LMFDB - Embedded newform 2450.4.a.cs.1.1

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			1.1
Root			$-6.02497$ 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)

$f(q)$	$=$	$q+2.00000 q^{2} -6.02497 q^{3} +4.00000 q^{4} -12.0499 q^{6} +8.00000 q^{8} +9.30030 q^{9} +O(q^{10})$	\(q+2.00000 q^{2} -6.02497 q^{3} +4.00000 q^{4} -12.0499 q^{6} +8.00000 q^{8} +9.30030 q^{9} -54.9009 q^{11} -24.0999 q^{12} +10.6723 q^{13} +16.0000 q^{16} +12.1327 q^{17} +18.6006 q^{18} -27.4523 q^{19} -109.802 q^{22} +181.003 q^{23} -48.1998 q^{24} +21.3446 q^{26} +106.640 q^{27} -124.300 q^{29} +334.478 q^{31} +32.0000 q^{32} +330.776 q^{33} +24.2654 q^{34} +37.2012 q^{36} -88.1982 q^{37} -54.9046 q^{38} -64.3003 q^{39} +134.110 q^{41} -190.402 q^{43} -219.604 q^{44} +362.006 q^{46} -177.746 q^{47} -96.3996 q^{48} -73.0991 q^{51} +42.6892 q^{52} +211.802 q^{53} +213.280 q^{54} +165.399 q^{57} -248.601 q^{58} -342.679 q^{59} +659.891 q^{61} +668.955 q^{62} +64.0000 q^{64} +661.553 q^{66} -572.805 q^{67} +48.5307 q^{68} -1090.54 q^{69} -743.405 q^{71} +74.4024 q^{72} -1163.33 q^{73} -176.396 q^{74} -109.809 q^{76} -128.601 q^{78} +1007.11 q^{79} -893.612 q^{81} +268.221 q^{82} +348.189 q^{83} -380.805 q^{86} +748.906 q^{87} -439.207 q^{88} +1060.06 q^{89} +724.012 q^{92} -2015.22 q^{93} -355.491 q^{94} -192.799 q^{96} -578.020 q^{97} -510.595 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−6.02497	−1.15951	−0.579753	−	0.814792i	$-0.696851\pi$
$3$	−6.02497	−1.15951	−0.579753	+	0.814792i	$0.696851\pi$
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−12.0499	−0.819895
$7$	0	0
$7$	0	0
$8$	8.00000	0.353553
$9$	9.30030	0.344455
$10$	0	0
$11$	−54.9009	−1.50484	−0.752420	−	0.658684i	$-0.771114\pi$
$11$	−54.9009	−1.50484	−0.752420	+	0.658684i	$0.771114\pi$
$12$	−24.0999	−0.579753
$13$	10.6723	0.227689	0.113845	−	0.993499i	$-0.463683\pi$
$13$	10.6723	0.227689	0.113845	+	0.993499i	$0.463683\pi$
$14$	0	0
$15$	0	0
$16$	16.0000	0.250000
$17$	12.1327	0.173095	0.0865473	−	0.996248i	$-0.472417\pi$
$17$	12.1327	0.173095	0.0865473	+	0.996248i	$0.472417\pi$
$18$	18.6006	0.243567
$19$	−27.4523	−0.331473	−0.165737	−	0.986170i	$-0.553000\pi$
$19$	−27.4523	−0.331473	−0.165737	+	0.986170i	$0.553000\pi$
$20$	0	0
$21$	0	0
$22$	−109.802	−1.06408
$23$	181.003	1.64094	0.820472	−	0.571686i	$-0.193711\pi$
$23$	181.003	1.64094	0.820472	+	0.571686i	$0.193711\pi$
$24$	−48.1998	−0.409947
$25$	0	0
$26$	21.3446	0.161001
$27$	106.640	0.760108
$28$	0	0
$29$	−124.300	−0.795931	−0.397965	−	0.917400i	$-0.630284\pi$
$29$	−124.300	−0.795931	−0.397965	+	0.917400i	$0.630284\pi$
$30$	0	0
$31$	334.478	1.93787	0.968935	−	0.247316i	$-0.0795485\pi$
$31$	334.478	1.93787	0.968935	+	0.247316i	$0.0795485\pi$
$32$	32.0000	0.176777
$33$	330.776	1.74487
$34$	24.2654	0.122396
$35$	0	0
$36$	37.2012	0.172228
$37$	−88.1982	−0.391884	−0.195942	−	0.980616i	$-0.562776\pi$
$37$	−88.1982	−0.391884	−0.195942	+	0.980616i	$0.562776\pi$
$38$	−54.9046	−0.234387
$39$	−64.3003	−0.264007
$40$	0	0
$41$	134.110	0.510842	0.255421	−	0.966830i	$-0.417786\pi$
$41$	134.110	0.510842	0.255421	+	0.966830i	$0.417786\pi$
$42$	0	0
$43$	−190.402	−0.675258	−0.337629	−	0.941279i	$-0.609625\pi$
$43$	−190.402	−0.675258	−0.337629	+	0.941279i	$0.609625\pi$
$44$	−219.604	−0.752420
$45$	0	0
$46$	362.006	1.16032
$47$	−177.746	−0.551636	−0.275818	−	0.961210i	$-0.588949\pi$
$47$	−177.746	−0.551636	−0.275818	+	0.961210i	$0.588949\pi$
$48$	−96.3996	−0.289877
$49$	0	0
$50$	0	0
$51$	−73.0991	−0.200704
$52$	42.6892	0.113845
$53$	211.802	0.548929	0.274464	−	0.961597i	$-0.411499\pi$
$53$	211.802	0.548929	0.274464	+	0.961597i	$0.411499\pi$
$54$	213.280	0.537478
$55$	0	0
$56$	0	0
$57$	165.399	0.384345
$58$	−248.601	−0.562808
$59$	−342.679	−0.756152	−0.378076	−	0.925775i	$-0.623414\pi$
$59$	−342.679	−0.756152	−0.378076	+	0.925775i	$0.623414\pi$
$60$	0	0
$61$	659.891	1.38509	0.692544	−	0.721375i	$-0.256490\pi$
$61$	659.891	1.38509	0.692544	+	0.721375i	$0.256490\pi$
$62$	668.955	1.37028
$63$	0	0
$64$	64.0000	0.125000
$65$	0	0
$66$	661.553	1.23381
$67$	−572.805	−1.04447	−0.522233	−	0.852803i	$-0.674901\pi$
$67$	−572.805	−1.04447	−0.522233	+	0.852803i	$0.674901\pi$
$68$	48.5307	0.0865473
$69$	−1090.54	−1.90269
$70$	0	0
$71$	−743.405	−1.24262	−0.621310	−	0.783565i	$-0.713399\pi$
$71$	−743.405	−1.24262	−0.621310	+	0.783565i	$0.713399\pi$
$72$	74.4024	0.121783
$73$	−1163.33	−1.86518	−0.932589	−	0.360941i	$-0.882456\pi$
$73$	−1163.33	−1.86518	−0.932589	+	0.360941i	$0.882456\pi$
$74$	−176.396	−0.277104
$75$	0	0
$76$	−109.809	−0.165737
$77$	0	0
$78$	−128.601	−0.186681
$79$	1007.11	1.43429	0.717145	−	0.696924i	$-0.245449\pi$
$79$	1007.11	1.43429	0.717145	+	0.696924i	$0.245449\pi$
$80$	0	0
$81$	−893.612	−1.22581
$82$	268.221	0.361220
$83$	348.189	0.460467	0.230233	−	0.973135i	$-0.426051\pi$
$83$	348.189	0.460467	0.230233	+	0.973135i	$0.426051\pi$
$84$	0	0
$85$	0	0
$86$	−380.805	−0.477479
$87$	748.906	0.922887
$88$	−439.207	−0.532041
$89$	1060.06	1.26255	0.631273	−	0.775561i	$-0.282533\pi$
$89$	1060.06	1.26255	0.631273	+	0.775561i	$0.282533\pi$
$90$	0	0
$91$	0	0
$92$	724.012	0.820472
$93$	−2015.22	−2.24697
$94$	−355.491	−0.390065
$95$	0	0
$96$	−192.799	−0.204974
$97$	−578.020	−0.605041	−0.302521	−	0.953143i	$-0.597828\pi$
$97$	−578.020	−0.605041	−0.302521	+	0.953143i	$0.597828\pi$
$98$	0	0
$99$	−510.595	−0.518350
$100$	0	0
$101$	1668.37	1.64366	0.821829	−	0.569734i	$-0.192954\pi$
$101$	1668.37	1.64366	0.821829	+	0.569734i	$0.192954\pi$
$102$	−146.198	−0.141919
$103$	869.571	0.831858	0.415929	−	0.909397i	$-0.363456\pi$
$103$	869.571	0.831858	0.415929	+	0.909397i	$0.363456\pi$
$104$	85.3784	0.0805004
$105$	0	0
$106$	423.604	0.388151
$107$	−1718.41	−1.55257	−0.776287	−	0.630380i	$-0.782899\pi$
$107$	−1718.41	−1.55257	−0.776287	+	0.630380i	$0.782899\pi$
$108$	426.561	0.380054
$109$	−1771.71	−1.55687	−0.778437	−	0.627723i	$-0.783987\pi$
$109$	−1771.71	−1.55687	−0.778437	+	0.627723i	$0.783987\pi$
$110$	0	0
$111$	531.392	0.454392
$112$	0	0
$113$	−270.985	−0.225594	−0.112797	−	0.993618i	$-0.535981\pi$
$113$	−270.985	−0.225594	−0.112797	+	0.993618i	$0.535981\pi$
$114$	330.799	0.271773
$115$	0	0
$116$	−497.201	−0.397965
$117$	99.2555	0.0784289
$118$	−685.358	−0.534680
$119$	0	0
$120$	0	0
$121$	1683.11	1.26454
$122$	1319.78	0.979405
$123$	−808.012	−0.592325
$124$	1337.91	0.968935
$125$	0	0
$126$	0	0
$127$	−1094.60	−0.764804	−0.382402	−	0.923996i	$-0.624903\pi$
$127$	−1094.60	−0.764804	−0.382402	+	0.923996i	$0.624903\pi$
$128$	128.000	0.0883883
$129$	1147.17	0.782966
$130$	0	0
$131$	2369.79	1.58053	0.790265	−	0.612765i	$-0.209943\pi$
$131$	2369.79	1.58053	0.790265	+	0.612765i	$0.209943\pi$
$132$	1323.11	0.872436
$133$	0	0
$134$	−1145.61	−0.738549
$135$	0	0
$136$	97.0615	0.0611982
$137$	1220.02	0.760830	0.380415	−	0.924816i	$-0.375781\pi$
$137$	1220.02	0.760830	0.380415	+	0.924816i	$0.375781\pi$
$138$	−2181.08	−1.34540
$139$	−681.602	−0.415919	−0.207960	−	0.978137i	$-0.566682\pi$
$139$	−681.602	−0.415919	−0.207960	+	0.978137i	$0.566682\pi$
$140$	0	0
$141$	1070.91	0.639625
$142$	−1486.81	−0.878665
$143$	−585.919	−0.342636
$144$	148.805	0.0861139
$145$	0	0
$146$	−2326.67	−1.31888
$147$	0	0
$148$	−352.793	−0.195942
$149$	−2008.44	−1.10428	−0.552140	−	0.833752i	$-0.686189\pi$
$149$	−2008.44	−1.10428	−0.552140	+	0.833752i	$0.686189\pi$
$150$	0	0
$151$	1749.34	0.942776	0.471388	−	0.881926i	$-0.343753\pi$
$151$	1749.34	0.942776	0.471388	+	0.881926i	$0.343753\pi$
$152$	−219.618	−0.117193
$153$	112.838	0.0596234
$154$	0	0
$155$	0	0
$156$	−257.201	−0.132004
$157$	−2701.44	−1.37324	−0.686619	−	0.727017i	$-0.740906\pi$
$157$	−2701.44	−1.37324	−0.686619	+	0.727017i	$0.740906\pi$
$158$	2014.22	1.01420
$159$	−1276.10	−0.636486
$160$	0	0
$161$	0	0
$162$	−1787.22	−0.866776
$163$	305.237	0.146675	0.0733374	−	0.997307i	$-0.476635\pi$
$163$	305.237	0.146675	0.0733374	+	0.997307i	$0.476635\pi$
$164$	536.442	0.255421
$165$	0	0
$166$	696.379	0.325599
$167$	−4034.39	−1.86940	−0.934702	−	0.355432i	$-0.884334\pi$
$167$	−4034.39	−1.86940	−0.934702	+	0.355432i	$0.884334\pi$
$168$	0	0
$169$	−2083.10	−0.948158
$170$	0	0
$171$	−255.315	−0.114178
$172$	−761.610	−0.337629
$173$	−1210.08	−0.531798	−0.265899	−	0.964001i	$-0.585669\pi$
$173$	−1210.08	−0.531798	−0.265899	+	0.964001i	$0.585669\pi$
$174$	1497.81	0.652579
$175$	0	0
$176$	−878.414	−0.376210
$177$	2064.63	0.876763
$178$	2120.13	0.892755
$179$	−1011.57	−0.422392	−0.211196	−	0.977444i	$-0.567736\pi$
$179$	−1011.57	−0.422392	−0.211196	+	0.977444i	$0.567736\pi$
$180$	0	0
$181$	−1386.21	−0.569261	−0.284631	−	0.958637i	$-0.591871\pi$
$181$	−1386.21	−0.569261	−0.284631	+	0.958637i	$0.591871\pi$
$182$	0	0
$183$	−3975.83	−1.60602
$184$	1448.02	0.580162
$185$	0	0
$186$	−4030.44	−1.58885
$187$	−666.095	−0.260480
$188$	−710.983	−0.275818
$189$	0	0
$190$	0	0
$191$	197.285	0.0747386	0.0373693	−	0.999302i	$-0.488102\pi$
$191$	197.285	0.0747386	0.0373693	+	0.999302i	$0.488102\pi$
$192$	−385.598	−0.144938
$193$	3711.83	1.38437	0.692184	−	0.721721i	$-0.256649\pi$
$193$	3711.83	1.38437	0.692184	+	0.721721i	$0.256649\pi$
$194$	−1156.04	−0.427829
$195$	0	0
$196$	0	0
$197$	509.423	0.184238	0.0921190	−	0.995748i	$-0.470636\pi$
$197$	509.423	0.184238	0.0921190	+	0.995748i	$0.470636\pi$
$198$	−1021.19	−0.366529
$199$	4504.16	1.60448	0.802240	−	0.597002i	$-0.203641\pi$
$199$	4504.16	1.60448	0.802240	+	0.597002i	$0.203641\pi$
$200$	0	0
$201$	3451.13	1.21107
$202$	3336.75	1.16224
$203$	0	0
$204$	−292.396	−0.100352
$205$	0	0
$206$	1739.14	0.588212
$207$	1683.38	0.565232
$208$	170.757	0.0569224
$209$	1507.16	0.498814
$210$	0	0
$211$	−2469.10	−0.805591	−0.402796	−	0.915290i	$-0.631961\pi$
$211$	−2469.10	−0.805591	−0.402796	+	0.915290i	$0.631961\pi$
$212$	847.207	0.274464
$213$	4479.00	1.44083
$214$	−3436.83	−1.09784
$215$	0	0
$216$	853.122	0.268739
$217$	0	0
$218$	−3543.42	−1.10088
$219$	7009.06	2.16269
$220$	0	0
$221$	129.484	0.0394118
$222$	1062.78	0.321303
$223$	−1826.84	−0.548584	−0.274292	−	0.961646i	$-0.588443\pi$
$223$	−1826.84	−0.548584	−0.274292	+	0.961646i	$0.588443\pi$
$224$	0	0
$225$	0	0
$226$	−541.970	−0.159519
$227$	−4245.55	−1.24135	−0.620676	−	0.784067i	$-0.713142\pi$
$227$	−4245.55	−1.24135	−0.620676	+	0.784067i	$0.713142\pi$
$228$	661.598	0.192173
$229$	−3252.06	−0.938438	−0.469219	−	0.883082i	$-0.655464\pi$
$229$	−3252.06	−0.938438	−0.469219	+	0.883082i	$0.655464\pi$
$230$	0	0
$231$	0	0
$232$	−994.402	−0.281404
$233$	−3894.39	−1.09498	−0.547489	−	0.836813i	$-0.684416\pi$
$233$	−3894.39	−1.09498	−0.547489	+	0.836813i	$0.684416\pi$
$234$	198.511	0.0554576
$235$	0	0
$236$	−1370.72	−0.378076
$237$	−6067.82	−1.66307
$238$	0	0
$239$	−641.099	−0.173512	−0.0867558	−	0.996230i	$-0.527650\pi$
$239$	−641.099	−0.173512	−0.0867558	+	0.996230i	$0.527650\pi$
$240$	0	0
$241$	−4403.72	−1.17705	−0.588524	−	0.808480i	$-0.700291\pi$
$241$	−4403.72	−1.17705	−0.588524	+	0.808480i	$0.700291\pi$
$242$	3366.22	0.894168
$243$	2504.70	0.661222
$244$	2639.56	0.692544
$245$	0	0
$246$	−1616.02	−0.418837
$247$	−292.979	−0.0754729
$248$	2675.82	0.685140
$249$	−2097.83	−0.533914
$250$	0	0
$251$	−4207.24	−1.05800	−0.529001	−	0.848621i	$-0.677433\pi$
$251$	−4207.24	−1.05800	−0.529001	+	0.848621i	$0.677433\pi$
$252$	0	0
$253$	−9937.22	−2.46936
$254$	−2189.20	−0.540798
$255$	0	0
$256$	256.000	0.0625000
$257$	−83.2133	−0.0201973	−0.0100986	−	0.999949i	$-0.503215\pi$
$257$	−83.2133	−0.0201973	−0.0100986	+	0.999949i	$0.503215\pi$
$258$	2294.34	0.553641
$259$	0	0
$260$	0	0
$261$	−1156.03	−0.274163
$262$	4739.58	1.11760
$263$	5040.00	1.18167	0.590836	−	0.806792i	$-0.298798\pi$
$263$	5040.00	1.18167	0.590836	+	0.806792i	$0.298798\pi$
$264$	2646.21	0.616905
$265$	0	0
$266$	0	0
$267$	−6386.86	−1.46393
$268$	−2291.22	−0.522233
$269$	−5090.54	−1.15381	−0.576907	−	0.816810i	$-0.695741\pi$
$269$	−5090.54	−1.15381	−0.576907	+	0.816810i	$0.695741\pi$
$270$	0	0
$271$	−1671.78	−0.374737	−0.187368	−	0.982290i	$-0.559996\pi$
$271$	−1671.78	−0.374737	−0.187368	+	0.982290i	$0.559996\pi$
$272$	194.123	0.0432737
$273$	0	0
$274$	2440.05	0.537988
$275$	0	0
$276$	−4362.15	−0.951343
$277$	−3064.68	−0.664760	−0.332380	−	0.943145i	$-0.607852\pi$
$277$	−3064.68	−0.664760	−0.332380	+	0.943145i	$0.607852\pi$
$278$	−1363.20	−0.294099
$279$	3110.74	0.667510
$280$	0	0
$281$	−1788.29	−0.379645	−0.189823	−	0.981818i	$-0.560791\pi$
$281$	−1788.29	−0.379645	−0.189823	+	0.981818i	$0.560791\pi$
$282$	2141.83	0.452283
$283$	−4343.06	−0.912255	−0.456128	−	0.889914i	$-0.650764\pi$
$283$	−4343.06	−0.912255	−0.456128	+	0.889914i	$0.650764\pi$
$284$	−2973.62	−0.621310
$285$	0	0
$286$	−1171.84	−0.242280
$287$	0	0
$288$	297.610	0.0608917
$289$	−4765.80	−0.970038
$290$	0	0
$291$	3482.55	0.701549
$292$	−4653.34	−0.932589
$293$	7302.49	1.45603	0.728013	−	0.685563i	$-0.240444\pi$
$293$	7302.49	1.45603	0.728013	+	0.685563i	$0.240444\pi$
$294$	0	0
$295$	0	0
$296$	−705.586	−0.138552
$297$	−5854.64	−1.14384
$298$	−4016.88	−0.780844
$299$	1931.72	0.373626
$300$	0	0
$301$	0	0
$302$	3498.68	0.666643
$303$	−10051.9	−1.90583
$304$	−439.237	−0.0828683
$305$	0	0
$306$	225.675	0.0421601
$307$	5837.81	1.08528	0.542641	−	0.839965i	$-0.317425\pi$
$307$	5837.81	1.08528	0.542641	+	0.839965i	$0.317425\pi$
$308$	0	0
$309$	−5239.14	−0.964545
$310$	0	0
$311$	−5809.92	−1.05933	−0.529663	−	0.848208i	$-0.677681\pi$
$311$	−5809.92	−1.05933	−0.529663	+	0.848208i	$0.677681\pi$
$312$	−514.402	−0.0933407
$313$	−5186.92	−0.936684	−0.468342	−	0.883547i	$-0.655148\pi$
$313$	−5186.92	−0.936684	−0.468342	+	0.883547i	$0.655148\pi$
$314$	−5402.88	−0.971026
$315$	0	0
$316$	4028.44	0.717145
$317$	−8132.32	−1.44087	−0.720436	−	0.693521i	$-0.756058\pi$
$317$	−8132.32	−1.44087	−0.720436	+	0.693521i	$0.756058\pi$
$318$	−2552.20	−0.450064
$319$	6824.20	1.19775
$320$	0	0
$321$	10353.4	1.80022
$322$	0	0
$323$	−333.070	−0.0573762
$324$	−3574.45	−0.612903
$325$	0	0
$326$	610.474	0.103715
$327$	10674.5	1.80520
$328$	1072.88	0.180610
$329$	0	0
$330$	0	0
$331$	5765.99	0.957485	0.478743	−	0.877955i	$-0.341093\pi$
$331$	5765.99	0.957485	0.478743	+	0.877955i	$0.341093\pi$
$332$	1392.76	0.230233
$333$	−820.270	−0.134986
$334$	−8068.78	−1.32187
$335$	0	0
$336$	0	0
$337$	−6697.82	−1.08265	−0.541326	−	0.840813i	$-0.682077\pi$
$337$	−6697.82	−1.08265	−0.541326	+	0.840813i	$0.682077\pi$
$338$	−4166.20	−0.670449
$339$	1632.68	0.261578
$340$	0	0
$341$	−18363.1	−2.91618
$342$	−510.629	−0.0807359
$343$	0	0
$344$	−1523.22	−0.238740
$345$	0	0
$346$	−2420.17	−0.376038
$347$	8793.32	1.36038	0.680188	−	0.733038i	$-0.261898\pi$
$347$	8793.32	1.36038	0.680188	+	0.733038i	$0.261898\pi$
$348$	2995.62	0.461443
$349$	−4678.89	−0.717637	−0.358818	−	0.933407i	$-0.616820\pi$
$349$	−4678.89	−0.717637	−0.358818	+	0.933407i	$0.616820\pi$
$350$	0	0
$351$	1138.10	0.173069
$352$	−1756.83	−0.266021
$353$	−5008.06	−0.755105	−0.377552	−	0.925988i	$-0.623234\pi$
$353$	−5008.06	−0.755105	−0.377552	+	0.925988i	$0.623234\pi$
$354$	4129.26	0.619965
$355$	0	0
$356$	4240.26	0.631273
$357$	0	0
$358$	−2023.14	−0.298676
$359$	4554.91	0.669634	0.334817	−	0.942283i	$-0.391325\pi$
$359$	4554.91	0.669634	0.334817	+	0.942283i	$0.391325\pi$
$360$	0	0
$361$	−6105.37	−0.890126
$362$	−2772.42	−0.402528
$363$	−10140.7	−1.46625
$364$	0	0
$365$	0	0
$366$	−7951.65	−1.13563
$367$	8341.17	1.18639	0.593195	−	0.805058i	$-0.297866\pi$
$367$	8341.17	1.18639	0.593195	+	0.805058i	$0.297866\pi$
$368$	2896.05	0.410236
$369$	1247.27	0.175962
$370$	0	0
$371$	0	0
$372$	−8060.88	−1.12349
$373$	−2842.93	−0.394641	−0.197321	−	0.980339i	$-0.563224\pi$
$373$	−2842.93	−0.394641	−0.197321	+	0.980339i	$0.563224\pi$
$374$	−1332.19	−0.184187
$375$	0	0
$376$	−1421.97	−0.195033
$377$	−1326.57	−0.181225
$378$	0	0
$379$	−1311.25	−0.177717	−0.0888584	−	0.996044i	$-0.528322\pi$
$379$	−1311.25	−0.177717	−0.0888584	+	0.996044i	$0.528322\pi$
$380$	0	0
$381$	6594.94	0.886795
$382$	394.571	0.0528482
$383$	11802.6	1.57464	0.787318	−	0.616547i	$-0.211469\pi$
$383$	11802.6	1.57464	0.787318	+	0.616547i	$0.211469\pi$
$384$	−771.197	−0.102487
$385$	0	0
$386$	7423.65	0.978896
$387$	−1770.80	−0.232596
$388$	−2312.08	−0.302521
$389$	−3508.24	−0.457262	−0.228631	−	0.973513i	$-0.573425\pi$
$389$	−3508.24	−0.457262	−0.228631	+	0.973513i	$0.573425\pi$
$390$	0	0
$391$	2196.05	0.284039
$392$	0	0
$393$	−14277.9	−1.83263
$394$	1018.85	0.130276
$395$	0	0
$396$	−2042.38	−0.259175
$397$	3728.41	0.471344	0.235672	−	0.971833i	$-0.424271\pi$
$397$	3728.41	0.471344	0.235672	+	0.971833i	$0.424271\pi$
$398$	9008.32	1.13454
$399$	0	0
$400$	0	0
$401$	8353.81	1.04032	0.520161	−	0.854068i	$-0.325872\pi$
$401$	8353.81	1.04032	0.520161	+	0.854068i	$0.325872\pi$
$402$	6902.27	0.856353
$403$	3569.65	0.441232
$404$	6673.50	0.821829
$405$	0	0
$406$	0	0
$407$	4842.16	0.589722
$408$	−584.793	−0.0709597
$409$	−11362.2	−1.37366	−0.686829	−	0.726819i	$-0.740998\pi$
$409$	−11362.2	−1.37366	−0.686829	+	0.726819i	$0.740998\pi$
$410$	0	0
$411$	−7350.61	−0.882187
$412$	3478.28	0.415929
$413$	0	0
$414$	3366.76	0.399680
$415$	0	0
$416$	341.513	0.0402502
$417$	4106.64	0.482261
$418$	3014.31	0.352715
$419$	7444.97	0.868044	0.434022	−	0.900902i	$-0.357094\pi$
$419$	7444.97	0.868044	0.434022	+	0.900902i	$0.357094\pi$
$420$	0	0
$421$	−9173.07	−1.06192	−0.530960	−	0.847397i	$-0.678168\pi$
$421$	−9173.07	−1.06192	−0.530960	+	0.847397i	$0.678168\pi$
$422$	−4938.20	−0.569639
$423$	−1653.09	−0.190014
$424$	1694.41	0.194076
$425$	0	0
$426$	8957.99	1.01882
$427$	0	0
$428$	−6873.66	−0.776287
$429$	3530.14	0.397289
$430$	0	0
$431$	6715.89	0.750563	0.375282	−	0.926911i	$-0.377546\pi$
$431$	6715.89	0.750563	0.375282	+	0.926911i	$0.377546\pi$
$432$	1706.24	0.190027
$433$	−4741.46	−0.526235	−0.263118	−	0.964764i	$-0.584751\pi$
$433$	−4741.46	−0.526235	−0.263118	+	0.964764i	$0.584751\pi$
$434$	0	0
$435$	0	0
$436$	−7086.85	−0.778437
$437$	−4968.95	−0.543929
$438$	14018.1	1.52925
$439$	−137.434	−0.0149416	−0.00747080	−	0.999972i	$-0.502378\pi$
$439$	−137.434	−0.0149416	−0.00747080	+	0.999972i	$0.502378\pi$
$440$	0	0
$441$	0	0
$442$	258.967	0.0278684
$443$	−11003.6	−1.18012	−0.590062	−	0.807358i	$-0.700897\pi$
$443$	−11003.6	−1.18012	−0.590062	+	0.807358i	$0.700897\pi$
$444$	2125.57	0.227196
$445$	0	0
$446$	−3653.68	−0.387907
$447$	12100.8	1.28042
$448$	0	0
$449$	16849.9	1.77104	0.885519	−	0.464604i	$-0.153803\pi$
$449$	16849.9	1.77104	0.885519	+	0.464604i	$0.153803\pi$
$450$	0	0
$451$	−7362.78	−0.768736
$452$	−1083.94	−0.112797
$453$	−10539.7	−1.09316
$454$	−8491.10	−0.877769
$455$	0	0
$456$	1323.20	0.135887
$457$	4262.78	0.436333	0.218167	−	0.975912i	$-0.429992\pi$
$457$	4262.78	0.436333	0.218167	+	0.975912i	$0.429992\pi$
$458$	−6504.12	−0.663576
$459$	1293.83	0.131571
$460$	0	0
$461$	−3616.84	−0.365408	−0.182704	−	0.983168i	$-0.558485\pi$
$461$	−3616.84	−0.365408	−0.182704	+	0.983168i	$0.558485\pi$
$462$	0	0
$463$	−15914.0	−1.59738	−0.798691	−	0.601741i	$-0.794474\pi$
$463$	−15914.0	−1.59738	−0.798691	+	0.601741i	$0.794474\pi$
$464$	−1988.80	−0.198983
$465$	0	0
$466$	−7788.78	−0.774267
$467$	−17975.7	−1.78119	−0.890597	−	0.454793i	$-0.849713\pi$
$467$	−17975.7	−1.78119	−0.890597	+	0.454793i	$0.849713\pi$
$468$	397.022	0.0392144
$469$	0	0
$470$	0	0
$471$	16276.1	1.59228
$472$	−2741.43	−0.267340
$473$	10453.3	1.01616
$474$	−12135.6	−1.17597
$475$	0	0
$476$	0	0
$477$	1969.82	0.189081
$478$	−1282.20	−0.122691
$479$	−12545.9	−1.19674	−0.598369	−	0.801221i	$-0.704184\pi$
$479$	−12545.9	−1.19674	−0.598369	+	0.801221i	$0.704184\pi$
$480$	0	0
$481$	−941.278	−0.0892278
$482$	−8807.44	−0.832298
$483$	0	0
$484$	6732.43	0.632272
$485$	0	0
$486$	5009.41	0.467554
$487$	−8373.44	−0.779131	−0.389565	−	0.920999i	$-0.627375\pi$
$487$	−8373.44	−0.779131	−0.389565	+	0.920999i	$0.627375\pi$
$488$	5279.13	0.489703
$489$	−1839.04	−0.170070
$490$	0	0
$491$	−978.998	−0.0899828	−0.0449914	−	0.998987i	$-0.514326\pi$
$491$	−978.998	−0.0899828	−0.0449914	+	0.998987i	$0.514326\pi$
$492$	−3232.05	−0.296163
$493$	−1508.10	−0.137771
$494$	−585.958	−0.0533674
$495$	0	0
$496$	5351.64	0.484467
$497$	0	0
$498$	−4195.66	−0.377534
$499$	−7427.49	−0.666333	−0.333166	−	0.942868i	$-0.608117\pi$
$499$	−7427.49	−0.666333	−0.333166	+	0.942868i	$0.608117\pi$
$500$	0	0
$501$	24307.1	2.16759
$502$	−8414.47	−0.748120
$503$	−7872.21	−0.697822	−0.348911	−	0.937156i	$-0.613448\pi$
$503$	−7872.21	−0.697822	−0.348911	+	0.937156i	$0.613448\pi$
$504$	0	0
$505$	0	0
$506$	−19874.4	−1.74610
$507$	12550.6	1.09939
$508$	−4378.40	−0.382402
$509$	12082.5	1.05216	0.526078	−	0.850436i	$-0.323662\pi$
$509$	12082.5	1.05216	0.526078	+	0.850436i	$0.323662\pi$
$510$	0	0
$511$	0	0
$512$	512.000	0.0441942
$513$	−2927.52	−0.251956
$514$	−166.427	−0.0142816
$515$	0	0
$516$	4588.68	0.391483
$517$	9758.40	0.830123
$518$	0	0
$519$	7290.73	0.616623
$520$	0	0
$521$	19728.8	1.65899	0.829497	−	0.558511i	$-0.188627\pi$
$521$	19728.8	1.65899	0.829497	+	0.558511i	$0.188627\pi$
$522$	−2312.06	−0.193862
$523$	90.0382	0.00752791	0.00376396	−	0.999993i	$-0.498802\pi$
$523$	90.0382	0.00752791	0.00376396	+	0.999993i	$0.498802\pi$
$524$	9479.16	0.790265
$525$	0	0
$526$	10080.0	0.835568
$527$	4058.11	0.335435
$528$	5292.42	0.436218
$529$	20595.1	1.69270
$530$	0	0
$531$	−3187.01	−0.260461
$532$	0	0
$533$	1431.27	0.116313
$534$	−12773.7	−1.03516
$535$	0	0
$536$	−4582.44	−0.369275
$537$	6094.67	0.489766
$538$	−10181.1	−0.815870
$539$	0	0
$540$	0	0
$541$	−4428.92	−0.351967	−0.175984	−	0.984393i	$-0.556311\pi$
$541$	−4428.92	−0.351967	−0.175984	+	0.984393i	$0.556311\pi$
$542$	−3343.57	−0.264979
$543$	8351.89	0.660062
$544$	388.246	0.0305991
$545$	0	0
$546$	0	0
$547$	−7292.17	−0.570001	−0.285000	−	0.958527i	$-0.591994\pi$
$547$	−7292.17	−0.570001	−0.285000	+	0.958527i	$0.591994\pi$
$548$	4880.10	0.380415
$549$	6137.18	0.477101
$550$	0	0
$551$	3412.33	0.263830
$552$	−8724.30	−0.672701
$553$	0	0
$554$	−6129.36	−0.470057
$555$	0	0
$556$	−2726.41	−0.207960
$557$	95.2982	0.00724940	0.00362470	−	0.999993i	$-0.498846\pi$
$557$	95.2982	0.00724940	0.00362470	+	0.999993i	$0.498846\pi$
$558$	6221.48	0.472001
$559$	−2032.03	−0.153749
$560$	0	0
$561$	4013.21	0.302028
$562$	−3576.58	−0.268450
$563$	15610.4	1.16856	0.584279	−	0.811553i	$-0.301377\pi$
$563$	15610.4	1.16856	0.584279	+	0.811553i	$0.301377\pi$
$564$	4283.65	0.319813
$565$	0	0
$566$	−8686.12	−0.645062
$567$	0	0
$568$	−5947.24	−0.439332
$569$	3870.40	0.285159	0.142580	−	0.989783i	$-0.454460\pi$
$569$	3870.40	0.285159	0.142580	+	0.989783i	$0.454460\pi$
$570$	0	0
$571$	7447.18	0.545805	0.272903	−	0.962042i	$-0.412016\pi$
$571$	7447.18	0.545805	0.272903	+	0.962042i	$0.412016\pi$
$572$	−2343.67	−0.171318
$573$	−1188.64	−0.0866599
$574$	0	0
$575$	0	0
$576$	595.219	0.0430569
$577$	−16956.1	−1.22338	−0.611691	−	0.791097i	$-0.709510\pi$
$577$	−16956.1	−1.22338	−0.611691	+	0.791097i	$0.709510\pi$
$578$	−9531.60	−0.685921
$579$	−22363.6	−1.60518
$580$	0	0
$581$	0	0
$582$	6965.10	0.496070
$583$	−11628.1	−0.826050
$584$	−9306.67	−0.659440
$585$	0	0
$586$	14605.0	1.02957
$587$	1037.11	0.0729238	0.0364619	−	0.999335i	$-0.488391\pi$
$587$	1037.11	0.0729238	0.0364619	+	0.999335i	$0.488391\pi$
$588$	0	0
$589$	−9182.18	−0.642352
$590$	0	0
$591$	−3069.26	−0.213625
$592$	−1411.17	−0.0979709
$593$	−11649.7	−0.806737	−0.403368	−	0.915038i	$-0.632161\pi$
$593$	−11649.7	−0.806737	−0.403368	+	0.915038i	$0.632161\pi$
$594$	−11709.3	−0.808818
$595$	0	0
$596$	−8033.75	−0.552140
$597$	−27137.5	−1.86041
$598$	3863.43	0.264193
$599$	18549.7	1.26531	0.632656	−	0.774433i	$-0.281965\pi$
$599$	18549.7	1.26531	0.632656	+	0.774433i	$0.281965\pi$
$600$	0	0
$601$	4808.55	0.326364	0.163182	−	0.986596i	$-0.447824\pi$
$601$	4808.55	0.326364	0.163182	+	0.986596i	$0.447824\pi$
$602$	0	0
$603$	−5327.25	−0.359772
$604$	6997.36	0.471388
$605$	0	0
$606$	−20103.8	−1.34763
$607$	7157.03	0.478575	0.239287	−	0.970949i	$-0.423086\pi$
$607$	7157.03	0.478575	0.239287	+	0.970949i	$0.423086\pi$
$608$	−878.474	−0.0585967
$609$	0	0
$610$	0	0
$611$	−1896.95	−0.125602
$612$	451.350	0.0298117
$613$	−24865.2	−1.63833	−0.819165	−	0.573557i	$-0.805563\pi$
$613$	−24865.2	−1.63833	−0.819165	+	0.573557i	$0.805563\pi$
$614$	11675.6	0.767410
$615$	0	0
$616$	0	0
$617$	−24953.7	−1.62820	−0.814099	−	0.580726i	$-0.802769\pi$
$617$	−24953.7	−1.62820	−0.814099	+	0.580726i	$0.802769\pi$
$618$	−10478.3	−0.682036
$619$	17423.5	1.13136	0.565678	−	0.824626i	$-0.308615\pi$
$619$	17423.5	1.13136	0.565678	+	0.824626i	$0.308615\pi$
$620$	0	0
$621$	19302.2	1.24730
$622$	−11619.8	−0.749056
$623$	0	0
$624$	−1028.80	−0.0660018
$625$	0	0
$626$	−10373.8	−0.662335
$627$	−9080.57	−0.578378
$628$	−10805.8	−0.686619
$629$	−1070.08	−0.0678330
$630$	0	0
$631$	14833.8	0.935858	0.467929	−	0.883766i	$-0.345000\pi$
$631$	14833.8	0.935858	0.467929	+	0.883766i	$0.345000\pi$
$632$	8056.89	0.507098
$633$	14876.3	0.934089
$634$	−16264.6	−1.01885
$635$	0	0
$636$	−5104.40	−0.318243
$637$	0	0
$638$	13648.4	0.846936
$639$	−6913.89	−0.428027
$640$	0	0
$641$	8833.77	0.544326	0.272163	−	0.962251i	$-0.412261\pi$
$641$	8833.77	0.544326	0.272163	+	0.962251i	$0.412261\pi$
$642$	20706.8	1.27295
$643$	−10067.6	−0.617462	−0.308731	−	0.951149i	$-0.599904\pi$
$643$	−10067.6	−0.617462	−0.308731	+	0.951149i	$0.599904\pi$
$644$	0	0
$645$	0	0
$646$	−666.140	−0.0405711
$647$	21455.5	1.30372	0.651858	−	0.758341i	$-0.273990\pi$
$647$	21455.5	1.30372	0.651858	+	0.758341i	$0.273990\pi$
$648$	−7148.90	−0.433388
$649$	18813.4	1.13789
$650$	0	0
$651$	0	0
$652$	1220.95	0.0733374
$653$	−8688.51	−0.520685	−0.260343	−	0.965516i	$-0.583836\pi$
$653$	−8688.51	−0.520685	−0.260343	+	0.965516i	$0.583836\pi$
$654$	21349.0	1.27647
$655$	0	0
$656$	2145.77	0.127711
$657$	−10819.4	−0.642471
$658$	0	0
$659$	3196.76	0.188965	0.0944827	−	0.995527i	$-0.469880\pi$
$659$	3196.76	0.188965	0.0944827	+	0.995527i	$0.469880\pi$
$660$	0	0
$661$	−4906.23	−0.288699	−0.144350	−	0.989527i	$-0.546109\pi$
$661$	−4906.23	−0.288699	−0.144350	+	0.989527i	$0.546109\pi$
$662$	11532.0	0.677044
$663$	−780.135	−0.0456983
$664$	2785.52	0.162800
$665$	0	0
$666$	−1640.54	−0.0954499
$667$	−22498.7	−1.30608
$668$	−16137.6	−0.934702
$669$	11006.7	0.636087
$670$	0	0
$671$	−36228.6	−2.08434
$672$	0	0
$673$	27838.8	1.59451	0.797255	−	0.603643i	$-0.206285\pi$
$673$	27838.8	1.59451	0.797255	+	0.603643i	$0.206285\pi$
$674$	−13395.6	−0.765550
$675$	0	0
$676$	−8332.41	−0.474079
$677$	−17110.3	−0.971350	−0.485675	−	0.874140i	$-0.661426\pi$
$677$	−17110.3	−0.971350	−0.485675	+	0.874140i	$0.661426\pi$
$678$	3265.36	0.184963
$679$	0	0
$680$	0	0
$681$	25579.3	1.43936
$682$	−36726.2	−2.06205
$683$	−3308.61	−0.185359	−0.0926797	−	0.995696i	$-0.529543\pi$
$683$	−3308.61	−0.185359	−0.0926797	+	0.995696i	$0.529543\pi$
$684$	−1021.26	−0.0570889
$685$	0	0
$686$	0	0
$687$	19593.6	1.08812
$688$	−3046.44	−0.168814
$689$	2260.41	0.124985
$690$	0	0
$691$	28127.2	1.54849	0.774246	−	0.632885i	$-0.218129\pi$
$691$	28127.2	1.54849	0.774246	+	0.632885i	$0.218129\pi$
$692$	−4840.34	−0.265899
$693$	0	0
$694$	17586.6	0.961930
$695$	0	0
$696$	5991.25	0.326290
$697$	1627.12	0.0884241
$698$	−9357.78	−0.507446
$699$	23463.6	1.26963
$700$	0	0
$701$	−21962.0	−1.18330	−0.591650	−	0.806195i	$-0.701523\pi$
$701$	−21962.0	−1.18330	−0.591650	+	0.806195i	$0.701523\pi$
$702$	2276.19	0.122378
$703$	2421.24	0.129899
$704$	−3513.66	−0.188105
$705$	0	0
$706$	−10016.1	−0.533940
$707$	0	0
$708$	8258.52	0.438382
$709$	16353.0	0.866222	0.433111	−	0.901341i	$-0.357416\pi$
$709$	16353.0	0.866222	0.433111	+	0.901341i	$0.357416\pi$
$710$	0	0
$711$	9366.43	0.494049
$712$	8480.51	0.446377
$713$	60541.5	3.17994
$714$	0	0
$715$	0	0
$716$	−4046.27	−0.211196
$717$	3862.60	0.201188
$718$	9109.81	0.473503
$719$	18095.8	0.938607	0.469303	−	0.883037i	$-0.344505\pi$
$719$	18095.8	0.938607	0.469303	+	0.883037i	$0.344505\pi$
$720$	0	0
$721$	0	0
$722$	−12210.7	−0.629414
$723$	26532.3	1.36479
$724$	−5544.84	−0.284631
$725$	0	0
$726$	−20281.4	−1.03679
$727$	−9732.61	−0.496510	−0.248255	−	0.968695i	$-0.579857\pi$
$727$	−9732.61	−0.496510	−0.248255	+	0.968695i	$0.579857\pi$
$728$	0	0
$729$	9036.76	0.459115
$730$	0	0
$731$	−2310.09	−0.116884
$732$	−15903.3	−0.803010
$733$	−31738.9	−1.59932	−0.799661	−	0.600452i	$-0.794987\pi$
$733$	−31738.9	−1.59932	−0.799661	+	0.600452i	$0.794987\pi$
$734$	16682.3	0.838905
$735$	0	0
$736$	5792.10	0.290081
$737$	31447.5	1.57175
$738$	2494.53	0.124424
$739$	−6082.66	−0.302780	−0.151390	−	0.988474i	$-0.548375\pi$
$739$	−6082.66	−0.302780	−0.151390	+	0.988474i	$0.548375\pi$
$740$	0	0
$741$	1765.19	0.0875114
$742$	0	0
$743$	−21355.5	−1.05445	−0.527225	−	0.849726i	$-0.676768\pi$
$743$	−21355.5	−1.05445	−0.527225	+	0.849726i	$0.676768\pi$
$744$	−16121.8	−0.794425
$745$	0	0
$746$	−5685.85	−0.279053
$747$	3238.26	0.158610
$748$	−2664.38	−0.130240
$749$	0	0
$750$	0	0
$751$	955.107	0.0464079	0.0232040	−	0.999731i	$-0.492613\pi$
$751$	955.107	0.0464079	0.0232040	+	0.999731i	$0.492613\pi$
$752$	−2843.93	−0.137909
$753$	25348.5	1.22676
$754$	−2653.14	−0.128145
$755$	0	0
$756$	0	0
$757$	32116.3	1.54199	0.770996	−	0.636840i	$-0.219759\pi$
$757$	32116.3	1.54199	0.770996	+	0.636840i	$0.219759\pi$
$758$	−2622.51	−0.125665
$759$	59871.5	2.86324
$760$	0	0
$761$	9234.12	0.439864	0.219932	−	0.975515i	$-0.429416\pi$
$761$	9234.12	0.439864	0.219932	+	0.975515i	$0.429416\pi$
$762$	13189.9	0.627059
$763$	0	0
$764$	789.142	0.0373693
$765$	0	0
$766$	23605.2	1.11344
$767$	−3657.17	−0.172168
$768$	−1542.39	−0.0724692
$769$	−29271.9	−1.37266	−0.686328	−	0.727292i	$-0.740778\pi$
$769$	−29271.9	−1.37266	−0.686328	+	0.727292i	$0.740778\pi$
$770$	0	0
$771$	501.358	0.0234189
$772$	14847.3	0.692184
$773$	−9256.13	−0.430686	−0.215343	−	0.976539i	$-0.569087\pi$
$773$	−9256.13	−0.430686	−0.215343	+	0.976539i	$0.569087\pi$
$774$	−3541.60	−0.164470
$775$	0	0
$776$	−4624.16	−0.213914
$777$	0	0
$778$	−7016.48	−0.323333
$779$	−3681.64	−0.169331
$780$	0	0
$781$	40813.6	1.86994
$782$	4392.10	0.200846
$783$	−13255.4	−0.604993
$784$	0	0
$785$	0	0
$786$	−28555.8	−1.29587
$787$	21454.9	0.971772	0.485886	−	0.874022i	$-0.338497\pi$
$787$	21454.9	0.971772	0.485886	+	0.874022i	$0.338497\pi$
$788$	2037.69	0.0921190
$789$	−30365.9	−1.37016
$790$	0	0
$791$	0	0
$792$	−4084.76	−0.183265
$793$	7042.55	0.315370
$794$	7456.81	0.333290
$795$	0	0
$796$	18016.6	0.802240
$797$	−9524.52	−0.423307	−0.211654	−	0.977345i	$-0.567885\pi$
$797$	−9524.52	−0.423307	−0.211654	+	0.977345i	$0.567885\pi$
$798$	0	0
$799$	−2156.53	−0.0954851
$800$	0	0
$801$	9858.91	0.434891
$802$	16707.6	0.735619
$803$	63868.1	2.80679
$804$	13804.5	0.605533
$805$	0	0
$806$	7139.29	0.311998
$807$	30670.4	1.33786
$808$	13347.0	0.581121
$809$	26558.5	1.15420	0.577099	−	0.816674i	$-0.304185\pi$
$809$	26558.5	1.15420	0.577099	+	0.816674i	$0.304185\pi$
$810$	0	0
$811$	5940.88	0.257229	0.128614	−	0.991695i	$-0.458947\pi$
$811$	5940.88	0.257229	0.128614	+	0.991695i	$0.458947\pi$
$812$	0	0
$813$	10072.5	0.434510
$814$	9684.32	0.416997
$815$	0	0
$816$	−1169.59	−0.0501761
$817$	5226.98	0.223830
$818$	−22724.5	−0.971323
$819$	0	0
$820$	0	0
$821$	−15988.9	−0.679679	−0.339839	−	0.940483i	$-0.610373\pi$
$821$	−15988.9	−0.679679	−0.339839	+	0.940483i	$0.610373\pi$
$822$	−14701.2	−0.623800
$823$	−15330.7	−0.649325	−0.324662	−	0.945830i	$-0.605251\pi$
$823$	−15330.7	−0.649325	−0.324662	+	0.945830i	$0.605251\pi$
$824$	6956.57	0.294106
$825$	0	0
$826$	0	0
$827$	7892.50	0.331861	0.165931	−	0.986137i	$-0.446937\pi$
$827$	7892.50	0.331861	0.165931	+	0.986137i	$0.446937\pi$
$828$	6733.53	0.282616
$829$	−17664.6	−0.740069	−0.370034	−	0.929018i	$-0.620654\pi$
$829$	−17664.6	−0.740069	−0.370034	+	0.929018i	$0.620654\pi$
$830$	0	0
$831$	18464.6	0.770794
$832$	683.027	0.0284612
$833$	0	0
$834$	8213.27	0.341010
$835$	0	0
$836$	6028.62	0.249407
$837$	35668.8	1.47299
$838$	14889.9	0.613800
$839$	−2826.25	−0.116297	−0.0581484	−	0.998308i	$-0.518520\pi$
$839$	−2826.25	−0.116297	−0.0581484	+	0.998308i	$0.518520\pi$
$840$	0	0
$841$	−8938.44	−0.366495
$842$	−18346.1	−0.750890
$843$	10774.4	0.440201
$844$	−9876.40	−0.402796
$845$	0	0
$846$	−3306.18	−0.134360
$847$	0	0
$848$	3388.83	0.137232
$849$	26166.8	1.05777
$850$	0	0
$851$	−15964.1	−0.643060
$852$	17916.0	0.720413
$853$	46441.9	1.86417	0.932087	−	0.362234i	$-0.117986\pi$
$853$	46441.9	1.86417	0.932087	+	0.362234i	$0.117986\pi$
$854$	0	0
$855$	0	0
$856$	−13747.3	−0.548918
$857$	−5995.48	−0.238975	−0.119488	−	0.992836i	$-0.538125\pi$
$857$	−5995.48	−0.238975	−0.119488	+	0.992836i	$0.538125\pi$
$858$	7060.29	0.280926
$859$	4671.33	0.185546	0.0927729	−	0.995687i	$-0.470427\pi$
$859$	4671.33	0.185546	0.0927729	+	0.995687i	$0.470427\pi$
$860$	0	0
$861$	0	0
$862$	13431.8	0.530729
$863$	40220.0	1.58645	0.793224	−	0.608930i	$-0.208401\pi$
$863$	40220.0	1.58645	0.793224	+	0.608930i	$0.208401\pi$
$864$	3412.49	0.134369
$865$	0	0
$866$	−9482.91	−0.372104
$867$	28713.8	1.12477
$868$	0	0
$869$	−55291.3	−2.15838
$870$	0	0
$871$	−6113.14	−0.237814
$872$	−14173.7	−0.550438
$873$	−5375.75	−0.208410
$874$	−9937.90	−0.384616
$875$	0	0
$876$	28036.2	1.08134
$877$	15820.2	0.609135	0.304567	−	0.952491i	$-0.401488\pi$
$877$	15820.2	0.609135	0.304567	+	0.952491i	$0.401488\pi$
$878$	−274.868	−0.0105653
$879$	−43997.3	−1.68827
$880$	0	0
$881$	−4103.34	−0.156918	−0.0784592	−	0.996917i	$-0.525000\pi$
$881$	−4103.34	−0.156918	−0.0784592	+	0.996917i	$0.525000\pi$
$882$	0	0
$883$	25725.3	0.980436	0.490218	−	0.871600i	$-0.336917\pi$
$883$	25725.3	0.980436	0.490218	+	0.871600i	$0.336917\pi$
$884$	517.935	0.0197059
$885$	0	0
$886$	−22007.1	−0.834473
$887$	−16215.9	−0.613840	−0.306920	−	0.951735i	$-0.599298\pi$
$887$	−16215.9	−0.613840	−0.306920	+	0.951735i	$0.599298\pi$
$888$	4251.13	0.160652
$889$	0	0
$890$	0	0
$891$	49060.1	1.84464
$892$	−7307.36	−0.274292
$893$	4879.53	0.182852
$894$	24201.6	0.905393
$895$	0	0
$896$	0	0
$897$	−11638.5	−0.433222
$898$	33699.8	1.25231
$899$	−41575.7	−1.54241
$900$	0	0
$901$	2569.72	0.0950166
$902$	−14725.6	−0.543578
$903$	0	0
$904$	−2167.88	−0.0797596
$905$	0	0
$906$	−21079.4	−0.772977
$907$	−633.923	−0.0232073	−0.0116037	−	0.999933i	$-0.503694\pi$
$907$	−633.923	−0.0232073	−0.0116037	+	0.999933i	$0.503694\pi$
$908$	−16982.2	−0.620676
$909$	15516.4	0.566167
$910$	0	0
$911$	−35153.6	−1.27848	−0.639238	−	0.769009i	$-0.720750\pi$
$911$	−35153.6	−1.27848	−0.639238	+	0.769009i	$0.720750\pi$
$912$	2646.39	0.0960863
$913$	−19115.9	−0.692929
$914$	8525.56	0.308534
$915$	0	0
$916$	−13008.2	−0.469219
$917$	0	0
$918$	2587.66	0.0930345
$919$	12758.7	0.457964	0.228982	−	0.973431i	$-0.426460\pi$
$919$	12758.7	0.457964	0.228982	+	0.973431i	$0.426460\pi$
$920$	0	0
$921$	−35172.6	−1.25839
$922$	−7233.68	−0.258382
$923$	−7933.84	−0.282931
$924$	0	0
$925$	0	0
$926$	−31828.1	−1.12952
$927$	8087.27	0.286538
$928$	−3977.61	−0.140702
$929$	−14821.9	−0.523457	−0.261728	−	0.965142i	$-0.584292\pi$
$929$	−14821.9	−0.523457	−0.261728	+	0.965142i	$0.584292\pi$
$930$	0	0
$931$	0	0
$932$	−15577.6	−0.547489
$933$	35004.6	1.22829
$934$	−35951.5	−1.25949
$935$	0	0
$936$	794.044	0.0277288
$937$	30747.5	1.07201	0.536006	−	0.844214i	$-0.319932\pi$
$937$	30747.5	1.07201	0.536006	+	0.844214i	$0.319932\pi$
$938$	0	0
$939$	31251.0	1.08609
$940$	0	0
$941$	14309.1	0.495711	0.247855	−	0.968797i	$-0.420274\pi$
$941$	14309.1	0.495711	0.247855	+	0.968797i	$0.420274\pi$
$942$	32552.2	1.12591
$943$	24274.4	0.838264
$944$	−5482.86	−0.189038
$945$	0	0
$946$	20906.5	0.718530
$947$	29708.6	1.01943	0.509715	−	0.860343i	$-0.329751\pi$
$947$	29708.6	1.01943	0.509715	+	0.860343i	$0.329751\pi$
$948$	−24271.3	−0.831534
$949$	−12415.4	−0.424681
$950$	0	0
$951$	48997.0	1.67070
$952$	0	0
$953$	31956.6	1.08623	0.543114	−	0.839659i	$-0.317245\pi$
$953$	31956.6	1.08623	0.543114	+	0.839659i	$0.317245\pi$
$954$	3939.64	0.133701
$955$	0	0
$956$	−2564.40	−0.0867558
$957$	−41115.6	−1.38880
$958$	−25091.8	−0.846221
$959$	0	0
$960$	0	0
$961$	82084.3	2.75534
$962$	−1882.56	−0.0630936
$963$	−15981.8	−0.534792
$964$	−17614.9	−0.588524
$965$	0	0
$966$	0	0
$967$	−32350.5	−1.07582	−0.537911	−	0.843001i	$-0.680786\pi$
$967$	−32350.5	−1.07582	−0.537911	+	0.843001i	$0.680786\pi$
$968$	13464.9	0.447084
$969$	2006.74	0.0665281
$970$	0	0
$971$	7988.02	0.264004	0.132002	−	0.991249i	$-0.457860\pi$
$971$	7988.02	0.264004	0.132002	+	0.991249i	$0.457860\pi$
$972$	10018.8	0.330611
$973$	0	0
$974$	−16746.9	−0.550928
$975$	0	0
$976$	10558.3	0.346272
$977$	−43075.2	−1.41054	−0.705270	−	0.708939i	$-0.749174\pi$
$977$	−43075.2	−1.41054	−0.705270	+	0.708939i	$0.749174\pi$
$978$	−3678.09	−0.120258
$979$	−58198.5	−1.89993
$980$	0	0
$981$	−16477.4	−0.536274
$982$	−1958.00	−0.0636275
$983$	−28618.9	−0.928587	−0.464294	−	0.885681i	$-0.653692\pi$
$983$	−28618.9	−0.928587	−0.464294	+	0.885681i	$0.653692\pi$
$984$	−6464.10	−0.209419
$985$	0	0
$986$	−3016.19	−0.0974190
$987$	0	0
$988$	−1171.92	−0.0377365
$989$	−34463.4	−1.10806
$990$	0	0
$991$	−13783.9	−0.441836	−0.220918	−	0.975292i	$-0.570905\pi$
$991$	−13783.9	−0.441836	−0.220918	+	0.975292i	$0.570905\pi$
$992$	10703.3	0.342570
$993$	−34740.0	−1.11021
$994$	0	0
$995$	0	0
$996$	−8391.33	−0.266957
$997$	8857.97	0.281379	0.140689	−	0.990054i	$-0.455068\pi$
$997$	8857.97	0.281379	0.140689	+	0.990054i	$0.455068\pi$
$998$	−14855.0	−0.471168
$999$	−9405.48	−0.297874

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2450.4.a.cs.1.1		4
5.2	odd	4		490.4.c.d.99.8	yes	8
5.3	odd	4		490.4.c.d.99.1	✓	8
5.4	even	2		2450.4.a.cm.1.4		4
7.6	odd	2	inner	2450.4.a.cs.1.4		4
35.13	even	4		490.4.c.d.99.4	yes	8
35.27	even	4		490.4.c.d.99.5	yes	8
35.34	odd	2		2450.4.a.cm.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
490.4.c.d.99.1	✓	8	5.3	odd	4
490.4.c.d.99.4	yes	8	35.13	even	4
490.4.c.d.99.5	yes	8	35.27	even	4
490.4.c.d.99.8	yes	8	5.2	odd	4
2450.4.a.cm.1.1		4	35.34	odd	2
2450.4.a.cm.1.4		4	5.4	even	2
2450.4.a.cs.1.1		4	1.1	even	1	trivial
2450.4.a.cs.1.4		4	7.6	odd	2	inner

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)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} -6.02497 q^{3} +4.00000 q^{4} -12.0499 q^{6} +8.00000 q^{8} +9.30030 q^{9} +O(q^{10})\)	\(q+2.00000 q^{2} -6.02497 q^{3} +4.00000 q^{4} -12.0499 q^{6} +8.00000 q^{8} +9.30030 q^{9} -54.9009 q^{11} -24.0999 q^{12} +10.6723 q^{13} +16.0000 q^{16} +12.1327 q^{17} +18.6006 q^{18} -27.4523 q^{19} -109.802 q^{22} +181.003 q^{23} -48.1998 q^{24} +21.3446 q^{26} +106.640 q^{27} -124.300 q^{29} +334.478 q^{31} +32.0000 q^{32} +330.776 q^{33} +24.2654 q^{34} +37.2012 q^{36} -88.1982 q^{37} -54.9046 q^{38} -64.3003 q^{39} +134.110 q^{41} -190.402 q^{43} -219.604 q^{44} +362.006 q^{46} -177.746 q^{47} -96.3996 q^{48} -73.0991 q^{51} +42.6892 q^{52} +211.802 q^{53} +213.280 q^{54} +165.399 q^{57} -248.601 q^{58} -342.679 q^{59} +659.891 q^{61} +668.955 q^{62} +64.0000 q^{64} +661.553 q^{66} -572.805 q^{67} +48.5307 q^{68} -1090.54 q^{69} -743.405 q^{71} +74.4024 q^{72} -1163.33 q^{73} -176.396 q^{74} -109.809 q^{76} -128.601 q^{78} +1007.11 q^{79} -893.612 q^{81} +268.221 q^{82} +348.189 q^{83} -380.805 q^{86} +748.906 q^{87} -439.207 q^{88} +1060.06 q^{89} +724.012 q^{92} -2015.22 q^{93} -355.491 q^{94} -192.799 q^{96} -578.020 q^{97} -510.595 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{2} + 16 q^{4} + 32 q^{8} - 30 q^{9}+O(q^{10}) \) 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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