LMFDB - Embedded newform 768.4.d.t.385.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,4,Mod(385,768)]

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([0, 1, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("768.385");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	768.d (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:	$45.3134668844$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{15})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 7x^{2} + 16 $ x^4 - 7*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 384)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			385.2
Root			$1.93649 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	768.385
Dual form			768.4.d.t.385.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} +11.4919i q^{5} -13.4919 q^{7} -9.00000 q^{9} +O(q^{10})$	$q-3.00000i q^{3} +11.4919i q^{5} -13.4919 q^{7} -9.00000 q^{9} -10.9839i q^{11} +2.00000i q^{13} +34.4758 q^{15} +106.952 q^{17} -86.9839i q^{19} +40.4758i q^{21} +64.9516 q^{23} -7.06453 q^{25} +27.0000i q^{27} +129.395i q^{29} -246.444 q^{31} -32.9516 q^{33} -155.048i q^{35} +259.016i q^{37} +6.00000 q^{39} -324.984 q^{41} +292.823i q^{43} -103.427i q^{45} -386.984 q^{47} -160.968 q^{49} -320.855i q^{51} +536.508i q^{53} +126.226 q^{55} -260.952 q^{57} -103.806i q^{59} -628.790i q^{61} +121.427 q^{63} -22.9839 q^{65} +719.548i q^{67} -194.855i q^{69} +200.952 q^{71} -987.581 q^{73} +21.1936i q^{75} +148.194i q^{77} -770.379 q^{79} +81.0000 q^{81} -1284.50i q^{83} +1229.08i q^{85} +388.185 q^{87} -844.161 q^{89} -26.9839i q^{91} +739.331i q^{93} +999.613 q^{95} -1259.84 q^{97} +98.8548i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{7} - 36 q^{9}+O(q^{10}) $ 4 * q + 8 * q^7 - 36 * q^9	$ 4 q + 8 q^{7} - 36 q^{9} - 48 q^{15} + 56 q^{17} - 112 q^{23} - 524 q^{25} - 552 q^{31} + 240 q^{33} + 24 q^{39} - 1176 q^{41} - 1424 q^{47} - 396 q^{49} + 2240 q^{55} - 672 q^{57} - 72 q^{63} + 32 q^{65} + 432 q^{71} - 728 q^{73} - 2152 q^{79} + 324 q^{81} - 864 q^{87} - 4616 q^{89} + 1024 q^{95} - 3800 q^{97}+O(q^{100}) $ 4 * q + 8 * q^7 - 36 * q^9 - 48 * q^15 + 56 * q^17 - 112 * q^23 - 524 * q^25 - 552 * q^31 + 240 * q^33 + 24 * q^39 - 1176 * q^41 - 1424 * q^47 - 396 * q^49 + 2240 * q^55 - 672 * q^57 - 72 * q^63 + 32 * q^65 + 432 * q^71 - 728 * q^73 - 2152 * q^79 + 324 * q^81 - 864 * q^87 - 4616 * q^89 + 1024 * q^95 - 3800 * 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
$3$	− 3.00000i	− 0.577350i
$3$	− 3.00000i	− 0.577350i
$4$	0	0
$5$	11.4919i	1.02787i	0.857829	+	0.513935i	$0.171813\pi$
$5$	11.4919i	1.02787i	−0.857829	+	0.513935i	$0.828187\pi$
$6$	0	0
$7$	−13.4919	−0.728496	−0.364248	−	0.931302i	$-0.618674\pi$
$7$	−13.4919	−0.728496	−0.364248	+	0.931302i	$0.618674\pi$
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	− 10.9839i	− 0.301069i	−0.988605	−	0.150535i	$-0.951901\pi$
$11$	− 10.9839i	− 0.301069i	0.988605	−	0.150535i	$-0.0480995\pi$
$12$	0	0
$13$	2.00000i	0.0426692i	0.999772	+	0.0213346i	$0.00679154\pi$
$13$	2.00000i	0.0426692i	−0.999772	+	0.0213346i	$0.993208\pi$
$14$	0	0
$15$	34.4758	0.593441
$16$	0	0
$17$	106.952	1.52586	0.762929	−	0.646483i	$-0.223761\pi$
$17$	106.952	1.52586	0.762929	+	0.646483i	$0.223761\pi$
$18$	0	0
$19$	− 86.9839i	− 1.05029i	−0.851013	−	0.525144i	$-0.824011\pi$
$19$	− 86.9839i	− 1.05029i	0.851013	−	0.525144i	$-0.175989\pi$
$20$	0	0
$21$	40.4758i	0.420597i
$22$	0	0
$23$	64.9516	0.588841	0.294421	−	0.955676i	$-0.404873\pi$
$23$	64.9516	0.588841	0.294421	+	0.955676i	$0.404873\pi$
$24$	0	0
$25$	−7.06453	−0.0565163
$26$	0	0
$27$	27.0000i	0.192450i
$28$	0	0
$29$	129.395i	0.828554i	0.910151	+	0.414277i	$0.135966\pi$
$29$	129.395i	0.828554i	−0.910151	+	0.414277i	$0.864034\pi$
$30$	0	0
$31$	−246.444	−1.42782	−0.713912	−	0.700235i	$-0.753079\pi$
$31$	−246.444	−1.42782	−0.713912	+	0.700235i	$0.753079\pi$
$32$	0	0
$33$	−32.9516	−0.173822
$34$	0	0
$35$	− 155.048i	− 0.748799i
$36$	0	0
$37$	259.016i	1.15086i	0.817849	+	0.575432i	$0.195166\pi$
$37$	259.016i	1.15086i	−0.817849	+	0.575432i	$0.804834\pi$
$38$	0	0
$39$	6.00000	0.0246351
$40$	0	0
$41$	−324.984	−1.23790	−0.618951	−	0.785430i	$-0.712442\pi$
$41$	−324.984	−1.23790	−0.618951	+	0.785430i	$0.712442\pi$
$42$	0	0
$43$	292.823i	1.03849i	0.854626	+	0.519244i	$0.173787\pi$
$43$	292.823i	1.03849i	−0.854626	+	0.519244i	$0.826213\pi$
$44$	0	0
$45$	− 103.427i	− 0.342623i
$46$	0	0
$47$	−386.984	−1.20101	−0.600504	−	0.799622i	$-0.705033\pi$
$47$	−386.984	−1.20101	−0.600504	+	0.799622i	$0.705033\pi$
$48$	0	0
$49$	−160.968	−0.469294
$50$	0	0
$51$	− 320.855i	− 0.880954i
$52$	0	0
$53$	536.508i	1.39047i	0.718781	+	0.695236i	$0.244700\pi$
$53$	536.508i	1.39047i	−0.718781	+	0.695236i	$0.755300\pi$
$54$	0	0
$55$	126.226	0.309460
$56$	0	0
$57$	−260.952	−0.606384
$58$	0	0
$59$	− 103.806i	− 0.229058i	−0.993420	−	0.114529i	$-0.963464\pi$
$59$	− 103.806i	− 0.229058i	0.993420	−	0.114529i	$-0.0365359\pi$
$60$	0	0
$61$	− 628.790i	− 1.31981i	−0.751350	−	0.659904i	$-0.770597\pi$
$61$	− 628.790i	− 1.31981i	0.751350	−	0.659904i	$-0.229403\pi$
$62$	0	0
$63$	121.427	0.242832
$64$	0	0
$65$	−22.9839	−0.0438584
$66$	0	0
$67$	719.548i	1.31204i	0.754743	+	0.656021i	$0.227762\pi$
$67$	719.548i	1.31204i	−0.754743	+	0.656021i	$0.772238\pi$
$68$	0	0
$69$	− 194.855i	− 0.339968i
$70$	0	0
$71$	200.952	0.335895	0.167948	−	0.985796i	$-0.446286\pi$
$71$	200.952	0.335895	0.167948	+	0.985796i	$0.446286\pi$
$72$	0	0
$73$	−987.581	−1.58339	−0.791696	−	0.610916i	$-0.790801\pi$
$73$	−987.581	−1.58339	−0.791696	+	0.610916i	$0.790801\pi$
$74$	0	0
$75$	21.1936i	0.0326297i
$76$	0	0
$77$	148.194i	0.219328i
$78$	0	0
$79$	−770.379	−1.09714	−0.548572	−	0.836103i	$-0.684828\pi$
$79$	−770.379	−1.09714	−0.548572	+	0.836103i	$0.684828\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	− 1284.50i	− 1.69870i	−0.527829	−	0.849350i	$-0.676994\pi$
$83$	− 1284.50i	− 1.69870i	0.527829	−	0.849350i	$-0.323006\pi$
$84$	0	0
$85$	1229.08i	1.56838i
$86$	0	0
$87$	388.185	0.478366
$88$	0	0
$89$	−844.161	−1.00540	−0.502702	−	0.864460i	$-0.667661\pi$
$89$	−844.161	−1.00540	−0.502702	+	0.864460i	$0.667661\pi$
$90$	0	0
$91$	− 26.9839i	− 0.0310844i
$92$	0	0
$93$	739.331i	0.824355i
$94$	0	0
$95$	999.613	1.07956
$96$	0	0
$97$	−1259.84	−1.31873	−0.659367	−	0.751821i	$-0.729176\pi$
$97$	−1259.84	−1.31873	−0.659367	+	0.751821i	$0.729176\pi$
$98$	0	0
$99$	98.8548i	0.100356i
$100$	0	0
$101$	50.4113i	0.0496644i	0.999692	+	0.0248322i	$0.00790515\pi$
$101$	50.4113i	0.0496644i	−0.999692	+	0.0248322i	$0.992095\pi$
$102$	0	0
$103$	1055.40	1.00962	0.504812	−	0.863230i	$-0.331562\pi$
$103$	1055.40	1.00962	0.504812	+	0.863230i	$0.331562\pi$
$104$	0	0
$105$	−465.145	−0.432319
$106$	0	0
$107$	1197.39i	1.08183i	0.841077	+	0.540915i	$0.181922\pi$
$107$	1197.39i	1.08183i	−0.841077	+	0.540915i	$0.818078\pi$
$108$	0	0
$109$	1583.32i	1.39133i	0.718367	+	0.695664i	$0.244890\pi$
$109$	1583.32i	1.39133i	−0.718367	+	0.695664i	$0.755110\pi$
$110$	0	0
$111$	777.048	0.664452
$112$	0	0
$113$	−907.710	−0.755665	−0.377832	−	0.925874i	$-0.623330\pi$
$113$	−907.710	−0.755665	−0.377832	+	0.925874i	$0.623330\pi$
$114$	0	0
$115$	746.419i	0.605252i
$116$	0	0
$117$	− 18.0000i	− 0.0142231i
$118$	0	0
$119$	−1442.98	−1.11158
$120$	0	0
$121$	1210.35	0.909357
$122$	0	0
$123$	974.952i	0.714703i
$124$	0	0
$125$	1355.31i	0.969778i
$126$	0	0
$127$	37.0566	0.0258917	0.0129458	−	0.999916i	$-0.495879\pi$
$127$	37.0566	0.0258917	0.0129458	+	0.999916i	$0.495879\pi$
$128$	0	0
$129$	878.468	0.599572
$130$	0	0
$131$	203.613i	0.135799i	0.997692	+	0.0678997i	$0.0216298\pi$
$131$	203.613i	0.135799i	−0.997692	+	0.0678997i	$0.978370\pi$
$132$	0	0
$133$	1173.58i	0.765130i
$134$	0	0
$135$	−310.282	−0.197814
$136$	0	0
$137$	−247.403	−0.154285	−0.0771427	−	0.997020i	$-0.524580\pi$
$137$	−247.403	−0.154285	−0.0771427	+	0.997020i	$0.524580\pi$
$138$	0	0
$139$	− 1631.68i	− 0.995662i	−0.867274	−	0.497831i	$-0.834130\pi$
$139$	− 1631.68i	− 0.995662i	0.867274	−	0.497831i	$-0.165870\pi$
$140$	0	0
$141$	1160.95i	0.693403i
$142$	0	0
$143$	21.9677	0.0128464
$144$	0	0
$145$	−1487.00	−0.851646
$146$	0	0
$147$	482.903i	0.270947i
$148$	0	0
$149$	495.298i	0.272325i	0.990687	+	0.136162i	$0.0434769\pi$
$149$	495.298i	0.272325i	−0.990687	+	0.136162i	$0.956523\pi$
$150$	0	0
$151$	576.605	0.310751	0.155376	−	0.987855i	$-0.450341\pi$
$151$	576.605	0.310751	0.155376	+	0.987855i	$0.450341\pi$
$152$	0	0
$153$	−962.564	−0.508619
$154$	0	0
$155$	− 2832.11i	− 1.46762i
$156$	0	0
$157$	212.855i	0.108202i	0.998535	+	0.0541008i	$0.0172292\pi$
$157$	212.855i	0.108202i	−0.998535	+	0.0541008i	$0.982771\pi$
$158$	0	0
$159$	1609.52	0.802790
$160$	0	0
$161$	−876.323	−0.428968
$162$	0	0
$163$	2991.11i	1.43731i	0.695365	+	0.718657i	$0.255243\pi$
$163$	2991.11i	1.43731i	−0.695365	+	0.718657i	$0.744757\pi$
$164$	0	0
$165$	− 378.678i	− 0.178667i
$166$	0	0
$167$	−3231.23	−1.49724	−0.748622	−	0.662997i	$-0.769284\pi$
$167$	−3231.23	−1.49724	−0.748622	+	0.662997i	$0.769284\pi$
$168$	0	0
$169$	2193.00	0.998179
$170$	0	0
$171$	782.855i	0.350096i
$172$	0	0
$173$	2815.93i	1.23752i	0.785580	+	0.618760i	$0.212365\pi$
$173$	2815.93i	1.23752i	−0.785580	+	0.618760i	$0.787635\pi$
$174$	0	0
$175$	95.3142	0.0411719
$176$	0	0
$177$	−311.419	−0.132247
$178$	0	0
$179$	630.903i	0.263441i	0.991287	+	0.131720i	$0.0420501\pi$
$179$	630.903i	0.263441i	−0.991287	+	0.131720i	$0.957950\pi$
$180$	0	0
$181$	1748.23i	0.717926i	0.933352	+	0.358963i	$0.116870\pi$
$181$	1748.23i	0.717926i	−0.933352	+	0.358963i	$0.883130\pi$
$182$	0	0
$183$	−1886.37	−0.761992
$184$	0	0
$185$	−2976.60	−1.18294
$186$	0	0
$187$	− 1174.74i	− 0.459389i
$188$	0	0
$189$	− 364.282i	− 0.140199i
$190$	0	0
$191$	−1450.52	−0.549506	−0.274753	−	0.961515i	$-0.588596\pi$
$191$	−1450.52	−0.549506	−0.274753	+	0.961515i	$0.588596\pi$
$192$	0	0
$193$	−4302.94	−1.60483	−0.802415	−	0.596767i	$-0.796452\pi$
$193$	−4302.94	−1.60483	−0.802415	+	0.596767i	$0.796452\pi$
$194$	0	0
$195$	68.9516i	0.0253217i
$196$	0	0
$197$	− 4336.23i	− 1.56824i	−0.620607	−	0.784121i	$-0.713114\pi$
$197$	− 4336.23i	− 1.56824i	0.620607	−	0.784121i	$-0.286886\pi$
$198$	0	0
$199$	4183.88	1.49039	0.745194	−	0.666848i	$-0.232357\pi$
$199$	4183.88	1.49039	0.745194	+	0.666848i	$0.232357\pi$
$200$	0	0
$201$	2158.64	0.757508
$202$	0	0
$203$	− 1745.79i	− 0.603598i
$204$	0	0
$205$	− 3734.69i	− 1.27240i
$206$	0	0
$207$	−584.564	−0.196280
$208$	0	0
$209$	−955.419	−0.316209
$210$	0	0
$211$	1476.65i	0.481784i	0.970552	+	0.240892i	$0.0774400\pi$
$211$	1476.65i	0.481784i	−0.970552	+	0.240892i	$0.922560\pi$
$212$	0	0
$213$	− 602.855i	− 0.193929i
$214$	0	0
$215$	−3365.10	−1.06743
$216$	0	0
$217$	3325.00	1.04016
$218$	0	0
$219$	2962.74i	0.914171i
$220$	0	0
$221$	213.903i	0.0651072i
$222$	0	0
$223$	6097.96	1.83116	0.915582	−	0.402131i	$-0.131731\pi$
$223$	6097.96	1.83116	0.915582	+	0.402131i	$0.131731\pi$
$224$	0	0
$225$	63.5808	0.0188388
$226$	0	0
$227$	− 6141.37i	− 1.79567i	−0.440332	−	0.897835i	$-0.645139\pi$
$227$	− 6141.37i	− 1.79567i	0.440332	−	0.897835i	$-0.354861\pi$
$228$	0	0
$229$	− 2168.84i	− 0.625855i	−0.949777	−	0.312928i	$-0.898690\pi$
$229$	− 2168.84i	− 0.625855i	0.949777	−	0.312928i	$-0.101310\pi$
$230$	0	0
$231$	444.581	0.126629
$232$	0	0
$233$	2142.61	0.602434	0.301217	−	0.953556i	$-0.402607\pi$
$233$	2142.61	0.602434	0.301217	+	0.953556i	$0.402607\pi$
$234$	0	0
$235$	− 4447.19i	− 1.23448i
$236$	0	0
$237$	2311.14i	0.633437i
$238$	0	0
$239$	−6721.00	−1.81902	−0.909509	−	0.415684i	$-0.863542\pi$
$239$	−6721.00	−1.81902	−0.909509	+	0.415684i	$0.863542\pi$
$240$	0	0
$241$	−193.226	−0.0516463	−0.0258231	−	0.999667i	$-0.508221\pi$
$241$	−193.226	−0.0516463	−0.0258231	+	0.999667i	$0.508221\pi$
$242$	0	0
$243$	− 243.000i	− 0.0641500i
$244$	0	0
$245$	− 1849.83i	− 0.482373i
$246$	0	0
$247$	173.968	0.0448150
$248$	0	0
$249$	−3853.50	−0.980745
$250$	0	0
$251$	− 1434.98i	− 0.360858i	−0.983588	−	0.180429i	$-0.942251\pi$
$251$	− 1434.98i	− 0.360858i	0.983588	−	0.180429i	$-0.0577486\pi$
$252$	0	0
$253$	− 713.420i	− 0.177282i
$254$	0	0
$255$	3687.24	0.905506
$256$	0	0
$257$	5333.03	1.29442	0.647209	−	0.762313i	$-0.275936\pi$
$257$	5333.03	1.29442	0.647209	+	0.762313i	$0.275936\pi$
$258$	0	0
$259$	− 3494.63i	− 0.838400i
$260$	0	0
$261$	− 1164.56i	− 0.276185i
$262$	0	0
$263$	−552.774	−0.129603	−0.0648014	−	0.997898i	$-0.520641\pi$
$263$	−552.774	−0.129603	−0.0648014	+	0.997898i	$0.520641\pi$
$264$	0	0
$265$	−6165.51	−1.42922
$266$	0	0
$267$	2532.48i	0.580470i
$268$	0	0
$269$	− 2364.31i	− 0.535891i	−0.963434	−	0.267946i	$-0.913655\pi$
$269$	− 2364.31i	− 0.535891i	0.963434	−	0.267946i	$-0.0863448\pi$
$270$	0	0
$271$	−7133.72	−1.59905	−0.799525	−	0.600633i	$-0.794915\pi$
$271$	−7133.72	−1.59905	−0.799525	+	0.600633i	$0.794915\pi$
$272$	0	0
$273$	−80.9516	−0.0179466
$274$	0	0
$275$	77.5959i	0.0170153i
$276$	0	0
$277$	2025.77i	0.439411i	0.975566	+	0.219706i	$0.0705097\pi$
$277$	2025.77i	0.439411i	−0.975566	+	0.219706i	$0.929490\pi$
$278$	0	0
$279$	2217.99	0.475942
$280$	0	0
$281$	2068.84	0.439205	0.219602	−	0.975589i	$-0.429524\pi$
$281$	2068.84	0.439205	0.219602	+	0.975589i	$0.429524\pi$
$282$	0	0
$283$	3408.19i	0.715887i	0.933743	+	0.357944i	$0.116522\pi$
$283$	3408.19i	0.715887i	−0.933743	+	0.357944i	$0.883478\pi$
$284$	0	0
$285$	− 2998.84i	− 0.623284i
$286$	0	0
$287$	4384.66	0.901806
$288$	0	0
$289$	6525.64	1.32824
$290$	0	0
$291$	3779.52i	0.761372i
$292$	0	0
$293$	4200.96i	0.837620i	0.908074	+	0.418810i	$0.137553\pi$
$293$	4200.96i	0.837620i	−0.908074	+	0.418810i	$0.862447\pi$
$294$	0	0
$295$	1192.94	0.235442
$296$	0	0
$297$	296.564	0.0579408
$298$	0	0
$299$	129.903i	0.0251254i
$300$	0	0
$301$	− 3950.74i	− 0.756535i
$302$	0	0
$303$	151.234	0.0286738
$304$	0	0
$305$	7226.02	1.35659
$306$	0	0
$307$	4876.19i	0.906511i	0.891381	+	0.453256i	$0.149738\pi$
$307$	4876.19i	0.906511i	−0.891381	+	0.453256i	$0.850262\pi$
$308$	0	0
$309$	− 3166.19i	− 0.582906i
$310$	0	0
$311$	−6881.71	−1.25475	−0.627373	−	0.778719i	$-0.715870\pi$
$311$	−6881.71	−1.25475	−0.627373	+	0.778719i	$0.715870\pi$
$312$	0	0
$313$	7419.45	1.33985	0.669924	−	0.742430i	$-0.266327\pi$
$313$	7419.45	1.33985	0.669924	+	0.742430i	$0.266327\pi$
$314$	0	0
$315$	1395.44i	0.249600i
$316$	0	0
$317$	− 2675.06i	− 0.473963i	−0.971514	−	0.236981i	$-0.923842\pi$
$317$	− 2675.06i	− 0.473963i	0.971514	−	0.236981i	$-0.0761580\pi$
$318$	0	0
$319$	1421.26	0.249452
$320$	0	0
$321$	3592.16	0.624595
$322$	0	0
$323$	− 9303.06i	− 1.60259i
$324$	0	0
$325$	− 14.1291i	− 0.00241151i
$326$	0	0
$327$	4749.97	0.803284
$328$	0	0
$329$	5221.16	0.874930
$330$	0	0
$331$	9878.74i	1.64044i	0.572050	+	0.820219i	$0.306148\pi$
$331$	9878.74i	1.64044i	−0.572050	+	0.820219i	$0.693852\pi$
$332$	0	0
$333$	− 2331.15i	− 0.383622i
$334$	0	0
$335$	−8269.00	−1.34861
$336$	0	0
$337$	7522.81	1.21600	0.608002	−	0.793935i	$-0.291971\pi$
$337$	7522.81	1.21600	0.608002	+	0.793935i	$0.291971\pi$
$338$	0	0
$339$	2723.13i	0.436283i
$340$	0	0
$341$	2706.90i	0.429874i
$342$	0	0
$343$	6799.50	1.07037
$344$	0	0
$345$	2239.26	0.349442
$346$	0	0
$347$	1762.56i	0.272678i	0.990662	+	0.136339i	$0.0435337\pi$
$347$	1762.56i	0.272678i	−0.990662	+	0.136339i	$0.956466\pi$
$348$	0	0
$349$	− 3913.89i	− 0.600303i	−0.953892	−	0.300151i	$-0.902963\pi$
$349$	− 3913.89i	− 0.600303i	0.953892	−	0.300151i	$-0.0970372\pi$
$350$	0	0
$351$	−54.0000	−0.00821170
$352$	0	0
$353$	848.742	0.127972	0.0639858	−	0.997951i	$-0.479619\pi$
$353$	848.742	0.127972	0.0639858	+	0.997951i	$0.479619\pi$
$354$	0	0
$355$	2309.32i	0.345257i
$356$	0	0
$357$	4328.95i	0.641771i
$358$	0	0
$359$	3899.98	0.573352	0.286676	−	0.958028i	$-0.407450\pi$
$359$	3899.98	0.573352	0.286676	+	0.958028i	$0.407450\pi$
$360$	0	0
$361$	−707.193	−0.103104
$362$	0	0
$363$	− 3631.06i	− 0.525018i
$364$	0	0
$365$	− 11349.2i	− 1.62752i
$366$	0	0
$367$	−4038.70	−0.574437	−0.287219	−	0.957865i	$-0.592731\pi$
$367$	−4038.70	−0.574437	−0.287219	+	0.957865i	$0.592731\pi$
$368$	0	0
$369$	2924.85	0.412634
$370$	0	0
$371$	− 7238.53i	− 1.01295i
$372$	0	0
$373$	− 11503.2i	− 1.59682i	−0.602115	−	0.798410i	$-0.705675\pi$
$373$	− 11503.2i	− 1.59682i	0.602115	−	0.798410i	$-0.294325\pi$
$374$	0	0
$375$	4065.92	0.559902
$376$	0	0
$377$	−258.790	−0.0353538
$378$	0	0
$379$	8121.63i	1.10074i	0.834921	+	0.550369i	$0.185513\pi$
$379$	8121.63i	1.10074i	−0.834921	+	0.550369i	$0.814487\pi$
$380$	0	0
$381$	− 111.170i	− 0.0149486i
$382$	0	0
$383$	−3528.45	−0.470745	−0.235373	−	0.971905i	$-0.575631\pi$
$383$	−3528.45	−0.470745	−0.235373	+	0.971905i	$0.575631\pi$
$384$	0	0
$385$	−1703.03	−0.225440
$386$	0	0
$387$	− 2635.40i	− 0.346163i
$388$	0	0
$389$	106.234i	0.0138465i	0.999976	+	0.00692324i	$0.00220375\pi$
$389$	106.234i	0.0138465i	−0.999976	+	0.00692324i	$0.997796\pi$
$390$	0	0
$391$	6946.68	0.898487
$392$	0	0
$393$	610.838	0.0784039
$394$	0	0
$395$	− 8853.14i	− 1.12772i
$396$	0	0
$397$	5479.72i	0.692744i	0.938097	+	0.346372i	$0.112587\pi$
$397$	5479.72i	0.692744i	−0.938097	+	0.346372i	$0.887413\pi$
$398$	0	0
$399$	3520.74	0.441748
$400$	0	0
$401$	−4977.08	−0.619809	−0.309905	−	0.950768i	$-0.600297\pi$
$401$	−4977.08	−0.619809	−0.309905	+	0.950768i	$0.600297\pi$
$402$	0	0
$403$	− 492.887i	− 0.0609242i
$404$	0	0
$405$	930.847i	0.114208i
$406$	0	0
$407$	2845.00	0.346490
$408$	0	0
$409$	−1272.35	−0.153824	−0.0769119	−	0.997038i	$-0.524506\pi$
$409$	−1272.35	−0.153824	−0.0769119	+	0.997038i	$0.524506\pi$
$410$	0	0
$411$	742.210i	0.0890767i
$412$	0	0
$413$	1400.55i	0.166868i
$414$	0	0
$415$	14761.4	1.74604
$416$	0	0
$417$	−4895.03	−0.574846
$418$	0	0
$419$	6921.14i	0.806969i	0.914986	+	0.403485i	$0.132201\pi$
$419$	6921.14i	0.806969i	−0.914986	+	0.403485i	$0.867799\pi$
$420$	0	0
$421$	1903.90i	0.220405i	0.993909	+	0.110203i	$0.0351500\pi$
$421$	1903.90i	0.220405i	−0.993909	+	0.110203i	$0.964850\pi$
$422$	0	0
$423$	3482.85	0.400336
$424$	0	0
$425$	−755.563	−0.0862358
$426$	0	0
$427$	8483.60i	0.961475i
$428$	0	0
$429$	− 65.9032i	− 0.00741687i
$430$	0	0
$431$	−13752.4	−1.53696	−0.768482	−	0.639871i	$-0.778988\pi$
$431$	−13752.4	−1.53696	−0.768482	+	0.639871i	$0.778988\pi$
$432$	0	0
$433$	12292.7	1.36432	0.682159	−	0.731204i	$-0.261041\pi$
$433$	12292.7	1.36432	0.682159	+	0.731204i	$0.261041\pi$
$434$	0	0
$435$	4461.00i	0.491698i
$436$	0	0
$437$	− 5649.74i	− 0.618453i
$438$	0	0
$439$	−16495.5	−1.79337	−0.896683	−	0.442673i	$-0.854030\pi$
$439$	−16495.5	−1.79337	−0.896683	+	0.442673i	$0.854030\pi$
$440$	0	0
$441$	1448.71	0.156431
$442$	0	0
$443$	− 15534.9i	− 1.66610i	−0.553196	−	0.833051i	$-0.686592\pi$
$443$	− 15534.9i	− 1.66610i	0.553196	−	0.833051i	$-0.313408\pi$
$444$	0	0
$445$	− 9701.05i	− 1.03342i
$446$	0	0
$447$	1485.90	0.157227
$448$	0	0
$449$	2147.08	0.225673	0.112836	−	0.993614i	$-0.464006\pi$
$449$	2147.08	0.225673	0.112836	+	0.993614i	$0.464006\pi$
$450$	0	0
$451$	3569.58i	0.372694i
$452$	0	0
$453$	− 1729.81i	− 0.179412i
$454$	0	0
$455$	310.097	0.0319507
$456$	0	0
$457$	−8907.65	−0.911777	−0.455888	−	0.890037i	$-0.650678\pi$
$457$	−8907.65	−0.911777	−0.455888	+	0.890037i	$0.650678\pi$
$458$	0	0
$459$	2887.69i	0.293651i
$460$	0	0
$461$	− 11673.7i	− 1.17939i	−0.807627	−	0.589694i	$-0.799248\pi$
$461$	− 11673.7i	− 1.17939i	0.807627	−	0.589694i	$-0.200752\pi$
$462$	0	0
$463$	416.121	0.0417685	0.0208842	−	0.999782i	$-0.493352\pi$
$463$	416.121	0.0417685	0.0208842	+	0.999782i	$0.493352\pi$
$464$	0	0
$465$	−8496.34	−0.847330
$466$	0	0
$467$	− 12272.9i	− 1.21611i	−0.793895	−	0.608055i	$-0.791950\pi$
$467$	− 12272.9i	− 1.21611i	0.793895	−	0.608055i	$-0.208050\pi$
$468$	0	0
$469$	− 9708.10i	− 0.955817i
$470$	0	0
$471$	638.564	0.0624703
$472$	0	0
$473$	3216.32	0.312657
$474$	0	0
$475$	614.500i	0.0593583i
$476$	0	0
$477$	− 4828.57i	− 0.463491i
$478$	0	0
$479$	19279.7	1.83906	0.919532	−	0.393015i	$-0.128568\pi$
$479$	19279.7	1.83906	0.919532	+	0.393015i	$0.128568\pi$
$480$	0	0
$481$	−518.032	−0.0491065
$482$	0	0
$483$	2628.97i	0.247665i
$484$	0	0
$485$	− 14478.0i	− 1.35549i
$486$	0	0
$487$	−14088.1	−1.31087	−0.655433	−	0.755254i	$-0.727514\pi$
$487$	−14088.1	−1.31087	−0.655433	+	0.755254i	$0.727514\pi$
$488$	0	0
$489$	8973.34	0.829833
$490$	0	0
$491$	− 6405.32i	− 0.588734i	−0.955693	−	0.294367i	$-0.904891\pi$
$491$	− 6405.32i	− 0.588734i	0.955693	−	0.294367i	$-0.0951087\pi$
$492$	0	0
$493$	13839.0i	1.26426i
$494$	0	0
$495$	−1136.03	−0.103153
$496$	0	0
$497$	−2711.23	−0.244698
$498$	0	0
$499$	− 4781.48i	− 0.428955i	−0.976729	−	0.214478i	$-0.931195\pi$
$499$	− 4781.48i	− 0.428955i	0.976729	−	0.214478i	$-0.0688049\pi$
$500$	0	0
$501$	9693.68i	0.864434i
$502$	0	0
$503$	−17222.8	−1.52669	−0.763346	−	0.645990i	$-0.776445\pi$
$503$	−17222.8	−1.52669	−0.763346	+	0.645990i	$0.776445\pi$
$504$	0	0
$505$	−579.323	−0.0510486
$506$	0	0
$507$	− 6579.00i	− 0.576299i
$508$	0	0
$509$	17597.9i	1.53244i	0.642578	+	0.766220i	$0.277865\pi$
$509$	17597.9i	1.53244i	−0.642578	+	0.766220i	$0.722135\pi$
$510$	0	0
$511$	13324.4	1.15349
$512$	0	0
$513$	2348.56	0.202128
$514$	0	0
$515$	12128.5i	1.03776i
$516$	0	0
$517$	4250.58i	0.361587i
$518$	0	0
$519$	8447.78	0.714483
$520$	0	0
$521$	−6205.24	−0.521798	−0.260899	−	0.965366i	$-0.584019\pi$
$521$	−6205.24	−0.521798	−0.260899	+	0.965366i	$0.584019\pi$
$522$	0	0
$523$	1164.66i	0.0973749i	0.998814	+	0.0486874i	$0.0155038\pi$
$523$	1164.66i	0.0973749i	−0.998814	+	0.0486874i	$0.984496\pi$
$524$	0	0
$525$	− 285.943i	− 0.0237706i
$526$	0	0
$527$	−26357.5	−2.17866
$528$	0	0
$529$	−7948.29	−0.653266
$530$	0	0
$531$	934.258i	0.0763528i
$532$	0	0
$533$	− 649.968i	− 0.0528203i
$534$	0	0
$535$	−13760.3	−1.11198
$536$	0	0
$537$	1892.71	0.152098
$538$	0	0
$539$	1768.05i	0.141290i
$540$	0	0
$541$	18467.1i	1.46759i	0.679373	+	0.733793i	$0.262252\pi$
$541$	18467.1i	1.46759i	−0.679373	+	0.733793i	$0.737748\pi$
$542$	0	0
$543$	5244.68	0.414495
$544$	0	0
$545$	−18195.4	−1.43010
$546$	0	0
$547$	− 4387.05i	− 0.342919i	−0.985191	−	0.171459i	$-0.945152\pi$
$547$	− 4387.05i	− 0.342919i	0.985191	−	0.171459i	$-0.0548483\pi$
$548$	0	0
$549$	5659.11i	0.439936i
$550$	0	0
$551$	11255.3	0.870220
$552$	0	0
$553$	10393.9	0.799265
$554$	0	0
$555$	8929.79i	0.682970i
$556$	0	0
$557$	2979.09i	0.226621i	0.993560	+	0.113311i	$0.0361455\pi$
$557$	2979.09i	0.226621i	−0.993560	+	0.113311i	$0.963854\pi$
$558$	0	0
$559$	−585.645	−0.0443115
$560$	0	0
$561$	−3524.23	−0.265228
$562$	0	0
$563$	13696.2i	1.02527i	0.858607	+	0.512634i	$0.171330\pi$
$563$	13696.2i	1.02527i	−0.858607	+	0.512634i	$0.828670\pi$
$564$	0	0
$565$	− 10431.3i	− 0.776725i
$566$	0	0
$567$	−1092.85	−0.0809440
$568$	0	0
$569$	6973.98	0.513822	0.256911	−	0.966435i	$-0.417295\pi$
$569$	6973.98	0.513822	0.256911	+	0.966435i	$0.417295\pi$
$570$	0	0
$571$	− 10571.3i	− 0.774774i	−0.921917	−	0.387387i	$-0.873378\pi$
$571$	− 10571.3i	− 0.774774i	0.921917	−	0.387387i	$-0.126622\pi$
$572$	0	0
$573$	4351.55i	0.317257i
$574$	0	0
$575$	−458.853	−0.0332791
$576$	0	0
$577$	9195.26	0.663438	0.331719	−	0.943378i	$-0.392371\pi$
$577$	9195.26	0.663438	0.331719	+	0.943378i	$0.392371\pi$
$578$	0	0
$579$	12908.8i	0.926549i
$580$	0	0
$581$	17330.4i	1.23750i
$582$	0	0
$583$	5892.93	0.418628
$584$	0	0
$585$	206.855	0.0146195
$586$	0	0
$587$	− 14538.3i	− 1.02225i	−0.859507	−	0.511124i	$-0.829229\pi$
$587$	− 14538.3i	− 1.02225i	0.859507	−	0.511124i	$-0.170771\pi$
$588$	0	0
$589$	21436.6i	1.49963i
$590$	0	0
$591$	−13008.7	−0.905425
$592$	0	0
$593$	−10422.9	−0.721785	−0.360893	−	0.932607i	$-0.617528\pi$
$593$	−10422.9	−0.721785	−0.360893	+	0.932607i	$0.617528\pi$
$594$	0	0
$595$	− 16582.7i	− 1.14256i
$596$	0	0
$597$	− 12551.6i	− 0.860476i
$598$	0	0
$599$	−7293.27	−0.497488	−0.248744	−	0.968569i	$-0.580018\pi$
$599$	−7293.27	−0.497488	−0.248744	+	0.968569i	$0.580018\pi$
$600$	0	0
$601$	−16685.1	−1.13245	−0.566223	−	0.824252i	$-0.691596\pi$
$601$	−16685.1	−1.13245	−0.566223	+	0.824252i	$0.691596\pi$
$602$	0	0
$603$	− 6475.93i	− 0.437347i
$604$	0	0
$605$	13909.3i	0.934701i
$606$	0	0
$607$	14554.1	0.973200	0.486600	−	0.873625i	$-0.338237\pi$
$607$	14554.1	0.973200	0.486600	+	0.873625i	$0.338237\pi$
$608$	0	0
$609$	−5237.37	−0.348488
$610$	0	0
$611$	− 773.968i	− 0.0512461i
$612$	0	0
$613$	− 3335.34i	− 0.219760i	−0.993945	−	0.109880i	$-0.964953\pi$
$613$	− 3335.34i	− 0.219760i	0.993945	−	0.109880i	$-0.0350467\pi$
$614$	0	0
$615$	−11204.1	−0.734621
$616$	0	0
$617$	18534.0	1.20932	0.604660	−	0.796484i	$-0.293309\pi$
$617$	18534.0	1.20932	0.604660	+	0.796484i	$0.293309\pi$
$618$	0	0
$619$	− 1292.45i	− 0.0839224i	−0.999119	−	0.0419612i	$-0.986639\pi$
$619$	− 1292.45i	− 0.0839224i	0.999119	−	0.0419612i	$-0.0133606\pi$
$620$	0	0
$621$	1753.69i	0.113323i
$622$	0	0
$623$	11389.4	0.732432
$624$	0	0
$625$	−16458.2	−1.05332
$626$	0	0
$627$	2866.26i	0.182563i
$628$	0	0
$629$	27702.2i	1.75606i
$630$	0	0
$631$	10708.0	0.675560	0.337780	−	0.941225i	$-0.390324\pi$
$631$	10708.0	0.675560	0.337780	+	0.941225i	$0.390324\pi$
$632$	0	0
$633$	4429.94	0.278158
$634$	0	0
$635$	425.852i	0.0266133i
$636$	0	0
$637$	− 321.935i	− 0.0200244i
$638$	0	0
$639$	−1808.56	−0.111965
$640$	0	0
$641$	−2773.50	−0.170900	−0.0854499	−	0.996342i	$-0.527233\pi$
$641$	−2773.50	−0.170900	−0.0854499	+	0.996342i	$0.527233\pi$
$642$	0	0
$643$	5474.50i	0.335759i	0.985808	+	0.167880i	$0.0536920\pi$
$643$	5474.50i	0.335759i	−0.985808	+	0.167880i	$0.946308\pi$
$644$	0	0
$645$	10095.3i	0.616282i
$646$	0	0
$647$	−19944.3	−1.21189	−0.605943	−	0.795508i	$-0.707204\pi$
$647$	−19944.3	−1.21189	−0.605943	+	0.795508i	$0.707204\pi$
$648$	0	0
$649$	−1140.20	−0.0689624
$650$	0	0
$651$	− 9975.00i	− 0.600539i
$652$	0	0
$653$	− 12550.2i	− 0.752111i	−0.926597	−	0.376056i	$-0.877280\pi$
$653$	− 12550.2i	− 0.752111i	0.926597	−	0.376056i	$-0.122720\pi$
$654$	0	0
$655$	−2339.90	−0.139584
$656$	0	0
$657$	8888.22	0.527797
$658$	0	0
$659$	− 6571.77i	− 0.388467i	−0.980955	−	0.194234i	$-0.937778\pi$
$659$	− 6571.77i	− 0.388467i	0.980955	−	0.194234i	$-0.0622220\pi$
$660$	0	0
$661$	11166.8i	0.657094i	0.944488	+	0.328547i	$0.106559\pi$
$661$	11166.8i	0.657094i	−0.944488	+	0.328547i	$0.893441\pi$
$662$	0	0
$663$	641.710	0.0375896
$664$	0	0
$665$	−13486.7	−0.786454
$666$	0	0
$667$	8404.42i	0.487887i
$668$	0	0
$669$	− 18293.9i	− 1.05722i
$670$	0	0
$671$	−6906.55	−0.397354
$672$	0	0
$673$	−4712.78	−0.269932	−0.134966	−	0.990850i	$-0.543093\pi$
$673$	−4712.78	−0.269932	−0.134966	+	0.990850i	$0.543093\pi$
$674$	0	0
$675$	− 190.742i	− 0.0108766i
$676$	0	0
$677$	16444.0i	0.933520i	0.884384	+	0.466760i	$0.154579\pi$
$677$	16444.0i	0.933520i	−0.884384	+	0.466760i	$0.845421\pi$
$678$	0	0
$679$	16997.7	0.960693
$680$	0	0
$681$	−18424.1	−1.03673
$682$	0	0
$683$	7109.18i	0.398280i	0.979971	+	0.199140i	$0.0638148\pi$
$683$	7109.18i	0.398280i	−0.979971	+	0.199140i	$0.936185\pi$
$684$	0	0
$685$	− 2843.14i	− 0.158585i
$686$	0	0
$687$	−6506.52	−0.361338
$688$	0	0
$689$	−1073.02	−0.0593304
$690$	0	0
$691$	7879.89i	0.433813i	0.976192	+	0.216907i	$0.0695967\pi$
$691$	7879.89i	0.433813i	−0.976192	+	0.216907i	$0.930403\pi$
$692$	0	0
$693$	− 1333.74i	− 0.0731092i
$694$	0	0
$695$	18751.1	1.02341
$696$	0	0
$697$	−34757.5	−1.88886
$698$	0	0
$699$	− 6427.84i	− 0.347816i
$700$	0	0
$701$	17218.0i	0.927698i	0.885914	+	0.463849i	$0.153532\pi$
$701$	17218.0i	0.927698i	−0.885914	+	0.463849i	$0.846468\pi$
$702$	0	0
$703$	22530.2	1.20874
$704$	0	0
$705$	−13341.6	−0.712728
$706$	0	0
$707$	− 680.145i	− 0.0361803i
$708$	0	0
$709$	3957.16i	0.209611i	0.994493	+	0.104806i	$0.0334220\pi$
$709$	3957.16i	0.209611i	−0.994493	+	0.104806i	$0.966578\pi$
$710$	0	0
$711$	6933.41	0.365715
$712$	0	0
$713$	−16006.9	−0.840762
$714$	0	0
$715$	252.452i	0.0132044i
$716$	0	0
$717$	20163.0i	1.05021i
$718$	0	0
$719$	23531.0	1.22053	0.610263	−	0.792199i	$-0.291064\pi$
$719$	23531.0	1.22053	0.610263	+	0.792199i	$0.291064\pi$
$720$	0	0
$721$	−14239.3	−0.735506
$722$	0	0
$723$	579.677i	0.0298180i
$724$	0	0
$725$	− 914.116i	− 0.0468268i
$726$	0	0
$727$	2462.54	0.125627	0.0628133	−	0.998025i	$-0.479993\pi$
$727$	2462.54	0.125627	0.0628133	+	0.998025i	$0.479993\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	31317.8i	1.58459i
$732$	0	0
$733$	7436.45i	0.374722i	0.982291	+	0.187361i	$0.0599935\pi$
$733$	7436.45i	0.374722i	−0.982291	+	0.187361i	$0.940007\pi$
$734$	0	0
$735$	−5549.49	−0.278498
$736$	0	0
$737$	7903.42	0.395015
$738$	0	0
$739$	− 7969.32i	− 0.396693i	−0.980132	−	0.198347i	$-0.936443\pi$
$739$	− 7969.32i	− 0.396693i	0.980132	−	0.198347i	$-0.0635571\pi$
$740$	0	0
$741$	− 521.903i	− 0.0258739i
$742$	0	0
$743$	13701.3	0.676515	0.338257	−	0.941054i	$-0.390163\pi$
$743$	13701.3	0.676515	0.338257	+	0.941054i	$0.390163\pi$
$744$	0	0
$745$	−5691.94	−0.279915
$746$	0	0
$747$	11560.5i	0.566234i
$748$	0	0
$749$	− 16155.1i	− 0.788108i
$750$	0	0
$751$	−13574.0	−0.659550	−0.329775	−	0.944060i	$-0.606973\pi$
$751$	−13574.0	−0.659550	−0.329775	+	0.944060i	$0.606973\pi$
$752$	0	0
$753$	−4304.95	−0.208342
$754$	0	0
$755$	6626.30i	0.319412i
$756$	0	0
$757$	19115.4i	0.917783i	0.888492	+	0.458891i	$0.151753\pi$
$757$	19115.4i	0.917783i	−0.888492	+	0.458891i	$0.848247\pi$
$758$	0	0
$759$	−2140.26	−0.102354
$760$	0	0
$761$	−647.405	−0.0308389	−0.0154195	−	0.999881i	$-0.504908\pi$
$761$	−647.405	−0.0308389	−0.0154195	+	0.999881i	$0.504908\pi$
$762$	0	0
$763$	− 21362.1i	− 1.01358i
$764$	0	0
$765$	− 11061.7i	− 0.522794i
$766$	0	0
$767$	207.613	0.00977375
$768$	0	0
$769$	13993.6	0.656208	0.328104	−	0.944642i	$-0.393590\pi$
$769$	13993.6	0.656208	0.328104	+	0.944642i	$0.393590\pi$
$770$	0	0
$771$	− 15999.1i	− 0.747333i
$772$	0	0
$773$	− 38664.6i	− 1.79905i	−0.436865	−	0.899527i	$-0.643911\pi$
$773$	− 38664.6i	− 1.79905i	0.436865	−	0.899527i	$-0.356089\pi$
$774$	0	0
$775$	1741.01	0.0806953
$776$	0	0
$777$	−10483.9	−0.484051
$778$	0	0
$779$	28268.4i	1.30015i
$780$	0	0
$781$	− 2207.23i	− 0.101128i
$782$	0	0
$783$	−3493.67	−0.159455
$784$	0	0
$785$	−2446.11	−0.111217
$786$	0	0
$787$	− 16965.4i	− 0.768426i	−0.923244	−	0.384213i	$-0.874473\pi$
$787$	− 16965.4i	− 0.768426i	0.923244	−	0.384213i	$-0.125527\pi$
$788$	0	0
$789$	1658.32i	0.0748262i
$790$	0	0
$791$	12246.8	0.550499
$792$	0	0
$793$	1257.58	0.0563153
$794$	0	0
$795$	18496.5i	0.825163i
$796$	0	0
$797$	− 10338.1i	− 0.459467i	−0.973254	−	0.229734i	$-0.926214\pi$
$797$	− 10338.1i	− 0.459467i	0.973254	−	0.229734i	$-0.0737855\pi$
$798$	0	0
$799$	−41388.5	−1.83257
$800$	0	0
$801$	7597.45	0.335135
$802$	0	0
$803$	10847.5i	0.476710i
$804$	0	0
$805$	− 10070.6i	− 0.440924i
$806$	0	0
$807$	−7092.94	−0.309397
$808$	0	0
$809$	−28418.7	−1.23504	−0.617520	−	0.786555i	$-0.711863\pi$
$809$	−28418.7	−1.23504	−0.617520	+	0.786555i	$0.711863\pi$
$810$	0	0
$811$	15766.2i	0.682645i	0.939946	+	0.341323i	$0.110875\pi$
$811$	15766.2i	0.682645i	−0.939946	+	0.341323i	$0.889125\pi$
$812$	0	0
$813$	21401.2i	0.923212i
$814$	0	0
$815$	−34373.7	−1.47737
$816$	0	0
$817$	25470.8	1.09071
$818$	0	0
$819$	242.855i	0.0103615i
$820$	0	0
$821$	32902.1i	1.39865i	0.714804	+	0.699325i	$0.246516\pi$
$821$	32902.1i	1.39865i	−0.714804	+	0.699325i	$0.753484\pi$
$822$	0	0
$823$	−17044.5	−0.721912	−0.360956	−	0.932583i	$-0.617550\pi$
$823$	−17044.5	−0.721912	−0.360956	+	0.932583i	$0.617550\pi$
$824$	0	0
$825$	232.788	0.00982379
$826$	0	0
$827$	18923.1i	0.795673i	0.917456	+	0.397837i	$0.130239\pi$
$827$	18923.1i	0.795673i	−0.917456	+	0.397837i	$0.869761\pi$
$828$	0	0
$829$	− 42319.4i	− 1.77300i	−0.462733	−	0.886498i	$-0.653131\pi$
$829$	− 42319.4i	− 1.77300i	0.462733	−	0.886498i	$-0.346869\pi$
$830$	0	0
$831$	6077.32	0.253694
$832$	0	0
$833$	−17215.8	−0.716075
$834$	0	0
$835$	− 37133.0i	− 1.53897i
$836$	0	0
$837$	− 6653.98i	− 0.274785i
$838$	0	0
$839$	18135.8	0.746267	0.373134	−	0.927778i	$-0.378283\pi$
$839$	18135.8	0.746267	0.373134	+	0.927778i	$0.378283\pi$
$840$	0	0
$841$	7645.90	0.313498
$842$	0	0
$843$	− 6206.52i	− 0.253575i
$844$	0	0
$845$	25201.8i	1.02600i
$846$	0	0
$847$	−16330.0	−0.662463
$848$	0	0
$849$	10224.6	0.413318
$850$	0	0
$851$	16823.5i	0.677676i
$852$	0	0
$853$	13903.8i	0.558097i	0.960277	+	0.279048i	$0.0900190\pi$
$853$	13903.8i	0.558097i	−0.960277	+	0.279048i	$0.909981\pi$
$854$	0	0
$855$	−8996.52	−0.359853
$856$	0	0
$857$	2097.53	0.0836059	0.0418030	−	0.999126i	$-0.486690\pi$
$857$	2097.53	0.0836059	0.0418030	+	0.999126i	$0.486690\pi$
$858$	0	0
$859$	19650.3i	0.780513i	0.920706	+	0.390257i	$0.127614\pi$
$859$	19650.3i	0.780513i	−0.920706	+	0.390257i	$0.872386\pi$
$860$	0	0
$861$	− 13154.0i	− 0.520658i
$862$	0	0
$863$	37574.9	1.48211	0.741057	−	0.671442i	$-0.234325\pi$
$863$	37574.9	1.48211	0.741057	+	0.671442i	$0.234325\pi$
$864$	0	0
$865$	−32360.4	−1.27201
$866$	0	0
$867$	− 19576.9i	− 0.766860i
$868$	0	0
$869$	8461.74i	0.330316i
$870$	0	0
$871$	−1439.10	−0.0559838
$872$	0	0
$873$	11338.5	0.439578
$874$	0	0
$875$	− 18285.7i	− 0.706480i
$876$	0	0
$877$	− 21977.0i	− 0.846194i	−0.906084	−	0.423097i	$-0.860943\pi$
$877$	− 21977.0i	− 0.846194i	0.906084	−	0.423097i	$-0.139057\pi$
$878$	0	0
$879$	12602.9	0.483600
$880$	0	0
$881$	45627.0	1.74485	0.872425	−	0.488747i	$-0.162546\pi$
$881$	45627.0	1.74485	0.872425	+	0.488747i	$0.162546\pi$
$882$	0	0
$883$	51068.7i	1.94632i	0.230138	+	0.973158i	$0.426082\pi$
$883$	51068.7i	1.94632i	−0.230138	+	0.973158i	$0.573918\pi$
$884$	0	0
$885$	− 3578.81i	− 0.135933i
$886$	0	0
$887$	31208.4	1.18137	0.590685	−	0.806902i	$-0.298858\pi$
$887$	31208.4	1.18137	0.590685	+	0.806902i	$0.298858\pi$
$888$	0	0
$889$	−499.965	−0.0188620
$890$	0	0
$891$	− 889.693i	− 0.0334521i
$892$	0	0
$893$	33661.4i	1.26140i
$894$	0	0
$895$	−7250.29	−0.270783
$896$	0	0
$897$	389.710	0.0145062
$898$	0	0
$899$	− 31888.6i	− 1.18303i
$900$	0	0
$901$	57380.4i	2.12166i
$902$	0	0
$903$	−11852.2	−0.436786
$904$	0	0
$905$	−20090.5	−0.737934
$906$	0	0
$907$	− 46768.6i	− 1.71216i	−0.516847	−	0.856078i	$-0.672895\pi$
$907$	− 46768.6i	− 1.71216i	0.516847	−	0.856078i	$-0.327105\pi$
$908$	0	0
$909$	− 453.701i	− 0.0165548i
$910$	0	0
$911$	21752.7	0.791107	0.395553	−	0.918443i	$-0.370553\pi$
$911$	21752.7	0.791107	0.395553	+	0.918443i	$0.370553\pi$
$912$	0	0
$913$	−14108.8	−0.511426
$914$	0	0
$915$	− 21678.0i	− 0.783229i
$916$	0	0
$917$	− 2747.13i	− 0.0989294i
$918$	0	0
$919$	3529.71	0.126697	0.0633483	−	0.997991i	$-0.479822\pi$
$919$	3529.71	0.126697	0.0633483	+	0.997991i	$0.479822\pi$
$920$	0	0
$921$	14628.6	0.523375
$922$	0	0
$923$	401.903i	0.0143324i
$924$	0	0
$925$	− 1829.83i	− 0.0650426i
$926$	0	0
$927$	−9498.56	−0.336541
$928$	0	0
$929$	29473.1	1.04088	0.520442	−	0.853897i	$-0.325767\pi$
$929$	29473.1	1.04088	0.520442	+	0.853897i	$0.325767\pi$
$930$	0	0
$931$	14001.6i	0.492893i
$932$	0	0
$933$	20645.1i	0.724428i
$934$	0	0
$935$	13500.1	0.472192
$936$	0	0
$937$	43332.6	1.51079	0.755397	−	0.655268i	$-0.227444\pi$
$937$	43332.6	1.51079	0.755397	+	0.655268i	$0.227444\pi$
$938$	0	0
$939$	− 22258.4i	− 0.773561i
$940$	0	0
$941$	− 48873.8i	− 1.69314i	−0.532281	−	0.846568i	$-0.678665\pi$
$941$	− 48873.8i	− 1.69314i	0.532281	−	0.846568i	$-0.321335\pi$
$942$	0	0
$943$	−21108.2	−0.728927
$944$	0	0
$945$	4186.31	0.144106
$946$	0	0
$947$	2596.71i	0.0891043i	0.999007	+	0.0445521i	$0.0141861\pi$
$947$	2596.71i	0.0891043i	−0.999007	+	0.0445521i	$0.985814\pi$
$948$	0	0
$949$	− 1975.16i	− 0.0675621i
$950$	0	0
$951$	−8025.17	−0.273642
$952$	0	0
$953$	6101.34	0.207389	0.103695	−	0.994609i	$-0.466934\pi$
$953$	6101.34	0.207389	0.103695	+	0.994609i	$0.466934\pi$
$954$	0	0
$955$	− 16669.2i	− 0.564821i
$956$	0	0
$957$	− 4263.78i	− 0.144021i
$958$	0	0
$959$	3337.95	0.112396
$960$	0	0
$961$	30943.4	1.03868
$962$	0	0
$963$	− 10776.5i	− 0.360610i
$964$	0	0
$965$	− 49449.0i	− 1.64956i
$966$	0	0
$967$	−7375.86	−0.245286	−0.122643	−	0.992451i	$-0.539137\pi$
$967$	−7375.86	−0.245286	−0.122643	+	0.992451i	$0.539137\pi$
$968$	0	0
$969$	−27909.2	−0.925255
$970$	0	0
$971$	34152.0i	1.12872i	0.825528	+	0.564361i	$0.190877\pi$
$971$	34152.0i	1.12872i	−0.825528	+	0.564361i	$0.809123\pi$
$972$	0	0
$973$	22014.5i	0.725336i
$974$	0	0
$975$	−42.3872	−0.00139228
$976$	0	0
$977$	−1170.76	−0.0383377	−0.0191689	−	0.999816i	$-0.506102\pi$
$977$	−1170.76	−0.0383377	−0.0191689	+	0.999816i	$0.506102\pi$
$978$	0	0
$979$	9272.16i	0.302696i
$980$	0	0
$981$	− 14249.9i	− 0.463776i
$982$	0	0
$983$	−31786.9	−1.03138	−0.515689	−	0.856776i	$-0.672464\pi$
$983$	−31786.9	−1.03138	−0.515689	+	0.856776i	$0.672464\pi$
$984$	0	0
$985$	49831.7	1.61195
$986$	0	0
$987$	− 15663.5i	− 0.505141i
$988$	0	0
$989$	19019.3i	0.611505i
$990$	0	0
$991$	−47078.3	−1.50907	−0.754537	−	0.656258i	$-0.772138\pi$
$991$	−47078.3	−1.50907	−0.754537	+	0.656258i	$0.772138\pi$
$992$	0	0
$993$	29636.2	0.947107
$994$	0	0
$995$	48080.9i	1.53193i
$996$	0	0
$997$	− 40091.9i	− 1.27354i	−0.771053	−	0.636771i	$-0.780270\pi$
$997$	− 40091.9i	− 1.27354i	0.771053	−	0.636771i	$-0.219730\pi$
$998$	0	0
$999$	−6993.44	−0.221484

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.4.d.t.385.2		4
4.3	odd	2		768.4.d.q.385.4		4
8.3	odd	2		768.4.d.q.385.1		4
8.5	even	2	inner	768.4.d.t.385.3		4
16.3	odd	4		384.4.a.j.1.2	✓	2
16.5	even	4		384.4.a.k.1.1	yes	2
16.11	odd	4		384.4.a.o.1.1	yes	2
16.13	even	4		384.4.a.n.1.2	yes	2
48.5	odd	4		1152.4.a.o.1.2		2
48.11	even	4		1152.4.a.p.1.2		2
48.29	odd	4		1152.4.a.u.1.1		2
48.35	even	4		1152.4.a.v.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.4.a.j.1.2	✓	2	16.3	odd	4
384.4.a.k.1.1	yes	2	16.5	even	4
384.4.a.n.1.2	yes	2	16.13	even	4
384.4.a.o.1.1	yes	2	16.11	odd	4
768.4.d.q.385.1		4	8.3	odd	2
768.4.d.q.385.4		4	4.3	odd	2
768.4.d.t.385.2		4	1.1	even	1	trivial
768.4.d.t.385.3		4	8.5	even	2	inner
1152.4.a.o.1.2		2	48.5	odd	4
1152.4.a.p.1.2		2	48.11	even	4
1152.4.a.u.1.1		2	48.29	odd	4
1152.4.a.v.1.1		2	48.35	even	4

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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	768.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} +11.4919i q^{5} -13.4919 q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q-3.00000i q^{3} +11.4919i q^{5} -13.4919 q^{7} -9.00000 q^{9} -10.9839i q^{11} +2.00000i q^{13} +34.4758 q^{15} +106.952 q^{17} -86.9839i q^{19} +40.4758i q^{21} +64.9516 q^{23} -7.06453 q^{25} +27.0000i q^{27} +129.395i q^{29} -246.444 q^{31} -32.9516 q^{33} -155.048i q^{35} +259.016i q^{37} +6.00000 q^{39} -324.984 q^{41} +292.823i q^{43} -103.427i q^{45} -386.984 q^{47} -160.968 q^{49} -320.855i q^{51} +536.508i q^{53} +126.226 q^{55} -260.952 q^{57} -103.806i q^{59} -628.790i q^{61} +121.427 q^{63} -22.9839 q^{65} +719.548i q^{67} -194.855i q^{69} +200.952 q^{71} -987.581 q^{73} +21.1936i q^{75} +148.194i q^{77} -770.379 q^{79} +81.0000 q^{81} -1284.50i q^{83} +1229.08i q^{85} +388.185 q^{87} -844.161 q^{89} -26.9839i q^{91} +739.331i q^{93} +999.613 q^{95} -1259.84 q^{97} +98.8548i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{7} - 36 q^{9}+O(q^{10}) \) 4 * q + 8 * q^7 - 36 * q^9	\( 4 q + 8 q^{7} - 36 q^{9} - 48 q^{15} + 56 q^{17} - 112 q^{23} - 524 q^{25} - 552 q^{31} + 240 q^{33} + 24 q^{39} - 1176 q^{41} - 1424 q^{47} - 396 q^{49} + 2240 q^{55} - 672 q^{57} - 72 q^{63} + 32 q^{65} + 432 q^{71} - 728 q^{73} - 2152 q^{79} + 324 q^{81} - 864 q^{87} - 4616 q^{89} + 1024 q^{95} - 3800 q^{97}+O(q^{100}) \) 4 * q + 8 * q^7 - 36 * q^9 - 48 * q^15 + 56 * q^17 - 112 * q^23 - 524 * q^25 - 552 * q^31 + 240 * q^33 + 24 * q^39 - 1176 * q^41 - 1424 * q^47 - 396 * q^49 + 2240 * q^55 - 672 * q^57 - 72 * q^63 + 32 * q^65 + 432 * q^71 - 728 * q^73 - 2152 * q^79 + 324 * q^81 - 864 * q^87 - 4616 * q^89 + 1024 * q^95 - 3800 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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