LMFDB - Embedded newform 2205.4.a.cb.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:	$8$
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} - 4x^{7} - 45x^{6} + 134x^{5} + 641x^{4} - 1130x^{3} - 2877x^{2} + 2584x + 3696 $ x^8 - 4x^7 - 45x^6 + 134x^5 + 641x^4 - 1130x^3 - 2877x^2 + 2584*x + 3696
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 315)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-4.59196$ 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)

$f(q)$	$=$	$q-5.59196 q^{2} +23.2700 q^{4} -5.00000 q^{5} -85.3894 q^{8} +O(q^{10})$	$q-5.59196 q^{2} +23.2700 q^{4} -5.00000 q^{5} -85.3894 q^{8} +27.9598 q^{10} +20.8488 q^{11} +41.4382 q^{13} +291.334 q^{16} +81.5764 q^{17} +15.9685 q^{19} -116.350 q^{20} -116.586 q^{22} -75.5448 q^{23} +25.0000 q^{25} -231.721 q^{26} -167.568 q^{29} -184.619 q^{31} -946.012 q^{32} -456.172 q^{34} +330.810 q^{37} -89.2954 q^{38} +426.947 q^{40} -478.334 q^{41} -14.5727 q^{43} +485.152 q^{44} +422.444 q^{46} +387.035 q^{47} -139.799 q^{50} +964.268 q^{52} +66.1892 q^{53} -104.244 q^{55} +937.035 q^{58} -553.410 q^{59} -277.492 q^{61} +1032.38 q^{62} +2959.39 q^{64} -207.191 q^{65} +872.023 q^{67} +1898.28 q^{68} -291.792 q^{71} -1139.81 q^{73} -1849.87 q^{74} +371.588 q^{76} -925.516 q^{79} -1456.67 q^{80} +2674.83 q^{82} -9.11942 q^{83} -407.882 q^{85} +81.4899 q^{86} -1780.26 q^{88} +125.703 q^{89} -1757.93 q^{92} -2164.28 q^{94} -79.8427 q^{95} +932.786 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{2} + 42 q^{4} - 40 q^{5} - 48 q^{8}+O(q^{10}) $ 8 * q - 4 * q^2 + 42 * q^4 - 40 * q^5 - 48 * q^8	$ 8 q - 4 q^{2} + 42 q^{4} - 40 q^{5} - 48 q^{8} + 20 q^{10} - 100 q^{11} + 102 q^{13} + 266 q^{16} + 56 q^{17} - 210 q^{20} - 70 q^{22} - 190 q^{23} + 200 q^{25} + 60 q^{26} - 296 q^{29} - 42 q^{31} - 718 q^{32} + 488 q^{34} + 314 q^{37} - 514 q^{38} + 240 q^{40} - 28 q^{41} + 714 q^{43} - 410 q^{44} - 650 q^{46} + 326 q^{47} - 100 q^{50} + 794 q^{52} - 1282 q^{53} + 500 q^{55} + 942 q^{58} - 924 q^{59} + 536 q^{61} + 50 q^{62} + 1902 q^{64} - 510 q^{65} + 2 q^{67} + 2690 q^{68} - 1516 q^{71} - 86 q^{73} - 4754 q^{74} - 30 q^{76} - 42 q^{79} - 1330 q^{80} + 3602 q^{82} + 572 q^{83} - 280 q^{85} - 1758 q^{86} - 4312 q^{88} + 940 q^{89} - 3844 q^{92} - 2866 q^{94} + 1720 q^{97}+O(q^{100}) $ 8 * q - 4 * q^2 + 42 * q^4 - 40 * q^5 - 48 * q^8 + 20 * q^10 - 100 * q^11 + 102 * q^13 + 266 * q^16 + 56 * q^17 - 210 * q^20 - 70 * q^22 - 190 * q^23 + 200 * q^25 + 60 * q^26 - 296 * q^29 - 42 * q^31 - 718 * q^32 + 488 * q^34 + 314 * q^37 - 514 * q^38 + 240 * q^40 - 28 * q^41 + 714 * q^43 - 410 * q^44 - 650 * q^46 + 326 * q^47 - 100 * q^50 + 794 * q^52 - 1282 * q^53 + 500 * q^55 + 942 * q^58 - 924 * q^59 + 536 * q^61 + 50 * q^62 + 1902 * q^64 - 510 * q^65 + 2 * q^67 + 2690 * q^68 - 1516 * q^71 - 86 * q^73 - 4754 * q^74 - 30 * q^76 - 42 * q^79 - 1330 * q^80 + 3602 * q^82 + 572 * q^83 - 280 * q^85 - 1758 * q^86 - 4312 * q^88 + 940 * q^89 - 3844 * q^92 - 2866 * q^94 + 1720 * 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 $

$2$	−5.59196	−1.97706	−0.988528	−	0.151036i	$-0.951739\pi$
$2$	−5.59196	−1.97706	−0.988528	+	0.151036i	$0.951739\pi$
$3$	0	0
$3$	0	0
$4$	23.2700	2.90875
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−85.3894	−3.77371
$9$	0	0
$10$	27.9598	0.884167
$11$	20.8488	0.571468	0.285734	−	0.958309i	$-0.407763\pi$
$11$	20.8488	0.571468	0.285734	+	0.958309i	$0.407763\pi$
$12$	0	0
$13$	41.4382	0.884069	0.442034	−	0.896998i	$-0.354257\pi$
$13$	41.4382	0.884069	0.442034	+	0.896998i	$0.354257\pi$
$14$	0	0
$15$	0	0
$16$	291.334	4.55209
$17$	81.5764	1.16383	0.581917	−	0.813248i	$-0.302303\pi$
$17$	81.5764	1.16383	0.581917	+	0.813248i	$0.302303\pi$
$18$	0	0
$19$	15.9685	0.192812	0.0964062	−	0.995342i	$-0.469265\pi$
$19$	15.9685	0.192812	0.0964062	+	0.995342i	$0.469265\pi$
$20$	−116.350	−1.30083
$21$	0	0
$22$	−116.586	−1.12982
$23$	−75.5448	−0.684878	−0.342439	−	0.939540i	$-0.611253\pi$
$23$	−75.5448	−0.684878	−0.342439	+	0.939540i	$0.611253\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−231.721	−1.74785
$27$	0	0
$28$	0	0
$29$	−167.568	−1.07299	−0.536494	−	0.843904i	$-0.680252\pi$
$29$	−167.568	−1.07299	−0.536494	+	0.843904i	$0.680252\pi$
$30$	0	0
$31$	−184.619	−1.06963	−0.534815	−	0.844969i	$-0.679619\pi$
$31$	−184.619	−1.06963	−0.534815	+	0.844969i	$0.679619\pi$
$32$	−946.012	−5.22603
$33$	0	0
$34$	−456.172	−2.30097
$35$	0	0
$36$	0	0
$37$	330.810	1.46986	0.734929	−	0.678144i	$-0.237215\pi$
$37$	330.810	1.46986	0.734929	+	0.678144i	$0.237215\pi$
$38$	−89.2954	−0.381201
$39$	0	0
$40$	426.947	1.68766
$41$	−478.334	−1.82203	−0.911015	−	0.412373i	$-0.864700\pi$
$41$	−478.334	−1.82203	−0.911015	+	0.412373i	$0.864700\pi$
$42$	0	0
$43$	−14.5727	−0.0516817	−0.0258408	−	0.999666i	$-0.508226\pi$
$43$	−14.5727	−0.0516817	−0.0258408	+	0.999666i	$0.508226\pi$
$44$	485.152	1.66226
$45$	0	0
$46$	422.444	1.35404
$47$	387.035	1.20117	0.600583	−	0.799562i	$-0.294935\pi$
$47$	387.035	1.20117	0.600583	+	0.799562i	$0.294935\pi$
$48$	0	0
$49$	0	0
$50$	−139.799	−0.395411
$51$	0	0
$52$	964.268	2.57154
$53$	66.1892	0.171543	0.0857716	−	0.996315i	$-0.472664\pi$
$53$	66.1892	0.171543	0.0857716	+	0.996315i	$0.472664\pi$
$54$	0	0
$55$	−104.244	−0.255568
$56$	0	0
$57$	0	0
$58$	937.035	2.12136
$59$	−553.410	−1.22115	−0.610575	−	0.791959i	$-0.709062\pi$
$59$	−553.410	−1.22115	−0.610575	+	0.791959i	$0.709062\pi$
$60$	0	0
$61$	−277.492	−0.582446	−0.291223	−	0.956655i	$-0.594062\pi$
$61$	−277.492	−0.582446	−0.291223	+	0.956655i	$0.594062\pi$
$62$	1032.38	2.11472
$63$	0	0
$64$	2959.39	5.78006
$65$	−207.191	−0.395367
$66$	0	0
$67$	872.023	1.59007	0.795034	−	0.606565i	$-0.207453\pi$
$67$	872.023	1.59007	0.795034	+	0.606565i	$0.207453\pi$
$68$	1898.28	3.38531
$69$	0	0
$70$	0	0
$71$	−291.792	−0.487737	−0.243869	−	0.969808i	$-0.578417\pi$
$71$	−291.792	−0.487737	−0.243869	+	0.969808i	$0.578417\pi$
$72$	0	0
$73$	−1139.81	−1.82746	−0.913729	−	0.406324i	$-0.866810\pi$
$73$	−1139.81	−1.82746	−0.913729	+	0.406324i	$0.866810\pi$
$74$	−1849.87	−2.90599
$75$	0	0
$76$	371.588	0.560843
$77$	0	0
$78$	0	0
$79$	−925.516	−1.31808	−0.659042	−	0.752106i	$-0.729038\pi$
$79$	−925.516	−1.31808	−0.659042	+	0.752106i	$0.729038\pi$
$80$	−1456.67	−2.03576
$81$	0	0
$82$	2674.83	3.60226
$83$	−9.11942	−0.0120601	−0.00603004	−	0.999982i	$-0.501919\pi$
$83$	−9.11942	−0.0120601	−0.00603004	+	0.999982i	$0.501919\pi$
$84$	0	0
$85$	−407.882	−0.520483
$86$	81.4899	0.102178
$87$	0	0
$88$	−1780.26	−2.15655
$89$	125.703	0.149713	0.0748566	−	0.997194i	$-0.476150\pi$
$89$	125.703	0.149713	0.0748566	+	0.997194i	$0.476150\pi$
$90$	0	0
$91$	0	0
$92$	−1757.93	−1.99214
$93$	0	0
$94$	−2164.28	−2.37477
$95$	−79.8427	−0.0862283
$96$	0	0
$97$	932.786	0.976392	0.488196	−	0.872734i	$-0.337655\pi$
$97$	932.786	0.976392	0.488196	+	0.872734i	$0.337655\pi$
$98$	0	0
$99$	0	0
$100$	581.751	0.581751
$101$	1383.36	1.36286	0.681432	−	0.731882i	$-0.261358\pi$
$101$	1383.36	1.36286	0.681432	+	0.731882i	$0.261358\pi$
$102$	0	0
$103$	−165.687	−0.158502	−0.0792508	−	0.996855i	$-0.525253\pi$
$103$	−165.687	−0.158502	−0.0792508	+	0.996855i	$0.525253\pi$
$104$	−3538.38	−3.33622
$105$	0	0
$106$	−370.128	−0.339151
$107$	−882.131	−0.796998	−0.398499	−	0.917169i	$-0.630469\pi$
$107$	−882.131	−0.796998	−0.398499	+	0.917169i	$0.630469\pi$
$108$	0	0
$109$	−1204.00	−1.05800	−0.529000	−	0.848622i	$-0.677433\pi$
$109$	−1204.00	−1.05800	−0.529000	+	0.848622i	$0.677433\pi$
$110$	582.928	0.505273
$111$	0	0
$112$	0	0
$113$	351.310	0.292464	0.146232	−	0.989250i	$-0.453285\pi$
$113$	351.310	0.292464	0.146232	+	0.989250i	$0.453285\pi$
$114$	0	0
$115$	377.724	0.306287
$116$	−3899.32	−3.12106
$117$	0	0
$118$	3094.65	2.41428
$119$	0	0
$120$	0	0
$121$	−896.328	−0.673425
$122$	1551.72	1.15153
$123$	0	0
$124$	−4296.09	−3.11129
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−1110.92	−0.776205	−0.388103	−	0.921616i	$-0.626869\pi$
$127$	−1110.92	−0.776205	−0.388103	+	0.921616i	$0.626869\pi$
$128$	−8980.70	−6.20148
$129$	0	0
$130$	1158.60	0.781664
$131$	463.687	0.309256	0.154628	−	0.987973i	$-0.450582\pi$
$131$	463.687	0.309256	0.154628	+	0.987973i	$0.450582\pi$
$132$	0	0
$133$	0	0
$134$	−4876.32	−3.14365
$135$	0	0
$136$	−6965.76	−4.39198
$137$	−196.601	−0.122604	−0.0613021	−	0.998119i	$-0.519525\pi$
$137$	−196.601	−0.122604	−0.0613021	+	0.998119i	$0.519525\pi$
$138$	0	0
$139$	2456.86	1.49919	0.749597	−	0.661894i	$-0.230247\pi$
$139$	2456.86	1.49919	0.749597	+	0.661894i	$0.230247\pi$
$140$	0	0
$141$	0	0
$142$	1631.69	0.964284
$143$	863.936	0.505217
$144$	0	0
$145$	837.841	0.479855
$146$	6373.76	3.61299
$147$	0	0
$148$	7697.95	4.27545
$149$	2617.95	1.43940	0.719701	−	0.694284i	$-0.244279\pi$
$149$	2617.95	1.43940	0.719701	+	0.694284i	$0.244279\pi$
$150$	0	0
$151$	1112.96	0.599809	0.299905	−	0.953969i	$-0.403045\pi$
$151$	1112.96	0.599809	0.299905	+	0.953969i	$0.403045\pi$
$152$	−1363.54	−0.727618
$153$	0	0
$154$	0	0
$155$	923.094	0.478353
$156$	0	0
$157$	53.8867	0.0273925	0.0136963	−	0.999906i	$-0.495640\pi$
$157$	53.8867	0.0273925	0.0136963	+	0.999906i	$0.495640\pi$
$158$	5175.45	2.60593
$159$	0	0
$160$	4730.06	2.33715
$161$	0	0
$162$	0	0
$163$	−1521.35	−0.731052	−0.365526	−	0.930801i	$-0.619111\pi$
$163$	−1521.35	−0.731052	−0.365526	+	0.930801i	$0.619111\pi$
$164$	−11130.8	−5.29984
$165$	0	0
$166$	50.9955	0.0238435
$167$	−3924.20	−1.81835	−0.909173	−	0.416418i	$-0.863285\pi$
$167$	−3924.20	−1.81835	−0.909173	+	0.416418i	$0.863285\pi$
$168$	0	0
$169$	−479.875	−0.218423
$170$	2280.86	1.02902
$171$	0	0
$172$	−339.107	−0.150329
$173$	2322.17	1.02053	0.510264	−	0.860018i	$-0.329548\pi$
$173$	2322.17	1.02053	0.510264	+	0.860018i	$0.329548\pi$
$174$	0	0
$175$	0	0
$176$	6073.95	2.60137
$177$	0	0
$178$	−702.926	−0.295992
$179$	705.203	0.294466	0.147233	−	0.989102i	$-0.452963\pi$
$179$	705.203	0.294466	0.147233	+	0.989102i	$0.452963\pi$
$180$	0	0
$181$	−2743.77	−1.12675	−0.563377	−	0.826200i	$-0.690498\pi$
$181$	−2743.77	−1.12675	−0.563377	+	0.826200i	$0.690498\pi$
$182$	0	0
$183$	0	0
$184$	6450.72	2.58453
$185$	−1654.05	−0.657341
$186$	0	0
$187$	1700.77	0.665094
$188$	9006.31	3.49390
$189$	0	0
$190$	446.477	0.170478
$191$	−561.459	−0.212700	−0.106350	−	0.994329i	$-0.533916\pi$
$191$	−561.459	−0.212700	−0.106350	+	0.994329i	$0.533916\pi$
$192$	0	0
$193$	−3005.92	−1.12109	−0.560546	−	0.828123i	$-0.689409\pi$
$193$	−3005.92	−1.12109	−0.560546	+	0.828123i	$0.689409\pi$
$194$	−5216.10	−1.93038
$195$	0	0
$196$	0	0
$197$	3152.12	1.14000	0.569999	−	0.821646i	$-0.306944\pi$
$197$	3152.12	1.14000	0.569999	+	0.821646i	$0.306944\pi$
$198$	0	0
$199$	−182.983	−0.0651825	−0.0325913	−	0.999469i	$-0.510376\pi$
$199$	−182.983	−0.0651825	−0.0325913	+	0.999469i	$0.510376\pi$
$200$	−2134.73	−0.754742
$201$	0	0
$202$	−7735.68	−2.69446
$203$	0	0
$204$	0	0
$205$	2391.67	0.814837
$206$	926.518	0.313367
$207$	0	0
$208$	12072.3	4.02436
$209$	332.925	0.110186
$210$	0	0
$211$	2034.81	0.663897	0.331949	−	0.943297i	$-0.392294\pi$
$211$	2034.81	0.663897	0.331949	+	0.943297i	$0.392294\pi$
$212$	1540.23	0.498977
$213$	0	0
$214$	4932.84	1.57571
$215$	72.8634	0.0231128
$216$	0	0
$217$	0	0
$218$	6732.70	2.09173
$219$	0	0
$220$	−2425.76	−0.743384
$221$	3380.38	1.02891
$222$	0	0
$223$	−1122.82	−0.337172	−0.168586	−	0.985687i	$-0.553920\pi$
$223$	−1122.82	−0.337172	−0.168586	+	0.985687i	$0.553920\pi$
$224$	0	0
$225$	0	0
$226$	−1964.51	−0.578218
$227$	1771.03	0.517831	0.258915	−	0.965900i	$-0.416635\pi$
$227$	1771.03	0.517831	0.258915	+	0.965900i	$0.416635\pi$
$228$	0	0
$229$	3405.97	0.982850	0.491425	−	0.870920i	$-0.336476\pi$
$229$	3405.97	0.982850	0.491425	+	0.870920i	$0.336476\pi$
$230$	−2112.22	−0.605546
$231$	0	0
$232$	14308.5	4.04915
$233$	−2241.36	−0.630199	−0.315099	−	0.949059i	$-0.602038\pi$
$233$	−2241.36	−0.630199	−0.315099	+	0.949059i	$0.602038\pi$
$234$	0	0
$235$	−1935.17	−0.537178
$236$	−12877.9	−3.55202
$237$	0	0
$238$	0	0
$239$	−1229.76	−0.332831	−0.166415	−	0.986056i	$-0.553219\pi$
$239$	−1229.76	−0.332831	−0.166415	+	0.986056i	$0.553219\pi$
$240$	0	0
$241$	3872.76	1.03513	0.517565	−	0.855644i	$-0.326839\pi$
$241$	3872.76	1.03513	0.517565	+	0.855644i	$0.326839\pi$
$242$	5012.23	1.33140
$243$	0	0
$244$	−6457.25	−1.69419
$245$	0	0
$246$	0	0
$247$	661.708	0.170459
$248$	15764.5	4.03648
$249$	0	0
$250$	698.995	0.176833
$251$	−3883.35	−0.976552	−0.488276	−	0.872689i	$-0.662374\pi$
$251$	−3883.35	−0.976552	−0.488276	+	0.872689i	$0.662374\pi$
$252$	0	0
$253$	−1575.02	−0.391385
$254$	6212.21	1.53460
$255$	0	0
$256$	26544.6	6.48062
$257$	3738.26	0.907340	0.453670	−	0.891170i	$-0.350114\pi$
$257$	3738.26	0.907340	0.453670	+	0.891170i	$0.350114\pi$
$258$	0	0
$259$	0	0
$260$	−4821.34	−1.15003
$261$	0	0
$262$	−2592.92	−0.611417
$263$	434.302	0.101826	0.0509129	−	0.998703i	$-0.483787\pi$
$263$	434.302	0.101826	0.0509129	+	0.998703i	$0.483787\pi$
$264$	0	0
$265$	−330.946	−0.0767165
$266$	0	0
$267$	0	0
$268$	20292.0	4.62511
$269$	2414.83	0.547342	0.273671	−	0.961823i	$-0.411762\pi$
$269$	2414.83	0.547342	0.273671	+	0.961823i	$0.411762\pi$
$270$	0	0
$271$	−628.272	−0.140830	−0.0704148	−	0.997518i	$-0.522432\pi$
$271$	−628.272	−0.140830	−0.0704148	+	0.997518i	$0.522432\pi$
$272$	23766.0	5.29788
$273$	0	0
$274$	1099.39	0.242395
$275$	521.220	0.114294
$276$	0	0
$277$	−2211.96	−0.479797	−0.239899	−	0.970798i	$-0.577114\pi$
$277$	−2211.96	−0.479797	−0.239899	+	0.970798i	$0.577114\pi$
$278$	−13738.7	−2.96399
$279$	0	0
$280$	0	0
$281$	−2364.30	−0.501930	−0.250965	−	0.967996i	$-0.580748\pi$
$281$	−2364.30	−0.501930	−0.250965	+	0.967996i	$0.580748\pi$
$282$	0	0
$283$	1823.83	0.383092	0.191546	−	0.981484i	$-0.438650\pi$
$283$	1823.83	0.383092	0.191546	+	0.981484i	$0.438650\pi$
$284$	−6790.01	−1.41871
$285$	0	0
$286$	−4831.10	−0.998842
$287$	0	0
$288$	0	0
$289$	1741.71	0.354510
$290$	−4685.17	−0.948700
$291$	0	0
$292$	−26523.4	−5.31562
$293$	6658.98	1.32772	0.663860	−	0.747857i	$-0.268917\pi$
$293$	6658.98	1.32772	0.663860	+	0.747857i	$0.268917\pi$
$294$	0	0
$295$	2767.05	0.546115
$296$	−28247.6	−5.54682
$297$	0	0
$298$	−14639.5	−2.84578
$299$	−3130.44	−0.605479
$300$	0	0
$301$	0	0
$302$	−6223.62	−1.18586
$303$	0	0
$304$	4652.17	0.877699
$305$	1387.46	0.260478
$306$	0	0
$307$	−8118.87	−1.50934	−0.754672	−	0.656102i	$-0.772204\pi$
$307$	−8118.87	−1.50934	−0.754672	+	0.656102i	$0.772204\pi$
$308$	0	0
$309$	0	0
$310$	−5161.91	−0.945731
$311$	1021.03	0.186165	0.0930827	−	0.995658i	$-0.470328\pi$
$311$	1021.03	0.186165	0.0930827	+	0.995658i	$0.470328\pi$
$312$	0	0
$313$	6777.91	1.22399	0.611997	−	0.790860i	$-0.290366\pi$
$313$	6777.91	1.22399	0.611997	+	0.790860i	$0.290366\pi$
$314$	−301.332	−0.0541566
$315$	0	0
$316$	−21536.8	−3.83398
$317$	−4124.03	−0.730689	−0.365344	−	0.930872i	$-0.619049\pi$
$317$	−4124.03	−0.730689	−0.365344	+	0.930872i	$0.619049\pi$
$318$	0	0
$319$	−3493.59	−0.613178
$320$	−14797.0	−2.58492
$321$	0	0
$322$	0	0
$323$	1302.66	0.224402
$324$	0	0
$325$	1035.96	0.176814
$326$	8507.34	1.44533
$327$	0	0
$328$	40844.7	6.87582
$329$	0	0
$330$	0	0
$331$	55.8328	0.00927145	0.00463572	−	0.999989i	$-0.498524\pi$
$331$	55.8328	0.00927145	0.00463572	+	0.999989i	$0.498524\pi$
$332$	−212.209	−0.0350798
$333$	0	0
$334$	21944.0	3.59497
$335$	−4360.11	−0.711100
$336$	0	0
$337$	2005.49	0.324171	0.162086	−	0.986777i	$-0.448178\pi$
$337$	2005.49	0.324171	0.162086	+	0.986777i	$0.448178\pi$
$338$	2683.44	0.431834
$339$	0	0
$340$	−9491.42	−1.51396
$341$	−3849.08	−0.611259
$342$	0	0
$343$	0	0
$344$	1244.35	0.195032
$345$	0	0
$346$	−12985.5	−2.01764
$347$	−4253.50	−0.658040	−0.329020	−	0.944323i	$-0.606718\pi$
$347$	−4253.50	−0.658040	−0.329020	+	0.944323i	$0.606718\pi$
$348$	0	0
$349$	−4012.38	−0.615408	−0.307704	−	0.951482i	$-0.599561\pi$
$349$	−4012.38	−0.615408	−0.307704	+	0.951482i	$0.599561\pi$
$350$	0	0
$351$	0	0
$352$	−19723.2	−2.98651
$353$	606.778	0.0914888	0.0457444	−	0.998953i	$-0.485434\pi$
$353$	606.778	0.0914888	0.0457444	+	0.998953i	$0.485434\pi$
$354$	0	0
$355$	1458.96	0.218123
$356$	2925.11	0.435479
$357$	0	0
$358$	−3943.47	−0.582175
$359$	9736.33	1.43138	0.715688	−	0.698421i	$-0.246113\pi$
$359$	9736.33	1.43138	0.715688	+	0.698421i	$0.246113\pi$
$360$	0	0
$361$	−6604.01	−0.962823
$362$	15343.0	2.22766
$363$	0	0
$364$	0	0
$365$	5699.04	0.817264
$366$	0	0
$367$	−8572.43	−1.21928	−0.609642	−	0.792677i	$-0.708687\pi$
$367$	−8572.43	−1.21928	−0.609642	+	0.792677i	$0.708687\pi$
$368$	−22008.8	−3.11762
$369$	0	0
$370$	9249.37	1.29960
$371$	0	0
$372$	0	0
$373$	−4640.81	−0.644215	−0.322108	−	0.946703i	$-0.604391\pi$
$373$	−4640.81	−0.644215	−0.322108	+	0.946703i	$0.604391\pi$
$374$	−9510.63	−1.31493
$375$	0	0
$376$	−33048.6	−4.53286
$377$	−6943.73	−0.948594
$378$	0	0
$379$	−11739.0	−1.59101	−0.795504	−	0.605948i	$-0.792794\pi$
$379$	−11739.0	−1.59101	−0.795504	+	0.605948i	$0.792794\pi$
$380$	−1857.94	−0.250817
$381$	0	0
$382$	3139.66	0.420521
$383$	−1010.96	−0.134877	−0.0674384	−	0.997723i	$-0.521483\pi$
$383$	−1010.96	−0.134877	−0.0674384	+	0.997723i	$0.521483\pi$
$384$	0	0
$385$	0	0
$386$	16809.0	2.21646
$387$	0	0
$388$	21705.9	2.84008
$389$	5219.67	0.680328	0.340164	−	0.940366i	$-0.389517\pi$
$389$	5219.67	0.680328	0.340164	+	0.940366i	$0.389517\pi$
$390$	0	0
$391$	−6162.67	−0.797084
$392$	0	0
$393$	0	0
$394$	−17626.5	−2.25384
$395$	4627.58	0.589465
$396$	0	0
$397$	−5388.03	−0.681153	−0.340576	−	0.940217i	$-0.610622\pi$
$397$	−5388.03	−0.681153	−0.340576	+	0.940217i	$0.610622\pi$
$398$	1023.23	0.128870
$399$	0	0
$400$	7283.34	0.910418
$401$	−2475.91	−0.308332	−0.154166	−	0.988045i	$-0.549269\pi$
$401$	−2475.91	−0.308332	−0.154166	+	0.988045i	$0.549269\pi$
$402$	0	0
$403$	−7650.28	−0.945626
$404$	32190.8	3.96423
$405$	0	0
$406$	0	0
$407$	6896.98	0.839977
$408$	0	0
$409$	10337.3	1.24974	0.624871	−	0.780728i	$-0.285152\pi$
$409$	10337.3	1.24974	0.624871	+	0.780728i	$0.285152\pi$
$410$	−13374.1	−1.61098
$411$	0	0
$412$	−3855.55	−0.461042
$413$	0	0
$414$	0	0
$415$	45.5971	0.00539343
$416$	−39201.0	−4.62017
$417$	0	0
$418$	−1861.70	−0.217844
$419$	9394.06	1.09530	0.547649	−	0.836708i	$-0.315523\pi$
$419$	9394.06	1.09530	0.547649	+	0.836708i	$0.315523\pi$
$420$	0	0
$421$	−4841.02	−0.560421	−0.280210	−	0.959939i	$-0.590404\pi$
$421$	−4841.02	−0.560421	−0.280210	+	0.959939i	$0.590404\pi$
$422$	−11378.6	−1.31256
$423$	0	0
$424$	−5651.86	−0.647355
$425$	2039.41	0.232767
$426$	0	0
$427$	0	0
$428$	−20527.2	−2.31827
$429$	0	0
$430$	−407.449	−0.0456952
$431$	−15394.9	−1.72053	−0.860265	−	0.509848i	$-0.829702\pi$
$431$	−15394.9	−1.72053	−0.860265	+	0.509848i	$0.829702\pi$
$432$	0	0
$433$	10103.9	1.12139	0.560694	−	0.828023i	$-0.310534\pi$
$433$	10103.9	1.12139	0.560694	+	0.828023i	$0.310534\pi$
$434$	0	0
$435$	0	0
$436$	−28017.0	−3.07746
$437$	−1206.34	−0.132053
$438$	0	0
$439$	−2718.17	−0.295515	−0.147758	−	0.989024i	$-0.547206\pi$
$439$	−2718.17	−0.295515	−0.147758	+	0.989024i	$0.547206\pi$
$440$	8901.32	0.964441
$441$	0	0
$442$	−18903.0	−2.03421
$443$	14688.4	1.57532	0.787661	−	0.616109i	$-0.211292\pi$
$443$	14688.4	1.57532	0.787661	+	0.616109i	$0.211292\pi$
$444$	0	0
$445$	−628.515	−0.0669538
$446$	6278.74	0.666607
$447$	0	0
$448$	0	0
$449$	−18360.9	−1.92985	−0.964925	−	0.262525i	$-0.915445\pi$
$449$	−18360.9	−1.92985	−0.964925	+	0.262525i	$0.915445\pi$
$450$	0	0
$451$	−9972.69	−1.04123
$452$	8174.99	0.850706
$453$	0	0
$454$	−9903.54	−1.02378
$455$	0	0
$456$	0	0
$457$	−16984.7	−1.73854	−0.869269	−	0.494340i	$-0.835410\pi$
$457$	−16984.7	−1.73854	−0.869269	+	0.494340i	$0.835410\pi$
$458$	−19046.0	−1.94315
$459$	0	0
$460$	8789.65	0.890912
$461$	253.996	0.0256611	0.0128306	−	0.999918i	$-0.495916\pi$
$461$	253.996	0.0256611	0.0128306	+	0.999918i	$0.495916\pi$
$462$	0	0
$463$	5216.53	0.523613	0.261806	−	0.965120i	$-0.415682\pi$
$463$	5216.53	0.523613	0.261806	+	0.965120i	$0.415682\pi$
$464$	−48818.3	−4.88434
$465$	0	0
$466$	12533.6	1.24594
$467$	4487.27	0.444638	0.222319	−	0.974974i	$-0.428637\pi$
$467$	4487.27	0.444638	0.222319	+	0.974974i	$0.428637\pi$
$468$	0	0
$469$	0	0
$470$	10821.4	1.06203
$471$	0	0
$472$	47255.3	4.60827
$473$	−303.823	−0.0295344
$474$	0	0
$475$	399.214	0.0385625
$476$	0	0
$477$	0	0
$478$	6876.77	0.658025
$479$	4840.19	0.461699	0.230850	−	0.972989i	$-0.425849\pi$
$479$	4840.19	0.461699	0.230850	+	0.972989i	$0.425849\pi$
$480$	0	0
$481$	13708.2	1.29946
$482$	−21656.3	−2.04651
$483$	0	0
$484$	−20857.6	−1.95883
$485$	−4663.93	−0.436656
$486$	0	0
$487$	−7743.50	−0.720517	−0.360258	−	0.932853i	$-0.617311\pi$
$487$	−7743.50	−0.720517	−0.360258	+	0.932853i	$0.617311\pi$
$488$	23694.9	2.19798
$489$	0	0
$490$	0	0
$491$	−9079.20	−0.834498	−0.417249	−	0.908792i	$-0.637006\pi$
$491$	−9079.20	−0.834498	−0.417249	+	0.908792i	$0.637006\pi$
$492$	0	0
$493$	−13669.6	−1.24878
$494$	−3700.24	−0.337008
$495$	0	0
$496$	−53785.7	−4.86905
$497$	0	0
$498$	0	0
$499$	799.489	0.0717235	0.0358618	−	0.999357i	$-0.488582\pi$
$499$	799.489	0.0717235	0.0358618	+	0.999357i	$0.488582\pi$
$500$	−2908.75	−0.260167
$501$	0	0
$502$	21715.5	1.93070
$503$	−20124.1	−1.78388	−0.891938	−	0.452158i	$-0.850654\pi$
$503$	−20124.1	−1.78388	−0.891938	+	0.452158i	$0.850654\pi$
$504$	0	0
$505$	−6916.79	−0.609491
$506$	8807.44	0.773791
$507$	0	0
$508$	−25851.1	−2.25779
$509$	−20475.1	−1.78299	−0.891497	−	0.453026i	$-0.850344\pi$
$509$	−20475.1	−1.78299	−0.891497	+	0.453026i	$0.850344\pi$
$510$	0	0
$511$	0	0
$512$	−76590.8	−6.61107
$513$	0	0
$514$	−20904.2	−1.79386
$515$	828.437	0.0708841
$516$	0	0
$517$	8069.20	0.686428
$518$	0	0
$519$	0	0
$520$	17691.9	1.49200
$521$	−10565.3	−0.888433	−0.444217	−	0.895919i	$-0.646518\pi$
$521$	−10565.3	−0.888433	−0.444217	+	0.895919i	$0.646518\pi$
$522$	0	0
$523$	18994.8	1.58812	0.794059	−	0.607840i	$-0.207964\pi$
$523$	18994.8	1.58812	0.794059	+	0.607840i	$0.207964\pi$
$524$	10790.0	0.899550
$525$	0	0
$526$	−2428.60	−0.201316
$527$	−15060.5	−1.24487
$528$	0	0
$529$	−6459.98	−0.530943
$530$	1850.64	0.151673
$531$	0	0
$532$	0	0
$533$	−19821.3	−1.61080
$534$	0	0
$535$	4410.65	0.356428
$536$	−74461.4	−6.00046
$537$	0	0
$538$	−13503.6	−1.08213
$539$	0	0
$540$	0	0
$541$	4744.42	0.377039	0.188520	−	0.982069i	$-0.439631\pi$
$541$	4744.42	0.377039	0.188520	+	0.982069i	$0.439631\pi$
$542$	3513.27	0.278428
$543$	0	0
$544$	−77172.3	−6.08223
$545$	6019.98	0.473152
$546$	0	0
$547$	−19922.0	−1.55723	−0.778614	−	0.627504i	$-0.784077\pi$
$547$	−19922.0	−1.55723	−0.778614	+	0.627504i	$0.784077\pi$
$548$	−4574.92	−0.356625
$549$	0	0
$550$	−2914.64	−0.225965
$551$	−2675.82	−0.206885
$552$	0	0
$553$	0	0
$554$	12369.2	0.948586
$555$	0	0
$556$	57171.1	4.36079
$557$	9535.10	0.725342	0.362671	−	0.931917i	$-0.381865\pi$
$557$	9535.10	0.725342	0.362671	+	0.931917i	$0.381865\pi$
$558$	0	0
$559$	−603.866	−0.0456902
$560$	0	0
$561$	0	0
$562$	13221.1	0.992344
$563$	−3679.15	−0.275413	−0.137707	−	0.990473i	$-0.543973\pi$
$563$	−3679.15	−0.275413	−0.137707	+	0.990473i	$0.543973\pi$
$564$	0	0
$565$	−1756.55	−0.130794
$566$	−10198.8	−0.757396
$567$	0	0
$568$	24915.9	1.84058
$569$	−1871.83	−0.137911	−0.0689554	−	0.997620i	$-0.521967\pi$
$569$	−1871.83	−0.137911	−0.0689554	+	0.997620i	$0.521967\pi$
$570$	0	0
$571$	7805.45	0.572063	0.286032	−	0.958220i	$-0.407664\pi$
$571$	7805.45	0.572063	0.286032	+	0.958220i	$0.407664\pi$
$572$	20103.8	1.46955
$573$	0	0
$574$	0	0
$575$	−1888.62	−0.136976
$576$	0	0
$577$	−9017.94	−0.650644	−0.325322	−	0.945603i	$-0.605473\pi$
$577$	−9017.94	−0.650644	−0.325322	+	0.945603i	$0.605473\pi$
$578$	−9739.57	−0.700887
$579$	0	0
$580$	19496.6	1.39578
$581$	0	0
$582$	0	0
$583$	1379.97	0.0980314
$584$	97327.5	6.89630
$585$	0	0
$586$	−37236.8	−2.62498
$587$	−7651.07	−0.537979	−0.268989	−	0.963143i	$-0.586690\pi$
$587$	−7651.07	−0.537979	−0.268989	+	0.963143i	$0.586690\pi$
$588$	0	0
$589$	−2948.09	−0.206238
$590$	−15473.2	−1.07970
$591$	0	0
$592$	96376.0	6.69093
$593$	−13181.8	−0.912839	−0.456420	−	0.889765i	$-0.650868\pi$
$593$	−13181.8	−0.912839	−0.456420	+	0.889765i	$0.650868\pi$
$594$	0	0
$595$	0	0
$596$	60919.8	4.18687
$597$	0	0
$598$	17505.3	1.19707
$599$	−1961.72	−0.133812	−0.0669061	−	0.997759i	$-0.521313\pi$
$599$	−1961.72	−0.133812	−0.0669061	+	0.997759i	$0.521313\pi$
$600$	0	0
$601$	−20692.3	−1.40442	−0.702210	−	0.711970i	$-0.747803\pi$
$601$	−20692.3	−1.40442	−0.702210	+	0.711970i	$0.747803\pi$
$602$	0	0
$603$	0	0
$604$	25898.6	1.74470
$605$	4481.64	0.301165
$606$	0	0
$607$	26491.9	1.77146	0.885728	−	0.464204i	$-0.153660\pi$
$607$	26491.9	1.77146	0.885728	+	0.464204i	$0.153660\pi$
$608$	−15106.4	−1.00764
$609$	0	0
$610$	−7758.62	−0.514979
$611$	16038.0	1.06191
$612$	0	0
$613$	2376.25	0.156567	0.0782837	−	0.996931i	$-0.475056\pi$
$613$	2376.25	0.156567	0.0782837	+	0.996931i	$0.475056\pi$
$614$	45400.4	2.98406
$615$	0	0
$616$	0	0
$617$	−188.499	−0.0122993	−0.00614966	−	0.999981i	$-0.501958\pi$
$617$	−188.499	−0.0122993	−0.00614966	+	0.999981i	$0.501958\pi$
$618$	0	0
$619$	−7405.37	−0.480851	−0.240426	−	0.970668i	$-0.577287\pi$
$619$	−7405.37	−0.480851	−0.240426	+	0.970668i	$0.577287\pi$
$620$	21480.4	1.39141
$621$	0	0
$622$	−5709.57	−0.368060
$623$	0	0
$624$	0	0
$625$	625.000	0.0400000
$626$	−37901.8	−2.41991
$627$	0	0
$628$	1253.95	0.0796781
$629$	26986.3	1.71067
$630$	0	0
$631$	−22413.1	−1.41402	−0.707012	−	0.707201i	$-0.749958\pi$
$631$	−22413.1	−1.41402	−0.707012	+	0.707201i	$0.749958\pi$
$632$	79029.2	4.97407
$633$	0	0
$634$	23061.4	1.44461
$635$	5554.59	0.347130
$636$	0	0
$637$	0	0
$638$	19536.0	1.21229
$639$	0	0
$640$	44903.5	2.77339
$641$	23999.4	1.47881	0.739407	−	0.673259i	$-0.235106\pi$
$641$	23999.4	1.47881	0.739407	+	0.673259i	$0.235106\pi$
$642$	0	0
$643$	−5365.32	−0.329063	−0.164531	−	0.986372i	$-0.552611\pi$
$643$	−5365.32	−0.329063	−0.164531	+	0.986372i	$0.552611\pi$
$644$	0	0
$645$	0	0
$646$	−7284.40	−0.443655
$647$	19079.6	1.15934	0.579672	−	0.814850i	$-0.303181\pi$
$647$	19079.6	1.15934	0.579672	+	0.814850i	$0.303181\pi$
$648$	0	0
$649$	−11537.9	−0.697847
$650$	−5793.02	−0.349571
$651$	0	0
$652$	−35401.9	−2.12645
$653$	−5131.22	−0.307504	−0.153752	−	0.988109i	$-0.549136\pi$
$653$	−5131.22	−0.307504	−0.153752	+	0.988109i	$0.549136\pi$
$654$	0	0
$655$	−2318.44	−0.138304
$656$	−139355.	−8.29405
$657$	0	0
$658$	0	0
$659$	15072.9	0.890984	0.445492	−	0.895286i	$-0.353029\pi$
$659$	15072.9	0.890984	0.445492	+	0.895286i	$0.353029\pi$
$660$	0	0
$661$	−25784.6	−1.51725	−0.758626	−	0.651526i	$-0.774129\pi$
$661$	−25784.6	−1.51725	−0.758626	+	0.651526i	$0.774129\pi$
$662$	−312.215	−0.0183302
$663$	0	0
$664$	778.702	0.0455113
$665$	0	0
$666$	0	0
$667$	12658.9	0.734865
$668$	−91316.3	−5.28912
$669$	0	0
$670$	24381.6	1.40588
$671$	−5785.37	−0.332849
$672$	0	0
$673$	−11350.3	−0.650104	−0.325052	−	0.945696i	$-0.605382\pi$
$673$	−11350.3	−0.650104	−0.325052	+	0.945696i	$0.605382\pi$
$674$	−11214.6	−0.640905
$675$	0	0
$676$	−11166.7	−0.635338
$677$	26770.2	1.51974	0.759869	−	0.650076i	$-0.225263\pi$
$677$	26770.2	1.51974	0.759869	+	0.650076i	$0.225263\pi$
$678$	0	0
$679$	0	0
$680$	34828.8	1.96415
$681$	0	0
$682$	21523.9	1.20849
$683$	5861.79	0.328397	0.164199	−	0.986427i	$-0.447496\pi$
$683$	5861.79	0.328397	0.164199	+	0.986427i	$0.447496\pi$
$684$	0	0
$685$	983.006	0.0548303
$686$	0	0
$687$	0	0
$688$	−4245.51	−0.235260
$689$	2742.76	0.151656
$690$	0	0
$691$	14734.8	0.811199	0.405599	−	0.914051i	$-0.367063\pi$
$691$	14734.8	0.811199	0.405599	+	0.914051i	$0.367063\pi$
$692$	54037.0	2.96846
$693$	0	0
$694$	23785.4	1.30098
$695$	−12284.3	−0.670460
$696$	0	0
$697$	−39020.8	−2.12054
$698$	22437.0	1.21670
$699$	0	0
$700$	0	0
$701$	−14998.3	−0.808101	−0.404051	−	0.914737i	$-0.632398\pi$
$701$	−14998.3	−0.808101	−0.404051	+	0.914737i	$0.632398\pi$
$702$	0	0
$703$	5282.55	0.283407
$704$	61699.7	3.30312
$705$	0	0
$706$	−3393.08	−0.180878
$707$	0	0
$708$	0	0
$709$	2967.62	0.157195	0.0785975	−	0.996906i	$-0.474956\pi$
$709$	2967.62	0.157195	0.0785975	+	0.996906i	$0.474956\pi$
$710$	−8158.45	−0.431241
$711$	0	0
$712$	−10733.7	−0.564975
$713$	13947.0	0.732566
$714$	0	0
$715$	−4319.68	−0.225940
$716$	16410.1	0.856528
$717$	0	0
$718$	−54445.2	−2.82991
$719$	−3902.35	−0.202410	−0.101205	−	0.994866i	$-0.532270\pi$
$719$	−3902.35	−0.202410	−0.101205	+	0.994866i	$0.532270\pi$
$720$	0	0
$721$	0	0
$722$	36929.3	1.90356
$723$	0	0
$724$	−63847.5	−3.27745
$725$	−4189.21	−0.214597
$726$	0	0
$727$	−3617.87	−0.184566	−0.0922830	−	0.995733i	$-0.529416\pi$
$727$	−3617.87	−0.184566	−0.0922830	+	0.995733i	$0.529416\pi$
$728$	0	0
$729$	0	0
$730$	−31868.8	−1.61578
$731$	−1188.79	−0.0601489
$732$	0	0
$733$	7471.28	0.376477	0.188239	−	0.982123i	$-0.439722\pi$
$733$	7471.28	0.376477	0.188239	+	0.982123i	$0.439722\pi$
$734$	47936.7	2.41059
$735$	0	0
$736$	71466.3	3.57919
$737$	18180.6	0.908672
$738$	0	0
$739$	−33974.7	−1.69118	−0.845589	−	0.533835i	$-0.820750\pi$
$739$	−33974.7	−1.69118	−0.845589	+	0.533835i	$0.820750\pi$
$740$	−38489.7	−1.91204
$741$	0	0
$742$	0	0
$743$	−23040.7	−1.13766	−0.568829	−	0.822456i	$-0.692603\pi$
$743$	−23040.7	−1.13766	−0.568829	+	0.822456i	$0.692603\pi$
$744$	0	0
$745$	−13089.8	−0.643721
$746$	25951.2	1.27365
$747$	0	0
$748$	39576.9	1.93459
$749$	0	0
$750$	0	0
$751$	−25925.5	−1.25970	−0.629850	−	0.776717i	$-0.716884\pi$
$751$	−25925.5	−1.25970	−0.629850	+	0.776717i	$0.716884\pi$
$752$	112756.	5.46782
$753$	0	0
$754$	38829.0	1.87543
$755$	−5564.79	−0.268243
$756$	0	0
$757$	−15626.4	−0.750266	−0.375133	−	0.926971i	$-0.622403\pi$
$757$	−15626.4	−0.750266	−0.375133	+	0.926971i	$0.622403\pi$
$758$	65644.1	3.14551
$759$	0	0
$760$	6817.72	0.325401
$761$	−19568.1	−0.932120	−0.466060	−	0.884753i	$-0.654327\pi$
$761$	−19568.1	−0.932120	−0.466060	+	0.884753i	$0.654327\pi$
$762$	0	0
$763$	0	0
$764$	−13065.2	−0.618693
$765$	0	0
$766$	5653.26	0.266659
$767$	−22932.3	−1.07958
$768$	0	0
$769$	−18078.1	−0.847742	−0.423871	−	0.905722i	$-0.639329\pi$
$769$	−18078.1	−0.847742	−0.423871	+	0.905722i	$0.639329\pi$
$770$	0	0
$771$	0	0
$772$	−69947.8	−3.26098
$773$	25350.9	1.17957	0.589785	−	0.807560i	$-0.299212\pi$
$773$	25350.9	1.17957	0.589785	+	0.807560i	$0.299212\pi$
$774$	0	0
$775$	−4615.47	−0.213926
$776$	−79650.0	−3.68462
$777$	0	0
$778$	−29188.2	−1.34505
$779$	−7638.30	−0.351310
$780$	0	0
$781$	−6083.51	−0.278726
$782$	34461.4	1.57588
$783$	0	0
$784$	0	0
$785$	−269.434	−0.0122503
$786$	0	0
$787$	−12366.7	−0.560135	−0.280068	−	0.959980i	$-0.590357\pi$
$787$	−12366.7	−0.560135	−0.280068	+	0.959980i	$0.590357\pi$
$788$	73350.0	3.31597
$789$	0	0
$790$	−25877.2	−1.16541
$791$	0	0
$792$	0	0
$793$	−11498.8	−0.514922
$794$	30129.7	1.34668
$795$	0	0
$796$	−4258.02	−0.189600
$797$	−17652.2	−0.784532	−0.392266	−	0.919852i	$-0.628309\pi$
$797$	−17652.2	−0.784532	−0.392266	+	0.919852i	$0.628309\pi$
$798$	0	0
$799$	31572.9	1.39796
$800$	−23650.3	−1.04521
$801$	0	0
$802$	13845.2	0.609589
$803$	−23763.6	−1.04433
$804$	0	0
$805$	0	0
$806$	42780.0	1.86956
$807$	0	0
$808$	−118124.	−5.14305
$809$	−7857.06	−0.341458	−0.170729	−	0.985318i	$-0.554612\pi$
$809$	−7857.06	−0.341458	−0.170729	+	0.985318i	$0.554612\pi$
$810$	0	0
$811$	23772.6	1.02931	0.514654	−	0.857398i	$-0.327921\pi$
$811$	23772.6	1.02931	0.514654	+	0.857398i	$0.327921\pi$
$812$	0	0
$813$	0	0
$814$	−38567.6	−1.66068
$815$	7606.76	0.326936
$816$	0	0
$817$	−232.704	−0.00996487
$818$	−57805.5	−2.47081
$819$	0	0
$820$	55654.2	2.37016
$821$	−17649.2	−0.750256	−0.375128	−	0.926973i	$-0.622401\pi$
$821$	−17649.2	−0.750256	−0.375128	+	0.926973i	$0.622401\pi$
$822$	0	0
$823$	−6119.28	−0.259180	−0.129590	−	0.991568i	$-0.541366\pi$
$823$	−6119.28	−0.259180	−0.129590	+	0.991568i	$0.541366\pi$
$824$	14147.9	0.598140
$825$	0	0
$826$	0	0
$827$	−27079.4	−1.13863	−0.569313	−	0.822121i	$-0.692791\pi$
$827$	−27079.4	−1.13863	−0.569313	+	0.822121i	$0.692791\pi$
$828$	0	0
$829$	21413.5	0.897131	0.448565	−	0.893750i	$-0.351935\pi$
$829$	21413.5	0.897131	0.448565	+	0.893750i	$0.351935\pi$
$830$	−254.977	−0.0106631
$831$	0	0
$832$	122632.	5.10997
$833$	0	0
$834$	0	0
$835$	19621.0	0.813189
$836$	7747.16	0.320504
$837$	0	0
$838$	−52531.2	−2.16547
$839$	16640.8	0.684747	0.342374	−	0.939564i	$-0.388769\pi$
$839$	16640.8	0.684747	0.342374	+	0.939564i	$0.388769\pi$
$840$	0	0
$841$	3690.11	0.151302
$842$	27070.8	1.10798
$843$	0	0
$844$	47350.1	1.93111
$845$	2399.37	0.0976816
$846$	0	0
$847$	0	0
$848$	19283.2	0.780880
$849$	0	0
$850$	−11404.3	−0.460193
$851$	−24990.9	−1.00667
$852$	0	0
$853$	−23477.1	−0.942368	−0.471184	−	0.882035i	$-0.656173\pi$
$853$	−23477.1	−0.942368	−0.471184	+	0.882035i	$0.656173\pi$
$854$	0	0
$855$	0	0
$856$	75324.6	3.00764
$857$	−26518.4	−1.05700	−0.528501	−	0.848933i	$-0.677246\pi$
$857$	−26518.4	−1.05700	−0.528501	+	0.848933i	$0.677246\pi$
$858$	0	0
$859$	26695.3	1.06034	0.530169	−	0.847892i	$-0.322128\pi$
$859$	26695.3	1.06034	0.530169	+	0.847892i	$0.322128\pi$
$860$	1695.53	0.0672293
$861$	0	0
$862$	86087.9	3.40158
$863$	−48714.8	−1.92152	−0.960759	−	0.277383i	$-0.910533\pi$
$863$	−48714.8	−1.92152	−0.960759	+	0.277383i	$0.910533\pi$
$864$	0	0
$865$	−11610.9	−0.456394
$866$	−56500.4	−2.21705
$867$	0	0
$868$	0	0
$869$	−19295.9	−0.753243
$870$	0	0
$871$	36135.1	1.40573
$872$	102809.	3.99259
$873$	0	0
$874$	6745.81	0.261076
$875$	0	0
$876$	0	0
$877$	−7810.79	−0.300743	−0.150372	−	0.988630i	$-0.548047\pi$
$877$	−7810.79	−0.300743	−0.150372	+	0.988630i	$0.548047\pi$
$878$	15199.9	0.584250
$879$	0	0
$880$	−30369.8	−1.16337
$881$	15185.1	0.580704	0.290352	−	0.956920i	$-0.406228\pi$
$881$	15185.1	0.580704	0.290352	+	0.956920i	$0.406228\pi$
$882$	0	0
$883$	−30595.3	−1.16604	−0.583020	−	0.812458i	$-0.698129\pi$
$883$	−30595.3	−1.16604	−0.583020	+	0.812458i	$0.698129\pi$
$884$	78661.5	2.99284
$885$	0	0
$886$	−82137.0	−3.11450
$887$	−44013.4	−1.66609	−0.833047	−	0.553202i	$-0.813406\pi$
$887$	−44013.4	−1.66609	−0.833047	+	0.553202i	$0.813406\pi$
$888$	0	0
$889$	0	0
$890$	3514.63	0.132371
$891$	0	0
$892$	−26127.9	−0.980749
$893$	6180.38	0.231600
$894$	0	0
$895$	−3526.01	−0.131689
$896$	0	0
$897$	0	0
$898$	102673.	3.81542
$899$	30936.3	1.14770
$900$	0	0
$901$	5399.48	0.199648
$902$	55766.9	2.05857
$903$	0	0
$904$	−29998.1	−1.10368
$905$	13718.8	0.503900
$906$	0	0
$907$	−18691.3	−0.684272	−0.342136	−	0.939651i	$-0.611150\pi$
$907$	−18691.3	−0.684272	−0.342136	+	0.939651i	$0.611150\pi$
$908$	41212.0	1.50624
$909$	0	0
$910$	0	0
$911$	31089.7	1.13068	0.565340	−	0.824858i	$-0.308745\pi$
$911$	31089.7	1.13068	0.565340	+	0.824858i	$0.308745\pi$
$912$	0	0
$913$	−190.129	−0.00689195
$914$	94977.9	3.43719
$915$	0	0
$916$	79257.0	2.85887
$917$	0	0
$918$	0	0
$919$	16771.7	0.602009	0.301005	−	0.953623i	$-0.402678\pi$
$919$	16771.7	0.602009	0.301005	+	0.953623i	$0.402678\pi$
$920$	−32253.6	−1.15584
$921$	0	0
$922$	−1420.34	−0.0507335
$923$	−12091.3	−0.431193
$924$	0	0
$925$	8270.24	0.293972
$926$	−29170.6	−1.03521
$927$	0	0
$928$	158522.	5.60746
$929$	31460.2	1.11106	0.555531	−	0.831496i	$-0.312515\pi$
$929$	31460.2	1.11106	0.555531	+	0.831496i	$0.312515\pi$
$930$	0	0
$931$	0	0
$932$	−52156.5	−1.83309
$933$	0	0
$934$	−25092.6	−0.879075
$935$	−8503.84	−0.297439
$936$	0	0
$937$	−29077.8	−1.01380	−0.506900	−	0.862005i	$-0.669209\pi$
$937$	−29077.8	−1.01380	−0.506900	+	0.862005i	$0.669209\pi$
$938$	0	0
$939$	0	0
$940$	−45031.5	−1.56252
$941$	10193.5	0.353134	0.176567	−	0.984289i	$-0.443501\pi$
$941$	10193.5	0.353134	0.176567	+	0.984289i	$0.443501\pi$
$942$	0	0
$943$	36135.7	1.24787
$944$	−161227.	−5.55878
$945$	0	0
$946$	1698.96	0.0583912
$947$	−36207.0	−1.24242	−0.621208	−	0.783646i	$-0.713358\pi$
$947$	−36207.0	−1.24242	−0.621208	+	0.783646i	$0.713358\pi$
$948$	0	0
$949$	−47231.6	−1.61560
$950$	−2232.39	−0.0762402
$951$	0	0
$952$	0	0
$953$	−15353.0	−0.521861	−0.260930	−	0.965358i	$-0.584029\pi$
$953$	−15353.0	−0.521861	−0.260930	+	0.965358i	$0.584029\pi$
$954$	0	0
$955$	2807.30	0.0951225
$956$	−28616.5	−0.968122
$957$	0	0
$958$	−27066.2	−0.912806
$959$	0	0
$960$	0	0
$961$	4293.13	0.144108
$962$	−76655.5	−2.56910
$963$	0	0
$964$	90119.2	3.01094
$965$	15029.6	0.501368
$966$	0	0
$967$	27658.2	0.919779	0.459890	−	0.887976i	$-0.347889\pi$
$967$	27658.2	0.919779	0.459890	+	0.887976i	$0.347889\pi$
$968$	76536.9	2.54131
$969$	0	0
$970$	26080.5	0.863293
$971$	33071.2	1.09300	0.546502	−	0.837458i	$-0.315959\pi$
$971$	33071.2	1.09300	0.546502	+	0.837458i	$0.315959\pi$
$972$	0	0
$973$	0	0
$974$	43301.4	1.42450
$975$	0	0
$976$	−80842.8	−2.65135
$977$	48650.1	1.59310	0.796548	−	0.604575i	$-0.206657\pi$
$977$	48650.1	1.59310	0.796548	+	0.604575i	$0.206657\pi$
$978$	0	0
$979$	2620.75	0.0855563
$980$	0	0
$981$	0	0
$982$	50770.5	1.64985
$983$	22824.8	0.740588	0.370294	−	0.928915i	$-0.379257\pi$
$983$	22824.8	0.740588	0.370294	+	0.928915i	$0.379257\pi$
$984$	0	0
$985$	−15760.6	−0.509822
$986$	76439.9	2.46891
$987$	0	0
$988$	15398.0	0.495824
$989$	1100.89	0.0353956
$990$	0	0
$991$	−7290.32	−0.233688	−0.116844	−	0.993150i	$-0.537278\pi$
$991$	−7290.32	−0.233688	−0.116844	+	0.993150i	$0.537278\pi$
$992$	174652.	5.58992
$993$	0	0
$994$	0	0
$995$	914.915	0.0291505
$996$	0	0
$997$	−3791.84	−0.120450	−0.0602251	−	0.998185i	$-0.519182\pi$
$997$	−3791.84	−0.120450	−0.0602251	+	0.998185i	$0.519182\pi$
$998$	−4470.71	−0.141801
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.4.a.cb.1.1		8
3.2	odd	2		2205.4.a.cg.1.8		8
7.3	odd	6		315.4.j.j.226.8	yes	16
7.5	odd	6		315.4.j.j.46.8	yes	16
7.6	odd	2		2205.4.a.cc.1.1		8
21.5	even	6		315.4.j.i.46.1	✓	16
21.17	even	6		315.4.j.i.226.1	yes	16
21.20	even	2		2205.4.a.cf.1.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.4.j.i.46.1	✓	16	21.5	even	6
315.4.j.i.226.1	yes	16	21.17	even	6
315.4.j.j.46.8	yes	16	7.5	odd	6
315.4.j.j.226.8	yes	16	7.3	odd	6
2205.4.a.cb.1.1		8	1.1	even	1	trivial
2205.4.a.cc.1.1		8	7.6	odd	2
2205.4.a.cf.1.8		8	21.20	even	2
2205.4.a.cg.1.8		8	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)

\(f(q)\)	\(=\)	\(q-5.59196 q^{2} +23.2700 q^{4} -5.00000 q^{5} -85.3894 q^{8} +O(q^{10})\)	\(q-5.59196 q^{2} +23.2700 q^{4} -5.00000 q^{5} -85.3894 q^{8} +27.9598 q^{10} +20.8488 q^{11} +41.4382 q^{13} +291.334 q^{16} +81.5764 q^{17} +15.9685 q^{19} -116.350 q^{20} -116.586 q^{22} -75.5448 q^{23} +25.0000 q^{25} -231.721 q^{26} -167.568 q^{29} -184.619 q^{31} -946.012 q^{32} -456.172 q^{34} +330.810 q^{37} -89.2954 q^{38} +426.947 q^{40} -478.334 q^{41} -14.5727 q^{43} +485.152 q^{44} +422.444 q^{46} +387.035 q^{47} -139.799 q^{50} +964.268 q^{52} +66.1892 q^{53} -104.244 q^{55} +937.035 q^{58} -553.410 q^{59} -277.492 q^{61} +1032.38 q^{62} +2959.39 q^{64} -207.191 q^{65} +872.023 q^{67} +1898.28 q^{68} -291.792 q^{71} -1139.81 q^{73} -1849.87 q^{74} +371.588 q^{76} -925.516 q^{79} -1456.67 q^{80} +2674.83 q^{82} -9.11942 q^{83} -407.882 q^{85} +81.4899 q^{86} -1780.26 q^{88} +125.703 q^{89} -1757.93 q^{92} -2164.28 q^{94} -79.8427 q^{95} +932.786 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{2} + 42 q^{4} - 40 q^{5} - 48 q^{8}+O(q^{10}) \) 8 * q - 4 * q^2 + 42 * q^4 - 40 * q^5 - 48 * q^8	\( 8 q - 4 q^{2} + 42 q^{4} - 40 q^{5} - 48 q^{8} + 20 q^{10} - 100 q^{11} + 102 q^{13} + 266 q^{16} + 56 q^{17} - 210 q^{20} - 70 q^{22} - 190 q^{23} + 200 q^{25} + 60 q^{26} - 296 q^{29} - 42 q^{31} - 718 q^{32} + 488 q^{34} + 314 q^{37} - 514 q^{38} + 240 q^{40} - 28 q^{41} + 714 q^{43} - 410 q^{44} - 650 q^{46} + 326 q^{47} - 100 q^{50} + 794 q^{52} - 1282 q^{53} + 500 q^{55} + 942 q^{58} - 924 q^{59} + 536 q^{61} + 50 q^{62} + 1902 q^{64} - 510 q^{65} + 2 q^{67} + 2690 q^{68} - 1516 q^{71} - 86 q^{73} - 4754 q^{74} - 30 q^{76} - 42 q^{79} - 1330 q^{80} + 3602 q^{82} + 572 q^{83} - 280 q^{85} - 1758 q^{86} - 4312 q^{88} + 940 q^{89} - 3844 q^{92} - 2866 q^{94} + 1720 q^{97}+O(q^{100}) \) 8 * q - 4 * q^2 + 42 * q^4 - 40 * q^5 - 48 * q^8 + 20 * q^10 - 100 * q^11 + 102 * q^13 + 266 * q^16 + 56 * q^17 - 210 * q^20 - 70 * q^22 - 190 * q^23 + 200 * q^25 + 60 * q^26 - 296 * q^29 - 42 * q^31 - 718 * q^32 + 488 * q^34 + 314 * q^37 - 514 * q^38 + 240 * q^40 - 28 * q^41 + 714 * q^43 - 410 * q^44 - 650 * q^46 + 326 * q^47 - 100 * q^50 + 794 * q^52 - 1282 * q^53 + 500 * q^55 + 942 * q^58 - 924 * q^59 + 536 * q^61 + 50 * q^62 + 1902 * q^64 - 510 * q^65 + 2 * q^67 + 2690 * q^68 - 1516 * q^71 - 86 * q^73 - 4754 * q^74 - 30 * q^76 - 42 * q^79 - 1330 * q^80 + 3602 * q^82 + 572 * q^83 - 280 * q^85 - 1758 * q^86 - 4312 * q^88 + 940 * q^89 - 3844 * q^92 - 2866 * q^94 + 1720 * q^97

Introduction

L-functions

Modular forms

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