LMFDB - Embedded newform 464.4.a.d.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:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 116)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.30278$ 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)

$f(q)$	$=$	$q+3.60555 q^{3} +2.21110 q^{5} -4.42221 q^{7} -14.0000 q^{9} +O(q^{10})$	$q+3.60555 q^{3} +2.21110 q^{5} -4.42221 q^{7} -14.0000 q^{9} +12.3944 q^{11} -48.6333 q^{13} +7.97224 q^{15} -101.322 q^{17} +59.2666 q^{19} -15.9445 q^{21} -18.0555 q^{23} -120.111 q^{25} -147.828 q^{27} -29.0000 q^{29} -20.8167 q^{31} +44.6888 q^{33} -9.77795 q^{35} +101.066 q^{37} -175.350 q^{39} +40.7889 q^{41} -152.450 q^{43} -30.9554 q^{45} -121.328 q^{47} -323.444 q^{49} -365.322 q^{51} +177.700 q^{53} +27.4054 q^{55} +213.689 q^{57} -109.611 q^{59} +61.7662 q^{61} +61.9109 q^{63} -107.533 q^{65} +471.532 q^{67} -65.1001 q^{69} -546.522 q^{71} +169.733 q^{73} -433.066 q^{75} -54.8108 q^{77} -184.916 q^{79} -155.000 q^{81} -210.500 q^{83} -224.034 q^{85} -104.561 q^{87} -1393.32 q^{89} +215.066 q^{91} -75.0555 q^{93} +131.045 q^{95} -1099.43 q^{97} -173.522 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 10 q^{5} + 20 q^{7} - 28 q^{9}+O(q^{10}) $ 2 * q - 10 * q^5 + 20 * q^7 - 28 * q^9	$ 2 q - 10 q^{5} + 20 q^{7} - 28 q^{9} + 32 q^{11} - 54 q^{13} + 52 q^{15} - 44 q^{17} + 32 q^{19} - 104 q^{21} + 36 q^{23} - 96 q^{25} - 58 q^{29} - 20 q^{31} - 26 q^{33} - 308 q^{35} - 144 q^{37} - 156 q^{39} + 96 q^{41} - 240 q^{43} + 140 q^{45} - 596 q^{47} - 70 q^{49} - 572 q^{51} - 34 q^{53} - 212 q^{55} + 312 q^{57} - 724 q^{59} - 612 q^{61} - 280 q^{63} - 42 q^{65} - 528 q^{67} - 260 q^{69} + 104 q^{71} - 872 q^{73} - 520 q^{75} + 424 q^{77} + 820 q^{79} - 310 q^{81} + 228 q^{83} - 924 q^{85} - 32 q^{89} + 84 q^{91} - 78 q^{93} + 464 q^{95} - 1896 q^{97} - 448 q^{99}+O(q^{100}) $ 2 * q - 10 * q^5 + 20 * q^7 - 28 * q^9 + 32 * q^11 - 54 * q^13 + 52 * q^15 - 44 * q^17 + 32 * q^19 - 104 * q^21 + 36 * q^23 - 96 * q^25 - 58 * q^29 - 20 * q^31 - 26 * q^33 - 308 * q^35 - 144 * q^37 - 156 * q^39 + 96 * q^41 - 240 * q^43 + 140 * q^45 - 596 * q^47 - 70 * q^49 - 572 * q^51 - 34 * q^53 - 212 * q^55 + 312 * q^57 - 724 * q^59 - 612 * q^61 - 280 * q^63 - 42 * q^65 - 528 * q^67 - 260 * q^69 + 104 * q^71 - 872 * q^73 - 520 * q^75 + 424 * q^77 + 820 * q^79 - 310 * q^81 + 228 * q^83 - 924 * q^85 - 32 * q^89 + 84 * q^91 - 78 * q^93 + 464 * q^95 - 1896 * q^97 - 448 * 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
$3$	3.60555	0.693889	0.346944	−	0.937886i	$-0.387219\pi$
$3$	3.60555	0.693889	0.346944	+	0.937886i	$0.387219\pi$
$4$	0	0
$5$	2.21110	0.197767	0.0988835	−	0.995099i	$-0.468473\pi$
$5$	2.21110	0.197767	0.0988835	+	0.995099i	$0.468473\pi$
$6$	0	0
$7$	−4.42221	−0.238777	−0.119388	−	0.992848i	$-0.538093\pi$
$7$	−4.42221	−0.238777	−0.119388	+	0.992848i	$0.538093\pi$
$8$	0	0
$9$	−14.0000	−0.518519
$10$	0	0
$11$	12.3944	0.339733	0.169867	−	0.985467i	$-0.445666\pi$
$11$	12.3944	0.339733	0.169867	+	0.985467i	$0.445666\pi$
$12$	0	0
$13$	−48.6333	−1.03757	−0.518787	−	0.854904i	$-0.673616\pi$
$13$	−48.6333	−1.03757	−0.518787	+	0.854904i	$0.673616\pi$
$14$	0	0
$15$	7.97224	0.137228
$16$	0	0
$17$	−101.322	−1.44554	−0.722771	−	0.691087i	$-0.757132\pi$
$17$	−101.322	−1.44554	−0.722771	+	0.691087i	$0.757132\pi$
$18$	0	0
$19$	59.2666	0.715615	0.357808	−	0.933795i	$-0.383524\pi$
$19$	59.2666	0.715615	0.357808	+	0.933795i	$0.383524\pi$
$20$	0	0
$21$	−15.9445	−0.165684
$22$	0	0
$23$	−18.0555	−0.163688	−0.0818442	−	0.996645i	$-0.526081\pi$
$23$	−18.0555	−0.163688	−0.0818442	+	0.996645i	$0.526081\pi$
$24$	0	0
$25$	−120.111	−0.960888
$26$	0	0
$27$	−147.828	−1.05368
$28$	0	0
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	0	0
$31$	−20.8167	−0.120606	−0.0603029	−	0.998180i	$-0.519207\pi$
$31$	−20.8167	−0.120606	−0.0603029	+	0.998180i	$0.519207\pi$
$32$	0	0
$33$	44.6888	0.235737
$34$	0	0
$35$	−9.77795	−0.0472221
$36$	0	0
$37$	101.066	0.449060	0.224530	−	0.974467i	$-0.427915\pi$
$37$	101.066	0.449060	0.224530	+	0.974467i	$0.427915\pi$
$38$	0	0
$39$	−175.350	−0.719960
$40$	0	0
$41$	40.7889	0.155370	0.0776848	−	0.996978i	$-0.475247\pi$
$41$	40.7889	0.155370	0.0776848	+	0.996978i	$0.475247\pi$
$42$	0	0
$43$	−152.450	−0.540660	−0.270330	−	0.962768i	$-0.587133\pi$
$43$	−152.450	−0.540660	−0.270330	+	0.962768i	$0.587133\pi$
$44$	0	0
$45$	−30.9554	−0.102546
$46$	0	0
$47$	−121.328	−0.376543	−0.188271	−	0.982117i	$-0.560288\pi$
$47$	−121.328	−0.376543	−0.188271	+	0.982117i	$0.560288\pi$
$48$	0	0
$49$	−323.444	−0.942986
$50$	0	0
$51$	−365.322	−1.00305
$52$	0	0
$53$	177.700	0.460546	0.230273	−	0.973126i	$-0.426038\pi$
$53$	177.700	0.460546	0.230273	+	0.973126i	$0.426038\pi$
$54$	0	0
$55$	27.4054	0.0671881
$56$	0	0
$57$	213.689	0.496557
$58$	0	0
$59$	−109.611	−0.241868	−0.120934	−	0.992661i	$-0.538589\pi$
$59$	−109.611	−0.241868	−0.120934	+	0.992661i	$0.538589\pi$
$60$	0	0
$61$	61.7662	0.129645	0.0648226	−	0.997897i	$-0.479352\pi$
$61$	61.7662	0.129645	0.0648226	+	0.997897i	$0.479352\pi$
$62$	0	0
$63$	61.9109	0.123810
$64$	0	0
$65$	−107.533	−0.205198
$66$	0	0
$67$	471.532	0.859804	0.429902	−	0.902876i	$-0.358548\pi$
$67$	471.532	0.859804	0.429902	+	0.902876i	$0.358548\pi$
$68$	0	0
$69$	−65.1001	−0.113582
$70$	0	0
$71$	−546.522	−0.913524	−0.456762	−	0.889589i	$-0.650991\pi$
$71$	−546.522	−0.913524	−0.456762	+	0.889589i	$0.650991\pi$
$72$	0	0
$73$	169.733	0.272133	0.136066	−	0.990700i	$-0.456554\pi$
$73$	169.733	0.272133	0.136066	+	0.990700i	$0.456554\pi$
$74$	0	0
$75$	−433.066	−0.666749
$76$	0	0
$77$	−54.8108	−0.0811204
$78$	0	0
$79$	−184.916	−0.263350	−0.131675	−	0.991293i	$-0.542036\pi$
$79$	−184.916	−0.263350	−0.131675	+	0.991293i	$0.542036\pi$
$80$	0	0
$81$	−155.000	−0.212620
$82$	0	0
$83$	−210.500	−0.278378	−0.139189	−	0.990266i	$-0.544449\pi$
$83$	−210.500	−0.278378	−0.139189	+	0.990266i	$0.544449\pi$
$84$	0	0
$85$	−224.034	−0.285881
$86$	0	0
$87$	−104.561	−0.128852
$88$	0	0
$89$	−1393.32	−1.65946	−0.829729	−	0.558167i	$-0.811505\pi$
$89$	−1393.32	−1.65946	−0.829729	+	0.558167i	$0.811505\pi$
$90$	0	0
$91$	215.066	0.247748
$92$	0	0
$93$	−75.0555	−0.0836870
$94$	0	0
$95$	131.045	0.141525
$96$	0	0
$97$	−1099.43	−1.15083	−0.575415	−	0.817862i	$-0.695159\pi$
$97$	−1099.43	−1.15083	−0.575415	+	0.817862i	$0.695159\pi$
$98$	0	0
$99$	−173.522	−0.176158
$100$	0	0
$101$	322.799	0.318017	0.159008	−	0.987277i	$-0.449170\pi$
$101$	322.799	0.318017	0.159008	+	0.987277i	$0.449170\pi$
$102$	0	0
$103$	1741.61	1.66608	0.833038	−	0.553215i	$-0.186599\pi$
$103$	1741.61	1.66608	0.833038	+	0.553215i	$0.186599\pi$
$104$	0	0
$105$	−35.2549	−0.0327669
$106$	0	0
$107$	−560.244	−0.506176	−0.253088	−	0.967443i	$-0.581446\pi$
$107$	−560.244	−0.506176	−0.253088	+	0.967443i	$0.581446\pi$
$108$	0	0
$109$	−987.700	−0.867931	−0.433966	−	0.900929i	$-0.642886\pi$
$109$	−987.700	−0.867931	−0.433966	+	0.900929i	$0.642886\pi$
$110$	0	0
$111$	364.400	0.311598
$112$	0	0
$113$	493.532	0.410864	0.205432	−	0.978671i	$-0.434140\pi$
$113$	493.532	0.410864	0.205432	+	0.978671i	$0.434140\pi$
$114$	0	0
$115$	−39.9226	−0.0323722
$116$	0	0
$117$	680.866	0.538001
$118$	0	0
$119$	448.067	0.345162
$120$	0	0
$121$	−1177.38	−0.884581
$122$	0	0
$123$	147.066	0.107809
$124$	0	0
$125$	−541.966	−0.387799
$126$	0	0
$127$	663.822	0.463816	0.231908	−	0.972738i	$-0.425503\pi$
$127$	663.822	0.463816	0.231908	+	0.972738i	$0.425503\pi$
$128$	0	0
$129$	−549.666	−0.375158
$130$	0	0
$131$	−1456.67	−0.971523	−0.485761	−	0.874091i	$-0.661458\pi$
$131$	−1456.67	−0.971523	−0.485761	+	0.874091i	$0.661458\pi$
$132$	0	0
$133$	−262.089	−0.170872
$134$	0	0
$135$	−326.862	−0.208384
$136$	0	0
$137$	2670.15	1.66516	0.832579	−	0.553907i	$-0.186864\pi$
$137$	2670.15	1.66516	0.832579	+	0.553907i	$0.186864\pi$
$138$	0	0
$139$	−958.476	−0.584870	−0.292435	−	0.956285i	$-0.594465\pi$
$139$	−958.476	−0.584870	−0.292435	+	0.956285i	$0.594465\pi$
$140$	0	0
$141$	−437.454	−0.261279
$142$	0	0
$143$	−602.783	−0.352498
$144$	0	0
$145$	−64.1220	−0.0367244
$146$	0	0
$147$	−1166.19	−0.654327
$148$	0	0
$149$	490.435	0.269651	0.134825	−	0.990869i	$-0.456953\pi$
$149$	490.435	0.269651	0.134825	+	0.990869i	$0.456953\pi$
$150$	0	0
$151$	1999.24	1.07746	0.538728	−	0.842480i	$-0.318905\pi$
$151$	1999.24	1.07746	0.538728	+	0.842480i	$0.318905\pi$
$152$	0	0
$153$	1418.51	0.749541
$154$	0	0
$155$	−46.0278	−0.0238519
$156$	0	0
$157$	1830.68	0.930598	0.465299	−	0.885154i	$-0.345947\pi$
$157$	1830.68	0.930598	0.465299	+	0.885154i	$0.345947\pi$
$158$	0	0
$159$	640.706	0.319568
$160$	0	0
$161$	79.8452	0.0390850
$162$	0	0
$163$	−2884.29	−1.38598	−0.692991	−	0.720946i	$-0.743708\pi$
$163$	−2884.29	−1.38598	−0.692991	+	0.720946i	$0.743708\pi$
$164$	0	0
$165$	98.8116	0.0466210
$166$	0	0
$167$	−1658.79	−0.768628	−0.384314	−	0.923203i	$-0.625562\pi$
$167$	−1658.79	−0.768628	−0.384314	+	0.923203i	$0.625562\pi$
$168$	0	0
$169$	168.199	0.0765583
$170$	0	0
$171$	−829.733	−0.371060
$172$	0	0
$173$	2925.53	1.28569	0.642844	−	0.765997i	$-0.277754\pi$
$173$	2925.53	1.28569	0.642844	+	0.765997i	$0.277754\pi$
$174$	0	0
$175$	531.156	0.229438
$176$	0	0
$177$	−395.210	−0.167829
$178$	0	0
$179$	4190.23	1.74968	0.874839	−	0.484413i	$-0.160967\pi$
$179$	4190.23	1.74968	0.874839	+	0.484413i	$0.160967\pi$
$180$	0	0
$181$	−321.969	−0.132220	−0.0661098	−	0.997812i	$-0.521059\pi$
$181$	−321.969	−0.132220	−0.0661098	+	0.997812i	$0.521059\pi$
$182$	0	0
$183$	222.701	0.0899593
$184$	0	0
$185$	223.468	0.0888093
$186$	0	0
$187$	−1255.83	−0.491099
$188$	0	0
$189$	653.724	0.251595
$190$	0	0
$191$	914.466	0.346432	0.173216	−	0.984884i	$-0.444584\pi$
$191$	914.466	0.346432	0.173216	+	0.984884i	$0.444584\pi$
$192$	0	0
$193$	−1055.21	−0.393552	−0.196776	−	0.980448i	$-0.563047\pi$
$193$	−1055.21	−0.393552	−0.196776	+	0.980448i	$0.563047\pi$
$194$	0	0
$195$	−387.717	−0.142384
$196$	0	0
$197$	549.869	0.198866	0.0994328	−	0.995044i	$-0.468297\pi$
$197$	549.869	0.198866	0.0994328	+	0.995044i	$0.468297\pi$
$198$	0	0
$199$	4225.60	1.50525	0.752625	−	0.658449i	$-0.228787\pi$
$199$	4225.60	1.50525	0.752625	+	0.658449i	$0.228787\pi$
$200$	0	0
$201$	1700.13	0.596608
$202$	0	0
$203$	128.244	0.0443397
$204$	0	0
$205$	90.1884	0.0307270
$206$	0	0
$207$	252.777	0.0848755
$208$	0	0
$209$	734.577	0.243118
$210$	0	0
$211$	692.728	0.226016	0.113008	−	0.993594i	$-0.463951\pi$
$211$	692.728	0.226016	0.113008	+	0.993594i	$0.463951\pi$
$212$	0	0
$213$	−1970.51	−0.633884
$214$	0	0
$215$	−337.082	−0.106925
$216$	0	0
$217$	92.0555	0.0287979
$218$	0	0
$219$	611.980	0.188830
$220$	0	0
$221$	4927.63	1.49986
$222$	0	0
$223$	3864.52	1.16048	0.580241	−	0.814445i	$-0.302958\pi$
$223$	3864.52	1.16048	0.580241	+	0.814445i	$0.302958\pi$
$224$	0	0
$225$	1681.55	0.498238
$226$	0	0
$227$	3410.81	0.997283	0.498642	−	0.866808i	$-0.333832\pi$
$227$	3410.81	0.997283	0.498642	+	0.866808i	$0.333832\pi$
$228$	0	0
$229$	−2695.50	−0.777833	−0.388916	−	0.921273i	$-0.627150\pi$
$229$	−2695.50	−0.777833	−0.388916	+	0.921273i	$0.627150\pi$
$230$	0	0
$231$	−197.623	−0.0562885
$232$	0	0
$233$	4829.82	1.35799	0.678995	−	0.734143i	$-0.262416\pi$
$233$	4829.82	1.35799	0.678995	+	0.734143i	$0.262416\pi$
$234$	0	0
$235$	−268.269	−0.0744677
$236$	0	0
$237$	−666.724	−0.182736
$238$	0	0
$239$	4869.12	1.31781	0.658906	−	0.752225i	$-0.271019\pi$
$239$	4869.12	1.31781	0.658906	+	0.752225i	$0.271019\pi$
$240$	0	0
$241$	−3096.69	−0.827698	−0.413849	−	0.910345i	$-0.635816\pi$
$241$	−3096.69	−0.827698	−0.413849	+	0.910345i	$0.635816\pi$
$242$	0	0
$243$	3432.48	0.906148
$244$	0	0
$245$	−715.168	−0.186491
$246$	0	0
$247$	−2882.33	−0.742503
$248$	0	0
$249$	−758.967	−0.193163
$250$	0	0
$251$	−2208.84	−0.555461	−0.277730	−	0.960659i	$-0.589582\pi$
$251$	−2208.84	−0.555461	−0.277730	+	0.960659i	$0.589582\pi$
$252$	0	0
$253$	−223.788	−0.0556104
$254$	0	0
$255$	−807.765	−0.198369
$256$	0	0
$257$	526.955	0.127901	0.0639504	−	0.997953i	$-0.479630\pi$
$257$	526.955	0.127901	0.0639504	+	0.997953i	$0.479630\pi$
$258$	0	0
$259$	−446.937	−0.107225
$260$	0	0
$261$	406.000	0.0962865
$262$	0	0
$263$	4892.69	1.14713	0.573567	−	0.819158i	$-0.305559\pi$
$263$	4892.69	1.14713	0.573567	+	0.819158i	$0.305559\pi$
$264$	0	0
$265$	392.912	0.0910808
$266$	0	0
$267$	−5023.69	−1.15148
$268$	0	0
$269$	−4744.74	−1.07543	−0.537717	−	0.843125i	$-0.680713\pi$
$269$	−4744.74	−1.07543	−0.537717	+	0.843125i	$0.680713\pi$
$270$	0	0
$271$	8187.57	1.83527	0.917637	−	0.397420i	$-0.130094\pi$
$271$	8187.57	1.83527	0.917637	+	0.397420i	$0.130094\pi$
$272$	0	0
$273$	775.433	0.171910
$274$	0	0
$275$	−1488.71	−0.326446
$276$	0	0
$277$	−4966.39	−1.07726	−0.538631	−	0.842542i	$-0.681058\pi$
$277$	−4966.39	−1.07726	−0.538631	+	0.842542i	$0.681058\pi$
$278$	0	0
$279$	291.433	0.0625364
$280$	0	0
$281$	81.1971	0.0172378	0.00861888	−	0.999963i	$-0.497256\pi$
$281$	81.1971	0.0172378	0.00861888	+	0.999963i	$0.497256\pi$
$282$	0	0
$283$	−213.170	−0.0447762	−0.0223881	−	0.999749i	$-0.507127\pi$
$283$	−213.170	−0.0447762	−0.0223881	+	0.999749i	$0.507127\pi$
$284$	0	0
$285$	472.488	0.0982027
$286$	0	0
$287$	−180.377	−0.0370986
$288$	0	0
$289$	5353.17	1.08959
$290$	0	0
$291$	−3964.06	−0.798548
$292$	0	0
$293$	−6346.40	−1.26539	−0.632697	−	0.774399i	$-0.718052\pi$
$293$	−6346.40	−1.26539	−0.632697	+	0.774399i	$0.718052\pi$
$294$	0	0
$295$	−242.362	−0.0478334
$296$	0	0
$297$	−1832.24	−0.357971
$298$	0	0
$299$	878.099	0.169839
$300$	0	0
$301$	674.165	0.129097
$302$	0	0
$303$	1163.87	0.220668
$304$	0	0
$305$	136.571	0.0256395
$306$	0	0
$307$	−5636.38	−1.04784	−0.523918	−	0.851769i	$-0.675530\pi$
$307$	−5636.38	−1.04784	−0.523918	+	0.851769i	$0.675530\pi$
$308$	0	0
$309$	6279.46	1.15607
$310$	0	0
$311$	586.998	0.107028	0.0535138	−	0.998567i	$-0.482958\pi$
$311$	586.998	0.107028	0.0535138	+	0.998567i	$0.482958\pi$
$312$	0	0
$313$	−1249.95	−0.225723	−0.112862	−	0.993611i	$-0.536002\pi$
$313$	−1249.95	−0.225723	−0.112862	+	0.993611i	$0.536002\pi$
$314$	0	0
$315$	136.891	0.0244856
$316$	0	0
$317$	4821.77	0.854314	0.427157	−	0.904177i	$-0.359515\pi$
$317$	4821.77	0.854314	0.427157	+	0.904177i	$0.359515\pi$
$318$	0	0
$319$	−359.439	−0.0630869
$320$	0	0
$321$	−2019.99	−0.351230
$322$	0	0
$323$	−6005.02	−1.03445
$324$	0	0
$325$	5841.40	0.996992
$326$	0	0
$327$	−3561.20	−0.602247
$328$	0	0
$329$	536.537	0.0899096
$330$	0	0
$331$	−8287.49	−1.37620	−0.688100	−	0.725616i	$-0.741555\pi$
$331$	−8287.49	−1.37620	−0.688100	+	0.725616i	$0.741555\pi$
$332$	0	0
$333$	−1414.93	−0.232846
$334$	0	0
$335$	1042.61	0.170041
$336$	0	0
$337$	9910.59	1.60197	0.800985	−	0.598684i	$-0.204309\pi$
$337$	9910.59	1.60197	0.800985	+	0.598684i	$0.204309\pi$
$338$	0	0
$339$	1779.46	0.285094
$340$	0	0
$341$	−258.011	−0.0409738
$342$	0	0
$343$	2947.15	0.463940
$344$	0	0
$345$	−143.943	−0.0224627
$346$	0	0
$347$	−9826.54	−1.52022	−0.760110	−	0.649794i	$-0.774855\pi$
$347$	−9826.54	−1.52022	−0.760110	+	0.649794i	$0.774855\pi$
$348$	0	0
$349$	−7438.96	−1.14097	−0.570485	−	0.821308i	$-0.693245\pi$
$349$	−7438.96	−1.14097	−0.570485	+	0.821308i	$0.693245\pi$
$350$	0	0
$351$	7189.35	1.09327
$352$	0	0
$353$	−9385.04	−1.41506	−0.707529	−	0.706684i	$-0.750190\pi$
$353$	−9385.04	−1.41506	−0.707529	+	0.706684i	$0.750190\pi$
$354$	0	0
$355$	−1208.42	−0.180665
$356$	0	0
$357$	1615.53	0.239504
$358$	0	0
$359$	−6673.24	−0.981059	−0.490529	−	0.871425i	$-0.663197\pi$
$359$	−6673.24	−0.981059	−0.490529	+	0.871425i	$0.663197\pi$
$360$	0	0
$361$	−3346.47	−0.487894
$362$	0	0
$363$	−4245.10	−0.613801
$364$	0	0
$365$	375.296	0.0538189
$366$	0	0
$367$	−12384.4	−1.76147	−0.880733	−	0.473613i	$-0.842949\pi$
$367$	−12384.4	−1.76147	−0.880733	+	0.473613i	$0.842949\pi$
$368$	0	0
$369$	−571.045	−0.0805620
$370$	0	0
$371$	−785.825	−0.109968
$372$	0	0
$373$	−5167.77	−0.717364	−0.358682	−	0.933460i	$-0.616774\pi$
$373$	−5167.77	−0.717364	−0.358682	+	0.933460i	$0.616774\pi$
$374$	0	0
$375$	−1954.08	−0.269089
$376$	0	0
$377$	1410.37	0.192673
$378$	0	0
$379$	−7398.92	−1.00279	−0.501395	−	0.865219i	$-0.667180\pi$
$379$	−7398.92	−1.00279	−0.501395	+	0.865219i	$0.667180\pi$
$380$	0	0
$381$	2393.44	0.321837
$382$	0	0
$383$	3262.55	0.435270	0.217635	−	0.976030i	$-0.430166\pi$
$383$	3262.55	0.435270	0.217635	+	0.976030i	$0.430166\pi$
$384$	0	0
$385$	−121.192	−0.0160429
$386$	0	0
$387$	2134.30	0.280342
$388$	0	0
$389$	−5819.82	−0.758552	−0.379276	−	0.925284i	$-0.623827\pi$
$389$	−5819.82	−0.758552	−0.379276	+	0.925284i	$0.623827\pi$
$390$	0	0
$391$	1829.42	0.236619
$392$	0	0
$393$	−5252.08	−0.674129
$394$	0	0
$395$	−408.868	−0.0520820
$396$	0	0
$397$	−7678.61	−0.970726	−0.485363	−	0.874313i	$-0.661313\pi$
$397$	−7678.61	−0.970726	−0.485363	+	0.874313i	$0.661313\pi$
$398$	0	0
$399$	−944.976	−0.118566
$400$	0	0
$401$	−1125.32	−0.140139	−0.0700693	−	0.997542i	$-0.522322\pi$
$401$	−1125.32	−0.140139	−0.0700693	+	0.997542i	$0.522322\pi$
$402$	0	0
$403$	1012.38	0.125137
$404$	0	0
$405$	−342.721	−0.0420492
$406$	0	0
$407$	1252.66	0.152561
$408$	0	0
$409$	−11937.5	−1.44321	−0.721605	−	0.692305i	$-0.756595\pi$
$409$	−11937.5	−1.44321	−0.721605	+	0.692305i	$0.756595\pi$
$410$	0	0
$411$	9627.37	1.15543
$412$	0	0
$413$	484.724	0.0577523
$414$	0	0
$415$	−465.436	−0.0550539
$416$	0	0
$417$	−3455.84	−0.405834
$418$	0	0
$419$	16082.3	1.87511	0.937554	−	0.347840i	$-0.113085\pi$
$419$	16082.3	1.87511	0.937554	+	0.347840i	$0.113085\pi$
$420$	0	0
$421$	6896.09	0.798326	0.399163	−	0.916880i	$-0.369301\pi$
$421$	6896.09	0.798326	0.399163	+	0.916880i	$0.369301\pi$
$422$	0	0
$423$	1698.59	0.195244
$424$	0	0
$425$	12169.9	1.38900
$426$	0	0
$427$	−273.143	−0.0309562
$428$	0	0
$429$	−2173.37	−0.244595
$430$	0	0
$431$	9632.03	1.07647	0.538235	−	0.842795i	$-0.319091\pi$
$431$	9632.03	1.07647	0.538235	+	0.842795i	$0.319091\pi$
$432$	0	0
$433$	6657.14	0.738850	0.369425	−	0.929261i	$-0.379555\pi$
$433$	6657.14	0.738850	0.369425	+	0.929261i	$0.379555\pi$
$434$	0	0
$435$	−231.195	−0.0254827
$436$	0	0
$437$	−1070.09	−0.117138
$438$	0	0
$439$	−16147.8	−1.75556	−0.877781	−	0.479061i	$-0.840977\pi$
$439$	−16147.8	−1.75556	−0.877781	+	0.479061i	$0.840977\pi$
$440$	0	0
$441$	4528.22	0.488956
$442$	0	0
$443$	6307.18	0.676440	0.338220	−	0.941067i	$-0.390175\pi$
$443$	6307.18	0.676440	0.338220	+	0.941067i	$0.390175\pi$
$444$	0	0
$445$	−3080.77	−0.328186
$446$	0	0
$447$	1768.29	0.187108
$448$	0	0
$449$	−11001.7	−1.15636	−0.578178	−	0.815911i	$-0.696236\pi$
$449$	−11001.7	−1.15636	−0.578178	+	0.815911i	$0.696236\pi$
$450$	0	0
$451$	505.556	0.0527843
$452$	0	0
$453$	7208.37	0.747635
$454$	0	0
$455$	475.534	0.0489964
$456$	0	0
$457$	1258.34	0.128802	0.0644011	−	0.997924i	$-0.479486\pi$
$457$	1258.34	0.128802	0.0644011	+	0.997924i	$0.479486\pi$
$458$	0	0
$459$	14978.2	1.52314
$460$	0	0
$461$	−7461.56	−0.753838	−0.376919	−	0.926246i	$-0.623016\pi$
$461$	−7461.56	−0.753838	−0.376919	+	0.926246i	$0.623016\pi$
$462$	0	0
$463$	−7744.02	−0.777311	−0.388655	−	0.921383i	$-0.627060\pi$
$463$	−7744.02	−0.777311	−0.388655	+	0.921383i	$0.627060\pi$
$464$	0	0
$465$	−165.955	−0.0165505
$466$	0	0
$467$	244.138	0.0241913	0.0120957	−	0.999927i	$-0.496150\pi$
$467$	244.138	0.0241913	0.0120957	+	0.999927i	$0.496150\pi$
$468$	0	0
$469$	−2085.21	−0.205301
$470$	0	0
$471$	6600.60	0.645731
$472$	0	0
$473$	−1889.53	−0.183680
$474$	0	0
$475$	−7118.57	−0.687626
$476$	0	0
$477$	−2487.80	−0.238802
$478$	0	0
$479$	4381.25	0.417922	0.208961	−	0.977924i	$-0.432992\pi$
$479$	4381.25	0.417922	0.208961	+	0.977924i	$0.432992\pi$
$480$	0	0
$481$	−4915.20	−0.465933
$482$	0	0
$483$	287.886	0.0271206
$484$	0	0
$485$	−2430.96	−0.227596
$486$	0	0
$487$	−15726.4	−1.46331	−0.731654	−	0.681676i	$-0.761251\pi$
$487$	−15726.4	−1.46331	−0.731654	+	0.681676i	$0.761251\pi$
$488$	0	0
$489$	−10399.5	−0.961718
$490$	0	0
$491$	−17159.0	−1.57714	−0.788568	−	0.614947i	$-0.789177\pi$
$491$	−17159.0	−1.57714	−0.788568	+	0.614947i	$0.789177\pi$
$492$	0	0
$493$	2938.34	0.268431
$494$	0	0
$495$	−383.676	−0.0348383
$496$	0	0
$497$	2416.83	0.218128
$498$	0	0
$499$	−21508.3	−1.92955	−0.964774	−	0.263081i	$-0.915261\pi$
$499$	−21508.3	−1.92955	−0.964774	+	0.263081i	$0.915261\pi$
$500$	0	0
$501$	−5980.84	−0.533342
$502$	0	0
$503$	22195.2	1.96746	0.983731	−	0.179648i	$-0.0574958\pi$
$503$	22195.2	1.96746	0.983731	+	0.179648i	$0.0574958\pi$
$504$	0	0
$505$	713.742	0.0628933
$506$	0	0
$507$	606.449	0.0531229
$508$	0	0
$509$	11767.9	1.02476	0.512381	−	0.858758i	$-0.328763\pi$
$509$	11767.9	1.02476	0.512381	+	0.858758i	$0.328763\pi$
$510$	0	0
$511$	−750.592	−0.0649790
$512$	0	0
$513$	−8761.24	−0.754032
$514$	0	0
$515$	3850.88	0.329495
$516$	0	0
$517$	−1503.79	−0.127924
$518$	0	0
$519$	10548.2	0.892125
$520$	0	0
$521$	8418.16	0.707881	0.353941	−	0.935268i	$-0.384842\pi$
$521$	8418.16	0.707881	0.353941	+	0.935268i	$0.384842\pi$
$522$	0	0
$523$	−3414.16	−0.285451	−0.142726	−	0.989762i	$-0.545587\pi$
$523$	−3414.16	−0.285451	−0.142726	+	0.989762i	$0.545587\pi$
$524$	0	0
$525$	1915.11	0.159204
$526$	0	0
$527$	2109.19	0.174341
$528$	0	0
$529$	−11841.0	−0.973206
$530$	0	0
$531$	1534.56	0.125413
$532$	0	0
$533$	−1983.70	−0.161207
$534$	0	0
$535$	−1238.76	−0.100105
$536$	0	0
$537$	15108.1	1.21408
$538$	0	0
$539$	−4008.91	−0.320364
$540$	0	0
$541$	15393.9	1.22335	0.611677	−	0.791108i	$-0.290495\pi$
$541$	15393.9	1.22335	0.611677	+	0.791108i	$0.290495\pi$
$542$	0	0
$543$	−1160.87	−0.0917457
$544$	0	0
$545$	−2183.91	−0.171648
$546$	0	0
$547$	−9547.99	−0.746330	−0.373165	−	0.927765i	$-0.621727\pi$
$547$	−9547.99	−0.746330	−0.373165	+	0.927765i	$0.621727\pi$
$548$	0	0
$549$	−864.727	−0.0672234
$550$	0	0
$551$	−1718.73	−0.132886
$552$	0	0
$553$	817.736	0.0628819
$554$	0	0
$555$	805.726	0.0616238
$556$	0	0
$557$	19667.1	1.49609	0.748044	−	0.663649i	$-0.230993\pi$
$557$	19667.1	1.49609	0.748044	+	0.663649i	$0.230993\pi$
$558$	0	0
$559$	7414.15	0.560975
$560$	0	0
$561$	−4527.97	−0.340768
$562$	0	0
$563$	−13151.8	−0.984517	−0.492258	−	0.870449i	$-0.663828\pi$
$563$	−13151.8	−0.984517	−0.492258	+	0.870449i	$0.663828\pi$
$564$	0	0
$565$	1091.25	0.0812553
$566$	0	0
$567$	685.442	0.0507687
$568$	0	0
$569$	17071.3	1.25776	0.628880	−	0.777502i	$-0.283514\pi$
$569$	17071.3	1.25776	0.628880	+	0.777502i	$0.283514\pi$
$570$	0	0
$571$	10724.9	0.786033	0.393017	−	0.919531i	$-0.371432\pi$
$571$	10724.9	0.786033	0.393017	+	0.919531i	$0.371432\pi$
$572$	0	0
$573$	3297.15	0.240385
$574$	0	0
$575$	2168.67	0.157286
$576$	0	0
$577$	−10700.0	−0.772008	−0.386004	−	0.922497i	$-0.626145\pi$
$577$	−10700.0	−0.772008	−0.386004	+	0.922497i	$0.626145\pi$
$578$	0	0
$579$	−3804.61	−0.273081
$580$	0	0
$581$	930.872	0.0664700
$582$	0	0
$583$	2202.49	0.156463
$584$	0	0
$585$	1505.47	0.106399
$586$	0	0
$587$	10143.6	0.713237	0.356618	−	0.934250i	$-0.383930\pi$
$587$	10143.6	0.713237	0.356618	+	0.934250i	$0.383930\pi$
$588$	0	0
$589$	−1233.73	−0.0863074
$590$	0	0
$591$	1982.58	0.137991
$592$	0	0
$593$	−4547.22	−0.314894	−0.157447	−	0.987527i	$-0.550326\pi$
$593$	−4547.22	−0.314894	−0.157447	+	0.987527i	$0.550326\pi$
$594$	0	0
$595$	990.723	0.0682616
$596$	0	0
$597$	15235.6	1.04448
$598$	0	0
$599$	−26519.1	−1.80892	−0.904458	−	0.426563i	$-0.859724\pi$
$599$	−26519.1	−1.80892	−0.904458	+	0.426563i	$0.859724\pi$
$600$	0	0
$601$	4927.20	0.334417	0.167209	−	0.985922i	$-0.446525\pi$
$601$	4927.20	0.334417	0.167209	+	0.985922i	$0.446525\pi$
$602$	0	0
$603$	−6601.45	−0.445824
$604$	0	0
$605$	−2603.30	−0.174941
$606$	0	0
$607$	−16406.6	−1.09707	−0.548535	−	0.836127i	$-0.684814\pi$
$607$	−16406.6	−1.09707	−0.548535	+	0.836127i	$0.684814\pi$
$608$	0	0
$609$	462.390	0.0307668
$610$	0	0
$611$	5900.58	0.390691
$612$	0	0
$613$	−4914.16	−0.323786	−0.161893	−	0.986808i	$-0.551760\pi$
$613$	−4914.16	−0.323786	−0.161893	+	0.986808i	$0.551760\pi$
$614$	0	0
$615$	325.179	0.0213211
$616$	0	0
$617$	−186.272	−0.0121540	−0.00607702	−	0.999982i	$-0.501934\pi$
$617$	−186.272	−0.0121540	−0.00607702	+	0.999982i	$0.501934\pi$
$618$	0	0
$619$	−408.125	−0.0265007	−0.0132503	−	0.999912i	$-0.504218\pi$
$619$	−408.125	−0.0265007	−0.0132503	+	0.999912i	$0.504218\pi$
$620$	0	0
$621$	2669.10	0.172476
$622$	0	0
$623$	6161.55	0.396240
$624$	0	0
$625$	13815.5	0.884194
$626$	0	0
$627$	2648.56	0.168697
$628$	0	0
$629$	−10240.3	−0.649136
$630$	0	0
$631$	−16001.4	−1.00952	−0.504760	−	0.863260i	$-0.668419\pi$
$631$	−16001.4	−1.00952	−0.504760	+	0.863260i	$0.668419\pi$
$632$	0	0
$633$	2497.66	0.156830
$634$	0	0
$635$	1467.78	0.0917275
$636$	0	0
$637$	15730.2	0.978417
$638$	0	0
$639$	7651.30	0.473679
$640$	0	0
$641$	−16188.4	−0.997510	−0.498755	−	0.866743i	$-0.666209\pi$
$641$	−16188.4	−0.997510	−0.498755	+	0.866743i	$0.666209\pi$
$642$	0	0
$643$	24550.3	1.50571	0.752854	−	0.658187i	$-0.228677\pi$
$643$	24550.3	1.50571	0.752854	+	0.658187i	$0.228677\pi$
$644$	0	0
$645$	−1215.37	−0.0741939
$646$	0	0
$647$	−4566.00	−0.277447	−0.138723	−	0.990331i	$-0.544300\pi$
$647$	−4566.00	−0.277447	−0.138723	+	0.990331i	$0.544300\pi$
$648$	0	0
$649$	−1358.57	−0.0821705
$650$	0	0
$651$	331.911	0.0199825
$652$	0	0
$653$	12985.7	0.778209	0.389105	−	0.921194i	$-0.372784\pi$
$653$	12985.7	0.778209	0.389105	+	0.921194i	$0.372784\pi$
$654$	0	0
$655$	−3220.84	−0.192135
$656$	0	0
$657$	−2376.26	−0.141106
$658$	0	0
$659$	22004.9	1.30074	0.650370	−	0.759618i	$-0.274614\pi$
$659$	22004.9	1.30074	0.650370	+	0.759618i	$0.274614\pi$
$660$	0	0
$661$	−10592.0	−0.623267	−0.311634	−	0.950202i	$-0.600876\pi$
$661$	−10592.0	−0.623267	−0.311634	+	0.950202i	$0.600876\pi$
$662$	0	0
$663$	17766.8	1.04073
$664$	0	0
$665$	−579.506	−0.0337929
$666$	0	0
$667$	523.610	0.0303962
$668$	0	0
$669$	13933.7	0.805246
$670$	0	0
$671$	765.558	0.0440448
$672$	0	0
$673$	−131.859	−0.00755242	−0.00377621	−	0.999993i	$-0.501202\pi$
$673$	−131.859	−0.00755242	−0.00377621	+	0.999993i	$0.501202\pi$
$674$	0	0
$675$	17755.7	1.01247
$676$	0	0
$677$	−15415.4	−0.875130	−0.437565	−	0.899187i	$-0.644159\pi$
$677$	−15415.4	−0.875130	−0.437565	+	0.899187i	$0.644159\pi$
$678$	0	0
$679$	4861.92	0.274791
$680$	0	0
$681$	12297.8	0.692003
$682$	0	0
$683$	26452.5	1.48196	0.740978	−	0.671530i	$-0.234362\pi$
$683$	26452.5	1.48196	0.740978	+	0.671530i	$0.234362\pi$
$684$	0	0
$685$	5903.98	0.329313
$686$	0	0
$687$	−9718.76	−0.539729
$688$	0	0
$689$	−8642.13	−0.477850
$690$	0	0
$691$	20331.6	1.11932	0.559661	−	0.828721i	$-0.310931\pi$
$691$	20331.6	1.11932	0.559661	+	0.828721i	$0.310931\pi$
$692$	0	0
$693$	767.351	0.0420624
$694$	0	0
$695$	−2119.29	−0.115668
$696$	0	0
$697$	−4132.82	−0.224593
$698$	0	0
$699$	17414.2	0.942294
$700$	0	0
$701$	−13325.5	−0.717973	−0.358987	−	0.933343i	$-0.616878\pi$
$701$	−13325.5	−0.717973	−0.358987	+	0.933343i	$0.616878\pi$
$702$	0	0
$703$	5989.87	0.321354
$704$	0	0
$705$	−967.256	−0.0516723
$706$	0	0
$707$	−1427.48	−0.0759350
$708$	0	0
$709$	−33177.3	−1.75741	−0.878703	−	0.477370i	$-0.841590\pi$
$709$	−33177.3	−1.75741	−0.878703	+	0.477370i	$0.841590\pi$
$710$	0	0
$711$	2588.82	0.136552
$712$	0	0
$713$	375.855	0.0197418
$714$	0	0
$715$	−1332.82	−0.0697125
$716$	0	0
$717$	17555.9	0.914415
$718$	0	0
$719$	−12840.3	−0.666013	−0.333007	−	0.942924i	$-0.608063\pi$
$719$	−12840.3	−0.666013	−0.333007	+	0.942924i	$0.608063\pi$
$720$	0	0
$721$	−7701.76	−0.397820
$722$	0	0
$723$	−11165.3	−0.574330
$724$	0	0
$725$	3483.22	0.178432
$726$	0	0
$727$	23615.7	1.20475	0.602377	−	0.798212i	$-0.294220\pi$
$727$	23615.7	1.20475	0.602377	+	0.798212i	$0.294220\pi$
$728$	0	0
$729$	16561.0	0.841386
$730$	0	0
$731$	15446.6	0.781548
$732$	0	0
$733$	23648.0	1.19162	0.595812	−	0.803124i	$-0.296830\pi$
$733$	23648.0	1.19162	0.595812	+	0.803124i	$0.296830\pi$
$734$	0	0
$735$	−2578.58	−0.129404
$736$	0	0
$737$	5844.38	0.292104
$738$	0	0
$739$	18418.8	0.916841	0.458420	−	0.888735i	$-0.348415\pi$
$739$	18418.8	0.916841	0.458420	+	0.888735i	$0.348415\pi$
$740$	0	0
$741$	−10392.4	−0.515215
$742$	0	0
$743$	−19005.1	−0.938396	−0.469198	−	0.883093i	$-0.655457\pi$
$743$	−19005.1	−0.938396	−0.469198	+	0.883093i	$0.655457\pi$
$744$	0	0
$745$	1084.40	0.0533280
$746$	0	0
$747$	2946.99	0.144344
$748$	0	0
$749$	2477.51	0.120863
$750$	0	0
$751$	−26136.1	−1.26993	−0.634967	−	0.772540i	$-0.718986\pi$
$751$	−26136.1	−1.26993	−0.634967	+	0.772540i	$0.718986\pi$
$752$	0	0
$753$	−7964.08	−0.385428
$754$	0	0
$755$	4420.53	0.213085
$756$	0	0
$757$	−19642.3	−0.943081	−0.471540	−	0.881844i	$-0.656302\pi$
$757$	−19642.3	−0.943081	−0.471540	+	0.881844i	$0.656302\pi$
$758$	0	0
$759$	−806.880	−0.0385874
$760$	0	0
$761$	36349.8	1.73151	0.865754	−	0.500469i	$-0.166839\pi$
$761$	36349.8	1.73151	0.865754	+	0.500469i	$0.166839\pi$
$762$	0	0
$763$	4367.81	0.207242
$764$	0	0
$765$	3136.47	0.148234
$766$	0	0
$767$	5330.77	0.250955
$768$	0	0
$769$	−4940.72	−0.231686	−0.115843	−	0.993268i	$-0.536957\pi$
$769$	−4940.72	−0.231686	−0.115843	+	0.993268i	$0.536957\pi$
$770$	0	0
$771$	1899.96	0.0887490
$772$	0	0
$773$	−4543.81	−0.211422	−0.105711	−	0.994397i	$-0.533712\pi$
$773$	−4543.81	−0.211422	−0.105711	+	0.994397i	$0.533712\pi$
$774$	0	0
$775$	2500.31	0.115889
$776$	0	0
$777$	−1611.45	−0.0744023
$778$	0	0
$779$	2417.42	0.111185
$780$	0	0
$781$	−6773.83	−0.310354
$782$	0	0
$783$	4287.00	0.195664
$784$	0	0
$785$	4047.81	0.184042
$786$	0	0
$787$	−18523.9	−0.839016	−0.419508	−	0.907752i	$-0.637797\pi$
$787$	−18523.9	−0.839016	−0.419508	+	0.907752i	$0.637797\pi$
$788$	0	0
$789$	17640.9	0.795984
$790$	0	0
$791$	−2182.50	−0.0981047
$792$	0	0
$793$	−3003.90	−0.134516
$794$	0	0
$795$	1416.67	0.0631999
$796$	0	0
$797$	9666.66	0.429624	0.214812	−	0.976655i	$-0.431086\pi$
$797$	9666.66	0.429624	0.214812	+	0.976655i	$0.431086\pi$
$798$	0	0
$799$	12293.2	0.544309
$800$	0	0
$801$	19506.5	0.860459
$802$	0	0
$803$	2103.74	0.0924526
$804$	0	0
$805$	176.546	0.00772972
$806$	0	0
$807$	−17107.4	−0.746232
$808$	0	0
$809$	8733.19	0.379533	0.189767	−	0.981829i	$-0.439227\pi$
$809$	8733.19	0.379533	0.189767	+	0.981829i	$0.439227\pi$
$810$	0	0
$811$	−14082.6	−0.609750	−0.304875	−	0.952392i	$-0.598615\pi$
$811$	−14082.6	−0.609750	−0.304875	+	0.952392i	$0.598615\pi$
$812$	0	0
$813$	29520.7	1.27348
$814$	0	0
$815$	−6377.47	−0.274102
$816$	0	0
$817$	−9035.19	−0.386905
$818$	0	0
$819$	−3010.93	−0.128462
$820$	0	0
$821$	−15741.2	−0.669149	−0.334574	−	0.942369i	$-0.608593\pi$
$821$	−15741.2	−0.669149	−0.334574	+	0.942369i	$0.608593\pi$
$822$	0	0
$823$	2136.76	0.0905014	0.0452507	−	0.998976i	$-0.485591\pi$
$823$	2136.76	0.0905014	0.0452507	+	0.998976i	$0.485591\pi$
$824$	0	0
$825$	−5367.62	−0.226517
$826$	0	0
$827$	−2966.17	−0.124721	−0.0623603	−	0.998054i	$-0.519863\pi$
$827$	−2966.17	−0.124721	−0.0623603	+	0.998054i	$0.519863\pi$
$828$	0	0
$829$	−28562.2	−1.19663	−0.598316	−	0.801260i	$-0.704163\pi$
$829$	−28562.2	−1.19663	−0.598316	+	0.801260i	$0.704163\pi$
$830$	0	0
$831$	−17906.6	−0.747500
$832$	0	0
$833$	32772.0	1.36313
$834$	0	0
$835$	−3667.75	−0.152009
$836$	0	0
$837$	3077.28	0.127080
$838$	0	0
$839$	9658.72	0.397445	0.198722	−	0.980056i	$-0.436321\pi$
$839$	9658.72	0.397445	0.198722	+	0.980056i	$0.436321\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	292.760	0.0119611
$844$	0	0
$845$	371.904	0.0151407
$846$	0	0
$847$	5206.61	0.211217
$848$	0	0
$849$	−768.597	−0.0310697
$850$	0	0
$851$	−1824.81	−0.0735060
$852$	0	0
$853$	−5702.59	−0.228901	−0.114451	−	0.993429i	$-0.536511\pi$
$853$	−5702.59	−0.228901	−0.114451	+	0.993429i	$0.536511\pi$
$854$	0	0
$855$	−1834.62	−0.0733834
$856$	0	0
$857$	−11141.0	−0.444071	−0.222035	−	0.975039i	$-0.571270\pi$
$857$	−11141.0	−0.444071	−0.222035	+	0.975039i	$0.571270\pi$
$858$	0	0
$859$	15527.7	0.616763	0.308382	−	0.951263i	$-0.400213\pi$
$859$	15527.7	0.616763	0.308382	+	0.951263i	$0.400213\pi$
$860$	0	0
$861$	−650.358	−0.0257423
$862$	0	0
$863$	−19168.3	−0.756080	−0.378040	−	0.925789i	$-0.623402\pi$
$863$	−19168.3	−0.756080	−0.378040	+	0.925789i	$0.623402\pi$
$864$	0	0
$865$	6468.65	0.254267
$866$	0	0
$867$	19301.1	0.756057
$868$	0	0
$869$	−2291.93	−0.0894689
$870$	0	0
$871$	−22932.2	−0.892110
$872$	0	0
$873$	15392.1	0.596727
$874$	0	0
$875$	2396.68	0.0925973
$876$	0	0
$877$	−33530.4	−1.29104	−0.645519	−	0.763744i	$-0.723359\pi$
$877$	−33530.4	−1.29104	−0.645519	+	0.763744i	$0.723359\pi$
$878$	0	0
$879$	−22882.3	−0.878043
$880$	0	0
$881$	−1779.67	−0.0680574	−0.0340287	−	0.999421i	$-0.510834\pi$
$881$	−1779.67	−0.0680574	−0.0340287	+	0.999421i	$0.510834\pi$
$882$	0	0
$883$	−47347.6	−1.80450	−0.902250	−	0.431212i	$-0.858086\pi$
$883$	−47347.6	−1.80450	−0.902250	+	0.431212i	$0.858086\pi$
$884$	0	0
$885$	−873.849	−0.0331911
$886$	0	0
$887$	−33076.7	−1.25210	−0.626048	−	0.779785i	$-0.715328\pi$
$887$	−33076.7	−1.25210	−0.626048	+	0.779785i	$0.715328\pi$
$888$	0	0
$889$	−2935.56	−0.110748
$890$	0	0
$891$	−1921.14	−0.0722341
$892$	0	0
$893$	−7190.70	−0.269460
$894$	0	0
$895$	9265.03	0.346029
$896$	0	0
$897$	3166.03	0.117849
$898$	0	0
$899$	603.683	0.0223959
$900$	0	0
$901$	−18004.9	−0.665739
$902$	0	0
$903$	2430.74	0.0895790
$904$	0	0
$905$	−711.906	−0.0261487
$906$	0	0
$907$	−2824.54	−0.103404	−0.0517019	−	0.998663i	$-0.516465\pi$
$907$	−2824.54	−0.103404	−0.0517019	+	0.998663i	$0.516465\pi$
$908$	0	0
$909$	−4519.19	−0.164898
$910$	0	0
$911$	−41606.9	−1.51317	−0.756586	−	0.653895i	$-0.773134\pi$
$911$	−41606.9	−1.51317	−0.756586	+	0.653895i	$0.773134\pi$
$912$	0	0
$913$	−2609.03	−0.0945741
$914$	0	0
$915$	492.415	0.0177910
$916$	0	0
$917$	6441.68	0.231977
$918$	0	0
$919$	−618.058	−0.0221848	−0.0110924	−	0.999938i	$-0.503531\pi$
$919$	−618.058	−0.0221848	−0.0110924	+	0.999938i	$0.503531\pi$
$920$	0	0
$921$	−20322.3	−0.727081
$922$	0	0
$923$	26579.1	0.947848
$924$	0	0
$925$	−12139.2	−0.431497
$926$	0	0
$927$	−24382.5	−0.863892
$928$	0	0
$929$	−8931.48	−0.315428	−0.157714	−	0.987485i	$-0.550412\pi$
$929$	−8931.48	−0.315428	−0.157714	+	0.987485i	$0.550412\pi$
$930$	0	0
$931$	−19169.4	−0.674815
$932$	0	0
$933$	2116.45	0.0742652
$934$	0	0
$935$	−2776.77	−0.0971232
$936$	0	0
$937$	−37657.4	−1.31293	−0.656464	−	0.754357i	$-0.727949\pi$
$937$	−37657.4	−1.31293	−0.656464	+	0.754357i	$0.727949\pi$
$938$	0	0
$939$	−4506.76	−0.156627
$940$	0	0
$941$	43174.4	1.49569	0.747846	−	0.663872i	$-0.231088\pi$
$941$	43174.4	1.49569	0.747846	+	0.663872i	$0.231088\pi$
$942$	0	0
$943$	−736.464	−0.0254322
$944$	0	0
$945$	1445.45	0.0497572
$946$	0	0
$947$	36963.1	1.26836	0.634181	−	0.773185i	$-0.281337\pi$
$947$	36963.1	1.26836	0.634181	+	0.773185i	$0.281337\pi$
$948$	0	0
$949$	−8254.66	−0.282358
$950$	0	0
$951$	17385.1	0.592799
$952$	0	0
$953$	32697.1	1.11140	0.555699	−	0.831384i	$-0.312451\pi$
$953$	32697.1	1.11140	0.555699	+	0.831384i	$0.312451\pi$
$954$	0	0
$955$	2021.98	0.0685127
$956$	0	0
$957$	−1295.98	−0.0437753
$958$	0	0
$959$	−11808.0	−0.397601
$960$	0	0
$961$	−29357.7	−0.985454
$962$	0	0
$963$	7843.42	0.262462
$964$	0	0
$965$	−2333.17	−0.0778316
$966$	0	0
$967$	−4649.77	−0.154629	−0.0773146	−	0.997007i	$-0.524635\pi$
$967$	−4649.77	−0.154629	−0.0773146	+	0.997007i	$0.524635\pi$
$968$	0	0
$969$	−21651.4	−0.717795
$970$	0	0
$971$	−27533.7	−0.909987	−0.454994	−	0.890495i	$-0.650358\pi$
$971$	−27533.7	−0.909987	−0.454994	+	0.890495i	$0.650358\pi$
$972$	0	0
$973$	4238.58	0.139653
$974$	0	0
$975$	21061.5	0.691801
$976$	0	0
$977$	−16474.7	−0.539482	−0.269741	−	0.962933i	$-0.586938\pi$
$977$	−16474.7	−0.539482	−0.269741	+	0.962933i	$0.586938\pi$
$978$	0	0
$979$	−17269.4	−0.563773
$980$	0	0
$981$	13827.8	0.450038
$982$	0	0
$983$	−26969.9	−0.875084	−0.437542	−	0.899198i	$-0.644151\pi$
$983$	−26969.9	−0.875084	−0.437542	+	0.899198i	$0.644151\pi$
$984$	0	0
$985$	1215.82	0.0393291
$986$	0	0
$987$	1934.51	0.0623872
$988$	0	0
$989$	2752.56	0.0884999
$990$	0	0
$991$	−6331.81	−0.202963	−0.101482	−	0.994837i	$-0.532358\pi$
$991$	−6331.81	−0.202963	−0.101482	+	0.994837i	$0.532358\pi$
$992$	0	0
$993$	−29881.0	−0.954929
$994$	0	0
$995$	9343.23	0.297689
$996$	0	0
$997$	−53115.7	−1.68725	−0.843626	−	0.536932i	$-0.819583\pi$
$997$	−53115.7	−1.68725	−0.843626	+	0.536932i	$0.819583\pi$
$998$	0	0
$999$	−14940.4	−0.473167

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	464.4.a.d.1.2		2
4.3	odd	2		116.4.a.a.1.1	✓	2
8.3	odd	2		1856.4.a.j.1.2		2
8.5	even	2		1856.4.a.k.1.1		2
12.11	even	2		1044.4.a.d.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
116.4.a.a.1.1	✓	2	4.3	odd	2
464.4.a.d.1.2		2	1.1	even	1	trivial
1044.4.a.d.1.1		2	12.11	even	2
1856.4.a.j.1.2		2	8.3	odd	2
1856.4.a.k.1.1		2	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)

\(f(q)\)	\(=\)	\(q+3.60555 q^{3} +2.21110 q^{5} -4.42221 q^{7} -14.0000 q^{9} +O(q^{10})\)	\(q+3.60555 q^{3} +2.21110 q^{5} -4.42221 q^{7} -14.0000 q^{9} +12.3944 q^{11} -48.6333 q^{13} +7.97224 q^{15} -101.322 q^{17} +59.2666 q^{19} -15.9445 q^{21} -18.0555 q^{23} -120.111 q^{25} -147.828 q^{27} -29.0000 q^{29} -20.8167 q^{31} +44.6888 q^{33} -9.77795 q^{35} +101.066 q^{37} -175.350 q^{39} +40.7889 q^{41} -152.450 q^{43} -30.9554 q^{45} -121.328 q^{47} -323.444 q^{49} -365.322 q^{51} +177.700 q^{53} +27.4054 q^{55} +213.689 q^{57} -109.611 q^{59} +61.7662 q^{61} +61.9109 q^{63} -107.533 q^{65} +471.532 q^{67} -65.1001 q^{69} -546.522 q^{71} +169.733 q^{73} -433.066 q^{75} -54.8108 q^{77} -184.916 q^{79} -155.000 q^{81} -210.500 q^{83} -224.034 q^{85} -104.561 q^{87} -1393.32 q^{89} +215.066 q^{91} -75.0555 q^{93} +131.045 q^{95} -1099.43 q^{97} -173.522 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 10 q^{5} + 20 q^{7} - 28 q^{9}+O(q^{10}) \) 2 * q - 10 * q^5 + 20 * q^7 - 28 * q^9	\( 2 q - 10 q^{5} + 20 q^{7} - 28 q^{9} + 32 q^{11} - 54 q^{13} + 52 q^{15} - 44 q^{17} + 32 q^{19} - 104 q^{21} + 36 q^{23} - 96 q^{25} - 58 q^{29} - 20 q^{31} - 26 q^{33} - 308 q^{35} - 144 q^{37} - 156 q^{39} + 96 q^{41} - 240 q^{43} + 140 q^{45} - 596 q^{47} - 70 q^{49} - 572 q^{51} - 34 q^{53} - 212 q^{55} + 312 q^{57} - 724 q^{59} - 612 q^{61} - 280 q^{63} - 42 q^{65} - 528 q^{67} - 260 q^{69} + 104 q^{71} - 872 q^{73} - 520 q^{75} + 424 q^{77} + 820 q^{79} - 310 q^{81} + 228 q^{83} - 924 q^{85} - 32 q^{89} + 84 q^{91} - 78 q^{93} + 464 q^{95} - 1896 q^{97} - 448 q^{99}+O(q^{100}) \) 2 * q - 10 * q^5 + 20 * q^7 - 28 * q^9 + 32 * q^11 - 54 * q^13 + 52 * q^15 - 44 * q^17 + 32 * q^19 - 104 * q^21 + 36 * q^23 - 96 * q^25 - 58 * q^29 - 20 * q^31 - 26 * q^33 - 308 * q^35 - 144 * q^37 - 156 * q^39 + 96 * q^41 - 240 * q^43 + 140 * q^45 - 596 * q^47 - 70 * q^49 - 572 * q^51 - 34 * q^53 - 212 * q^55 + 312 * q^57 - 724 * q^59 - 612 * q^61 - 280 * q^63 - 42 * q^65 - 528 * q^67 - 260 * q^69 + 104 * q^71 - 872 * q^73 - 520 * q^75 + 424 * q^77 + 820 * q^79 - 310 * q^81 + 228 * q^83 - 924 * q^85 - 32 * q^89 + 84 * q^91 - 78 * q^93 + 464 * q^95 - 1896 * q^97 - 448 * q^99

Introduction

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