LMFDB - Embedded newform 2205.4.a.bx.1.1

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:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 34x^{4} + 241x^{2} - 64 $ x^6 - 34x^4 + 241x^2 - 64
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-4.91092$ 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)

$f(q)$	$=$	$q-4.91092 q^{2} +16.1171 q^{4} -5.00000 q^{5} -39.8626 q^{8} +O(q^{10})$	$q-4.91092 q^{2} +16.1171 q^{4} -5.00000 q^{5} -39.8626 q^{8} +24.5546 q^{10} -12.0023 q^{11} +46.9540 q^{13} +66.8251 q^{16} -44.7080 q^{17} -31.7414 q^{19} -80.5857 q^{20} +58.9423 q^{22} +39.4080 q^{23} +25.0000 q^{25} -230.587 q^{26} -87.2713 q^{29} +304.718 q^{31} -9.27181 q^{32} +219.557 q^{34} -151.061 q^{37} +155.879 q^{38} +199.313 q^{40} -282.927 q^{41} -143.177 q^{43} -193.443 q^{44} -193.530 q^{46} -88.8029 q^{47} -122.773 q^{50} +756.764 q^{52} -149.940 q^{53} +60.0114 q^{55} +428.582 q^{58} +712.934 q^{59} +65.0347 q^{61} -1496.45 q^{62} -489.068 q^{64} -234.770 q^{65} +698.725 q^{67} -720.565 q^{68} +291.400 q^{71} +476.241 q^{73} +741.849 q^{74} -511.580 q^{76} +256.947 q^{79} -334.126 q^{80} +1389.43 q^{82} -1177.61 q^{83} +223.540 q^{85} +703.129 q^{86} +478.443 q^{88} -1009.96 q^{89} +635.145 q^{92} +436.104 q^{94} +158.707 q^{95} -1149.16 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 20 q^{4} - 30 q^{5}+O(q^{10}) $ 6 * q + 20 * q^4 - 30 * q^5	$ 6 q + 20 q^{4} - 30 q^{5} + 100 q^{16} - 44 q^{17} - 100 q^{20} - 24 q^{22} + 150 q^{25} - 168 q^{26} + 380 q^{37} - 56 q^{38} - 612 q^{41} - 328 q^{43} + 432 q^{46} - 120 q^{47} + 1120 q^{58} - 136 q^{59} - 2264 q^{62} - 1052 q^{64} + 1112 q^{67} - 264 q^{68} + 1400 q^{79} - 500 q^{80} - 2912 q^{83} + 220 q^{85} + 1384 q^{88} - 372 q^{89}+O(q^{100}) $ 6 * q + 20 * q^4 - 30 * q^5 + 100 * q^16 - 44 * q^17 - 100 * q^20 - 24 * q^22 + 150 * q^25 - 168 * q^26 + 380 * q^37 - 56 * q^38 - 612 * q^41 - 328 * q^43 + 432 * q^46 - 120 * q^47 + 1120 * q^58 - 136 * q^59 - 2264 * q^62 - 1052 * q^64 + 1112 * q^67 - 264 * q^68 + 1400 * q^79 - 500 * q^80 - 2912 * q^83 + 220 * q^85 + 1384 * q^88 - 372 * q^89

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 $

$2$	−4.91092	−1.73627	−0.868136	−	0.496326i	$-0.834682\pi$
$2$	−4.91092	−1.73627	−0.868136	+	0.496326i	$0.834682\pi$
$3$	0	0
$3$	0	0
$4$	16.1171	2.01464
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−39.8626	−1.76170
$9$	0	0
$10$	24.5546	0.776485
$11$	−12.0023	−0.328984	−0.164492	−	0.986378i	$-0.552598\pi$
$11$	−12.0023	−0.328984	−0.164492	+	0.986378i	$0.552598\pi$
$12$	0	0
$13$	46.9540	1.00175	0.500873	−	0.865521i	$-0.333012\pi$
$13$	46.9540	1.00175	0.500873	+	0.865521i	$0.333012\pi$
$14$	0	0
$15$	0	0
$16$	66.8251	1.04414
$17$	−44.7080	−0.637840	−0.318920	−	0.947782i	$-0.603320\pi$
$17$	−44.7080	−0.637840	−0.318920	+	0.947782i	$0.603320\pi$
$18$	0	0
$19$	−31.7414	−0.383262	−0.191631	−	0.981467i	$-0.561378\pi$
$19$	−31.7414	−0.383262	−0.191631	+	0.981467i	$0.561378\pi$
$20$	−80.5857	−0.900976
$21$	0	0
$22$	58.9423	0.571206
$23$	39.4080	0.357267	0.178634	−	0.983916i	$-0.442832\pi$
$23$	39.4080	0.357267	0.178634	+	0.983916i	$0.442832\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−230.587	−1.73930
$27$	0	0
$28$	0	0
$29$	−87.2713	−0.558823	−0.279412	−	0.960171i	$-0.590139\pi$
$29$	−87.2713	−0.558823	−0.279412	+	0.960171i	$0.590139\pi$
$30$	0	0
$31$	304.718	1.76545	0.882726	−	0.469887i	$-0.155705\pi$
$31$	304.718	1.76545	0.882726	+	0.469887i	$0.155705\pi$
$32$	−9.27181	−0.0512200
$33$	0	0
$34$	219.557	1.10746
$35$	0	0
$36$	0	0
$37$	−151.061	−0.671197	−0.335599	−	0.942005i	$-0.608939\pi$
$37$	−151.061	−0.671197	−0.335599	+	0.942005i	$0.608939\pi$
$38$	155.879	0.665447
$39$	0	0
$40$	199.313	0.787855
$41$	−282.927	−1.07770	−0.538851	−	0.842401i	$-0.681141\pi$
$41$	−282.927	−1.07770	−0.538851	+	0.842401i	$0.681141\pi$
$42$	0	0
$43$	−143.177	−0.507773	−0.253886	−	0.967234i	$-0.581709\pi$
$43$	−143.177	−0.507773	−0.253886	+	0.967234i	$0.581709\pi$
$44$	−193.443	−0.662786
$45$	0	0
$46$	−193.530	−0.620313
$47$	−88.8029	−0.275601	−0.137800	−	0.990460i	$-0.544003\pi$
$47$	−88.8029	−0.275601	−0.137800	+	0.990460i	$0.544003\pi$
$48$	0	0
$49$	0	0
$50$	−122.773	−0.347255
$51$	0	0
$52$	756.764	2.01816
$53$	−149.940	−0.388601	−0.194301	−	0.980942i	$-0.562244\pi$
$53$	−149.940	−0.388601	−0.194301	+	0.980942i	$0.562244\pi$
$54$	0	0
$55$	60.0114	0.147126
$56$	0	0
$57$	0	0
$58$	428.582	0.970269
$59$	712.934	1.57315	0.786577	−	0.617493i	$-0.211851\pi$
$59$	712.934	1.57315	0.786577	+	0.617493i	$0.211851\pi$
$60$	0	0
$61$	65.0347	0.136506	0.0682528	−	0.997668i	$-0.478258\pi$
$61$	65.0347	0.136506	0.0682528	+	0.997668i	$0.478258\pi$
$62$	−1496.45	−3.06531
$63$	0	0
$64$	−489.068	−0.955211
$65$	−234.770	−0.447994
$66$	0	0
$67$	698.725	1.27407	0.637036	−	0.770834i	$-0.280160\pi$
$67$	698.725	1.27407	0.637036	+	0.770834i	$0.280160\pi$
$68$	−720.565	−1.28502
$69$	0	0
$70$	0	0
$71$	291.400	0.487082	0.243541	−	0.969891i	$-0.421691\pi$
$71$	291.400	0.487082	0.243541	+	0.969891i	$0.421691\pi$
$72$	0	0
$73$	476.241	0.763560	0.381780	−	0.924253i	$-0.375311\pi$
$73$	476.241	0.763560	0.381780	+	0.924253i	$0.375311\pi$
$74$	741.849	1.16538
$75$	0	0
$76$	−511.580	−0.772135
$77$	0	0
$78$	0	0
$79$	256.947	0.365935	0.182967	−	0.983119i	$-0.441430\pi$
$79$	256.947	0.365935	0.182967	+	0.983119i	$0.441430\pi$
$80$	−334.126	−0.466955
$81$	0	0
$82$	1389.43	1.87118
$83$	−1177.61	−1.55735	−0.778673	−	0.627430i	$-0.784107\pi$
$83$	−1177.61	−1.55735	−0.778673	+	0.627430i	$0.784107\pi$
$84$	0	0
$85$	223.540	0.285251
$86$	703.129	0.881632
$87$	0	0
$88$	478.443	0.579570
$89$	−1009.96	−1.20287	−0.601433	−	0.798923i	$-0.705403\pi$
$89$	−1009.96	−1.20287	−0.601433	+	0.798923i	$0.705403\pi$
$90$	0	0
$91$	0	0
$92$	635.145	0.719765
$93$	0	0
$94$	436.104	0.478518
$95$	158.707	0.171400
$96$	0	0
$97$	−1149.16	−1.20288	−0.601441	−	0.798917i	$-0.705407\pi$
$97$	−1149.16	−1.20288	−0.601441	+	0.798917i	$0.705407\pi$
$98$	0	0
$99$	0	0
$100$	402.929	0.402929
$101$	−288.883	−0.284604	−0.142302	−	0.989823i	$-0.545450\pi$
$101$	−288.883	−0.284604	−0.142302	+	0.989823i	$0.545450\pi$
$102$	0	0
$103$	1113.21	1.06493	0.532464	−	0.846453i	$-0.321266\pi$
$103$	1113.21	1.06493	0.532464	+	0.846453i	$0.321266\pi$
$104$	−1871.71	−1.76477
$105$	0	0
$106$	736.345	0.674718
$107$	−263.840	−0.238377	−0.119189	−	0.992872i	$-0.538029\pi$
$107$	−263.840	−0.238377	−0.119189	+	0.992872i	$0.538029\pi$
$108$	0	0
$109$	−1246.96	−1.09575	−0.547877	−	0.836559i	$-0.684564\pi$
$109$	−1246.96	−1.09575	−0.547877	+	0.836559i	$0.684564\pi$
$110$	−294.711	−0.255451
$111$	0	0
$112$	0	0
$113$	2034.45	1.69367	0.846835	−	0.531855i	$-0.178505\pi$
$113$	2034.45	1.69367	0.846835	+	0.531855i	$0.178505\pi$
$114$	0	0
$115$	−197.040	−0.159775
$116$	−1406.56	−1.12583
$117$	0	0
$118$	−3501.16	−2.73142
$119$	0	0
$120$	0	0
$121$	−1186.95	−0.891769
$122$	−319.380	−0.237011
$123$	0	0
$124$	4911.19	3.55676
$125$	−125.000	−0.0894427
$126$	0	0
$127$	1143.07	0.798670	0.399335	−	0.916805i	$-0.369241\pi$
$127$	1143.07	0.798670	0.399335	+	0.916805i	$0.369241\pi$
$128$	2475.95	1.70973
$129$	0	0
$130$	1152.94	0.777841
$131$	−1544.70	−1.03024	−0.515118	−	0.857119i	$-0.672252\pi$
$131$	−1544.70	−1.03024	−0.515118	+	0.857119i	$0.672252\pi$
$132$	0	0
$133$	0	0
$134$	−3431.38	−2.21214
$135$	0	0
$136$	1782.18	1.12368
$137$	−243.730	−0.151995	−0.0759974	−	0.997108i	$-0.524214\pi$
$137$	−243.730	−0.151995	−0.0759974	+	0.997108i	$0.524214\pi$
$138$	0	0
$139$	−624.857	−0.381293	−0.190646	−	0.981659i	$-0.561058\pi$
$139$	−624.857	−0.381293	−0.190646	+	0.981659i	$0.561058\pi$
$140$	0	0
$141$	0	0
$142$	−1431.04	−0.845707
$143$	−563.555	−0.329559
$144$	0	0
$145$	436.356	0.249913
$146$	−2338.78	−1.32575
$147$	0	0
$148$	−2434.67	−1.35222
$149$	−3082.32	−1.69472	−0.847359	−	0.531020i	$-0.821809\pi$
$149$	−3082.32	−1.69472	−0.847359	+	0.531020i	$0.821809\pi$
$150$	0	0
$151$	−1840.16	−0.991720	−0.495860	−	0.868402i	$-0.665147\pi$
$151$	−1840.16	−0.991720	−0.495860	+	0.868402i	$0.665147\pi$
$152$	1265.30	0.675191
$153$	0	0
$154$	0	0
$155$	−1523.59	−0.789535
$156$	0	0
$157$	129.576	0.0658679	0.0329339	−	0.999458i	$-0.489515\pi$
$157$	129.576	0.0658679	0.0329339	+	0.999458i	$0.489515\pi$
$158$	−1261.85	−0.635362
$159$	0	0
$160$	46.3591	0.0229063
$161$	0	0
$162$	0	0
$163$	3575.75	1.71825	0.859124	−	0.511768i	$-0.171009\pi$
$163$	3575.75	1.71825	0.859124	+	0.511768i	$0.171009\pi$
$164$	−4559.97	−2.17118
$165$	0	0
$166$	5783.16	2.70398
$167$	2054.63	0.952049	0.476025	−	0.879432i	$-0.342077\pi$
$167$	2054.63	0.952049	0.476025	+	0.879432i	$0.342077\pi$
$168$	0	0
$169$	7.67889	0.00349517
$170$	−1097.79	−0.495273
$171$	0	0
$172$	−2307.60	−1.02298
$173$	3279.38	1.44120	0.720598	−	0.693353i	$-0.243867\pi$
$173$	3279.38	1.44120	0.720598	+	0.693353i	$0.243867\pi$
$174$	0	0
$175$	0	0
$176$	−802.054	−0.343506
$177$	0	0
$178$	4959.81	2.08850
$179$	3287.59	1.37277	0.686385	−	0.727239i	$-0.259197\pi$
$179$	3287.59	1.37277	0.686385	+	0.727239i	$0.259197\pi$
$180$	0	0
$181$	−2709.27	−1.11259	−0.556293	−	0.830986i	$-0.687777\pi$
$181$	−2709.27	−1.11259	−0.556293	+	0.830986i	$0.687777\pi$
$182$	0	0
$183$	0	0
$184$	−1570.91	−0.629396
$185$	755.306	0.300169
$186$	0	0
$187$	536.598	0.209839
$188$	−1431.25	−0.555237
$189$	0	0
$190$	−779.397	−0.297597
$191$	2106.69	0.798087	0.399044	−	0.916932i	$-0.369342\pi$
$191$	2106.69	0.798087	0.399044	+	0.916932i	$0.369342\pi$
$192$	0	0
$193$	3956.67	1.47569	0.737844	−	0.674972i	$-0.235844\pi$
$193$	3956.67	1.47569	0.737844	+	0.674972i	$0.235844\pi$
$194$	5643.44	2.08853
$195$	0	0
$196$	0	0
$197$	1892.16	0.684319	0.342160	−	0.939642i	$-0.388842\pi$
$197$	1892.16	0.684319	0.342160	+	0.939642i	$0.388842\pi$
$198$	0	0
$199$	4422.23	1.57529	0.787646	−	0.616128i	$-0.211299\pi$
$199$	4422.23	1.57529	0.787646	+	0.616128i	$0.211299\pi$
$200$	−996.566	−0.352339
$201$	0	0
$202$	1418.68	0.494150
$203$	0	0
$204$	0	0
$205$	1414.63	0.481963
$206$	−5466.87	−1.84901
$207$	0	0
$208$	3137.71	1.04597
$209$	380.969	0.126087
$210$	0	0
$211$	1402.47	0.457584	0.228792	−	0.973475i	$-0.426522\pi$
$211$	1402.47	0.457584	0.228792	+	0.973475i	$0.426522\pi$
$212$	−2416.61	−0.782893
$213$	0	0
$214$	1295.70	0.413888
$215$	715.883	0.227083
$216$	0	0
$217$	0	0
$218$	6123.73	1.90253
$219$	0	0
$220$	967.213	0.296407
$221$	−2099.22	−0.638954
$222$	0	0
$223$	1111.09	0.333651	0.166826	−	0.985986i	$-0.446648\pi$
$223$	1111.09	0.333651	0.166826	+	0.985986i	$0.446648\pi$
$224$	0	0
$225$	0	0
$226$	−9991.01	−2.94067
$227$	−1715.41	−0.501567	−0.250784	−	0.968043i	$-0.580688\pi$
$227$	−1715.41	−0.501567	−0.250784	+	0.968043i	$0.580688\pi$
$228$	0	0
$229$	934.139	0.269562	0.134781	−	0.990875i	$-0.456967\pi$
$229$	934.139	0.269562	0.134781	+	0.990875i	$0.456967\pi$
$230$	967.648	0.277412
$231$	0	0
$232$	3478.86	0.984477
$233$	3585.12	1.00802	0.504010	−	0.863698i	$-0.331857\pi$
$233$	3585.12	1.00802	0.504010	+	0.863698i	$0.331857\pi$
$234$	0	0
$235$	444.014	0.123252
$236$	11490.5	3.16934
$237$	0	0
$238$	0	0
$239$	−5944.92	−1.60897	−0.804487	−	0.593970i	$-0.797560\pi$
$239$	−5944.92	−1.60897	−0.804487	+	0.593970i	$0.797560\pi$
$240$	0	0
$241$	5606.87	1.49863	0.749316	−	0.662212i	$-0.230382\pi$
$241$	5606.87	1.49863	0.749316	+	0.662212i	$0.230382\pi$
$242$	5828.99	1.54835
$243$	0	0
$244$	1048.17	0.275010
$245$	0	0
$246$	0	0
$247$	−1490.39	−0.383931
$248$	−12146.9	−3.11019
$249$	0	0
$250$	613.865	0.155297
$251$	5182.08	1.30315	0.651574	−	0.758585i	$-0.274109\pi$
$251$	5182.08	1.30315	0.651574	+	0.758585i	$0.274109\pi$
$252$	0	0
$253$	−472.986	−0.117535
$254$	−5613.52	−1.38671
$255$	0	0
$256$	−8246.64	−2.01334
$257$	−37.7581	−0.00916453	−0.00458226	−	0.999990i	$-0.501459\pi$
$257$	−37.7581	−0.00916453	−0.00458226	+	0.999990i	$0.501459\pi$
$258$	0	0
$259$	0	0
$260$	−3783.82	−0.902549
$261$	0	0
$262$	7585.90	1.78877
$263$	−1147.11	−0.268950	−0.134475	−	0.990917i	$-0.542935\pi$
$263$	−1147.11	−0.268950	−0.134475	+	0.990917i	$0.542935\pi$
$264$	0	0
$265$	749.701	0.173788
$266$	0	0
$267$	0	0
$268$	11261.5	2.56680
$269$	−5417.07	−1.22782	−0.613912	−	0.789374i	$-0.710405\pi$
$269$	−5417.07	−1.22782	−0.613912	+	0.789374i	$0.710405\pi$
$270$	0	0
$271$	−6108.76	−1.36930	−0.684651	−	0.728871i	$-0.740045\pi$
$271$	−6108.76	−1.36930	−0.684651	+	0.728871i	$0.740045\pi$
$272$	−2987.62	−0.665996
$273$	0	0
$274$	1196.94	0.263904
$275$	−300.057	−0.0657968
$276$	0	0
$277$	−7247.04	−1.57196	−0.785979	−	0.618253i	$-0.787841\pi$
$277$	−7247.04	−1.57196	−0.785979	+	0.618253i	$0.787841\pi$
$278$	3068.62	0.662028
$279$	0	0
$280$	0	0
$281$	563.644	0.119659	0.0598295	−	0.998209i	$-0.480944\pi$
$281$	563.644	0.119659	0.0598295	+	0.998209i	$0.480944\pi$
$282$	0	0
$283$	−49.6198	−0.0104226	−0.00521129	−	0.999986i	$-0.501659\pi$
$283$	−49.6198	−0.0104226	−0.00521129	+	0.999986i	$0.501659\pi$
$284$	4696.54	0.981296
$285$	0	0
$286$	2767.58	0.572204
$287$	0	0
$288$	0	0
$289$	−2914.20	−0.593160
$290$	−2142.91	−0.433918
$291$	0	0
$292$	7675.65	1.53830
$293$	5681.57	1.13284	0.566418	−	0.824118i	$-0.308329\pi$
$293$	5681.57	1.13284	0.566418	+	0.824118i	$0.308329\pi$
$294$	0	0
$295$	−3564.67	−0.703536
$296$	6021.70	1.18245
$297$	0	0
$298$	15137.0	2.94249
$299$	1850.36	0.357891
$300$	0	0
$301$	0	0
$302$	9036.86	1.72190
$303$	0	0
$304$	−2121.12	−0.400180
$305$	−325.173	−0.0610471
$306$	0	0
$307$	−9112.83	−1.69413	−0.847063	−	0.531493i	$-0.821631\pi$
$307$	−9112.83	−1.69413	−0.847063	+	0.531493i	$0.821631\pi$
$308$	0	0
$309$	0	0
$310$	7482.24	1.37085
$311$	−2520.08	−0.459487	−0.229743	−	0.973251i	$-0.573789\pi$
$311$	−2520.08	−0.459487	−0.229743	+	0.973251i	$0.573789\pi$
$312$	0	0
$313$	−1041.28	−0.188040	−0.0940200	−	0.995570i	$-0.529972\pi$
$313$	−1041.28	−0.188040	−0.0940200	+	0.995570i	$0.529972\pi$
$314$	−636.335	−0.114365
$315$	0	0
$316$	4141.26	0.737228
$317$	1294.04	0.229277	0.114638	−	0.993407i	$-0.463429\pi$
$317$	1294.04	0.229277	0.114638	+	0.993407i	$0.463429\pi$
$318$	0	0
$319$	1047.45	0.183844
$320$	2445.34	0.427183
$321$	0	0
$322$	0	0
$323$	1419.09	0.244460
$324$	0	0
$325$	1173.85	0.200349
$326$	−17560.2	−2.98335
$327$	0	0
$328$	11278.2	1.89858
$329$	0	0
$330$	0	0
$331$	−7053.87	−1.17135	−0.585674	−	0.810547i	$-0.699170\pi$
$331$	−7053.87	−1.17135	−0.585674	+	0.810547i	$0.699170\pi$
$332$	−18979.7	−3.13750
$333$	0	0
$334$	−10090.1	−1.65302
$335$	−3493.63	−0.569783
$336$	0	0
$337$	−1616.01	−0.261216	−0.130608	−	0.991434i	$-0.541693\pi$
$337$	−1616.01	−0.261216	−0.130608	+	0.991434i	$0.541693\pi$
$338$	−37.7104	−0.00606857
$339$	0	0
$340$	3602.83	0.574678
$341$	−3657.32	−0.580806
$342$	0	0
$343$	0	0
$344$	5707.40	0.894541
$345$	0	0
$346$	−16104.8	−2.50231
$347$	2320.46	0.358987	0.179494	−	0.983759i	$-0.442554\pi$
$347$	2320.46	0.358987	0.179494	+	0.983759i	$0.442554\pi$
$348$	0	0
$349$	−8883.26	−1.36249	−0.681247	−	0.732054i	$-0.738562\pi$
$349$	−8883.26	−1.36249	−0.681247	+	0.732054i	$0.738562\pi$
$350$	0	0
$351$	0	0
$352$	111.283	0.0168506
$353$	7213.82	1.08769	0.543843	−	0.839187i	$-0.316969\pi$
$353$	7213.82	1.08769	0.543843	+	0.839187i	$0.316969\pi$
$354$	0	0
$355$	−1457.00	−0.217830
$356$	−16277.6	−2.42334
$357$	0	0
$358$	−16145.1	−2.38350
$359$	−571.681	−0.0840451	−0.0420225	−	0.999117i	$-0.513380\pi$
$359$	−571.681	−0.0840451	−0.0420225	+	0.999117i	$0.513380\pi$
$360$	0	0
$361$	−5851.48	−0.853110
$362$	13305.0	1.93175
$363$	0	0
$364$	0	0
$365$	−2381.21	−0.341474
$366$	0	0
$367$	−7595.02	−1.08026	−0.540132	−	0.841580i	$-0.681626\pi$
$367$	−7595.02	−1.08026	−0.540132	+	0.841580i	$0.681626\pi$
$368$	2633.45	0.373038
$369$	0	0
$370$	−3709.25	−0.521174
$371$	0	0
$372$	0	0
$373$	12739.0	1.76837	0.884184	−	0.467139i	$-0.154715\pi$
$373$	12739.0	1.76837	0.884184	+	0.467139i	$0.154715\pi$
$374$	−2635.19	−0.364338
$375$	0	0
$376$	3539.92	0.485525
$377$	−4097.74	−0.559799
$378$	0	0
$379$	2726.44	0.369519	0.184759	−	0.982784i	$-0.440849\pi$
$379$	2726.44	0.369519	0.184759	+	0.982784i	$0.440849\pi$
$380$	2557.90	0.345309
$381$	0	0
$382$	−10345.8	−1.38570
$383$	44.4732	0.00593336	0.00296668	−	0.999996i	$-0.499056\pi$
$383$	44.4732	0.00593336	0.00296668	+	0.999996i	$0.499056\pi$
$384$	0	0
$385$	0	0
$386$	−19430.9	−2.56220
$387$	0	0
$388$	−18521.2	−2.42338
$389$	2160.33	0.281576	0.140788	−	0.990040i	$-0.455036\pi$
$389$	2160.33	0.281576	0.140788	+	0.990040i	$0.455036\pi$
$390$	0	0
$391$	−1761.85	−0.227879
$392$	0	0
$393$	0	0
$394$	−9292.25	−1.18816
$395$	−1284.74	−0.163651
$396$	0	0
$397$	−10915.6	−1.37994	−0.689970	−	0.723838i	$-0.742376\pi$
$397$	−10915.6	−1.37994	−0.689970	+	0.723838i	$0.742376\pi$
$398$	−21717.2	−2.73514
$399$	0	0
$400$	1670.63	0.208829
$401$	7119.55	0.886617	0.443308	−	0.896369i	$-0.353805\pi$
$401$	7119.55	0.886617	0.443308	+	0.896369i	$0.353805\pi$
$402$	0	0
$403$	14307.8	1.76854
$404$	−4655.97	−0.573375
$405$	0	0
$406$	0	0
$407$	1813.08	0.220813
$408$	0	0
$409$	−10435.9	−1.26166	−0.630832	−	0.775920i	$-0.717286\pi$
$409$	−10435.9	−1.26166	−0.630832	+	0.775920i	$0.717286\pi$
$410$	−6947.16	−0.836819
$411$	0	0
$412$	17941.7	2.14545
$413$	0	0
$414$	0	0
$415$	5888.06	0.696467
$416$	−435.349	−0.0513094
$417$	0	0
$418$	−1870.91	−0.218921
$419$	−6763.41	−0.788579	−0.394289	−	0.918986i	$-0.629009\pi$
$419$	−6763.41	−0.788579	−0.394289	+	0.918986i	$0.629009\pi$
$420$	0	0
$421$	12708.5	1.47119	0.735597	−	0.677419i	$-0.236902\pi$
$421$	12708.5	1.47119	0.735597	+	0.677419i	$0.236902\pi$
$422$	−6887.44	−0.794491
$423$	0	0
$424$	5977.01	0.684598
$425$	−1117.70	−0.127568
$426$	0	0
$427$	0	0
$428$	−4252.34	−0.480245
$429$	0	0
$430$	−3515.64	−0.394278
$431$	−8478.80	−0.947585	−0.473793	−	0.880636i	$-0.657115\pi$
$431$	−8478.80	−0.947585	−0.473793	+	0.880636i	$0.657115\pi$
$432$	0	0
$433$	−13442.6	−1.49194	−0.745968	−	0.665982i	$-0.768013\pi$
$433$	−13442.6	−1.49194	−0.745968	+	0.665982i	$0.768013\pi$
$434$	0	0
$435$	0	0
$436$	−20097.4	−2.20755
$437$	−1250.87	−0.136927
$438$	0	0
$439$	−11890.4	−1.29270	−0.646352	−	0.763040i	$-0.723706\pi$
$439$	−11890.4	−1.29270	−0.646352	+	0.763040i	$0.723706\pi$
$440$	−2392.21	−0.259192
$441$	0	0
$442$	10309.1	1.10940
$443$	−11118.4	−1.19245	−0.596223	−	0.802819i	$-0.703333\pi$
$443$	−11118.4	−1.19245	−0.596223	+	0.802819i	$0.703333\pi$
$444$	0	0
$445$	5049.78	0.537938
$446$	−5456.48	−0.579309
$447$	0	0
$448$	0	0
$449$	14077.5	1.47964	0.739818	−	0.672807i	$-0.234912\pi$
$449$	14077.5	1.47964	0.739818	+	0.672807i	$0.234912\pi$
$450$	0	0
$451$	3395.77	0.354547
$452$	32789.5	3.41214
$453$	0	0
$454$	8424.24	0.870858
$455$	0	0
$456$	0	0
$457$	−6557.10	−0.671178	−0.335589	−	0.942009i	$-0.608935\pi$
$457$	−6557.10	−0.671178	−0.335589	+	0.942009i	$0.608935\pi$
$458$	−4587.48	−0.468033
$459$	0	0
$460$	−3175.72	−0.321889
$461$	−5157.65	−0.521075	−0.260538	−	0.965464i	$-0.583900\pi$
$461$	−5157.65	−0.521075	−0.260538	+	0.965464i	$0.583900\pi$
$462$	0	0
$463$	−6069.13	−0.609193	−0.304596	−	0.952482i	$-0.598522\pi$
$463$	−6069.13	−0.609193	−0.304596	+	0.952482i	$0.598522\pi$
$464$	−5831.91	−0.583491
$465$	0	0
$466$	−17606.2	−1.75020
$467$	−16351.0	−1.62020	−0.810101	−	0.586290i	$-0.800588\pi$
$467$	−16351.0	−1.62020	−0.810101	+	0.586290i	$0.800588\pi$
$468$	0	0
$469$	0	0
$470$	−2180.52	−0.214000
$471$	0	0
$472$	−28419.4	−2.77142
$473$	1718.45	0.167049
$474$	0	0
$475$	−793.535	−0.0766523
$476$	0	0
$477$	0	0
$478$	29195.0	2.79362
$479$	9345.33	0.891439	0.445719	−	0.895173i	$-0.352948\pi$
$479$	9345.33	0.891439	0.445719	+	0.895173i	$0.352948\pi$
$480$	0	0
$481$	−7092.93	−0.672369
$482$	−27534.9	−2.60204
$483$	0	0
$484$	−19130.2	−1.79660
$485$	5745.81	0.537946
$486$	0	0
$487$	−648.574	−0.0603485	−0.0301742	−	0.999545i	$-0.509606\pi$
$487$	−648.574	−0.0603485	−0.0301742	+	0.999545i	$0.509606\pi$
$488$	−2592.45	−0.240481
$489$	0	0
$490$	0	0
$491$	−16109.9	−1.48071	−0.740354	−	0.672217i	$-0.765342\pi$
$491$	−16109.9	−1.48071	−0.740354	+	0.672217i	$0.765342\pi$
$492$	0	0
$493$	3901.72	0.356440
$494$	7319.16	0.666609
$495$	0	0
$496$	20362.8	1.84338
$497$	0	0
$498$	0	0
$499$	6357.21	0.570316	0.285158	−	0.958481i	$-0.407954\pi$
$499$	6357.21	0.570316	0.285158	+	0.958481i	$0.407954\pi$
$500$	−2014.64	−0.180195
$501$	0	0
$502$	−25448.8	−2.26262
$503$	−3100.43	−0.274833	−0.137417	−	0.990513i	$-0.543880\pi$
$503$	−3100.43	−0.274833	−0.137417	+	0.990513i	$0.543880\pi$
$504$	0	0
$505$	1444.42	0.127279
$506$	2322.80	0.204073
$507$	0	0
$508$	18423.0	1.60903
$509$	−16682.7	−1.45275	−0.726374	−	0.687300i	$-0.758796\pi$
$509$	−16682.7	−1.45275	−0.726374	+	0.687300i	$0.758796\pi$
$510$	0	0
$511$	0	0
$512$	20691.0	1.78598
$513$	0	0
$514$	185.427	0.0159121
$515$	−5566.04	−0.476250
$516$	0	0
$517$	1065.84	0.0906682
$518$	0	0
$519$	0	0
$520$	9358.55	0.789230
$521$	−11268.9	−0.947597	−0.473798	−	0.880633i	$-0.657117\pi$
$521$	−11268.9	−0.947597	−0.473798	+	0.880633i	$0.657117\pi$
$522$	0	0
$523$	−21181.0	−1.77090	−0.885448	−	0.464738i	$-0.846148\pi$
$523$	−21181.0	−1.77090	−0.885448	+	0.464738i	$0.846148\pi$
$524$	−24896.1	−2.07556
$525$	0	0
$526$	5633.37	0.466970
$527$	−13623.3	−1.12608
$528$	0	0
$529$	−10614.0	−0.872360
$530$	−3681.72	−0.301743
$531$	0	0
$532$	0	0
$533$	−13284.5	−1.07958
$534$	0	0
$535$	1319.20	0.106606
$536$	−27853.0	−2.24453
$537$	0	0
$538$	26602.8	2.13184
$539$	0	0
$540$	0	0
$541$	−13392.8	−1.06433	−0.532163	−	0.846642i	$-0.678621\pi$
$541$	−13392.8	−1.06433	−0.532163	+	0.846642i	$0.678621\pi$
$542$	29999.7	2.37748
$543$	0	0
$544$	414.524	0.0326702
$545$	6234.81	0.490036
$546$	0	0
$547$	14270.9	1.11550	0.557752	−	0.830008i	$-0.311664\pi$
$547$	14270.9	1.11550	0.557752	+	0.830008i	$0.311664\pi$
$548$	−3928.23	−0.306215
$549$	0	0
$550$	1473.56	0.114241
$551$	2770.11	0.214175
$552$	0	0
$553$	0	0
$554$	35589.6	2.72935
$555$	0	0
$556$	−10070.9	−0.768169
$557$	12713.6	0.967130	0.483565	−	0.875309i	$-0.339342\pi$
$557$	12713.6	0.967130	0.483565	+	0.875309i	$0.339342\pi$
$558$	0	0
$559$	−6722.71	−0.508659
$560$	0	0
$561$	0	0
$562$	−2768.01	−0.207761
$563$	−16455.5	−1.23183	−0.615913	−	0.787814i	$-0.711213\pi$
$563$	−16455.5	−1.23183	−0.615913	+	0.787814i	$0.711213\pi$
$564$	0	0
$565$	−10172.2	−0.757433
$566$	243.679	0.0180964
$567$	0	0
$568$	−11616.0	−0.858091
$569$	22665.7	1.66994	0.834968	−	0.550298i	$-0.185486\pi$
$569$	22665.7	1.66994	0.834968	+	0.550298i	$0.185486\pi$
$570$	0	0
$571$	−12436.3	−0.911460	−0.455730	−	0.890118i	$-0.650622\pi$
$571$	−12436.3	−0.911460	−0.455730	+	0.890118i	$0.650622\pi$
$572$	−9082.90	−0.663943
$573$	0	0
$574$	0	0
$575$	985.201	0.0714534
$576$	0	0
$577$	15784.4	1.13884	0.569421	−	0.822046i	$-0.307168\pi$
$577$	15784.4	1.13884	0.569421	+	0.822046i	$0.307168\pi$
$578$	14311.4	1.02989
$579$	0	0
$580$	7032.82	0.503486
$581$	0	0
$582$	0	0
$583$	1799.63	0.127844
$584$	−18984.2	−1.34516
$585$	0	0
$586$	−27901.7	−1.96691
$587$	5373.31	0.377820	0.188910	−	0.981994i	$-0.439505\pi$
$587$	5373.31	0.377820	0.188910	+	0.981994i	$0.439505\pi$
$588$	0	0
$589$	−9672.18	−0.676631
$590$	17505.8	1.22153
$591$	0	0
$592$	−10094.7	−0.700826
$593$	−19885.4	−1.37706	−0.688530	−	0.725208i	$-0.741744\pi$
$593$	−19885.4	−1.37706	−0.688530	+	0.725208i	$0.741744\pi$
$594$	0	0
$595$	0	0
$596$	−49678.1	−3.41425
$597$	0	0
$598$	−9086.99	−0.621396
$599$	−5568.32	−0.379825	−0.189913	−	0.981801i	$-0.560820\pi$
$599$	−5568.32	−0.379825	−0.189913	+	0.981801i	$0.560820\pi$
$600$	0	0
$601$	5796.37	0.393409	0.196704	−	0.980463i	$-0.436976\pi$
$601$	5796.37	0.393409	0.196704	+	0.980463i	$0.436976\pi$
$602$	0	0
$603$	0	0
$604$	−29658.0	−1.99796
$605$	5934.73	0.398811
$606$	0	0
$607$	−9808.47	−0.655871	−0.327935	−	0.944700i	$-0.606353\pi$
$607$	−9808.47	−0.655871	−0.327935	+	0.944700i	$0.606353\pi$
$608$	294.300	0.0196307
$609$	0	0
$610$	1596.90	0.105994
$611$	−4169.65	−0.276082
$612$	0	0
$613$	2745.09	0.180870	0.0904350	−	0.995902i	$-0.471174\pi$
$613$	2745.09	0.180870	0.0904350	+	0.995902i	$0.471174\pi$
$614$	44752.4	2.94146
$615$	0	0
$616$	0	0
$617$	15727.9	1.02623	0.513114	−	0.858321i	$-0.328492\pi$
$617$	15727.9	1.02623	0.513114	+	0.858321i	$0.328492\pi$
$618$	0	0
$619$	13005.2	0.844462	0.422231	−	0.906488i	$-0.361247\pi$
$619$	13005.2	0.844462	0.422231	+	0.906488i	$0.361247\pi$
$620$	−24555.9	−1.59063
$621$	0	0
$622$	12375.9	0.797794
$623$	0	0
$624$	0	0
$625$	625.000	0.0400000
$626$	5113.64	0.326489
$627$	0	0
$628$	2088.39	0.132700
$629$	6753.64	0.428116
$630$	0	0
$631$	−10019.2	−0.632102	−0.316051	−	0.948742i	$-0.602357\pi$
$631$	−10019.2	−0.632102	−0.316051	+	0.948742i	$0.602357\pi$
$632$	−10242.6	−0.644666
$633$	0	0
$634$	−6354.95	−0.398087
$635$	−5715.35	−0.357176
$636$	0	0
$637$	0	0
$638$	−5143.97	−0.319203
$639$	0	0
$640$	−12379.7	−0.764613
$641$	−28621.9	−1.76365	−0.881824	−	0.471579i	$-0.843684\pi$
$641$	−28621.9	−1.76365	−0.881824	+	0.471579i	$0.843684\pi$
$642$	0	0
$643$	−21100.2	−1.29411	−0.647054	−	0.762444i	$-0.723999\pi$
$643$	−21100.2	−1.29411	−0.647054	+	0.762444i	$0.723999\pi$
$644$	0	0
$645$	0	0
$646$	−6969.06	−0.424449
$647$	−3060.93	−0.185993	−0.0929965	−	0.995666i	$-0.529645\pi$
$647$	−3060.93	−0.185993	−0.0929965	+	0.995666i	$0.529645\pi$
$648$	0	0
$649$	−8556.83	−0.517543
$650$	−5764.69	−0.347861
$651$	0	0
$652$	57630.9	3.46165
$653$	−23083.9	−1.38337	−0.691687	−	0.722197i	$-0.743132\pi$
$653$	−23083.9	−1.38337	−0.691687	+	0.722197i	$0.743132\pi$
$654$	0	0
$655$	7723.50	0.460736
$656$	−18906.6	−1.12527
$657$	0	0
$658$	0	0
$659$	19482.0	1.15161	0.575806	−	0.817586i	$-0.304688\pi$
$659$	19482.0	1.15161	0.575806	+	0.817586i	$0.304688\pi$
$660$	0	0
$661$	2297.05	0.135166	0.0675831	−	0.997714i	$-0.478471\pi$
$661$	2297.05	0.135166	0.0675831	+	0.997714i	$0.478471\pi$
$662$	34641.0	2.03378
$663$	0	0
$664$	46942.7	2.74357
$665$	0	0
$666$	0	0
$667$	−3439.19	−0.199649
$668$	33114.8	1.91804
$669$	0	0
$670$	17156.9	0.989298
$671$	−780.565	−0.0449081
$672$	0	0
$673$	−17610.4	−1.00866	−0.504332	−	0.863510i	$-0.668261\pi$
$673$	−17610.4	−1.00866	−0.504332	+	0.863510i	$0.668261\pi$
$674$	7936.10	0.453542
$675$	0	0
$676$	123.762	0.00704152
$677$	25472.5	1.44607	0.723033	−	0.690814i	$-0.242748\pi$
$677$	25472.5	1.44607	0.723033	+	0.690814i	$0.242748\pi$
$678$	0	0
$679$	0	0
$680$	−8910.89	−0.502525
$681$	0	0
$682$	17960.8	1.00844
$683$	6821.30	0.382152	0.191076	−	0.981575i	$-0.438802\pi$
$683$	6821.30	0.382152	0.191076	+	0.981575i	$0.438802\pi$
$684$	0	0
$685$	1218.65	0.0679741
$686$	0	0
$687$	0	0
$688$	−9567.79	−0.530187
$689$	−7040.29	−0.389280
$690$	0	0
$691$	−16765.6	−0.923003	−0.461502	−	0.887139i	$-0.652689\pi$
$691$	−16765.6	−0.923003	−0.461502	+	0.887139i	$0.652689\pi$
$692$	52854.3	2.90350
$693$	0	0
$694$	−11395.6	−0.623300
$695$	3124.29	0.170519
$696$	0	0
$697$	12649.1	0.687401
$698$	43625.0	2.36566
$699$	0	0
$700$	0	0
$701$	−14137.8	−0.761736	−0.380868	−	0.924629i	$-0.624375\pi$
$701$	−14137.8	−0.761736	−0.380868	+	0.924629i	$0.624375\pi$
$702$	0	0
$703$	4794.89	0.257244
$704$	5869.93	0.314249
$705$	0	0
$706$	−35426.5	−1.88852
$707$	0	0
$708$	0	0
$709$	2483.11	0.131531	0.0657654	−	0.997835i	$-0.479051\pi$
$709$	2483.11	0.131531	0.0657654	+	0.997835i	$0.479051\pi$
$710$	7155.21	0.378212
$711$	0	0
$712$	40259.5	2.11908
$713$	12008.4	0.630738
$714$	0	0
$715$	2817.78	0.147383
$716$	52986.5	2.76564
$717$	0	0
$718$	2807.48	0.145925
$719$	−9074.07	−0.470662	−0.235331	−	0.971915i	$-0.575617\pi$
$719$	−9074.07	−0.470662	−0.235331	+	0.971915i	$0.575617\pi$
$720$	0	0
$721$	0	0
$722$	28736.2	1.48123
$723$	0	0
$724$	−43665.6	−2.24146
$725$	−2181.78	−0.111765
$726$	0	0
$727$	−27924.6	−1.42457	−0.712286	−	0.701889i	$-0.752340\pi$
$727$	−27924.6	−1.42457	−0.712286	+	0.701889i	$0.752340\pi$
$728$	0	0
$729$	0	0
$730$	11693.9	0.592892
$731$	6401.14	0.323878
$732$	0	0
$733$	−11177.6	−0.563238	−0.281619	−	0.959526i	$-0.590871\pi$
$733$	−11177.6	−0.563238	−0.281619	+	0.959526i	$0.590871\pi$
$734$	37298.5	1.87563
$735$	0	0
$736$	−365.384	−0.0182992
$737$	−8386.30	−0.419150
$738$	0	0
$739$	12147.5	0.604671	0.302335	−	0.953202i	$-0.402234\pi$
$739$	12147.5	0.604671	0.302335	+	0.953202i	$0.402234\pi$
$740$	12173.4	0.604732
$741$	0	0
$742$	0	0
$743$	−8190.00	−0.404390	−0.202195	−	0.979345i	$-0.564808\pi$
$743$	−8190.00	−0.404390	−0.202195	+	0.979345i	$0.564808\pi$
$744$	0	0
$745$	15411.6	0.757901
$746$	−62560.3	−3.07037
$747$	0	0
$748$	8648.43	0.422751
$749$	0	0
$750$	0	0
$751$	−11143.0	−0.541432	−0.270716	−	0.962659i	$-0.587260\pi$
$751$	−11143.0	−0.541432	−0.270716	+	0.962659i	$0.587260\pi$
$752$	−5934.26	−0.287766
$753$	0	0
$754$	20123.7	0.971963
$755$	9200.78	0.443511
$756$	0	0
$757$	−20072.7	−0.963746	−0.481873	−	0.876241i	$-0.660043\pi$
$757$	−20072.7	−0.963746	−0.481873	+	0.876241i	$0.660043\pi$
$758$	−13389.3	−0.641585
$759$	0	0
$760$	−6326.48	−0.301955
$761$	5593.19	0.266430	0.133215	−	0.991087i	$-0.457470\pi$
$761$	5593.19	0.266430	0.133215	+	0.991087i	$0.457470\pi$
$762$	0	0
$763$	0	0
$764$	33953.8	1.60786
$765$	0	0
$766$	−218.405	−0.0103019
$767$	33475.1	1.57590
$768$	0	0
$769$	−35021.4	−1.64227	−0.821133	−	0.570737i	$-0.806658\pi$
$769$	−35021.4	−1.64227	−0.821133	+	0.570737i	$0.806658\pi$
$770$	0	0
$771$	0	0
$772$	63770.3	2.97298
$773$	−36389.0	−1.69317	−0.846585	−	0.532254i	$-0.821345\pi$
$773$	−36389.0	−1.69317	−0.846585	+	0.532254i	$0.821345\pi$
$774$	0	0
$775$	7617.96	0.353091
$776$	45808.6	2.11911
$777$	0	0
$778$	−10609.2	−0.488893
$779$	8980.49	0.413042
$780$	0	0
$781$	−3497.47	−0.160242
$782$	8652.32	0.395660
$783$	0	0
$784$	0	0
$785$	−647.878	−0.0294570
$786$	0	0
$787$	−23542.2	−1.06632	−0.533158	−	0.846016i	$-0.678995\pi$
$787$	−23542.2	−1.06632	−0.533158	+	0.846016i	$0.678995\pi$
$788$	30496.2	1.37866
$789$	0	0
$790$	6309.24	0.284143
$791$	0	0
$792$	0	0
$793$	3053.64	0.136744
$794$	53605.4	2.39595
$795$	0	0
$796$	71273.6	3.17365
$797$	9662.12	0.429423	0.214711	−	0.976678i	$-0.431119\pi$
$797$	9662.12	0.429423	0.214711	+	0.976678i	$0.431119\pi$
$798$	0	0
$799$	3970.20	0.175789
$800$	−231.795	−0.0102440
$801$	0	0
$802$	−34963.5	−1.53941
$803$	−5715.99	−0.251199
$804$	0	0
$805$	0	0
$806$	−70264.2	−3.07066
$807$	0	0
$808$	11515.7	0.501385
$809$	24010.3	1.04346	0.521728	−	0.853112i	$-0.325287\pi$
$809$	24010.3	1.04346	0.521728	+	0.853112i	$0.325287\pi$
$810$	0	0
$811$	−13412.7	−0.580746	−0.290373	−	0.956913i	$-0.593779\pi$
$811$	−13412.7	−0.580746	−0.290373	+	0.956913i	$0.593779\pi$
$812$	0	0
$813$	0	0
$814$	−8903.89	−0.383392
$815$	−17878.7	−0.768423
$816$	0	0
$817$	4544.62	0.194610
$818$	51249.7	2.19059
$819$	0	0
$820$	22799.9	0.970983
$821$	−17462.1	−0.742304	−0.371152	−	0.928572i	$-0.621037\pi$
$821$	−17462.1	−0.742304	−0.371152	+	0.928572i	$0.621037\pi$
$822$	0	0
$823$	−14905.4	−0.631311	−0.315655	−	0.948874i	$-0.602224\pi$
$823$	−14905.4	−0.631311	−0.315655	+	0.948874i	$0.602224\pi$
$824$	−44375.4	−1.87608
$825$	0	0
$826$	0	0
$827$	29123.5	1.22458	0.612288	−	0.790635i	$-0.290249\pi$
$827$	29123.5	1.22458	0.612288	+	0.790635i	$0.290249\pi$
$828$	0	0
$829$	−9272.75	−0.388487	−0.194244	−	0.980953i	$-0.562225\pi$
$829$	−9272.75	−0.388487	−0.194244	+	0.980953i	$0.562225\pi$
$830$	−28915.8	−1.20926
$831$	0	0
$832$	−22963.7	−0.956879
$833$	0	0
$834$	0	0
$835$	−10273.2	−0.425769
$836$	6140.13	0.254020
$837$	0	0
$838$	33214.6	1.36919
$839$	20794.4	0.855664	0.427832	−	0.903858i	$-0.359277\pi$
$839$	20794.4	0.855664	0.427832	+	0.903858i	$0.359277\pi$
$840$	0	0
$841$	−16772.7	−0.687717
$842$	−62410.3	−2.55439
$843$	0	0
$844$	22603.9	0.921869
$845$	−38.3944	−0.00156309
$846$	0	0
$847$	0	0
$848$	−10019.8	−0.405755
$849$	0	0
$850$	5488.94	0.221493
$851$	−5953.02	−0.239797
$852$	0	0
$853$	−31567.2	−1.26711	−0.633553	−	0.773699i	$-0.718404\pi$
$853$	−31567.2	−1.26711	−0.633553	+	0.773699i	$0.718404\pi$
$854$	0	0
$855$	0	0
$856$	10517.4	0.419948
$857$	−21907.7	−0.873223	−0.436612	−	0.899650i	$-0.643822\pi$
$857$	−21907.7	−0.873223	−0.436612	+	0.899650i	$0.643822\pi$
$858$	0	0
$859$	12141.1	0.482247	0.241124	−	0.970494i	$-0.422484\pi$
$859$	12141.1	0.482247	0.241124	+	0.970494i	$0.422484\pi$
$860$	11538.0	0.457491
$861$	0	0
$862$	41638.7	1.64527
$863$	−7598.88	−0.299732	−0.149866	−	0.988706i	$-0.547884\pi$
$863$	−7598.88	−0.299732	−0.149866	+	0.988706i	$0.547884\pi$
$864$	0	0
$865$	−16396.9	−0.644523
$866$	66015.4	2.59041
$867$	0	0
$868$	0	0
$869$	−3083.96	−0.120387
$870$	0	0
$871$	32807.9	1.27630
$872$	49707.2	1.93039
$873$	0	0
$874$	6142.90	0.237742
$875$	0	0
$876$	0	0
$877$	5560.67	0.214105	0.107053	−	0.994253i	$-0.465859\pi$
$877$	5560.67	0.214105	0.107053	+	0.994253i	$0.465859\pi$
$878$	58392.7	2.24449
$879$	0	0
$880$	4010.27	0.153621
$881$	37010.1	1.41532	0.707662	−	0.706551i	$-0.249750\pi$
$881$	37010.1	1.41532	0.707662	+	0.706551i	$0.249750\pi$
$882$	0	0
$883$	38430.3	1.46465	0.732323	−	0.680958i	$-0.238436\pi$
$883$	38430.3	1.46465	0.732323	+	0.680958i	$0.238436\pi$
$884$	−33833.4	−1.28726
$885$	0	0
$886$	54601.8	2.07041
$887$	−21019.9	−0.795694	−0.397847	−	0.917452i	$-0.630242\pi$
$887$	−21019.9	−0.795694	−0.397847	+	0.917452i	$0.630242\pi$
$888$	0	0
$889$	0	0
$890$	−24799.1	−0.934007
$891$	0	0
$892$	17907.6	0.672188
$893$	2818.73	0.105627
$894$	0	0
$895$	−16437.9	−0.613921
$896$	0	0
$897$	0	0
$898$	−69133.3	−2.56905
$899$	−26593.2	−0.986576
$900$	0	0
$901$	6703.53	0.247866
$902$	−16676.4	−0.615590
$903$	0	0
$904$	−81098.5	−2.98373
$905$	13546.3	0.497564
$906$	0	0
$907$	−14566.9	−0.533281	−0.266641	−	0.963796i	$-0.585914\pi$
$907$	−14566.9	−0.533281	−0.266641	+	0.963796i	$0.585914\pi$
$908$	−27647.5	−1.01048
$909$	0	0
$910$	0	0
$911$	−329.028	−0.0119662	−0.00598309	−	0.999982i	$-0.501904\pi$
$911$	−329.028	−0.0119662	−0.00598309	+	0.999982i	$0.501904\pi$
$912$	0	0
$913$	14134.0	0.512342
$914$	32201.4	1.16535
$915$	0	0
$916$	15055.6	0.543070
$917$	0	0
$918$	0	0
$919$	−26414.9	−0.948147	−0.474073	−	0.880485i	$-0.657217\pi$
$919$	−26414.9	−0.948147	−0.474073	+	0.880485i	$0.657217\pi$
$920$	7854.54	0.281474
$921$	0	0
$922$	25328.8	0.904729
$923$	13682.4	0.487932
$924$	0	0
$925$	−3776.53	−0.134239
$926$	29805.0	1.05772
$927$	0	0
$928$	809.163	0.0286229
$929$	18244.8	0.644342	0.322171	−	0.946682i	$-0.395587\pi$
$929$	18244.8	0.644342	0.322171	+	0.946682i	$0.395587\pi$
$930$	0	0
$931$	0	0
$932$	57781.8	2.03080
$933$	0	0
$934$	80298.5	2.81311
$935$	−2682.99	−0.0938430
$936$	0	0
$937$	140.951	0.00491427	0.00245713	−	0.999997i	$-0.499218\pi$
$937$	140.951	0.00491427	0.00245713	+	0.999997i	$0.499218\pi$
$938$	0	0
$939$	0	0
$940$	7156.24	0.248309
$941$	21674.8	0.750879	0.375440	−	0.926847i	$-0.377492\pi$
$941$	21674.8	0.750879	0.375440	+	0.926847i	$0.377492\pi$
$942$	0	0
$943$	−11149.6	−0.385027
$944$	47641.9	1.64260
$945$	0	0
$946$	−8439.15	−0.290043
$947$	22150.2	0.760069	0.380035	−	0.924972i	$-0.375912\pi$
$947$	22150.2	0.760069	0.380035	+	0.924972i	$0.375912\pi$
$948$	0	0
$949$	22361.4	0.764893
$950$	3896.99	0.133089
$951$	0	0
$952$	0	0
$953$	15980.9	0.543203	0.271602	−	0.962410i	$-0.412447\pi$
$953$	15980.9	0.543203	0.271602	+	0.962410i	$0.412447\pi$
$954$	0	0
$955$	−10533.4	−0.356916
$956$	−95815.1	−3.24151
$957$	0	0
$958$	−45894.2	−1.54778
$959$	0	0
$960$	0	0
$961$	63062.3	2.11682
$962$	34832.8	1.16742
$963$	0	0
$964$	90366.8	3.01921
$965$	−19783.4	−0.659947
$966$	0	0
$967$	−13804.2	−0.459064	−0.229532	−	0.973301i	$-0.573720\pi$
$967$	−13804.2	−0.459064	−0.229532	+	0.973301i	$0.573720\pi$
$968$	47314.8	1.57103
$969$	0	0
$970$	−28217.2	−0.934020
$971$	−24806.1	−0.819841	−0.409921	−	0.912121i	$-0.634444\pi$
$971$	−24806.1	−0.819841	−0.409921	+	0.912121i	$0.634444\pi$
$972$	0	0
$973$	0	0
$974$	3185.10	0.104781
$975$	0	0
$976$	4345.95	0.142531
$977$	−2069.03	−0.0677524	−0.0338762	−	0.999426i	$-0.510785\pi$
$977$	−2069.03	−0.0677524	−0.0338762	+	0.999426i	$0.510785\pi$
$978$	0	0
$979$	12121.8	0.395724
$980$	0	0
$981$	0	0
$982$	79114.2	2.57091
$983$	22474.1	0.729210	0.364605	−	0.931162i	$-0.381204\pi$
$983$	22474.1	0.729210	0.364605	+	0.931162i	$0.381204\pi$
$984$	0	0
$985$	−9460.80	−0.306037
$986$	−19161.1	−0.618877
$987$	0	0
$988$	−24020.7	−0.773484
$989$	−5642.31	−0.181410
$990$	0	0
$991$	40924.6	1.31182	0.655910	−	0.754839i	$-0.272285\pi$
$991$	40924.6	1.31182	0.655910	+	0.754839i	$0.272285\pi$
$992$	−2825.29	−0.0904265
$993$	0	0
$994$	0	0
$995$	−22111.1	−0.704492
$996$	0	0
$997$	60402.8	1.91873	0.959366	−	0.282164i	$-0.0910522\pi$
$997$	60402.8	1.91873	0.959366	+	0.282164i	$0.0910522\pi$
$998$	−31219.7	−0.990224
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.4.a.bx.1.1	✓	6
3.2	odd	2		2205.4.a.by.1.6	yes	6
7.6	odd	2		2205.4.a.by.1.1	yes	6
21.20	even	2	inner	2205.4.a.bx.1.6	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2205.4.a.bx.1.1	✓	6	1.1	even	1	trivial
2205.4.a.bx.1.6	yes	6	21.20	even	2	inner
2205.4.a.by.1.1	yes	6	7.6	odd	2
2205.4.a.by.1.6	yes	6	3.2	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)

\(f(q)\)	\(=\)	\(q-4.91092 q^{2} +16.1171 q^{4} -5.00000 q^{5} -39.8626 q^{8} +O(q^{10})\)	\(q-4.91092 q^{2} +16.1171 q^{4} -5.00000 q^{5} -39.8626 q^{8} +24.5546 q^{10} -12.0023 q^{11} +46.9540 q^{13} +66.8251 q^{16} -44.7080 q^{17} -31.7414 q^{19} -80.5857 q^{20} +58.9423 q^{22} +39.4080 q^{23} +25.0000 q^{25} -230.587 q^{26} -87.2713 q^{29} +304.718 q^{31} -9.27181 q^{32} +219.557 q^{34} -151.061 q^{37} +155.879 q^{38} +199.313 q^{40} -282.927 q^{41} -143.177 q^{43} -193.443 q^{44} -193.530 q^{46} -88.8029 q^{47} -122.773 q^{50} +756.764 q^{52} -149.940 q^{53} +60.0114 q^{55} +428.582 q^{58} +712.934 q^{59} +65.0347 q^{61} -1496.45 q^{62} -489.068 q^{64} -234.770 q^{65} +698.725 q^{67} -720.565 q^{68} +291.400 q^{71} +476.241 q^{73} +741.849 q^{74} -511.580 q^{76} +256.947 q^{79} -334.126 q^{80} +1389.43 q^{82} -1177.61 q^{83} +223.540 q^{85} +703.129 q^{86} +478.443 q^{88} -1009.96 q^{89} +635.145 q^{92} +436.104 q^{94} +158.707 q^{95} -1149.16 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 20 q^{4} - 30 q^{5}+O(q^{10}) \) 6 * q + 20 * q^4 - 30 * q^5	\( 6 q + 20 q^{4} - 30 q^{5} + 100 q^{16} - 44 q^{17} - 100 q^{20} - 24 q^{22} + 150 q^{25} - 168 q^{26} + 380 q^{37} - 56 q^{38} - 612 q^{41} - 328 q^{43} + 432 q^{46} - 120 q^{47} + 1120 q^{58} - 136 q^{59} - 2264 q^{62} - 1052 q^{64} + 1112 q^{67} - 264 q^{68} + 1400 q^{79} - 500 q^{80} - 2912 q^{83} + 220 q^{85} + 1384 q^{88} - 372 q^{89}+O(q^{100}) \) 6 * q + 20 * q^4 - 30 * q^5 + 100 * q^16 - 44 * q^17 - 100 * q^20 - 24 * q^22 + 150 * q^25 - 168 * q^26 + 380 * q^37 - 56 * q^38 - 612 * q^41 - 328 * q^43 + 432 * q^46 - 120 * q^47 + 1120 * q^58 - 136 * q^59 - 2264 * q^62 - 1052 * q^64 + 1112 * q^67 - 264 * q^68 + 1400 * q^79 - 500 * q^80 - 2912 * q^83 + 220 * q^85 + 1384 * q^88 - 372 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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