LMFDB - Embedded newform 2205.4.a.cd.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2205,4,Mod(1,2205)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2205, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2205.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2205 = 3^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2205.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:	$130.099211563$
Analytic rank:	$1$
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} - 2x^{7} - 55x^{6} + 80x^{5} + 969x^{4} - 866x^{3} - 5783x^{2} + 2328x + 9992 $ x^8 - 2x^7 - 55x^6 + 80x^5 + 969x^4 - 866x^3 - 5783x^2 + 2328*x + 9992
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 735)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.52943$ of defining polynomial
Character	$\chi$	$=$	2205.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.52943 q^{2} -5.66083 q^{4} -5.00000 q^{5} -20.8933 q^{8} +O(q^{10})$	$q+1.52943 q^{2} -5.66083 q^{4} -5.00000 q^{5} -20.8933 q^{8} -7.64717 q^{10} -34.3826 q^{11} -29.4418 q^{13} +13.3317 q^{16} +1.20008 q^{17} +117.794 q^{19} +28.3042 q^{20} -52.5858 q^{22} +181.026 q^{23} +25.0000 q^{25} -45.0293 q^{26} -34.8875 q^{29} +211.797 q^{31} +187.537 q^{32} +1.83544 q^{34} -81.8762 q^{37} +180.157 q^{38} +104.467 q^{40} +232.665 q^{41} +37.4767 q^{43} +194.634 q^{44} +276.867 q^{46} +35.2539 q^{47} +38.2359 q^{50} +166.665 q^{52} -603.278 q^{53} +171.913 q^{55} -53.3582 q^{58} -536.847 q^{59} -805.493 q^{61} +323.929 q^{62} +180.172 q^{64} +147.209 q^{65} +219.919 q^{67} -6.79343 q^{68} -104.873 q^{71} +934.665 q^{73} -125.224 q^{74} -666.809 q^{76} +638.150 q^{79} -66.6583 q^{80} +355.846 q^{82} -60.3503 q^{83} -6.00038 q^{85} +57.3182 q^{86} +718.366 q^{88} -1372.92 q^{89} -1024.76 q^{92} +53.9185 q^{94} -588.968 q^{95} -1562.27 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{2} + 50 q^{4} - 40 q^{5} - 66 q^{8}+O(q^{10}) $ 8 * q - 2 * q^2 + 50 * q^4 - 40 * q^5 - 66 * q^8	$ 8 q - 2 q^{2} + 50 q^{4} - 40 q^{5} - 66 q^{8} + 10 q^{10} - 64 q^{11} + 206 q^{16} - 48 q^{17} + 80 q^{19} - 250 q^{20} + 452 q^{22} - 120 q^{23} + 200 q^{25} - 272 q^{26} - 76 q^{29} + 20 q^{31} - 770 q^{32} + 320 q^{34} + 348 q^{37} + 236 q^{38} + 330 q^{40} - 944 q^{41} + 1116 q^{43} - 172 q^{44} + 496 q^{46} + 208 q^{47} - 50 q^{50} - 2272 q^{52} - 1144 q^{53} + 320 q^{55} + 560 q^{58} - 596 q^{59} + 740 q^{61} + 1184 q^{62} + 1298 q^{64} + 1964 q^{67} - 96 q^{68} + 4 q^{71} - 1500 q^{73} - 3368 q^{74} - 2912 q^{76} - 460 q^{79} - 1030 q^{80} - 5644 q^{82} - 700 q^{83} + 240 q^{85} + 1396 q^{86} + 6892 q^{88} - 644 q^{92} - 6692 q^{94} - 400 q^{95} - 2052 q^{97}+O(q^{100}) $ 8 * q - 2 * q^2 + 50 * q^4 - 40 * q^5 - 66 * q^8 + 10 * q^10 - 64 * q^11 + 206 * q^16 - 48 * q^17 + 80 * q^19 - 250 * q^20 + 452 * q^22 - 120 * q^23 + 200 * q^25 - 272 * q^26 - 76 * q^29 + 20 * q^31 - 770 * q^32 + 320 * q^34 + 348 * q^37 + 236 * q^38 + 330 * q^40 - 944 * q^41 + 1116 * q^43 - 172 * q^44 + 496 * q^46 + 208 * q^47 - 50 * q^50 - 2272 * q^52 - 1144 * q^53 + 320 * q^55 + 560 * q^58 - 596 * q^59 + 740 * q^61 + 1184 * q^62 + 1298 * q^64 + 1964 * q^67 - 96 * q^68 + 4 * q^71 - 1500 * q^73 - 3368 * q^74 - 2912 * q^76 - 460 * q^79 - 1030 * q^80 - 5644 * q^82 - 700 * q^83 + 240 * q^85 + 1396 * q^86 + 6892 * q^88 - 644 * q^92 - 6692 * q^94 - 400 * q^95 - 2052 * q^97

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 $

$2$	1.52943	0.540737	0.270368	−	0.962757i	$-0.412855\pi$
$2$	1.52943	0.540737	0.270368	+	0.962757i	$0.412855\pi$
$3$	0	0
$3$	0	0
$4$	−5.66083	−0.707604
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−20.8933	−0.923364
$9$	0	0
$10$	−7.64717	−0.241825
$11$	−34.3826	−0.942430	−0.471215	−	0.882018i	$-0.656184\pi$
$11$	−34.3826	−0.942430	−0.471215	+	0.882018i	$0.656184\pi$
$12$	0	0
$13$	−29.4418	−0.628130	−0.314065	−	0.949402i	$-0.601691\pi$
$13$	−29.4418	−0.628130	−0.314065	+	0.949402i	$0.601691\pi$
$14$	0	0
$15$	0	0
$16$	13.3317	0.208307
$17$	1.20008	0.0171212	0.00856062	−	0.999963i	$-0.497275\pi$
$17$	1.20008	0.0171212	0.00856062	+	0.999963i	$0.497275\pi$
$18$	0	0
$19$	117.794	1.42230	0.711150	−	0.703040i	$-0.248175\pi$
$19$	117.794	1.42230	0.711150	+	0.703040i	$0.248175\pi$
$20$	28.3042	0.316450
$21$	0	0
$22$	−52.5858	−0.509606
$23$	181.026	1.64115	0.820576	−	0.571537i	$-0.193653\pi$
$23$	181.026	1.64115	0.820576	+	0.571537i	$0.193653\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−45.0293	−0.339653
$27$	0	0
$28$	0	0
$29$	−34.8875	−0.223395	−0.111697	−	0.993742i	$-0.535629\pi$
$29$	−34.8875	−0.223395	−0.111697	+	0.993742i	$0.535629\pi$
$30$	0	0
$31$	211.797	1.22709	0.613545	−	0.789660i	$-0.289743\pi$
$31$	211.797	1.22709	0.613545	+	0.789660i	$0.289743\pi$
$32$	187.537	1.03600
$33$	0	0
$34$	1.83544	0.00925808
$35$	0	0
$36$	0	0
$37$	−81.8762	−0.363794	−0.181897	−	0.983318i	$-0.558224\pi$
$37$	−81.8762	−0.363794	−0.181897	+	0.983318i	$0.558224\pi$
$38$	180.157	0.769090
$39$	0	0
$40$	104.467	0.412941
$41$	232.665	0.886250	0.443125	−	0.896460i	$-0.353870\pi$
$41$	232.665	0.886250	0.443125	+	0.896460i	$0.353870\pi$
$42$	0	0
$43$	37.4767	0.132910	0.0664552	−	0.997789i	$-0.478831\pi$
$43$	37.4767	0.132910	0.0664552	+	0.997789i	$0.478831\pi$
$44$	194.634	0.666867
$45$	0	0
$46$	276.867	0.887431
$47$	35.2539	0.109411	0.0547054	−	0.998503i	$-0.482578\pi$
$47$	35.2539	0.109411	0.0547054	+	0.998503i	$0.482578\pi$
$48$	0	0
$49$	0	0
$50$	38.2359	0.108147
$51$	0	0
$52$	166.665	0.444467
$53$	−603.278	−1.56352	−0.781761	−	0.623578i	$-0.785678\pi$
$53$	−603.278	−1.56352	−0.781761	+	0.623578i	$0.785678\pi$
$54$	0	0
$55$	171.913	0.421468
$56$	0	0
$57$	0	0
$58$	−53.3582	−0.120798
$59$	−536.847	−1.18460	−0.592301	−	0.805717i	$-0.701780\pi$
$59$	−536.847	−1.18460	−0.592301	+	0.805717i	$0.701780\pi$
$60$	0	0
$61$	−805.493	−1.69070	−0.845351	−	0.534212i	$-0.820608\pi$
$61$	−805.493	−1.69070	−0.845351	+	0.534212i	$0.820608\pi$
$62$	323.929	0.663533
$63$	0	0
$64$	180.172	0.351898
$65$	147.209	0.280908
$66$	0	0
$67$	219.919	0.401005	0.200503	−	0.979693i	$-0.435742\pi$
$67$	219.919	0.401005	0.200503	+	0.979693i	$0.435742\pi$
$68$	−6.79343	−0.0121151
$69$	0	0
$70$	0	0
$71$	−104.873	−0.175298	−0.0876490	−	0.996151i	$-0.527935\pi$
$71$	−104.873	−0.175298	−0.0876490	+	0.996151i	$0.527935\pi$
$72$	0	0
$73$	934.665	1.49855	0.749275	−	0.662258i	$-0.230402\pi$
$73$	934.665	1.49855	0.749275	+	0.662258i	$0.230402\pi$
$74$	−125.224	−0.196717
$75$	0	0
$76$	−666.809	−1.00642
$77$	0	0
$78$	0	0
$79$	638.150	0.908829	0.454414	−	0.890790i	$-0.349849\pi$
$79$	638.150	0.908829	0.454414	+	0.890790i	$0.349849\pi$
$80$	−66.6583	−0.0931578
$81$	0	0
$82$	355.846	0.479228
$83$	−60.3503	−0.0798109	−0.0399054	−	0.999203i	$-0.512706\pi$
$83$	−60.3503	−0.0798109	−0.0399054	+	0.999203i	$0.512706\pi$
$84$	0	0
$85$	−6.00038	−0.00765685
$86$	57.3182	0.0718695
$87$	0	0
$88$	718.366	0.870206
$89$	−1372.92	−1.63516	−0.817578	−	0.575818i	$-0.804684\pi$
$89$	−1372.92	−1.63516	−0.817578	+	0.575818i	$0.804684\pi$
$90$	0	0
$91$	0	0
$92$	−1024.76	−1.16129
$93$	0	0
$94$	53.9185	0.0591624
$95$	−588.968	−0.636072
$96$	0	0
$97$	−1562.27	−1.63530	−0.817652	−	0.575713i	$-0.804725\pi$
$97$	−1562.27	−1.63530	−0.817652	+	0.575713i	$0.804725\pi$
$98$	0	0
$99$	0	0
$100$	−141.521	−0.141521
$101$	−575.810	−0.567280	−0.283640	−	0.958931i	$-0.591542\pi$
$101$	−575.810	−0.567280	−0.283640	+	0.958931i	$0.591542\pi$
$102$	0	0
$103$	170.176	0.162795	0.0813976	−	0.996682i	$-0.474062\pi$
$103$	170.176	0.162795	0.0813976	+	0.996682i	$0.474062\pi$
$104$	615.137	0.579992
$105$	0	0
$106$	−922.675	−0.845454
$107$	−391.545	−0.353758	−0.176879	−	0.984233i	$-0.556600\pi$
$107$	−391.545	−0.353758	−0.176879	+	0.984233i	$0.556600\pi$
$108$	0	0
$109$	822.858	0.723078	0.361539	−	0.932357i	$-0.382251\pi$
$109$	822.858	0.723078	0.361539	+	0.932357i	$0.382251\pi$
$110$	262.929	0.227903
$111$	0	0
$112$	0	0
$113$	−1156.23	−0.962556	−0.481278	−	0.876568i	$-0.659827\pi$
$113$	−1156.23	−0.962556	−0.481278	+	0.876568i	$0.659827\pi$
$114$	0	0
$115$	−905.130	−0.733946
$116$	197.493	0.158075
$117$	0	0
$118$	−821.072	−0.640558
$119$	0	0
$120$	0	0
$121$	−148.840	−0.111826
$122$	−1231.95	−0.914224
$123$	0	0
$124$	−1198.95	−0.868294
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−53.5419	−0.0374100	−0.0187050	−	0.999825i	$-0.505954\pi$
$127$	−53.5419	−0.0374100	−0.0187050	+	0.999825i	$0.505954\pi$
$128$	−1224.73	−0.845719
$129$	0	0
$130$	225.146	0.151897
$131$	−819.592	−0.546627	−0.273313	−	0.961925i	$-0.588120\pi$
$131$	−819.592	−0.546627	−0.273313	+	0.961925i	$0.588120\pi$
$132$	0	0
$133$	0	0
$134$	336.351	0.216838
$135$	0	0
$136$	−25.0736	−0.0158091
$137$	1107.59	0.690716	0.345358	−	0.938471i	$-0.387758\pi$
$137$	1107.59	0.690716	0.345358	+	0.938471i	$0.387758\pi$
$138$	0	0
$139$	1421.02	0.867118	0.433559	−	0.901125i	$-0.357258\pi$
$139$	1421.02	0.867118	0.433559	+	0.901125i	$0.357258\pi$
$140$	0	0
$141$	0	0
$142$	−160.397	−0.0947901
$143$	1012.28	0.591968
$144$	0	0
$145$	174.438	0.0999053
$146$	1429.51	0.810321
$147$	0	0
$148$	463.487	0.257422
$149$	25.3563	0.0139414	0.00697069	−	0.999976i	$-0.497781\pi$
$149$	25.3563	0.0139414	0.00697069	+	0.999976i	$0.497781\pi$
$150$	0	0
$151$	3505.27	1.88910	0.944552	−	0.328360i	$-0.106496\pi$
$151$	3505.27	1.88910	0.944552	+	0.328360i	$0.106496\pi$
$152$	−2461.10	−1.31330
$153$	0	0
$154$	0	0
$155$	−1058.98	−0.548771
$156$	0	0
$157$	−770.714	−0.391781	−0.195891	−	0.980626i	$-0.562760\pi$
$157$	−770.714	−0.391781	−0.195891	+	0.980626i	$0.562760\pi$
$158$	976.008	0.491437
$159$	0	0
$160$	−937.683	−0.463315
$161$	0	0
$162$	0	0
$163$	4001.44	1.92281	0.961403	−	0.275145i	$-0.0887258\pi$
$163$	4001.44	1.92281	0.961403	+	0.275145i	$0.0887258\pi$
$164$	−1317.08	−0.627114
$165$	0	0
$166$	−92.3017	−0.0431567
$167$	−959.298	−0.444507	−0.222254	−	0.974989i	$-0.571341\pi$
$167$	−959.298	−0.444507	−0.222254	+	0.974989i	$0.571341\pi$
$168$	0	0
$169$	−1330.18	−0.605453
$170$	−9.17719	−0.00414034
$171$	0	0
$172$	−212.149	−0.0940479
$173$	−367.321	−0.161427	−0.0807135	−	0.996737i	$-0.525720\pi$
$173$	−367.321	−0.161427	−0.0807135	+	0.996737i	$0.525720\pi$
$174$	0	0
$175$	0	0
$176$	−458.377	−0.196315
$177$	0	0
$178$	−2099.79	−0.884189
$179$	−3348.02	−1.39801	−0.699003	−	0.715119i	$-0.746373\pi$
$179$	−3348.02	−1.39801	−0.699003	+	0.715119i	$0.746373\pi$
$180$	0	0
$181$	−1649.24	−0.677276	−0.338638	−	0.940917i	$-0.609966\pi$
$181$	−1649.24	−0.677276	−0.338638	+	0.940917i	$0.609966\pi$
$182$	0	0
$183$	0	0
$184$	−3782.24	−1.51538
$185$	409.381	0.162693
$186$	0	0
$187$	−41.2617	−0.0161356
$188$	−199.566	−0.0774195
$189$	0	0
$190$	−900.787	−0.343947
$191$	2868.03	1.08651	0.543254	−	0.839568i	$-0.317192\pi$
$191$	2868.03	1.08651	0.543254	+	0.839568i	$0.317192\pi$
$192$	0	0
$193$	4103.05	1.53028	0.765139	−	0.643865i	$-0.222670\pi$
$193$	4103.05	1.53028	0.765139	+	0.643865i	$0.222670\pi$
$194$	−2389.39	−0.884268
$195$	0	0
$196$	0	0
$197$	−2744.26	−0.992488	−0.496244	−	0.868183i	$-0.665288\pi$
$197$	−2744.26	−0.992488	−0.496244	+	0.868183i	$0.665288\pi$
$198$	0	0
$199$	343.049	0.122202	0.0611008	−	0.998132i	$-0.480539\pi$
$199$	343.049	0.122202	0.0611008	+	0.998132i	$0.480539\pi$
$200$	−522.334	−0.184673
$201$	0	0
$202$	−880.664	−0.306749
$203$	0	0
$204$	0	0
$205$	−1163.33	−0.396343
$206$	260.272	0.0880293
$207$	0	0
$208$	−392.508	−0.130844
$209$	−4050.04	−1.34042
$210$	0	0
$211$	2190.98	0.714850	0.357425	−	0.933942i	$-0.383655\pi$
$211$	2190.98	0.714850	0.357425	+	0.933942i	$0.383655\pi$
$212$	3415.06	1.10635
$213$	0	0
$214$	−598.842	−0.191290
$215$	−187.384	−0.0594393
$216$	0	0
$217$	0	0
$218$	1258.51	0.390995
$219$	0	0
$220$	−973.169	−0.298232
$221$	−35.3324	−0.0107544
$222$	0	0
$223$	−3697.58	−1.11035	−0.555175	−	0.831734i	$-0.687349\pi$
$223$	−3697.58	−1.11035	−0.555175	+	0.831734i	$0.687349\pi$
$224$	0	0
$225$	0	0
$226$	−1768.38	−0.520489
$227$	3301.41	0.965297	0.482649	−	0.875814i	$-0.339675\pi$
$227$	3301.41	0.965297	0.482649	+	0.875814i	$0.339675\pi$
$228$	0	0
$229$	−3122.57	−0.901071	−0.450536	−	0.892758i	$-0.648767\pi$
$229$	−3122.57	−0.901071	−0.450536	+	0.892758i	$0.648767\pi$
$230$	−1384.34	−0.396871
$231$	0	0
$232$	728.917	0.206275
$233$	−5807.23	−1.63281	−0.816404	−	0.577481i	$-0.804036\pi$
$233$	−5807.23	−1.63281	−0.816404	+	0.577481i	$0.804036\pi$
$234$	0	0
$235$	−176.269	−0.0489300
$236$	3039.00	0.838229
$237$	0	0
$238$	0	0
$239$	−2762.90	−0.747772	−0.373886	−	0.927475i	$-0.621975\pi$
$239$	−2762.90	−0.747772	−0.373886	+	0.927475i	$0.621975\pi$
$240$	0	0
$241$	−2308.16	−0.616935	−0.308468	−	0.951235i	$-0.599816\pi$
$241$	−2308.16	−0.616935	−0.308468	+	0.951235i	$0.599816\pi$
$242$	−227.641	−0.0604682
$243$	0	0
$244$	4559.76	1.19635
$245$	0	0
$246$	0	0
$247$	−3468.05	−0.893389
$248$	−4425.14	−1.13305
$249$	0	0
$250$	−191.179	−0.0483650
$251$	−4140.26	−1.04116	−0.520579	−	0.853813i	$-0.674284\pi$
$251$	−4140.26	−1.04116	−0.520579	+	0.853813i	$0.674284\pi$
$252$	0	0
$253$	−6224.13	−1.54667
$254$	−81.8888	−0.0202290
$255$	0	0
$256$	−3314.52	−0.809209
$257$	4394.46	1.06661	0.533305	−	0.845923i	$-0.320950\pi$
$257$	4394.46	1.06661	0.533305	+	0.845923i	$0.320950\pi$
$258$	0	0
$259$	0	0
$260$	−833.325	−0.198772
$261$	0	0
$262$	−1253.51	−0.295581
$263$	3884.60	0.910777	0.455389	−	0.890293i	$-0.349500\pi$
$263$	3884.60	0.910777	0.455389	+	0.890293i	$0.349500\pi$
$264$	0	0
$265$	3016.39	0.699228
$266$	0	0
$267$	0	0
$268$	−1244.92	−0.283753
$269$	−4485.41	−1.01665	−0.508327	−	0.861164i	$-0.669736\pi$
$269$	−4485.41	−1.01665	−0.508327	+	0.861164i	$0.669736\pi$
$270$	0	0
$271$	−6617.26	−1.48328	−0.741642	−	0.670795i	$-0.765953\pi$
$271$	−6617.26	−1.48328	−0.741642	+	0.670795i	$0.765953\pi$
$272$	15.9990	0.00356648
$273$	0	0
$274$	1693.99	0.373495
$275$	−859.564	−0.188486
$276$	0	0
$277$	−1252.30	−0.271638	−0.135819	−	0.990734i	$-0.543367\pi$
$277$	−1252.30	−0.271638	−0.135819	+	0.990734i	$0.543367\pi$
$278$	2173.36	0.468883
$279$	0	0
$280$	0	0
$281$	7438.29	1.57912	0.789558	−	0.613676i	$-0.210310\pi$
$281$	7438.29	1.57912	0.789558	+	0.613676i	$0.210310\pi$
$282$	0	0
$283$	−8792.00	−1.84675	−0.923375	−	0.383900i	$-0.874581\pi$
$283$	−8792.00	−1.84675	−0.923375	+	0.383900i	$0.874581\pi$
$284$	593.670	0.124042
$285$	0	0
$286$	1548.22	0.320099
$287$	0	0
$288$	0	0
$289$	−4911.56	−0.999707
$290$	266.791	0.0540224
$291$	0	0
$292$	−5290.98	−1.06038
$293$	−3071.61	−0.612441	−0.306220	−	0.951961i	$-0.599064\pi$
$293$	−3071.61	−0.612441	−0.306220	+	0.951961i	$0.599064\pi$
$294$	0	0
$295$	2684.24	0.529770
$296$	1710.67	0.335914
$297$	0	0
$298$	38.7807	0.00753861
$299$	−5329.73	−1.03086
$300$	0	0
$301$	0	0
$302$	5361.08	1.02151
$303$	0	0
$304$	1570.38	0.296275
$305$	4027.46	0.756105
$306$	0	0
$307$	−2546.63	−0.473434	−0.236717	−	0.971579i	$-0.576071\pi$
$307$	−2546.63	−0.473434	−0.236717	+	0.971579i	$0.576071\pi$
$308$	0	0
$309$	0	0
$310$	−1619.64	−0.296741
$311$	110.229	0.0200982	0.0100491	−	0.999950i	$-0.496801\pi$
$311$	110.229	0.0200982	0.0100491	+	0.999950i	$0.496801\pi$
$312$	0	0
$313$	−5954.74	−1.07534	−0.537670	−	0.843155i	$-0.680695\pi$
$313$	−5954.74	−1.07534	−0.537670	+	0.843155i	$0.680695\pi$
$314$	−1178.76	−0.211851
$315$	0	0
$316$	−3612.46	−0.643091
$317$	−5630.91	−0.997676	−0.498838	−	0.866695i	$-0.666240\pi$
$317$	−5630.91	−0.997676	−0.498838	+	0.866695i	$0.666240\pi$
$318$	0	0
$319$	1199.52	0.210534
$320$	−900.858	−0.157373
$321$	0	0
$322$	0	0
$323$	141.361	0.0243515
$324$	0	0
$325$	−736.045	−0.125626
$326$	6119.95	1.03973
$327$	0	0
$328$	−4861.16	−0.818331
$329$	0	0
$330$	0	0
$331$	9745.44	1.61830	0.809150	−	0.587602i	$-0.199928\pi$
$331$	9745.44	1.61830	0.809150	+	0.587602i	$0.199928\pi$
$332$	341.633	0.0564745
$333$	0	0
$334$	−1467.18	−0.240361
$335$	−1099.59	−0.179335
$336$	0	0
$337$	202.614	0.0327511	0.0163755	−	0.999866i	$-0.494787\pi$
$337$	202.614	0.0327511	0.0163755	+	0.999866i	$0.494787\pi$
$338$	−2034.42	−0.327391
$339$	0	0
$340$	33.9671	0.00541802
$341$	−7282.11	−1.15645
$342$	0	0
$343$	0	0
$344$	−783.014	−0.122725
$345$	0	0
$346$	−561.793	−0.0872895
$347$	11018.5	1.70462	0.852308	−	0.523039i	$-0.175202\pi$
$347$	11018.5	1.70462	0.852308	+	0.523039i	$0.175202\pi$
$348$	0	0
$349$	9472.82	1.45292	0.726459	−	0.687210i	$-0.241165\pi$
$349$	9472.82	1.45292	0.726459	+	0.687210i	$0.241165\pi$
$350$	0	0
$351$	0	0
$352$	−6447.99	−0.976361
$353$	1448.54	0.218409	0.109204	−	0.994019i	$-0.465170\pi$
$353$	1448.54	0.218409	0.109204	+	0.994019i	$0.465170\pi$
$354$	0	0
$355$	524.366	0.0783957
$356$	7771.85	1.15704
$357$	0	0
$358$	−5120.58	−0.755953
$359$	11909.9	1.75092	0.875460	−	0.483291i	$-0.160559\pi$
$359$	11909.9	1.75092	0.875460	+	0.483291i	$0.160559\pi$
$360$	0	0
$361$	7016.32	1.02294
$362$	−2522.40	−0.366228
$363$	0	0
$364$	0	0
$365$	−4673.32	−0.670172
$366$	0	0
$367$	−7770.80	−1.10527	−0.552633	−	0.833425i	$-0.686377\pi$
$367$	−7770.80	−1.10527	−0.552633	+	0.833425i	$0.686377\pi$
$368$	2413.38	0.341864
$369$	0	0
$370$	626.121	0.0879743
$371$	0	0
$372$	0	0
$373$	−4016.23	−0.557513	−0.278757	−	0.960362i	$-0.589922\pi$
$373$	−4016.23	−0.557513	−0.278757	+	0.960362i	$0.589922\pi$
$374$	−63.1070	−0.00872510
$375$	0	0
$376$	−736.571	−0.101026
$377$	1027.15	0.140321
$378$	0	0
$379$	−11210.8	−1.51942	−0.759711	−	0.650261i	$-0.774660\pi$
$379$	−11210.8	−1.51942	−0.759711	+	0.650261i	$0.774660\pi$
$380$	3334.05	0.450087
$381$	0	0
$382$	4386.46	0.587515
$383$	3468.05	0.462687	0.231344	−	0.972872i	$-0.425688\pi$
$383$	3468.05	0.462687	0.231344	+	0.972872i	$0.425688\pi$
$384$	0	0
$385$	0	0
$386$	6275.34	0.827478
$387$	0	0
$388$	8843.75	1.15715
$389$	1264.76	0.164848	0.0824240	−	0.996597i	$-0.473734\pi$
$389$	1264.76	0.164848	0.0824240	+	0.996597i	$0.473734\pi$
$390$	0	0
$391$	217.245	0.0280986
$392$	0	0
$393$	0	0
$394$	−4197.16	−0.536675
$395$	−3190.75	−0.406441
$396$	0	0
$397$	−15074.9	−1.90576	−0.952881	−	0.303344i	$-0.901897\pi$
$397$	−15074.9	−1.90576	−0.952881	+	0.303344i	$0.901897\pi$
$398$	524.671	0.0660789
$399$	0	0
$400$	333.292	0.0416615
$401$	−12220.4	−1.52184	−0.760922	−	0.648843i	$-0.775253\pi$
$401$	−12220.4	−1.52184	−0.760922	+	0.648843i	$0.775253\pi$
$402$	0	0
$403$	−6235.67	−0.770772
$404$	3259.56	0.401409
$405$	0	0
$406$	0	0
$407$	2815.11	0.342850
$408$	0	0
$409$	−10168.0	−1.22928	−0.614640	−	0.788808i	$-0.710699\pi$
$409$	−10168.0	−1.22928	−0.614640	+	0.788808i	$0.710699\pi$
$410$	−1779.23	−0.214317
$411$	0	0
$412$	−963.335	−0.115194
$413$	0	0
$414$	0	0
$415$	301.751	0.0356925
$416$	−5521.41	−0.650744
$417$	0	0
$418$	−6194.27	−0.724813
$419$	−8908.44	−1.03868	−0.519339	−	0.854568i	$-0.673822\pi$
$419$	−8908.44	−1.03868	−0.519339	+	0.854568i	$0.673822\pi$
$420$	0	0
$421$	−4730.78	−0.547658	−0.273829	−	0.961778i	$-0.588290\pi$
$421$	−4730.78	−0.547658	−0.273829	+	0.961778i	$0.588290\pi$
$422$	3350.96	0.386545
$423$	0	0
$424$	12604.5	1.44370
$425$	30.0019	0.00342425
$426$	0	0
$427$	0	0
$428$	2216.47	0.250320
$429$	0	0
$430$	−286.591	−0.0321410
$431$	14592.4	1.63083	0.815417	−	0.578875i	$-0.196508\pi$
$431$	14592.4	1.63083	0.815417	+	0.578875i	$0.196508\pi$
$432$	0	0
$433$	3230.32	0.358520	0.179260	−	0.983802i	$-0.442630\pi$
$433$	3230.32	0.358520	0.179260	+	0.983802i	$0.442630\pi$
$434$	0	0
$435$	0	0
$436$	−4658.06	−0.511653
$437$	21323.7	2.33421
$438$	0	0
$439$	11011.0	1.19710	0.598548	−	0.801087i	$-0.295745\pi$
$439$	11011.0	1.19710	0.598548	+	0.801087i	$0.295745\pi$
$440$	−3591.83	−0.389168
$441$	0	0
$442$	−54.0386	−0.00581528
$443$	−16424.2	−1.76149	−0.880744	−	0.473592i	$-0.842957\pi$
$443$	−16424.2	−1.76149	−0.880744	+	0.473592i	$0.842957\pi$
$444$	0	0
$445$	6864.59	0.731264
$446$	−5655.20	−0.600407
$447$	0	0
$448$	0	0
$449$	−9215.14	−0.968573	−0.484287	−	0.874909i	$-0.660921\pi$
$449$	−9215.14	−0.968573	−0.484287	+	0.874909i	$0.660921\pi$
$450$	0	0
$451$	−7999.63	−0.835228
$452$	6545.22	0.681109
$453$	0	0
$454$	5049.29	0.521972
$455$	0	0
$456$	0	0
$457$	2946.45	0.301596	0.150798	−	0.988565i	$-0.451816\pi$
$457$	2946.45	0.301596	0.150798	+	0.988565i	$0.451816\pi$
$458$	−4775.77	−0.487242
$459$	0	0
$460$	5123.79	0.519343
$461$	13313.6	1.34507	0.672533	−	0.740067i	$-0.265206\pi$
$461$	13313.6	1.34507	0.672533	+	0.740067i	$0.265206\pi$
$462$	0	0
$463$	−10458.2	−1.04975	−0.524874	−	0.851180i	$-0.675887\pi$
$463$	−10458.2	−1.04975	−0.524874	+	0.851180i	$0.675887\pi$
$464$	−465.109	−0.0465348
$465$	0	0
$466$	−8881.78	−0.882920
$467$	−6421.20	−0.636269	−0.318135	−	0.948046i	$-0.603056\pi$
$467$	−6421.20	−0.636269	−0.318135	+	0.948046i	$0.603056\pi$
$468$	0	0
$469$	0	0
$470$	−269.592	−0.0264582
$471$	0	0
$472$	11216.5	1.09382
$473$	−1288.55	−0.125259
$474$	0	0
$475$	2944.84	0.284460
$476$	0	0
$477$	0	0
$478$	−4225.68	−0.404348
$479$	7777.09	0.741846	0.370923	−	0.928664i	$-0.379041\pi$
$479$	7777.09	0.741846	0.370923	+	0.928664i	$0.379041\pi$
$480$	0	0
$481$	2410.58	0.228510
$482$	−3530.17	−0.333599
$483$	0	0
$484$	842.558	0.0791283
$485$	7811.35	0.731330
$486$	0	0
$487$	13708.5	1.27555	0.637773	−	0.770225i	$-0.279856\pi$
$487$	13708.5	1.27555	0.637773	+	0.770225i	$0.279856\pi$
$488$	16829.4	1.56113
$489$	0	0
$490$	0	0
$491$	−10769.1	−0.989821	−0.494911	−	0.868944i	$-0.664799\pi$
$491$	−10769.1	−0.989821	−0.494911	+	0.868944i	$0.664799\pi$
$492$	0	0
$493$	−41.8677	−0.00382480
$494$	−5304.16	−0.483088
$495$	0	0
$496$	2823.60	0.255612
$497$	0	0
$498$	0	0
$499$	−17314.2	−1.55329	−0.776645	−	0.629939i	$-0.783080\pi$
$499$	−17314.2	−1.55329	−0.776645	+	0.629939i	$0.783080\pi$
$500$	707.604	0.0632900
$501$	0	0
$502$	−6332.25	−0.562992
$503$	13889.3	1.23120	0.615598	−	0.788060i	$-0.288915\pi$
$503$	13889.3	1.23120	0.615598	+	0.788060i	$0.288915\pi$
$504$	0	0
$505$	2879.05	0.253695
$506$	−9519.40	−0.836342
$507$	0	0
$508$	303.092	0.0264715
$509$	−11133.8	−0.969540	−0.484770	−	0.874642i	$-0.661097\pi$
$509$	−11133.8	−0.969540	−0.484770	+	0.874642i	$0.661097\pi$
$510$	0	0
$511$	0	0
$512$	4728.52	0.408150
$513$	0	0
$514$	6721.03	0.576755
$515$	−850.878	−0.0728042
$516$	0	0
$517$	−1212.12	−0.103112
$518$	0	0
$519$	0	0
$520$	−3075.69	−0.259380
$521$	7332.47	0.616586	0.308293	−	0.951291i	$-0.400242\pi$
$521$	7332.47	0.616586	0.308293	+	0.951291i	$0.400242\pi$
$522$	0	0
$523$	1748.66	0.146202	0.0731008	−	0.997325i	$-0.476711\pi$
$523$	1748.66	0.146202	0.0731008	+	0.997325i	$0.476711\pi$
$524$	4639.57	0.386795
$525$	0	0
$526$	5941.23	0.492491
$527$	254.172	0.0210093
$528$	0	0
$529$	20603.4	1.69338
$530$	4613.37	0.378098
$531$	0	0
$532$	0	0
$533$	−6850.09	−0.556680
$534$	0	0
$535$	1957.72	0.158205
$536$	−4594.84	−0.370274
$537$	0	0
$538$	−6860.13	−0.549742
$539$	0	0
$540$	0	0
$541$	22483.4	1.78676	0.893379	−	0.449304i	$-0.148328\pi$
$541$	22483.4	1.78676	0.893379	+	0.449304i	$0.148328\pi$
$542$	−10120.7	−0.802066
$543$	0	0
$544$	225.058	0.0177377
$545$	−4114.29	−0.323370
$546$	0	0
$547$	14066.5	1.09952	0.549761	−	0.835322i	$-0.314719\pi$
$547$	14066.5	1.09952	0.549761	+	0.835322i	$0.314719\pi$
$548$	−6269.90	−0.488753
$549$	0	0
$550$	−1314.65	−0.101921
$551$	−4109.53	−0.317735
$552$	0	0
$553$	0	0
$554$	−1915.32	−0.146884
$555$	0	0
$556$	−8044.16	−0.613576
$557$	−17208.2	−1.30904	−0.654519	−	0.756046i	$-0.727129\pi$
$557$	−17208.2	−1.30904	−0.654519	+	0.756046i	$0.727129\pi$
$558$	0	0
$559$	−1103.38	−0.0834849
$560$	0	0
$561$	0	0
$562$	11376.4	0.853885
$563$	−7504.93	−0.561803	−0.280901	−	0.959737i	$-0.590633\pi$
$563$	−7504.93	−0.561803	−0.280901	+	0.959737i	$0.590633\pi$
$564$	0	0
$565$	5781.15	0.430468
$566$	−13446.8	−0.998605
$567$	0	0
$568$	2191.15	0.161864
$569$	2222.49	0.163746	0.0818731	−	0.996643i	$-0.473910\pi$
$569$	2222.49	0.163746	0.0818731	+	0.996643i	$0.473910\pi$
$570$	0	0
$571$	17106.5	1.25374	0.626868	−	0.779126i	$-0.284337\pi$
$571$	17106.5	1.25374	0.626868	+	0.779126i	$0.284337\pi$
$572$	−5730.37	−0.418879
$573$	0	0
$574$	0	0
$575$	4525.65	0.328231
$576$	0	0
$577$	18382.1	1.32627	0.663135	−	0.748500i	$-0.269226\pi$
$577$	18382.1	1.32627	0.663135	+	0.748500i	$0.269226\pi$
$578$	−7511.91	−0.540578
$579$	0	0
$580$	−987.463	−0.0706934
$581$	0	0
$582$	0	0
$583$	20742.3	1.47351
$584$	−19528.3	−1.38371
$585$	0	0
$586$	−4697.82	−0.331169
$587$	−5053.51	−0.355333	−0.177667	−	0.984091i	$-0.556855\pi$
$587$	−5053.51	−0.355333	−0.177667	+	0.984091i	$0.556855\pi$
$588$	0	0
$589$	24948.3	1.74529
$590$	4105.36	0.286466
$591$	0	0
$592$	−1091.55	−0.0757809
$593$	8083.46	0.559777	0.279889	−	0.960032i	$-0.409702\pi$
$593$	8083.46	0.559777	0.279889	+	0.960032i	$0.409702\pi$
$594$	0	0
$595$	0	0
$596$	−143.537	−0.00986497
$597$	0	0
$598$	−8151.47	−0.557422
$599$	−21389.0	−1.45898	−0.729490	−	0.683991i	$-0.760243\pi$
$599$	−21389.0	−1.45898	−0.729490	+	0.683991i	$0.760243\pi$
$600$	0	0
$601$	4040.80	0.274256	0.137128	−	0.990553i	$-0.456213\pi$
$601$	4040.80	0.274256	0.137128	+	0.990553i	$0.456213\pi$
$602$	0	0
$603$	0	0
$604$	−19842.7	−1.33674
$605$	744.200	0.0500099
$606$	0	0
$607$	6852.45	0.458208	0.229104	−	0.973402i	$-0.426420\pi$
$607$	6852.45	0.458208	0.229104	+	0.973402i	$0.426420\pi$
$608$	22090.6	1.47351
$609$	0	0
$610$	6159.74	0.408853
$611$	−1037.94	−0.0687241
$612$	0	0
$613$	−27139.0	−1.78815	−0.894073	−	0.447921i	$-0.852165\pi$
$613$	−27139.0	−1.78815	−0.894073	+	0.447921i	$0.852165\pi$
$614$	−3894.91	−0.256003
$615$	0	0
$616$	0	0
$617$	2590.37	0.169018	0.0845091	−	0.996423i	$-0.473068\pi$
$617$	2590.37	0.169018	0.0845091	+	0.996423i	$0.473068\pi$
$618$	0	0
$619$	15787.7	1.02514	0.512569	−	0.858646i	$-0.328694\pi$
$619$	15787.7	1.02514	0.512569	+	0.858646i	$0.328694\pi$
$620$	5994.73	0.388313
$621$	0	0
$622$	168.588	0.0108678
$623$	0	0
$624$	0	0
$625$	625.000	0.0400000
$626$	−9107.38	−0.581476
$627$	0	0
$628$	4362.88	0.277226
$629$	−98.2577	−0.00622860
$630$	0	0
$631$	−8956.22	−0.565042	−0.282521	−	0.959261i	$-0.591171\pi$
$631$	−8956.22	−0.565042	−0.282521	+	0.959261i	$0.591171\pi$
$632$	−13333.1	−0.839180
$633$	0	0
$634$	−8612.10	−0.539480
$635$	267.709	0.0167303
$636$	0	0
$637$	0	0
$638$	1834.59	0.113844
$639$	0	0
$640$	6123.66	0.378217
$641$	−11495.7	−0.708349	−0.354175	−	0.935179i	$-0.615238\pi$
$641$	−11495.7	−0.708349	−0.354175	+	0.935179i	$0.615238\pi$
$642$	0	0
$643$	−25643.1	−1.57273	−0.786365	−	0.617763i	$-0.788039\pi$
$643$	−25643.1	−1.57273	−0.786365	+	0.617763i	$0.788039\pi$
$644$	0	0
$645$	0	0
$646$	216.203	0.0131678
$647$	798.201	0.0485016	0.0242508	−	0.999706i	$-0.492280\pi$
$647$	798.201	0.0485016	0.0242508	+	0.999706i	$0.492280\pi$
$648$	0	0
$649$	18458.2	1.11640
$650$	−1125.73	−0.0679305
$651$	0	0
$652$	−22651.5	−1.36058
$653$	−2848.47	−0.170703	−0.0853516	−	0.996351i	$-0.527201\pi$
$653$	−2848.47	−0.170703	−0.0853516	+	0.996351i	$0.527201\pi$
$654$	0	0
$655$	4097.96	0.244459
$656$	3101.82	0.184612
$657$	0	0
$658$	0	0
$659$	−14906.9	−0.881166	−0.440583	−	0.897712i	$-0.645228\pi$
$659$	−14906.9	−0.881166	−0.440583	+	0.897712i	$0.645228\pi$
$660$	0	0
$661$	−16425.3	−0.966522	−0.483261	−	0.875476i	$-0.660548\pi$
$661$	−16425.3	−0.966522	−0.483261	+	0.875476i	$0.660548\pi$
$662$	14905.0	0.875074
$663$	0	0
$664$	1260.92	0.0736945
$665$	0	0
$666$	0	0
$667$	−6315.55	−0.366625
$668$	5430.42	0.314535
$669$	0	0
$670$	−1681.76	−0.0969730
$671$	27694.9	1.59337
$672$	0	0
$673$	−23242.8	−1.33127	−0.665635	−	0.746277i	$-0.731839\pi$
$673$	−23242.8	−1.33127	−0.665635	+	0.746277i	$0.731839\pi$
$674$	309.886	0.0177097
$675$	0	0
$676$	7529.93	0.428421
$677$	22229.6	1.26197	0.630984	−	0.775796i	$-0.282651\pi$
$677$	22229.6	1.26197	0.630984	+	0.775796i	$0.282651\pi$
$678$	0	0
$679$	0	0
$680$	125.368	0.00707006
$681$	0	0
$682$	−11137.5	−0.625333
$683$	−3231.37	−0.181032	−0.0905162	−	0.995895i	$-0.528852\pi$
$683$	−3231.37	−0.181032	−0.0905162	+	0.995895i	$0.528852\pi$
$684$	0	0
$685$	−5537.97	−0.308898
$686$	0	0
$687$	0	0
$688$	499.627	0.0276862
$689$	17761.6	0.982094
$690$	0	0
$691$	14101.9	0.776355	0.388177	−	0.921585i	$-0.373105\pi$
$691$	14101.9	0.776355	0.388177	+	0.921585i	$0.373105\pi$
$692$	2079.34	0.114226
$693$	0	0
$694$	16852.0	0.921749
$695$	−7105.10	−0.387787
$696$	0	0
$697$	279.216	0.0151737
$698$	14488.0	0.785646
$699$	0	0
$700$	0	0
$701$	14291.6	0.770026	0.385013	−	0.922911i	$-0.374197\pi$
$701$	14291.6	0.770026	0.385013	+	0.922911i	$0.374197\pi$
$702$	0	0
$703$	−9644.49	−0.517424
$704$	−6194.76	−0.331639
$705$	0	0
$706$	2215.45	0.118102
$707$	0	0
$708$	0	0
$709$	−9013.18	−0.477429	−0.238715	−	0.971090i	$-0.576726\pi$
$709$	−9013.18	−0.477429	−0.238715	+	0.971090i	$0.576726\pi$
$710$	801.983	0.0423914
$711$	0	0
$712$	28684.8	1.50984
$713$	38340.7	2.01384
$714$	0	0
$715$	−5061.42	−0.264736
$716$	18952.6	0.989234
$717$	0	0
$718$	18215.4	0.946786
$719$	−34651.5	−1.79734	−0.898668	−	0.438629i	$-0.855464\pi$
$719$	−34651.5	−1.79734	−0.898668	+	0.438629i	$0.855464\pi$
$720$	0	0
$721$	0	0
$722$	10731.0	0.553139
$723$	0	0
$724$	9336.06	0.479243
$725$	−872.189	−0.0446790
$726$	0	0
$727$	−21912.3	−1.11786	−0.558929	−	0.829216i	$-0.688788\pi$
$727$	−21912.3	−1.11786	−0.558929	+	0.829216i	$0.688788\pi$
$728$	0	0
$729$	0	0
$730$	−7147.54	−0.362387
$731$	44.9749	0.00227559
$732$	0	0
$733$	−29429.4	−1.48295	−0.741474	−	0.670982i	$-0.765873\pi$
$733$	−29429.4	−1.48295	−0.741474	+	0.670982i	$0.765873\pi$
$734$	−11884.9	−0.597658
$735$	0	0
$736$	33949.0	1.70024
$737$	−7561.37	−0.377919
$738$	0	0
$739$	−27900.3	−1.38881	−0.694404	−	0.719585i	$-0.744332\pi$
$739$	−27900.3	−1.38881	−0.694404	+	0.719585i	$0.744332\pi$
$740$	−2317.44	−0.115123
$741$	0	0
$742$	0	0
$743$	−4450.09	−0.219728	−0.109864	−	0.993947i	$-0.535042\pi$
$743$	−4450.09	−0.219728	−0.109864	+	0.993947i	$0.535042\pi$
$744$	0	0
$745$	−126.781	−0.00623477
$746$	−6142.56	−0.301468
$747$	0	0
$748$	233.575	0.0114176
$749$	0	0
$750$	0	0
$751$	17191.6	0.835327	0.417664	−	0.908602i	$-0.362849\pi$
$751$	17191.6	0.835327	0.417664	+	0.908602i	$0.362849\pi$
$752$	469.993	0.0227911
$753$	0	0
$754$	1570.96	0.0758767
$755$	−17526.4	−0.844833
$756$	0	0
$757$	8146.19	0.391120	0.195560	−	0.980692i	$-0.437347\pi$
$757$	8146.19	0.391120	0.195560	+	0.980692i	$0.437347\pi$
$758$	−17146.2	−0.821607
$759$	0	0
$760$	12305.5	0.587326
$761$	26162.5	1.24624	0.623121	−	0.782125i	$-0.285864\pi$
$761$	26162.5	1.24624	0.623121	+	0.782125i	$0.285864\pi$
$762$	0	0
$763$	0	0
$764$	−16235.4	−0.768818
$765$	0	0
$766$	5304.16	0.250192
$767$	15805.7	0.744084
$768$	0	0
$769$	−21315.1	−0.999536	−0.499768	−	0.866159i	$-0.666581\pi$
$769$	−21315.1	−0.999536	−0.499768	+	0.866159i	$0.666581\pi$
$770$	0	0
$771$	0	0
$772$	−23226.7	−1.08283
$773$	−21283.0	−0.990293	−0.495147	−	0.868809i	$-0.664886\pi$
$773$	−21283.0	−0.990293	−0.495147	+	0.868809i	$0.664886\pi$
$774$	0	0
$775$	5294.92	0.245418
$776$	32641.0	1.50998
$777$	0	0
$778$	1934.37	0.0891394
$779$	27406.5	1.26051
$780$	0	0
$781$	3605.81	0.165206
$782$	332.262	0.0151939
$783$	0	0
$784$	0	0
$785$	3853.57	0.175210
$786$	0	0
$787$	32965.8	1.49314	0.746571	−	0.665306i	$-0.231699\pi$
$787$	32965.8	1.49314	0.746571	+	0.665306i	$0.231699\pi$
$788$	15534.8	0.702289
$789$	0	0
$790$	−4880.04	−0.219777
$791$	0	0
$792$	0	0
$793$	23715.2	1.06198
$794$	−23056.1	−1.03052
$795$	0	0
$796$	−1941.94	−0.0864703
$797$	−34809.8	−1.54708	−0.773542	−	0.633745i	$-0.781517\pi$
$797$	−34809.8	−1.54708	−0.773542	+	0.633745i	$0.781517\pi$
$798$	0	0
$799$	42.3073	0.00187325
$800$	4688.42	0.207201
$801$	0	0
$802$	−18690.4	−0.822917
$803$	−32136.2	−1.41228
$804$	0	0
$805$	0	0
$806$	−9537.05	−0.416784
$807$	0	0
$808$	12030.6	0.523806
$809$	−22685.5	−0.985882	−0.492941	−	0.870063i	$-0.664078\pi$
$809$	−22685.5	−0.985882	−0.492941	+	0.870063i	$0.664078\pi$
$810$	0	0
$811$	−19076.8	−0.825987	−0.412993	−	0.910734i	$-0.635517\pi$
$811$	−19076.8	−0.825987	−0.412993	+	0.910734i	$0.635517\pi$
$812$	0	0
$813$	0	0
$814$	4305.53	0.185392
$815$	−20007.2	−0.859905
$816$	0	0
$817$	4414.52	0.189038
$818$	−15551.3	−0.664717
$819$	0	0
$820$	6585.40	0.280454
$821$	25551.5	1.08618	0.543090	−	0.839674i	$-0.317254\pi$
$821$	25551.5	1.08618	0.543090	+	0.839674i	$0.317254\pi$
$822$	0	0
$823$	−32208.9	−1.36420	−0.682098	−	0.731261i	$-0.738932\pi$
$823$	−32208.9	−1.36420	−0.682098	+	0.731261i	$0.738932\pi$
$824$	−3555.54	−0.150319
$825$	0	0
$826$	0	0
$827$	4573.91	0.192322	0.0961611	−	0.995366i	$-0.469344\pi$
$827$	4573.91	0.192322	0.0961611	+	0.995366i	$0.469344\pi$
$828$	0	0
$829$	2064.08	0.0864757	0.0432378	−	0.999065i	$-0.486233\pi$
$829$	2064.08	0.0864757	0.0432378	+	0.999065i	$0.486233\pi$
$830$	461.509	0.0193002
$831$	0	0
$832$	−5304.57	−0.221037
$833$	0	0
$834$	0	0
$835$	4796.49	0.198790
$836$	22926.6	0.948485
$837$	0	0
$838$	−13624.9	−0.561651
$839$	15171.8	0.624300	0.312150	−	0.950033i	$-0.398951\pi$
$839$	15171.8	0.624300	0.312150	+	0.950033i	$0.398951\pi$
$840$	0	0
$841$	−23171.9	−0.950095
$842$	−7235.41	−0.296139
$843$	0	0
$844$	−12402.8	−0.505831
$845$	6650.90	0.270767
$846$	0	0
$847$	0	0
$848$	−8042.71	−0.325693
$849$	0	0
$850$	45.8859	0.00185162
$851$	−14821.7	−0.597041
$852$	0	0
$853$	28288.9	1.13552	0.567758	−	0.823196i	$-0.307811\pi$
$853$	28288.9	1.13552	0.567758	+	0.823196i	$0.307811\pi$
$854$	0	0
$855$	0	0
$856$	8180.68	0.326647
$857$	10423.5	0.415472	0.207736	−	0.978185i	$-0.433390\pi$
$857$	10423.5	0.415472	0.207736	+	0.978185i	$0.433390\pi$
$858$	0	0
$859$	−28258.3	−1.12242	−0.561211	−	0.827673i	$-0.689664\pi$
$859$	−28258.3	−1.12242	−0.561211	+	0.827673i	$0.689664\pi$
$860$	1060.75	0.0420595
$861$	0	0
$862$	22318.0	0.881851
$863$	19333.9	0.762611	0.381306	−	0.924449i	$-0.375475\pi$
$863$	19333.9	0.762611	0.381306	+	0.924449i	$0.375475\pi$
$864$	0	0
$865$	1836.60	0.0721923
$866$	4940.55	0.193865
$867$	0	0
$868$	0	0
$869$	−21941.2	−0.856508
$870$	0	0
$871$	−6474.80	−0.251883
$872$	−17192.3	−0.667664
$873$	0	0
$874$	32613.2	1.26219
$875$	0	0
$876$	0	0
$877$	33656.1	1.29588	0.647940	−	0.761691i	$-0.275631\pi$
$877$	33656.1	1.29588	0.647940	+	0.761691i	$0.275631\pi$
$878$	16840.5	0.647313
$879$	0	0
$880$	2291.88	0.0877947
$881$	−43732.7	−1.67241	−0.836204	−	0.548418i	$-0.815230\pi$
$881$	−43732.7	−1.67241	−0.836204	+	0.548418i	$0.815230\pi$
$882$	0	0
$883$	31357.2	1.19508	0.597539	−	0.801840i	$-0.296145\pi$
$883$	31357.2	1.19508	0.597539	+	0.801840i	$0.296145\pi$
$884$	200.011	0.00760983
$885$	0	0
$886$	−25119.8	−0.952501
$887$	−12457.9	−0.471584	−0.235792	−	0.971803i	$-0.575768\pi$
$887$	−12457.9	−0.471584	−0.235792	+	0.971803i	$0.575768\pi$
$888$	0	0
$889$	0	0
$890$	10498.9	0.395421
$891$	0	0
$892$	20931.3	0.785688
$893$	4152.68	0.155615
$894$	0	0
$895$	16740.1	0.625207
$896$	0	0
$897$	0	0
$898$	−14094.0	−0.523743
$899$	−7389.06	−0.274126
$900$	0	0
$901$	−723.980	−0.0267694
$902$	−12234.9	−0.451639
$903$	0	0
$904$	24157.5	0.888790
$905$	8246.19	0.302887
$906$	0	0
$907$	−33590.7	−1.22973	−0.614863	−	0.788634i	$-0.710789\pi$
$907$	−33590.7	−1.22973	−0.614863	+	0.788634i	$0.710789\pi$
$908$	−18688.7	−0.683048
$909$	0	0
$910$	0	0
$911$	34316.8	1.24804	0.624022	−	0.781407i	$-0.285498\pi$
$911$	34316.8	1.24804	0.624022	+	0.781407i	$0.285498\pi$
$912$	0	0
$913$	2075.00	0.0752162
$914$	4506.40	0.163084
$915$	0	0
$916$	17676.4	0.637602
$917$	0	0
$918$	0	0
$919$	336.846	0.0120909	0.00604545	−	0.999982i	$-0.498076\pi$
$919$	336.846	0.0120909	0.00604545	+	0.999982i	$0.498076\pi$
$920$	18911.2	0.677699
$921$	0	0
$922$	20362.2	0.727326
$923$	3087.66	0.110110
$924$	0	0
$925$	−2046.91	−0.0727587
$926$	−15995.1	−0.567637
$927$	0	0
$928$	−6542.69	−0.231438
$929$	23909.9	0.844411	0.422206	−	0.906500i	$-0.361256\pi$
$929$	23909.9	0.844411	0.422206	+	0.906500i	$0.361256\pi$
$930$	0	0
$931$	0	0
$932$	32873.8	1.15538
$933$	0	0
$934$	−9820.80	−0.344054
$935$	206.308	0.00721605
$936$	0	0
$937$	−28534.2	−0.994848	−0.497424	−	0.867507i	$-0.665721\pi$
$937$	−28534.2	−0.994848	−0.497424	+	0.867507i	$0.665721\pi$
$938$	0	0
$939$	0	0
$940$	997.831	0.0346230
$941$	−31169.8	−1.07981	−0.539907	−	0.841724i	$-0.681541\pi$
$941$	−31169.8	−1.07981	−0.539907	+	0.841724i	$0.681541\pi$
$942$	0	0
$943$	42118.5	1.45447
$944$	−7157.07	−0.246761
$945$	0	0
$946$	−1970.75	−0.0677320
$947$	34374.4	1.17953	0.589766	−	0.807574i	$-0.299220\pi$
$947$	34374.4	1.17953	0.589766	+	0.807574i	$0.299220\pi$
$948$	0	0
$949$	−27518.2	−0.941284
$950$	4503.94	0.153818
$951$	0	0
$952$	0	0
$953$	−9032.47	−0.307020	−0.153510	−	0.988147i	$-0.549058\pi$
$953$	−9032.47	−0.307020	−0.153510	+	0.988147i	$0.549058\pi$
$954$	0	0
$955$	−14340.1	−0.485901
$956$	15640.3	0.529126
$957$	0	0
$958$	11894.5	0.401143
$959$	0	0
$960$	0	0
$961$	15066.8	0.505750
$962$	3686.83	0.123563
$963$	0	0
$964$	13066.1	0.436546
$965$	−20515.2	−0.684362
$966$	0	0
$967$	−44896.8	−1.49306	−0.746528	−	0.665354i	$-0.768280\pi$
$967$	−44896.8	−1.49306	−0.746528	+	0.665354i	$0.768280\pi$
$968$	3109.76	0.103256
$969$	0	0
$970$	11946.9	0.395457
$971$	26866.3	0.887932	0.443966	−	0.896044i	$-0.353571\pi$
$971$	26866.3	0.887932	0.443966	+	0.896044i	$0.353571\pi$
$972$	0	0
$973$	0	0
$974$	20966.2	0.689734
$975$	0	0
$976$	−10738.6	−0.352185
$977$	13588.6	0.444973	0.222487	−	0.974936i	$-0.428583\pi$
$977$	13588.6	0.444973	0.222487	+	0.974936i	$0.428583\pi$
$978$	0	0
$979$	47204.4	1.54102
$980$	0	0
$981$	0	0
$982$	−16470.6	−0.535233
$983$	40429.1	1.31179	0.655894	−	0.754853i	$-0.272292\pi$
$983$	40429.1	1.31179	0.655894	+	0.754853i	$0.272292\pi$
$984$	0	0
$985$	13721.3	0.443854
$986$	−64.0339	−0.00206821
$987$	0	0
$988$	19632.1	0.632165
$989$	6784.26	0.218126
$990$	0	0
$991$	−42072.6	−1.34862	−0.674308	−	0.738450i	$-0.735558\pi$
$991$	−42072.6	−1.34862	−0.674308	+	0.738450i	$0.735558\pi$
$992$	39719.6	1.27127
$993$	0	0
$994$	0	0
$995$	−1715.25	−0.0546502
$996$	0	0
$997$	40874.2	1.29839	0.649197	−	0.760620i	$-0.275105\pi$
$997$	40874.2	1.29839	0.649197	+	0.760620i	$0.275105\pi$
$998$	−26481.0	−0.839921
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.4.a.cd.1.5		8
3.2	odd	2		735.4.a.bc.1.4	yes	8
7.6	odd	2		2205.4.a.ce.1.5		8
21.20	even	2		735.4.a.bb.1.4	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
735.4.a.bb.1.4	✓	8	21.20	even	2
735.4.a.bc.1.4	yes	8	3.2	odd	2
2205.4.a.cd.1.5		8	1.1	even	1	trivial
2205.4.a.ce.1.5		8	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2205.a (trivial)

\(f(q)\)	\(=\)	\(q+1.52943 q^{2} -5.66083 q^{4} -5.00000 q^{5} -20.8933 q^{8} +O(q^{10})\)	\(q+1.52943 q^{2} -5.66083 q^{4} -5.00000 q^{5} -20.8933 q^{8} -7.64717 q^{10} -34.3826 q^{11} -29.4418 q^{13} +13.3317 q^{16} +1.20008 q^{17} +117.794 q^{19} +28.3042 q^{20} -52.5858 q^{22} +181.026 q^{23} +25.0000 q^{25} -45.0293 q^{26} -34.8875 q^{29} +211.797 q^{31} +187.537 q^{32} +1.83544 q^{34} -81.8762 q^{37} +180.157 q^{38} +104.467 q^{40} +232.665 q^{41} +37.4767 q^{43} +194.634 q^{44} +276.867 q^{46} +35.2539 q^{47} +38.2359 q^{50} +166.665 q^{52} -603.278 q^{53} +171.913 q^{55} -53.3582 q^{58} -536.847 q^{59} -805.493 q^{61} +323.929 q^{62} +180.172 q^{64} +147.209 q^{65} +219.919 q^{67} -6.79343 q^{68} -104.873 q^{71} +934.665 q^{73} -125.224 q^{74} -666.809 q^{76} +638.150 q^{79} -66.6583 q^{80} +355.846 q^{82} -60.3503 q^{83} -6.00038 q^{85} +57.3182 q^{86} +718.366 q^{88} -1372.92 q^{89} -1024.76 q^{92} +53.9185 q^{94} -588.968 q^{95} -1562.27 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} + 50 q^{4} - 40 q^{5} - 66 q^{8}+O(q^{10}) \) 8 * q - 2 * q^2 + 50 * q^4 - 40 * q^5 - 66 * q^8	\( 8 q - 2 q^{2} + 50 q^{4} - 40 q^{5} - 66 q^{8} + 10 q^{10} - 64 q^{11} + 206 q^{16} - 48 q^{17} + 80 q^{19} - 250 q^{20} + 452 q^{22} - 120 q^{23} + 200 q^{25} - 272 q^{26} - 76 q^{29} + 20 q^{31} - 770 q^{32} + 320 q^{34} + 348 q^{37} + 236 q^{38} + 330 q^{40} - 944 q^{41} + 1116 q^{43} - 172 q^{44} + 496 q^{46} + 208 q^{47} - 50 q^{50} - 2272 q^{52} - 1144 q^{53} + 320 q^{55} + 560 q^{58} - 596 q^{59} + 740 q^{61} + 1184 q^{62} + 1298 q^{64} + 1964 q^{67} - 96 q^{68} + 4 q^{71} - 1500 q^{73} - 3368 q^{74} - 2912 q^{76} - 460 q^{79} - 1030 q^{80} - 5644 q^{82} - 700 q^{83} + 240 q^{85} + 1396 q^{86} + 6892 q^{88} - 644 q^{92} - 6692 q^{94} - 400 q^{95} - 2052 q^{97}+O(q^{100}) \) 8 * q - 2 * q^2 + 50 * q^4 - 40 * q^5 - 66 * q^8 + 10 * q^10 - 64 * q^11 + 206 * q^16 - 48 * q^17 + 80 * q^19 - 250 * q^20 + 452 * q^22 - 120 * q^23 + 200 * q^25 - 272 * q^26 - 76 * q^29 + 20 * q^31 - 770 * q^32 + 320 * q^34 + 348 * q^37 + 236 * q^38 + 330 * q^40 - 944 * q^41 + 1116 * q^43 - 172 * q^44 + 496 * q^46 + 208 * q^47 - 50 * q^50 - 2272 * q^52 - 1144 * q^53 + 320 * q^55 + 560 * q^58 - 596 * q^59 + 740 * q^61 + 1184 * q^62 + 1298 * q^64 + 1964 * q^67 - 96 * q^68 + 4 * q^71 - 1500 * q^73 - 3368 * q^74 - 2912 * q^76 - 460 * q^79 - 1030 * q^80 - 5644 * q^82 - 700 * q^83 + 240 * q^85 + 1396 * q^86 + 6892 * q^88 - 644 * q^92 - 6692 * q^94 - 400 * q^95 - 2052 * q^97

Introduction

L-functions

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Database

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