LMFDB - Embedded newform 464.4.a.l.1.5

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			1.5
Root			$3.03898$ 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)

$f(q)$	$=$	$q+6.46343 q^{3} +2.14270 q^{5} -20.3573 q^{7} +14.7760 q^{9} +O(q^{10})$	$q+6.46343 q^{3} +2.14270 q^{5} -20.3573 q^{7} +14.7760 q^{9} -52.0703 q^{11} +7.04574 q^{13} +13.8492 q^{15} +28.7724 q^{17} -76.4208 q^{19} -131.578 q^{21} -59.7251 q^{23} -120.409 q^{25} -79.0092 q^{27} -29.0000 q^{29} +3.25229 q^{31} -336.553 q^{33} -43.6196 q^{35} +150.673 q^{37} +45.5397 q^{39} -92.3254 q^{41} +100.703 q^{43} +31.6605 q^{45} -324.003 q^{47} +71.4197 q^{49} +185.969 q^{51} +374.774 q^{53} -111.571 q^{55} -493.941 q^{57} -489.567 q^{59} +221.508 q^{61} -300.799 q^{63} +15.0969 q^{65} +427.538 q^{67} -386.029 q^{69} +898.999 q^{71} -1087.35 q^{73} -778.254 q^{75} +1060.01 q^{77} +798.018 q^{79} -909.622 q^{81} +436.713 q^{83} +61.6507 q^{85} -187.440 q^{87} +456.763 q^{89} -143.432 q^{91} +21.0209 q^{93} -163.747 q^{95} +803.714 q^{97} -769.390 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	6.46343	1.24389	0.621944	−	0.783062i	$-0.286343\pi$
$3$	6.46343	1.24389	0.621944	+	0.783062i	$0.286343\pi$
$4$	0	0
$5$	2.14270	0.191649	0.0958246	−	0.995398i	$-0.469451\pi$
$5$	2.14270	0.191649	0.0958246	+	0.995398i	$0.469451\pi$
$6$	0	0
$7$	−20.3573	−1.09919	−0.549595	−	0.835431i	$-0.685218\pi$
$7$	−20.3573	−1.09919	−0.549595	+	0.835431i	$0.685218\pi$
$8$	0	0
$9$	14.7760	0.547258
$10$	0	0
$11$	−52.0703	−1.42725	−0.713627	−	0.700526i	$-0.752949\pi$
$11$	−52.0703	−1.42725	−0.713627	+	0.700526i	$0.752949\pi$
$12$	0	0
$13$	7.04574	0.150318	0.0751591	−	0.997172i	$-0.476054\pi$
$13$	7.04574	0.150318	0.0751591	+	0.997172i	$0.476054\pi$
$14$	0	0
$15$	13.8492	0.238390
$16$	0	0
$17$	28.7724	0.410490	0.205245	−	0.978711i	$-0.434201\pi$
$17$	28.7724	0.410490	0.205245	+	0.978711i	$0.434201\pi$
$18$	0	0
$19$	−76.4208	−0.922744	−0.461372	−	0.887207i	$-0.652643\pi$
$19$	−76.4208	−0.922744	−0.461372	+	0.887207i	$0.652643\pi$
$20$	0	0
$21$	−131.578	−1.36727
$22$	0	0
$23$	−59.7251	−0.541458	−0.270729	−	0.962656i	$-0.587265\pi$
$23$	−59.7251	−0.541458	−0.270729	+	0.962656i	$0.587265\pi$
$24$	0	0
$25$	−120.409	−0.963271
$26$	0	0
$27$	−79.0092	−0.563160
$28$	0	0
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	0	0
$31$	3.25229	0.0188428	0.00942142	−	0.999956i	$-0.497001\pi$
$31$	3.25229	0.0188428	0.00942142	+	0.999956i	$0.497001\pi$
$32$	0	0
$33$	−336.553	−1.77534
$34$	0	0
$35$	−43.6196	−0.210659
$36$	0	0
$37$	150.673	0.669471	0.334736	−	0.942312i	$-0.391353\pi$
$37$	150.673	0.669471	0.334736	+	0.942312i	$0.391353\pi$
$38$	0	0
$39$	45.5397	0.186979
$40$	0	0
$41$	−92.3254	−0.351678	−0.175839	−	0.984419i	$-0.556264\pi$
$41$	−92.3254	−0.351678	−0.175839	+	0.984419i	$0.556264\pi$
$42$	0	0
$43$	100.703	0.357142	0.178571	−	0.983927i	$-0.442852\pi$
$43$	100.703	0.357142	0.178571	+	0.983927i	$0.442852\pi$
$44$	0	0
$45$	31.6605	0.104882
$46$	0	0
$47$	−324.003	−1.00555	−0.502774	−	0.864418i	$-0.667687\pi$
$47$	−324.003	−1.00555	−0.502774	+	0.864418i	$0.667687\pi$
$48$	0	0
$49$	71.4197	0.208221
$50$	0	0
$51$	185.969	0.510604
$52$	0	0
$53$	374.774	0.971305	0.485653	−	0.874152i	$-0.338582\pi$
$53$	374.774	0.971305	0.485653	+	0.874152i	$0.338582\pi$
$54$	0	0
$55$	−111.571	−0.273532
$56$	0	0
$57$	−493.941	−1.14779
$58$	0	0
$59$	−489.567	−1.08027	−0.540137	−	0.841577i	$-0.681628\pi$
$59$	−489.567	−1.08027	−0.540137	+	0.841577i	$0.681628\pi$
$60$	0	0
$61$	221.508	0.464937	0.232468	−	0.972604i	$-0.425320\pi$
$61$	221.508	0.464937	0.232468	+	0.972604i	$0.425320\pi$
$62$	0	0
$63$	−300.799	−0.601541
$64$	0	0
$65$	15.0969	0.0288083
$66$	0	0
$67$	427.538	0.779584	0.389792	−	0.920903i	$-0.372547\pi$
$67$	427.538	0.779584	0.389792	+	0.920903i	$0.372547\pi$
$68$	0	0
$69$	−386.029	−0.673514
$70$	0	0
$71$	898.999	1.50270	0.751349	−	0.659905i	$-0.229404\pi$
$71$	898.999	1.50270	0.751349	+	0.659905i	$0.229404\pi$
$72$	0	0
$73$	−1087.35	−1.74335	−0.871673	−	0.490087i	$-0.836965\pi$
$73$	−1087.35	−1.74335	−0.871673	+	0.490087i	$0.836965\pi$
$74$	0	0
$75$	−778.254	−1.19820
$76$	0	0
$77$	1060.01	1.56882
$78$	0	0
$79$	798.018	1.13651	0.568253	−	0.822854i	$-0.307619\pi$
$79$	798.018	1.13651	0.568253	+	0.822854i	$0.307619\pi$
$80$	0	0
$81$	−909.622	−1.24777
$82$	0	0
$83$	436.713	0.577536	0.288768	−	0.957399i	$-0.406754\pi$
$83$	436.713	0.577536	0.288768	+	0.957399i	$0.406754\pi$
$84$	0	0
$85$	61.6507	0.0786701
$86$	0	0
$87$	−187.440	−0.230984
$88$	0	0
$89$	456.763	0.544009	0.272004	−	0.962296i	$-0.412313\pi$
$89$	456.763	0.544009	0.272004	+	0.962296i	$0.412313\pi$
$90$	0	0
$91$	−143.432	−0.165228
$92$	0	0
$93$	21.0209	0.0234384
$94$	0	0
$95$	−163.747	−0.176843
$96$	0	0
$97$	803.714	0.841287	0.420643	−	0.907226i	$-0.361804\pi$
$97$	803.714	0.841287	0.420643	+	0.907226i	$0.361804\pi$
$98$	0	0
$99$	−769.390	−0.781077
$100$	0	0
$101$	−738.800	−0.727855	−0.363928	−	0.931427i	$-0.618564\pi$
$101$	−738.800	−0.727855	−0.363928	+	0.931427i	$0.618564\pi$
$102$	0	0
$103$	−2031.60	−1.94349	−0.971743	−	0.236042i	$-0.924150\pi$
$103$	−2031.60	−1.94349	−0.971743	+	0.236042i	$0.924150\pi$
$104$	0	0
$105$	−281.933	−0.262036
$106$	0	0
$107$	−1594.33	−1.44047	−0.720233	−	0.693732i	$-0.755965\pi$
$107$	−1594.33	−1.44047	−0.720233	+	0.693732i	$0.755965\pi$
$108$	0	0
$109$	−229.668	−0.201818	−0.100909	−	0.994896i	$-0.532175\pi$
$109$	−229.668	−0.201818	−0.100909	+	0.994896i	$0.532175\pi$
$110$	0	0
$111$	973.863	0.832748
$112$	0	0
$113$	−1584.73	−1.31928	−0.659640	−	0.751582i	$-0.729291\pi$
$113$	−1584.73	−1.31928	−0.659640	+	0.751582i	$0.729291\pi$
$114$	0	0
$115$	−127.973	−0.103770
$116$	0	0
$117$	104.108	0.0822629
$118$	0	0
$119$	−585.728	−0.451207
$120$	0	0
$121$	1380.32	1.03705
$122$	0	0
$123$	−596.739	−0.437448
$124$	0	0
$125$	−525.838	−0.376259
$126$	0	0
$127$	−621.184	−0.434025	−0.217013	−	0.976169i	$-0.569631\pi$
$127$	−621.184	−0.434025	−0.217013	+	0.976169i	$0.569631\pi$
$128$	0	0
$129$	650.890	0.444245
$130$	0	0
$131$	1504.34	1.00332	0.501659	−	0.865066i	$-0.332723\pi$
$131$	1504.34	1.00332	0.501659	+	0.865066i	$0.332723\pi$
$132$	0	0
$133$	1555.72	1.01427
$134$	0	0
$135$	−169.293	−0.107929
$136$	0	0
$137$	−29.3812	−0.0183227	−0.00916135	−	0.999958i	$-0.502916\pi$
$137$	−29.3812	−0.0183227	−0.00916135	+	0.999958i	$0.502916\pi$
$138$	0	0
$139$	−1262.60	−0.770450	−0.385225	−	0.922823i	$-0.625876\pi$
$139$	−1262.60	−0.770450	−0.385225	+	0.922823i	$0.625876\pi$
$140$	0	0
$141$	−2094.17	−1.25079
$142$	0	0
$143$	−366.874	−0.214542
$144$	0	0
$145$	−62.1384	−0.0355883
$146$	0	0
$147$	461.616	0.259003
$148$	0	0
$149$	826.526	0.454441	0.227220	−	0.973843i	$-0.427036\pi$
$149$	826.526	0.454441	0.227220	+	0.973843i	$0.427036\pi$
$150$	0	0
$151$	2632.03	1.41849	0.709243	−	0.704964i	$-0.249037\pi$
$151$	2632.03	1.41849	0.709243	+	0.704964i	$0.249037\pi$
$152$	0	0
$153$	425.140	0.224644
$154$	0	0
$155$	6.96868	0.00361121
$156$	0	0
$157$	234.540	0.119225	0.0596124	−	0.998222i	$-0.481014\pi$
$157$	234.540	0.119225	0.0596124	+	0.998222i	$0.481014\pi$
$158$	0	0
$159$	2422.33	1.20820
$160$	0	0
$161$	1215.84	0.595166
$162$	0	0
$163$	1650.78	0.793246	0.396623	−	0.917982i	$-0.370182\pi$
$163$	1650.78	0.793246	0.396623	+	0.917982i	$0.370182\pi$
$164$	0	0
$165$	−721.133	−0.340243
$166$	0	0
$167$	3684.28	1.70717	0.853587	−	0.520950i	$-0.174422\pi$
$167$	3684.28	1.70717	0.853587	+	0.520950i	$0.174422\pi$
$168$	0	0
$169$	−2147.36	−0.977404
$170$	0	0
$171$	−1129.19	−0.504979
$172$	0	0
$173$	−3073.58	−1.35075	−0.675375	−	0.737474i	$-0.736018\pi$
$173$	−3073.58	−1.35075	−0.675375	+	0.737474i	$0.736018\pi$
$174$	0	0
$175$	2451.20	1.05882
$176$	0	0
$177$	−3164.28	−1.34374
$178$	0	0
$179$	−2980.70	−1.24463	−0.622313	−	0.782769i	$-0.713807\pi$
$179$	−2980.70	−1.24463	−0.622313	+	0.782769i	$0.713807\pi$
$180$	0	0
$181$	3145.07	1.29155	0.645777	−	0.763526i	$-0.276533\pi$
$181$	3145.07	1.29155	0.645777	+	0.763526i	$0.276533\pi$
$182$	0	0
$183$	1431.70	0.578329
$184$	0	0
$185$	322.847	0.128304
$186$	0	0
$187$	−1498.19	−0.585874
$188$	0	0
$189$	1608.41	0.619021
$190$	0	0
$191$	2125.56	0.805236	0.402618	−	0.915368i	$-0.368100\pi$
$191$	2125.56	0.805236	0.402618	+	0.915368i	$0.368100\pi$
$192$	0	0
$193$	3085.29	1.15069	0.575347	−	0.817909i	$-0.304867\pi$
$193$	3085.29	1.15069	0.575347	+	0.817909i	$0.304867\pi$
$194$	0	0
$195$	97.5779	0.0358344
$196$	0	0
$197$	−2484.45	−0.898526	−0.449263	−	0.893400i	$-0.648313\pi$
$197$	−2484.45	−0.898526	−0.449263	+	0.893400i	$0.648313\pi$
$198$	0	0
$199$	−5489.57	−1.95550	−0.977752	−	0.209764i	$-0.932730\pi$
$199$	−5489.57	−1.95550	−0.977752	+	0.209764i	$0.932730\pi$
$200$	0	0
$201$	2763.37	0.969715
$202$	0	0
$203$	590.362	0.204115
$204$	0	0
$205$	−197.826	−0.0673988
$206$	0	0
$207$	−882.496	−0.296318
$208$	0	0
$209$	3979.26	1.31699
$210$	0	0
$211$	1884.64	0.614902	0.307451	−	0.951564i	$-0.400524\pi$
$211$	1884.64	0.614902	0.307451	+	0.951564i	$0.400524\pi$
$212$	0	0
$213$	5810.62	1.86919
$214$	0	0
$215$	215.777	0.0684460
$216$	0	0
$217$	−66.2078	−0.0207119
$218$	0	0
$219$	−7027.99	−2.16853
$220$	0	0
$221$	202.723	0.0617041
$222$	0	0
$223$	5208.50	1.56407	0.782033	−	0.623237i	$-0.214183\pi$
$223$	5208.50	1.56407	0.782033	+	0.623237i	$0.214183\pi$
$224$	0	0
$225$	−1779.16	−0.527158
$226$	0	0
$227$	5243.29	1.53308	0.766540	−	0.642196i	$-0.221977\pi$
$227$	5243.29	1.53308	0.766540	+	0.642196i	$0.221977\pi$
$228$	0	0
$229$	−846.348	−0.244228	−0.122114	−	0.992516i	$-0.538967\pi$
$229$	−846.348	−0.244228	−0.122114	+	0.992516i	$0.538967\pi$
$230$	0	0
$231$	6851.31	1.95144
$232$	0	0
$233$	823.620	0.231576	0.115788	−	0.993274i	$-0.463061\pi$
$233$	823.620	0.231576	0.115788	+	0.993274i	$0.463061\pi$
$234$	0	0
$235$	−694.243	−0.192712
$236$	0	0
$237$	5157.93	1.41369
$238$	0	0
$239$	2471.50	0.668903	0.334452	−	0.942413i	$-0.391449\pi$
$239$	2471.50	0.668903	0.334452	+	0.942413i	$0.391449\pi$
$240$	0	0
$241$	−1148.79	−0.307055	−0.153527	−	0.988144i	$-0.549063\pi$
$241$	−1148.79	−0.307055	−0.153527	+	0.988144i	$0.549063\pi$
$242$	0	0
$243$	−3746.03	−0.988922
$244$	0	0
$245$	153.031	0.0399053
$246$	0	0
$247$	−538.441	−0.138705
$248$	0	0
$249$	2822.67	0.718391
$250$	0	0
$251$	1686.20	0.424032	0.212016	−	0.977266i	$-0.431997\pi$
$251$	1686.20	0.424032	0.212016	+	0.977266i	$0.431997\pi$
$252$	0	0
$253$	3109.91	0.772799
$254$	0	0
$255$	398.475	0.0978568
$256$	0	0
$257$	3593.97	0.872318	0.436159	−	0.899870i	$-0.356338\pi$
$257$	3593.97	0.872318	0.436159	+	0.899870i	$0.356338\pi$
$258$	0	0
$259$	−3067.29	−0.735877
$260$	0	0
$261$	−428.503	−0.101623
$262$	0	0
$263$	2321.87	0.544382	0.272191	−	0.962243i	$-0.412252\pi$
$263$	2321.87	0.544382	0.272191	+	0.962243i	$0.412252\pi$
$264$	0	0
$265$	803.029	0.186150
$266$	0	0
$267$	2952.26	0.676686
$268$	0	0
$269$	2365.41	0.536140	0.268070	−	0.963399i	$-0.413614\pi$
$269$	2365.41	0.536140	0.268070	+	0.963399i	$0.413614\pi$
$270$	0	0
$271$	−1732.51	−0.388348	−0.194174	−	0.980967i	$-0.562203\pi$
$271$	−1732.51	−0.388348	−0.194174	+	0.980967i	$0.562203\pi$
$272$	0	0
$273$	−927.065	−0.205526
$274$	0	0
$275$	6269.73	1.37483
$276$	0	0
$277$	−688.616	−0.149368	−0.0746839	−	0.997207i	$-0.523795\pi$
$277$	−688.616	−0.149368	−0.0746839	+	0.997207i	$0.523795\pi$
$278$	0	0
$279$	48.0557	0.0103119
$280$	0	0
$281$	2224.44	0.472238	0.236119	−	0.971724i	$-0.424125\pi$
$281$	2224.44	0.472238	0.236119	+	0.971724i	$0.424125\pi$
$282$	0	0
$283$	−5037.52	−1.05813	−0.529063	−	0.848582i	$-0.677457\pi$
$283$	−5037.52	−1.05813	−0.529063	+	0.848582i	$0.677457\pi$
$284$	0	0
$285$	−1058.37	−0.219973
$286$	0	0
$287$	1879.50	0.386561
$288$	0	0
$289$	−4085.15	−0.831498
$290$	0	0
$291$	5194.75	1.04647
$292$	0	0
$293$	−6761.01	−1.34806	−0.674032	−	0.738702i	$-0.735439\pi$
$293$	−6761.01	−1.34806	−0.674032	+	0.738702i	$0.735439\pi$
$294$	0	0
$295$	−1049.00	−0.207034
$296$	0	0
$297$	4114.03	0.803773
$298$	0	0
$299$	−420.807	−0.0813910
$300$	0	0
$301$	−2050.05	−0.392568
$302$	0	0
$303$	−4775.19	−0.905371
$304$	0	0
$305$	474.625	0.0891047
$306$	0	0
$307$	−8860.99	−1.64731	−0.823653	−	0.567093i	$-0.808068\pi$
$307$	−8860.99	−1.64731	−0.823653	+	0.567093i	$0.808068\pi$
$308$	0	0
$309$	−13131.1	−2.41748
$310$	0	0
$311$	−4127.77	−0.752619	−0.376309	−	0.926494i	$-0.622807\pi$
$311$	−4127.77	−0.752619	−0.376309	+	0.926494i	$0.622807\pi$
$312$	0	0
$313$	−5384.53	−0.972369	−0.486184	−	0.873856i	$-0.661612\pi$
$313$	−5384.53	−0.972369	−0.486184	+	0.873856i	$0.661612\pi$
$314$	0	0
$315$	−644.522	−0.115285
$316$	0	0
$317$	−5379.49	−0.953129	−0.476565	−	0.879139i	$-0.658118\pi$
$317$	−5379.49	−0.953129	−0.476565	+	0.879139i	$0.658118\pi$
$318$	0	0
$319$	1510.04	0.265034
$320$	0	0
$321$	−10304.9	−1.79178
$322$	0	0
$323$	−2198.81	−0.378778
$324$	0	0
$325$	−848.369	−0.144797
$326$	0	0
$327$	−1484.44	−0.251039
$328$	0	0
$329$	6595.83	1.10529
$330$	0	0
$331$	−5825.09	−0.967298	−0.483649	−	0.875262i	$-0.660689\pi$
$331$	−5825.09	−0.967298	−0.483649	+	0.875262i	$0.660689\pi$
$332$	0	0
$333$	2226.34	0.366374
$334$	0	0
$335$	916.087	0.149407
$336$	0	0
$337$	−3627.26	−0.586318	−0.293159	−	0.956064i	$-0.594707\pi$
$337$	−3627.26	−0.586318	−0.293159	+	0.956064i	$0.594707\pi$
$338$	0	0
$339$	−10242.8	−1.64104
$340$	0	0
$341$	−169.348	−0.0268935
$342$	0	0
$343$	5528.64	0.870317
$344$	0	0
$345$	−827.145	−0.129078
$346$	0	0
$347$	−1172.68	−0.181421	−0.0907104	−	0.995877i	$-0.528914\pi$
$347$	−1172.68	−0.181421	−0.0907104	+	0.995877i	$0.528914\pi$
$348$	0	0
$349$	−173.078	−0.0265463	−0.0132731	−	0.999912i	$-0.504225\pi$
$349$	−173.078	−0.0265463	−0.0132731	+	0.999912i	$0.504225\pi$
$350$	0	0
$351$	−556.678	−0.0846532
$352$	0	0
$353$	13008.4	1.96137	0.980687	−	0.195582i	$-0.0626596\pi$
$353$	13008.4	1.96137	0.980687	+	0.195582i	$0.0626596\pi$
$354$	0	0
$355$	1926.29	0.287991
$356$	0	0
$357$	−3785.82	−0.561251
$358$	0	0
$359$	4487.40	0.659710	0.329855	−	0.944032i	$-0.393000\pi$
$359$	4487.40	0.659710	0.329855	+	0.944032i	$0.393000\pi$
$360$	0	0
$361$	−1018.85	−0.148543
$362$	0	0
$363$	8921.60	1.28998
$364$	0	0
$365$	−2329.86	−0.334111
$366$	0	0
$367$	−1674.11	−0.238114	−0.119057	−	0.992887i	$-0.537987\pi$
$367$	−1674.11	−0.238114	−0.119057	+	0.992887i	$0.537987\pi$
$368$	0	0
$369$	−1364.20	−0.192459
$370$	0	0
$371$	−7629.39	−1.06765
$372$	0	0
$373$	−6800.89	−0.944066	−0.472033	−	0.881581i	$-0.656480\pi$
$373$	−6800.89	−0.944066	−0.472033	+	0.881581i	$0.656480\pi$
$374$	0	0
$375$	−3398.72	−0.468024
$376$	0	0
$377$	−204.326	−0.0279134
$378$	0	0
$379$	−10216.7	−1.38468	−0.692341	−	0.721571i	$-0.743421\pi$
$379$	−10216.7	−1.38468	−0.692341	+	0.721571i	$0.743421\pi$
$380$	0	0
$381$	−4014.98	−0.539879
$382$	0	0
$383$	7184.47	0.958509	0.479255	−	0.877676i	$-0.340907\pi$
$383$	7184.47	0.958509	0.479255	+	0.877676i	$0.340907\pi$
$384$	0	0
$385$	2271.29	0.300664
$386$	0	0
$387$	1487.99	0.195449
$388$	0	0
$389$	−3848.67	−0.501633	−0.250817	−	0.968035i	$-0.580699\pi$
$389$	−3848.67	−0.501633	−0.250817	+	0.968035i	$0.580699\pi$
$390$	0	0
$391$	−1718.43	−0.222263
$392$	0	0
$393$	9723.18	1.24801
$394$	0	0
$395$	1709.91	0.217810
$396$	0	0
$397$	−1622.51	−0.205117	−0.102559	−	0.994727i	$-0.532703\pi$
$397$	−1622.51	−0.205117	−0.102559	+	0.994727i	$0.532703\pi$
$398$	0	0
$399$	10055.3	1.26164
$400$	0	0
$401$	13842.9	1.72389	0.861947	−	0.506999i	$-0.169245\pi$
$401$	13842.9	1.72389	0.861947	+	0.506999i	$0.169245\pi$
$402$	0	0
$403$	22.9148	0.00283242
$404$	0	0
$405$	−1949.05	−0.239133
$406$	0	0
$407$	−7845.58	−0.955506
$408$	0	0
$409$	3357.24	0.405880	0.202940	−	0.979191i	$-0.434950\pi$
$409$	3357.24	0.405880	0.202940	+	0.979191i	$0.434950\pi$
$410$	0	0
$411$	−189.904	−0.0227914
$412$	0	0
$413$	9966.26	1.18743
$414$	0	0
$415$	935.746	0.110684
$416$	0	0
$417$	−8160.74	−0.958353
$418$	0	0
$419$	−1652.11	−0.192627	−0.0963134	−	0.995351i	$-0.530705\pi$
$419$	−1652.11	−0.192627	−0.0963134	+	0.995351i	$0.530705\pi$
$420$	0	0
$421$	−6405.13	−0.741490	−0.370745	−	0.928735i	$-0.620898\pi$
$421$	−6405.13	−0.741490	−0.370745	+	0.928735i	$0.620898\pi$
$422$	0	0
$423$	−4787.46	−0.550294
$424$	0	0
$425$	−3464.45	−0.395413
$426$	0	0
$427$	−4509.30	−0.511054
$428$	0	0
$429$	−2371.27	−0.266867
$430$	0	0
$431$	−252.536	−0.0282233	−0.0141116	−	0.999900i	$-0.504492\pi$
$431$	−252.536	−0.0282233	−0.0141116	+	0.999900i	$0.504492\pi$
$432$	0	0
$433$	−12779.1	−1.41831	−0.709153	−	0.705055i	$-0.750922\pi$
$433$	−12779.1	−1.41831	−0.709153	+	0.705055i	$0.750922\pi$
$434$	0	0
$435$	−401.627	−0.0442679
$436$	0	0
$437$	4564.24	0.499628
$438$	0	0
$439$	3760.84	0.408873	0.204437	−	0.978880i	$-0.434464\pi$
$439$	3760.84	0.408873	0.204437	+	0.978880i	$0.434464\pi$
$440$	0	0
$441$	1055.29	0.113950
$442$	0	0
$443$	−3713.77	−0.398299	−0.199149	−	0.979969i	$-0.563818\pi$
$443$	−3713.77	−0.398299	−0.199149	+	0.979969i	$0.563818\pi$
$444$	0	0
$445$	978.707	0.104259
$446$	0	0
$447$	5342.20	0.565274
$448$	0	0
$449$	−12043.9	−1.26589	−0.632947	−	0.774195i	$-0.718155\pi$
$449$	−12043.9	−1.26589	−0.632947	+	0.774195i	$0.718155\pi$
$450$	0	0
$451$	4807.41	0.501934
$452$	0	0
$453$	17011.9	1.76444
$454$	0	0
$455$	−307.333	−0.0316659
$456$	0	0
$457$	−12995.7	−1.33023	−0.665113	−	0.746743i	$-0.731617\pi$
$457$	−12995.7	−1.33023	−0.665113	+	0.746743i	$0.731617\pi$
$458$	0	0
$459$	−2273.28	−0.231172
$460$	0	0
$461$	7874.30	0.795538	0.397769	−	0.917486i	$-0.369785\pi$
$461$	7874.30	0.795538	0.397769	+	0.917486i	$0.369785\pi$
$462$	0	0
$463$	3466.08	0.347910	0.173955	−	0.984754i	$-0.444345\pi$
$463$	3466.08	0.347910	0.173955	+	0.984754i	$0.444345\pi$
$464$	0	0
$465$	45.0416	0.00449195
$466$	0	0
$467$	−14835.9	−1.47007	−0.735034	−	0.678030i	$-0.762834\pi$
$467$	−14835.9	−1.47007	−0.735034	+	0.678030i	$0.762834\pi$
$468$	0	0
$469$	−8703.53	−0.856912
$470$	0	0
$471$	1515.93	0.148302
$472$	0	0
$473$	−5243.66	−0.509733
$474$	0	0
$475$	9201.74	0.888853
$476$	0	0
$477$	5537.65	0.531555
$478$	0	0
$479$	−8119.86	−0.774542	−0.387271	−	0.921966i	$-0.626582\pi$
$479$	−8119.86	−0.774542	−0.387271	+	0.921966i	$0.626582\pi$
$480$	0	0
$481$	1061.60	0.100634
$482$	0	0
$483$	7858.51	0.740320
$484$	0	0
$485$	1722.12	0.161232
$486$	0	0
$487$	−9698.92	−0.902464	−0.451232	−	0.892407i	$-0.649015\pi$
$487$	−9698.92	−0.902464	−0.451232	+	0.892407i	$0.649015\pi$
$488$	0	0
$489$	10669.7	0.986709
$490$	0	0
$491$	6844.25	0.629076	0.314538	−	0.949245i	$-0.398150\pi$
$491$	6844.25	0.629076	0.314538	+	0.949245i	$0.398150\pi$
$492$	0	0
$493$	−834.400	−0.0762261
$494$	0	0
$495$	−1648.57	−0.149693
$496$	0	0
$497$	−18301.2	−1.65175
$498$	0	0
$499$	18603.4	1.66895	0.834473	−	0.551049i	$-0.185772\pi$
$499$	18603.4	1.66895	0.834473	+	0.551049i	$0.185772\pi$
$500$	0	0
$501$	23813.1	2.12353
$502$	0	0
$503$	−10624.8	−0.941822	−0.470911	−	0.882181i	$-0.656075\pi$
$503$	−10624.8	−0.941822	−0.470911	+	0.882181i	$0.656075\pi$
$504$	0	0
$505$	−1583.03	−0.139493
$506$	0	0
$507$	−13879.3	−1.21578
$508$	0	0
$509$	−7931.22	−0.690658	−0.345329	−	0.938482i	$-0.612233\pi$
$509$	−7931.22	−0.690658	−0.345329	+	0.938482i	$0.612233\pi$
$510$	0	0
$511$	22135.4	1.91627
$512$	0	0
$513$	6037.95	0.519653
$514$	0	0
$515$	−4353.10	−0.372467
$516$	0	0
$517$	16871.0	1.43517
$518$	0	0
$519$	−19865.9	−1.68018
$520$	0	0
$521$	13771.1	1.15801	0.579005	−	0.815324i	$-0.303441\pi$
$521$	13771.1	1.15801	0.579005	+	0.815324i	$0.303441\pi$
$522$	0	0
$523$	−20397.1	−1.70536	−0.852680	−	0.522433i	$-0.825024\pi$
$523$	−20397.1	−1.70536	−0.852680	+	0.522433i	$0.825024\pi$
$524$	0	0
$525$	15843.2	1.31705
$526$	0	0
$527$	93.5761	0.00773480
$528$	0	0
$529$	−8599.91	−0.706823
$530$	0	0
$531$	−7233.83	−0.591189
$532$	0	0
$533$	−650.501	−0.0528636
$534$	0	0
$535$	−3416.18	−0.276064
$536$	0	0
$537$	−19265.6	−1.54817
$538$	0	0
$539$	−3718.85	−0.297184
$540$	0	0
$541$	1883.72	0.149699	0.0748496	−	0.997195i	$-0.476152\pi$
$541$	1883.72	0.149699	0.0748496	+	0.997195i	$0.476152\pi$
$542$	0	0
$543$	20328.0	1.60655
$544$	0	0
$545$	−492.110	−0.0386783
$546$	0	0
$547$	9079.06	0.709676	0.354838	−	0.934928i	$-0.384536\pi$
$547$	9079.06	0.709676	0.354838	+	0.934928i	$0.384536\pi$
$548$	0	0
$549$	3272.99	0.254440
$550$	0	0
$551$	2216.20	0.171349
$552$	0	0
$553$	−16245.5	−1.24924
$554$	0	0
$555$	2086.70	0.159595
$556$	0	0
$557$	−5982.38	−0.455084	−0.227542	−	0.973768i	$-0.573069\pi$
$557$	−5982.38	−0.455084	−0.227542	+	0.973768i	$0.573069\pi$
$558$	0	0
$559$	709.530	0.0536850
$560$	0	0
$561$	−9683.44	−0.728762
$562$	0	0
$563$	7837.88	0.586727	0.293363	−	0.956001i	$-0.405225\pi$
$563$	7837.88	0.586727	0.293363	+	0.956001i	$0.405225\pi$
$564$	0	0
$565$	−3395.60	−0.252839
$566$	0	0
$567$	18517.4	1.37153
$568$	0	0
$569$	10019.2	0.738182	0.369091	−	0.929393i	$-0.379669\pi$
$569$	10019.2	0.738182	0.369091	+	0.929393i	$0.379669\pi$
$570$	0	0
$571$	−8832.38	−0.647327	−0.323663	−	0.946172i	$-0.604915\pi$
$571$	−8832.38	−0.647327	−0.323663	+	0.946172i	$0.604915\pi$
$572$	0	0
$573$	13738.4	1.00162
$574$	0	0
$575$	7191.43	0.521571
$576$	0	0
$577$	−17583.3	−1.26863	−0.634317	−	0.773073i	$-0.718719\pi$
$577$	−17583.3	−1.26863	−0.634317	+	0.773073i	$0.718719\pi$
$578$	0	0
$579$	19941.6	1.43133
$580$	0	0
$581$	−8890.30	−0.634823
$582$	0	0
$583$	−19514.6	−1.38630
$584$	0	0
$585$	223.072	0.0157656
$586$	0	0
$587$	10120.7	0.711630	0.355815	−	0.934556i	$-0.384203\pi$
$587$	10120.7	0.711630	0.355815	+	0.934556i	$0.384203\pi$
$588$	0	0
$589$	−248.543	−0.0173871
$590$	0	0
$591$	−16058.1	−1.11767
$592$	0	0
$593$	3597.06	0.249095	0.124548	−	0.992214i	$-0.460252\pi$
$593$	3597.06	0.249095	0.124548	+	0.992214i	$0.460252\pi$
$594$	0	0
$595$	−1255.04	−0.0864734
$596$	0	0
$597$	−35481.5	−2.43243
$598$	0	0
$599$	7987.90	0.544870	0.272435	−	0.962174i	$-0.412171\pi$
$599$	7987.90	0.544870	0.272435	+	0.962174i	$0.412171\pi$
$600$	0	0
$601$	2646.60	0.179629	0.0898147	−	0.995958i	$-0.471373\pi$
$601$	2646.60	0.179629	0.0898147	+	0.995958i	$0.471373\pi$
$602$	0	0
$603$	6317.29	0.426634
$604$	0	0
$605$	2957.61	0.198751
$606$	0	0
$607$	15181.8	1.01517	0.507587	−	0.861600i	$-0.330537\pi$
$607$	15181.8	1.01517	0.507587	+	0.861600i	$0.330537\pi$
$608$	0	0
$609$	3815.76	0.253896
$610$	0	0
$611$	−2282.84	−0.151152
$612$	0	0
$613$	−9721.86	−0.640558	−0.320279	−	0.947323i	$-0.603777\pi$
$613$	−9721.86	−0.640558	−0.320279	+	0.947323i	$0.603777\pi$
$614$	0	0
$615$	−1278.63	−0.0838366
$616$	0	0
$617$	−14150.2	−0.923282	−0.461641	−	0.887067i	$-0.652739\pi$
$617$	−14150.2	−0.923282	−0.461641	+	0.887067i	$0.652739\pi$
$618$	0	0
$619$	10804.4	0.701560	0.350780	−	0.936458i	$-0.385917\pi$
$619$	10804.4	0.701560	0.350780	+	0.936458i	$0.385917\pi$
$620$	0	0
$621$	4718.83	0.304928
$622$	0	0
$623$	−9298.46	−0.597970
$624$	0	0
$625$	13924.4	0.891161
$626$	0	0
$627$	25719.7	1.63819
$628$	0	0
$629$	4335.22	0.274811
$630$	0	0
$631$	−20692.1	−1.30545	−0.652725	−	0.757595i	$-0.726374\pi$
$631$	−20692.1	−1.30545	−0.652725	+	0.757595i	$0.726374\pi$
$632$	0	0
$633$	12181.3	0.764869
$634$	0	0
$635$	−1331.01	−0.0831805
$636$	0	0
$637$	503.204	0.0312993
$638$	0	0
$639$	13283.6	0.822364
$640$	0	0
$641$	−1995.59	−0.122966	−0.0614828	−	0.998108i	$-0.519583\pi$
$641$	−1995.59	−0.122966	−0.0614828	+	0.998108i	$0.519583\pi$
$642$	0	0
$643$	−10911.2	−0.669203	−0.334602	−	0.942360i	$-0.608602\pi$
$643$	−10911.2	−0.669203	−0.334602	+	0.942360i	$0.608602\pi$
$644$	0	0
$645$	1394.66	0.0851392
$646$	0	0
$647$	−20046.3	−1.21808	−0.609042	−	0.793138i	$-0.708446\pi$
$647$	−20046.3	−1.21808	−0.609042	+	0.793138i	$0.708446\pi$
$648$	0	0
$649$	25491.9	1.54183
$650$	0	0
$651$	−427.930	−0.0257633
$652$	0	0
$653$	20205.6	1.21088	0.605441	−	0.795890i	$-0.292997\pi$
$653$	20205.6	1.21088	0.605441	+	0.795890i	$0.292997\pi$
$654$	0	0
$655$	3223.35	0.192285
$656$	0	0
$657$	−16066.6	−0.954061
$658$	0	0
$659$	9268.73	0.547888	0.273944	−	0.961746i	$-0.411672\pi$
$659$	9268.73	0.547888	0.273944	+	0.961746i	$0.411672\pi$
$660$	0	0
$661$	−18209.9	−1.07153	−0.535765	−	0.844367i	$-0.679977\pi$
$661$	−18209.9	−1.07153	−0.535765	+	0.844367i	$0.679977\pi$
$662$	0	0
$663$	1310.29	0.0767531
$664$	0	0
$665$	3333.45	0.194384
$666$	0	0
$667$	1732.03	0.100546
$668$	0	0
$669$	33664.8	1.94552
$670$	0	0
$671$	−11534.0	−0.663583
$672$	0	0
$673$	−63.9154	−0.00366086	−0.00183043	−	0.999998i	$-0.500583\pi$
$673$	−63.9154	−0.00366086	−0.00183043	+	0.999998i	$0.500583\pi$
$674$	0	0
$675$	9513.40	0.542476
$676$	0	0
$677$	11436.2	0.649231	0.324616	−	0.945846i	$-0.394765\pi$
$677$	11436.2	0.649231	0.324616	+	0.945846i	$0.394765\pi$
$678$	0	0
$679$	−16361.5	−0.924735
$680$	0	0
$681$	33889.6	1.90698
$682$	0	0
$683$	22719.5	1.27282	0.636410	−	0.771351i	$-0.280419\pi$
$683$	22719.5	1.27282	0.636410	+	0.771351i	$0.280419\pi$
$684$	0	0
$685$	−62.9553	−0.00351153
$686$	0	0
$687$	−5470.31	−0.303793
$688$	0	0
$689$	2640.56	0.146005
$690$	0	0
$691$	−10495.0	−0.577783	−0.288891	−	0.957362i	$-0.593287\pi$
$691$	−10495.0	−0.577783	−0.288891	+	0.957362i	$0.593287\pi$
$692$	0	0
$693$	15662.7	0.858552
$694$	0	0
$695$	−2705.38	−0.147656
$696$	0	0
$697$	−2656.42	−0.144360
$698$	0	0
$699$	5323.41	0.288054
$700$	0	0
$701$	−6530.49	−0.351859	−0.175930	−	0.984403i	$-0.556293\pi$
$701$	−6530.49	−0.351859	−0.175930	+	0.984403i	$0.556293\pi$
$702$	0	0
$703$	−11514.5	−0.617751
$704$	0	0
$705$	−4487.19	−0.239713
$706$	0	0
$707$	15040.0	0.800052
$708$	0	0
$709$	−16597.7	−0.879183	−0.439591	−	0.898198i	$-0.644877\pi$
$709$	−16597.7	−0.879183	−0.439591	+	0.898198i	$0.644877\pi$
$710$	0	0
$711$	11791.5	0.621962
$712$	0	0
$713$	−194.243	−0.0102026
$714$	0	0
$715$	−786.102	−0.0411168
$716$	0	0
$717$	15974.4	0.832041
$718$	0	0
$719$	−6525.25	−0.338457	−0.169229	−	0.985577i	$-0.554128\pi$
$719$	−6525.25	−0.338457	−0.169229	+	0.985577i	$0.554128\pi$
$720$	0	0
$721$	41357.8	2.13626
$722$	0	0
$723$	−7425.14	−0.381942
$724$	0	0
$725$	3491.86	0.178875
$726$	0	0
$727$	−1066.84	−0.0544249	−0.0272125	−	0.999630i	$-0.508663\pi$
$727$	−1066.84	−0.0544249	−0.0272125	+	0.999630i	$0.508663\pi$
$728$	0	0
$729$	347.561	0.0176579
$730$	0	0
$731$	2897.48	0.146603
$732$	0	0
$733$	−6226.75	−0.313765	−0.156883	−	0.987617i	$-0.550144\pi$
$733$	−6226.75	−0.313765	−0.156883	+	0.987617i	$0.550144\pi$
$734$	0	0
$735$	989.106	0.0496377
$736$	0	0
$737$	−22262.1	−1.11266
$738$	0	0
$739$	−31226.3	−1.55437	−0.777183	−	0.629274i	$-0.783352\pi$
$739$	−31226.3	−1.55437	−0.777183	+	0.629274i	$0.783352\pi$
$740$	0	0
$741$	−3480.18	−0.172534
$742$	0	0
$743$	30790.4	1.52031	0.760156	−	0.649741i	$-0.225123\pi$
$743$	30790.4	1.52031	0.760156	+	0.649741i	$0.225123\pi$
$744$	0	0
$745$	1771.00	0.0870932
$746$	0	0
$747$	6452.86	0.316061
$748$	0	0
$749$	32456.3	1.58335
$750$	0	0
$751$	−11908.0	−0.578598	−0.289299	−	0.957239i	$-0.593422\pi$
$751$	−11908.0	−0.578598	−0.289299	+	0.957239i	$0.593422\pi$
$752$	0	0
$753$	10898.6	0.527448
$754$	0	0
$755$	5639.65	0.271852
$756$	0	0
$757$	−7197.74	−0.345583	−0.172792	−	0.984958i	$-0.555279\pi$
$757$	−7197.74	−0.345583	−0.172792	+	0.984958i	$0.555279\pi$
$758$	0	0
$759$	20100.7	0.961275
$760$	0	0
$761$	9185.20	0.437534	0.218767	−	0.975777i	$-0.429797\pi$
$761$	9185.20	0.437534	0.218767	+	0.975777i	$0.429797\pi$
$762$	0	0
$763$	4675.42	0.221837
$764$	0	0
$765$	910.949	0.0430528
$766$	0	0
$767$	−3449.36	−0.162385
$768$	0	0
$769$	32791.1	1.53768	0.768840	−	0.639441i	$-0.220834\pi$
$769$	32791.1	1.53768	0.768840	+	0.639441i	$0.220834\pi$
$770$	0	0
$771$	23229.4	1.08507
$772$	0	0
$773$	−28237.1	−1.31386	−0.656932	−	0.753950i	$-0.728146\pi$
$773$	−28237.1	−1.31386	−0.656932	+	0.753950i	$0.728146\pi$
$774$	0	0
$775$	−391.604	−0.0181508
$776$	0	0
$777$	−19825.2	−0.915349
$778$	0	0
$779$	7055.58	0.324509
$780$	0	0
$781$	−46811.2	−2.14473
$782$	0	0
$783$	2291.27	0.104576
$784$	0	0
$785$	502.548	0.0228493
$786$	0	0
$787$	−12082.3	−0.547254	−0.273627	−	0.961836i	$-0.588223\pi$
$787$	−12082.3	−0.547254	−0.273627	+	0.961836i	$0.588223\pi$
$788$	0	0
$789$	15007.2	0.677151
$790$	0	0
$791$	32260.8	1.45014
$792$	0	0
$793$	1560.68	0.0698884
$794$	0	0
$795$	5190.33	0.231550
$796$	0	0
$797$	−1248.33	−0.0554807	−0.0277404	−	0.999615i	$-0.508831\pi$
$797$	−1248.33	−0.0554807	−0.0277404	+	0.999615i	$0.508831\pi$
$798$	0	0
$799$	−9322.35	−0.412767
$800$	0	0
$801$	6749.12	0.297713
$802$	0	0
$803$	56618.5	2.48820
$804$	0	0
$805$	2605.19	0.114063
$806$	0	0
$807$	15288.7	0.666898
$808$	0	0
$809$	−23512.6	−1.02183	−0.510914	−	0.859632i	$-0.670693\pi$
$809$	−23512.6	−1.02183	−0.510914	+	0.859632i	$0.670693\pi$
$810$	0	0
$811$	−23248.4	−1.00661	−0.503306	−	0.864108i	$-0.667883\pi$
$811$	−23248.4	−1.00661	−0.503306	+	0.864108i	$0.667883\pi$
$812$	0	0
$813$	−11197.9	−0.483062
$814$	0	0
$815$	3537.13	0.152025
$816$	0	0
$817$	−7695.84	−0.329551
$818$	0	0
$819$	−2119.35	−0.0904226
$820$	0	0
$821$	26898.0	1.14342	0.571708	−	0.820457i	$-0.306281\pi$
$821$	26898.0	1.14342	0.571708	+	0.820457i	$0.306281\pi$
$822$	0	0
$823$	−12422.7	−0.526158	−0.263079	−	0.964774i	$-0.584738\pi$
$823$	−12422.7	−0.526158	−0.263079	+	0.964774i	$0.584738\pi$
$824$	0	0
$825$	40524.0	1.71014
$826$	0	0
$827$	4915.14	0.206670	0.103335	−	0.994647i	$-0.467049\pi$
$827$	4915.14	0.206670	0.103335	+	0.994647i	$0.467049\pi$
$828$	0	0
$829$	−45226.4	−1.89478	−0.947392	−	0.320074i	$-0.896292\pi$
$829$	−45226.4	−1.89478	−0.947392	+	0.320074i	$0.896292\pi$
$830$	0	0
$831$	−4450.82	−0.185797
$832$	0	0
$833$	2054.92	0.0854725
$834$	0	0
$835$	7894.31	0.327178
$836$	0	0
$837$	−256.961	−0.0106115
$838$	0	0
$839$	−3982.56	−0.163877	−0.0819387	−	0.996637i	$-0.526111\pi$
$839$	−3982.56	−0.163877	−0.0819387	+	0.996637i	$0.526111\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	14377.5	0.587411
$844$	0	0
$845$	−4601.15	−0.187319
$846$	0	0
$847$	−28099.6	−1.13992
$848$	0	0
$849$	−32559.7	−1.31619
$850$	0	0
$851$	−8998.94	−0.362491
$852$	0	0
$853$	−37142.6	−1.49090	−0.745450	−	0.666562i	$-0.767765\pi$
$853$	−37142.6	−1.49090	−0.745450	+	0.666562i	$0.767765\pi$
$854$	0	0
$855$	−2419.52	−0.0967789
$856$	0	0
$857$	18690.5	0.744990	0.372495	−	0.928034i	$-0.378502\pi$
$857$	18690.5	0.744990	0.372495	+	0.928034i	$0.378502\pi$
$858$	0	0
$859$	22852.1	0.907687	0.453843	−	0.891081i	$-0.350053\pi$
$859$	22852.1	0.907687	0.453843	+	0.891081i	$0.350053\pi$
$860$	0	0
$861$	12148.0	0.480839
$862$	0	0
$863$	−41976.4	−1.65573	−0.827865	−	0.560928i	$-0.810445\pi$
$863$	−41976.4	−1.65573	−0.827865	+	0.560928i	$0.810445\pi$
$864$	0	0
$865$	−6585.77	−0.258870
$866$	0	0
$867$	−26404.1	−1.03429
$868$	0	0
$869$	−41553.0	−1.62208
$870$	0	0
$871$	3012.32	0.117186
$872$	0	0
$873$	11875.7	0.460401
$874$	0	0
$875$	10704.6	0.413581
$876$	0	0
$877$	44394.0	1.70932	0.854662	−	0.519184i	$-0.173764\pi$
$877$	44394.0	1.70932	0.854662	+	0.519184i	$0.173764\pi$
$878$	0	0
$879$	−43699.4	−1.67684
$880$	0	0
$881$	−6337.13	−0.242342	−0.121171	−	0.992632i	$-0.538665\pi$
$881$	−6337.13	−0.242342	−0.121171	+	0.992632i	$0.538665\pi$
$882$	0	0
$883$	2834.24	0.108018	0.0540090	−	0.998540i	$-0.482800\pi$
$883$	2834.24	0.108018	0.0540090	+	0.998540i	$0.482800\pi$
$884$	0	0
$885$	−6780.12	−0.257527
$886$	0	0
$887$	−76.3532	−0.00289029	−0.00144515	−	0.999999i	$-0.500460\pi$
$887$	−76.3532	−0.00289029	−0.00144515	+	0.999999i	$0.500460\pi$
$888$	0	0
$889$	12645.6	0.477077
$890$	0	0
$891$	47364.3	1.78088
$892$	0	0
$893$	24760.6	0.927863
$894$	0	0
$895$	−6386.75	−0.238531
$896$	0	0
$897$	−2719.86	−0.101241
$898$	0	0
$899$	−94.3163	−0.00349903
$900$	0	0
$901$	10783.2	0.398711
$902$	0	0
$903$	−13250.4	−0.488310
$904$	0	0
$905$	6738.95	0.247525
$906$	0	0
$907$	−18209.7	−0.666641	−0.333320	−	0.942814i	$-0.608169\pi$
$907$	−18209.7	−0.666641	−0.333320	+	0.942814i	$0.608169\pi$
$908$	0	0
$909$	−10916.5	−0.398325
$910$	0	0
$911$	−33348.8	−1.21284	−0.606419	−	0.795145i	$-0.707395\pi$
$911$	−33348.8	−1.21284	−0.606419	+	0.795145i	$0.707395\pi$
$912$	0	0
$913$	−22739.8	−0.824291
$914$	0	0
$915$	3067.71	0.110836
$916$	0	0
$917$	−30624.2	−1.10284
$918$	0	0
$919$	−20190.6	−0.724730	−0.362365	−	0.932036i	$-0.618031\pi$
$919$	−20190.6	−0.724730	−0.362365	+	0.932036i	$0.618031\pi$
$920$	0	0
$921$	−57272.4	−2.04907
$922$	0	0
$923$	6334.11	0.225883
$924$	0	0
$925$	−18142.3	−0.644882
$926$	0	0
$927$	−30018.8	−1.06359
$928$	0	0
$929$	9089.58	0.321011	0.160506	−	0.987035i	$-0.448688\pi$
$929$	9089.58	0.321011	0.160506	+	0.987035i	$0.448688\pi$
$930$	0	0
$931$	−5457.95	−0.192134
$932$	0	0
$933$	−26679.6	−0.936173
$934$	0	0
$935$	−3210.17	−0.112282
$936$	0	0
$937$	39646.5	1.38228	0.691140	−	0.722721i	$-0.257109\pi$
$937$	39646.5	1.38228	0.691140	+	0.722721i	$0.257109\pi$
$938$	0	0
$939$	−34802.5	−1.20952
$940$	0	0
$941$	−51013.4	−1.76726	−0.883629	−	0.468188i	$-0.844907\pi$
$941$	−51013.4	−1.76726	−0.883629	+	0.468188i	$0.844907\pi$
$942$	0	0
$943$	5514.14	0.190419
$944$	0	0
$945$	3446.35	0.118635
$946$	0	0
$947$	−43867.1	−1.50527	−0.752634	−	0.658439i	$-0.771217\pi$
$947$	−43867.1	−1.50527	−0.752634	+	0.658439i	$0.771217\pi$
$948$	0	0
$949$	−7661.16	−0.262057
$950$	0	0
$951$	−34769.9	−1.18559
$952$	0	0
$953$	−4167.10	−0.141643	−0.0708214	−	0.997489i	$-0.522562\pi$
$953$	−4167.10	−0.141643	−0.0708214	+	0.997489i	$0.522562\pi$
$954$	0	0
$955$	4554.44	0.154323
$956$	0	0
$957$	9760.04	0.329673
$958$	0	0
$959$	598.123	0.0201401
$960$	0	0
$961$	−29780.4	−0.999645
$962$	0	0
$963$	−23557.8	−0.788307
$964$	0	0
$965$	6610.85	0.220529
$966$	0	0
$967$	21935.0	0.729454	0.364727	−	0.931115i	$-0.381162\pi$
$967$	21935.0	0.729454	0.364727	+	0.931115i	$0.381162\pi$
$968$	0	0
$969$	−14211.9	−0.471157
$970$	0	0
$971$	24169.6	0.798805	0.399403	−	0.916776i	$-0.369218\pi$
$971$	24169.6	0.798805	0.399403	+	0.916776i	$0.369218\pi$
$972$	0	0
$973$	25703.2	0.846871
$974$	0	0
$975$	−5483.38	−0.180111
$976$	0	0
$977$	16106.2	0.527415	0.263707	−	0.964603i	$-0.415055\pi$
$977$	16106.2	0.527415	0.263707	+	0.964603i	$0.415055\pi$
$978$	0	0
$979$	−23783.8	−0.776439
$980$	0	0
$981$	−3393.57	−0.110447
$982$	0	0
$983$	22527.2	0.730933	0.365466	−	0.930825i	$-0.380910\pi$
$983$	22527.2	0.730933	0.365466	+	0.930825i	$0.380910\pi$
$984$	0	0
$985$	−5323.43	−0.172202
$986$	0	0
$987$	42631.7	1.37486
$988$	0	0
$989$	−6014.52	−0.193378
$990$	0	0
$991$	−8337.94	−0.267269	−0.133634	−	0.991031i	$-0.542665\pi$
$991$	−8337.94	−0.267269	−0.133634	+	0.991031i	$0.542665\pi$
$992$	0	0
$993$	−37650.1	−1.20321
$994$	0	0
$995$	−11762.5	−0.374771
$996$	0	0
$997$	−18827.2	−0.598058	−0.299029	−	0.954244i	$-0.596663\pi$
$997$	−18827.2	−0.598058	−0.299029	+	0.954244i	$0.596663\pi$
$998$	0	0
$999$	−11904.5	−0.377020

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	464.4.a.l.1.5		5
4.3	odd	2		29.4.a.b.1.1	✓	5
8.3	odd	2		1856.4.a.y.1.5		5
8.5	even	2		1856.4.a.bb.1.1		5
12.11	even	2		261.4.a.f.1.5		5
20.19	odd	2		725.4.a.c.1.5		5
28.27	even	2		1421.4.a.e.1.1		5
116.115	odd	2		841.4.a.b.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.4.a.b.1.1	✓	5	4.3	odd	2
261.4.a.f.1.5		5	12.11	even	2
464.4.a.l.1.5		5	1.1	even	1	trivial
725.4.a.c.1.5		5	20.19	odd	2
841.4.a.b.1.5		5	116.115	odd	2
1421.4.a.e.1.1		5	28.27	even	2
1856.4.a.y.1.5		5	8.3	odd	2
1856.4.a.bb.1.1		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)

\(f(q)\)	\(=\)	\(q+6.46343 q^{3} +2.14270 q^{5} -20.3573 q^{7} +14.7760 q^{9} +O(q^{10})\)	\(q+6.46343 q^{3} +2.14270 q^{5} -20.3573 q^{7} +14.7760 q^{9} -52.0703 q^{11} +7.04574 q^{13} +13.8492 q^{15} +28.7724 q^{17} -76.4208 q^{19} -131.578 q^{21} -59.7251 q^{23} -120.409 q^{25} -79.0092 q^{27} -29.0000 q^{29} +3.25229 q^{31} -336.553 q^{33} -43.6196 q^{35} +150.673 q^{37} +45.5397 q^{39} -92.3254 q^{41} +100.703 q^{43} +31.6605 q^{45} -324.003 q^{47} +71.4197 q^{49} +185.969 q^{51} +374.774 q^{53} -111.571 q^{55} -493.941 q^{57} -489.567 q^{59} +221.508 q^{61} -300.799 q^{63} +15.0969 q^{65} +427.538 q^{67} -386.029 q^{69} +898.999 q^{71} -1087.35 q^{73} -778.254 q^{75} +1060.01 q^{77} +798.018 q^{79} -909.622 q^{81} +436.713 q^{83} +61.6507 q^{85} -187.440 q^{87} +456.763 q^{89} -143.432 q^{91} +21.0209 q^{93} -163.747 q^{95} +803.714 q^{97} -769.390 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 8 q^{3} + 10 q^{5} - 40 q^{7} + 33 q^{9}+O(q^{10}) \) 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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