LMFDB - Embedded newform 3528.2.s.bk.361.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3528,2,Mod(361,3528)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3528, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))

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

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

chi := DirichletCharacter("3528.361");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3528.s (of order $3$, degree $2$, not 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:	$28.1712218331$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-19})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} - 5x + 25 $ x^4 - x^3 - 4x^2 - 5x + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.2
Root			$2.13746 + 0.656712i$ of defining polynomial
Character	$\chi$	$=$	3528.361
Dual form			3528.2.s.bk.3313.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.13746 + 3.70219i) q^{5} +O(q^{10})$	$q+(2.13746 + 3.70219i) q^{5} +(-2.13746 + 3.70219i) q^{11} +1.27492 q^{13} +(2.00000 - 3.46410i) q^{17} +(0.637459 + 1.10411i) q^{19} +(2.00000 + 3.46410i) q^{23} +(-6.63746 + 11.4964i) q^{25} +2.27492 q^{29} +(-0.500000 + 0.866025i) q^{31} +(-2.63746 - 4.56821i) q^{37} +10.5498 q^{41} -7.27492 q^{43} +(3.00000 + 5.19615i) q^{47} +(0.862541 - 1.49397i) q^{53} -18.2749 q^{55} +(-3.13746 + 5.43424i) q^{59} +(5.00000 + 8.66025i) q^{61} +(2.72508 + 4.71998i) q^{65} +(-3.63746 + 6.30026i) q^{67} -2.00000 q^{71} +(1.63746 - 2.83616i) q^{73} +(1.77492 + 3.07425i) q^{79} -0.274917 q^{83} +17.0997 q^{85} +(-2.27492 - 3.94027i) q^{89} +(-2.72508 + 4.71998i) q^{95} -16.2749 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{5}+O(q^{10}) $ 4 * q + q^5	$ 4 q + q^{5} - q^{11} - 10 q^{13} + 8 q^{17} - 5 q^{19} + 8 q^{23} - 19 q^{25} - 6 q^{29} - 2 q^{31} - 3 q^{37} + 12 q^{41} - 14 q^{43} + 12 q^{47} + 11 q^{53} - 58 q^{55} - 5 q^{59} + 20 q^{61} + 26 q^{65} - 7 q^{67} - 8 q^{71} - q^{73} - 8 q^{79} + 14 q^{83} + 8 q^{85} + 6 q^{89} - 26 q^{95} - 50 q^{97}+O(q^{100}) $ 4 * q + q^5 - q^11 - 10 * q^13 + 8 * q^17 - 5 * q^19 + 8 * q^23 - 19 * q^25 - 6 * q^29 - 2 * q^31 - 3 * q^37 + 12 * q^41 - 14 * q^43 + 12 * q^47 + 11 * q^53 - 58 * q^55 - 5 * q^59 + 20 * q^61 + 26 * q^65 - 7 * q^67 - 8 * q^71 - q^73 - 8 * q^79 + 14 * q^83 + 8 * q^85 + 6 * q^89 - 26 * q^95 - 50 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3528\mathbb{Z}\right)^\times$.

$n$	$785$	$1081$	$1765$	$2647$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$	$1$

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
$5$	2.13746	+	3.70219i	0.955901	+	1.65567i	0.732294	+	0.680989i	$0.238450\pi$
$5$	2.13746	+	3.70219i	0.955901	+	1.65567i	0.223607	+	0.974679i	$0.428217\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.13746	+	3.70219i	−0.644468	+	1.11625i	0.339956	+	0.940441i	$0.389588\pi$
$11$	−2.13746	+	3.70219i	−0.644468	+	1.11625i	−0.984424	+	0.175810i	$0.943746\pi$
$12$		0			0
$13$	1.27492			0.353598			0.176799	−	0.984247i	$-0.443426\pi$
$13$	1.27492			0.353598			0.176799	+	0.984247i	$0.443426\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.00000	−	3.46410i	0.485071	−	0.840168i	−0.514782	−	0.857321i	$-0.672127\pi$
$17$	2.00000	−	3.46410i	0.485071	−	0.840168i	0.999853	+	0.0171533i	$0.00546033\pi$
$18$		0			0
$19$	0.637459	+	1.10411i	0.146243	+	0.253300i	0.929836	−	0.367974i	$-0.119949\pi$
$19$	0.637459	+	1.10411i	0.146243	+	0.253300i	−0.783593	+	0.621275i	$0.786615\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.00000	+	3.46410i	0.417029	+	0.722315i	0.995639	−	0.0932891i	$-0.0297381\pi$
$23$	2.00000	+	3.46410i	0.417029	+	0.722315i	−0.578610	+	0.815604i	$0.696405\pi$
$24$		0			0
$25$	−6.63746	+	11.4964i	−1.32749	+	2.29928i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	2.27492			0.422442			0.211221	−	0.977438i	$-0.432256\pi$
$29$	2.27492			0.422442			0.211221	+	0.977438i	$0.432256\pi$
$30$		0			0
$31$	−0.500000	+	0.866025i	−0.0898027	+	0.155543i	−0.907428	−	0.420208i	$-0.861957\pi$
$31$	−0.500000	+	0.866025i	−0.0898027	+	0.155543i	0.817625	+	0.575751i	$0.195290\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−2.63746	−	4.56821i	−0.433596	−	0.751009i	0.563584	−	0.826059i	$-0.309422\pi$
$37$	−2.63746	−	4.56821i	−0.433596	−	0.751009i	−0.997180	+	0.0750491i	$0.976089\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	10.5498			1.64761			0.823804	−	0.566875i	$-0.191848\pi$
$41$	10.5498			1.64761			0.823804	+	0.566875i	$0.191848\pi$
$42$		0			0
$43$	−7.27492			−1.10941			−0.554707	−	0.832046i	$-0.687170\pi$
$43$	−7.27492			−1.10941			−0.554707	+	0.832046i	$0.687170\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.00000	+	5.19615i	0.437595	+	0.757937i	0.997503	−	0.0706177i	$-0.0224970\pi$
$47$	3.00000	+	5.19615i	0.437595	+	0.757937i	−0.559908	+	0.828554i	$0.689164\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	0.862541	−	1.49397i	0.118479	−	0.205212i	−0.800686	−	0.599084i	$-0.795531\pi$
$53$	0.862541	−	1.49397i	0.118479	−	0.205212i	0.919165	+	0.393872i	$0.128865\pi$
$54$		0			0
$55$	−18.2749			−2.46419
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−3.13746	+	5.43424i	−0.408462	+	0.707477i	−0.994718	−	0.102649i	$-0.967268\pi$
$59$	−3.13746	+	5.43424i	−0.408462	+	0.707477i	0.586255	+	0.810126i	$0.300602\pi$
$60$		0			0
$61$	5.00000	+	8.66025i	0.640184	+	1.10883i	0.985391	+	0.170305i	$0.0544754\pi$
$61$	5.00000	+	8.66025i	0.640184	+	1.10883i	−0.345207	+	0.938527i	$0.612191\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	2.72508	+	4.71998i	0.338005	+	0.585442i
$66$		0			0
$67$	−3.63746	+	6.30026i	−0.444386	+	0.769700i	−0.998009	−	0.0630678i	$-0.979912\pi$
$67$	−3.63746	+	6.30026i	−0.444386	+	0.769700i	0.553623	+	0.832767i	$0.313245\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−2.00000			−0.237356			−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000			−0.237356			−0.118678	+	0.992933i	$0.537866\pi$
$72$		0			0
$73$	1.63746	−	2.83616i	0.191650	−	0.331948i	−0.754147	−	0.656705i	$-0.771949\pi$
$73$	1.63746	−	2.83616i	0.191650	−	0.331948i	0.945797	+	0.324758i	$0.105283\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	1.77492	+	3.07425i	0.199694	+	0.345880i	0.948429	−	0.316989i	$-0.102672\pi$
$79$	1.77492	+	3.07425i	0.199694	+	0.345880i	−0.748735	+	0.662869i	$0.769339\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−0.274917			−0.0301761			−0.0150880	−	0.999886i	$-0.504803\pi$
$83$	−0.274917			−0.0301761			−0.0150880	+	0.999886i	$0.504803\pi$
$84$		0			0
$85$	17.0997			1.85472
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−2.27492	−	3.94027i	−0.241141	−	0.417668i	0.719899	−	0.694079i	$-0.244188\pi$
$89$	−2.27492	−	3.94027i	−0.241141	−	0.417668i	−0.961040	+	0.276411i	$0.910855\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2.72508	+	4.71998i	−0.279588	+	0.484260i
$96$		0			0
$97$	−16.2749			−1.65247			−0.826234	−	0.563327i	$-0.809521\pi$
$97$	−16.2749			−1.65247			−0.826234	+	0.563327i	$0.809521\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	3.00000	−	5.19615i	0.298511	−	0.517036i	−0.677284	−	0.735721i	$-0.736843\pi$
$101$	3.00000	−	5.19615i	0.298511	−	0.517036i	0.975796	+	0.218685i	$0.0701767\pi$
$102$		0			0
$103$	−5.91238	−	10.2405i	−0.582564	−	1.00903i	−0.995174	−	0.0981224i	$-0.968716\pi$
$103$	−5.91238	−	10.2405i	−0.582564	−	1.00903i	0.412611	−	0.910907i	$-0.364617\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	3.41238	+	5.91041i	0.329887	+	0.571381i	0.982489	−	0.186320i	$-0.0596561\pi$
$107$	3.41238	+	5.91041i	0.329887	+	0.571381i	−0.652602	+	0.757701i	$0.726323\pi$
$108$		0			0
$109$	2.91238	−	5.04438i	0.278955	−	0.483164i	−0.692170	−	0.721734i	$-0.743345\pi$
$109$	2.91238	−	5.04438i	0.278955	−	0.483164i	0.971125	+	0.238570i	$0.0766786\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−10.5498			−0.992445			−0.496222	−	0.868195i	$-0.665280\pi$
$113$	−10.5498			−0.992445			−0.496222	+	0.868195i	$0.665280\pi$
$114$		0			0
$115$	−8.54983	+	14.8087i	−0.797276	+	1.38092i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−3.63746	−	6.30026i	−0.330678	−	0.572751i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−35.3746			−3.16400
$126$		0			0
$127$	−21.5498			−1.91224			−0.956119	−	0.292978i	$-0.905354\pi$
$127$	−21.5498			−1.91224			−0.956119	+	0.292978i	$0.905354\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	0.137459	+	0.238085i	0.0120098	+	0.0208016i	0.871968	−	0.489563i	$-0.162844\pi$
$131$	0.137459	+	0.238085i	0.0120098	+	0.0208016i	−0.859958	+	0.510365i	$0.829510\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	8.27492	−	14.3326i	0.706974	−	1.22451i	−0.259001	−	0.965877i	$-0.583393\pi$
$137$	8.27492	−	14.3326i	0.706974	−	1.22451i	0.965974	−	0.258638i	$-0.0832735\pi$
$138$		0			0
$139$	−0.725083			−0.0615007			−0.0307504	−	0.999527i	$-0.509790\pi$
$139$	−0.725083			−0.0615007			−0.0307504	+	0.999527i	$0.509790\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−2.72508	+	4.71998i	−0.227883	+	0.394705i
$144$		0			0
$145$	4.86254	+	8.42217i	0.403812	+	0.699423i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	0.274917	+	0.476171i	0.0225221	+	0.0390094i	0.877067	−	0.480368i	$-0.159497\pi$
$149$	0.274917	+	0.476171i	0.0225221	+	0.0390094i	−0.854545	+	0.519378i	$0.826164\pi$
$150$		0			0
$151$	7.68729	−	13.3148i	0.625583	−	1.08354i	−0.362845	−	0.931850i	$-0.618194\pi$
$151$	7.68729	−	13.3148i	0.625583	−	1.08354i	0.988428	−	0.151692i	$-0.0484722\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−4.27492			−0.343370
$156$		0			0
$157$	7.27492	−	12.6005i	0.580602	−	1.00563i	−0.414807	−	0.909910i	$-0.636151\pi$
$157$	7.27492	−	12.6005i	0.580602	−	1.00563i	0.995408	−	0.0957218i	$-0.0305159\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	6.00000	+	10.3923i	0.469956	+	0.813988i	0.999410	−	0.0343508i	$-0.0109363\pi$
$163$	6.00000	+	10.3923i	0.469956	+	0.813988i	−0.529454	+	0.848339i	$0.677603\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−6.00000			−0.464294			−0.232147	−	0.972681i	$-0.574575\pi$
$167$	−6.00000			−0.464294			−0.232147	+	0.972681i	$0.574575\pi$
$168$		0			0
$169$	−11.3746			−0.874968
$170$		0			0
$171$		0			0
$172$		0			0
$173$	3.72508	+	6.45203i	0.283213	+	0.490539i	0.972174	−	0.234259i	$-0.0752664\pi$
$173$	3.72508	+	6.45203i	0.283213	+	0.490539i	−0.688961	+	0.724798i	$0.741933\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−0.725083	+	1.25588i	−0.0541952	+	0.0938689i	−0.891850	−	0.452331i	$-0.850593\pi$
$179$	−0.725083	+	1.25588i	−0.0541952	+	0.0938689i	0.837655	+	0.546200i	$0.183926\pi$
$180$		0			0
$181$	−3.82475			−0.284292			−0.142146	−	0.989846i	$-0.545400\pi$
$181$	−3.82475			−0.284292			−0.142146	+	0.989846i	$0.545400\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	11.2749	−	19.5287i	0.828948	−	1.43578i
$186$		0			0
$187$	8.54983	+	14.8087i	0.625226	+	1.08292i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−5.27492	−	9.13642i	−0.381680	−	0.661088i	0.609623	−	0.792692i	$-0.291321\pi$
$191$	−5.27492	−	9.13642i	−0.381680	−	0.661088i	−0.991302	+	0.131603i	$0.957988\pi$
$192$		0			0
$193$	−7.77492	+	13.4666i	−0.559651	+	0.969344i	0.437875	+	0.899036i	$0.355731\pi$
$193$	−7.77492	+	13.4666i	−0.559651	+	0.969344i	−0.997525	+	0.0703075i	$0.977602\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	16.5498			1.17913			0.589563	−	0.807722i	$-0.299300\pi$
$197$	16.5498			1.17913			0.589563	+	0.807722i	$0.299300\pi$
$198$		0			0
$199$	12.5498	−	21.7370i	0.889634	−	1.54089i	0.0493259	−	0.998783i	$-0.484293\pi$
$199$	12.5498	−	21.7370i	0.889634	−	1.54089i	0.840308	−	0.542109i	$-0.182374\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	22.5498	+	39.0575i	1.57495	+	2.72789i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−5.45017			−0.376996
$210$		0			0
$211$	17.6495			1.21504			0.607521	−	0.794304i	$-0.292164\pi$
$211$	17.6495			1.21504			0.607521	+	0.794304i	$0.292164\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−15.5498	−	26.9331i	−1.06049	−	1.83682i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	2.54983	−	4.41644i	0.171520	−	0.297082i
$222$		0			0
$223$	−6.27492			−0.420200			−0.210100	−	0.977680i	$-0.567379\pi$
$223$	−6.27492			−0.420200			−0.210100	+	0.977680i	$0.567379\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	1.86254	−	3.22602i	0.123621	−	0.214118i	−0.797572	−	0.603224i	$-0.793883\pi$
$227$	1.86254	−	3.22602i	0.123621	−	0.214118i	0.921193	+	0.389106i	$0.127216\pi$
$228$		0			0
$229$	5.36254	+	9.28819i	0.354367	+	0.613781i	0.987009	−	0.160663i	$-0.0513631\pi$
$229$	5.36254	+	9.28819i	0.354367	+	0.613781i	−0.632643	+	0.774444i	$0.718030\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	7.27492	+	12.6005i	0.476596	+	0.825488i	0.999640	−	0.0268173i	$-0.00853725\pi$
$233$	7.27492	+	12.6005i	0.476596	+	0.825488i	−0.523045	+	0.852305i	$0.675204\pi$
$234$		0			0
$235$	−12.8248	+	22.2131i	−0.836595	+	1.44902i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−30.5498			−1.97610			−0.988052	−	0.154119i	$-0.950746\pi$
$239$	−30.5498			−1.97610			−0.988052	+	0.154119i	$0.950746\pi$
$240$		0			0
$241$	−6.41238	+	11.1066i	−0.413057	+	0.715436i	−0.995222	−	0.0976343i	$-0.968872\pi$
$241$	−6.41238	+	11.1066i	−0.413057	+	0.715436i	0.582165	+	0.813071i	$0.302206\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	0.812707	+	1.40765i	0.0517113	+	0.0895666i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	19.3746			1.22291			0.611457	−	0.791278i	$-0.290584\pi$
$251$	19.3746			1.22291			0.611457	+	0.791278i	$0.290584\pi$
$252$		0			0
$253$	−17.0997			−1.07505
$254$		0			0
$255$		0			0
$256$		0			0
$257$	9.54983	+	16.5408i	0.595702	+	1.03179i	0.993447	+	0.114291i	$0.0364595\pi$
$257$	9.54983	+	16.5408i	0.595702	+	1.03179i	−0.397745	+	0.917496i	$0.630207\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−12.2749	+	21.2608i	−0.756904	+	1.31100i	0.187518	+	0.982261i	$0.439956\pi$
$263$	−12.2749	+	21.2608i	−0.756904	+	1.31100i	−0.944422	+	0.328735i	$0.893378\pi$
$264$		0			0
$265$	7.37459			0.453017
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−14.1375	+	24.4868i	−0.861976	+	1.49299i	0.00804266	+	0.999968i	$0.497440\pi$
$269$	−14.1375	+	24.4868i	−0.861976	+	1.49299i	−0.870019	+	0.493019i	$0.835893\pi$
$270$		0			0
$271$	−3.13746	−	5.43424i	−0.190587	−	0.330106i	0.754858	−	0.655888i	$-0.227706\pi$
$271$	−3.13746	−	5.43424i	−0.190587	−	0.330106i	−0.945445	+	0.325782i	$0.894372\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−28.3746	−	49.1462i	−1.71105	−	2.96363i
$276$		0			0
$277$	2.08762	−	3.61587i	0.125433	−	0.217257i	−0.796469	−	0.604679i	$-0.793301\pi$
$277$	2.08762	−	3.61587i	0.125433	−	0.217257i	0.921902	+	0.387423i	$0.126635\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	11.4502			0.683060			0.341530	−	0.939871i	$-0.389055\pi$
$281$	11.4502			0.683060			0.341530	+	0.939871i	$0.389055\pi$
$282$		0			0
$283$	13.4622	−	23.3172i	0.800245	−	1.38607i	−0.119209	−	0.992869i	$-0.538036\pi$
$283$	13.4622	−	23.3172i	0.800245	−	1.38607i	0.919454	−	0.393196i	$-0.128631\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.500000	+	0.866025i	0.0294118	+	0.0509427i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−5.17525			−0.302341			−0.151171	−	0.988508i	$-0.548304\pi$
$293$	−5.17525			−0.302341			−0.151171	+	0.988508i	$0.548304\pi$
$294$		0			0
$295$	−26.8248			−1.56180
$296$		0			0
$297$		0			0
$298$		0			0
$299$	2.54983	+	4.41644i	0.147461	+	0.255409i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−21.3746	+	37.0219i	−1.22391	+	2.11987i
$306$		0			0
$307$	26.3746			1.50528			0.752639	−	0.658434i	$-0.228781\pi$
$307$	26.3746			1.50528			0.752639	+	0.658434i	$0.228781\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−5.27492	+	9.13642i	−0.299113	+	0.518079i	−0.975933	−	0.218069i	$-0.930024\pi$
$311$	−5.27492	+	9.13642i	−0.299113	+	0.518079i	0.676820	+	0.736148i	$0.263357\pi$
$312$		0			0
$313$	−2.22508	−	3.85396i	−0.125769	−	0.217838i	0.796264	−	0.604949i	$-0.206807\pi$
$313$	−2.22508	−	3.85396i	−0.125769	−	0.217838i	−0.922033	+	0.387111i	$0.873473\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−2.58762	−	4.48190i	−0.145335	−	0.251728i	0.784163	−	0.620555i	$-0.213093\pi$
$317$	−2.58762	−	4.48190i	−0.145335	−	0.251728i	−0.929498	+	0.368827i	$0.879759\pi$
$318$		0			0
$319$	−4.86254	+	8.42217i	−0.272250	+	0.471551i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	5.09967			0.283753
$324$		0			0
$325$	−8.46221	+	14.6570i	−0.469399	+	0.813023i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−11.9124	−	20.6328i	−0.654763	−	1.13408i	−0.981953	−	0.189125i	$-0.939435\pi$
$331$	−11.9124	−	20.6328i	−0.654763	−	1.13408i	0.327190	−	0.944959i	$-0.393898\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−31.0997			−1.69916
$336$		0			0
$337$	6.09967			0.332270			0.166135	−	0.986103i	$-0.446871\pi$
$337$	6.09967			0.332270			0.166135	+	0.986103i	$0.446871\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−2.13746	−	3.70219i	−0.115750	−	0.200485i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−15.0997	+	26.1534i	−0.810593	+	1.40399i	0.101857	+	0.994799i	$0.467522\pi$
$347$	−15.0997	+	26.1534i	−0.810593	+	1.40399i	−0.912450	+	0.409189i	$0.865812\pi$
$348$		0			0
$349$	6.00000			0.321173			0.160586	−	0.987022i	$-0.448662\pi$
$349$	6.00000			0.321173			0.160586	+	0.987022i	$0.448662\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	2.72508	−	4.71998i	0.145042	−	0.251219i	−0.784347	−	0.620322i	$-0.787002\pi$
$353$	2.72508	−	4.71998i	0.145042	−	0.251219i	0.929388	+	0.369103i	$0.120335\pi$
$354$		0			0
$355$	−4.27492	−	7.40437i	−0.226889	−	0.392983i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	12.8248	+	22.2131i	0.676865	+	1.17236i	0.975920	+	0.218128i	$0.0699950\pi$
$359$	12.8248	+	22.2131i	0.676865	+	1.17236i	−0.299056	+	0.954236i	$0.596672\pi$
$360$		0			0
$361$	8.68729	−	15.0468i	0.457226	−	0.791939i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	14.0000			0.732793
$366$		0			0
$367$	−4.04983	+	7.01452i	−0.211400	+	0.366155i	−0.952153	−	0.305622i	$-0.901135\pi$
$367$	−4.04983	+	7.01452i	−0.211400	+	0.366155i	0.740753	+	0.671777i	$0.234469\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−0.637459	−	1.10411i	−0.0330064	−	0.0571687i	0.849050	−	0.528312i	$-0.177175\pi$
$373$	−0.637459	−	1.10411i	−0.0330064	−	0.0571687i	−0.882057	+	0.471143i	$0.843841\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	2.90033			0.149375
$378$		0			0
$379$	35.8248			1.84019			0.920097	−	0.391691i	$-0.128110\pi$
$379$	35.8248			1.84019			0.920097	+	0.391691i	$0.128110\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	2.27492	+	3.94027i	0.116243	+	0.201339i	0.918276	−	0.395941i	$-0.129582\pi$
$383$	2.27492	+	3.94027i	0.116243	+	0.201339i	−0.802033	+	0.597280i	$0.796248\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−1.00000	+	1.73205i	−0.0507020	+	0.0878185i	−0.890263	−	0.455448i	$-0.849479\pi$
$389$	−1.00000	+	1.73205i	−0.0507020	+	0.0878185i	0.839561	+	0.543266i	$0.182813\pi$
$390$		0			0
$391$	16.0000			0.809155
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−7.58762	+	13.1422i	−0.381775	+	0.661253i
$396$		0			0
$397$	17.1873	+	29.7693i	0.862606	+	1.49408i	0.869405	+	0.494100i	$0.164502\pi$
$397$	17.1873	+	29.7693i	0.862606	+	1.49408i	−0.00679974	+	0.999977i	$0.502164\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	12.0000	+	20.7846i	0.599251	+	1.03793i	0.992932	+	0.118686i	$0.0378683\pi$
$401$	12.0000	+	20.7846i	0.599251	+	1.03793i	−0.393680	+	0.919247i	$0.628798\pi$
$402$		0			0
$403$	−0.637459	+	1.10411i	−0.0317541	+	0.0549997i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	22.5498			1.11775
$408$		0			0
$409$	12.7749	−	22.1268i	0.631679	−	1.09410i	−0.355529	−	0.934665i	$-0.615699\pi$
$409$	12.7749	−	22.1268i	0.631679	−	1.09410i	0.987208	−	0.159435i	$-0.0509673\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−0.587624	−	1.01779i	−0.0288453	−	0.0499616i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	13.4502			0.657084			0.328542	−	0.944489i	$-0.393443\pi$
$419$	13.4502			0.657084			0.328542	+	0.944489i	$0.393443\pi$
$420$		0			0
$421$	−13.8248			−0.673777			−0.336889	−	0.941545i	$-0.609375\pi$
$421$	−13.8248			−0.673777			−0.336889	+	0.941545i	$0.609375\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	26.5498	+	45.9857i	1.28786	+	2.23063i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	13.8248	−	23.9452i	0.665915	−	1.15340i	−0.313122	−	0.949713i	$-0.601375\pi$
$431$	13.8248	−	23.9452i	0.665915	−	1.15340i	0.979036	−	0.203685i	$-0.0652919\pi$
$432$		0			0
$433$	−25.8248			−1.24106			−0.620529	−	0.784183i	$-0.713082\pi$
$433$	−25.8248			−1.24106			−0.620529	+	0.784183i	$0.713082\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−2.54983	+	4.41644i	−0.121975	+	0.211267i
$438$		0			0
$439$	4.86254	+	8.42217i	0.232076	+	0.401968i	0.958419	−	0.285365i	$-0.0921147\pi$
$439$	4.86254	+	8.42217i	0.232076	+	0.401968i	−0.726343	+	0.687333i	$0.758781\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	15.6873	+	27.1712i	0.745326	+	1.29094i	0.950042	+	0.312121i	$0.101039\pi$
$443$	15.6873	+	27.1712i	0.745326	+	1.29094i	−0.204717	+	0.978821i	$0.565627\pi$
$444$		0			0
$445$	9.72508	−	16.8443i	0.461013	−	0.798498i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−5.45017			−0.257209			−0.128605	−	0.991696i	$-0.541050\pi$
$449$	−5.45017			−0.257209			−0.128605	+	0.991696i	$0.541050\pi$
$450$		0			0
$451$	−22.5498	+	39.0575i	−1.06183	+	1.83914i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	4.32475	+	7.49069i	0.202303	+	0.350400i	0.949270	−	0.314462i	$-0.101824\pi$
$457$	4.32475	+	7.49069i	0.202303	+	0.350400i	−0.746967	+	0.664861i	$0.768491\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	41.6495			1.93981			0.969905	−	0.243482i	$-0.0782897\pi$
$461$	41.6495			1.93981			0.969905	+	0.243482i	$0.0782897\pi$
$462$		0			0
$463$	35.8248			1.66492			0.832459	−	0.554087i	$-0.186933\pi$
$463$	35.8248			1.66492			0.832459	+	0.554087i	$0.186933\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−12.7251	−	22.0405i	−0.588847	−	1.01991i	−0.994384	−	0.105834i	$-0.966249\pi$
$467$	−12.7251	−	22.0405i	−0.588847	−	1.01991i	0.405537	−	0.914079i	$-0.367084\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	15.5498	−	26.9331i	0.714982	−	1.23839i
$474$		0			0
$475$	−16.9244			−0.776546
$476$		0			0
$477$		0			0
$478$		0			0
$479$	10.2749	−	17.7967i	0.469473	−	0.813151i	−0.529918	−	0.848049i	$-0.677777\pi$
$479$	10.2749	−	17.7967i	0.469473	−	0.813151i	0.999391	+	0.0348979i	$0.0111106\pi$
$480$		0			0
$481$	−3.36254	−	5.82409i	−0.153319	−	0.265556i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−34.7870	−	60.2528i	−1.57959	−	2.73594i
$486$		0			0
$487$	−0.500000	+	0.866025i	−0.0226572	+	0.0392434i	−0.877132	−	0.480250i	$-0.840546\pi$
$487$	−0.500000	+	0.866025i	−0.0226572	+	0.0392434i	0.854475	+	0.519493i	$0.173879\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	15.9244			0.718659			0.359330	−	0.933211i	$-0.383005\pi$
$491$	15.9244			0.718659			0.359330	+	0.933211i	$0.383005\pi$
$492$		0			0
$493$	4.54983	−	7.88054i	0.204914	−	0.354922i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−12.3625	−	21.4125i	−0.553423	−	0.958557i	−0.998024	−	0.0628286i	$-0.979988\pi$
$499$	−12.3625	−	21.4125i	−0.553423	−	0.958557i	0.444601	−	0.895729i	$-0.353345\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	7.64950			0.341074			0.170537	−	0.985351i	$-0.445450\pi$
$503$	7.64950			0.341074			0.170537	+	0.985351i	$0.445450\pi$
$504$		0			0
$505$	25.6495			1.14139
$506$		0			0
$507$		0			0
$508$		0			0
$509$	1.86254	+	3.22602i	0.0825557	+	0.142991i	0.904347	−	0.426798i	$-0.140358\pi$
$509$	1.86254	+	3.22602i	0.0825557	+	0.142991i	−0.821791	+	0.569789i	$0.807025\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	25.2749	−	43.7774i	1.11375	−	1.92906i
$516$		0			0
$517$	−25.6495			−1.12806
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−0.274917	+	0.476171i	−0.0120443	+	0.0208614i	−0.871985	−	0.489533i	$-0.837167\pi$
$521$	−0.274917	+	0.476171i	−0.0120443	+	0.0208614i	0.859940	+	0.510394i	$0.170501\pi$
$522$		0			0
$523$	−12.6375	−	21.8887i	−0.552597	−	0.957127i	−0.998086	−	0.0618393i	$-0.980303\pi$
$523$	−12.6375	−	21.8887i	−0.552597	−	0.957127i	0.445489	−	0.895288i	$-0.353030\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	2.00000	+	3.46410i	0.0871214	+	0.150899i
$528$		0			0
$529$	3.50000	−	6.06218i	0.152174	−	0.263573i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	13.4502			0.582591
$534$		0			0
$535$	−14.5876	+	25.2665i	−0.630678	+	1.09237i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	0.362541	+	0.627940i	0.0155869	+	0.0269973i	0.873714	−	0.486441i	$-0.161705\pi$
$541$	0.362541	+	0.627940i	0.0155869	+	0.0269973i	−0.858127	+	0.513438i	$0.828372\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	24.9003			1.06661
$546$		0			0
$547$	−28.0000			−1.19719			−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000			−1.19719			−0.598597	+	0.801050i	$0.704275\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	1.45017	+	2.51176i	0.0617791	+	0.107005i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−3.58762	+	6.21395i	−0.152013	+	0.263293i	−0.931967	−	0.362543i	$-0.881909\pi$
$557$	−3.58762	+	6.21395i	−0.152013	+	0.263293i	0.779955	+	0.625836i	$0.215242\pi$
$558$		0			0
$559$	−9.27492			−0.392287
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−3.86254	+	6.69012i	−0.162787	+	0.281955i	−0.935867	−	0.352353i	$-0.885382\pi$
$563$	−3.86254	+	6.69012i	−0.162787	+	0.281955i	0.773080	+	0.634308i	$0.218715\pi$
$564$		0			0
$565$	−22.5498	−	39.0575i	−0.948679	−	1.64316i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	13.2749	+	22.9928i	0.556513	+	0.963910i	0.997784	+	0.0665355i	$0.0211946\pi$
$569$	13.2749	+	22.9928i	0.556513	+	0.963910i	−0.441271	+	0.897374i	$0.645472\pi$
$570$		0			0
$571$	−0.362541	+	0.627940i	−0.0151719	+	0.0262785i	−0.873512	−	0.486803i	$-0.838163\pi$
$571$	−0.362541	+	0.627940i	−0.0151719	+	0.0262785i	0.858340	+	0.513082i	$0.171496\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−53.0997			−2.21441
$576$		0			0
$577$	−12.5000	+	21.6506i	−0.520382	+	0.901328i	0.479337	+	0.877631i	$0.340877\pi$
$577$	−12.5000	+	21.6506i	−0.520382	+	0.901328i	−0.999719	+	0.0236970i	$0.992456\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	3.68729	+	6.38658i	0.152712	+	0.264505i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	1.72508			0.0712018			0.0356009	−	0.999366i	$-0.488665\pi$
$587$	1.72508			0.0712018			0.0356009	+	0.999366i	$0.488665\pi$
$588$		0			0
$589$	−1.27492			−0.0525320
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−7.27492	−	12.6005i	−0.298745	−	0.517442i	0.677104	−	0.735887i	$-0.263235\pi$
$593$	−7.27492	−	12.6005i	−0.298745	−	0.517442i	−0.975849	+	0.218446i	$0.929901\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	3.72508	−	6.45203i	0.152203	−	0.263623i	−0.779834	−	0.625986i	$-0.784697\pi$
$599$	3.72508	−	6.45203i	0.152203	−	0.263623i	0.932037	+	0.362363i	$0.118030\pi$
$600$		0			0
$601$	26.0997			1.06463			0.532314	−	0.846547i	$-0.321323\pi$
$601$	26.0997			1.06463			0.532314	+	0.846547i	$0.321323\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	15.5498	−	26.9331i	0.632191	−	1.09499i
$606$		0			0
$607$	−3.50000	−	6.06218i	−0.142061	−	0.246056i	0.786212	−	0.617957i	$-0.212039\pi$
$607$	−3.50000	−	6.06218i	−0.142061	−	0.246056i	−0.928272	+	0.371901i	$0.878706\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	3.82475	+	6.62466i	0.154733	+	0.268005i
$612$		0			0
$613$	−3.27492	+	5.67232i	−0.132273	+	0.229103i	−0.924552	−	0.381055i	$-0.875561\pi$
$613$	−3.27492	+	5.67232i	−0.132273	+	0.229103i	0.792280	+	0.610158i	$0.208894\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	6.00000			0.241551			0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000			0.241551			0.120775	+	0.992680i	$0.461462\pi$
$618$		0			0
$619$	3.08762	−	5.34792i	0.124102	−	0.214951i	−0.797280	−	0.603610i	$-0.793728\pi$
$619$	3.08762	−	5.34792i	0.124102	−	0.214951i	0.921382	+	0.388659i	$0.127062\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−42.4244	−	73.4813i	−1.69698	−	2.93925i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−21.0997			−0.841299
$630$		0			0
$631$	−2.82475			−0.112452			−0.0562258	−	0.998418i	$-0.517907\pi$
$631$	−2.82475			−0.112452			−0.0562258	+	0.998418i	$0.517907\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−46.0619	−	79.7815i	−1.82791	−	3.16603i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	20.8248	−	36.0695i	0.822528	−	1.42466i	−0.0812655	−	0.996692i	$-0.525896\pi$
$641$	20.8248	−	36.0695i	0.822528	−	1.42466i	0.903794	−	0.427968i	$-0.140770\pi$
$642$		0			0
$643$	32.3746			1.27673			0.638365	−	0.769734i	$-0.279611\pi$
$643$	32.3746			1.27673			0.638365	+	0.769734i	$0.279611\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	17.0000	−	29.4449i	0.668339	−	1.15760i	−0.310029	−	0.950727i	$-0.600339\pi$
$647$	17.0000	−	29.4449i	0.668339	−	1.15760i	0.978368	−	0.206870i	$-0.0663277\pi$
$648$		0			0
$649$	−13.4124	−	23.2309i	−0.526482	−	0.911893i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	22.9622	+	39.7717i	0.898581	+	1.55639i	0.829309	+	0.558790i	$0.188734\pi$
$653$	22.9622	+	39.7717i	0.898581	+	1.55639i	0.0692713	+	0.997598i	$0.477933\pi$
$654$		0			0
$655$	−0.587624	+	1.01779i	−0.0229604	+	0.0397685i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−18.1993			−0.708946			−0.354473	−	0.935066i	$-0.615340\pi$
$659$	−18.1993			−0.708946			−0.354473	+	0.935066i	$0.615340\pi$
$660$		0			0
$661$	14.9124	−	25.8290i	0.580024	−	1.00463i	−0.415452	−	0.909615i	$-0.636376\pi$
$661$	14.9124	−	25.8290i	0.580024	−	1.00463i	0.995476	−	0.0950161i	$-0.0302902\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	4.54983	+	7.88054i	0.176170	+	0.305136i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−42.7492			−1.65031
$672$		0			0
$673$	26.4502			1.01958			0.509789	−	0.860299i	$-0.329723\pi$
$673$	26.4502			1.01958			0.509789	+	0.860299i	$0.329723\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	2.86254	+	4.95807i	0.110016	+	0.190554i	0.915777	−	0.401688i	$-0.131576\pi$
$677$	2.86254	+	4.95807i	0.110016	+	0.190554i	−0.805760	+	0.592242i	$0.798243\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	17.4124	−	30.1591i	0.666266	−	1.15401i	−0.312674	−	0.949860i	$-0.601225\pi$
$683$	17.4124	−	30.1591i	0.666266	−	1.15401i	0.978940	−	0.204146i	$-0.0654418\pi$
$684$		0			0
$685$	70.7492			2.70319
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1.09967	−	1.90468i	0.0418940	−	0.0725626i
$690$		0			0
$691$	5.91238	+	10.2405i	0.224917	+	0.389568i	0.956295	−	0.292405i	$-0.0944554\pi$
$691$	5.91238	+	10.2405i	0.224917	+	0.389568i	−0.731377	+	0.681973i	$0.761122\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1.54983	−	2.68439i	−0.0587886	−	0.101825i
$696$		0			0
$697$	21.0997	−	36.5457i	0.799207	−	1.38427i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−39.9244			−1.50792			−0.753962	−	0.656918i	$-0.771860\pi$
$701$	−39.9244			−1.50792			−0.753962	+	0.656918i	$0.771860\pi$
$702$		0			0
$703$	3.36254	−	5.82409i	0.126821	−	0.219660i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−16.0997	−	27.8854i	−0.604636	−	1.04726i	−0.992109	−	0.125379i	$-0.959985\pi$
$709$	−16.0997	−	27.8854i	−0.604636	−	1.04726i	0.387473	−	0.921881i	$-0.373348\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−4.00000			−0.149801
$714$		0			0
$715$	−23.2990			−0.871333
$716$		0			0
$717$		0			0
$718$		0			0
$719$	16.0997	+	27.8854i	0.600416	+	1.03995i	0.992758	+	0.120132i	$0.0383318\pi$
$719$	16.0997	+	27.8854i	0.600416	+	1.03995i	−0.392342	+	0.919820i	$0.628335\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−15.0997	+	26.1534i	−0.560788	+	0.971313i
$726$		0			0
$727$	−16.4502			−0.610103			−0.305051	−	0.952336i	$-0.598674\pi$
$727$	−16.4502			−0.610103			−0.305051	+	0.952336i	$0.598674\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−14.5498	+	25.2011i	−0.538145	+	0.932095i
$732$		0			0
$733$	−23.4622	−	40.6377i	−0.866597	−	1.50099i	−0.865453	−	0.500990i	$-0.832969\pi$
$733$	−23.4622	−	40.6377i	−0.866597	−	1.50099i	−0.00114334	−	0.999999i	$-0.500364\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−15.5498	−	26.9331i	−0.572786	−	0.992094i
$738$		0			0
$739$	18.1873	−	31.5013i	0.669030	−	1.15879i	−0.309145	−	0.951015i	$-0.600043\pi$
$739$	18.1873	−	31.5013i	0.669030	−	1.15879i	0.978176	−	0.207780i	$-0.0666237\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−16.1993			−0.594296			−0.297148	−	0.954831i	$-0.596035\pi$
$743$	−16.1993			−0.594296			−0.297148	+	0.954831i	$0.596035\pi$
$744$		0			0
$745$	−1.17525	+	2.03559i	−0.0430578	+	0.0745782i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	12.7749	+	22.1268i	0.466163	+	0.807419i	0.999253	−	0.0386400i	$-0.0123026\pi$
$751$	12.7749	+	22.1268i	0.466163	+	0.807419i	−0.533090	+	0.846059i	$0.678969\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	65.7251			2.39198
$756$		0			0
$757$	−22.0000			−0.799604			−0.399802	−	0.916602i	$-0.630921\pi$
$757$	−22.0000			−0.799604			−0.399802	+	0.916602i	$0.630921\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−2.54983	−	4.41644i	−0.0924314	−	0.160096i	0.816102	−	0.577907i	$-0.196131\pi$
$761$	−2.54983	−	4.41644i	−0.0924314	−	0.160096i	−0.908534	+	0.417812i	$0.862797\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−4.00000	+	6.92820i	−0.144432	+	0.250163i
$768$		0			0
$769$	−12.6495			−0.456153			−0.228076	−	0.973643i	$-0.573244\pi$
$769$	−12.6495			−0.456153			−0.228076	+	0.973643i	$0.573244\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	13.5498	−	23.4690i	0.487354	−	0.844121i	−0.512541	−	0.858663i	$-0.671296\pi$
$773$	13.5498	−	23.4690i	0.487354	−	0.844121i	0.999894	+	0.0145417i	$0.00462892\pi$
$774$		0			0
$775$	−6.63746	−	11.4964i	−0.238425	−	0.412963i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	6.72508	+	11.6482i	0.240951	+	0.417340i
$780$		0			0
$781$	4.27492	−	7.40437i	0.152969	−	0.264949i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	62.1993			2.21999
$786$		0			0
$787$	−6.27492	+	10.8685i	−0.223677	+	0.387419i	−0.955922	−	0.293622i	$-0.905139\pi$
$787$	−6.27492	+	10.8685i	−0.223677	+	0.387419i	0.732245	+	0.681041i	$0.238473\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	6.37459	+	11.0411i	0.226368	+	0.392081i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	36.4743			1.29198			0.645992	−	0.763344i	$-0.276444\pi$
$797$	36.4743			1.29198			0.645992	+	0.763344i	$0.276444\pi$
$798$		0			0
$799$	24.0000			0.849059
$800$		0			0
$801$		0			0
$802$		0			0
$803$	7.00000	+	12.1244i	0.247025	+	0.427859i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	7.82475	−	13.5529i	0.275104	−	0.476493i	−0.695058	−	0.718954i	$-0.744621\pi$
$809$	7.82475	−	13.5529i	0.275104	−	0.476493i	0.970161	+	0.242461i	$0.0779545\pi$
$810$		0			0
$811$	27.4502			0.963906			0.481953	−	0.876197i	$-0.339928\pi$
$811$	27.4502			0.963906			0.481953	+	0.876197i	$0.339928\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−25.6495	+	44.4262i	−0.898463	+	1.55618i
$816$		0			0
$817$	−4.63746	−	8.03231i	−0.162244	−	0.281015i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	10.1375	+	17.5586i	0.353800	+	0.612799i	0.986912	−	0.161261i	$-0.0515561\pi$
$821$	10.1375	+	17.5586i	0.353800	+	0.612799i	−0.633112	+	0.774060i	$0.718223\pi$
$822$		0			0
$823$	−13.0997	+	22.6893i	−0.456626	+	0.790899i	−0.998780	−	0.0493799i	$-0.984275\pi$
$823$	−13.0997	+	22.6893i	−0.456626	+	0.790899i	0.542154	+	0.840279i	$0.317609\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−39.0241			−1.35700			−0.678500	−	0.734600i	$-0.737370\pi$
$827$	−39.0241			−1.35700			−0.678500	+	0.734600i	$0.737370\pi$
$828$		0			0
$829$	11.3625	−	19.6805i	0.394637	−	0.683532i	−0.598417	−	0.801184i	$-0.704204\pi$
$829$	11.3625	−	19.6805i	0.394637	−	0.683532i	0.993055	+	0.117652i	$0.0375369\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−12.8248	−	22.2131i	−0.443819	−	0.768717i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	30.1993			1.04260			0.521298	−	0.853374i	$-0.325448\pi$
$839$	30.1993			1.04260			0.521298	+	0.853374i	$0.325448\pi$
$840$		0			0
$841$	−23.8248			−0.821543
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−24.3127	−	42.1108i	−0.836383	−	1.44866i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	10.5498	−	18.2728i	0.361644	−	0.626385i
$852$		0			0
$853$	24.3746			0.834570			0.417285	−	0.908776i	$-0.362982\pi$
$853$	24.3746			0.834570			0.417285	+	0.908776i	$0.362982\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	14.2749	−	24.7249i	0.487622	−	0.844586i	−0.512277	−	0.858820i	$-0.671198\pi$
$857$	14.2749	−	24.7249i	0.487622	−	0.844586i	0.999899	+	0.0142345i	$0.00453114\pi$
$858$		0			0
$859$	−5.17525	−	8.96379i	−0.176577	−	0.305841i	0.764129	−	0.645064i	$-0.223169\pi$
$859$	−5.17525	−	8.96379i	−0.176577	−	0.305841i	−0.940706	+	0.339223i	$0.889836\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−5.54983	−	9.61260i	−0.188919	−	0.327217i	0.755971	−	0.654605i	$-0.227165\pi$
$863$	−5.54983	−	9.61260i	−0.188919	−	0.327217i	−0.944890	+	0.327388i	$0.893832\pi$
$864$		0			0
$865$	−15.9244	+	27.5819i	−0.541447	+	0.937813i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−15.1752			−0.514785
$870$		0			0
$871$	−4.63746	+	8.03231i	−0.157134	+	0.272165i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	14.3746	+	24.8975i	0.485395	+	0.840729i	0.999859	−	0.0167828i	$-0.00534237\pi$
$877$	14.3746	+	24.8975i	0.485395	+	0.840729i	−0.514464	+	0.857512i	$0.672009\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	24.5498			0.827105			0.413552	−	0.910480i	$-0.364288\pi$
$881$	24.5498			0.827105			0.413552	+	0.910480i	$0.364288\pi$
$882$		0			0
$883$	−30.3746			−1.02219			−0.511093	−	0.859525i	$-0.670759\pi$
$883$	−30.3746			−1.02219			−0.511093	+	0.859525i	$0.670759\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	5.82475	+	10.0888i	0.195576	+	0.338748i	0.947089	−	0.320970i	$-0.104009\pi$
$887$	5.82475	+	10.0888i	0.195576	+	0.338748i	−0.751513	+	0.659718i	$0.770676\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−3.82475	+	6.62466i	−0.127990	+	0.221686i
$894$		0			0
$895$	−6.19934			−0.207221
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−1.13746	+	1.97014i	−0.0379364	+	0.0657077i
$900$		0			0
$901$	−3.45017	−	5.97586i	−0.114942	−	0.199085i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−8.17525	−	14.1599i	−0.271754	−	0.470693i
$906$		0			0
$907$	18.6375	−	32.2810i	0.618847	−	1.07187i	−0.370850	−	0.928693i	$-0.620934\pi$
$907$	18.6375	−	32.2810i	0.618847	−	1.07187i	0.989697	−	0.143181i	$-0.0457331\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	51.8488			1.71783			0.858914	−	0.512119i	$-0.171139\pi$
$911$	51.8488			1.71783			0.858914	+	0.512119i	$0.171139\pi$
$912$		0			0
$913$	0.587624	−	1.01779i	0.0194475	−	0.0336841i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−9.91238	−	17.1687i	−0.326979	−	0.566344i	0.654932	−	0.755688i	$-0.272697\pi$
$919$	−9.91238	−	17.1687i	−0.326979	−	0.566344i	−0.981911	+	0.189344i	$0.939364\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−2.54983			−0.0839288
$924$		0			0
$925$	70.0241			2.30238
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−10.0997	−	17.4931i	−0.331359	−	0.573931i	0.651419	−	0.758718i	$-0.274174\pi$
$929$	−10.0997	−	17.4931i	−0.331359	−	0.573931i	−0.982779	+	0.184787i	$0.940841\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−36.5498	+	63.3062i	−1.19531	+	2.07033i
$936$		0			0
$937$	−24.0997			−0.787302			−0.393651	−	0.919260i	$-0.628788\pi$
$937$	−24.0997			−0.787302			−0.393651	+	0.919260i	$0.628788\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−14.5876	+	25.2665i	−0.475543	+	0.823665i	−0.999608	−	0.0280140i	$-0.991082\pi$
$941$	−14.5876	+	25.2665i	−0.475543	+	0.823665i	0.524065	+	0.851679i	$0.324415\pi$
$942$		0			0
$943$	21.0997	+	36.5457i	0.687100	+	1.19009i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−17.2749	−	29.9210i	−0.561359	−	0.972303i	−0.997378	−	0.0723654i	$-0.976945\pi$
$947$	−17.2749	−	29.9210i	−0.561359	−	0.972303i	0.436019	−	0.899938i	$-0.356388\pi$
$948$		0			0
$949$	2.08762	−	3.61587i	0.0677671	−	0.117376i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−10.3505			−0.335285			−0.167643	−	0.985848i	$-0.553616\pi$
$953$	−10.3505			−0.335285			−0.167643	+	0.985848i	$0.553616\pi$
$954$		0			0
$955$	22.5498	−	39.0575i	0.729696	−	1.26387i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	15.0000	+	25.9808i	0.483871	+	0.838089i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−66.4743			−2.13988
$966$		0			0
$967$	−38.4502			−1.23647			−0.618237	−	0.785992i	$-0.712153\pi$
$967$	−38.4502			−1.23647			−0.618237	+	0.785992i	$0.712153\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	20.7870	+	36.0041i	0.667085	+	1.15543i	0.978715	+	0.205222i	$0.0657917\pi$
$971$	20.7870	+	36.0041i	0.667085	+	1.15543i	−0.311630	+	0.950204i	$0.600875\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	19.5498	−	33.8613i	0.625455	−	1.08332i	−0.362998	−	0.931790i	$-0.618247\pi$
$977$	19.5498	−	33.8613i	0.625455	−	1.08332i	0.988453	−	0.151529i	$-0.0484198\pi$
$978$		0			0
$979$	19.4502			0.621630
$980$		0			0
$981$		0			0
$982$		0			0
$983$	25.6495	−	44.4262i	0.818092	−	1.41698i	−0.0889941	−	0.996032i	$-0.528365\pi$
$983$	25.6495	−	44.4262i	0.818092	−	1.41698i	0.907086	−	0.420945i	$-0.138301\pi$
$984$		0			0
$985$	35.3746	+	61.2706i	1.12713	+	1.95224i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−14.5498	−	25.2011i	−0.462658	−	0.801347i
$990$		0			0
$991$	−8.04983	+	13.9427i	−0.255711	+	0.442905i	−0.965089	−	0.261924i	$-0.915643\pi$
$991$	−8.04983	+	13.9427i	−0.255711	+	0.442905i	0.709377	+	0.704829i	$0.248976\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	107.299			3.40161
$996$		0			0
$997$	−11.0876	+	19.2043i	−0.351149	+	0.608207i	−0.986451	−	0.164056i	$-0.947542\pi$
$997$	−11.0876	+	19.2043i	−0.351149	+	0.608207i	0.635302	+	0.772263i	$0.280875\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3528.2.s.bk.361.2		4
3.2	odd	2		1176.2.q.l.361.1		4
7.2	even	3	inner	3528.2.s.bk.3313.2		4
7.3	odd	6		3528.2.a.bk.1.2		2
7.4	even	3		3528.2.a.bd.1.1		2
7.5	odd	6		504.2.s.i.289.1		4
7.6	odd	2		504.2.s.i.361.1		4
12.11	even	2		2352.2.q.bf.1537.1		4
21.2	odd	6		1176.2.q.l.961.1		4
21.5	even	6		168.2.q.c.121.2	yes	4
21.11	odd	6		1176.2.a.n.1.2		2
21.17	even	6		1176.2.a.k.1.1		2
21.20	even	2		168.2.q.c.25.2	✓	4
28.3	even	6		7056.2.a.cu.1.2		2
28.11	odd	6		7056.2.a.ch.1.1		2
28.19	even	6		1008.2.s.r.289.1		4
28.27	even	2		1008.2.s.r.865.1		4
84.11	even	6		2352.2.a.ba.1.2		2
84.23	even	6		2352.2.q.bf.961.1		4
84.47	odd	6		336.2.q.g.289.2		4
84.59	odd	6		2352.2.a.bf.1.1		2
84.83	odd	2		336.2.q.g.193.2		4
168.5	even	6		1344.2.q.w.961.1		4
168.11	even	6		9408.2.a.dw.1.1		2
168.53	odd	6		9408.2.a.dj.1.1		2
168.59	odd	6		9408.2.a.dp.1.2		2
168.83	odd	2		1344.2.q.x.193.1		4
168.101	even	6		9408.2.a.ec.1.2		2
168.125	even	2		1344.2.q.w.193.1		4
168.131	odd	6		1344.2.q.x.961.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.2.q.c.25.2	✓	4	21.20	even	2
168.2.q.c.121.2	yes	4	21.5	even	6
336.2.q.g.193.2		4	84.83	odd	2
336.2.q.g.289.2		4	84.47	odd	6
504.2.s.i.289.1		4	7.5	odd	6
504.2.s.i.361.1		4	7.6	odd	2
1008.2.s.r.289.1		4	28.19	even	6
1008.2.s.r.865.1		4	28.27	even	2
1176.2.a.k.1.1		2	21.17	even	6
1176.2.a.n.1.2		2	21.11	odd	6
1176.2.q.l.361.1		4	3.2	odd	2
1176.2.q.l.961.1		4	21.2	odd	6
1344.2.q.w.193.1		4	168.125	even	2
1344.2.q.w.961.1		4	168.5	even	6
1344.2.q.x.193.1		4	168.83	odd	2
1344.2.q.x.961.1		4	168.131	odd	6
2352.2.a.ba.1.2		2	84.11	even	6
2352.2.a.bf.1.1		2	84.59	odd	6
2352.2.q.bf.961.1		4	84.23	even	6
2352.2.q.bf.1537.1		4	12.11	even	2
3528.2.a.bd.1.1		2	7.4	even	3
3528.2.a.bk.1.2		2	7.3	odd	6
3528.2.s.bk.361.2		4	1.1	even	1	trivial
3528.2.s.bk.3313.2		4	7.2	even	3	inner
7056.2.a.ch.1.1		2	28.11	odd	6
7056.2.a.cu.1.2		2	28.3	even	6
9408.2.a.dj.1.1		2	168.53	odd	6
9408.2.a.dp.1.2		2	168.59	odd	6
9408.2.a.dw.1.1		2	168.11	even	6
9408.2.a.ec.1.2		2	168.101	even	6

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:	\( N \)	\(=\)	\( 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3528.s (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.13746 + 3.70219i) q^{5} +O(q^{10})\)	\(q+(2.13746 + 3.70219i) q^{5} +(-2.13746 + 3.70219i) q^{11} +1.27492 q^{13} +(2.00000 - 3.46410i) q^{17} +(0.637459 + 1.10411i) q^{19} +(2.00000 + 3.46410i) q^{23} +(-6.63746 + 11.4964i) q^{25} +2.27492 q^{29} +(-0.500000 + 0.866025i) q^{31} +(-2.63746 - 4.56821i) q^{37} +10.5498 q^{41} -7.27492 q^{43} +(3.00000 + 5.19615i) q^{47} +(0.862541 - 1.49397i) q^{53} -18.2749 q^{55} +(-3.13746 + 5.43424i) q^{59} +(5.00000 + 8.66025i) q^{61} +(2.72508 + 4.71998i) q^{65} +(-3.63746 + 6.30026i) q^{67} -2.00000 q^{71} +(1.63746 - 2.83616i) q^{73} +(1.77492 + 3.07425i) q^{79} -0.274917 q^{83} +17.0997 q^{85} +(-2.27492 - 3.94027i) q^{89} +(-2.72508 + 4.71998i) q^{95} -16.2749 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{5}+O(q^{10}) \) 4 * q + q^5	\( 4 q + q^{5} - q^{11} - 10 q^{13} + 8 q^{17} - 5 q^{19} + 8 q^{23} - 19 q^{25} - 6 q^{29} - 2 q^{31} - 3 q^{37} + 12 q^{41} - 14 q^{43} + 12 q^{47} + 11 q^{53} - 58 q^{55} - 5 q^{59} + 20 q^{61} + 26 q^{65} - 7 q^{67} - 8 q^{71} - q^{73} - 8 q^{79} + 14 q^{83} + 8 q^{85} + 6 q^{89} - 26 q^{95} - 50 q^{97}+O(q^{100}) \) 4 * q + q^5 - q^11 - 10 * q^13 + 8 * q^17 - 5 * q^19 + 8 * q^23 - 19 * q^25 - 6 * q^29 - 2 * q^31 - 3 * q^37 + 12 * q^41 - 14 * q^43 + 12 * q^47 + 11 * q^53 - 58 * q^55 - 5 * q^59 + 20 * q^61 + 26 * q^65 - 7 * q^67 - 8 * q^71 - q^73 - 8 * q^79 + 14 * q^83 + 8 * q^85 + 6 * q^89 - 26 * q^95 - 50 * q^97

Introduction

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