LMFDB - Embedded newform 2205.4.a.by.1.4

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:	$0$
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			1.4
Root			$0.525584$ 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)

$f(q)$	$=$	$q+0.525584 q^{2} -7.72376 q^{4} +5.00000 q^{5} -8.26415 q^{8} +O(q^{10})$	$q+0.525584 q^{2} -7.72376 q^{4} +5.00000 q^{5} -8.26415 q^{8} +2.62792 q^{10} +48.9415 q^{11} +40.1517 q^{13} +57.4466 q^{16} +59.1704 q^{17} +129.508 q^{19} -38.6188 q^{20} +25.7228 q^{22} +93.5609 q^{23} +25.0000 q^{25} +21.1031 q^{26} +117.184 q^{29} -162.945 q^{31} +96.3062 q^{32} +31.0990 q^{34} -3.72097 q^{37} +68.0673 q^{38} -41.3208 q^{40} -222.816 q^{41} -62.2753 q^{43} -378.012 q^{44} +49.1741 q^{46} -460.327 q^{47} +13.1396 q^{50} -310.122 q^{52} -492.977 q^{53} +244.707 q^{55} +61.5902 q^{58} +153.928 q^{59} +194.916 q^{61} -85.6412 q^{62} -408.956 q^{64} +200.759 q^{65} +667.084 q^{67} -457.018 q^{68} -256.941 q^{71} +1219.59 q^{73} -1.95568 q^{74} -1000.29 q^{76} +381.236 q^{79} +287.233 q^{80} -117.108 q^{82} -281.766 q^{83} +295.852 q^{85} -32.7309 q^{86} -404.460 q^{88} -926.242 q^{89} -722.642 q^{92} -241.940 q^{94} +647.540 q^{95} -634.683 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.525584	0.185822	0.0929109	−	0.995674i	$-0.470383\pi$
$2$	0.525584	0.185822	0.0929109	+	0.995674i	$0.470383\pi$
$3$	0	0
$3$	0	0
$4$	−7.72376	−0.965470
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−8.26415	−0.365227
$9$	0	0
$10$	2.62792	0.0831021
$11$	48.9415	1.34149	0.670746	−	0.741687i	$-0.265974\pi$
$11$	48.9415	1.34149	0.670746	+	0.741687i	$0.265974\pi$
$12$	0	0
$13$	40.1517	0.856622	0.428311	−	0.903631i	$-0.359109\pi$
$13$	40.1517	0.856622	0.428311	+	0.903631i	$0.359109\pi$
$14$	0	0
$15$	0	0
$16$	57.4466	0.897603
$17$	59.1704	0.844172	0.422086	−	0.906556i	$-0.361298\pi$
$17$	59.1704	0.844172	0.422086	+	0.906556i	$0.361298\pi$
$18$	0	0
$19$	129.508	1.56375	0.781873	−	0.623438i	$-0.214265\pi$
$19$	129.508	1.56375	0.781873	+	0.623438i	$0.214265\pi$
$20$	−38.6188	−0.431771
$21$	0	0
$22$	25.7228	0.249278
$23$	93.5609	0.848209	0.424104	−	0.905613i	$-0.360589\pi$
$23$	93.5609	0.848209	0.424104	+	0.905613i	$0.360589\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	21.1031	0.159179
$27$	0	0
$28$	0	0
$29$	117.184	0.750365	0.375183	−	0.926951i	$-0.377580\pi$
$29$	117.184	0.750365	0.375183	+	0.926951i	$0.377580\pi$
$30$	0	0
$31$	−162.945	−0.944057	−0.472029	−	0.881583i	$-0.656478\pi$
$31$	−162.945	−0.944057	−0.472029	+	0.881583i	$0.656478\pi$
$32$	96.3062	0.532022
$33$	0	0
$34$	31.0990	0.156866
$35$	0	0
$36$	0	0
$37$	−3.72097	−0.0165331	−0.00826654	−	0.999966i	$-0.502631\pi$
$37$	−3.72097	−0.0165331	−0.00826654	+	0.999966i	$0.502631\pi$
$38$	68.0673	0.290578
$39$	0	0
$40$	−41.3208	−0.163335
$41$	−222.816	−0.848732	−0.424366	−	0.905491i	$-0.639503\pi$
$41$	−222.816	−0.848732	−0.424366	+	0.905491i	$0.639503\pi$
$42$	0	0
$43$	−62.2753	−0.220858	−0.110429	−	0.993884i	$-0.535222\pi$
$43$	−62.2753	−0.220858	−0.110429	+	0.993884i	$0.535222\pi$
$44$	−378.012	−1.29517
$45$	0	0
$46$	49.1741	0.157616
$47$	−460.327	−1.42863	−0.714315	−	0.699824i	$-0.753262\pi$
$47$	−460.327	−1.42863	−0.714315	+	0.699824i	$0.753262\pi$
$48$	0	0
$49$	0	0
$50$	13.1396	0.0371644
$51$	0	0
$52$	−310.122	−0.827043
$53$	−492.977	−1.27765	−0.638826	−	0.769351i	$-0.720580\pi$
$53$	−492.977	−1.27765	−0.638826	+	0.769351i	$0.720580\pi$
$54$	0	0
$55$	244.707	0.599933
$56$	0	0
$57$	0	0
$58$	61.5902	0.139434
$59$	153.928	0.339657	0.169829	−	0.985474i	$-0.445679\pi$
$59$	153.928	0.339657	0.169829	+	0.985474i	$0.445679\pi$
$60$	0	0
$61$	194.916	0.409122	0.204561	−	0.978854i	$-0.434423\pi$
$61$	194.916	0.409122	0.204561	+	0.978854i	$0.434423\pi$
$62$	−85.6412	−0.175427
$63$	0	0
$64$	−408.956	−0.798742
$65$	200.759	0.383093
$66$	0	0
$67$	667.084	1.21638	0.608189	−	0.793793i	$-0.291896\pi$
$67$	667.084	1.21638	0.608189	+	0.793793i	$0.291896\pi$
$68$	−457.018	−0.815023
$69$	0	0
$70$	0	0
$71$	−256.941	−0.429483	−0.214741	−	0.976671i	$-0.568891\pi$
$71$	−256.941	−0.429483	−0.214741	+	0.976671i	$0.568891\pi$
$72$	0	0
$73$	1219.59	1.95537	0.977686	−	0.210073i	$-0.0673703\pi$
$73$	1219.59	1.95537	0.977686	+	0.210073i	$0.0673703\pi$
$74$	−1.95568	−0.00307221
$75$	0	0
$76$	−1000.29	−1.50975
$77$	0	0
$78$	0	0
$79$	381.236	0.542941	0.271471	−	0.962447i	$-0.412490\pi$
$79$	381.236	0.542941	0.271471	+	0.962447i	$0.412490\pi$
$80$	287.233	0.401420
$81$	0	0
$82$	−117.108	−0.157713
$83$	−281.766	−0.372625	−0.186313	−	0.982491i	$-0.559654\pi$
$83$	−281.766	−0.372625	−0.186313	+	0.982491i	$0.559654\pi$
$84$	0	0
$85$	295.852	0.377525
$86$	−32.7309	−0.0410402
$87$	0	0
$88$	−404.460	−0.489949
$89$	−926.242	−1.10316	−0.551581	−	0.834121i	$-0.685975\pi$
$89$	−926.242	−1.10316	−0.551581	+	0.834121i	$0.685975\pi$
$90$	0	0
$91$	0	0
$92$	−722.642	−0.818920
$93$	0	0
$94$	−241.940	−0.265471
$95$	647.540	0.699328
$96$	0	0
$97$	−634.683	−0.664354	−0.332177	−	0.943217i	$-0.607783\pi$
$97$	−634.683	−0.664354	−0.332177	+	0.943217i	$0.607783\pi$
$98$	0	0
$99$	0	0
$100$	−193.094	−0.193094
$101$	1921.81	1.89334	0.946670	−	0.322205i	$-0.104424\pi$
$101$	1921.81	1.89334	0.946670	+	0.322205i	$0.104424\pi$
$102$	0	0
$103$	−1310.57	−1.25373	−0.626865	−	0.779128i	$-0.715662\pi$
$103$	−1310.57	−1.25373	−0.626865	+	0.779128i	$0.715662\pi$
$104$	−331.820	−0.312862
$105$	0	0
$106$	−259.101	−0.237416
$107$	1866.73	1.68657	0.843286	−	0.537465i	$-0.180618\pi$
$107$	1866.73	1.68657	0.843286	+	0.537465i	$0.180618\pi$
$108$	0	0
$109$	−649.011	−0.570312	−0.285156	−	0.958481i	$-0.592045\pi$
$109$	−649.011	−0.570312	−0.285156	+	0.958481i	$0.592045\pi$
$110$	128.614	0.111481
$111$	0	0
$112$	0	0
$113$	857.830	0.714140	0.357070	−	0.934078i	$-0.383776\pi$
$113$	857.830	0.714140	0.357070	+	0.934078i	$0.383776\pi$
$114$	0	0
$115$	467.805	0.379330
$116$	−905.104	−0.724455
$117$	0	0
$118$	80.9022	0.0631157
$119$	0	0
$120$	0	0
$121$	1064.27	0.799600
$122$	102.445	0.0760238
$123$	0	0
$124$	1258.55	0.911459
$125$	125.000	0.0894427
$126$	0	0
$127$	972.678	0.679616	0.339808	−	0.940495i	$-0.389638\pi$
$127$	972.678	0.679616	0.339808	+	0.940495i	$0.389638\pi$
$128$	−985.390	−0.680445
$129$	0	0
$130$	105.515	0.0711871
$131$	−131.175	−0.0874873	−0.0437436	−	0.999043i	$-0.513928\pi$
$131$	−131.175	−0.0874873	−0.0437436	+	0.999043i	$0.513928\pi$
$132$	0	0
$133$	0	0
$134$	350.608	0.226029
$135$	0	0
$136$	−488.993	−0.308315
$137$	2160.79	1.34751	0.673754	−	0.738956i	$-0.264681\pi$
$137$	2160.79	1.34751	0.673754	+	0.738956i	$0.264681\pi$
$138$	0	0
$139$	1791.95	1.09346	0.546732	−	0.837308i	$-0.315872\pi$
$139$	1791.95	1.09346	0.546732	+	0.837308i	$0.315872\pi$
$140$	0	0
$141$	0	0
$142$	−135.044	−0.0798073
$143$	1965.09	1.14915
$144$	0	0
$145$	585.922	0.335573
$146$	640.996	0.363351
$147$	0	0
$148$	28.7399	0.0159622
$149$	−3022.74	−1.66196	−0.830981	−	0.556301i	$-0.812220\pi$
$149$	−3022.74	−1.66196	−0.830981	+	0.556301i	$0.812220\pi$
$150$	0	0
$151$	908.211	0.489464	0.244732	−	0.969591i	$-0.421300\pi$
$151$	908.211	0.489464	0.244732	+	0.969591i	$0.421300\pi$
$152$	−1070.27	−0.571123
$153$	0	0
$154$	0	0
$155$	−814.725	−0.422195
$156$	0	0
$157$	1387.49	0.705310	0.352655	−	0.935753i	$-0.385279\pi$
$157$	1387.49	0.705310	0.352655	+	0.935753i	$0.385279\pi$
$158$	200.371	0.100890
$159$	0	0
$160$	481.531	0.237927
$161$	0	0
$162$	0	0
$163$	−2637.35	−1.26732	−0.633660	−	0.773612i	$-0.718448\pi$
$163$	−2637.35	−1.26732	−0.633660	+	0.773612i	$0.718448\pi$
$164$	1720.98	0.819425
$165$	0	0
$166$	−148.092	−0.0692419
$167$	2525.54	1.17025	0.585126	−	0.810943i	$-0.301045\pi$
$167$	2525.54	1.17025	0.585126	+	0.810943i	$0.301045\pi$
$168$	0	0
$169$	−584.838	−0.266198
$170$	155.495	0.0701524
$171$	0	0
$172$	481.000	0.213232
$173$	−649.874	−0.285601	−0.142801	−	0.989751i	$-0.545611\pi$
$173$	−649.874	−0.285601	−0.142801	+	0.989751i	$0.545611\pi$
$174$	0	0
$175$	0	0
$176$	2811.52	1.20413
$177$	0	0
$178$	−486.817	−0.204992
$179$	−3434.35	−1.43405	−0.717026	−	0.697047i	$-0.754497\pi$
$179$	−3434.35	−1.43405	−0.717026	+	0.697047i	$0.754497\pi$
$180$	0	0
$181$	1187.20	0.487537	0.243768	−	0.969834i	$-0.421616\pi$
$181$	1187.20	0.487537	0.243768	+	0.969834i	$0.421616\pi$
$182$	0	0
$183$	0	0
$184$	−773.201	−0.309789
$185$	−18.6049	−0.00739382
$186$	0	0
$187$	2895.88	1.13245
$188$	3555.46	1.37930
$189$	0	0
$190$	340.336	0.129950
$191$	−180.895	−0.0685293	−0.0342646	−	0.999413i	$-0.510909\pi$
$191$	−180.895	−0.0685293	−0.0342646	+	0.999413i	$0.510909\pi$
$192$	0	0
$193$	−3100.23	−1.15627	−0.578134	−	0.815942i	$-0.696219\pi$
$193$	−3100.23	−1.15627	−0.578134	+	0.815942i	$0.696219\pi$
$194$	−333.579	−0.123451
$195$	0	0
$196$	0	0
$197$	−611.701	−0.221228	−0.110614	−	0.993863i	$-0.535282\pi$
$197$	−611.701	−0.221228	−0.110614	+	0.993863i	$0.535282\pi$
$198$	0	0
$199$	−1387.39	−0.494218	−0.247109	−	0.968988i	$-0.579481\pi$
$199$	−1387.39	−0.494218	−0.247109	+	0.968988i	$0.579481\pi$
$200$	−206.604	−0.0730455
$201$	0	0
$202$	1010.07	0.351824
$203$	0	0
$204$	0	0
$205$	−1114.08	−0.379564
$206$	−688.814	−0.232970
$207$	0	0
$208$	2306.58	0.768907
$209$	6338.31	2.09775
$210$	0	0
$211$	−1415.49	−0.461830	−0.230915	−	0.972974i	$-0.574172\pi$
$211$	−1415.49	−0.461830	−0.230915	+	0.972974i	$0.574172\pi$
$212$	3807.64	1.23354
$213$	0	0
$214$	981.121	0.313402
$215$	−311.377	−0.0987707
$216$	0	0
$217$	0	0
$218$	−341.109	−0.105976
$219$	0	0
$220$	−1890.06	−0.579218
$221$	2375.79	0.723136
$222$	0	0
$223$	−4213.75	−1.26535	−0.632676	−	0.774417i	$-0.718043\pi$
$223$	−4213.75	−1.26535	−0.632676	+	0.774417i	$0.718043\pi$
$224$	0	0
$225$	0	0
$226$	450.861	0.132703
$227$	6180.32	1.80706	0.903529	−	0.428526i	$-0.140967\pi$
$227$	6180.32	1.80706	0.903529	+	0.428526i	$0.140967\pi$
$228$	0	0
$229$	−1196.83	−0.345365	−0.172683	−	0.984978i	$-0.555244\pi$
$229$	−1196.83	−0.345365	−0.172683	+	0.984978i	$0.555244\pi$
$230$	245.870	0.0704879
$231$	0	0
$232$	−968.429	−0.274054
$233$	927.278	0.260721	0.130360	−	0.991467i	$-0.458387\pi$
$233$	927.278	0.260721	0.130360	+	0.991467i	$0.458387\pi$
$234$	0	0
$235$	−2301.64	−0.638903
$236$	−1188.91	−0.327929
$237$	0	0
$238$	0	0
$239$	−3429.57	−0.928202	−0.464101	−	0.885782i	$-0.653622\pi$
$239$	−3429.57	−0.928202	−0.464101	+	0.885782i	$0.653622\pi$
$240$	0	0
$241$	4748.69	1.26925	0.634626	−	0.772819i	$-0.281154\pi$
$241$	4748.69	1.26925	0.634626	+	0.772819i	$0.281154\pi$
$242$	559.362	0.148583
$243$	0	0
$244$	−1505.49	−0.394995
$245$	0	0
$246$	0	0
$247$	5199.97	1.33954
$248$	1346.60	0.344796
$249$	0	0
$250$	65.6979	0.0166204
$251$	7273.04	1.82896	0.914482	−	0.404626i	$-0.132598\pi$
$251$	7273.04	1.82896	0.914482	+	0.404626i	$0.132598\pi$
$252$	0	0
$253$	4579.01	1.13786
$254$	511.223	0.126287
$255$	0	0
$256$	2753.74	0.672300
$257$	1613.28	0.391570	0.195785	−	0.980647i	$-0.437275\pi$
$257$	1613.28	0.391570	0.195785	+	0.980647i	$0.437275\pi$
$258$	0	0
$259$	0	0
$260$	−1550.61	−0.369865
$261$	0	0
$262$	−68.9436	−0.0162571
$263$	−2593.26	−0.608013	−0.304007	−	0.952670i	$-0.598324\pi$
$263$	−2593.26	−0.608013	−0.304007	+	0.952670i	$0.598324\pi$
$264$	0	0
$265$	−2464.88	−0.571384
$266$	0	0
$267$	0	0
$268$	−5152.40	−1.17438
$269$	5922.82	1.34246	0.671228	−	0.741251i	$-0.265767\pi$
$269$	5922.82	1.34246	0.671228	+	0.741251i	$0.265767\pi$
$270$	0	0
$271$	−1783.59	−0.399798	−0.199899	−	0.979816i	$-0.564061\pi$
$271$	−1783.59	−0.399798	−0.199899	+	0.979816i	$0.564061\pi$
$272$	3399.14	0.757731
$273$	0	0
$274$	1135.68	0.250397
$275$	1223.54	0.268298
$276$	0	0
$277$	−3042.89	−0.660035	−0.330017	−	0.943975i	$-0.607055\pi$
$277$	−3042.89	−0.660035	−0.330017	+	0.943975i	$0.607055\pi$
$278$	941.821	0.203190
$279$	0	0
$280$	0	0
$281$	−1812.14	−0.384708	−0.192354	−	0.981326i	$-0.561612\pi$
$281$	−1812.14	−0.384708	−0.192354	+	0.981326i	$0.561612\pi$
$282$	0	0
$283$	−7062.64	−1.48350	−0.741750	−	0.670677i	$-0.766004\pi$
$283$	−7062.64	−1.48350	−0.741750	+	0.670677i	$0.766004\pi$
$284$	1984.55	0.414653
$285$	0	0
$286$	1032.82	0.213537
$287$	0	0
$288$	0	0
$289$	−1411.87	−0.287374
$290$	307.951	0.0623569
$291$	0	0
$292$	−9419.82	−1.88785
$293$	6252.90	1.24675	0.623376	−	0.781922i	$-0.285760\pi$
$293$	6252.90	1.24675	0.623376	+	0.781922i	$0.285760\pi$
$294$	0	0
$295$	769.642	0.151899
$296$	30.7507	0.00603833
$297$	0	0
$298$	−1588.70	−0.308829
$299$	3756.63	0.726594
$300$	0	0
$301$	0	0
$302$	477.341	0.0909532
$303$	0	0
$304$	7439.79	1.40362
$305$	974.580	0.182965
$306$	0	0
$307$	−946.955	−0.176044	−0.0880221	−	0.996119i	$-0.528055\pi$
$307$	−946.955	−0.176044	−0.0880221	+	0.996119i	$0.528055\pi$
$308$	0	0
$309$	0	0
$310$	−428.206	−0.0784531
$311$	7114.71	1.29723	0.648614	−	0.761117i	$-0.275349\pi$
$311$	7114.71	1.29723	0.648614	+	0.761117i	$0.275349\pi$
$312$	0	0
$313$	2357.90	0.425803	0.212901	−	0.977074i	$-0.431709\pi$
$313$	2357.90	0.425803	0.212901	+	0.977074i	$0.431709\pi$
$314$	729.241	0.131062
$315$	0	0
$316$	−2944.57	−0.524194
$317$	−9260.69	−1.64079	−0.820397	−	0.571794i	$-0.806248\pi$
$317$	−9260.69	−1.64079	−0.820397	+	0.571794i	$0.806248\pi$
$318$	0	0
$319$	5735.17	1.00661
$320$	−2044.78	−0.357208
$321$	0	0
$322$	0	0
$323$	7663.03	1.32007
$324$	0	0
$325$	1003.79	0.171324
$326$	−1386.15	−0.235496
$327$	0	0
$328$	1841.38	0.309980
$329$	0	0
$330$	0	0
$331$	10664.3	1.77089	0.885443	−	0.464748i	$-0.153855\pi$
$331$	10664.3	1.77089	0.885443	+	0.464748i	$0.153855\pi$
$332$	2176.30	0.359758
$333$	0	0
$334$	1327.38	0.217458
$335$	3335.42	0.543980
$336$	0	0
$337$	−865.728	−0.139938	−0.0699691	−	0.997549i	$-0.522290\pi$
$337$	−865.728	−0.139938	−0.0699691	+	0.997549i	$0.522290\pi$
$338$	−307.381	−0.0494655
$339$	0	0
$340$	−2285.09	−0.364489
$341$	−7974.77	−1.26645
$342$	0	0
$343$	0	0
$344$	514.653	0.0806634
$345$	0	0
$346$	−341.563	−0.0530710
$347$	−8164.72	−1.26313	−0.631563	−	0.775324i	$-0.717586\pi$
$347$	−8164.72	−1.26313	−0.631563	+	0.775324i	$0.717586\pi$
$348$	0	0
$349$	−6227.30	−0.955128	−0.477564	−	0.878597i	$-0.658480\pi$
$349$	−6227.30	−0.955128	−0.477564	+	0.878597i	$0.658480\pi$
$350$	0	0
$351$	0	0
$352$	4713.37	0.713703
$353$	11121.3	1.67685	0.838425	−	0.545017i	$-0.183477\pi$
$353$	11121.3	1.67685	0.838425	+	0.545017i	$0.183477\pi$
$354$	0	0
$355$	−1284.70	−0.192071
$356$	7154.07	1.06507
$357$	0	0
$358$	−1805.04	−0.266478
$359$	3362.46	0.494328	0.247164	−	0.968974i	$-0.420501\pi$
$359$	3362.46	0.494328	0.247164	+	0.968974i	$0.420501\pi$
$360$	0	0
$361$	9913.32	1.44530
$362$	623.974	0.0905949
$363$	0	0
$364$	0	0
$365$	6097.95	0.874469
$366$	0	0
$367$	−1447.20	−0.205839	−0.102920	−	0.994690i	$-0.532818\pi$
$367$	−1447.20	−0.205839	−0.102920	+	0.994690i	$0.532818\pi$
$368$	5374.76	0.761355
$369$	0	0
$370$	−9.77841	−0.00137393
$371$	0	0
$372$	0	0
$373$	−13117.6	−1.82092	−0.910460	−	0.413597i	$-0.864272\pi$
$373$	−13117.6	−1.82092	−0.910460	+	0.413597i	$0.864272\pi$
$374$	1522.03	0.210434
$375$	0	0
$376$	3804.21	0.521775
$377$	4705.15	0.642779
$378$	0	0
$379$	−4102.23	−0.555983	−0.277991	−	0.960584i	$-0.589669\pi$
$379$	−4102.23	−0.555983	−0.277991	+	0.960584i	$0.589669\pi$
$380$	−5001.44	−0.675181
$381$	0	0
$382$	−95.0754	−0.0127342
$383$	−7371.61	−0.983477	−0.491739	−	0.870743i	$-0.663638\pi$
$383$	−7371.61	−0.983477	−0.491739	+	0.870743i	$0.663638\pi$
$384$	0	0
$385$	0	0
$386$	−1629.43	−0.214860
$387$	0	0
$388$	4902.14	0.641414
$389$	8529.66	1.11175	0.555875	−	0.831266i	$-0.312383\pi$
$389$	8529.66	1.11175	0.555875	+	0.831266i	$0.312383\pi$
$390$	0	0
$391$	5536.03	0.716034
$392$	0	0
$393$	0	0
$394$	−321.500	−0.0411090
$395$	1906.18	0.242811
$396$	0	0
$397$	10274.2	1.29886	0.649428	−	0.760423i	$-0.275008\pi$
$397$	10274.2	1.29886	0.649428	+	0.760423i	$0.275008\pi$
$398$	−729.189	−0.0918365
$399$	0	0
$400$	1436.16	0.179521
$401$	254.845	0.0317366	0.0158683	−	0.999874i	$-0.494949\pi$
$401$	254.845	0.0317366	0.0158683	+	0.999874i	$0.494949\pi$
$402$	0	0
$403$	−6542.52	−0.808701
$404$	−14843.6	−1.82796
$405$	0	0
$406$	0	0
$407$	−182.110	−0.0221790
$408$	0	0
$409$	−6790.11	−0.820903	−0.410451	−	0.911882i	$-0.634629\pi$
$409$	−6790.11	−0.820903	−0.410451	+	0.911882i	$0.634629\pi$
$410$	−585.542	−0.0705314
$411$	0	0
$412$	10122.5	1.21044
$413$	0	0
$414$	0	0
$415$	−1408.83	−0.166643
$416$	3866.86	0.455741
$417$	0	0
$418$	3331.31	0.389808
$419$	2717.47	0.316843	0.158421	−	0.987372i	$-0.449360\pi$
$419$	2717.47	0.316843	0.158421	+	0.987372i	$0.449360\pi$
$420$	0	0
$421$	−3455.49	−0.400024	−0.200012	−	0.979793i	$-0.564098\pi$
$421$	−3455.49	−0.400024	−0.200012	+	0.979793i	$0.564098\pi$
$422$	−743.957	−0.0858181
$423$	0	0
$424$	4074.04	0.466634
$425$	1479.26	0.168834
$426$	0	0
$427$	0	0
$428$	−14418.1	−1.62834
$429$	0	0
$430$	−163.654	−0.0183538
$431$	5333.95	0.596119	0.298060	−	0.954547i	$-0.403661\pi$
$431$	5333.95	0.596119	0.298060	+	0.954547i	$0.403661\pi$
$432$	0	0
$433$	14032.3	1.55739	0.778696	−	0.627401i	$-0.215881\pi$
$433$	14032.3	1.55739	0.778696	+	0.627401i	$0.215881\pi$
$434$	0	0
$435$	0	0
$436$	5012.81	0.550619
$437$	12116.9	1.32638
$438$	0	0
$439$	−13158.0	−1.43052	−0.715258	−	0.698860i	$-0.753691\pi$
$439$	−13158.0	−1.43052	−0.715258	+	0.698860i	$0.753691\pi$
$440$	−2022.30	−0.219112
$441$	0	0
$442$	1248.68	0.134375
$443$	14922.9	1.60047	0.800237	−	0.599683i	$-0.204707\pi$
$443$	14922.9	1.60047	0.800237	+	0.599683i	$0.204707\pi$
$444$	0	0
$445$	−4631.21	−0.493349
$446$	−2214.68	−0.235130
$447$	0	0
$448$	0	0
$449$	640.776	0.0673498	0.0336749	−	0.999433i	$-0.489279\pi$
$449$	640.776	0.0673498	0.0336749	+	0.999433i	$0.489279\pi$
$450$	0	0
$451$	−10904.9	−1.13857
$452$	−6625.67	−0.689481
$453$	0	0
$454$	3248.28	0.335791
$455$	0	0
$456$	0	0
$457$	14469.0	1.48103	0.740515	−	0.672039i	$-0.234581\pi$
$457$	14469.0	1.48103	0.740515	+	0.672039i	$0.234581\pi$
$458$	−629.033	−0.0641764
$459$	0	0
$460$	−3613.21	−0.366232
$461$	−10275.8	−1.03816	−0.519082	−	0.854725i	$-0.673726\pi$
$461$	−10275.8	−1.03816	−0.519082	+	0.854725i	$0.673726\pi$
$462$	0	0
$463$	4435.77	0.445244	0.222622	−	0.974905i	$-0.428538\pi$
$463$	4435.77	0.445244	0.222622	+	0.974905i	$0.428538\pi$
$464$	6731.84	0.673530
$465$	0	0
$466$	487.362	0.0484476
$467$	−12909.3	−1.27917	−0.639585	−	0.768720i	$-0.720894\pi$
$467$	−12909.3	−1.27917	−0.639585	+	0.768720i	$0.720894\pi$
$468$	0	0
$469$	0	0
$470$	−1209.70	−0.118722
$471$	0	0
$472$	−1272.09	−0.124052
$473$	−3047.85	−0.296279
$474$	0	0
$475$	3237.70	0.312749
$476$	0	0
$477$	0	0
$478$	−1802.52	−0.172480
$479$	15622.8	1.49023	0.745117	−	0.666934i	$-0.232394\pi$
$479$	15622.8	1.49023	0.745117	+	0.666934i	$0.232394\pi$
$480$	0	0
$481$	−149.403	−0.0141626
$482$	2495.83	0.235855
$483$	0	0
$484$	−8220.15	−0.771990
$485$	−3173.42	−0.297108
$486$	0	0
$487$	7817.06	0.727361	0.363680	−	0.931524i	$-0.381520\pi$
$487$	7817.06	0.727361	0.363680	+	0.931524i	$0.381520\pi$
$488$	−1610.82	−0.149423
$489$	0	0
$490$	0	0
$491$	1057.33	0.0971822	0.0485911	−	0.998819i	$-0.484527\pi$
$491$	1057.33	0.0971822	0.0485911	+	0.998819i	$0.484527\pi$
$492$	0	0
$493$	6933.84	0.633437
$494$	2733.02	0.248916
$495$	0	0
$496$	−9360.63	−0.847389
$497$	0	0
$498$	0	0
$499$	−4434.30	−0.397809	−0.198904	−	0.980019i	$-0.563738\pi$
$499$	−4434.30	−0.397809	−0.198904	+	0.980019i	$0.563738\pi$
$500$	−965.470	−0.0863543
$501$	0	0
$502$	3822.59	0.339862
$503$	−3705.19	−0.328442	−0.164221	−	0.986424i	$-0.552511\pi$
$503$	−3705.19	−0.328442	−0.164221	+	0.986424i	$0.552511\pi$
$504$	0	0
$505$	9609.05	0.846727
$506$	2406.65	0.211440
$507$	0	0
$508$	−7512.73	−0.656149
$509$	1670.83	0.145498	0.0727490	−	0.997350i	$-0.476823\pi$
$509$	1670.83	0.145498	0.0727490	+	0.997350i	$0.476823\pi$
$510$	0	0
$511$	0	0
$512$	9330.44	0.805373
$513$	0	0
$514$	847.911	0.0727622
$515$	−6552.85	−0.560685
$516$	0	0
$517$	−22529.1	−1.91649
$518$	0	0
$519$	0	0
$520$	−1659.10	−0.139916
$521$	9521.12	0.800629	0.400315	−	0.916378i	$-0.368901\pi$
$521$	9521.12	0.800629	0.400315	+	0.916378i	$0.368901\pi$
$522$	0	0
$523$	−14460.0	−1.20897	−0.604487	−	0.796615i	$-0.706622\pi$
$523$	−14460.0	−1.20897	−0.604487	+	0.796615i	$0.706622\pi$
$524$	1013.17	0.0844664
$525$	0	0
$526$	−1362.98	−0.112982
$527$	−9641.51	−0.796947
$528$	0	0
$529$	−3413.36	−0.280542
$530$	−1295.50	−0.106176
$531$	0	0
$532$	0	0
$533$	−8946.45	−0.727042
$534$	0	0
$535$	9333.63	0.754258
$536$	−5512.88	−0.444254
$537$	0	0
$538$	3112.93	0.249458
$539$	0	0
$540$	0	0
$541$	5461.17	0.434000	0.217000	−	0.976172i	$-0.430373\pi$
$541$	5461.17	0.434000	0.217000	+	0.976172i	$0.430373\pi$
$542$	−937.425	−0.0742913
$543$	0	0
$544$	5698.47	0.449118
$545$	−3245.05	−0.255051
$546$	0	0
$547$	20483.5	1.60112	0.800559	−	0.599254i	$-0.204536\pi$
$547$	20483.5	1.60112	0.800559	+	0.599254i	$0.204536\pi$
$548$	−16689.4	−1.30098
$549$	0	0
$550$	643.071	0.0498557
$551$	15176.3	1.17338
$552$	0	0
$553$	0	0
$554$	−1599.29	−0.122649
$555$	0	0
$556$	−13840.6	−1.05571
$557$	−8382.62	−0.637671	−0.318836	−	0.947810i	$-0.603292\pi$
$557$	−8382.62	−0.637671	−0.318836	+	0.947810i	$0.603292\pi$
$558$	0	0
$559$	−2500.46	−0.189192
$560$	0	0
$561$	0	0
$562$	−952.429	−0.0714872
$563$	10678.5	0.799369	0.399685	−	0.916653i	$-0.369120\pi$
$563$	10678.5	0.799369	0.399685	+	0.916653i	$0.369120\pi$
$564$	0	0
$565$	4289.15	0.319373
$566$	−3712.01	−0.275667
$567$	0	0
$568$	2123.40	0.156859
$569$	17416.3	1.28318	0.641590	−	0.767048i	$-0.278275\pi$
$569$	17416.3	1.28318	0.641590	+	0.767048i	$0.278275\pi$
$570$	0	0
$571$	7072.55	0.518348	0.259174	−	0.965831i	$-0.416550\pi$
$571$	7072.55	0.518348	0.259174	+	0.965831i	$0.416550\pi$
$572$	−15177.8	−1.10947
$573$	0	0
$574$	0	0
$575$	2339.02	0.169642
$576$	0	0
$577$	−2399.09	−0.173094	−0.0865471	−	0.996248i	$-0.527583\pi$
$577$	−2399.09	−0.173094	−0.0865471	+	0.996248i	$0.527583\pi$
$578$	−742.055	−0.0534004
$579$	0	0
$580$	−4525.52	−0.323986
$581$	0	0
$582$	0	0
$583$	−24127.0	−1.71396
$584$	−10078.9	−0.714155
$585$	0	0
$586$	3286.42	0.231674
$587$	−10345.7	−0.727451	−0.363725	−	0.931506i	$-0.618495\pi$
$587$	−10345.7	−0.727451	−0.363725	+	0.931506i	$0.618495\pi$
$588$	0	0
$589$	−21102.7	−1.47627
$590$	404.511	0.0282262
$591$	0	0
$592$	−213.757	−0.0148401
$593$	−3622.84	−0.250881	−0.125440	−	0.992101i	$-0.540034\pi$
$593$	−3622.84	−0.250881	−0.125440	+	0.992101i	$0.540034\pi$
$594$	0	0
$595$	0	0
$596$	23346.9	1.60457
$597$	0	0
$598$	1974.42	0.135017
$599$	−1509.74	−0.102982	−0.0514910	−	0.998673i	$-0.516397\pi$
$599$	−1509.74	−0.102982	−0.0514910	+	0.998673i	$0.516397\pi$
$600$	0	0
$601$	−10522.7	−0.714194	−0.357097	−	0.934067i	$-0.616233\pi$
$601$	−10522.7	−0.714194	−0.357097	+	0.934067i	$0.616233\pi$
$602$	0	0
$603$	0	0
$604$	−7014.80	−0.472563
$605$	5321.34	0.357592
$606$	0	0
$607$	−15525.1	−1.03813	−0.519065	−	0.854735i	$-0.673720\pi$
$607$	−15525.1	−1.03813	−0.519065	+	0.854735i	$0.673720\pi$
$608$	12472.4	0.831946
$609$	0	0
$610$	512.223	0.0339989
$611$	−18482.9	−1.22380
$612$	0	0
$613$	−10339.1	−0.681227	−0.340614	−	0.940203i	$-0.610635\pi$
$613$	−10339.1	−0.681227	−0.340614	+	0.940203i	$0.610635\pi$
$614$	−497.704	−0.0327129
$615$	0	0
$616$	0	0
$617$	−12652.5	−0.825562	−0.412781	−	0.910830i	$-0.635442\pi$
$617$	−12652.5	−0.825562	−0.412781	+	0.910830i	$0.635442\pi$
$618$	0	0
$619$	−4188.31	−0.271959	−0.135979	−	0.990712i	$-0.543418\pi$
$619$	−4188.31	−0.271959	−0.135979	+	0.990712i	$0.543418\pi$
$620$	6292.74	0.407617
$621$	0	0
$622$	3739.37	0.241053
$623$	0	0
$624$	0	0
$625$	625.000	0.0400000
$626$	1239.27	0.0791235
$627$	0	0
$628$	−10716.6	−0.680956
$629$	−220.171	−0.0139568
$630$	0	0
$631$	−12886.7	−0.813016	−0.406508	−	0.913647i	$-0.633254\pi$
$631$	−12886.7	−0.813016	−0.406508	+	0.913647i	$0.633254\pi$
$632$	−3150.59	−0.198297
$633$	0	0
$634$	−4867.26	−0.304896
$635$	4863.39	0.303933
$636$	0	0
$637$	0	0
$638$	3014.31	0.187050
$639$	0	0
$640$	−4926.95	−0.304304
$641$	23195.4	1.42927	0.714635	−	0.699498i	$-0.246593\pi$
$641$	23195.4	1.42927	0.714635	+	0.699498i	$0.246593\pi$
$642$	0	0
$643$	−9691.46	−0.594391	−0.297196	−	0.954817i	$-0.596051\pi$
$643$	−9691.46	−0.594391	−0.297196	+	0.954817i	$0.596051\pi$
$644$	0	0
$645$	0	0
$646$	4027.56	0.245298
$647$	−15711.6	−0.954694	−0.477347	−	0.878715i	$-0.658401\pi$
$647$	−15711.6	−0.954694	−0.477347	+	0.878715i	$0.658401\pi$
$648$	0	0
$649$	7533.48	0.455647
$650$	527.577	0.0318358
$651$	0	0
$652$	20370.3	1.22356
$653$	−5596.58	−0.335392	−0.167696	−	0.985839i	$-0.553633\pi$
$653$	−5596.58	−0.335392	−0.167696	+	0.985839i	$0.553633\pi$
$654$	0	0
$655$	−655.876	−0.0391255
$656$	−12800.0	−0.761824
$657$	0	0
$658$	0	0
$659$	−490.979	−0.0290225	−0.0145112	−	0.999895i	$-0.504619\pi$
$659$	−490.979	−0.0290225	−0.0145112	+	0.999895i	$0.504619\pi$
$660$	0	0
$661$	26400.3	1.55348	0.776741	−	0.629820i	$-0.216872\pi$
$661$	26400.3	1.55348	0.776741	+	0.629820i	$0.216872\pi$
$662$	5604.98	0.329069
$663$	0	0
$664$	2328.56	0.136093
$665$	0	0
$666$	0	0
$667$	10963.9	0.636466
$668$	−19506.7	−1.12984
$669$	0	0
$670$	1753.04	0.101083
$671$	9539.48	0.548834
$672$	0	0
$673$	−22727.4	−1.30175	−0.650875	−	0.759185i	$-0.725598\pi$
$673$	−22727.4	−1.30175	−0.650875	+	0.759185i	$0.725598\pi$
$674$	−455.012	−0.0260036
$675$	0	0
$676$	4517.15	0.257007
$677$	26717.2	1.51673	0.758365	−	0.651830i	$-0.225999\pi$
$677$	26717.2	1.51673	0.758365	+	0.651830i	$0.225999\pi$
$678$	0	0
$679$	0	0
$680$	−2444.96	−0.137882
$681$	0	0
$682$	−4191.41	−0.235333
$683$	19578.8	1.09687	0.548433	−	0.836194i	$-0.315224\pi$
$683$	19578.8	1.09687	0.548433	+	0.836194i	$0.315224\pi$
$684$	0	0
$685$	10803.9	0.602624
$686$	0	0
$687$	0	0
$688$	−3577.50	−0.198243
$689$	−19793.9	−1.09447
$690$	0	0
$691$	−13856.7	−0.762854	−0.381427	−	0.924399i	$-0.624567\pi$
$691$	−13856.7	−0.762854	−0.381427	+	0.924399i	$0.624567\pi$
$692$	5019.47	0.275740
$693$	0	0
$694$	−4291.24	−0.234717
$695$	8959.77	0.489012
$696$	0	0
$697$	−13184.1	−0.716475
$698$	−3272.97	−0.177484
$699$	0	0
$700$	0	0
$701$	−28106.3	−1.51435	−0.757176	−	0.653210i	$-0.773422\pi$
$701$	−28106.3	−1.51435	−0.757176	+	0.653210i	$0.773422\pi$
$702$	0	0
$703$	−481.896	−0.0258535
$704$	−20014.9	−1.07151
$705$	0	0
$706$	5845.18	0.311595
$707$	0	0
$708$	0	0
$709$	−15866.5	−0.840449	−0.420225	−	0.907420i	$-0.638049\pi$
$709$	−15866.5	−0.840449	−0.420225	+	0.907420i	$0.638049\pi$
$710$	−675.220	−0.0356909
$711$	0	0
$712$	7654.60	0.402905
$713$	−15245.3	−0.800758
$714$	0	0
$715$	9825.43	0.513916
$716$	26526.1	1.38453
$717$	0	0
$718$	1767.25	0.0918569
$719$	9593.84	0.497621	0.248811	−	0.968552i	$-0.419960\pi$
$719$	9593.84	0.497621	0.248811	+	0.968552i	$0.419960\pi$
$720$	0	0
$721$	0	0
$722$	5210.28	0.268568
$723$	0	0
$724$	−9169.67	−0.470702
$725$	2929.61	0.150073
$726$	0	0
$727$	29844.5	1.52252	0.761259	−	0.648448i	$-0.224581\pi$
$727$	29844.5	1.52252	0.761259	+	0.648448i	$0.224581\pi$
$728$	0	0
$729$	0	0
$730$	3204.98	0.162495
$731$	−3684.85	−0.186442
$732$	0	0
$733$	26065.3	1.31343	0.656714	−	0.754139i	$-0.271946\pi$
$733$	26065.3	1.31343	0.656714	+	0.754139i	$0.271946\pi$
$734$	−760.623	−0.0382495
$735$	0	0
$736$	9010.49	0.451265
$737$	32648.1	1.63176
$738$	0	0
$739$	11751.1	0.584941	0.292470	−	0.956275i	$-0.405523\pi$
$739$	11751.1	0.584941	0.292470	+	0.956275i	$0.405523\pi$
$740$	143.699	0.00713851
$741$	0	0
$742$	0	0
$743$	31874.5	1.57384	0.786919	−	0.617057i	$-0.211675\pi$
$743$	31874.5	1.57384	0.786919	+	0.617057i	$0.211675\pi$
$744$	0	0
$745$	−15113.7	−0.743252
$746$	−6894.39	−0.338367
$747$	0	0
$748$	−22367.1	−1.09335
$749$	0	0
$750$	0	0
$751$	−12981.1	−0.630741	−0.315370	−	0.948969i	$-0.602129\pi$
$751$	−12981.1	−0.630741	−0.315370	+	0.948969i	$0.602129\pi$
$752$	−26444.2	−1.28234
$753$	0	0
$754$	2472.95	0.119442
$755$	4541.05	0.218895
$756$	0	0
$757$	31766.3	1.52518	0.762592	−	0.646880i	$-0.223927\pi$
$757$	31766.3	1.52518	0.762592	+	0.646880i	$0.223927\pi$
$758$	−2156.06	−0.103314
$759$	0	0
$760$	−5351.37	−0.255414
$761$	21164.7	1.00817	0.504087	−	0.863653i	$-0.331829\pi$
$761$	21164.7	1.00817	0.504087	+	0.863653i	$0.331829\pi$
$762$	0	0
$763$	0	0
$764$	1397.19	0.0661630
$765$	0	0
$766$	−3874.40	−0.182752
$767$	6180.49	0.290958
$768$	0	0
$769$	29643.8	1.39009	0.695047	−	0.718965i	$-0.255384\pi$
$769$	29643.8	1.39009	0.695047	+	0.718965i	$0.255384\pi$
$770$	0	0
$771$	0	0
$772$	23945.5	1.11634
$773$	−15627.2	−0.727128	−0.363564	−	0.931569i	$-0.618440\pi$
$773$	−15627.2	−0.727128	−0.363564	+	0.931569i	$0.618440\pi$
$774$	0	0
$775$	−4073.62	−0.188811
$776$	5245.12	0.242640
$777$	0	0
$778$	4483.05	0.206587
$779$	−28856.4	−1.32720
$780$	0	0
$781$	−12575.1	−0.576148
$782$	2909.65	0.133055
$783$	0	0
$784$	0	0
$785$	6937.44	0.315424
$786$	0	0
$787$	4641.64	0.210237	0.105118	−	0.994460i	$-0.466478\pi$
$787$	4641.64	0.210237	0.105118	+	0.994460i	$0.466478\pi$
$788$	4724.63	0.213589
$789$	0	0
$790$	1001.86	0.0451196
$791$	0	0
$792$	0	0
$793$	7826.22	0.350463
$794$	5399.93	0.241356
$795$	0	0
$796$	10715.9	0.477153
$797$	4097.25	0.182098	0.0910489	−	0.995846i	$-0.470978\pi$
$797$	4097.25	0.182098	0.0910489	+	0.995846i	$0.470978\pi$
$798$	0	0
$799$	−27237.7	−1.20601
$800$	2407.65	0.106404
$801$	0	0
$802$	133.943	0.00589735
$803$	59688.5	2.62311
$804$	0	0
$805$	0	0
$806$	−3438.64	−0.150274
$807$	0	0
$808$	−15882.1	−0.691499
$809$	−36787.7	−1.59875	−0.799374	−	0.600834i	$-0.794835\pi$
$809$	−36787.7	−1.59875	−0.799374	+	0.600834i	$0.794835\pi$
$810$	0	0
$811$	23507.4	1.01782	0.508912	−	0.860819i	$-0.330048\pi$
$811$	23507.4	1.01782	0.508912	+	0.860819i	$0.330048\pi$
$812$	0	0
$813$	0	0
$814$	−95.7139	−0.00412134
$815$	−13186.7	−0.566762
$816$	0	0
$817$	−8065.15	−0.345366
$818$	−3568.77	−0.152542
$819$	0	0
$820$	8604.89	0.366458
$821$	−34565.8	−1.46937	−0.734686	−	0.678407i	$-0.762670\pi$
$821$	−34565.8	−1.46937	−0.734686	+	0.678407i	$0.762670\pi$
$822$	0	0
$823$	26917.7	1.14009	0.570044	−	0.821614i	$-0.306926\pi$
$823$	26917.7	1.14009	0.570044	+	0.821614i	$0.306926\pi$
$824$	10830.7	0.457896
$825$	0	0
$826$	0	0
$827$	−35448.4	−1.49052	−0.745261	−	0.666773i	$-0.767675\pi$
$827$	−35448.4	−1.49052	−0.745261	+	0.666773i	$0.767675\pi$
$828$	0	0
$829$	24726.0	1.03591	0.517955	−	0.855408i	$-0.326694\pi$
$829$	24726.0	1.03591	0.517955	+	0.855408i	$0.326694\pi$
$830$	−740.459	−0.0309659
$831$	0	0
$832$	−16420.3	−0.684220
$833$	0	0
$834$	0	0
$835$	12627.7	0.523352
$836$	−48955.6	−2.02532
$837$	0	0
$838$	1428.26	0.0588764
$839$	−21114.3	−0.868829	−0.434415	−	0.900713i	$-0.643045\pi$
$839$	−21114.3	−0.868829	−0.434415	+	0.900713i	$0.643045\pi$
$840$	0	0
$841$	−10656.8	−0.436952
$842$	−1816.15	−0.0743332
$843$	0	0
$844$	10932.9	0.445883
$845$	−2924.19	−0.119048
$846$	0	0
$847$	0	0
$848$	−28319.8	−1.14682
$849$	0	0
$850$	777.474	0.0313731
$851$	−348.138	−0.0140235
$852$	0	0
$853$	−19801.4	−0.794828	−0.397414	−	0.917640i	$-0.630092\pi$
$853$	−19801.4	−0.794828	−0.397414	+	0.917640i	$0.630092\pi$
$854$	0	0
$855$	0	0
$856$	−15426.9	−0.615982
$857$	13201.0	0.526180	0.263090	−	0.964771i	$-0.415258\pi$
$857$	13201.0	0.526180	0.263090	+	0.964771i	$0.415258\pi$
$858$	0	0
$859$	43020.3	1.70877	0.854385	−	0.519641i	$-0.173934\pi$
$859$	43020.3	1.70877	0.854385	+	0.519641i	$0.173934\pi$
$860$	2405.00	0.0953602
$861$	0	0
$862$	2803.44	0.110772
$863$	6633.09	0.261637	0.130819	−	0.991406i	$-0.458239\pi$
$863$	6633.09	0.261637	0.130819	+	0.991406i	$0.458239\pi$
$864$	0	0
$865$	−3249.37	−0.127725
$866$	7375.16	0.289397
$867$	0	0
$868$	0	0
$869$	18658.2	0.728351
$870$	0	0
$871$	26784.6	1.04198
$872$	5363.52	0.208293
$873$	0	0
$874$	6368.44	0.246471
$875$	0	0
$876$	0	0
$877$	41119.1	1.58323	0.791616	−	0.611019i	$-0.209240\pi$
$877$	41119.1	1.58323	0.791616	+	0.611019i	$0.209240\pi$
$878$	−6915.62	−0.265821
$879$	0	0
$880$	14057.6	0.538502
$881$	−37501.8	−1.43413	−0.717064	−	0.697007i	$-0.754515\pi$
$881$	−37501.8	−1.43413	−0.717064	+	0.697007i	$0.754515\pi$
$882$	0	0
$883$	−10396.0	−0.396208	−0.198104	−	0.980181i	$-0.563478\pi$
$883$	−10396.0	−0.396208	−0.198104	+	0.980181i	$0.563478\pi$
$884$	−18350.1	−0.698166
$885$	0	0
$886$	7843.25	0.297403
$887$	42954.0	1.62599	0.812996	−	0.582270i	$-0.197835\pi$
$887$	42954.0	1.62599	0.812996	+	0.582270i	$0.197835\pi$
$888$	0	0
$889$	0	0
$890$	−2434.09	−0.0916750
$891$	0	0
$892$	32546.0	1.22166
$893$	−59616.0	−2.23401
$894$	0	0
$895$	−17171.7	−0.641327
$896$	0	0
$897$	0	0
$898$	336.781	0.0125151
$899$	−19094.6	−0.708388
$900$	0	0
$901$	−29169.6	−1.07856
$902$	−5731.46	−0.211571
$903$	0	0
$904$	−7089.23	−0.260823
$905$	5936.02	0.218033
$906$	0	0
$907$	−45162.6	−1.65336	−0.826681	−	0.562670i	$-0.809774\pi$
$907$	−45162.6	−1.65336	−0.826681	+	0.562670i	$0.809774\pi$
$908$	−47735.3	−1.74466
$909$	0	0
$910$	0	0
$911$	40669.4	1.47908	0.739538	−	0.673115i	$-0.235044\pi$
$911$	40669.4	1.47908	0.739538	+	0.673115i	$0.235044\pi$
$912$	0	0
$913$	−13790.1	−0.499873
$914$	7604.67	0.275208
$915$	0	0
$916$	9244.02	0.333440
$917$	0	0
$918$	0	0
$919$	−50310.3	−1.80586	−0.902930	−	0.429788i	$-0.858588\pi$
$919$	−50310.3	−1.80586	−0.902930	+	0.429788i	$0.858588\pi$
$920$	−3866.01	−0.138542
$921$	0	0
$922$	−5400.81	−0.192914
$923$	−10316.6	−0.367905
$924$	0	0
$925$	−93.0243	−0.00330662
$926$	2331.37	0.0827360
$927$	0	0
$928$	11285.6	0.399210
$929$	24058.7	0.849668	0.424834	−	0.905271i	$-0.360332\pi$
$929$	24058.7	0.849668	0.424834	+	0.905271i	$0.360332\pi$
$930$	0	0
$931$	0	0
$932$	−7162.07	−0.251718
$933$	0	0
$934$	−6784.93	−0.237698
$935$	14479.4	0.506447
$936$	0	0
$937$	−11252.2	−0.392310	−0.196155	−	0.980573i	$-0.562846\pi$
$937$	−11252.2	−0.392310	−0.196155	+	0.980573i	$0.562846\pi$
$938$	0	0
$939$	0	0
$940$	17777.3	0.616841
$941$	−40002.5	−1.38581	−0.692903	−	0.721030i	$-0.743669\pi$
$941$	−40002.5	−1.38581	−0.692903	+	0.721030i	$0.743669\pi$
$942$	0	0
$943$	−20846.9	−0.719902
$944$	8842.66	0.304877
$945$	0	0
$946$	−1601.90	−0.0550552
$947$	42875.0	1.47123	0.735613	−	0.677402i	$-0.236894\pi$
$947$	42875.0	1.47123	0.735613	+	0.677402i	$0.236894\pi$
$948$	0	0
$949$	48968.6	1.67501
$950$	1701.68	0.0581156
$951$	0	0
$952$	0	0
$953$	7241.41	0.246141	0.123070	−	0.992398i	$-0.460726\pi$
$953$	7241.41	0.246141	0.123070	+	0.992398i	$0.460726\pi$
$954$	0	0
$955$	−904.474	−0.0306472
$956$	26489.2	0.896151
$957$	0	0
$958$	8211.06	0.276918
$959$	0	0
$960$	0	0
$961$	−3239.93	−0.108755
$962$	−78.5240	−0.00263172
$963$	0	0
$964$	−36677.7	−1.22543
$965$	−15501.2	−0.517099
$966$	0	0
$967$	32573.3	1.08323	0.541617	−	0.840625i	$-0.317812\pi$
$967$	32573.3	1.08323	0.541617	+	0.840625i	$0.317812\pi$
$968$	−8795.27	−0.292036
$969$	0	0
$970$	−1667.90	−0.0552092
$971$	−1233.95	−0.0407820	−0.0203910	−	0.999792i	$-0.506491\pi$
$971$	−1233.95	−0.0407820	−0.0203910	+	0.999792i	$0.506491\pi$
$972$	0	0
$973$	0	0
$974$	4108.52	0.135159
$975$	0	0
$976$	11197.3	0.367229
$977$	35891.5	1.17530	0.587651	−	0.809115i	$-0.300053\pi$
$977$	35891.5	1.17530	0.587651	+	0.809115i	$0.300053\pi$
$978$	0	0
$979$	−45331.6	−1.47988
$980$	0	0
$981$	0	0
$982$	555.713	0.0180586
$983$	−26119.1	−0.847476	−0.423738	−	0.905785i	$-0.639282\pi$
$983$	−26119.1	−0.847476	−0.423738	+	0.905785i	$0.639282\pi$
$984$	0	0
$985$	−3058.51	−0.0989361
$986$	3644.31	0.117706
$987$	0	0
$988$	−40163.3	−1.29329
$989$	−5826.53	−0.187334
$990$	0	0
$991$	−7910.14	−0.253556	−0.126778	−	0.991931i	$-0.540464\pi$
$991$	−7910.14	−0.253556	−0.126778	+	0.991931i	$0.540464\pi$
$992$	−15692.6	−0.502259
$993$	0	0
$994$	0	0
$995$	−6936.95	−0.221021
$996$	0	0
$997$	−22952.3	−0.729094	−0.364547	−	0.931185i	$-0.618776\pi$
$997$	−22952.3	−0.729094	−0.364547	+	0.931185i	$0.618776\pi$
$998$	−2330.60	−0.0739215
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.4.a.by.1.4	yes	6
3.2	odd	2		2205.4.a.bx.1.3	✓	6
7.6	odd	2		2205.4.a.bx.1.4	yes	6
21.20	even	2	inner	2205.4.a.by.1.3	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2205.4.a.bx.1.3	✓	6	3.2	odd	2
2205.4.a.bx.1.4	yes	6	7.6	odd	2
2205.4.a.by.1.3	yes	6	21.20	even	2	inner
2205.4.a.by.1.4	yes	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+0.525584 q^{2} -7.72376 q^{4} +5.00000 q^{5} -8.26415 q^{8} +O(q^{10})\)	\(q+0.525584 q^{2} -7.72376 q^{4} +5.00000 q^{5} -8.26415 q^{8} +2.62792 q^{10} +48.9415 q^{11} +40.1517 q^{13} +57.4466 q^{16} +59.1704 q^{17} +129.508 q^{19} -38.6188 q^{20} +25.7228 q^{22} +93.5609 q^{23} +25.0000 q^{25} +21.1031 q^{26} +117.184 q^{29} -162.945 q^{31} +96.3062 q^{32} +31.0990 q^{34} -3.72097 q^{37} +68.0673 q^{38} -41.3208 q^{40} -222.816 q^{41} -62.2753 q^{43} -378.012 q^{44} +49.1741 q^{46} -460.327 q^{47} +13.1396 q^{50} -310.122 q^{52} -492.977 q^{53} +244.707 q^{55} +61.5902 q^{58} +153.928 q^{59} +194.916 q^{61} -85.6412 q^{62} -408.956 q^{64} +200.759 q^{65} +667.084 q^{67} -457.018 q^{68} -256.941 q^{71} +1219.59 q^{73} -1.95568 q^{74} -1000.29 q^{76} +381.236 q^{79} +287.233 q^{80} -117.108 q^{82} -281.766 q^{83} +295.852 q^{85} -32.7309 q^{86} -404.460 q^{88} -926.242 q^{89} -722.642 q^{92} -241.940 q^{94} +647.540 q^{95} -634.683 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 20 q^{4} + 30 q^{5}+O(q^{10}) \) 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

Properties

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