LMFDB - Embedded newform 2450.4.a.db.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2450,4,Mod(1,2450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2450, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2450.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2450 = 2 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2450.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:	$144.554679514$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 214x^{8} + 15801x^{6} - 479776x^{4} + 5017216x^{2} - 1411200 $ x^10 - 214x^8 + 15801x^6 - 479776x^4 + 5017216x^2 - 1411200
Coefficient ring:	$\Z[a_1, \ldots, a_{41}]$
Coefficient ring index:	$ 2^{6}\cdot 5^{2}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 490)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.537767$ of defining polynomial
Character	$\chi$	$=$	2450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} +0.537767 q^{3} +4.00000 q^{4} -1.07553 q^{6} -8.00000 q^{8} -26.7108 q^{9} +O(q^{10})$	\(q-2.00000 q^{2} +0.537767 q^{3} +4.00000 q^{4} -1.07553 q^{6} -8.00000 q^{8} -26.7108 q^{9} +14.1793 q^{11} +2.15107 q^{12} +13.9177 q^{13} +16.0000 q^{16} -65.7823 q^{17} +53.4216 q^{18} +95.1968 q^{19} -28.3586 q^{22} -69.4781 q^{23} -4.30213 q^{24} -27.8353 q^{26} -28.8839 q^{27} +127.840 q^{29} -98.3654 q^{31} -32.0000 q^{32} +7.62517 q^{33} +131.565 q^{34} -106.843 q^{36} +287.582 q^{37} -190.394 q^{38} +7.48445 q^{39} -310.112 q^{41} -197.986 q^{43} +56.7173 q^{44} +138.956 q^{46} +538.883 q^{47} +8.60427 q^{48} -35.3755 q^{51} +55.6706 q^{52} +314.867 q^{53} +57.7678 q^{54} +51.1937 q^{57} -255.680 q^{58} +242.143 q^{59} -440.612 q^{61} +196.731 q^{62} +64.0000 q^{64} -15.2503 q^{66} -858.375 q^{67} -263.129 q^{68} -37.3630 q^{69} +142.246 q^{71} +213.686 q^{72} -459.680 q^{73} -575.164 q^{74} +380.787 q^{76} -14.9689 q^{78} +1193.09 q^{79} +705.659 q^{81} +620.223 q^{82} +403.628 q^{83} +395.973 q^{86} +68.7480 q^{87} -113.435 q^{88} +1084.30 q^{89} -277.912 q^{92} -52.8977 q^{93} -1077.77 q^{94} -17.2085 q^{96} -166.723 q^{97} -378.741 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 20 q^{2} + 40 q^{4} - 80 q^{8} + 158 q^{9}+O(q^{10}) $ 10 * q - 20 * q^2 + 40 * q^4 - 80 * q^8 + 158 * q^9	$ 10 q - 20 q^{2} + 40 q^{4} - 80 q^{8} + 158 q^{9} + 52 q^{11} + 160 q^{16} - 316 q^{18} - 104 q^{22} - 400 q^{23} + 108 q^{29} - 320 q^{32} + 632 q^{36} - 1492 q^{37} + 252 q^{39} - 904 q^{43} + 208 q^{44} + 800 q^{46} - 148 q^{51} - 968 q^{53} - 3024 q^{57} - 216 q^{58} + 640 q^{64} - 1880 q^{67} - 936 q^{71} - 1264 q^{72} + 2984 q^{74} - 504 q^{78} + 3212 q^{79} + 1010 q^{81} + 1808 q^{86} - 416 q^{88} - 1600 q^{92} - 304 q^{93} - 312 q^{99}+O(q^{100}) $ 10 * q - 20 * q^2 + 40 * q^4 - 80 * q^8 + 158 * q^9 + 52 * q^11 + 160 * q^16 - 316 * q^18 - 104 * q^22 - 400 * q^23 + 108 * q^29 - 320 * q^32 + 632 * q^36 - 1492 * q^37 + 252 * q^39 - 904 * q^43 + 208 * q^44 + 800 * q^46 - 148 * q^51 - 968 * q^53 - 3024 * q^57 - 216 * q^58 + 640 * q^64 - 1880 * q^67 - 936 * q^71 - 1264 * q^72 + 2984 * q^74 - 504 * q^78 + 3212 * q^79 + 1010 * q^81 + 1808 * q^86 - 416 * q^88 - 1600 * q^92 - 304 * q^93 - 312 * q^99

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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	0.537767	0.103493	0.0517466	−	0.998660i	$-0.483521\pi$
$3$	0.537767	0.103493	0.0517466	+	0.998660i	$0.483521\pi$
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.07553	−0.0731808
$7$	0	0
$7$	0	0
$8$	−8.00000	−0.353553
$9$	−26.7108	−0.989289
$10$	0	0
$11$	14.1793	0.388657	0.194328	−	0.980937i	$-0.437747\pi$
$11$	14.1793	0.388657	0.194328	+	0.980937i	$0.437747\pi$
$12$	2.15107	0.0517466
$13$	13.9177	0.296928	0.148464	−	0.988918i	$-0.452567\pi$
$13$	13.9177	0.296928	0.148464	+	0.988918i	$0.452567\pi$
$14$	0	0
$15$	0	0
$16$	16.0000	0.250000
$17$	−65.7823	−0.938503	−0.469251	−	0.883065i	$-0.655476\pi$
$17$	−65.7823	−0.938503	−0.469251	+	0.883065i	$0.655476\pi$
$18$	53.4216	0.699533
$19$	95.1968	1.14945	0.574727	−	0.818345i	$-0.305108\pi$
$19$	95.1968	1.14945	0.574727	+	0.818345i	$0.305108\pi$
$20$	0	0
$21$	0	0
$22$	−28.3586	−0.274822
$23$	−69.4781	−0.629877	−0.314939	−	0.949112i	$-0.601984\pi$
$23$	−69.4781	−0.629877	−0.314939	+	0.949112i	$0.601984\pi$
$24$	−4.30213	−0.0365904
$25$	0	0
$26$	−27.8353	−0.209960
$27$	−28.8839	−0.205878
$28$	0	0
$29$	127.840	0.818595	0.409298	−	0.912401i	$-0.365774\pi$
$29$	127.840	0.818595	0.409298	+	0.912401i	$0.365774\pi$
$30$	0	0
$31$	−98.3654	−0.569902	−0.284951	−	0.958542i	$-0.591977\pi$
$31$	−98.3654	−0.569902	−0.284951	+	0.958542i	$0.591977\pi$
$32$	−32.0000	−0.176777
$33$	7.62517	0.0402234
$34$	131.565	0.663622
$35$	0	0
$36$	−106.843	−0.494645
$37$	287.582	1.27779	0.638895	−	0.769294i	$-0.279392\pi$
$37$	287.582	1.27779	0.638895	+	0.769294i	$0.279392\pi$
$38$	−190.394	−0.812787
$39$	7.48445	0.0307300
$40$	0	0
$41$	−310.112	−1.18125	−0.590625	−	0.806946i	$-0.701119\pi$
$41$	−310.112	−1.18125	−0.590625	+	0.806946i	$0.701119\pi$
$42$	0	0
$43$	−197.986	−0.702154	−0.351077	−	0.936347i	$-0.614184\pi$
$43$	−197.986	−0.702154	−0.351077	+	0.936347i	$0.614184\pi$
$44$	56.7173	0.194328
$45$	0	0
$46$	138.956	0.445391
$47$	538.883	1.67243	0.836215	−	0.548402i	$-0.184763\pi$
$47$	538.883	1.67243	0.836215	+	0.548402i	$0.184763\pi$
$48$	8.60427	0.0258733
$49$	0	0
$50$	0	0
$51$	−35.3755	−0.0971287
$52$	55.6706	0.148464
$53$	314.867	0.816043	0.408021	−	0.912972i	$-0.366219\pi$
$53$	314.867	0.816043	0.408021	+	0.912972i	$0.366219\pi$
$54$	57.7678	0.145578
$55$	0	0
$56$	0	0
$57$	51.1937	0.118961
$58$	−255.680	−0.578834
$59$	242.143	0.534311	0.267156	−	0.963653i	$-0.413916\pi$
$59$	242.143	0.534311	0.267156	+	0.963653i	$0.413916\pi$
$60$	0	0
$61$	−440.612	−0.924830	−0.462415	−	0.886664i	$-0.653017\pi$
$61$	−440.612	−0.924830	−0.462415	+	0.886664i	$0.653017\pi$
$62$	196.731	0.402981
$63$	0	0
$64$	64.0000	0.125000
$65$	0	0
$66$	−15.2503	−0.0284422
$67$	−858.375	−1.56518	−0.782591	−	0.622536i	$-0.786102\pi$
$67$	−858.375	−1.56518	−0.782591	+	0.622536i	$0.786102\pi$
$68$	−263.129	−0.469251
$69$	−37.3630	−0.0651881
$70$	0	0
$71$	142.246	0.237768	0.118884	−	0.992908i	$-0.462068\pi$
$71$	142.246	0.237768	0.118884	+	0.992908i	$0.462068\pi$
$72$	213.686	0.349767
$73$	−459.680	−0.737007	−0.368503	−	0.929626i	$-0.620130\pi$
$73$	−459.680	−0.737007	−0.368503	+	0.929626i	$0.620130\pi$
$74$	−575.164	−0.903534
$75$	0	0
$76$	380.787	0.574727
$77$	0	0
$78$	−14.9689	−0.0217294
$79$	1193.09	1.69915	0.849574	−	0.527470i	$-0.176859\pi$
$79$	1193.09	1.69915	0.849574	+	0.527470i	$0.176859\pi$
$80$	0	0
$81$	705.659	0.967982
$82$	620.223	0.835271
$83$	403.628	0.533782	0.266891	−	0.963727i	$-0.414004\pi$
$83$	403.628	0.533782	0.266891	+	0.963727i	$0.414004\pi$
$84$	0	0
$85$	0	0
$86$	395.973	0.496498
$87$	68.7480	0.0847191
$88$	−113.435	−0.137411
$89$	1084.30	1.29141	0.645704	−	0.763587i	$-0.276564\pi$
$89$	1084.30	1.29141	0.645704	+	0.763587i	$0.276564\pi$
$90$	0	0
$91$	0	0
$92$	−277.912	−0.314939
$93$	−52.8977	−0.0589810
$94$	−1077.77	−1.18259
$95$	0	0
$96$	−17.2085	−0.0182952
$97$	−166.723	−0.174517	−0.0872583	−	0.996186i	$-0.527811\pi$
$97$	−166.723	−0.174517	−0.0872583	+	0.996186i	$0.527811\pi$
$98$	0	0
$99$	−378.741	−0.384494
$100$	0	0
$101$	−834.364	−0.822003	−0.411002	−	0.911635i	$-0.634821\pi$
$101$	−834.364	−0.822003	−0.411002	+	0.911635i	$0.634821\pi$
$102$	70.7510	0.0686804
$103$	−1415.11	−1.35373	−0.676867	−	0.736106i	$-0.736663\pi$
$103$	−1415.11	−1.35373	−0.676867	+	0.736106i	$0.736663\pi$
$104$	−111.341	−0.104980
$105$	0	0
$106$	−629.733	−0.577029
$107$	−1569.05	−1.41763	−0.708813	−	0.705396i	$-0.750769\pi$
$107$	−1569.05	−1.41763	−0.708813	+	0.705396i	$0.750769\pi$
$108$	−115.536	−0.102939
$109$	750.399	0.659405	0.329703	−	0.944085i	$-0.393052\pi$
$109$	750.399	0.659405	0.329703	+	0.944085i	$0.393052\pi$
$110$	0	0
$111$	154.652	0.132243
$112$	0	0
$113$	−2017.09	−1.67922	−0.839611	−	0.543188i	$-0.817217\pi$
$113$	−2017.09	−1.67922	−0.839611	+	0.543188i	$0.817217\pi$
$114$	−102.387	−0.0841180
$115$	0	0
$116$	511.359	0.409298
$117$	−371.752	−0.293748
$118$	−484.287	−0.377815
$119$	0	0
$120$	0	0
$121$	−1129.95	−0.848946
$122$	881.225	0.653953
$123$	−166.768	−0.122252
$124$	−393.462	−0.284951
$125$	0	0
$126$	0	0
$127$	89.7776	0.0627281	0.0313641	−	0.999508i	$-0.490015\pi$
$127$	89.7776	0.0627281	0.0313641	+	0.999508i	$0.490015\pi$
$128$	−128.000	−0.0883883
$129$	−106.470	−0.0726682
$130$	0	0
$131$	1406.47	0.938042	0.469021	−	0.883187i	$-0.344607\pi$
$131$	1406.47	0.938042	0.469021	+	0.883187i	$0.344607\pi$
$132$	30.5007	0.0201117
$133$	0	0
$134$	1716.75	1.10675
$135$	0	0
$136$	526.258	0.331811
$137$	−646.886	−0.403410	−0.201705	−	0.979446i	$-0.564648\pi$
$137$	−646.886	−0.403410	−0.201705	+	0.979446i	$0.564648\pi$
$138$	74.7260	0.0460949
$139$	1309.84	0.799273	0.399636	−	0.916674i	$-0.369136\pi$
$139$	1309.84	0.799273	0.399636	+	0.916674i	$0.369136\pi$
$140$	0	0
$141$	289.794	0.173085
$142$	−284.493	−0.168127
$143$	197.343	0.115403
$144$	−427.373	−0.247322
$145$	0	0
$146$	919.360	0.521142
$147$	0	0
$148$	1150.33	0.638895
$149$	1457.54	0.801385	0.400692	−	0.916213i	$-0.368770\pi$
$149$	1457.54	0.801385	0.400692	+	0.916213i	$0.368770\pi$
$150$	0	0
$151$	−1943.86	−1.04761	−0.523804	−	0.851839i	$-0.675488\pi$
$151$	−1943.86	−1.04761	−0.523804	+	0.851839i	$0.675488\pi$
$152$	−761.574	−0.406394
$153$	1757.10	0.928451
$154$	0	0
$155$	0	0
$156$	29.9378	0.0153650
$157$	−1322.67	−0.672361	−0.336181	−	0.941798i	$-0.609135\pi$
$157$	−1322.67	−0.672361	−0.336181	+	0.941798i	$0.609135\pi$
$158$	−2386.17	−1.20148
$159$	169.325	0.0844549
$160$	0	0
$161$	0	0
$162$	−1411.32	−0.684467
$163$	−1133.89	−0.544866	−0.272433	−	0.962175i	$-0.587828\pi$
$163$	−1133.89	−0.544866	−0.272433	+	0.962175i	$0.587828\pi$
$164$	−1240.45	−0.590625
$165$	0	0
$166$	−807.256	−0.377441
$167$	483.314	0.223952	0.111976	−	0.993711i	$-0.464282\pi$
$167$	483.314	0.223952	0.111976	+	0.993711i	$0.464282\pi$
$168$	0	0
$169$	−2003.30	−0.911834
$170$	0	0
$171$	−2542.78	−1.13714
$172$	−791.945	−0.351077
$173$	1289.72	0.566795	0.283398	−	0.959003i	$-0.408538\pi$
$173$	1289.72	0.566795	0.283398	+	0.959003i	$0.408538\pi$
$174$	−137.496	−0.0599054
$175$	0	0
$176$	226.869	0.0971642
$177$	130.217	0.0552976
$178$	−2168.60	−0.913164
$179$	2735.61	1.14229	0.571143	−	0.820851i	$-0.306500\pi$
$179$	2735.61	1.14229	0.571143	+	0.820851i	$0.306500\pi$
$180$	0	0
$181$	−1146.35	−0.470761	−0.235380	−	0.971903i	$-0.575634\pi$
$181$	−1146.35	−0.470761	−0.235380	+	0.971903i	$0.575634\pi$
$182$	0	0
$183$	−236.947	−0.0957137
$184$	555.825	0.222695
$185$	0	0
$186$	105.795	0.0417059
$187$	−932.748	−0.364755
$188$	2155.53	0.836215
$189$	0	0
$190$	0	0
$191$	4247.74	1.60919	0.804597	−	0.593822i	$-0.202382\pi$
$191$	4247.74	1.60919	0.804597	+	0.593822i	$0.202382\pi$
$192$	34.4171	0.0129367
$193$	−5298.59	−1.97617	−0.988086	−	0.153903i	$-0.950816\pi$
$193$	−5298.59	−1.97617	−0.988086	+	0.153903i	$0.950816\pi$
$194$	333.445	0.123402
$195$	0	0
$196$	0	0
$197$	−653.948	−0.236507	−0.118253	−	0.992983i	$-0.537730\pi$
$197$	−653.948	−0.236507	−0.118253	+	0.992983i	$0.537730\pi$
$198$	757.482	0.271878
$199$	−4484.94	−1.59763	−0.798817	−	0.601574i	$-0.794540\pi$
$199$	−4484.94	−1.59763	−0.798817	+	0.601574i	$0.794540\pi$
$200$	0	0
$201$	−461.606	−0.161986
$202$	1668.73	0.581244
$203$	0	0
$204$	−141.502	−0.0485644
$205$	0	0
$206$	2830.21	0.957234
$207$	1855.82	0.623131
$208$	222.683	0.0742320
$209$	1349.83	0.446743
$210$	0	0
$211$	2160.14	0.704787	0.352394	−	0.935852i	$-0.385368\pi$
$211$	2160.14	0.704787	0.352394	+	0.935852i	$0.385368\pi$
$212$	1259.47	0.408021
$213$	76.4953	0.0246074
$214$	3138.11	1.00241
$215$	0	0
$216$	231.071	0.0727889
$217$	0	0
$218$	−1500.80	−0.466270
$219$	−247.201	−0.0762752
$220$	0	0
$221$	−915.535	−0.278668
$222$	−309.304	−0.0935096
$223$	6026.45	1.80969	0.904845	−	0.425742i	$-0.139987\pi$
$223$	6026.45	1.80969	0.904845	+	0.425742i	$0.139987\pi$
$224$	0	0
$225$	0	0
$226$	4034.19	1.18739
$227$	−4659.04	−1.36225	−0.681127	−	0.732165i	$-0.738510\pi$
$227$	−4659.04	−1.36225	−0.681127	+	0.732165i	$0.738510\pi$
$228$	204.775	0.0594804
$229$	356.000	0.102730	0.0513649	−	0.998680i	$-0.483643\pi$
$229$	356.000	0.102730	0.0513649	+	0.998680i	$0.483643\pi$
$230$	0	0
$231$	0	0
$232$	−1022.72	−0.289417
$233$	3308.76	0.930318	0.465159	−	0.885227i	$-0.345997\pi$
$233$	3308.76	0.930318	0.465159	+	0.885227i	$0.345997\pi$
$234$	743.504	0.207711
$235$	0	0
$236$	968.573	0.267156
$237$	641.602	0.175850
$238$	0	0
$239$	985.684	0.266772	0.133386	−	0.991064i	$-0.457415\pi$
$239$	985.684	0.266772	0.133386	+	0.991064i	$0.457415\pi$
$240$	0	0
$241$	−351.514	−0.0939544	−0.0469772	−	0.998896i	$-0.514959\pi$
$241$	−351.514	−0.0939544	−0.0469772	+	0.998896i	$0.514959\pi$
$242$	2259.89	0.600295
$243$	1159.34	0.306058
$244$	−1762.45	−0.462415
$245$	0	0
$246$	333.535	0.0864449
$247$	1324.92	0.341305
$248$	786.923	0.201491
$249$	217.058	0.0552429
$250$	0	0
$251$	16.6514	0.00418737	0.00209368	−	0.999998i	$-0.499334\pi$
$251$	16.6514	0.00418737	0.00209368	+	0.999998i	$0.499334\pi$
$252$	0	0
$253$	−985.152	−0.244806
$254$	−179.555	−0.0443555
$255$	0	0
$256$	256.000	0.0625000
$257$	−7095.34	−1.72216	−0.861081	−	0.508469i	$-0.830212\pi$
$257$	−7095.34	−1.72216	−0.861081	+	0.508469i	$0.830212\pi$
$258$	212.941	0.0513842
$259$	0	0
$260$	0	0
$261$	−3414.70	−0.809827
$262$	−2812.93	−0.663296
$263$	−4943.46	−1.15904	−0.579519	−	0.814959i	$-0.696760\pi$
$263$	−4943.46	−1.15904	−0.579519	+	0.814959i	$0.696760\pi$
$264$	−61.0013	−0.0142211
$265$	0	0
$266$	0	0
$267$	583.100	0.133652
$268$	−3433.50	−0.782591
$269$	−2145.29	−0.486248	−0.243124	−	0.969995i	$-0.578172\pi$
$269$	−2145.29	−0.486248	−0.243124	+	0.969995i	$0.578172\pi$
$270$	0	0
$271$	8339.99	1.86944	0.934720	−	0.355385i	$-0.115650\pi$
$271$	8339.99	1.86944	0.934720	+	0.355385i	$0.115650\pi$
$272$	−1052.52	−0.234626
$273$	0	0
$274$	1293.77	0.285254
$275$	0	0
$276$	−149.452	−0.0325940
$277$	1094.05	0.237311	0.118655	−	0.992935i	$-0.462142\pi$
$277$	1094.05	0.237311	0.118655	+	0.992935i	$0.462142\pi$
$278$	−2619.67	−0.565171
$279$	2627.42	0.563798
$280$	0	0
$281$	−5699.48	−1.20997	−0.604987	−	0.796235i	$-0.706822\pi$
$281$	−5699.48	−1.20997	−0.604987	+	0.796235i	$0.706822\pi$
$282$	−579.587	−0.122390
$283$	−6101.50	−1.28161	−0.640806	−	0.767703i	$-0.721400\pi$
$283$	−6101.50	−1.28161	−0.640806	+	0.767703i	$0.721400\pi$
$284$	568.985	0.118884
$285$	0	0
$286$	−394.686	−0.0816023
$287$	0	0
$288$	854.746	0.174883
$289$	−585.692	−0.119213
$290$	0	0
$291$	−89.6579	−0.0180613
$292$	−1838.72	−0.368503
$293$	9195.72	1.83351	0.916757	−	0.399446i	$-0.130797\pi$
$293$	9195.72	1.83351	0.916757	+	0.399446i	$0.130797\pi$
$294$	0	0
$295$	0	0
$296$	−2300.66	−0.451767
$297$	−409.554	−0.0800159
$298$	−2915.08	−0.566665
$299$	−966.972	−0.187028
$300$	0	0
$301$	0	0
$302$	3887.72	0.740771
$303$	−448.693	−0.0850718
$304$	1523.15	0.287364
$305$	0	0
$306$	−3514.20	−0.656514
$307$	7888.57	1.46653	0.733264	−	0.679944i	$-0.237996\pi$
$307$	7888.57	1.46653	0.733264	+	0.679944i	$0.237996\pi$
$308$	0	0
$309$	−760.997	−0.140102
$310$	0	0
$311$	1745.55	0.318267	0.159133	−	0.987257i	$-0.449130\pi$
$311$	1745.55	0.318267	0.159133	+	0.987257i	$0.449130\pi$
$312$	−59.8756	−0.0108647
$313$	−7968.41	−1.43898	−0.719491	−	0.694502i	$-0.755625\pi$
$313$	−7968.41	−1.43898	−0.719491	+	0.694502i	$0.755625\pi$
$314$	2645.34	0.475431
$315$	0	0
$316$	4772.34	0.849574
$317$	−3587.39	−0.635608	−0.317804	−	0.948156i	$-0.602945\pi$
$317$	−3587.39	−0.635608	−0.317804	+	0.948156i	$0.602945\pi$
$318$	−338.650	−0.0597187
$319$	1812.68	0.318153
$320$	0	0
$321$	−843.784	−0.146715
$322$	0	0
$323$	−6262.26	−1.07877
$324$	2822.64	0.483991
$325$	0	0
$326$	2267.78	0.385278
$327$	403.540	0.0682440
$328$	2480.89	0.417635
$329$	0	0
$330$	0	0
$331$	−1442.47	−0.239532	−0.119766	−	0.992802i	$-0.538214\pi$
$331$	−1442.47	−0.239532	−0.119766	+	0.992802i	$0.538214\pi$
$332$	1614.51	0.266891
$333$	−7681.55	−1.26410
$334$	−966.627	−0.158358
$335$	0	0
$336$	0	0
$337$	−5613.50	−0.907380	−0.453690	−	0.891160i	$-0.649893\pi$
$337$	−5613.50	−0.907380	−0.453690	+	0.891160i	$0.649893\pi$
$338$	4006.60	0.644764
$339$	−1084.73	−0.173788
$340$	0	0
$341$	−1394.75	−0.221496
$342$	5085.57	0.804082
$343$	0	0
$344$	1583.89	0.248249
$345$	0	0
$346$	−2579.44	−0.400785
$347$	−11217.9	−1.73547	−0.867734	−	0.497029i	$-0.834424\pi$
$347$	−11217.9	−1.73547	−0.867734	+	0.497029i	$0.834424\pi$
$348$	274.992	0.0423595
$349$	−5462.91	−0.837888	−0.418944	−	0.908012i	$-0.637600\pi$
$349$	−5462.91	−0.837888	−0.418944	+	0.908012i	$0.637600\pi$
$350$	0	0
$351$	−401.996	−0.0611309
$352$	−453.738	−0.0687055
$353$	−1434.34	−0.216268	−0.108134	−	0.994136i	$-0.534487\pi$
$353$	−1434.34	−0.216268	−0.108134	+	0.994136i	$0.534487\pi$
$354$	−260.433	−0.0391013
$355$	0	0
$356$	4337.19	0.645704
$357$	0	0
$358$	−5471.22	−0.807718
$359$	174.484	0.0256515	0.0128258	−	0.999918i	$-0.495917\pi$
$359$	174.484	0.0256515	0.0128258	+	0.999918i	$0.495917\pi$
$360$	0	0
$361$	2203.43	0.321246
$362$	2292.70	0.332878
$363$	−607.648	−0.0878602
$364$	0	0
$365$	0	0
$366$	473.893	0.0676798
$367$	12258.5	1.74356	0.871782	−	0.489893i	$-0.162964\pi$
$367$	12258.5	1.74356	0.871782	+	0.489893i	$0.162964\pi$
$368$	−1111.65	−0.157469
$369$	8283.33	1.16860
$370$	0	0
$371$	0	0
$372$	−211.591	−0.0294905
$373$	−5921.30	−0.821966	−0.410983	−	0.911643i	$-0.634814\pi$
$373$	−5921.30	−0.821966	−0.410983	+	0.911643i	$0.634814\pi$
$374$	1865.50	0.257921
$375$	0	0
$376$	−4311.07	−0.591293
$377$	1779.23	0.243064
$378$	0	0
$379$	−6322.18	−0.856857	−0.428428	−	0.903576i	$-0.640933\pi$
$379$	−6322.18	−0.856857	−0.428428	+	0.903576i	$0.640933\pi$
$380$	0	0
$381$	48.2794	0.00649194
$382$	−8495.49	−1.13787
$383$	−11997.0	−1.60057	−0.800285	−	0.599619i	$-0.795319\pi$
$383$	−11997.0	−1.60057	−0.800285	+	0.599619i	$0.795319\pi$
$384$	−68.8341	−0.00914760
$385$	0	0
$386$	10597.2	1.39736
$387$	5288.37	0.694633
$388$	−666.891	−0.0872583
$389$	−14636.9	−1.90776	−0.953879	−	0.300191i	$-0.902949\pi$
$389$	−14636.9	−1.90776	−0.953879	+	0.300191i	$0.902949\pi$
$390$	0	0
$391$	4570.43	0.591142
$392$	0	0
$393$	756.351	0.0970810
$394$	1307.90	0.167236
$395$	0	0
$396$	−1514.96	−0.192247
$397$	6298.50	0.796254	0.398127	−	0.917330i	$-0.369660\pi$
$397$	6298.50	0.796254	0.398127	+	0.917330i	$0.369660\pi$
$398$	8969.88	1.12970
$399$	0	0
$400$	0	0
$401$	−8289.93	−1.03237	−0.516183	−	0.856478i	$-0.672648\pi$
$401$	−8289.93	−1.03237	−0.516183	+	0.856478i	$0.672648\pi$
$402$	923.211	0.114541
$403$	−1369.02	−0.169220
$404$	−3337.46	−0.411002
$405$	0	0
$406$	0	0
$407$	4077.72	0.496622
$408$	283.004	0.0343402
$409$	9522.06	1.15119	0.575593	−	0.817736i	$-0.304771\pi$
$409$	9522.06	1.15119	0.575593	+	0.817736i	$0.304771\pi$
$410$	0	0
$411$	−347.874	−0.0417503
$412$	−5660.42	−0.676867
$413$	0	0
$414$	−3711.63	−0.440620
$415$	0	0
$416$	−445.365	−0.0524899
$417$	704.387	0.0827194
$418$	−2699.65	−0.315895
$419$	−8936.51	−1.04195	−0.520975	−	0.853572i	$-0.674432\pi$
$419$	−8936.51	−1.04195	−0.520975	+	0.853572i	$0.674432\pi$
$420$	0	0
$421$	−10741.8	−1.24352	−0.621760	−	0.783207i	$-0.713582\pi$
$421$	−10741.8	−1.24352	−0.621760	+	0.783207i	$0.713582\pi$
$422$	−4320.28	−0.498360
$423$	−14394.0	−1.65452
$424$	−2518.93	−0.288515
$425$	0	0
$426$	−152.991	−0.0174001
$427$	0	0
$428$	−6276.21	−0.708813
$429$	106.124	0.0119434
$430$	0	0
$431$	7886.50	0.881391	0.440695	−	0.897657i	$-0.354732\pi$
$431$	7886.50	0.881391	0.440695	+	0.897657i	$0.354732\pi$
$432$	−462.142	−0.0514695
$433$	3756.12	0.416877	0.208438	−	0.978035i	$-0.433162\pi$
$433$	3756.12	0.416877	0.208438	+	0.978035i	$0.433162\pi$
$434$	0	0
$435$	0	0
$436$	3001.60	0.329703
$437$	−6614.09	−0.724016
$438$	494.401	0.0539347
$439$	−6920.65	−0.752402	−0.376201	−	0.926538i	$-0.622770\pi$
$439$	−6920.65	−0.752402	−0.376201	+	0.926538i	$0.622770\pi$
$440$	0	0
$441$	0	0
$442$	1831.07	0.197048
$443$	−11618.9	−1.24611	−0.623057	−	0.782176i	$-0.714110\pi$
$443$	−11618.9	−1.24611	−0.623057	+	0.782176i	$0.714110\pi$
$444$	618.608	0.0661213
$445$	0	0
$446$	−12052.9	−1.27964
$447$	783.817	0.0829379
$448$	0	0
$449$	17715.7	1.86204	0.931018	−	0.364973i	$-0.118922\pi$
$449$	17715.7	1.86204	0.931018	+	0.364973i	$0.118922\pi$
$450$	0	0
$451$	−4397.17	−0.459101
$452$	−8068.37	−0.839611
$453$	−1045.34	−0.108420
$454$	9318.09	0.963259
$455$	0	0
$456$	−409.549	−0.0420590
$457$	−2724.63	−0.278890	−0.139445	−	0.990230i	$-0.544532\pi$
$457$	−2724.63	−0.278890	−0.139445	+	0.990230i	$0.544532\pi$
$458$	−712.000	−0.0726410
$459$	1900.05	0.193217
$460$	0	0
$461$	−11108.3	−1.12227	−0.561135	−	0.827724i	$-0.689635\pi$
$461$	−11108.3	−1.12227	−0.561135	+	0.827724i	$0.689635\pi$
$462$	0	0
$463$	−14640.7	−1.46957	−0.734784	−	0.678301i	$-0.762716\pi$
$463$	−14640.7	−1.46957	−0.734784	+	0.678301i	$0.762716\pi$
$464$	2045.44	0.204649
$465$	0	0
$466$	−6617.52	−0.657834
$467$	1242.76	0.123143	0.0615716	−	0.998103i	$-0.480389\pi$
$467$	1242.76	0.123143	0.0615716	+	0.998103i	$0.480389\pi$
$468$	−1487.01	−0.146874
$469$	0	0
$470$	0	0
$471$	−711.289	−0.0695849
$472$	−1937.15	−0.188908
$473$	−2807.31	−0.272897
$474$	−1283.20	−0.124345
$475$	0	0
$476$	0	0
$477$	−8410.34	−0.807302
$478$	−1971.37	−0.188637
$479$	−3197.61	−0.305015	−0.152508	−	0.988302i	$-0.548735\pi$
$479$	−3197.61	−0.305015	−0.152508	+	0.988302i	$0.548735\pi$
$480$	0	0
$481$	4002.47	0.379411
$482$	703.028	0.0664358
$483$	0	0
$484$	−4519.79	−0.424473
$485$	0	0
$486$	−2318.69	−0.216415
$487$	−7448.42	−0.693060	−0.346530	−	0.938039i	$-0.612640\pi$
$487$	−7448.42	−0.693060	−0.346530	+	0.938039i	$0.612640\pi$
$488$	3524.90	0.326977
$489$	−609.769	−0.0563900
$490$	0	0
$491$	−20139.2	−1.85106	−0.925528	−	0.378679i	$-0.876378\pi$
$491$	−20139.2	−1.85106	−0.925528	+	0.378679i	$0.876378\pi$
$492$	−667.071	−0.0611258
$493$	−8409.59	−0.768254
$494$	−2649.83	−0.241339
$495$	0	0
$496$	−1573.85	−0.142475
$497$	0	0
$498$	−434.115	−0.0390626
$499$	−17794.2	−1.59635	−0.798174	−	0.602427i	$-0.794201\pi$
$499$	−17794.2	−1.59635	−0.798174	+	0.602427i	$0.794201\pi$
$500$	0	0
$501$	259.910	0.0231775
$502$	−33.3029	−0.00296092
$503$	16316.2	1.44633	0.723166	−	0.690674i	$-0.242686\pi$
$503$	16316.2	1.44633	0.723166	+	0.690674i	$0.242686\pi$
$504$	0	0
$505$	0	0
$506$	1970.30	0.173104
$507$	−1077.31	−0.0943687
$508$	359.110	0.0313641
$509$	8586.36	0.747709	0.373854	−	0.927487i	$-0.378036\pi$
$509$	8586.36	0.747709	0.373854	+	0.927487i	$0.378036\pi$
$510$	0	0
$511$	0	0
$512$	−512.000	−0.0441942
$513$	−2749.65	−0.236647
$514$	14190.7	1.21775
$515$	0	0
$516$	−425.882	−0.0363341
$517$	7641.00	0.650001
$518$	0	0
$519$	693.568	0.0586595
$520$	0	0
$521$	8784.84	0.738715	0.369358	−	0.929287i	$-0.379578\pi$
$521$	8784.84	0.738715	0.369358	+	0.929287i	$0.379578\pi$
$522$	6829.41	0.572634
$523$	−6156.83	−0.514760	−0.257380	−	0.966310i	$-0.582859\pi$
$523$	−6156.83	−0.514760	−0.257380	+	0.966310i	$0.582859\pi$
$524$	5625.86	0.469021
$525$	0	0
$526$	9886.93	0.819564
$527$	6470.70	0.534854
$528$	122.003	0.0100558
$529$	−7339.80	−0.603254
$530$	0	0
$531$	−6467.84	−0.528588
$532$	0	0
$533$	−4316.03	−0.350746
$534$	−1166.20	−0.0945063
$535$	0	0
$536$	6867.00	0.553375
$537$	1471.12	0.118219
$538$	4290.58	0.343829
$539$	0	0
$540$	0	0
$541$	−9921.09	−0.788431	−0.394215	−	0.919018i	$-0.628984\pi$
$541$	−9921.09	−0.788431	−0.394215	+	0.919018i	$0.628984\pi$
$542$	−16680.0	−1.32189
$543$	−616.470	−0.0487206
$544$	2105.03	0.165905
$545$	0	0
$546$	0	0
$547$	−17559.1	−1.37253	−0.686264	−	0.727353i	$-0.740751\pi$
$547$	−17559.1	−1.37253	−0.686264	+	0.727353i	$0.740751\pi$
$548$	−2587.54	−0.201705
$549$	11769.1	0.914924
$550$	0	0
$551$	12169.9	0.940938
$552$	298.904	0.0230475
$553$	0	0
$554$	−2188.10	−0.167804
$555$	0	0
$556$	5239.35	0.399636
$557$	642.121	0.0488466	0.0244233	−	0.999702i	$-0.492225\pi$
$557$	642.121	0.0488466	0.0244233	+	0.999702i	$0.492225\pi$
$558$	−5254.84	−0.398665
$559$	−2755.51	−0.208489
$560$	0	0
$561$	−501.601	−0.0377497
$562$	11399.0	0.855581
$563$	−3933.27	−0.294436	−0.147218	−	0.989104i	$-0.547032\pi$
$563$	−3933.27	−0.294436	−0.147218	+	0.989104i	$0.547032\pi$
$564$	1159.17	0.0865426
$565$	0	0
$566$	12203.0	0.906237
$567$	0	0
$568$	−1137.97	−0.0840637
$569$	−15117.4	−1.11380	−0.556902	−	0.830578i	$-0.688010\pi$
$569$	−15117.4	−1.11380	−0.556902	+	0.830578i	$0.688010\pi$
$570$	0	0
$571$	−20883.6	−1.53056	−0.765282	−	0.643695i	$-0.777400\pi$
$571$	−20883.6	−1.53056	−0.765282	+	0.643695i	$0.777400\pi$
$572$	789.372	0.0577015
$573$	2284.30	0.166541
$574$	0	0
$575$	0	0
$576$	−1709.49	−0.123661
$577$	−738.564	−0.0532874	−0.0266437	−	0.999645i	$-0.508482\pi$
$577$	−738.564	−0.0532874	−0.0266437	+	0.999645i	$0.508482\pi$
$578$	1171.38	0.0842961
$579$	−2849.41	−0.204520
$580$	0	0
$581$	0	0
$582$	179.316	0.0127713
$583$	4464.60	0.317161
$584$	3677.44	0.260571
$585$	0	0
$586$	−18391.4	−1.29649
$587$	−9348.74	−0.657349	−0.328674	−	0.944443i	$-0.606602\pi$
$587$	−9348.74	−0.657349	−0.328674	+	0.944443i	$0.606602\pi$
$588$	0	0
$589$	−9364.07	−0.655076
$590$	0	0
$591$	−351.671	−0.0244769
$592$	4601.31	0.319447
$593$	20470.1	1.41755	0.708775	−	0.705434i	$-0.249248\pi$
$593$	20470.1	1.41755	0.708775	+	0.705434i	$0.249248\pi$
$594$	819.108	0.0565798
$595$	0	0
$596$	5830.16	0.400692
$597$	−2411.85	−0.165344
$598$	1933.94	0.132249
$599$	2394.09	0.163306	0.0816528	−	0.996661i	$-0.473980\pi$
$599$	2394.09	0.163306	0.0816528	+	0.996661i	$0.473980\pi$
$600$	0	0
$601$	−5007.10	−0.339840	−0.169920	−	0.985458i	$-0.554351\pi$
$601$	−5007.10	−0.339840	−0.169920	+	0.985458i	$0.554351\pi$
$602$	0	0
$603$	22927.9	1.54842
$604$	−7775.43	−0.523804
$605$	0	0
$606$	897.387	0.0601549
$607$	15037.7	1.00554	0.502768	−	0.864422i	$-0.332315\pi$
$607$	15037.7	1.00554	0.502768	+	0.864422i	$0.332315\pi$
$608$	−3046.30	−0.203197
$609$	0	0
$610$	0	0
$611$	7500.00	0.496591
$612$	7028.39	0.464225
$613$	−29396.1	−1.93687	−0.968433	−	0.249276i	$-0.919807\pi$
$613$	−29396.1	−1.93687	−0.968433	+	0.249276i	$0.919807\pi$
$614$	−15777.1	−1.03699
$615$	0	0
$616$	0	0
$617$	14648.4	0.955788	0.477894	−	0.878417i	$-0.341400\pi$
$617$	14648.4	0.955788	0.477894	+	0.878417i	$0.341400\pi$
$618$	1521.99	0.0990673
$619$	−8336.14	−0.541289	−0.270644	−	0.962679i	$-0.587237\pi$
$619$	−8336.14	−0.541289	−0.270644	+	0.962679i	$0.587237\pi$
$620$	0	0
$621$	2006.80	0.129678
$622$	−3491.10	−0.225049
$623$	0	0
$624$	119.751	0.00768251
$625$	0	0
$626$	15936.8	1.01751
$627$	725.891	0.0462349
$628$	−5290.69	−0.336181
$629$	−18917.8	−1.19921
$630$	0	0
$631$	5599.86	0.353292	0.176646	−	0.984274i	$-0.443475\pi$
$631$	5599.86	0.353292	0.176646	+	0.984274i	$0.443475\pi$
$632$	−9544.69	−0.600739
$633$	1161.65	0.0729407
$634$	7174.77	0.449442
$635$	0	0
$636$	677.299	0.0422275
$637$	0	0
$638$	−3625.36	−0.224968
$639$	−3799.51	−0.235221
$640$	0	0
$641$	18625.8	1.14770	0.573850	−	0.818960i	$-0.305449\pi$
$641$	18625.8	1.14770	0.573850	+	0.818960i	$0.305449\pi$
$642$	1687.57	0.103743
$643$	−9741.95	−0.597489	−0.298744	−	0.954333i	$-0.596568\pi$
$643$	−9741.95	−0.597489	−0.298744	+	0.954333i	$0.596568\pi$
$644$	0	0
$645$	0	0
$646$	12524.5	0.762803
$647$	−15071.8	−0.915820	−0.457910	−	0.888999i	$-0.651402\pi$
$647$	−15071.8	−0.915820	−0.457910	+	0.888999i	$0.651402\pi$
$648$	−5645.27	−0.342233
$649$	3433.43	0.207664
$650$	0	0
$651$	0	0
$652$	−4535.56	−0.272433
$653$	22142.9	1.32698	0.663489	−	0.748186i	$-0.269075\pi$
$653$	22142.9	1.32698	0.663489	+	0.748186i	$0.269075\pi$
$654$	−807.079	−0.0482558
$655$	0	0
$656$	−4961.79	−0.295313
$657$	12278.4	0.729113
$658$	0	0
$659$	−12874.3	−0.761022	−0.380511	−	0.924776i	$-0.624252\pi$
$659$	−12874.3	−0.761022	−0.380511	+	0.924776i	$0.624252\pi$
$660$	0	0
$661$	−13184.1	−0.775799	−0.387899	−	0.921702i	$-0.626799\pi$
$661$	−13184.1	−0.775799	−0.387899	+	0.921702i	$0.626799\pi$
$662$	2884.94	0.169375
$663$	−492.344	−0.0288402
$664$	−3229.02	−0.188720
$665$	0	0
$666$	15363.1	0.893856
$667$	−8882.06	−0.515615
$668$	1933.25	0.111976
$669$	3240.82	0.187291
$670$	0	0
$671$	−6247.58	−0.359441
$672$	0	0
$673$	5699.50	0.326448	0.163224	−	0.986589i	$-0.447811\pi$
$673$	5699.50	0.326448	0.163224	+	0.986589i	$0.447811\pi$
$674$	11227.0	0.641615
$675$	0	0
$676$	−8013.20	−0.455917
$677$	−23186.5	−1.31629	−0.658147	−	0.752889i	$-0.728660\pi$
$677$	−23186.5	−1.31629	−0.658147	+	0.752889i	$0.728660\pi$
$678$	2169.45	0.122887
$679$	0	0
$680$	0	0
$681$	−2505.48	−0.140984
$682$	2789.51	0.156621
$683$	9258.04	0.518666	0.259333	−	0.965788i	$-0.416497\pi$
$683$	9258.04	0.518666	0.259333	+	0.965788i	$0.416497\pi$
$684$	−10171.1	−0.568572
$685$	0	0
$686$	0	0
$687$	191.445	0.0106318
$688$	−3167.78	−0.175539
$689$	4382.21	0.242306
$690$	0	0
$691$	30149.5	1.65983	0.829913	−	0.557892i	$-0.188390\pi$
$691$	30149.5	1.65983	0.829913	+	0.557892i	$0.188390\pi$
$692$	5158.88	0.283398
$693$	0	0
$694$	22435.8	1.22716
$695$	0	0
$696$	−549.984	−0.0299527
$697$	20399.8	1.10861
$698$	10925.8	0.592476
$699$	1779.34	0.0962817
$700$	0	0
$701$	−9299.00	−0.501025	−0.250512	−	0.968113i	$-0.580599\pi$
$701$	−9299.00	−0.501025	−0.250512	+	0.968113i	$0.580599\pi$
$702$	803.992	0.0432261
$703$	27376.9	1.46876
$704$	907.476	0.0485821
$705$	0	0
$706$	2868.69	0.152924
$707$	0	0
$708$	520.867	0.0276488
$709$	26855.2	1.42252	0.711262	−	0.702927i	$-0.248124\pi$
$709$	26855.2	1.42252	0.711262	+	0.702927i	$0.248124\pi$
$710$	0	0
$711$	−31868.3	−1.68095
$712$	−8674.39	−0.456582
$713$	6834.24	0.358968
$714$	0	0
$715$	0	0
$716$	10942.4	0.571143
$717$	530.068	0.0276091
$718$	−348.967	−0.0181384
$719$	−29252.7	−1.51730	−0.758651	−	0.651498i	$-0.774141\pi$
$719$	−29252.7	−1.51730	−0.758651	+	0.651498i	$0.774141\pi$
$720$	0	0
$721$	0	0
$722$	−4406.86	−0.227155
$723$	−189.033	−0.00972365
$724$	−4585.41	−0.235380
$725$	0	0
$726$	1215.30	0.0621265
$727$	−2271.26	−0.115869	−0.0579343	−	0.998320i	$-0.518451\pi$
$727$	−2271.26	−0.115869	−0.0579343	+	0.998320i	$0.518451\pi$
$728$	0	0
$729$	−18429.3	−0.936307
$730$	0	0
$731$	13024.0	0.658974
$732$	−947.787	−0.0478568
$733$	7665.14	0.386246	0.193123	−	0.981175i	$-0.438138\pi$
$733$	7665.14	0.386246	0.193123	+	0.981175i	$0.438138\pi$
$734$	−24517.0	−1.23289
$735$	0	0
$736$	2223.30	0.111348
$737$	−12171.2	−0.608319
$738$	−16566.7	−0.826324
$739$	−4464.70	−0.222242	−0.111121	−	0.993807i	$-0.535444\pi$
$739$	−4464.70	−0.222242	−0.111121	+	0.993807i	$0.535444\pi$
$740$	0	0
$741$	712.496	0.0353228
$742$	0	0
$743$	−26438.3	−1.30542	−0.652710	−	0.757608i	$-0.726368\pi$
$743$	−26438.3	−1.30542	−0.652710	+	0.757608i	$0.726368\pi$
$744$	423.181	0.0208529
$745$	0	0
$746$	11842.6	0.581217
$747$	−10781.2	−0.528065
$748$	−3730.99	−0.182378
$749$	0	0
$750$	0	0
$751$	−30714.1	−1.49238	−0.746188	−	0.665735i	$-0.768118\pi$
$751$	−30714.1	−1.49238	−0.746188	+	0.665735i	$0.768118\pi$
$752$	8622.13	0.418108
$753$	8.95459	0.000433364 0
$754$	−3558.46	−0.171872
$755$	0	0
$756$	0	0
$757$	−7199.44	−0.345665	−0.172832	−	0.984951i	$-0.555292\pi$
$757$	−7199.44	−0.345665	−0.172832	+	0.984951i	$0.555292\pi$
$758$	12644.4	0.605889
$759$	−529.782	−0.0253358
$760$	0	0
$761$	16957.6	0.807768	0.403884	−	0.914810i	$-0.367660\pi$
$761$	16957.6	0.807768	0.403884	+	0.914810i	$0.367660\pi$
$762$	−96.5588	−0.00459049
$763$	0	0
$764$	16991.0	0.804597
$765$	0	0
$766$	23994.0	1.13177
$767$	3370.07	0.158652
$768$	137.668	0.00646833
$769$	9879.39	0.463277	0.231638	−	0.972802i	$-0.425591\pi$
$769$	9879.39	0.463277	0.231638	+	0.972802i	$0.425591\pi$
$770$	0	0
$771$	−3815.64	−0.178232
$772$	−21194.4	−0.988086
$773$	21797.7	1.01424	0.507121	−	0.861875i	$-0.330710\pi$
$773$	21797.7	1.01424	0.507121	+	0.861875i	$0.330710\pi$
$774$	−10576.7	−0.491180
$775$	0	0
$776$	1333.78	0.0617010
$777$	0	0
$778$	29273.7	1.34899
$779$	−29521.6	−1.35779
$780$	0	0
$781$	2016.96	0.0924102
$782$	−9140.85	−0.418000
$783$	−3692.51	−0.168531
$784$	0	0
$785$	0	0
$786$	−1512.70	−0.0686467
$787$	5954.44	0.269699	0.134849	−	0.990866i	$-0.456945\pi$
$787$	5954.44	0.269699	0.134849	+	0.990866i	$0.456945\pi$
$788$	−2615.79	−0.118253
$789$	−2658.43	−0.119953
$790$	0	0
$791$	0	0
$792$	3029.93	0.135939
$793$	−6132.29	−0.274608
$794$	−12597.0	−0.563037
$795$	0	0
$796$	−17939.8	−0.798817
$797$	−30426.4	−1.35227	−0.676134	−	0.736779i	$-0.736346\pi$
$797$	−30426.4	−1.35227	−0.676134	+	0.736779i	$0.736346\pi$
$798$	0	0
$799$	−35449.0	−1.56958
$800$	0	0
$801$	−28962.5	−1.27758
$802$	16579.9	0.729994
$803$	−6517.95	−0.286443
$804$	−1846.42	−0.0809929
$805$	0	0
$806$	2738.03	0.119656
$807$	−1153.67	−0.0503234
$808$	6674.91	0.290622
$809$	12450.3	0.541074	0.270537	−	0.962710i	$-0.412799\pi$
$809$	12450.3	0.541074	0.270537	+	0.962710i	$0.412799\pi$
$810$	0	0
$811$	−3330.98	−0.144225	−0.0721126	−	0.997397i	$-0.522974\pi$
$811$	−3330.98	−0.144225	−0.0721126	+	0.997397i	$0.522974\pi$
$812$	0	0
$813$	4484.97	0.193474
$814$	−8155.44	−0.351165
$815$	0	0
$816$	−566.008	−0.0242822
$817$	−18847.7	−0.807094
$818$	−19044.1	−0.814012
$819$	0	0
$820$	0	0
$821$	4257.43	0.180981	0.0904905	−	0.995897i	$-0.471157\pi$
$821$	4257.43	0.180981	0.0904905	+	0.995897i	$0.471157\pi$
$822$	695.748	0.0295219
$823$	9845.13	0.416986	0.208493	−	0.978024i	$-0.433144\pi$
$823$	9845.13	0.416986	0.208493	+	0.978024i	$0.433144\pi$
$824$	11320.8	0.478617
$825$	0	0
$826$	0	0
$827$	31609.5	1.32911	0.664553	−	0.747242i	$-0.268622\pi$
$827$	31609.5	1.32911	0.664553	+	0.747242i	$0.268622\pi$
$828$	7423.26	0.311565
$829$	448.283	0.0187811	0.00939054	−	0.999956i	$-0.497011\pi$
$829$	448.283	0.0187811	0.00939054	+	0.999956i	$0.497011\pi$
$830$	0	0
$831$	588.344	0.0245601
$832$	890.730	0.0371160
$833$	0	0
$834$	−1408.77	−0.0584914
$835$	0	0
$836$	5399.30	0.223372
$837$	2841.18	0.117330
$838$	17873.0	0.736770
$839$	14956.1	0.615424	0.307712	−	0.951480i	$-0.400437\pi$
$839$	14956.1	0.615424	0.307712	+	0.951480i	$0.400437\pi$
$840$	0	0
$841$	−8045.98	−0.329902
$842$	21483.6	0.879302
$843$	−3064.99	−0.125224
$844$	8640.55	0.352394
$845$	0	0
$846$	28788.0	1.16992
$847$	0	0
$848$	5037.87	0.204011
$849$	−3281.18	−0.132638
$850$	0	0
$851$	−19980.7	−0.804851
$852$	305.981	0.0123037
$853$	−45363.9	−1.82090	−0.910452	−	0.413615i	$-0.864266\pi$
$853$	−45363.9	−1.82090	−0.910452	+	0.413615i	$0.864266\pi$
$854$	0	0
$855$	0	0
$856$	12552.4	0.501207
$857$	−1714.18	−0.0683258	−0.0341629	−	0.999416i	$-0.510877\pi$
$857$	−1714.18	−0.0683258	−0.0341629	+	0.999416i	$0.510877\pi$
$858$	−212.249	−0.00844529
$859$	34820.1	1.38306	0.691529	−	0.722348i	$-0.256937\pi$
$859$	34820.1	1.38306	0.691529	+	0.722348i	$0.256937\pi$
$860$	0	0
$861$	0	0
$862$	−15773.0	−0.623237
$863$	7309.71	0.288326	0.144163	−	0.989554i	$-0.453951\pi$
$863$	7309.71	0.288326	0.144163	+	0.989554i	$0.453951\pi$
$864$	924.284	0.0363944
$865$	0	0
$866$	−7512.24	−0.294776
$867$	−314.966	−0.0123377
$868$	0	0
$869$	16917.1	0.660385
$870$	0	0
$871$	−11946.6	−0.464746
$872$	−6003.19	−0.233135
$873$	4453.30	0.172647
$874$	13228.2	0.511956
$875$	0	0
$876$	−988.803	−0.0381376
$877$	−2992.90	−0.115237	−0.0576186	−	0.998339i	$-0.518351\pi$
$877$	−2992.90	−0.115237	−0.0576186	+	0.998339i	$0.518351\pi$
$878$	13841.3	0.532029
$879$	4945.15	0.189756
$880$	0	0
$881$	−28667.6	−1.09629	−0.548147	−	0.836382i	$-0.684667\pi$
$881$	−28667.6	−1.09629	−0.548147	+	0.836382i	$0.684667\pi$
$882$	0	0
$883$	23488.0	0.895169	0.447585	−	0.894242i	$-0.352284\pi$
$883$	23488.0	0.895169	0.447585	+	0.894242i	$0.352284\pi$
$884$	−3662.14	−0.139334
$885$	0	0
$886$	23237.7	0.881136
$887$	22944.5	0.868546	0.434273	−	0.900781i	$-0.357005\pi$
$887$	22944.5	0.868546	0.434273	+	0.900781i	$0.357005\pi$
$888$	−1237.22	−0.0467548
$889$	0	0
$890$	0	0
$891$	10005.8	0.376213
$892$	24105.8	0.904845
$893$	51300.0	1.92238
$894$	−1567.63	−0.0586460
$895$	0	0
$896$	0	0
$897$	−520.005	−0.0193562
$898$	−35431.3	−1.31666
$899$	−12575.0	−0.466519
$900$	0	0
$901$	−20712.7	−0.765858
$902$	8794.34	0.324634
$903$	0	0
$904$	16136.7	0.593695
$905$	0	0
$906$	2090.68	0.0766648
$907$	−14063.6	−0.514855	−0.257428	−	0.966298i	$-0.582875\pi$
$907$	−14063.6	−0.514855	−0.257428	+	0.966298i	$0.582875\pi$
$908$	−18636.2	−0.681127
$909$	22286.5	0.813199
$910$	0	0
$911$	−1591.22	−0.0578700	−0.0289350	−	0.999581i	$-0.509212\pi$
$911$	−1591.22	−0.0578700	−0.0289350	+	0.999581i	$0.509212\pi$
$912$	819.099	0.0297402
$913$	5723.17	0.207458
$914$	5449.26	0.197205
$915$	0	0
$916$	1424.00	0.0513649
$917$	0	0
$918$	−3800.10	−0.136625
$919$	33319.1	1.19597	0.597984	−	0.801508i	$-0.295969\pi$
$919$	33319.1	1.19597	0.597984	+	0.801508i	$0.295969\pi$
$920$	0	0
$921$	4242.21	0.151776
$922$	22216.7	0.793565
$923$	1979.74	0.0706000
$924$	0	0
$925$	0	0
$926$	29281.4	1.03914
$927$	37798.6	1.33923
$928$	−4090.87	−0.144709
$929$	44409.8	1.56839	0.784197	−	0.620512i	$-0.213075\pi$
$929$	44409.8	1.56839	0.784197	+	0.620512i	$0.213075\pi$
$930$	0	0
$931$	0	0
$932$	13235.0	0.465159
$933$	938.698	0.0329385
$934$	−2485.51	−0.0870754
$935$	0	0
$936$	2974.02	0.103855
$937$	−25069.7	−0.874055	−0.437028	−	0.899448i	$-0.643969\pi$
$937$	−25069.7	−0.874055	−0.437028	+	0.899448i	$0.643969\pi$
$938$	0	0
$939$	−4285.15	−0.148925
$940$	0	0
$941$	−27867.4	−0.965409	−0.482704	−	0.875783i	$-0.660345\pi$
$941$	−27867.4	−0.965409	−0.482704	+	0.875783i	$0.660345\pi$
$942$	1422.58	0.0492039
$943$	21546.0	0.744043
$944$	3874.29	0.133578
$945$	0	0
$946$	5614.62	0.192967
$947$	31947.8	1.09627	0.548133	−	0.836391i	$-0.315339\pi$
$947$	31947.8	1.09627	0.548133	+	0.836391i	$0.315339\pi$
$948$	2566.41	0.0879252
$949$	−6397.67	−0.218838
$950$	0	0
$951$	−1929.18	−0.0657811
$952$	0	0
$953$	13026.8	0.442792	0.221396	−	0.975184i	$-0.428939\pi$
$953$	13026.8	0.442792	0.221396	+	0.975184i	$0.428939\pi$
$954$	16820.7	0.570849
$955$	0	0
$956$	3942.74	0.133386
$957$	974.800	0.0329266
$958$	6395.21	0.215678
$959$	0	0
$960$	0	0
$961$	−20115.2	−0.675212
$962$	−8004.94	−0.268284
$963$	41910.7	1.40244
$964$	−1406.06	−0.0469772
$965$	0	0
$966$	0	0
$967$	25825.8	0.858844	0.429422	−	0.903104i	$-0.358717\pi$
$967$	25825.8	0.858844	0.429422	+	0.903104i	$0.358717\pi$
$968$	9039.58	0.300148
$969$	−3367.64	−0.111645
$970$	0	0
$971$	27715.9	0.916011	0.458005	−	0.888949i	$-0.348564\pi$
$971$	27715.9	0.916011	0.458005	+	0.888949i	$0.348564\pi$
$972$	4637.38	0.153029
$973$	0	0
$974$	14896.8	0.490067
$975$	0	0
$976$	−7049.80	−0.231207
$977$	−16522.4	−0.541043	−0.270521	−	0.962714i	$-0.587196\pi$
$977$	−16522.4	−0.541043	−0.270521	+	0.962714i	$0.587196\pi$
$978$	1219.54	0.0398737
$979$	15374.6	0.501915
$980$	0	0
$981$	−20043.8	−0.652342
$982$	40278.4	1.30889
$983$	16046.6	0.520659	0.260330	−	0.965520i	$-0.416169\pi$
$983$	16046.6	0.520659	0.260330	+	0.965520i	$0.416169\pi$
$984$	1334.14	0.0432224
$985$	0	0
$986$	16819.2	0.543237
$987$	0	0
$988$	5299.67	0.170653
$989$	13755.7	0.442271
$990$	0	0
$991$	7725.20	0.247628	0.123814	−	0.992305i	$-0.460487\pi$
$991$	7725.20	0.247628	0.123814	+	0.992305i	$0.460487\pi$
$992$	3147.69	0.100745
$993$	−775.711	−0.0247900
$994$	0	0
$995$	0	0
$996$	868.231	0.0276214
$997$	−11167.4	−0.354738	−0.177369	−	0.984144i	$-0.556759\pi$
$997$	−11167.4	−0.354738	−0.177369	+	0.984144i	$0.556759\pi$
$998$	35588.4	1.12879
$999$	−8306.49	−0.263069

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2450.4.a.db.1.6		10
5.2	odd	4		490.4.c.g.99.5	✓	20
5.3	odd	4		490.4.c.g.99.16	yes	20
5.4	even	2		2450.4.a.dc.1.5		10
7.6	odd	2	inner	2450.4.a.db.1.5		10
35.13	even	4		490.4.c.g.99.15	yes	20
35.27	even	4		490.4.c.g.99.6	yes	20
35.34	odd	2		2450.4.a.dc.1.6		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
490.4.c.g.99.5	✓	20	5.2	odd	4
490.4.c.g.99.6	yes	20	35.27	even	4
490.4.c.g.99.15	yes	20	35.13	even	4
490.4.c.g.99.16	yes	20	5.3	odd	4
2450.4.a.db.1.5		10	7.6	odd	2	inner
2450.4.a.db.1.6		10	1.1	even	1	trivial
2450.4.a.dc.1.5		10	5.4	even	2
2450.4.a.dc.1.6		10	35.34	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:	\( N \)	\(=\)	\( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2450.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +0.537767 q^{3} +4.00000 q^{4} -1.07553 q^{6} -8.00000 q^{8} -26.7108 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} +0.537767 q^{3} +4.00000 q^{4} -1.07553 q^{6} -8.00000 q^{8} -26.7108 q^{9} +14.1793 q^{11} +2.15107 q^{12} +13.9177 q^{13} +16.0000 q^{16} -65.7823 q^{17} +53.4216 q^{18} +95.1968 q^{19} -28.3586 q^{22} -69.4781 q^{23} -4.30213 q^{24} -27.8353 q^{26} -28.8839 q^{27} +127.840 q^{29} -98.3654 q^{31} -32.0000 q^{32} +7.62517 q^{33} +131.565 q^{34} -106.843 q^{36} +287.582 q^{37} -190.394 q^{38} +7.48445 q^{39} -310.112 q^{41} -197.986 q^{43} +56.7173 q^{44} +138.956 q^{46} +538.883 q^{47} +8.60427 q^{48} -35.3755 q^{51} +55.6706 q^{52} +314.867 q^{53} +57.7678 q^{54} +51.1937 q^{57} -255.680 q^{58} +242.143 q^{59} -440.612 q^{61} +196.731 q^{62} +64.0000 q^{64} -15.2503 q^{66} -858.375 q^{67} -263.129 q^{68} -37.3630 q^{69} +142.246 q^{71} +213.686 q^{72} -459.680 q^{73} -575.164 q^{74} +380.787 q^{76} -14.9689 q^{78} +1193.09 q^{79} +705.659 q^{81} +620.223 q^{82} +403.628 q^{83} +395.973 q^{86} +68.7480 q^{87} -113.435 q^{88} +1084.30 q^{89} -277.912 q^{92} -52.8977 q^{93} -1077.77 q^{94} -17.2085 q^{96} -166.723 q^{97} -378.741 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 20 q^{2} + 40 q^{4} - 80 q^{8} + 158 q^{9}+O(q^{10}) \) 10 * q - 20 * q^2 + 40 * q^4 - 80 * q^8 + 158 * q^9	\( 10 q - 20 q^{2} + 40 q^{4} - 80 q^{8} + 158 q^{9} + 52 q^{11} + 160 q^{16} - 316 q^{18} - 104 q^{22} - 400 q^{23} + 108 q^{29} - 320 q^{32} + 632 q^{36} - 1492 q^{37} + 252 q^{39} - 904 q^{43} + 208 q^{44} + 800 q^{46} - 148 q^{51} - 968 q^{53} - 3024 q^{57} - 216 q^{58} + 640 q^{64} - 1880 q^{67} - 936 q^{71} - 1264 q^{72} + 2984 q^{74} - 504 q^{78} + 3212 q^{79} + 1010 q^{81} + 1808 q^{86} - 416 q^{88} - 1600 q^{92} - 304 q^{93} - 312 q^{99}+O(q^{100}) \) 10 * q - 20 * q^2 + 40 * q^4 - 80 * q^8 + 158 * q^9 + 52 * q^11 + 160 * q^16 - 316 * q^18 - 104 * q^22 - 400 * q^23 + 108 * q^29 - 320 * q^32 + 632 * q^36 - 1492 * q^37 + 252 * q^39 - 904 * q^43 + 208 * q^44 + 800 * q^46 - 148 * q^51 - 968 * q^53 - 3024 * q^57 - 216 * q^58 + 640 * q^64 - 1880 * q^67 - 936 * q^71 - 1264 * q^72 + 2984 * q^74 - 504 * q^78 + 3212 * q^79 + 1010 * q^81 + 1808 * q^86 - 416 * q^88 - 1600 * q^92 - 304 * q^93 - 312 * q^99

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