LMFDB - Embedded newform 464.4.a.l.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [464,4,Mod(1,464)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(464, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("464.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 464 = 2^{4} \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	464.a (trivial)

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:	yes
Analytic conductor:	$27.3768862427$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.13458092.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 14x^{3} + 18x^{2} + 20x - 8 $ x^5 - x^4 - 14x^3 + 18x^2 + 20*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 29)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.957567$ of defining polynomial
Character	$\chi$	$=$	464.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.64574 q^{3} +12.8729 q^{5} -26.0540 q^{7} -5.41713 q^{9} +O(q^{10})$	$q-4.64574 q^{3} +12.8729 q^{5} -26.0540 q^{7} -5.41713 q^{9} +62.8274 q^{11} +22.3936 q^{13} -59.8039 q^{15} -57.9808 q^{17} -71.3143 q^{19} +121.040 q^{21} +49.5307 q^{23} +40.7104 q^{25} +150.601 q^{27} -29.0000 q^{29} -62.9198 q^{31} -291.880 q^{33} -335.389 q^{35} +119.123 q^{37} -104.035 q^{39} -414.916 q^{41} +348.009 q^{43} -69.7340 q^{45} -553.259 q^{47} +335.808 q^{49} +269.364 q^{51} -107.308 q^{53} +808.768 q^{55} +331.308 q^{57} -136.881 q^{59} -579.408 q^{61} +141.138 q^{63} +288.269 q^{65} -919.959 q^{67} -230.106 q^{69} -781.802 q^{71} -133.237 q^{73} -189.130 q^{75} -1636.90 q^{77} -868.196 q^{79} -553.392 q^{81} +83.3560 q^{83} -746.379 q^{85} +134.726 q^{87} -357.919 q^{89} -583.442 q^{91} +292.309 q^{93} -918.019 q^{95} -187.105 q^{97} -340.344 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 8 q^{3} + 10 q^{5} - 40 q^{7} + 33 q^{9}+O(q^{10}) $ 5 * q - 8 * q^3 + 10 * q^5 - 40 * q^7 + 33 * q^9	$ 5 q - 8 q^{3} + 10 q^{5} - 40 q^{7} + 33 q^{9} - 12 q^{11} + 14 q^{13} + 74 q^{15} + 66 q^{17} - 214 q^{19} - 164 q^{23} + 207 q^{25} - 362 q^{27} - 145 q^{29} - 420 q^{31} - 576 q^{33} + 52 q^{35} + 378 q^{37} + 374 q^{39} - 1158 q^{41} + 204 q^{43} - 1506 q^{45} - 248 q^{47} - 283 q^{49} - 228 q^{51} - 554 q^{53} - 546 q^{55} + 44 q^{57} - 440 q^{59} + 618 q^{61} - 804 q^{63} - 1656 q^{65} - 1164 q^{67} - 1968 q^{69} + 692 q^{71} - 1950 q^{73} - 3074 q^{75} - 1616 q^{77} - 272 q^{79} + 1801 q^{81} - 512 q^{83} - 1628 q^{85} + 232 q^{87} + 866 q^{89} - 2580 q^{91} - 40 q^{93} - 2244 q^{95} + 1562 q^{97} + 238 q^{99}+O(q^{100}) $ 5 * q - 8 * q^3 + 10 * q^5 - 40 * q^7 + 33 * q^9 - 12 * q^11 + 14 * q^13 + 74 * q^15 + 66 * q^17 - 214 * q^19 - 164 * q^23 + 207 * q^25 - 362 * q^27 - 145 * q^29 - 420 * q^31 - 576 * q^33 + 52 * q^35 + 378 * q^37 + 374 * q^39 - 1158 * q^41 + 204 * q^43 - 1506 * q^45 - 248 * q^47 - 283 * q^49 - 228 * q^51 - 554 * q^53 - 546 * q^55 + 44 * q^57 - 440 * q^59 + 618 * q^61 - 804 * q^63 - 1656 * q^65 - 1164 * q^67 - 1968 * q^69 + 692 * q^71 - 1950 * q^73 - 3074 * q^75 - 1616 * q^77 - 272 * q^79 + 1801 * q^81 - 512 * q^83 - 1628 * q^85 + 232 * q^87 + 866 * q^89 - 2580 * q^91 - 40 * q^93 - 2244 * q^95 + 1562 * q^97 + 238 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−4.64574	−0.894072	−0.447036	−	0.894516i	$-0.647520\pi$
$3$	−4.64574	−0.894072	−0.447036	+	0.894516i	$0.647520\pi$
$4$	0	0
$5$	12.8729	1.15138	0.575692	−	0.817667i	$-0.304733\pi$
$5$	12.8729	1.15138	0.575692	+	0.817667i	$0.304733\pi$
$6$	0	0
$7$	−26.0540	−1.40678	−0.703391	−	0.710804i	$-0.748331\pi$
$7$	−26.0540	−1.40678	−0.703391	+	0.710804i	$0.748331\pi$
$8$	0	0
$9$	−5.41713	−0.200635
$10$	0	0
$11$	62.8274	1.72211	0.861053	−	0.508515i	$-0.169805\pi$
$11$	62.8274	1.72211	0.861053	+	0.508515i	$0.169805\pi$
$12$	0	0
$13$	22.3936	0.477759	0.238879	−	0.971049i	$-0.423220\pi$
$13$	22.3936	0.477759	0.238879	+	0.971049i	$0.423220\pi$
$14$	0	0
$15$	−59.8039	−1.02942
$16$	0	0
$17$	−57.9808	−0.827201	−0.413601	−	0.910458i	$-0.635729\pi$
$17$	−57.9808	−0.827201	−0.413601	+	0.910458i	$0.635729\pi$
$18$	0	0
$19$	−71.3143	−0.861086	−0.430543	−	0.902570i	$-0.641678\pi$
$19$	−71.3143	−0.861086	−0.430543	+	0.902570i	$0.641678\pi$
$20$	0	0
$21$	121.040	1.25776
$22$	0	0
$23$	49.5307	0.449037	0.224519	−	0.974470i	$-0.427919\pi$
$23$	49.5307	0.449037	0.224519	+	0.974470i	$0.427919\pi$
$24$	0	0
$25$	40.7104	0.325683
$26$	0	0
$27$	150.601	1.07345
$28$	0	0
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	0	0
$31$	−62.9198	−0.364540	−0.182270	−	0.983249i	$-0.558344\pi$
$31$	−62.9198	−0.364540	−0.182270	+	0.983249i	$0.558344\pi$
$32$	0	0
$33$	−291.880	−1.53969
$34$	0	0
$35$	−335.389	−1.61974
$36$	0	0
$37$	119.123	0.529288	0.264644	−	0.964346i	$-0.414746\pi$
$37$	119.123	0.529288	0.264644	+	0.964346i	$0.414746\pi$
$38$	0	0
$39$	−104.035	−0.427151
$40$	0	0
$41$	−414.916	−1.58046	−0.790231	−	0.612810i	$-0.790039\pi$
$41$	−414.916	−1.58046	−0.790231	+	0.612810i	$0.790039\pi$
$42$	0	0
$43$	348.009	1.23421	0.617103	−	0.786882i	$-0.288306\pi$
$43$	348.009	1.23421	0.617103	+	0.786882i	$0.288306\pi$
$44$	0	0
$45$	−69.7340	−0.231007
$46$	0	0
$47$	−553.259	−1.71705	−0.858523	−	0.512775i	$-0.828618\pi$
$47$	−553.259	−1.71705	−0.858523	+	0.512775i	$0.828618\pi$
$48$	0	0
$49$	335.808	0.979033
$50$	0	0
$51$	269.364	0.739578
$52$	0	0
$53$	−107.308	−0.278111	−0.139055	−	0.990285i	$-0.544407\pi$
$53$	−107.308	−0.278111	−0.139055	+	0.990285i	$0.544407\pi$
$54$	0	0
$55$	808.768	1.98280
$56$	0	0
$57$	331.308	0.769873
$58$	0	0
$59$	−136.881	−0.302041	−0.151020	−	0.988531i	$-0.548256\pi$
$59$	−136.881	−0.302041	−0.151020	+	0.988531i	$0.548256\pi$
$60$	0	0
$61$	−579.408	−1.21616	−0.608078	−	0.793877i	$-0.708059\pi$
$61$	−579.408	−1.21616	−0.608078	+	0.793877i	$0.708059\pi$
$62$	0	0
$63$	141.138	0.282249
$64$	0	0
$65$	288.269	0.550084
$66$	0	0
$67$	−919.959	−1.67748	−0.838738	−	0.544535i	$-0.816706\pi$
$67$	−919.959	−1.67748	−0.838738	+	0.544535i	$0.816706\pi$
$68$	0	0
$69$	−230.106	−0.401472
$70$	0	0
$71$	−781.802	−1.30680	−0.653400	−	0.757013i	$-0.726658\pi$
$71$	−781.802	−1.30680	−0.653400	+	0.757013i	$0.726658\pi$
$72$	0	0
$73$	−133.237	−0.213619	−0.106810	−	0.994279i	$-0.534064\pi$
$73$	−133.237	−0.213619	−0.106810	+	0.994279i	$0.534064\pi$
$74$	0	0
$75$	−189.130	−0.291185
$76$	0	0
$77$	−1636.90	−2.42263
$78$	0	0
$79$	−868.196	−1.23645	−0.618225	−	0.786001i	$-0.712148\pi$
$79$	−868.196	−1.23645	−0.618225	+	0.786001i	$0.712148\pi$
$80$	0	0
$81$	−553.392	−0.759111
$82$	0	0
$83$	83.3560	0.110235	0.0551175	−	0.998480i	$-0.482447\pi$
$83$	83.3560	0.110235	0.0551175	+	0.998480i	$0.482447\pi$
$84$	0	0
$85$	−746.379	−0.952425
$86$	0	0
$87$	134.726	0.166025
$88$	0	0
$89$	−357.919	−0.426284	−0.213142	−	0.977021i	$-0.568370\pi$
$89$	−357.919	−0.426284	−0.213142	+	0.977021i	$0.568370\pi$
$90$	0	0
$91$	−583.442	−0.672102
$92$	0	0
$93$	292.309	0.325925
$94$	0	0
$95$	−918.019	−0.991440
$96$	0	0
$97$	−187.105	−0.195852	−0.0979260	−	0.995194i	$-0.531221\pi$
$97$	−187.105	−0.195852	−0.0979260	+	0.995194i	$0.531221\pi$
$98$	0	0
$99$	−340.344	−0.345514
$100$	0	0
$101$	−959.423	−0.945209	−0.472605	−	0.881275i	$-0.656686\pi$
$101$	−959.423	−0.945209	−0.472605	+	0.881275i	$0.656686\pi$
$102$	0	0
$103$	−78.8738	−0.0754531	−0.0377265	−	0.999288i	$-0.512012\pi$
$103$	−78.8738	−0.0754531	−0.0377265	+	0.999288i	$0.512012\pi$
$104$	0	0
$105$	1558.13	1.44817
$106$	0	0
$107$	−713.851	−0.644959	−0.322479	−	0.946577i	$-0.604516\pi$
$107$	−713.851	−0.644959	−0.322479	+	0.946577i	$0.604516\pi$
$108$	0	0
$109$	536.561	0.471497	0.235749	−	0.971814i	$-0.424246\pi$
$109$	536.561	0.471497	0.235749	+	0.971814i	$0.424246\pi$
$110$	0	0
$111$	−553.412	−0.473222
$112$	0	0
$113$	1946.36	1.62034	0.810170	−	0.586195i	$-0.199375\pi$
$113$	1946.36	1.62034	0.810170	+	0.586195i	$0.199375\pi$
$114$	0	0
$115$	637.601	0.517014
$116$	0	0
$117$	−121.309	−0.0958549
$118$	0	0
$119$	1510.63	1.16369
$120$	0	0
$121$	2616.28	1.96565
$122$	0	0
$123$	1927.59	1.41305
$124$	0	0
$125$	−1085.05	−0.776397
$126$	0	0
$127$	1995.14	1.39402	0.697009	−	0.717062i	$-0.254514\pi$
$127$	1995.14	1.39402	0.697009	+	0.717062i	$0.254514\pi$
$128$	0	0
$129$	−1616.76	−1.10347
$130$	0	0
$131$	−1544.84	−1.03033	−0.515164	−	0.857092i	$-0.672269\pi$
$131$	−1544.84	−1.03033	−0.515164	+	0.857092i	$0.672269\pi$
$132$	0	0
$133$	1858.02	1.21136
$134$	0	0
$135$	1938.67	1.23596
$136$	0	0
$137$	−1294.93	−0.807543	−0.403771	−	0.914860i	$-0.632301\pi$
$137$	−1294.93	−0.807543	−0.403771	+	0.914860i	$0.632301\pi$
$138$	0	0
$139$	−1999.66	−1.22021	−0.610105	−	0.792320i	$-0.708873\pi$
$139$	−1999.66	−1.22021	−0.610105	+	0.792320i	$0.708873\pi$
$140$	0	0
$141$	2570.30	1.53516
$142$	0	0
$143$	1406.93	0.822752
$144$	0	0
$145$	−373.313	−0.213807
$146$	0	0
$147$	−1560.08	−0.875326
$148$	0	0
$149$	1187.63	0.652984	0.326492	−	0.945200i	$-0.394133\pi$
$149$	1187.63	0.652984	0.326492	+	0.945200i	$0.394133\pi$
$150$	0	0
$151$	2257.61	1.21670	0.608350	−	0.793669i	$-0.291832\pi$
$151$	2257.61	1.21670	0.608350	+	0.793669i	$0.291832\pi$
$152$	0	0
$153$	314.090	0.165965
$154$	0	0
$155$	−809.958	−0.419725
$156$	0	0
$157$	1188.18	0.603995	0.301997	−	0.953309i	$-0.402347\pi$
$157$	1188.18	0.603995	0.301997	+	0.953309i	$0.402347\pi$
$158$	0	0
$159$	498.524	0.248651
$160$	0	0
$161$	−1290.47	−0.631697
$162$	0	0
$163$	2452.33	1.17841	0.589207	−	0.807982i	$-0.299440\pi$
$163$	2452.33	1.17841	0.589207	+	0.807982i	$0.299440\pi$
$164$	0	0
$165$	−3757.32	−1.77277
$166$	0	0
$167$	2020.14	0.936067	0.468034	−	0.883711i	$-0.344963\pi$
$167$	2020.14	0.936067	0.468034	+	0.883711i	$0.344963\pi$
$168$	0	0
$169$	−1695.53	−0.771746
$170$	0	0
$171$	386.319	0.172764
$172$	0	0
$173$	2862.12	1.25782	0.628910	−	0.777478i	$-0.283501\pi$
$173$	2862.12	1.25782	0.628910	+	0.777478i	$0.283501\pi$
$174$	0	0
$175$	−1060.67	−0.458165
$176$	0	0
$177$	635.914	0.270046
$178$	0	0
$179$	232.651	0.0971460	0.0485730	−	0.998820i	$-0.484533\pi$
$179$	232.651	0.0971460	0.0485730	+	0.998820i	$0.484533\pi$
$180$	0	0
$181$	−2607.67	−1.07086	−0.535432	−	0.844578i	$-0.679851\pi$
$181$	−2607.67	−1.07086	−0.535432	+	0.844578i	$0.679851\pi$
$182$	0	0
$183$	2691.78	1.08733
$184$	0	0
$185$	1533.45	0.609413
$186$	0	0
$187$	−3642.78	−1.42453
$188$	0	0
$189$	−3923.76	−1.51012
$190$	0	0
$191$	−1528.90	−0.579202	−0.289601	−	0.957147i	$-0.593523\pi$
$191$	−1528.90	−0.579202	−0.289601	+	0.957147i	$0.593523\pi$
$192$	0	0
$193$	1017.58	0.379518	0.189759	−	0.981831i	$-0.439229\pi$
$193$	1017.58	0.379518	0.189759	+	0.981831i	$0.439229\pi$
$194$	0	0
$195$	−1339.22	−0.491815
$196$	0	0
$197$	3290.20	1.18994	0.594968	−	0.803749i	$-0.297165\pi$
$197$	3290.20	1.18994	0.594968	+	0.803749i	$0.297165\pi$
$198$	0	0
$199$	29.9190	0.0106578	0.00532891	−	0.999986i	$-0.498304\pi$
$199$	29.9190	0.0106578	0.00532891	+	0.999986i	$0.498304\pi$
$200$	0	0
$201$	4273.89	1.49979
$202$	0	0
$203$	755.565	0.261233
$204$	0	0
$205$	−5341.15	−1.81972
$206$	0	0
$207$	−268.314	−0.0900924
$208$	0	0
$209$	−4480.49	−1.48288
$210$	0	0
$211$	−2267.20	−0.739717	−0.369859	−	0.929088i	$-0.620594\pi$
$211$	−2267.20	−0.739717	−0.369859	+	0.929088i	$0.620594\pi$
$212$	0	0
$213$	3632.05	1.16837
$214$	0	0
$215$	4479.87	1.42105
$216$	0	0
$217$	1639.31	0.512828
$218$	0	0
$219$	618.984	0.190991
$220$	0	0
$221$	−1298.40	−0.395203
$222$	0	0
$223$	−4945.03	−1.48495	−0.742474	−	0.669875i	$-0.766348\pi$
$223$	−4945.03	−1.48495	−0.742474	+	0.669875i	$0.766348\pi$
$224$	0	0
$225$	−220.534	−0.0653433
$226$	0	0
$227$	3559.46	1.04075	0.520374	−	0.853938i	$-0.325792\pi$
$227$	3559.46	1.04075	0.520374	+	0.853938i	$0.325792\pi$
$228$	0	0
$229$	6143.40	1.77278	0.886391	−	0.462937i	$-0.153204\pi$
$229$	6143.40	1.77278	0.886391	+	0.462937i	$0.153204\pi$
$230$	0	0
$231$	7604.61	2.16600
$232$	0	0
$233$	−1087.06	−0.305648	−0.152824	−	0.988253i	$-0.548837\pi$
$233$	−1087.06	−0.305648	−0.152824	+	0.988253i	$0.548837\pi$
$234$	0	0
$235$	−7122.03	−1.97698
$236$	0	0
$237$	4033.41	1.10548
$238$	0	0
$239$	−1079.00	−0.292027	−0.146013	−	0.989283i	$-0.546644\pi$
$239$	−1079.00	−0.292027	−0.146013	+	0.989283i	$0.546644\pi$
$240$	0	0
$241$	−989.224	−0.264405	−0.132202	−	0.991223i	$-0.542205\pi$
$241$	−989.224	−0.264405	−0.132202	+	0.991223i	$0.542205\pi$
$242$	0	0
$243$	−1495.33	−0.394754
$244$	0	0
$245$	4322.81	1.12724
$246$	0	0
$247$	−1596.98	−0.411391
$248$	0	0
$249$	−387.250	−0.0985581
$250$	0	0
$251$	−900.246	−0.226386	−0.113193	−	0.993573i	$-0.536108\pi$
$251$	−900.246	−0.226386	−0.113193	+	0.993573i	$0.536108\pi$
$252$	0	0
$253$	3111.88	0.773290
$254$	0	0
$255$	3467.48	0.851537
$256$	0	0
$257$	−3125.18	−0.758534	−0.379267	−	0.925287i	$-0.623824\pi$
$257$	−3125.18	−0.758534	−0.379267	+	0.925287i	$0.623824\pi$
$258$	0	0
$259$	−3103.62	−0.744592
$260$	0	0
$261$	157.097	0.0372569
$262$	0	0
$263$	−2814.89	−0.659976	−0.329988	−	0.943985i	$-0.607045\pi$
$263$	−2814.89	−0.659976	−0.329988	+	0.943985i	$0.607045\pi$
$264$	0	0
$265$	−1381.36	−0.320212
$266$	0	0
$267$	1662.80	0.381129
$268$	0	0
$269$	4409.28	0.999400	0.499700	−	0.866199i	$-0.333444\pi$
$269$	4409.28	0.999400	0.499700	+	0.866199i	$0.333444\pi$
$270$	0	0
$271$	4417.97	0.990304	0.495152	−	0.868806i	$-0.335112\pi$
$271$	4417.97	0.990304	0.495152	+	0.868806i	$0.335112\pi$
$272$	0	0
$273$	2710.52	0.600908
$274$	0	0
$275$	2557.73	0.560862
$276$	0	0
$277$	−887.577	−0.192525	−0.0962624	−	0.995356i	$-0.530689\pi$
$277$	−887.577	−0.192525	−0.0962624	+	0.995356i	$0.530689\pi$
$278$	0	0
$279$	340.845	0.0731393
$280$	0	0
$281$	−4286.58	−0.910021	−0.455010	−	0.890486i	$-0.650364\pi$
$281$	−4286.58	−0.910021	−0.455010	+	0.890486i	$0.650364\pi$
$282$	0	0
$283$	1709.59	0.359097	0.179549	−	0.983749i	$-0.442536\pi$
$283$	1709.59	0.359097	0.179549	+	0.983749i	$0.442536\pi$
$284$	0	0
$285$	4264.88	0.886419
$286$	0	0
$287$	10810.2	2.22336
$288$	0	0
$289$	−1551.22	−0.315738
$290$	0	0
$291$	869.241	0.175106
$292$	0	0
$293$	1145.01	0.228301	0.114151	−	0.993463i	$-0.463585\pi$
$293$	1145.01	0.228301	0.114151	+	0.993463i	$0.463585\pi$
$294$	0	0
$295$	−1762.05	−0.347765
$296$	0	0
$297$	9461.90	1.84860
$298$	0	0
$299$	1109.17	0.214532
$300$	0	0
$301$	−9067.01	−1.73626
$302$	0	0
$303$	4457.23	0.845086
$304$	0	0
$305$	−7458.63	−1.40026
$306$	0	0
$307$	−1079.39	−0.200665	−0.100333	−	0.994954i	$-0.531991\pi$
$307$	−1079.39	−0.200665	−0.100333	+	0.994954i	$0.531991\pi$
$308$	0	0
$309$	366.427	0.0674605
$310$	0	0
$311$	−2313.90	−0.421895	−0.210947	−	0.977497i	$-0.567655\pi$
$311$	−2313.90	−0.421895	−0.210947	+	0.977497i	$0.567655\pi$
$312$	0	0
$313$	−7653.19	−1.38206	−0.691029	−	0.722827i	$-0.742842\pi$
$313$	−7653.19	−1.38206	−0.691029	+	0.722827i	$0.742842\pi$
$314$	0	0
$315$	1816.85	0.324977
$316$	0	0
$317$	−3657.23	−0.647982	−0.323991	−	0.946060i	$-0.605025\pi$
$317$	−3657.23	−0.647982	−0.323991	+	0.946060i	$0.605025\pi$
$318$	0	0
$319$	−1821.99	−0.319787
$320$	0	0
$321$	3316.36	0.576640
$322$	0	0
$323$	4134.86	0.712291
$324$	0	0
$325$	911.653	0.155598
$326$	0	0
$327$	−2492.72	−0.421553
$328$	0	0
$329$	14414.6	2.41551
$330$	0	0
$331$	−3237.92	−0.537681	−0.268841	−	0.963185i	$-0.586640\pi$
$331$	−3237.92	−0.537681	−0.268841	+	0.963185i	$0.586640\pi$
$332$	0	0
$333$	−645.303	−0.106193
$334$	0	0
$335$	−11842.5	−1.93142
$336$	0	0
$337$	−7976.89	−1.28940	−0.644702	−	0.764434i	$-0.723018\pi$
$337$	−7976.89	−1.28940	−0.644702	+	0.764434i	$0.723018\pi$
$338$	0	0
$339$	−9042.29	−1.44870
$340$	0	0
$341$	−3953.09	−0.627777
$342$	0	0
$343$	187.371	0.0294959
$344$	0	0
$345$	−2962.13	−0.462248
$346$	0	0
$347$	8355.93	1.29271	0.646354	−	0.763037i	$-0.276293\pi$
$347$	8355.93	1.29271	0.646354	+	0.763037i	$0.276293\pi$
$348$	0	0
$349$	5544.56	0.850411	0.425205	−	0.905097i	$-0.360202\pi$
$349$	5544.56	0.850411	0.425205	+	0.905097i	$0.360202\pi$
$350$	0	0
$351$	3372.51	0.512852
$352$	0	0
$353$	−1682.79	−0.253727	−0.126864	−	0.991920i	$-0.540491\pi$
$353$	−1682.79	−0.253727	−0.126864	+	0.991920i	$0.540491\pi$
$354$	0	0
$355$	−10064.0	−1.50463
$356$	0	0
$357$	−7017.99	−1.04042
$358$	0	0
$359$	−7143.13	−1.05014	−0.525069	−	0.851059i	$-0.675961\pi$
$359$	−7143.13	−1.05014	−0.525069	+	0.851059i	$0.675961\pi$
$360$	0	0
$361$	−1773.27	−0.258531
$362$	0	0
$363$	−12154.6	−1.75743
$364$	0	0
$365$	−1715.14	−0.245958
$366$	0	0
$367$	4456.16	0.633814	0.316907	−	0.948457i	$-0.397356\pi$
$367$	4456.16	0.633814	0.316907	+	0.948457i	$0.397356\pi$
$368$	0	0
$369$	2247.65	0.317095
$370$	0	0
$371$	2795.79	0.391241
$372$	0	0
$373$	2508.28	0.348187	0.174094	−	0.984729i	$-0.444300\pi$
$373$	2508.28	0.348187	0.174094	+	0.984729i	$0.444300\pi$
$374$	0	0
$375$	5040.85	0.694155
$376$	0	0
$377$	−649.414	−0.0887176
$378$	0	0
$379$	12733.9	1.72585	0.862926	−	0.505331i	$-0.168629\pi$
$379$	12733.9	1.72585	0.862926	+	0.505331i	$0.168629\pi$
$380$	0	0
$381$	−9268.91	−1.24635
$382$	0	0
$383$	1027.19	0.137042	0.0685209	−	0.997650i	$-0.478172\pi$
$383$	1027.19	0.137042	0.0685209	+	0.997650i	$0.478172\pi$
$384$	0	0
$385$	−21071.6	−2.78937
$386$	0	0
$387$	−1885.21	−0.247624
$388$	0	0
$389$	5153.35	0.671684	0.335842	−	0.941918i	$-0.390979\pi$
$389$	5153.35	0.671684	0.335842	+	0.941918i	$0.390979\pi$
$390$	0	0
$391$	−2871.83	−0.371444
$392$	0	0
$393$	7176.90	0.921187
$394$	0	0
$395$	−11176.2	−1.42363
$396$	0	0
$397$	6250.95	0.790242	0.395121	−	0.918629i	$-0.370703\pi$
$397$	6250.95	0.790242	0.395121	+	0.918629i	$0.370703\pi$
$398$	0	0
$399$	−8631.87	−1.08304
$400$	0	0
$401$	11083.8	1.38029	0.690145	−	0.723671i	$-0.257547\pi$
$401$	11083.8	1.38029	0.690145	+	0.723671i	$0.257547\pi$
$402$	0	0
$403$	−1409.00	−0.174162
$404$	0	0
$405$	−7123.74	−0.874028
$406$	0	0
$407$	7484.17	0.911490
$408$	0	0
$409$	−3375.43	−0.408079	−0.204039	−	0.978963i	$-0.565407\pi$
$409$	−3375.43	−0.408079	−0.204039	+	0.978963i	$0.565407\pi$
$410$	0	0
$411$	6015.91	0.722002
$412$	0	0
$413$	3566.29	0.424905
$414$	0	0
$415$	1073.03	0.126923
$416$	0	0
$417$	9289.91	1.09096
$418$	0	0
$419$	−6486.50	−0.756292	−0.378146	−	0.925746i	$-0.623438\pi$
$419$	−6486.50	−0.756292	−0.378146	+	0.925746i	$0.623438\pi$
$420$	0	0
$421$	−10938.6	−1.26631	−0.633153	−	0.774026i	$-0.718240\pi$
$421$	−10938.6	−1.26631	−0.633153	+	0.774026i	$0.718240\pi$
$422$	0	0
$423$	2997.08	0.344499
$424$	0	0
$425$	−2360.42	−0.269406
$426$	0	0
$427$	15095.9	1.71087
$428$	0	0
$429$	−6536.23	−0.735600
$430$	0	0
$431$	−4124.34	−0.460934	−0.230467	−	0.973080i	$-0.574025\pi$
$431$	−4124.34	−0.460934	−0.230467	+	0.973080i	$0.574025\pi$
$432$	0	0
$433$	−1561.41	−0.173295	−0.0866473	−	0.996239i	$-0.527615\pi$
$433$	−1561.41	−0.173295	−0.0866473	+	0.996239i	$0.527615\pi$
$434$	0	0
$435$	1734.31	0.191158
$436$	0	0
$437$	−3532.25	−0.386660
$438$	0	0
$439$	15712.7	1.70826	0.854130	−	0.520060i	$-0.174091\pi$
$439$	15712.7	1.70826	0.854130	+	0.520060i	$0.174091\pi$
$440$	0	0
$441$	−1819.12	−0.196428
$442$	0	0
$443$	−12763.5	−1.36888	−0.684439	−	0.729070i	$-0.739953\pi$
$443$	−12763.5	−1.36888	−0.684439	+	0.729070i	$0.739953\pi$
$444$	0	0
$445$	−4607.44	−0.490817
$446$	0	0
$447$	−5517.42	−0.583815
$448$	0	0
$449$	3117.88	0.327710	0.163855	−	0.986484i	$-0.447607\pi$
$449$	3117.88	0.327710	0.163855	+	0.986484i	$0.447607\pi$
$450$	0	0
$451$	−26068.1	−2.72172
$452$	0	0
$453$	−10488.3	−1.08782
$454$	0	0
$455$	−7510.56	−0.773847
$456$	0	0
$457$	−8479.41	−0.867943	−0.433972	−	0.900927i	$-0.642888\pi$
$457$	−8479.41	−0.867943	−0.433972	+	0.900927i	$0.642888\pi$
$458$	0	0
$459$	−8732.00	−0.887962
$460$	0	0
$461$	9253.32	0.934859	0.467430	−	0.884030i	$-0.345180\pi$
$461$	9253.32	0.934859	0.467430	+	0.884030i	$0.345180\pi$
$462$	0	0
$463$	−521.395	−0.0523354	−0.0261677	−	0.999658i	$-0.508330\pi$
$463$	−521.395	−0.0523354	−0.0261677	+	0.999658i	$0.508330\pi$
$464$	0	0
$465$	3762.85	0.375265
$466$	0	0
$467$	−4337.35	−0.429783	−0.214891	−	0.976638i	$-0.568940\pi$
$467$	−4337.35	−0.429783	−0.214891	+	0.976638i	$0.568940\pi$
$468$	0	0
$469$	23968.6	2.35984
$470$	0	0
$471$	−5519.98	−0.540015
$472$	0	0
$473$	21864.5	2.12544
$474$	0	0
$475$	−2903.24	−0.280441
$476$	0	0
$477$	581.300	0.0557986
$478$	0	0
$479$	11258.1	1.07390	0.536949	−	0.843615i	$-0.319577\pi$
$479$	11258.1	1.07390	0.536949	+	0.843615i	$0.319577\pi$
$480$	0	0
$481$	2667.58	0.252872
$482$	0	0
$483$	5995.18	0.564783
$484$	0	0
$485$	−2408.58	−0.225501
$486$	0	0
$487$	−4353.54	−0.405088	−0.202544	−	0.979273i	$-0.564921\pi$
$487$	−4353.54	−0.405088	−0.202544	+	0.979273i	$0.564921\pi$
$488$	0	0
$489$	−11392.9	−1.05359
$490$	0	0
$491$	8458.77	0.777472	0.388736	−	0.921349i	$-0.372912\pi$
$491$	8458.77	0.777472	0.388736	+	0.921349i	$0.372912\pi$
$492$	0	0
$493$	1681.44	0.153607
$494$	0	0
$495$	−4381.20	−0.397819
$496$	0	0
$497$	20369.0	1.83838
$498$	0	0
$499$	−14850.9	−1.33230	−0.666149	−	0.745819i	$-0.732058\pi$
$499$	−14850.9	−1.33230	−0.666149	+	0.745819i	$0.732058\pi$
$500$	0	0
$501$	−9385.04	−0.836912
$502$	0	0
$503$	1686.45	0.149493	0.0747464	−	0.997203i	$-0.476185\pi$
$503$	1686.45	0.149493	0.0747464	+	0.997203i	$0.476185\pi$
$504$	0	0
$505$	−12350.5	−1.08830
$506$	0	0
$507$	7876.97	0.689997
$508$	0	0
$509$	−11113.5	−0.967773	−0.483887	−	0.875131i	$-0.660775\pi$
$509$	−11113.5	−0.967773	−0.483887	+	0.875131i	$0.660775\pi$
$510$	0	0
$511$	3471.35	0.300516
$512$	0	0
$513$	−10740.0	−0.924336
$514$	0	0
$515$	−1015.33	−0.0868754
$516$	0	0
$517$	−34759.8	−2.95694
$518$	0	0
$519$	−13296.7	−1.12458
$520$	0	0
$521$	−15931.1	−1.33964	−0.669820	−	0.742523i	$-0.733629\pi$
$521$	−15931.1	−1.33964	−0.669820	+	0.742523i	$0.733629\pi$
$522$	0	0
$523$	7960.43	0.665555	0.332778	−	0.943005i	$-0.392014\pi$
$523$	7960.43	0.665555	0.332778	+	0.943005i	$0.392014\pi$
$524$	0	0
$525$	4927.58	0.409633
$526$	0	0
$527$	3648.14	0.301548
$528$	0	0
$529$	−9713.71	−0.798366
$530$	0	0
$531$	741.503	0.0605998
$532$	0	0
$533$	−9291.45	−0.755079
$534$	0	0
$535$	−9189.30	−0.742594
$536$	0	0
$537$	−1080.83	−0.0868555
$538$	0	0
$539$	21098.0	1.68600
$540$	0	0
$541$	13818.9	1.09819	0.549096	−	0.835760i	$-0.314972\pi$
$541$	13818.9	1.09819	0.549096	+	0.835760i	$0.314972\pi$
$542$	0	0
$543$	12114.5	0.957430
$544$	0	0
$545$	6907.07	0.542874
$546$	0	0
$547$	−22093.3	−1.72695	−0.863474	−	0.504393i	$-0.831716\pi$
$547$	−22093.3	−1.72695	−0.863474	+	0.504393i	$0.831716\pi$
$548$	0	0
$549$	3138.73	0.244003
$550$	0	0
$551$	2068.12	0.159900
$552$	0	0
$553$	22619.9	1.73942
$554$	0	0
$555$	−7124.00	−0.544859
$556$	0	0
$557$	−3110.34	−0.236606	−0.118303	−	0.992978i	$-0.537745\pi$
$557$	−3110.34	−0.236606	−0.118303	+	0.992978i	$0.537745\pi$
$558$	0	0
$559$	7793.17	0.589653
$560$	0	0
$561$	16923.4	1.27363
$562$	0	0
$563$	13284.6	0.994455	0.497227	−	0.867620i	$-0.334351\pi$
$563$	13284.6	0.994455	0.497227	+	0.867620i	$0.334351\pi$
$564$	0	0
$565$	25055.2	1.86563
$566$	0	0
$567$	14418.1	1.06790
$568$	0	0
$569$	−6809.18	−0.501680	−0.250840	−	0.968029i	$-0.580707\pi$
$569$	−6809.18	−0.501680	−0.250840	+	0.968029i	$0.580707\pi$
$570$	0	0
$571$	13471.8	0.987352	0.493676	−	0.869646i	$-0.335653\pi$
$571$	13471.8	0.987352	0.493676	+	0.869646i	$0.335653\pi$
$572$	0	0
$573$	7102.88	0.517849
$574$	0	0
$575$	2016.41	0.146244
$576$	0	0
$577$	5331.06	0.384636	0.192318	−	0.981333i	$-0.438399\pi$
$577$	5331.06	0.384636	0.192318	+	0.981333i	$0.438399\pi$
$578$	0	0
$579$	−4727.41	−0.339317
$580$	0	0
$581$	−2171.75	−0.155077
$582$	0	0
$583$	−6741.87	−0.478936
$584$	0	0
$585$	−1561.59	−0.110366
$586$	0	0
$587$	−3333.96	−0.234425	−0.117212	−	0.993107i	$-0.537396\pi$
$587$	−3333.96	−0.234425	−0.117212	+	0.993107i	$0.537396\pi$
$588$	0	0
$589$	4487.09	0.313900
$590$	0	0
$591$	−15285.4	−1.06389
$592$	0	0
$593$	−23405.3	−1.62081	−0.810404	−	0.585872i	$-0.800752\pi$
$593$	−23405.3	−1.62081	−0.810404	+	0.585872i	$0.800752\pi$
$594$	0	0
$595$	19446.1	1.33985
$596$	0	0
$597$	−138.996	−0.00952886
$598$	0	0
$599$	10352.1	0.706133	0.353067	−	0.935598i	$-0.385139\pi$
$599$	10352.1	0.706133	0.353067	+	0.935598i	$0.385139\pi$
$600$	0	0
$601$	15171.1	1.02969	0.514843	−	0.857285i	$-0.327850\pi$
$601$	15171.1	1.02969	0.514843	+	0.857285i	$0.327850\pi$
$602$	0	0
$603$	4983.54	0.336560
$604$	0	0
$605$	33679.0	2.26322
$606$	0	0
$607$	20823.0	1.39239	0.696193	−	0.717854i	$-0.254876\pi$
$607$	20823.0	1.39239	0.696193	+	0.717854i	$0.254876\pi$
$608$	0	0
$609$	−3510.15	−0.233561
$610$	0	0
$611$	−12389.5	−0.820334
$612$	0	0
$613$	−19071.6	−1.25660	−0.628299	−	0.777972i	$-0.716249\pi$
$613$	−19071.6	−1.25660	−0.628299	+	0.777972i	$0.716249\pi$
$614$	0	0
$615$	24813.6	1.62696
$616$	0	0
$617$	15200.4	0.991806	0.495903	−	0.868378i	$-0.334837\pi$
$617$	15200.4	0.991806	0.495903	+	0.868378i	$0.334837\pi$
$618$	0	0
$619$	−4358.76	−0.283026	−0.141513	−	0.989936i	$-0.545197\pi$
$619$	−4358.76	−0.283026	−0.141513	+	0.989936i	$0.545197\pi$
$620$	0	0
$621$	7459.39	0.482021
$622$	0	0
$623$	9325.20	0.599689
$624$	0	0
$625$	−19056.5	−1.21961
$626$	0	0
$627$	20815.2	1.32580
$628$	0	0
$629$	−6906.83	−0.437827
$630$	0	0
$631$	2580.66	0.162812	0.0814062	−	0.996681i	$-0.474059\pi$
$631$	2580.66	0.162812	0.0814062	+	0.996681i	$0.474059\pi$
$632$	0	0
$633$	10532.8	0.661361
$634$	0	0
$635$	25683.2	1.60505
$636$	0	0
$637$	7519.96	0.467742
$638$	0	0
$639$	4235.12	0.262189
$640$	0	0
$641$	19858.9	1.22368	0.611840	−	0.790982i	$-0.290430\pi$
$641$	19858.9	1.22368	0.611840	+	0.790982i	$0.290430\pi$
$642$	0	0
$643$	−17371.9	−1.06545	−0.532723	−	0.846290i	$-0.678831\pi$
$643$	−17371.9	−1.06545	−0.532723	+	0.846290i	$0.678831\pi$
$644$	0	0
$645$	−20812.3	−1.27052
$646$	0	0
$647$	−3275.07	−0.199005	−0.0995024	−	0.995037i	$-0.531725\pi$
$647$	−3275.07	−0.199005	−0.0995024	+	0.995037i	$0.531725\pi$
$648$	0	0
$649$	−8599.88	−0.520146
$650$	0	0
$651$	−7615.80	−0.458505
$652$	0	0
$653$	20726.5	1.24210	0.621049	−	0.783772i	$-0.286707\pi$
$653$	20726.5	1.24210	0.621049	+	0.783772i	$0.286707\pi$
$654$	0	0
$655$	−19886.4	−1.18630
$656$	0	0
$657$	721.762	0.0428594
$658$	0	0
$659$	18404.6	1.08792	0.543960	−	0.839111i	$-0.316924\pi$
$659$	18404.6	1.08792	0.543960	+	0.839111i	$0.316924\pi$
$660$	0	0
$661$	−7146.10	−0.420501	−0.210250	−	0.977648i	$-0.567428\pi$
$661$	−7146.10	−0.420501	−0.210250	+	0.977648i	$0.567428\pi$
$662$	0	0
$663$	6032.02	0.353340
$664$	0	0
$665$	23918.0	1.39474
$666$	0	0
$667$	−1436.39	−0.0833841
$668$	0	0
$669$	22973.3	1.32765
$670$	0	0
$671$	−36402.7	−2.09435
$672$	0	0
$673$	−26819.0	−1.53610	−0.768051	−	0.640389i	$-0.778773\pi$
$673$	−26819.0	−1.53610	−0.768051	+	0.640389i	$0.778773\pi$
$674$	0	0
$675$	6131.05	0.349606
$676$	0	0
$677$	−20093.1	−1.14068	−0.570340	−	0.821408i	$-0.693189\pi$
$677$	−20093.1	−1.14068	−0.570340	+	0.821408i	$0.693189\pi$
$678$	0	0
$679$	4874.82	0.275521
$680$	0	0
$681$	−16536.3	−0.930505
$682$	0	0
$683$	6876.22	0.385229	0.192614	−	0.981275i	$-0.438303\pi$
$683$	6876.22	0.385229	0.192614	+	0.981275i	$0.438303\pi$
$684$	0	0
$685$	−16669.5	−0.929791
$686$	0	0
$687$	−28540.6	−1.58500
$688$	0	0
$689$	−2403.01	−0.132870
$690$	0	0
$691$	−17332.7	−0.954224	−0.477112	−	0.878842i	$-0.658316\pi$
$691$	−17332.7	−0.954224	−0.477112	+	0.878842i	$0.658316\pi$
$692$	0	0
$693$	8867.31	0.486063
$694$	0	0
$695$	−25741.4	−1.40493
$696$	0	0
$697$	24057.1	1.30736
$698$	0	0
$699$	5050.21	0.273271
$700$	0	0
$701$	11127.5	0.599545	0.299772	−	0.954011i	$-0.403089\pi$
$701$	11127.5	0.599545	0.299772	+	0.954011i	$0.403089\pi$
$702$	0	0
$703$	−8495.15	−0.455762
$704$	0	0
$705$	33087.1	1.76756
$706$	0	0
$707$	24996.8	1.32970
$708$	0	0
$709$	17432.0	0.923374	0.461687	−	0.887043i	$-0.347244\pi$
$709$	17432.0	0.923374	0.461687	+	0.887043i	$0.347244\pi$
$710$	0	0
$711$	4703.13	0.248075
$712$	0	0
$713$	−3116.46	−0.163692
$714$	0	0
$715$	18111.2	0.947302
$716$	0	0
$717$	5012.73	0.261093
$718$	0	0
$719$	−21082.7	−1.09353	−0.546767	−	0.837285i	$-0.684142\pi$
$719$	−21082.7	−1.09353	−0.546767	+	0.837285i	$0.684142\pi$
$720$	0	0
$721$	2054.97	0.106146
$722$	0	0
$723$	4595.68	0.236397
$724$	0	0
$725$	−1180.60	−0.0604779
$726$	0	0
$727$	25839.1	1.31818	0.659091	−	0.752063i	$-0.270941\pi$
$727$	25839.1	1.31818	0.659091	+	0.752063i	$0.270941\pi$
$728$	0	0
$729$	21888.5	1.11205
$730$	0	0
$731$	−20177.9	−1.02094
$732$	0	0
$733$	1278.54	0.0644256	0.0322128	−	0.999481i	$-0.489745\pi$
$733$	1278.54	0.0644256	0.0322128	+	0.999481i	$0.489745\pi$
$734$	0	0
$735$	−20082.6	−1.00784
$736$	0	0
$737$	−57798.6	−2.88879
$738$	0	0
$739$	−4224.54	−0.210287	−0.105144	−	0.994457i	$-0.533530\pi$
$739$	−4224.54	−0.210287	−0.105144	+	0.994457i	$0.533530\pi$
$740$	0	0
$741$	7419.17	0.367814
$742$	0	0
$743$	17992.3	0.888390	0.444195	−	0.895930i	$-0.353490\pi$
$743$	17992.3	0.888390	0.444195	+	0.895930i	$0.353490\pi$
$744$	0	0
$745$	15288.2	0.751835
$746$	0	0
$747$	−451.550	−0.0221170
$748$	0	0
$749$	18598.6	0.907316
$750$	0	0
$751$	10082.4	0.489895	0.244948	−	0.969536i	$-0.421229\pi$
$751$	10082.4	0.489895	0.244948	+	0.969536i	$0.421229\pi$
$752$	0	0
$753$	4182.30	0.202406
$754$	0	0
$755$	29061.9	1.40089
$756$	0	0
$757$	10806.5	0.518851	0.259425	−	0.965763i	$-0.416467\pi$
$757$	10806.5	0.518851	0.259425	+	0.965763i	$0.416467\pi$
$758$	0	0
$759$	−14457.0	−0.691377
$760$	0	0
$761$	−30710.4	−1.46288	−0.731439	−	0.681907i	$-0.761151\pi$
$761$	−30710.4	−1.46288	−0.731439	+	0.681907i	$0.761151\pi$
$762$	0	0
$763$	−13979.5	−0.663293
$764$	0	0
$765$	4043.23	0.191089
$766$	0	0
$767$	−3065.26	−0.144303
$768$	0	0
$769$	10757.1	0.504436	0.252218	−	0.967670i	$-0.418840\pi$
$769$	10757.1	0.504436	0.252218	+	0.967670i	$0.418840\pi$
$770$	0	0
$771$	14518.8	0.678185
$772$	0	0
$773$	−18077.2	−0.841125	−0.420563	−	0.907263i	$-0.638167\pi$
$773$	−18077.2	−0.841125	−0.420563	+	0.907263i	$0.638167\pi$
$774$	0	0
$775$	−2561.49	−0.118725
$776$	0	0
$777$	14418.6	0.665719
$778$	0	0
$779$	29589.4	1.36091
$780$	0	0
$781$	−49118.6	−2.25045
$782$	0	0
$783$	−4367.44	−0.199335
$784$	0	0
$785$	15295.3	0.695430
$786$	0	0
$787$	31543.7	1.42873	0.714366	−	0.699773i	$-0.246715\pi$
$787$	31543.7	1.42873	0.714366	+	0.699773i	$0.246715\pi$
$788$	0	0
$789$	13077.3	0.590066
$790$	0	0
$791$	−50710.4	−2.27946
$792$	0	0
$793$	−12975.0	−0.581030
$794$	0	0
$795$	6417.42	0.286293
$796$	0	0
$797$	18280.9	0.812477	0.406239	−	0.913767i	$-0.366840\pi$
$797$	18280.9	0.812477	0.406239	+	0.913767i	$0.366840\pi$
$798$	0	0
$799$	32078.4	1.42034
$800$	0	0
$801$	1938.89	0.0855273
$802$	0	0
$803$	−8370.93	−0.367875
$804$	0	0
$805$	−16612.0	−0.727326
$806$	0	0
$807$	−20484.4	−0.893536
$808$	0	0
$809$	21776.3	0.946372	0.473186	−	0.880963i	$-0.343104\pi$
$809$	21776.3	0.946372	0.473186	+	0.880963i	$0.343104\pi$
$810$	0	0
$811$	−17035.7	−0.737612	−0.368806	−	0.929506i	$-0.620233\pi$
$811$	−17035.7	−0.737612	−0.368806	+	0.929506i	$0.620233\pi$
$812$	0	0
$813$	−20524.7	−0.885404
$814$	0	0
$815$	31568.5	1.35681
$816$	0	0
$817$	−24818.0	−1.06276
$818$	0	0
$819$	3160.58	0.134847
$820$	0	0
$821$	−45120.7	−1.91805	−0.959027	−	0.283314i	$-0.908566\pi$
$821$	−45120.7	−1.91805	−0.959027	+	0.283314i	$0.908566\pi$
$822$	0	0
$823$	−22261.7	−0.942883	−0.471442	−	0.881897i	$-0.656266\pi$
$823$	−22261.7	−0.942883	−0.471442	+	0.881897i	$0.656266\pi$
$824$	0	0
$825$	−11882.5	−0.501451
$826$	0	0
$827$	−10280.4	−0.432266	−0.216133	−	0.976364i	$-0.569345\pi$
$827$	−10280.4	−0.432266	−0.216133	+	0.976364i	$0.569345\pi$
$828$	0	0
$829$	28509.2	1.19441	0.597206	−	0.802088i	$-0.296278\pi$
$829$	28509.2	1.19441	0.597206	+	0.802088i	$0.296278\pi$
$830$	0	0
$831$	4123.45	0.172131
$832$	0	0
$833$	−19470.4	−0.809857
$834$	0	0
$835$	26005.0	1.07777
$836$	0	0
$837$	−9475.82	−0.391317
$838$	0	0
$839$	−4746.97	−0.195332	−0.0976661	−	0.995219i	$-0.531138\pi$
$839$	−4746.97	−0.195332	−0.0976661	+	0.995219i	$0.531138\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	19914.3	0.813625
$844$	0	0
$845$	−21826.3	−0.888576
$846$	0	0
$847$	−68164.5	−2.76524
$848$	0	0
$849$	−7942.30	−0.321059
$850$	0	0
$851$	5900.22	0.237670
$852$	0	0
$853$	35313.3	1.41747	0.708736	−	0.705473i	$-0.249266\pi$
$853$	35313.3	1.41747	0.708736	+	0.705473i	$0.249266\pi$
$854$	0	0
$855$	4973.03	0.198917
$856$	0	0
$857$	32142.8	1.28119	0.640594	−	0.767880i	$-0.278688\pi$
$857$	32142.8	1.28119	0.640594	+	0.767880i	$0.278688\pi$
$858$	0	0
$859$	−37568.2	−1.49221	−0.746107	−	0.665826i	$-0.768079\pi$
$859$	−37568.2	−1.49221	−0.746107	+	0.665826i	$0.768079\pi$
$860$	0	0
$861$	−50221.3	−1.98785
$862$	0	0
$863$	−3416.95	−0.134779	−0.0673896	−	0.997727i	$-0.521467\pi$
$863$	−3416.95	−0.134779	−0.0673896	+	0.997727i	$0.521467\pi$
$864$	0	0
$865$	36843.6	1.44823
$866$	0	0
$867$	7206.57	0.282293
$868$	0	0
$869$	−54546.5	−2.12930
$870$	0	0
$871$	−20601.2	−0.801429
$872$	0	0
$873$	1013.57	0.0392947
$874$	0	0
$875$	28269.8	1.09222
$876$	0	0
$877$	−11891.0	−0.457847	−0.228923	−	0.973444i	$-0.573521\pi$
$877$	−11891.0	−0.457847	−0.228923	+	0.973444i	$0.573521\pi$
$878$	0	0
$879$	−5319.42	−0.204118
$880$	0	0
$881$	20042.7	0.766464	0.383232	−	0.923652i	$-0.374811\pi$
$881$	20042.7	0.766464	0.383232	+	0.923652i	$0.374811\pi$
$882$	0	0
$883$	18042.8	0.687641	0.343821	−	0.939035i	$-0.388279\pi$
$883$	18042.8	0.687641	0.343821	+	0.939035i	$0.388279\pi$
$884$	0	0
$885$	8186.02	0.310927
$886$	0	0
$887$	−2247.91	−0.0850929	−0.0425464	−	0.999094i	$-0.513547\pi$
$887$	−2247.91	−0.0850929	−0.0425464	+	0.999094i	$0.513547\pi$
$888$	0	0
$889$	−51981.4	−1.96108
$890$	0	0
$891$	−34768.2	−1.30727
$892$	0	0
$893$	39455.3	1.47852
$894$	0	0
$895$	2994.88	0.111852
$896$	0	0
$897$	−5152.91	−0.191807
$898$	0	0
$899$	1824.68	0.0676934
$900$	0	0
$901$	6221.80	0.230053
$902$	0	0
$903$	42123.0	1.55234
$904$	0	0
$905$	−33568.1	−1.23298
$906$	0	0
$907$	1798.34	0.0658354	0.0329177	−	0.999458i	$-0.489520\pi$
$907$	1798.34	0.0658354	0.0329177	+	0.999458i	$0.489520\pi$
$908$	0	0
$909$	5197.32	0.189642
$910$	0	0
$911$	−4602.02	−0.167367	−0.0836837	−	0.996492i	$-0.526669\pi$
$911$	−4602.02	−0.167367	−0.0836837	+	0.996492i	$0.526669\pi$
$912$	0	0
$913$	5237.04	0.189836
$914$	0	0
$915$	34650.8	1.25194
$916$	0	0
$917$	40249.1	1.44944
$918$	0	0
$919$	−20622.0	−0.740215	−0.370107	−	0.928989i	$-0.620679\pi$
$919$	−20622.0	−0.740215	−0.370107	+	0.928989i	$0.620679\pi$
$920$	0	0
$921$	5014.58	0.179409
$922$	0	0
$923$	−17507.4	−0.624335
$924$	0	0
$925$	4849.53	0.172380
$926$	0	0
$927$	427.270	0.0151385
$928$	0	0
$929$	−24061.8	−0.849777	−0.424889	−	0.905246i	$-0.639687\pi$
$929$	−24061.8	−0.849777	−0.424889	+	0.905246i	$0.639687\pi$
$930$	0	0
$931$	−23947.9	−0.843032
$932$	0	0
$933$	10749.8	0.377204
$934$	0	0
$935$	−46893.0	−1.64018
$936$	0	0
$937$	−6581.96	−0.229481	−0.114740	−	0.993396i	$-0.536604\pi$
$937$	−6581.96	−0.229481	−0.114740	+	0.993396i	$0.536604\pi$
$938$	0	0
$939$	35554.7	1.23566
$940$	0	0
$941$	23579.7	0.816873	0.408436	−	0.912787i	$-0.366074\pi$
$941$	23579.7	0.816873	0.408436	+	0.912787i	$0.366074\pi$
$942$	0	0
$943$	−20551.0	−0.709686
$944$	0	0
$945$	−50510.0	−1.73872
$946$	0	0
$947$	36368.6	1.24796	0.623982	−	0.781439i	$-0.285514\pi$
$947$	36368.6	1.24796	0.623982	+	0.781439i	$0.285514\pi$
$948$	0	0
$949$	−2983.65	−0.102058
$950$	0	0
$951$	16990.5	0.579343
$952$	0	0
$953$	44404.3	1.50933	0.754667	−	0.656108i	$-0.227798\pi$
$953$	44404.3	1.50933	0.754667	+	0.656108i	$0.227798\pi$
$954$	0	0
$955$	−19681.4	−0.666884
$956$	0	0
$957$	8464.51	0.285913
$958$	0	0
$959$	33738.1	1.13604
$960$	0	0
$961$	−25832.1	−0.867111
$962$	0	0
$963$	3867.02	0.129401
$964$	0	0
$965$	13099.2	0.436971
$966$	0	0
$967$	−21928.3	−0.729230	−0.364615	−	0.931158i	$-0.618799\pi$
$967$	−21928.3	−0.729230	−0.364615	+	0.931158i	$0.618799\pi$
$968$	0	0
$969$	−19209.5	−0.636840
$970$	0	0
$971$	6352.95	0.209965	0.104982	−	0.994474i	$-0.466521\pi$
$971$	6352.95	0.209965	0.104982	+	0.994474i	$0.466521\pi$
$972$	0	0
$973$	52099.1	1.71657
$974$	0	0
$975$	−4235.30	−0.139116
$976$	0	0
$977$	12792.8	0.418913	0.209456	−	0.977818i	$-0.432831\pi$
$977$	12792.8	0.418913	0.209456	+	0.977818i	$0.432831\pi$
$978$	0	0
$979$	−22487.1	−0.734107
$980$	0	0
$981$	−2906.62	−0.0945986
$982$	0	0
$983$	−7536.73	−0.244542	−0.122271	−	0.992497i	$-0.539018\pi$
$983$	−7536.73	−0.244542	−0.122271	+	0.992497i	$0.539018\pi$
$984$	0	0
$985$	42354.3	1.37007
$986$	0	0
$987$	−66966.4	−2.15964
$988$	0	0
$989$	17237.1	0.554205
$990$	0	0
$991$	28088.3	0.900357	0.450179	−	0.892938i	$-0.351360\pi$
$991$	28088.3	0.900357	0.450179	+	0.892938i	$0.351360\pi$
$992$	0	0
$993$	15042.5	0.480726
$994$	0	0
$995$	385.144	0.0122712
$996$	0	0
$997$	−44951.9	−1.42793	−0.713963	−	0.700184i	$-0.753101\pi$
$997$	−44951.9	−1.42793	−0.713963	+	0.700184i	$0.753101\pi$
$998$	0	0
$999$	17940.0	0.568166

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	464.4.a.l.1.2		5
4.3	odd	2		29.4.a.b.1.2	✓	5
8.3	odd	2		1856.4.a.y.1.2		5
8.5	even	2		1856.4.a.bb.1.4		5
12.11	even	2		261.4.a.f.1.4		5
20.19	odd	2		725.4.a.c.1.4		5
28.27	even	2		1421.4.a.e.1.2		5
116.115	odd	2		841.4.a.b.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.4.a.b.1.2	✓	5	4.3	odd	2
261.4.a.f.1.4		5	12.11	even	2
464.4.a.l.1.2		5	1.1	even	1	trivial
725.4.a.c.1.4		5	20.19	odd	2
841.4.a.b.1.4		5	116.115	odd	2
1421.4.a.e.1.2		5	28.27	even	2
1856.4.a.y.1.2		5	8.3	odd	2
1856.4.a.bb.1.4		5	8.5	even	2

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:	\( N \)	\(=\)	\( 464 = 2^{4} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	464.a (trivial)

\(f(q)\)	\(=\)	\(q-4.64574 q^{3} +12.8729 q^{5} -26.0540 q^{7} -5.41713 q^{9} +O(q^{10})\)	\(q-4.64574 q^{3} +12.8729 q^{5} -26.0540 q^{7} -5.41713 q^{9} +62.8274 q^{11} +22.3936 q^{13} -59.8039 q^{15} -57.9808 q^{17} -71.3143 q^{19} +121.040 q^{21} +49.5307 q^{23} +40.7104 q^{25} +150.601 q^{27} -29.0000 q^{29} -62.9198 q^{31} -291.880 q^{33} -335.389 q^{35} +119.123 q^{37} -104.035 q^{39} -414.916 q^{41} +348.009 q^{43} -69.7340 q^{45} -553.259 q^{47} +335.808 q^{49} +269.364 q^{51} -107.308 q^{53} +808.768 q^{55} +331.308 q^{57} -136.881 q^{59} -579.408 q^{61} +141.138 q^{63} +288.269 q^{65} -919.959 q^{67} -230.106 q^{69} -781.802 q^{71} -133.237 q^{73} -189.130 q^{75} -1636.90 q^{77} -868.196 q^{79} -553.392 q^{81} +83.3560 q^{83} -746.379 q^{85} +134.726 q^{87} -357.919 q^{89} -583.442 q^{91} +292.309 q^{93} -918.019 q^{95} -187.105 q^{97} -340.344 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 8 q^{3} + 10 q^{5} - 40 q^{7} + 33 q^{9}+O(q^{10}) \) 5 * q - 8 * q^3 + 10 * q^5 - 40 * q^7 + 33 * q^9	\( 5 q - 8 q^{3} + 10 q^{5} - 40 q^{7} + 33 q^{9} - 12 q^{11} + 14 q^{13} + 74 q^{15} + 66 q^{17} - 214 q^{19} - 164 q^{23} + 207 q^{25} - 362 q^{27} - 145 q^{29} - 420 q^{31} - 576 q^{33} + 52 q^{35} + 378 q^{37} + 374 q^{39} - 1158 q^{41} + 204 q^{43} - 1506 q^{45} - 248 q^{47} - 283 q^{49} - 228 q^{51} - 554 q^{53} - 546 q^{55} + 44 q^{57} - 440 q^{59} + 618 q^{61} - 804 q^{63} - 1656 q^{65} - 1164 q^{67} - 1968 q^{69} + 692 q^{71} - 1950 q^{73} - 3074 q^{75} - 1616 q^{77} - 272 q^{79} + 1801 q^{81} - 512 q^{83} - 1628 q^{85} + 232 q^{87} + 866 q^{89} - 2580 q^{91} - 40 q^{93} - 2244 q^{95} + 1562 q^{97} + 238 q^{99}+O(q^{100}) \) 5 * q - 8 * q^3 + 10 * q^5 - 40 * q^7 + 33 * q^9 - 12 * q^11 + 14 * q^13 + 74 * q^15 + 66 * q^17 - 214 * q^19 - 164 * q^23 + 207 * q^25 - 362 * q^27 - 145 * q^29 - 420 * q^31 - 576 * q^33 + 52 * q^35 + 378 * q^37 + 374 * q^39 - 1158 * q^41 + 204 * q^43 - 1506 * q^45 - 248 * q^47 - 283 * q^49 - 228 * q^51 - 554 * q^53 - 546 * q^55 + 44 * q^57 - 440 * q^59 + 618 * q^61 - 804 * q^63 - 1656 * q^65 - 1164 * q^67 - 1968 * q^69 + 692 * q^71 - 1950 * q^73 - 3074 * q^75 - 1616 * q^77 - 272 * q^79 + 1801 * q^81 - 512 * q^83 - 1628 * q^85 + 232 * q^87 + 866 * q^89 - 2580 * q^91 - 40 * q^93 - 2244 * q^95 + 1562 * q^97 + 238 * q^99

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