LMFDB - Embedded newform 675.4.a.y.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [675,4,Mod(1,675)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("675.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(675, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	675.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,1,0,19,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$39.8262892539$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.183945.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 12x^{2} + 3x + 18 $ x^4 - x^3 - 12x^2 + 3x + 18
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{3}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.67875$ of defining polynomial
Character	$\chi$	$=$	675.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-4.72651 q^{2} +14.3399 q^{4} -17.6256 q^{7} -29.9655 q^{8} +34.2390 q^{11} -53.8784 q^{13} +83.3075 q^{14} +26.9130 q^{16} +74.7605 q^{17} -89.5599 q^{19} -161.831 q^{22} -176.169 q^{23} +254.657 q^{26} -252.749 q^{28} +194.211 q^{29} +107.939 q^{31} +112.519 q^{32} -353.356 q^{34} +430.818 q^{37} +423.305 q^{38} -108.894 q^{41} -409.261 q^{43} +490.982 q^{44} +832.666 q^{46} -409.188 q^{47} -32.3387 q^{49} -772.610 q^{52} +24.7760 q^{53} +528.159 q^{56} -917.940 q^{58} -295.748 q^{59} +305.325 q^{61} -510.176 q^{62} -747.127 q^{64} -915.415 q^{67} +1072.06 q^{68} +228.340 q^{71} +158.720 q^{73} -2036.27 q^{74} -1284.28 q^{76} -603.482 q^{77} -319.140 q^{79} +514.686 q^{82} +936.446 q^{83} +1934.38 q^{86} -1025.99 q^{88} +920.893 q^{89} +949.639 q^{91} -2526.25 q^{92} +1934.03 q^{94} +914.533 q^{97} +152.849 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{2} + 19 q^{4} - 4 q^{7} - 15 q^{8} + 52 q^{11} + 2 q^{13} + 138 q^{14} - 5 q^{16} + 64 q^{17} - 46 q^{19} - 87 q^{22} - 90 q^{23} + 469 q^{26} + 110 q^{28} + 470 q^{29} - 262 q^{31} - 199 q^{32}+ \cdots + 4301 q^{98}+O(q^{100}) $ 4 * q + q^2 + 19 * q^4 - 4 * q^7 - 15 * q^8 + 52 * q^11 + 2 * q^13 + 138 * q^14 - 5 * q^16 + 64 * q^17 - 46 * q^19 - 87 * q^22 - 90 * q^23 + 469 * q^26 + 110 * q^28 + 470 * q^29 - 262 * q^31 - 199 * q^32 - 42 * q^34 + 542 * q^37 + 532 * q^38 + 698 * q^41 - 142 * q^43 + 419 * q^44 + 537 * q^46 - 542 * q^47 + 780 * q^49 - 409 * q^52 + 910 * q^53 + 2034 * q^56 + 576 * q^58 + 100 * q^59 + 74 * q^61 - 2406 * q^62 - 965 * q^64 - 928 * q^67 + 2810 * q^68 + 1622 * q^71 + 536 * q^73 - 253 * q^74 - 2068 * q^76 - 1932 * q^77 - 508 * q^79 + 1782 * q^82 + 1524 * q^83 + 3940 * q^86 - 2247 * q^88 + 756 * q^89 + 1120 * q^91 - 3645 * q^92 + 2847 * q^94 - 892 * q^97 + 4301 * q^98

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 $

$2$	−4.72651	−1.67107	−0.835536	−	0.549435i	$-0.814843\pi$
$2$	−4.72651	−1.67107	−0.835536	+	0.549435i	$0.814843\pi$
$3$	0	0
$3$	0	0
$4$	14.3399	1.79248
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−17.6256	−0.951692	−0.475846	−	0.879529i	$-0.657858\pi$
$7$	−17.6256	−0.951692	−0.475846	+	0.879529i	$0.657858\pi$
$8$	−29.9655	−1.32430
$9$	0	0
$10$	0	0
$11$	34.2390	0.938494	0.469247	−	0.883067i	$-0.344525\pi$
$11$	34.2390	0.938494	0.469247	+	0.883067i	$0.344525\pi$
$12$	0	0
$13$	−53.8784	−1.14948	−0.574738	−	0.818337i	$-0.694896\pi$
$13$	−53.8784	−1.14948	−0.574738	+	0.818337i	$0.694896\pi$
$14$	83.3075	1.59035
$15$	0	0
$16$	26.9130	0.420515
$17$	74.7605	1.06659	0.533296	−	0.845928i	$-0.320953\pi$
$17$	74.7605	1.06659	0.533296	+	0.845928i	$0.320953\pi$
$18$	0	0
$19$	−89.5599	−1.08139	−0.540696	−	0.841218i	$-0.681839\pi$
$19$	−89.5599	−1.08139	−0.540696	+	0.841218i	$0.681839\pi$
$20$	0	0
$21$	0	0
$22$	−161.831	−1.56829
$23$	−176.169	−1.59712	−0.798562	−	0.601913i	$-0.794405\pi$
$23$	−176.169	−1.59712	−0.798562	+	0.601913i	$0.794405\pi$
$24$	0	0
$25$	0	0
$26$	254.657	1.92086
$27$	0	0
$28$	−252.749	−1.70589
$29$	194.211	1.24359	0.621795	−	0.783180i	$-0.286404\pi$
$29$	194.211	1.24359	0.621795	+	0.783180i	$0.286404\pi$
$30$	0	0
$31$	107.939	0.625370	0.312685	−	0.949857i	$-0.398772\pi$
$31$	107.939	0.625370	0.312685	+	0.949857i	$0.398772\pi$
$32$	112.519	0.621587
$33$	0	0
$34$	−353.356	−1.78235
$35$	0	0
$36$	0	0
$37$	430.818	1.91422	0.957110	−	0.289726i	$-0.0935642\pi$
$37$	430.818	1.91422	0.957110	+	0.289726i	$0.0935642\pi$
$38$	423.305	1.80708
$39$	0	0
$40$	0	0
$41$	−108.894	−0.414788	−0.207394	−	0.978257i	$-0.566498\pi$
$41$	−108.894	−0.414788	−0.207394	+	0.978257i	$0.566498\pi$
$42$	0	0
$43$	−409.261	−1.45144	−0.725718	−	0.687992i	$-0.758492\pi$
$43$	−409.261	−1.45144	−0.725718	+	0.687992i	$0.758492\pi$
$44$	490.982	1.68224
$45$	0	0
$46$	832.666	2.66891
$47$	−409.188	−1.26992	−0.634959	−	0.772546i	$-0.718983\pi$
$47$	−409.188	−1.26992	−0.634959	+	0.772546i	$0.718983\pi$
$48$	0	0
$49$	−32.3387	−0.0942820
$50$	0	0
$51$	0	0
$52$	−772.610	−2.06042
$53$	24.7760	0.0642121	0.0321061	−	0.999484i	$-0.489779\pi$
$53$	24.7760	0.0642121	0.0321061	+	0.999484i	$0.489779\pi$
$54$	0	0
$55$	0	0
$56$	528.159	1.26032
$57$	0	0
$58$	−917.940	−2.07813
$59$	−295.748	−0.652596	−0.326298	−	0.945267i	$-0.605801\pi$
$59$	−295.748	−0.652596	−0.326298	+	0.945267i	$0.605801\pi$
$60$	0	0
$61$	305.325	0.640868	0.320434	−	0.947271i	$-0.396171\pi$
$61$	305.325	0.640868	0.320434	+	0.947271i	$0.396171\pi$
$62$	−510.176	−1.04504
$63$	0	0
$64$	−747.127	−1.45923
$65$	0	0
$66$	0	0
$67$	−915.415	−1.66919	−0.834595	−	0.550864i	$-0.814298\pi$
$67$	−915.415	−1.66919	−0.834595	+	0.550864i	$0.814298\pi$
$68$	1072.06	1.91185
$69$	0	0
$70$	0	0
$71$	228.340	0.381675	0.190838	−	0.981622i	$-0.438880\pi$
$71$	228.340	0.381675	0.190838	+	0.981622i	$0.438880\pi$
$72$	0	0
$73$	158.720	0.254476	0.127238	−	0.991872i	$-0.459389\pi$
$73$	158.720	0.254476	0.127238	+	0.991872i	$0.459389\pi$
$74$	−2036.27	−3.19880
$75$	0	0
$76$	−1284.28	−1.93838
$77$	−603.482	−0.893157
$78$	0	0
$79$	−319.140	−0.454507	−0.227254	−	0.973836i	$-0.572975\pi$
$79$	−319.140	−0.454507	−0.227254	+	0.973836i	$0.572975\pi$
$80$	0	0
$81$	0	0
$82$	514.686	0.693141
$83$	936.446	1.23841	0.619207	−	0.785228i	$-0.287454\pi$
$83$	936.446	1.23841	0.619207	+	0.785228i	$0.287454\pi$
$84$	0	0
$85$	0	0
$86$	1934.38	2.42545
$87$	0	0
$88$	−1025.99	−1.24285
$89$	920.893	1.09679	0.548396	−	0.836219i	$-0.315239\pi$
$89$	920.893	1.09679	0.548396	+	0.836219i	$0.315239\pi$
$90$	0	0
$91$	949.639	1.09395
$92$	−2526.25	−2.86282
$93$	0	0
$94$	1934.03	2.12213
$95$	0	0
$96$	0	0
$97$	914.533	0.957286	0.478643	−	0.878010i	$-0.341129\pi$
$97$	914.533	0.957286	0.478643	+	0.878010i	$0.341129\pi$
$98$	152.849	0.157552
$99$	0	0
$100$	0	0
$101$	−942.162	−0.928205	−0.464102	−	0.885782i	$-0.653623\pi$
$101$	−942.162	−0.928205	−0.464102	+	0.885782i	$0.653623\pi$
$102$	0	0
$103$	1204.14	1.15192	0.575960	−	0.817478i	$-0.304628\pi$
$103$	1204.14	1.15192	0.575960	+	0.817478i	$0.304628\pi$
$104$	1614.49	1.52225
$105$	0	0
$106$	−117.104	−0.107303
$107$	1416.82	1.28008	0.640042	−	0.768340i	$-0.278917\pi$
$107$	1416.82	1.28008	0.640042	+	0.768340i	$0.278917\pi$
$108$	0	0
$109$	−1432.65	−1.25893	−0.629464	−	0.777029i	$-0.716726\pi$
$109$	−1432.65	−1.25893	−0.629464	+	0.777029i	$0.716726\pi$
$110$	0	0
$111$	0	0
$112$	−474.357	−0.400201
$113$	−713.619	−0.594085	−0.297043	−	0.954864i	$-0.596000\pi$
$113$	−713.619	−0.594085	−0.297043	+	0.954864i	$0.596000\pi$
$114$	0	0
$115$	0	0
$116$	2784.96	2.22911
$117$	0	0
$118$	1397.86	1.09054
$119$	−1317.70	−1.01507
$120$	0	0
$121$	−158.694	−0.119229
$122$	−1443.12	−1.07094
$123$	0	0
$124$	1547.84	1.12097
$125$	0	0
$126$	0	0
$127$	2507.42	1.75195	0.875974	−	0.482359i	$-0.160220\pi$
$127$	2507.42	1.75195	0.875974	+	0.482359i	$0.160220\pi$
$128$	2631.15	1.81690
$129$	0	0
$130$	0	0
$131$	1003.74	0.669444	0.334722	−	0.942317i	$-0.391358\pi$
$131$	1003.74	0.669444	0.334722	+	0.942317i	$0.391358\pi$
$132$	0	0
$133$	1578.55	1.02915
$134$	4326.72	2.78934
$135$	0	0
$136$	−2240.23	−1.41249
$137$	821.572	0.512347	0.256174	−	0.966631i	$-0.417538\pi$
$137$	821.572	0.512347	0.256174	+	0.966631i	$0.417538\pi$
$138$	0	0
$139$	−434.511	−0.265142	−0.132571	−	0.991174i	$-0.542323\pi$
$139$	−434.511	−0.265142	−0.132571	+	0.991174i	$0.542323\pi$
$140$	0	0
$141$	0	0
$142$	−1079.25	−0.637807
$143$	−1844.74	−1.07878
$144$	0	0
$145$	0	0
$146$	−750.191	−0.425248
$147$	0	0
$148$	6177.88	3.43121
$149$	3379.68	1.85822	0.929109	−	0.369807i	$-0.120576\pi$
$149$	3379.68	1.85822	0.929109	+	0.369807i	$0.120576\pi$
$150$	0	0
$151$	1003.54	0.540838	0.270419	−	0.962743i	$-0.412838\pi$
$151$	1003.54	0.540838	0.270419	+	0.962743i	$0.412838\pi$
$152$	2683.70	1.43209
$153$	0	0
$154$	2852.36	1.49253
$155$	0	0
$156$	0	0
$157$	3652.95	1.85693	0.928463	−	0.371426i	$-0.121131\pi$
$157$	3652.95	1.85693	0.928463	+	0.371426i	$0.121131\pi$
$158$	1508.42	0.759514
$159$	0	0
$160$	0	0
$161$	3105.09	1.51997
$162$	0	0
$163$	705.127	0.338833	0.169416	−	0.985545i	$-0.445812\pi$
$163$	705.127	0.338833	0.169416	+	0.985545i	$0.445812\pi$
$164$	−1561.52	−0.743501
$165$	0	0
$166$	−4426.12	−2.06948
$167$	2471.73	1.14532	0.572658	−	0.819794i	$-0.305912\pi$
$167$	2471.73	1.14532	0.572658	+	0.819794i	$0.305912\pi$
$168$	0	0
$169$	705.886	0.321296
$170$	0	0
$171$	0	0
$172$	−5868.75	−2.60168
$173$	384.262	0.168872	0.0844360	−	0.996429i	$-0.473091\pi$
$173$	384.262	0.168872	0.0844360	+	0.996429i	$0.473091\pi$
$174$	0	0
$175$	0	0
$176$	921.472	0.394651
$177$	0	0
$178$	−4352.61	−1.83282
$179$	2775.75	1.15904	0.579522	−	0.814956i	$-0.303239\pi$
$179$	2775.75	1.15904	0.579522	+	0.814956i	$0.303239\pi$
$180$	0	0
$181$	−3653.40	−1.50030	−0.750151	−	0.661266i	$-0.770019\pi$
$181$	−3653.40	−1.50030	−0.750151	+	0.661266i	$0.770019\pi$
$182$	−4488.48	−1.82807
$183$	0	0
$184$	5278.99	2.11507
$185$	0	0
$186$	0	0
$187$	2559.72	1.00099
$188$	−5867.70	−2.27631
$189$	0	0
$190$	0	0
$191$	1045.38	0.396028	0.198014	−	0.980199i	$-0.436551\pi$
$191$	1045.38	0.396028	0.198014	+	0.980199i	$0.436551\pi$
$192$	0	0
$193$	1442.74	0.538087	0.269043	−	0.963128i	$-0.413292\pi$
$193$	1442.74	0.538087	0.269043	+	0.963128i	$0.413292\pi$
$194$	−4322.55	−1.59969
$195$	0	0
$196$	−463.733	−0.168999
$197$	−1363.99	−0.493300	−0.246650	−	0.969105i	$-0.579330\pi$
$197$	−1363.99	−0.493300	−0.246650	+	0.969105i	$0.579330\pi$
$198$	0	0
$199$	921.087	0.328111	0.164056	−	0.986451i	$-0.447542\pi$
$199$	921.087	0.328111	0.164056	+	0.986451i	$0.447542\pi$
$200$	0	0
$201$	0	0
$202$	4453.14	1.55110
$203$	−3423.08	−1.18351
$204$	0	0
$205$	0	0
$206$	−5691.39	−1.92494
$207$	0	0
$208$	−1450.03	−0.483372
$209$	−3066.44	−1.01488
$210$	0	0
$211$	4461.60	1.45568	0.727842	−	0.685745i	$-0.240523\pi$
$211$	4461.60	1.45568	0.727842	+	0.685745i	$0.240523\pi$
$212$	355.284	0.115099
$213$	0	0
$214$	−6696.60	−2.13911
$215$	0	0
$216$	0	0
$217$	−1902.49	−0.595160
$218$	6771.44	2.10376
$219$	0	0
$220$	0	0
$221$	−4027.98	−1.22602
$222$	0	0
$223$	2623.53	0.787822	0.393911	−	0.919149i	$-0.371122\pi$
$223$	2623.53	0.787822	0.393911	+	0.919149i	$0.371122\pi$
$224$	−1983.22	−0.591559
$225$	0	0
$226$	3372.92	0.992759
$227$	−2571.02	−0.751739	−0.375870	−	0.926673i	$-0.622656\pi$
$227$	−2571.02	−0.751739	−0.375870	+	0.926673i	$0.622656\pi$
$228$	0	0
$229$	−3814.54	−1.10075	−0.550375	−	0.834918i	$-0.685515\pi$
$229$	−3814.54	−1.10075	−0.550375	+	0.834918i	$0.685515\pi$
$230$	0	0
$231$	0	0
$232$	−5819.62	−1.64688
$233$	4827.35	1.35730	0.678649	−	0.734463i	$-0.262566\pi$
$233$	4827.35	1.35730	0.678649	+	0.734463i	$0.262566\pi$
$234$	0	0
$235$	0	0
$236$	−4240.99	−1.16977
$237$	0	0
$238$	6228.11	1.69625
$239$	−4165.97	−1.12751	−0.563753	−	0.825943i	$-0.690643\pi$
$239$	−4165.97	−1.12751	−0.563753	+	0.825943i	$0.690643\pi$
$240$	0	0
$241$	5232.03	1.39844	0.699221	−	0.714906i	$-0.253530\pi$
$241$	5232.03	1.39844	0.699221	+	0.714906i	$0.253530\pi$
$242$	750.069	0.199241
$243$	0	0
$244$	4378.33	1.14874
$245$	0	0
$246$	0	0
$247$	4825.35	1.24303
$248$	−3234.45	−0.828177
$249$	0	0
$250$	0	0
$251$	4222.11	1.06174	0.530871	−	0.847453i	$-0.321865\pi$
$251$	4222.11	1.06174	0.530871	+	0.847453i	$0.321865\pi$
$252$	0	0
$253$	−6031.85	−1.49889
$254$	−11851.3	−2.92763
$255$	0	0
$256$	−6459.12	−1.57693
$257$	6104.54	1.48168	0.740838	−	0.671684i	$-0.234429\pi$
$257$	6104.54	1.48168	0.740838	+	0.671684i	$0.234429\pi$
$258$	0	0
$259$	−7593.43	−1.82175
$260$	0	0
$261$	0	0
$262$	−4744.18	−1.11869
$263$	−4328.06	−1.01475	−0.507376	−	0.861725i	$-0.669384\pi$
$263$	−4328.06	−1.01475	−0.507376	+	0.861725i	$0.669384\pi$
$264$	0	0
$265$	0	0
$266$	−7461.01	−1.71979
$267$	0	0
$268$	−13126.9	−2.99200
$269$	4131.70	0.936484	0.468242	−	0.883600i	$-0.344888\pi$
$269$	4131.70	0.936484	0.468242	+	0.883600i	$0.344888\pi$
$270$	0	0
$271$	3274.65	0.734024	0.367012	−	0.930216i	$-0.380381\pi$
$271$	3274.65	0.734024	0.367012	+	0.930216i	$0.380381\pi$
$272$	2012.03	0.448519
$273$	0	0
$274$	−3883.16	−0.856170
$275$	0	0
$276$	0	0
$277$	2779.74	0.602955	0.301478	−	0.953473i	$-0.402520\pi$
$277$	2779.74	0.602955	0.301478	+	0.953473i	$0.402520\pi$
$278$	2053.72	0.443072
$279$	0	0
$280$	0	0
$281$	−1386.05	−0.294252	−0.147126	−	0.989118i	$-0.547002\pi$
$281$	−1386.05	−0.294252	−0.147126	+	0.989118i	$0.547002\pi$
$282$	0	0
$283$	−743.439	−0.156159	−0.0780793	−	0.996947i	$-0.524879\pi$
$283$	−743.439	−0.156159	−0.0780793	+	0.996947i	$0.524879\pi$
$284$	3274.36	0.684147
$285$	0	0
$286$	8719.18	1.80271
$287$	1919.31	0.394751
$288$	0	0
$289$	676.129	0.137620
$290$	0	0
$291$	0	0
$292$	2276.02	0.456145
$293$	4366.87	0.870701	0.435350	−	0.900261i	$-0.356624\pi$
$293$	4366.87	0.870701	0.435350	+	0.900261i	$0.356624\pi$
$294$	0	0
$295$	0	0
$296$	−12909.7	−2.53500
$297$	0	0
$298$	−15974.1	−3.10522
$299$	9491.73	1.83586
$300$	0	0
$301$	7213.47	1.38132
$302$	−4743.22	−0.903780
$303$	0	0
$304$	−2410.32	−0.454742
$305$	0	0
$306$	0	0
$307$	−2887.01	−0.536712	−0.268356	−	0.963320i	$-0.586480\pi$
$307$	−2887.01	−0.536712	−0.268356	+	0.963320i	$0.586480\pi$
$308$	−8653.85	−1.60097
$309$	0	0
$310$	0	0
$311$	−4549.32	−0.829481	−0.414740	−	0.909940i	$-0.636128\pi$
$311$	−4549.32	−0.829481	−0.414740	+	0.909940i	$0.636128\pi$
$312$	0	0
$313$	−3385.58	−0.611387	−0.305694	−	0.952130i	$-0.598888\pi$
$313$	−3385.58	−0.611387	−0.305694	+	0.952130i	$0.598888\pi$
$314$	−17265.7	−3.10306
$315$	0	0
$316$	−4576.43	−0.814697
$317$	5150.39	0.912539	0.456270	−	0.889842i	$-0.349185\pi$
$317$	5150.39	0.912539	0.456270	+	0.889842i	$0.349185\pi$
$318$	0	0
$319$	6649.58	1.16710
$320$	0	0
$321$	0	0
$322$	−14676.2	−2.53998
$323$	−6695.54	−1.15340
$324$	0	0
$325$	0	0
$326$	−3332.79	−0.566215
$327$	0	0
$328$	3263.05	0.549303
$329$	7212.17	1.20857
$330$	0	0
$331$	−5835.19	−0.968976	−0.484488	−	0.874798i	$-0.660994\pi$
$331$	−5835.19	−0.968976	−0.484488	+	0.874798i	$0.660994\pi$
$332$	13428.5	2.21984
$333$	0	0
$334$	−11682.6	−1.91391
$335$	0	0
$336$	0	0
$337$	−3879.29	−0.627057	−0.313529	−	0.949579i	$-0.601511\pi$
$337$	−3879.29	−0.627057	−0.313529	+	0.949579i	$0.601511\pi$
$338$	−3336.38	−0.536908
$339$	0	0
$340$	0	0
$341$	3695.73	0.586906
$342$	0	0
$343$	6615.56	1.04142
$344$	12263.7	1.92213
$345$	0	0
$346$	−1816.22	−0.282198
$347$	−2674.45	−0.413752	−0.206876	−	0.978367i	$-0.566330\pi$
$347$	−2674.45	−0.413752	−0.206876	+	0.978367i	$0.566330\pi$
$348$	0	0
$349$	−4887.15	−0.749579	−0.374790	−	0.927110i	$-0.622285\pi$
$349$	−4887.15	−0.749579	−0.374790	+	0.927110i	$0.622285\pi$
$350$	0	0
$351$	0	0
$352$	3852.54	0.583356
$353$	−11808.5	−1.78046	−0.890232	−	0.455508i	$-0.849458\pi$
$353$	−11808.5	−1.78046	−0.890232	+	0.455508i	$0.849458\pi$
$354$	0	0
$355$	0	0
$356$	13205.5	1.96598
$357$	0	0
$358$	−13119.6	−1.93685
$359$	8464.47	1.24440	0.622198	−	0.782860i	$-0.286240\pi$
$359$	8464.47	1.24440	0.622198	+	0.782860i	$0.286240\pi$
$360$	0	0
$361$	1161.97	0.169408
$362$	17267.8	2.50712
$363$	0	0
$364$	13617.7	1.96088
$365$	0	0
$366$	0	0
$367$	692.882	0.0985508	0.0492754	−	0.998785i	$-0.484309\pi$
$367$	692.882	0.0985508	0.0492754	+	0.998785i	$0.484309\pi$
$368$	−4741.24	−0.671615
$369$	0	0
$370$	0	0
$371$	−436.691	−0.0611102
$372$	0	0
$373$	5892.36	0.817948	0.408974	−	0.912546i	$-0.365887\pi$
$373$	5892.36	0.817948	0.408974	+	0.912546i	$0.365887\pi$
$374$	−12098.5	−1.67273
$375$	0	0
$376$	12261.5	1.68175
$377$	−10463.8	−1.42948
$378$	0	0
$379$	−9962.11	−1.35018	−0.675091	−	0.737734i	$-0.735896\pi$
$379$	−9962.11	−1.35018	−0.675091	+	0.737734i	$0.735896\pi$
$380$	0	0
$381$	0	0
$382$	−4941.01	−0.661791
$383$	−9038.77	−1.20590	−0.602950	−	0.797779i	$-0.706008\pi$
$383$	−9038.77	−1.20590	−0.602950	+	0.797779i	$0.706008\pi$
$384$	0	0
$385$	0	0
$386$	−6819.13	−0.899182
$387$	0	0
$388$	13114.3	1.71592
$389$	9387.09	1.22351	0.611754	−	0.791048i	$-0.290464\pi$
$389$	9387.09	1.22351	0.611754	+	0.791048i	$0.290464\pi$
$390$	0	0
$391$	−13170.5	−1.70348
$392$	969.045	0.124858
$393$	0	0
$394$	6446.90	0.824340
$395$	0	0
$396$	0	0
$397$	8786.21	1.11075	0.555374	−	0.831600i	$-0.312575\pi$
$397$	8786.21	1.11075	0.555374	+	0.831600i	$0.312575\pi$
$398$	−4353.53	−0.548298
$399$	0	0
$400$	0	0
$401$	4867.79	0.606199	0.303099	−	0.952959i	$-0.401979\pi$
$401$	4867.79	0.606199	0.303099	+	0.952959i	$0.401979\pi$
$402$	0	0
$403$	−5815.60	−0.718848
$404$	−13510.5	−1.66379
$405$	0	0
$406$	16179.2	1.97774
$407$	14750.8	1.79648
$408$	0	0
$409$	−2001.50	−0.241975	−0.120988	−	0.992654i	$-0.538606\pi$
$409$	−2001.50	−0.241975	−0.120988	+	0.992654i	$0.538606\pi$
$410$	0	0
$411$	0	0
$412$	17267.3	2.06480
$413$	5212.74	0.621070
$414$	0	0
$415$	0	0
$416$	−6062.36	−0.714499
$417$	0	0
$418$	14493.5	1.69594
$419$	−2270.88	−0.264772	−0.132386	−	0.991198i	$-0.542264\pi$
$419$	−2270.88	−0.264772	−0.132386	+	0.991198i	$0.542264\pi$
$420$	0	0
$421$	7383.89	0.854795	0.427397	−	0.904064i	$-0.359431\pi$
$421$	7383.89	0.854795	0.427397	+	0.904064i	$0.359431\pi$
$422$	−21087.8	−2.43255
$423$	0	0
$424$	−742.424	−0.0850360
$425$	0	0
$426$	0	0
$427$	−5381.54	−0.609909
$428$	20317.0	2.29453
$429$	0	0
$430$	0	0
$431$	962.054	0.107519	0.0537593	−	0.998554i	$-0.482880\pi$
$431$	962.054	0.107519	0.0537593	+	0.998554i	$0.482880\pi$
$432$	0	0
$433$	2416.32	0.268178	0.134089	−	0.990969i	$-0.457189\pi$
$433$	2416.32	0.268178	0.134089	+	0.990969i	$0.457189\pi$
$434$	8992.15	0.994555
$435$	0	0
$436$	−20544.1	−2.25661
$437$	15777.7	1.72712
$438$	0	0
$439$	6535.97	0.710580	0.355290	−	0.934756i	$-0.384382\pi$
$439$	6535.97	0.710580	0.355290	+	0.934756i	$0.384382\pi$
$440$	0	0
$441$	0	0
$442$	19038.3	2.04877
$443$	5684.43	0.609651	0.304826	−	0.952408i	$-0.401402\pi$
$443$	5684.43	0.609651	0.304826	+	0.952408i	$0.401402\pi$
$444$	0	0
$445$	0	0
$446$	−12400.1	−1.31651
$447$	0	0
$448$	13168.6	1.38874
$449$	−7486.60	−0.786892	−0.393446	−	0.919348i	$-0.628717\pi$
$449$	−7486.60	−0.786892	−0.393446	+	0.919348i	$0.628717\pi$
$450$	0	0
$451$	−3728.40	−0.389276
$452$	−10233.2	−1.06489
$453$	0	0
$454$	12152.0	1.25621
$455$	0	0
$456$	0	0
$457$	−2773.87	−0.283931	−0.141965	−	0.989872i	$-0.545342\pi$
$457$	−2773.87	−0.283931	−0.141965	+	0.989872i	$0.545342\pi$
$458$	18029.4	1.83943
$459$	0	0
$460$	0	0
$461$	17868.7	1.80527	0.902633	−	0.430411i	$-0.141631\pi$
$461$	17868.7	1.80527	0.902633	+	0.430411i	$0.141631\pi$
$462$	0	0
$463$	−18349.1	−1.84180	−0.920900	−	0.389799i	$-0.872544\pi$
$463$	−18349.1	−1.84180	−0.920900	+	0.389799i	$0.872544\pi$
$464$	5226.80	0.522948
$465$	0	0
$466$	−22816.5	−2.26814
$467$	−15896.5	−1.57517	−0.787584	−	0.616207i	$-0.788668\pi$
$467$	−15896.5	−1.57517	−0.787584	+	0.616207i	$0.788668\pi$
$468$	0	0
$469$	16134.7	1.58856
$470$	0	0
$471$	0	0
$472$	8862.24	0.864232
$473$	−14012.7	−1.36216
$474$	0	0
$475$	0	0
$476$	−18895.6	−1.81949
$477$	0	0
$478$	19690.5	1.88415
$479$	−331.824	−0.0316522	−0.0158261	−	0.999875i	$-0.505038\pi$
$479$	−331.824	−0.0316522	−0.0158261	+	0.999875i	$0.505038\pi$
$480$	0	0
$481$	−23211.8	−2.20035
$482$	−24729.2	−2.33690
$483$	0	0
$484$	−2275.65	−0.213717
$485$	0	0
$486$	0	0
$487$	−9442.70	−0.878623	−0.439312	−	0.898335i	$-0.644778\pi$
$487$	−9442.70	−0.878623	−0.439312	+	0.898335i	$0.644778\pi$
$488$	−9149.22	−0.848700
$489$	0	0
$490$	0	0
$491$	−15333.4	−1.40934	−0.704671	−	0.709534i	$-0.748905\pi$
$491$	−15333.4	−1.40934	−0.704671	+	0.709534i	$0.748905\pi$
$492$	0	0
$493$	14519.3	1.32640
$494$	−22807.0	−2.07720
$495$	0	0
$496$	2904.97	0.262978
$497$	−4024.62	−0.363238
$498$	0	0
$499$	12847.3	1.15256	0.576278	−	0.817254i	$-0.304505\pi$
$499$	12847.3	1.15256	0.576278	+	0.817254i	$0.304505\pi$
$500$	0	0
$501$	0	0
$502$	−19955.8	−1.77425
$503$	19623.4	1.73949	0.869747	−	0.493498i	$-0.164282\pi$
$503$	19623.4	1.73949	0.869747	+	0.493498i	$0.164282\pi$
$504$	0	0
$505$	0	0
$506$	28509.6	2.50476
$507$	0	0
$508$	35956.1	3.14034
$509$	−7398.97	−0.644310	−0.322155	−	0.946687i	$-0.604407\pi$
$509$	−7398.97	−0.644310	−0.322155	+	0.946687i	$0.604407\pi$
$510$	0	0
$511$	−2797.53	−0.242183
$512$	9479.91	0.818275
$513$	0	0
$514$	−28853.1	−2.47599
$515$	0	0
$516$	0	0
$517$	−14010.2	−1.19181
$518$	35890.4	3.04427
$519$	0	0
$520$	0	0
$521$	19995.7	1.68144	0.840718	−	0.541473i	$-0.182133\pi$
$521$	19995.7	1.68144	0.840718	+	0.541473i	$0.182133\pi$
$522$	0	0
$523$	1340.93	0.112113	0.0560564	−	0.998428i	$-0.482147\pi$
$523$	1340.93	0.112113	0.0560564	+	0.998428i	$0.482147\pi$
$524$	14393.5	1.19997
$525$	0	0
$526$	20456.6	1.69572
$527$	8069.59	0.667015
$528$	0	0
$529$	18868.6	1.55080
$530$	0	0
$531$	0	0
$532$	22636.1	1.84474
$533$	5867.01	0.476789
$534$	0	0
$535$	0	0
$536$	27430.8	2.21051
$537$	0	0
$538$	−19528.5	−1.56493
$539$	−1107.24	−0.0884831
$540$	0	0
$541$	−588.601	−0.0467762	−0.0233881	−	0.999726i	$-0.507445\pi$
$541$	−588.601	−0.0467762	−0.0233881	+	0.999726i	$0.507445\pi$
$542$	−15477.6	−1.22661
$543$	0	0
$544$	8411.99	0.662980
$545$	0	0
$546$	0	0
$547$	5606.58	0.438245	0.219123	−	0.975697i	$-0.429681\pi$
$547$	5606.58	0.438245	0.219123	+	0.975697i	$0.429681\pi$
$548$	11781.2	0.918375
$549$	0	0
$550$	0	0
$551$	−17393.5	−1.34481
$552$	0	0
$553$	5625.03	0.432551
$554$	−13138.5	−1.00758
$555$	0	0
$556$	−6230.83	−0.475263
$557$	9216.20	0.701083	0.350541	−	0.936547i	$-0.385998\pi$
$557$	9216.20	0.701083	0.350541	+	0.936547i	$0.385998\pi$
$558$	0	0
$559$	22050.4	1.66839
$560$	0	0
$561$	0	0
$562$	6551.17	0.491716
$563$	−26712.3	−1.99963	−0.999814	−	0.0192945i	$-0.993858\pi$
$563$	−26712.3	−1.99963	−0.999814	+	0.0192945i	$0.993858\pi$
$564$	0	0
$565$	0	0
$566$	3513.87	0.260952
$567$	0	0
$568$	−6842.31	−0.505452
$569$	19903.6	1.46643	0.733217	−	0.679994i	$-0.238018\pi$
$569$	19903.6	1.46643	0.733217	+	0.679994i	$0.238018\pi$
$570$	0	0
$571$	9848.73	0.721815	0.360908	−	0.932602i	$-0.382467\pi$
$571$	9848.73	0.721815	0.360908	+	0.932602i	$0.382467\pi$
$572$	−26453.4	−1.93369
$573$	0	0
$574$	−9071.65	−0.659657
$575$	0	0
$576$	0	0
$577$	−20534.4	−1.48155	−0.740777	−	0.671751i	$-0.765543\pi$
$577$	−20534.4	−1.48155	−0.740777	+	0.671751i	$0.765543\pi$
$578$	−3195.73	−0.229974
$579$	0	0
$580$	0	0
$581$	−16505.4	−1.17859
$582$	0	0
$583$	848.304	0.0602627
$584$	−4756.12	−0.337003
$585$	0	0
$586$	−20640.1	−1.45500
$587$	17036.4	1.19790	0.598950	−	0.800786i	$-0.295585\pi$
$587$	17036.4	1.19790	0.598950	+	0.800786i	$0.295585\pi$
$588$	0	0
$589$	−9667.03	−0.676270
$590$	0	0
$591$	0	0
$592$	11594.6	0.804959
$593$	5045.69	0.349413	0.174706	−	0.984621i	$-0.444102\pi$
$593$	5045.69	0.349413	0.174706	+	0.984621i	$0.444102\pi$
$594$	0	0
$595$	0	0
$596$	48464.2	3.33083
$597$	0	0
$598$	−44862.7	−3.06785
$599$	1428.92	0.0974693	0.0487346	−	0.998812i	$-0.484481\pi$
$599$	1428.92	0.0974693	0.0487346	+	0.998812i	$0.484481\pi$
$600$	0	0
$601$	4679.36	0.317596	0.158798	−	0.987311i	$-0.449238\pi$
$601$	4679.36	0.317596	0.158798	+	0.987311i	$0.449238\pi$
$602$	−34094.5	−2.30829
$603$	0	0
$604$	14390.6	0.969444
$605$	0	0
$606$	0	0
$607$	4793.69	0.320544	0.160272	−	0.987073i	$-0.448763\pi$
$607$	4793.69	0.320544	0.160272	+	0.987073i	$0.448763\pi$
$608$	−10077.2	−0.672179
$609$	0	0
$610$	0	0
$611$	22046.4	1.45974
$612$	0	0
$613$	−4528.68	−0.298387	−0.149194	−	0.988808i	$-0.547668\pi$
$613$	−4528.68	−0.298387	−0.149194	+	0.988808i	$0.547668\pi$
$614$	13645.5	0.896885
$615$	0	0
$616$	18083.6	1.18281
$617$	−3749.88	−0.244675	−0.122337	−	0.992489i	$-0.539039\pi$
$617$	−3749.88	−0.244675	−0.122337	+	0.992489i	$0.539039\pi$
$618$	0	0
$619$	−23973.1	−1.55664	−0.778320	−	0.627868i	$-0.783928\pi$
$619$	−23973.1	−1.55664	−0.778320	+	0.627868i	$0.783928\pi$
$620$	0	0
$621$	0	0
$622$	21502.4	1.38612
$623$	−16231.3	−1.04381
$624$	0	0
$625$	0	0
$626$	16002.0	1.02167
$627$	0	0
$628$	52382.9	3.32851
$629$	32208.2	2.04169
$630$	0	0
$631$	3635.10	0.229336	0.114668	−	0.993404i	$-0.463420\pi$
$631$	3635.10	0.229336	0.114668	+	0.993404i	$0.463420\pi$
$632$	9563.18	0.601903
$633$	0	0
$634$	−24343.4	−1.52492
$635$	0	0
$636$	0	0
$637$	1742.36	0.108375
$638$	−31429.3	−1.95031
$639$	0	0
$640$	0	0
$641$	6743.48	0.415525	0.207762	−	0.978179i	$-0.433382\pi$
$641$	6743.48	0.415525	0.207762	+	0.978179i	$0.433382\pi$
$642$	0	0
$643$	7401.83	0.453965	0.226983	−	0.973899i	$-0.427114\pi$
$643$	7401.83	0.453965	0.226983	+	0.973899i	$0.427114\pi$
$644$	44526.6	2.72452
$645$	0	0
$646$	31646.5	1.92742
$647$	15410.5	0.936395	0.468197	−	0.883624i	$-0.344904\pi$
$647$	15410.5	0.936395	0.468197	+	0.883624i	$0.344904\pi$
$648$	0	0
$649$	−10126.1	−0.612457
$650$	0	0
$651$	0	0
$652$	10111.4	0.607353
$653$	12798.9	0.767013	0.383507	−	0.923538i	$-0.374716\pi$
$653$	12798.9	0.767013	0.383507	+	0.923538i	$0.374716\pi$
$654$	0	0
$655$	0	0
$656$	−2930.65	−0.174425
$657$	0	0
$658$	−34088.4	−2.01961
$659$	2194.59	0.129726	0.0648628	−	0.997894i	$-0.479339\pi$
$659$	2194.59	0.129726	0.0648628	+	0.997894i	$0.479339\pi$
$660$	0	0
$661$	−14915.1	−0.877653	−0.438826	−	0.898572i	$-0.644606\pi$
$661$	−14915.1	−0.877653	−0.438826	+	0.898572i	$0.644606\pi$
$662$	27580.1	1.61923
$663$	0	0
$664$	−28061.0	−1.64003
$665$	0	0
$666$	0	0
$667$	−34214.0	−1.98617
$668$	35444.2	2.05296
$669$	0	0
$670$	0	0
$671$	10454.0	0.601450
$672$	0	0
$673$	−27650.0	−1.58370	−0.791848	−	0.610718i	$-0.790881\pi$
$673$	−27650.0	−1.58370	−0.791848	+	0.610718i	$0.790881\pi$
$674$	18335.5	1.04786
$675$	0	0
$676$	10122.3	0.575917
$677$	−17963.4	−1.01978	−0.509889	−	0.860240i	$-0.670314\pi$
$677$	−17963.4	−1.01978	−0.509889	+	0.860240i	$0.670314\pi$
$678$	0	0
$679$	−16119.2	−0.911042
$680$	0	0
$681$	0	0
$682$	−17467.9	−0.980763
$683$	22747.4	1.27438	0.637192	−	0.770705i	$-0.280096\pi$
$683$	22747.4	1.27438	0.637192	+	0.770705i	$0.280096\pi$
$684$	0	0
$685$	0	0
$686$	−31268.5	−1.74029
$687$	0	0
$688$	−11014.4	−0.610351
$689$	−1334.89	−0.0738103
$690$	0	0
$691$	−16494.4	−0.908071	−0.454036	−	0.890984i	$-0.650016\pi$
$691$	−16494.4	−0.908071	−0.454036	+	0.890984i	$0.650016\pi$
$692$	5510.26	0.302701
$693$	0	0
$694$	12640.8	0.691410
$695$	0	0
$696$	0	0
$697$	−8140.93	−0.442410
$698$	23099.2	1.25260
$699$	0	0
$700$	0	0
$701$	23463.2	1.26418	0.632092	−	0.774894i	$-0.282197\pi$
$701$	23463.2	1.26418	0.632092	+	0.774894i	$0.282197\pi$
$702$	0	0
$703$	−38584.0	−2.07002
$704$	−25580.8	−1.36948
$705$	0	0
$706$	55813.0	2.97528
$707$	16606.2	0.883365
$708$	0	0
$709$	16436.0	0.870615	0.435308	−	0.900282i	$-0.356640\pi$
$709$	16436.0	0.870615	0.435308	+	0.900282i	$0.356640\pi$
$710$	0	0
$711$	0	0
$712$	−27595.0	−1.45248
$713$	−19015.6	−0.998793
$714$	0	0
$715$	0	0
$716$	39803.9	2.07757
$717$	0	0
$718$	−40007.4	−2.07947
$719$	543.595	0.0281957	0.0140978	−	0.999901i	$-0.495512\pi$
$719$	543.595	0.0281957	0.0140978	+	0.999901i	$0.495512\pi$
$720$	0	0
$721$	−21223.7	−1.09627
$722$	−5492.06	−0.283093
$723$	0	0
$724$	−52389.2	−2.68927
$725$	0	0
$726$	0	0
$727$	−1097.53	−0.0559905	−0.0279953	−	0.999608i	$-0.508912\pi$
$727$	−1097.53	−0.0559905	−0.0279953	+	0.999608i	$0.508912\pi$
$728$	−28456.4	−1.44871
$729$	0	0
$730$	0	0
$731$	−30596.6	−1.54809
$732$	0	0
$733$	−26094.8	−1.31492	−0.657459	−	0.753490i	$-0.728369\pi$
$733$	−26094.8	−1.31492	−0.657459	+	0.753490i	$0.728369\pi$
$734$	−3274.91	−0.164685
$735$	0	0
$736$	−19822.4	−0.992751
$737$	−31342.9	−1.56652
$738$	0	0
$739$	17477.5	0.869985	0.434992	−	0.900434i	$-0.356751\pi$
$739$	17477.5	0.869985	0.434992	+	0.900434i	$0.356751\pi$
$740$	0	0
$741$	0	0
$742$	2064.02	0.102120
$743$	11534.6	0.569533	0.284766	−	0.958597i	$-0.408084\pi$
$743$	11534.6	0.569533	0.284766	+	0.958597i	$0.408084\pi$
$744$	0	0
$745$	0	0
$746$	−27850.3	−1.36685
$747$	0	0
$748$	36706.1	1.79426
$749$	−24972.2	−1.21825
$750$	0	0
$751$	22616.9	1.09894	0.549470	−	0.835514i	$-0.314830\pi$
$751$	22616.9	1.09894	0.549470	+	0.835514i	$0.314830\pi$
$752$	−11012.5	−0.534020
$753$	0	0
$754$	49457.2	2.38876
$755$	0	0
$756$	0	0
$757$	−39919.4	−1.91664	−0.958320	−	0.285699i	$-0.907774\pi$
$757$	−39919.4	−1.91664	−0.958320	+	0.285699i	$0.907774\pi$
$758$	47086.0	2.25625
$759$	0	0
$760$	0	0
$761$	−5317.72	−0.253308	−0.126654	−	0.991947i	$-0.540424\pi$
$761$	−5317.72	−0.253308	−0.126654	+	0.991947i	$0.540424\pi$
$762$	0	0
$763$	25251.3	1.19811
$764$	14990.7	0.709873
$765$	0	0
$766$	42721.8	2.01515
$767$	15934.5	0.750143
$768$	0	0
$769$	19615.7	0.919846	0.459923	−	0.887959i	$-0.347877\pi$
$769$	19615.7	0.919846	0.459923	+	0.887959i	$0.347877\pi$
$770$	0	0
$771$	0	0
$772$	20688.7	0.964512
$773$	−27548.0	−1.28180	−0.640901	−	0.767624i	$-0.721439\pi$
$773$	−27548.0	−1.28180	−0.640901	+	0.767624i	$0.721439\pi$
$774$	0	0
$775$	0	0
$776$	−27404.4	−1.26773
$777$	0	0
$778$	−44368.2	−2.04457
$779$	9752.49	0.448549
$780$	0	0
$781$	7818.12	0.358200
$782$	62250.5	2.84664
$783$	0	0
$784$	−870.331	−0.0396470
$785$	0	0
$786$	0	0
$787$	−362.900	−0.0164371	−0.00821854	−	0.999966i	$-0.502616\pi$
$787$	−362.900	−0.0164371	−0.00821854	+	0.999966i	$0.502616\pi$
$788$	−19559.4	−0.884233
$789$	0	0
$790$	0	0
$791$	12577.9	0.565386
$792$	0	0
$793$	−16450.5	−0.736662
$794$	−41528.1	−1.85614
$795$	0	0
$796$	13208.3	0.588134
$797$	−35122.3	−1.56097	−0.780487	−	0.625172i	$-0.785029\pi$
$797$	−35122.3	−1.56097	−0.780487	+	0.625172i	$0.785029\pi$
$798$	0	0
$799$	−30591.1	−1.35449
$800$	0	0
$801$	0	0
$802$	−23007.6	−1.01300
$803$	5434.41	0.238824
$804$	0	0
$805$	0	0
$806$	27487.5	1.20125
$807$	0	0
$808$	28232.3	1.22922
$809$	−20667.7	−0.898194	−0.449097	−	0.893483i	$-0.648254\pi$
$809$	−20667.7	−0.898194	−0.449097	+	0.893483i	$0.648254\pi$
$810$	0	0
$811$	−45901.0	−1.98743	−0.993713	−	0.111958i	$-0.964288\pi$
$811$	−45901.0	−1.98743	−0.993713	+	0.111958i	$0.964288\pi$
$812$	−49086.6	−2.12143
$813$	0	0
$814$	−69719.6	−3.00205
$815$	0	0
$816$	0	0
$817$	36653.4	1.56957
$818$	9460.11	0.404358
$819$	0	0
$820$	0	0
$821$	265.370	0.0112807	0.00564037	−	0.999984i	$-0.498205\pi$
$821$	265.370	0.0112807	0.00564037	+	0.999984i	$0.498205\pi$
$822$	0	0
$823$	960.254	0.0406711	0.0203356	−	0.999793i	$-0.493527\pi$
$823$	960.254	0.0406711	0.0203356	+	0.999793i	$0.493527\pi$
$824$	−36082.7	−1.52549
$825$	0	0
$826$	−24638.0	−1.03785
$827$	−8116.35	−0.341273	−0.170637	−	0.985334i	$-0.554582\pi$
$827$	−8116.35	−0.341273	−0.170637	+	0.985334i	$0.554582\pi$
$828$	0	0
$829$	2257.09	0.0945620	0.0472810	−	0.998882i	$-0.484944\pi$
$829$	2257.09	0.0945620	0.0472810	+	0.998882i	$0.484944\pi$
$830$	0	0
$831$	0	0
$832$	40254.0	1.67735
$833$	−2417.66	−0.100560
$834$	0	0
$835$	0	0
$836$	−43972.3	−1.81916
$837$	0	0
$838$	10733.3	0.442454
$839$	−33981.9	−1.39831	−0.699156	−	0.714969i	$-0.746441\pi$
$839$	−33981.9	−1.39831	−0.699156	+	0.714969i	$0.746441\pi$
$840$	0	0
$841$	13328.9	0.546514
$842$	−34900.0	−1.42842
$843$	0	0
$844$	63978.8	2.60929
$845$	0	0
$846$	0	0
$847$	2797.08	0.113470
$848$	666.796	0.0270022
$849$	0	0
$850$	0	0
$851$	−75897.0	−3.05724
$852$	0	0
$853$	8331.29	0.334417	0.167209	−	0.985922i	$-0.446525\pi$
$853$	8331.29	0.334417	0.167209	+	0.985922i	$0.446525\pi$
$854$	25435.9	1.01920
$855$	0	0
$856$	−42455.6	−1.69521
$857$	−25427.2	−1.01351	−0.506753	−	0.862091i	$-0.669155\pi$
$857$	−25427.2	−1.01351	−0.506753	+	0.862091i	$0.669155\pi$
$858$	0	0
$859$	22758.3	0.903961	0.451981	−	0.892028i	$-0.350718\pi$
$859$	22758.3	0.903961	0.451981	+	0.892028i	$0.350718\pi$
$860$	0	0
$861$	0	0
$862$	−4547.16	−0.179671
$863$	−4016.10	−0.158412	−0.0792060	−	0.996858i	$-0.525238\pi$
$863$	−4016.10	−0.158412	−0.0792060	+	0.996858i	$0.525238\pi$
$864$	0	0
$865$	0	0
$866$	−11420.8	−0.448145
$867$	0	0
$868$	−27281.5	−1.06681
$869$	−10927.0	−0.426552
$870$	0	0
$871$	49321.1	1.91869
$872$	42930.1	1.66720
$873$	0	0
$874$	−74573.4	−2.88614
$875$	0	0
$876$	0	0
$877$	2913.80	0.112192	0.0560958	−	0.998425i	$-0.482135\pi$
$877$	2913.80	0.112192	0.0560958	+	0.998425i	$0.482135\pi$
$878$	−30892.3	−1.18743
$879$	0	0
$880$	0	0
$881$	15832.1	0.605446	0.302723	−	0.953079i	$-0.402104\pi$
$881$	15832.1	0.605446	0.302723	+	0.953079i	$0.402104\pi$
$882$	0	0
$883$	41471.6	1.58056	0.790278	−	0.612748i	$-0.209936\pi$
$883$	41471.6	1.58056	0.790278	+	0.612748i	$0.209936\pi$
$884$	−57760.7	−2.19763
$885$	0	0
$886$	−26867.5	−1.01877
$887$	41362.0	1.56573	0.782864	−	0.622193i	$-0.213758\pi$
$887$	41362.0	1.56573	0.782864	+	0.622193i	$0.213758\pi$
$888$	0	0
$889$	−44194.7	−1.66732
$890$	0	0
$891$	0	0
$892$	37621.0	1.41216
$893$	36646.8	1.37328
$894$	0	0
$895$	0	0
$896$	−46375.5	−1.72913
$897$	0	0
$898$	35385.5	1.31495
$899$	20963.0	0.777703
$900$	0	0
$901$	1852.26	0.0684882
$902$	17622.3	0.650509
$903$	0	0
$904$	21383.9	0.786746
$905$	0	0
$906$	0	0
$907$	−6539.32	−0.239399	−0.119699	−	0.992810i	$-0.538193\pi$
$907$	−6539.32	−0.239399	−0.119699	+	0.992810i	$0.538193\pi$
$908$	−36868.1	−1.34748
$909$	0	0
$910$	0	0
$911$	7793.75	0.283445	0.141723	−	0.989906i	$-0.454736\pi$
$911$	7793.75	0.283445	0.141723	+	0.989906i	$0.454736\pi$
$912$	0	0
$913$	32062.9	1.16224
$914$	13110.7	0.474469
$915$	0	0
$916$	−54700.0	−1.97308
$917$	−17691.5	−0.637104
$918$	0	0
$919$	38128.9	1.36862	0.684308	−	0.729193i	$-0.260104\pi$
$919$	38128.9	1.36862	0.684308	+	0.729193i	$0.260104\pi$
$920$	0	0
$921$	0	0
$922$	−84456.5	−3.01673
$923$	−12302.6	−0.438727
$924$	0	0
$925$	0	0
$926$	86727.0	3.07778
$927$	0	0
$928$	21852.5	0.772999
$929$	−35726.2	−1.26172	−0.630860	−	0.775897i	$-0.717298\pi$
$929$	−35726.2	−1.26172	−0.630860	+	0.775897i	$0.717298\pi$
$930$	0	0
$931$	2896.25	0.101956
$932$	69223.6	2.43293
$933$	0	0
$934$	75135.1	2.63222
$935$	0	0
$936$	0	0
$937$	49634.6	1.73052	0.865258	−	0.501327i	$-0.167155\pi$
$937$	49634.6	1.73052	0.865258	+	0.501327i	$0.167155\pi$
$938$	−76260.9	−2.65459
$939$	0	0
$940$	0	0
$941$	7852.19	0.272024	0.136012	−	0.990707i	$-0.456571\pi$
$941$	7852.19	0.272024	0.136012	+	0.990707i	$0.456571\pi$
$942$	0	0
$943$	19183.7	0.662468
$944$	−7959.47	−0.274427
$945$	0	0
$946$	66231.0	2.27627
$947$	−45715.9	−1.56871	−0.784355	−	0.620313i	$-0.787006\pi$
$947$	−45715.9	−1.56871	−0.784355	+	0.620313i	$0.787006\pi$
$948$	0	0
$949$	−8551.59	−0.292515
$950$	0	0
$951$	0	0
$952$	39485.4	1.34425
$953$	41726.6	1.41832	0.709160	−	0.705048i	$-0.249075\pi$
$953$	41726.6	1.41832	0.709160	+	0.705048i	$0.249075\pi$
$954$	0	0
$955$	0	0
$956$	−59739.4	−2.02104
$957$	0	0
$958$	1568.37	0.0528932
$959$	−14480.7	−0.487597
$960$	0	0
$961$	−18140.1	−0.608912
$962$	109711.	3.67694
$963$	0	0
$964$	75026.6	2.50669
$965$	0	0
$966$	0	0
$967$	−4802.46	−0.159707	−0.0798535	−	0.996807i	$-0.525445\pi$
$967$	−4802.46	−0.159707	−0.0798535	+	0.996807i	$0.525445\pi$
$968$	4755.34	0.157895
$969$	0	0
$970$	0	0
$971$	−50899.2	−1.68222	−0.841109	−	0.540865i	$-0.818097\pi$
$971$	−50899.2	−1.68222	−0.841109	+	0.540865i	$0.818097\pi$
$972$	0	0
$973$	7658.51	0.252334
$974$	44631.0	1.46824
$975$	0	0
$976$	8217.22	0.269495
$977$	18227.1	0.596865	0.298433	−	0.954431i	$-0.403536\pi$
$977$	18227.1	0.596865	0.298433	+	0.954431i	$0.403536\pi$
$978$	0	0
$979$	31530.4	1.02933
$980$	0	0
$981$	0	0
$982$	72473.4	2.35511
$983$	−9327.86	−0.302658	−0.151329	−	0.988483i	$-0.548355\pi$
$983$	−9327.86	−0.302658	−0.151329	+	0.988483i	$0.548355\pi$
$984$	0	0
$985$	0	0
$986$	−68625.6	−2.21652
$987$	0	0
$988$	69194.9	2.22812
$989$	72099.2	2.31812
$990$	0	0
$991$	−20920.4	−0.670593	−0.335296	−	0.942113i	$-0.608836\pi$
$991$	−20920.4	−0.670593	−0.335296	+	0.942113i	$0.608836\pi$
$992$	12145.3	0.388722
$993$	0	0
$994$	19022.4	0.606996
$995$	0	0
$996$	0	0
$997$	39491.7	1.25448	0.627239	−	0.778827i	$-0.284185\pi$
$997$	39491.7	1.25448	0.627239	+	0.778827i	$0.284185\pi$
$998$	−60723.0	−1.92600
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	675.4.a.y.1.1	yes	4
3.2	odd	2		675.4.a.u.1.4	✓	4
5.2	odd	4		675.4.b.q.649.1		8
5.3	odd	4		675.4.b.q.649.8		8
5.4	even	2		675.4.a.v.1.4	yes	4
15.2	even	4		675.4.b.p.649.8		8
15.8	even	4		675.4.b.p.649.1		8
15.14	odd	2		675.4.a.z.1.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
675.4.a.u.1.4	✓	4	3.2	odd	2
675.4.a.v.1.4	yes	4	5.4	even	2
675.4.a.y.1.1	yes	4	1.1	even	1	trivial
675.4.a.z.1.1	yes	4	15.14	odd	2
675.4.b.p.649.1		8	15.8	even	4
675.4.b.p.649.8		8	15.2	even	4
675.4.b.q.649.1		8	5.2	odd	4
675.4.b.q.649.8		8	5.3	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	675.a (trivial)

\(f(q)\)	\(=\)	\(q-4.72651 q^{2} +14.3399 q^{4} -17.6256 q^{7} -29.9655 q^{8} +34.2390 q^{11} -53.8784 q^{13} +83.3075 q^{14} +26.9130 q^{16} +74.7605 q^{17} -89.5599 q^{19} -161.831 q^{22} -176.169 q^{23} +254.657 q^{26} -252.749 q^{28} +194.211 q^{29} +107.939 q^{31} +112.519 q^{32} -353.356 q^{34} +430.818 q^{37} +423.305 q^{38} -108.894 q^{41} -409.261 q^{43} +490.982 q^{44} +832.666 q^{46} -409.188 q^{47} -32.3387 q^{49} -772.610 q^{52} +24.7760 q^{53} +528.159 q^{56} -917.940 q^{58} -295.748 q^{59} +305.325 q^{61} -510.176 q^{62} -747.127 q^{64} -915.415 q^{67} +1072.06 q^{68} +228.340 q^{71} +158.720 q^{73} -2036.27 q^{74} -1284.28 q^{76} -603.482 q^{77} -319.140 q^{79} +514.686 q^{82} +936.446 q^{83} +1934.38 q^{86} -1025.99 q^{88} +920.893 q^{89} +949.639 q^{91} -2526.25 q^{92} +1934.03 q^{94} +914.533 q^{97} +152.849 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{2} + 19 q^{4} - 4 q^{7} - 15 q^{8} + 52 q^{11} + 2 q^{13} + 138 q^{14} - 5 q^{16} + 64 q^{17} - 46 q^{19} - 87 q^{22} - 90 q^{23} + 469 q^{26} + 110 q^{28} + 470 q^{29} - 262 q^{31} - 199 q^{32}+ \cdots + 4301 q^{98}+O(q^{100}) \) 4 * q + q^2 + 19 * q^4 - 4 * q^7 - 15 * q^8 + 52 * q^11 + 2 * q^13 + 138 * q^14 - 5 * q^16 + 64 * q^17 - 46 * q^19 - 87 * q^22 - 90 * q^23 + 469 * q^26 + 110 * q^28 + 470 * q^29 - 262 * q^31 - 199 * q^32 - 42 * q^34 + 542 * q^37 + 532 * q^38 + 698 * q^41 - 142 * q^43 + 419 * q^44 + 537 * q^46 - 542 * q^47 + 780 * q^49 - 409 * q^52 + 910 * q^53 + 2034 * q^56 + 576 * q^58 + 100 * q^59 + 74 * q^61 - 2406 * q^62 - 965 * q^64 - 928 * q^67 + 2810 * q^68 + 1622 * q^71 + 536 * q^73 - 253 * q^74 - 2068 * q^76 - 1932 * q^77 - 508 * q^79 + 1782 * q^82 + 1524 * q^83 + 3940 * q^86 - 2247 * q^88 + 756 * q^89 + 1120 * q^91 - 3645 * q^92 + 2847 * q^94 - 892 * q^97 + 4301 * q^98

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