LMFDB - Embedded newform 768.5.g.k.511.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,5,Mod(511,768)]

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

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$79.3881316484$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 48x^{14} + 858x^{12} + 7028x^{10} + 25803x^{8} + 34572x^{6} + 14794x^{4} + 708x^{2} + 9 $ x^16 + 48x^14 + 858x^12 + 7028x^10 + 25803x^8 + 34572x^6 + 14794x^4 + 708*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{88}\cdot 3^{6} $
Twist minimal:	no (minimal twist has level 24)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			511.1
Root			$-4.12169i$ of defining polynomial
Character	$\chi$	$=$	768.511
Dual form			768.5.g.k.511.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.19615i q^{3} -38.1617 q^{5} -42.0542i q^{7} -27.0000 q^{9} +O(q^{10})$	$q-5.19615i q^{3} -38.1617 q^{5} -42.0542i q^{7} -27.0000 q^{9} +100.141i q^{11} -171.607 q^{13} +198.294i q^{15} -351.156 q^{17} -494.248i q^{19} -218.520 q^{21} +254.258i q^{23} +831.317 q^{25} +140.296i q^{27} -1388.76 q^{29} +603.769i q^{31} +520.347 q^{33} +1604.86i q^{35} -538.756 q^{37} +891.695i q^{39} -540.370 q^{41} +2047.07i q^{43} +1030.37 q^{45} -1830.56i q^{47} +632.448 q^{49} +1824.66i q^{51} -3473.31 q^{53} -3821.54i q^{55} -2568.19 q^{57} -1577.30i q^{59} -1057.51 q^{61} +1135.46i q^{63} +6548.81 q^{65} +517.305i q^{67} +1321.16 q^{69} +1950.41i q^{71} +8306.47 q^{73} -4319.65i q^{75} +4211.34 q^{77} -11240.5i q^{79} +729.000 q^{81} +719.298i q^{83} +13400.7 q^{85} +7216.23i q^{87} +4185.54 q^{89} +7216.78i q^{91} +3137.27 q^{93} +18861.4i q^{95} -4891.42 q^{97} -2703.80i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 432 q^{9}+O(q^{10}) $ 16 * q - 432 * q^9	$ 16 q - 432 q^{9} + 480 q^{17} + 1328 q^{25} - 1440 q^{41} - 2480 q^{49} - 7488 q^{57} - 2688 q^{65} + 33760 q^{73} + 11664 q^{81} - 31200 q^{89} - 24352 q^{97}+O(q^{100}) $ 16 * q - 432 * q^9 + 480 * q^17 + 1328 * q^25 - 1440 * q^41 - 2480 * q^49 - 7488 * q^57 - 2688 * q^65 + 33760 * q^73 + 11664 * q^81 - 31200 * q^89 - 24352 * q^97

Display 100 coefficients 10 coefficients

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
$5$	−38.1617	−1.52647	−0.763234	−	0.646122i	$-0.776390\pi$
$5$	−38.1617	−1.52647	−0.763234	+	0.646122i	$0.776390\pi$
$6$	0	0
$7$	− 42.0542i	− 0.858248i	−0.903246	−	0.429124i	$-0.858822\pi$
$7$	− 42.0542i	− 0.858248i	0.903246	−	0.429124i	$-0.141178\pi$
$8$	0	0
$9$	−27.0000	−0.333333
$10$	0	0
$11$	100.141i	0.827610i	0.910366	+	0.413805i	$0.135800\pi$
$11$	100.141i	0.827610i	−0.910366	+	0.413805i	$0.864200\pi$
$12$	0	0
$13$	−171.607	−1.01543	−0.507713	−	0.861527i	$-0.669509\pi$
$13$	−171.607	−1.01543	−0.507713	+	0.861527i	$0.669509\pi$
$14$	0	0
$15$	198.294i	0.881307i
$16$	0	0
$17$	−351.156	−1.21507	−0.607537	−	0.794292i	$-0.707842\pi$
$17$	−351.156	−1.21507	−0.607537	+	0.794292i	$0.707842\pi$
$18$	0	0
$19$	− 494.248i	− 1.36911i	−0.728962	−	0.684555i	$-0.759997\pi$
$19$	− 494.248i	− 1.36911i	0.728962	−	0.684555i	$-0.240003\pi$
$20$	0	0
$21$	−218.520	−0.495510
$22$	0	0
$23$	254.258i	0.480638i	0.970694	+	0.240319i	$0.0772521\pi$
$23$	254.258i	0.480638i	−0.970694	+	0.240319i	$0.922748\pi$
$24$	0	0
$25$	831.317	1.33011
$26$	0	0
$27$	140.296i	0.192450i
$28$	0	0
$29$	−1388.76	−1.65132	−0.825662	−	0.564165i	$-0.809198\pi$
$29$	−1388.76	−1.65132	−0.825662	+	0.564165i	$0.809198\pi$
$30$	0	0
$31$	603.769i	0.628271i	0.949378	+	0.314136i	$0.101715\pi$
$31$	603.769i	0.628271i	−0.949378	+	0.314136i	$0.898285\pi$
$32$	0	0
$33$	520.347	0.477821
$34$	0	0
$35$	1604.86i	1.31009i
$36$	0	0
$37$	−538.756	−0.393540	−0.196770	−	0.980450i	$-0.563045\pi$
$37$	−538.756	−0.393540	−0.196770	+	0.980450i	$0.563045\pi$
$38$	0	0
$39$	891.695i	0.586256i
$40$	0	0
$41$	−540.370	−0.321458	−0.160729	−	0.986999i	$-0.551384\pi$
$41$	−540.370	−0.321458	−0.160729	+	0.986999i	$0.551384\pi$
$42$	0	0
$43$	2047.07i	1.10712i	0.832808	+	0.553561i	$0.186732\pi$
$43$	2047.07i	1.10712i	−0.832808	+	0.553561i	$0.813268\pi$
$44$	0	0
$45$	1030.37	0.508823
$46$	0	0
$47$	− 1830.56i	− 0.828685i	−0.910121	−	0.414342i	$-0.864012\pi$
$47$	− 1830.56i	− 0.828685i	0.910121	−	0.414342i	$-0.135988\pi$
$48$	0	0
$49$	632.448	0.263410
$50$	0	0
$51$	1824.66i	0.701523i
$52$	0	0
$53$	−3473.31	−1.23649	−0.618247	−	0.785984i	$-0.712157\pi$
$53$	−3473.31	−1.23649	−0.618247	+	0.785984i	$0.712157\pi$
$54$	0	0
$55$	− 3821.54i	− 1.26332i
$56$	0	0
$57$	−2568.19	−0.790456
$58$	0	0
$59$	− 1577.30i	− 0.453116i	−0.973998	−	0.226558i	$-0.927253\pi$
$59$	− 1577.30i	− 0.453116i	0.973998	−	0.226558i	$-0.0727472\pi$
$60$	0	0
$61$	−1057.51	−0.284201	−0.142100	−	0.989852i	$-0.545386\pi$
$61$	−1057.51	−0.284201	−0.142100	+	0.989852i	$0.545386\pi$
$62$	0	0
$63$	1135.46i	0.286083i
$64$	0	0
$65$	6548.81	1.55001
$66$	0	0
$67$	517.305i	0.115238i	0.998339	+	0.0576191i	$0.0183509\pi$
$67$	517.305i	0.115238i	−0.998339	+	0.0576191i	$0.981649\pi$
$68$	0	0
$69$	1321.16	0.277497
$70$	0	0
$71$	1950.41i	0.386909i	0.981109	+	0.193454i	$0.0619692\pi$
$71$	1950.41i	0.386909i	−0.981109	+	0.193454i	$0.938031\pi$
$72$	0	0
$73$	8306.47	1.55873	0.779364	−	0.626571i	$-0.215542\pi$
$73$	8306.47	1.55873	0.779364	+	0.626571i	$0.215542\pi$
$74$	0	0
$75$	− 4319.65i	− 0.767938i
$76$	0	0
$77$	4211.34	0.710295
$78$	0	0
$79$	− 11240.5i	− 1.80107i	−0.434782	−	0.900536i	$-0.643175\pi$
$79$	− 11240.5i	− 1.80107i	0.434782	−	0.900536i	$-0.356825\pi$
$80$	0	0
$81$	729.000	0.111111
$82$	0	0
$83$	719.298i	0.104413i	0.998636	+	0.0522063i	$0.0166253\pi$
$83$	719.298i	0.104413i	−0.998636	+	0.0522063i	$0.983375\pi$
$84$	0	0
$85$	13400.7	1.85477
$86$	0	0
$87$	7216.23i	0.953393i
$88$	0	0
$89$	4185.54	0.528410	0.264205	−	0.964467i	$-0.414890\pi$
$89$	4185.54	0.528410	0.264205	+	0.964467i	$0.414890\pi$
$90$	0	0
$91$	7216.78i	0.871487i
$92$	0	0
$93$	3137.27	0.362733
$94$	0	0
$95$	18861.4i	2.08990i
$96$	0	0
$97$	−4891.42	−0.519866	−0.259933	−	0.965627i	$-0.583701\pi$
$97$	−4891.42	−0.519866	−0.259933	+	0.965627i	$0.583701\pi$
$98$	0	0
$99$	− 2703.80i	− 0.275870i
$100$	0	0
$101$	3582.12	0.351154	0.175577	−	0.984466i	$-0.443821\pi$
$101$	3582.12	0.351154	0.175577	+	0.984466i	$0.443821\pi$
$102$	0	0
$103$	− 17391.9i	− 1.63935i	−0.572826	−	0.819677i	$-0.694153\pi$
$103$	− 17391.9i	− 1.63935i	0.572826	−	0.819677i	$-0.305847\pi$
$104$	0	0
$105$	8339.09	0.756380
$106$	0	0
$107$	11793.1i	1.03005i	0.857174	+	0.515027i	$0.172218\pi$
$107$	11793.1i	1.03005i	−0.857174	+	0.515027i	$0.827782\pi$
$108$	0	0
$109$	10564.5	0.889192	0.444596	−	0.895731i	$-0.353347\pi$
$109$	10564.5	0.889192	0.444596	+	0.895731i	$0.353347\pi$
$110$	0	0
$111$	2799.46i	0.227210i
$112$	0	0
$113$	7408.35	0.580183	0.290091	−	0.956999i	$-0.406314\pi$
$113$	7408.35	0.580183	0.290091	+	0.956999i	$0.406314\pi$
$114$	0	0
$115$	− 9702.91i	− 0.733680i
$116$	0	0
$117$	4633.38	0.338475
$118$	0	0
$119$	14767.6i	1.04283i
$120$	0	0
$121$	4612.82	0.315062
$122$	0	0
$123$	2807.85i	0.185594i
$124$	0	0
$125$	−7873.41	−0.503898
$126$	0	0
$127$	− 443.504i	− 0.0274973i	−0.999905	−	0.0137487i	$-0.995624\pi$
$127$	− 443.504i	− 0.0274973i	0.999905	−	0.0137487i	$-0.00437647\pi$
$128$	0	0
$129$	10636.9	0.639198
$130$	0	0
$131$	− 106.777i	− 0.00622209i	−0.999995	−	0.00311105i	$-0.999010\pi$
$131$	− 106.777i	− 0.00622209i	0.999995	−	0.00311105i	$-0.000990279\pi$
$132$	0	0
$133$	−20785.2	−1.17504
$134$	0	0
$135$	− 5353.94i	− 0.293769i
$136$	0	0
$137$	29064.5	1.54854	0.774268	−	0.632858i	$-0.218118\pi$
$137$	29064.5	1.54854	0.774268	+	0.632858i	$0.218118\pi$
$138$	0	0
$139$	− 32437.6i	− 1.67888i	−0.543453	−	0.839440i	$-0.682883\pi$
$139$	− 32437.6i	− 1.67888i	0.543453	−	0.839440i	$-0.317117\pi$
$140$	0	0
$141$	−9511.89	−0.478441
$142$	0	0
$143$	− 17184.8i	− 0.840376i
$144$	0	0
$145$	52997.6	2.52070
$146$	0	0
$147$	− 3286.29i	− 0.152080i
$148$	0	0
$149$	−5874.77	−0.264617	−0.132309	−	0.991209i	$-0.542239\pi$
$149$	−5874.77	−0.264617	−0.132309	+	0.991209i	$0.542239\pi$
$150$	0	0
$151$	13831.1i	0.606599i	0.952895	+	0.303300i	$0.0980883\pi$
$151$	13831.1i	0.606599i	−0.952895	+	0.303300i	$0.901912\pi$
$152$	0	0
$153$	9481.22	0.405024
$154$	0	0
$155$	− 23040.8i	− 0.959036i
$156$	0	0
$157$	−22719.2	−0.921708	−0.460854	−	0.887476i	$-0.652457\pi$
$157$	−22719.2	−0.921708	−0.460854	+	0.887476i	$0.652457\pi$
$158$	0	0
$159$	18047.9i	0.713890i
$160$	0	0
$161$	10692.6	0.412507
$162$	0	0
$163$	14332.3i	0.539438i	0.962939	+	0.269719i	$0.0869308\pi$
$163$	14332.3i	0.539438i	−0.962939	+	0.269719i	$0.913069\pi$
$164$	0	0
$165$	−19857.3	−0.729379
$166$	0	0
$167$	− 2863.09i	− 0.102660i	−0.998682	−	0.0513301i	$-0.983654\pi$
$167$	− 2863.09i	− 0.102660i	0.998682	−	0.0513301i	$-0.0163461\pi$
$168$	0	0
$169$	887.905	0.0310880
$170$	0	0
$171$	13344.7i	0.456370i
$172$	0	0
$173$	−14515.8	−0.485008	−0.242504	−	0.970150i	$-0.577969\pi$
$173$	−14515.8	−0.485008	−0.242504	+	0.970150i	$0.577969\pi$
$174$	0	0
$175$	− 34960.3i	− 1.14156i
$176$	0	0
$177$	−8195.87	−0.261606
$178$	0	0
$179$	8072.26i	0.251935i	0.992034	+	0.125968i	$0.0402036\pi$
$179$	8072.26i	0.251935i	−0.992034	+	0.125968i	$0.959796\pi$
$180$	0	0
$181$	−36860.1	−1.12512	−0.562561	−	0.826756i	$-0.690184\pi$
$181$	−36860.1	−1.12512	−0.562561	+	0.826756i	$0.690184\pi$
$182$	0	0
$183$	5494.99i	0.164083i
$184$	0	0
$185$	20559.8	0.600726
$186$	0	0
$187$	− 35165.1i	− 1.00561i
$188$	0	0
$189$	5900.04	0.165170
$190$	0	0
$191$	2132.11i	0.0584444i	0.999573	+	0.0292222i	$0.00930304\pi$
$191$	2132.11i	0.0584444i	−0.999573	+	0.0292222i	$0.990697\pi$
$192$	0	0
$193$	33387.8	0.896341	0.448170	−	0.893948i	$-0.352076\pi$
$193$	33387.8	0.896341	0.448170	+	0.893948i	$0.352076\pi$
$194$	0	0
$195$	− 34028.6i	− 0.894901i
$196$	0	0
$197$	40238.1	1.03682	0.518412	−	0.855131i	$-0.326523\pi$
$197$	40238.1	1.03682	0.518412	+	0.855131i	$0.326523\pi$
$198$	0	0
$199$	− 2284.56i	− 0.0576894i	−0.999584	−	0.0288447i	$-0.990817\pi$
$199$	− 2284.56i	− 0.0576894i	0.999584	−	0.0288447i	$-0.00918283\pi$
$200$	0	0
$201$	2687.99	0.0665328
$202$	0	0
$203$	58403.3i	1.41725i
$204$	0	0
$205$	20621.5	0.490695
$206$	0	0
$207$	− 6864.96i	− 0.160213i
$208$	0	0
$209$	49494.4	1.13309
$210$	0	0
$211$	− 13228.5i	− 0.297129i	−0.988903	−	0.148565i	$-0.952535\pi$
$211$	− 13228.5i	− 0.297129i	0.988903	−	0.148565i	$-0.0474653\pi$
$212$	0	0
$213$	10134.6	0.223382
$214$	0	0
$215$	− 78119.7i	− 1.68999i
$216$	0	0
$217$	25391.0	0.539213
$218$	0	0
$219$	− 43161.7i	− 0.899932i
$220$	0	0
$221$	60260.8	1.23382
$222$	0	0
$223$	59636.9i	1.19924i	0.800286	+	0.599619i	$0.204681\pi$
$223$	59636.9i	1.19924i	−0.800286	+	0.599619i	$0.795319\pi$
$224$	0	0
$225$	−22445.6	−0.443369
$226$	0	0
$227$	10099.9i	0.196005i	0.995186	+	0.0980024i	$0.0312453\pi$
$227$	10099.9i	0.196005i	−0.995186	+	0.0980024i	$0.968755\pi$
$228$	0	0
$229$	33647.7	0.641630	0.320815	−	0.947142i	$-0.396043\pi$
$229$	33647.7	0.641630	0.320815	+	0.947142i	$0.396043\pi$
$230$	0	0
$231$	− 21882.7i	− 0.410089i
$232$	0	0
$233$	74833.7	1.37843	0.689216	−	0.724556i	$-0.257955\pi$
$233$	74833.7	1.37843	0.689216	+	0.724556i	$0.257955\pi$
$234$	0	0
$235$	69857.5i	1.26496i
$236$	0	0
$237$	−58407.3	−1.03985
$238$	0	0
$239$	− 24812.5i	− 0.434384i	−0.976129	−	0.217192i	$-0.930310\pi$
$239$	− 24812.5i	− 0.434384i	0.976129	−	0.217192i	$-0.0696898\pi$
$240$	0	0
$241$	−15222.1	−0.262084	−0.131042	−	0.991377i	$-0.541832\pi$
$241$	−15222.1	−0.262084	−0.131042	+	0.991377i	$0.541832\pi$
$242$	0	0
$243$	− 3788.00i	− 0.0641500i
$244$	0	0
$245$	−24135.3	−0.402087
$246$	0	0
$247$	84816.4i	1.39023i
$248$	0	0
$249$	3737.58	0.0602827
$250$	0	0
$251$	91146.1i	1.44674i	0.690461	+	0.723370i	$0.257408\pi$
$251$	91146.1i	1.44674i	−0.690461	+	0.723370i	$0.742592\pi$
$252$	0	0
$253$	−25461.6	−0.397781
$254$	0	0
$255$	− 69632.2i	− 1.07085i
$256$	0	0
$257$	−24844.7	−0.376156	−0.188078	−	0.982154i	$-0.560226\pi$
$257$	−24844.7	−0.376156	−0.188078	+	0.982154i	$0.560226\pi$
$258$	0	0
$259$	22656.9i	0.337755i
$260$	0	0
$261$	37496.6	0.550442
$262$	0	0
$263$	65761.8i	0.950741i	0.879786	+	0.475371i	$0.157686\pi$
$263$	65761.8i	0.950741i	−0.879786	+	0.475371i	$0.842314\pi$
$264$	0	0
$265$	132548.	1.88747
$266$	0	0
$267$	− 21748.7i	− 0.305078i
$268$	0	0
$269$	−3022.19	−0.0417654	−0.0208827	−	0.999782i	$-0.506648\pi$
$269$	−3022.19	−0.0417654	−0.0208827	+	0.999782i	$0.506648\pi$
$270$	0	0
$271$	145358.i	1.97925i	0.143665	+	0.989626i	$0.454111\pi$
$271$	145358.i	1.97925i	−0.143665	+	0.989626i	$0.545889\pi$
$272$	0	0
$273$	37499.5	0.503153
$274$	0	0
$275$	83248.7i	1.10081i
$276$	0	0
$277$	120910.	1.57580	0.787901	−	0.615802i	$-0.211168\pi$
$277$	120910.	1.57580	0.787901	+	0.615802i	$0.211168\pi$
$278$	0	0
$279$	− 16301.8i	− 0.209424i
$280$	0	0
$281$	−139506.	−1.76677	−0.883383	−	0.468653i	$-0.844740\pi$
$281$	−139506.	−1.76677	−0.883383	+	0.468653i	$0.844740\pi$
$282$	0	0
$283$	− 52948.2i	− 0.661117i	−0.943786	−	0.330558i	$-0.892763\pi$
$283$	− 52948.2i	− 0.661117i	0.943786	−	0.330558i	$-0.107237\pi$
$284$	0	0
$285$	98006.6	1.20661
$286$	0	0
$287$	22724.8i	0.275890i
$288$	0	0
$289$	39789.6	0.476403
$290$	0	0
$291$	25416.6i	0.300145i
$292$	0	0
$293$	22036.7	0.256691	0.128346	−	0.991730i	$-0.459033\pi$
$293$	22036.7	0.256691	0.128346	+	0.991730i	$0.459033\pi$
$294$	0	0
$295$	60192.3i	0.691667i
$296$	0	0
$297$	−14049.4	−0.159274
$298$	0	0
$299$	− 43632.4i	− 0.488052i
$300$	0	0
$301$	86087.8	0.950186
$302$	0	0
$303$	− 18613.3i	− 0.202739i
$304$	0	0
$305$	40356.5	0.433824
$306$	0	0
$307$	− 144674.i	− 1.53502i	−0.641037	−	0.767510i	$-0.721496\pi$
$307$	− 144674.i	− 1.53502i	0.641037	−	0.767510i	$-0.278504\pi$
$308$	0	0
$309$	−90371.0	−0.946481
$310$	0	0
$311$	− 58811.2i	− 0.608050i	−0.952664	−	0.304025i	$-0.901669\pi$
$311$	− 58811.2i	− 0.608050i	0.952664	−	0.304025i	$-0.0983306\pi$
$312$	0	0
$313$	−91670.3	−0.935707	−0.467853	−	0.883806i	$-0.654972\pi$
$313$	−91670.3	−0.935707	−0.467853	+	0.883806i	$0.654972\pi$
$314$	0	0
$315$	− 43331.2i	− 0.436696i
$316$	0	0
$317$	68445.4	0.681124	0.340562	−	0.940222i	$-0.389383\pi$
$317$	68445.4	0.681124	0.340562	+	0.940222i	$0.389383\pi$
$318$	0	0
$319$	− 139072.i	− 1.36665i
$320$	0	0
$321$	61278.7	0.594702
$322$	0	0
$323$	173558.i	1.66357i
$324$	0	0
$325$	−142660.	−1.35062
$326$	0	0
$327$	− 54894.7i	− 0.513375i
$328$	0	0
$329$	−76982.8	−0.711217
$330$	0	0
$331$	45633.5i	0.416513i	0.978074	+	0.208256i	$0.0667788\pi$
$331$	45633.5i	0.416513i	−0.978074	+	0.208256i	$0.933221\pi$
$332$	0	0
$333$	14546.4	0.131180
$334$	0	0
$335$	− 19741.2i	− 0.175908i
$336$	0	0
$337$	−178174.	−1.56886	−0.784430	−	0.620217i	$-0.787045\pi$
$337$	−178174.	−1.56886	−0.784430	+	0.620217i	$0.787045\pi$
$338$	0	0
$339$	− 38494.9i	− 0.334969i
$340$	0	0
$341$	−60461.9	−0.519963
$342$	0	0
$343$	− 127569.i	− 1.08432i
$344$	0	0
$345$	−50417.8	−0.423590
$346$	0	0
$347$	22112.6i	0.183646i	0.995775	+	0.0918228i	$0.0292693\pi$
$347$	22112.6i	0.183646i	−0.995775	+	0.0918228i	$0.970731\pi$
$348$	0	0
$349$	−186409.	−1.53044	−0.765218	−	0.643772i	$-0.777369\pi$
$349$	−186409.	−1.53044	−0.765218	+	0.643772i	$0.777369\pi$
$350$	0	0
$351$	− 24075.8i	− 0.195419i
$352$	0	0
$353$	−15985.3	−0.128284	−0.0641420	−	0.997941i	$-0.520431\pi$
$353$	−15985.3	−0.128284	−0.0641420	+	0.997941i	$0.520431\pi$
$354$	0	0
$355$	− 74430.9i	− 0.590604i
$356$	0	0
$357$	76734.6	0.602081
$358$	0	0
$359$	130882.i	1.01553i	0.861497	+	0.507763i	$0.169527\pi$
$359$	130882.i	1.01553i	−0.861497	+	0.507763i	$0.830473\pi$
$360$	0	0
$361$	−113961.	−0.874460
$362$	0	0
$363$	− 23968.9i	− 0.181901i
$364$	0	0
$365$	−316989.	−2.37935
$366$	0	0
$367$	157598.i	1.17009i	0.811001	+	0.585045i	$0.198923\pi$
$367$	157598.i	1.17009i	−0.811001	+	0.585045i	$0.801077\pi$
$368$	0	0
$369$	14590.0	0.107153
$370$	0	0
$371$	146067.i	1.06122i
$372$	0	0
$373$	43982.6	0.316128	0.158064	−	0.987429i	$-0.449475\pi$
$373$	43982.6	0.316128	0.158064	+	0.987429i	$0.449475\pi$
$374$	0	0
$375$	40911.4i	0.290926i
$376$	0	0
$377$	238321.	1.67680
$378$	0	0
$379$	− 163953.i	− 1.14141i	−0.821156	−	0.570703i	$-0.806671\pi$
$379$	− 163953.i	− 1.14141i	0.821156	−	0.570703i	$-0.193329\pi$
$380$	0	0
$381$	−2304.51	−0.0158756
$382$	0	0
$383$	− 274654.i	− 1.87236i	−0.351525	−	0.936179i	$-0.614337\pi$
$383$	− 274654.i	− 1.87236i	0.351525	−	0.936179i	$-0.385663\pi$
$384$	0	0
$385$	−160712.	−1.08424
$386$	0	0
$387$	− 55270.9i	− 0.369041i
$388$	0	0
$389$	−192557.	−1.27251	−0.636253	−	0.771480i	$-0.719517\pi$
$389$	−192557.	−1.27251	−0.636253	+	0.771480i	$0.719517\pi$
$390$	0	0
$391$	− 89284.2i	− 0.584011i
$392$	0	0
$393$	−554.831	−0.00359233
$394$	0	0
$395$	428956.i	2.74928i
$396$	0	0
$397$	152788.	0.969410	0.484705	−	0.874678i	$-0.338927\pi$
$397$	152788.	0.969410	0.484705	+	0.874678i	$0.338927\pi$
$398$	0	0
$399$	108003.i	0.678407i
$400$	0	0
$401$	−181363.	−1.12787	−0.563936	−	0.825819i	$-0.690713\pi$
$401$	−181363.	−1.12787	−0.563936	+	0.825819i	$0.690713\pi$
$402$	0	0
$403$	− 103611.i	− 0.637962i
$404$	0	0
$405$	−27819.9	−0.169608
$406$	0	0
$407$	− 53951.4i	− 0.325697i
$408$	0	0
$409$	−115126.	−0.688217	−0.344109	−	0.938930i	$-0.611819\pi$
$409$	−115126.	−0.688217	−0.344109	+	0.938930i	$0.611819\pi$
$410$	0	0
$411$	− 151023.i	− 0.894048i
$412$	0	0
$413$	−66331.8	−0.388886
$414$	0	0
$415$	− 27449.7i	− 0.159383i
$416$	0	0
$417$	−168551.	−0.969301
$418$	0	0
$419$	62276.3i	0.354727i	0.984145	+	0.177364i	$0.0567569\pi$
$419$	62276.3i	0.354727i	−0.984145	+	0.177364i	$0.943243\pi$
$420$	0	0
$421$	−28425.3	−0.160377	−0.0801884	−	0.996780i	$-0.525552\pi$
$421$	−28425.3	−0.160377	−0.0801884	+	0.996780i	$0.525552\pi$
$422$	0	0
$423$	49425.2i	0.276228i
$424$	0	0
$425$	−291922.	−1.61618
$426$	0	0
$427$	44472.8i	0.243915i
$428$	0	0
$429$	−89295.1	−0.485191
$430$	0	0
$431$	337494.i	1.81682i	0.418081	+	0.908410i	$0.362703\pi$
$431$	337494.i	1.81682i	−0.418081	+	0.908410i	$0.637297\pi$
$432$	0	0
$433$	−27404.4	−0.146165	−0.0730827	−	0.997326i	$-0.523284\pi$
$433$	−27404.4	−0.146165	−0.0730827	+	0.997326i	$0.523284\pi$
$434$	0	0
$435$	− 275384.i	− 1.45532i
$436$	0	0
$437$	125666.	0.658047
$438$	0	0
$439$	152979.i	0.793787i	0.917865	+	0.396894i	$0.129912\pi$
$439$	152979.i	0.793787i	−0.917865	+	0.396894i	$0.870088\pi$
$440$	0	0
$441$	−17076.1	−0.0878034
$442$	0	0
$443$	− 206035.i	− 1.04987i	−0.851143	−	0.524933i	$-0.824090\pi$
$443$	− 206035.i	− 1.04987i	0.851143	−	0.524933i	$-0.175910\pi$
$444$	0	0
$445$	−159727.	−0.806601
$446$	0	0
$447$	30526.2i	0.152777i
$448$	0	0
$449$	−189831.	−0.941615	−0.470808	−	0.882236i	$-0.656037\pi$
$449$	−189831.	−0.941615	−0.470808	+	0.882236i	$0.656037\pi$
$450$	0	0
$451$	− 54113.1i	− 0.266042i
$452$	0	0
$453$	71868.3	0.350220
$454$	0	0
$455$	− 275405.i	− 1.33030i
$456$	0	0
$457$	−89739.8	−0.429687	−0.214844	−	0.976648i	$-0.568924\pi$
$457$	−89739.8	−0.429687	−0.214844	+	0.976648i	$0.568924\pi$
$458$	0	0
$459$	− 49265.8i	− 0.233841i
$460$	0	0
$461$	282353.	1.32859	0.664294	−	0.747472i	$-0.268732\pi$
$461$	282353.	1.32859	0.664294	+	0.747472i	$0.268732\pi$
$462$	0	0
$463$	− 276727.i	− 1.29089i	−0.763807	−	0.645445i	$-0.776672\pi$
$463$	− 276727.i	− 1.29089i	0.763807	−	0.645445i	$-0.223328\pi$
$464$	0	0
$465$	−119724.	−0.553700
$466$	0	0
$467$	− 367707.i	− 1.68604i	−0.537881	−	0.843021i	$-0.680775\pi$
$467$	− 367707.i	− 1.68604i	0.537881	−	0.843021i	$-0.319225\pi$
$468$	0	0
$469$	21754.8	0.0989030
$470$	0	0
$471$	118052.i	0.532149i
$472$	0	0
$473$	−204995.	−0.916266
$474$	0	0
$475$	− 410877.i	− 1.82106i
$476$	0	0
$477$	93779.4	0.412165
$478$	0	0
$479$	286130.i	1.24708i	0.781793	+	0.623538i	$0.214305\pi$
$479$	286130.i	1.24708i	−0.781793	+	0.623538i	$0.785695\pi$
$480$	0	0
$481$	92454.1	0.399610
$482$	0	0
$483$	− 55560.4i	− 0.238161i
$484$	0	0
$485$	186665.	0.793560
$486$	0	0
$487$	61310.2i	0.258508i	0.991611	+	0.129254i	$0.0412583\pi$
$487$	61310.2i	0.258508i	−0.991611	+	0.129254i	$0.958742\pi$
$488$	0	0
$489$	74472.9	0.311445
$490$	0	0
$491$	422941.i	1.75435i	0.480170	+	0.877175i	$0.340575\pi$
$491$	422941.i	1.75435i	−0.480170	+	0.877175i	$0.659425\pi$
$492$	0	0
$493$	487673.	2.00648
$494$	0	0
$495$	103182.i	0.421107i
$496$	0	0
$497$	82022.7	0.332064
$498$	0	0
$499$	324563.i	1.30346i	0.758451	+	0.651730i	$0.225957\pi$
$499$	324563.i	1.30346i	−0.758451	+	0.651730i	$0.774043\pi$
$500$	0	0
$501$	−14877.1	−0.0592709
$502$	0	0
$503$	185687.i	0.733915i	0.930238	+	0.366958i	$0.119601\pi$
$503$	185687.i	0.733915i	−0.930238	+	0.366958i	$0.880399\pi$
$504$	0	0
$505$	−136700.	−0.536026
$506$	0	0
$507$	− 4613.69i	− 0.0179487i
$508$	0	0
$509$	401954.	1.55146	0.775731	−	0.631064i	$-0.217382\pi$
$509$	401954.	1.55146	0.775731	+	0.631064i	$0.217382\pi$
$510$	0	0
$511$	− 349321.i	− 1.33778i
$512$	0	0
$513$	69341.1	0.263485
$514$	0	0
$515$	663705.i	2.50242i
$516$	0	0
$517$	183314.	0.685828
$518$	0	0
$519$	75426.3i	0.280020i
$520$	0	0
$521$	−63089.4	−0.232424	−0.116212	−	0.993224i	$-0.537075\pi$
$521$	−63089.4	−0.232424	−0.116212	+	0.993224i	$0.537075\pi$
$522$	0	0
$523$	− 183735.i	− 0.671722i	−0.941912	−	0.335861i	$-0.890973\pi$
$523$	− 183735.i	− 0.671722i	0.941912	−	0.335861i	$-0.109027\pi$
$524$	0	0
$525$	−181659.	−0.659081
$526$	0	0
$527$	− 212017.i	− 0.763395i
$528$	0	0
$529$	215194.	0.768987
$530$	0	0
$531$	42587.0i	0.151039i
$532$	0	0
$533$	92731.2	0.326416
$534$	0	0
$535$	− 450045.i	− 1.57235i
$536$	0	0
$537$	41944.7	0.145455
$538$	0	0
$539$	63333.8i	0.218001i
$540$	0	0
$541$	−436991.	−1.49306	−0.746531	−	0.665350i	$-0.768282\pi$
$541$	−436991.	−1.49306	−0.746531	+	0.665350i	$0.768282\pi$
$542$	0	0
$543$	191531.i	0.649590i
$544$	0	0
$545$	−403159.	−1.35732
$546$	0	0
$547$	− 44994.9i	− 0.150379i	−0.997169	−	0.0751897i	$-0.976044\pi$
$547$	− 44994.9i	− 0.150379i	0.997169	−	0.0751897i	$-0.0239562\pi$
$548$	0	0
$549$	28552.8	0.0947336
$550$	0	0
$551$	686394.i	2.26084i
$552$	0	0
$553$	−472709.	−1.54577
$554$	0	0
$555$	− 106832.i	− 0.346829i
$556$	0	0
$557$	−181185.	−0.583999	−0.291999	−	0.956419i	$-0.594320\pi$
$557$	−181185.	−0.583999	−0.291999	+	0.956419i	$0.594320\pi$
$558$	0	0
$559$	− 351291.i	− 1.12420i
$560$	0	0
$561$	−182723.	−0.580587
$562$	0	0
$563$	252429.i	0.796382i	0.917302	+	0.398191i	$0.130362\pi$
$563$	252429.i	0.796382i	−0.917302	+	0.398191i	$0.869638\pi$
$564$	0	0
$565$	−282715.	−0.885631
$566$	0	0
$567$	− 30657.5i	− 0.0953609i
$568$	0	0
$569$	−244576.	−0.755423	−0.377711	−	0.925923i	$-0.623289\pi$
$569$	−244576.	−0.755423	−0.377711	+	0.925923i	$0.623289\pi$
$570$	0	0
$571$	− 320074.i	− 0.981699i	−0.871244	−	0.490849i	$-0.836687\pi$
$571$	− 320074.i	− 0.981699i	0.871244	−	0.490849i	$-0.163313\pi$
$572$	0	0
$573$	11078.8	0.0337429
$574$	0	0
$575$	211369.i	0.639301i
$576$	0	0
$577$	−214829.	−0.645269	−0.322635	−	0.946524i	$-0.604569\pi$
$577$	−214829.	−0.645269	−0.322635	+	0.946524i	$0.604569\pi$
$578$	0	0
$579$	− 173488.i	− 0.517503i
$580$	0	0
$581$	30249.5	0.0896119
$582$	0	0
$583$	− 347820.i	− 1.02333i
$584$	0	0
$585$	−176818.	−0.516672
$586$	0	0
$587$	674424.i	1.95730i	0.205537	+	0.978649i	$0.434106\pi$
$587$	674424.i	1.95730i	−0.205537	+	0.978649i	$0.565894\pi$
$588$	0	0
$589$	298412.	0.860172
$590$	0	0
$591$	− 209083.i	− 0.598611i
$592$	0	0
$593$	−337443.	−0.959601	−0.479800	−	0.877378i	$-0.659291\pi$
$593$	−337443.	−0.959601	−0.479800	+	0.877378i	$0.659291\pi$
$594$	0	0
$595$	− 563556.i	− 1.59185i
$596$	0	0
$597$	−11870.9	−0.0333070
$598$	0	0
$599$	− 308879.i	− 0.860864i	−0.902623	−	0.430432i	$-0.858361\pi$
$599$	− 308879.i	− 0.860864i	0.902623	−	0.430432i	$-0.141639\pi$
$600$	0	0
$601$	505644.	1.39990	0.699948	−	0.714193i	$-0.253206\pi$
$601$	505644.	1.39990	0.699948	+	0.714193i	$0.253206\pi$
$602$	0	0
$603$	− 13967.2i	− 0.0384128i
$604$	0	0
$605$	−176033.	−0.480932
$606$	0	0
$607$	662175.i	1.79720i	0.438773	+	0.898598i	$0.355413\pi$
$607$	662175.i	1.79720i	−0.438773	+	0.898598i	$0.644587\pi$
$608$	0	0
$609$	303472.	0.818248
$610$	0	0
$611$	314137.i	0.841467i
$612$	0	0
$613$	−470942.	−1.25328	−0.626638	−	0.779311i	$-0.715569\pi$
$613$	−470942.	−1.25328	−0.626638	+	0.779311i	$0.715569\pi$
$614$	0	0
$615$	− 107152.i	− 0.283303i
$616$	0	0
$617$	239921.	0.630229	0.315114	−	0.949054i	$-0.397957\pi$
$617$	239921.	0.630229	0.315114	+	0.949054i	$0.397957\pi$
$618$	0	0
$619$	730806.i	1.90731i	0.300907	+	0.953653i	$0.402710\pi$
$619$	730806.i	1.90731i	−0.300907	+	0.953653i	$0.597290\pi$
$620$	0	0
$621$	−35671.4	−0.0924989
$622$	0	0
$623$	− 176019.i	− 0.453507i
$624$	0	0
$625$	−219110.	−0.560922
$626$	0	0
$627$	− 257181.i	− 0.654189i
$628$	0	0
$629$	189187.	0.478179
$630$	0	0
$631$	242358.i	0.608693i	0.952561	+	0.304346i	$0.0984380\pi$
$631$	242358.i	0.608693i	−0.952561	+	0.304346i	$0.901562\pi$
$632$	0	0
$633$	−68737.3	−0.171548
$634$	0	0
$635$	16924.9i	0.0419738i
$636$	0	0
$637$	−108532.	−0.267473
$638$	0	0
$639$	− 52661.0i	− 0.128970i
$640$	0	0
$641$	684663.	1.66633	0.833165	−	0.553025i	$-0.186527\pi$
$641$	684663.	1.66633	0.833165	+	0.553025i	$0.186527\pi$
$642$	0	0
$643$	− 415456.i	− 1.00485i	−0.864620	−	0.502427i	$-0.832441\pi$
$643$	− 415456.i	− 1.00485i	0.864620	−	0.502427i	$-0.167559\pi$
$644$	0	0
$645$	−405922.	−0.975715
$646$	0	0
$647$	− 743379.i	− 1.77583i	−0.460007	−	0.887915i	$-0.652153\pi$
$647$	− 743379.i	− 1.77583i	0.460007	−	0.887915i	$-0.347847\pi$
$648$	0	0
$649$	157952.	0.375003
$650$	0	0
$651$	− 131935.i	− 0.311315i
$652$	0	0
$653$	−318695.	−0.747393	−0.373697	−	0.927551i	$-0.621910\pi$
$653$	−318695.	−0.747393	−0.373697	+	0.927551i	$0.621910\pi$
$654$	0	0
$655$	4074.81i	0.00949783i
$656$	0	0
$657$	−224275.	−0.519576
$658$	0	0
$659$	661853.i	1.52402i	0.647566	+	0.762010i	$0.275787\pi$
$659$	661853.i	1.52402i	−0.647566	+	0.762010i	$0.724213\pi$
$660$	0	0
$661$	477462.	1.09279	0.546394	−	0.837529i	$-0.316000\pi$
$661$	477462.	1.09279	0.546394	+	0.837529i	$0.316000\pi$
$662$	0	0
$663$	− 313124.i	− 0.712344i
$664$	0	0
$665$	793199.	1.79366
$666$	0	0
$667$	− 353104.i	− 0.793690i
$668$	0	0
$669$	309882.	0.692380
$670$	0	0
$671$	− 105900.i	− 0.235207i
$672$	0	0
$673$	7442.28	0.0164314	0.00821572	−	0.999966i	$-0.497385\pi$
$673$	7442.28	0.0164314	0.00821572	+	0.999966i	$0.497385\pi$
$674$	0	0
$675$	116631.i	0.255979i
$676$	0	0
$677$	−55653.5	−0.121427	−0.0607135	−	0.998155i	$-0.519338\pi$
$677$	−55653.5	−0.121427	−0.0607135	+	0.998155i	$0.519338\pi$
$678$	0	0
$679$	205705.i	0.446174i
$680$	0	0
$681$	52480.8	0.113163
$682$	0	0
$683$	323036.i	0.692484i	0.938145	+	0.346242i	$0.112542\pi$
$683$	323036.i	0.692484i	−0.938145	+	0.346242i	$0.887458\pi$
$684$	0	0
$685$	−1.10915e6	−2.36379
$686$	0	0
$687$	− 174839.i	− 0.370445i
$688$	0	0
$689$	596044.	1.25557
$690$	0	0
$691$	− 154529.i	− 0.323633i	−0.986821	−	0.161816i	$-0.948265\pi$
$691$	− 154529.i	− 0.323633i	0.986821	−	0.161816i	$-0.0517353\pi$
$692$	0	0
$693$	−113706.	−0.236765
$694$	0	0
$695$	1.23788e6i	2.56276i
$696$	0	0
$697$	189754.	0.390595
$698$	0	0
$699$	− 388847.i	− 0.795838i
$700$	0	0
$701$	−53163.0	−0.108187	−0.0540933	−	0.998536i	$-0.517227\pi$
$701$	−53163.0	−0.108187	−0.0540933	+	0.998536i	$0.517227\pi$
$702$	0	0
$703$	266279.i	0.538799i
$704$	0	0
$705$	362990.	0.730326
$706$	0	0
$707$	− 150643.i	− 0.301377i
$708$	0	0
$709$	280896.	0.558795	0.279397	−	0.960176i	$-0.409865\pi$
$709$	280896.	0.558795	0.279397	+	0.960176i	$0.409865\pi$
$710$	0	0
$711$	303493.i	0.600357i
$712$	0	0
$713$	−153513.	−0.301971
$714$	0	0
$715$	655803.i	1.28281i
$716$	0	0
$717$	−128929.	−0.250792
$718$	0	0
$719$	− 367010.i	− 0.709937i	−0.934878	−	0.354968i	$-0.884492\pi$
$719$	− 367010.i	− 0.709937i	0.934878	−	0.354968i	$-0.115508\pi$
$720$	0	0
$721$	−731402.	−1.40697
$722$	0	0
$723$	79096.3i	0.151314i
$724$	0	0
$725$	−1.15450e6	−2.19644
$726$	0	0
$727$	340510.i	0.644261i	0.946695	+	0.322130i	$0.104399\pi$
$727$	340510.i	0.644261i	−0.946695	+	0.322130i	$0.895601\pi$
$728$	0	0
$729$	−19683.0	−0.0370370
$730$	0	0
$731$	− 718841.i	− 1.34524i
$732$	0	0
$733$	815103.	1.51707	0.758533	−	0.651635i	$-0.225916\pi$
$733$	815103.	1.51707	0.758533	+	0.651635i	$0.225916\pi$
$734$	0	0
$735$	125411.i	0.232145i
$736$	0	0
$737$	−51803.3	−0.0953723
$738$	0	0
$739$	994330.i	1.82071i	0.413824	+	0.910357i	$0.364193\pi$
$739$	994330.i	1.82071i	−0.413824	+	0.910357i	$0.635807\pi$
$740$	0	0
$741$	440719.	0.802648
$742$	0	0
$743$	− 917401.i	− 1.66181i	−0.556414	−	0.830905i	$-0.687823\pi$
$743$	− 917401.i	− 1.66181i	0.556414	−	0.830905i	$-0.312177\pi$
$744$	0	0
$745$	224191.	0.403930
$746$	0	0
$747$	− 19421.1i	− 0.0348042i
$748$	0	0
$749$	495948.	0.884042
$750$	0	0
$751$	507595.i	0.899989i	0.893031	+	0.449994i	$0.148574\pi$
$751$	507595.i	0.899989i	−0.893031	+	0.449994i	$0.851426\pi$
$752$	0	0
$753$	473609.	0.835276
$754$	0	0
$755$	− 527817.i	− 0.925954i
$756$	0	0
$757$	71291.0	0.124406	0.0622032	−	0.998064i	$-0.480187\pi$
$757$	71291.0	0.124406	0.0622032	+	0.998064i	$0.480187\pi$
$758$	0	0
$759$	132302.i	0.229659i
$760$	0	0
$761$	−206970.	−0.357387	−0.178693	−	0.983905i	$-0.557187\pi$
$761$	−206970.	−0.357387	−0.178693	+	0.983905i	$0.557187\pi$
$762$	0	0
$763$	− 444281.i	− 0.763147i
$764$	0	0
$765$	−361820.	−0.618257
$766$	0	0
$767$	270675.i	0.460105i
$768$	0	0
$769$	931878.	1.57582	0.787910	−	0.615791i	$-0.211163\pi$
$769$	931878.	1.57582	0.787910	+	0.615791i	$0.211163\pi$
$770$	0	0
$771$	129097.i	0.217174i
$772$	0	0
$773$	184225.	0.308312	0.154156	−	0.988047i	$-0.450734\pi$
$773$	184225.	0.308312	0.154156	+	0.988047i	$0.450734\pi$
$774$	0	0
$775$	501923.i	0.835668i
$776$	0	0
$777$	117729.	0.195003
$778$	0	0
$779$	267077.i	0.440111i
$780$	0	0
$781$	−195315.	−0.320210
$782$	0	0
$783$	− 194838.i	− 0.317798i
$784$	0	0
$785$	867003.	1.40696
$786$	0	0
$787$	− 936929.i	− 1.51271i	−0.654158	−	0.756357i	$-0.726977\pi$
$787$	− 936929.i	− 1.51271i	0.654158	−	0.756357i	$-0.273023\pi$
$788$	0	0
$789$	341709.	0.548911
$790$	0	0
$791$	− 311552.i	− 0.497941i
$792$	0	0
$793$	181476.	0.288585
$794$	0	0
$795$	− 688737.i	− 1.08973i
$796$	0	0
$797$	239720.	0.377388	0.188694	−	0.982036i	$-0.439575\pi$
$797$	239720.	0.377388	0.188694	+	0.982036i	$0.439575\pi$
$798$	0	0
$799$	642814.i	1.00691i
$800$	0	0
$801$	−113009.	−0.176137
$802$	0	0
$803$	831816.i	1.29002i
$804$	0	0
$805$	−408048.	−0.629679
$806$	0	0
$807$	15703.7i	0.0241133i
$808$	0	0
$809$	−598514.	−0.914486	−0.457243	−	0.889342i	$-0.651163\pi$
$809$	−598514.	−0.914486	−0.457243	+	0.889342i	$0.651163\pi$
$810$	0	0
$811$	17550.1i	0.0266831i	0.999911	+	0.0133416i	$0.00424688\pi$
$811$	17550.1i	0.0266831i	−0.999911	+	0.0133416i	$0.995753\pi$
$812$	0	0
$813$	755304.	1.14272
$814$	0	0
$815$	− 546946.i	− 0.823435i
$816$	0	0
$817$	1.01176e6	1.51577
$818$	0	0
$819$	− 194853.i	− 0.290496i
$820$	0	0
$821$	18495.4	0.0274396	0.0137198	−	0.999906i	$-0.495633\pi$
$821$	18495.4	0.0274396	0.0137198	+	0.999906i	$0.495633\pi$
$822$	0	0
$823$	425107.i	0.627622i	0.949485	+	0.313811i	$0.101606\pi$
$823$	425107.i	0.627622i	−0.949485	+	0.313811i	$0.898394\pi$
$824$	0	0
$825$	432573.	0.635553
$826$	0	0
$827$	239745.i	0.350540i	0.984520	+	0.175270i	$0.0560799\pi$
$827$	239745.i	0.350540i	−0.984520	+	0.175270i	$0.943920\pi$
$828$	0	0
$829$	904750.	1.31650	0.658248	−	0.752801i	$-0.271298\pi$
$829$	904750.	1.31650	0.658248	+	0.752801i	$0.271298\pi$
$830$	0	0
$831$	− 628265.i	− 0.909790i
$832$	0	0
$833$	−222088.	−0.320063
$834$	0	0
$835$	109260.i	0.156708i
$836$	0	0
$837$	−84706.4	−0.120911
$838$	0	0
$839$	− 376178.i	− 0.534403i	−0.963641	−	0.267202i	$-0.913901\pi$
$839$	− 376178.i	− 0.534403i	0.963641	−	0.267202i	$-0.0860990\pi$
$840$	0	0
$841$	1.22138e6	1.72687
$842$	0	0
$843$	724892.i	1.02004i
$844$	0	0
$845$	−33884.0	−0.0474549
$846$	0	0
$847$	− 193988.i	− 0.270401i
$848$	0	0
$849$	−275127.	−0.381696
$850$	0	0
$851$	− 136983.i	− 0.189150i
$852$	0	0
$853$	662654.	0.910729	0.455364	−	0.890305i	$-0.349509\pi$
$853$	662654.	0.910729	0.455364	+	0.890305i	$0.349509\pi$
$854$	0	0
$855$	− 509257.i	− 0.696634i
$856$	0	0
$857$	767295.	1.04472	0.522361	−	0.852724i	$-0.325051\pi$
$857$	767295.	1.04472	0.522361	+	0.852724i	$0.325051\pi$
$858$	0	0
$859$	− 823262.i	− 1.11571i	−0.829938	−	0.557856i	$-0.811624\pi$
$859$	− 823262.i	− 1.11571i	0.829938	−	0.557856i	$-0.188376\pi$
$860$	0	0
$861$	118082.	0.159285
$862$	0	0
$863$	− 605487.i	− 0.812987i	−0.913654	−	0.406493i	$-0.866751\pi$
$863$	− 605487.i	− 0.812987i	0.913654	−	0.406493i	$-0.133249\pi$
$864$	0	0
$865$	553948.	0.740350
$866$	0	0
$867$	− 206753.i	− 0.275051i
$868$	0	0
$869$	1.12563e6	1.49058
$870$	0	0
$871$	− 88773.0i	− 0.117016i
$872$	0	0
$873$	132068.	0.173289
$874$	0	0
$875$	331110.i	0.432470i
$876$	0	0
$877$	578472.	0.752114	0.376057	−	0.926597i	$-0.377280\pi$
$877$	578472.	0.752114	0.376057	+	0.926597i	$0.377280\pi$
$878$	0	0
$879$	− 114506.i	− 0.148201i
$880$	0	0
$881$	218679.	0.281745	0.140872	−	0.990028i	$-0.455009\pi$
$881$	218679.	0.281745	0.140872	+	0.990028i	$0.455009\pi$
$882$	0	0
$883$	− 302510.i	− 0.387988i	−0.981003	−	0.193994i	$-0.937856\pi$
$883$	− 302510.i	− 0.387988i	0.981003	−	0.193994i	$-0.0621443\pi$
$884$	0	0
$885$	312768.	0.399334
$886$	0	0
$887$	− 932038.i	− 1.18464i	−0.805703	−	0.592320i	$-0.798212\pi$
$887$	− 932038.i	− 1.18464i	0.805703	−	0.592320i	$-0.201788\pi$
$888$	0	0
$889$	−18651.2	−0.0235995
$890$	0	0
$891$	73002.6i	0.0919566i
$892$	0	0
$893$	−904754.	−1.13456
$894$	0	0
$895$	− 308051.i	− 0.384571i
$896$	0	0
$897$	−226720.	−0.281777
$898$	0	0
$899$	− 838492.i	− 1.03748i
$900$	0	0
$901$	1.21968e6	1.50243
$902$	0	0
$903$	− 447325.i	− 0.548590i
$904$	0	0
$905$	1.40665e6	1.71746
$906$	0	0
$907$	− 553592.i	− 0.672938i	−0.941695	−	0.336469i	$-0.890767\pi$
$907$	− 553592.i	− 0.672938i	0.941695	−	0.336469i	$-0.109233\pi$
$908$	0	0
$909$	−96717.3	−0.117051
$910$	0	0
$911$	1.42062e6i	1.71175i	0.517184	+	0.855874i	$0.326980\pi$
$911$	1.42062e6i	1.71175i	−0.517184	+	0.855874i	$0.673020\pi$
$912$	0	0
$913$	−72031.1	−0.0864129
$914$	0	0
$915$	− 209698.i	− 0.250468i
$916$	0	0
$917$	−4490.43	−0.00534010
$918$	0	0
$919$	− 212699.i	− 0.251845i	−0.992040	−	0.125923i	$-0.959811\pi$
$919$	− 212699.i	− 0.251845i	0.992040	−	0.125923i	$-0.0401891\pi$
$920$	0	0
$921$	−751748.	−0.886244
$922$	0	0
$923$	− 334703.i	− 0.392877i
$924$	0	0
$925$	−447877.	−0.523450
$926$	0	0
$927$	469581.i	0.546451i
$928$	0	0
$929$	−1.13677e6	−1.31717	−0.658585	−	0.752506i	$-0.728845\pi$
$929$	−1.13677e6	−1.31717	−0.658585	+	0.752506i	$0.728845\pi$
$930$	0	0
$931$	− 312586.i	− 0.360637i
$932$	0	0
$933$	−305592.	−0.351058
$934$	0	0
$935$	1.34196e6i	1.53503i
$936$	0	0
$937$	285636.	0.325337	0.162669	−	0.986681i	$-0.447990\pi$
$937$	285636.	0.325337	0.162669	+	0.986681i	$0.447990\pi$
$938$	0	0
$939$	476333.i	0.540231i
$940$	0	0
$941$	184285.	0.208118	0.104059	−	0.994571i	$-0.466817\pi$
$941$	184285.	0.208118	0.104059	+	0.994571i	$0.466817\pi$
$942$	0	0
$943$	− 137393.i	− 0.154505i
$944$	0	0
$945$	−225156.	−0.252127
$946$	0	0
$947$	− 477705.i	− 0.532672i	−0.963880	−	0.266336i	$-0.914187\pi$
$947$	− 477705.i	− 0.532672i	0.963880	−	0.266336i	$-0.0858130\pi$
$948$	0	0
$949$	−1.42545e6	−1.58277
$950$	0	0
$951$	− 355653.i	− 0.393247i
$952$	0	0
$953$	1.40115e6	1.54276	0.771380	−	0.636375i	$-0.219567\pi$
$953$	1.40115e6	1.54276	0.771380	+	0.636375i	$0.219567\pi$
$954$	0	0
$955$	− 81365.0i	− 0.0892135i
$956$	0	0
$957$	−722639.	−0.789037
$958$	0	0
$959$	− 1.22228e6i	− 1.32903i
$960$	0	0
$961$	558984.	0.605275
$962$	0	0
$963$	− 318413.i	− 0.343351i
$964$	0	0
$965$	−1.27414e6	−1.36824
$966$	0	0
$967$	877370.i	0.938275i	0.883125	+	0.469137i	$0.155435\pi$
$967$	877370.i	0.938275i	−0.883125	+	0.469137i	$0.844565\pi$
$968$	0	0
$969$	901836.	0.960461
$970$	0	0
$971$	29888.2i	0.0317002i	0.999874	+	0.0158501i	$0.00504545\pi$
$971$	29888.2i	0.0317002i	−0.999874	+	0.0158501i	$0.994955\pi$
$972$	0	0
$973$	−1.36414e6	−1.44090
$974$	0	0
$975$	741281.i	0.779783i
$976$	0	0
$977$	1.28420e6	1.34537	0.672687	−	0.739927i	$-0.265140\pi$
$977$	1.28420e6	1.34537	0.672687	+	0.739927i	$0.265140\pi$
$978$	0	0
$979$	419143.i	0.437317i
$980$	0	0
$981$	−285241.	−0.296397
$982$	0	0
$983$	1.30863e6i	1.35428i	0.735852	+	0.677142i	$0.236782\pi$
$983$	1.30863e6i	1.35428i	−0.735852	+	0.677142i	$0.763218\pi$
$984$	0	0
$985$	−1.53556e6	−1.58268
$986$	0	0
$987$	400015.i	0.410621i
$988$	0	0
$989$	−520483.	−0.532126
$990$	0	0
$991$	− 958452.i	− 0.975940i	−0.872860	−	0.487970i	$-0.837737\pi$
$991$	− 958452.i	− 0.975940i	0.872860	−	0.487970i	$-0.162263\pi$
$992$	0	0
$993$	237119.	0.240474
$994$	0	0
$995$	87182.7i	0.0880611i
$996$	0	0
$997$	−1.30519e6	−1.31306	−0.656529	−	0.754301i	$-0.727976\pi$
$997$	−1.30519e6	−1.31306	−0.656529	+	0.754301i	$0.727976\pi$
$998$	0	0
$999$	− 75585.3i	− 0.0757367i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.5.g.k.511.1		16
4.3	odd	2	inner	768.5.g.k.511.9		16
8.3	odd	2	inner	768.5.g.k.511.8		16
8.5	even	2	inner	768.5.g.k.511.16		16
16.3	odd	4		96.5.b.a.79.4		8
16.5	even	4		96.5.b.a.79.1		8
16.11	odd	4		24.5.b.a.19.1	✓	8
16.13	even	4		24.5.b.a.19.2	yes	8
48.5	odd	4		288.5.b.d.271.8		8
48.11	even	4		72.5.b.d.19.8		8
48.29	odd	4		72.5.b.d.19.7		8
48.35	even	4		288.5.b.d.271.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.5.b.a.19.1	✓	8	16.11	odd	4
24.5.b.a.19.2	yes	8	16.13	even	4
72.5.b.d.19.7		8	48.29	odd	4
72.5.b.d.19.8		8	48.11	even	4
96.5.b.a.79.1		8	16.5	even	4
96.5.b.a.79.4		8	16.3	odd	4
288.5.b.d.271.1		8	48.35	even	4
288.5.b.d.271.8		8	48.5	odd	4
768.5.g.k.511.1		16	1.1	even	1	trivial
768.5.g.k.511.8		16	8.3	odd	2	inner
768.5.g.k.511.9		16	4.3	odd	2	inner
768.5.g.k.511.16		16	8.5	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:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	768.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-5.19615i q^{3} -38.1617 q^{5} -42.0542i q^{7} -27.0000 q^{9} +O(q^{10})\)	\(q-5.19615i q^{3} -38.1617 q^{5} -42.0542i q^{7} -27.0000 q^{9} +100.141i q^{11} -171.607 q^{13} +198.294i q^{15} -351.156 q^{17} -494.248i q^{19} -218.520 q^{21} +254.258i q^{23} +831.317 q^{25} +140.296i q^{27} -1388.76 q^{29} +603.769i q^{31} +520.347 q^{33} +1604.86i q^{35} -538.756 q^{37} +891.695i q^{39} -540.370 q^{41} +2047.07i q^{43} +1030.37 q^{45} -1830.56i q^{47} +632.448 q^{49} +1824.66i q^{51} -3473.31 q^{53} -3821.54i q^{55} -2568.19 q^{57} -1577.30i q^{59} -1057.51 q^{61} +1135.46i q^{63} +6548.81 q^{65} +517.305i q^{67} +1321.16 q^{69} +1950.41i q^{71} +8306.47 q^{73} -4319.65i q^{75} +4211.34 q^{77} -11240.5i q^{79} +729.000 q^{81} +719.298i q^{83} +13400.7 q^{85} +7216.23i q^{87} +4185.54 q^{89} +7216.78i q^{91} +3137.27 q^{93} +18861.4i q^{95} -4891.42 q^{97} -2703.80i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 432 q^{9}+O(q^{10}) \) 16 * q - 432 * q^9	\( 16 q - 432 q^{9} + 480 q^{17} + 1328 q^{25} - 1440 q^{41} - 2480 q^{49} - 7488 q^{57} - 2688 q^{65} + 33760 q^{73} + 11664 q^{81} - 31200 q^{89} - 24352 q^{97}+O(q^{100}) \) 16 * q - 432 * q^9 + 480 * q^17 + 1328 * q^25 - 1440 * q^41 - 2480 * q^49 - 7488 * q^57 - 2688 * q^65 + 33760 * q^73 + 11664 * q^81 - 31200 * q^89 - 24352 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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