LMFDB - Embedded newform 768.5.g.e.511.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [768,5,Mod(511,768)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("768.511"); S:= CuspForms(chi, 5); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(768, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0])) N = Newforms(chi, 5, names="a")

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	768.g (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,-108,0,0,0,0,0,0,0,-1224] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	no
Analytic conductor:	$79.3881316484$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 192)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			511.4
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	768.511
Dual form			768.5.g.e.511.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+5.19615i q^{3} +24.2487 q^{5} -58.0000i q^{7} -27.0000 q^{9} +13.8564i q^{11} +20.7846 q^{13} +126.000i q^{15} -306.000 q^{17} -602.754i q^{19} +301.377 q^{21} +468.000i q^{23} -37.0000 q^{25} -140.296i q^{27} -1465.31 q^{29} +110.000i q^{31} -72.0000 q^{33} -1406.43i q^{35} +1039.23 q^{37} +108.000i q^{39} -2970.00 q^{41} +2889.06i q^{43} -654.715 q^{45} +396.000i q^{47} -963.000 q^{49} -1590.02i q^{51} +1125.83 q^{53} +336.000i q^{55} +3132.00 q^{57} +2681.21i q^{59} -5985.97 q^{61} +1566.00i q^{63} +504.000 q^{65} +4801.24i q^{67} -2431.80 q^{69} +6588.00i q^{71} +5894.00 q^{73} -192.258i q^{75} +803.672 q^{77} +8486.00i q^{79} +729.000 q^{81} +13.8564i q^{83} -7420.11 q^{85} -7614.00i q^{87} -8766.00 q^{89} -1205.51i q^{91} -571.577 q^{93} -14616.0i q^{95} +5918.00 q^{97} -374.123i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 108 q^{9} - 1224 q^{17} - 148 q^{25} - 288 q^{33} - 11880 q^{41} - 3852 q^{49} + 12528 q^{57} + 2016 q^{65} + 23576 q^{73} + 2916 q^{81} - 35064 q^{89} + 23672 q^{97}+O(q^{100}) $ 4 * q - 108 * q^9 - 1224 * q^17 - 148 * q^25 - 288 * q^33 - 11880 * q^41 - 3852 * q^49 + 12528 * q^57 + 2016 * q^65 + 23576 * q^73 + 2916 * q^81 - 35064 * q^89 + 23672 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/768\mathbb{Z}\right)^\times$.

$n$	$257$	$511$	$517$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	5.19615i	0.577350i
$3$	5.19615i	0.577350i
$4$	0	0
$5$	24.2487	0.969948	0.484974	−	0.874528i	$-0.338829\pi$
$5$	24.2487	0.969948	0.484974	+	0.874528i	$0.338829\pi$
$6$	0	0
$7$	− 58.0000i	− 1.18367i	−0.806058	−	0.591837i	$-0.798403\pi$
$7$	− 58.0000i	− 1.18367i	0.806058	−	0.591837i	$-0.201597\pi$
$8$	0	0
$9$	−27.0000	−0.333333
$10$	0	0
$11$	13.8564i	0.114516i	0.998359	+	0.0572579i	$0.0182357\pi$
$11$	13.8564i	0.114516i	−0.998359	+	0.0572579i	$0.981764\pi$
$12$	0	0
$13$	20.7846	0.122986	0.0614929	−	0.998108i	$-0.480414\pi$
$13$	20.7846	0.122986	0.0614929	+	0.998108i	$0.480414\pi$
$14$	0	0
$15$	126.000i	0.560000i
$16$	0	0
$17$	−306.000	−1.05882	−0.529412	−	0.848365i	$-0.677587\pi$
$17$	−306.000	−1.05882	−0.529412	+	0.848365i	$0.677587\pi$
$18$	0	0
$19$	− 602.754i	− 1.66968i	−0.550494	−	0.834839i	$-0.685561\pi$
$19$	− 602.754i	− 1.66968i	0.550494	−	0.834839i	$-0.314439\pi$
$20$	0	0
$21$	301.377	0.683394
$22$	0	0
$23$	468.000i	0.884688i	0.896845	+	0.442344i	$0.145853\pi$
$23$	468.000i	0.884688i	−0.896845	+	0.442344i	$0.854147\pi$
$24$	0	0
$25$	−37.0000	−0.0592000
$26$	0	0
$27$	− 140.296i	− 0.192450i
$28$	0	0
$29$	−1465.31	−1.74235	−0.871174	−	0.490974i	$-0.836641\pi$
$29$	−1465.31	−1.74235	−0.871174	+	0.490974i	$0.836641\pi$
$30$	0	0
$31$	110.000i	0.114464i	0.998361	+	0.0572320i	$0.0182275\pi$
$31$	110.000i	0.114464i	−0.998361	+	0.0572320i	$0.981773\pi$
$32$	0	0
$33$	−72.0000	−0.0661157
$34$	0	0
$35$	− 1406.43i	− 1.14810i
$36$	0	0
$37$	1039.23	0.759116	0.379558	−	0.925168i	$-0.376076\pi$
$37$	1039.23	0.759116	0.379558	+	0.925168i	$0.376076\pi$
$38$	0	0
$39$	108.000i	0.0710059i
$40$	0	0
$41$	−2970.00	−1.76681	−0.883403	−	0.468615i	$-0.844753\pi$
$41$	−2970.00	−1.76681	−0.883403	+	0.468615i	$0.844753\pi$
$42$	0	0
$43$	2889.06i	1.56250i	0.624219	+	0.781250i	$0.285417\pi$
$43$	2889.06i	1.56250i	−0.624219	+	0.781250i	$0.714583\pi$
$44$	0	0
$45$	−654.715	−0.323316
$46$	0	0
$47$	396.000i	0.179267i	0.995975	+	0.0896333i	$0.0285695\pi$
$47$	396.000i	0.179267i	−0.995975	+	0.0896333i	$0.971430\pi$
$48$	0	0
$49$	−963.000	−0.401083
$50$	0	0
$51$	− 1590.02i	− 0.611312i
$52$	0	0
$53$	1125.83	0.400795	0.200397	−	0.979715i	$-0.435777\pi$
$53$	1125.83	0.400795	0.200397	+	0.979715i	$0.435777\pi$
$54$	0	0
$55$	336.000i	0.111074i
$56$	0	0
$57$	3132.00	0.963989
$58$	0	0
$59$	2681.21i	0.770243i	0.922866	+	0.385121i	$0.125840\pi$
$59$	2681.21i	0.770243i	−0.922866	+	0.385121i	$0.874160\pi$
$60$	0	0
$61$	−5985.97	−1.60870	−0.804349	−	0.594157i	$-0.797486\pi$
$61$	−5985.97	−1.60870	−0.804349	+	0.594157i	$0.797486\pi$
$62$	0	0
$63$	1566.00i	0.394558i
$64$	0	0
$65$	504.000	0.119290
$66$	0	0
$67$	4801.24i	1.06956i	0.844992	+	0.534779i	$0.179605\pi$
$67$	4801.24i	1.06956i	−0.844992	+	0.534779i	$0.820395\pi$
$68$	0	0
$69$	−2431.80	−0.510775
$70$	0	0
$71$	6588.00i	1.30688i	0.756977	+	0.653442i	$0.226676\pi$
$71$	6588.00i	1.30688i	−0.756977	+	0.653442i	$0.773324\pi$
$72$	0	0
$73$	5894.00	1.10602	0.553012	−	0.833173i	$-0.313478\pi$
$73$	5894.00	1.10602	0.553012	+	0.833173i	$0.313478\pi$
$74$	0	0
$75$	− 192.258i	− 0.0341791i
$76$	0	0
$77$	803.672	0.135549
$78$	0	0
$79$	8486.00i	1.35972i	0.733343	+	0.679859i	$0.237959\pi$
$79$	8486.00i	1.35972i	−0.733343	+	0.679859i	$0.762041\pi$
$80$	0	0
$81$	729.000	0.111111
$82$	0	0
$83$	13.8564i	0.00201138i	0.999999	+	0.00100569i	$0.000320121\pi$
$83$	13.8564i	0.00201138i	−0.999999	+	0.00100569i	$0.999680\pi$
$84$	0	0
$85$	−7420.11	−1.02700
$86$	0	0
$87$	− 7614.00i	− 1.00595i
$88$	0	0
$89$	−8766.00	−1.10668	−0.553339	−	0.832956i	$-0.686647\pi$
$89$	−8766.00	−1.10668	−0.553339	+	0.832956i	$0.686647\pi$
$90$	0	0
$91$	− 1205.51i	− 0.145575i
$92$	0	0
$93$	−571.577	−0.0660859
$94$	0	0
$95$	− 14616.0i	− 1.61950i
$96$	0	0
$97$	5918.00	0.628972	0.314486	−	0.949262i	$-0.398168\pi$
$97$	5918.00	0.628972	0.314486	+	0.949262i	$0.398168\pi$
$98$	0	0
$99$	− 374.123i	− 0.0381719i
$100$	0	0
$101$	−16416.4	−1.60929	−0.804646	−	0.593756i	$-0.797645\pi$
$101$	−16416.4	−1.60929	−0.804646	+	0.593756i	$0.797645\pi$
$102$	0	0
$103$	− 5342.00i	− 0.503535i	−0.967788	−	0.251767i	$-0.918988\pi$
$103$	− 5342.00i	− 0.503535i	0.967788	−	0.251767i	$-0.0810118\pi$
$104$	0	0
$105$	7308.00	0.662857
$106$	0	0
$107$	− 21733.8i	− 1.89831i	−0.314806	−	0.949156i	$-0.601940\pi$
$107$	− 21733.8i	− 1.89831i	0.314806	−	0.949156i	$-0.398060\pi$
$108$	0	0
$109$	8126.78	0.684015	0.342008	−	0.939697i	$-0.388893\pi$
$109$	8126.78	0.684015	0.342008	+	0.939697i	$0.388893\pi$
$110$	0	0
$111$	5400.00i	0.438276i
$112$	0	0
$113$	15354.0	1.20244	0.601222	−	0.799082i	$-0.294681\pi$
$113$	15354.0	1.20244	0.601222	+	0.799082i	$0.294681\pi$
$114$	0	0
$115$	11348.4i	0.858102i
$116$	0	0
$117$	−561.184	−0.0409953
$118$	0	0
$119$	17748.0i	1.25330i
$120$	0	0
$121$	14449.0	0.986886
$122$	0	0
$123$	− 15432.6i	− 1.02007i
$124$	0	0
$125$	−16052.6	−1.02737
$126$	0	0
$127$	− 9998.00i	− 0.619877i	−0.950757	−	0.309939i	$-0.899691\pi$
$127$	− 9998.00i	− 0.619877i	0.950757	−	0.309939i	$-0.100309\pi$
$128$	0	0
$129$	−15012.0	−0.902109
$130$	0	0
$131$	1877.54i	0.109408i	0.998503	+	0.0547038i	$0.0174214\pi$
$131$	1877.54i	0.109408i	−0.998503	+	0.0547038i	$0.982579\pi$
$132$	0	0
$133$	−34959.7	−1.97635
$134$	0	0
$135$	− 3402.00i	− 0.186667i
$136$	0	0
$137$	−27882.0	−1.48553	−0.742767	−	0.669550i	$-0.766487\pi$
$137$	−27882.0	−1.48553	−0.742767	+	0.669550i	$0.766487\pi$
$138$	0	0
$139$	11327.6i	0.586285i	0.956069	+	0.293142i	$0.0947010\pi$
$139$	11327.6i	0.586285i	−0.956069	+	0.293142i	$0.905299\pi$
$140$	0	0
$141$	−2057.68	−0.103500
$142$	0	0
$143$	288.000i	0.0140838i
$144$	0	0
$145$	−35532.0	−1.68999
$146$	0	0
$147$	− 5003.89i	− 0.231565i
$148$	0	0
$149$	−33134.1	−1.49246	−0.746231	−	0.665688i	$-0.768138\pi$
$149$	−33134.1	−1.49246	−0.746231	+	0.665688i	$0.768138\pi$
$150$	0	0
$151$	− 38794.0i	− 1.70142i	−0.525638	−	0.850708i	$-0.676173\pi$
$151$	− 38794.0i	− 1.70142i	0.525638	−	0.850708i	$-0.323827\pi$
$152$	0	0
$153$	8262.00	0.352941
$154$	0	0
$155$	2667.36i	0.111024i
$156$	0	0
$157$	−35957.4	−1.45878	−0.729388	−	0.684100i	$-0.760195\pi$
$157$	−35957.4	−1.45878	−0.729388	+	0.684100i	$0.760195\pi$
$158$	0	0
$159$	5850.00i	0.231399i
$160$	0	0
$161$	27144.0	1.04718
$162$	0	0
$163$	3013.77i	0.113432i	0.998390	+	0.0567159i	$0.0180629\pi$
$163$	3013.77i	0.113432i	−0.998390	+	0.0567159i	$0.981937\pi$
$164$	0	0
$165$	−1745.91	−0.0641288
$166$	0	0
$167$	11304.0i	0.405321i	0.979249	+	0.202661i	$0.0649588\pi$
$167$	11304.0i	0.405321i	−0.979249	+	0.202661i	$0.935041\pi$
$168$	0	0
$169$	−28129.0	−0.984874
$170$	0	0
$171$	16274.3i	0.556559i
$172$	0	0
$173$	−35309.6	−1.17978	−0.589889	−	0.807484i	$-0.700829\pi$
$173$	−35309.6	−1.17978	−0.589889	+	0.807484i	$0.700829\pi$
$174$	0	0
$175$	2146.00i	0.0700735i
$176$	0	0
$177$	−13932.0	−0.444700
$178$	0	0
$179$	− 6699.57i	− 0.209094i	−0.994520	−	0.104547i	$-0.966661\pi$
$179$	− 6699.57i	− 0.209094i	0.994520	−	0.104547i	$-0.0333392\pi$
$180$	0	0
$181$	14445.3	0.440930	0.220465	−	0.975395i	$-0.429243\pi$
$181$	14445.3	0.440930	0.220465	+	0.975395i	$0.429243\pi$
$182$	0	0
$183$	− 31104.0i	− 0.928783i
$184$	0	0
$185$	25200.0	0.736304
$186$	0	0
$187$	− 4240.06i	− 0.121252i
$188$	0	0
$189$	−8137.17	−0.227798
$190$	0	0
$191$	− 35064.0i	− 0.961158i	−0.876951	−	0.480579i	$-0.840427\pi$
$191$	− 35064.0i	− 0.961158i	0.876951	−	0.480579i	$-0.159573\pi$
$192$	0	0
$193$	−11230.0	−0.301485	−0.150742	−	0.988573i	$-0.548166\pi$
$193$	−11230.0	−0.301485	−0.150742	+	0.988573i	$0.548166\pi$
$194$	0	0
$195$	2618.86i	0.0688721i
$196$	0	0
$197$	28991.1	0.747019	0.373510	−	0.927626i	$-0.378154\pi$
$197$	28991.1	0.747019	0.373510	+	0.927626i	$0.378154\pi$
$198$	0	0
$199$	18226.0i	0.460241i	0.973162	+	0.230120i	$0.0739120\pi$
$199$	18226.0i	0.460241i	−0.973162	+	0.230120i	$0.926088\pi$
$200$	0	0
$201$	−24948.0	−0.617509
$202$	0	0
$203$	84988.3i	2.06237i
$204$	0	0
$205$	−72018.7	−1.71371
$206$	0	0
$207$	− 12636.0i	− 0.294896i
$208$	0	0
$209$	8352.00	0.191204
$210$	0	0
$211$	− 37266.8i	− 0.837061i	−0.908203	−	0.418531i	$-0.862545\pi$
$211$	− 37266.8i	− 0.837061i	0.908203	−	0.418531i	$-0.137455\pi$
$212$	0	0
$213$	−34232.3	−0.754530
$214$	0	0
$215$	70056.0i	1.51554i
$216$	0	0
$217$	6380.00	0.135488
$218$	0	0
$219$	30626.1i	0.638563i
$220$	0	0
$221$	−6360.09	−0.130220
$222$	0	0
$223$	10162.0i	0.204348i	0.994767	+	0.102174i	$0.0325798\pi$
$223$	10162.0i	0.204348i	−0.994767	+	0.102174i	$0.967420\pi$
$224$	0	0
$225$	999.000	0.0197333
$226$	0	0
$227$	− 15214.3i	− 0.295258i	−0.989043	−	0.147629i	$-0.952836\pi$
$227$	− 15214.3i	− 0.295258i	0.989043	−	0.147629i	$-0.0471641\pi$
$228$	0	0
$229$	−7711.09	−0.147043	−0.0735216	−	0.997294i	$-0.523424\pi$
$229$	−7711.09	−0.147043	−0.0735216	+	0.997294i	$0.523424\pi$
$230$	0	0
$231$	4176.00i	0.0782594i
$232$	0	0
$233$	21258.0	0.391571	0.195786	−	0.980647i	$-0.437274\pi$
$233$	21258.0	0.391571	0.195786	+	0.980647i	$0.437274\pi$
$234$	0	0
$235$	9602.49i	0.173879i
$236$	0	0
$237$	−44094.5	−0.785034
$238$	0	0
$239$	97056.0i	1.69913i	0.527484	+	0.849565i	$0.323135\pi$
$239$	97056.0i	1.69913i	−0.527484	+	0.849565i	$0.676865\pi$
$240$	0	0
$241$	47242.0	0.813381	0.406691	−	0.913566i	$-0.366683\pi$
$241$	47242.0	0.813381	0.406691	+	0.913566i	$0.366683\pi$
$242$	0	0
$243$	3788.00i	0.0641500i
$244$	0	0
$245$	−23351.5	−0.389030
$246$	0	0
$247$	− 12528.0i	− 0.205347i
$248$	0	0
$249$	−72.0000	−0.00116127
$250$	0	0
$251$	− 78191.7i	− 1.24112i	−0.784160	−	0.620559i	$-0.786906\pi$
$251$	− 78191.7i	− 1.24112i	0.784160	−	0.620559i	$-0.213094\pi$
$252$	0	0
$253$	−6484.80	−0.101311
$254$	0	0
$255$	− 38556.0i	− 0.592941i
$256$	0	0
$257$	23922.0	0.362186	0.181093	−	0.983466i	$-0.442037\pi$
$257$	23922.0	0.362186	0.181093	+	0.983466i	$0.442037\pi$
$258$	0	0
$259$	− 60275.4i	− 0.898546i
$260$	0	0
$261$	39563.5	0.580783
$262$	0	0
$263$	− 84528.0i	− 1.22205i	−0.791611	−	0.611025i	$-0.790757\pi$
$263$	− 84528.0i	− 1.22205i	0.791611	−	0.611025i	$-0.209243\pi$
$264$	0	0
$265$	27300.0	0.388750
$266$	0	0
$267$	− 45549.5i	− 0.638941i
$268$	0	0
$269$	−92751.3	−1.28179	−0.640893	−	0.767630i	$-0.721436\pi$
$269$	−92751.3	−1.28179	−0.640893	+	0.767630i	$0.721436\pi$
$270$	0	0
$271$	− 61118.0i	− 0.832205i	−0.909318	−	0.416103i	$-0.863396\pi$
$271$	− 61118.0i	− 0.832205i	0.909318	−	0.416103i	$-0.136604\pi$
$272$	0	0
$273$	6264.00	0.0840478
$274$	0	0
$275$	− 512.687i	− 0.00677933i
$276$	0	0
$277$	−69441.4	−0.905021	−0.452511	−	0.891759i	$-0.649472\pi$
$277$	−69441.4	−0.905021	−0.452511	+	0.891759i	$0.649472\pi$
$278$	0	0
$279$	− 2970.00i	− 0.0381547i
$280$	0	0
$281$	60570.0	0.767088	0.383544	−	0.923523i	$-0.374704\pi$
$281$	60570.0	0.767088	0.383544	+	0.923523i	$0.374704\pi$
$282$	0	0
$283$	− 87565.6i	− 1.09335i	−0.837344	−	0.546677i	$-0.815893\pi$
$283$	− 87565.6i	− 1.09335i	0.837344	−	0.546677i	$-0.184107\pi$
$284$	0	0
$285$	75947.0	0.935020
$286$	0	0
$287$	172260.i	2.09132i
$288$	0	0
$289$	10115.0	0.121107
$290$	0	0
$291$	30750.8i	0.363137i
$292$	0	0
$293$	−77162.9	−0.898821	−0.449410	−	0.893325i	$-0.648366\pi$
$293$	−77162.9	−0.898821	−0.449410	+	0.893325i	$0.648366\pi$
$294$	0	0
$295$	65016.0i	0.747096i
$296$	0	0
$297$	1944.00	0.0220386
$298$	0	0
$299$	9727.20i	0.108804i
$300$	0	0
$301$	167566.	1.84949
$302$	0	0
$303$	− 85302.0i	− 0.929125i
$304$	0	0
$305$	−145152.	−1.56035
$306$	0	0
$307$	− 12657.8i	− 0.134302i	−0.997743	−	0.0671510i	$-0.978609\pi$
$307$	− 12657.8i	− 0.134302i	0.997743	−	0.0671510i	$-0.0213909\pi$
$308$	0	0
$309$	27757.8	0.290716
$310$	0	0
$311$	137592.i	1.42257i	0.702906	+	0.711283i	$0.251886\pi$
$311$	137592.i	1.42257i	−0.702906	+	0.711283i	$0.748114\pi$
$312$	0	0
$313$	−13198.0	−0.134716	−0.0673580	−	0.997729i	$-0.521457\pi$
$313$	−13198.0	−0.134716	−0.0673580	+	0.997729i	$0.521457\pi$
$314$	0	0
$315$	37973.5i	0.382701i
$316$	0	0
$317$	134106.	1.33453	0.667266	−	0.744820i	$-0.267464\pi$
$317$	134106.	1.33453	0.667266	+	0.744820i	$0.267464\pi$
$318$	0	0
$319$	− 20304.0i	− 0.199526i
$320$	0	0
$321$	112932.	1.09599
$322$	0	0
$323$	184443.i	1.76789i
$324$	0	0
$325$	−769.031	−0.00728076
$326$	0	0
$327$	42228.0i	0.394916i
$328$	0	0
$329$	22968.0	0.212193
$330$	0	0
$331$	− 52107.0i	− 0.475598i	−0.971314	−	0.237799i	$-0.923574\pi$
$331$	− 52107.0i	− 0.475598i	0.971314	−	0.237799i	$-0.0764260\pi$
$332$	0	0
$333$	−28059.2	−0.253039
$334$	0	0
$335$	116424.i	1.03742i
$336$	0	0
$337$	84470.0	0.743777	0.371888	−	0.928277i	$-0.378710\pi$
$337$	84470.0	0.743777	0.371888	+	0.928277i	$0.378710\pi$
$338$	0	0
$339$	79781.7i	0.694231i
$340$	0	0
$341$	−1524.20	−0.0131079
$342$	0	0
$343$	− 83404.0i	− 0.708922i
$344$	0	0
$345$	−58968.0	−0.495425
$346$	0	0
$347$	− 57670.4i	− 0.478954i	−0.970902	−	0.239477i	$-0.923024\pi$
$347$	− 57670.4i	− 0.478954i	0.970902	−	0.239477i	$-0.0769760\pi$
$348$	0	0
$349$	−43315.1	−0.355622	−0.177811	−	0.984065i	$-0.556902\pi$
$349$	−43315.1	−0.355622	−0.177811	+	0.984065i	$0.556902\pi$
$350$	0	0
$351$	− 2916.00i	− 0.0236686i
$352$	0	0
$353$	−188118.	−1.50967	−0.754833	−	0.655917i	$-0.772282\pi$
$353$	−188118.	−1.50967	−0.754833	+	0.655917i	$0.772282\pi$
$354$	0	0
$355$	159751.i	1.26761i
$356$	0	0
$357$	−92221.3	−0.723594
$358$	0	0
$359$	− 14148.0i	− 0.109776i	−0.998493	−	0.0548878i	$-0.982520\pi$
$359$	− 14148.0i	− 0.109776i	0.998493	−	0.0548878i	$-0.0174801\pi$
$360$	0	0
$361$	−232991.	−1.78782
$362$	0	0
$363$	75079.2i	0.569779i
$364$	0	0
$365$	142922.	1.07279
$366$	0	0
$367$	− 265810.i	− 1.97351i	−0.162220	−	0.986755i	$-0.551865\pi$
$367$	− 265810.i	− 1.97351i	0.162220	−	0.986755i	$-0.448135\pi$
$368$	0	0
$369$	80190.0	0.588935
$370$	0	0
$371$	− 65298.3i	− 0.474410i
$372$	0	0
$373$	131774.	0.947138	0.473569	−	0.880757i	$-0.342965\pi$
$373$	131774.	0.947138	0.473569	+	0.880757i	$0.342965\pi$
$374$	0	0
$375$	− 83412.0i	− 0.593152i
$376$	0	0
$377$	−30456.0	−0.214284
$378$	0	0
$379$	− 91930.3i	− 0.640001i	−0.947417	−	0.320000i	$-0.896317\pi$
$379$	− 91930.3i	− 0.640001i	0.947417	−	0.320000i	$-0.103683\pi$
$380$	0	0
$381$	51951.1	0.357886
$382$	0	0
$383$	189000.i	1.28844i	0.764840	+	0.644220i	$0.222818\pi$
$383$	189000.i	1.28844i	−0.764840	+	0.644220i	$0.777182\pi$
$384$	0	0
$385$	19488.0	0.131476
$386$	0	0
$387$	− 78004.6i	− 0.520833i
$388$	0	0
$389$	180906.	1.19551	0.597755	−	0.801679i	$-0.296060\pi$
$389$	180906.	1.19551	0.597755	+	0.801679i	$0.296060\pi$
$390$	0	0
$391$	− 143208.i	− 0.936729i
$392$	0	0
$393$	−9756.00	−0.0631665
$394$	0	0
$395$	205775.i	1.31886i
$396$	0	0
$397$	140005.	0.888307	0.444153	−	0.895951i	$-0.353505\pi$
$397$	140005.	0.888307	0.444153	+	0.895951i	$0.353505\pi$
$398$	0	0
$399$	− 181656.i	− 1.14105i
$400$	0	0
$401$	−208674.	−1.29772	−0.648858	−	0.760910i	$-0.724753\pi$
$401$	−208674.	−1.29772	−0.648858	+	0.760910i	$0.724753\pi$
$402$	0	0
$403$	2286.31i	0.0140775i
$404$	0	0
$405$	17677.3	0.107772
$406$	0	0
$407$	14400.0i	0.0869308i
$408$	0	0
$409$	194078.	1.16019	0.580096	−	0.814548i	$-0.303015\pi$
$409$	194078.	1.16019	0.580096	+	0.814548i	$0.303015\pi$
$410$	0	0
$411$	− 144879.i	− 0.857674i
$412$	0	0
$413$	155510.	0.911716
$414$	0	0
$415$	336.000i	0.00195094i
$416$	0	0
$417$	−58860.0	−0.338492
$418$	0	0
$419$	− 72829.3i	− 0.414837i	−0.978252	−	0.207419i	$-0.933494\pi$
$419$	− 72829.3i	− 0.414837i	0.978252	−	0.207419i	$-0.0665062\pi$
$420$	0	0
$421$	107893.	0.608736	0.304368	−	0.952555i	$-0.401555\pi$
$421$	107893.	0.608736	0.304368	+	0.952555i	$0.401555\pi$
$422$	0	0
$423$	− 10692.0i	− 0.0597555i
$424$	0	0
$425$	11322.0	0.0626824
$426$	0	0
$427$	347186.i	1.90417i
$428$	0	0
$429$	−1496.49	−0.00813130
$430$	0	0
$431$	151380.i	0.814918i	0.913223	+	0.407459i	$0.133585\pi$
$431$	151380.i	0.814918i	−0.913223	+	0.407459i	$0.866415\pi$
$432$	0	0
$433$	−13922.0	−0.0742550	−0.0371275	−	0.999311i	$-0.511821\pi$
$433$	−13922.0	−0.0742550	−0.0371275	+	0.999311i	$0.511821\pi$
$434$	0	0
$435$	− 184630.i	− 0.975715i
$436$	0	0
$437$	282089.	1.47714
$438$	0	0
$439$	171130.i	0.887968i	0.896035	+	0.443984i	$0.146435\pi$
$439$	171130.i	0.887968i	−0.896035	+	0.443984i	$0.853565\pi$
$440$	0	0
$441$	26001.0	0.133694
$442$	0	0
$443$	− 226719.i	− 1.15526i	−0.816299	−	0.577630i	$-0.803978\pi$
$443$	− 226719.i	− 1.15526i	0.816299	−	0.577630i	$-0.196022\pi$
$444$	0	0
$445$	−212564.	−1.07342
$446$	0	0
$447$	− 172170.i	− 0.861673i
$448$	0	0
$449$	160830.	0.797764	0.398882	−	0.917002i	$-0.369398\pi$
$449$	160830.	0.797764	0.398882	+	0.917002i	$0.369398\pi$
$450$	0	0
$451$	− 41153.5i	− 0.202327i
$452$	0	0
$453$	201580.	0.982313
$454$	0	0
$455$	− 29232.0i	− 0.141200i
$456$	0	0
$457$	146030.	0.699213	0.349607	−	0.936897i	$-0.386315\pi$
$457$	146030.	0.699213	0.349607	+	0.936897i	$0.386315\pi$
$458$	0	0
$459$	42930.6i	0.203771i
$460$	0	0
$461$	−99561.7	−0.468480	−0.234240	−	0.972179i	$-0.575260\pi$
$461$	−99561.7	−0.468480	−0.234240	+	0.972179i	$0.575260\pi$
$462$	0	0
$463$	47194.0i	0.220153i	0.993923	+	0.110077i	$0.0351096\pi$
$463$	47194.0i	0.220153i	−0.993923	+	0.110077i	$0.964890\pi$
$464$	0	0
$465$	−13860.0	−0.0640999
$466$	0	0
$467$	− 279872.i	− 1.28329i	−0.767001	−	0.641646i	$-0.778252\pi$
$467$	− 279872.i	− 1.28329i	0.767001	−	0.641646i	$-0.221748\pi$
$468$	0	0
$469$	278472.	1.26601
$470$	0	0
$471$	− 186840.i	− 0.842225i
$472$	0	0
$473$	−40032.0	−0.178931
$474$	0	0
$475$	22301.9i	0.0988449i
$476$	0	0
$477$	−30397.5	−0.133598
$478$	0	0
$479$	126828.i	0.552770i	0.961047	+	0.276385i	$0.0891364\pi$
$479$	126828.i	0.552770i	−0.961047	+	0.276385i	$0.910864\pi$
$480$	0	0
$481$	21600.0	0.0933606
$482$	0	0
$483$	141044.i	0.604591i
$484$	0	0
$485$	143504.	0.610071
$486$	0	0
$487$	− 177010.i	− 0.746345i	−0.927762	−	0.373173i	$-0.878270\pi$
$487$	− 177010.i	− 0.746345i	0.927762	−	0.373173i	$-0.121730\pi$
$488$	0	0
$489$	−15660.0	−0.0654899
$490$	0	0
$491$	85667.2i	0.355346i	0.984090	+	0.177673i	$0.0568570\pi$
$491$	85667.2i	0.355346i	−0.984090	+	0.177673i	$0.943143\pi$
$492$	0	0
$493$	448386.	1.84484
$494$	0	0
$495$	− 9072.00i	− 0.0370248i
$496$	0	0
$497$	382104.	1.54692
$498$	0	0
$499$	− 318566.i	− 1.27938i	−0.768635	−	0.639688i	$-0.779064\pi$
$499$	− 318566.i	− 1.27938i	0.768635	−	0.639688i	$-0.220936\pi$
$500$	0	0
$501$	−58737.3	−0.234012
$502$	0	0
$503$	379404.i	1.49957i	0.661683	+	0.749784i	$0.269842\pi$
$503$	379404.i	1.49957i	−0.661683	+	0.749784i	$0.730158\pi$
$504$	0	0
$505$	−398076.	−1.56093
$506$	0	0
$507$	− 146163.i	− 0.568618i
$508$	0	0
$509$	−234973.	−0.906950	−0.453475	−	0.891269i	$-0.649816\pi$
$509$	−234973.	−0.906950	−0.453475	+	0.891269i	$0.649816\pi$
$510$	0	0
$511$	− 341852.i	− 1.30917i
$512$	0	0
$513$	−84564.0	−0.321330
$514$	0	0
$515$	− 129537.i	− 0.488403i
$516$	0	0
$517$	−5487.14	−0.0205289
$518$	0	0
$519$	− 183474.i	− 0.681145i
$520$	0	0
$521$	289422.	1.06624	0.533121	−	0.846039i	$-0.321019\pi$
$521$	289422.	1.06624	0.533121	+	0.846039i	$0.321019\pi$
$522$	0	0
$523$	382790.i	1.39945i	0.714412	+	0.699725i	$0.246694\pi$
$523$	382790.i	1.39945i	−0.714412	+	0.699725i	$0.753306\pi$
$524$	0	0
$525$	−11150.9	−0.0404569
$526$	0	0
$527$	− 33660.0i	− 0.121197i
$528$	0	0
$529$	60817.0	0.217327
$530$	0	0
$531$	− 72392.8i	− 0.256748i
$532$	0	0
$533$	−61730.3	−0.217292
$534$	0	0
$535$	− 527016.i	− 1.84126i
$536$	0	0
$537$	34812.0	0.120720
$538$	0	0
$539$	− 13343.7i	− 0.0459303i
$540$	0	0
$541$	79043.9	0.270068	0.135034	−	0.990841i	$-0.456886\pi$
$541$	79043.9	0.270068	0.135034	+	0.990841i	$0.456886\pi$
$542$	0	0
$543$	75060.0i	0.254571i
$544$	0	0
$545$	197064.	0.663459
$546$	0	0
$547$	305970.i	1.02260i	0.859403	+	0.511299i	$0.170835\pi$
$547$	305970.i	1.02260i	−0.859403	+	0.511299i	$0.829165\pi$
$548$	0	0
$549$	161621.	0.536233
$550$	0	0
$551$	883224.i	2.90916i
$552$	0	0
$553$	492188.	1.60946
$554$	0	0
$555$	130943.i	0.425105i
$556$	0	0
$557$	333714.	1.07563	0.537817	−	0.843062i	$-0.319249\pi$
$557$	333714.	1.07563	0.537817	+	0.843062i	$0.319249\pi$
$558$	0	0
$559$	60048.0i	0.192165i
$560$	0	0
$561$	22032.0	0.0700049
$562$	0	0
$563$	472088.i	1.48938i	0.667410	+	0.744691i	$0.267403\pi$
$563$	472088.i	1.48938i	−0.667410	+	0.744691i	$0.732597\pi$
$564$	0	0
$565$	372315.	1.16631
$566$	0	0
$567$	− 42282.0i	− 0.131519i
$568$	0	0
$569$	−21258.0	−0.0656595	−0.0328298	−	0.999461i	$-0.510452\pi$
$569$	−21258.0	−0.0656595	−0.0328298	+	0.999461i	$0.510452\pi$
$570$	0	0
$571$	− 490205.i	− 1.50351i	−0.659444	−	0.751754i	$-0.729208\pi$
$571$	− 490205.i	− 1.50351i	0.659444	−	0.751754i	$-0.270792\pi$
$572$	0	0
$573$	182198.	0.554925
$574$	0	0
$575$	− 17316.0i	− 0.0523735i
$576$	0	0
$577$	−454610.	−1.36549	−0.682743	−	0.730658i	$-0.739213\pi$
$577$	−454610.	−1.36549	−0.682743	+	0.730658i	$0.739213\pi$
$578$	0	0
$579$	− 58352.8i	− 0.174062i
$580$	0	0
$581$	803.672	0.00238082
$582$	0	0
$583$	15600.0i	0.0458973i
$584$	0	0
$585$	−13608.0	−0.0397633
$586$	0	0
$587$	− 35243.8i	− 0.102284i	−0.998691	−	0.0511418i	$-0.983714\pi$
$587$	− 35243.8i	− 0.102284i	0.998691	−	0.0511418i	$-0.0162861\pi$
$588$	0	0
$589$	66302.9	0.191118
$590$	0	0
$591$	150642.i	0.431292i
$592$	0	0
$593$	197658.	0.562089	0.281044	−	0.959695i	$-0.409319\pi$
$593$	197658.	0.562089	0.281044	+	0.959695i	$0.409319\pi$
$594$	0	0
$595$	430366.i	1.21564i
$596$	0	0
$597$	−94705.1	−0.265720
$598$	0	0
$599$	204156.i	0.568995i	0.958677	+	0.284498i	$0.0918268\pi$
$599$	204156.i	0.568995i	−0.958677	+	0.284498i	$0.908173\pi$
$600$	0	0
$601$	−294242.	−0.814621	−0.407311	−	0.913290i	$-0.633533\pi$
$601$	−294242.	−0.814621	−0.407311	+	0.913290i	$0.633533\pi$
$602$	0	0
$603$	− 129634.i	− 0.356519i
$604$	0	0
$605$	350370.	0.957229
$606$	0	0
$607$	331762.i	0.900429i	0.892921	+	0.450214i	$0.148652\pi$
$607$	331762.i	0.900429i	−0.892921	+	0.450214i	$0.851348\pi$
$608$	0	0
$609$	−441612.	−1.19071
$610$	0	0
$611$	8230.71i	0.0220473i
$612$	0	0
$613$	−448698.	−1.19408	−0.597040	−	0.802212i	$-0.703657\pi$
$613$	−448698.	−1.19408	−0.597040	+	0.802212i	$0.703657\pi$
$614$	0	0
$615$	− 374220.i	− 0.989411i
$616$	0	0
$617$	−358470.	−0.941635	−0.470817	−	0.882231i	$-0.656041\pi$
$617$	−358470.	−0.941635	−0.470817	+	0.882231i	$0.656041\pi$
$618$	0	0
$619$	321808.i	0.839877i	0.907553	+	0.419939i	$0.137948\pi$
$619$	321808.i	0.839877i	−0.907553	+	0.419939i	$0.862052\pi$
$620$	0	0
$621$	65658.6	0.170258
$622$	0	0
$623$	508428.i	1.30995i
$624$	0	0
$625$	−366131.	−0.937295
$626$	0	0
$627$	43398.3i	0.110392i
$628$	0	0
$629$	−318005.	−0.803770
$630$	0	0
$631$	− 26002.0i	− 0.0653052i	−0.999467	−	0.0326526i	$-0.989605\pi$
$631$	− 26002.0i	− 0.0653052i	0.999467	−	0.0326526i	$-0.0103955\pi$
$632$	0	0
$633$	193644.	0.483278
$634$	0	0
$635$	− 242439.i	− 0.601249i
$636$	0	0
$637$	−20015.6	−0.0493275
$638$	0	0
$639$	− 177876.i	− 0.435628i
$640$	0	0
$641$	322758.	0.785527	0.392763	−	0.919640i	$-0.371519\pi$
$641$	322758.	0.785527	0.392763	+	0.919640i	$0.371519\pi$
$642$	0	0
$643$	596788.i	1.44344i	0.692186	+	0.721720i	$0.256648\pi$
$643$	596788.i	1.44344i	−0.692186	+	0.721720i	$0.743352\pi$
$644$	0	0
$645$	−364022.	−0.874999
$646$	0	0
$647$	345348.i	0.824989i	0.910960	+	0.412495i	$0.135342\pi$
$647$	345348.i	0.824989i	−0.910960	+	0.412495i	$0.864658\pi$
$648$	0	0
$649$	−37152.0	−0.0882049
$650$	0	0
$651$	33151.5i	0.0782241i
$652$	0	0
$653$	76095.9	0.178458	0.0892288	−	0.996011i	$-0.471560\pi$
$653$	76095.9	0.178458	0.0892288	+	0.996011i	$0.471560\pi$
$654$	0	0
$655$	45528.0i	0.106120i
$656$	0	0
$657$	−159138.	−0.368675
$658$	0	0
$659$	− 303712.i	− 0.699344i	−0.936872	−	0.349672i	$-0.886293\pi$
$659$	− 303712.i	− 0.699344i	0.936872	−	0.349672i	$-0.113707\pi$
$660$	0	0
$661$	−595978.	−1.36404	−0.682020	−	0.731333i	$-0.738898\pi$
$661$	−595978.	−1.36404	−0.682020	+	0.731333i	$0.738898\pi$
$662$	0	0
$663$	− 33048.0i	− 0.0751827i
$664$	0	0
$665$	−847728.	−1.91696
$666$	0	0
$667$	− 685767.i	− 1.54143i
$668$	0	0
$669$	−52803.3	−0.117980
$670$	0	0
$671$	− 82944.0i	− 0.184221i
$672$	0	0
$673$	362542.	0.800439	0.400219	−	0.916419i	$-0.368934\pi$
$673$	362542.	0.800439	0.400219	+	0.916419i	$0.368934\pi$
$674$	0	0
$675$	5190.96i	0.0113930i
$676$	0	0
$677$	−109026.	−0.237876	−0.118938	−	0.992902i	$-0.537949\pi$
$677$	−109026.	−0.237876	−0.118938	+	0.992902i	$0.537949\pi$
$678$	0	0
$679$	− 343244.i	− 0.744498i
$680$	0	0
$681$	79056.0	0.170467
$682$	0	0
$683$	210382.i	0.450990i	0.974244	+	0.225495i	$0.0723999\pi$
$683$	210382.i	0.450990i	−0.974244	+	0.225495i	$0.927600\pi$
$684$	0	0
$685$	−676103.	−1.44089
$686$	0	0
$687$	− 40068.0i	− 0.0848954i
$688$	0	0
$689$	23400.0	0.0492921
$690$	0	0
$691$	− 521673.i	− 1.09255i	−0.837605	−	0.546276i	$-0.816045\pi$
$691$	− 521673.i	− 1.09255i	0.837605	−	0.546276i	$-0.183955\pi$
$692$	0	0
$693$	−21699.1	−0.0451831
$694$	0	0
$695$	274680.i	0.568666i
$696$	0	0
$697$	908820.	1.87074
$698$	0	0
$699$	110460.i	0.226074i
$700$	0	0
$701$	264813.	0.538894	0.269447	−	0.963015i	$-0.413159\pi$
$701$	264813.	0.538894	0.269447	+	0.963015i	$0.413159\pi$
$702$	0	0
$703$	− 626400.i	− 1.26748i
$704$	0	0
$705$	−49896.0	−0.100389
$706$	0	0
$707$	952150.i	1.90488i
$708$	0	0
$709$	143767.	0.286001	0.143000	−	0.989723i	$-0.454325\pi$
$709$	143767.	0.286001	0.143000	+	0.989723i	$0.454325\pi$
$710$	0	0
$711$	− 229122.i	− 0.453239i
$712$	0	0
$713$	−51480.0	−0.101265
$714$	0	0
$715$	6983.63i	0.0136606i
$716$	0	0
$717$	−504318.	−0.980993
$718$	0	0
$719$	− 283212.i	− 0.547840i	−0.961752	−	0.273920i	$-0.911680\pi$
$719$	− 283212.i	− 0.547840i	0.961752	−	0.273920i	$-0.0883204\pi$
$720$	0	0
$721$	−309836.	−0.596021
$722$	0	0
$723$	245477.i	0.469606i
$724$	0	0
$725$	54216.7	0.103147
$726$	0	0
$727$	225086.i	0.425873i	0.977066	+	0.212936i	$0.0683027\pi$
$727$	225086.i	0.425873i	−0.977066	+	0.212936i	$0.931697\pi$
$728$	0	0
$729$	−19683.0	−0.0370370
$730$	0	0
$731$	− 884053.i	− 1.65441i
$732$	0	0
$733$	368158.	0.685214	0.342607	−	0.939479i	$-0.388690\pi$
$733$	368158.	0.685214	0.342607	+	0.939479i	$0.388690\pi$
$734$	0	0
$735$	− 121338.i	− 0.224606i
$736$	0	0
$737$	−66528.0	−0.122481
$738$	0	0
$739$	− 1.06623e6i	− 1.95237i	−0.216942	−	0.976184i	$-0.569608\pi$
$739$	− 1.06623e6i	− 1.95237i	0.216942	−	0.976184i	$-0.430392\pi$
$740$	0	0
$741$	65097.4	0.118557
$742$	0	0
$743$	903024.i	1.63577i	0.575383	+	0.817884i	$0.304853\pi$
$743$	903024.i	1.63577i	−0.575383	+	0.817884i	$0.695147\pi$
$744$	0	0
$745$	−803460.	−1.44761
$746$	0	0
$747$	− 374.123i	0 0.000670460i
$748$	0	0
$749$	−1.26056e6	−2.24698
$750$	0	0
$751$	765038.i	1.35645i	0.734855	+	0.678224i	$0.237250\pi$
$751$	765038.i	1.35645i	−0.734855	+	0.678224i	$0.762750\pi$
$752$	0	0
$753$	406296.	0.716560
$754$	0	0
$755$	− 940705.i	− 1.65029i
$756$	0	0
$757$	−583944.	−1.01901	−0.509506	−	0.860467i	$-0.670172\pi$
$757$	−583944.	−1.01901	−0.509506	+	0.860467i	$0.670172\pi$
$758$	0	0
$759$	− 33696.0i	− 0.0584918i
$760$	0	0
$761$	20430.0	0.0352776	0.0176388	−	0.999844i	$-0.494385\pi$
$761$	20430.0	0.0352776	0.0176388	+	0.999844i	$0.494385\pi$
$762$	0	0
$763$	− 471353.i	− 0.809650i
$764$	0	0
$765$	200343.	0.342335
$766$	0	0
$767$	55728.0i	0.0947290i
$768$	0	0
$769$	−172654.	−0.291960	−0.145980	−	0.989288i	$-0.546634\pi$
$769$	−172654.	−0.291960	−0.145980	+	0.989288i	$0.546634\pi$
$770$	0	0
$771$	124302.i	0.209108i
$772$	0	0
$773$	−402525.	−0.673650	−0.336825	−	0.941567i	$-0.609353\pi$
$773$	−402525.	−0.673650	−0.336825	+	0.941567i	$0.609353\pi$
$774$	0	0
$775$	− 4070.00i	− 0.00677627i
$776$	0	0
$777$	313200.	0.518776
$778$	0	0
$779$	1.79018e6i	2.95000i
$780$	0	0
$781$	−91286.0	−0.149659
$782$	0	0
$783$	205578.i	0.335315i
$784$	0	0
$785$	−871920.	−1.41494
$786$	0	0
$787$	− 938986.i	− 1.51604i	−0.652233	−	0.758018i	$-0.726168\pi$
$787$	− 938986.i	− 1.51604i	0.652233	−	0.758018i	$-0.273832\pi$
$788$	0	0
$789$	439220.	0.705551
$790$	0	0
$791$	− 890532.i	− 1.42330i
$792$	0	0
$793$	−124416.	−0.197847
$794$	0	0
$795$	141855.i	0.224445i
$796$	0	0
$797$	−1.01547e6	−1.59864	−0.799320	−	0.600906i	$-0.794807\pi$
$797$	−1.01547e6	−1.59864	−0.799320	+	0.600906i	$0.794807\pi$
$798$	0	0
$799$	− 121176.i	− 0.189812i
$800$	0	0
$801$	236682.	0.368893
$802$	0	0
$803$	81669.7i	0.126657i
$804$	0	0
$805$	658207.	1.01571
$806$	0	0
$807$	− 481950.i	− 0.740040i
$808$	0	0
$809$	572742.	0.875109	0.437554	−	0.899192i	$-0.355845\pi$
$809$	572742.	0.875109	0.437554	+	0.899192i	$0.355845\pi$
$810$	0	0
$811$	503590.i	0.765659i	0.923819	+	0.382830i	$0.125050\pi$
$811$	503590.i	0.765659i	−0.923819	+	0.382830i	$0.874950\pi$
$812$	0	0
$813$	317578.	0.480474
$814$	0	0
$815$	73080.0i	0.110023i
$816$	0	0
$817$	1.74139e6	2.60887
$818$	0	0
$819$	32548.7i	0.0485250i
$820$	0	0
$821$	−345783.	−0.513000	−0.256500	−	0.966544i	$-0.582569\pi$
$821$	−345783.	−0.513000	−0.256500	+	0.966544i	$0.582569\pi$
$822$	0	0
$823$	− 739546.i	− 1.09186i	−0.837832	−	0.545928i	$-0.816177\pi$
$823$	− 739546.i	− 1.09186i	0.837832	−	0.545928i	$-0.183823\pi$
$824$	0	0
$825$	2664.00	0.00391405
$826$	0	0
$827$	− 551769.i	− 0.806764i	−0.915032	−	0.403382i	$-0.867835\pi$
$827$	− 551769.i	− 0.806764i	0.915032	−	0.403382i	$-0.132165\pi$
$828$	0	0
$829$	−885944.	−1.28913	−0.644566	−	0.764549i	$-0.722962\pi$
$829$	−885944.	−1.28913	−0.644566	+	0.764549i	$0.722962\pi$
$830$	0	0
$831$	− 360828.i	− 0.522514i
$832$	0	0
$833$	294678.	0.424676
$834$	0	0
$835$	274107.i	0.393141i
$836$	0	0
$837$	15432.6	0.0220286
$838$	0	0
$839$	− 418140.i	− 0.594016i	−0.954875	−	0.297008i	$-0.904011\pi$
$839$	− 418140.i	− 0.594016i	0.954875	−	0.297008i	$-0.0959887\pi$
$840$	0	0
$841$	1.43987e6	2.03578
$842$	0	0
$843$	314731.i	0.442878i
$844$	0	0
$845$	−682092.	−0.955277
$846$	0	0
$847$	− 838042.i	− 1.16815i
$848$	0	0
$849$	455004.	0.631248
$850$	0	0
$851$	486360.i	0.671581i
$852$	0	0
$853$	−422634.	−0.580854	−0.290427	−	0.956897i	$-0.593797\pi$
$853$	−422634.	−0.580854	−0.290427	+	0.956897i	$0.593797\pi$
$854$	0	0
$855$	394632.i	0.539834i
$856$	0	0
$857$	−49914.0	−0.0679612	−0.0339806	−	0.999422i	$-0.510818\pi$
$857$	−49914.0	−0.0679612	−0.0339806	+	0.999422i	$0.510818\pi$
$858$	0	0
$859$	726443.i	0.984499i	0.870454	+	0.492249i	$0.163825\pi$
$859$	726443.i	0.984499i	−0.870454	+	0.492249i	$0.836175\pi$
$860$	0	0
$861$	−895089.	−1.20742
$862$	0	0
$863$	375912.i	0.504736i	0.967631	+	0.252368i	$0.0812094\pi$
$863$	375912.i	0.504736i	−0.967631	+	0.252368i	$0.918791\pi$
$864$	0	0
$865$	−856212.	−1.14432
$866$	0	0
$867$	52559.1i	0.0699213i
$868$	0	0
$869$	−117585.	−0.155709
$870$	0	0
$871$	99792.0i	0.131540i
$872$	0	0
$873$	−159786.	−0.209657
$874$	0	0
$875$	931054.i	1.21607i
$876$	0	0
$877$	−1.41335e6	−1.83760	−0.918801	−	0.394720i	$-0.870841\pi$
$877$	−1.41335e6	−1.83760	−0.918801	+	0.394720i	$0.870841\pi$
$878$	0	0
$879$	− 400950.i	− 0.518934i
$880$	0	0
$881$	−689742.	−0.888658	−0.444329	−	0.895864i	$-0.646558\pi$
$881$	−689742.	−0.888658	−0.444329	+	0.895864i	$0.646558\pi$
$882$	0	0
$883$	707695.i	0.907663i	0.891087	+	0.453832i	$0.149943\pi$
$883$	707695.i	0.907663i	−0.891087	+	0.453832i	$0.850057\pi$
$884$	0	0
$885$	−337833.	−0.431336
$886$	0	0
$887$	− 706032.i	− 0.897382i	−0.893687	−	0.448691i	$-0.851890\pi$
$887$	− 706032.i	− 0.897382i	0.893687	−	0.448691i	$-0.148110\pi$
$888$	0	0
$889$	−579884.	−0.733732
$890$	0	0
$891$	10101.3i	0.0127240i
$892$	0	0
$893$	238690.	0.299318
$894$	0	0
$895$	− 162456.i	− 0.202810i
$896$	0	0
$897$	−50544.0	−0.0628181
$898$	0	0
$899$	− 161185.i	− 0.199436i
$900$	0	0
$901$	−344505.	−0.424371
$902$	0	0
$903$	870696.i	1.06780i
$904$	0	0
$905$	350280.	0.427679
$906$	0	0
$907$	13364.5i	0.0162457i	0.999967	+	0.00812285i	$0.00258561\pi$
$907$	13364.5i	0.0162457i	−0.999967	+	0.00812285i	$0.997414\pi$
$908$	0	0
$909$	443242.	0.536430
$910$	0	0
$911$	194544.i	0.234413i	0.993108	+	0.117206i	$0.0373939\pi$
$911$	194544.i	0.234413i	−0.993108	+	0.117206i	$0.962606\pi$
$912$	0	0
$913$	−192.000	−0.000230335 0
$914$	0	0
$915$	− 754232.i	− 0.900871i
$916$	0	0
$917$	108897.	0.129503
$918$	0	0
$919$	1.17935e6i	1.39641i	0.715900	+	0.698203i	$0.246017\pi$
$919$	1.17935e6i	1.39641i	−0.715900	+	0.698203i	$0.753983\pi$
$920$	0	0
$921$	65772.0	0.0775393
$922$	0	0
$923$	136929.i	0.160728i
$924$	0	0
$925$	−38451.5	−0.0449397
$926$	0	0
$927$	144234.i	0.167845i
$928$	0	0
$929$	232110.	0.268944	0.134472	−	0.990917i	$-0.457066\pi$
$929$	232110.	0.268944	0.134472	+	0.990917i	$0.457066\pi$
$930$	0	0
$931$	580452.i	0.669679i
$932$	0	0
$933$	−714949.	−0.821319
$934$	0	0
$935$	− 102816.i	− 0.117608i
$936$	0	0
$937$	−1.21008e6	−1.37827	−0.689135	−	0.724633i	$-0.742009\pi$
$937$	−1.21008e6	−1.37827	−0.689135	+	0.724633i	$0.742009\pi$
$938$	0	0
$939$	− 68578.8i	− 0.0777784i
$940$	0	0
$941$	1.19574e6	1.35038	0.675190	−	0.737644i	$-0.264062\pi$
$941$	1.19574e6	1.35038	0.675190	+	0.737644i	$0.264062\pi$
$942$	0	0
$943$	− 1.38996e6i	− 1.56307i
$944$	0	0
$945$	−197316.	−0.220952
$946$	0	0
$947$	− 986985.i	− 1.10055i	−0.834983	−	0.550276i	$-0.814523\pi$
$947$	− 986985.i	− 1.10055i	0.834983	−	0.550276i	$-0.185477\pi$
$948$	0	0
$949$	122504.	0.136025
$950$	0	0
$951$	696834.i	0.770492i
$952$	0	0
$953$	156078.	0.171853	0.0859263	−	0.996301i	$-0.472615\pi$
$953$	156078.	0.171853	0.0859263	+	0.996301i	$0.472615\pi$
$954$	0	0
$955$	− 850257.i	− 0.932274i
$956$	0	0
$957$	105503.	0.115197
$958$	0	0
$959$	1.61716e6i	1.75839i
$960$	0	0
$961$	911421.	0.986898
$962$	0	0
$963$	586812.i	0.632771i
$964$	0	0
$965$	−272313.	−0.292425
$966$	0	0
$967$	− 1.04281e6i	− 1.11520i	−0.830109	−	0.557601i	$-0.811722\pi$
$967$	− 1.04281e6i	− 1.11520i	0.830109	−	0.557601i	$-0.188278\pi$
$968$	0	0
$969$	−958392.	−1.02069
$970$	0	0
$971$	− 1.15445e6i	− 1.22443i	−0.790690	−	0.612217i	$-0.790278\pi$
$971$	− 1.15445e6i	− 1.22443i	0.790690	−	0.612217i	$-0.209722\pi$
$972$	0	0
$973$	657002.	0.693970
$974$	0	0
$975$	− 3996.00i	− 0.00420355i
$976$	0	0
$977$	−1.24965e6	−1.30918	−0.654590	−	0.755984i	$-0.727159\pi$
$977$	−1.24965e6	−1.30918	−0.654590	+	0.755984i	$0.727159\pi$
$978$	0	0
$979$	− 121465.i	− 0.126732i
$980$	0	0
$981$	−219423.	−0.228005
$982$	0	0
$983$	− 839088.i	− 0.868361i	−0.900826	−	0.434181i	$-0.857038\pi$
$983$	− 839088.i	− 0.868361i	0.900826	−	0.434181i	$-0.142962\pi$
$984$	0	0
$985$	702996.	0.724570
$986$	0	0
$987$	119345.i	0.122510i
$988$	0	0
$989$	−1.35208e6	−1.38232
$990$	0	0
$991$	422558.i	0.430268i	0.976585	+	0.215134i	$0.0690188\pi$
$991$	422558.i	0.430268i	−0.976585	+	0.215134i	$0.930981\pi$
$992$	0	0
$993$	270756.	0.274587
$994$	0	0
$995$	441957.i	0.446410i
$996$	0	0
$997$	−945533.	−0.951232	−0.475616	−	0.879653i	$-0.657775\pi$
$997$	−945533.	−0.951232	−0.475616	+	0.879653i	$0.657775\pi$
$998$	0	0
$999$	− 145800.i	− 0.146092i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.5.g.e.511.4		4
4.3	odd	2	inner	768.5.g.e.511.2		4
8.3	odd	2	inner	768.5.g.e.511.3		4
8.5	even	2	inner	768.5.g.e.511.1		4
16.3	odd	4		192.5.b.a.31.3	yes	4
16.5	even	4		192.5.b.a.31.4	yes	4
16.11	odd	4		192.5.b.a.31.2	yes	4
16.13	even	4		192.5.b.a.31.1	✓	4
48.5	odd	4		576.5.b.f.415.2		4
48.11	even	4		576.5.b.f.415.1		4
48.29	odd	4		576.5.b.f.415.4		4
48.35	even	4		576.5.b.f.415.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
192.5.b.a.31.1	✓	4	16.13	even	4
192.5.b.a.31.2	yes	4	16.11	odd	4
192.5.b.a.31.3	yes	4	16.3	odd	4
192.5.b.a.31.4	yes	4	16.5	even	4
576.5.b.f.415.1		4	48.11	even	4
576.5.b.f.415.2		4	48.5	odd	4
576.5.b.f.415.3		4	48.35	even	4
576.5.b.f.415.4		4	48.29	odd	4
768.5.g.e.511.1		4	8.5	even	2	inner
768.5.g.e.511.2		4	4.3	odd	2	inner
768.5.g.e.511.3		4	8.3	odd	2	inner
768.5.g.e.511.4		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	768.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+5.19615i q^{3} +24.2487 q^{5} -58.0000i q^{7} -27.0000 q^{9} +13.8564i q^{11} +20.7846 q^{13} +126.000i q^{15} -306.000 q^{17} -602.754i q^{19} +301.377 q^{21} +468.000i q^{23} -37.0000 q^{25} -140.296i q^{27} -1465.31 q^{29} +110.000i q^{31} -72.0000 q^{33} -1406.43i q^{35} +1039.23 q^{37} +108.000i q^{39} -2970.00 q^{41} +2889.06i q^{43} -654.715 q^{45} +396.000i q^{47} -963.000 q^{49} -1590.02i q^{51} +1125.83 q^{53} +336.000i q^{55} +3132.00 q^{57} +2681.21i q^{59} -5985.97 q^{61} +1566.00i q^{63} +504.000 q^{65} +4801.24i q^{67} -2431.80 q^{69} +6588.00i q^{71} +5894.00 q^{73} -192.258i q^{75} +803.672 q^{77} +8486.00i q^{79} +729.000 q^{81} +13.8564i q^{83} -7420.11 q^{85} -7614.00i q^{87} -8766.00 q^{89} -1205.51i q^{91} -571.577 q^{93} -14616.0i q^{95} +5918.00 q^{97} -374.123i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 108 q^{9} - 1224 q^{17} - 148 q^{25} - 288 q^{33} - 11880 q^{41} - 3852 q^{49} + 12528 q^{57} + 2016 q^{65} + 23576 q^{73} + 2916 q^{81} - 35064 q^{89} + 23672 q^{97}+O(q^{100}) \) 4 * q - 108 * q^9 - 1224 * q^17 - 148 * q^25 - 288 * q^33 - 11880 * q^41 - 3852 * q^49 + 12528 * q^57 + 2016 * q^65 + 23576 * q^73 + 2916 * q^81 - 35064 * q^89 + 23672 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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