LMFDB - Embedded newform 1323.4.a.bm.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1323,4,Mod(1,1323)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1323, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1323.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1323 = 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1323.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:	$78.0595269376$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 54x^{6} + 887x^{4} - 4176x^{2} + 3136 $ x^8 - 54x^6 + 887x^4 - 4176*x^2 + 3136
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 7 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.959848$ of defining polynomial
Character	$\chi$	$=$	1323.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.959848 q^{2} -7.07869 q^{4} +10.2898 q^{5} -14.4733 q^{8} +9.87663 q^{10} -28.3265 q^{11} -5.96269 q^{13} +42.7374 q^{16} -104.664 q^{17} -34.0521 q^{19} -72.8381 q^{20} -27.1891 q^{22} +103.629 q^{23} -19.1205 q^{25} -5.72328 q^{26} +195.695 q^{29} -9.17704 q^{31} +156.807 q^{32} -100.462 q^{34} +245.968 q^{37} -32.6849 q^{38} -148.927 q^{40} +366.457 q^{41} -366.354 q^{43} +200.514 q^{44} +99.4681 q^{46} +244.288 q^{47} -18.3528 q^{50} +42.2080 q^{52} -281.226 q^{53} -291.473 q^{55} +187.838 q^{58} -181.642 q^{59} +24.1701 q^{61} -8.80857 q^{62} -191.388 q^{64} -61.3547 q^{65} -336.788 q^{67} +740.886 q^{68} -196.781 q^{71} +683.417 q^{73} +236.092 q^{74} +241.044 q^{76} +619.337 q^{79} +439.758 q^{80} +351.743 q^{82} -176.475 q^{83} -1076.97 q^{85} -351.644 q^{86} +409.976 q^{88} -761.046 q^{89} -733.557 q^{92} +234.480 q^{94} -350.389 q^{95} +1277.59 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 44 q^{4} + 132 q^{10} + 336 q^{13} + 204 q^{16} - 288 q^{19} + 484 q^{22} + 152 q^{25} + 120 q^{31} + 1008 q^{34} + 592 q^{37} + 1620 q^{40} - 1872 q^{43} - 1644 q^{46} + 2400 q^{52} + 1344 q^{55}+ \cdots + 5712 q^{97}+O(q^{100}) $ 8 * q + 44 * q^4 + 132 * q^10 + 336 * q^13 + 204 * q^16 - 288 * q^19 + 484 * q^22 + 152 * q^25 + 120 * q^31 + 1008 * q^34 + 592 * q^37 + 1620 * q^40 - 1872 * q^43 - 1644 * q^46 + 2400 * q^52 + 1344 * q^55 - 1200 * q^58 + 2400 * q^61 - 1388 * q^64 + 1824 * q^73 + 2844 * q^76 + 2368 * q^79 + 2436 * q^82 + 3512 * q^85 + 3780 * q^88 + 4368 * q^94 + 5712 * q^97

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 $

$2$	0.959848	0.339358	0.169679	−	0.985499i	$-0.445727\pi$
$2$	0.959848	0.339358	0.169679	+	0.985499i	$0.445727\pi$
$3$	0	0
$3$	0	0
$4$	−7.07869	−0.884836
$5$	10.2898	0.920346	0.460173	−	0.887829i	$-0.347787\pi$
$5$	10.2898	0.920346	0.460173	+	0.887829i	$0.347787\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−14.4733	−0.639634
$9$	0	0
$10$	9.87663	0.312326
$11$	−28.3265	−0.776432	−0.388216	−	0.921568i	$-0.626909\pi$
$11$	−28.3265	−0.776432	−0.388216	+	0.921568i	$0.626909\pi$
$12$	0	0
$13$	−5.96269	−0.127212	−0.0636058	−	0.997975i	$-0.520260\pi$
$13$	−5.96269	−0.127212	−0.0636058	+	0.997975i	$0.520260\pi$
$14$	0	0
$15$	0	0
$16$	42.7374	0.667772
$17$	−104.664	−1.49323	−0.746613	−	0.665259i	$-0.768321\pi$
$17$	−104.664	−1.49323	−0.746613	+	0.665259i	$0.768321\pi$
$18$	0	0
$19$	−34.0521	−0.411163	−0.205581	−	0.978640i	$-0.565909\pi$
$19$	−34.0521	−0.411163	−0.205581	+	0.978640i	$0.565909\pi$
$20$	−72.8381	−0.814355
$21$	0	0
$22$	−27.1891	−0.263488
$23$	103.629	0.939484	0.469742	−	0.882804i	$-0.344347\pi$
$23$	103.629	0.939484	0.469742	+	0.882804i	$0.344347\pi$
$24$	0	0
$25$	−19.1205	−0.152964
$26$	−5.72328	−0.0431703
$27$	0	0
$28$	0	0
$29$	195.695	1.25309	0.626547	−	0.779384i	$-0.284468\pi$
$29$	195.695	1.25309	0.626547	+	0.779384i	$0.284468\pi$
$30$	0	0
$31$	−9.17704	−0.0531692	−0.0265846	−	0.999647i	$-0.508463\pi$
$31$	−9.17704	−0.0531692	−0.0265846	+	0.999647i	$0.508463\pi$
$32$	156.807	0.866247
$33$	0	0
$34$	−100.462	−0.506737
$35$	0	0
$36$	0	0
$37$	245.968	1.09289	0.546444	−	0.837496i	$-0.315981\pi$
$37$	245.968	1.09289	0.546444	+	0.837496i	$0.315981\pi$
$38$	−32.6849	−0.139531
$39$	0	0
$40$	−148.927	−0.588684
$41$	366.457	1.39588	0.697939	−	0.716157i	$-0.254101\pi$
$41$	366.457	1.39588	0.697939	+	0.716157i	$0.254101\pi$
$42$	0	0
$43$	−366.354	−1.29927	−0.649633	−	0.760248i	$-0.725078\pi$
$43$	−366.354	−1.29927	−0.649633	+	0.760248i	$0.725078\pi$
$44$	200.514	0.687015
$45$	0	0
$46$	99.4681	0.318821
$47$	244.288	0.758152	0.379076	−	0.925366i	$-0.376242\pi$
$47$	244.288	0.758152	0.379076	+	0.925366i	$0.376242\pi$
$48$	0	0
$49$	0	0
$50$	−18.3528	−0.0519095
$51$	0	0
$52$	42.2080	0.112562
$53$	−281.226	−0.728857	−0.364428	−	0.931231i	$-0.618736\pi$
$53$	−281.226	−0.728857	−0.364428	+	0.931231i	$0.618736\pi$
$54$	0	0
$55$	−291.473	−0.714586
$56$	0	0
$57$	0	0
$58$	187.838	0.425247
$59$	−181.642	−0.400810	−0.200405	−	0.979713i	$-0.564226\pi$
$59$	−181.642	−0.400810	−0.200405	+	0.979713i	$0.564226\pi$
$60$	0	0
$61$	24.1701	0.0507323	0.0253661	−	0.999678i	$-0.491925\pi$
$61$	24.1701	0.0507323	0.0253661	+	0.999678i	$0.491925\pi$
$62$	−8.80857	−0.0180434
$63$	0	0
$64$	−191.388	−0.373804
$65$	−61.3547	−0.117079
$66$	0	0
$67$	−336.788	−0.614108	−0.307054	−	0.951692i	$-0.599343\pi$
$67$	−336.788	−0.614108	−0.307054	+	0.951692i	$0.599343\pi$
$68$	740.886	1.32126
$69$	0	0
$70$	0	0
$71$	−196.781	−0.328924	−0.164462	−	0.986383i	$-0.552589\pi$
$71$	−196.781	−0.328924	−0.164462	+	0.986383i	$0.552589\pi$
$72$	0	0
$73$	683.417	1.09573	0.547863	−	0.836568i	$-0.315442\pi$
$73$	683.417	1.09573	0.547863	+	0.836568i	$0.315442\pi$
$74$	236.092	0.370880
$75$	0	0
$76$	241.044	0.363812
$77$	0	0
$78$	0	0
$79$	619.337	0.882036	0.441018	−	0.897498i	$-0.354617\pi$
$79$	619.337	0.882036	0.441018	+	0.897498i	$0.354617\pi$
$80$	439.758	0.614581
$81$	0	0
$82$	351.743	0.473702
$83$	−176.475	−0.233381	−0.116690	−	0.993168i	$-0.537229\pi$
$83$	−176.475	−0.233381	−0.116690	+	0.993168i	$0.537229\pi$
$84$	0	0
$85$	−1076.97	−1.37428
$86$	−351.644	−0.440916
$87$	0	0
$88$	409.976	0.496632
$89$	−761.046	−0.906413	−0.453207	−	0.891406i	$-0.649720\pi$
$89$	−761.046	−0.906413	−0.453207	+	0.891406i	$0.649720\pi$
$90$	0	0
$91$	0	0
$92$	−733.557	−0.831290
$93$	0	0
$94$	234.480	0.257285
$95$	−350.389	−0.378412
$96$	0	0
$97$	1277.59	1.33732	0.668660	−	0.743568i	$-0.266868\pi$
$97$	1277.59	1.33732	0.668660	+	0.743568i	$0.266868\pi$
$98$	0	0
$99$	0	0
$100$	135.348	0.135348
$101$	1555.82	1.53277	0.766387	−	0.642379i	$-0.222052\pi$
$101$	1555.82	1.53277	0.766387	+	0.642379i	$0.222052\pi$
$102$	0	0
$103$	−246.314	−0.235632	−0.117816	−	0.993035i	$-0.537589\pi$
$103$	−246.314	−0.235632	−0.117816	+	0.993035i	$0.537589\pi$
$104$	86.2995	0.0813689
$105$	0	0
$106$	−269.935	−0.247343
$107$	1621.50	1.46501	0.732504	−	0.680762i	$-0.238351\pi$
$107$	1621.50	1.46501	0.732504	+	0.680762i	$0.238351\pi$
$108$	0	0
$109$	1906.69	1.67548	0.837742	−	0.546066i	$-0.183875\pi$
$109$	1906.69	1.67548	0.837742	+	0.546066i	$0.183875\pi$
$110$	−279.770	−0.242500
$111$	0	0
$112$	0	0
$113$	1886.64	1.57062	0.785312	−	0.619100i	$-0.212502\pi$
$113$	1886.64	1.57062	0.785312	+	0.619100i	$0.212502\pi$
$114$	0	0
$115$	1066.32	0.864650
$116$	−1385.27	−1.10878
$117$	0	0
$118$	−174.349	−0.136018
$119$	0	0
$120$	0	0
$121$	−528.611	−0.397153
$122$	23.1997	0.0172164
$123$	0	0
$124$	64.9614	0.0470460
$125$	−1482.97	−1.06113
$126$	0	0
$127$	−594.201	−0.415172	−0.207586	−	0.978217i	$-0.566561\pi$
$127$	−594.201	−0.415172	−0.207586	+	0.978217i	$0.566561\pi$
$128$	−1438.16	−0.993100
$129$	0	0
$130$	−58.8912	−0.0397316
$131$	−494.116	−0.329551	−0.164775	−	0.986331i	$-0.552690\pi$
$131$	−494.116	−0.329551	−0.164775	+	0.986331i	$0.552690\pi$
$132$	0	0
$133$	0	0
$134$	−323.266	−0.208402
$135$	0	0
$136$	1514.83	0.955117
$137$	−1807.01	−1.12689	−0.563443	−	0.826155i	$-0.690523\pi$
$137$	−1807.01	−1.12689	−0.563443	+	0.826155i	$0.690523\pi$
$138$	0	0
$139$	3149.34	1.92175	0.960876	−	0.276979i	$-0.0893331\pi$
$139$	3149.34	1.92175	0.960876	+	0.276979i	$0.0893331\pi$
$140$	0	0
$141$	0	0
$142$	−188.880	−0.111623
$143$	168.902	0.0987712
$144$	0	0
$145$	2013.66	1.15328
$146$	655.977	0.371843
$147$	0	0
$148$	−1741.13	−0.967027
$149$	−1899.54	−1.04441	−0.522203	−	0.852821i	$-0.674890\pi$
$149$	−1899.54	−1.04441	−0.522203	+	0.852821i	$0.674890\pi$
$150$	0	0
$151$	1521.53	0.820004	0.410002	−	0.912085i	$-0.365528\pi$
$151$	1521.53	0.820004	0.410002	+	0.912085i	$0.365528\pi$
$152$	492.845	0.262994
$153$	0	0
$154$	0	0
$155$	−94.4297	−0.0489340
$156$	0	0
$157$	3606.96	1.83355	0.916773	−	0.399408i	$-0.130784\pi$
$157$	3606.96	1.83355	0.916773	+	0.399408i	$0.130784\pi$
$158$	594.470	0.299326
$159$	0	0
$160$	1613.51	0.797247
$161$	0	0
$162$	0	0
$163$	887.302	0.426373	0.213187	−	0.977011i	$-0.431616\pi$
$163$	887.302	0.426373	0.213187	+	0.977011i	$0.431616\pi$
$164$	−2594.04	−1.23512
$165$	0	0
$166$	−169.389	−0.0791996
$167$	1248.96	0.578729	0.289364	−	0.957219i	$-0.406556\pi$
$167$	1248.96	0.578729	0.289364	+	0.957219i	$0.406556\pi$
$168$	0	0
$169$	−2161.45	−0.983817
$170$	−1033.73	−0.466374
$171$	0	0
$172$	2593.31	1.14964
$173$	2130.55	0.936318	0.468159	−	0.883644i	$-0.344917\pi$
$173$	2130.55	0.936318	0.468159	+	0.883644i	$0.344917\pi$
$174$	0	0
$175$	0	0
$176$	−1210.60	−0.518479
$177$	0	0
$178$	−730.489	−0.307598
$179$	2756.11	1.15084	0.575422	−	0.817857i	$-0.304838\pi$
$179$	2756.11	1.15084	0.575422	+	0.817857i	$0.304838\pi$
$180$	0	0
$181$	734.801	0.301753	0.150877	−	0.988553i	$-0.451790\pi$
$181$	734.801	0.301753	0.150877	+	0.988553i	$0.451790\pi$
$182$	0	0
$183$	0	0
$184$	−1499.85	−0.600926
$185$	2530.95	1.00583
$186$	0	0
$187$	2964.77	1.15939
$188$	−1729.24	−0.670840
$189$	0	0
$190$	−336.320	−0.128417
$191$	2034.49	0.770736	0.385368	−	0.922763i	$-0.374075\pi$
$191$	2034.49	0.770736	0.385368	+	0.922763i	$0.374075\pi$
$192$	0	0
$193$	−664.168	−0.247709	−0.123855	−	0.992300i	$-0.539526\pi$
$193$	−664.168	−0.247709	−0.123855	+	0.992300i	$0.539526\pi$
$194$	1226.30	0.453830
$195$	0	0
$196$	0	0
$197$	1682.40	0.608455	0.304228	−	0.952599i	$-0.401602\pi$
$197$	1682.40	0.608455	0.304228	+	0.952599i	$0.401602\pi$
$198$	0	0
$199$	3778.06	1.34583	0.672913	−	0.739721i	$-0.265043\pi$
$199$	3778.06	1.34583	0.672913	+	0.739721i	$0.265043\pi$
$200$	276.736	0.0978410
$201$	0	0
$202$	1493.35	0.520159
$203$	0	0
$204$	0	0
$205$	3770.76	1.28469
$206$	−236.425	−0.0799635
$207$	0	0
$208$	−254.830	−0.0849484
$209$	964.576	0.319240
$210$	0	0
$211$	−5122.91	−1.67145	−0.835725	−	0.549148i	$-0.814952\pi$
$211$	−5122.91	−1.67145	−0.835725	+	0.549148i	$0.814952\pi$
$212$	1990.71	0.644919
$213$	0	0
$214$	1556.39	0.497162
$215$	−3769.70	−1.19577
$216$	0	0
$217$	0	0
$218$	1830.13	0.568589
$219$	0	0
$220$	2063.25	0.632291
$221$	624.081	0.189956
$222$	0	0
$223$	2118.71	0.636229	0.318115	−	0.948052i	$-0.396950\pi$
$223$	2118.71	0.636229	0.318115	+	0.948052i	$0.396950\pi$
$224$	0	0
$225$	0	0
$226$	1810.89	0.533004
$227$	−4239.57	−1.23960	−0.619802	−	0.784758i	$-0.712787\pi$
$227$	−4239.57	−1.23960	−0.619802	+	0.784758i	$0.712787\pi$
$228$	0	0
$229$	−1518.18	−0.438098	−0.219049	−	0.975714i	$-0.570295\pi$
$229$	−1518.18	−0.438098	−0.219049	+	0.975714i	$0.570295\pi$
$230$	1023.50	0.293426
$231$	0	0
$232$	−2832.35	−0.801521
$233$	4596.37	1.29235	0.646177	−	0.763188i	$-0.276367\pi$
$233$	4596.37	1.29235	0.646177	+	0.763188i	$0.276367\pi$
$234$	0	0
$235$	2513.67	0.697761
$236$	1285.79	0.354651
$237$	0	0
$238$	0	0
$239$	−2956.83	−0.800256	−0.400128	−	0.916459i	$-0.631034\pi$
$239$	−2956.83	−0.800256	−0.400128	+	0.916459i	$0.631034\pi$
$240$	0	0
$241$	113.007	0.0302050	0.0151025	−	0.999886i	$-0.495193\pi$
$241$	113.007	0.0302050	0.0151025	+	0.999886i	$0.495193\pi$
$242$	−507.387	−0.134777
$243$	0	0
$244$	−171.093	−0.0448897
$245$	0	0
$246$	0	0
$247$	203.042	0.0523047
$248$	132.822	0.0340088
$249$	0	0
$250$	−1423.42	−0.360101
$251$	6988.70	1.75746	0.878731	−	0.477318i	$-0.158391\pi$
$251$	6988.70	1.75746	0.878731	+	0.477318i	$0.158391\pi$
$252$	0	0
$253$	−2935.44	−0.729445
$254$	−570.343	−0.140892
$255$	0	0
$256$	150.683	0.0367880
$257$	−2969.04	−0.720637	−0.360318	−	0.932829i	$-0.617332\pi$
$257$	−2969.04	−0.720637	−0.360318	+	0.932829i	$0.617332\pi$
$258$	0	0
$259$	0	0
$260$	434.311	0.103595
$261$	0	0
$262$	−474.277	−0.111836
$263$	4545.72	1.06578	0.532892	−	0.846183i	$-0.321105\pi$
$263$	4545.72	1.06578	0.532892	+	0.846183i	$0.321105\pi$
$264$	0	0
$265$	−2893.76	−0.670800
$266$	0	0
$267$	0	0
$268$	2384.02	0.543385
$269$	−1438.64	−0.326079	−0.163040	−	0.986620i	$-0.552130\pi$
$269$	−1438.64	−0.326079	−0.163040	+	0.986620i	$0.552130\pi$
$270$	0	0
$271$	−583.420	−0.130776	−0.0653879	−	0.997860i	$-0.520828\pi$
$271$	−583.420	−0.130776	−0.0653879	+	0.997860i	$0.520828\pi$
$272$	−4473.08	−0.997134
$273$	0	0
$274$	−1734.46	−0.382417
$275$	541.616	0.118766
$276$	0	0
$277$	1559.79	0.338335	0.169168	−	0.985587i	$-0.445892\pi$
$277$	1559.79	0.338335	0.169168	+	0.985587i	$0.445892\pi$
$278$	3022.89	0.652161
$279$	0	0
$280$	0	0
$281$	−8809.34	−1.87018	−0.935091	−	0.354407i	$-0.884683\pi$
$281$	−8809.34	−1.87018	−0.935091	+	0.354407i	$0.884683\pi$
$282$	0	0
$283$	8501.27	1.78568	0.892841	−	0.450371i	$-0.148708\pi$
$283$	8501.27	1.78568	0.892841	+	0.450371i	$0.148708\pi$
$284$	1392.95	0.291044
$285$	0	0
$286$	162.120	0.0335188
$287$	0	0
$288$	0	0
$289$	6041.62	1.22972
$290$	1932.81	0.391374
$291$	0	0
$292$	−4837.70	−0.969537
$293$	−8571.41	−1.70903	−0.854517	−	0.519423i	$-0.826147\pi$
$293$	−8571.41	−1.70903	−0.854517	+	0.519423i	$0.826147\pi$
$294$	0	0
$295$	−1869.06	−0.368883
$296$	−3559.95	−0.699048
$297$	0	0
$298$	−1823.27	−0.354427
$299$	−617.907	−0.119513
$300$	0	0
$301$	0	0
$302$	1460.44	0.278275
$303$	0	0
$304$	−1455.30	−0.274563
$305$	248.705	0.0466912
$306$	0	0
$307$	1751.31	0.325578	0.162789	−	0.986661i	$-0.447951\pi$
$307$	1751.31	0.325578	0.162789	+	0.986661i	$0.447951\pi$
$308$	0	0
$309$	0	0
$310$	−90.6382	−0.0166061
$311$	−6681.74	−1.21829	−0.609143	−	0.793061i	$-0.708486\pi$
$311$	−6681.74	−1.21829	−0.609143	+	0.793061i	$0.708486\pi$
$312$	0	0
$313$	6049.69	1.09249	0.546244	−	0.837626i	$-0.316057\pi$
$313$	6049.69	1.09249	0.546244	+	0.837626i	$0.316057\pi$
$314$	3462.14	0.622228
$315$	0	0
$316$	−4384.10	−0.780458
$317$	−3678.59	−0.651767	−0.325884	−	0.945410i	$-0.605662\pi$
$317$	−3678.59	−0.651767	−0.325884	+	0.945410i	$0.605662\pi$
$318$	0	0
$319$	−5543.36	−0.972942
$320$	−1969.34	−0.344029
$321$	0	0
$322$	0	0
$323$	3564.04	0.613959
$324$	0	0
$325$	114.010	0.0194588
$326$	851.675	0.144693
$327$	0	0
$328$	−5303.83	−0.892850
$329$	0	0
$330$	0	0
$331$	10462.0	1.73729	0.868647	−	0.495432i	$-0.164990\pi$
$331$	10462.0	1.73729	0.868647	+	0.495432i	$0.164990\pi$
$332$	1249.21	0.206504
$333$	0	0
$334$	1198.82	0.196396
$335$	−3465.48	−0.565192
$336$	0	0
$337$	1072.17	0.173308	0.0866539	−	0.996238i	$-0.472383\pi$
$337$	1072.17	0.173308	0.0866539	+	0.996238i	$0.472383\pi$
$338$	−2074.66	−0.333866
$339$	0	0
$340$	7623.56	1.21602
$341$	259.953	0.0412823
$342$	0	0
$343$	0	0
$344$	5302.34	0.831055
$345$	0	0
$346$	2045.01	0.317747
$347$	−9562.41	−1.47936	−0.739679	−	0.672960i	$-0.765023\pi$
$347$	−9562.41	−1.47936	−0.739679	+	0.672960i	$0.765023\pi$
$348$	0	0
$349$	4159.34	0.637950	0.318975	−	0.947763i	$-0.396661\pi$
$349$	4159.34	0.637950	0.318975	+	0.947763i	$0.396661\pi$
$350$	0	0
$351$	0	0
$352$	−4441.80	−0.672582
$353$	6003.78	0.905238	0.452619	−	0.891704i	$-0.350490\pi$
$353$	6003.78	0.905238	0.452619	+	0.891704i	$0.350490\pi$
$354$	0	0
$355$	−2024.83	−0.302723
$356$	5387.21	0.802027
$357$	0	0
$358$	2645.44	0.390548
$359$	−11515.2	−1.69290	−0.846448	−	0.532472i	$-0.821263\pi$
$359$	−11515.2	−1.69290	−0.846448	+	0.532472i	$0.821263\pi$
$360$	0	0
$361$	−5699.45	−0.830945
$362$	705.298	0.102402
$363$	0	0
$364$	0	0
$365$	7032.21	1.00845
$366$	0	0
$367$	−8631.92	−1.22774	−0.613872	−	0.789405i	$-0.710389\pi$
$367$	−8631.92	−1.22774	−0.613872	+	0.789405i	$0.710389\pi$
$368$	4428.83	0.627361
$369$	0	0
$370$	2429.33	0.341338
$371$	0	0
$372$	0	0
$373$	−11586.5	−1.60838	−0.804189	−	0.594373i	$-0.797400\pi$
$373$	−11586.5	−1.60838	−0.804189	+	0.594373i	$0.797400\pi$
$374$	2845.73	0.393447
$375$	0	0
$376$	−3535.65	−0.484939
$377$	−1166.87	−0.159408
$378$	0	0
$379$	11.3954	0.00154443	0.000772216	−	1.00000i	$-0.499754\pi$
$379$	11.3954	0.00154443	0.000772216		1.00000i	$0.499754\pi$
$380$	2480.29	0.334833
$381$	0	0
$382$	1952.80	0.261555
$383$	−7649.91	−1.02061	−0.510303	−	0.859995i	$-0.670467\pi$
$383$	−7649.91	−1.02061	−0.510303	+	0.859995i	$0.670467\pi$
$384$	0	0
$385$	0	0
$386$	−637.501	−0.0840620
$387$	0	0
$388$	−9043.69	−1.18331
$389$	−1018.92	−0.132805	−0.0664027	−	0.997793i	$-0.521152\pi$
$389$	−1018.92	−0.132805	−0.0664027	+	0.997793i	$0.521152\pi$
$390$	0	0
$391$	−10846.3	−1.40286
$392$	0	0
$393$	0	0
$394$	1614.84	0.206484
$395$	6372.84	0.811778
$396$	0	0
$397$	1213.68	0.153433	0.0767164	−	0.997053i	$-0.475556\pi$
$397$	1213.68	0.153433	0.0767164	+	0.997053i	$0.475556\pi$
$398$	3626.36	0.456717
$399$	0	0
$400$	−817.161	−0.102145
$401$	7176.55	0.893715	0.446858	−	0.894605i	$-0.352543\pi$
$401$	7176.55	0.893715	0.446858	+	0.894605i	$0.352543\pi$
$402$	0	0
$403$	54.7198	0.00676374
$404$	−11013.2	−1.35625
$405$	0	0
$406$	0	0
$407$	−6967.40	−0.848553
$408$	0	0
$409$	14888.8	1.80001	0.900006	−	0.435878i	$-0.143562\pi$
$409$	14888.8	1.80001	0.900006	+	0.435878i	$0.143562\pi$
$410$	3619.36	0.435969
$411$	0	0
$412$	1743.58	0.208496
$413$	0	0
$414$	0	0
$415$	−1815.89	−0.214791
$416$	−934.994	−0.110197
$417$	0	0
$418$	925.847	0.108337
$419$	−7504.96	−0.875039	−0.437519	−	0.899209i	$-0.644143\pi$
$419$	−7504.96	−0.875039	−0.437519	+	0.899209i	$0.644143\pi$
$420$	0	0
$421$	8568.94	0.991983	0.495991	−	0.868327i	$-0.334805\pi$
$421$	8568.94	0.991983	0.495991	+	0.868327i	$0.334805\pi$
$422$	−4917.22	−0.567219
$423$	0	0
$424$	4070.26	0.466201
$425$	2001.23	0.228410
$426$	0	0
$427$	0	0
$428$	−11478.1	−1.29629
$429$	0	0
$430$	−3618.34	−0.405795
$431$	−1869.85	−0.208974	−0.104487	−	0.994526i	$-0.533320\pi$
$431$	−1869.85	−0.208974	−0.104487	+	0.994526i	$0.533320\pi$
$432$	0	0
$433$	11202.1	1.24327	0.621636	−	0.783306i	$-0.286468\pi$
$433$	11202.1	1.24327	0.621636	+	0.783306i	$0.286468\pi$
$434$	0	0
$435$	0	0
$436$	−13496.9	−1.48253
$437$	−3528.79	−0.386281
$438$	0	0
$439$	9815.68	1.06715	0.533573	−	0.845754i	$-0.320849\pi$
$439$	9815.68	1.06715	0.533573	+	0.845754i	$0.320849\pi$
$440$	4218.56	0.457073
$441$	0	0
$442$	599.023	0.0644629
$443$	−5807.00	−0.622797	−0.311399	−	0.950279i	$-0.600797\pi$
$443$	−5807.00	−0.622797	−0.311399	+	0.950279i	$0.600797\pi$
$444$	0	0
$445$	−7831.00	−0.834213
$446$	2033.64	0.215909
$447$	0	0
$448$	0	0
$449$	257.367	0.0270510	0.0135255	−	0.999909i	$-0.495695\pi$
$449$	257.367	0.0270510	0.0135255	+	0.999909i	$0.495695\pi$
$450$	0	0
$451$	−10380.4	−1.08380
$452$	−13355.0	−1.38975
$453$	0	0
$454$	−4069.34	−0.420669
$455$	0	0
$456$	0	0
$457$	1448.74	0.148292	0.0741458	−	0.997247i	$-0.476377\pi$
$457$	1448.74	0.148292	0.0741458	+	0.997247i	$0.476377\pi$
$458$	−1457.23	−0.148672
$459$	0	0
$460$	−7548.14	−0.765074
$461$	−8713.18	−0.880289	−0.440144	−	0.897927i	$-0.645073\pi$
$461$	−8713.18	−0.880289	−0.440144	+	0.897927i	$0.645073\pi$
$462$	0	0
$463$	17766.2	1.78329	0.891646	−	0.452734i	$-0.149551\pi$
$463$	17766.2	1.78329	0.891646	+	0.452734i	$0.149551\pi$
$464$	8363.51	0.836780
$465$	0	0
$466$	4411.82	0.438570
$467$	13735.0	1.36098	0.680492	−	0.732755i	$-0.261766\pi$
$467$	13735.0	1.36098	0.680492	+	0.732755i	$0.261766\pi$
$468$	0	0
$469$	0	0
$470$	2412.74	0.236791
$471$	0	0
$472$	2628.95	0.256371
$473$	10377.5	1.00879
$474$	0	0
$475$	651.094	0.0628931
$476$	0	0
$477$	0	0
$478$	−2838.11	−0.271573
$479$	−4821.63	−0.459929	−0.229964	−	0.973199i	$-0.573861\pi$
$479$	−4821.63	−0.459929	−0.229964	+	0.973199i	$0.573861\pi$
$480$	0	0
$481$	−1466.63	−0.139028
$482$	108.469	0.0102503
$483$	0	0
$484$	3741.88	0.351416
$485$	13146.2	1.23080
$486$	0	0
$487$	−13603.0	−1.26573	−0.632864	−	0.774263i	$-0.718121\pi$
$487$	−13603.0	−1.26573	−0.632864	+	0.774263i	$0.718121\pi$
$488$	−349.821	−0.0324501
$489$	0	0
$490$	0	0
$491$	−6137.42	−0.564110	−0.282055	−	0.959398i	$-0.591016\pi$
$491$	−6137.42	−0.564110	−0.282055	+	0.959398i	$0.591016\pi$
$492$	0	0
$493$	−20482.3	−1.87115
$494$	194.890	0.0177500
$495$	0	0
$496$	−392.203	−0.0355049
$497$	0	0
$498$	0	0
$499$	−13674.1	−1.22673	−0.613363	−	0.789801i	$-0.710184\pi$
$499$	−13674.1	−1.22673	−0.613363	+	0.789801i	$0.710184\pi$
$500$	10497.5	0.938922
$501$	0	0
$502$	6708.09	0.596408
$503$	−4346.13	−0.385257	−0.192629	−	0.981272i	$-0.561701\pi$
$503$	−4346.13	−0.385257	−0.192629	+	0.981272i	$0.561701\pi$
$504$	0	0
$505$	16009.1	1.41068
$506$	−2817.58	−0.247543
$507$	0	0
$508$	4206.17	0.367359
$509$	19414.9	1.69067	0.845336	−	0.534235i	$-0.179400\pi$
$509$	19414.9	1.69067	0.845336	+	0.534235i	$0.179400\pi$
$510$	0	0
$511$	0	0
$512$	11649.9	1.00558
$513$	0	0
$514$	−2849.83	−0.244554
$515$	−2534.52	−0.216863
$516$	0	0
$517$	−6919.83	−0.588653
$518$	0	0
$519$	0	0
$520$	888.003	0.0748875
$521$	2749.25	0.231184	0.115592	−	0.993297i	$-0.463124\pi$
$521$	2749.25	0.231184	0.115592	+	0.993297i	$0.463124\pi$
$522$	0	0
$523$	3313.72	0.277054	0.138527	−	0.990359i	$-0.455763\pi$
$523$	3313.72	0.277054	0.138527	+	0.990359i	$0.455763\pi$
$524$	3497.70	0.291598
$525$	0	0
$526$	4363.21	0.361682
$527$	960.509	0.0793936
$528$	0	0
$529$	−1428.04	−0.117370
$530$	−2777.57	−0.227641
$531$	0	0
$532$	0	0
$533$	−2185.07	−0.177572
$534$	0	0
$535$	16684.8	1.34831
$536$	4874.42	0.392804
$537$	0	0
$538$	−1380.87	−0.110657
$539$	0	0
$540$	0	0
$541$	−1490.04	−0.118414	−0.0592069	−	0.998246i	$-0.518857\pi$
$541$	−1490.04	−0.118414	−0.0592069	+	0.998246i	$0.518857\pi$
$542$	−559.995	−0.0443798
$543$	0	0
$544$	−16412.1	−1.29350
$545$	19619.4	1.54202
$546$	0	0
$547$	18962.2	1.48220	0.741102	−	0.671392i	$-0.234303\pi$
$547$	18962.2	1.48220	0.741102	+	0.671392i	$0.234303\pi$
$548$	12791.3	0.997109
$549$	0	0
$550$	519.870	0.0403042
$551$	−6663.84	−0.515225
$552$	0	0
$553$	0	0
$554$	1497.16	0.114817
$555$	0	0
$556$	−22293.2	−1.70044
$557$	4846.56	0.368681	0.184341	−	0.982862i	$-0.440985\pi$
$557$	4846.56	0.368681	0.184341	+	0.982862i	$0.440985\pi$
$558$	0	0
$559$	2184.45	0.165282
$560$	0	0
$561$	0	0
$562$	−8455.63	−0.634661
$563$	−12390.7	−0.927540	−0.463770	−	0.885956i	$-0.653504\pi$
$563$	−12390.7	−0.927540	−0.463770	+	0.885956i	$0.653504\pi$
$564$	0	0
$565$	19413.2	1.44552
$566$	8159.93	0.605985
$567$	0	0
$568$	2848.06	0.210391
$569$	8706.19	0.641445	0.320723	−	0.947173i	$-0.396074\pi$
$569$	8706.19	0.641445	0.320723	+	0.947173i	$0.396074\pi$
$570$	0	0
$571$	−22592.6	−1.65582	−0.827909	−	0.560862i	$-0.810470\pi$
$571$	−22592.6	−1.65582	−0.827909	+	0.560862i	$0.810470\pi$
$572$	−1195.60	−0.0873964
$573$	0	0
$574$	0	0
$575$	−1981.44	−0.143707
$576$	0	0
$577$	−11721.9	−0.845734	−0.422867	−	0.906192i	$-0.638976\pi$
$577$	−11721.9	−0.845734	−0.422867	+	0.906192i	$0.638976\pi$
$578$	5799.04	0.417315
$579$	0	0
$580$	−14254.1	−1.02046
$581$	0	0
$582$	0	0
$583$	7966.15	0.565908
$584$	−9891.27	−0.700863
$585$	0	0
$586$	−8227.25	−0.579974
$587$	18102.6	1.27287	0.636433	−	0.771332i	$-0.280409\pi$
$587$	18102.6	1.27287	0.636433	+	0.771332i	$0.280409\pi$
$588$	0	0
$589$	312.498	0.0218612
$590$	−1794.01	−0.125183
$591$	0	0
$592$	10512.0	0.729800
$593$	538.675	0.0373031	0.0186516	−	0.999826i	$-0.494063\pi$
$593$	538.675	0.0373031	0.0186516	+	0.999826i	$0.494063\pi$
$594$	0	0
$595$	0	0
$596$	13446.3	0.924128
$597$	0	0
$598$	−593.097	−0.0405578
$599$	−23628.2	−1.61172	−0.805861	−	0.592105i	$-0.798297\pi$
$599$	−23628.2	−1.61172	−0.805861	+	0.592105i	$0.798297\pi$
$600$	0	0
$601$	−3930.39	−0.266762	−0.133381	−	0.991065i	$-0.542583\pi$
$601$	−3930.39	−0.266762	−0.133381	+	0.991065i	$0.542583\pi$
$602$	0	0
$603$	0	0
$604$	−10770.5	−0.725569
$605$	−5439.29	−0.365518
$606$	0	0
$607$	−9149.06	−0.611778	−0.305889	−	0.952067i	$-0.598954\pi$
$607$	−9149.06	−0.611778	−0.305889	+	0.952067i	$0.598954\pi$
$608$	−5339.63	−0.356169
$609$	0	0
$610$	238.719	0.0158450
$611$	−1456.62	−0.0964457
$612$	0	0
$613$	−16844.1	−1.10983	−0.554917	−	0.831905i	$-0.687250\pi$
$613$	−16844.1	−1.10983	−0.554917	+	0.831905i	$0.687250\pi$
$614$	1680.99	0.110487
$615$	0	0
$616$	0	0
$617$	25080.5	1.63647	0.818235	−	0.574884i	$-0.194953\pi$
$617$	25080.5	1.63647	0.818235	+	0.574884i	$0.194953\pi$
$618$	0	0
$619$	−23686.3	−1.53802	−0.769009	−	0.639238i	$-0.779250\pi$
$619$	−23686.3	−1.53802	−0.769009	+	0.639238i	$0.779250\pi$
$620$	668.439	0.0432986
$621$	0	0
$622$	−6413.46	−0.413434
$623$	0	0
$624$	0	0
$625$	−12869.3	−0.823638
$626$	5806.78	0.370744
$627$	0	0
$628$	−25532.6	−1.62239
$629$	−25744.0	−1.63193
$630$	0	0
$631$	11732.0	0.740162	0.370081	−	0.929000i	$-0.379330\pi$
$631$	11732.0	0.740162	0.370081	+	0.929000i	$0.379330\pi$
$632$	−8963.82	−0.564180
$633$	0	0
$634$	−3530.89	−0.221182
$635$	−6114.20	−0.382102
$636$	0	0
$637$	0	0
$638$	−5320.78	−0.330175
$639$	0	0
$640$	−14798.4	−0.913996
$641$	9158.83	0.564356	0.282178	−	0.959362i	$-0.408943\pi$
$641$	9158.83	0.564356	0.282178	+	0.959362i	$0.408943\pi$
$642$	0	0
$643$	−29594.0	−1.81504	−0.907522	−	0.420003i	$-0.862029\pi$
$643$	−29594.0	−1.81504	−0.907522	+	0.420003i	$0.862029\pi$
$644$	0	0
$645$	0	0
$646$	3420.94	0.208352
$647$	4569.57	0.277663	0.138832	−	0.990316i	$-0.455665\pi$
$647$	4569.57	0.277663	0.138832	+	0.990316i	$0.455665\pi$
$648$	0	0
$649$	5145.28	0.311201
$650$	109.432	0.00660350
$651$	0	0
$652$	−6280.94	−0.377271
$653$	−21264.7	−1.27435	−0.637176	−	0.770718i	$-0.719898\pi$
$653$	−21264.7	−1.27435	−0.637176	+	0.770718i	$0.719898\pi$
$654$	0	0
$655$	−5084.35	−0.303300
$656$	15661.4	0.932128
$657$	0	0
$658$	0	0
$659$	−9214.64	−0.544691	−0.272345	−	0.962200i	$-0.587799\pi$
$659$	−9214.64	−0.544691	−0.272345	+	0.962200i	$0.587799\pi$
$660$	0	0
$661$	7730.00	0.454859	0.227430	−	0.973794i	$-0.426968\pi$
$661$	7730.00	0.454859	0.227430	+	0.973794i	$0.426968\pi$
$662$	10041.9	0.589564
$663$	0	0
$664$	2554.16	0.149278
$665$	0	0
$666$	0	0
$667$	20279.7	1.17726
$668$	−8841.03	−0.512080
$669$	0	0
$670$	−3326.33	−0.191802
$671$	−684.654	−0.0393901
$672$	0	0
$673$	14627.9	0.837835	0.418917	−	0.908024i	$-0.362410\pi$
$673$	14627.9	0.837835	0.418917	+	0.908024i	$0.362410\pi$
$674$	1029.12	0.0588133
$675$	0	0
$676$	15300.2	0.870517
$677$	27179.8	1.54299	0.771496	−	0.636234i	$-0.219509\pi$
$677$	27179.8	1.54299	0.771496	+	0.636234i	$0.219509\pi$
$678$	0	0
$679$	0	0
$680$	15587.3	0.879038
$681$	0	0
$682$	249.516	0.0140095
$683$	−34448.5	−1.92992	−0.964959	−	0.262402i	$-0.915485\pi$
$683$	−34448.5	−1.92992	−0.964959	+	0.262402i	$0.915485\pi$
$684$	0	0
$685$	−18593.7	−1.03712
$686$	0	0
$687$	0	0
$688$	−15657.0	−0.867614
$689$	1676.86	0.0927191
$690$	0	0
$691$	13766.2	0.757872	0.378936	−	0.925423i	$-0.376290\pi$
$691$	13766.2	0.757872	0.378936	+	0.925423i	$0.376290\pi$
$692$	−15081.5	−0.828488
$693$	0	0
$694$	−9178.46	−0.502031
$695$	32406.0	1.76868
$696$	0	0
$697$	−38355.0	−2.08436
$698$	3992.34	0.216493
$699$	0	0
$700$	0	0
$701$	−28107.2	−1.51440	−0.757200	−	0.653183i	$-0.773433\pi$
$701$	−28107.2	−1.51440	−0.757200	+	0.653183i	$0.773433\pi$
$702$	0	0
$703$	−8375.72	−0.449355
$704$	5421.34	0.290234
$705$	0	0
$706$	5762.72	0.307200
$707$	0	0
$708$	0	0
$709$	29866.8	1.58205	0.791023	−	0.611786i	$-0.209549\pi$
$709$	29866.8	1.58205	0.791023	+	0.611786i	$0.209549\pi$
$710$	−1943.53	−0.102732
$711$	0	0
$712$	11014.8	0.579772
$713$	−951.007	−0.0499516
$714$	0	0
$715$	1737.96	0.0909037
$716$	−19509.6	−1.01831
$717$	0	0
$718$	−11052.9	−0.574497
$719$	−20815.8	−1.07969	−0.539845	−	0.841764i	$-0.681517\pi$
$719$	−20815.8	−1.07969	−0.539845	+	0.841764i	$0.681517\pi$
$720$	0	0
$721$	0	0
$722$	−5470.61	−0.281988
$723$	0	0
$724$	−5201.43	−0.267002
$725$	−3741.79	−0.191678
$726$	0	0
$727$	−2638.00	−0.134578	−0.0672888	−	0.997734i	$-0.521435\pi$
$727$	−2638.00	−0.134578	−0.0672888	+	0.997734i	$0.521435\pi$
$728$	0	0
$729$	0	0
$730$	6749.86	0.342224
$731$	38344.2	1.94010
$732$	0	0
$733$	−19483.1	−0.981754	−0.490877	−	0.871229i	$-0.663323\pi$
$733$	−19483.1	−0.981754	−0.490877	+	0.871229i	$0.663323\pi$
$734$	−8285.33	−0.416645
$735$	0	0
$736$	16249.8	0.813825
$737$	9540.02	0.476813
$738$	0	0
$739$	−2190.08	−0.109017	−0.0545084	−	0.998513i	$-0.517359\pi$
$739$	−2190.08	−0.109017	−0.0545084	+	0.998513i	$0.517359\pi$
$740$	−17915.8	−0.889999
$741$	0	0
$742$	0	0
$743$	−21777.5	−1.07529	−0.537644	−	0.843172i	$-0.680685\pi$
$743$	−21777.5	−1.07529	−0.537644	+	0.843172i	$0.680685\pi$
$744$	0	0
$745$	−19545.8	−0.961214
$746$	−11121.3	−0.545815
$747$	0	0
$748$	−20986.7	−1.02587
$749$	0	0
$750$	0	0
$751$	−14790.8	−0.718672	−0.359336	−	0.933208i	$-0.616997\pi$
$751$	−14790.8	−0.718672	−0.359336	+	0.933208i	$0.616997\pi$
$752$	10440.2	0.506272
$753$	0	0
$754$	−1120.02	−0.0540964
$755$	15656.2	0.754687
$756$	0	0
$757$	30374.8	1.45838	0.729188	−	0.684313i	$-0.239898\pi$
$757$	30374.8	1.45838	0.729188	+	0.684313i	$0.239898\pi$
$758$	10.9378	0.000524115 0
$759$	0	0
$760$	5071.27	0.242045
$761$	−14854.4	−0.707584	−0.353792	−	0.935324i	$-0.615108\pi$
$761$	−14854.4	−0.707584	−0.353792	+	0.935324i	$0.615108\pi$
$762$	0	0
$763$	0	0
$764$	−14401.5	−0.681975
$765$	0	0
$766$	−7342.75	−0.346350
$767$	1083.07	0.0509877
$768$	0	0
$769$	3016.48	0.141453	0.0707264	−	0.997496i	$-0.477468\pi$
$769$	3016.48	0.141453	0.0707264	+	0.997496i	$0.477468\pi$
$770$	0	0
$771$	0	0
$772$	4701.44	0.219182
$773$	−5047.57	−0.234862	−0.117431	−	0.993081i	$-0.537466\pi$
$773$	−5047.57	−0.234862	−0.117431	+	0.993081i	$0.537466\pi$
$774$	0	0
$775$	175.470	0.00813298
$776$	−18490.9	−0.855395
$777$	0	0
$778$	−978.008	−0.0450685
$779$	−12478.6	−0.573933
$780$	0	0
$781$	5574.11	0.255387
$782$	−10410.8	−0.476072
$783$	0	0
$784$	0	0
$785$	37114.8	1.68750
$786$	0	0
$787$	19962.3	0.904166	0.452083	−	0.891976i	$-0.350681\pi$
$787$	19962.3	0.904166	0.452083	+	0.891976i	$0.350681\pi$
$788$	−11909.2	−0.538383
$789$	0	0
$790$	6116.96	0.275483
$791$	0	0
$792$	0	0
$793$	−144.119	−0.00645374
$794$	1164.95	0.0520686
$795$	0	0
$796$	−26743.7	−1.19084
$797$	806.334	0.0358367	0.0179183	−	0.999839i	$-0.494296\pi$
$797$	806.334	0.0358367	0.0179183	+	0.999839i	$0.494296\pi$
$798$	0	0
$799$	−25568.3	−1.13209
$800$	−2998.24	−0.132505
$801$	0	0
$802$	6888.40	0.303289
$803$	−19358.8	−0.850756
$804$	0	0
$805$	0	0
$806$	52.5227	0.00229533
$807$	0	0
$808$	−22517.8	−0.980414
$809$	42013.3	1.82584	0.912922	−	0.408134i	$-0.133820\pi$
$809$	42013.3	1.82584	0.912922	+	0.408134i	$0.133820\pi$
$810$	0	0
$811$	−2938.48	−0.127231	−0.0636153	−	0.997974i	$-0.520263\pi$
$811$	−2938.48	−0.127231	−0.0636153	+	0.997974i	$0.520263\pi$
$812$	0	0
$813$	0	0
$814$	−6687.64	−0.287963
$815$	9130.14	0.392411
$816$	0	0
$817$	12475.1	0.534210
$818$	14291.0	0.610848
$819$	0	0
$820$	−26692.1	−1.13674
$821$	−24232.8	−1.03012	−0.515060	−	0.857154i	$-0.672230\pi$
$821$	−24232.8	−1.03012	−0.515060	+	0.857154i	$0.672230\pi$
$822$	0	0
$823$	−27959.8	−1.18422	−0.592112	−	0.805856i	$-0.701706\pi$
$823$	−27959.8	−1.18422	−0.592112	+	0.805856i	$0.701706\pi$
$824$	3564.97	0.150718
$825$	0	0
$826$	0	0
$827$	11855.3	0.498487	0.249244	−	0.968441i	$-0.419818\pi$
$827$	11855.3	0.498487	0.249244	+	0.968441i	$0.419818\pi$
$828$	0	0
$829$	43362.5	1.81670	0.908349	−	0.418213i	$-0.137343\pi$
$829$	43362.5	1.81670	0.908349	+	0.418213i	$0.137343\pi$
$830$	−1742.97	−0.0728910
$831$	0	0
$832$	1141.19	0.0475523
$833$	0	0
$834$	0	0
$835$	12851.6	0.532630
$836$	−6827.94	−0.282475
$837$	0	0
$838$	−7203.62	−0.296951
$839$	20467.7	0.842221	0.421111	−	0.907009i	$-0.361640\pi$
$839$	20467.7	0.842221	0.421111	+	0.907009i	$0.361640\pi$
$840$	0	0
$841$	13907.7	0.570243
$842$	8224.89	0.336637
$843$	0	0
$844$	36263.5	1.47896
$845$	−22240.8	−0.905452
$846$	0	0
$847$	0	0
$848$	−12018.9	−0.486710
$849$	0	0
$850$	1920.88	0.0775126
$851$	25489.4	1.02675
$852$	0	0
$853$	17910.2	0.718914	0.359457	−	0.933162i	$-0.382962\pi$
$853$	17910.2	0.718914	0.359457	+	0.933162i	$0.382962\pi$
$854$	0	0
$855$	0	0
$856$	−23468.3	−0.937069
$857$	4494.31	0.179140	0.0895698	−	0.995981i	$-0.471451\pi$
$857$	4494.31	0.179140	0.0895698	+	0.995981i	$0.471451\pi$
$858$	0	0
$859$	14367.1	0.570663	0.285331	−	0.958429i	$-0.407896\pi$
$859$	14367.1	0.570663	0.285331	+	0.958429i	$0.407896\pi$
$860$	26684.5	1.05806
$861$	0	0
$862$	−1794.78	−0.0709169
$863$	−12240.1	−0.482802	−0.241401	−	0.970425i	$-0.577607\pi$
$863$	−12240.1	−0.482802	−0.241401	+	0.970425i	$0.577607\pi$
$864$	0	0
$865$	21922.9	0.861736
$866$	10752.3	0.421914
$867$	0	0
$868$	0	0
$869$	−17543.6	−0.684841
$870$	0	0
$871$	2008.16	0.0781217
$872$	−27596.0	−1.07170
$873$	0	0
$874$	−3387.10	−0.131087
$875$	0	0
$876$	0	0
$877$	−18127.6	−0.697978	−0.348989	−	0.937127i	$-0.613475\pi$
$877$	−18127.6	−0.697978	−0.348989	+	0.937127i	$0.613475\pi$
$878$	9421.57	0.362144
$879$	0	0
$880$	−12456.8	−0.477180
$881$	−28737.3	−1.09896	−0.549480	−	0.835507i	$-0.685174\pi$
$881$	−28737.3	−1.09896	−0.549480	+	0.835507i	$0.685174\pi$
$882$	0	0
$883$	29899.7	1.13953	0.569765	−	0.821807i	$-0.307034\pi$
$883$	29899.7	1.13953	0.569765	+	0.821807i	$0.307034\pi$
$884$	−4417.67	−0.168080
$885$	0	0
$886$	−5573.84	−0.211351
$887$	35540.7	1.34537	0.672683	−	0.739931i	$-0.265142\pi$
$887$	35540.7	1.34537	0.672683	+	0.739931i	$0.265142\pi$
$888$	0	0
$889$	0	0
$890$	−7516.57	−0.283097
$891$	0	0
$892$	−14997.7	−0.562959
$893$	−8318.54	−0.311724
$894$	0	0
$895$	28359.7	1.05917
$896$	0	0
$897$	0	0
$898$	247.034	0.00917998
$899$	−1795.90	−0.0666260
$900$	0	0
$901$	29434.4	1.08835
$902$	−9963.64	−0.367797
$903$	0	0
$904$	−27305.9	−1.00462
$905$	7560.94	0.277717
$906$	0	0
$907$	−16816.0	−0.615617	−0.307808	−	0.951448i	$-0.599596\pi$
$907$	−16816.0	−0.615617	−0.307808	+	0.951448i	$0.599596\pi$
$908$	30010.6	1.09685
$909$	0	0
$910$	0	0
$911$	21626.8	0.786527	0.393264	−	0.919426i	$-0.371346\pi$
$911$	21626.8	0.786527	0.393264	+	0.919426i	$0.371346\pi$
$912$	0	0
$913$	4998.91	0.181204
$914$	1390.57	0.0503239
$915$	0	0
$916$	10746.8	0.387645
$917$	0	0
$918$	0	0
$919$	31512.2	1.13111	0.565556	−	0.824710i	$-0.308662\pi$
$919$	31512.2	1.13111	0.565556	+	0.824710i	$0.308662\pi$
$920$	−15433.1	−0.553059
$921$	0	0
$922$	−8363.33	−0.298733
$923$	1173.34	0.0418429
$924$	0	0
$925$	−4703.03	−0.167173
$926$	17052.8	0.605174
$927$	0	0
$928$	30686.5	1.08549
$929$	−25267.9	−0.892373	−0.446186	−	0.894940i	$-0.647218\pi$
$929$	−25267.9	−0.892373	−0.446186	+	0.894940i	$0.647218\pi$
$930$	0	0
$931$	0	0
$932$	−32536.3	−1.14352
$933$	0	0
$934$	13183.5	0.461861
$935$	30506.8	1.06704
$936$	0	0
$937$	14694.4	0.512321	0.256161	−	0.966634i	$-0.417542\pi$
$937$	14694.4	0.512321	0.256161	+	0.966634i	$0.417542\pi$
$938$	0	0
$939$	0	0
$940$	−17793.5	−0.617405
$941$	−7004.20	−0.242647	−0.121323	−	0.992613i	$-0.538714\pi$
$941$	−7004.20	−0.242647	−0.121323	+	0.992613i	$0.538714\pi$
$942$	0	0
$943$	37975.6	1.31140
$944$	−7762.91	−0.267649
$945$	0	0
$946$	9960.84	0.342341
$947$	−10065.2	−0.345379	−0.172690	−	0.984976i	$-0.555246\pi$
$947$	−10065.2	−0.345379	−0.172690	+	0.984976i	$0.555246\pi$
$948$	0	0
$949$	−4075.00	−0.139389
$950$	624.951	0.0213433
$951$	0	0
$952$	0	0
$953$	15265.2	0.518876	0.259438	−	0.965760i	$-0.416463\pi$
$953$	15265.2	0.518876	0.259438	+	0.965760i	$0.416463\pi$
$954$	0	0
$955$	20934.5	0.709343
$956$	20930.5	0.708096
$957$	0	0
$958$	−4628.03	−0.156080
$959$	0	0
$960$	0	0
$961$	−29706.8	−0.997173
$962$	−1407.74	−0.0471802
$963$	0	0
$964$	−799.940	−0.0267265
$965$	−6834.14	−0.227978
$966$	0	0
$967$	−32705.8	−1.08764	−0.543820	−	0.839202i	$-0.683022\pi$
$967$	−32705.8	−1.08764	−0.543820	+	0.839202i	$0.683022\pi$
$968$	7650.73	0.254033
$969$	0	0
$970$	12618.3	0.417680
$971$	48338.6	1.59759	0.798795	−	0.601603i	$-0.205471\pi$
$971$	48338.6	1.59759	0.798795	+	0.601603i	$0.205471\pi$
$972$	0	0
$973$	0	0
$974$	−13056.8	−0.429535
$975$	0	0
$976$	1032.97	0.0338776
$977$	21230.1	0.695200	0.347600	−	0.937643i	$-0.386997\pi$
$977$	21230.1	0.695200	0.347600	+	0.937643i	$0.386997\pi$
$978$	0	0
$979$	21557.8	0.703768
$980$	0	0
$981$	0	0
$982$	−5890.99	−0.191435
$983$	−54979.2	−1.78389	−0.891946	−	0.452142i	$-0.850660\pi$
$983$	−54979.2	−1.78389	−0.891946	+	0.452142i	$0.850660\pi$
$984$	0	0
$985$	17311.5	0.559989
$986$	−19659.9	−0.634989
$987$	0	0
$988$	−1437.27	−0.0462811
$989$	−37964.9	−1.22064
$990$	0	0
$991$	12954.3	0.415245	0.207622	−	0.978209i	$-0.433427\pi$
$991$	12954.3	0.415245	0.207622	+	0.978209i	$0.433427\pi$
$992$	−1439.03	−0.0460577
$993$	0	0
$994$	0	0
$995$	38875.4	1.23863
$996$	0	0
$997$	13382.0	0.425088	0.212544	−	0.977151i	$-0.431825\pi$
$997$	13382.0	0.425088	0.212544	+	0.977151i	$0.431825\pi$
$998$	−13125.0	−0.416299
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1323.4.a.bm.1.5	yes	8
3.2	odd	2	inner	1323.4.a.bm.1.4	yes	8
7.6	odd	2		1323.4.a.bl.1.5	yes	8
21.20	even	2		1323.4.a.bl.1.4	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1323.4.a.bl.1.4	✓	8	21.20	even	2
1323.4.a.bl.1.5	yes	8	7.6	odd	2
1323.4.a.bm.1.4	yes	8	3.2	odd	2	inner
1323.4.a.bm.1.5	yes	8	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1323 = 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1323.a (trivial)

\(f(q)\)	\(=\)	\(q+0.959848 q^{2} -7.07869 q^{4} +10.2898 q^{5} -14.4733 q^{8} +9.87663 q^{10} -28.3265 q^{11} -5.96269 q^{13} +42.7374 q^{16} -104.664 q^{17} -34.0521 q^{19} -72.8381 q^{20} -27.1891 q^{22} +103.629 q^{23} -19.1205 q^{25} -5.72328 q^{26} +195.695 q^{29} -9.17704 q^{31} +156.807 q^{32} -100.462 q^{34} +245.968 q^{37} -32.6849 q^{38} -148.927 q^{40} +366.457 q^{41} -366.354 q^{43} +200.514 q^{44} +99.4681 q^{46} +244.288 q^{47} -18.3528 q^{50} +42.2080 q^{52} -281.226 q^{53} -291.473 q^{55} +187.838 q^{58} -181.642 q^{59} +24.1701 q^{61} -8.80857 q^{62} -191.388 q^{64} -61.3547 q^{65} -336.788 q^{67} +740.886 q^{68} -196.781 q^{71} +683.417 q^{73} +236.092 q^{74} +241.044 q^{76} +619.337 q^{79} +439.758 q^{80} +351.743 q^{82} -176.475 q^{83} -1076.97 q^{85} -351.644 q^{86} +409.976 q^{88} -761.046 q^{89} -733.557 q^{92} +234.480 q^{94} -350.389 q^{95} +1277.59 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 44 q^{4} + 132 q^{10} + 336 q^{13} + 204 q^{16} - 288 q^{19} + 484 q^{22} + 152 q^{25} + 120 q^{31} + 1008 q^{34} + 592 q^{37} + 1620 q^{40} - 1872 q^{43} - 1644 q^{46} + 2400 q^{52} + 1344 q^{55}+ \cdots + 5712 q^{97}+O(q^{100}) \) 8 * q + 44 * q^4 + 132 * q^10 + 336 * q^13 + 204 * q^16 - 288 * q^19 + 484 * q^22 + 152 * q^25 + 120 * q^31 + 1008 * q^34 + 592 * q^37 + 1620 * q^40 - 1872 * q^43 - 1644 * q^46 + 2400 * q^52 + 1344 * q^55 - 1200 * q^58 + 2400 * q^61 - 1388 * q^64 + 1824 * q^73 + 2844 * q^76 + 2368 * q^79 + 2436 * q^82 + 3512 * q^85 + 3780 * q^88 + 4368 * q^94 + 5712 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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