LMFDB - Embedded newform 816.2.e.b.239.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [816,2,Mod(239,816)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(816, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("816.239");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 816 = 2^{4} \cdot 3 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	816.e (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:	$6.51579280494$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.40282095616.8
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{6} + 8x^{4} - 36x^{2} + 81 $ x^8 - 4x^6 + 8x^4 - 36*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			239.5
Root			$-1.71331 - 0.254137i$ of defining polynomial
Character	$\chi$	$=$	816.239
Dual form			816.2.e.b.239.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.254137 - 1.71331i) q^{3} -2.00000i q^{5} +0.508274i q^{7} +(-2.87083 - 0.870829i) q^{9} +O(q^{10})$	\(q+(0.254137 - 1.71331i) q^{3} -2.00000i q^{5} +0.508274i q^{7} +(-2.87083 - 0.870829i) q^{9} -4.44316 q^{11} -1.74166 q^{13} +(-3.42661 - 0.508274i) q^{15} -1.00000i q^{17} +1.01655i q^{19} +(0.870829 + 0.129171i) q^{21} -6.34495 q^{23} +1.00000 q^{25} +(-2.22158 + 4.69730i) q^{27} -9.48331i q^{29} +4.44316i q^{31} +(-1.12917 + 7.61249i) q^{33} +1.01655 q^{35} +5.48331 q^{37} +(-0.442620 + 2.98399i) q^{39} +5.48331i q^{41} -8.88632i q^{43} +(-1.74166 + 5.74166i) q^{45} -5.83667 q^{47} +6.74166 q^{49} +(-1.71331 - 0.254137i) q^{51} -2.00000i q^{53} +8.88632i q^{55} +(1.74166 + 0.258343i) q^{57} -1.01655 q^{59} -9.48331 q^{61} +(0.442620 - 1.45917i) q^{63} +3.48331i q^{65} -2.03310i q^{67} +(-1.61249 + 10.8708i) q^{69} -12.3129 q^{71} -6.00000 q^{73} +(0.254137 - 1.71331i) q^{75} -2.25834i q^{77} -9.26328i q^{79} +(7.48331 + 5.00000i) q^{81} +14.7230 q^{83} -2.00000 q^{85} +(-16.2478 - 2.41006i) q^{87} -9.22497i q^{89} -0.885239i q^{91} +(7.61249 + 1.12917i) q^{93} +2.03310 q^{95} +5.48331 q^{97} +(12.7555 + 3.86923i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^9	$ 8 q - 8 q^{9} + 16 q^{13} - 8 q^{21} + 8 q^{25} - 24 q^{33} - 16 q^{37} + 16 q^{45} + 24 q^{49} - 16 q^{57} - 16 q^{61} + 32 q^{69} - 48 q^{73} - 16 q^{85} + 16 q^{93} - 16 q^{97}+O(q^{100}) $ 8 * q - 8 * q^9 + 16 * q^13 - 8 * q^21 + 8 * q^25 - 24 * q^33 - 16 * q^37 + 16 * q^45 + 24 * q^49 - 16 * q^57 - 16 * q^61 + 32 * q^69 - 48 * q^73 - 16 * q^85 + 16 * q^93 - 16 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/816\mathbb{Z}\right)^\times$.

$n$	$241$	$511$	$545$	$613$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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
$3$	0.254137	−	1.71331i	0.146726	−	0.989177i
$3$	0.254137	−	1.71331i	0.146726	−	0.989177i
$4$		0			0
$5$		−	2.00000i		−	0.894427i	−0.894427	−	0.447214i	$-0.852416\pi$
$5$		−	2.00000i		−	0.894427i	0.894427	−	0.447214i	$-0.147584\pi$
$6$		0			0
$7$			0.508274i			0.192110i	0.995376	+	0.0960548i	$0.0306224\pi$
$7$			0.508274i			0.192110i	−0.995376	+	0.0960548i	$0.969378\pi$
$8$		0			0
$9$	−2.87083	−	0.870829i	−0.956943	−	0.290276i
$10$		0			0
$11$	−4.44316			−1.33966			−0.669831	−	0.742513i	$-0.733634\pi$
$11$	−4.44316			−1.33966			−0.669831	+	0.742513i	$0.733634\pi$
$12$		0			0
$13$	−1.74166			−0.483049			−0.241524	−	0.970395i	$-0.577647\pi$
$13$	−1.74166			−0.483049			−0.241524	+	0.970395i	$0.577647\pi$
$14$		0			0
$15$	−3.42661	−	0.508274i	−0.884747	−	0.131236i
$16$		0			0
$17$		−	1.00000i		−	0.242536i
$17$		−	1.00000i		−	0.242536i
$18$		0			0
$19$			1.01655i			0.233212i	0.993178	+	0.116606i	$0.0372015\pi$
$19$			1.01655i			0.233212i	−0.993178	+	0.116606i	$0.962798\pi$
$20$		0			0
$21$	0.870829	+	0.129171i	0.190030	+	0.0281875i
$22$		0			0
$23$	−6.34495			−1.32301			−0.661506	−	0.749940i	$-0.730083\pi$
$23$	−6.34495			−1.32301			−0.661506	+	0.749940i	$0.730083\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	−2.22158	+	4.69730i	−0.427543	+	0.903995i
$28$		0			0
$29$		−	9.48331i		−	1.76101i	−0.474039	−	0.880504i	$-0.657205\pi$
$29$		−	9.48331i		−	1.76101i	0.474039	−	0.880504i	$-0.342795\pi$
$30$		0			0
$31$			4.44316i			0.798015i	0.916948	+	0.399007i	$0.130645\pi$
$31$			4.44316i			0.798015i	−0.916948	+	0.399007i	$0.869355\pi$
$32$		0			0
$33$	−1.12917	+	7.61249i	−0.196563	+	1.32516i
$34$		0			0
$35$	1.01655			0.171828
$36$		0			0
$37$	5.48331			0.901451			0.450726	−	0.892663i	$-0.351165\pi$
$37$	5.48331			0.901451			0.450726	+	0.892663i	$0.351165\pi$
$38$		0			0
$39$	−0.442620	+	2.98399i	−0.0708759	+	0.477821i
$40$		0			0
$41$			5.48331i			0.856350i	0.903696	+	0.428175i	$0.140843\pi$
$41$			5.48331i			0.856350i	−0.903696	+	0.428175i	$0.859157\pi$
$42$		0			0
$43$		−	8.88632i		−	1.35515i	−0.735453	−	0.677575i	$-0.763031\pi$
$43$		−	8.88632i		−	1.35515i	0.735453	−	0.677575i	$-0.236969\pi$
$44$		0			0
$45$	−1.74166	+	5.74166i	−0.259631	+	0.855916i
$46$		0			0
$47$	−5.83667			−0.851366			−0.425683	−	0.904872i	$-0.639966\pi$
$47$	−5.83667			−0.851366			−0.425683	+	0.904872i	$0.639966\pi$
$48$		0			0
$49$	6.74166			0.963094
$50$		0			0
$51$	−1.71331	−	0.254137i	−0.239911	−	0.0355863i
$52$		0			0
$53$		−	2.00000i		−	0.274721i	−0.990521	−	0.137361i	$-0.956138\pi$
$53$		−	2.00000i		−	0.274721i	0.990521	−	0.137361i	$-0.0438619\pi$
$54$		0			0
$55$			8.88632i			1.19823i
$56$		0			0
$57$	1.74166	+	0.258343i	0.230688	+	0.0342183i
$58$		0			0
$59$	−1.01655			−0.132343			−0.0661717	−	0.997808i	$-0.521078\pi$
$59$	−1.01655			−0.132343			−0.0661717	+	0.997808i	$0.521078\pi$
$60$		0			0
$61$	−9.48331			−1.21421			−0.607107	−	0.794620i	$-0.707670\pi$
$61$	−9.48331			−1.21421			−0.607107	+	0.794620i	$0.707670\pi$
$62$		0			0
$63$	0.442620	−	1.45917i	0.0557648	−	0.183838i
$64$		0			0
$65$			3.48331i			0.432052i
$66$		0			0
$67$		−	2.03310i		−	0.248382i	−0.992258	−	0.124191i	$-0.960366\pi$
$67$		−	2.03310i		−	0.248382i	0.992258	−	0.124191i	$-0.0396336\pi$
$68$		0			0
$69$	−1.61249	+	10.8708i	−0.194121	+	1.30869i
$70$		0			0
$71$	−12.3129			−1.46128			−0.730638	−	0.682765i	$-0.760777\pi$
$71$	−12.3129			−1.46128			−0.730638	+	0.682765i	$0.760777\pi$
$72$		0			0
$73$	−6.00000			−0.702247			−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000			−0.702247			−0.351123	+	0.936329i	$0.614200\pi$
$74$		0			0
$75$	0.254137	−	1.71331i	0.0293452	−	0.197835i
$76$		0			0
$77$		−	2.25834i		−	0.257362i
$78$		0			0
$79$		−	9.26328i		−	1.04220i	−0.853495	−	0.521100i	$-0.825522\pi$
$79$		−	9.26328i		−	1.04220i	0.853495	−	0.521100i	$-0.174478\pi$
$80$		0			0
$81$	7.48331	+	5.00000i	0.831479	+	0.555556i
$82$		0			0
$83$	14.7230			1.61606			0.808029	−	0.589143i	$-0.200534\pi$
$83$	14.7230			1.61606			0.808029	+	0.589143i	$0.200534\pi$
$84$		0			0
$85$	−2.00000			−0.216930
$86$		0			0
$87$	−16.2478	−	2.41006i	−1.74195	−	0.258386i
$88$		0			0
$89$		−	9.22497i		−	0.977845i	−0.872327	−	0.488923i	$-0.837390\pi$
$89$		−	9.22497i		−	0.977845i	0.872327	−	0.488923i	$-0.162610\pi$
$90$		0			0
$91$		−	0.885239i		−	0.0927983i
$92$		0			0
$93$	7.61249	+	1.12917i	0.789378	+	0.117090i
$94$		0			0
$95$	2.03310			0.208591
$96$		0			0
$97$	5.48331			0.556746			0.278373	−	0.960473i	$-0.410205\pi$
$97$	5.48331			0.556746			0.278373	+	0.960473i	$0.410205\pi$
$98$		0			0
$99$	12.7555	+	3.86923i	1.28198	+	0.388872i
$100$		0			0
$101$		−	10.2583i		−	1.02074i	−0.859954	−	0.510372i	$-0.829508\pi$
$101$		−	10.2583i		−	1.02074i	0.859954	−	0.510372i	$-0.170492\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$	0.258343	−	1.74166i	0.0252117	−	0.169968i
$106$		0			0
$107$	−1.52482			−0.147410			−0.0737051	−	0.997280i	$-0.523482\pi$
$107$	−1.52482			−0.147410			−0.0737051	+	0.997280i	$0.523482\pi$
$108$		0			0
$109$	−9.48331			−0.908337			−0.454168	−	0.890916i	$-0.650064\pi$
$109$	−9.48331			−0.908337			−0.454168	+	0.890916i	$0.650064\pi$
$110$		0			0
$111$	1.39351	−	9.39459i	0.132266	−	0.891695i
$112$		0			0
$113$			2.00000i			0.188144i	0.995565	+	0.0940721i	$0.0299884\pi$
$113$			2.00000i			0.188144i	−0.995565	+	0.0940721i	$0.970012\pi$
$114$		0			0
$115$			12.6899i			1.18334i
$116$		0			0
$117$	5.00000	+	1.51669i	0.462250	+	0.140218i
$118$		0			0
$119$	0.508274			0.0465934
$120$		0			0
$121$	8.74166			0.794696
$122$		0			0
$123$	9.39459	+	1.39351i	0.847082	+	0.125649i
$124$		0			0
$125$		−	12.0000i		−	1.07331i
$126$		0			0
$127$		−	1.01655i		−	0.0902041i	−0.998982	−	0.0451020i	$-0.985639\pi$
$127$		−	1.01655i		−	0.0902041i	0.998982	−	0.0451020i	$-0.0143613\pi$
$128$		0			0
$129$	−15.2250	−	2.25834i	−1.34048	−	0.198836i
$130$		0			0
$131$	15.2313			1.33076			0.665381	−	0.746504i	$-0.268269\pi$
$131$	15.2313			1.33076			0.665381	+	0.746504i	$0.268269\pi$
$132$		0			0
$133$	−0.516685			−0.0448023
$134$		0			0
$135$	9.39459	+	4.44316i	0.808558	+	0.382406i
$136$		0			0
$137$		−	13.7417i		−	1.17403i	−0.809576	−	0.587015i	$-0.800303\pi$
$137$		−	13.7417i		−	1.17403i	0.809576	−	0.587015i	$-0.199697\pi$
$138$		0			0
$139$		−	9.39459i		−	0.796839i	−0.917203	−	0.398419i	$-0.869559\pi$
$139$		−	9.39459i		−	0.796839i	0.917203	−	0.398419i	$-0.130441\pi$
$140$		0			0
$141$	−1.48331	+	10.0000i	−0.124918	+	0.842152i
$142$		0			0
$143$	7.73846			0.647123
$144$		0			0
$145$	−18.9666			−1.57509
$146$		0			0
$147$	1.71331	−	11.5505i	0.141311	−	0.952670i
$148$		0			0
$149$		−	4.00000i		−	0.327693i	−0.986486	−	0.163846i	$-0.947610\pi$
$149$		−	4.00000i		−	0.327693i	0.986486	−	0.163846i	$-0.0523901\pi$
$150$		0			0
$151$			4.82012i			0.392256i	0.980578	+	0.196128i	$0.0628368\pi$
$151$			4.82012i			0.392256i	−0.980578	+	0.196128i	$0.937163\pi$
$152$		0			0
$153$	−0.870829	+	2.87083i	−0.0704023	+	0.232093i
$154$		0			0
$155$	8.88632			0.713766
$156$		0			0
$157$	12.0000			0.957704			0.478852	−	0.877896i	$-0.341053\pi$
$157$	12.0000			0.957704			0.478852	+	0.877896i	$0.341053\pi$
$158$		0			0
$159$	−3.42661	−	0.508274i	−0.271748	−	0.0403088i
$160$		0			0
$161$		−	3.22497i		−	0.254163i
$162$		0			0
$163$			20.0514i			1.57055i	0.619150	+	0.785273i	$0.287477\pi$
$163$			20.0514i			1.57055i	−0.619150	+	0.785273i	$0.712523\pi$
$164$		0			0
$165$	15.2250	+	2.25834i	1.18526	+	0.175812i
$166$		0			0
$167$	5.45971			0.422485			0.211242	−	0.977434i	$-0.432249\pi$
$167$	5.45971			0.422485			0.211242	+	0.977434i	$0.432249\pi$
$168$		0			0
$169$	−9.96663			−0.766664
$170$		0			0
$171$	0.885239	−	2.91834i	0.0676960	−	0.223171i
$172$		0			0
$173$		−	9.48331i		−	0.721003i	−0.932759	−	0.360502i	$-0.882606\pi$
$173$		−	9.48331i		−	0.721003i	0.932759	−	0.360502i	$-0.117394\pi$
$174$		0			0
$175$			0.508274i			0.0384219i
$176$		0			0
$177$	−0.258343	+	1.74166i	−0.0194182	+	0.130911i
$178$		0			0
$179$	18.5266			1.38474			0.692370	−	0.721542i	$-0.256567\pi$
$179$	18.5266			1.38474			0.692370	+	0.721542i	$0.256567\pi$
$180$		0			0
$181$	−13.4833			−1.00221			−0.501103	−	0.865387i	$-0.667072\pi$
$181$	−13.4833			−1.00221			−0.501103	+	0.865387i	$0.667072\pi$
$182$		0			0
$183$	−2.41006	+	16.2478i	−0.178157	+	1.20107i
$184$		0			0
$185$		−	10.9666i		−	0.806283i
$186$		0			0
$187$			4.44316i			0.324916i
$188$		0			0
$189$	−2.38751	−	1.12917i	−0.173666	−	0.0821351i
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$	14.0000			1.00774			0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000			1.00774			0.503871	+	0.863779i	$0.331909\pi$
$194$		0			0
$195$	5.96798	+	0.885239i	0.427376	+	0.0633933i
$196$		0			0
$197$			20.4499i			1.45700i	0.685047	+	0.728499i	$0.259782\pi$
$197$			20.4499i			1.45700i	−0.685047	+	0.728499i	$0.740218\pi$
$198$		0			0
$199$		−	22.2158i		−	1.57484i	−0.616420	−	0.787418i	$-0.711417\pi$
$199$		−	22.2158i		−	1.57484i	0.616420	−	0.787418i	$-0.288583\pi$
$200$		0			0
$201$	−3.48331	−	0.516685i	−0.245694	−	0.0364442i
$202$		0			0
$203$	4.82012			0.338306
$204$		0			0
$205$	10.9666			0.765943
$206$		0			0
$207$	18.2153	+	5.52536i	1.26605	+	0.384039i
$208$		0			0
$209$		−	4.51669i		−	0.312426i
$210$		0			0
$211$		−	19.1661i		−	1.31945i	−0.751506	−	0.659726i	$-0.770672\pi$
$211$		−	19.1661i		−	1.31945i	0.751506	−	0.659726i	$-0.229328\pi$
$212$		0			0
$213$	−3.12917	+	21.0958i	−0.214407	+	1.44546i
$214$		0			0
$215$	−17.7726			−1.21208
$216$		0			0
$217$	−2.25834			−0.153306
$218$		0			0
$219$	−1.52482	+	10.2798i	−0.103038	+	0.694647i
$220$		0			0
$221$			1.74166i			0.117157i
$222$		0			0
$223$		−	28.4294i		−	1.90378i	−0.306447	−	0.951888i	$-0.599140\pi$
$223$		−	28.4294i		−	1.90378i	0.306447	−	0.951888i	$-0.400860\pi$
$224$		0			0
$225$	−2.87083	−	0.870829i	−0.191389	−	0.0580552i
$226$		0			0
$227$	−15.3626			−1.01965			−0.509825	−	0.860278i	$-0.670290\pi$
$227$	−15.3626			−1.01965			−0.509825	+	0.860278i	$0.670290\pi$
$228$		0			0
$229$	5.74166			0.379419			0.189710	−	0.981840i	$-0.439245\pi$
$229$	5.74166			0.379419			0.189710	+	0.981840i	$0.439245\pi$
$230$		0			0
$231$	−3.86923	−	0.573929i	−0.254577	−	0.0377617i
$232$		0			0
$233$			24.9666i			1.63562i	0.575490	+	0.817809i	$0.304812\pi$
$233$			24.9666i			1.63562i	−0.575490	+	0.817809i	$0.695188\pi$
$234$		0			0
$235$			11.6733i			0.761485i
$236$		0			0
$237$	−15.8708	−	2.35414i	−1.03092	−	0.152918i
$238$		0			0
$239$	11.6733			0.755086			0.377543	−	0.925992i	$-0.376769\pi$
$239$	11.6733			0.755086			0.377543	+	0.925992i	$0.376769\pi$
$240$		0			0
$241$	24.9666			1.60824			0.804121	−	0.594466i	$-0.202636\pi$
$241$	24.9666			1.60824			0.804121	+	0.594466i	$0.202636\pi$
$242$		0			0
$243$	10.4683	−	11.5505i	0.671543	−	0.740966i
$244$		0			0
$245$		−	13.4833i		−	0.861417i
$246$		0			0
$247$		−	1.77048i		−	0.112653i
$248$		0			0
$249$	3.74166	−	25.2250i	0.237118	−	1.59857i
$250$		0			0
$251$	19.8057			1.25013			0.625063	−	0.780574i	$-0.285073\pi$
$251$	19.8057			1.25013			0.625063	+	0.780574i	$0.285073\pi$
$252$		0			0
$253$	28.1916			1.77239
$254$		0			0
$255$	−0.508274	+	3.42661i	−0.0318294	+	0.214583i
$256$		0			0
$257$			14.2583i			0.889411i	0.895677	+	0.444705i	$0.146692\pi$
$257$			14.2583i			0.889411i	−0.895677	+	0.444705i	$0.853308\pi$
$258$		0			0
$259$			2.78703i			0.173177i
$260$		0			0
$261$	−8.25834	+	27.2250i	−0.511179	+	1.68518i
$262$		0			0
$263$	10.9194			0.673320			0.336660	−	0.941626i	$-0.390703\pi$
$263$	10.9194			0.673320			0.336660	+	0.941626i	$0.390703\pi$
$264$		0			0
$265$	−4.00000			−0.245718
$266$		0			0
$267$	−15.8052	−	2.34441i	−0.967262	−	0.143475i
$268$		0			0
$269$		−	8.96663i		−	0.546705i	−0.961914	−	0.273353i	$-0.911867\pi$
$269$		−	8.96663i		−	0.546705i	0.961914	−	0.273353i	$-0.0881326\pi$
$270$		0			0
$271$			21.5762i			1.31066i	0.755342	+	0.655331i	$0.227471\pi$
$271$			21.5762i			1.31066i	−0.755342	+	0.655331i	$0.772529\pi$
$272$		0			0
$273$	−1.51669	−	0.224972i	−0.0917940	−	0.0136159i
$274$		0			0
$275$	−4.44316			−0.267933
$276$		0			0
$277$	−24.9666			−1.50010			−0.750050	−	0.661381i	$-0.769970\pi$
$277$	−24.9666			−1.50010			−0.750050	+	0.661381i	$0.769970\pi$
$278$		0			0
$279$	3.86923	−	12.7555i	0.231645	−	0.763655i
$280$		0			0
$281$			2.96663i			0.176974i	0.996077	+	0.0884871i	$0.0282032\pi$
$281$			2.96663i			0.176974i	−0.996077	+	0.0884871i	$0.971797\pi$
$282$		0			0
$283$		−	21.0679i		−	1.25236i	−0.779679	−	0.626179i	$-0.784618\pi$
$283$		−	21.0679i		−	1.25236i	0.779679	−	0.626179i	$-0.215382\pi$
$284$		0			0
$285$	0.516685	−	3.48331i	0.0306058	−	0.206334i
$286$		0			0
$287$	−2.78703			−0.164513
$288$		0			0
$289$	−1.00000			−0.0588235
$290$		0			0
$291$	1.39351	−	9.39459i	0.0816892	−	0.550721i
$292$		0			0
$293$			26.9666i			1.57541i	0.616054	+	0.787704i	$0.288730\pi$
$293$			26.9666i			1.57541i	−0.616054	+	0.787704i	$0.711270\pi$
$294$		0			0
$295$			2.03310i			0.118371i
$296$		0			0
$297$	9.87083	−	20.8708i	0.572764	−	1.21105i
$298$		0			0
$299$	11.0507			0.639080
$300$		0			0
$301$	4.51669			0.260337
$302$		0			0
$303$	−17.5757	−	2.60703i	−1.00970	−	0.149770i
$304$		0			0
$305$			18.9666i			1.08603i
$306$		0			0
$307$		−	19.5431i		−	1.11538i	−0.830048	−	0.557692i	$-0.811687\pi$
$307$		−	19.5431i		−	1.11538i	0.830048	−	0.557692i	$-0.188313\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−19.1661			−1.08681			−0.543406	−	0.839470i	$-0.682866\pi$
$311$	−19.1661			−1.08681			−0.543406	+	0.839470i	$0.682866\pi$
$312$		0			0
$313$	−20.9666			−1.18510			−0.592552	−	0.805532i	$-0.701880\pi$
$313$	−20.9666			−1.18510			−0.592552	+	0.805532i	$0.701880\pi$
$314$		0			0
$315$	−2.91834	−	0.885239i	−0.164430	−	0.0498776i
$316$		0			0
$317$			5.48331i			0.307974i	0.988073	+	0.153987i	$0.0492113\pi$
$317$			5.48331i			0.307974i	−0.988073	+	0.153987i	$0.950789\pi$
$318$		0			0
$319$			42.1359i			2.35916i
$320$		0			0
$321$	−0.387514	+	2.61249i	−0.0216289	+	0.145815i
$322$		0			0
$323$	1.01655			0.0565623
$324$		0			0
$325$	−1.74166			−0.0966098
$326$		0			0
$327$	−2.41006	+	16.2478i	−0.133277	+	0.898506i
$328$		0			0
$329$		−	2.96663i		−	0.163556i
$330$		0			0
$331$			28.6920i			1.57706i	0.614998	+	0.788529i	$0.289157\pi$
$331$			28.6920i			1.57706i	−0.614998	+	0.788529i	$0.710843\pi$
$332$		0			0
$333$	−15.7417	−	4.77503i	−0.862638	−	0.261670i
$334$		0			0
$335$	−4.06619			−0.222160
$336$		0			0
$337$	−16.9666			−0.924231			−0.462116	−	0.886820i	$-0.652909\pi$
$337$	−16.9666			−0.924231			−0.462116	+	0.886820i	$0.652909\pi$
$338$		0			0
$339$	3.42661	+	0.508274i	0.186108	+	0.0276057i
$340$		0			0
$341$		−	19.7417i		−	1.06907i
$342$		0			0
$343$			6.98453i			0.377129i
$344$		0			0
$345$	21.7417	+	3.22497i	1.17053	+	0.173627i
$346$		0			0
$347$	−31.8560			−1.71012			−0.855061	−	0.518528i	$-0.826480\pi$
$347$	−31.8560			−1.71012			−0.855061	+	0.518528i	$0.826480\pi$
$348$		0			0
$349$	8.00000			0.428230			0.214115	−	0.976808i	$-0.431313\pi$
$349$	8.00000			0.428230			0.214115	+	0.976808i	$0.431313\pi$
$350$		0			0
$351$	3.86923	−	8.18108i	0.206524	−	0.436674i
$352$		0			0
$353$		−	22.9666i		−	1.22239i	−0.791480	−	0.611195i	$-0.790689\pi$
$353$		−	22.9666i		−	1.22239i	0.791480	−	0.611195i	$-0.209311\pi$
$354$		0			0
$355$			24.6259i			1.30700i
$356$		0			0
$357$	0.129171	−	0.870829i	0.00683647	−	0.0460891i
$358$		0			0
$359$	−18.7892			−0.991655			−0.495828	−	0.868421i	$-0.665135\pi$
$359$	−18.7892			−0.991655			−0.495828	+	0.868421i	$0.665135\pi$
$360$		0			0
$361$	17.9666			0.945612
$362$		0			0
$363$	2.22158	−	14.9771i	0.116603	−	0.786095i
$364$		0			0
$365$			12.0000i			0.628109i
$366$		0			0
$367$			11.1651i			0.582812i	0.956599	+	0.291406i	$0.0941230\pi$
$367$			11.1651i			0.582812i	−0.956599	+	0.291406i	$0.905877\pi$
$368$		0			0
$369$	4.77503	−	15.7417i	0.248578	−	0.819478i
$370$		0			0
$371$	1.01655			0.0527766
$372$		0			0
$373$	21.2250			1.09899			0.549494	−	0.835498i	$-0.314821\pi$
$373$	21.2250			1.09899			0.549494	+	0.835498i	$0.314821\pi$
$374$		0			0
$375$	−20.5597	−	3.04964i	−1.06170	−	0.157483i
$376$		0			0
$377$			16.5167i			0.850653i
$378$		0			0
$379$		−	28.0525i		−	1.44096i	−0.693477	−	0.720479i	$-0.743922\pi$
$379$		−	28.0525i		−	1.44096i	0.693477	−	0.720479i	$-0.256078\pi$
$380$		0			0
$381$	−1.74166	−	0.258343i	−0.0892278	−	0.0132353i
$382$		0			0
$383$	−20.5597			−1.05055			−0.525275	−	0.850933i	$-0.676037\pi$
$383$	−20.5597			−1.05055			−0.525275	+	0.850933i	$0.676037\pi$
$384$		0			0
$385$	−4.51669			−0.230192
$386$		0			0
$387$	−7.73846	+	25.5111i	−0.393368	+	1.29680i
$388$		0			0
$389$		−	21.2250i		−	1.07615i	−0.842897	−	0.538074i	$-0.819152\pi$
$389$		−	21.2250i		−	1.07615i	0.842897	−	0.538074i	$-0.180848\pi$
$390$		0			0
$391$			6.34495i			0.320878i
$392$		0			0
$393$	3.87083	−	26.0958i	0.195257	−	1.31636i
$394$		0			0
$395$	−18.5266			−0.932173
$396$		0			0
$397$	−1.48331			−0.0744454			−0.0372227	−	0.999307i	$-0.511851\pi$
$397$	−1.48331			−0.0744454			−0.0372227	+	0.999307i	$0.511851\pi$
$398$		0			0
$399$	−0.131309	+	0.885239i	−0.00657367	+	0.0443174i
$400$		0			0
$401$			14.5167i			0.724929i	0.931998	+	0.362464i	$0.118065\pi$
$401$			14.5167i			0.724929i	−0.931998	+	0.362464i	$0.881935\pi$
$402$		0			0
$403$		−	7.73846i		−	0.385480i
$404$		0			0
$405$	10.0000	−	14.9666i	0.496904	−	0.743698i
$406$		0			0
$407$	−24.3632			−1.20764
$408$		0			0
$409$	−6.96663			−0.344478			−0.172239	−	0.985055i	$-0.555100\pi$
$409$	−6.96663			−0.344478			−0.172239	+	0.985055i	$0.555100\pi$
$410$		0			0
$411$	−23.5437	−	3.49226i	−1.16132	−	0.172261i
$412$		0			0
$413$		−	0.516685i		−	0.0254244i
$414$		0			0
$415$		−	29.4460i		−	1.44545i
$416$		0			0
$417$	−16.0958	−	2.38751i	−0.788215	−	0.116917i
$418$		0			0
$419$	23.8550			1.16539			0.582696	−	0.812691i	$-0.301998\pi$
$419$	23.8550			1.16539			0.582696	+	0.812691i	$0.301998\pi$
$420$		0			0
$421$	−14.2583			−0.694909			−0.347455	−	0.937697i	$-0.612954\pi$
$421$	−14.2583			−0.694909			−0.347455	+	0.937697i	$0.612954\pi$
$422$		0			0
$423$	16.7561	+	5.08274i	0.814709	+	0.247131i
$424$		0			0
$425$		−	1.00000i		−	0.0485071i
$426$		0			0
$427$		−	4.82012i		−	0.233262i
$428$		0			0
$429$	1.96663	−	13.2583i	0.0949498	−	0.640119i
$430$		0			0
$431$	38.8406			1.87088			0.935442	−	0.353480i	$-0.115002\pi$
$431$	38.8406			1.87088			0.935442	+	0.353480i	$0.115002\pi$
$432$		0			0
$433$	−6.77503			−0.325587			−0.162794	−	0.986660i	$-0.552050\pi$
$433$	−6.77503			−0.325587			−0.162794	+	0.986660i	$0.552050\pi$
$434$		0			0
$435$	−4.82012	+	32.4956i	−0.231107	+	1.55805i
$436$		0			0
$437$		−	6.44994i		−	0.308543i
$438$		0			0
$439$			20.3140i			0.969535i	0.874643	+	0.484767i	$0.161096\pi$
$439$			20.3140i			0.969535i	−0.874643	+	0.484767i	$0.838904\pi$
$440$		0			0
$441$	−19.3541	−	5.87083i	−0.921626	−	0.279563i
$442$		0			0
$443$	−25.6424			−1.21831			−0.609154	−	0.793052i	$-0.708491\pi$
$443$	−25.6424			−1.21831			−0.609154	+	0.793052i	$0.708491\pi$
$444$		0			0
$445$	−18.4499			−0.874611
$446$		0			0
$447$	−6.85322	−	1.01655i	−0.324146	−	0.0480811i
$448$		0			0
$449$			10.0000i			0.471929i	0.971762	+	0.235965i	$0.0758249\pi$
$449$			10.0000i			0.471929i	−0.971762	+	0.235965i	$0.924175\pi$
$450$		0			0
$451$		−	24.3632i		−	1.14722i
$452$		0			0
$453$	8.25834	+	1.22497i	0.388011	+	0.0575542i
$454$		0			0
$455$	−1.77048			−0.0830013
$456$		0			0
$457$	−12.7083			−0.594469			−0.297234	−	0.954805i	$-0.596064\pi$
$457$	−12.7083			−0.594469			−0.297234	+	0.954805i	$0.596064\pi$
$458$		0			0
$459$	4.69730	+	2.22158i	0.219251	+	0.103694i
$460$		0			0
$461$			2.00000i			0.0931493i	0.998915	+	0.0465746i	$0.0148305\pi$
$461$			2.00000i			0.0931493i	−0.998915	+	0.0465746i	$0.985169\pi$
$462$		0			0
$463$			13.7064i			0.636992i	0.947924	+	0.318496i	$0.103178\pi$
$463$			13.7064i			0.636992i	−0.947924	+	0.318496i	$0.896822\pi$
$464$		0			0
$465$	2.25834	−	15.2250i	0.104728	−	0.706041i
$466$		0			0
$467$	−36.2992			−1.67973			−0.839863	−	0.542798i	$-0.817365\pi$
$467$	−36.2992			−1.67973			−0.839863	+	0.542798i	$0.817365\pi$
$468$		0			0
$469$	1.03337			0.0477166
$470$		0			0
$471$	3.04964	−	20.5597i	0.140520	−	0.947339i
$472$		0			0
$473$			39.4833i			1.81544i
$474$		0			0
$475$			1.01655i			0.0466424i
$476$		0			0
$477$	−1.74166	+	5.74166i	−0.0797450	+	0.262892i
$478$		0			0
$479$	20.0514			0.916171			0.458086	−	0.888908i	$-0.348535\pi$
$479$	20.0514			0.916171			0.458086	+	0.888908i	$0.348535\pi$
$480$		0			0
$481$	−9.55006			−0.435445
$482$		0			0
$483$	−5.52536	−	0.819585i	−0.251413	−	0.0372924i
$484$		0			0
$485$		−	10.9666i		−	0.497969i
$486$		0			0
$487$		−	42.6441i		−	1.93239i	−0.257811	−	0.966195i	$-0.583001\pi$
$487$		−	42.6441i		−	1.93239i	0.257811	−	0.966195i	$-0.416999\pi$
$488$		0			0
$489$	34.3541	+	5.09580i	1.55355	+	0.230440i
$490$		0			0
$491$	−24.3632			−1.09950			−0.549749	−	0.835330i	$-0.685277\pi$
$491$	−24.3632			−1.09950			−0.549749	+	0.835330i	$0.685277\pi$
$492$		0			0
$493$	−9.48331			−0.427107
$494$		0			0
$495$	7.73846	−	25.5111i	0.347818	−	1.14664i
$496$		0			0
$497$		−	6.25834i		−	0.280725i
$498$		0			0
$499$		−	30.0856i		−	1.34681i	−0.739272	−	0.673407i	$-0.764830\pi$
$499$		−	30.0856i		−	1.34681i	0.739272	−	0.673407i	$-0.235170\pi$
$500$		0			0
$501$	1.38751	−	9.35414i	0.0619896	−	0.417912i
$502$		0			0
$503$	−2.67268			−0.119169			−0.0595844	−	0.998223i	$-0.518978\pi$
$503$	−2.67268			−0.119169			−0.0595844	+	0.998223i	$0.518978\pi$
$504$		0			0
$505$	−20.5167			−0.912981
$506$		0			0
$507$	−2.53289	+	17.0759i	−0.112490	+	0.758366i
$508$		0			0
$509$		−	10.9666i		−	0.486087i	−0.970015	−	0.243044i	$-0.921854\pi$
$509$		−	10.9666i		−	0.486087i	0.970015	−	0.243044i	$-0.0781458\pi$
$510$		0			0
$511$		−	3.04964i		−	0.134908i
$512$		0			0
$513$	−4.77503	−	2.25834i	−0.210823	−	0.0997083i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	25.9333			1.14054
$518$		0			0
$519$	−16.2478	−	2.41006i	−0.713200	−	0.105790i
$520$		0			0
$521$			33.4833i			1.46693i	0.679727	+	0.733465i	$0.262098\pi$
$521$			33.4833i			1.46693i	−0.679727	+	0.733465i	$0.737902\pi$
$522$		0			0
$523$			31.2165i			1.36500i	0.730885	+	0.682500i	$0.239107\pi$
$523$			31.2165i			1.36500i	−0.730885	+	0.682500i	$0.760893\pi$
$524$		0			0
$525$	0.870829	+	0.129171i	0.0380061	+	0.00563750i
$526$		0			0
$527$	4.44316			0.193547
$528$		0			0
$529$	17.2583			0.750363
$530$		0			0
$531$	2.91834	+	0.885239i	0.126645	+	0.0384161i
$532$		0			0
$533$		−	9.55006i		−	0.413659i
$534$		0			0
$535$			3.04964i			0.131848i
$536$		0			0
$537$	4.70829	−	31.7417i	0.203178	−	1.36975i
$538$		0			0
$539$	−29.9543			−1.29022
$540$		0			0
$541$	−1.48331			−0.0637727			−0.0318863	−	0.999492i	$-0.510151\pi$
$541$	−1.48331			−0.0637727			−0.0318863	+	0.999492i	$0.510151\pi$
$542$		0			0
$543$	−3.42661	+	23.1010i	−0.147050	+	0.991360i
$544$		0			0
$545$			18.9666i			0.812441i
$546$		0			0
$547$		−	1.39351i		−	0.0595823i	−0.999556	−	0.0297912i	$-0.990516\pi$
$547$		−	1.39351i		−	0.0595823i	0.999556	−	0.0297912i	$-0.00948423\pi$
$548$		0			0
$549$	27.2250	+	8.25834i	1.16193	+	0.352457i
$550$		0			0
$551$	9.64025			0.410688
$552$		0			0
$553$	4.70829			0.200217
$554$		0			0
$555$	−18.7892	−	2.78703i	−0.797556	−	0.118303i
$556$		0			0
$557$			36.7083i			1.55538i	0.628648	+	0.777690i	$0.283609\pi$
$557$			36.7083i			1.55538i	−0.628648	+	0.777690i	$0.716391\pi$
$558$		0			0
$559$			15.4769i			0.654604i
$560$		0			0
$561$	7.61249	+	1.12917i	0.321399	+	0.0476736i
$562$		0			0
$563$	14.4604			0.609432			0.304716	−	0.952443i	$-0.401438\pi$
$563$	14.4604			0.609432			0.304716	+	0.952443i	$0.401438\pi$
$564$		0			0
$565$	4.00000			0.168281
$566$		0			0
$567$	−2.54137	+	3.80358i	−0.106728	+	0.159735i
$568$		0			0
$569$			39.9333i			1.67409i	0.547135	+	0.837045i	$0.315719\pi$
$569$			39.9333i			1.67409i	−0.547135	+	0.837045i	$0.684281\pi$
$570$		0			0
$571$			46.5790i			1.94927i	0.223797	+	0.974636i	$0.428155\pi$
$571$			46.5790i			1.94927i	−0.223797	+	0.974636i	$0.571845\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−6.34495			−0.264603
$576$		0			0
$577$	10.7750			0.448570			0.224285	−	0.974524i	$-0.427995\pi$
$577$	10.7750			0.448570			0.224285	+	0.974524i	$0.427995\pi$
$578$		0			0
$579$	3.55792	−	23.9863i	0.147862	−	0.996835i
$580$		0			0
$581$			7.48331i			0.310460i
$582$		0			0
$583$			8.88632i			0.368034i
$584$		0			0
$585$	3.03337	−	10.0000i	0.125414	−	0.413449i
$586$		0			0
$587$	−28.6920			−1.18425			−0.592124	−	0.805847i	$-0.701710\pi$
$587$	−28.6920			−1.18425			−0.592124	+	0.805847i	$0.701710\pi$
$588$		0			0
$589$	−4.51669			−0.186107
$590$		0			0
$591$	35.0370	+	5.19709i	1.44123	+	0.213780i
$592$		0			0
$593$			43.9333i			1.80412i	0.431608	+	0.902061i	$0.357946\pi$
$593$			43.9333i			1.80412i	−0.431608	+	0.902061i	$0.642054\pi$
$594$		0			0
$595$		−	1.01655i		−	0.0416744i
$596$		0			0
$597$	−38.0624	−	5.64586i	−1.55779	−	0.231070i
$598$		0			0
$599$	27.6755			1.13079			0.565395	−	0.824820i	$-0.308724\pi$
$599$	27.6755			1.13079			0.565395	+	0.824820i	$0.308724\pi$
$600$		0			0
$601$	31.9333			1.30258			0.651292	−	0.758827i	$-0.274227\pi$
$601$	31.9333			1.30258			0.651292	+	0.758827i	$0.274227\pi$
$602$		0			0
$603$	−1.77048	+	5.83667i	−0.0720995	+	0.237688i
$604$		0			0
$605$		−	17.4833i		−	0.710798i
$606$		0			0
$607$		−	14.0834i		−	0.571628i	−0.958285	−	0.285814i	$-0.907736\pi$
$607$		−	14.0834i		−	0.571628i	0.958285	−	0.285814i	$-0.0922639\pi$
$608$		0			0
$609$	1.22497	−	8.25834i	0.0496384	−	0.334645i
$610$		0			0
$611$	10.1655			0.411251
$612$		0			0
$613$	10.0000			0.403896			0.201948	−	0.979396i	$-0.435273\pi$
$613$	10.0000			0.403896			0.201948	+	0.979396i	$0.435273\pi$
$614$		0			0
$615$	2.78703	−	18.7892i	0.112384	−	0.757653i
$616$		0			0
$617$			15.0334i			0.605221i	0.953114	+	0.302610i	$0.0978581\pi$
$617$			15.0334i			0.605221i	−0.953114	+	0.302610i	$0.902142\pi$
$618$		0			0
$619$		−	1.26220i		−	0.0507323i	−0.999678	−	0.0253661i	$-0.991925\pi$
$619$		−	1.26220i		−	0.0507323i	0.999678	−	0.0253661i	$-0.00807516\pi$
$620$		0			0
$621$	14.0958	−	29.8041i	0.565645	−	1.19600i
$622$		0			0
$623$	4.68881			0.187853
$624$		0			0
$625$	−19.0000			−0.760000
$626$		0			0
$627$	−7.73846	−	1.14786i	−0.309044	−	0.0458410i
$628$		0			0
$629$		−	5.48331i		−	0.218634i
$630$		0			0
$631$		−	34.2661i		−	1.36411i	−0.731300	−	0.682056i	$-0.761086\pi$
$631$		−	34.2661i		−	1.36411i	0.731300	−	0.682056i	$-0.238914\pi$
$632$		0			0
$633$	−32.8375	−	4.87083i	−1.30517	−	0.193598i
$634$		0			0
$635$	−2.03310			−0.0806810
$636$		0			0
$637$	−11.7417			−0.465221
$638$		0			0
$639$	35.3483	+	10.7225i	1.39836	+	0.424174i
$640$		0			0
$641$		−	32.9666i		−	1.30210i	−0.759033	−	0.651052i	$-0.774328\pi$
$641$		−	32.9666i		−	1.30210i	0.759033	−	0.651052i	$-0.225672\pi$
$642$		0			0
$643$			26.7733i			1.05584i	0.849295	+	0.527918i	$0.177027\pi$
$643$			26.7733i			1.05584i	−0.849295	+	0.527918i	$0.822973\pi$
$644$		0			0
$645$	−4.51669	+	30.4499i	−0.177844	+	1.19897i
$646$		0			0
$647$	23.8719			0.938502			0.469251	−	0.883065i	$-0.344524\pi$
$647$	23.8719			0.938502			0.469251	+	0.883065i	$0.344524\pi$
$648$		0			0
$649$	4.51669			0.177295
$650$		0			0
$651$	−0.573929	+	3.86923i	−0.0224940	+	0.151647i
$652$		0			0
$653$			45.4833i			1.77990i	0.456058	+	0.889950i	$0.349261\pi$
$653$			45.4833i			1.77990i	−0.456058	+	0.889950i	$0.650739\pi$
$654$		0			0
$655$		−	30.4625i		−	1.19027i
$656$		0			0
$657$	17.2250	+	5.22497i	0.672010	+	0.203846i
$658$		0			0
$659$	23.6093			0.919688			0.459844	−	0.888000i	$-0.347905\pi$
$659$	23.6093			0.919688			0.459844	+	0.888000i	$0.347905\pi$
$660$		0			0
$661$	−15.9333			−0.619732			−0.309866	−	0.950780i	$-0.600284\pi$
$661$	−15.9333			−0.619732			−0.309866	+	0.950780i	$0.600284\pi$
$662$		0			0
$663$	2.98399	+	0.442620i	0.115889	+	0.0171899i
$664$		0			0
$665$			1.03337i			0.0400724i
$666$		0			0
$667$			60.1711i			2.32984i
$668$		0			0
$669$	−48.7083	−	7.22497i	−1.88317	−	0.279334i
$670$		0			0
$671$	42.1359			1.62664
$672$		0			0
$673$	45.4833			1.75325			0.876626	−	0.481172i	$-0.159789\pi$
$673$	45.4833			1.75325			0.876626	+	0.481172i	$0.159789\pi$
$674$		0			0
$675$	−2.22158	+	4.69730i	−0.0855086	+	0.180799i
$676$		0			0
$677$			1.48331i			0.0570084i	0.999594	+	0.0285042i	$0.00907440\pi$
$677$			1.48331i			0.0570084i	−0.999594	+	0.0285042i	$0.990926\pi$
$678$		0			0
$679$			2.78703i			0.106956i
$680$		0			0
$681$	−3.90420	+	26.3208i	−0.149609	+	1.00861i
$682$		0			0
$683$	29.9543			1.14617			0.573084	−	0.819497i	$-0.305747\pi$
$683$	29.9543			1.14617			0.573084	+	0.819497i	$0.305747\pi$
$684$		0			0
$685$	−27.4833			−1.05008
$686$		0			0
$687$	1.45917	−	9.83721i	0.0556707	−	0.375313i
$688$		0			0
$689$			3.48331i			0.132704i
$690$		0			0
$691$		−	28.1838i		−	1.07216i	−0.844167	−	0.536081i	$-0.819904\pi$
$691$		−	28.1838i		−	1.07216i	0.844167	−	0.536081i	$-0.180096\pi$
$692$		0			0
$693$	−1.96663	+	6.48331i	−0.0747061	+	0.246281i
$694$		0			0
$695$	−18.7892			−0.712714
$696$		0			0
$697$	5.48331			0.207695
$698$		0			0
$699$	42.7755	+	6.34495i	1.61792	+	0.239988i
$700$		0			0
$701$		−	24.1916i		−	0.913704i	−0.889543	−	0.456852i	$-0.848977\pi$
$701$		−	24.1916i		−	0.913704i	0.889543	−	0.456852i	$-0.151023\pi$
$702$		0			0
$703$			5.57405i			0.210229i
$704$		0			0
$705$	20.0000	+	2.96663i	0.753244	+	0.111730i
$706$		0			0
$707$	5.21405			0.196095
$708$		0			0
$709$	−11.0334			−0.414367			−0.207183	−	0.978302i	$-0.566430\pi$
$709$	−11.0334			−0.414367			−0.207183	+	0.978302i	$0.566430\pi$
$710$		0			0
$711$	−8.06673	+	26.5933i	−0.302526	+	0.997327i
$712$		0			0
$713$		−	28.1916i		−	1.05578i
$714$		0			0
$715$		−	15.4769i		−	0.578804i
$716$		0			0
$717$	2.96663	−	20.0000i	0.110791	−	0.746914i
$718$		0			0
$719$	50.6452			1.88875			0.944374	−	0.328873i	$-0.106669\pi$
$719$	50.6452			1.88875			0.944374	+	0.328873i	$0.106669\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	6.34495	−	42.7755i	0.235971	−	1.59084i
$724$		0			0
$725$		−	9.48331i		−	0.352201i
$726$		0			0
$727$			39.0862i			1.44963i	0.688945	+	0.724814i	$0.258074\pi$
$727$			39.0862i			1.44963i	−0.688945	+	0.724814i	$0.741926\pi$
$728$		0			0
$729$	−17.1292	−	20.8708i	−0.634414	−	0.772994i
$730$		0			0
$731$	−8.88632			−0.328672
$732$		0			0
$733$	2.00000			0.0738717			0.0369358	−	0.999318i	$-0.488240\pi$
$733$	2.00000			0.0738717			0.0369358	+	0.999318i	$0.488240\pi$
$734$		0			0
$735$	−23.1010	−	3.42661i	−0.852094	−	0.126392i
$736$		0			0
$737$			9.03337i			0.332748i
$738$		0			0
$739$			0.753931i			0.0277338i	0.999904	+	0.0138669i	$0.00441411\pi$
$739$			0.753931i			0.0277338i	−0.999904	+	0.0138669i	$0.995586\pi$
$740$		0			0
$741$	−3.03337	−	0.449944i	−0.111434	−	0.0165291i
$742$		0			0
$743$	−17.5270			−0.643002			−0.321501	−	0.946909i	$-0.604187\pi$
$743$	−17.5270			−0.643002			−0.321501	+	0.946909i	$0.604187\pi$
$744$		0			0
$745$	−8.00000			−0.293097
$746$		0			0
$747$	−42.2672	−	12.8212i	−1.54648	−	0.469103i
$748$		0			0
$749$		−	0.775028i		−	0.0283189i
$750$		0			0
$751$		−	38.0866i		−	1.38980i	−0.719106	−	0.694901i	$-0.755448\pi$
$751$		−	38.0866i		−	1.38980i	0.719106	−	0.694901i	$-0.244552\pi$
$752$		0			0
$753$	5.03337	−	33.9333i	0.183426	−	1.23660i
$754$		0			0
$755$	9.64025			0.350845
$756$		0			0
$757$	33.2250			1.20758			0.603791	−	0.797143i	$-0.293656\pi$
$757$	33.2250			1.20758			0.603791	+	0.797143i	$0.293656\pi$
$758$		0			0
$759$	7.16453	−	48.3008i	0.260056	−	1.75321i
$760$		0			0
$761$			9.22497i			0.334405i	0.985923	+	0.167202i	$0.0534734\pi$
$761$			9.22497i			0.334405i	−0.985923	+	0.167202i	$0.946527\pi$
$762$		0			0
$763$		−	4.82012i		−	0.174500i
$764$		0			0
$765$	5.74166	+	1.74166i	0.207590	+	0.0629698i
$766$		0			0
$767$	1.77048			0.0639283
$768$		0			0
$769$	−8.70829			−0.314029			−0.157014	−	0.987596i	$-0.550187\pi$
$769$	−8.70829			−0.314029			−0.157014	+	0.987596i	$0.550187\pi$
$770$		0			0
$771$	24.4289	+	3.62357i	0.879785	+	0.130500i
$772$		0			0
$773$		−	36.1916i		−	1.30172i	−0.759197	−	0.650861i	$-0.774408\pi$
$773$		−	36.1916i		−	1.30172i	0.759197	−	0.650861i	$-0.225592\pi$
$774$		0			0
$775$			4.44316i			0.159603i
$776$		0			0
$777$	4.77503	+	0.708287i	0.171303	+	0.0254097i
$778$		0			0
$779$	−5.57405			−0.199711
$780$		0			0
$781$	54.7083			1.95762
$782$		0			0
$783$	44.5459	+	21.0679i	1.59194	+	0.752907i
$784$		0			0
$785$		−	24.0000i		−	0.856597i
$786$		0			0
$787$		−	2.27875i		−	0.0812288i	−0.999175	−	0.0406144i	$-0.987068\pi$
$787$		−	2.27875i		−	0.0812288i	0.999175	−	0.0406144i	$-0.0129315\pi$
$788$		0			0
$789$	2.77503	−	18.7083i	0.0987936	−	0.666033i
$790$		0			0
$791$	−1.01655			−0.0361443
$792$		0			0
$793$	16.5167			0.586525
$794$		0			0
$795$	−1.01655	+	6.85322i	−0.0360533	+	0.243059i
$796$		0			0
$797$			14.9666i			0.530145i	0.964228	+	0.265073i	$0.0853959\pi$
$797$			14.9666i			0.530145i	−0.964228	+	0.265073i	$0.914604\pi$
$798$		0			0
$799$			5.83667i			0.206487i
$800$		0			0
$801$	−8.03337	+	26.4833i	−0.283845	+	0.935742i
$802$		0			0
$803$	26.6590			0.940774
$804$		0			0
$805$	−6.44994			−0.227331
$806$		0			0
$807$	−15.3626	−	2.27875i	−0.540788	−	0.0802159i
$808$		0			0
$809$			23.0334i			0.809810i	0.914359	+	0.404905i	$0.132695\pi$
$809$			23.0334i			0.809810i	−0.914359	+	0.404905i	$0.867305\pi$
$810$		0			0
$811$		−	22.0845i		−	0.775491i	−0.921766	−	0.387746i	$-0.873254\pi$
$811$		−	22.0845i		−	0.775491i	0.921766	−	0.387746i	$-0.126746\pi$
$812$		0			0
$813$	36.9666	+	5.48331i	1.29648	+	0.192308i
$814$		0			0
$815$	40.1028			1.40474
$816$		0			0
$817$	9.03337			0.316038
$818$		0			0
$819$	−0.770892	+	2.54137i	−0.0269371	+	0.0888027i
$820$		0			0
$821$		−	52.4499i		−	1.83052i	−0.402869	−	0.915258i	$-0.631987\pi$
$821$		−	52.4499i		−	1.83052i	0.402869	−	0.915258i	$-0.368013\pi$
$822$		0			0
$823$		−	13.3295i		−	0.464636i	−0.972640	−	0.232318i	$-0.925369\pi$
$823$		−	13.3295i		−	0.464636i	0.972640	−	0.232318i	$-0.0746311\pi$
$824$		0			0
$825$	−1.12917	+	7.61249i	−0.0393127	+	0.265033i
$826$		0			0
$827$	−15.2313			−0.529643			−0.264821	−	0.964297i	$-0.585313\pi$
$827$	−15.2313			−0.529643			−0.264821	+	0.964297i	$0.585313\pi$
$828$		0			0
$829$	−20.0000			−0.694629			−0.347314	−	0.937749i	$-0.612906\pi$
$829$	−20.0000			−0.694629			−0.347314	+	0.937749i	$0.612906\pi$
$830$		0			0
$831$	−6.34495	+	42.7755i	−0.220104	+	1.48386i
$832$		0			0
$833$		−	6.74166i		−	0.233585i
$834$		0			0
$835$		−	10.9194i		−	0.377882i
$836$		0			0
$837$	−20.8708	−	9.87083i	−0.721401	−	0.341186i
$838$		0			0
$839$	21.9532			0.757908			0.378954	−	0.925416i	$-0.376284\pi$
$839$	21.9532			0.757908			0.378954	+	0.925416i	$0.376284\pi$
$840$		0			0
$841$	−60.9333			−2.10115
$842$		0			0
$843$	5.08274	+	0.753931i	0.175059	+	0.0259667i
$844$		0			0
$845$			19.9333i			0.685725i
$846$		0			0
$847$			4.44316i			0.152669i
$848$		0			0
$849$	−36.0958	−	5.35414i	−1.23880	−	0.183754i
$850$		0			0
$851$	−34.7913			−1.19263
$852$		0			0
$853$	−24.4499			−0.837150			−0.418575	−	0.908182i	$-0.637470\pi$
$853$	−24.4499			−0.837150			−0.418575	+	0.908182i	$0.637470\pi$
$854$		0			0
$855$	−5.83667	−	1.77048i	−0.199610	−	0.0605491i
$856$		0			0
$857$		−	22.5167i		−	0.769155i	−0.923093	−	0.384578i	$-0.874347\pi$
$857$		−	22.5167i		−	0.769155i	0.923093	−	0.384578i	$-0.125653\pi$
$858$		0			0
$859$		−	32.4956i		−	1.10874i	−0.832272	−	0.554368i	$-0.812960\pi$
$859$		−	32.4956i		−	1.10874i	0.832272	−	0.554368i	$-0.187040\pi$
$860$		0			0
$861$	−0.708287	+	4.77503i	−0.0241384	+	0.162733i
$862$		0			0
$863$	−3.80358			−0.129475			−0.0647376	−	0.997902i	$-0.520621\pi$
$863$	−3.80358			−0.129475			−0.0647376	+	0.997902i	$0.520621\pi$
$864$		0			0
$865$	−18.9666			−0.644885
$866$		0			0
$867$	−0.254137	+	1.71331i	−0.00863095	+	0.0581869i
$868$		0			0
$869$			41.1582i			1.39620i
$870$		0			0
$871$			3.54096i			0.119981i
$872$		0			0
$873$	−15.7417	−	4.77503i	−0.532774	−	0.161610i
$874$		0			0
$875$	6.09929			0.206194
$876$		0			0
$877$	50.3832			1.70132			0.850660	−	0.525716i	$-0.176203\pi$
$877$	50.3832			1.70132			0.850660	+	0.525716i	$0.176203\pi$
$878$		0			0
$879$	46.2021	+	6.85322i	1.55836	+	0.231153i
$880$		0			0
$881$		−	33.4833i		−	1.12808i	−0.825747	−	0.564041i	$-0.809246\pi$
$881$		−	33.4833i		−	1.12808i	0.825747	−	0.564041i	$-0.190754\pi$
$882$		0			0
$883$			9.90287i			0.333258i	0.986020	+	0.166629i	$0.0532882\pi$
$883$			9.90287i			0.333258i	−0.986020	+	0.166629i	$0.946712\pi$
$884$		0			0
$885$	3.48331	+	0.516685i	0.117090	+	0.0173682i
$886$		0			0
$887$	−18.0183			−0.604995			−0.302497	−	0.953150i	$-0.597820\pi$
$887$	−18.0183			−0.604995			−0.302497	+	0.953150i	$0.597820\pi$
$888$		0			0
$889$	0.516685			0.0173291
$890$		0			0
$891$	−33.2496	−	22.2158i	−1.11390	−	0.744257i
$892$		0			0
$893$		−	5.93326i		−	0.198549i
$894$		0			0
$895$		−	37.0531i		−	1.23855i
$896$		0			0
$897$	2.80840	−	18.9333i	0.0937697	−	0.632163i
$898$		0			0
$899$	42.1359			1.40531
$900$		0			0
$901$	−2.00000			−0.0666297
$902$		0			0
$903$	1.14786	−	7.73846i	0.0381983	−	0.257520i
$904$		0			0
$905$			26.9666i			0.896401i
$906$		0			0
$907$		−	29.6916i		−	0.985895i	−0.870059	−	0.492947i	$-0.835920\pi$
$907$		−	29.6916i		−	0.985895i	0.870059	−	0.492947i	$-0.164080\pi$
$908$		0			0
$909$	−8.93326	+	29.4499i	−0.296298	+	0.976793i
$910$		0			0
$911$	14.3460			0.475305			0.237652	−	0.971350i	$-0.423622\pi$
$911$	14.3460			0.475305			0.237652	+	0.971350i	$0.423622\pi$
$912$		0			0
$913$	−65.4166			−2.16497
$914$		0			0
$915$	32.4956	+	4.82012i	1.07427	+	0.159348i
$916$		0			0
$917$			7.74166i			0.255652i
$918$		0			0
$919$			47.2186i			1.55760i	0.627273	+	0.778799i	$0.284171\pi$
$919$			47.2186i			1.55760i	−0.627273	+	0.778799i	$0.715829\pi$
$920$		0			0
$921$	−33.4833	−	4.96663i	−1.10331	−	0.163656i
$922$		0			0
$923$	21.4449			0.705868
$924$		0			0
$925$	5.48331			0.180290
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	47.4166i		−	1.55569i	−0.628458	−	0.777844i	$-0.716314\pi$
$929$		−	47.4166i		−	1.55569i	0.628458	−	0.777844i	$-0.283686\pi$
$930$		0			0
$931$			6.85322i			0.224605i
$932$		0			0
$933$	−4.87083	+	32.8375i	−0.159464	+	1.07505i
$934$		0			0
$935$	8.88632			0.290614
$936$		0			0
$937$	33.9333			1.10855			0.554276	−	0.832333i	$-0.312995\pi$
$937$	33.9333			1.10855			0.554276	+	0.832333i	$0.312995\pi$
$938$		0			0
$939$	−5.32840	+	35.9222i	−0.173886	+	1.17228i
$940$		0			0
$941$			16.4499i			0.536253i	0.963384	+	0.268126i	$0.0864045\pi$
$941$			16.4499i			0.536253i	−0.963384	+	0.268126i	$0.913596\pi$
$942$		0			0
$943$		−	34.7913i		−	1.13296i
$944$		0			0
$945$	−2.25834	+	4.77503i	−0.0734639	+	0.155332i
$946$		0			0
$947$	−7.23019			−0.234949			−0.117475	−	0.993076i	$-0.537480\pi$
$947$	−7.23019			−0.234949			−0.117475	+	0.993076i	$0.537480\pi$
$948$		0			0
$949$	10.4499			0.339220
$950$		0			0
$951$	9.39459	+	1.39351i	0.304640	+	0.0451878i
$952$		0			0
$953$		−	12.7083i		−	0.411662i	−0.978588	−	0.205831i	$-0.934010\pi$
$953$		−	12.7083i		−	0.411662i	0.978588	−	0.205831i	$-0.0659897\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	72.1916	+	10.7083i	2.33362	+	0.346150i
$958$		0			0
$959$	6.98453			0.225542
$960$		0			0
$961$	11.2583			0.363172
$962$		0			0
$963$	4.37750	+	1.32786i	0.141063	+	0.0427897i
$964$		0			0
$965$		−	28.0000i		−	0.901352i
$966$		0			0
$967$			6.85322i			0.220385i	0.993910	+	0.110192i	$0.0351467\pi$
$967$			6.85322i			0.220385i	−0.993910	+	0.110192i	$0.964853\pi$
$968$		0			0
$969$	0.258343	−	1.74166i	0.00829916	−	0.0559501i
$970$		0			0
$971$	−44.4316			−1.42588			−0.712939	−	0.701226i	$-0.752636\pi$
$971$	−44.4316			−1.42588			−0.712939	+	0.701226i	$0.752636\pi$
$972$		0			0
$973$	4.77503			0.153080
$974$		0			0
$975$	−0.442620	+	2.98399i	−0.0141752	+	0.0955642i
$976$		0			0
$977$		−	6.00000i		−	0.191957i	−0.995383	−	0.0959785i	$-0.969402\pi$
$977$		−	6.00000i		−	0.191957i	0.995383	−	0.0959785i	$-0.0305980\pi$
$978$		0			0
$979$			40.9880i			1.30998i
$980$		0			0
$981$	27.2250	+	8.25834i	0.869226	+	0.263669i
$982$		0			0
$983$	20.4453			0.652104			0.326052	−	0.945352i	$-0.394282\pi$
$983$	20.4453			0.652104			0.326052	+	0.945352i	$0.394282\pi$
$984$		0			0
$985$	40.8999			1.30318
$986$		0			0
$987$	−5.08274	−	0.753931i	−0.161785	−	0.0239979i
$988$		0			0
$989$			56.3832i			1.79288i
$990$		0			0
$991$			26.9046i			0.854653i	0.904097	+	0.427327i	$0.140545\pi$
$991$			26.9046i			0.854653i	−0.904097	+	0.427327i	$0.859455\pi$
$992$		0			0
$993$	49.1582	+	7.29171i	1.55999	+	0.231396i
$994$		0			0
$995$	−44.4316			−1.40858
$996$		0			0
$997$	−48.9666			−1.55079			−0.775394	−	0.631477i	$-0.782449\pi$
$997$	−48.9666			−1.55079			−0.775394	+	0.631477i	$0.782449\pi$
$998$		0			0
$999$	−12.1816	+	25.7568i	−0.385409	+	0.814908i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	816.2.e.b.239.5	yes	8
3.2	odd	2	inner	816.2.e.b.239.3	✓	8
4.3	odd	2	inner	816.2.e.b.239.4	yes	8
12.11	even	2	inner	816.2.e.b.239.6	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
816.2.e.b.239.3	✓	8	3.2	odd	2	inner
816.2.e.b.239.4	yes	8	4.3	odd	2	inner
816.2.e.b.239.5	yes	8	1.1	even	1	trivial
816.2.e.b.239.6	yes	8	12.11	even	2	inner

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:	\( N \)	\(=\)	\( 816 = 2^{4} \cdot 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	816.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(0.254137 - 1.71331i) q^{3} -2.00000i q^{5} +0.508274i q^{7} +(-2.87083 - 0.870829i) q^{9} +O(q^{10})\)	\(q+(0.254137 - 1.71331i) q^{3} -2.00000i q^{5} +0.508274i q^{7} +(-2.87083 - 0.870829i) q^{9} -4.44316 q^{11} -1.74166 q^{13} +(-3.42661 - 0.508274i) q^{15} -1.00000i q^{17} +1.01655i q^{19} +(0.870829 + 0.129171i) q^{21} -6.34495 q^{23} +1.00000 q^{25} +(-2.22158 + 4.69730i) q^{27} -9.48331i q^{29} +4.44316i q^{31} +(-1.12917 + 7.61249i) q^{33} +1.01655 q^{35} +5.48331 q^{37} +(-0.442620 + 2.98399i) q^{39} +5.48331i q^{41} -8.88632i q^{43} +(-1.74166 + 5.74166i) q^{45} -5.83667 q^{47} +6.74166 q^{49} +(-1.71331 - 0.254137i) q^{51} -2.00000i q^{53} +8.88632i q^{55} +(1.74166 + 0.258343i) q^{57} -1.01655 q^{59} -9.48331 q^{61} +(0.442620 - 1.45917i) q^{63} +3.48331i q^{65} -2.03310i q^{67} +(-1.61249 + 10.8708i) q^{69} -12.3129 q^{71} -6.00000 q^{73} +(0.254137 - 1.71331i) q^{75} -2.25834i q^{77} -9.26328i q^{79} +(7.48331 + 5.00000i) q^{81} +14.7230 q^{83} -2.00000 q^{85} +(-16.2478 - 2.41006i) q^{87} -9.22497i q^{89} -0.885239i q^{91} +(7.61249 + 1.12917i) q^{93} +2.03310 q^{95} +5.48331 q^{97} +(12.7555 + 3.86923i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^9	\( 8 q - 8 q^{9} + 16 q^{13} - 8 q^{21} + 8 q^{25} - 24 q^{33} - 16 q^{37} + 16 q^{45} + 24 q^{49} - 16 q^{57} - 16 q^{61} + 32 q^{69} - 48 q^{73} - 16 q^{85} + 16 q^{93} - 16 q^{97}+O(q^{100}) \) 8 * q - 8 * q^9 + 16 * q^13 - 8 * q^21 + 8 * q^25 - 24 * q^33 - 16 * q^37 + 16 * q^45 + 24 * q^49 - 16 * q^57 - 16 * q^61 + 32 * q^69 - 48 * q^73 - 16 * q^85 + 16 * q^93 - 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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