$307$	117.282			0.382025			0.191013	−	0.981588i	$-0.438823\pi$
$307$	117.282			0.382025			0.191013	+	0.981588i	$0.438823\pi$
$308$		0			0
$309$	−190.873	−	65.1159i	−0.617714	−	0.210731i
$310$		0			0
$311$		−	406.259i		−	1.30630i	−0.757229	−	0.653149i	$-0.773447\pi$
$311$		−	406.259i		−	1.30630i	0.757229	−	0.653149i	$-0.226553\pi$
$312$		0			0
$313$	186.759			0.596675			0.298338	−	0.954460i	$-0.403568\pi$
$313$	186.759			0.596675			0.298338	+	0.954460i	$0.403568\pi$
$314$		0			0
$315$	120.061	−	155.488i	0.381147	−	0.493612i
$316$		0			0
$317$		−	297.699i		−	0.939114i	−0.882902	−	0.469557i	$-0.844414\pi$
$317$		−	297.699i		−	0.939114i	0.882902	−	0.469557i	$-0.155586\pi$
$318$		0			0
$319$	−619.614			−1.94236
$320$		0			0
$321$	−80.7363	+	236.661i	−0.251515	+	0.737263i
$322$		0			0
$323$		−	57.7119i		−	0.178675i
$324$		0			0
$325$	−22.4389			−0.0690427
$326$		0			0
$327$	433.515	+	147.892i	1.32573	+	0.452270i
$328$		0			0
$329$			132.072i			0.401436i
$330$		0			0
$331$	306.702			0.926592			0.463296	−	0.886203i	$-0.346667\pi$
$331$	306.702			0.926592			0.463296	+	0.886203i	$0.346667\pi$
$332$		0			0
$333$	25.3309	+	19.5594i	0.0760686	+	0.0587371i
$334$		0			0
$335$			628.995i			1.87760i
$336$		0			0
$337$	−67.4134			−0.200040			−0.100020	−	0.994985i	$-0.531891\pi$
$337$	−67.4134			−0.200040			−0.100020	+	0.994985i	$0.531891\pi$
$338$		0			0
$339$	−171.201	+	501.840i	−0.505019	+	1.48035i
$340$		0			0
$341$			344.843i			1.01127i
$342$		0			0
$343$	−323.208			−0.942298
$344$		0			0
$345$	277.568	+	94.6917i	0.804546	+	0.274469i
$346$		0			0
$347$		−	222.331i		−	0.640724i	−0.947295	−	0.320362i	$-0.896196\pi$
$347$		−	222.331i		−	0.640724i	0.947295	−	0.320362i	$-0.103804\pi$
$348$		0			0
$349$	−128.360			−0.367795			−0.183897	−	0.982945i	$-0.558871\pi$
$349$	−128.360			−0.367795			−0.183897	+	0.982945i	$0.558871\pi$
$350$		0			0
$351$	53.7158	+	81.1888i	0.153036	+	0.231307i
$352$		0			0
$353$		−	101.336i		−	0.287069i	−0.989645	−	0.143535i	$-0.954153\pi$
$353$		−	101.336i		−	0.287069i	0.989645	−	0.143535i	$-0.0458469\pi$
$354$		0			0
$355$	−524.944			−1.47872
$356$		0			0
$357$	−100.963	+	295.950i	−0.282808	+	0.828992i
$358$		0			0
$359$			26.7975i			0.0746449i	0.999303	+	0.0373225i	$0.0118829\pi$
$359$			26.7975i			0.0746449i	−0.999303	+	0.0373225i	$0.988117\pi$
$360$		0			0
$361$	−356.322			−0.987042
$362$		0			0
$363$	−239.926	−	81.8501i	−0.660954	−	0.225482i
$364$		0			0
$365$		−	134.509i		−	0.368517i
$366$		0			0
$367$	57.9441			0.157886			0.0789429	−	0.996879i	$-0.474846\pi$
$367$	57.9441			0.157886			0.0789429	+	0.996879i	$0.474846\pi$
$368$		0			0
$369$	−265.729	+	344.137i	−0.720132	+	0.932622i
$370$		0			0
$371$			36.1259i			0.0973744i
$372$		0			0
$373$	−132.001			−0.353890			−0.176945	−	0.984221i	$-0.556621\pi$
$373$	−132.001			−0.353890			−0.176945	+	0.984221i	$0.556621\pi$
$374$		0			0
$375$	101.628	−	297.901i	0.271008	−	0.794402i
$376$		0			0
$377$		−	155.842i		−	0.413375i
$378$		0			0
$379$	−531.530			−1.40245			−0.701227	−	0.712938i	$-0.747364\pi$
$379$	−531.530			−1.40245			−0.701227	+	0.712938i	$0.747364\pi$
$380$		0			0
$381$	322.125	+	109.892i	0.845473	+	0.288431i
$382$		0			0
$383$			515.418i			1.34574i	0.739761	+	0.672869i	$0.234938\pi$
$383$			515.418i			1.34574i	−0.739761	+	0.672869i	$0.765062\pi$
$384$		0			0
$385$	−312.902			−0.812733
$386$		0			0
$387$	−602.832	−	465.482i	−1.55771	−	1.20280i
$388$		0			0
$389$			501.541i			1.28931i	0.764474	+	0.644655i	$0.222999\pi$
$389$			501.541i			1.28931i	−0.764474	+	0.644655i	$0.777001\pi$
$390$		0			0
$391$	−466.829			−1.19394
$392$		0			0
$393$	177.698	−	520.885i	0.452159	−	1.32541i
$394$		0			0
$395$		−	248.314i		−	0.628642i
$396$		0			0
$397$	398.734			1.00437			0.502184	−	0.864761i	$-0.332530\pi$
$397$	398.734			1.00437			0.502184	+	0.864761i	$0.332530\pi$
$398$		0			0
$399$	−23.9883	−	8.18352i	−0.0601210	−	0.0205101i
$400$		0			0
$401$			36.1233i			0.0900830i	0.998985	+	0.0450415i	$0.0143420\pi$
$401$			36.1233i			0.0900830i	−0.998985	+	0.0450415i	$0.985658\pi$
$402$		0			0
$403$	−86.7334			−0.215219
$404$		0			0
$405$	437.892	−	114.489i	1.08121	−	0.282690i
$406$		0			0
$407$		−	50.9756i		−	0.125247i
$408$		0			0
$409$	413.134			1.01011			0.505054	−	0.863088i	$-0.331473\pi$
$409$	413.134			1.01011			0.505054	+	0.863088i	$0.331473\pi$
$410$		0			0
$411$	234.795	−	688.251i	0.571277	−	1.67458i
$412$		0			0
$413$			141.557i			0.342753i
$414$		0			0
$415$	308.351			0.743014
$416$		0			0
$417$	−23.9425	−	8.16792i	−0.0574161	−	0.0195873i
$418$		0			0
$419$			760.160i			1.81422i	0.420889	+	0.907112i	$0.361718\pi$
$419$			760.160i			1.81422i	−0.420889	+	0.907112i	$0.638282\pi$
$420$		0			0
$421$	−360.755			−0.856901			−0.428451	−	0.903565i	$-0.640940\pi$
$421$	−360.755			−0.856901			−0.428451	+	0.903565i	$0.640940\pi$
$422$		0			0
$423$	−185.974	+	240.850i	−0.439655	+	0.569385i
$424$		0			0
$425$		−	166.063i		−	0.390736i
$426$		0			0
$427$	93.3310			0.218574
$428$		0			0
$429$	50.0651	−	146.755i	0.116702	−	0.342087i
$430$		0			0
$431$		−	566.479i		−	1.31434i	−0.753744	−	0.657168i	$-0.771754\pi$
$431$		−	566.479i		−	1.31434i	0.753744	−	0.657168i	$-0.228246\pi$
$432$		0			0
$433$	51.3193			0.118520			0.0592602	−	0.998243i	$-0.481126\pi$
$433$	51.3193			0.118520			0.0592602	+	0.998243i	$0.481126\pi$
$434$		0			0
$435$	−685.755	−	233.943i	−1.57645	−	0.537801i
$436$		0			0
$437$		−	37.8388i		−	0.0865877i
$438$		0			0
$439$	344.469			0.784667			0.392333	−	0.919823i	$-0.371668\pi$
$439$	344.469			0.784667			0.392333	+	0.919823i	$0.371668\pi$
$440$		0			0
$441$	−240.356	−	185.593i	−0.545025	−	0.420846i
$442$		0			0
$443$		−	208.015i		−	0.469559i	−0.972049	−	0.234780i	$-0.924563\pi$
$443$		−	208.015i		−	0.469559i	0.972049	−	0.234780i	$-0.0754368\pi$
$444$		0			0
$445$	−432.577			−0.972082
$446$		0			0
$447$	−233.175	+	683.503i	−0.521644	+	1.52909i
$448$		0			0
$449$			352.003i			0.783971i	0.919971	+	0.391986i	$0.128212\pi$
$449$			352.003i			0.783971i	−0.919971	+	0.391986i	$0.871788\pi$
$450$		0			0
$451$	692.539			1.53556
$452$		0			0
$453$	376.282	+	128.367i	0.830644	+	0.283372i
$454$		0			0
$455$		−	78.6997i		−	0.172966i
$456$		0			0
$457$	791.879			1.73278			0.866389	−	0.499370i	$-0.166435\pi$
$457$	791.879			1.73278			0.866389	+	0.499370i	$0.166435\pi$
$458$		0			0
$459$	−600.851	+	397.533i	−1.30904	+	0.866084i
$460$		0			0
$461$		−	803.733i		−	1.74346i	−0.489990	−	0.871728i	$-0.663000\pi$
$461$		−	803.733i		−	1.74346i	0.489990	−	0.871728i	$-0.337000\pi$
$462$		0			0
$463$	−486.244			−1.05020			−0.525101	−	0.851040i	$-0.675973\pi$
$463$	−486.244			−1.05020			−0.525101	+	0.851040i	$0.675973\pi$
$464$		0			0
$465$	−130.200	+	381.654i	−0.280000	+	0.820761i
$466$		0			0
$467$			20.1644i			0.0431786i	0.999767	+	0.0215893i	$0.00687262\pi$
$467$			20.1644i			0.0431786i	−0.999767	+	0.0215893i	$0.993127\pi$
$468$		0			0
$469$	−439.712			−0.937552
$470$		0			0
$471$	794.766	+	271.132i	1.68740	+	0.575652i
$472$		0			0
$473$			1213.13i			2.56476i
$474$		0			0
$475$	13.4602			0.0283373
$476$		0			0
$477$	−50.8697	+	65.8799i	−0.106645	+	0.138113i
$478$		0			0
$479$			90.7190i			0.189392i	0.995506	+	0.0946962i	$0.0301880\pi$
$479$			90.7190i			0.189392i	−0.995506	+	0.0946962i	$0.969812\pi$
$480$		0			0
$481$	12.8211			0.0266552
$482$		0			0
$483$	−66.1961	+	194.040i	−0.137052	+	0.401739i
$484$		0			0
$485$		−	799.853i		−	1.64918i
$486$		0			0
$487$	496.649			1.01981			0.509906	−	0.860230i	$-0.329680\pi$
$487$	496.649			1.01981			0.509906	+	0.860230i	$0.329680\pi$
$488$		0			0
$489$	−74.6389	−	25.4628i	−0.152636	−	0.0520712i
$490$		0			0
$491$			499.699i			1.01772i	0.860850	+	0.508858i	$0.169932\pi$
$491$			499.699i			1.01772i	−0.860850	+	0.508858i	$0.830068\pi$
$492$		0			0
$493$	1153.34			2.33943
$494$		0			0
$495$	−570.614	−	440.605i	−1.15276	−	0.890110i
$496$		0			0
$497$		−	366.973i		−	0.738376i
$498$		0			0
$499$	−446.942			−0.895675			−0.447838	−	0.894115i	$-0.647806\pi$
$499$	−446.942			−0.895675			−0.447838	+	0.894115i	$0.647806\pi$
$500$		0			0
$501$	−5.75254	+	16.8624i	−0.0114821	+	0.0336574i
$502$		0			0
$503$		−	692.921i		−	1.37758i	−0.724963	−	0.688788i	$-0.758143\pi$
$503$		−	692.921i		−	1.37758i	0.724963	−	0.688788i	$-0.241857\pi$
$504$		0			0
$505$	−22.2974			−0.0441532
$506$		0			0
$507$	36.9112	+	12.5921i	0.0728032	+	0.0248366i
$508$		0			0
$509$		−	425.139i		−	0.835244i	−0.908621	−	0.417622i	$-0.862864\pi$
$509$		−	425.139i		−	0.835244i	0.908621	−	0.417622i	$-0.137136\pi$
$510$		0			0
$511$	94.0310			0.184014
$512$		0			0
$513$	−32.2220	−	48.7020i	−0.0628110	−	0.0949357i
$514$		0			0
$515$			375.639i			0.729397i
$516$		0			0
$517$	484.684			0.937493
$518$		0			0
$519$	62.1975	−	182.319i	0.119841	−	0.351289i
$520$		0			0
$521$			32.5207i			0.0624197i	0.999513	+	0.0312099i	$0.00993602\pi$
$521$			32.5207i			0.0624197i	−0.999513	+	0.0312099i	$0.990064\pi$
$522$		0			0
$523$	357.817			0.684162			0.342081	−	0.939670i	$-0.388868\pi$
$523$	357.817			0.684162			0.342081	+	0.939670i	$0.388868\pi$
$524$		0			0
$525$	−69.0249	−	23.5476i	−0.131476	−	0.0448526i
$526$		0			0
$527$		−	641.885i		−	1.21800i
$528$		0			0
$529$	222.924			0.421406
$530$		0			0
$531$	−199.330	+	258.146i	−0.375386	+	0.486151i
$532$		0			0
$533$			174.184i			0.326800i
$534$		0			0
$535$	465.750			0.870561
$536$		0			0
$537$	243.006	−	712.321i	0.452526	−	1.32648i
$538$		0			0
$539$			483.690i			0.897384i
$540$		0			0
$541$	−776.332			−1.43499			−0.717497	−	0.696561i	$-0.754712\pi$
$541$	−776.332			−1.43499			−0.717497	+	0.696561i	$0.754712\pi$
$542$		0			0
$543$	−537.895	−	183.501i	−0.990598	−	0.337940i
$544$		0			0
$545$		−	853.157i		−	1.56543i
$546$		0			0
$547$	292.992			0.535634			0.267817	−	0.963470i	$-0.413698\pi$
$547$	292.992			0.535634			0.267817	+	0.963470i	$0.413698\pi$
$548$		0			0
$549$	170.200	+	131.421i	0.310018	+	0.239383i
$550$		0			0
$551$			93.4839i			0.169662i
$552$		0			0
$553$	173.589			0.313903
$554$		0			0
$555$	19.2465	−	56.4170i	0.0346784	−	0.101652i
$556$		0			0
$557$			466.191i			0.836968i	0.908224	+	0.418484i	$0.137438\pi$
$557$			466.191i			0.836968i	−0.908224	+	0.418484i	$0.862562\pi$
$558$		0			0
$559$	−305.122			−0.545835
$560$		0			0
$561$	1086.09	+	370.516i	1.93599	+	0.660455i
$562$		0			0
$563$			636.308i			1.13021i	0.825019	+	0.565105i	$0.191164\pi$
$563$			636.308i			1.13021i	−0.825019	+	0.565105i	$0.808836\pi$
$564$		0			0
$565$	987.622			1.74800
$566$		0			0
$567$	80.0361	+	306.117i	0.141157	+	0.539889i
$568$		0			0
$569$		−	900.253i		−	1.58217i	−0.611708	−	0.791083i	$-0.709517\pi$
$569$		−	900.253i		−	1.58217i	0.611708	−	0.791083i	$-0.290483\pi$
$570$		0			0
$571$	1023.62			1.79268			0.896342	−	0.443363i	$-0.146215\pi$
$571$	1023.62			1.79268			0.896342	+	0.443363i	$0.146215\pi$
$572$		0			0
$573$	−79.5932	+	233.310i	−0.138906	+	0.407174i
$574$		0			0
$575$		−	108.879i		−	0.189355i
$576$		0			0
$577$	−44.2803			−0.0767423			−0.0383711	−	0.999264i	$-0.512217\pi$
$577$	−44.2803			−0.0767423			−0.0383711	+	0.999264i	$0.512217\pi$
$578$		0			0
$579$	−837.840	−	285.826i	−1.44705	−	0.493655i
$580$		0			0
$581$			215.559i			0.371013i
$582$		0			0
$583$	132.576			0.227403
$584$		0			0
$585$	110.819	−	143.518i	0.189434	−	0.245330i
$586$		0			0
$587$			254.621i			0.433766i	0.976198	+	0.216883i	$0.0695890\pi$
$587$			254.621i			0.433766i	−0.976198	+	0.216883i	$0.930411\pi$
$588$		0			0
$589$	52.0280			0.0883328
$590$		0			0
$591$	−67.8415	+	198.863i	−0.114791	+	0.336486i
$592$		0			0
$593$		−	597.882i		−	1.00823i	−0.863636	−	0.504116i	$-0.831818\pi$
$593$		−	597.882i		−	1.00823i	0.863636	−	0.504116i	$-0.168182\pi$
$594$		0			0
$595$	582.430			0.978875
$596$		0			0
$597$	−479.842	−	163.697i	−0.803755	−	0.274199i
$598$		0			0
$599$			508.042i			0.848151i	0.905627	+	0.424075i	$0.139401\pi$
$599$			508.042i			0.848151i	−0.905627	+	0.424075i	$0.860599\pi$
$600$		0			0
$601$	−518.398			−0.862559			−0.431279	−	0.902218i	$-0.641938\pi$
$601$	−518.398			−0.862559			−0.431279	+	0.902218i	$0.641938\pi$
$602$		0			0
$603$	−801.866	−	619.168i	−1.32980	−	1.02681i
$604$		0			0
$605$			472.175i			0.780455i
$606$		0			0
$607$	742.148			1.22265			0.611325	−	0.791380i	$-0.290637\pi$
$607$	742.148			1.22265			0.611325	+	0.791380i	$0.290637\pi$
$608$		0			0
$609$	163.543	−	479.391i	0.268543	−	0.787177i
$610$		0			0
$611$			121.905i			0.199518i
$612$		0			0
$613$	−156.917			−0.255982			−0.127991	−	0.991775i	$-0.540853\pi$
$613$	−156.917			−0.255982			−0.127991	+	0.991775i	$0.540853\pi$
$614$		0			0
$615$	766.465	+	261.477i	1.24628	+	0.425166i
$616$		0			0
$617$			57.0811i			0.0925140i	0.998930	+	0.0462570i	$0.0147293\pi$
$617$			57.0811i			0.0925140i	−0.998930	+	0.0462570i	$0.985271\pi$
$618$		0			0
$619$	474.558			0.766652			0.383326	−	0.923613i	$-0.374779\pi$
$619$	474.558			0.766652			0.383326	+	0.923613i	$0.374779\pi$
$620$		0			0
$621$	−393.948	+	260.642i	−0.634377	+	0.419714i
$622$		0			0
$623$		−	302.401i		−	0.485395i
$624$		0			0
$625$	−741.855			−1.18697
$626$		0			0
$627$	−30.0322	+	88.0329i	−0.0478982	+	0.140403i
$628$		0			0
$629$			94.8850i			0.150851i
$630$		0			0
$631$	−740.771			−1.17396			−0.586982	−	0.809600i	$-0.699684\pi$
$631$	−740.771			−1.17396			−0.586982	+	0.809600i	$0.699684\pi$
$632$		0			0
$633$	−611.240	−	208.523i	−0.965624	−	0.329420i
$634$		0			0
$635$		−	633.943i		−	0.998336i
$636$		0			0
$637$	−121.655			−0.190982
$638$		0			0
$639$	516.743	−	669.218i	0.808674	−	1.04729i
$640$		0			0
$641$			734.853i			1.14642i	0.819410	+	0.573208i	$0.194301\pi$
$641$			734.853i			1.14642i	−0.819410	+	0.573208i	$0.805699\pi$
$642$		0			0
$643$	−457.234			−0.711095			−0.355547	−	0.934658i	$-0.615706\pi$
$643$	−457.234			−0.711095			−0.355547	+	0.934658i	$0.615706\pi$
$644$		0			0
$645$	−458.035	+	1342.63i	−0.710131	+	2.08160i
$646$		0			0
$647$		−	466.711i		−	0.721346i	−0.932692	−	0.360673i	$-0.882547\pi$
$647$		−	466.711i		−	0.721346i	0.932692	−	0.360673i	$-0.117453\pi$
$648$		0			0
$649$	519.491			0.800448
$650$		0			0
$651$	−266.803	−	91.0190i	−0.409836	−	0.139814i
$652$		0			0
$653$			554.439i			0.849064i	0.905413	+	0.424532i	$0.139561\pi$
$653$			554.439i			0.849064i	−0.905413	+	0.424532i	$0.860439\pi$
$654$		0			0
$655$	−1025.10			−1.56504
$656$		0			0
$657$	171.477	+	132.407i	0.261000	+	0.201533i
$658$		0			0
$659$			246.481i			0.374022i	0.982358	+	0.187011i	$0.0598801\pi$
$659$			246.481i			0.374022i	−0.982358	+	0.187011i	$0.940120\pi$
$660$		0			0
$661$	−992.484			−1.50149			−0.750745	−	0.660592i	$-0.770305\pi$
$661$	−992.484			−1.50149			−0.750745	+	0.660592i	$0.770305\pi$
$662$		0			0
$663$	−93.1903	+	273.168i	−0.140559	+	0.412018i
$664$		0			0
$665$			47.2089i			0.0709909i
$666$		0			0
$667$	756.186			1.13371
$668$		0			0
$669$	−1146.79	−	391.225i	−1.71419	−	0.584791i
$670$		0			0
$671$		−	342.509i		−	0.510446i
$672$		0			0
$673$	−24.4852			−0.0363821			−0.0181911	−	0.999835i	$-0.505791\pi$
$673$	−24.4852			−0.0363821			−0.0181911	+	0.999835i	$0.505791\pi$
$674$		0			0
$675$	−92.7171	−	140.137i	−0.137359	−	0.207611i
$676$		0			0
$677$			1246.59i			1.84135i	0.390333	+	0.920674i	$0.372360\pi$
$677$			1246.59i			1.84135i	−0.390333	+	0.920674i	$0.627640\pi$
$678$		0			0
$679$	559.153			0.823495
$680$		0			0
$681$	174.538	−	511.622i	0.256297	−	0.751280i
$682$		0			0
$683$			107.372i			0.157206i	0.996906	+	0.0786031i	$0.0250460\pi$
$683$			107.372i			0.157206i	−0.996906	+	0.0786031i	$0.974954\pi$
$684$		0			0
$685$	−1354.48			−1.97734
$686$		0			0
$687$	−420.292	−	143.381i	−0.611779	−	0.208707i
$688$		0			0
$689$			33.3449i			0.0483961i
$690$		0			0
$691$	−230.718			−0.333890			−0.166945	−	0.985966i	$-0.553390\pi$
$691$	−230.718			−0.333890			−0.166945	+	0.985966i	$0.553390\pi$
$692$		0			0
$693$	308.013	−	398.899i	0.444464	−	0.575612i
$694$		0			0
$695$			47.1189i			0.0677970i
$696$		0			0
$697$	−1289.08			−1.84947
$698$		0			0
$699$	−240.259	+	704.268i	−0.343718	+	1.00754i
$700$		0			0
$701$			842.822i			1.20231i	0.799131	+	0.601157i	$0.205293\pi$
$701$			842.822i			1.20231i	−0.799131	+	0.601157i	$0.794707\pi$
$702$		0			0
$703$	−7.69091			−0.0109401
$704$		0			0
$705$	536.422	+	182.999i	0.760882	+	0.259573i
$706$		0			0
$707$		−	15.5874i		−	0.0220473i
$708$		0			0
$709$	−719.643			−1.01501			−0.507505	−	0.861649i	$-0.669432\pi$
$709$	−719.643			−1.01501			−0.507505	+	0.861649i	$0.669432\pi$
$710$		0			0
$711$	316.559	+	244.434i	0.445231	+	0.343789i
$712$		0			0
$713$		−	420.852i		−	0.590255i
$714$		0			0
$715$	−288.815			−0.403936
$716$		0			0
$717$	350.503	−	1027.42i	0.488846	−	1.43295i
$718$		0			0
$719$		−	782.261i		−	1.08799i	−0.839090	−	0.543993i	$-0.816912\pi$
$719$		−	782.261i		−	1.08799i	0.839090	−	0.543993i	$-0.183088\pi$
$720$		0			0
$721$	−262.598			−0.364214
$722$		0			0
$723$	778.175	+	265.472i	1.07631	+	0.367181i
$724$		0			0
$725$			268.995i			0.371027i
$726$		0			0
$727$	849.689			1.16876			0.584381	−	0.811480i	$-0.301338\pi$
$727$	849.689			1.16876			0.584381	+	0.811480i	$0.301338\pi$
$728$		0			0
$729$	−285.095	+	670.941i	−0.391077	+	0.920358i
$730$		0			0
$731$		−	2258.11i		−	3.08906i
$732$		0			0
$733$	−307.893			−0.420046			−0.210023	−	0.977696i	$-0.567354\pi$
$733$	−307.893			−0.420046			−0.210023	+	0.977696i	$0.567354\pi$
$734$		0			0
$735$	−182.623	+	535.322i	−0.248467	+	0.728329i
$736$		0			0
$737$			1613.67i			2.18951i
$738$		0			0
$739$	99.2662			0.134325			0.0671625	−	0.997742i	$-0.478605\pi$
$739$	99.2662			0.134325			0.0671625	+	0.997742i	$0.478605\pi$
$740$		0			0
$741$	−22.1416	−	7.55355i	−0.0298807	−	0.0101937i
$742$		0			0
$743$			354.754i			0.477462i	0.971086	+	0.238731i	$0.0767314\pi$
$743$			354.754i			0.477462i	−0.971086	+	0.238731i	$0.923269\pi$
$744$		0			0
$745$	1345.14			1.80555
$746$		0			0
$747$	−303.533	+	393.097i	−0.406336	+	0.526234i
$748$		0			0
$749$			325.592i			0.434702i
$750$		0			0
$751$	−32.6013			−0.0434105			−0.0217053	−	0.999764i	$-0.506910\pi$
$751$	−32.6013			−0.0434105			−0.0217053	+	0.999764i	$0.506910\pi$
$752$		0			0
$753$	−151.576	+	444.312i	−0.201296	+	0.590056i
$754$		0			0
$755$		−	740.523i		−	0.980825i
$756$		0			0
$757$	−1430.86			−1.89017			−0.945086	−	0.326821i	$-0.894023\pi$
$757$	−1430.86			−1.89017			−0.945086	+	0.326821i	$0.894023\pi$
$758$		0			0
$759$	712.094	+	242.929i	0.938200	+	0.320064i
$760$		0			0
$761$			393.429i			0.516990i	0.966013	+	0.258495i	$0.0832265\pi$
$761$			393.429i			0.516990i	−0.966013	+	0.258495i	$0.916774\pi$
$762$		0			0
$763$	596.417			0.781673
$764$		0			0
$765$	1062.13	+	820.133i	1.38841	+	1.07207i
$766$		0			0
$767$			130.660i			0.170352i
$768$		0			0
$769$	1000.26			1.30073			0.650365	−	0.759622i	$-0.274616\pi$
$769$	1000.26			1.30073			0.650365	+	0.759622i	$0.274616\pi$
$770$		0			0
$771$	105.037	−	307.893i	0.136235	−	0.399343i
$772$		0			0
$773$		−	251.703i		−	0.325618i	−0.986658	−	0.162809i	$-0.947945\pi$
$773$		−	251.703i		−	0.325618i	0.986658	−	0.162809i	$-0.0520554\pi$
$774$		0			0
$775$	149.708			0.193171
$776$		0			0
$777$	39.4395	+	13.4547i	0.0507586	+	0.0173162i
$778$		0			0
$779$		−	104.486i		−	0.134129i
$780$		0			0
$781$	−1346.73			−1.72436
$782$		0			0
$783$	973.281	−	643.938i	1.24302	−	0.822398i
$784$		0			0
$785$		−	1564.10i		−	1.99248i
$786$		0			0
$787$	−56.7081			−0.0720561			−0.0360280	−	0.999351i	$-0.511471\pi$
$787$	−56.7081			−0.0720561			−0.0360280	+	0.999351i	$0.511471\pi$
$788$		0			0
$789$	307.048	−	900.046i	0.389161	−	1.14074i
$790$		0			0
$791$			690.417i			0.872841i
$792$		0			0
$793$	86.1463			0.108633
$794$		0			0
$795$	146.728	+	50.0558i	0.184564	+	0.0629633i
$796$		0			0
$797$		−	827.587i		−	1.03838i	−0.854659	−	0.519189i	$-0.826234\pi$
$797$		−	827.587i		−	1.03838i	0.854659	−	0.519189i	$-0.173766\pi$
$798$		0			0
$799$	−902.182			−1.12914
$800$		0			0
$801$	425.818	−	551.465i	0.531608	−	0.688470i
$802$		0			0
$803$		−	345.078i		−	0.429736i
$804$		0			0
$805$	381.871			0.474373
$806$		0			0
$807$	155.920	−	457.045i	0.193209	−	0.566351i
$808$		0			0
$809$		−	949.841i		−	1.17409i	−0.809553	−	0.587046i	$-0.800291\pi$
$809$		−	949.841i		−	1.17409i	0.809553	−	0.587046i	$-0.199709\pi$
$810$		0			0
$811$	658.962			0.812531			0.406265	−	0.913755i	$-0.366831\pi$
$811$	658.962			0.812531			0.406265	+	0.913755i	$0.366831\pi$
$812$		0			0
$813$	−520.966	−	177.726i	−0.640794	−	0.218605i
$814$		0			0
$815$			146.889i			0.180232i
$816$		0			0
$817$	183.031			0.224028
$818$		0			0
$819$	100.329	+	77.4701i	0.122502	+	0.0945911i
$820$		0			0
$821$		−	520.430i		−	0.633897i	−0.948443	−	0.316949i	$-0.897342\pi$
$821$		−	520.430i		−	0.633897i	0.948443	−	0.316949i	$-0.102658\pi$
$822$		0			0
$823$	174.921			0.212540			0.106270	−	0.994337i	$-0.466109\pi$
$823$	174.921			0.212540			0.106270	+	0.994337i	$0.466109\pi$
$824$		0			0
$825$	−86.4158	+	253.310i	−0.104746	+	0.307042i
$826$		0			0
$827$			693.472i			0.838539i	0.907862	+	0.419270i	$0.137714\pi$
$827$			693.472i			0.838539i	−0.907862	+	0.419270i	$0.862286\pi$
$828$		0			0
$829$	790.929			0.954076			0.477038	−	0.878883i	$-0.341710\pi$
$829$	790.929			0.954076			0.477038	+	0.878883i	$0.341710\pi$
$830$		0			0
$831$	1058.21	+	361.004i	1.27341	+	0.434421i
$832$		0			0
$833$		−	900.332i		−	1.08083i
$834$		0			0
$835$	33.1851			0.0397427
$836$		0			0
$837$	−358.381	−	541.675i	−0.428173	−	0.647163i
$838$		0			0
$839$			647.323i			0.771541i	0.922595	+	0.385770i	$0.126064\pi$
$839$			647.323i			0.771541i	−0.922595	+	0.385770i	$0.873936\pi$
$840$		0			0
$841$	−1027.22			−1.22143
$842$		0			0
$843$	−241.192	+	707.004i	−0.286112	+	0.838676i
$844$		0			0
$845$		−	72.6413i		−	0.0859660i
$846$		0			0
$847$	−330.083			−0.389709
$848$		0			0
$849$	1054.79	+	359.837i	1.24239	+	0.423836i
$850$		0			0
$851$			62.2114i			0.0731039i
$852$		0			0
$853$	−733.400			−0.859789			−0.429895	−	0.902879i	$-0.641449\pi$
$853$	−733.400			−0.859789			−0.429895	+	0.902879i	$0.641449\pi$
$854$		0			0
$855$	−66.4760	+	86.0910i	−0.0777497	+	0.100691i
$856$		0			0
$857$			532.786i			0.621687i	0.950461	+	0.310844i	$0.100612\pi$
$857$			532.786i			0.621687i	−0.950461	+	0.310844i	$0.899388\pi$
$858$		0			0
$859$	1287.13			1.49840			0.749201	−	0.662342i	$-0.230438\pi$
$859$	1287.13			1.49840			0.749201	+	0.662342i	$0.230438\pi$
$860$		0			0
$861$	−182.791	+	535.813i	−0.212301	+	0.622314i
$862$		0			0
$863$			1709.02i			1.98033i	0.139916	+	0.990163i	$0.455317\pi$
$863$			1709.02i			1.98033i	−0.139916	+	0.990163i	$0.544683\pi$
$864$		0			0
$865$	−358.804			−0.414802
$866$		0			0
$867$	−1201.06	−	409.738i	−1.38531	−	0.472593i
$868$		0			0
$869$		−	637.041i		−	0.733074i
$870$		0			0
$871$	−405.862			−0.465973
$872$		0			0
$873$	1019.68	+	787.356i	1.16802	+	0.901897i
$874$		0			0
$875$		−	409.843i		−	0.468392i
$876$		0			0
$877$	1108.70			1.26420			0.632100	−	0.774887i	$-0.282193\pi$
$877$	1108.70			1.26420			0.632100	+	0.774887i	$0.282193\pi$
$878$		0			0
$879$	512.891	−	1503.43i	0.583494	−	1.71039i
$880$		0			0
$881$			58.7418i			0.0666763i	0.999444	+	0.0333381i	$0.0106138\pi$
$881$			58.7418i			0.0666763i	−0.999444	+	0.0333381i	$0.989386\pi$
$882$		0			0
$883$	−531.631			−0.602073			−0.301037	−	0.953613i	$-0.597333\pi$
$883$	−531.631			−0.602073			−0.301037	+	0.953613i	$0.597333\pi$
$884$		0			0
$885$	574.945	+	196.141i	0.649655	+	0.221628i
$886$		0			0
$887$			786.215i			0.886375i	0.896429	+	0.443188i	$0.146152\pi$
$887$			786.215i			0.886375i	−0.896429	+	0.443188i	$0.853848\pi$
$888$		0			0
$889$	443.171			0.498505
$890$		0			0
$891$	1123.40	−	293.719i	1.26083	−	0.329651i
$892$		0			0
$893$		−	73.1264i		−	0.0818884i
$894$		0			0
$895$	−1401.85			−1.56631
$896$		0			0
$897$	−61.1003	+	179.102i	−0.0681162	+	0.199668i
$898$		0			0
$899$			1039.75i			1.15656i
$900$		0			0
$901$	−246.775			−0.273890
$902$		0			0
$903$	−938.593	−	320.198i	−1.03942	−	0.354594i
$904$		0			0
$905$			1058.58i			1.16970i
$906$		0			0
$907$	239.054			0.263566			0.131783	−	0.991279i	$-0.457930\pi$
$907$	239.054			0.263566			0.131783	+	0.991279i	$0.457930\pi$
$908$		0			0
$909$	21.9490	−	28.4255i	0.0241463	−	0.0312712i
$910$		0			0
$911$			1586.80i			1.74182i	0.491439	+	0.870912i	$0.336471\pi$
$911$			1586.80i			1.74182i	−0.491439	+	0.870912i	$0.663529\pi$
$912$		0			0
$913$	791.064			0.866445
$914$		0			0
$915$	129.319	−	379.071i	0.141332	−	0.414285i
$916$		0			0
$917$		−	716.619i		−	0.781482i
$918$		0			0
$919$	1183.01			1.28728			0.643638	−	0.765330i	$-0.277424\pi$
$919$	1183.01			1.28728			0.643638	+	0.765330i	$0.277424\pi$
$920$		0			0
$921$	333.001	+	113.602i	0.361565	+	0.123347i
$922$		0			0
$923$		−	338.723i		−	0.366981i
$924$		0			0
$925$	−22.1302			−0.0239245
$926$		0			0
$927$	−478.879	−	369.770i	−0.516590	−	0.398889i
$928$		0			0
$929$			603.858i			0.650008i	0.945713	+	0.325004i	$0.105366\pi$
$929$			603.858i			0.650008i	−0.945713	+	0.325004i	$0.894634\pi$
$930$		0			0
$931$	72.9764			0.0783850
$932$		0			0
$933$	393.513	−	1153.50i	0.421772	−	1.23634i
$934$		0			0
$935$		−	2137.42i		−	2.28601i
$936$		0			0
$937$	145.480			0.155261			0.0776305	−	0.996982i	$-0.475265\pi$
$937$	145.480			0.155261			0.0776305	+	0.996982i	$0.475265\pi$
$938$		0			0
$939$	530.270	+	180.900i	0.564718	+	0.192652i
$940$		0			0
$941$			406.855i			0.432364i	0.976353	+	0.216182i	$0.0693605\pi$
$941$			406.855i			0.432364i	−0.976353	+	0.216182i	$0.930640\pi$
$942$		0			0
$943$	−845.185			−0.896273
$944$		0			0
$945$	491.502	−	325.186i	0.520108	−	0.344112i
$946$		0			0
$947$		−	494.131i		−	0.521785i	−0.965368	−	0.260893i	$-0.915983\pi$
$947$		−	494.131i		−	0.521785i	0.965368	−	0.260893i	$-0.0840169\pi$
$948$		0			0
$949$	86.7924			0.0914567
$950$		0			0
$951$	288.359	−	845.265i	0.303217	−	0.888817i
$952$		0			0
$953$		−	263.106i		−	0.276082i	−0.990427	−	0.138041i	$-0.955919\pi$
$953$		−	263.106i		−	0.276082i	0.990427	−	0.138041i	$-0.0440806\pi$
$954$		0			0
$955$	459.155			0.480791
$956$		0			0
$957$	−1759.28	−	600.175i	−1.83833	−	0.627142i
$958$		0			0
$959$		−	946.876i		−	0.987357i
$960$		0			0
$961$	−382.332			−0.397848
$962$		0			0
$963$	−458.473	+	593.755i	−0.476088	+	0.616568i
$964$		0			0
$965$			1648.87i			1.70867i
$966$		0			0
$967$	−1053.74			−1.08970			−0.544852	−	0.838532i	$-0.683414\pi$
$967$	−1053.74			−1.08970			−0.544852	+	0.838532i	$0.683414\pi$
$968$		0			0
$969$	55.9013	−	163.863i	0.0576897	−	0.169105i
$970$		0			0
$971$		−	46.1117i		−	0.0474889i	−0.999718	−	0.0237444i	$-0.992441\pi$
$971$		−	46.1117i		−	0.0474889i	0.999718	−	0.0237444i	$-0.00755880\pi$
$972$		0			0
$973$	−32.9394			−0.0338534
$974$		0			0
$975$	−63.7113	−	21.7349i	−0.0653449	−	0.0222922i
$976$		0			0
$977$		−	1142.60i		−	1.16950i	−0.811215	−	0.584748i	$-0.801194\pi$
$977$		−	1142.60i		−	1.16950i	0.811215	−	0.584748i	$-0.198806\pi$
$978$		0			0
$979$	−1109.76			−1.13357
$980$		0			0
$981$	1087.64	+	839.828i	1.10870	+	0.856094i
$982$		0			0
$983$			266.414i			0.271021i	0.990776	+	0.135511i	$0.0432675\pi$
$983$			266.414i			0.271021i	−0.990776	+	0.135511i	$0.956732\pi$
$984$		0			0
$985$	391.363			0.397322
$986$		0			0
$987$	−127.929	+	374.997i	−0.129614	+	0.379936i
$988$		0			0
$989$		−	1480.53i		−	1.49699i
$990$		0			0
$991$	80.5974			0.0813293			0.0406647	−	0.999173i	$-0.487052\pi$
$991$	80.5974			0.0813293			0.0406647	+	0.999173i	$0.487052\pi$
$992$		0			0
$993$	870.827	+	297.080i	0.876965	+	0.299174i
$994$		0			0
$995$			944.329i			0.949074i
$996$		0			0
$997$	−348.827			−0.349876			−0.174938	−	0.984579i	$-0.555973\pi$
$997$	−348.827			−0.349876			−0.174938	+	0.984579i	$0.555973\pi$
$998$		0			0
$999$	52.9767	+	80.0717i	0.0530297	+	0.0801519i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	156.3.d.a.53.8	yes	8
3.2	odd	2	inner	156.3.d.a.53.7	✓	8
4.3	odd	2		624.3.f.a.209.1		8
12.11	even	2		624.3.f.a.209.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
156.3.d.a.53.7	✓	8	3.2	odd	2	inner
156.3.d.a.53.8	yes	8	1.1	even	1	trivial
624.3.f.a.209.1		8	4.3	odd	2
624.3.f.a.209.2		8	12.11	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:	\( N \)	\(=\)	\( 156 = 2^{2} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	156.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(2.83932 + 0.968627i) q^{3} -5.58779i q^{5} +3.90626 q^{7} +(7.12352 + 5.50049i) q^{9} +O(q^{10})\)	\(q+(2.83932 + 0.968627i) q^{3} -5.58779i q^{5} +3.90626 q^{7} +(7.12352 + 5.50049i) q^{9} -14.3353i q^{11} +3.60555 q^{13} +(5.41249 - 15.8656i) q^{15} +26.6835i q^{17} -2.16283 q^{19} +(11.0911 + 3.78371i) q^{21} +17.4950i q^{23} -6.22343 q^{25} +(14.8981 + 22.5177i) q^{27} -43.2229i q^{29} -24.0555 q^{31} +(13.8856 - 40.7026i) q^{33} -21.8274i q^{35} +3.55594 q^{37} +(10.2373 + 3.49243i) q^{39} +48.3100i q^{41} -84.6256 q^{43} +(30.7356 - 39.8048i) q^{45} +33.8105i q^{47} -33.7411 q^{49} +(-25.8464 + 75.7631i) q^{51} +9.24821i q^{53} -80.1028 q^{55} +(-6.14098 - 2.09498i) q^{57} +36.2385i q^{59} +23.8927 q^{61} +(27.8263 + 21.4863i) q^{63} -20.1471i q^{65} -112.566 q^{67} +(-16.9462 + 49.6741i) q^{69} -93.9449i q^{71} +24.0719 q^{73} +(-17.6703 - 6.02818i) q^{75} -55.9974i q^{77} +44.4386 q^{79} +(20.4892 + 78.3658i) q^{81} +55.1829i q^{83} +149.102 q^{85} +(41.8669 - 122.724i) q^{87} -77.4146i q^{89} +14.0842 q^{91} +(-68.3014 - 23.3008i) q^{93} +12.0855i q^{95} +143.143 q^{97} +(78.8513 - 102.118i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 6 q^{3} - 8 q^{7} - 22 q^{9}+O(q^{10}) \) 8 * q + 6 * q^3 - 8 * q^7 - 22 * q^9	\( 8 q + 6 q^{3} - 8 q^{7} - 22 q^{9} + 4 q^{15} - 24 q^{19} + 16 q^{21} - 28 q^{25} + 36 q^{27} + 96 q^{31} + 4 q^{33} - 96 q^{37} + 76 q^{43} - 136 q^{45} + 204 q^{49} - 62 q^{51} - 80 q^{55} + 184 q^{57} - 104 q^{61} + 36 q^{63} - 384 q^{67} - 8 q^{73} - 100 q^{75} + 328 q^{79} + 50 q^{81} - 64 q^{85} - 204 q^{87} - 52 q^{91} + 72 q^{93} + 416 q^{97} + 284 q^{99}+O(q^{100}) \) 8 * q + 6 * q^3 - 8 * q^7 - 22 * q^9 + 4 * q^15 - 24 * q^19 + 16 * q^21 - 28 * q^25 + 36 * q^27 + 96 * q^31 + 4 * q^33 - 96 * q^37 + 76 * q^43 - 136 * q^45 + 204 * q^49 - 62 * q^51 - 80 * q^55 + 184 * q^57 - 104 * q^61 + 36 * q^63 - 384 * q^67 - 8 * q^73 - 100 * q^75 + 328 * q^79 + 50 * q^81 - 64 * q^85 - 204 * q^87 - 52 * q^91 + 72 * q^93 + 416 * q^97 + 284 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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