LMFDB - Embedded newform 1620.2.d.d.649.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,2,Mod(649,1620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1620.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1620.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:	$12.9357651274$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.301925376.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 14x^{4} + 43x^{2} + 36 $ x^6 + 14x^4 + 43x^2 + 36
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 180)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.1
Root			$1.20590i$ of defining polynomial
Character	$\chi$	$=$	1620.649
Dual form			1620.2.d.d.649.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.83216 - 1.28187i) q^{5} +3.76963i q^{7} +3.54580 q^{11} +1.00351i q^{13} +1.56023i q^{17} -7.21013 q^{19} -6.18143i q^{23} +(1.71364 + 4.69717i) q^{25} -3.00000 q^{29} -5.78285 q^{31} +(4.83216 - 6.90658i) q^{35} -0.851576i q^{37} +3.11852 q^{41} -2.70666i q^{43} +9.74867i q^{47} -7.21013 q^{49} +11.2494i q^{53} +(-6.49649 - 4.54524i) q^{55} -9.66433 q^{59} -8.21013 q^{61} +(1.28636 - 1.83859i) q^{65} +2.91806i q^{67} -7.21013 q^{71} +16.7817i q^{73} +13.3664i q^{77} -10.9930 q^{79} -10.1955i q^{83} +(2.00000 - 2.85859i) q^{85} -5.33567 q^{89} -3.78285 q^{91} +(13.2101 + 9.24242i) q^{95} +3.86209i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + q^{5} + 2 q^{11} + 3 q^{25} - 18 q^{29} - 6 q^{31} + 17 q^{35} + 14 q^{41} - 3 q^{55} - 34 q^{59} - 6 q^{61} + 15 q^{65} + 6 q^{79} + 12 q^{85} - 56 q^{89} + 6 q^{91} + 36 q^{95}+O(q^{100}) $ 6 * q + q^5 + 2 * q^11 + 3 * q^25 - 18 * q^29 - 6 * q^31 + 17 * q^35 + 14 * q^41 - 3 * q^55 - 34 * q^59 - 6 * q^61 + 15 * q^65 + 6 * q^79 + 12 * q^85 - 56 * q^89 + 6 * q^91 + 36 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times$.

$n$	$811$	$1297$	$1541$
$\chi(n)$	$1$	$-1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.83216	−	1.28187i	−0.819368	−	0.573268i
$5$	−1.83216	−	1.28187i	−0.819368	−	0.573268i
$6$		0			0
$7$			3.76963i			1.42479i	0.701780	+	0.712394i	$0.252389\pi$
$7$			3.76963i			1.42479i	−0.701780	+	0.712394i	$0.747611\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	3.54580			1.06910			0.534550	−	0.845137i	$-0.320481\pi$
$11$	3.54580			1.06910			0.534550	+	0.845137i	$0.320481\pi$
$12$		0			0
$13$			1.00351i			0.278322i	0.990270	+	0.139161i	$0.0444406\pi$
$13$			1.00351i			0.278322i	−0.990270	+	0.139161i	$0.955559\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			1.56023i			0.378410i	0.981938	+	0.189205i	$0.0605911\pi$
$17$			1.56023i			0.378410i	−0.981938	+	0.189205i	$0.939409\pi$
$18$		0			0
$19$	−7.21013			−1.65412			−0.827058	−	0.562116i	$-0.809987\pi$
$19$	−7.21013			−1.65412			−0.827058	+	0.562116i	$0.809987\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	6.18143i		−	1.28892i	−0.764639	−	0.644459i	$-0.777083\pi$
$23$		−	6.18143i		−	1.28892i	0.764639	−	0.644459i	$-0.222917\pi$
$24$		0			0
$25$	1.71364	+	4.69717i	0.342728	+	0.939435i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3.00000			−0.557086			−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000			−0.557086			−0.278543	+	0.960424i	$0.589851\pi$
$30$		0			0
$31$	−5.78285			−1.03863			−0.519315	−	0.854583i	$-0.673813\pi$
$31$	−5.78285			−1.03863			−0.519315	+	0.854583i	$0.673813\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	4.83216	−	6.90658i	0.816785	−	1.16743i
$36$		0			0
$37$		−	0.851576i		−	0.139998i	−0.997547	−	0.0699991i	$-0.977700\pi$
$37$		−	0.851576i		−	0.139998i	0.997547	−	0.0699991i	$-0.0222996\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	3.11852			0.487031			0.243516	−	0.969897i	$-0.421699\pi$
$41$	3.11852			0.487031			0.243516	+	0.969897i	$0.421699\pi$
$42$		0			0
$43$		−	2.70666i		−	0.412761i	−0.978472	−	0.206381i	$-0.933831\pi$
$43$		−	2.70666i		−	0.412761i	0.978472	−	0.206381i	$-0.0661685\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			9.74867i			1.42199i	0.703197	+	0.710995i	$0.251755\pi$
$47$			9.74867i			1.42199i	−0.703197	+	0.710995i	$0.748245\pi$
$48$		0			0
$49$	−7.21013			−1.03002
$50$		0			0
$51$		0			0
$52$		0			0
$53$			11.2494i			1.54523i	0.634876	+	0.772614i	$0.281051\pi$
$53$			11.2494i			1.54523i	−0.634876	+	0.772614i	$0.718949\pi$
$54$		0			0
$55$	−6.49649	−	4.54524i	−0.875986	−	0.612881i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−9.66433			−1.25819			−0.629094	−	0.777329i	$-0.716574\pi$
$59$	−9.66433			−1.25819			−0.629094	+	0.777329i	$0.716574\pi$
$60$		0			0
$61$	−8.21013			−1.05120			−0.525600	−	0.850732i	$-0.676159\pi$
$61$	−8.21013			−1.05120			−0.525600	+	0.850732i	$0.676159\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.28636	−	1.83859i	0.159553	−	0.228049i
$66$		0			0
$67$			2.91806i			0.356497i	0.983985	+	0.178249i	$0.0570431\pi$
$67$			2.91806i			0.356497i	−0.983985	+	0.178249i	$0.942957\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−7.21013			−0.855685			−0.427842	−	0.903853i	$-0.640726\pi$
$71$	−7.21013			−0.855685			−0.427842	+	0.903853i	$0.640726\pi$
$72$		0			0
$73$			16.7817i			1.96415i	0.188498	+	0.982074i	$0.439638\pi$
$73$			16.7817i			1.96415i	−0.188498	+	0.982074i	$0.560362\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			13.3664i			1.52324i
$78$		0			0
$79$	−10.9930			−1.23681			−0.618403	−	0.785861i	$-0.712220\pi$
$79$	−10.9930			−1.23681			−0.618403	+	0.785861i	$0.712220\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	10.1955i		−	1.11910i	−0.828798	−	0.559548i	$-0.810975\pi$
$83$		−	10.1955i		−	1.11910i	0.828798	−	0.559548i	$-0.189025\pi$
$84$		0			0
$85$	2.00000	−	2.85859i	0.216930	−	0.310057i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−5.33567			−0.565580			−0.282790	−	0.959182i	$-0.591260\pi$
$89$	−5.33567			−0.565580			−0.282790	+	0.959182i	$0.591260\pi$
$90$		0			0
$91$	−3.78285			−0.396550
$92$		0			0
$93$		0			0
$94$		0			0
$95$	13.2101	+	9.24242i	1.35533	+	0.948252i
$96$		0			0
$97$			3.86209i			0.392136i	0.980590	+	0.196068i	$0.0628174\pi$
$97$			3.86209i			0.392136i	−0.980590	+	0.196068i	$0.937183\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−4.75593			−0.473233			−0.236616	−	0.971603i	$-0.576038\pi$
$101$	−4.75593			−0.473233			−0.236616	+	0.971603i	$0.576038\pi$
$102$		0			0
$103$			18.0890i			1.78237i	0.453643	+	0.891183i	$0.350124\pi$
$103$			18.0890i			1.78237i	−0.453643	+	0.891183i	$0.649876\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	3.61770i		−	0.349737i	−0.984592	−	0.174868i	$-0.944050\pi$
$107$		−	3.61770i		−	0.349737i	0.984592	−	0.174868i	$-0.0559500\pi$
$108$		0			0
$109$	9.63741			0.923096			0.461548	−	0.887115i	$-0.347294\pi$
$109$	9.63741			0.923096			0.461548	+	0.887115i	$0.347294\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	3.55823i		−	0.334730i	−0.985895	−	0.167365i	$-0.946474\pi$
$113$		−	3.55823i		−	0.334730i	0.985895	−	0.167365i	$-0.0535259\pi$
$114$		0			0
$115$	−7.92377	+	11.3254i	−0.738895	+	1.05610i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−5.88148			−0.539154
$120$		0			0
$121$	1.57272			0.142974
$122$		0			0
$123$		0			0
$124$		0			0
$125$	2.88148	−	10.8026i	0.257727	−	0.966218i
$126$		0			0
$127$			6.78015i			0.601641i	0.953681	+	0.300820i	$0.0972605\pi$
$127$			6.78015i			0.601641i	−0.953681	+	0.300820i	$0.902739\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	3.42728			0.299443			0.149721	−	0.988728i	$-0.452162\pi$
$131$	3.42728			0.299443			0.149721	+	0.988728i	$0.452162\pi$
$132$		0			0
$133$		−	27.1795i		−	2.35676i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	14.9266i		−	1.27527i	−0.770340	−	0.637633i	$-0.779914\pi$
$137$		−	14.9266i		−	1.27527i	0.770340	−	0.637633i	$-0.220086\pi$
$138$		0			0
$139$	5.42728			0.460336			0.230168	−	0.973151i	$-0.426072\pi$
$139$	5.42728			0.460336			0.230168	+	0.973151i	$0.426072\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			3.55823i			0.297554i
$144$		0			0
$145$	5.49649	+	3.84560i	0.456458	+	0.319359i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	5.53878			0.453754			0.226877	−	0.973923i	$-0.427148\pi$
$149$	5.53878			0.453754			0.226877	+	0.973923i	$0.427148\pi$
$150$		0			0
$151$	0.217152			0.0176716			0.00883580	−	0.999961i	$-0.497187\pi$
$151$	0.217152			0.0176716			0.00883580	+	0.999961i	$0.497187\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	10.5951	+	7.41283i	0.851020	+	0.595413i
$156$		0			0
$157$		−	2.15894i		−	0.172302i	−0.996282	−	0.0861511i	$-0.972543\pi$
$157$		−	2.15894i		−	0.172302i	0.996282	−	0.0861511i	$-0.0274568\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	23.3017			1.83643
$162$		0			0
$163$			8.39084i			0.657221i	0.944465	+	0.328611i	$0.106580\pi$
$163$			8.39084i			0.657221i	−0.944465	+	0.328611i	$0.893420\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	4.62121i		−	0.357600i	−0.983885	−	0.178800i	$-0.942779\pi$
$167$		−	4.62121i		−	0.357600i	0.983885	−	0.178800i	$-0.0572215\pi$
$168$		0			0
$169$	11.9930			0.922537
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	1.45029i		−	0.110264i	−0.998479	−	0.0551318i	$-0.982442\pi$
$173$		−	1.45029i		−	0.110264i	0.998479	−	0.0551318i	$-0.0175579\pi$
$174$		0			0
$175$	−17.7066	+	6.45979i	−1.33849	+	0.488315i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−19.4472			−1.45355			−0.726775	−	0.686876i	$-0.758982\pi$
$179$	−19.4472			−1.45355			−0.726775	+	0.686876i	$0.758982\pi$
$180$		0			0
$181$	−10.4273			−0.775054			−0.387527	−	0.921858i	$-0.626671\pi$
$181$	−10.4273			−0.775054			−0.387527	+	0.921858i	$0.626671\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1.09161	+	1.56023i	−0.0802565	+	0.114710i
$186$		0			0
$187$			5.53225i			0.404558i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	3.78285			0.273717			0.136859	−	0.990591i	$-0.456299\pi$
$191$	3.78285			0.273717			0.136859	+	0.990591i	$0.456299\pi$
$192$		0			0
$193$		−	14.0750i		−	1.01314i	−0.862198	−	0.506571i	$-0.830913\pi$
$193$		−	14.0750i		−	1.01314i	0.862198	−	0.506571i	$-0.169087\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	21.5044i		−	1.53212i	−0.642768	−	0.766061i	$-0.722214\pi$
$197$		−	21.5044i		−	1.53212i	0.642768	−	0.766061i	$-0.277786\pi$
$198$		0			0
$199$	−7.21013			−0.511112			−0.255556	−	0.966794i	$-0.582259\pi$
$199$	−7.21013			−0.511112			−0.255556	+	0.966794i	$0.582259\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	11.3089i		−	0.793729i
$204$		0			0
$205$	−5.71364	−	3.99753i	−0.399058	−	0.279199i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−25.5657			−1.76842
$210$		0			0
$211$	3.78285			0.260422			0.130211	−	0.991486i	$-0.458435\pi$
$211$	3.78285			0.260422			0.130211	+	0.991486i	$0.458435\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−3.46957	+	4.95904i	−0.236623	+	0.338204i
$216$		0			0
$217$		−	21.7992i		−	1.47983i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1.56570			−0.105320
$222$		0			0
$223$			2.91806i			0.195408i	0.995216	+	0.0977038i	$0.0311498\pi$
$223$			2.91806i			0.195408i	−0.995216	+	0.0977038i	$0.968850\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			5.11846i			0.339724i	0.985468	+	0.169862i	$0.0543322\pi$
$227$			5.11846i			0.339724i	−0.985468	+	0.169862i	$0.945668\pi$
$228$		0			0
$229$	9.42026			0.622508			0.311254	−	0.950327i	$-0.399251\pi$
$229$	9.42026			0.622508			0.311254	+	0.950327i	$0.399251\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			26.1850i			1.71544i	0.514118	+	0.857719i	$0.328119\pi$
$233$			26.1850i			1.71544i	−0.514118	+	0.857719i	$0.671881\pi$
$234$		0			0
$235$	12.4965	−	17.8612i	0.815181	−	1.16513i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−22.7559			−1.47196			−0.735979	−	0.677004i	$-0.763278\pi$
$239$	−22.7559			−1.47196			−0.735979	+	0.677004i	$0.763278\pi$
$240$		0			0
$241$	16.6304			1.07126			0.535629	−	0.844454i	$-0.320075\pi$
$241$	16.6304			1.07126			0.535629	+	0.844454i	$0.320075\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	13.2101	+	9.24242i	0.843964	+	0.590476i
$246$		0			0
$247$		−	7.23541i		−	0.460378i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	20.7758			1.31136			0.655679	−	0.755040i	$-0.272382\pi$
$251$	20.7758			1.31136			0.655679	+	0.755040i	$0.272382\pi$
$252$		0			0
$253$		−	21.9181i		−	1.37798i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	19.0596i		−	1.18890i	−0.804132	−	0.594451i	$-0.797369\pi$
$257$		−	19.0596i		−	1.18890i	0.804132	−	0.594451i	$-0.202631\pi$
$258$		0			0
$259$	3.21013			0.199468
$260$		0			0
$261$		0			0
$262$		0			0
$263$			8.13798i			0.501809i	0.968012	+	0.250905i	$0.0807281\pi$
$263$			8.13798i			0.501809i	−0.968012	+	0.250905i	$0.919272\pi$
$264$		0			0
$265$	14.4203	−	20.6108i	0.885829	−	1.26611i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−25.6374			−1.56314			−0.781570	−	0.623817i	$-0.785581\pi$
$269$	−25.6374			−1.56314			−0.781570	+	0.623817i	$0.785581\pi$
$270$		0			0
$271$	−3.21013			−0.195001			−0.0975007	−	0.995235i	$-0.531085\pi$
$271$	−3.21013			−0.195001			−0.0975007	+	0.995235i	$0.531085\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	6.07623	+	16.6553i	0.366411	+	1.00435i
$276$		0			0
$277$			0.699646i			0.0420377i	0.999779	+	0.0210188i	$0.00669100\pi$
$277$			0.699646i			0.0420377i	−0.999779	+	0.0210188i	$0.993309\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	30.6304			1.82726			0.913628	−	0.406552i	$-0.133269\pi$
$281$	30.6304			1.82726			0.913628	+	0.406552i	$0.133269\pi$
$282$		0			0
$283$		−	9.60575i		−	0.571002i	−0.958378	−	0.285501i	$-0.907840\pi$
$283$		−	9.60575i		−	0.571002i	0.958378	−	0.285501i	$-0.0921600\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			11.7557i			0.693916i
$288$		0			0
$289$	14.5657			0.856806
$290$		0			0
$291$		0			0
$292$		0			0
$293$			3.01052i			0.175876i	0.996126	+	0.0879381i	$0.0280278\pi$
$293$			3.01052i			0.175876i	−0.996126	+	0.0879381i	$0.971972\pi$
$294$		0			0
$295$	17.7066	+	12.3884i	1.03092	+	0.721279i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	6.20310			0.358735
$300$		0			0
$301$	10.2031			0.588097
$302$		0			0
$303$		0			0
$304$		0			0
$305$	15.0423	+	10.5243i	0.861319	+	0.602619i
$306$		0			0
$307$			14.8671i			0.848512i	0.905542	+	0.424256i	$0.139464\pi$
$307$			14.8671i			0.848512i	−0.905542	+	0.424256i	$0.860536\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−3.07171			−0.174181			−0.0870905	−	0.996200i	$-0.527757\pi$
$311$	−3.07171			−0.174181			−0.0870905	+	0.996200i	$0.527757\pi$
$312$		0			0
$313$			23.0136i			1.30080i	0.759590	+	0.650402i	$0.225400\pi$
$313$			23.0136i			1.30080i	−0.759590	+	0.650402i	$0.774600\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			0.00900354i			0.000505689i	1.00000		0.000252844i	$8.04829e-5\pi$
$317$			0.00900354i			0.000505689i	−1.00000		0.000252844i	$0.999920\pi$
$318$		0			0
$319$	−10.6374			−0.595581
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	11.2494i		−	0.625935i
$324$		0			0
$325$	−4.71364	+	1.71965i	−0.261466	+	0.0953889i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−36.7489			−2.02603
$330$		0			0
$331$	8.99298			0.494299			0.247149	−	0.968977i	$-0.420506\pi$
$331$	8.99298			0.494299			0.247149	+	0.968977i	$0.420506\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	3.74056	−	5.34635i	0.204368	−	0.292103i
$336$		0			0
$337$			5.26139i			0.286606i	0.989679	+	0.143303i	$0.0457723\pi$
$337$			5.26139i			0.286606i	−0.989679	+	0.143303i	$0.954228\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−20.5048			−1.11040
$342$		0			0
$343$		−	0.792107i		−	0.0427698i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	21.4954i		−	1.15393i	−0.816769	−	0.576965i	$-0.804237\pi$
$347$		−	21.4954i		−	1.15393i	0.816769	−	0.576965i	$-0.195763\pi$
$348$		0			0
$349$	12.5657			0.672626			0.336313	−	0.941750i	$-0.390820\pi$
$349$	12.5657			0.672626			0.336313	+	0.941750i	$0.390820\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			25.0626i			1.33395i	0.745081	+	0.666973i	$0.232411\pi$
$353$			25.0626i			1.33395i	−0.745081	+	0.666973i	$0.767589\pi$
$354$		0			0
$355$	13.2101	+	9.24242i	0.701121	+	0.490537i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	23.1961			1.22424			0.612121	−	0.790764i	$-0.290316\pi$
$359$	23.1961			1.22424			0.612121	+	0.790764i	$0.290316\pi$
$360$		0			0
$361$	32.9860			1.73610
$362$		0			0
$363$		0			0
$364$		0			0
$365$	21.5119	−	30.7468i	1.12598	−	1.60936i
$366$		0			0
$367$		−	17.7852i		−	0.928379i	−0.885736	−	0.464190i	$-0.846346\pi$
$367$		−	17.7852i		−	0.928379i	0.885736	−	0.464190i	$-0.153654\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−42.4062			−2.20162
$372$		0			0
$373$			12.9196i			0.668951i	0.942404	+	0.334475i	$0.108559\pi$
$373$			12.9196i			0.668951i	−0.942404	+	0.334475i	$0.891441\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	3.01052i		−	0.155050i
$378$		0			0
$379$	5.56570			0.285891			0.142945	−	0.989731i	$-0.454343\pi$
$379$	5.56570			0.285891			0.142945	+	0.989731i	$0.454343\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			29.7433i			1.51981i	0.650034	+	0.759905i	$0.274755\pi$
$383$			29.7433i			1.51981i	−0.650034	+	0.759905i	$0.725245\pi$
$384$		0			0
$385$	17.1339	−	24.4894i	0.873224	−	1.24809i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−7.55283			−0.382944			−0.191472	−	0.981498i	$-0.561326\pi$
$389$	−7.55283			−0.382944			−0.191472	+	0.981498i	$0.561326\pi$
$390$		0			0
$391$	9.64443			0.487740
$392$		0			0
$393$		0			0
$394$		0			0
$395$	20.1409	+	14.0915i	1.01340	+	0.709021i
$396$		0			0
$397$		−	30.1571i		−	1.51354i	−0.653682	−	0.756770i	$-0.726776\pi$
$397$		−	30.1571i		−	1.51354i	0.653682	−	0.756770i	$-0.273224\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	21.6643			1.08186			0.540932	−	0.841066i	$-0.318071\pi$
$401$	21.6643			1.08186			0.540932	+	0.841066i	$0.318071\pi$
$402$		0			0
$403$		−	5.80312i		−	0.289074i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	3.01952i		−	0.149672i
$408$		0			0
$409$	8.13841			0.402419			0.201209	−	0.979548i	$-0.435513\pi$
$409$	8.13841			0.402419			0.201209	+	0.979548i	$0.435513\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	36.4310i		−	1.79265i
$414$		0			0
$415$	−13.0692	+	18.6797i	−0.641542	+	0.916952i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	13.2947			0.649489			0.324745	−	0.945802i	$-0.394722\pi$
$419$	13.2947			0.649489			0.324745	+	0.945802i	$0.394722\pi$
$420$		0			0
$421$	20.1384			0.981486			0.490743	−	0.871304i	$-0.336725\pi$
$421$	20.1384			0.981486			0.490743	+	0.871304i	$0.336725\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−7.32865	+	2.67367i	−0.355492	+	0.129692i
$426$		0			0
$427$		−	30.9492i		−	1.49774i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−25.8027			−1.24287			−0.621437	−	0.783464i	$-0.713451\pi$
$431$	−25.8027			−1.24287			−0.621437	+	0.783464i	$0.713451\pi$
$432$		0			0
$433$		−	6.38383i		−	0.306787i	−0.988165	−	0.153394i	$-0.950980\pi$
$433$		−	6.38383i		−	0.306787i	0.988165	−	0.153394i	$-0.0490202\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			44.5689i			2.13202i
$438$		0			0
$439$	−10.9930			−0.524666			−0.262333	−	0.964977i	$-0.584492\pi$
$439$	−10.9930			−0.524666			−0.262333	+	0.964977i	$0.584492\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	25.2320i		−	1.19881i	−0.800447	−	0.599404i	$-0.795404\pi$
$443$		−	25.2320i		−	1.19881i	0.800447	−	0.599404i	$-0.204596\pi$
$444$		0			0
$445$	9.77582	+	6.83962i	0.463419	+	0.324229i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−19.5657			−0.923362			−0.461681	−	0.887046i	$-0.652754\pi$
$449$	−19.5657			−0.923362			−0.461681	+	0.887046i	$0.652754\pi$
$450$		0			0
$451$	11.0577			0.520685
$452$		0			0
$453$		0			0
$454$		0			0
$455$	6.93079	+	4.84910i	0.324921	+	0.227329i
$456$		0			0
$457$		−	6.53576i		−	0.305730i	−0.988247	−	0.152865i	$-0.951150\pi$
$457$		−	6.53576i		−	0.305730i	0.988247	−	0.152865i	$-0.0488500\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	35.4203			1.64969			0.824843	−	0.565362i	$-0.191264\pi$
$461$	35.4203			1.64969			0.824843	+	0.565362i	$0.191264\pi$
$462$		0			0
$463$		−	16.2670i		−	0.755990i	−0.925808	−	0.377995i	$-0.876614\pi$
$463$		−	16.2670i		−	0.755990i	0.925808	−	0.377995i	$-0.123386\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			15.8112i			0.731654i	0.930683	+	0.365827i	$0.119214\pi$
$467$			15.8112i			0.731654i	−0.930683	+	0.365827i	$0.880786\pi$
$468$		0			0
$469$	−11.0000			−0.507933
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	9.59728i		−	0.441283i
$474$		0			0
$475$	−12.3556	−	33.8672i	−0.566912	−	1.55393i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−32.7419			−1.49601			−0.748007	−	0.663690i	$-0.768989\pi$
$479$	−32.7419			−1.49601			−0.748007	+	0.663690i	$0.768989\pi$
$480$		0			0
$481$	0.854561			0.0389646
$482$		0			0
$483$		0			0
$484$		0			0
$485$	4.95069	−	7.07598i	0.224799	−	0.321304i
$486$		0			0
$487$			36.4220i			1.65044i	0.564814	+	0.825218i	$0.308948\pi$
$487$			36.4220i			1.65044i	−0.564814	+	0.825218i	$0.691052\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	3.42728			0.154671			0.0773355	−	0.997005i	$-0.475359\pi$
$491$	3.42728			0.154671			0.0773355	+	0.997005i	$0.475359\pi$
$492$		0			0
$493$		−	4.68068i		−	0.210807i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	27.1795i		−	1.21917i
$498$		0			0
$499$	−3.42728			−0.153426			−0.0767131	−	0.997053i	$-0.524443\pi$
$499$	−3.42728			−0.153426			−0.0767131	+	0.997053i	$0.524443\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	17.4219i		−	0.776802i	−0.921490	−	0.388401i	$-0.873027\pi$
$503$		−	17.4219i		−	0.776802i	0.921490	−	0.388401i	$-0.126973\pi$
$504$		0			0
$505$	8.71364	+	6.09647i	0.387752	+	0.271289i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−14.1454			−0.626986			−0.313493	−	0.949591i	$-0.601499\pi$
$509$	−14.1454			−0.626986			−0.313493	+	0.949591i	$0.601499\pi$
$510$		0			0
$511$	−63.2608			−2.79849
$512$		0			0
$513$		0			0
$514$		0			0
$515$	23.1877	−	33.1421i	1.02177	−	1.46041i
$516$		0			0
$517$			34.5669i			1.52025i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	20.2300			0.886293			0.443147	−	0.896449i	$-0.353862\pi$
$521$	20.2300			0.886293			0.443147	+	0.896449i	$0.353862\pi$
$522$		0			0
$523$		−	36.3295i		−	1.58858i	−0.607540	−	0.794289i	$-0.707844\pi$
$523$		−	36.3295i		−	1.58858i	0.607540	−	0.794289i	$-0.292156\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	9.02255i		−	0.393028i
$528$		0			0
$529$	−15.2101			−0.661310
$530$		0			0
$531$		0			0
$532$		0			0
$533$			3.12946i			0.135552i
$534$		0			0
$535$	−4.63741	+	6.62822i	−0.200493	+	0.286563i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−25.5657			−1.10119
$540$		0			0
$541$	24.8475			1.06828			0.534140	−	0.845396i	$-0.320636\pi$
$541$	24.8475			1.06828			0.534140	+	0.845396i	$0.320636\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−17.6573	−	12.3539i	−0.756356	−	0.529181i
$546$		0			0
$547$		−	32.1046i		−	1.37269i	−0.727274	−	0.686347i	$-0.759213\pi$
$547$		−	32.1046i		−	1.37269i	0.727274	−	0.686347i	$-0.240787\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	21.6304			0.921485
$552$		0			0
$553$		−	41.4395i		−	1.76219i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			17.3894i			0.736813i	0.929665	+	0.368406i	$0.120096\pi$
$557$			17.3894i			0.736813i	−0.929665	+	0.368406i	$0.879904\pi$
$558$		0			0
$559$	2.71615			0.114881
$560$		0			0
$561$		0			0
$562$		0			0
$563$			27.3399i			1.15224i	0.817365	+	0.576120i	$0.195434\pi$
$563$			27.3399i			1.15224i	−0.817365	+	0.576120i	$0.804566\pi$
$564$		0			0
$565$	−4.56118	+	6.51926i	−0.191890	+	0.274267i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−15.1902			−0.636808			−0.318404	−	0.947955i	$-0.603147\pi$
$569$	−15.1902			−0.636808			−0.318404	+	0.947955i	$0.603147\pi$
$570$		0			0
$571$	−38.7021			−1.61963			−0.809816	−	0.586684i	$-0.800433\pi$
$571$	−38.7021			−1.61963			−0.809816	+	0.586684i	$0.800433\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	29.0353	−	10.5928i	1.21085	−	0.441748i
$576$		0			0
$577$			3.82910i			0.159408i	0.996819	+	0.0797038i	$0.0253974\pi$
$577$			3.82910i			0.159408i	−0.996819	+	0.0797038i	$0.974603\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	38.4331			1.59447
$582$		0			0
$583$			39.8883i			1.65200i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			24.6843i			1.01883i	0.860521	+	0.509415i	$0.170138\pi$
$587$			24.6843i			1.01883i	−0.860521	+	0.509415i	$0.829862\pi$
$588$		0			0
$589$	41.6951			1.71802
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	38.4290i		−	1.57809i	−0.614336	−	0.789044i	$-0.710576\pi$
$593$		−	38.4290i		−	1.57809i	0.614336	−	0.789044i	$-0.289424\pi$
$594$		0			0
$595$	10.7758	+	7.53926i	0.441766	+	0.309080i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	15.4273			0.630342			0.315171	−	0.949035i	$-0.397938\pi$
$599$	15.4273			0.630342			0.315171	+	0.949035i	$0.397938\pi$
$600$		0			0
$601$	−0.572719			−0.0233617			−0.0116809	−	0.999932i	$-0.503718\pi$
$601$	−0.572719			−0.0233617			−0.0116809	+	0.999932i	$0.503718\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−2.88148	−	2.01602i	−0.117149	−	0.0819627i
$606$		0			0
$607$			9.05803i			0.367654i	0.982959	+	0.183827i	$0.0588486\pi$
$607$			9.05803i			0.367654i	−0.982959	+	0.183827i	$0.941151\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−9.78285			−0.395772
$612$		0			0
$613$			6.99155i			0.282386i	0.989982	+	0.141193i	$0.0450938\pi$
$613$			6.99155i			0.282386i	−0.989982	+	0.141193i	$0.954906\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			19.8342i			0.798495i	0.916843	+	0.399247i	$0.130729\pi$
$617$			19.8342i			0.798495i	−0.916843	+	0.399247i	$0.869271\pi$
$618$		0			0
$619$	28.2031			1.13358			0.566789	−	0.823863i	$-0.308185\pi$
$619$	28.2031			1.13358			0.566789	+	0.823863i	$0.308185\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	20.1135i		−	0.805832i
$624$		0			0
$625$	−19.1269	+	16.0985i	−0.765075	+	0.643941i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	1.32865			0.0529768
$630$		0			0
$631$	−7.27482			−0.289606			−0.144803	−	0.989461i	$-0.546255\pi$
$631$	−7.27482			−0.289606			−0.144803	+	0.989461i	$0.546255\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	8.69124	−	12.4223i	0.344901	−	0.492965i
$636$		0			0
$637$		−	7.23541i		−	0.286677i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−12.8674			−0.508233			−0.254116	−	0.967174i	$-0.581785\pi$
$641$	−12.8674			−0.508233			−0.254116	+	0.967174i	$0.581785\pi$
$642$		0			0
$643$			25.2320i			0.995053i	0.867449	+	0.497526i	$0.165758\pi$
$643$			25.2320i			0.995053i	−0.867449	+	0.497526i	$0.834242\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			23.5618i			0.926311i	0.886277	+	0.463156i	$0.153283\pi$
$647$			23.5618i			0.926311i	−0.886277	+	0.463156i	$0.846717\pi$
$648$		0			0
$649$	−34.2678			−1.34513
$650$		0			0
$651$		0			0
$652$		0			0
$653$			17.3804i			0.680147i	0.940399	+	0.340074i	$0.110452\pi$
$653$			17.3804i			0.680147i	−0.940399	+	0.340074i	$0.889548\pi$
$654$		0			0
$655$	−6.27934	−	4.39331i	−0.245354	−	0.171661i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−15.6643			−0.610195			−0.305098	−	0.952321i	$-0.598689\pi$
$659$	−15.6643			−0.610195			−0.305098	+	0.952321i	$0.598689\pi$
$660$		0			0
$661$	23.4273			0.911216			0.455608	−	0.890181i	$-0.349422\pi$
$661$	23.4273			0.911216			0.455608	+	0.890181i	$0.349422\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−34.8405	+	49.7973i	−1.35106	+	1.93106i
$666$		0			0
$667$			18.5443i			0.718038i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−29.1115			−1.12384
$672$		0			0
$673$			28.7308i			1.10749i	0.832687	+	0.553745i	$0.186802\pi$
$673$			28.7308i			1.10749i	−0.832687	+	0.553745i	$0.813198\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			50.3540i			1.93526i	0.252367	+	0.967632i	$0.418791\pi$
$677$			50.3540i			1.93526i	−0.252367	+	0.967632i	$0.581209\pi$
$678$		0			0
$679$	−14.5587			−0.558711
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	11.5953i		−	0.443681i	−0.975083	−	0.221841i	$-0.928794\pi$
$683$		−	11.5953i		−	0.443681i	0.975083	−	0.221841i	$-0.0712065\pi$
$684$		0			0
$685$	−19.1339	+	27.3480i	−0.731069	+	1.04491i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−11.2889			−0.430072
$690$		0			0
$691$	45.8475			1.74412			0.872061	−	0.489397i	$-0.162783\pi$
$691$	45.8475			1.74412			0.872061	+	0.489397i	$0.162783\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−9.94366	−	6.95705i	−0.377185	−	0.263896i
$696$		0			0
$697$			4.86560i			0.184298i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	38.1115			1.43945			0.719726	−	0.694259i	$-0.244268\pi$
$701$	38.1115			1.43945			0.719726	+	0.694259i	$0.244268\pi$
$702$		0			0
$703$			6.13997i			0.231573i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	17.9281i		−	0.674256i
$708$		0			0
$709$	2.28887			0.0859602			0.0429801	−	0.999076i	$-0.486315\pi$
$709$	2.28887			0.0859602			0.0429801	+	0.999076i	$0.486315\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			35.7463i			1.33871i
$714$		0			0
$715$	4.56118	−	6.51926i	0.170578	−	0.243807i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−16.6035			−0.619205			−0.309602	−	0.950866i	$-0.600196\pi$
$719$	−16.6035			−0.619205			−0.309602	+	0.950866i	$0.600196\pi$
$720$		0			0
$721$	−68.1891			−2.53949
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−5.14092	−	14.0915i	−0.190929	−	0.523346i
$726$		0			0
$727$			38.6074i			1.43187i	0.698168	+	0.715934i	$0.253999\pi$
$727$			38.6074i			1.43187i	−0.698168	+	0.715934i	$0.746001\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	4.22300			0.156193
$732$		0			0
$733$			33.9002i			1.25213i	0.779770	+	0.626066i	$0.215336\pi$
$733$			33.9002i			1.25213i	−0.779770	+	0.626066i	$0.784664\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			10.3469i			0.381131i
$738$		0			0
$739$	36.8405			1.35520			0.677600	−	0.735431i	$-0.263020\pi$
$739$	36.8405			1.35520			0.677600	+	0.735431i	$0.263020\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	11.4278i		−	0.419247i	−0.977782	−	0.209623i	$-0.932776\pi$
$743$		−	11.4278i		−	0.419247i	0.977782	−	0.209623i	$-0.0672238\pi$
$744$		0			0
$745$	−10.1479	−	7.09997i	−0.371792	−	0.260123i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	13.6374			0.498300
$750$		0			0
$751$	−16.2031			−0.591260			−0.295630	−	0.955303i	$-0.595529\pi$
$751$	−16.2031			−0.591260			−0.295630	+	0.955303i	$0.595529\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−0.397858	−	0.278360i	−0.0144795	−	0.0101306i
$756$		0			0
$757$		−	8.08698i		−	0.293926i	−0.989142	−	0.146963i	$-0.953050\pi$
$757$		−	8.08698i		−	0.293926i	0.989142	−	0.146963i	$-0.0469499\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−21.4062			−0.775974			−0.387987	−	0.921665i	$-0.626830\pi$
$761$	−21.4062			−0.775974			−0.387987	+	0.921665i	$0.626830\pi$
$762$		0			0
$763$			36.3295i			1.31522i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	9.69821i		−	0.350182i
$768$		0			0
$769$	−51.4062			−1.85376			−0.926878	−	0.375364i	$-0.877518\pi$
$769$	−51.4062			−1.85376			−0.926878	+	0.375364i	$0.877518\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$	−9.90972	−	27.1630i	−0.355968	−	0.975725i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−22.4849			−0.805607
$780$		0			0
$781$	−25.5657			−0.914813
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−2.76747	+	3.95553i	−0.0987753	+	0.141179i
$786$		0			0
$787$			4.71367i			0.168024i	0.996465	+	0.0840121i	$0.0267734\pi$
$787$			4.71367i			0.168024i	−0.996465	+	0.0840121i	$0.973227\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	13.4132			0.476920
$792$		0			0
$793$		−	8.23891i		−	0.292572i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	10.2639i		−	0.363567i	−0.983339	−	0.181784i	$-0.941813\pi$
$797$		−	10.2639i		−	0.363567i	0.983339	−	0.181784i	$-0.0581870\pi$
$798$		0			0
$799$	−15.2101			−0.538096
$800$		0			0
$801$		0			0
$802$		0			0
$803$			59.5045i			2.09987i
$804$		0			0
$805$	−42.6926	−	29.8697i	−1.50472	−	1.05277i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	21.9860			0.772985			0.386492	−	0.922293i	$-0.373687\pi$
$809$	21.9860			0.772985			0.386492	+	0.922293i	$0.373687\pi$
$810$		0			0
$811$	−25.6951			−0.902276			−0.451138	−	0.892454i	$-0.648982\pi$
$811$	−25.6951			−0.902276			−0.451138	+	0.892454i	$0.648982\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	10.7559	−	15.3734i	0.376764	−	0.538506i
$816$		0			0
$817$			19.5153i			0.682756i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−24.3933			−0.851333			−0.425667	−	0.904880i	$-0.639960\pi$
$821$	−24.3933			−0.851333			−0.425667	+	0.904880i	$0.639960\pi$
$822$		0			0
$823$		−	2.61420i		−	0.0911252i	−0.998961	−	0.0455626i	$-0.985492\pi$
$823$		−	2.61420i		−	0.0911252i	0.998961	−	0.0455626i	$-0.0145080\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			0.0684725i			0.00238102i	0.999999	+	0.00119051i	$0.000378951\pi$
$827$			0.0684725i			0.00238102i	−0.999999	+	0.00119051i	$0.999621\pi$
$828$		0			0
$829$	24.7828			0.860744			0.430372	−	0.902652i	$-0.358382\pi$
$829$	24.7828			0.860744			0.430372	+	0.902652i	$0.358382\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	11.2494i		−	0.389770i
$834$		0			0
$835$	−5.92377	+	8.46681i	−0.205000	+	0.293006i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	45.2409			1.56189			0.780944	−	0.624601i	$-0.214738\pi$
$839$	45.2409			1.56189			0.780944	+	0.624601i	$0.214738\pi$
$840$		0			0
$841$	−20.0000			−0.689655
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−21.9731	−	15.3734i	−0.755897	−	0.528861i
$846$		0			0
$847$			5.92857i			0.203708i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−5.26396			−0.180446
$852$		0			0
$853$			20.2150i			0.692148i	0.938207	+	0.346074i	$0.112485\pi$
$853$			20.2150i			0.692148i	−0.938207	+	0.346074i	$0.887515\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	26.8427i		−	0.916929i	−0.888713	−	0.458464i	$-0.848400\pi$
$857$		−	26.8427i		−	0.916929i	0.888713	−	0.458464i	$-0.151600\pi$
$858$		0			0
$859$	−3.36259			−0.114730			−0.0573651	−	0.998353i	$-0.518270\pi$
$859$	−3.36259			−0.114730			−0.0573651	+	0.998353i	$0.518270\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			18.4524i			0.628126i	0.949402	+	0.314063i	$0.101690\pi$
$863$			18.4524i			0.628126i	−0.949402	+	0.314063i	$0.898310\pi$
$864$		0			0
$865$	−1.85908	+	2.65717i	−0.0632106	+	0.0903465i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−38.9789			−1.32227
$870$		0			0
$871$	−2.92829			−0.0992212
$872$		0			0
$873$		0			0
$874$		0			0
$875$	40.7220	+	10.8621i	1.37665	+	0.367206i
$876$		0			0
$877$		−	23.1985i		−	0.783358i	−0.920102	−	0.391679i	$-0.871894\pi$
$877$		−	23.1985i		−	0.783358i	0.920102	−	0.391679i	$-0.128106\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−2.34854			−0.0791244			−0.0395622	−	0.999217i	$-0.512596\pi$
$881$	−2.34854			−0.0791244			−0.0395622	+	0.999217i	$0.512596\pi$
$882$		0			0
$883$		−	16.3264i		−	0.549428i	−0.961526	−	0.274714i	$-0.911417\pi$
$883$		−	16.3264i		−	0.549428i	0.961526	−	0.274714i	$-0.0885832\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	26.8427i		−	0.901289i	−0.892703	−	0.450645i	$-0.851194\pi$
$887$		−	26.8427i		−	0.901289i	0.892703	−	0.450645i	$-0.148806\pi$
$888$		0			0
$889$	−25.5587			−0.857210
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	70.2892i		−	2.35214i
$894$		0			0
$895$	35.6304	+	24.9287i	1.19099	+	0.833273i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	17.3485			0.578606
$900$		0			0
$901$	−17.5516			−0.584730
$902$		0			0
$903$		0			0
$904$		0			0
$905$	19.1045	+	13.3664i	0.635054	+	0.444313i
$906$		0			0
$907$			7.84259i			0.260409i	0.991487	+	0.130204i	$0.0415634\pi$
$907$			7.84259i			0.260409i	−0.991487	+	0.130204i	$0.958437\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−2.57272			−0.0852380			−0.0426190	−	0.999091i	$-0.513570\pi$
$911$	−2.57272			−0.0852380			−0.0426190	+	0.999091i	$0.513570\pi$
$912$		0			0
$913$		−	36.1511i		−	1.19643i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			12.9196i			0.426642i
$918$		0			0
$919$	−42.4062			−1.39885			−0.699426	−	0.714705i	$-0.746561\pi$
$919$	−42.4062			−1.39885			−0.699426	+	0.714705i	$0.746561\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	7.23541i		−	0.238156i
$924$		0			0
$925$	4.00000	−	1.45929i	0.131519	−	0.0479813i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	51.8335			1.70060			0.850301	−	0.526297i	$-0.176420\pi$
$929$	51.8335			1.70060			0.850301	+	0.526297i	$0.176420\pi$
$930$		0			0
$931$	51.9860			1.70377
$932$		0			0
$933$		0			0
$934$		0			0
$935$	7.09161	−	10.1360i	0.231920	−	0.331482i
$936$		0			0
$937$		−	16.9666i		−	0.554275i	−0.960830	−	0.277137i	$-0.910614\pi$
$937$		−	16.9666i		−	0.554275i	0.960830	−	0.277137i	$-0.0893857\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−5.65730			−0.184423			−0.0922114	−	0.995739i	$-0.529394\pi$
$941$	−5.65730			−0.184423			−0.0922114	+	0.995739i	$0.529394\pi$
$942$		0			0
$943$		−	19.2769i		−	0.627744i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	35.3860i		−	1.14989i	−0.818192	−	0.574945i	$-0.805023\pi$
$947$		−	35.3860i		−	1.14989i	0.818192	−	0.574945i	$-0.194977\pi$
$948$		0			0
$949$	−16.8405			−0.546666
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	1.66116i		−	0.0538101i	−0.999638	−	0.0269051i	$-0.991435\pi$
$953$		−	1.66116i		−	0.0538101i	0.999638	−	0.0269051i	$-0.00856518\pi$
$954$		0			0
$955$	−6.93079	−	4.84910i	−0.224275	−	0.156913i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	56.2678			1.81698
$960$		0			0
$961$	2.44133			0.0787525
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−18.0423	+	25.7877i	−0.580802	+	0.830137i
$966$		0			0
$967$			20.7362i			0.666832i	0.942780	+	0.333416i	$0.108201\pi$
$967$			20.7362i			0.666832i	−0.942780	+	0.333416i	$0.891799\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−20.9193			−0.671331			−0.335665	−	0.941981i	$-0.608961\pi$
$971$	−20.9193			−0.671331			−0.335665	+	0.941981i	$0.608961\pi$
$972$		0			0
$973$			20.4589i			0.655881i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	2.89158i		−	0.0925098i	−0.998930	−	0.0462549i	$-0.985271\pi$
$977$		−	2.89158i		−	0.0925098i	0.998930	−	0.0462549i	$-0.0147287\pi$
$978$		0			0
$979$	−18.9193			−0.604662
$980$		0			0
$981$		0			0
$982$		0			0
$983$			6.30037i			0.200951i	0.994940	+	0.100475i	$0.0320363\pi$
$983$			6.30037i			0.200951i	−0.994940	+	0.100475i	$0.967964\pi$
$984$		0			0
$985$	−27.5657	+	39.3995i	−0.878316	+	1.25537i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−16.7310			−0.532016
$990$		0			0
$991$	−32.0000			−1.01651			−0.508257	−	0.861206i	$-0.669710\pi$
$991$	−32.0000			−1.01651			−0.508257	+	0.861206i	$0.669710\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	13.2101	+	9.24242i	0.418789	+	0.293004i
$996$		0			0
$997$			9.57927i			0.303378i	0.988428	+	0.151689i	$0.0484713\pi$
$997$			9.57927i			0.303378i	−0.988428	+	0.151689i	$0.951529\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.2.d.d.649.1		6
3.2	odd	2		1620.2.d.c.649.6		6
5.2	odd	4		8100.2.a.bd.1.1		6
5.3	odd	4		8100.2.a.bd.1.6		6
5.4	even	2	inner	1620.2.d.d.649.2		6
9.2	odd	6		180.2.r.a.49.6	yes	12
9.4	even	3		540.2.r.a.289.4		12
9.5	odd	6		180.2.r.a.169.1	yes	12
9.7	even	3		540.2.r.a.469.6		12
15.2	even	4		8100.2.a.bc.1.1		6
15.8	even	4		8100.2.a.bc.1.6		6
15.14	odd	2		1620.2.d.c.649.5		6
36.7	odd	6		2160.2.by.e.1009.6		12
36.11	even	6		720.2.by.e.49.1		12
36.23	even	6		720.2.by.e.529.6		12
36.31	odd	6		2160.2.by.e.289.4		12
45.2	even	12		900.2.i.f.301.4		12
45.4	even	6		540.2.r.a.289.6		12
45.7	odd	12		2700.2.i.f.901.6		12
45.13	odd	12		2700.2.i.f.1801.1		12
45.14	odd	6		180.2.r.a.169.6	yes	12
45.22	odd	12		2700.2.i.f.1801.6		12
45.23	even	12		900.2.i.f.601.3		12
45.29	odd	6		180.2.r.a.49.1	✓	12
45.32	even	12		900.2.i.f.601.4		12
45.34	even	6		540.2.r.a.469.4		12
45.38	even	12		900.2.i.f.301.3		12
45.43	odd	12		2700.2.i.f.901.1		12
180.59	even	6		720.2.by.e.529.1		12
180.79	odd	6		2160.2.by.e.1009.4		12
180.119	even	6		720.2.by.e.49.6		12
180.139	odd	6		2160.2.by.e.289.6		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.2.r.a.49.1	✓	12	45.29	odd	6
180.2.r.a.49.6	yes	12	9.2	odd	6
180.2.r.a.169.1	yes	12	9.5	odd	6
180.2.r.a.169.6	yes	12	45.14	odd	6
540.2.r.a.289.4		12	9.4	even	3
540.2.r.a.289.6		12	45.4	even	6
540.2.r.a.469.4		12	45.34	even	6
540.2.r.a.469.6		12	9.7	even	3
720.2.by.e.49.1		12	36.11	even	6
720.2.by.e.49.6		12	180.119	even	6
720.2.by.e.529.1		12	180.59	even	6
720.2.by.e.529.6		12	36.23	even	6
900.2.i.f.301.3		12	45.38	even	12
900.2.i.f.301.4		12	45.2	even	12
900.2.i.f.601.3		12	45.23	even	12
900.2.i.f.601.4		12	45.32	even	12
1620.2.d.c.649.5		6	15.14	odd	2
1620.2.d.c.649.6		6	3.2	odd	2
1620.2.d.d.649.1		6	1.1	even	1	trivial
1620.2.d.d.649.2		6	5.4	even	2	inner
2160.2.by.e.289.4		12	36.31	odd	6
2160.2.by.e.289.6		12	180.139	odd	6
2160.2.by.e.1009.4		12	180.79	odd	6
2160.2.by.e.1009.6		12	36.7	odd	6
2700.2.i.f.901.1		12	45.43	odd	12
2700.2.i.f.901.6		12	45.7	odd	12
2700.2.i.f.1801.1		12	45.13	odd	12
2700.2.i.f.1801.6		12	45.22	odd	12
8100.2.a.bc.1.1		6	15.2	even	4
8100.2.a.bc.1.6		6	15.8	even	4
8100.2.a.bd.1.1		6	5.2	odd	4
8100.2.a.bd.1.6		6	5.3	odd	4

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:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1620.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.83216 - 1.28187i) q^{5} +3.76963i q^{7} +3.54580 q^{11} +1.00351i q^{13} +1.56023i q^{17} -7.21013 q^{19} -6.18143i q^{23} +(1.71364 + 4.69717i) q^{25} -3.00000 q^{29} -5.78285 q^{31} +(4.83216 - 6.90658i) q^{35} -0.851576i q^{37} +3.11852 q^{41} -2.70666i q^{43} +9.74867i q^{47} -7.21013 q^{49} +11.2494i q^{53} +(-6.49649 - 4.54524i) q^{55} -9.66433 q^{59} -8.21013 q^{61} +(1.28636 - 1.83859i) q^{65} +2.91806i q^{67} -7.21013 q^{71} +16.7817i q^{73} +13.3664i q^{77} -10.9930 q^{79} -10.1955i q^{83} +(2.00000 - 2.85859i) q^{85} -5.33567 q^{89} -3.78285 q^{91} +(13.2101 + 9.24242i) q^{95} +3.86209i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + q^{5} + 2 q^{11} + 3 q^{25} - 18 q^{29} - 6 q^{31} + 17 q^{35} + 14 q^{41} - 3 q^{55} - 34 q^{59} - 6 q^{61} + 15 q^{65} + 6 q^{79} + 12 q^{85} - 56 q^{89} + 6 q^{91} + 36 q^{95}+O(q^{100}) \) 6 * q + q^5 + 2 * q^11 + 3 * q^25 - 18 * q^29 - 6 * q^31 + 17 * q^35 + 14 * q^41 - 3 * q^55 - 34 * q^59 - 6 * q^61 + 15 * q^65 + 6 * q^79 + 12 * q^85 - 56 * q^89 + 6 * q^91 + 36 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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