LMFDB - Embedded newform 3360.2.d.a.2911.12

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3360,2,Mod(2911,3360)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3360.2911"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3360, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3360.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$26.8297350792$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 24x^{14} + 238x^{12} + 1262x^{10} + 3861x^{8} + 6834x^{6} + 6589x^{4} + 2916x^{2} + 324 $ x^16 + 24x^14 + 238x^12 + 1262x^10 + 3861x^8 + 6834x^6 + 6589x^4 + 2916*x^2 + 324
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2911.12
Root			$0.400519i$ of defining polynomial
Character	$\chi$	$=$	3360.2911
Dual form			3360.2.d.a.2911.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000i q^{5} +(-0.400519 - 2.61526i) q^{7} +1.00000 q^{9} -3.87335i q^{11} -2.15545i q^{13} -1.00000i q^{15} +4.79356i q^{17} +5.31954 q^{19} +(0.400519 + 2.61526i) q^{21} -5.67917i q^{23} -1.00000 q^{25} -1.00000 q^{27} +1.79858 q^{29} -0.221571 q^{31} +3.87335i q^{33} +(2.61526 - 0.400519i) q^{35} -7.50006 q^{37} +2.15545i q^{39} -11.1072i q^{41} +11.2593i q^{43} +1.00000i q^{45} +7.11469 q^{47} +(-6.67917 + 2.09492i) q^{49} -4.79356i q^{51} -1.97985 q^{53} +3.87335 q^{55} -5.31954 q^{57} -5.27574 q^{59} +7.99669i q^{61} +(-0.400519 - 2.61526i) q^{63} +2.15545 q^{65} -3.00246i q^{67} +5.67917i q^{69} -6.50426i q^{71} +3.16961i q^{73} +1.00000 q^{75} +(-10.1298 + 1.55135i) q^{77} -7.99118i q^{79} +1.00000 q^{81} -10.4453 q^{83} -4.79356 q^{85} -1.79858 q^{87} -11.4560i q^{89} +(-5.63706 + 0.863298i) q^{91} +0.221571 q^{93} +5.31954i q^{95} -0.716023i q^{97} -3.87335i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{3} + 16 q^{9} + 16 q^{19} - 16 q^{25} - 16 q^{27} + 8 q^{29} - 8 q^{31} - 8 q^{37} + 16 q^{47} - 16 q^{49} - 48 q^{53} + 8 q^{55} - 16 q^{57} + 16 q^{59} - 8 q^{65} + 16 q^{75} + 16 q^{77}+ \cdots + 8 q^{93}+O(q^{100}) $ 16 * q - 16 * q^3 + 16 * q^9 + 16 * q^19 - 16 * q^25 - 16 * q^27 + 8 * q^29 - 8 * q^31 - 8 * q^37 + 16 * q^47 - 16 * q^49 - 48 * q^53 + 8 * q^55 - 16 * q^57 + 16 * q^59 - 8 * q^65 + 16 * q^75 + 16 * q^77 + 16 * q^81 - 24 * q^83 + 8 * q^85 - 8 * q^87 + 24 * q^91 + 8 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3360\mathbb{Z}\right)^\times$.

$n$	$421$	$1121$	$1471$	$1921$	$2017$
$\chi(n)$	$1$	$1$	$-1$	$-1$	$1$

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
$3$	−1.00000			−0.577350
$3$	−1.00000			−0.577350
$4$		0			0
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$	−0.400519	−	2.61526i	−0.151382	−	0.988475i
$7$	−0.400519	−	2.61526i	−0.151382	−	0.988475i
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$		−	3.87335i		−	1.16786i	−0.811804	−	0.583930i	$-0.801514\pi$
$11$		−	3.87335i		−	1.16786i	0.811804	−	0.583930i	$-0.198486\pi$
$12$		0			0
$13$		−	2.15545i		−	0.597814i	−0.954282	−	0.298907i	$-0.903378\pi$
$13$		−	2.15545i		−	0.597814i	0.954282	−	0.298907i	$-0.0966220\pi$
$14$		0			0
$15$		−	1.00000i		−	0.258199i
$16$		0			0
$17$			4.79356i			1.16261i	0.813686	+	0.581304i	$0.197457\pi$
$17$			4.79356i			1.16261i	−0.813686	+	0.581304i	$0.802543\pi$
$18$		0			0
$19$	5.31954			1.22039			0.610193	−	0.792253i	$-0.291092\pi$
$19$	5.31954			1.22039			0.610193	+	0.792253i	$0.291092\pi$
$20$		0			0
$21$	0.400519	+	2.61526i	0.0874004	+	0.570697i
$22$		0			0
$23$		−	5.67917i		−	1.18419i	−0.805869	−	0.592094i	$-0.798301\pi$
$23$		−	5.67917i		−	1.18419i	0.805869	−	0.592094i	$-0.201699\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$	1.79858			0.333987			0.166994	−	0.985958i	$-0.446594\pi$
$29$	1.79858			0.333987			0.166994	+	0.985958i	$0.446594\pi$
$30$		0			0
$31$	−0.221571			−0.0397954			−0.0198977	−	0.999802i	$-0.506334\pi$
$31$	−0.221571			−0.0397954			−0.0198977	+	0.999802i	$0.506334\pi$
$32$		0			0
$33$			3.87335i			0.674264i
$34$		0			0
$35$	2.61526	−	0.400519i	0.442060	−	0.0677001i
$36$		0			0
$37$	−7.50006			−1.23300			−0.616501	−	0.787354i	$-0.711450\pi$
$37$	−7.50006			−1.23300			−0.616501	+	0.787354i	$0.711450\pi$
$38$		0			0
$39$			2.15545i			0.345148i
$40$		0			0
$41$		−	11.1072i		−	1.73465i	−0.497738	−	0.867327i	$-0.665836\pi$
$41$		−	11.1072i		−	1.73465i	0.497738	−	0.867327i	$-0.334164\pi$
$42$		0			0
$43$			11.2593i			1.71703i	0.512788	+	0.858515i	$0.328613\pi$
$43$			11.2593i			1.71703i	−0.512788	+	0.858515i	$0.671387\pi$
$44$		0			0
$45$			1.00000i			0.149071i
$46$		0			0
$47$	7.11469			1.03778			0.518892	−	0.854840i	$-0.326345\pi$
$47$	7.11469			1.03778			0.518892	+	0.854840i	$0.326345\pi$
$48$		0			0
$49$	−6.67917	+	2.09492i	−0.954167	+	0.299275i
$50$		0			0
$51$		−	4.79356i		−	0.671232i
$52$		0			0
$53$	−1.97985			−0.271954			−0.135977	−	0.990712i	$-0.543417\pi$
$53$	−1.97985			−0.271954			−0.135977	+	0.990712i	$0.543417\pi$
$54$		0			0
$55$	3.87335			0.522283
$56$		0			0
$57$	−5.31954			−0.704590
$58$		0			0
$59$	−5.27574			−0.686843			−0.343421	−	0.939181i	$-0.611586\pi$
$59$	−5.27574			−0.686843			−0.343421	+	0.939181i	$0.611586\pi$
$60$		0			0
$61$			7.99669i			1.02387i	0.859024	+	0.511935i	$0.171071\pi$
$61$			7.99669i			1.02387i	−0.859024	+	0.511935i	$0.828929\pi$
$62$		0			0
$63$	−0.400519	−	2.61526i	−0.0504607	−	0.329492i
$64$		0			0
$65$	2.15545			0.267350
$66$		0			0
$67$		−	3.00246i		−	0.366809i	−0.983038	−	0.183405i	$-0.941288\pi$
$67$		−	3.00246i		−	0.366809i	0.983038	−	0.183405i	$-0.0587118\pi$
$68$		0			0
$69$			5.67917i			0.683692i
$70$		0			0
$71$		−	6.50426i		−	0.771913i	−0.922517	−	0.385957i	$-0.873871\pi$
$71$		−	6.50426i		−	0.771913i	0.922517	−	0.385957i	$-0.126129\pi$
$72$		0			0
$73$			3.16961i			0.370974i	0.982647	+	0.185487i	$0.0593863\pi$
$73$			3.16961i			0.370974i	−0.982647	+	0.185487i	$0.940614\pi$
$74$		0			0
$75$	1.00000			0.115470
$76$		0			0
$77$	−10.1298	+	1.55135i	−1.15440	+	0.176793i
$78$		0			0
$79$		−	7.99118i		−	0.899078i	−0.893261	−	0.449539i	$-0.851588\pi$
$79$		−	7.99118i		−	0.899078i	0.893261	−	0.449539i	$-0.148412\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	−10.4453			−1.14652			−0.573262	−	0.819372i	$-0.694322\pi$
$83$	−10.4453			−1.14652			−0.573262	+	0.819372i	$0.694322\pi$
$84$		0			0
$85$	−4.79356			−0.519934
$86$		0			0
$87$	−1.79858			−0.192828
$88$		0			0
$89$		−	11.4560i		−	1.21434i	−0.794574	−	0.607168i	$-0.792306\pi$
$89$		−	11.4560i		−	1.21434i	0.794574	−	0.607168i	$-0.207694\pi$
$90$		0			0
$91$	−5.63706	+	0.863298i	−0.590924	+	0.0904982i
$92$		0			0
$93$	0.221571			0.0229759
$94$		0			0
$95$			5.31954i			0.545773i
$96$		0			0
$97$		−	0.716023i		−	0.0727012i	−0.999339	−	0.0363506i	$-0.988427\pi$
$97$		−	0.716023i		−	0.0727012i	0.999339	−	0.0363506i	$-0.0115733\pi$
$98$		0			0
$99$		−	3.87335i		−	0.389286i
$100$		0			0
$101$		−	5.18641i		−	0.516067i	−0.966136	−	0.258033i	$-0.916926\pi$
$101$		−	5.18641i		−	0.516067i	0.966136	−	0.258033i	$-0.0830744\pi$
$102$		0			0
$103$	−3.19896			−0.315203			−0.157602	−	0.987503i	$-0.550376\pi$
$103$	−3.19896			−0.315203			−0.157602	+	0.987503i	$0.550376\pi$
$104$		0			0
$105$	−2.61526	+	0.400519i	−0.255223	+	0.0390867i
$106$		0			0
$107$			8.49311i			0.821060i	0.911847	+	0.410530i	$0.134656\pi$
$107$			8.49311i			0.821060i	−0.911847	+	0.410530i	$0.865344\pi$
$108$		0			0
$109$	−8.39403			−0.804002			−0.402001	−	0.915639i	$-0.631685\pi$
$109$	−8.39403			−0.804002			−0.402001	+	0.915639i	$0.631685\pi$
$110$		0			0
$111$	7.50006			0.711874
$112$		0			0
$113$	−10.0556			−0.945952			−0.472976	−	0.881075i	$-0.656820\pi$
$113$	−10.0556			−0.945952			−0.472976	+	0.881075i	$0.656820\pi$
$114$		0			0
$115$	5.67917			0.529585
$116$		0			0
$117$		−	2.15545i		−	0.199271i
$118$		0			0
$119$	12.5364	−	1.91991i	1.14921	−	0.175998i
$120$		0			0
$121$	−4.00285			−0.363895
$122$		0			0
$123$			11.1072i			1.00150i
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$		−	19.4226i		−	1.72347i	−0.507356	−	0.861737i	$-0.669377\pi$
$127$		−	19.4226i		−	1.72347i	0.507356	−	0.861737i	$-0.330623\pi$
$128$		0			0
$129$		−	11.2593i		−	0.991328i
$130$		0			0
$131$	−4.52170			−0.395063			−0.197531	−	0.980297i	$-0.563292\pi$
$131$	−4.52170			−0.395063			−0.197531	+	0.980297i	$0.563292\pi$
$132$		0			0
$133$	−2.13058	−	13.9120i	−0.184744	−	1.20632i
$134$		0			0
$135$		−	1.00000i		−	0.0860663i
$136$		0			0
$137$	12.7163			1.08643			0.543215	−	0.839594i	$-0.317207\pi$
$137$	12.7163			1.08643			0.543215	+	0.839594i	$0.317207\pi$
$138$		0			0
$139$	2.52657			0.214301			0.107150	−	0.994243i	$-0.465827\pi$
$139$	2.52657			0.214301			0.107150	+	0.994243i	$0.465827\pi$
$140$		0			0
$141$	−7.11469			−0.599165
$142$		0			0
$143$	−8.34881			−0.698162
$144$		0			0
$145$			1.79858i			0.149364i
$146$		0			0
$147$	6.67917	−	2.09492i	0.550889	−	0.172786i
$148$		0			0
$149$	−3.27960			−0.268675			−0.134338	−	0.990936i	$-0.542891\pi$
$149$	−3.27960			−0.268675			−0.134338	+	0.990936i	$0.542891\pi$
$150$		0			0
$151$		−	4.74673i		−	0.386284i	−0.981171	−	0.193142i	$-0.938132\pi$
$151$		−	4.74673i		−	0.386284i	0.981171	−	0.193142i	$-0.0618677\pi$
$152$		0			0
$153$			4.79356i			0.387536i
$154$		0			0
$155$		−	0.221571i		−	0.0177970i
$156$		0			0
$157$			15.5590i			1.24174i	0.783912	+	0.620872i	$0.213221\pi$
$157$			15.5590i			1.24174i	−0.783912	+	0.620872i	$0.786779\pi$
$158$		0			0
$159$	1.97985			0.157012
$160$		0			0
$161$	−14.8525	+	2.27462i	−1.17054	+	0.179265i
$162$		0			0
$163$		−	21.4106i		−	1.67701i	−0.544894	−	0.838505i	$-0.683430\pi$
$163$		−	21.4106i		−	1.67701i	0.544894	−	0.838505i	$-0.316570\pi$
$164$		0			0
$165$	−3.87335			−0.301540
$166$		0			0
$167$	6.86529			0.531252			0.265626	−	0.964076i	$-0.414421\pi$
$167$	6.86529			0.531252			0.265626	+	0.964076i	$0.414421\pi$
$168$		0			0
$169$	8.35404			0.642619
$170$		0			0
$171$	5.31954			0.406795
$172$		0			0
$173$		−	13.9225i		−	1.05851i	−0.848463	−	0.529255i	$-0.822471\pi$
$173$		−	13.9225i		−	1.05851i	0.848463	−	0.529255i	$-0.177529\pi$
$174$		0			0
$175$	0.400519	+	2.61526i	0.0302764	+	0.197695i
$176$		0			0
$177$	5.27574			0.396549
$178$		0			0
$179$		−	6.99942i		−	0.523162i	−0.965182	−	0.261581i	$-0.915756\pi$
$179$		−	6.99942i		−	0.523162i	0.965182	−	0.261581i	$-0.0842438\pi$
$180$		0			0
$181$			1.97879i			0.147082i	0.997292	+	0.0735412i	$0.0234300\pi$
$181$			1.97879i			0.147082i	−0.997292	+	0.0735412i	$0.976570\pi$
$182$		0			0
$183$		−	7.99669i		−	0.591132i
$184$		0			0
$185$		−	7.50006i		−	0.551415i
$186$		0			0
$187$	18.5671			1.35776
$188$		0			0
$189$	0.400519	+	2.61526i	0.0291335	+	0.190232i
$190$		0			0
$191$		−	20.5288i		−	1.48541i	−0.669616	−	0.742707i	$-0.733541\pi$
$191$		−	20.5288i		−	1.48541i	0.669616	−	0.742707i	$-0.266459\pi$
$192$		0			0
$193$	−13.1050			−0.943321			−0.471661	−	0.881780i	$-0.656345\pi$
$193$	−13.1050			−0.943321			−0.471661	+	0.881780i	$0.656345\pi$
$194$		0			0
$195$	−2.15545			−0.154355
$196$		0			0
$197$	6.78520			0.483426			0.241713	−	0.970348i	$-0.422291\pi$
$197$	6.78520			0.483426			0.241713	+	0.970348i	$0.422291\pi$
$198$		0			0
$199$	6.62783			0.469835			0.234917	−	0.972015i	$-0.424518\pi$
$199$	6.62783			0.469835			0.234917	+	0.972015i	$0.424518\pi$
$200$		0			0
$201$			3.00246i			0.211777i
$202$		0			0
$203$	−0.720364	−	4.70375i	−0.0505597	−	0.330138i
$204$		0			0
$205$	11.1072			0.775761
$206$		0			0
$207$		−	5.67917i		−	0.394730i
$208$		0			0
$209$		−	20.6044i		−	1.42524i
$210$		0			0
$211$			10.7000i			0.736618i	0.929703	+	0.368309i	$0.120063\pi$
$211$			10.7000i			0.736618i	−0.929703	+	0.368309i	$0.879937\pi$
$212$		0			0
$213$			6.50426i			0.445664i
$214$		0			0
$215$	−11.2593			−0.767879
$216$		0			0
$217$	0.0887435	+	0.579467i	0.00602430	+	0.0393368i
$218$		0			0
$219$		−	3.16961i		−	0.214182i
$220$		0			0
$221$	10.3323			0.695023
$222$		0			0
$223$	−26.0191			−1.74236			−0.871182	−	0.490960i	$-0.836646\pi$
$223$	−26.0191			−1.74236			−0.871182	+	0.490960i	$0.836646\pi$
$224$		0			0
$225$	−1.00000			−0.0666667
$226$		0			0
$227$	−13.0050			−0.863173			−0.431586	−	0.902072i	$-0.642046\pi$
$227$	−13.0050			−0.863173			−0.431586	+	0.902072i	$0.642046\pi$
$228$		0			0
$229$		−	21.0268i		−	1.38949i	−0.719255	−	0.694746i	$-0.755517\pi$
$229$		−	21.0268i		−	1.38949i	0.719255	−	0.694746i	$-0.244483\pi$
$230$		0			0
$231$	10.1298	−	1.55135i	0.666493	−	0.102071i
$232$		0			0
$233$	−22.2270			−1.45614			−0.728070	−	0.685503i	$-0.759583\pi$
$233$	−22.2270			−1.45614			−0.728070	+	0.685503i	$0.759583\pi$
$234$		0			0
$235$			7.11469i			0.464111i
$236$		0			0
$237$			7.99118i			0.519083i
$238$		0			0
$239$		−	13.6516i		−	0.883048i	−0.897249	−	0.441524i	$-0.854438\pi$
$239$		−	13.6516i		−	0.883048i	0.897249	−	0.441524i	$-0.145562\pi$
$240$		0			0
$241$			18.7440i			1.20741i	0.797209	+	0.603704i	$0.206309\pi$
$241$			18.7440i			1.20741i	−0.797209	+	0.603704i	$0.793691\pi$
$242$		0			0
$243$	−1.00000			−0.0641500
$244$		0			0
$245$	−2.09492	−	6.67917i	−0.133840	−	0.426716i
$246$		0			0
$247$		−	11.4660i		−	0.729564i
$248$		0			0
$249$	10.4453			0.661946
$250$		0			0
$251$	−13.1144			−0.827773			−0.413887	−	0.910328i	$-0.635829\pi$
$251$	−13.1144			−0.827773			−0.413887	+	0.910328i	$0.635829\pi$
$252$		0			0
$253$	−21.9974			−1.38297
$254$		0			0
$255$	4.79356			0.300184
$256$		0			0
$257$		−	10.4950i		−	0.654663i	−0.944910	−	0.327331i	$-0.893851\pi$
$257$		−	10.4950i		−	0.654663i	0.944910	−	0.327331i	$-0.106149\pi$
$258$		0			0
$259$	3.00392	+	19.6146i	0.186654	+	1.21879i
$260$		0			0
$261$	1.79858			0.111329
$262$		0			0
$263$		−	27.9160i		−	1.72138i	−0.509134	−	0.860688i	$-0.670034\pi$
$263$		−	27.9160i		−	1.72138i	0.509134	−	0.860688i	$-0.329966\pi$
$264$		0			0
$265$		−	1.97985i		−	0.121621i
$266$		0			0
$267$			11.4560i			0.701097i
$268$		0			0
$269$			1.37287i			0.0837051i	0.999124	+	0.0418525i	$0.0133260\pi$
$269$			1.37287i			0.0837051i	−0.999124	+	0.0418525i	$0.986674\pi$
$270$		0			0
$271$	−2.00052			−0.121523			−0.0607616	−	0.998152i	$-0.519353\pi$
$271$	−2.00052			−0.121523			−0.0607616	+	0.998152i	$0.519353\pi$
$272$		0			0
$273$	5.63706	−	0.863298i	0.341170	−	0.0522492i
$274$		0			0
$275$			3.87335i			0.233572i
$276$		0			0
$277$	−3.87788			−0.232999			−0.116500	−	0.993191i	$-0.537167\pi$
$277$	−3.87788			−0.232999			−0.116500	+	0.993191i	$0.537167\pi$
$278$		0			0
$279$	−0.221571			−0.0132651
$280$		0			0
$281$	−15.4583			−0.922167			−0.461083	−	0.887357i	$-0.652539\pi$
$281$	−15.4583			−0.922167			−0.461083	+	0.887357i	$0.652539\pi$
$282$		0			0
$283$	22.1130			1.31448			0.657240	−	0.753681i	$-0.271724\pi$
$283$	22.1130			1.31448			0.657240	+	0.753681i	$0.271724\pi$
$284$		0			0
$285$		−	5.31954i		−	0.315102i
$286$		0			0
$287$	−29.0482	+	4.44865i	−1.71466	+	0.262595i
$288$		0			0
$289$	−5.97818			−0.351658
$290$		0			0
$291$			0.716023i			0.0419740i
$292$		0			0
$293$			24.2993i			1.41958i	0.704412	+	0.709792i	$0.251211\pi$
$293$			24.2993i			1.41958i	−0.704412	+	0.709792i	$0.748789\pi$
$294$		0			0
$295$		−	5.27574i		−	0.307165i
$296$		0			0
$297$			3.87335i			0.224755i
$298$		0			0
$299$	−12.2412			−0.707924
$300$		0			0
$301$	29.4460	−	4.50957i	1.69724	−	0.259927i
$302$		0			0
$303$			5.18641i			0.297951i
$304$		0			0
$305$	−7.99669			−0.457889
$306$		0			0
$307$	−29.4220			−1.67920			−0.839602	−	0.543202i	$-0.817212\pi$
$307$	−29.4220			−1.67920			−0.839602	+	0.543202i	$0.817212\pi$
$308$		0			0
$309$	3.19896			0.181983
$310$		0			0
$311$	−26.2914			−1.49085			−0.745424	−	0.666590i	$-0.767753\pi$
$311$	−26.2914			−1.49085			−0.745424	+	0.666590i	$0.767753\pi$
$312$		0			0
$313$			31.6179i			1.78715i	0.448914	+	0.893575i	$0.351811\pi$
$313$			31.6179i			1.78715i	−0.448914	+	0.893575i	$0.648189\pi$
$314$		0			0
$315$	2.61526	−	0.400519i	0.147353	−	0.0225667i
$316$		0			0
$317$	26.4489			1.48552			0.742760	−	0.669558i	$-0.233516\pi$
$317$	26.4489			1.48552			0.742760	+	0.669558i	$0.233516\pi$
$318$		0			0
$319$		−	6.96652i		−	0.390050i
$320$		0			0
$321$		−	8.49311i		−	0.474039i
$322$		0			0
$323$			25.4995i			1.41883i
$324$		0			0
$325$			2.15545i			0.119563i
$326$		0			0
$327$	8.39403			0.464191
$328$		0			0
$329$	−2.84957	−	18.6068i	−0.157102	−	1.02582i
$330$		0			0
$331$		−	8.39519i		−	0.461441i	−0.973020	−	0.230721i	$-0.925892\pi$
$331$		−	8.39519i		−	0.461441i	0.973020	−	0.230721i	$-0.0741084\pi$
$332$		0			0
$333$	−7.50006			−0.411001
$334$		0			0
$335$	3.00246			0.164042
$336$		0			0
$337$	−23.9891			−1.30677			−0.653384	−	0.757027i	$-0.726651\pi$
$337$	−23.9891			−1.30677			−0.653384	+	0.757027i	$0.726651\pi$
$338$		0			0
$339$	10.0556			0.546146
$340$		0			0
$341$			0.858224i			0.0464754i
$342$		0			0
$343$	8.15390	+	16.6287i	0.440269	+	0.897866i
$344$		0			0
$345$	−5.67917			−0.305756
$346$		0			0
$347$		−	11.0986i		−	0.595803i	−0.954597	−	0.297901i	$-0.903713\pi$
$347$		−	11.0986i		−	0.595803i	0.954597	−	0.297901i	$-0.0962867\pi$
$348$		0			0
$349$		−	10.0912i		−	0.540168i	−0.962837	−	0.270084i	$-0.912949\pi$
$349$		−	10.0912i		−	0.540168i	0.962837	−	0.270084i	$-0.0870515\pi$
$350$		0			0
$351$			2.15545i			0.115049i
$352$		0			0
$353$			11.3661i			0.604959i	0.953156	+	0.302479i	$0.0978143\pi$
$353$			11.3661i			0.604959i	−0.953156	+	0.302479i	$0.902186\pi$
$354$		0			0
$355$	6.50426			0.345210
$356$		0			0
$357$	−12.5364	+	1.91991i	−0.663496	+	0.101612i
$358$		0			0
$359$		−	32.1591i		−	1.69729i	−0.528960	−	0.848647i	$-0.677418\pi$
$359$		−	32.1591i		−	1.69729i	0.528960	−	0.848647i	$-0.322582\pi$
$360$		0			0
$361$	9.29750			0.489342
$362$		0			0
$363$	4.00285			0.210095
$364$		0			0
$365$	−3.16961			−0.165905
$366$		0			0
$367$	16.4005			0.856101			0.428050	−	0.903755i	$-0.359201\pi$
$367$	16.4005			0.856101			0.428050	+	0.903755i	$0.359201\pi$
$368$		0			0
$369$		−	11.1072i		−	0.578218i
$370$		0			0
$371$	0.792968	+	5.17783i	0.0411689	+	0.268819i
$372$		0			0
$373$	9.52954			0.493421			0.246710	−	0.969089i	$-0.420650\pi$
$373$	9.52954			0.493421			0.246710	+	0.969089i	$0.420650\pi$
$374$		0			0
$375$			1.00000i			0.0516398i
$376$		0			0
$377$		−	3.87674i		−	0.199662i
$378$		0			0
$379$			15.1323i			0.777296i	0.921386	+	0.388648i	$0.127058\pi$
$379$			15.1323i			0.777296i	−0.921386	+	0.388648i	$0.872942\pi$
$380$		0			0
$381$			19.4226i			0.995048i
$382$		0			0
$383$	−17.1555			−0.876607			−0.438304	−	0.898827i	$-0.644420\pi$
$383$	−17.1555			−0.876607			−0.438304	+	0.898827i	$0.644420\pi$
$384$		0			0
$385$	−1.55135	−	10.1298i	−0.0790642	−	0.516263i
$386$		0			0
$387$			11.2593i			0.572343i
$388$		0			0
$389$	−23.7357			−1.20345			−0.601724	−	0.798704i	$-0.705519\pi$
$389$	−23.7357			−1.20345			−0.601724	+	0.798704i	$0.705519\pi$
$390$		0			0
$391$	27.2234			1.37675
$392$		0			0
$393$	4.52170			0.228090
$394$		0			0
$395$	7.99118			0.402080
$396$		0			0
$397$		−	15.2336i		−	0.764554i	−0.924048	−	0.382277i	$-0.875140\pi$
$397$		−	15.2336i		−	0.764554i	0.924048	−	0.382277i	$-0.124860\pi$
$398$		0			0
$399$	2.13058	+	13.9120i	0.106662	+	0.696470i
$400$		0			0
$401$	30.1291			1.50458			0.752288	−	0.658834i	$-0.228950\pi$
$401$	30.1291			1.50458			0.752288	+	0.658834i	$0.228950\pi$
$402$		0			0
$403$			0.477586i			0.0237902i
$404$		0			0
$405$			1.00000i			0.0496904i
$406$		0			0
$407$			29.0504i			1.43997i
$408$		0			0
$409$		−	9.56863i		−	0.473138i	−0.971615	−	0.236569i	$-0.923977\pi$
$409$		−	9.56863i		−	0.473138i	0.971615	−	0.236569i	$-0.0760230\pi$
$410$		0			0
$411$	−12.7163			−0.627250
$412$		0			0
$413$	2.11303	+	13.7974i	0.103976	+	0.678927i
$414$		0			0
$415$		−	10.4453i		−	0.512741i
$416$		0			0
$417$	−2.52657			−0.123727
$418$		0			0
$419$	22.4745			1.09795			0.548975	−	0.835839i	$-0.315018\pi$
$419$	22.4745			1.09795			0.548975	+	0.835839i	$0.315018\pi$
$420$		0			0
$421$	−14.5415			−0.708710			−0.354355	−	0.935111i	$-0.615299\pi$
$421$	−14.5415			−0.708710			−0.354355	+	0.935111i	$0.615299\pi$
$422$		0			0
$423$	7.11469			0.345928
$424$		0			0
$425$		−	4.79356i		−	0.232522i
$426$		0			0
$427$	20.9134	−	3.20283i	1.01207	−	0.154996i
$428$		0			0
$429$	8.34881			0.403084
$430$		0			0
$431$			0.493919i			0.0237912i	0.999929	+	0.0118956i	$0.00378658\pi$
$431$			0.493919i			0.0237912i	−0.999929	+	0.0118956i	$0.996213\pi$
$432$		0			0
$433$			38.4919i			1.84980i	0.380204	+	0.924902i	$0.375854\pi$
$433$			38.4919i			1.84980i	−0.380204	+	0.924902i	$0.624146\pi$
$434$		0			0
$435$		−	1.79858i		−	0.0862352i
$436$		0			0
$437$		−	30.2106i		−	1.44517i
$438$		0			0
$439$	38.0565			1.81634			0.908170	−	0.418602i	$-0.137480\pi$
$439$	38.0565			1.81634			0.908170	+	0.418602i	$0.137480\pi$
$440$		0			0
$441$	−6.67917	+	2.09492i	−0.318056	+	0.0997582i
$442$		0			0
$443$			29.2504i			1.38973i	0.719141	+	0.694864i	$0.244535\pi$
$443$			29.2504i			1.38973i	−0.719141	+	0.694864i	$0.755465\pi$
$444$		0			0
$445$	11.4560			0.543067
$446$		0			0
$447$	3.27960			0.155120
$448$		0			0
$449$	23.6966			1.11831			0.559155	−	0.829063i	$-0.311126\pi$
$449$	23.6966			1.11831			0.559155	+	0.829063i	$0.311126\pi$
$450$		0			0
$451$	−43.0221			−2.02583
$452$		0			0
$453$			4.74673i			0.223021i
$454$		0			0
$455$	−0.863298	−	5.63706i	−0.0404720	−	0.264269i
$456$		0			0
$457$	38.9971			1.82421			0.912103	−	0.409961i	$-0.134458\pi$
$457$	38.9971			1.82421			0.912103	+	0.409961i	$0.134458\pi$
$458$		0			0
$459$		−	4.79356i		−	0.223744i
$460$		0			0
$461$		−	15.4289i		−	0.718595i	−0.933223	−	0.359298i	$-0.883016\pi$
$461$		−	15.4289i		−	0.718595i	0.933223	−	0.359298i	$-0.116984\pi$
$462$		0			0
$463$			15.5650i			0.723365i	0.932301	+	0.361682i	$0.117798\pi$
$463$			15.5650i			0.723365i	−0.932301	+	0.361682i	$0.882202\pi$
$464$		0			0
$465$			0.221571i			0.0102751i
$466$		0			0
$467$	−34.9489			−1.61724			−0.808621	−	0.588330i	$-0.799786\pi$
$467$	−34.9489			−1.61724			−0.808621	+	0.588330i	$0.799786\pi$
$468$		0			0
$469$	−7.85222	+	1.20254i	−0.362582	+	0.0555283i
$470$		0			0
$471$		−	15.5590i		−	0.716921i
$472$		0			0
$473$	43.6113			2.00525
$474$		0			0
$475$	−5.31954			−0.244077
$476$		0			0
$477$	−1.97985			−0.0906512
$478$		0			0
$479$	−12.0326			−0.549782			−0.274891	−	0.961475i	$-0.588642\pi$
$479$	−12.0326			−0.549782			−0.274891	+	0.961475i	$0.588642\pi$
$480$		0			0
$481$			16.1660i			0.737106i
$482$		0			0
$483$	14.8525	−	2.27462i	0.675812	−	0.103499i
$484$		0			0
$485$	0.716023			0.0325130
$486$		0			0
$487$		−	33.1682i		−	1.50299i	−0.659736	−	0.751497i	$-0.729332\pi$
$487$		−	33.1682i		−	1.50299i	0.659736	−	0.751497i	$-0.270668\pi$
$488$		0			0
$489$			21.4106i			0.968222i
$490$		0			0
$491$		−	3.74110i		−	0.168834i	−0.996431	−	0.0844169i	$-0.973097\pi$
$491$		−	3.74110i		−	0.168834i	0.996431	−	0.0844169i	$-0.0269027\pi$
$492$		0			0
$493$			8.62158i			0.388296i
$494$		0			0
$495$	3.87335			0.174094
$496$		0			0
$497$	−17.0103	+	2.60508i	−0.763017	+	0.116854i
$498$		0			0
$499$			28.7989i			1.28921i	0.764514	+	0.644607i	$0.222979\pi$
$499$			28.7989i			1.28921i	−0.764514	+	0.644607i	$0.777021\pi$
$500$		0			0
$501$	−6.86529			−0.306719
$502$		0			0
$503$	−30.5630			−1.36274			−0.681369	−	0.731940i	$-0.738615\pi$
$503$	−30.5630			−1.36274			−0.681369	+	0.731940i	$0.738615\pi$
$504$		0			0
$505$	5.18641			0.230792
$506$		0			0
$507$	−8.35404			−0.371016
$508$		0			0
$509$		−	24.5626i		−	1.08872i	−0.838852	−	0.544359i	$-0.816773\pi$
$509$		−	24.5626i		−	1.08872i	0.838852	−	0.544359i	$-0.183227\pi$
$510$		0			0
$511$	8.28934	−	1.26949i	0.366699	−	0.0561588i
$512$		0			0
$513$	−5.31954			−0.234863
$514$		0			0
$515$		−	3.19896i		−	0.140963i
$516$		0			0
$517$		−	27.5577i		−	1.21199i
$518$		0			0
$519$			13.9225i			0.611131i
$520$		0			0
$521$			32.4634i			1.42225i	0.703066	+	0.711125i	$0.251814\pi$
$521$			32.4634i			1.42225i	−0.703066	+	0.711125i	$0.748186\pi$
$522$		0			0
$523$	−33.8522			−1.48026			−0.740128	−	0.672466i	$-0.765235\pi$
$523$	−33.8522			−1.48026			−0.740128	+	0.672466i	$0.765235\pi$
$524$		0			0
$525$	−0.400519	−	2.61526i	−0.0174801	−	0.114139i
$526$		0			0
$527$		−	1.06211i		−	0.0462664i
$528$		0			0
$529$	−9.25296			−0.402303
$530$		0			0
$531$	−5.27574			−0.228948
$532$		0			0
$533$	−23.9410			−1.03700
$534$		0			0
$535$	−8.49311			−0.367189
$536$		0			0
$537$			6.99942i			0.302047i
$538$		0			0
$539$	8.11437	+	25.8708i	0.349511	+	1.11433i
$540$		0			0
$541$	32.2557			1.38678			0.693390	−	0.720563i	$-0.256116\pi$
$541$	32.2557			1.38678			0.693390	+	0.720563i	$0.256116\pi$
$542$		0			0
$543$		−	1.97879i		−	0.0849180i
$544$		0			0
$545$		−	8.39403i		−	0.359561i
$546$		0			0
$547$		−	10.1007i		−	0.431877i	−0.976407	−	0.215938i	$-0.930719\pi$
$547$		−	10.1007i		−	0.431877i	0.976407	−	0.215938i	$-0.0692810\pi$
$548$		0			0
$549$			7.99669i			0.341290i
$550$		0			0
$551$	9.56760			0.407593
$552$		0			0
$553$	−20.8990	+	3.20062i	−0.888716	+	0.136104i
$554$		0			0
$555$			7.50006i			0.318360i
$556$		0			0
$557$	−30.5437			−1.29418			−0.647089	−	0.762414i	$-0.724014\pi$
$557$	−30.5437			−1.29418			−0.647089	+	0.762414i	$0.724014\pi$
$558$		0			0
$559$	24.2689			1.02646
$560$		0			0
$561$	−18.5671			−0.783905
$562$		0			0
$563$	−1.16461			−0.0490825			−0.0245413	−	0.999699i	$-0.507813\pi$
$563$	−1.16461			−0.0490825			−0.0245413	+	0.999699i	$0.507813\pi$
$564$		0			0
$565$		−	10.0556i		−	0.423043i
$566$		0			0
$567$	−0.400519	−	2.61526i	−0.0168202	−	0.109831i
$568$		0			0
$569$	33.8001			1.41697			0.708487	−	0.705724i	$-0.249378\pi$
$569$	33.8001			1.41697			0.708487	+	0.705724i	$0.249378\pi$
$570$		0			0
$571$			20.7283i			0.867452i	0.901045	+	0.433726i	$0.142801\pi$
$571$			20.7283i			0.867452i	−0.901045	+	0.433726i	$0.857199\pi$
$572$		0			0
$573$			20.5288i			0.857604i
$574$		0			0
$575$			5.67917i			0.236838i
$576$		0			0
$577$			14.3167i			0.596011i	0.954564	+	0.298005i	$0.0963213\pi$
$577$			14.3167i			0.596011i	−0.954564	+	0.298005i	$0.903679\pi$
$578$		0			0
$579$	13.1050			0.544627
$580$		0			0
$581$	4.18356	+	27.3173i	0.173563	+	1.13331i
$582$		0			0
$583$			7.66866i			0.317604i
$584$		0			0
$585$	2.15545			0.0891168
$586$		0			0
$587$	10.4316			0.430558			0.215279	−	0.976553i	$-0.430934\pi$
$587$	10.4316			0.430558			0.215279	+	0.976553i	$0.430934\pi$
$588$		0			0
$589$	−1.17866			−0.0485657
$590$		0			0
$591$	−6.78520			−0.279106
$592$		0			0
$593$		−	42.2732i		−	1.73595i	−0.496606	−	0.867976i	$-0.665421\pi$
$593$		−	42.2732i		−	1.73595i	0.496606	−	0.867976i	$-0.334579\pi$
$594$		0			0
$595$	1.91991	+	12.5364i	0.0787087	+	0.513942i
$596$		0			0
$597$	−6.62783			−0.271259
$598$		0			0
$599$			39.9371i			1.63179i	0.578202	+	0.815894i	$0.303755\pi$
$599$			39.9371i			1.63179i	−0.578202	+	0.815894i	$0.696245\pi$
$600$		0			0
$601$			27.0538i			1.10355i	0.833994	+	0.551773i	$0.186049\pi$
$601$			27.0538i			1.10355i	−0.833994	+	0.551773i	$0.813951\pi$
$602$		0			0
$603$		−	3.00246i		−	0.122270i
$604$		0			0
$605$		−	4.00285i		−	0.162739i
$606$		0			0
$607$	18.6004			0.754968			0.377484	−	0.926016i	$-0.376789\pi$
$607$	18.6004			0.754968			0.377484	+	0.926016i	$0.376789\pi$
$608$		0			0
$609$	0.720364	+	4.70375i	0.0291906	+	0.190605i
$610$		0			0
$611$		−	15.3354i		−	0.620402i
$612$		0			0
$613$	−2.72099			−0.109900			−0.0549499	−	0.998489i	$-0.517500\pi$
$613$	−2.72099			−0.109900			−0.0549499	+	0.998489i	$0.517500\pi$
$614$		0			0
$615$	−11.1072			−0.447886
$616$		0			0
$617$	9.75305			0.392643			0.196321	−	0.980540i	$-0.437100\pi$
$617$	9.75305			0.392643			0.196321	+	0.980540i	$0.437100\pi$
$618$		0			0
$619$	46.0457			1.85073			0.925367	−	0.379073i	$-0.123757\pi$
$619$	46.0457			1.85073			0.925367	+	0.379073i	$0.123757\pi$
$620$		0			0
$621$			5.67917i			0.227897i
$622$		0			0
$623$	−29.9605	+	4.58835i	−1.20034	+	0.183828i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$			20.6044i			0.822862i
$628$		0			0
$629$		−	35.9520i		−	1.43350i
$630$		0			0
$631$			13.4133i			0.533973i	0.963700	+	0.266987i	$0.0860280\pi$
$631$			13.4133i			0.533973i	−0.963700	+	0.266987i	$0.913972\pi$
$632$		0			0
$633$		−	10.7000i		−	0.425287i
$634$		0			0
$635$	19.4226			0.770761
$636$		0			0
$637$	4.51550	+	14.3966i	0.178911	+	0.570414i
$638$		0			0
$639$		−	6.50426i		−	0.257304i
$640$		0			0
$641$	−20.6816			−0.816876			−0.408438	−	0.912786i	$-0.633926\pi$
$641$	−20.6816			−0.816876			−0.408438	+	0.912786i	$0.633926\pi$
$642$		0			0
$643$	9.99396			0.394123			0.197062	−	0.980391i	$-0.436860\pi$
$643$	9.99396			0.394123			0.197062	+	0.980391i	$0.436860\pi$
$644$		0			0
$645$	11.2593			0.443335
$646$		0			0
$647$	−14.4620			−0.568558			−0.284279	−	0.958742i	$-0.591754\pi$
$647$	−14.4620			−0.568558			−0.284279	+	0.958742i	$0.591754\pi$
$648$		0			0
$649$			20.4348i			0.802136i
$650$		0			0
$651$	−0.0887435	−	0.579467i	−0.00347813	−	0.0227111i
$652$		0			0
$653$	43.4639			1.70087			0.850437	−	0.526076i	$-0.176337\pi$
$653$	43.4639			1.70087			0.850437	+	0.526076i	$0.176337\pi$
$654$		0			0
$655$		−	4.52170i		−	0.176677i
$656$		0			0
$657$			3.16961i			0.123658i
$658$		0			0
$659$			43.7090i			1.70266i	0.524631	+	0.851330i	$0.324203\pi$
$659$			43.7090i			1.70266i	−0.524631	+	0.851330i	$0.675797\pi$
$660$		0			0
$661$		−	32.8390i		−	1.27729i	−0.769503	−	0.638644i	$-0.779496\pi$
$661$		−	32.8390i		−	1.27729i	0.769503	−	0.638644i	$-0.220504\pi$
$662$		0			0
$663$	−10.3323			−0.401272
$664$		0			0
$665$	13.9120	−	2.13058i	0.539483	−	0.0826202i
$666$		0			0
$667$		−	10.2144i		−	0.395504i
$668$		0			0
$669$	26.0191			1.00595
$670$		0			0
$671$	30.9740			1.19574
$672$		0			0
$673$	19.6011			0.755568			0.377784	−	0.925894i	$-0.376686\pi$
$673$	19.6011			0.755568			0.377784	+	0.925894i	$0.376686\pi$
$674$		0			0
$675$	1.00000			0.0384900
$676$		0			0
$677$			30.8816i			1.18688i	0.804880	+	0.593438i	$0.202230\pi$
$677$			30.8816i			1.18688i	−0.804880	+	0.593438i	$0.797770\pi$
$678$		0			0
$679$	−1.87259	+	0.286781i	−0.0718633	+	0.0110056i
$680$		0			0
$681$	13.0050			0.498353
$682$		0			0
$683$		−	13.7409i		−	0.525781i	−0.964826	−	0.262891i	$-0.915324\pi$
$683$		−	13.7409i		−	0.525781i	0.964826	−	0.262891i	$-0.0846758\pi$
$684$		0			0
$685$			12.7163i			0.485866i
$686$		0			0
$687$			21.0268i			0.802224i
$688$		0			0
$689$			4.26747i			0.162578i
$690$		0			0
$691$	47.6561			1.81292			0.906462	−	0.422288i	$-0.138773\pi$
$691$	47.6561			1.81292			0.906462	+	0.422288i	$0.138773\pi$
$692$		0			0
$693$	−10.1298	+	1.55135i	−0.384800	+	0.0589309i
$694$		0			0
$695$			2.52657i			0.0958383i
$696$		0			0
$697$	53.2430			2.01672
$698$		0			0
$699$	22.2270			0.840703
$700$		0			0
$701$	10.7523			0.406109			0.203055	−	0.979167i	$-0.434913\pi$
$701$	10.7523			0.406109			0.203055	+	0.979167i	$0.434913\pi$
$702$		0			0
$703$	−39.8969			−1.50474
$704$		0			0
$705$		−	7.11469i		−	0.267955i
$706$		0			0
$707$	−13.5638	+	2.07725i	−0.510119	+	0.0781232i
$708$		0			0
$709$	18.9409			0.711340			0.355670	−	0.934612i	$-0.384253\pi$
$709$	18.9409			0.711340			0.355670	+	0.934612i	$0.384253\pi$
$710$		0			0
$711$		−	7.99118i		−	0.299693i
$712$		0			0
$713$			1.25834i			0.0471252i
$714$		0			0
$715$		−	8.34881i		−	0.312228i
$716$		0			0
$717$			13.6516i			0.509828i
$718$		0			0
$719$	38.4552			1.43414			0.717068	−	0.697003i	$-0.245484\pi$
$719$	38.4552			1.43414			0.717068	+	0.697003i	$0.245484\pi$
$720$		0			0
$721$	1.28125	+	8.36612i	0.0477161	+	0.311570i
$722$		0			0
$723$		−	18.7440i		−	0.697097i
$724$		0			0
$725$	−1.79858			−0.0667975
$726$		0			0
$727$	5.37304			0.199275			0.0996375	−	0.995024i	$-0.468232\pi$
$727$	5.37304			0.199275			0.0996375	+	0.995024i	$0.468232\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	−53.9722			−1.99623
$732$		0			0
$733$			19.9952i			0.738540i	0.929322	+	0.369270i	$0.120392\pi$
$733$			19.9952i			0.738540i	−0.929322	+	0.369270i	$0.879608\pi$
$734$		0			0
$735$	2.09492	+	6.67917i	0.0772724	+	0.246365i
$736$		0			0
$737$	−11.6296			−0.428381
$738$		0			0
$739$		−	21.4299i		−	0.788312i	−0.919044	−	0.394156i	$-0.871037\pi$
$739$		−	21.4299i		−	0.788312i	0.919044	−	0.394156i	$-0.128963\pi$
$740$		0			0
$741$			11.4660i			0.421214i
$742$		0			0
$743$			1.35726i			0.0497930i	0.999690	+	0.0248965i	$0.00792563\pi$
$743$			1.35726i			0.0497930i	−0.999690	+	0.0248965i	$0.992074\pi$
$744$		0			0
$745$		−	3.27960i		−	0.120155i
$746$		0			0
$747$	−10.4453			−0.382175
$748$		0			0
$749$	22.2117	−	3.40165i	0.811598	−	0.124294i
$750$		0			0
$751$			23.5955i			0.861012i	0.902588	+	0.430506i	$0.141665\pi$
$751$			23.5955i			0.861012i	−0.902588	+	0.430506i	$0.858335\pi$
$752$		0			0
$753$	13.1144			0.477915
$754$		0			0
$755$	4.74673			0.172751
$756$		0			0
$757$	22.2907			0.810170			0.405085	−	0.914279i	$-0.367242\pi$
$757$	22.2907			0.810170			0.405085	+	0.914279i	$0.367242\pi$
$758$		0			0
$759$	21.9974			0.798456
$760$		0			0
$761$		−	27.8234i		−	1.00860i	−0.863530	−	0.504298i	$-0.831751\pi$
$761$		−	27.8234i		−	1.00860i	0.863530	−	0.504298i	$-0.168249\pi$
$762$		0			0
$763$	3.36197	+	21.9526i	0.121711	+	0.794736i
$764$		0			0
$765$	−4.79356			−0.173311
$766$		0			0
$767$			11.3716i			0.410604i
$768$		0			0
$769$		−	17.5470i		−	0.632760i	−0.948633	−	0.316380i	$-0.897533\pi$
$769$		−	17.5470i		−	0.632760i	0.948633	−	0.316380i	$-0.102467\pi$
$770$		0			0
$771$			10.4950i			0.377970i
$772$		0			0
$773$			27.3646i			0.984235i	0.870529	+	0.492118i	$0.163777\pi$
$773$			27.3646i			0.984235i	−0.870529	+	0.492118i	$0.836223\pi$
$774$		0			0
$775$	0.221571			0.00795908
$776$		0			0
$777$	−3.00392	−	19.6146i	−0.107765	−	0.703670i
$778$		0			0
$779$		−	59.0852i		−	2.11695i
$780$		0			0
$781$	−25.1933			−0.901486
$782$		0			0
$783$	−1.79858			−0.0642759
$784$		0			0
$785$	−15.5590			−0.555325
$786$		0			0
$787$	−37.1840			−1.32547			−0.662733	−	0.748855i	$-0.730604\pi$
$787$	−37.1840			−1.32547			−0.662733	+	0.748855i	$0.730604\pi$
$788$		0			0
$789$			27.9160i			0.993836i
$790$		0			0
$791$	4.02746	+	26.2980i	0.143200	+	0.935050i
$792$		0			0
$793$	17.2364			0.612084
$794$		0			0
$795$			1.97985i			0.0702181i
$796$		0			0
$797$		−	39.4892i		−	1.39878i	−0.714741	−	0.699389i	$-0.753456\pi$
$797$		−	39.4892i		−	1.39878i	0.714741	−	0.699389i	$-0.246544\pi$
$798$		0			0
$799$			34.1047i			1.20654i
$800$		0			0
$801$		−	11.4560i		−	0.404779i
$802$		0			0
$803$	12.2770			0.433246
$804$		0			0
$805$	−2.27462	−	14.8525i	−0.0801697	−	0.523482i
$806$		0			0
$807$		−	1.37287i		−	0.0483272i
$808$		0			0
$809$	44.5833			1.56746			0.783732	−	0.621099i	$-0.213314\pi$
$809$	44.5833			1.56746			0.783732	+	0.621099i	$0.213314\pi$
$810$		0			0
$811$	−42.6704			−1.49836			−0.749181	−	0.662365i	$-0.769553\pi$
$811$	−42.6704			−1.49836			−0.749181	+	0.662365i	$0.769553\pi$
$812$		0			0
$813$	2.00052			0.0701614
$814$		0			0
$815$	21.4106			0.749982
$816$		0			0
$817$			59.8944i			2.09544i
$818$		0			0
$819$	−5.63706	+	0.863298i	−0.196975	+	0.0301661i
$820$		0			0
$821$	2.16342			0.0755039			0.0377519	−	0.999287i	$-0.487980\pi$
$821$	2.16342			0.0755039			0.0377519	+	0.999287i	$0.487980\pi$
$822$		0			0
$823$		−	24.8339i		−	0.865655i	−0.901477	−	0.432828i	$-0.857516\pi$
$823$		−	24.8339i		−	0.865655i	0.901477	−	0.432828i	$-0.142484\pi$
$824$		0			0
$825$		−	3.87335i		−	0.134853i
$826$		0			0
$827$			15.1462i			0.526685i	0.964702	+	0.263343i	$0.0848250\pi$
$827$			15.1462i			0.526685i	−0.964702	+	0.263343i	$0.915175\pi$
$828$		0			0
$829$		−	38.5463i		−	1.33877i	−0.742917	−	0.669383i	$-0.766558\pi$
$829$		−	38.5463i		−	1.33877i	0.742917	−	0.669383i	$-0.233442\pi$
$830$		0			0
$831$	3.87788			0.134522
$832$		0			0
$833$	−10.0421	−	32.0170i	−0.347939	−	1.10932i
$834$		0			0
$835$			6.86529i			0.237583i
$836$		0			0
$837$	0.221571			0.00765863
$838$		0			0
$839$	26.9316			0.929783			0.464892	−	0.885368i	$-0.346093\pi$
$839$	26.9316			0.929783			0.464892	+	0.885368i	$0.346093\pi$
$840$		0			0
$841$	−25.7651			−0.888452
$842$		0			0
$843$	15.4583			0.532413
$844$		0			0
$845$			8.35404i			0.287388i
$846$		0			0
$847$	1.60322	+	10.4685i	0.0550872	+	0.359702i
$848$		0			0
$849$	−22.1130			−0.758915
$850$		0			0
$851$			42.5941i			1.46011i
$852$		0			0
$853$			28.2964i			0.968852i	0.874832	+	0.484426i	$0.160971\pi$
$853$			28.2964i			0.968852i	−0.874832	+	0.484426i	$0.839029\pi$
$854$		0			0
$855$			5.31954i			0.181924i
$856$		0			0
$857$		−	44.2477i		−	1.51147i	−0.654875	−	0.755737i	$-0.727279\pi$
$857$		−	44.2477i		−	1.51147i	0.654875	−	0.755737i	$-0.272721\pi$
$858$		0			0
$859$	−5.70764			−0.194742			−0.0973710	−	0.995248i	$-0.531043\pi$
$859$	−5.70764			−0.194742			−0.0973710	+	0.995248i	$0.531043\pi$
$860$		0			0
$861$	29.0482	−	4.44865i	0.989962	−	0.151610i
$862$		0			0
$863$		−	22.5353i		−	0.767110i	−0.923518	−	0.383555i	$-0.874700\pi$
$863$		−	22.5353i		−	0.767110i	0.923518	−	0.383555i	$-0.125300\pi$
$864$		0			0
$865$	13.9225			0.473380
$866$		0			0
$867$	5.97818			0.203030
$868$		0			0
$869$	−30.9526			−1.05000
$870$		0			0
$871$	−6.47165			−0.219283
$872$		0			0
$873$		−	0.716023i		−	0.0242337i
$874$		0			0
$875$	−2.61526	+	0.400519i	−0.0884119	+	0.0135400i
$876$		0			0
$877$	42.9633			1.45077			0.725384	−	0.688344i	$-0.241662\pi$
$877$	42.9633			1.45077			0.725384	+	0.688344i	$0.241662\pi$
$878$		0			0
$879$		−	24.2993i		−	0.819597i
$880$		0			0
$881$		−	37.3812i		−	1.25940i	−0.776837	−	0.629702i	$-0.783177\pi$
$881$		−	37.3812i		−	1.25940i	0.776837	−	0.629702i	$-0.216823\pi$
$882$		0			0
$883$			22.2753i			0.749624i	0.927101	+	0.374812i	$0.122293\pi$
$883$			22.2753i			0.749624i	−0.927101	+	0.374812i	$0.877707\pi$
$884$		0			0
$885$			5.27574i			0.177342i
$886$		0			0
$887$	10.0896			0.338774			0.169387	−	0.985550i	$-0.445821\pi$
$887$	10.0896			0.338774			0.169387	+	0.985550i	$0.445821\pi$
$888$		0			0
$889$	−50.7950	+	7.77910i	−1.70361	+	0.260903i
$890$		0			0
$891$		−	3.87335i		−	0.129762i
$892$		0			0
$893$	37.8469			1.26650
$894$		0			0
$895$	6.99942			0.233965
$896$		0			0
$897$	12.2412			0.408720
$898$		0			0
$899$	−0.398513			−0.0132912
$900$		0			0
$901$		−	9.49053i		−	0.316175i
$902$		0			0
$903$	−29.4460	+	4.50957i	−0.979903	+	0.150069i
$904$		0			0
$905$	−1.97879			−0.0657772
$906$		0			0
$907$		−	9.27869i		−	0.308094i	−0.988064	−	0.154047i	$-0.950769\pi$
$907$		−	9.27869i		−	0.308094i	0.988064	−	0.154047i	$-0.0492307\pi$
$908$		0			0
$909$		−	5.18641i		−	0.172022i
$910$		0			0
$911$		−	12.0695i		−	0.399880i	−0.979808	−	0.199940i	$-0.935925\pi$
$911$		−	12.0695i		−	0.399880i	0.979808	−	0.199940i	$-0.0640748\pi$
$912$		0			0
$913$			40.4585i			1.33898i
$914$		0			0
$915$	7.99669			0.264362
$916$		0			0
$917$	1.81103	+	11.8254i	0.0598054	+	0.390510i
$918$		0			0
$919$			41.5233i			1.36973i	0.728672	+	0.684863i	$0.240138\pi$
$919$			41.5233i			1.36973i	−0.728672	+	0.684863i	$0.759862\pi$
$920$		0			0
$921$	29.4220			0.969489
$922$		0			0
$923$	−14.0196			−0.461460
$924$		0			0
$925$	7.50006			0.246600
$926$		0			0
$927$	−3.19896			−0.105068
$928$		0			0
$929$			1.38568i			0.0454625i	0.999742	+	0.0227313i	$0.00723621\pi$
$929$			1.38568i			0.0454625i	−0.999742	+	0.0227313i	$0.992764\pi$
$930$		0			0
$931$	−35.5301	+	11.1440i	−1.16445	+	0.365231i
$932$		0			0
$933$	26.2914			0.860742
$934$		0			0
$935$			18.5671i			0.607210i
$936$		0			0
$937$			20.3283i			0.664096i	0.943263	+	0.332048i	$0.107740\pi$
$937$			20.3283i			0.664096i	−0.943263	+	0.332048i	$0.892260\pi$
$938$		0			0
$939$		−	31.6179i		−	1.03181i
$940$		0			0
$941$			17.6899i			0.576675i	0.957529	+	0.288338i	$0.0931026\pi$
$941$			17.6899i			0.576675i	−0.957529	+	0.288338i	$0.906897\pi$
$942$		0			0
$943$	−63.0797			−2.05416
$944$		0			0
$945$	−2.61526	+	0.400519i	−0.0850744	+	0.0130289i
$946$		0			0
$947$		−	39.4890i		−	1.28322i	−0.767031	−	0.641610i	$-0.778267\pi$
$947$		−	39.4890i		−	1.28322i	0.767031	−	0.641610i	$-0.221733\pi$
$948$		0			0
$949$	6.83192			0.221774
$950$		0			0
$951$	−26.4489			−0.857666
$952$		0			0
$953$	8.25093			0.267274			0.133637	−	0.991030i	$-0.457334\pi$
$953$	8.25093			0.267274			0.133637	+	0.991030i	$0.457334\pi$
$954$		0			0
$955$	20.5288			0.664297
$956$		0			0
$957$			6.96652i			0.225196i
$958$		0			0
$959$	−5.09313	−	33.2565i	−0.164466	−	1.07391i
$960$		0			0
$961$	−30.9509			−0.998416
$962$		0			0
$963$			8.49311i			0.273687i
$964$		0			0
$965$		−	13.1050i		−	0.421866i
$966$		0			0
$967$			49.0779i			1.57824i	0.614241	+	0.789119i	$0.289462\pi$
$967$			49.0779i			1.57824i	−0.614241	+	0.789119i	$0.710538\pi$
$968$		0			0
$969$		−	25.4995i		−	0.819162i
$970$		0			0
$971$	−9.07094			−0.291100			−0.145550	−	0.989351i	$-0.546495\pi$
$971$	−9.07094			−0.291100			−0.145550	+	0.989351i	$0.546495\pi$
$972$		0			0
$973$	−1.01194	−	6.60764i	−0.0324413	−	0.211831i
$974$		0			0
$975$		−	2.15545i		−	0.0690296i
$976$		0			0
$977$	19.3704			0.619713			0.309857	−	0.950783i	$-0.399719\pi$
$977$	19.3704			0.619713			0.309857	+	0.950783i	$0.399719\pi$
$978$		0			0
$979$	−44.3732			−1.41817
$980$		0			0
$981$	−8.39403			−0.268001
$982$		0			0
$983$	−31.0078			−0.988995			−0.494497	−	0.869179i	$-0.664648\pi$
$983$	−31.0078			−0.988995			−0.494497	+	0.869179i	$0.664648\pi$
$984$		0			0
$985$			6.78520i			0.216194i
$986$		0			0
$987$	2.84957	+	18.6068i	0.0907028	+	0.592260i
$988$		0			0
$989$	63.9436			2.03329
$990$		0			0
$991$			27.8278i			0.883981i	0.897020	+	0.441990i	$0.145727\pi$
$991$			27.8278i			0.883981i	−0.897020	+	0.441990i	$0.854273\pi$
$992$		0			0
$993$			8.39519i			0.266413i
$994$		0			0
$995$			6.62783i			0.210116i
$996$		0			0
$997$		−	6.73019i		−	0.213147i	−0.994305	−	0.106574i	$-0.966012\pi$
$997$		−	6.73019i		−	0.213147i	0.994305	−	0.106574i	$-0.0339880\pi$
$998$		0			0
$999$	7.50006			0.237291

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3360.2.d.a.2911.12	yes	16
4.3	odd	2		3360.2.d.d.2911.13	yes	16
7.6	odd	2		3360.2.d.d.2911.5	yes	16
28.27	even	2	inner	3360.2.d.a.2911.4	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3360.2.d.a.2911.4	✓	16	28.27	even	2	inner
3360.2.d.a.2911.12	yes	16	1.1	even	1	trivial
3360.2.d.d.2911.5	yes	16	7.6	odd	2
3360.2.d.d.2911.13	yes	16	4.3	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:	\( N \)	\(=\)	\( 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3360.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000i q^{5} +(-0.400519 - 2.61526i) q^{7} +1.00000 q^{9} -3.87335i q^{11} -2.15545i q^{13} -1.00000i q^{15} +4.79356i q^{17} +5.31954 q^{19} +(0.400519 + 2.61526i) q^{21} -5.67917i q^{23} -1.00000 q^{25} -1.00000 q^{27} +1.79858 q^{29} -0.221571 q^{31} +3.87335i q^{33} +(2.61526 - 0.400519i) q^{35} -7.50006 q^{37} +2.15545i q^{39} -11.1072i q^{41} +11.2593i q^{43} +1.00000i q^{45} +7.11469 q^{47} +(-6.67917 + 2.09492i) q^{49} -4.79356i q^{51} -1.97985 q^{53} +3.87335 q^{55} -5.31954 q^{57} -5.27574 q^{59} +7.99669i q^{61} +(-0.400519 - 2.61526i) q^{63} +2.15545 q^{65} -3.00246i q^{67} +5.67917i q^{69} -6.50426i q^{71} +3.16961i q^{73} +1.00000 q^{75} +(-10.1298 + 1.55135i) q^{77} -7.99118i q^{79} +1.00000 q^{81} -10.4453 q^{83} -4.79356 q^{85} -1.79858 q^{87} -11.4560i q^{89} +(-5.63706 + 0.863298i) q^{91} +0.221571 q^{93} +5.31954i q^{95} -0.716023i q^{97} -3.87335i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{3} + 16 q^{9} + 16 q^{19} - 16 q^{25} - 16 q^{27} + 8 q^{29} - 8 q^{31} - 8 q^{37} + 16 q^{47} - 16 q^{49} - 48 q^{53} + 8 q^{55} - 16 q^{57} + 16 q^{59} - 8 q^{65} + 16 q^{75} + 16 q^{77}+ \cdots + 8 q^{93}+O(q^{100}) \) 16 * q - 16 * q^3 + 16 * q^9 + 16 * q^19 - 16 * q^25 - 16 * q^27 + 8 * q^29 - 8 * q^31 - 8 * q^37 + 16 * q^47 - 16 * q^49 - 48 * q^53 + 8 * q^55 - 16 * q^57 + 16 * q^59 - 8 * q^65 + 16 * q^75 + 16 * q^77 + 16 * q^81 - 24 * q^83 + 8 * q^85 - 8 * q^87 + 24 * q^91 + 8 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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