LMFDB - Embedded newform 2205.4.a.ca.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2205,4,Mod(1,2205)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2205.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage: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")

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:traces = [6,2,0,14,30,0,0,66,0,10,16,0,168] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$130.099211563$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.1163891200.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 23x^{4} + 12x^{3} + 154x^{2} + 152x + 28 $ x^6 - 2x^5 - 23x^4 + 12x^3 + 154x^2 + 152*x + 28
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2\cdot 7 $
Twist minimal:	no (minimal twist has level 245)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$4.10376$ 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)

$f(q)$	$=$	$q+5.51797 q^{2} +22.4480 q^{4} +5.00000 q^{5} +79.7239 q^{8} +27.5899 q^{10} -34.5211 q^{11} +68.8935 q^{13} +260.330 q^{16} +91.4346 q^{17} +11.8278 q^{19} +112.240 q^{20} -190.486 q^{22} +0.104165 q^{23} +25.0000 q^{25} +380.152 q^{26} -190.863 q^{29} +159.802 q^{31} +798.703 q^{32} +504.534 q^{34} -177.908 q^{37} +65.2657 q^{38} +398.619 q^{40} -145.247 q^{41} +8.25729 q^{43} -774.930 q^{44} +0.574782 q^{46} -260.529 q^{47} +137.949 q^{50} +1546.52 q^{52} -353.107 q^{53} -172.605 q^{55} -1053.18 q^{58} -240.495 q^{59} +778.188 q^{61} +881.783 q^{62} +2324.58 q^{64} +344.467 q^{65} +151.945 q^{67} +2052.53 q^{68} +311.449 q^{71} +639.888 q^{73} -981.690 q^{74} +265.512 q^{76} +391.186 q^{79} +1301.65 q^{80} -801.472 q^{82} -493.205 q^{83} +457.173 q^{85} +45.5635 q^{86} -2752.15 q^{88} -473.850 q^{89} +2.33831 q^{92} -1437.59 q^{94} +59.1392 q^{95} +839.005 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{2} + 14 q^{4} + 30 q^{5} + 66 q^{8} + 10 q^{10} + 16 q^{11} + 168 q^{13} + 298 q^{16} + 4 q^{17} + 308 q^{19} + 70 q^{20} - 236 q^{22} + 336 q^{23} + 150 q^{25} - 56 q^{26} - 176 q^{29} + 392 q^{31}+ \cdots + 516 q^{97}+O(q^{100}) $ 6 * q + 2 * q^2 + 14 * q^4 + 30 * q^5 + 66 * q^8 + 10 * q^10 + 16 * q^11 + 168 * q^13 + 298 * q^16 + 4 * q^17 + 308 * q^19 + 70 * q^20 - 236 * q^22 + 336 * q^23 + 150 * q^25 - 56 * q^26 - 176 * q^29 + 392 * q^31 + 770 * q^32 + 812 * q^34 - 140 * q^37 - 20 * q^38 + 330 * q^40 - 656 * q^41 - 388 * q^43 + 160 * q^44 - 388 * q^46 - 628 * q^47 + 50 * q^50 + 1520 * q^52 + 676 * q^53 + 80 * q^55 - 2012 * q^58 - 996 * q^59 + 740 * q^61 + 364 * q^62 + 1426 * q^64 + 840 * q^65 + 1768 * q^67 + 2940 * q^68 + 224 * q^71 + 2640 * q^73 - 928 * q^74 - 1340 * q^76 + 1636 * q^79 + 1490 * q^80 - 1756 * q^82 - 140 * q^83 + 20 * q^85 - 1180 * q^86 - 5652 * q^88 + 1904 * q^89 + 1952 * q^92 - 3332 * q^94 + 1540 * q^95 + 516 * q^97

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	5.51797	1.95090	0.975449	−	0.220225i	$-0.0706790\pi$
$2$	5.51797	1.95090	0.975449	+	0.220225i	$0.0706790\pi$
$3$	0	0
$3$	0	0
$4$	22.4480	2.80600
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	79.7239	3.52333
$9$	0	0
$10$	27.5899	0.872468
$11$	−34.5211	−0.946227	−0.473113	−	0.881002i	$-0.656870\pi$
$11$	−34.5211	−0.946227	−0.473113	+	0.881002i	$0.656870\pi$
$12$	0	0
$13$	68.8935	1.46982	0.734908	−	0.678167i	$-0.237225\pi$
$13$	68.8935	1.46982	0.734908	+	0.678167i	$0.237225\pi$
$14$	0	0
$15$	0	0
$16$	260.330	4.06766
$17$	91.4346	1.30448	0.652239	−	0.758013i	$-0.273830\pi$
$17$	91.4346	1.30448	0.652239	+	0.758013i	$0.273830\pi$
$18$	0	0
$19$	11.8278	0.142815	0.0714077	−	0.997447i	$-0.477251\pi$
$19$	11.8278	0.142815	0.0714077	+	0.997447i	$0.477251\pi$
$20$	112.240	1.25488
$21$	0	0
$22$	−190.486	−1.84599
$23$	0.104165	0.000944348 0	0.000472174	−	1.00000i	$-0.499850\pi$
$23$	0.104165	0.000944348 0	0.000472174		1.00000i	$0.499850\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	380.152	2.86746
$27$	0	0
$28$	0	0
$29$	−190.863	−1.22215	−0.611076	−	0.791572i	$-0.709263\pi$
$29$	−190.863	−1.22215	−0.611076	+	0.791572i	$0.709263\pi$
$30$	0	0
$31$	159.802	0.925847	0.462924	−	0.886398i	$-0.346800\pi$
$31$	159.802	0.925847	0.462924	+	0.886398i	$0.346800\pi$
$32$	798.703	4.41225
$33$	0	0
$34$	504.534	2.54491
$35$	0	0
$36$	0	0
$37$	−177.908	−0.790482	−0.395241	−	0.918577i	$-0.629339\pi$
$37$	−177.908	−0.790482	−0.395241	+	0.918577i	$0.629339\pi$
$38$	65.2657	0.278618
$39$	0	0
$40$	398.619	1.57568
$41$	−145.247	−0.553264	−0.276632	−	0.960976i	$-0.589218\pi$
$41$	−145.247	−0.553264	−0.276632	+	0.960976i	$0.589218\pi$
$42$	0	0
$43$	8.25729	0.0292843	0.0146421	−	0.999893i	$-0.495339\pi$
$43$	8.25729	0.0292843	0.0146421	+	0.999893i	$0.495339\pi$
$44$	−774.930	−2.65512
$45$	0	0
$46$	0.574782	0.00184233
$47$	−260.529	−0.808553	−0.404277	−	0.914637i	$-0.632477\pi$
$47$	−260.529	−0.808553	−0.404277	+	0.914637i	$0.632477\pi$
$48$	0	0
$49$	0	0
$50$	137.949	0.390180
$51$	0	0
$52$	1546.52	4.12431
$53$	−353.107	−0.915151	−0.457576	−	0.889171i	$-0.651282\pi$
$53$	−353.107	−0.915151	−0.457576	+	0.889171i	$0.651282\pi$
$54$	0	0
$55$	−172.605	−0.423165
$56$	0	0
$57$	0	0
$58$	−1053.18	−2.38429
$59$	−240.495	−0.530673	−0.265337	−	0.964156i	$-0.585483\pi$
$59$	−240.495	−0.530673	−0.265337	+	0.964156i	$0.585483\pi$
$60$	0	0
$61$	778.188	1.63339	0.816695	−	0.577069i	$-0.195804\pi$
$61$	778.188	1.63339	0.816695	+	0.577069i	$0.195804\pi$
$62$	881.783	1.80623
$63$	0	0
$64$	2324.58	4.54020
$65$	344.467	0.657322
$66$	0	0
$67$	151.945	0.277059	0.138530	−	0.990358i	$-0.455762\pi$
$67$	151.945	0.277059	0.138530	+	0.990358i	$0.455762\pi$
$68$	2052.53	3.66037
$69$	0	0
$70$	0	0
$71$	311.449	0.520594	0.260297	−	0.965529i	$-0.416180\pi$
$71$	311.449	0.520594	0.260297	+	0.965529i	$0.416180\pi$
$72$	0	0
$73$	639.888	1.02593	0.512967	−	0.858408i	$-0.328546\pi$
$73$	639.888	1.02593	0.512967	+	0.858408i	$0.328546\pi$
$74$	−981.690	−1.54215
$75$	0	0
$76$	265.512	0.400740
$77$	0	0
$78$	0	0
$79$	391.186	0.557112	0.278556	−	0.960420i	$-0.410144\pi$
$79$	391.186	0.557112	0.278556	+	0.960420i	$0.410144\pi$
$80$	1301.65	1.81911
$81$	0	0
$82$	−801.472	−1.07936
$83$	−493.205	−0.652245	−0.326122	−	0.945328i	$-0.605742\pi$
$83$	−493.205	−0.652245	−0.326122	+	0.945328i	$0.605742\pi$
$84$	0	0
$85$	457.173	0.583381
$86$	45.5635	0.0571307
$87$	0	0
$88$	−2752.15	−3.33387
$89$	−473.850	−0.564359	−0.282180	−	0.959362i	$-0.591057\pi$
$89$	−473.850	−0.564359	−0.282180	+	0.959362i	$0.591057\pi$
$90$	0	0
$91$	0	0
$92$	2.33831	0.00264984
$93$	0	0
$94$	−1437.59	−1.57741
$95$	59.1392	0.0638690
$96$	0	0
$97$	839.005	0.878227	0.439114	−	0.898431i	$-0.355293\pi$
$97$	839.005	0.878227	0.439114	+	0.898431i	$0.355293\pi$
$98$	0	0
$99$	0	0
$100$	561.201	0.561201
$101$	−887.440	−0.874293	−0.437146	−	0.899390i	$-0.644011\pi$
$101$	−887.440	−0.874293	−0.437146	+	0.899390i	$0.644011\pi$
$102$	0	0
$103$	619.087	0.592238	0.296119	−	0.955151i	$-0.404308\pi$
$103$	619.087	0.592238	0.296119	+	0.955151i	$0.404308\pi$
$104$	5492.46	5.17865
$105$	0	0
$106$	−1948.44	−1.78537
$107$	2151.30	1.94368	0.971841	−	0.235639i	$-0.0757183\pi$
$107$	2151.30	1.94368	0.971841	+	0.235639i	$0.0757183\pi$
$108$	0	0
$109$	−407.076	−0.357714	−0.178857	−	0.983875i	$-0.557240\pi$
$109$	−407.076	−0.357714	−0.178857	+	0.983875i	$0.557240\pi$
$110$	−952.432	−0.825553
$111$	0	0
$112$	0	0
$113$	349.581	0.291025	0.145513	−	0.989356i	$-0.453517\pi$
$113$	349.581	0.291025	0.145513	+	0.989356i	$0.453517\pi$
$114$	0	0
$115$	0.520827	0.000422325 0
$116$	−4284.50	−3.42936
$117$	0	0
$118$	−1327.04	−1.03529
$119$	0	0
$120$	0	0
$121$	−139.296	−0.104655
$122$	4294.02	3.18658
$123$	0	0
$124$	3587.24	2.59793
$125$	125.000	0.0894427
$126$	0	0
$127$	1183.78	0.827114	0.413557	−	0.910478i	$-0.364286\pi$
$127$	1183.78	0.827114	0.413557	+	0.910478i	$0.364286\pi$
$128$	6437.36	4.44522
$129$	0	0
$130$	1900.76	1.28237
$131$	−223.357	−0.148968	−0.0744840	−	0.997222i	$-0.523731\pi$
$131$	−223.357	−0.148968	−0.0744840	+	0.997222i	$0.523731\pi$
$132$	0	0
$133$	0	0
$134$	838.426	0.540515
$135$	0	0
$136$	7289.52	4.59611
$137$	−2036.66	−1.27010	−0.635050	−	0.772471i	$-0.719021\pi$
$137$	−2036.66	−1.27010	−0.635050	+	0.772471i	$0.719021\pi$
$138$	0	0
$139$	2687.00	1.63963	0.819815	−	0.572629i	$-0.194076\pi$
$139$	2687.00	1.63963	0.819815	+	0.572629i	$0.194076\pi$
$140$	0	0
$141$	0	0
$142$	1718.57	1.01563
$143$	−2378.28	−1.39078
$144$	0	0
$145$	−954.316	−0.546563
$146$	3530.88	2.00149
$147$	0	0
$148$	−3993.68	−2.21810
$149$	−673.500	−0.370304	−0.185152	−	0.982710i	$-0.559278\pi$
$149$	−673.500	−0.370304	−0.185152	+	0.982710i	$0.559278\pi$
$150$	0	0
$151$	−2125.18	−1.14533	−0.572664	−	0.819790i	$-0.694090\pi$
$151$	−2125.18	−1.14533	−0.572664	+	0.819790i	$0.694090\pi$
$152$	942.961	0.503186
$153$	0	0
$154$	0	0
$155$	799.010	0.414052
$156$	0	0
$157$	−2813.03	−1.42997	−0.714983	−	0.699142i	$-0.753565\pi$
$157$	−2813.03	−1.42997	−0.714983	+	0.699142i	$0.753565\pi$
$158$	2158.55	1.08687
$159$	0	0
$160$	3993.52	1.97322
$161$	0	0
$162$	0	0
$163$	−1344.42	−0.646032	−0.323016	−	0.946394i	$-0.604697\pi$
$163$	−1344.42	−0.646032	−0.323016	+	0.946394i	$0.604697\pi$
$164$	−3260.52	−1.55246
$165$	0	0
$166$	−2721.49	−1.27246
$167$	−1451.24	−0.672456	−0.336228	−	0.941781i	$-0.609151\pi$
$167$	−1451.24	−0.672456	−0.336228	+	0.941781i	$0.609151\pi$
$168$	0	0
$169$	2549.31	1.16036
$170$	2522.67	1.13812
$171$	0	0
$172$	185.360	0.0821718
$173$	1979.16	0.869784	0.434892	−	0.900483i	$-0.356787\pi$
$173$	1979.16	0.869784	0.434892	+	0.900483i	$0.356787\pi$
$174$	0	0
$175$	0	0
$176$	−8986.87	−3.84893
$177$	0	0
$178$	−2614.69	−1.10101
$179$	4358.66	1.82001	0.910005	−	0.414598i	$-0.136078\pi$
$179$	4358.66	1.82001	0.910005	+	0.414598i	$0.136078\pi$
$180$	0	0
$181$	−377.923	−0.155198	−0.0775988	−	0.996985i	$-0.524725\pi$
$181$	−377.923	−0.155198	−0.0775988	+	0.996985i	$0.524725\pi$
$182$	0	0
$183$	0	0
$184$	8.30447	0.00332725
$185$	−889.538	−0.353514
$186$	0	0
$187$	−3156.42	−1.23433
$188$	−5848.36	−2.26880
$189$	0	0
$190$	326.328	0.124602
$191$	2425.95	0.919033	0.459517	−	0.888169i	$-0.348023\pi$
$191$	2425.95	0.919033	0.459517	+	0.888169i	$0.348023\pi$
$192$	0	0
$193$	−622.923	−0.232326	−0.116163	−	0.993230i	$-0.537060\pi$
$193$	−622.923	−0.232326	−0.116163	+	0.993230i	$0.537060\pi$
$194$	4629.61	1.71333
$195$	0	0
$196$	0	0
$197$	−2842.29	−1.02794	−0.513971	−	0.857807i	$-0.671826\pi$
$197$	−2842.29	−1.02794	−0.513971	+	0.857807i	$0.671826\pi$
$198$	0	0
$199$	−867.364	−0.308974	−0.154487	−	0.987995i	$-0.549372\pi$
$199$	−867.364	−0.308974	−0.154487	+	0.987995i	$0.549372\pi$
$200$	1993.10	0.704666
$201$	0	0
$202$	−4896.87	−1.70566
$203$	0	0
$204$	0	0
$205$	−726.237	−0.247427
$206$	3416.11	1.15540
$207$	0	0
$208$	17935.0	5.97871
$209$	−408.309	−0.135136
$210$	0	0
$211$	−5975.92	−1.94976	−0.974880	−	0.222730i	$-0.928503\pi$
$211$	−5975.92	−1.94976	−0.974880	+	0.222730i	$0.928503\pi$
$212$	−7926.57	−2.56792
$213$	0	0
$214$	11870.8	3.79192
$215$	41.2864	0.0130963
$216$	0	0
$217$	0	0
$218$	−2246.24	−0.697864
$219$	0	0
$220$	−3874.65	−1.18740
$221$	6299.25	1.91734
$222$	0	0
$223$	−5181.58	−1.55598	−0.777992	−	0.628274i	$-0.783762\pi$
$223$	−5181.58	−1.55598	−0.777992	+	0.628274i	$0.783762\pi$
$224$	0	0
$225$	0	0
$226$	1928.98	0.567761
$227$	−3753.06	−1.09735	−0.548677	−	0.836035i	$-0.684868\pi$
$227$	−3753.06	−1.09735	−0.548677	+	0.836035i	$0.684868\pi$
$228$	0	0
$229$	−6258.14	−1.80589	−0.902947	−	0.429752i	$-0.858601\pi$
$229$	−6258.14	−1.80589	−0.902947	+	0.429752i	$0.858601\pi$
$230$	2.87391	0.000823913 0
$231$	0	0
$232$	−15216.4	−4.30605
$233$	−1779.96	−0.500469	−0.250234	−	0.968185i	$-0.580508\pi$
$233$	−1779.96	−0.500469	−0.250234	+	0.968185i	$0.580508\pi$
$234$	0	0
$235$	−1302.64	−0.361596
$236$	−5398.63	−1.48907
$237$	0	0
$238$	0	0
$239$	−3519.46	−0.952532	−0.476266	−	0.879301i	$-0.658010\pi$
$239$	−3519.46	−0.952532	−0.476266	+	0.879301i	$0.658010\pi$
$240$	0	0
$241$	362.930	0.0970058	0.0485029	−	0.998823i	$-0.484555\pi$
$241$	362.930	0.0970058	0.0485029	+	0.998823i	$0.484555\pi$
$242$	−768.629	−0.204171
$243$	0	0
$244$	17468.8	4.58330
$245$	0	0
$246$	0	0
$247$	814.861	0.209912
$248$	12740.0	3.26207
$249$	0	0
$250$	689.747	0.174494
$251$	5333.85	1.34131	0.670656	−	0.741768i	$-0.266012\pi$
$251$	5333.85	1.34131	0.670656	+	0.741768i	$0.266012\pi$
$252$	0	0
$253$	−3.59590	−0.000893567 0
$254$	6532.06	1.61361
$255$	0	0
$256$	16924.5	4.13197
$257$	2438.78	0.591933	0.295966	−	0.955198i	$-0.404358\pi$
$257$	2438.78	0.591933	0.295966	+	0.955198i	$0.404358\pi$
$258$	0	0
$259$	0	0
$260$	7732.62	1.84445
$261$	0	0
$262$	−1232.48	−0.290621
$263$	−1526.25	−0.357843	−0.178921	−	0.983863i	$-0.557261\pi$
$263$	−1526.25	−0.357843	−0.178921	+	0.983863i	$0.557261\pi$
$264$	0	0
$265$	−1765.54	−0.409268
$266$	0	0
$267$	0	0
$268$	3410.86	0.777430
$269$	7564.12	1.71447	0.857235	−	0.514925i	$-0.172180\pi$
$269$	7564.12	1.71447	0.857235	+	0.514925i	$0.172180\pi$
$270$	0	0
$271$	4282.68	0.959980	0.479990	−	0.877274i	$-0.340640\pi$
$271$	4282.68	0.959980	0.479990	+	0.877274i	$0.340640\pi$
$272$	23803.2	5.30617
$273$	0	0
$274$	−11238.2	−2.47784
$275$	−863.027	−0.189245
$276$	0	0
$277$	−4008.41	−0.869465	−0.434732	−	0.900560i	$-0.643157\pi$
$277$	−4008.41	−0.869465	−0.434732	+	0.900560i	$0.643157\pi$
$278$	14826.8	3.19875
$279$	0	0
$280$	0	0
$281$	−6935.44	−1.47236	−0.736181	−	0.676785i	$-0.763373\pi$
$281$	−6935.44	−1.47236	−0.736181	+	0.676785i	$0.763373\pi$
$282$	0	0
$283$	−2666.64	−0.560124	−0.280062	−	0.959982i	$-0.590355\pi$
$283$	−2666.64	−0.560124	−0.280062	+	0.959982i	$0.590355\pi$
$284$	6991.41	1.46079
$285$	0	0
$286$	−13123.3	−2.71327
$287$	0	0
$288$	0	0
$289$	3447.28	0.701665
$290$	−5265.89	−1.06629
$291$	0	0
$292$	14364.2	2.87878
$293$	−5939.10	−1.18418	−0.592092	−	0.805870i	$-0.701698\pi$
$293$	−5939.10	−1.18418	−0.592092	+	0.805870i	$0.701698\pi$
$294$	0	0
$295$	−1202.47	−0.237324
$296$	−14183.5	−2.78513
$297$	0	0
$298$	−3716.36	−0.722425
$299$	7.17632	0.00138802
$300$	0	0
$301$	0	0
$302$	−11726.7	−2.23442
$303$	0	0
$304$	3079.14	0.580924
$305$	3890.94	0.730474
$306$	0	0
$307$	10381.5	1.92998	0.964992	−	0.262278i	$-0.0844737\pi$
$307$	10381.5	1.92998	0.964992	+	0.262278i	$0.0844737\pi$
$308$	0	0
$309$	0	0
$310$	4408.91	0.807772
$311$	−4240.54	−0.773181	−0.386590	−	0.922252i	$-0.626347\pi$
$311$	−4240.54	−0.773181	−0.386590	+	0.922252i	$0.626347\pi$
$312$	0	0
$313$	−283.903	−0.0512688	−0.0256344	−	0.999671i	$-0.508161\pi$
$313$	−283.903	−0.0512688	−0.0256344	+	0.999671i	$0.508161\pi$
$314$	−15522.2	−2.78972
$315$	0	0
$316$	8781.36	1.56326
$317$	−1739.49	−0.308201	−0.154100	−	0.988055i	$-0.549248\pi$
$317$	−1739.49	−0.308201	−0.154100	+	0.988055i	$0.549248\pi$
$318$	0	0
$319$	6588.80	1.15643
$320$	11622.9	2.03044
$321$	0	0
$322$	0	0
$323$	1081.47	0.186300
$324$	0	0
$325$	1722.34	0.293963
$326$	−7418.48	−1.26034
$327$	0	0
$328$	−11579.7	−1.94933
$329$	0	0
$330$	0	0
$331$	5606.96	0.931077	0.465538	−	0.885028i	$-0.345861\pi$
$331$	5606.96	0.931077	0.465538	+	0.885028i	$0.345861\pi$
$332$	−11071.5	−1.83020
$333$	0	0
$334$	−8007.89	−1.31189
$335$	759.723	0.123905
$336$	0	0
$337$	−9427.44	−1.52387	−0.761937	−	0.647652i	$-0.775751\pi$
$337$	−9427.44	−1.52387	−0.761937	+	0.647652i	$0.775751\pi$
$338$	14067.0	2.26375
$339$	0	0
$340$	10262.6	1.63697
$341$	−5516.53	−0.876062
$342$	0	0
$343$	0	0
$344$	658.303	0.103178
$345$	0	0
$346$	10920.9	1.69686
$347$	−11634.2	−1.79988	−0.899939	−	0.436017i	$-0.856389\pi$
$347$	−11634.2	−1.79988	−0.899939	+	0.436017i	$0.856389\pi$
$348$	0	0
$349$	−1317.10	−0.202013	−0.101006	−	0.994886i	$-0.532206\pi$
$349$	−1317.10	−0.202013	−0.101006	+	0.994886i	$0.532206\pi$
$350$	0	0
$351$	0	0
$352$	−27572.1	−4.17499
$353$	−5848.82	−0.881874	−0.440937	−	0.897538i	$-0.645354\pi$
$353$	−5848.82	−0.881874	−0.440937	+	0.897538i	$0.645354\pi$
$354$	0	0
$355$	1557.24	0.232817
$356$	−10637.0	−1.58359
$357$	0	0
$358$	24051.0	3.55065
$359$	12422.0	1.82621	0.913104	−	0.407727i	$-0.133679\pi$
$359$	12422.0	1.82621	0.913104	+	0.407727i	$0.133679\pi$
$360$	0	0
$361$	−6719.10	−0.979604
$362$	−2085.37	−0.302775
$363$	0	0
$364$	0	0
$365$	3199.44	0.458812
$366$	0	0
$367$	−6590.78	−0.937427	−0.468713	−	0.883350i	$-0.655282\pi$
$367$	−6590.78	−0.937427	−0.468713	+	0.883350i	$0.655282\pi$
$368$	27.1174	0.00384128
$369$	0	0
$370$	−4908.45	−0.689670
$371$	0	0
$372$	0	0
$373$	344.278	0.0477910	0.0238955	−	0.999714i	$-0.492393\pi$
$373$	344.278	0.0477910	0.0238955	+	0.999714i	$0.492393\pi$
$374$	−17417.0	−2.40806
$375$	0	0
$376$	−20770.4	−2.84880
$377$	−13149.2	−1.79634
$378$	0	0
$379$	5241.23	0.710353	0.355177	−	0.934799i	$-0.384421\pi$
$379$	5241.23	0.710353	0.355177	+	0.934799i	$0.384421\pi$
$380$	1327.56	0.179217
$381$	0	0
$382$	13386.3	1.79294
$383$	−7597.37	−1.01360	−0.506798	−	0.862065i	$-0.669171\pi$
$383$	−7597.37	−1.01360	−0.506798	+	0.862065i	$0.669171\pi$
$384$	0	0
$385$	0	0
$386$	−3437.27	−0.453245
$387$	0	0
$388$	18834.0	2.46431
$389$	−3101.84	−0.404291	−0.202146	−	0.979355i	$-0.564791\pi$
$389$	−3101.84	−0.404291	−0.202146	+	0.979355i	$0.564791\pi$
$390$	0	0
$391$	9.52432	0.00123188
$392$	0	0
$393$	0	0
$394$	−15683.7	−2.00541
$395$	1955.93	0.249148
$396$	0	0
$397$	−2932.06	−0.370669	−0.185335	−	0.982675i	$-0.559337\pi$
$397$	−2932.06	−0.370669	−0.185335	+	0.982675i	$0.559337\pi$
$398$	−4786.09	−0.602777
$399$	0	0
$400$	6508.25	0.813531
$401$	−89.2375	−0.0111130	−0.00555649	−	0.999985i	$-0.501769\pi$
$401$	−89.2375	−0.0111130	−0.00555649	+	0.999985i	$0.501769\pi$
$402$	0	0
$403$	11009.3	1.36083
$404$	−19921.3	−2.45327
$405$	0	0
$406$	0	0
$407$	6141.56	0.747975
$408$	0	0
$409$	147.388	0.0178188	0.00890940	−	0.999960i	$-0.497164\pi$
$409$	147.388	0.0178188	0.00890940	+	0.999960i	$0.497164\pi$
$410$	−4007.36	−0.482706
$411$	0	0
$412$	13897.3	1.66182
$413$	0	0
$414$	0	0
$415$	−2466.03	−0.291693
$416$	55025.4	6.48520
$417$	0	0
$418$	−2253.04	−0.263636
$419$	3781.67	0.440923	0.220462	−	0.975396i	$-0.429244\pi$
$419$	3781.67	0.440923	0.220462	+	0.975396i	$0.429244\pi$
$420$	0	0
$421$	−10899.2	−1.26175	−0.630874	−	0.775885i	$-0.717304\pi$
$421$	−10899.2	−1.26175	−0.630874	+	0.775885i	$0.717304\pi$
$422$	−32975.0	−3.80378
$423$	0	0
$424$	−28151.1	−3.22438
$425$	2285.86	0.260896
$426$	0	0
$427$	0	0
$428$	48292.4	5.45398
$429$	0	0
$430$	227.817	0.0255496
$431$	11283.4	1.26102	0.630512	−	0.776180i	$-0.282845\pi$
$431$	11283.4	1.26102	0.630512	+	0.776180i	$0.282845\pi$
$432$	0	0
$433$	−8906.19	−0.988462	−0.494231	−	0.869331i	$-0.664550\pi$
$433$	−8906.19	−0.988462	−0.494231	+	0.869331i	$0.664550\pi$
$434$	0	0
$435$	0	0
$436$	−9138.07	−1.00375
$437$	1.23205	0.000134867 0
$438$	0	0
$439$	8149.49	0.886000	0.443000	−	0.896522i	$-0.353914\pi$
$439$	8149.49	0.886000	0.443000	+	0.896522i	$0.353914\pi$
$440$	−13760.8	−1.49095
$441$	0	0
$442$	34759.1	3.74054
$443$	11472.8	1.23045	0.615223	−	0.788353i	$-0.289066\pi$
$443$	11472.8	1.23045	0.615223	+	0.788353i	$0.289066\pi$
$444$	0	0
$445$	−2369.25	−0.252389
$446$	−28591.8	−3.03557
$447$	0	0
$448$	0	0
$449$	−1963.80	−0.206409	−0.103204	−	0.994660i	$-0.532910\pi$
$449$	−1963.80	−0.206409	−0.103204	+	0.994660i	$0.532910\pi$
$450$	0	0
$451$	5014.10	0.523514
$452$	7847.42	0.816618
$453$	0	0
$454$	−20709.3	−2.14083
$455$	0	0
$456$	0	0
$457$	5589.85	0.572171	0.286086	−	0.958204i	$-0.407646\pi$
$457$	5589.85	0.572171	0.286086	+	0.958204i	$0.407646\pi$
$458$	−34532.3	−3.52311
$459$	0	0
$460$	11.6915	0.00118505
$461$	18790.9	1.89844	0.949219	−	0.314618i	$-0.101876\pi$
$461$	18790.9	1.89844	0.949219	+	0.314618i	$0.101876\pi$
$462$	0	0
$463$	−7892.22	−0.792187	−0.396094	−	0.918210i	$-0.629634\pi$
$463$	−7892.22	−0.792187	−0.396094	+	0.918210i	$0.629634\pi$
$464$	−49687.4	−4.97130
$465$	0	0
$466$	−9821.78	−0.976363
$467$	−385.511	−0.0381998	−0.0190999	−	0.999818i	$-0.506080\pi$
$467$	−385.511	−0.0381998	−0.0190999	+	0.999818i	$0.506080\pi$
$468$	0	0
$469$	0	0
$470$	−7187.95	−0.705437
$471$	0	0
$472$	−19173.2	−1.86974
$473$	−285.050	−0.0277096
$474$	0	0
$475$	295.696	0.0285631
$476$	0	0
$477$	0	0
$478$	−19420.3	−1.85829
$479$	−1694.46	−0.161632	−0.0808160	−	0.996729i	$-0.525753\pi$
$479$	−1694.46	−0.161632	−0.0808160	+	0.996729i	$0.525753\pi$
$480$	0	0
$481$	−12256.7	−1.16186
$482$	2002.64	0.189248
$483$	0	0
$484$	−3126.91	−0.293662
$485$	4195.03	0.392755
$486$	0	0
$487$	15711.2	1.46189	0.730945	−	0.682436i	$-0.239079\pi$
$487$	15711.2	1.46189	0.730945	+	0.682436i	$0.239079\pi$
$488$	62040.2	5.75498
$489$	0	0
$490$	0	0
$491$	2716.34	0.249667	0.124834	−	0.992178i	$-0.460160\pi$
$491$	2716.34	0.249667	0.124834	+	0.992178i	$0.460160\pi$
$492$	0	0
$493$	−17451.5	−1.59427
$494$	4496.38	0.409518
$495$	0	0
$496$	41601.2	3.76603
$497$	0	0
$498$	0	0
$499$	4295.34	0.385342	0.192671	−	0.981263i	$-0.438285\pi$
$499$	4295.34	0.385342	0.192671	+	0.981263i	$0.438285\pi$
$500$	2806.00	0.250977
$501$	0	0
$502$	29432.0	2.61677
$503$	−6515.26	−0.577537	−0.288769	−	0.957399i	$-0.593246\pi$
$503$	−6515.26	−0.577537	−0.288769	+	0.957399i	$0.593246\pi$
$504$	0	0
$505$	−4437.20	−0.390996
$506$	−19.8421	−0.00174326
$507$	0	0
$508$	26573.5	2.32088
$509$	−14391.8	−1.25325	−0.626624	−	0.779321i	$-0.715564\pi$
$509$	−14391.8	−1.25325	−0.626624	+	0.779321i	$0.715564\pi$
$510$	0	0
$511$	0	0
$512$	41890.2	3.61583
$513$	0	0
$514$	13457.1	1.15480
$515$	3095.44	0.264857
$516$	0	0
$517$	8993.73	0.765075
$518$	0	0
$519$	0	0
$520$	27462.3	2.31596
$521$	−2913.36	−0.244984	−0.122492	−	0.992469i	$-0.539089\pi$
$521$	−2913.36	−0.244984	−0.122492	+	0.992469i	$0.539089\pi$
$522$	0	0
$523$	16870.2	1.41048	0.705242	−	0.708967i	$-0.250838\pi$
$523$	16870.2	1.41048	0.705242	+	0.708967i	$0.250838\pi$
$524$	−5013.93	−0.418005
$525$	0	0
$526$	−8421.82	−0.698115
$527$	14611.4	1.20775
$528$	0	0
$529$	−12167.0	−0.999999
$530$	−9742.18	−0.798441
$531$	0	0
$532$	0	0
$533$	−10006.6	−0.813197
$534$	0	0
$535$	10756.5	0.869241
$536$	12113.6	0.976172
$537$	0	0
$538$	41738.6	3.34476
$539$	0	0
$540$	0	0
$541$	−2229.59	−0.177186	−0.0885929	−	0.996068i	$-0.528237\pi$
$541$	−2229.59	−0.177186	−0.0885929	+	0.996068i	$0.528237\pi$
$542$	23631.7	1.87282
$543$	0	0
$544$	73029.1	5.75569
$545$	−2035.38	−0.159975
$546$	0	0
$547$	−1218.73	−0.0952639	−0.0476319	−	0.998865i	$-0.515167\pi$
$547$	−1218.73	−0.0952639	−0.0476319	+	0.998865i	$0.515167\pi$
$548$	−45719.1	−3.56391
$549$	0	0
$550$	−4762.16	−0.369198
$551$	−2257.50	−0.174542
$552$	0	0
$553$	0	0
$554$	−22118.3	−1.69624
$555$	0	0
$556$	60317.9	4.60081
$557$	22734.5	1.72943	0.864714	−	0.502264i	$-0.167499\pi$
$557$	22734.5	1.72943	0.864714	+	0.502264i	$0.167499\pi$
$558$	0	0
$559$	568.873	0.0430425
$560$	0	0
$561$	0	0
$562$	−38269.6	−2.87243
$563$	−4302.11	−0.322047	−0.161023	−	0.986951i	$-0.551479\pi$
$563$	−4302.11	−0.322047	−0.161023	+	0.986951i	$0.551479\pi$
$564$	0	0
$565$	1747.91	0.130150
$566$	−14714.4	−1.09275
$567$	0	0
$568$	24829.9	1.83422
$569$	−14866.9	−1.09535	−0.547675	−	0.836691i	$-0.684487\pi$
$569$	−14866.9	−1.09535	−0.547675	+	0.836691i	$0.684487\pi$
$570$	0	0
$571$	16514.5	1.21035	0.605174	−	0.796093i	$-0.293103\pi$
$571$	16514.5	1.21035	0.605174	+	0.796093i	$0.293103\pi$
$572$	−53387.6	−3.90253
$573$	0	0
$574$	0	0
$575$	2.60414	0.000188870 0
$576$	0	0
$577$	1867.97	0.134774	0.0673870	−	0.997727i	$-0.478534\pi$
$577$	1867.97	0.134774	0.0673870	+	0.997727i	$0.478534\pi$
$578$	19022.0	1.36888
$579$	0	0
$580$	−21422.5	−1.53366
$581$	0	0
$582$	0	0
$583$	12189.6	0.865941
$584$	51014.3	3.61471
$585$	0	0
$586$	−32771.8	−2.31022
$587$	−3926.15	−0.276064	−0.138032	−	0.990428i	$-0.544078\pi$
$587$	−3926.15	−0.276064	−0.138032	+	0.990428i	$0.544078\pi$
$588$	0	0
$589$	1890.11	0.132225
$590$	−6635.21	−0.462996
$591$	0	0
$592$	−46314.7	−3.21541
$593$	7554.18	0.523125	0.261562	−	0.965187i	$-0.415762\pi$
$593$	7554.18	0.523125	0.261562	+	0.965187i	$0.415762\pi$
$594$	0	0
$595$	0	0
$596$	−15118.8	−1.03907
$597$	0	0
$598$	39.5988	0.00270788
$599$	−28210.9	−1.92432	−0.962160	−	0.272487i	$-0.912154\pi$
$599$	−28210.9	−1.92432	−0.962160	+	0.272487i	$0.912154\pi$
$600$	0	0
$601$	−23181.4	−1.57336	−0.786679	−	0.617363i	$-0.788201\pi$
$601$	−23181.4	−1.57336	−0.786679	+	0.617363i	$0.788201\pi$
$602$	0	0
$603$	0	0
$604$	−47706.1	−3.21380
$605$	−696.478	−0.0468031
$606$	0	0
$607$	−24863.6	−1.66257	−0.831287	−	0.555844i	$-0.812395\pi$
$607$	−24863.6	−1.66257	−0.831287	+	0.555844i	$0.812395\pi$
$608$	9446.93	0.630137
$609$	0	0
$610$	21470.1	1.42508
$611$	−17948.7	−1.18843
$612$	0	0
$613$	−12792.7	−0.842893	−0.421446	−	0.906853i	$-0.638477\pi$
$613$	−12792.7	−0.842893	−0.421446	+	0.906853i	$0.638477\pi$
$614$	57285.0	3.76520
$615$	0	0
$616$	0	0
$617$	−19793.3	−1.29149	−0.645744	−	0.763554i	$-0.723453\pi$
$617$	−19793.3	−1.29149	−0.645744	+	0.763554i	$0.723453\pi$
$618$	0	0
$619$	−20951.7	−1.36045	−0.680226	−	0.733002i	$-0.738118\pi$
$619$	−20951.7	−1.36045	−0.680226	+	0.733002i	$0.738118\pi$
$620$	17936.2	1.16183
$621$	0	0
$622$	−23399.2	−1.50840
$623$	0	0
$624$	0	0
$625$	625.000	0.0400000
$626$	−1566.57	−0.100020
$627$	0	0
$628$	−63147.1	−4.01249
$629$	−16266.9	−1.03117
$630$	0	0
$631$	23273.5	1.46831	0.734156	−	0.678981i	$-0.237578\pi$
$631$	23273.5	1.46831	0.734156	+	0.678981i	$0.237578\pi$
$632$	31186.9	1.96289
$633$	0	0
$634$	−9598.47	−0.601268
$635$	5918.90	0.369896
$636$	0	0
$637$	0	0
$638$	36356.8	2.25608
$639$	0	0
$640$	32186.8	1.98796
$641$	22117.7	1.36287	0.681434	−	0.731880i	$-0.261357\pi$
$641$	22117.7	1.36287	0.681434	+	0.731880i	$0.261357\pi$
$642$	0	0
$643$	20269.9	1.24318	0.621591	−	0.783342i	$-0.286487\pi$
$643$	20269.9	1.24318	0.621591	+	0.783342i	$0.286487\pi$
$644$	0	0
$645$	0	0
$646$	5967.54	0.363451
$647$	3145.25	0.191117	0.0955583	−	0.995424i	$-0.469536\pi$
$647$	3145.25	0.191117	0.0955583	+	0.995424i	$0.469536\pi$
$648$	0	0
$649$	8302.13	0.502137
$650$	9503.81	0.573493
$651$	0	0
$652$	−30179.6	−1.81277
$653$	5953.35	0.356773	0.178386	−	0.983961i	$-0.442912\pi$
$653$	5953.35	0.356773	0.178386	+	0.983961i	$0.442912\pi$
$654$	0	0
$655$	−1116.79	−0.0666205
$656$	−37812.3	−2.25049
$657$	0	0
$658$	0	0
$659$	−26277.5	−1.55330	−0.776652	−	0.629930i	$-0.783084\pi$
$659$	−26277.5	−1.55330	−0.776652	+	0.629930i	$0.783084\pi$
$660$	0	0
$661$	−24004.3	−1.41250	−0.706248	−	0.707964i	$-0.749614\pi$
$661$	−24004.3	−1.41250	−0.706248	+	0.707964i	$0.749614\pi$
$662$	30939.1	1.81644
$663$	0	0
$664$	−39320.3	−2.29808
$665$	0	0
$666$	0	0
$667$	−19.8814	−0.00115414
$668$	−32577.4	−1.88691
$669$	0	0
$670$	4192.13	0.241725
$671$	−26863.9	−1.54556
$672$	0	0
$673$	9205.36	0.527252	0.263626	−	0.964625i	$-0.415082\pi$
$673$	9205.36	0.527252	0.263626	+	0.964625i	$0.415082\pi$
$674$	−52020.4	−2.97292
$675$	0	0
$676$	57227.1	3.25598
$677$	18773.1	1.06575	0.532873	−	0.846195i	$-0.321112\pi$
$677$	18773.1	1.06575	0.532873	+	0.846195i	$0.321112\pi$
$678$	0	0
$679$	0	0
$680$	36447.6	2.05544
$681$	0	0
$682$	−30440.1	−1.70911
$683$	10222.2	0.572684	0.286342	−	0.958127i	$-0.407561\pi$
$683$	10222.2	0.572684	0.286342	+	0.958127i	$0.407561\pi$
$684$	0	0
$685$	−10183.3	−0.568006
$686$	0	0
$687$	0	0
$688$	2149.62	0.119118
$689$	−24326.8	−1.34510
$690$	0	0
$691$	22355.1	1.23072	0.615361	−	0.788246i	$-0.289010\pi$
$691$	22355.1	1.23072	0.615361	+	0.788246i	$0.289010\pi$
$692$	44428.2	2.44062
$693$	0	0
$694$	−64197.3	−3.51138
$695$	13435.0	0.733264
$696$	0	0
$697$	−13280.6	−0.721722
$698$	−7267.70	−0.394107
$699$	0	0
$700$	0	0
$701$	−16217.3	−0.873779	−0.436890	−	0.899515i	$-0.643920\pi$
$701$	−16217.3	−0.873779	−0.436890	+	0.899515i	$0.643920\pi$
$702$	0	0
$703$	−2104.26	−0.112893
$704$	−80247.1	−4.29606
$705$	0	0
$706$	−32273.7	−1.72045
$707$	0	0
$708$	0	0
$709$	3922.92	0.207798	0.103899	−	0.994588i	$-0.466868\pi$
$709$	3922.92	0.207798	0.103899	+	0.994588i	$0.466868\pi$
$710$	8592.83	0.454202
$711$	0	0
$712$	−37777.1	−1.98842
$713$	16.6458	0.000874322 0
$714$	0	0
$715$	−11891.4	−0.621976
$716$	97843.4	5.10695
$717$	0	0
$718$	68544.3	3.56275
$719$	−23908.6	−1.24011	−0.620055	−	0.784558i	$-0.712890\pi$
$719$	−23908.6	−1.24011	−0.620055	+	0.784558i	$0.712890\pi$
$720$	0	0
$721$	0	0
$722$	−37075.8	−1.91111
$723$	0	0
$724$	−8483.62	−0.435485
$725$	−4771.58	−0.244430
$726$	0	0
$727$	−26906.4	−1.37263	−0.686315	−	0.727305i	$-0.740773\pi$
$727$	−26906.4	−1.37263	−0.686315	+	0.727305i	$0.740773\pi$
$728$	0	0
$729$	0	0
$730$	17654.4	0.895095
$731$	755.001	0.0382007
$732$	0	0
$733$	27637.5	1.39265	0.696325	−	0.717726i	$-0.254817\pi$
$733$	27637.5	1.39265	0.696325	+	0.717726i	$0.254817\pi$
$734$	−36367.7	−1.82882
$735$	0	0
$736$	83.1973	0.00416670
$737$	−5245.29	−0.262161
$738$	0	0
$739$	−14038.9	−0.698821	−0.349410	−	0.936970i	$-0.613618\pi$
$739$	−14038.9	−0.698821	−0.349410	+	0.936970i	$0.613618\pi$
$740$	−19968.4	−0.991963
$741$	0	0
$742$	0	0
$743$	−17698.7	−0.873893	−0.436946	−	0.899488i	$-0.643940\pi$
$743$	−17698.7	−0.873893	−0.436946	+	0.899488i	$0.643940\pi$
$744$	0	0
$745$	−3367.50	−0.165605
$746$	1899.72	0.0932354
$747$	0	0
$748$	−70855.4	−3.46354
$749$	0	0
$750$	0	0
$751$	17199.7	0.835719	0.417859	−	0.908512i	$-0.362780\pi$
$751$	17199.7	0.835719	0.417859	+	0.908512i	$0.362780\pi$
$752$	−67823.4	−3.28892
$753$	0	0
$754$	−72557.1	−3.50448
$755$	−10625.9	−0.512206
$756$	0	0
$757$	−19200.7	−0.921879	−0.460939	−	0.887432i	$-0.652487\pi$
$757$	−19200.7	−0.921879	−0.460939	+	0.887432i	$0.652487\pi$
$758$	28921.0	1.38583
$759$	0	0
$760$	4714.80	0.225031
$761$	27918.7	1.32990	0.664949	−	0.746889i	$-0.268453\pi$
$761$	27918.7	1.32990	0.664949	+	0.746889i	$0.268453\pi$
$762$	0	0
$763$	0	0
$764$	54457.7	2.57881
$765$	0	0
$766$	−41922.1	−1.97742
$767$	−16568.5	−0.779992
$768$	0	0
$769$	−7436.29	−0.348712	−0.174356	−	0.984683i	$-0.555784\pi$
$769$	−7436.29	−0.348712	−0.174356	+	0.984683i	$0.555784\pi$
$770$	0	0
$771$	0	0
$772$	−13983.4	−0.651908
$773$	−4989.84	−0.232176	−0.116088	−	0.993239i	$-0.537035\pi$
$773$	−4989.84	−0.232176	−0.116088	+	0.993239i	$0.537035\pi$
$774$	0	0
$775$	3995.05	0.185169
$776$	66888.7	3.09429
$777$	0	0
$778$	−17115.9	−0.788731
$779$	−1717.96	−0.0790146
$780$	0	0
$781$	−10751.5	−0.492600
$782$	52.5550	0.00240328
$783$	0	0
$784$	0	0
$785$	−14065.2	−0.639500
$786$	0	0
$787$	−2870.69	−0.130024	−0.0650122	−	0.997884i	$-0.520709\pi$
$787$	−2870.69	−0.130024	−0.0650122	+	0.997884i	$0.520709\pi$
$788$	−63803.8	−2.88441
$789$	0	0
$790$	10792.8	0.486063
$791$	0	0
$792$	0	0
$793$	53612.1	2.40078
$794$	−16179.0	−0.723138
$795$	0	0
$796$	−19470.6	−0.866982
$797$	−4676.61	−0.207847	−0.103923	−	0.994585i	$-0.533140\pi$
$797$	−4676.61	−0.207847	−0.103923	+	0.994585i	$0.533140\pi$
$798$	0	0
$799$	−23821.3	−1.05474
$800$	19967.6	0.882451
$801$	0	0
$802$	−492.410	−0.0216803
$803$	−22089.6	−0.970766
$804$	0	0
$805$	0	0
$806$	60749.1	2.65483
$807$	0	0
$808$	−70750.2	−3.08042
$809$	15376.9	0.668261	0.334131	−	0.942527i	$-0.391557\pi$
$809$	15376.9	0.668261	0.334131	+	0.942527i	$0.391557\pi$
$810$	0	0
$811$	36422.9	1.57704	0.788522	−	0.615007i	$-0.210847\pi$
$811$	36422.9	1.57704	0.788522	+	0.615007i	$0.210847\pi$
$812$	0	0
$813$	0	0
$814$	33889.0	1.45922
$815$	−6722.10	−0.288914
$816$	0	0
$817$	97.6658	0.00418225
$818$	813.285	0.0347627
$819$	0	0
$820$	−16302.6	−0.694282
$821$	25479.0	1.08310	0.541550	−	0.840669i	$-0.317838\pi$
$821$	25479.0	1.08310	0.541550	+	0.840669i	$0.317838\pi$
$822$	0	0
$823$	−17933.3	−0.759557	−0.379779	−	0.925077i	$-0.624000\pi$
$823$	−17933.3	−0.759557	−0.379779	+	0.925077i	$0.624000\pi$
$824$	49356.1	2.08665
$825$	0	0
$826$	0	0
$827$	−13021.6	−0.547529	−0.273764	−	0.961797i	$-0.588269\pi$
$827$	−13021.6	−0.547529	−0.273764	+	0.961797i	$0.588269\pi$
$828$	0	0
$829$	−11397.6	−0.477509	−0.238754	−	0.971080i	$-0.576739\pi$
$829$	−11397.6	−0.477509	−0.238754	+	0.971080i	$0.576739\pi$
$830$	−13607.5	−0.569063
$831$	0	0
$832$	160149.	6.67326
$833$	0	0
$834$	0	0
$835$	−7256.19	−0.300731
$836$	−9165.75	−0.379191
$837$	0	0
$838$	20867.2	0.860196
$839$	37681.2	1.55053	0.775267	−	0.631633i	$-0.217615\pi$
$839$	37681.2	1.55053	0.775267	+	0.631633i	$0.217615\pi$
$840$	0	0
$841$	12039.8	0.493656
$842$	−60141.7	−2.46154
$843$	0	0
$844$	−134148.	−5.47104
$845$	12746.6	0.518929
$846$	0	0
$847$	0	0
$848$	−91924.4	−3.72252
$849$	0	0
$850$	12613.3	0.508981
$851$	−18.5318	−0.000746490 0
$852$	0	0
$853$	21771.6	0.873911	0.436956	−	0.899483i	$-0.356057\pi$
$853$	21771.6	0.873911	0.436956	+	0.899483i	$0.356057\pi$
$854$	0	0
$855$	0	0
$856$	171510.	6.84823
$857$	−29860.2	−1.19021	−0.595103	−	0.803649i	$-0.702889\pi$
$857$	−29860.2	−1.19021	−0.595103	+	0.803649i	$0.702889\pi$
$858$	0	0
$859$	18530.6	0.736039	0.368019	−	0.929818i	$-0.380036\pi$
$859$	18530.6	0.736039	0.368019	+	0.929818i	$0.380036\pi$
$860$	926.799	0.0367484
$861$	0	0
$862$	62261.4	2.46013
$863$	21296.5	0.840024	0.420012	−	0.907519i	$-0.362026\pi$
$863$	21296.5	0.840024	0.420012	+	0.907519i	$0.362026\pi$
$864$	0	0
$865$	9895.79	0.388979
$866$	−49144.1	−1.92839
$867$	0	0
$868$	0	0
$869$	−13504.2	−0.527155
$870$	0	0
$871$	10468.0	0.407226
$872$	−32453.7	−1.26035
$873$	0	0
$874$	6.79843	0.000263112 0
$875$	0	0
$876$	0	0
$877$	−18793.4	−0.723614	−0.361807	−	0.932253i	$-0.617840\pi$
$877$	−18793.4	−0.723614	−0.361807	+	0.932253i	$0.617840\pi$
$878$	44968.7	1.72850
$879$	0	0
$880$	−44934.4	−1.72129
$881$	−1638.62	−0.0626634	−0.0313317	−	0.999509i	$-0.509975\pi$
$881$	−1638.62	−0.0626634	−0.0313317	+	0.999509i	$0.509975\pi$
$882$	0	0
$883$	−35424.1	−1.35008	−0.675038	−	0.737783i	$-0.735873\pi$
$883$	−35424.1	−1.35008	−0.675038	+	0.737783i	$0.735873\pi$
$884$	141406.	5.38008
$885$	0	0
$886$	63306.5	2.40048
$887$	5131.41	0.194246	0.0971229	−	0.995272i	$-0.469036\pi$
$887$	5131.41	0.194246	0.0971229	+	0.995272i	$0.469036\pi$
$888$	0	0
$889$	0	0
$890$	−13073.5	−0.492386
$891$	0	0
$892$	−116316.	−4.36610
$893$	−3081.49	−0.115474
$894$	0	0
$895$	21793.3	0.813933
$896$	0	0
$897$	0	0
$898$	−10836.2	−0.402682
$899$	−30500.3	−1.13153
$900$	0	0
$901$	−32286.2	−1.19380
$902$	27667.7	1.02132
$903$	0	0
$904$	27870.0	1.02538
$905$	−1889.61	−0.0694065
$906$	0	0
$907$	−19934.6	−0.729787	−0.364893	−	0.931049i	$-0.618895\pi$
$907$	−19934.6	−0.729787	−0.364893	+	0.931049i	$0.618895\pi$
$908$	−84248.8	−3.07918
$909$	0	0
$910$	0	0
$911$	−48387.4	−1.75976	−0.879882	−	0.475193i	$-0.842378\pi$
$911$	−48387.4	−1.75976	−0.879882	+	0.475193i	$0.842378\pi$
$912$	0	0
$913$	17026.0	0.617172
$914$	30844.7	1.11625
$915$	0	0
$916$	−140483.	−5.06735
$917$	0	0
$918$	0	0
$919$	−14431.7	−0.518019	−0.259009	−	0.965875i	$-0.583396\pi$
$919$	−14431.7	−0.518019	−0.259009	+	0.965875i	$0.583396\pi$
$920$	41.5224	0.00148799
$921$	0	0
$922$	103688.	3.70366
$923$	21456.8	0.765177
$924$	0	0
$925$	−4447.69	−0.158096
$926$	−43549.1	−1.54548
$927$	0	0
$928$	−152443.	−5.39244
$929$	−5549.82	−0.196000	−0.0979998	−	0.995186i	$-0.531244\pi$
$929$	−5549.82	−0.196000	−0.0979998	+	0.995186i	$0.531244\pi$
$930$	0	0
$931$	0	0
$932$	−39956.7	−1.40432
$933$	0	0
$934$	−2127.24	−0.0745239
$935$	−15782.1	−0.552010
$936$	0	0
$937$	26231.9	0.914576	0.457288	−	0.889319i	$-0.348821\pi$
$937$	26231.9	0.914576	0.457288	+	0.889319i	$0.348821\pi$
$938$	0	0
$939$	0	0
$940$	−29241.8	−1.01464
$941$	−2836.80	−0.0982753	−0.0491376	−	0.998792i	$-0.515647\pi$
$941$	−2836.80	−0.0982753	−0.0491376	+	0.998792i	$0.515647\pi$
$942$	0	0
$943$	−15.1298	−0.000522474 0
$944$	−62607.9	−2.15860
$945$	0	0
$946$	−1572.90	−0.0540586
$947$	40272.9	1.38193	0.690967	−	0.722886i	$-0.257185\pi$
$947$	40272.9	1.38193	0.690967	+	0.722886i	$0.257185\pi$
$948$	0	0
$949$	44084.1	1.50793
$950$	1631.64	0.0557236
$951$	0	0
$952$	0	0
$953$	−17770.1	−0.604019	−0.302010	−	0.953305i	$-0.597657\pi$
$953$	−17770.1	−0.604019	−0.302010	+	0.953305i	$0.597657\pi$
$954$	0	0
$955$	12129.7	0.411004
$956$	−79005.0	−2.67281
$957$	0	0
$958$	−9349.97	−0.315328
$959$	0	0
$960$	0	0
$961$	−4254.35	−0.142807
$962$	−67632.0	−2.26668
$963$	0	0
$964$	8147.07	0.272199
$965$	−3114.61	−0.103899
$966$	0	0
$967$	−3530.14	−0.117396	−0.0586978	−	0.998276i	$-0.518695\pi$
$967$	−3530.14	−0.117396	−0.0586978	+	0.998276i	$0.518695\pi$
$968$	−11105.2	−0.368734
$969$	0	0
$970$	23148.0	0.766226
$971$	−17650.7	−0.583356	−0.291678	−	0.956517i	$-0.594214\pi$
$971$	−17650.7	−0.583356	−0.291678	+	0.956517i	$0.594214\pi$
$972$	0	0
$973$	0	0
$974$	86693.8	2.85200
$975$	0	0
$976$	202586.	6.64407
$977$	14477.6	0.474085	0.237042	−	0.971499i	$-0.423822\pi$
$977$	14477.6	0.474085	0.237042	+	0.971499i	$0.423822\pi$
$978$	0	0
$979$	16357.8	0.534012
$980$	0	0
$981$	0	0
$982$	14988.7	0.487075
$983$	8764.09	0.284365	0.142183	−	0.989840i	$-0.454588\pi$
$983$	8764.09	0.284365	0.142183	+	0.989840i	$0.454588\pi$
$984$	0	0
$985$	−14211.4	−0.459710
$986$	−96296.9	−3.11026
$987$	0	0
$988$	18292.0	0.589015
$989$	0.860124	2.76546e−5 0
$990$	0	0
$991$	33624.1	1.07781	0.538903	−	0.842368i	$-0.318839\pi$
$991$	33624.1	1.07781	0.538903	+	0.842368i	$0.318839\pi$
$992$	127634.	4.08507
$993$	0	0
$994$	0	0
$995$	−4336.82	−0.138177
$996$	0	0
$997$	16631.3	0.528302	0.264151	−	0.964481i	$-0.414908\pi$
$997$	16631.3	0.528302	0.264151	+	0.964481i	$0.414908\pi$
$998$	23701.6	0.751764
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.4.a.ca.1.6		6
3.2	odd	2		245.4.a.p.1.1	yes	6
7.6	odd	2		2205.4.a.bz.1.6		6
15.14	odd	2		1225.4.a.bi.1.6		6
21.2	odd	6		245.4.e.p.116.6		12
21.5	even	6		245.4.e.q.116.6		12
21.11	odd	6		245.4.e.p.226.6		12
21.17	even	6		245.4.e.q.226.6		12
21.20	even	2		245.4.a.o.1.1	✓	6
105.104	even	2		1225.4.a.bj.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.4.a.o.1.1	✓	6	21.20	even	2
245.4.a.p.1.1	yes	6	3.2	odd	2
245.4.e.p.116.6		12	21.2	odd	6
245.4.e.p.226.6		12	21.11	odd	6
245.4.e.q.116.6		12	21.5	even	6
245.4.e.q.226.6		12	21.17	even	6
1225.4.a.bi.1.6		6	15.14	odd	2
1225.4.a.bj.1.6		6	105.104	even	2
2205.4.a.bz.1.6		6	7.6	odd	2
2205.4.a.ca.1.6		6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+5.51797 q^{2} +22.4480 q^{4} +5.00000 q^{5} +79.7239 q^{8} +27.5899 q^{10} -34.5211 q^{11} +68.8935 q^{13} +260.330 q^{16} +91.4346 q^{17} +11.8278 q^{19} +112.240 q^{20} -190.486 q^{22} +0.104165 q^{23} +25.0000 q^{25} +380.152 q^{26} -190.863 q^{29} +159.802 q^{31} +798.703 q^{32} +504.534 q^{34} -177.908 q^{37} +65.2657 q^{38} +398.619 q^{40} -145.247 q^{41} +8.25729 q^{43} -774.930 q^{44} +0.574782 q^{46} -260.529 q^{47} +137.949 q^{50} +1546.52 q^{52} -353.107 q^{53} -172.605 q^{55} -1053.18 q^{58} -240.495 q^{59} +778.188 q^{61} +881.783 q^{62} +2324.58 q^{64} +344.467 q^{65} +151.945 q^{67} +2052.53 q^{68} +311.449 q^{71} +639.888 q^{73} -981.690 q^{74} +265.512 q^{76} +391.186 q^{79} +1301.65 q^{80} -801.472 q^{82} -493.205 q^{83} +457.173 q^{85} +45.5635 q^{86} -2752.15 q^{88} -473.850 q^{89} +2.33831 q^{92} -1437.59 q^{94} +59.1392 q^{95} +839.005 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{2} + 14 q^{4} + 30 q^{5} + 66 q^{8} + 10 q^{10} + 16 q^{11} + 168 q^{13} + 298 q^{16} + 4 q^{17} + 308 q^{19} + 70 q^{20} - 236 q^{22} + 336 q^{23} + 150 q^{25} - 56 q^{26} - 176 q^{29} + 392 q^{31}+ \cdots + 516 q^{97}+O(q^{100}) \) 6 * q + 2 * q^2 + 14 * q^4 + 30 * q^5 + 66 * q^8 + 10 * q^10 + 16 * q^11 + 168 * q^13 + 298 * q^16 + 4 * q^17 + 308 * q^19 + 70 * q^20 - 236 * q^22 + 336 * q^23 + 150 * q^25 - 56 * q^26 - 176 * q^29 + 392 * q^31 + 770 * q^32 + 812 * q^34 - 140 * q^37 - 20 * q^38 + 330 * q^40 - 656 * q^41 - 388 * q^43 + 160 * q^44 - 388 * q^46 - 628 * q^47 + 50 * q^50 + 1520 * q^52 + 676 * q^53 + 80 * q^55 - 2012 * q^58 - 996 * q^59 + 740 * q^61 + 364 * q^62 + 1426 * q^64 + 840 * q^65 + 1768 * q^67 + 2940 * q^68 + 224 * q^71 + 2640 * q^73 - 928 * q^74 - 1340 * q^76 + 1636 * q^79 + 1490 * q^80 - 1756 * q^82 - 140 * q^83 + 20 * q^85 - 1180 * q^86 - 5652 * q^88 + 1904 * q^89 + 1952 * q^92 - 3332 * q^94 + 1540 * q^95 + 516 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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