LMFDB - Embedded newform 2040.2.m.h.409.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2040,2,Mod(409,2040)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2040, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2040.409"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2040 = 2^{3} \cdot 3 \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2040.m (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [10,0,0,0,-2,0,0,0,-10,0,10,0,0,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

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

Self dual:	no
Analytic conductor:	$16.2894820123$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 8x^{8} + 12x^{7} + 17x^{6} - 28x^{5} + 85x^{4} + 300x^{3} - 1000x^{2} - 1250x + 3125 $ x^10 - 2x^9 - 8x^8 + 12x^7 + 17x^6 - 28x^5 + 85x^4 + 300x^3 - 1000x^2 - 1250*x + 3125
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			409.8
Root			$-2.21950 - 0.271668i$ of defining polynomial
Character	$\chi$	$=$	2040.409
Dual form			2040.2.m.h.409.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +(-0.271668 - 2.21950i) q^{5} -1.68096i q^{7} -1.00000 q^{9} +4.22430 q^{11} +6.41187i q^{13} +(2.21950 - 0.271668i) q^{15} +1.00000i q^{17} -3.51721 q^{19} +1.68096 q^{21} +0.922922i q^{23} +(-4.85239 + 1.20594i) q^{25} -1.00000i q^{27} +6.59429 q^{29} +5.86854 q^{31} +4.22430i q^{33} +(-3.73091 + 0.456664i) q^{35} -9.19705i q^{37} -6.41187 q^{39} +7.74039 q^{41} +4.95006i q^{43} +(0.271668 + 2.21950i) q^{45} +10.1523i q^{47} +4.17436 q^{49} -1.00000 q^{51} -8.46283i q^{53} +(-1.14761 - 9.37585i) q^{55} -3.51721i q^{57} -4.44860 q^{59} +7.86854 q^{61} +1.68096i q^{63} +(14.2312 - 1.74190i) q^{65} -10.2753i q^{67} -0.922922 q^{69} -1.49339 q^{71} -10.4275i q^{73} +(-1.20594 - 4.85239i) q^{75} -7.10090i q^{77} +15.8332 q^{79} +1.00000 q^{81} +8.60276i q^{83} +(2.21950 - 0.271668i) q^{85} +6.59429i q^{87} +10.3398 q^{89} +10.7781 q^{91} +5.86854i q^{93} +(0.955514 + 7.80647i) q^{95} -6.06457i q^{97} -4.22430 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 2 q^{5} - 10 q^{9} + 10 q^{11} - 2 q^{15} + 6 q^{19} - 14 q^{21} - 20 q^{25} + 38 q^{29} + 12 q^{31} - 20 q^{35} - 16 q^{39} + 10 q^{41} + 2 q^{45} - 4 q^{49} - 10 q^{51} - 40 q^{55} + 20 q^{59}+ \cdots - 10 q^{99}+O(q^{100}) $ 10 * q - 2 * q^5 - 10 * q^9 + 10 * q^11 - 2 * q^15 + 6 * q^19 - 14 * q^21 - 20 * q^25 + 38 * q^29 + 12 * q^31 - 20 * q^35 - 16 * q^39 + 10 * q^41 + 2 * q^45 - 4 * q^49 - 10 * q^51 - 40 * q^55 + 20 * q^59 + 32 * q^61 - 24 * q^65 + 4 * q^69 + 12 * q^75 + 12 * q^79 + 10 * q^81 - 2 * q^85 - 28 * q^89 - 16 * q^91 - 22 * q^95 - 10 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2040\mathbb{Z}\right)^\times$.

$n$	$241$	$511$	$817$	$1021$	$1361$
$\chi(n)$	$1$	$1$	$-1$	$1$	$1$

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
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$		0			0
$5$	−0.271668	−	2.21950i	−0.121494	−	0.992592i
$5$	−0.271668	−	2.21950i	−0.121494	−	0.992592i
$6$		0			0
$7$		−	1.68096i		−	0.635345i	−0.948201	−	0.317672i	$-0.897099\pi$
$7$		−	1.68096i		−	0.635345i	0.948201	−	0.317672i	$-0.102901\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	4.22430			1.27367			0.636837	−	0.770998i	$-0.280242\pi$
$11$	4.22430			1.27367			0.636837	+	0.770998i	$0.280242\pi$
$12$		0			0
$13$			6.41187i			1.77833i	0.457584	+	0.889167i	$0.348715\pi$
$13$			6.41187i			1.77833i	−0.457584	+	0.889167i	$0.651285\pi$
$14$		0			0
$15$	2.21950	−	0.271668i	0.573073	−	0.0701444i
$16$		0			0
$17$			1.00000i			0.242536i
$17$			1.00000i			0.242536i
$18$		0			0
$19$	−3.51721			−0.806904			−0.403452	−	0.915001i	$-0.632190\pi$
$19$	−3.51721			−0.806904			−0.403452	+	0.915001i	$0.632190\pi$
$20$		0			0
$21$	1.68096			0.366817
$22$		0			0
$23$			0.922922i			0.192443i	0.995360	+	0.0962213i	$0.0306756\pi$
$23$			0.922922i			0.192443i	−0.995360	+	0.0962213i	$0.969324\pi$
$24$		0			0
$25$	−4.85239	+	1.20594i	−0.970479	+	0.241187i
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	6.59429			1.22453			0.612265	−	0.790653i	$-0.290259\pi$
$29$	6.59429			1.22453			0.612265	+	0.790653i	$0.290259\pi$
$30$		0			0
$31$	5.86854			1.05402			0.527010	−	0.849859i	$-0.323313\pi$
$31$	5.86854			1.05402			0.527010	+	0.849859i	$0.323313\pi$
$32$		0			0
$33$			4.22430i			0.735356i
$34$		0			0
$35$	−3.73091	+	0.456664i	−0.630638	+	0.0771903i
$36$		0			0
$37$		−	9.19705i		−	1.51199i	−0.654580	−	0.755993i	$-0.727154\pi$
$37$		−	9.19705i		−	1.51199i	0.654580	−	0.755993i	$-0.272846\pi$
$38$		0			0
$39$	−6.41187			−1.02672
$40$		0			0
$41$	7.74039			1.20885			0.604423	−	0.796664i	$-0.293404\pi$
$41$	7.74039			1.20885			0.604423	+	0.796664i	$0.293404\pi$
$42$		0			0
$43$			4.95006i			0.754877i	0.926035	+	0.377438i	$0.123195\pi$
$43$			4.95006i			0.754877i	−0.926035	+	0.377438i	$0.876805\pi$
$44$		0			0
$45$	0.271668	+	2.21950i	0.0404979	+	0.330864i
$46$		0			0
$47$			10.1523i			1.48086i	0.672134	+	0.740429i	$0.265378\pi$
$47$			10.1523i			1.48086i	−0.672134	+	0.740429i	$0.734622\pi$
$48$		0			0
$49$	4.17436			0.596337
$50$		0			0
$51$	−1.00000			−0.140028
$52$		0			0
$53$		−	8.46283i		−	1.16246i	−0.813740	−	0.581229i	$-0.802572\pi$
$53$		−	8.46283i		−	1.16246i	0.813740	−	0.581229i	$-0.197428\pi$
$54$		0			0
$55$	−1.14761	−	9.37585i	−0.154743	−	1.26424i
$56$		0			0
$57$		−	3.51721i		−	0.465866i
$58$		0			0
$59$	−4.44860			−0.579159			−0.289579	−	0.957154i	$-0.593515\pi$
$59$	−4.44860			−0.579159			−0.289579	+	0.957154i	$0.593515\pi$
$60$		0			0
$61$	7.86854			1.00746			0.503731	−	0.863860i	$-0.331960\pi$
$61$	7.86854			1.00746			0.503731	+	0.863860i	$0.331960\pi$
$62$		0			0
$63$			1.68096i			0.211782i
$64$		0			0
$65$	14.2312	−	1.74190i	1.76516	−	0.216056i
$66$		0			0
$67$		−	10.2753i		−	1.25532i	−0.778487	−	0.627661i	$-0.784012\pi$
$67$		−	10.2753i		−	1.25532i	0.778487	−	0.627661i	$-0.215988\pi$
$68$		0			0
$69$	−0.922922			−0.111107
$70$		0			0
$71$	−1.49339			−0.177233			−0.0886166	−	0.996066i	$-0.528245\pi$
$71$	−1.49339			−0.177233			−0.0886166	+	0.996066i	$0.528245\pi$
$72$		0			0
$73$		−	10.4275i		−	1.22045i	−0.792229	−	0.610224i	$-0.791079\pi$
$73$		−	10.4275i		−	1.22045i	0.792229	−	0.610224i	$-0.208921\pi$
$74$		0			0
$75$	−1.20594	−	4.85239i	−0.139249	−	0.560306i
$76$		0			0
$77$		−	7.10090i		−	0.809223i
$78$		0			0
$79$	15.8332			1.78138			0.890688	−	0.454614i	$-0.150223\pi$
$79$	15.8332			1.78138			0.890688	+	0.454614i	$0.150223\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			8.60276i			0.944275i	0.881525	+	0.472138i	$0.156517\pi$
$83$			8.60276i			0.944275i	−0.881525	+	0.472138i	$0.843483\pi$
$84$		0			0
$85$	2.21950	−	0.271668i	0.240739	−	0.0294665i
$86$		0			0
$87$			6.59429i			0.706982i
$88$		0			0
$89$	10.3398			1.09602			0.548010	−	0.836472i	$-0.315386\pi$
$89$	10.3398			1.09602			0.548010	+	0.836472i	$0.315386\pi$
$90$		0			0
$91$	10.7781			1.12985
$92$		0			0
$93$			5.86854i			0.608539i
$94$		0			0
$95$	0.955514	+	7.80647i	0.0980337	+	0.800927i
$96$		0			0
$97$		−	6.06457i		−	0.615764i	−0.951425	−	0.307882i	$-0.900380\pi$
$97$		−	6.06457i		−	0.615764i	0.951425	−	0.307882i	$-0.0996202\pi$
$98$		0			0
$99$	−4.22430			−0.424558
$100$		0			0
$101$	3.51609			0.349864			0.174932	−	0.984581i	$-0.444030\pi$
$101$	3.51609			0.349864			0.174932	+	0.984581i	$0.444030\pi$
$102$		0			0
$103$			0.739371i			0.0728524i	0.999336	+	0.0364262i	$0.0115974\pi$
$103$			0.739371i			0.0728524i	−0.999336	+	0.0364262i	$0.988403\pi$
$104$		0			0
$105$	−0.456664	−	3.73091i	−0.0445659	−	0.364099i
$106$		0			0
$107$			5.06672i			0.489818i	0.969546	+	0.244909i	$0.0787581\pi$
$107$			5.06672i			0.489818i	−0.969546	+	0.244909i	$0.921242\pi$
$108$		0			0
$109$	1.28271			0.122861			0.0614305	−	0.998111i	$-0.480434\pi$
$109$	1.28271			0.122861			0.0614305	+	0.998111i	$0.480434\pi$
$110$		0			0
$111$	9.19705			0.872945
$112$		0			0
$113$		−	10.7561i		−	1.01185i	−0.862577	−	0.505926i	$-0.831151\pi$
$113$		−	10.7561i		−	1.01185i	0.862577	−	0.505926i	$-0.168849\pi$
$114$		0			0
$115$	2.04843	−	0.250728i	0.191017	−	0.0233805i
$116$		0			0
$117$		−	6.41187i		−	0.592778i
$118$		0			0
$119$	1.68096			0.154094
$120$		0			0
$121$	6.84471			0.622247
$122$		0			0
$123$			7.74039i			0.697927i
$124$		0			0
$125$	3.99482	+	10.4423i	0.357307	+	0.933987i
$126$		0			0
$127$		−	0.722443i		−	0.0641064i	−0.999486	−	0.0320532i	$-0.989795\pi$
$127$		−	0.722443i		−	0.0641064i	0.999486	−	0.0320532i	$-0.0102046\pi$
$128$		0			0
$129$	−4.95006			−0.435828
$130$		0			0
$131$	−11.0146			−0.962353			−0.481176	−	0.876624i	$-0.659790\pi$
$131$	−11.0146			−0.962353			−0.481176	+	0.876624i	$0.659790\pi$
$132$		0			0
$133$			5.91231i			0.512663i
$134$		0			0
$135$	−2.21950	+	0.271668i	−0.191024	+	0.0233815i
$136$		0			0
$137$			9.72284i			0.830679i	0.909667	+	0.415339i	$0.136337\pi$
$137$			9.72284i			0.830679i	−0.909667	+	0.415339i	$0.863663\pi$
$138$		0			0
$139$	3.67250			0.311497			0.155749	−	0.987797i	$-0.450221\pi$
$139$	3.67250			0.311497			0.155749	+	0.987797i	$0.450221\pi$
$140$		0			0
$141$	−10.1523			−0.854974
$142$		0			0
$143$			27.0857i			2.26502i
$144$		0			0
$145$	−1.79146	−	14.6361i	−0.148772	−	1.21546i
$146$		0			0
$147$			4.17436i			0.344295i
$148$		0			0
$149$	−7.15124			−0.585853			−0.292926	−	0.956135i	$-0.594629\pi$
$149$	−7.15124			−0.585853			−0.292926	+	0.956135i	$0.594629\pi$
$150$		0			0
$151$	−11.8601			−0.965159			−0.482579	−	0.875852i	$-0.660300\pi$
$151$	−11.8601			−0.965159			−0.482579	+	0.875852i	$0.660300\pi$
$152$		0			0
$153$		−	1.00000i		−	0.0808452i
$154$		0			0
$155$	−1.59429	−	13.0252i	−0.128057	−	1.04621i
$156$		0			0
$157$			18.1489i			1.44844i	0.689567	+	0.724222i	$0.257801\pi$
$157$			18.1489i			1.44844i	−0.689567	+	0.724222i	$0.742199\pi$
$158$		0			0
$159$	8.46283			0.671146
$160$		0			0
$161$	1.55140			0.122267
$162$		0			0
$163$			7.54950i			0.591323i	0.955293	+	0.295661i	$0.0955400\pi$
$163$			7.54950i			0.591323i	−0.955293	+	0.295661i	$0.904460\pi$
$164$		0			0
$165$	9.37585	−	1.14761i	0.729909	−	0.0893411i
$166$		0			0
$167$			4.65999i			0.360601i	0.983612	+	0.180301i	$0.0577070\pi$
$167$			4.65999i			0.360601i	−0.983612	+	0.180301i	$0.942293\pi$
$168$		0			0
$169$	−28.1121			−2.16247
$170$		0			0
$171$	3.51721			0.268968
$172$		0			0
$173$			12.8141i			0.974242i	0.873335	+	0.487121i	$0.161953\pi$
$173$			12.8141i			0.974242i	−0.873335	+	0.487121i	$0.838047\pi$
$174$		0			0
$175$	2.02714	+	8.15670i	0.153237	+	0.616589i
$176$		0			0
$177$		−	4.44860i		−	0.334377i
$178$		0			0
$179$	7.18859			0.537300			0.268650	−	0.963238i	$-0.413422\pi$
$179$	7.18859			0.537300			0.268650	+	0.963238i	$0.413422\pi$
$180$		0			0
$181$	8.58006			0.637751			0.318876	−	0.947797i	$-0.396695\pi$
$181$	8.58006			0.637751			0.318876	+	0.947797i	$0.396695\pi$
$182$		0			0
$183$			7.86854i			0.581659i
$184$		0			0
$185$	−20.4129	+	2.49854i	−1.50079	+	0.183697i
$186$		0			0
$187$			4.22430i			0.308911i
$188$		0			0
$189$	−1.68096			−0.122272
$190$		0			0
$191$	18.8427			1.36341			0.681705	−	0.731627i	$-0.261239\pi$
$191$	18.8427			1.36341			0.681705	+	0.731627i	$0.261239\pi$
$192$		0			0
$193$		−	9.03217i		−	0.650150i	−0.945688	−	0.325075i	$-0.894610\pi$
$193$		−	9.03217i		−	0.650150i	0.945688	−	0.325075i	$-0.105390\pi$
$194$		0			0
$195$	1.74190	+	14.2312i	0.124740	+	1.01912i
$196$		0			0
$197$			20.9895i			1.49544i	0.664013	+	0.747721i	$0.268852\pi$
$197$			20.9895i			1.49544i	−0.664013	+	0.747721i	$0.731148\pi$
$198$		0			0
$199$	10.9552			0.776594			0.388297	−	0.921534i	$-0.373063\pi$
$199$	10.9552			0.776594			0.388297	+	0.921534i	$0.373063\pi$
$200$		0			0
$201$	10.2753			0.724761
$202$		0			0
$203$		−	11.0848i		−	0.777999i
$204$		0			0
$205$	−2.10281	−	17.1798i	−0.146867	−	1.19989i
$206$		0			0
$207$		−	0.922922i		−	0.0641475i
$208$		0			0
$209$	−14.8578			−1.02773
$210$		0			0
$211$	−25.3912			−1.74800			−0.874001	−	0.485925i	$-0.838483\pi$
$211$	−25.3912			−1.74800			−0.874001	+	0.485925i	$0.838483\pi$
$212$		0			0
$213$		−	1.49339i		−	0.102326i
$214$		0			0
$215$	10.9867	−	1.34477i	0.749285	−	0.0917127i
$216$		0			0
$217$		−	9.86480i		−	0.669666i
$218$		0			0
$219$	10.4275			0.704626
$220$		0			0
$221$	−6.41187			−0.431309
$222$		0			0
$223$		−	13.6717i		−	0.915522i	−0.889075	−	0.457761i	$-0.848652\pi$
$223$		−	13.6717i		−	0.915522i	0.889075	−	0.457761i	$-0.151348\pi$
$224$		0			0
$225$	4.85239	−	1.20594i	0.323493	−	0.0803957i
$226$		0			0
$227$		−	10.7372i		−	0.712652i	−0.934362	−	0.356326i	$-0.884029\pi$
$227$		−	10.7372i		−	0.712652i	0.934362	−	0.356326i	$-0.115971\pi$
$228$		0			0
$229$	26.1646			1.72900			0.864502	−	0.502629i	$-0.167634\pi$
$229$	26.1646			1.72900			0.864502	+	0.502629i	$0.167634\pi$
$230$		0			0
$231$	7.10090			0.467205
$232$		0			0
$233$		−	9.24006i		−	0.605336i	−0.953096	−	0.302668i	$-0.902123\pi$
$233$		−	9.24006i		−	0.605336i	0.953096	−	0.302668i	$-0.0978774\pi$
$234$		0			0
$235$	22.5330	−	2.75804i	1.46989	−	0.179915i
$236$		0			0
$237$			15.8332i			1.02848i
$238$		0			0
$239$	−1.91042			−0.123574			−0.0617872	−	0.998089i	$-0.519680\pi$
$239$	−1.91042			−0.123574			−0.0617872	+	0.998089i	$0.519680\pi$
$240$		0			0
$241$	−3.63719			−0.234292			−0.117146	−	0.993115i	$-0.537375\pi$
$241$	−3.63719			−0.234292			−0.117146	+	0.993115i	$0.537375\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	−1.13404	−	9.26500i	−0.0724511	−	0.591919i
$246$		0			0
$247$		−	22.5519i		−	1.43494i
$248$		0			0
$249$	−8.60276			−0.545178
$250$		0			0
$251$	−5.74370			−0.362539			−0.181270	−	0.983433i	$-0.558021\pi$
$251$	−5.74370			−0.362539			−0.181270	+	0.983433i	$0.558021\pi$
$252$		0			0
$253$			3.89870i			0.245109i
$254$		0			0
$255$	0.271668	+	2.21950i	0.0170125	+	0.138991i
$256$		0			0
$257$		−	4.16649i		−	0.259898i	−0.991521	−	0.129949i	$-0.958519\pi$
$257$		−	4.16649i		−	0.259898i	0.991521	−	0.129949i	$-0.0414814\pi$
$258$		0			0
$259$	−15.4599			−0.960632
$260$		0			0
$261$	−6.59429			−0.408176
$262$		0			0
$263$			18.0877i			1.11533i	0.830065	+	0.557667i	$0.188303\pi$
$263$			18.0877i			1.11533i	−0.830065	+	0.557667i	$0.811697\pi$
$264$		0			0
$265$	−18.7833	+	2.29908i	−1.15385	+	0.141231i
$266$		0			0
$267$			10.3398i			0.632787i
$268$		0			0
$269$	24.1574			1.47290			0.736452	−	0.676490i	$-0.236500\pi$
$269$	24.1574			1.47290			0.736452	+	0.676490i	$0.236500\pi$
$270$		0			0
$271$	−24.1181			−1.46507			−0.732534	−	0.680730i	$-0.761663\pi$
$271$	−24.1181			−1.46507			−0.732534	+	0.680730i	$0.761663\pi$
$272$		0			0
$273$			10.7781i			0.652322i
$274$		0			0
$275$	−20.4980	+	5.09424i	−1.23607	+	0.307194i
$276$		0			0
$277$		−	28.6047i		−	1.71869i	−0.511395	−	0.859346i	$-0.670871\pi$
$277$		−	28.6047i		−	1.71869i	0.511395	−	0.859346i	$-0.329129\pi$
$278$		0			0
$279$	−5.86854			−0.351340
$280$		0			0
$281$	−1.44263			−0.0860599			−0.0430300	−	0.999074i	$-0.513701\pi$
$281$	−1.44263			−0.0860599			−0.0430300	+	0.999074i	$0.513701\pi$
$282$		0			0
$283$			12.9467i			0.769604i	0.922999	+	0.384802i	$0.125730\pi$
$283$			12.9467i			0.769604i	−0.922999	+	0.384802i	$0.874270\pi$
$284$		0			0
$285$	−7.80647	+	0.955514i	−0.462415	+	0.0565998i
$286$		0			0
$287$		−	13.0113i		−	0.768034i
$288$		0			0
$289$	−1.00000			−0.0588235
$290$		0			0
$291$	6.06457			0.355511
$292$		0			0
$293$			15.1390i			0.884432i	0.896908	+	0.442216i	$0.145808\pi$
$293$			15.1390i			0.884432i	−0.896908	+	0.442216i	$0.854192\pi$
$294$		0			0
$295$	1.20854	+	9.87369i	0.0703640	+	0.574868i
$296$		0			0
$297$		−	4.22430i		−	0.245119i
$298$		0			0
$299$	−5.91766			−0.342227
$300$		0			0
$301$	8.32087			0.479607
$302$		0			0
$303$			3.51609i			0.201994i
$304$		0			0
$305$	−2.13763	−	17.4642i	−0.122400	−	1.00000i
$306$		0			0
$307$			12.8450i			0.733101i	0.930398	+	0.366550i	$0.119461\pi$
$307$			12.8450i			0.733101i	−0.930398	+	0.366550i	$0.880539\pi$
$308$		0			0
$309$	−0.739371			−0.0420614
$310$		0			0
$311$	−22.9341			−1.30047			−0.650237	−	0.759731i	$-0.725331\pi$
$311$	−22.9341			−1.30047			−0.650237	+	0.759731i	$0.725331\pi$
$312$		0			0
$313$		−	12.2734i		−	0.693731i	−0.937915	−	0.346866i	$-0.887246\pi$
$313$		−	12.2734i		−	0.693731i	0.937915	−	0.346866i	$-0.112754\pi$
$314$		0			0
$315$	3.73091	−	0.456664i	0.210213	−	0.0257301i
$316$		0			0
$317$		−	27.6791i		−	1.55461i	−0.629123	−	0.777306i	$-0.716586\pi$
$317$		−	27.6791i		−	1.55461i	0.629123	−	0.777306i	$-0.283414\pi$
$318$		0			0
$319$	27.8563			1.55965
$320$		0			0
$321$	−5.06672			−0.282796
$322$		0			0
$323$		−	3.51721i		−	0.195703i
$324$		0			0
$325$	−7.73231	−	31.1129i	−0.428911	−	1.72583i
$326$		0			0
$327$			1.28271i			0.0709339i
$328$		0			0
$329$	17.0656			0.940856
$330$		0			0
$331$	17.0083			0.934860			0.467430	−	0.884030i	$-0.345180\pi$
$331$	17.0083			0.934860			0.467430	+	0.884030i	$0.345180\pi$
$332$		0			0
$333$			9.19705i			0.503995i
$334$		0			0
$335$	−22.8060	+	2.79146i	−1.24602	+	0.152514i
$336$		0			0
$337$		−	19.1199i		−	1.04152i	−0.853702	−	0.520762i	$-0.825648\pi$
$337$		−	19.1199i		−	1.04152i	0.853702	−	0.520762i	$-0.174352\pi$
$338$		0			0
$339$	10.7561			0.584194
$340$		0			0
$341$	24.7905			1.34248
$342$		0			0
$343$		−	18.7837i		−	1.01422i
$344$		0			0
$345$	0.250728	+	2.04843i	0.0134988	+	0.110284i
$346$		0			0
$347$			20.8550i			1.11956i	0.828642	+	0.559778i	$0.189114\pi$
$347$			20.8550i			1.11956i	−0.828642	+	0.559778i	$0.810886\pi$
$348$		0			0
$349$	−15.8901			−0.850577			−0.425289	−	0.905058i	$-0.639827\pi$
$349$	−15.8901			−0.850577			−0.425289	+	0.905058i	$0.639827\pi$
$350$		0			0
$351$	6.41187			0.342240
$352$		0			0
$353$		−	34.9950i		−	1.86259i	−0.364261	−	0.931297i	$-0.618678\pi$
$353$		−	34.9950i		−	1.86259i	0.364261	−	0.931297i	$-0.381322\pi$
$354$		0			0
$355$	0.405707	+	3.31459i	0.0215327	+	0.175920i
$356$		0			0
$357$			1.68096i			0.0889661i
$358$		0			0
$359$	−8.54163			−0.450810			−0.225405	−	0.974265i	$-0.572371\pi$
$359$	−8.54163			−0.450810			−0.225405	+	0.974265i	$0.572371\pi$
$360$		0			0
$361$	−6.62920			−0.348905
$362$		0			0
$363$			6.84471i			0.359254i
$364$		0			0
$365$	−23.1439	+	2.83282i	−1.21141	+	0.148277i
$366$		0			0
$367$		−	34.8134i		−	1.81724i	−0.417620	−	0.908622i	$-0.637136\pi$
$367$		−	34.8134i		−	1.81724i	0.417620	−	0.908622i	$-0.362864\pi$
$368$		0			0
$369$	−7.74039			−0.402948
$370$		0			0
$371$	−14.2257			−0.738562
$372$		0			0
$373$			37.0306i			1.91737i	0.284464	+	0.958687i	$0.408184\pi$
$373$			37.0306i			1.91737i	−0.284464	+	0.958687i	$0.591816\pi$
$374$		0			0
$375$	−10.4423	+	3.99482i	−0.539238	+	0.206292i
$376$		0			0
$377$			42.2818i			2.17762i
$378$		0			0
$379$	−33.8602			−1.73928			−0.869641	−	0.493685i	$-0.835650\pi$
$379$	−33.8602			−1.73928			−0.869641	+	0.493685i	$0.835650\pi$
$380$		0			0
$381$	0.722443			0.0370119
$382$		0			0
$383$			33.1493i			1.69385i	0.531710	+	0.846926i	$0.321549\pi$
$383$			33.1493i			1.69385i	−0.531710	+	0.846926i	$0.678451\pi$
$384$		0			0
$385$	−15.7605	+	1.92909i	−0.803228	+	0.0983154i
$386$		0			0
$387$		−	4.95006i		−	0.251626i
$388$		0			0
$389$	−17.2178			−0.872979			−0.436490	−	0.899709i	$-0.643778\pi$
$389$	−17.2178			−0.872979			−0.436490	+	0.899709i	$0.643778\pi$
$390$		0			0
$391$	−0.922922			−0.0466742
$392$		0			0
$393$		−	11.0146i		−	0.555615i
$394$		0			0
$395$	−4.30138	−	35.1419i	−0.216426	−	1.76818i
$396$		0			0
$397$			4.42095i			0.221881i	0.993827	+	0.110941i	$0.0353863\pi$
$397$			4.42095i			0.221881i	−0.993827	+	0.110941i	$0.964614\pi$
$398$		0			0
$399$	−5.91231			−0.295986
$400$		0			0
$401$	10.4330			0.520997			0.260499	−	0.965474i	$-0.416113\pi$
$401$	10.4330			0.520997			0.260499	+	0.965474i	$0.416113\pi$
$402$		0			0
$403$			37.6283i			1.87440i
$404$		0			0
$405$	−0.271668	−	2.21950i	−0.0134993	−	0.110288i
$406$		0			0
$407$		−	38.8511i		−	1.92578i
$408$		0			0
$409$	−6.98566			−0.345419			−0.172709	−	0.984973i	$-0.555252\pi$
$409$	−6.98566			−0.345419			−0.172709	+	0.984973i	$0.555252\pi$
$410$		0			0
$411$	−9.72284			−0.479593
$412$		0			0
$413$			7.47794i			0.367965i
$414$		0			0
$415$	19.0939	−	2.33709i	0.937280	−	0.114723i
$416$		0			0
$417$			3.67250i			0.179843i
$418$		0			0
$419$	−22.7197			−1.10993			−0.554965	−	0.831874i	$-0.687268\pi$
$419$	−22.7197			−1.10993			−0.554965	+	0.831874i	$0.687268\pi$
$420$		0			0
$421$	−5.27324			−0.257002			−0.128501	−	0.991709i	$-0.541017\pi$
$421$	−5.27324			−0.257002			−0.128501	+	0.991709i	$0.541017\pi$
$422$		0			0
$423$		−	10.1523i		−	0.493620i
$424$		0			0
$425$	−1.20594	−	4.85239i	−0.0584965	−	0.235376i
$426$		0			0
$427$		−	13.2267i		−	0.640086i
$428$		0			0
$429$	−27.0857			−1.30771
$430$		0			0
$431$	−27.7191			−1.33518			−0.667591	−	0.744528i	$-0.732675\pi$
$431$	−27.7191			−1.33518			−0.667591	+	0.744528i	$0.732675\pi$
$432$		0			0
$433$		−	2.20547i		−	0.105988i	−0.998595	−	0.0529941i	$-0.983124\pi$
$433$		−	2.20547i		−	0.105988i	0.998595	−	0.0529941i	$-0.0168765\pi$
$434$		0			0
$435$	14.6361	−	1.79146i	0.701745	−	0.0858938i
$436$		0			0
$437$		−	3.24611i		−	0.155283i
$438$		0			0
$439$	13.6960			0.653674			0.326837	−	0.945081i	$-0.394017\pi$
$439$	13.6960			0.653674			0.326837	+	0.945081i	$0.394017\pi$
$440$		0			0
$441$	−4.17436			−0.198779
$442$		0			0
$443$			14.0999i			0.669906i	0.942235	+	0.334953i	$0.108720\pi$
$443$			14.0999i			0.669906i	−0.942235	+	0.334953i	$0.891280\pi$
$444$		0			0
$445$	−2.80900	−	22.9493i	−0.133159	−	1.08790i
$446$		0			0
$447$		−	7.15124i		−	0.338242i
$448$		0			0
$449$	−13.0514			−0.615932			−0.307966	−	0.951397i	$-0.599648\pi$
$449$	−13.0514			−0.615932			−0.307966	+	0.951397i	$0.599648\pi$
$450$		0			0
$451$	32.6977			1.53968
$452$		0			0
$453$		−	11.8601i		−	0.557235i
$454$		0			0
$455$	−2.92807	−	23.9221i	−0.137270	−	1.12149i
$456$		0			0
$457$		−	41.6466i		−	1.94815i	−0.226230	−	0.974074i	$-0.572640\pi$
$457$		−	41.6466i		−	1.94815i	0.226230	−	0.974074i	$-0.427360\pi$
$458$		0			0
$459$	1.00000			0.0466760
$460$		0			0
$461$	6.16649			0.287202			0.143601	−	0.989636i	$-0.454132\pi$
$461$	6.16649			0.287202			0.143601	+	0.989636i	$0.454132\pi$
$462$		0			0
$463$		−	6.29219i		−	0.292423i	−0.989253	−	0.146211i	$-0.953292\pi$
$463$		−	6.29219i		−	0.292423i	0.989253	−	0.146211i	$-0.0467080\pi$
$464$		0			0
$465$	13.0252	−	1.59429i	0.604031	−	0.0739336i
$466$		0			0
$467$		−	25.2392i		−	1.16793i	−0.811779	−	0.583964i	$-0.801501\pi$
$467$		−	25.2392i		−	1.16793i	0.811779	−	0.583964i	$-0.198499\pi$
$468$		0			0
$469$	−17.2723			−0.797563
$470$		0			0
$471$	−18.1489			−0.836259
$472$		0			0
$473$			20.9105i			0.961467i
$474$		0			0
$475$	17.0669	−	4.24154i	0.783083	−	0.194615i
$476$		0			0
$477$			8.46283i			0.387486i
$478$		0			0
$479$	30.8501			1.40958			0.704789	−	0.709417i	$-0.251042\pi$
$479$	30.8501			1.40958			0.704789	+	0.709417i	$0.251042\pi$
$480$		0			0
$481$	58.9703			2.68881
$482$		0			0
$483$			1.55140i			0.0705911i
$484$		0			0
$485$	−13.4603	+	1.64755i	−0.611202	+	0.0748114i
$486$		0			0
$487$			18.1983i			0.824643i	0.911038	+	0.412322i	$0.135282\pi$
$487$			18.1983i			0.824643i	−0.911038	+	0.412322i	$0.864718\pi$
$488$		0			0
$489$	−7.54950			−0.341400
$490$		0			0
$491$	−30.5959			−1.38077			−0.690387	−	0.723440i	$-0.742560\pi$
$491$	−30.5959			−1.38077			−0.690387	+	0.723440i	$0.742560\pi$
$492$		0			0
$493$			6.59429i			0.296992i
$494$		0			0
$495$	1.14761	+	9.37585i	0.0515811	+	0.421413i
$496$		0			0
$497$			2.51034i			0.112604i
$498$		0			0
$499$	−29.7663			−1.33252			−0.666262	−	0.745718i	$-0.732107\pi$
$499$	−29.7663			−1.33252			−0.666262	+	0.745718i	$0.732107\pi$
$500$		0			0
$501$	−4.65999			−0.208193
$502$		0			0
$503$			30.4160i			1.35618i	0.734977	+	0.678092i	$0.237193\pi$
$503$			30.4160i			1.35618i	−0.734977	+	0.678092i	$0.762807\pi$
$504$		0			0
$505$	−0.955208	−	7.80396i	−0.0425062	−	0.347272i
$506$		0			0
$507$		−	28.1121i		−	1.24850i
$508$		0			0
$509$	24.7047			1.09502			0.547508	−	0.836801i	$-0.315577\pi$
$509$	24.7047			1.09502			0.547508	+	0.836801i	$0.315577\pi$
$510$		0			0
$511$	−17.5283			−0.775406
$512$		0			0
$513$			3.51721i			0.155289i
$514$		0			0
$515$	1.64104	−	0.200863i	0.0723127	−	0.00885110i
$516$		0			0
$517$			42.8862i			1.88613i
$518$		0			0
$519$	−12.8141			−0.562479
$520$		0			0
$521$	3.25670			0.142679			0.0713393	−	0.997452i	$-0.477273\pi$
$521$	3.25670			0.142679			0.0713393	+	0.997452i	$0.477273\pi$
$522$		0			0
$523$		−	24.4733i		−	1.07014i	−0.844807	−	0.535070i	$-0.820285\pi$
$523$		−	24.4733i		−	1.07014i	0.844807	−	0.535070i	$-0.179715\pi$
$524$		0			0
$525$	−8.15670	+	2.02714i	−0.355988	+	0.0884714i
$526$		0			0
$527$			5.86854i			0.255637i
$528$		0			0
$529$	22.1482			0.962966
$530$		0			0
$531$	4.44860			0.193053
$532$		0			0
$533$			49.6304i			2.14973i
$534$		0			0
$535$	11.2456	−	1.37646i	0.486189	−	0.0595097i
$536$		0			0
$537$			7.18859i			0.310210i
$538$		0			0
$539$	17.6337			0.759539
$540$		0			0
$541$	−18.3010			−0.786822			−0.393411	−	0.919363i	$-0.628705\pi$
$541$	−18.3010			−0.786822			−0.393411	+	0.919363i	$0.628705\pi$
$542$		0			0
$543$			8.58006i			0.368206i
$544$		0			0
$545$	−0.348470	−	2.84697i	−0.0149268	−	0.121951i
$546$		0			0
$547$			20.8400i			0.891054i	0.895268	+	0.445527i	$0.146984\pi$
$547$			20.8400i			0.891054i	−0.895268	+	0.445527i	$0.853016\pi$
$548$		0			0
$549$	−7.86854			−0.335821
$550$		0			0
$551$	−23.1935			−0.988078
$552$		0			0
$553$		−	26.6151i		−	1.13179i
$554$		0			0
$555$	−2.49854	−	20.4129i	−0.106057	−	0.866479i
$556$		0			0
$557$			38.0878i			1.61383i	0.590666	+	0.806916i	$0.298865\pi$
$557$			38.0878i			1.61383i	−0.590666	+	0.806916i	$0.701135\pi$
$558$		0			0
$559$	−31.7391			−1.34242
$560$		0			0
$561$	−4.22430			−0.178350
$562$		0			0
$563$			40.6382i			1.71270i	0.516399	+	0.856348i	$0.327272\pi$
$563$			40.6382i			1.71270i	−0.516399	+	0.856348i	$0.672728\pi$
$564$		0			0
$565$	−23.8733	+	2.92210i	−1.00436	+	0.122934i
$566$		0			0
$567$		−	1.68096i		−	0.0705939i
$568$		0			0
$569$	17.2098			0.721472			0.360736	−	0.932668i	$-0.382525\pi$
$569$	17.2098			0.721472			0.360736	+	0.932668i	$0.382525\pi$
$570$		0			0
$571$	22.7239			0.950964			0.475482	−	0.879725i	$-0.342274\pi$
$571$	22.7239			0.950964			0.475482	+	0.879725i	$0.342274\pi$
$572$		0			0
$573$			18.8427i			0.787165i
$574$		0			0
$575$	−1.11298	−	4.47838i	−0.0464147	−	0.186761i
$576$		0			0
$577$		−	19.5198i		−	0.812618i	−0.913736	−	0.406309i	$-0.866816\pi$
$577$		−	19.5198i		−	0.812618i	0.913736	−	0.406309i	$-0.133184\pi$
$578$		0			0
$579$	9.03217			0.375364
$580$		0			0
$581$	14.4609			0.599940
$582$		0			0
$583$		−	35.7495i		−	1.48059i
$584$		0			0
$585$	−14.2312	+	1.74190i	−0.588387	+	0.0720187i
$586$		0			0
$587$			13.1324i			0.542033i	0.962575	+	0.271016i	$0.0873598\pi$
$587$			13.1324i			0.542033i	−0.962575	+	0.271016i	$0.912640\pi$
$588$		0			0
$589$	−20.6409			−0.850494
$590$		0			0
$591$	−20.9895			−0.863394
$592$		0			0
$593$			36.6132i			1.50352i	0.659435	+	0.751762i	$0.270796\pi$
$593$			36.6132i			1.50352i	−0.659435	+	0.751762i	$0.729204\pi$
$594$		0			0
$595$	−0.456664	−	3.73091i	−0.0187214	−	0.152952i
$596$		0			0
$597$			10.9552i			0.448367i
$598$		0			0
$599$	13.9624			0.570489			0.285245	−	0.958455i	$-0.407925\pi$
$599$	13.9624			0.570489			0.285245	+	0.958455i	$0.407925\pi$
$600$		0			0
$601$	−2.13803			−0.0872121			−0.0436060	−	0.999049i	$-0.513885\pi$
$601$	−2.13803			−0.0872121			−0.0436060	+	0.999049i	$0.513885\pi$
$602$		0			0
$603$			10.2753i			0.418441i
$604$		0			0
$605$	−1.85949	−	15.1919i	−0.0755990	−	0.617637i
$606$		0			0
$607$			26.2085i			1.06377i	0.846817	+	0.531885i	$0.178516\pi$
$607$			26.2085i			1.06377i	−0.846817	+	0.531885i	$0.821484\pi$
$608$		0			0
$609$	11.0848			0.449178
$610$		0			0
$611$	−65.0950			−2.63346
$612$		0			0
$613$			35.8777i			1.44909i	0.689228	+	0.724544i	$0.257950\pi$
$613$			35.8777i			1.44909i	−0.689228	+	0.724544i	$0.742050\pi$
$614$		0			0
$615$	17.1798	−	2.10281i	0.692757	−	0.0847937i
$616$		0			0
$617$		−	25.1444i		−	1.01227i	−0.862453	−	0.506137i	$-0.831073\pi$
$617$		−	25.1444i		−	1.01227i	0.862453	−	0.506137i	$-0.168927\pi$
$618$		0			0
$619$	−22.1050			−0.888474			−0.444237	−	0.895909i	$-0.646525\pi$
$619$	−22.1050			−0.888474			−0.444237	+	0.895909i	$0.646525\pi$
$620$		0			0
$621$	0.922922			0.0370356
$622$		0			0
$623$		−	17.3809i		−	0.696351i
$624$		0			0
$625$	22.0914	−	11.7033i	0.883657	−	0.468134i
$626$		0			0
$627$		−	14.8578i		−	0.593362i
$628$		0			0
$629$	9.19705			0.366710
$630$		0			0
$631$	−34.5052			−1.37363			−0.686815	−	0.726832i	$-0.740992\pi$
$631$	−34.5052			−1.37363			−0.686815	+	0.726832i	$0.740992\pi$
$632$		0			0
$633$		−	25.3912i		−	1.00921i
$634$		0			0
$635$	−1.60346	+	0.196265i	−0.0636316	+	0.00778852i
$636$		0			0
$637$			26.7654i			1.06049i
$638$		0			0
$639$	1.49339			0.0590777
$640$		0			0
$641$	−16.9108			−0.667937			−0.333969	−	0.942584i	$-0.608388\pi$
$641$	−16.9108			−0.667937			−0.333969	+	0.942584i	$0.608388\pi$
$642$		0			0
$643$		−	35.5434i		−	1.40170i	−0.713310	−	0.700848i	$-0.752805\pi$
$643$		−	35.5434i		−	1.40170i	0.713310	−	0.700848i	$-0.247195\pi$
$644$		0			0
$645$	1.34477	+	10.9867i	0.0529503	+	0.432600i
$646$		0			0
$647$		−	2.92222i		−	0.114884i	−0.998349	−	0.0574421i	$-0.981706\pi$
$647$		−	2.92222i		−	0.114884i	0.998349	−	0.0574421i	$-0.0182944\pi$
$648$		0			0
$649$	−18.7922			−0.737659
$650$		0			0
$651$	9.86480			0.386632
$652$		0			0
$653$			4.95881i			0.194053i	0.995282	+	0.0970266i	$0.0309332\pi$
$653$			4.95881i			0.194053i	−0.995282	+	0.0970266i	$0.969067\pi$
$654$		0			0
$655$	2.99232	+	24.4470i	0.116920	+	0.955224i
$656$		0			0
$657$			10.4275i			0.406816i
$658$		0			0
$659$	24.2732			0.945551			0.472775	−	0.881183i	$-0.343252\pi$
$659$	24.2732			0.945551			0.472775	+	0.881183i	$0.343252\pi$
$660$		0			0
$661$	−40.2637			−1.56608			−0.783038	−	0.621974i	$-0.786331\pi$
$661$	−40.2637			−1.56608			−0.783038	+	0.621974i	$0.786331\pi$
$662$		0			0
$663$		−	6.41187i		−	0.249016i
$664$		0			0
$665$	13.1224	−	1.60619i	0.508865	−	0.0622852i
$666$		0			0
$667$			6.08602i			0.235652i
$668$		0			0
$669$	13.6717			0.528577
$670$		0			0
$671$	33.2391			1.28318
$672$		0			0
$673$			31.1010i			1.19886i	0.800428	+	0.599428i	$0.204605\pi$
$673$			31.1010i			1.19886i	−0.800428	+	0.599428i	$0.795395\pi$
$674$		0			0
$675$	1.20594	+	4.85239i	0.0464165	+	0.186769i
$676$		0			0
$677$		−	34.3264i		−	1.31927i	−0.751585	−	0.659636i	$-0.770710\pi$
$677$		−	34.3264i		−	1.31927i	0.751585	−	0.659636i	$-0.229290\pi$
$678$		0			0
$679$	−10.1943			−0.391222
$680$		0			0
$681$	10.7372			0.411450
$682$		0			0
$683$			13.1144i			0.501807i	0.968012	+	0.250903i	$0.0807277\pi$
$683$			13.1144i			0.501807i	−0.968012	+	0.250903i	$0.919272\pi$
$684$		0			0
$685$	21.5799	−	2.64138i	0.824525	−	0.100922i
$686$		0			0
$687$			26.1646i			0.998241i
$688$		0			0
$689$	54.2626			2.06724
$690$		0			0
$691$	−29.3846			−1.11784			−0.558921	−	0.829221i	$-0.688784\pi$
$691$	−29.3846			−1.11784			−0.558921	+	0.829221i	$0.688784\pi$
$692$		0			0
$693$			7.10090i			0.269741i
$694$		0			0
$695$	−0.997701	−	8.15113i	−0.0378449	−	0.309190i
$696$		0			0
$697$			7.74039i			0.293188i
$698$		0			0
$699$	9.24006			0.349491
$700$		0			0
$701$	13.3427			0.503948			0.251974	−	0.967734i	$-0.418920\pi$
$701$	13.3427			0.503948			0.251974	+	0.967734i	$0.418920\pi$
$702$		0			0
$703$			32.3480i			1.22003i
$704$		0			0
$705$	2.75804	+	22.5330i	0.103874	+	0.848641i
$706$		0			0
$707$		−	5.91042i		−	0.222284i
$708$		0			0
$709$	−43.3788			−1.62912			−0.814562	−	0.580076i	$-0.803023\pi$
$709$	−43.3788			−1.62912			−0.814562	+	0.580076i	$0.803023\pi$
$710$		0			0
$711$	−15.8332			−0.593792
$712$		0			0
$713$			5.41620i			0.202838i
$714$		0			0
$715$	60.1168	−	7.35831i	2.24824	−	0.275185i
$716$		0			0
$717$		−	1.91042i		−	0.0713458i
$718$		0			0
$719$	34.8997			1.30154			0.650769	−	0.759276i	$-0.274447\pi$
$719$	34.8997			1.30154			0.650769	+	0.759276i	$0.274447\pi$
$720$		0			0
$721$	1.24286			0.0462864
$722$		0			0
$723$		−	3.63719i		−	0.135268i
$724$		0			0
$725$	−31.9981	+	7.95229i	−1.18838	+	0.295341i
$726$		0			0
$727$		−	18.8258i		−	0.698209i	−0.937084	−	0.349105i	$-0.886486\pi$
$727$		−	18.8258i		−	0.698209i	0.937084	−	0.349105i	$-0.113514\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−4.95006			−0.183084
$732$		0			0
$733$		−	8.20406i		−	0.303024i	−0.988455	−	0.151512i	$-0.951586\pi$
$733$		−	8.20406i		−	0.303024i	0.988455	−	0.151512i	$-0.0484142\pi$
$734$		0			0
$735$	9.26500	−	1.13404i	0.341745	−	0.0418297i
$736$		0			0
$737$		−	43.4058i		−	1.59887i
$738$		0			0
$739$	4.69134			0.172574			0.0862869	−	0.996270i	$-0.472500\pi$
$739$	4.69134			0.172574			0.0862869	+	0.996270i	$0.472500\pi$
$740$		0			0
$741$	22.5519			0.828466
$742$		0			0
$743$		−	27.2313i		−	0.999018i	−0.866309	−	0.499509i	$-0.833514\pi$
$743$		−	27.2313i		−	0.999018i	0.866309	−	0.499509i	$-0.166486\pi$
$744$		0			0
$745$	1.94276	+	15.8722i	0.0711773	+	0.581513i
$746$		0			0
$747$		−	8.60276i		−	0.314758i
$748$		0			0
$749$	8.51697			0.311203
$750$		0			0
$751$	−20.8723			−0.761639			−0.380820	−	0.924649i	$-0.624358\pi$
$751$	−20.8723			−0.761639			−0.380820	+	0.924649i	$0.624358\pi$
$752$		0			0
$753$		−	5.74370i		−	0.209312i
$754$		0			0
$755$	3.22200	+	26.3235i	0.117261	+	0.958009i
$756$		0			0
$757$			0.903828i			0.0328502i	0.999865	+	0.0164251i	$0.00522850\pi$
$757$			0.903828i			0.0328502i	−0.999865	+	0.0164251i	$0.994771\pi$
$758$		0			0
$759$	−3.89870			−0.141514
$760$		0			0
$761$	22.7597			0.825040			0.412520	−	0.910949i	$-0.364649\pi$
$761$	22.7597			0.825040			0.412520	+	0.910949i	$0.364649\pi$
$762$		0			0
$763$		−	2.15619i		−	0.0780592i
$764$		0			0
$765$	−2.21950	+	0.271668i	−0.0802463	+	0.00982217i
$766$		0			0
$767$		−	28.5239i		−	1.02994i
$768$		0			0
$769$	−33.9781			−1.22528			−0.612642	−	0.790361i	$-0.709893\pi$
$769$	−33.9781			−1.22528			−0.612642	+	0.790361i	$0.709893\pi$
$770$		0			0
$771$	4.16649			0.150052
$772$		0			0
$773$			42.6844i			1.53525i	0.640899	+	0.767625i	$0.278562\pi$
$773$			42.6844i			1.53525i	−0.640899	+	0.767625i	$0.721438\pi$
$774$		0			0
$775$	−28.4764	+	7.07708i	−1.02290	+	0.254216i
$776$		0			0
$777$		−	15.4599i		−	0.554621i
$778$		0			0
$779$	−27.2246			−0.975423
$780$		0			0
$781$	−6.30854			−0.225737
$782$		0			0
$783$		−	6.59429i		−	0.235661i
$784$		0			0
$785$	40.2816	−	4.93049i	1.43771	−	0.175977i
$786$		0			0
$787$		−	39.5265i		−	1.40897i	−0.709720	−	0.704484i	$-0.751178\pi$
$787$		−	39.5265i		−	1.40897i	0.709720	−	0.704484i	$-0.248822\pi$
$788$		0			0
$789$	−18.0877			−0.643939
$790$		0			0
$791$	−18.0807			−0.642876
$792$		0			0
$793$			50.4520i			1.79160i
$794$		0			0
$795$	−2.29908	−	18.7833i	−0.0815399	−	0.666174i
$796$		0			0
$797$		−	25.4335i		−	0.900900i	−0.892802	−	0.450450i	$-0.851263\pi$
$797$		−	25.4335i		−	0.900900i	0.892802	−	0.450450i	$-0.148737\pi$
$798$		0			0
$799$	−10.1523			−0.359161
$800$		0			0
$801$	−10.3398			−0.365340
$802$		0			0
$803$		−	44.0490i		−	1.55445i
$804$		0			0
$805$	−0.421465	−	3.44334i	−0.0148547	−	0.121362i
$806$		0			0
$807$			24.1574i			0.850381i
$808$		0			0
$809$	−35.3507			−1.24286			−0.621432	−	0.783468i	$-0.713449\pi$
$809$	−35.3507			−1.24286			−0.621432	+	0.783468i	$0.713449\pi$
$810$		0			0
$811$	16.2979			0.572296			0.286148	−	0.958185i	$-0.407625\pi$
$811$	16.2979			0.572296			0.286148	+	0.958185i	$0.407625\pi$
$812$		0			0
$813$		−	24.1181i		−	0.845858i
$814$		0			0
$815$	16.7561	−	2.05096i	0.586942	−	0.0718419i
$816$		0			0
$817$		−	17.4104i		−	0.609113i
$818$		0			0
$819$	−10.7781			−0.376618
$820$		0			0
$821$	−46.2052			−1.61257			−0.806286	−	0.591526i	$-0.798526\pi$
$821$	−46.2052			−1.61257			−0.806286	+	0.591526i	$0.798526\pi$
$822$		0			0
$823$			27.2657i			0.950422i	0.879872	+	0.475211i	$0.157628\pi$
$823$			27.2657i			0.950422i	−0.879872	+	0.475211i	$0.842372\pi$
$824$		0			0
$825$	−5.09424	−	20.4980i	−0.177359	−	0.713648i
$826$		0			0
$827$		−	3.00191i		−	0.104387i	−0.998637	−	0.0521934i	$-0.983379\pi$
$827$		−	3.00191i		−	0.104387i	0.998637	−	0.0521934i	$-0.0166212\pi$
$828$		0			0
$829$	−31.7915			−1.10416			−0.552082	−	0.833790i	$-0.686166\pi$
$829$	−31.7915			−1.10416			−0.552082	+	0.833790i	$0.686166\pi$
$830$		0			0
$831$	28.6047			0.992287
$832$		0			0
$833$			4.17436i			0.144633i
$834$		0			0
$835$	10.3429	−	1.26597i	0.357930	−	0.0438107i
$836$		0			0
$837$		−	5.86854i		−	0.202846i
$838$		0			0
$839$	20.9043			0.721695			0.360847	−	0.932625i	$-0.382488\pi$
$839$	20.9043			0.721695			0.360847	+	0.932625i	$0.382488\pi$
$840$		0			0
$841$	14.4847			0.499472
$842$		0			0
$843$		−	1.44263i		−	0.0496867i
$844$		0			0
$845$	7.63716	+	62.3949i	0.262726	+	2.14645i
$846$		0			0
$847$		−	11.5057i		−	0.395341i
$848$		0			0
$849$	−12.9467			−0.444331
$850$		0			0
$851$	8.48816			0.290970
$852$		0			0
$853$		−	12.8134i		−	0.438722i	−0.975644	−	0.219361i	$-0.929603\pi$
$853$		−	12.8134i		−	0.438722i	0.975644	−	0.219361i	$-0.0703972\pi$
$854$		0			0
$855$	−0.955514	−	7.80647i	−0.0326779	−	0.266976i
$856$		0			0
$857$			14.2761i			0.487661i	0.969818	+	0.243830i	$0.0784040\pi$
$857$			14.2761i			0.487661i	−0.969818	+	0.243830i	$0.921596\pi$
$858$		0			0
$859$	41.7915			1.42591			0.712953	−	0.701211i	$-0.247357\pi$
$859$	41.7915			1.42591			0.712953	+	0.701211i	$0.247357\pi$
$860$		0			0
$861$	13.0113			0.443424
$862$		0			0
$863$		−	44.5448i		−	1.51632i	−0.652067	−	0.758162i	$-0.726098\pi$
$863$		−	44.5448i		−	1.51632i	0.652067	−	0.758162i	$-0.273902\pi$
$864$		0			0
$865$	28.4411	−	3.48119i	0.967025	−	0.118364i
$866$		0			0
$867$		−	1.00000i		−	0.0339618i
$868$		0			0
$869$	66.8843			2.26889
$870$		0			0
$871$	65.8836			2.23238
$872$		0			0
$873$			6.06457i			0.205255i
$874$		0			0
$875$	17.5531	−	6.71515i	0.593404	−	0.227013i
$876$		0			0
$877$		−	27.1382i		−	0.916391i	−0.888851	−	0.458196i	$-0.848496\pi$
$877$		−	27.1382i		−	0.916391i	0.888851	−	0.458196i	$-0.151504\pi$
$878$		0			0
$879$	−15.1390			−0.510627
$880$		0			0
$881$	24.0691			0.810908			0.405454	−	0.914116i	$-0.367114\pi$
$881$	24.0691			0.810908			0.405454	+	0.914116i	$0.367114\pi$
$882$		0			0
$883$			4.27005i			0.143698i	0.997416	+	0.0718492i	$0.0228900\pi$
$883$			4.27005i			0.143698i	−0.997416	+	0.0718492i	$0.977110\pi$
$884$		0			0
$885$	−9.87369	+	1.20854i	−0.331900	+	0.0406247i
$886$		0			0
$887$			20.0426i			0.672966i	0.941689	+	0.336483i	$0.109238\pi$
$887$			20.0426i			0.672966i	−0.941689	+	0.336483i	$0.890762\pi$
$888$		0			0
$889$	−1.21440			−0.0407297
$890$		0			0
$891$	4.22430			0.141519
$892$		0			0
$893$		−	35.7077i		−	1.19491i
$894$		0			0
$895$	−1.95291	−	15.9551i	−0.0652785	−	0.533320i
$896$		0			0
$897$		−	5.91766i		−	0.197585i
$898$		0			0
$899$	38.6988			1.29068
$900$		0			0
$901$	8.46283			0.281938
$902$		0			0
$903$			8.32087i			0.276901i
$904$		0			0
$905$	−2.33093	−	19.0435i	−0.0774827	−	0.633027i
$906$		0			0
$907$			35.6421i			1.18348i	0.806130	+	0.591739i	$0.201558\pi$
$907$			35.6421i			1.18348i	−0.806130	+	0.591739i	$0.798442\pi$
$908$		0			0
$909$	−3.51609			−0.116621
$910$		0			0
$911$	−50.1910			−1.66290			−0.831450	−	0.555599i	$-0.812489\pi$
$911$	−50.1910			−1.66290			−0.831450	+	0.555599i	$0.812489\pi$
$912$		0			0
$913$			36.3406i			1.20270i
$914$		0			0
$915$	17.4642	−	2.13763i	0.577350	−	0.0706678i
$916$		0			0
$917$			18.5152i			0.611426i
$918$		0			0
$919$	31.4216			1.03650			0.518251	−	0.855229i	$-0.326583\pi$
$919$	31.4216			1.03650			0.518251	+	0.855229i	$0.326583\pi$
$920$		0			0
$921$	−12.8450			−0.423256
$922$		0			0
$923$		−	9.57545i		−	0.315180i
$924$		0			0
$925$	11.0911	+	44.6277i	0.364672	+	1.46735i
$926$		0			0
$927$		−	0.739371i		−	0.0242841i
$928$		0			0
$929$	−9.18527			−0.301359			−0.150679	−	0.988583i	$-0.548146\pi$
$929$	−9.18527			−0.301359			−0.150679	+	0.988583i	$0.548146\pi$
$930$		0			0
$931$	−14.6821			−0.481187
$932$		0			0
$933$		−	22.9341i		−	0.750829i
$934$		0			0
$935$	9.37585	−	1.14761i	0.306623	−	0.0375308i
$936$		0			0
$937$			9.82689i			0.321030i	0.987033	+	0.160515i	$0.0513156\pi$
$937$			9.82689i			0.321030i	−0.987033	+	0.160515i	$0.948684\pi$
$938$		0			0
$939$	12.2734			0.400526
$940$		0			0
$941$	−13.9208			−0.453804			−0.226902	−	0.973918i	$-0.572860\pi$
$941$	−13.9208			−0.453804			−0.226902	+	0.973918i	$0.572860\pi$
$942$		0			0
$943$			7.14377i			0.232633i
$944$		0			0
$945$	0.456664	+	3.73091i	0.0148553	+	0.121366i
$946$		0			0
$947$		−	44.0416i		−	1.43116i	−0.698531	−	0.715580i	$-0.746163\pi$
$947$		−	44.0416i		−	1.43116i	0.698531	−	0.715580i	$-0.253837\pi$
$948$		0			0
$949$	66.8599			2.17036
$950$		0			0
$951$	27.6791			0.897555
$952$		0			0
$953$			12.8239i			0.415406i	0.978192	+	0.207703i	$0.0665988\pi$
$953$			12.8239i			0.415406i	−0.978192	+	0.207703i	$0.933401\pi$
$954$		0			0
$955$	−5.11896	−	41.8214i	−0.165646	−	1.35331i
$956$		0			0
$957$			27.8563i			0.900465i
$958$		0			0
$959$	16.3438			0.527767
$960$		0			0
$961$	3.43971			0.110959
$962$		0			0
$963$		−	5.06672i		−	0.163273i
$964$		0			0
$965$	−20.0469	+	2.45375i	−0.645334	+	0.0789890i
$966$		0			0
$967$			49.1816i			1.58157i	0.612092	+	0.790787i	$0.290328\pi$
$967$			49.1816i			1.58157i	−0.612092	+	0.790787i	$0.709672\pi$
$968$		0			0
$969$	3.51721			0.112989
$970$		0			0
$971$	−1.82892			−0.0586927			−0.0293463	−	0.999569i	$-0.509343\pi$
$971$	−1.82892			−0.0586927			−0.0293463	+	0.999569i	$0.509343\pi$
$972$		0			0
$973$		−	6.17334i		−	0.197908i
$974$		0			0
$975$	31.1129	−	7.73231i	0.996411	−	0.247632i
$976$		0			0
$977$			28.2399i			0.903476i	0.892151	+	0.451738i	$0.149196\pi$
$977$			28.2399i			0.903476i	−0.892151	+	0.451738i	$0.850804\pi$
$978$		0			0
$979$	43.6785			1.39597
$980$		0			0
$981$	−1.28271			−0.0409537
$982$		0			0
$983$		−	43.4502i		−	1.38585i	−0.721011	−	0.692923i	$-0.756322\pi$
$983$		−	43.4502i		−	1.38585i	0.721011	−	0.692923i	$-0.243678\pi$
$984$		0			0
$985$	46.5863	−	5.70218i	1.48436	−	0.181687i
$986$		0			0
$987$			17.0656i			0.543204i
$988$		0			0
$989$	−4.56852			−0.145270
$990$		0			0
$991$	46.3587			1.47263			0.736317	−	0.676637i	$-0.236563\pi$
$991$	46.3587			1.47263			0.736317	+	0.676637i	$0.236563\pi$
$992$		0			0
$993$			17.0083i			0.539742i
$994$		0			0
$995$	−2.97618	−	24.3151i	−0.0943512	−	0.770841i
$996$		0			0
$997$		−	6.20099i		−	0.196387i	−0.995167	−	0.0981937i	$-0.968694\pi$
$997$		−	6.20099i		−	0.196387i	0.995167	−	0.0981937i	$-0.0313065\pi$
$998$		0			0
$999$	−9.19705			−0.290982

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2040.2.m.h.409.8	yes	10
4.3	odd	2		4080.2.m.q.2449.3		10
5.4	even	2	inner	2040.2.m.h.409.3	✓	10
20.19	odd	2		4080.2.m.q.2449.8		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2040.2.m.h.409.3	✓	10	5.4	even	2	inner
2040.2.m.h.409.8	yes	10	1.1	even	1	trivial
4080.2.m.q.2449.3		10	4.3	odd	2
4080.2.m.q.2449.8		10	20.19	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2040 = 2^{3} \cdot 3 \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2040.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +(-0.271668 - 2.21950i) q^{5} -1.68096i q^{7} -1.00000 q^{9} +4.22430 q^{11} +6.41187i q^{13} +(2.21950 - 0.271668i) q^{15} +1.00000i q^{17} -3.51721 q^{19} +1.68096 q^{21} +0.922922i q^{23} +(-4.85239 + 1.20594i) q^{25} -1.00000i q^{27} +6.59429 q^{29} +5.86854 q^{31} +4.22430i q^{33} +(-3.73091 + 0.456664i) q^{35} -9.19705i q^{37} -6.41187 q^{39} +7.74039 q^{41} +4.95006i q^{43} +(0.271668 + 2.21950i) q^{45} +10.1523i q^{47} +4.17436 q^{49} -1.00000 q^{51} -8.46283i q^{53} +(-1.14761 - 9.37585i) q^{55} -3.51721i q^{57} -4.44860 q^{59} +7.86854 q^{61} +1.68096i q^{63} +(14.2312 - 1.74190i) q^{65} -10.2753i q^{67} -0.922922 q^{69} -1.49339 q^{71} -10.4275i q^{73} +(-1.20594 - 4.85239i) q^{75} -7.10090i q^{77} +15.8332 q^{79} +1.00000 q^{81} +8.60276i q^{83} +(2.21950 - 0.271668i) q^{85} +6.59429i q^{87} +10.3398 q^{89} +10.7781 q^{91} +5.86854i q^{93} +(0.955514 + 7.80647i) q^{95} -6.06457i q^{97} -4.22430 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 2 q^{5} - 10 q^{9} + 10 q^{11} - 2 q^{15} + 6 q^{19} - 14 q^{21} - 20 q^{25} + 38 q^{29} + 12 q^{31} - 20 q^{35} - 16 q^{39} + 10 q^{41} + 2 q^{45} - 4 q^{49} - 10 q^{51} - 40 q^{55} + 20 q^{59}+ \cdots - 10 q^{99}+O(q^{100}) \) 10 * q - 2 * q^5 - 10 * q^9 + 10 * q^11 - 2 * q^15 + 6 * q^19 - 14 * q^21 - 20 * q^25 + 38 * q^29 + 12 * q^31 - 20 * q^35 - 16 * q^39 + 10 * q^41 + 2 * q^45 - 4 * q^49 - 10 * q^51 - 40 * q^55 + 20 * q^59 + 32 * q^61 - 24 * q^65 + 4 * q^69 + 12 * q^75 + 12 * q^79 + 10 * q^81 - 2 * q^85 - 28 * q^89 - 16 * q^91 - 22 * q^95 - 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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