LMFDB - Embedded newform 816.2.e.b.239.4

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			239.4
Root			$1.71331 + 0.254137i$ of defining polynomial
Character	$\chi$	$=$	816.239
Dual form			816.2.e.b.239.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.254137 + 1.71331i) q^{3} -2.00000i q^{5} -0.508274i q^{7} +(-2.87083 - 0.870829i) q^{9} +O(q^{10})$	\(q+(-0.254137 + 1.71331i) q^{3} -2.00000i q^{5} -0.508274i q^{7} +(-2.87083 - 0.870829i) q^{9} +4.44316 q^{11} -1.74166 q^{13} +(3.42661 + 0.508274i) q^{15} -1.00000i q^{17} -1.01655i q^{19} +(0.870829 + 0.129171i) q^{21} +6.34495 q^{23} +1.00000 q^{25} +(2.22158 - 4.69730i) q^{27} -9.48331i q^{29} -4.44316i q^{31} +(-1.12917 + 7.61249i) q^{33} -1.01655 q^{35} +5.48331 q^{37} +(0.442620 - 2.98399i) q^{39} +5.48331i q^{41} +8.88632i q^{43} +(-1.74166 + 5.74166i) q^{45} +5.83667 q^{47} +6.74166 q^{49} +(1.71331 + 0.254137i) q^{51} -2.00000i q^{53} -8.88632i q^{55} +(1.74166 + 0.258343i) q^{57} +1.01655 q^{59} -9.48331 q^{61} +(-0.442620 + 1.45917i) q^{63} +3.48331i q^{65} +2.03310i q^{67} +(-1.61249 + 10.8708i) q^{69} +12.3129 q^{71} -6.00000 q^{73} +(-0.254137 + 1.71331i) q^{75} -2.25834i q^{77} +9.26328i q^{79} +(7.48331 + 5.00000i) q^{81} -14.7230 q^{83} -2.00000 q^{85} +(16.2478 + 2.41006i) q^{87} -9.22497i q^{89} +0.885239i q^{91} +(7.61249 + 1.12917i) q^{93} -2.03310 q^{95} +5.48331 q^{97} +(-12.7555 - 3.86923i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$-0.969378\pi$
$7$		−	0.508274i		−	0.192110i	0.995376	−	0.0960548i	$-0.0306224\pi$
$8$		0			0
$9$	−2.87083	−	0.870829i	−0.956943	−	0.290276i
$10$		0			0
$11$	4.44316			1.33966			0.669831	−	0.742513i	$-0.266366\pi$
$11$	4.44316			1.33966			0.669831	+	0.742513i	$0.266366\pi$
$12$		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	$-0.962798\pi$
$19$		−	1.01655i		−	0.233212i	0.993178	−	0.116606i	$-0.0372015\pi$
$20$		0			0
$21$	0.870829	+	0.129171i	0.190030	+	0.0281875i
$22$		0			0
$23$	6.34495			1.32301			0.661506	−	0.749940i	$-0.269917\pi$
$23$	6.34495			1.32301			0.661506	+	0.749940i	$0.269917\pi$
$24$		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	$-0.869355\pi$
$31$		−	4.44316i		−	0.798015i	0.916948	−	0.399007i	$-0.130645\pi$
$32$		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	$0.236969\pi$
$43$			8.88632i			1.35515i	−0.735453	+	0.677575i	$0.763031\pi$
$44$		0			0
$45$	−1.74166	+	5.74166i	−0.259631	+	0.855916i
$46$		0			0
$47$	5.83667			0.851366			0.425683	−	0.904872i	$-0.360034\pi$
$47$	5.83667			0.851366			0.425683	+	0.904872i	$0.360034\pi$
$48$		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	$-0.478922\pi$
$59$	1.01655			0.132343			0.0661717	+	0.997808i	$0.478922\pi$
$60$		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	$0.0396336\pi$
$67$			2.03310i			0.248382i	−0.992258	+	0.124191i	$0.960366\pi$
$68$		0			0
$69$	−1.61249	+	10.8708i	−0.194121	+	1.30869i
$70$		0			0
$71$	12.3129			1.46128			0.730638	−	0.682765i	$-0.239223\pi$
$71$	12.3129			1.46128			0.730638	+	0.682765i	$0.239223\pi$
$72$		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	$0.174478\pi$
$79$			9.26328i			1.04220i	−0.853495	+	0.521100i	$0.825522\pi$
$80$		0			0
$81$	7.48331	+	5.00000i	0.831479	+	0.555556i
$82$		0			0
$83$	−14.7230			−1.61606			−0.808029	−	0.589143i	$-0.799466\pi$
$83$	−14.7230			−1.61606			−0.808029	+	0.589143i	$0.799466\pi$
$84$		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	$-0.476518\pi$
$107$	1.52482			0.147410			0.0737051	+	0.997280i	$0.476518\pi$
$108$		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	$0.0143613\pi$
$127$			1.01655i			0.0902041i	−0.998982	+	0.0451020i	$0.985639\pi$
$128$		0			0
$129$	−15.2250	−	2.25834i	−1.34048	−	0.198836i
$130$		0			0
$131$	−15.2313			−1.33076			−0.665381	−	0.746504i	$-0.731731\pi$
$131$	−15.2313			−1.33076			−0.665381	+	0.746504i	$0.731731\pi$
$132$		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	$0.130441\pi$
$139$			9.39459i			0.796839i	−0.917203	+	0.398419i	$0.869559\pi$
$140$		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	$-0.937163\pi$
$151$		−	4.82012i		−	0.392256i	0.980578	−	0.196128i	$-0.0628368\pi$
$152$		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	$-0.712523\pi$
$163$		−	20.0514i		−	1.57055i	0.619150	−	0.785273i	$-0.287477\pi$
$164$		0			0
$165$	15.2250	+	2.25834i	1.18526	+	0.175812i
$166$		0			0
$167$	−5.45971			−0.422485			−0.211242	−	0.977434i	$-0.567751\pi$
$167$	−5.45971			−0.422485			−0.211242	+	0.977434i	$0.567751\pi$
$168$		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	$-0.743433\pi$
$179$	−18.5266			−1.38474			−0.692370	+	0.721542i	$0.743433\pi$
$180$		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	$0.288583\pi$
$199$			22.2158i			1.57484i	−0.616420	+	0.787418i	$0.711417\pi$
$200$		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	$0.229328\pi$
$211$			19.1661i			1.31945i	−0.751506	+	0.659726i	$0.770672\pi$
$212$		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	$0.400860\pi$
$223$			28.4294i			1.90378i	−0.306447	+	0.951888i	$0.599140\pi$
$224$		0			0
$225$	−2.87083	−	0.870829i	−0.191389	−	0.0580552i
$226$		0			0
$227$	15.3626			1.01965			0.509825	−	0.860278i	$-0.329710\pi$
$227$	15.3626			1.01965			0.509825	+	0.860278i	$0.329710\pi$
$228$		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	$-0.623231\pi$
$239$	−11.6733			−0.755086			−0.377543	+	0.925992i	$0.623231\pi$
$240$		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	$-0.714927\pi$
$251$	−19.8057			−1.25013			−0.625063	+	0.780574i	$0.714927\pi$
$252$		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	$-0.609297\pi$
$263$	−10.9194			−0.673320			−0.336660	+	0.941626i	$0.609297\pi$
$264$		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	$-0.772529\pi$
$271$		−	21.5762i		−	1.31066i	0.755342	−	0.655331i	$-0.227471\pi$
$272$		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	$0.215382\pi$
$283$			21.0679i			1.25236i	−0.779679	+	0.626179i	$0.784618\pi$
$284$		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	$0.188313\pi$
$307$			19.5431i			1.11538i	−0.830048	+	0.557692i	$0.811687\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	19.1661			1.08681			0.543406	−	0.839470i	$-0.317134\pi$
$311$	19.1661			1.08681			0.543406	+	0.839470i	$0.317134\pi$
$312$		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	$-0.710843\pi$
$331$		−	28.6920i		−	1.57706i	0.614998	−	0.788529i	$-0.289157\pi$
$332$		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	$-0.173520\pi$
$347$	31.8560			1.71012			0.855061	+	0.518528i	$0.173520\pi$
$348$		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	$-0.334865\pi$
$359$	18.7892			0.991655			0.495828	+	0.868421i	$0.334865\pi$
$360$		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	$-0.905877\pi$
$367$		−	11.1651i		−	0.582812i	0.956599	−	0.291406i	$-0.0941230\pi$
$368$		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	$0.256078\pi$
$379$			28.0525i			1.44096i	−0.693477	+	0.720479i	$0.743922\pi$
$380$		0			0
$381$	−1.74166	−	0.258343i	−0.0892278	−	0.0132353i
$382$		0			0
$383$	20.5597			1.05055			0.525275	−	0.850933i	$-0.323963\pi$
$383$	20.5597			1.05055			0.525275	+	0.850933i	$0.323963\pi$
$384$		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	$-0.698002\pi$
$419$	−23.8550			−1.16539			−0.582696	+	0.812691i	$0.698002\pi$
$420$		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	$-0.884998\pi$
$431$	−38.8406			−1.87088			−0.935442	+	0.353480i	$0.884998\pi$
$432$		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	$-0.838904\pi$
$439$		−	20.3140i		−	0.969535i	0.874643	−	0.484767i	$-0.161096\pi$
$440$		0			0
$441$	−19.3541	−	5.87083i	−0.921626	−	0.279563i
$442$		0			0
$443$	25.6424			1.21831			0.609154	−	0.793052i	$-0.291509\pi$
$443$	25.6424			1.21831			0.609154	+	0.793052i	$0.291509\pi$
$444$		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	$-0.896822\pi$
$463$		−	13.7064i		−	0.636992i	0.947924	−	0.318496i	$-0.103178\pi$
$464$		0			0
$465$	2.25834	−	15.2250i	0.104728	−	0.706041i
$466$		0			0
$467$	36.2992			1.67973			0.839863	−	0.542798i	$-0.182635\pi$
$467$	36.2992			1.67973			0.839863	+	0.542798i	$0.182635\pi$
$468$		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	$-0.651465\pi$
$479$	−20.0514			−0.916171			−0.458086	+	0.888908i	$0.651465\pi$
$480$		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	$0.416999\pi$
$487$			42.6441i			1.93239i	−0.257811	+	0.966195i	$0.583001\pi$
$488$		0			0
$489$	34.3541	+	5.09580i	1.55355	+	0.230440i
$490$		0			0
$491$	24.3632			1.09950			0.549749	−	0.835330i	$-0.314723\pi$
$491$	24.3632			1.09950			0.549749	+	0.835330i	$0.314723\pi$
$492$		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	$0.235170\pi$
$499$			30.0856i			1.34681i	−0.739272	+	0.673407i	$0.764830\pi$
$500$		0			0
$501$	1.38751	−	9.35414i	0.0619896	−	0.417912i
$502$		0			0
$503$	2.67268			0.119169			0.0595844	−	0.998223i	$-0.481022\pi$
$503$	2.67268			0.119169			0.0595844	+	0.998223i	$0.481022\pi$
$504$		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	$-0.760893\pi$
$523$		−	31.2165i		−	1.36500i	0.730885	−	0.682500i	$-0.239107\pi$
$524$		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	$0.00948423\pi$
$547$			1.39351i			0.0595823i	−0.999556	+	0.0297912i	$0.990516\pi$
$548$		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	$-0.598562\pi$
$563$	−14.4604			−0.609432			−0.304716	+	0.952443i	$0.598562\pi$
$564$		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	$-0.571845\pi$
$571$		−	46.5790i		−	1.94927i	0.223797	−	0.974636i	$-0.428155\pi$
$572$		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	$-0.298290\pi$
$587$	28.6920			1.18425			0.592124	+	0.805847i	$0.298290\pi$
$588$		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	$-0.691276\pi$
$599$	−27.6755			−1.13079			−0.565395	+	0.824820i	$0.691276\pi$
$600$		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	$0.0922639\pi$
$607$			14.0834i			0.571628i	−0.958285	+	0.285814i	$0.907736\pi$
$608$		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	$0.00807516\pi$
$619$			1.26220i			0.0507323i	−0.999678	+	0.0253661i	$0.991925\pi$
$620$		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	$0.238914\pi$
$631$			34.2661i			1.36411i	−0.731300	+	0.682056i	$0.761086\pi$
$632$		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	$-0.822973\pi$
$643$		−	26.7733i		−	1.05584i	0.849295	−	0.527918i	$-0.177027\pi$
$644$		0			0
$645$	−4.51669	+	30.4499i	−0.177844	+	1.19897i
$646$		0			0
$647$	−23.8719			−0.938502			−0.469251	−	0.883065i	$-0.655476\pi$
$647$	−23.8719			−0.938502			−0.469251	+	0.883065i	$0.655476\pi$
$648$		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	$-0.652095\pi$
$659$	−23.6093			−0.919688			−0.459844	+	0.888000i	$0.652095\pi$
$660$		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	$-0.694253\pi$
$683$	−29.9543			−1.14617			−0.573084	+	0.819497i	$0.694253\pi$
$684$		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	$0.180096\pi$
$691$			28.1838i			1.07216i	−0.844167	+	0.536081i	$0.819904\pi$
$692$		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	$-0.893331\pi$
$719$	−50.6452			−1.88875			−0.944374	+	0.328873i	$0.893331\pi$
$720$		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	$-0.741926\pi$
$727$		−	39.0862i		−	1.44963i	0.688945	−	0.724814i	$-0.258074\pi$
$728$		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	$-0.995586\pi$
$739$		−	0.753931i		−	0.0277338i	0.999904	−	0.0138669i	$-0.00441411\pi$
$740$		0			0
$741$	−3.03337	−	0.449944i	−0.111434	−	0.0165291i
$742$		0			0
$743$	17.5270			0.643002			0.321501	−	0.946909i	$-0.395813\pi$
$743$	17.5270			0.643002			0.321501	+	0.946909i	$0.395813\pi$
$744$		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	$0.244552\pi$
$751$			38.0866i			1.38980i	−0.719106	+	0.694901i	$0.755448\pi$
$752$		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	$0.0129315\pi$
$787$			2.27875i			0.0812288i	−0.999175	+	0.0406144i	$0.987068\pi$
$788$		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	$0.126746\pi$
$811$			22.0845i			0.775491i	−0.921766	+	0.387746i	$0.873254\pi$
$812$		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	$0.0746311\pi$
$823$			13.3295i			0.464636i	−0.972640	+	0.232318i	$0.925369\pi$
$824$		0			0
$825$	−1.12917	+	7.61249i	−0.0393127	+	0.265033i
$826$		0			0
$827$	15.2313			0.529643			0.264821	−	0.964297i	$-0.414687\pi$
$827$	15.2313			0.529643			0.264821	+	0.964297i	$0.414687\pi$
$828$		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	$-0.623716\pi$
$839$	−21.9532			−0.757908			−0.378954	+	0.925416i	$0.623716\pi$
$840$		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	$0.187040\pi$
$859$			32.4956i			1.10874i	−0.832272	+	0.554368i	$0.812960\pi$
$860$		0			0
$861$	−0.708287	+	4.77503i	−0.0241384	+	0.162733i
$862$		0			0
$863$	3.80358			0.129475			0.0647376	−	0.997902i	$-0.479379\pi$
$863$	3.80358			0.129475			0.0647376	+	0.997902i	$0.479379\pi$
$864$		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	$-0.946712\pi$
$883$		−	9.90287i		−	0.333258i	0.986020	−	0.166629i	$-0.0532882\pi$
$884$		0			0
$885$	3.48331	+	0.516685i	0.117090	+	0.0173682i
$886$		0			0
$887$	18.0183			0.604995			0.302497	−	0.953150i	$-0.402180\pi$
$887$	18.0183			0.604995			0.302497	+	0.953150i	$0.402180\pi$
$888$		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	$0.164080\pi$
$907$			29.6916i			0.985895i	−0.870059	+	0.492947i	$0.835920\pi$
$908$		0			0
$909$	−8.93326	+	29.4499i	−0.296298	+	0.976793i
$910$		0			0
$911$	−14.3460			−0.475305			−0.237652	−	0.971350i	$-0.576378\pi$
$911$	−14.3460			−0.475305			−0.237652	+	0.971350i	$0.576378\pi$
$912$		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	$-0.715829\pi$
$919$		−	47.2186i		−	1.55760i	0.627273	−	0.778799i	$-0.284171\pi$
$920$		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	$-0.462520\pi$
$947$	7.23019			0.234949			0.117475	+	0.993076i	$0.462520\pi$
$948$		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	$-0.964853\pi$
$967$		−	6.85322i		−	0.220385i	0.993910	−	0.110192i	$-0.0351467\pi$
$968$		0			0
$969$	0.258343	−	1.74166i	0.00829916	−	0.0559501i
$970$		0			0
$971$	44.4316			1.42588			0.712939	−	0.701226i	$-0.247364\pi$
$971$	44.4316			1.42588			0.712939	+	0.701226i	$0.247364\pi$
$972$		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	$-0.605718\pi$
$983$	−20.4453			−0.652104			−0.326052	+	0.945352i	$0.605718\pi$
$984$		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	$-0.859455\pi$
$991$		−	26.9046i		−	0.854653i	0.904097	−	0.427327i	$-0.140545\pi$
$992$		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.239.4	yes	8
3.2	odd	2	inner	816.2.e.b.239.6	yes	8
4.3	odd	2	inner	816.2.e.b.239.5	yes	8
12.11	even	2	inner	816.2.e.b.239.3	✓	8

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

Related objects

Downloads

Learn more