LMFDB - Embedded newform 2450.4.a.cs.1.3

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:	$4$
Coefficient field:	4.4.10197128.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 39x^{2} + 98 $ x^4 - 39*x^2 + 98
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 490)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.64308$ 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)

$f(q)$	$=$	$q+2.00000 q^{2} +1.64308 q^{3} +4.00000 q^{4} +3.28615 q^{6} +8.00000 q^{8} -24.3003 q^{9} +O(q^{10})$	\(q+2.00000 q^{2} +1.64308 q^{3} +4.00000 q^{4} +3.28615 q^{6} +8.00000 q^{8} -24.3003 q^{9} +45.9009 q^{11} +6.57231 q^{12} -18.6843 q^{13} +16.0000 q^{16} -105.839 q^{17} -48.6006 q^{18} +141.564 q^{19} +91.8018 q^{22} -155.003 q^{23} +13.1446 q^{24} -37.3685 q^{26} -84.2903 q^{27} -90.6997 q^{29} +82.2962 q^{31} +32.0000 q^{32} +75.4187 q^{33} -211.677 q^{34} -97.2012 q^{36} -289.802 q^{37} +283.128 q^{38} -30.6997 q^{39} +326.224 q^{41} +78.4024 q^{43} +183.604 q^{44} -310.006 q^{46} -227.569 q^{47} +26.2892 q^{48} -173.901 q^{51} -74.7371 q^{52} +10.1982 q^{53} -168.581 q^{54} +232.601 q^{57} -181.399 q^{58} -829.316 q^{59} -661.061 q^{61} +164.592 q^{62} +64.0000 q^{64} +150.837 q^{66} -35.1952 q^{67} -423.354 q^{68} -254.682 q^{69} -138.595 q^{71} -194.402 q^{72} +380.349 q^{73} -579.604 q^{74} +566.256 q^{76} -61.3994 q^{78} -236.111 q^{79} +517.612 q^{81} +652.449 q^{82} +890.909 q^{83} +156.805 q^{86} -149.027 q^{87} +367.207 q^{88} +121.028 q^{89} -620.012 q^{92} +135.219 q^{93} -455.137 q^{94} +52.5785 q^{96} -1459.18 q^{97} -1115.41 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{2} + 16 q^{4} + 32 q^{8} - 30 q^{9}+O(q^{10}) $ 4 * q + 8 * q^2 + 16 * q^4 + 32 * q^8 - 30 * q^9	$ 4 q + 8 q^{2} + 16 q^{4} + 32 q^{8} - 30 q^{9} - 18 q^{11} + 64 q^{16} - 60 q^{18} - 36 q^{22} + 52 q^{23} - 430 q^{29} + 128 q^{32} - 120 q^{36} - 756 q^{37} - 190 q^{39} - 224 q^{43} - 72 q^{44} + 104 q^{46} - 494 q^{51} + 444 q^{53} + 796 q^{57} - 860 q^{58} + 256 q^{64} - 1216 q^{67} - 1764 q^{71} - 240 q^{72} - 1512 q^{74} - 380 q^{78} + 1542 q^{79} - 752 q^{81} - 448 q^{86} - 144 q^{88} + 208 q^{92} - 3760 q^{93} - 3252 q^{99}+O(q^{100}) $ 4 * q + 8 * q^2 + 16 * q^4 + 32 * q^8 - 30 * q^9 - 18 * q^11 + 64 * q^16 - 60 * q^18 - 36 * q^22 + 52 * q^23 - 430 * q^29 + 128 * q^32 - 120 * q^36 - 756 * q^37 - 190 * q^39 - 224 * q^43 - 72 * q^44 + 104 * q^46 - 494 * q^51 + 444 * q^53 + 796 * q^57 - 860 * q^58 + 256 * q^64 - 1216 * q^67 - 1764 * q^71 - 240 * q^72 - 1512 * q^74 - 380 * q^78 + 1542 * q^79 - 752 * q^81 - 448 * q^86 - 144 * q^88 + 208 * q^92 - 3760 * q^93 - 3252 * 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
$3$	1.64308	0.316210	0.158105	−	0.987422i	$-0.449461\pi$
$3$	1.64308	0.316210	0.158105	+	0.987422i	$0.449461\pi$
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	3.28615	0.223594
$7$	0	0
$7$	0	0
$8$	8.00000	0.353553
$9$	−24.3003	−0.900011
$10$	0	0
$11$	45.9009	1.25815	0.629075	−	0.777345i	$-0.283434\pi$
$11$	45.9009	1.25815	0.629075	+	0.777345i	$0.283434\pi$
$12$	6.57231	0.158105
$13$	−18.6843	−0.398622	−0.199311	−	0.979936i	$-0.563870\pi$
$13$	−18.6843	−0.398622	−0.199311	+	0.979936i	$0.563870\pi$
$14$	0	0
$15$	0	0
$16$	16.0000	0.250000
$17$	−105.839	−1.50998	−0.754989	−	0.655738i	$-0.772358\pi$
$17$	−105.839	−1.50998	−0.754989	+	0.655738i	$0.772358\pi$
$18$	−48.6006	−0.636404
$19$	141.564	1.70932	0.854658	−	0.519191i	$-0.173767\pi$
$19$	141.564	1.70932	0.854658	+	0.519191i	$0.173767\pi$
$20$	0	0
$21$	0	0
$22$	91.8018	0.889646
$23$	−155.003	−1.40523	−0.702616	−	0.711569i	$-0.747985\pi$
$23$	−155.003	−1.40523	−0.702616	+	0.711569i	$0.747985\pi$
$24$	13.1446	0.111797
$25$	0	0
$26$	−37.3685	−0.281868
$27$	−84.2903	−0.600803
$28$	0	0
$29$	−90.6997	−0.580776	−0.290388	−	0.956909i	$-0.593784\pi$
$29$	−90.6997	−0.580776	−0.290388	+	0.956909i	$0.593784\pi$
$30$	0	0
$31$	82.2962	0.476801	0.238401	−	0.971167i	$-0.423377\pi$
$31$	82.2962	0.476801	0.238401	+	0.971167i	$0.423377\pi$
$32$	32.0000	0.176777
$33$	75.4187	0.397840
$34$	−211.677	−1.06772
$35$	0	0
$36$	−97.2012	−0.450006
$37$	−289.802	−1.28765	−0.643826	−	0.765172i	$-0.722654\pi$
$37$	−289.802	−1.28765	−0.643826	+	0.765172i	$0.722654\pi$
$38$	283.128	1.20867
$39$	−30.6997	−0.126048
$40$	0	0
$41$	326.224	1.24263	0.621313	−	0.783562i	$-0.286599\pi$
$41$	326.224	1.24263	0.621313	+	0.783562i	$0.286599\pi$
$42$	0	0
$43$	78.4024	0.278052	0.139026	−	0.990289i	$-0.455603\pi$
$43$	78.4024	0.278052	0.139026	+	0.990289i	$0.455603\pi$
$44$	183.604	0.629075
$45$	0	0
$46$	−310.006	−0.993650
$47$	−227.569	−0.706262	−0.353131	−	0.935574i	$-0.614883\pi$
$47$	−227.569	−0.706262	−0.353131	+	0.935574i	$0.614883\pi$
$48$	26.2892	0.0790526
$49$	0	0
$50$	0	0
$51$	−173.901	−0.477471
$52$	−74.7371	−0.199311
$53$	10.1982	0.0264308	0.0132154	−	0.999913i	$-0.495793\pi$
$53$	10.1982	0.0264308	0.0132154	+	0.999913i	$0.495793\pi$
$54$	−168.581	−0.424832
$55$	0	0
$56$	0	0
$57$	232.601	0.540504
$58$	−181.399	−0.410671
$59$	−829.316	−1.82996	−0.914981	−	0.403497i	$-0.867795\pi$
$59$	−829.316	−1.82996	−0.914981	+	0.403497i	$0.867795\pi$
$60$	0	0
$61$	−661.061	−1.38754	−0.693772	−	0.720195i	$-0.744053\pi$
$61$	−661.061	−1.38754	−0.693772	+	0.720195i	$0.744053\pi$
$62$	164.592	0.337149
$63$	0	0
$64$	64.0000	0.125000
$65$	0	0
$66$	150.837	0.281315
$67$	−35.1952	−0.0641759	−0.0320879	−	0.999485i	$-0.510216\pi$
$67$	−35.1952	−0.0641759	−0.0320879	+	0.999485i	$0.510216\pi$
$68$	−423.354	−0.754989
$69$	−254.682	−0.444349
$70$	0	0
$71$	−138.595	−0.231664	−0.115832	−	0.993269i	$-0.536953\pi$
$71$	−138.595	−0.231664	−0.115832	+	0.993269i	$0.536953\pi$
$72$	−194.402	−0.318202
$73$	380.349	0.609816	0.304908	−	0.952382i	$-0.401374\pi$
$73$	380.349	0.609816	0.304908	+	0.952382i	$0.401374\pi$
$74$	−579.604	−0.910507
$75$	0	0
$76$	566.256	0.854658
$77$	0	0
$78$	−61.3994	−0.0891297
$79$	−236.111	−0.336260	−0.168130	−	0.985765i	$-0.553773\pi$
$79$	−236.111	−0.336260	−0.168130	+	0.985765i	$0.553773\pi$
$80$	0	0
$81$	517.612	0.710031
$82$	652.449	0.878670
$83$	890.909	1.17819	0.589096	−	0.808063i	$-0.299484\pi$
$83$	890.909	1.17819	0.589096	+	0.808063i	$0.299484\pi$
$84$	0	0
$85$	0	0
$86$	156.805	0.196613
$87$	−149.027	−0.183647
$88$	367.207	0.444823
$89$	121.028	0.144145	0.0720727	−	0.997399i	$-0.477039\pi$
$89$	121.028	0.144145	0.0720727	+	0.997399i	$0.477039\pi$
$90$	0	0
$91$	0	0
$92$	−620.012	−0.702616
$93$	135.219	0.150769
$94$	−455.137	−0.499402
$95$	0	0
$96$	52.5785	0.0558986
$97$	−1459.18	−1.52740	−0.763700	−	0.645572i	$-0.776619\pi$
$97$	−1459.18	−1.52740	−0.763700	+	0.645572i	$0.776619\pi$
$98$	0	0
$99$	−1115.41	−1.13235
$100$	0	0
$101$	1366.89	1.34664	0.673321	−	0.739350i	$-0.264867\pi$
$101$	1366.89	1.34664	0.673321	+	0.739350i	$0.264867\pi$
$102$	−347.802	−0.337623
$103$	−386.993	−0.370209	−0.185105	−	0.982719i	$-0.559262\pi$
$103$	−386.993	−0.370209	−0.185105	+	0.982719i	$0.559262\pi$
$104$	−149.474	−0.140934
$105$	0	0
$106$	20.3964	0.0186894
$107$	−105.586	−0.0953959	−0.0476979	−	0.998862i	$-0.515188\pi$
$107$	−105.586	−0.0953959	−0.0476979	+	0.998862i	$0.515188\pi$
$108$	−337.161	−0.300402
$109$	−461.288	−0.405352	−0.202676	−	0.979246i	$-0.564964\pi$
$109$	−461.288	−0.405352	−0.202676	+	0.979246i	$0.564964\pi$
$110$	0	0
$111$	−476.167	−0.407169
$112$	0	0
$113$	−1951.01	−1.62421	−0.812106	−	0.583509i	$-0.801679\pi$
$113$	−1951.01	−1.62421	−0.812106	+	0.583509i	$0.801679\pi$
$114$	465.201	0.382194
$115$	0	0
$116$	−362.799	−0.290388
$117$	454.033	0.358764
$118$	−1658.63	−1.29398
$119$	0	0
$120$	0	0
$121$	775.892	0.582939
$122$	−1322.12	−0.981142
$123$	536.012	0.392931
$124$	329.185	0.238401
$125$	0	0
$126$	0	0
$127$	−1027.40	−0.717850	−0.358925	−	0.933366i	$-0.616857\pi$
$127$	−1027.40	−0.717850	−0.358925	+	0.933366i	$0.616857\pi$
$128$	128.000	0.0883883
$129$	128.821	0.0879230
$130$	0	0
$131$	−796.119	−0.530971	−0.265486	−	0.964115i	$-0.585532\pi$
$131$	−796.119	−0.530971	−0.265486	+	0.964115i	$0.585532\pi$
$132$	301.675	0.198920
$133$	0	0
$134$	−70.3905	−0.0453792
$135$	0	0
$136$	−846.708	−0.533858
$137$	−1468.02	−0.915487	−0.457744	−	0.889084i	$-0.651342\pi$
$137$	−1468.02	−0.915487	−0.457744	+	0.889084i	$0.651342\pi$
$138$	−509.364	−0.314202
$139$	4.48165	0.00273474	0.00136737	−	0.999999i	$-0.499565\pi$
$139$	4.48165	0.00273474	0.00136737	+	0.999999i	$0.499565\pi$
$140$	0	0
$141$	−373.913	−0.223327
$142$	−277.189	−0.163811
$143$	−857.625	−0.501526
$144$	−388.805	−0.225003
$145$	0	0
$146$	760.699	0.431205
$147$	0	0
$148$	−1159.21	−0.643826
$149$	2292.44	1.26043	0.630214	−	0.776421i	$-0.282967\pi$
$149$	2292.44	1.26043	0.630214	+	0.776421i	$0.282967\pi$
$150$	0	0
$151$	−2652.34	−1.42943	−0.714716	−	0.699415i	$-0.753444\pi$
$151$	−2652.34	−1.42943	−0.714716	+	0.699415i	$0.753444\pi$
$152$	1132.51	0.604335
$153$	2571.91	1.35900
$154$	0	0
$155$	0	0
$156$	−122.799	−0.0630242
$157$	2952.93	1.50108	0.750541	−	0.660824i	$-0.229793\pi$
$157$	2952.93	1.50108	0.750541	+	0.660824i	$0.229793\pi$
$158$	−472.222	−0.237772
$159$	16.7565	0.00835769
$160$	0	0
$161$	0	0
$162$	1035.22	0.502068
$163$	−3861.24	−1.85543	−0.927716	−	0.373287i	$-0.878231\pi$
$163$	−3861.24	−1.85543	−0.927716	+	0.373287i	$0.878231\pi$
$164$	1304.90	0.621313
$165$	0	0
$166$	1781.82	0.833107
$167$	1644.42	0.761970	0.380985	−	0.924581i	$-0.375585\pi$
$167$	1644.42	0.761970	0.380985	+	0.924581i	$0.375585\pi$
$168$	0	0
$169$	−1847.90	−0.841101
$170$	0	0
$171$	−3440.05	−1.53840
$172$	313.610	0.139026
$173$	2617.21	1.15019	0.575094	−	0.818087i	$-0.304965\pi$
$173$	2617.21	1.15019	0.575094	+	0.818087i	$0.304965\pi$
$174$	−298.053	−0.129858
$175$	0	0
$176$	734.414	0.314537
$177$	−1362.63	−0.578653
$178$	242.056	0.101926
$179$	−4640.43	−1.93767	−0.968833	−	0.247715i	$-0.920320\pi$
$179$	−4640.43	−1.93767	−0.968833	+	0.247715i	$0.920320\pi$
$180$	0	0
$181$	−765.567	−0.314387	−0.157194	−	0.987568i	$-0.550245\pi$
$181$	−765.567	−0.314387	−0.157194	+	0.987568i	$0.550245\pi$
$182$	0	0
$183$	−1086.17	−0.438756
$184$	−1240.02	−0.496825
$185$	0	0
$186$	270.438	0.106610
$187$	−4858.08	−1.89978
$188$	−910.274	−0.353131
$189$	0	0
$190$	0	0
$191$	1843.71	0.698463	0.349232	−	0.937036i	$-0.386443\pi$
$191$	1843.71	0.698463	0.349232	+	0.937036i	$0.386443\pi$
$192$	105.157	0.0395263
$193$	822.174	0.306639	0.153320	−	0.988177i	$-0.451004\pi$
$193$	822.174	0.306639	0.153320	+	0.988177i	$0.451004\pi$
$194$	−2918.37	−1.08003
$195$	0	0
$196$	0	0
$197$	−2111.42	−0.763618	−0.381809	−	0.924241i	$-0.624699\pi$
$197$	−2111.42	−0.763618	−0.381809	+	0.924241i	$0.624699\pi$
$198$	−2230.81	−0.800691
$199$	743.392	0.264813	0.132406	−	0.991196i	$-0.457730\pi$
$199$	743.392	0.264813	0.132406	+	0.991196i	$0.457730\pi$
$200$	0	0
$201$	−57.8285	−0.0202931
$202$	2733.78	0.952220
$203$	0	0
$204$	−695.604	−0.238735
$205$	0	0
$206$	−773.986	−0.261777
$207$	3766.62	1.26473
$208$	−298.948	−0.0996555
$209$	6497.91	2.15057
$210$	0	0
$211$	−2569.90	−0.838480	−0.419240	−	0.907875i	$-0.637703\pi$
$211$	−2569.90	−0.838480	−0.419240	+	0.907875i	$0.637703\pi$
$212$	40.7929	0.0132154
$213$	−227.722	−0.0732546
$214$	−211.171	−0.0674551
$215$	0	0
$216$	−674.323	−0.212416
$217$	0	0
$218$	−922.577	−0.286627
$219$	624.943	0.192830
$220$	0	0
$221$	1977.52	0.601910
$222$	−952.333	−0.287912
$223$	−4241.83	−1.27379	−0.636893	−	0.770952i	$-0.719781\pi$
$223$	−4241.83	−1.27379	−0.636893	+	0.770952i	$0.719781\pi$
$224$	0	0
$225$	0	0
$226$	−3902.03	−1.14849
$227$	905.427	0.264737	0.132369	−	0.991201i	$-0.457742\pi$
$227$	905.427	0.264737	0.132369	+	0.991201i	$0.457742\pi$
$228$	930.402	0.270252
$229$	−304.050	−0.0877389	−0.0438694	−	0.999037i	$-0.513969\pi$
$229$	−304.050	−0.0877389	−0.0438694	+	0.999037i	$0.513969\pi$
$230$	0	0
$231$	0	0
$232$	−725.598	−0.205335
$233$	−4969.61	−1.39730	−0.698648	−	0.715466i	$-0.746215\pi$
$233$	−4969.61	−1.39730	−0.698648	+	0.715466i	$0.746215\pi$
$234$	908.067	0.253685
$235$	0	0
$236$	−3317.26	−0.914981
$237$	−387.949	−0.106329
$238$	0	0
$239$	−741.901	−0.200793	−0.100397	−	0.994947i	$-0.532011\pi$
$239$	−741.901	−0.200793	−0.100397	+	0.994947i	$0.532011\pi$
$240$	0	0
$241$	−3483.88	−0.931189	−0.465594	−	0.884998i	$-0.654159\pi$
$241$	−3483.88	−0.931189	−0.465594	+	0.884998i	$0.654159\pi$
$242$	1551.78	0.412200
$243$	3126.32	0.825322
$244$	−2644.24	−0.693772
$245$	0	0
$246$	1072.02	0.277844
$247$	−2645.02	−0.681371
$248$	658.370	0.168575
$249$	1463.83	0.372556
$250$	0	0
$251$	−4081.66	−1.02642	−0.513211	−	0.858262i	$-0.671544\pi$
$251$	−4081.66	−1.02642	−0.513211	+	0.858262i	$0.671544\pi$
$252$	0	0
$253$	−7114.78	−1.76799
$254$	−2054.80	−0.507597
$255$	0	0
$256$	256.000	0.0625000
$257$	3209.00	0.778880	0.389440	−	0.921052i	$-0.372669\pi$
$257$	3209.00	0.778880	0.389440	+	0.921052i	$0.372669\pi$
$258$	257.642	0.0621710
$259$	0	0
$260$	0	0
$261$	2204.03	0.522705
$262$	−1592.24	−0.375453
$263$	5040.00	1.18167	0.590836	−	0.806792i	$-0.298798\pi$
$263$	5040.00	1.18167	0.590836	+	0.806792i	$0.298798\pi$
$264$	603.350	0.140658
$265$	0	0
$266$	0	0
$267$	198.858	0.0455803
$268$	−140.781	−0.0320879
$269$	5292.27	1.19954	0.599768	−	0.800174i	$-0.295259\pi$
$269$	5292.27	1.19954	0.599768	+	0.800174i	$0.295259\pi$
$270$	0	0
$271$	2285.68	0.512343	0.256172	−	0.966631i	$-0.417539\pi$
$271$	2285.68	0.512343	0.256172	+	0.966631i	$0.417539\pi$
$272$	−1693.42	−0.377494
$273$	0	0
$274$	−2936.05	−0.647347
$275$	0	0
$276$	−1018.73	−0.222175
$277$	5738.68	1.24478	0.622389	−	0.782708i	$-0.286162\pi$
$277$	5738.68	1.24478	0.622389	+	0.782708i	$0.286162\pi$
$278$	8.96329	0.00193375
$279$	−1999.82	−0.429126
$280$	0	0
$281$	−3098.71	−0.657842	−0.328921	−	0.944357i	$-0.606685\pi$
$281$	−3098.71	−0.657842	−0.328921	+	0.944357i	$0.606685\pi$
$282$	−747.826	−0.157916
$283$	−6876.02	−1.44430	−0.722150	−	0.691736i	$-0.756846\pi$
$283$	−6876.02	−1.44430	−0.722150	+	0.691736i	$0.756846\pi$
$284$	−554.379	−0.115832
$285$	0	0
$286$	−1715.25	−0.354632
$287$	0	0
$288$	−777.610	−0.159101
$289$	6288.80	1.28003
$290$	0	0
$291$	−2397.55	−0.482979
$292$	1521.40	0.304908
$293$	−540.278	−0.107725	−0.0538624	−	0.998548i	$-0.517153\pi$
$293$	−540.278	−0.107725	−0.0538624	+	0.998548i	$0.517153\pi$
$294$	0	0
$295$	0	0
$296$	−2318.41	−0.455254
$297$	−3869.00	−0.755900
$298$	4584.88	0.891258
$299$	2896.12	0.560157
$300$	0	0
$301$	0	0
$302$	−5304.68	−1.01076
$303$	2245.91	0.425822
$304$	2265.02	0.427329
$305$	0	0
$306$	5143.82	0.960956
$307$	2335.65	0.434210	0.217105	−	0.976148i	$-0.430339\pi$
$307$	2335.65	0.434210	0.217105	+	0.976148i	$0.430339\pi$
$308$	0	0
$309$	−635.859	−0.117064
$310$	0	0
$311$	8461.81	1.54285	0.771424	−	0.636322i	$-0.219545\pi$
$311$	8461.81	1.54285	0.771424	+	0.636322i	$0.219545\pi$
$312$	−245.598	−0.0445648
$313$	−4476.99	−0.808481	−0.404240	−	0.914653i	$-0.632464\pi$
$313$	−4476.99	−0.808481	−0.404240	+	0.914653i	$0.632464\pi$
$314$	5905.87	1.06143
$315$	0	0
$316$	−944.444	−0.168130
$317$	5106.32	0.904730	0.452365	−	0.891833i	$-0.350580\pi$
$317$	5106.32	0.904730	0.452365	+	0.891833i	$0.350580\pi$
$318$	33.5129	0.00590978
$319$	−4163.20	−0.730703
$320$	0	0
$321$	−173.485	−0.0301652
$322$	0	0
$323$	−14982.9	−2.58103
$324$	2070.45	0.355015
$325$	0	0
$326$	−7722.47	−1.31199
$327$	−757.932	−0.128177
$328$	2609.80	0.439335
$329$	0	0
$330$	0	0
$331$	6438.01	1.06908	0.534539	−	0.845144i	$-0.320485\pi$
$331$	6438.01	1.06908	0.534539	+	0.845144i	$0.320485\pi$
$332$	3563.63	0.589096
$333$	7042.27	1.15890
$334$	3288.84	0.538794
$335$	0	0
$336$	0	0
$337$	−4480.18	−0.724187	−0.362093	−	0.932142i	$-0.617938\pi$
$337$	−4480.18	−0.724187	−0.362093	+	0.932142i	$0.617938\pi$
$338$	−3695.80	−0.594748
$339$	−3205.67	−0.513593
$340$	0	0
$341$	3777.47	0.599887
$342$	−6880.10	−1.08782
$343$	0	0
$344$	627.219	0.0983064
$345$	0	0
$346$	5234.41	0.813306
$347$	−4781.32	−0.739697	−0.369848	−	0.929092i	$-0.620590\pi$
$347$	−4781.32	−0.739697	−0.369848	+	0.929092i	$0.620590\pi$
$348$	−596.106	−0.0918237
$349$	−8732.50	−1.33937	−0.669685	−	0.742645i	$-0.733571\pi$
$349$	−8732.50	−1.33937	−0.669685	+	0.742645i	$0.733571\pi$
$350$	0	0
$351$	1574.90	0.239493
$352$	1468.83	0.222411
$353$	11595.1	1.74828	0.874141	−	0.485672i	$-0.161425\pi$
$353$	11595.1	1.74828	0.874141	+	0.485672i	$0.161425\pi$
$354$	−2725.26	−0.409169
$355$	0	0
$356$	484.112	0.0720727
$357$	0	0
$358$	−9280.86	−1.37014
$359$	−7406.91	−1.08892	−0.544459	−	0.838787i	$-0.683265\pi$
$359$	−7406.91	−1.08892	−0.544459	+	0.838787i	$0.683265\pi$
$360$	0	0
$361$	13181.4	1.92176
$362$	−1531.13	−0.222305
$363$	1274.85	0.184331
$364$	0	0
$365$	0	0
$366$	−2172.35	−0.310247
$367$	11724.5	1.66762	0.833809	−	0.552053i	$-0.186155\pi$
$367$	11724.5	1.66762	0.833809	+	0.552053i	$0.186155\pi$
$368$	−2480.05	−0.351308
$369$	−7927.35	−1.11838
$370$	0	0
$371$	0	0
$372$	540.876	0.0753847
$373$	−11243.1	−1.56071	−0.780354	−	0.625338i	$-0.784961\pi$
$373$	−11243.1	−1.56071	−0.780354	+	0.625338i	$0.784961\pi$
$374$	−9716.17	−1.34335
$375$	0	0
$376$	−1820.55	−0.249701
$377$	1694.66	0.231510
$378$	0	0
$379$	4871.25	0.660210	0.330105	−	0.943944i	$-0.392916\pi$
$379$	4871.25	0.660210	0.330105	+	0.943944i	$0.392916\pi$
$380$	0	0
$381$	−1688.10	−0.226992
$382$	3687.43	0.493888
$383$	−12604.1	−1.68157	−0.840784	−	0.541370i	$-0.817906\pi$
$383$	−12604.1	−1.68157	−0.840784	+	0.541370i	$0.817906\pi$
$384$	210.314	0.0279493
$385$	0	0
$386$	1644.35	0.216827
$387$	−1905.20	−0.250250
$388$	−5836.74	−0.763700
$389$	12183.2	1.58796	0.793978	−	0.607947i	$-0.208007\pi$
$389$	12183.2	1.58796	0.793978	+	0.607947i	$0.208007\pi$
$390$	0	0
$391$	16405.3	2.12187
$392$	0	0
$393$	−1308.09	−0.167899
$394$	−4222.85	−0.539959
$395$	0	0
$396$	−4461.62	−0.566174
$397$	3731.14	0.471689	0.235845	−	0.971791i	$-0.424214\pi$
$397$	3731.14	0.471689	0.235845	+	0.971791i	$0.424214\pi$
$398$	1486.78	0.187251
$399$	0	0
$400$	0	0
$401$	−3708.81	−0.461868	−0.230934	−	0.972969i	$-0.574178\pi$
$401$	−3708.81	−0.461868	−0.230934	+	0.972969i	$0.574178\pi$
$402$	−115.657	−0.0143494
$403$	−1537.65	−0.190063
$404$	5467.57	0.673321
$405$	0	0
$406$	0	0
$407$	−13302.2	−1.62006
$408$	−1391.21	−0.168811
$409$	−7170.15	−0.866849	−0.433424	−	0.901190i	$-0.642695\pi$
$409$	−7170.15	−0.866849	−0.433424	+	0.901190i	$0.642695\pi$
$410$	0	0
$411$	−2412.08	−0.289486
$412$	−1547.97	−0.185105
$413$	0	0
$414$	7533.24	0.894296
$415$	0	0
$416$	−597.897	−0.0704671
$417$	7.36369	0.000864752 0
$418$	12995.8	1.52069
$419$	−4845.95	−0.565013	−0.282506	−	0.959265i	$-0.591166\pi$
$419$	−4845.95	−0.565013	−0.282506	+	0.959265i	$0.591166\pi$
$420$	0	0
$421$	9744.07	1.12802	0.564010	−	0.825768i	$-0.309258\pi$
$421$	9744.07	1.12802	0.564010	+	0.825768i	$0.309258\pi$
$422$	−5139.80	−0.592895
$423$	5529.99	0.635643
$424$	81.5857	0.00934470
$425$	0	0
$426$	−455.443	−0.0517988
$427$	0	0
$428$	−422.343	−0.0476979
$429$	−1409.14	−0.158588
$430$	0	0
$431$	8295.11	0.927057	0.463529	−	0.886082i	$-0.346583\pi$
$431$	8295.11	0.927057	0.463529	+	0.886082i	$0.346583\pi$
$432$	−1348.65	−0.150201
$433$	7365.97	0.817519	0.408760	−	0.912642i	$-0.365961\pi$
$433$	7365.97	0.817519	0.408760	+	0.912642i	$0.365961\pi$
$434$	0	0
$435$	0	0
$436$	−1845.15	−0.202676
$437$	−21942.8	−2.40199
$438$	1249.89	0.136351
$439$	−1950.02	−0.212003	−0.106002	−	0.994366i	$-0.533805\pi$
$439$	−1950.02	−0.212003	−0.106002	+	0.994366i	$0.533805\pi$
$440$	0	0
$441$	0	0
$442$	3955.03	0.425615
$443$	−15976.4	−1.71346	−0.856731	−	0.515763i	$-0.827508\pi$
$443$	−15976.4	−1.71346	−0.856731	+	0.515763i	$0.827508\pi$
$444$	−1904.67	−0.203584
$445$	0	0
$446$	−8483.67	−0.900702
$447$	3766.65	0.398561
$448$	0	0
$449$	−5292.90	−0.556319	−0.278159	−	0.960535i	$-0.589724\pi$
$449$	−5292.90	−0.556319	−0.278159	+	0.960535i	$0.589724\pi$
$450$	0	0
$451$	14974.0	1.56341
$452$	−7804.06	−0.812106
$453$	−4358.00	−0.452001
$454$	1810.85	0.187197
$455$	0	0
$456$	1860.80	0.191097
$457$	−15964.8	−1.63414	−0.817069	−	0.576540i	$-0.804402\pi$
$457$	−15964.8	−1.63414	−0.817069	+	0.576540i	$0.804402\pi$
$458$	−608.100	−0.0620408
$459$	8921.17	0.907199
$460$	0	0
$461$	−11072.7	−1.11867	−0.559337	−	0.828940i	$-0.688944\pi$
$461$	−11072.7	−1.11867	−0.559337	+	0.828940i	$0.688944\pi$
$462$	0	0
$463$	−11210.0	−1.12521	−0.562604	−	0.826727i	$-0.690200\pi$
$463$	−11210.0	−1.12521	−0.562604	+	0.826727i	$0.690200\pi$
$464$	−1451.20	−0.145194
$465$	0	0
$466$	−9939.22	−0.988037
$467$	11211.7	1.11096	0.555478	−	0.831532i	$-0.312535\pi$
$467$	11211.7	1.11096	0.555478	+	0.831532i	$0.312535\pi$
$468$	1816.13	0.179382
$469$	0	0
$470$	0	0
$471$	4851.90	0.474658
$472$	−6634.53	−0.646989
$473$	3598.74	0.349831
$474$	−775.897	−0.0751859
$475$	0	0
$476$	0	0
$477$	−247.820	−0.0237880
$478$	−1483.80	−0.141982
$479$	−7983.06	−0.761494	−0.380747	−	0.924679i	$-0.624333\pi$
$479$	−7983.06	−0.761494	−0.380747	+	0.924679i	$0.624333\pi$
$480$	0	0
$481$	5414.74	0.513286
$482$	−6967.76	−0.658450
$483$	0	0
$484$	3103.57	0.291470
$485$	0	0
$486$	6252.63	0.583591
$487$	−4408.56	−0.410208	−0.205104	−	0.978740i	$-0.565753\pi$
$487$	−4408.56	−0.410208	−0.205104	+	0.978740i	$0.565753\pi$
$488$	−5288.49	−0.490571
$489$	−6344.31	−0.586707
$490$	0	0
$491$	−12504.0	−1.14928	−0.574641	−	0.818405i	$-0.694858\pi$
$491$	−12504.0	−1.14928	−0.574641	+	0.818405i	$0.694858\pi$
$492$	2144.05	0.196466
$493$	9599.52	0.876959
$494$	−5290.04	−0.481802
$495$	0	0
$496$	1316.74	0.119200
$497$	0	0
$498$	2927.66	0.263437
$499$	13774.5	1.23573	0.617866	−	0.786283i	$-0.287997\pi$
$499$	13774.5	1.23573	0.617866	+	0.786283i	$0.287997\pi$
$500$	0	0
$501$	2701.91	0.240943
$502$	−8163.32	−0.725791
$503$	−20969.7	−1.85883	−0.929416	−	0.369034i	$-0.879689\pi$
$503$	−20969.7	−1.85883	−0.929416	+	0.369034i	$0.879689\pi$
$504$	0	0
$505$	0	0
$506$	−14229.6	−1.25016
$507$	−3036.24	−0.265965
$508$	−4109.60	−0.358925
$509$	21256.9	1.85107	0.925537	−	0.378658i	$-0.123614\pi$
$509$	21256.9	1.85107	0.925537	+	0.378658i	$0.123614\pi$
$510$	0	0
$511$	0	0
$512$	512.000	0.0441942
$513$	−11932.5	−1.02696
$514$	6418.01	0.550751
$515$	0	0
$516$	515.285	0.0439615
$517$	−10445.6	−0.888582
$518$	0	0
$519$	4300.27	0.363701
$520$	0	0
$521$	−11484.7	−0.965750	−0.482875	−	0.875689i	$-0.660407\pi$
$521$	−11484.7	−0.965750	−0.482875	+	0.875689i	$0.660407\pi$
$522$	4408.06	0.369608
$523$	18770.0	1.56932	0.784658	−	0.619928i	$-0.212838\pi$
$523$	18770.0	1.56932	0.784658	+	0.619928i	$0.212838\pi$
$524$	−3184.48	−0.265486
$525$	0	0
$526$	10080.0	0.835568
$527$	−8710.11	−0.719959
$528$	1206.70	0.0994599
$529$	11858.9	0.974679
$530$	0	0
$531$	20152.6	1.64699
$532$	0	0
$533$	−6095.27	−0.495338
$534$	397.717	0.0322301
$535$	0	0
$536$	−281.562	−0.0226896
$537$	−7624.59	−0.612710
$538$	10584.5	0.848200
$539$	0	0
$540$	0	0
$541$	20737.9	1.64805	0.824023	−	0.566556i	$-0.191725\pi$
$541$	20737.9	1.64805	0.824023	+	0.566556i	$0.191725\pi$
$542$	4571.35	0.362281
$543$	−1257.89	−0.0994126
$544$	−3386.83	−0.266929
$545$	0	0
$546$	0	0
$547$	11524.2	0.900800	0.450400	−	0.892827i	$-0.351281\pi$
$547$	11524.2	0.900800	0.450400	+	0.892827i	$0.351281\pi$
$548$	−5872.10	−0.457744
$549$	16064.0	1.24881
$550$	0	0
$551$	−12839.8	−0.992730
$552$	−2037.45	−0.157101
$553$	0	0
$554$	11477.4	0.880192
$555$	0	0
$556$	17.9266	0.00136737
$557$	11586.7	0.881408	0.440704	−	0.897652i	$-0.354729\pi$
$557$	11586.7	0.881408	0.440704	+	0.897652i	$0.354729\pi$
$558$	−3999.65	−0.303438
$559$	−1464.89	−0.110838
$560$	0	0
$561$	−7982.21	−0.600729
$562$	−6197.42	−0.465165
$563$	−11883.8	−0.889593	−0.444797	−	0.895632i	$-0.646724\pi$
$563$	−11883.8	−0.889593	−0.444797	+	0.895632i	$0.646724\pi$
$564$	−1495.65	−0.111664
$565$	0	0
$566$	−13752.0	−1.02127
$567$	0	0
$568$	−1108.76	−0.0819057
$569$	−18776.4	−1.38339	−0.691694	−	0.722191i	$-0.743135\pi$
$569$	−18776.4	−1.38339	−0.691694	+	0.722191i	$0.743135\pi$
$570$	0	0
$571$	9328.82	0.683711	0.341855	−	0.939753i	$-0.388945\pi$
$571$	9328.82	0.683711	0.341855	+	0.939753i	$0.388945\pi$
$572$	−3430.50	−0.250763
$573$	3029.37	0.220861
$574$	0	0
$575$	0	0
$576$	−1555.22	−0.112501
$577$	−1156.99	−0.0834767	−0.0417384	−	0.999129i	$-0.513290\pi$
$577$	−1156.99	−0.0834767	−0.0417384	+	0.999129i	$0.513290\pi$
$578$	12577.6	0.905119
$579$	1350.90	0.0969626
$580$	0	0
$581$	0	0
$582$	−4795.10	−0.341518
$583$	468.107	0.0332539
$584$	3042.80	0.215602
$585$	0	0
$586$	−1080.56	−0.0761729
$587$	22250.1	1.56450	0.782248	−	0.622968i	$-0.214073\pi$
$587$	22250.1	1.56450	0.782248	+	0.622968i	$0.214073\pi$
$588$	0	0
$589$	11650.2	0.815004
$590$	0	0
$591$	−3469.23	−0.241464
$592$	−4636.83	−0.321913
$593$	−3518.99	−0.243689	−0.121845	−	0.992549i	$-0.538881\pi$
$593$	−3518.99	−0.243689	−0.121845	+	0.992549i	$0.538881\pi$
$594$	−7738.00	−0.534502
$595$	0	0
$596$	9169.75	0.630214
$597$	1221.45	0.0837365
$598$	5792.24	0.396091
$599$	13879.3	0.946730	0.473365	−	0.880866i	$-0.343039\pi$
$599$	13879.3	0.946730	0.473365	+	0.880866i	$0.343039\pi$
$600$	0	0
$601$	−3235.75	−0.219616	−0.109808	−	0.993953i	$-0.535024\pi$
$601$	−3235.75	−0.219616	−0.109808	+	0.993953i	$0.535024\pi$
$602$	0	0
$603$	855.255	0.0577590
$604$	−10609.4	−0.714716
$605$	0	0
$606$	4491.82	0.301102
$607$	2283.47	0.152691	0.0763453	−	0.997081i	$-0.475675\pi$
$607$	2283.47	0.152691	0.0763453	+	0.997081i	$0.475675\pi$
$608$	4530.05	0.302167
$609$	0	0
$610$	0	0
$611$	4251.95	0.281531
$612$	10287.6	0.679498
$613$	−336.783	−0.0221901	−0.0110950	−	0.999938i	$-0.503532\pi$
$613$	−336.783	−0.0221901	−0.0110950	+	0.999938i	$0.503532\pi$
$614$	4671.29	0.307033
$615$	0	0
$616$	0	0
$617$	9923.71	0.647510	0.323755	−	0.946141i	$-0.395055\pi$
$617$	9923.71	0.647510	0.323755	+	0.946141i	$0.395055\pi$
$618$	−1271.72	−0.0827767
$619$	7070.90	0.459133	0.229567	−	0.973293i	$-0.426269\pi$
$619$	7070.90	0.459133	0.229567	+	0.973293i	$0.426269\pi$
$620$	0	0
$621$	13065.3	0.844268
$622$	16923.6	1.09096
$623$	0	0
$624$	−491.195	−0.0315121
$625$	0	0
$626$	−8953.98	−0.571682
$627$	10676.6	0.680034
$628$	11811.7	0.750541
$629$	30672.2	1.94433
$630$	0	0
$631$	−1932.85	−0.121942	−0.0609711	−	0.998140i	$-0.519420\pi$
$631$	−1932.85	−0.121942	−0.0609711	+	0.998140i	$0.519420\pi$
$632$	−1888.89	−0.118886
$633$	−4222.55	−0.265136
$634$	10212.6	0.639741
$635$	0	0
$636$	67.0258	0.00417885
$637$	0	0
$638$	−8326.39	−0.516685
$639$	3367.89	0.208500
$640$	0	0
$641$	12664.2	0.780354	0.390177	−	0.920740i	$-0.372414\pi$
$641$	12664.2	0.780354	0.390177	+	0.920740i	$0.372414\pi$
$642$	−346.971	−0.0213300
$643$	6058.05	0.371550	0.185775	−	0.982592i	$-0.440521\pi$
$643$	6058.05	0.371550	0.185775	+	0.982592i	$0.440521\pi$
$644$	0	0
$645$	0	0
$646$	−29965.9	−1.82506
$647$	23819.4	1.44735	0.723676	−	0.690140i	$-0.242451\pi$
$647$	23819.4	1.44735	0.723676	+	0.690140i	$0.242451\pi$
$648$	4140.90	0.251034
$649$	−38066.4	−2.30237
$650$	0	0
$651$	0	0
$652$	−15444.9	−0.927716
$653$	26054.5	1.56140	0.780698	−	0.624908i	$-0.214864\pi$
$653$	26054.5	1.56140	0.780698	+	0.624908i	$0.214864\pi$
$654$	−1515.86	−0.0906346
$655$	0	0
$656$	5219.59	0.310657
$657$	−9242.60	−0.548841
$658$	0	0
$659$	−3825.76	−0.226146	−0.113073	−	0.993587i	$-0.536069\pi$
$659$	−3825.76	−0.226146	−0.113073	+	0.993587i	$0.536069\pi$
$660$	0	0
$661$	−17551.2	−1.03277	−0.516385	−	0.856356i	$-0.672723\pi$
$661$	−17551.2	−1.03277	−0.516385	+	0.856356i	$0.672723\pi$
$662$	12876.0	0.755952
$663$	3249.21	0.190330
$664$	7127.27	0.416554
$665$	0	0
$666$	14084.5	0.819467
$667$	14058.7	0.816126
$668$	6577.68	0.380985
$669$	−6969.66	−0.402784
$670$	0	0
$671$	−30343.3	−1.74574
$672$	0	0
$673$	−12750.8	−0.730320	−0.365160	−	0.930945i	$-0.618986\pi$
$673$	−12750.8	−0.730320	−0.365160	+	0.930945i	$0.618986\pi$
$674$	−8960.36	−0.512077
$675$	0	0
$676$	−7391.59	−0.420550
$677$	6811.42	0.386683	0.193341	−	0.981132i	$-0.438068\pi$
$677$	6811.42	0.386683	0.193341	+	0.981132i	$0.438068\pi$
$678$	−6411.34	−0.363165
$679$	0	0
$680$	0	0
$681$	1487.69	0.0837126
$682$	7554.94	0.424184
$683$	20480.6	1.14739	0.573696	−	0.819068i	$-0.305509\pi$
$683$	20480.6	1.14739	0.573696	+	0.819068i	$0.305509\pi$
$684$	−13760.2	−0.769202
$685$	0	0
$686$	0	0
$687$	−499.578	−0.0277439
$688$	1254.44	0.0695131
$689$	−190.546	−0.0105359
$690$	0	0
$691$	5776.59	0.318020	0.159010	−	0.987277i	$-0.449170\pi$
$691$	5776.59	0.318020	0.159010	+	0.987277i	$0.449170\pi$
$692$	10468.8	0.575094
$693$	0	0
$694$	−9562.64	−0.523044
$695$	0	0
$696$	−1192.21	−0.0649292
$697$	−34527.1	−1.87634
$698$	−17465.0	−0.947078
$699$	−8165.45	−0.441839
$700$	0	0
$701$	−11445.0	−0.616650	−0.308325	−	0.951281i	$-0.599768\pi$
$701$	−11445.0	−0.616650	−0.308325	+	0.951281i	$0.599768\pi$
$702$	3149.81	0.169347
$703$	−41025.5	−2.20100
$704$	2937.66	0.157269
$705$	0	0
$706$	23190.1	1.23622
$707$	0	0
$708$	−5450.52	−0.289326
$709$	123.956	0.00656597	0.00328299	−	0.999995i	$-0.498955\pi$
$709$	123.956	0.00656597	0.00328299	+	0.999995i	$0.498955\pi$
$710$	0	0
$711$	5737.57	0.302638
$712$	968.224	0.0509631
$713$	−12756.2	−0.670017
$714$	0	0
$715$	0	0
$716$	−18561.7	−0.968833
$717$	−1219.00	−0.0634929
$718$	−14813.8	−0.769981
$719$	−4351.29	−0.225696	−0.112848	−	0.993612i	$-0.535997\pi$
$719$	−4351.29	−0.225696	−0.112848	+	0.993612i	$0.535997\pi$
$720$	0	0
$721$	0	0
$722$	26362.7	1.35889
$723$	−5724.29	−0.294452
$724$	−3062.27	−0.157194
$725$	0	0
$726$	2549.70	0.130342
$727$	−14618.1	−0.745745	−0.372873	−	0.927883i	$-0.621627\pi$
$727$	−14618.1	−0.745745	−0.372873	+	0.927883i	$0.621627\pi$
$728$	0	0
$729$	−8838.76	−0.449055
$730$	0	0
$731$	−8297.99	−0.419853
$732$	−4344.70	−0.219378
$733$	−13412.0	−0.675831	−0.337916	−	0.941176i	$-0.609722\pi$
$733$	−13412.0	−0.675831	−0.337916	+	0.941176i	$0.609722\pi$
$734$	23449.1	1.17918
$735$	0	0
$736$	−4960.10	−0.248412
$737$	−1615.49	−0.0807428
$738$	−15854.7	−0.790813
$739$	11893.7	0.592037	0.296018	−	0.955182i	$-0.404341\pi$
$739$	11893.7	0.592037	0.296018	+	0.955182i	$0.404341\pi$
$740$	0	0
$741$	−4345.97	−0.215457
$742$	0	0
$743$	33413.5	1.64983	0.824914	−	0.565259i	$-0.191224\pi$
$743$	33413.5	1.64983	0.824914	+	0.565259i	$0.191224\pi$
$744$	1081.75	0.0533051
$745$	0	0
$746$	−22486.1	−1.10359
$747$	−21649.3	−1.06039
$748$	−19432.3	−0.949888
$749$	0	0
$750$	0	0
$751$	22761.9	1.10598	0.552992	−	0.833187i	$-0.313486\pi$
$751$	22761.9	1.10598	0.552992	+	0.833187i	$0.313486\pi$
$752$	−3641.10	−0.176565
$753$	−6706.48	−0.324566
$754$	3389.32	0.163702
$755$	0	0
$756$	0	0
$757$	−27222.3	−1.30702	−0.653509	−	0.756919i	$-0.726704\pi$
$757$	−27222.3	−1.30702	−0.653509	+	0.756919i	$0.726704\pi$
$758$	9742.51	0.466839
$759$	−11690.1	−0.559057
$760$	0	0
$761$	−26825.7	−1.27783	−0.638917	−	0.769276i	$-0.720617\pi$
$761$	−26825.7	−1.27783	−0.638917	+	0.769276i	$0.720617\pi$
$762$	−3376.19	−0.160507
$763$	0	0
$764$	7374.86	0.349232
$765$	0	0
$766$	−25208.3	−1.18905
$767$	15495.2	0.729463
$768$	420.628	0.0197631
$769$	18093.8	0.848477	0.424239	−	0.905550i	$-0.360542\pi$
$769$	18093.8	0.848477	0.424239	+	0.905550i	$0.360542\pi$
$770$	0	0
$771$	5272.64	0.246290
$772$	3288.70	0.153320
$773$	16783.8	0.780945	0.390473	−	0.920615i	$-0.372312\pi$
$773$	16783.8	0.780945	0.390473	+	0.920615i	$0.372312\pi$
$774$	−3810.40	−0.176954
$775$	0	0
$776$	−11673.5	−0.540017
$777$	0	0
$778$	24366.5	1.12285
$779$	46181.6	2.12404
$780$	0	0
$781$	−6361.62	−0.291468
$782$	32810.6	1.50039
$783$	7645.11	0.348932
$784$	0	0
$785$	0	0
$786$	−2616.17	−0.118722
$787$	−1229.26	−0.0556779	−0.0278389	−	0.999612i	$-0.508863\pi$
$787$	−1229.26	−0.0556779	−0.0278389	+	0.999612i	$0.508863\pi$
$788$	−8445.69	−0.381809
$789$	8281.11	0.373657
$790$	0	0
$791$	0	0
$792$	−8923.24	−0.400345
$793$	12351.4	0.553106
$794$	7462.28	0.333535
$795$	0	0
$796$	2973.57	0.132406
$797$	−10084.7	−0.448204	−0.224102	−	0.974566i	$-0.571945\pi$
$797$	−10084.7	−0.448204	−0.224102	+	0.974566i	$0.571945\pi$
$798$	0	0
$799$	24085.5	1.06644
$800$	0	0
$801$	−2941.02	−0.129732
$802$	−7417.61	−0.326590
$803$	17458.4	0.767239
$804$	−231.314	−0.0101465
$805$	0	0
$806$	−3075.29	−0.134395
$807$	8695.61	0.379306
$808$	10935.1	0.476110
$809$	−14669.5	−0.637517	−0.318758	−	0.947836i	$-0.603266\pi$
$809$	−14669.5	−0.637517	−0.318758	+	0.947836i	$0.603266\pi$
$810$	0	0
$811$	−7101.55	−0.307483	−0.153742	−	0.988111i	$-0.549132\pi$
$811$	−7101.55	−0.307483	−0.153742	+	0.988111i	$0.549132\pi$
$812$	0	0
$813$	3755.54	0.162008
$814$	−26604.3	−1.14555
$815$	0	0
$816$	−2782.41	−0.119368
$817$	11099.0	0.475279
$818$	−14340.3	−0.612955
$819$	0	0
$820$	0	0
$821$	28867.9	1.22716	0.613579	−	0.789633i	$-0.289729\pi$
$821$	28867.9	1.22716	0.613579	+	0.789633i	$0.289729\pi$
$822$	−4824.15	−0.204698
$823$	−27561.3	−1.16735	−0.583674	−	0.811988i	$-0.698385\pi$
$823$	−27561.3	−1.16735	−0.583674	+	0.811988i	$0.698385\pi$
$824$	−3095.94	−0.130889
$825$	0	0
$826$	0	0
$827$	−3800.50	−0.159802	−0.0799011	−	0.996803i	$-0.525460\pi$
$827$	−3800.50	−0.159802	−0.0799011	+	0.996803i	$0.525460\pi$
$828$	15066.5	0.632363
$829$	26451.1	1.10819	0.554093	−	0.832455i	$-0.313065\pi$
$829$	26451.1	1.10819	0.554093	+	0.832455i	$0.313065\pi$
$830$	0	0
$831$	9429.09	0.393612
$832$	−1195.79	−0.0498277
$833$	0	0
$834$	14.7274	0.000611472 0
$835$	0	0
$836$	25991.7	1.07529
$837$	−6936.78	−0.286464
$838$	−9691.90	−0.399524
$839$	5455.57	0.224490	0.112245	−	0.993681i	$-0.464196\pi$
$839$	5455.57	0.224490	0.112245	+	0.993681i	$0.464196\pi$
$840$	0	0
$841$	−16162.6	−0.662699
$842$	19488.1	0.797631
$843$	−5091.42	−0.208016
$844$	−10279.6	−0.419240
$845$	0	0
$846$	11060.0	0.449468
$847$	0	0
$848$	163.171	0.00660770
$849$	−11297.8	−0.456703
$850$	0	0
$851$	44920.1	1.80945
$852$	−910.887	−0.0366273
$853$	28638.5	1.14955	0.574774	−	0.818312i	$-0.305090\pi$
$853$	28638.5	1.14955	0.574774	+	0.818312i	$0.305090\pi$
$854$	0	0
$855$	0	0
$856$	−844.686	−0.0337275
$857$	−7340.27	−0.292577	−0.146289	−	0.989242i	$-0.546733\pi$
$857$	−7340.27	−0.292577	−0.146289	+	0.989242i	$0.546733\pi$
$858$	−2818.29	−0.112138
$859$	14649.7	0.581889	0.290945	−	0.956740i	$-0.406030\pi$
$859$	14649.7	0.581889	0.290945	+	0.956740i	$0.406030\pi$
$860$	0	0
$861$	0	0
$862$	16590.2	0.655528
$863$	−5880.01	−0.231933	−0.115966	−	0.993253i	$-0.536996\pi$
$863$	−5880.01	−0.231933	−0.115966	+	0.993253i	$0.536996\pi$
$864$	−2697.29	−0.106208
$865$	0	0
$866$	14731.9	0.578073
$867$	10333.0	0.404759
$868$	0	0
$869$	−10837.7	−0.423066
$870$	0	0
$871$	657.597	0.0255819
$872$	−3690.31	−0.143314
$873$	35458.6	1.37468
$874$	−43885.7	−1.69846
$875$	0	0
$876$	2499.77	0.0964150
$877$	11989.8	0.461648	0.230824	−	0.972995i	$-0.425858\pi$
$877$	11989.8	0.461648	0.230824	+	0.972995i	$0.425858\pi$
$878$	−3900.04	−0.149909
$879$	−887.718	−0.0340637
$880$	0	0
$881$	6702.96	0.256332	0.128166	−	0.991753i	$-0.459091\pi$
$881$	6702.96	0.256332	0.128166	+	0.991753i	$0.459091\pi$
$882$	0	0
$883$	16182.7	0.616752	0.308376	−	0.951265i	$-0.400215\pi$
$883$	16182.7	0.616752	0.308376	+	0.951265i	$0.400215\pi$
$884$	7910.07	0.300955
$885$	0	0
$886$	−31952.9	−1.21160
$887$	−35343.6	−1.33790	−0.668952	−	0.743306i	$-0.733257\pi$
$887$	−35343.6	−1.33790	−0.668952	+	0.743306i	$0.733257\pi$
$888$	−3809.33	−0.143956
$889$	0	0
$890$	0	0
$891$	23758.9	0.893325
$892$	−16967.3	−0.636893
$893$	−32215.5	−1.20722
$894$	7533.31	0.281825
$895$	0	0
$896$	0	0
$897$	4758.55	0.177127
$898$	−10585.8	−0.393377
$899$	−7464.24	−0.276915
$900$	0	0
$901$	−1079.36	−0.0399099
$902$	29948.0	1.10550
$903$	0	0
$904$	−15608.1	−0.574246
$905$	0	0
$906$	−8715.99	−0.319613
$907$	−9370.08	−0.343030	−0.171515	−	0.985181i	$-0.554866\pi$
$907$	−9370.08	−0.343030	−0.171515	+	0.985181i	$0.554866\pi$
$908$	3621.71	0.132369
$909$	−33215.9	−1.21199
$910$	0	0
$911$	13365.6	0.486085	0.243042	−	0.970016i	$-0.421855\pi$
$911$	13365.6	0.486085	0.243042	+	0.970016i	$0.421855\pi$
$912$	3721.61	0.135126
$913$	40893.5	1.48234
$914$	−31929.6	−1.15551
$915$	0	0
$916$	−1216.20	−0.0438694
$917$	0	0
$918$	17842.3	0.641487
$919$	17832.3	0.640081	0.320041	−	0.947404i	$-0.396303\pi$
$919$	17832.3	0.640081	0.320041	+	0.947404i	$0.396303\pi$
$920$	0	0
$921$	3837.65	0.137302
$922$	−22145.5	−0.791022
$923$	2589.54	0.0923464
$924$	0	0
$925$	0	0
$926$	−22419.9	−0.795642
$927$	9404.04	0.333192
$928$	−2902.39	−0.102668
$929$	26456.7	0.934355	0.467178	−	0.884163i	$-0.345271\pi$
$929$	26456.7	0.934355	0.467178	+	0.884163i	$0.345271\pi$
$930$	0	0
$931$	0	0
$932$	−19878.4	−0.698648
$933$	13903.4	0.487864
$934$	22423.4	0.785564
$935$	0	0
$936$	3632.27	0.126842
$937$	−33299.9	−1.16100	−0.580502	−	0.814259i	$-0.697144\pi$
$937$	−33299.9	−1.16100	−0.580502	+	0.814259i	$0.697144\pi$
$938$	0	0
$939$	−7356.04	−0.255650
$940$	0	0
$941$	45730.1	1.58423	0.792114	−	0.610374i	$-0.208981\pi$
$941$	45730.1	1.58423	0.792114	+	0.610374i	$0.208981\pi$
$942$	9703.80	0.335634
$943$	−50565.8	−1.74618
$944$	−13269.1	−0.457491
$945$	0	0
$946$	7197.48	0.247368
$947$	5919.39	0.203120	0.101560	−	0.994829i	$-0.467617\pi$
$947$	5919.39	0.203120	0.101560	+	0.994829i	$0.467617\pi$
$948$	−1551.79	−0.0531645
$949$	−7106.55	−0.243086
$950$	0	0
$951$	8390.07	0.286085
$952$	0	0
$953$	54267.4	1.84459	0.922295	−	0.386488i	$-0.126312\pi$
$953$	54267.4	1.84459	0.922295	+	0.386488i	$0.126312\pi$
$954$	−495.639	−0.0168207
$955$	0	0
$956$	−2967.60	−0.100397
$957$	−6840.45	−0.231056
$958$	−15966.1	−0.538457
$959$	0	0
$960$	0	0
$961$	−23018.3	−0.772661
$962$	10829.5	0.362948
$963$	2565.76	0.0858573
$964$	−13935.5	−0.465594
$965$	0	0
$966$	0	0
$967$	41100.5	1.36681	0.683403	−	0.730041i	$-0.260499\pi$
$967$	41100.5	1.36681	0.683403	+	0.730041i	$0.260499\pi$
$968$	6207.14	0.206100
$969$	−24618.1	−0.816148
$970$	0	0
$971$	−43892.3	−1.45064	−0.725320	−	0.688412i	$-0.758308\pi$
$971$	−43892.3	−1.45064	−0.725320	+	0.688412i	$0.758308\pi$
$972$	12505.3	0.412661
$973$	0	0
$974$	−8817.13	−0.290061
$975$	0	0
$976$	−10577.0	−0.346886
$977$	44555.2	1.45900	0.729502	−	0.683979i	$-0.239752\pi$
$977$	44555.2	1.45900	0.729502	+	0.683979i	$0.239752\pi$
$978$	−12688.6	−0.414864
$979$	5555.29	0.181356
$980$	0	0
$981$	11209.4	0.364822
$982$	−25008.0	−0.812665
$983$	7008.12	0.227390	0.113695	−	0.993516i	$-0.463731\pi$
$983$	7008.12	0.227390	0.113695	+	0.993516i	$0.463731\pi$
$984$	4288.10	0.138922
$985$	0	0
$986$	19199.0	0.620104
$987$	0	0
$988$	−10580.1	−0.340686
$989$	−12152.6	−0.390728
$990$	0	0
$991$	−2830.10	−0.0907176	−0.0453588	−	0.998971i	$-0.514443\pi$
$991$	−2830.10	−0.0907176	−0.0453588	+	0.998971i	$0.514443\pi$
$992$	2633.48	0.0842873
$993$	10578.1	0.338054
$994$	0	0
$995$	0	0
$996$	5855.33	0.186278
$997$	4635.23	0.147241	0.0736205	−	0.997286i	$-0.476545\pi$
$997$	4635.23	0.147241	0.0736205	+	0.997286i	$0.476545\pi$
$998$	27549.0	0.873795
$999$	24427.5	0.773625

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2450.4.a.cs.1.3		4
5.2	odd	4		490.4.c.d.99.6	yes	8
5.3	odd	4		490.4.c.d.99.3	yes	8
5.4	even	2		2450.4.a.cm.1.2		4
7.6	odd	2	inner	2450.4.a.cs.1.2		4
35.13	even	4		490.4.c.d.99.2	✓	8
35.27	even	4		490.4.c.d.99.7	yes	8
35.34	odd	2		2450.4.a.cm.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
490.4.c.d.99.2	✓	8	35.13	even	4
490.4.c.d.99.3	yes	8	5.3	odd	4
490.4.c.d.99.6	yes	8	5.2	odd	4
490.4.c.d.99.7	yes	8	35.27	even	4
2450.4.a.cm.1.2		4	5.4	even	2
2450.4.a.cm.1.3		4	35.34	odd	2
2450.4.a.cs.1.2		4	7.6	odd	2	inner
2450.4.a.cs.1.3		4	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:	\( N \)	\(=\)	\( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2450.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +1.64308 q^{3} +4.00000 q^{4} +3.28615 q^{6} +8.00000 q^{8} -24.3003 q^{9} +O(q^{10})\)	\(q+2.00000 q^{2} +1.64308 q^{3} +4.00000 q^{4} +3.28615 q^{6} +8.00000 q^{8} -24.3003 q^{9} +45.9009 q^{11} +6.57231 q^{12} -18.6843 q^{13} +16.0000 q^{16} -105.839 q^{17} -48.6006 q^{18} +141.564 q^{19} +91.8018 q^{22} -155.003 q^{23} +13.1446 q^{24} -37.3685 q^{26} -84.2903 q^{27} -90.6997 q^{29} +82.2962 q^{31} +32.0000 q^{32} +75.4187 q^{33} -211.677 q^{34} -97.2012 q^{36} -289.802 q^{37} +283.128 q^{38} -30.6997 q^{39} +326.224 q^{41} +78.4024 q^{43} +183.604 q^{44} -310.006 q^{46} -227.569 q^{47} +26.2892 q^{48} -173.901 q^{51} -74.7371 q^{52} +10.1982 q^{53} -168.581 q^{54} +232.601 q^{57} -181.399 q^{58} -829.316 q^{59} -661.061 q^{61} +164.592 q^{62} +64.0000 q^{64} +150.837 q^{66} -35.1952 q^{67} -423.354 q^{68} -254.682 q^{69} -138.595 q^{71} -194.402 q^{72} +380.349 q^{73} -579.604 q^{74} +566.256 q^{76} -61.3994 q^{78} -236.111 q^{79} +517.612 q^{81} +652.449 q^{82} +890.909 q^{83} +156.805 q^{86} -149.027 q^{87} +367.207 q^{88} +121.028 q^{89} -620.012 q^{92} +135.219 q^{93} -455.137 q^{94} +52.5785 q^{96} -1459.18 q^{97} -1115.41 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{2} + 16 q^{4} + 32 q^{8} - 30 q^{9}+O(q^{10}) \) 4 * q + 8 * q^2 + 16 * q^4 + 32 * q^8 - 30 * q^9	\( 4 q + 8 q^{2} + 16 q^{4} + 32 q^{8} - 30 q^{9} - 18 q^{11} + 64 q^{16} - 60 q^{18} - 36 q^{22} + 52 q^{23} - 430 q^{29} + 128 q^{32} - 120 q^{36} - 756 q^{37} - 190 q^{39} - 224 q^{43} - 72 q^{44} + 104 q^{46} - 494 q^{51} + 444 q^{53} + 796 q^{57} - 860 q^{58} + 256 q^{64} - 1216 q^{67} - 1764 q^{71} - 240 q^{72} - 1512 q^{74} - 380 q^{78} + 1542 q^{79} - 752 q^{81} - 448 q^{86} - 144 q^{88} + 208 q^{92} - 3760 q^{93} - 3252 q^{99}+O(q^{100}) \) 4 * q + 8 * q^2 + 16 * q^4 + 32 * q^8 - 30 * q^9 - 18 * q^11 + 64 * q^16 - 60 * q^18 - 36 * q^22 + 52 * q^23 - 430 * q^29 + 128 * q^32 - 120 * q^36 - 756 * q^37 - 190 * q^39 - 224 * q^43 - 72 * q^44 + 104 * q^46 - 494 * q^51 + 444 * q^53 + 796 * q^57 - 860 * q^58 + 256 * q^64 - 1216 * q^67 - 1764 * q^71 - 240 * q^72 - 1512 * q^74 - 380 * q^78 + 1542 * q^79 - 752 * q^81 - 448 * q^86 - 144 * q^88 + 208 * q^92 - 3760 * q^93 - 3252 * q^99

Introduction

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