LMFDB - Embedded newform 675.4.a.z.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [675,4,Mod(1,675)]

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

lf = mfeigenbasis(mf)

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")

//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);

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	675.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:	$39.8262892539$
Analytic rank:	$1$
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)

$f(q)$	$=$	$q-4.72651 q^{2} +14.3399 q^{4} +17.6256 q^{7} -29.9655 q^{8} +O(q^{10})$	$q-4.72651 q^{2} +14.3399 q^{4} +17.6256 q^{7} -29.9655 q^{8} -34.2390 q^{11} +53.8784 q^{13} -83.3075 q^{14} +26.9130 q^{16} +74.7605 q^{17} -89.5599 q^{19} +161.831 q^{22} -176.169 q^{23} -254.657 q^{26} +252.749 q^{28} -194.211 q^{29} +107.939 q^{31} +112.519 q^{32} -353.356 q^{34} -430.818 q^{37} +423.305 q^{38} +108.894 q^{41} +409.261 q^{43} -490.982 q^{44} +832.666 q^{46} -409.188 q^{47} -32.3387 q^{49} +772.610 q^{52} +24.7760 q^{53} -528.159 q^{56} +917.940 q^{58} +295.748 q^{59} +305.325 q^{61} -510.176 q^{62} -747.127 q^{64} +915.415 q^{67} +1072.06 q^{68} -228.340 q^{71} -158.720 q^{73} +2036.27 q^{74} -1284.28 q^{76} -603.482 q^{77} -319.140 q^{79} -514.686 q^{82} +936.446 q^{83} -1934.38 q^{86} +1025.99 q^{88} -920.893 q^{89} +949.639 q^{91} -2526.25 q^{92} +1934.03 q^{94} -914.533 q^{97} +152.849 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{2} + 19 q^{4} + 4 q^{7} - 15 q^{8}+O(q^{10}) $ 4 * q + q^2 + 19 * q^4 + 4 * q^7 - 15 * q^8	$ 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}+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

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 $

$2$	−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	$-0.342142\pi$
$7$	17.6256	0.951692	0.475846	+	0.879529i	$0.342142\pi$
$8$	−29.9655	−1.32430
$9$	0	0
$10$	0	0
$11$	−34.2390	−0.938494	−0.469247	−	0.883067i	$-0.655475\pi$
$11$	−34.2390	−0.938494	−0.469247	+	0.883067i	$0.655475\pi$
$12$	0	0
$13$	53.8784	1.14948	0.574738	−	0.818337i	$-0.305104\pi$
$13$	53.8784	1.14948	0.574738	+	0.818337i	$0.305104\pi$
$14$	−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	$-0.713596\pi$
$29$	−194.211	−1.24359	−0.621795	+	0.783180i	$0.713596\pi$
$30$	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	$-0.906436\pi$
$37$	−430.818	−1.91422	−0.957110	+	0.289726i	$0.906436\pi$
$38$	423.305	1.80708
$39$	0	0
$40$	0	0
$41$	108.894	0.414788	0.207394	−	0.978257i	$-0.433502\pi$
$41$	108.894	0.414788	0.207394	+	0.978257i	$0.433502\pi$
$42$	0	0
$43$	409.261	1.45144	0.725718	−	0.687992i	$-0.241508\pi$
$43$	409.261	1.45144	0.725718	+	0.687992i	$0.241508\pi$
$44$	−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	$-0.394199\pi$
$59$	295.748	0.652596	0.326298	+	0.945267i	$0.394199\pi$
$60$	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	$-0.185702\pi$
$67$	915.415	1.66919	0.834595	+	0.550864i	$0.185702\pi$
$68$	1072.06	1.91185
$69$	0	0
$70$	0	0
$71$	−228.340	−0.381675	−0.190838	−	0.981622i	$-0.561120\pi$
$71$	−228.340	−0.381675	−0.190838	+	0.981622i	$0.561120\pi$
$72$	0	0
$73$	−158.720	−0.254476	−0.127238	−	0.991872i	$-0.540611\pi$
$73$	−158.720	−0.254476	−0.127238	+	0.991872i	$0.540611\pi$
$74$	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	$-0.684761\pi$
$89$	−920.893	−1.09679	−0.548396	+	0.836219i	$0.684761\pi$
$90$	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	$-0.658871\pi$
$97$	−914.533	−0.957286	−0.478643	+	0.878010i	$0.658871\pi$
$98$	152.849	0.157552
$99$	0	0
$100$	0	0
$101$	942.162	0.928205	0.464102	−	0.885782i	$-0.346377\pi$
$101$	942.162	0.928205	0.464102	+	0.885782i	$0.346377\pi$
$102$	0	0
$103$	−1204.14	−1.15192	−0.575960	−	0.817478i	$-0.695372\pi$
$103$	−1204.14	−1.15192	−0.575960	+	0.817478i	$0.695372\pi$
$104$	−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	$-0.839780\pi$
$127$	−2507.42	−1.75195	−0.875974	+	0.482359i	$0.839780\pi$
$128$	2631.15	1.81690
$129$	0	0
$130$	0	0
$131$	−1003.74	−0.669444	−0.334722	−	0.942317i	$-0.608642\pi$
$131$	−1003.74	−0.669444	−0.334722	+	0.942317i	$0.608642\pi$
$132$	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	$-0.879424\pi$
$149$	−3379.68	−1.85822	−0.929109	+	0.369807i	$0.879424\pi$
$150$	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	$-0.878869\pi$
$157$	−3652.95	−1.85693	−0.928463	+	0.371426i	$0.878869\pi$
$158$	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	$-0.554188\pi$
$163$	−705.127	−0.338833	−0.169416	+	0.985545i	$0.554188\pi$
$164$	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	$-0.696761\pi$
$179$	−2775.75	−1.15904	−0.579522	+	0.814956i	$0.696761\pi$
$180$	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	$-0.563449\pi$
$191$	−1045.38	−0.396028	−0.198014	+	0.980199i	$0.563449\pi$
$192$	0	0
$193$	−1442.74	−0.538087	−0.269043	−	0.963128i	$-0.586708\pi$
$193$	−1442.74	−0.538087	−0.269043	+	0.963128i	$0.586708\pi$
$194$	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	$-0.628878\pi$
$223$	−2623.53	−0.787822	−0.393911	+	0.919149i	$0.628878\pi$
$224$	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	$-0.309357\pi$
$239$	4165.97	1.12751	0.563753	+	0.825943i	$0.309357\pi$
$240$	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	$-0.678135\pi$
$251$	−4222.11	−1.06174	−0.530871	+	0.847453i	$0.678135\pi$
$252$	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	$-0.655112\pi$
$269$	−4131.70	−0.936484	−0.468242	+	0.883600i	$0.655112\pi$
$270$	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	$-0.597480\pi$
$277$	−2779.74	−0.602955	−0.301478	+	0.953473i	$0.597480\pi$
$278$	2053.72	0.443072
$279$	0	0
$280$	0	0
$281$	1386.05	0.294252	0.147126	−	0.989118i	$-0.452998\pi$
$281$	1386.05	0.294252	0.147126	+	0.989118i	$0.452998\pi$
$282$	0	0
$283$	743.439	0.156159	0.0780793	−	0.996947i	$-0.475121\pi$
$283$	743.439	0.156159	0.0780793	+	0.996947i	$0.475121\pi$
$284$	−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	$-0.413520\pi$
$307$	2887.01	0.536712	0.268356	+	0.963320i	$0.413520\pi$
$308$	−8653.85	−1.60097
$309$	0	0
$310$	0	0
$311$	4549.32	0.829481	0.414740	−	0.909940i	$-0.363872\pi$
$311$	4549.32	0.829481	0.414740	+	0.909940i	$0.363872\pi$
$312$	0	0
$313$	3385.58	0.611387	0.305694	−	0.952130i	$-0.401112\pi$
$313$	3385.58	0.611387	0.305694	+	0.952130i	$0.401112\pi$
$314$	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	$-0.398489\pi$
$337$	3879.29	0.627057	0.313529	+	0.949579i	$0.398489\pi$
$338$	−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	$-0.713760\pi$
$359$	−8464.47	−1.24440	−0.622198	+	0.782860i	$0.713760\pi$
$360$	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	$-0.515691\pi$
$367$	−692.882	−0.0985508	−0.0492754	+	0.998785i	$0.515691\pi$
$368$	−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	$-0.634113\pi$
$373$	−5892.36	−0.817948	−0.408974	+	0.912546i	$0.634113\pi$
$374$	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	$-0.709536\pi$
$389$	−9387.09	−1.22351	−0.611754	+	0.791048i	$0.709536\pi$
$390$	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	$-0.687425\pi$
$397$	−8786.21	−1.11075	−0.555374	+	0.831600i	$0.687425\pi$
$398$	−4353.53	−0.548298
$399$	0	0
$400$	0	0
$401$	−4867.79	−0.606199	−0.303099	−	0.952959i	$-0.598021\pi$
$401$	−4867.79	−0.606199	−0.303099	+	0.952959i	$0.598021\pi$
$402$	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	$-0.457736\pi$
$419$	2270.88	0.264772	0.132386	+	0.991198i	$0.457736\pi$
$420$	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	$-0.517120\pi$
$431$	−962.054	−0.107519	−0.0537593	+	0.998554i	$0.517120\pi$
$432$	0	0
$433$	−2416.32	−0.268178	−0.134089	−	0.990969i	$-0.542811\pi$
$433$	−2416.32	−0.268178	−0.134089	+	0.990969i	$0.542811\pi$
$434$	−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	$-0.371283\pi$
$449$	7486.60	0.786892	0.393446	+	0.919348i	$0.371283\pi$
$450$	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	$-0.454658\pi$
$457$	2773.87	0.283931	0.141965	+	0.989872i	$0.454658\pi$
$458$	18029.4	1.83943
$459$	0	0
$460$	0	0
$461$	−17868.7	−1.80527	−0.902633	−	0.430411i	$-0.858369\pi$
$461$	−17868.7	−1.80527	−0.902633	+	0.430411i	$0.858369\pi$
$462$	0	0
$463$	18349.1	1.84180	0.920900	−	0.389799i	$-0.127456\pi$
$463$	18349.1	1.84180	0.920900	+	0.389799i	$0.127456\pi$
$464$	−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	$-0.494962\pi$
$479$	331.824	0.0316522	0.0158261	+	0.999875i	$0.494962\pi$
$480$	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	$-0.355222\pi$
$487$	9442.70	0.878623	0.439312	+	0.898335i	$0.355222\pi$
$488$	−9149.22	−0.848700
$489$	0	0
$490$	0	0
$491$	15333.4	1.40934	0.704671	−	0.709534i	$-0.251095\pi$
$491$	15333.4	1.40934	0.704671	+	0.709534i	$0.251095\pi$
$492$	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	$-0.395593\pi$
$509$	7398.97	0.644310	0.322155	+	0.946687i	$0.395593\pi$
$510$	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	$-0.817867\pi$
$521$	−19995.7	−1.68144	−0.840718	+	0.541473i	$0.817867\pi$
$522$	0	0
$523$	−1340.93	−0.112113	−0.0560564	−	0.998428i	$-0.517853\pi$
$523$	−1340.93	−0.112113	−0.0560564	+	0.998428i	$0.517853\pi$
$524$	−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	$-0.570319\pi$
$547$	−5606.58	−0.438245	−0.219123	+	0.975697i	$0.570319\pi$
$548$	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	$-0.761982\pi$
$569$	−19903.6	−1.46643	−0.733217	+	0.679994i	$0.761982\pi$
$570$	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	$-0.234457\pi$
$577$	20534.4	1.48155	0.740777	+	0.671751i	$0.234457\pi$
$578$	−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	$-0.515519\pi$
$599$	−1428.92	−0.0974693	−0.0487346	+	0.998812i	$0.515519\pi$
$600$	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	$-0.551237\pi$
$607$	−4793.69	−0.320544	−0.160272	+	0.987073i	$0.551237\pi$
$608$	−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	$-0.452332\pi$
$613$	4528.68	0.298387	0.149194	+	0.988808i	$0.452332\pi$
$614$	−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	$-0.566618\pi$
$641$	−6743.48	−0.415525	−0.207762	+	0.978179i	$0.566618\pi$
$642$	0	0
$643$	−7401.83	−0.453965	−0.226983	−	0.973899i	$-0.572886\pi$
$643$	−7401.83	−0.453965	−0.226983	+	0.973899i	$0.572886\pi$
$644$	−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	$-0.520661\pi$
$659$	−2194.59	−0.129726	−0.0648628	+	0.997894i	$0.520661\pi$
$660$	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	$-0.209119\pi$
$673$	27650.0	1.58370	0.791848	+	0.610718i	$0.209119\pi$
$674$	−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	$-0.717803\pi$
$701$	−23463.2	−1.26418	−0.632092	+	0.774894i	$0.717803\pi$
$702$	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	$-0.504488\pi$
$719$	−543.595	−0.0281957	−0.0140978	+	0.999901i	$0.504488\pi$
$720$	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	$-0.491088\pi$
$727$	1097.53	0.0559905	0.0279953	+	0.999608i	$0.491088\pi$
$728$	−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	$-0.271631\pi$
$733$	26094.8	1.31492	0.657459	+	0.753490i	$0.271631\pi$
$734$	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	$-0.0922256\pi$
$757$	39919.4	1.91664	0.958320	+	0.285699i	$0.0922256\pi$
$758$	47086.0	2.25625
$759$	0	0
$760$	0	0
$761$	5317.72	0.253308	0.126654	−	0.991947i	$-0.459576\pi$
$761$	5317.72	0.253308	0.126654	+	0.991947i	$0.459576\pi$
$762$	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	$-0.497384\pi$
$787$	362.900	0.0164371	0.00821854	+	0.999966i	$0.497384\pi$
$788$	−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	$-0.351746\pi$
$809$	20667.7	0.898194	0.449097	+	0.893483i	$0.351746\pi$
$810$	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	$-0.501795\pi$
$821$	−265.370	−0.0112807	−0.00564037	+	0.999984i	$0.501795\pi$
$822$	0	0
$823$	−960.254	−0.0406711	−0.0203356	−	0.999793i	$-0.506473\pi$
$823$	−960.254	−0.0406711	−0.0203356	+	0.999793i	$0.506473\pi$
$824$	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	$-0.253559\pi$
$839$	33981.9	1.39831	0.699156	+	0.714969i	$0.253559\pi$
$840$	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	$-0.553475\pi$
$853$	−8331.29	−0.334417	−0.167209	+	0.985922i	$0.553475\pi$
$854$	−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	$-0.517865\pi$
$877$	−2913.80	−0.112192	−0.0560958	+	0.998425i	$0.517865\pi$
$878$	−30892.3	−1.18743
$879$	0	0
$880$	0	0
$881$	−15832.1	−0.605446	−0.302723	−	0.953079i	$-0.597896\pi$
$881$	−15832.1	−0.605446	−0.302723	+	0.953079i	$0.597896\pi$
$882$	0	0
$883$	−41471.6	−1.58056	−0.790278	−	0.612748i	$-0.790064\pi$
$883$	−41471.6	−1.58056	−0.790278	+	0.612748i	$0.790064\pi$
$884$	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	$-0.461807\pi$
$907$	6539.32	0.239399	0.119699	+	0.992810i	$0.461807\pi$
$908$	−36868.1	−1.34748
$909$	0	0
$910$	0	0
$911$	−7793.75	−0.283445	−0.141723	−	0.989906i	$-0.545264\pi$
$911$	−7793.75	−0.283445	−0.141723	+	0.989906i	$0.545264\pi$
$912$	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	$-0.282702\pi$
$929$	35726.2	1.26172	0.630860	+	0.775897i	$0.282702\pi$
$930$	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	$-0.832845\pi$
$937$	−49634.6	−1.73052	−0.865258	+	0.501327i	$0.832845\pi$
$938$	−76260.9	−2.65459
$939$	0	0
$940$	0	0
$941$	−7852.19	−0.272024	−0.136012	−	0.990707i	$-0.543429\pi$
$941$	−7852.19	−0.272024	−0.136012	+	0.990707i	$0.543429\pi$
$942$	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	$-0.474555\pi$
$967$	4802.46	0.159707	0.0798535	+	0.996807i	$0.474555\pi$
$968$	4755.34	0.157895
$969$	0	0
$970$	0	0
$971$	50899.2	1.68222	0.841109	−	0.540865i	$-0.181903\pi$
$971$	50899.2	1.68222	0.841109	+	0.540865i	$0.181903\pi$
$972$	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	$-0.715815\pi$
$997$	−39491.7	−1.25448	−0.627239	+	0.778827i	$0.715815\pi$
$998$	−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.z.1.1	yes	4
3.2	odd	2		675.4.a.v.1.4	yes	4
5.2	odd	4		675.4.b.p.649.1		8
5.3	odd	4		675.4.b.p.649.8		8
5.4	even	2		675.4.a.u.1.4	✓	4
15.2	even	4		675.4.b.q.649.8		8
15.8	even	4		675.4.b.q.649.1		8
15.14	odd	2		675.4.a.y.1.1	yes	4

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

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

\(f(q)\)	\(=\)	\(q-4.72651 q^{2} +14.3399 q^{4} +17.6256 q^{7} -29.9655 q^{8} +O(q^{10})\)	\(q-4.72651 q^{2} +14.3399 q^{4} +17.6256 q^{7} -29.9655 q^{8} -34.2390 q^{11} +53.8784 q^{13} -83.3075 q^{14} +26.9130 q^{16} +74.7605 q^{17} -89.5599 q^{19} +161.831 q^{22} -176.169 q^{23} -254.657 q^{26} +252.749 q^{28} -194.211 q^{29} +107.939 q^{31} +112.519 q^{32} -353.356 q^{34} -430.818 q^{37} +423.305 q^{38} +108.894 q^{41} +409.261 q^{43} -490.982 q^{44} +832.666 q^{46} -409.188 q^{47} -32.3387 q^{49} +772.610 q^{52} +24.7760 q^{53} -528.159 q^{56} +917.940 q^{58} +295.748 q^{59} +305.325 q^{61} -510.176 q^{62} -747.127 q^{64} +915.415 q^{67} +1072.06 q^{68} -228.340 q^{71} -158.720 q^{73} +2036.27 q^{74} -1284.28 q^{76} -603.482 q^{77} -319.140 q^{79} -514.686 q^{82} +936.446 q^{83} -1934.38 q^{86} +1025.99 q^{88} -920.893 q^{89} +949.639 q^{91} -2526.25 q^{92} +1934.03 q^{94} -914.533 q^{97} +152.849 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{2} + 19 q^{4} + 4 q^{7} - 15 q^{8}+O(q^{10}) \) 4 * q + q^2 + 19 * q^4 + 4 * q^7 - 15 * q^8	\( 4 q + q^{2} + 19 q^{4} + 4 q^{7} - 15 q^{8} - 52 q^{11} - 2 q^{13} - 138 q^{14} - 5 q^{16} + 64 q^{17} - 46 q^{19} + 87 q^{22} - 90 q^{23} - 469 q^{26} - 110 q^{28} - 470 q^{29} - 262 q^{31} - 199 q^{32} - 42 q^{34} - 542 q^{37} + 532 q^{38} - 698 q^{41} + 142 q^{43} - 419 q^{44} + 537 q^{46} - 542 q^{47} + 780 q^{49} + 409 q^{52} + 910 q^{53} - 2034 q^{56} - 576 q^{58} - 100 q^{59} + 74 q^{61} - 2406 q^{62} - 965 q^{64} + 928 q^{67} + 2810 q^{68} - 1622 q^{71} - 536 q^{73} + 253 q^{74} - 2068 q^{76} - 1932 q^{77} - 508 q^{79} - 1782 q^{82} + 1524 q^{83} - 3940 q^{86} + 2247 q^{88} - 756 q^{89} + 1120 q^{91} - 3645 q^{92} + 2847 q^{94} + 892 q^{97} + 4301 q^{98}+O(q^{100}) \) 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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