LMFDB - Embedded newform 4450.2.a.bj.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4450,2,Mod(1,4450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4450, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4450.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4450 = 2 \cdot 5^{2} \cdot 89 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4450.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:	$35.5334288995$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 4x^{6} + 26x^{5} - 3x^{4} - 42x^{3} + 20x^{2} + 4x - 2 $ x^8 - 4x^7 - 4x^6 + 26x^5 - 3x^4 - 42x^3 + 20x^2 + 4*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 890)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.85072$ of defining polynomial
Character	$\chi$	$=$	4450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} -2.85072 q^{3} +1.00000 q^{4} -2.85072 q^{6} +4.52360 q^{7} +1.00000 q^{8} +5.12662 q^{9} -1.22864 q^{11} -2.85072 q^{12} +1.37273 q^{13} +4.52360 q^{14} +1.00000 q^{16} -4.61484 q^{17} +5.12662 q^{18} -4.47801 q^{19} -12.8955 q^{21} -1.22864 q^{22} -4.38002 q^{23} -2.85072 q^{24} +1.37273 q^{26} -6.06241 q^{27} +4.52360 q^{28} -5.55434 q^{29} +6.27071 q^{31} +1.00000 q^{32} +3.50252 q^{33} -4.61484 q^{34} +5.12662 q^{36} -7.27164 q^{37} -4.47801 q^{38} -3.91327 q^{39} -3.98348 q^{41} -12.8955 q^{42} -6.25152 q^{43} -1.22864 q^{44} -4.38002 q^{46} -10.6104 q^{47} -2.85072 q^{48} +13.4629 q^{49} +13.1556 q^{51} +1.37273 q^{52} -10.4611 q^{53} -6.06241 q^{54} +4.52360 q^{56} +12.7656 q^{57} -5.55434 q^{58} -6.84371 q^{59} +13.9786 q^{61} +6.27071 q^{62} +23.1908 q^{63} +1.00000 q^{64} +3.50252 q^{66} +0.657004 q^{67} -4.61484 q^{68} +12.4862 q^{69} -1.09617 q^{71} +5.12662 q^{72} +9.58493 q^{73} -7.27164 q^{74} -4.47801 q^{76} -5.55787 q^{77} -3.91327 q^{78} +15.9984 q^{79} +1.90240 q^{81} -3.98348 q^{82} +11.4611 q^{83} -12.8955 q^{84} -6.25152 q^{86} +15.8339 q^{87} -1.22864 q^{88} -1.00000 q^{89} +6.20967 q^{91} -4.38002 q^{92} -17.8761 q^{93} -10.6104 q^{94} -2.85072 q^{96} -15.1680 q^{97} +13.4629 q^{98} -6.29878 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} - 4 q^{3} + 8 q^{4} - 4 q^{6} - 2 q^{7} + 8 q^{8} - 4 q^{11} - 4 q^{12} - 4 q^{13} - 2 q^{14} + 8 q^{16} - 14 q^{17} - 8 q^{19} - 10 q^{21} - 4 q^{22} - 24 q^{23} - 4 q^{24} - 4 q^{26} - 10 q^{27}+ \cdots + 12 q^{99}+O(q^{100}) $ 8 * q + 8 * q^2 - 4 * q^3 + 8 * q^4 - 4 * q^6 - 2 * q^7 + 8 * q^8 - 4 * q^11 - 4 * q^12 - 4 * q^13 - 2 * q^14 + 8 * q^16 - 14 * q^17 - 8 * q^19 - 10 * q^21 - 4 * q^22 - 24 * q^23 - 4 * q^24 - 4 * q^26 - 10 * q^27 - 2 * q^28 - 10 * q^29 + 8 * q^32 - 10 * q^33 - 14 * q^34 - 14 * q^37 - 8 * q^38 - 10 * q^39 - 10 * q^42 - 4 * q^43 - 4 * q^44 - 24 * q^46 - 28 * q^47 - 4 * q^48 - 4 * q^49 + 4 * q^51 - 4 * q^52 - 42 * q^53 - 10 * q^54 - 2 * q^56 + 20 * q^57 - 10 * q^58 - 4 * q^59 - 2 * q^61 + 6 * q^63 + 8 * q^64 - 10 * q^66 - 18 * q^67 - 14 * q^68 - 18 * q^69 + 20 * q^71 - 18 * q^73 - 14 * q^74 - 8 * q^76 - 34 * q^77 - 10 * q^78 + 2 * q^79 - 4 * q^81 - 20 * q^83 - 10 * q^84 - 4 * q^86 + 30 * q^87 - 4 * q^88 - 8 * q^89 + 20 * q^91 - 24 * q^92 - 46 * q^93 - 28 * q^94 - 4 * q^96 - 16 * q^97 - 4 * q^98 + 12 * q^99

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−2.85072	−1.64587	−0.822933	−	0.568139i	$-0.807664\pi$
$3$	−2.85072	−1.64587	−0.822933	+	0.568139i	$0.807664\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−2.85072	−1.16380
$7$	4.52360	1.70976	0.854879	−	0.518827i	$-0.173631\pi$
$7$	4.52360	1.70976	0.854879	+	0.518827i	$0.173631\pi$
$8$	1.00000	0.353553
$9$	5.12662	1.70887
$10$	0	0
$11$	−1.22864	−0.370449	−0.185225	−	0.982696i	$-0.559301\pi$
$11$	−1.22864	−0.370449	−0.185225	+	0.982696i	$0.559301\pi$
$12$	−2.85072	−0.822933
$13$	1.37273	0.380726	0.190363	−	0.981714i	$-0.439033\pi$
$13$	1.37273	0.380726	0.190363	+	0.981714i	$0.439033\pi$
$14$	4.52360	1.20898
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.61484	−1.11926	−0.559632	−	0.828741i	$-0.689058\pi$
$17$	−4.61484	−1.11926	−0.559632	+	0.828741i	$0.689058\pi$
$18$	5.12662	1.20836
$19$	−4.47801	−1.02733	−0.513664	−	0.857992i	$-0.671712\pi$
$19$	−4.47801	−1.02733	−0.513664	+	0.857992i	$0.671712\pi$
$20$	0	0
$21$	−12.8955	−2.81403
$22$	−1.22864	−0.261947
$23$	−4.38002	−0.913298	−0.456649	−	0.889647i	$-0.650950\pi$
$23$	−4.38002	−0.913298	−0.456649	+	0.889647i	$0.650950\pi$
$24$	−2.85072	−0.581901
$25$	0	0
$26$	1.37273	0.269214
$27$	−6.06241	−1.16671
$28$	4.52360	0.854879
$29$	−5.55434	−1.03142	−0.515708	−	0.856764i	$-0.672471\pi$
$29$	−5.55434	−1.03142	−0.515708	+	0.856764i	$0.672471\pi$
$30$	0	0
$31$	6.27071	1.12625	0.563126	−	0.826371i	$-0.309598\pi$
$31$	6.27071	1.12625	0.563126	+	0.826371i	$0.309598\pi$
$32$	1.00000	0.176777
$33$	3.50252	0.609710
$34$	−4.61484	−0.791439
$35$	0	0
$36$	5.12662	0.854437
$37$	−7.27164	−1.19545	−0.597725	−	0.801701i	$-0.703929\pi$
$37$	−7.27164	−1.19545	−0.597725	+	0.801701i	$0.703929\pi$
$38$	−4.47801	−0.726430
$39$	−3.91327	−0.626624
$40$	0	0
$41$	−3.98348	−0.622114	−0.311057	−	0.950391i	$-0.600683\pi$
$41$	−3.98348	−0.622114	−0.311057	+	0.950391i	$0.600683\pi$
$42$	−12.8955	−1.98982
$43$	−6.25152	−0.953348	−0.476674	−	0.879080i	$-0.658158\pi$
$43$	−6.25152	−0.953348	−0.476674	+	0.879080i	$0.658158\pi$
$44$	−1.22864	−0.185225
$45$	0	0
$46$	−4.38002	−0.645799
$47$	−10.6104	−1.54769	−0.773845	−	0.633374i	$-0.781669\pi$
$47$	−10.6104	−1.54769	−0.773845	+	0.633374i	$0.781669\pi$
$48$	−2.85072	−0.411466
$49$	13.4629	1.92327
$50$	0	0
$51$	13.1556	1.84216
$52$	1.37273	0.190363
$53$	−10.4611	−1.43694	−0.718469	−	0.695559i	$-0.755157\pi$
$53$	−10.4611	−1.43694	−0.718469	+	0.695559i	$0.755157\pi$
$54$	−6.06241	−0.824990
$55$	0	0
$56$	4.52360	0.604491
$57$	12.7656	1.69084
$58$	−5.55434	−0.729321
$59$	−6.84371	−0.890976	−0.445488	−	0.895288i	$-0.646970\pi$
$59$	−6.84371	−0.890976	−0.445488	+	0.895288i	$0.646970\pi$
$60$	0	0
$61$	13.9786	1.78977	0.894887	−	0.446293i	$-0.147256\pi$
$61$	13.9786	1.78977	0.894887	+	0.446293i	$0.147256\pi$
$62$	6.27071	0.796381
$63$	23.1908	2.92176
$64$	1.00000	0.125000
$65$	0	0
$66$	3.50252	0.431130
$67$	0.657004	0.0802658	0.0401329	−	0.999194i	$-0.487222\pi$
$67$	0.657004	0.0802658	0.0401329	+	0.999194i	$0.487222\pi$
$68$	−4.61484	−0.559632
$69$	12.4862	1.50317
$70$	0	0
$71$	−1.09617	−0.130092	−0.0650460	−	0.997882i	$-0.520719\pi$
$71$	−1.09617	−0.130092	−0.0650460	+	0.997882i	$0.520719\pi$
$72$	5.12662	0.604178
$73$	9.58493	1.12183	0.560916	−	0.827873i	$-0.310449\pi$
$73$	9.58493	1.12183	0.560916	+	0.827873i	$0.310449\pi$
$74$	−7.27164	−0.845311
$75$	0	0
$76$	−4.47801	−0.513664
$77$	−5.55787	−0.633378
$78$	−3.91327	−0.443090
$79$	15.9984	1.79996	0.899979	−	0.435934i	$-0.143582\pi$
$79$	15.9984	1.79996	0.899979	+	0.435934i	$0.143582\pi$
$80$	0	0
$81$	1.90240	0.211377
$82$	−3.98348	−0.439901
$83$	11.4611	1.25801	0.629007	−	0.777400i	$-0.283462\pi$
$83$	11.4611	1.25801	0.629007	+	0.777400i	$0.283462\pi$
$84$	−12.8955	−1.40702
$85$	0	0
$86$	−6.25152	−0.674119
$87$	15.8339	1.69757
$88$	−1.22864	−0.130974
$89$	−1.00000	−0.106000
$89$	−1.00000	−0.106000
$90$	0	0
$91$	6.20967	0.650950
$92$	−4.38002	−0.456649
$93$	−17.8761	−1.85366
$94$	−10.6104	−1.09438
$95$	0	0
$96$	−2.85072	−0.290951
$97$	−15.1680	−1.54007	−0.770036	−	0.638000i	$-0.779762\pi$
$97$	−15.1680	−1.54007	−0.770036	+	0.638000i	$0.779762\pi$
$98$	13.4629	1.35996
$99$	−6.29878	−0.633051
$100$	0	0
$101$	−4.96914	−0.494448	−0.247224	−	0.968958i	$-0.579518\pi$
$101$	−4.96914	−0.494448	−0.247224	+	0.968958i	$0.579518\pi$
$102$	13.1556	1.30260
$103$	−12.8655	−1.26767	−0.633837	−	0.773467i	$-0.718521\pi$
$103$	−12.8655	−1.26767	−0.633837	+	0.773467i	$0.718521\pi$
$104$	1.37273	0.134607
$105$	0	0
$106$	−10.4611	−1.01607
$107$	−13.6498	−1.31958	−0.659790	−	0.751450i	$-0.729355\pi$
$107$	−13.6498	−1.31958	−0.659790	+	0.751450i	$0.729355\pi$
$108$	−6.06241	−0.583356
$109$	−3.90012	−0.373563	−0.186782	−	0.982401i	$-0.559806\pi$
$109$	−3.90012	−0.373563	−0.186782	+	0.982401i	$0.559806\pi$
$110$	0	0
$111$	20.7294	1.96755
$112$	4.52360	0.427440
$113$	−1.60019	−0.150533	−0.0752665	−	0.997163i	$-0.523981\pi$
$113$	−1.60019	−0.150533	−0.0752665	+	0.997163i	$0.523981\pi$
$114$	12.7656	1.19561
$115$	0	0
$116$	−5.55434	−0.515708
$117$	7.03746	0.650613
$118$	−6.84371	−0.630015
$119$	−20.8757	−1.91367
$120$	0	0
$121$	−9.49044	−0.862767
$122$	13.9786	1.26556
$123$	11.3558	1.02392
$124$	6.27071	0.563126
$125$	0	0
$126$	23.1908	2.06600
$127$	8.82600	0.783181	0.391590	−	0.920140i	$-0.371925\pi$
$127$	8.82600	0.783181	0.391590	+	0.920140i	$0.371925\pi$
$128$	1.00000	0.0883883
$129$	17.8214	1.56908
$130$	0	0
$131$	5.31775	0.464614	0.232307	−	0.972642i	$-0.425373\pi$
$131$	5.31775	0.464614	0.232307	+	0.972642i	$0.425373\pi$
$132$	3.50252	0.304855
$133$	−20.2567	−1.75648
$134$	0.657004	0.0567565
$135$	0	0
$136$	−4.61484	−0.395720
$137$	0.978495	0.0835985	0.0417992	−	0.999126i	$-0.486691\pi$
$137$	0.978495	0.0835985	0.0417992	+	0.999126i	$0.486691\pi$
$138$	12.4862	1.06290
$139$	4.23577	0.359273	0.179637	−	0.983733i	$-0.442508\pi$
$139$	4.23577	0.359273	0.179637	+	0.983733i	$0.442508\pi$
$140$	0	0
$141$	30.2474	2.54729
$142$	−1.09617	−0.0919890
$143$	−1.68659	−0.141040
$144$	5.12662	0.427219
$145$	0	0
$146$	9.58493	0.793255
$147$	−38.3790	−3.16545
$148$	−7.27164	−0.597725
$149$	8.49351	0.695815	0.347908	−	0.937529i	$-0.386892\pi$
$149$	8.49351	0.695815	0.347908	+	0.937529i	$0.386892\pi$
$150$	0	0
$151$	9.09426	0.740080	0.370040	−	0.929016i	$-0.379344\pi$
$151$	9.09426	0.740080	0.370040	+	0.929016i	$0.379344\pi$
$152$	−4.47801	−0.363215
$153$	−23.6586	−1.91268
$154$	−5.55787	−0.447866
$155$	0	0
$156$	−3.91327	−0.313312
$157$	16.5202	1.31845	0.659227	−	0.751944i	$-0.270884\pi$
$157$	16.5202	1.31845	0.659227	+	0.751944i	$0.270884\pi$
$158$	15.9984	1.27276
$159$	29.8216	2.36501
$160$	0	0
$161$	−19.8134	−1.56152
$162$	1.90240	0.149466
$163$	−17.0813	−1.33791	−0.668954	−	0.743304i	$-0.733257\pi$
$163$	−17.0813	−1.33791	−0.668954	+	0.743304i	$0.733257\pi$
$164$	−3.98348	−0.311057
$165$	0	0
$166$	11.4611	0.889550
$167$	−2.94829	−0.228146	−0.114073	−	0.993472i	$-0.536390\pi$
$167$	−2.94829	−0.228146	−0.114073	+	0.993472i	$0.536390\pi$
$168$	−12.8955	−0.994911
$169$	−11.1156	−0.855048
$170$	0	0
$171$	−22.9571	−1.75557
$172$	−6.25152	−0.476674
$173$	−5.36147	−0.407625	−0.203813	−	0.979010i	$-0.565333\pi$
$173$	−5.36147	−0.407625	−0.203813	+	0.979010i	$0.565333\pi$
$174$	15.8339	1.20036
$175$	0	0
$176$	−1.22864	−0.0926123
$177$	19.5095	1.46643
$178$	−1.00000	−0.0749532
$179$	11.5470	0.863067	0.431533	−	0.902097i	$-0.357973\pi$
$179$	11.5470	0.863067	0.431533	+	0.902097i	$0.357973\pi$
$180$	0	0
$181$	−25.3052	−1.88092	−0.940459	−	0.339907i	$-0.889604\pi$
$181$	−25.3052	−1.88092	−0.940459	+	0.339907i	$0.889604\pi$
$182$	6.20967	0.460291
$183$	−39.8491	−2.94573
$184$	−4.38002	−0.322900
$185$	0	0
$186$	−17.8761	−1.31074
$187$	5.66999	0.414630
$188$	−10.6104	−0.773845
$189$	−27.4239	−1.99480
$190$	0	0
$191$	15.7942	1.14283	0.571413	−	0.820662i	$-0.306395\pi$
$191$	15.7942	1.14283	0.571413	+	0.820662i	$0.306395\pi$
$192$	−2.85072	−0.205733
$193$	−7.69294	−0.553750	−0.276875	−	0.960906i	$-0.589299\pi$
$193$	−7.69294	−0.553750	−0.276875	+	0.960906i	$0.589299\pi$
$194$	−15.1680	−1.08900
$195$	0	0
$196$	13.4629	0.961637
$197$	0.115358	0.00821893	0.00410946	−	0.999992i	$-0.498692\pi$
$197$	0.115358	0.00821893	0.00410946	+	0.999992i	$0.498692\pi$
$198$	−6.29878	−0.447635
$199$	0.584508	0.0414347	0.0207173	−	0.999785i	$-0.493405\pi$
$199$	0.584508	0.0414347	0.0207173	+	0.999785i	$0.493405\pi$
$200$	0	0
$201$	−1.87294	−0.132107
$202$	−4.96914	−0.349627
$203$	−25.1256	−1.76347
$204$	13.1556	0.921079
$205$	0	0
$206$	−12.8655	−0.896381
$207$	−22.4547	−1.56071
$208$	1.37273	0.0951816
$209$	5.50187	0.380572
$210$	0	0
$211$	−15.0506	−1.03613	−0.518063	−	0.855343i	$-0.673347\pi$
$211$	−15.0506	−1.03613	−0.518063	+	0.855343i	$0.673347\pi$
$212$	−10.4611	−0.718469
$213$	3.12489	0.214114
$214$	−13.6498	−0.933083
$215$	0	0
$216$	−6.06241	−0.412495
$217$	28.3662	1.92562
$218$	−3.90012	−0.264149
$219$	−27.3240	−1.84638
$220$	0	0
$221$	−6.33493	−0.426133
$222$	20.7294	1.39127
$223$	−11.2778	−0.755219	−0.377609	−	0.925965i	$-0.623254\pi$
$223$	−11.2778	−0.755219	−0.377609	+	0.925965i	$0.623254\pi$
$224$	4.52360	0.302245
$225$	0	0
$226$	−1.60019	−0.106443
$227$	−19.6413	−1.30364	−0.651818	−	0.758375i	$-0.725994\pi$
$227$	−19.6413	−1.30364	−0.651818	+	0.758375i	$0.725994\pi$
$228$	12.7656	0.845421
$229$	−13.2889	−0.878152	−0.439076	−	0.898450i	$-0.644694\pi$
$229$	−13.2889	−0.878152	−0.439076	+	0.898450i	$0.644694\pi$
$230$	0	0
$231$	15.8440	1.04246
$232$	−5.55434	−0.364661
$233$	−19.0035	−1.24496	−0.622480	−	0.782636i	$-0.713875\pi$
$233$	−19.0035	−1.24496	−0.622480	+	0.782636i	$0.713875\pi$
$234$	7.03746	0.460053
$235$	0	0
$236$	−6.84371	−0.445488
$237$	−45.6069	−2.96249
$238$	−20.8757	−1.35317
$239$	1.66519	0.107712	0.0538561	−	0.998549i	$-0.482849\pi$
$239$	1.66519	0.107712	0.0538561	+	0.998549i	$0.482849\pi$
$240$	0	0
$241$	5.69530	0.366867	0.183433	−	0.983032i	$-0.441279\pi$
$241$	5.69530	0.366867	0.183433	+	0.983032i	$0.441279\pi$
$242$	−9.49044	−0.610069
$243$	12.7640	0.818813
$244$	13.9786	0.894887
$245$	0	0
$246$	11.3558	0.724019
$247$	−6.14710	−0.391130
$248$	6.27071	0.398191
$249$	−32.6723	−2.07052
$250$	0	0
$251$	−5.89313	−0.371971	−0.185986	−	0.982552i	$-0.559548\pi$
$251$	−5.89313	−0.371971	−0.185986	+	0.982552i	$0.559548\pi$
$252$	23.1908	1.46088
$253$	5.38147	0.338330
$254$	8.82600	0.553792
$255$	0	0
$256$	1.00000	0.0625000
$257$	0.854848	0.0533240	0.0266620	−	0.999645i	$-0.491512\pi$
$257$	0.854848	0.0533240	0.0266620	+	0.999645i	$0.491512\pi$
$258$	17.8214	1.10951
$259$	−32.8940	−2.04393
$260$	0	0
$261$	−28.4750	−1.76256
$262$	5.31775	0.328532
$263$	18.3357	1.13063	0.565315	−	0.824875i	$-0.308755\pi$
$263$	18.3357	1.13063	0.565315	+	0.824875i	$0.308755\pi$
$264$	3.50252	0.215565
$265$	0	0
$266$	−20.2567	−1.24202
$267$	2.85072	0.174461
$268$	0.657004	0.0401329
$269$	−12.3327	−0.751941	−0.375970	−	0.926632i	$-0.622691\pi$
$269$	−12.3327	−0.751941	−0.375970	+	0.926632i	$0.622691\pi$
$270$	0	0
$271$	21.1708	1.28603	0.643017	−	0.765852i	$-0.277683\pi$
$271$	21.1708	1.28603	0.643017	+	0.765852i	$0.277683\pi$
$272$	−4.61484	−0.279816
$273$	−17.7020	−1.07138
$274$	0.978495	0.0591131
$275$	0	0
$276$	12.4862	0.751583
$277$	−19.7804	−1.18849	−0.594246	−	0.804283i	$-0.702549\pi$
$277$	−19.7804	−1.18849	−0.594246	+	0.804283i	$0.702549\pi$
$278$	4.23577	0.254044
$279$	32.1476	1.92462
$280$	0	0
$281$	−12.0499	−0.718837	−0.359419	−	0.933176i	$-0.617025\pi$
$281$	−12.0499	−0.718837	−0.359419	+	0.933176i	$0.617025\pi$
$282$	30.2474	1.80121
$283$	−14.5109	−0.862583	−0.431291	−	0.902213i	$-0.641942\pi$
$283$	−14.5109	−0.862583	−0.431291	+	0.902213i	$0.641942\pi$
$284$	−1.09617	−0.0650460
$285$	0	0
$286$	−1.68659	−0.0997301
$287$	−18.0196	−1.06367
$288$	5.12662	0.302089
$289$	4.29679	0.252752
$290$	0	0
$291$	43.2396	2.53475
$292$	9.58493	0.560916
$293$	14.8147	0.865485	0.432743	−	0.901518i	$-0.357546\pi$
$293$	14.8147	0.865485	0.432743	+	0.901518i	$0.357546\pi$
$294$	−38.3790	−2.23831
$295$	0	0
$296$	−7.27164	−0.422655
$297$	7.44853	0.432208
$298$	8.49351	0.492016
$299$	−6.01258	−0.347716
$300$	0	0
$301$	−28.2793	−1.62999
$302$	9.09426	0.523316
$303$	14.1656	0.813795
$304$	−4.47801	−0.256832
$305$	0	0
$306$	−23.6586	−1.35247
$307$	33.2294	1.89650	0.948252	−	0.317520i	$-0.102850\pi$
$307$	33.2294	1.89650	0.948252	+	0.317520i	$0.102850\pi$
$308$	−5.55787	−0.316689
$309$	36.6759	2.08642
$310$	0	0
$311$	13.6843	0.775966	0.387983	−	0.921667i	$-0.373172\pi$
$311$	13.6843	0.775966	0.387983	+	0.921667i	$0.373172\pi$
$312$	−3.91327	−0.221545
$313$	8.07789	0.456589	0.228295	−	0.973592i	$-0.426685\pi$
$313$	8.07789	0.456589	0.228295	+	0.973592i	$0.426685\pi$
$314$	16.5202	0.932288
$315$	0	0
$316$	15.9984	0.899979
$317$	−30.2354	−1.69819	−0.849096	−	0.528239i	$-0.822853\pi$
$317$	−30.2354	−1.69819	−0.849096	+	0.528239i	$0.822853\pi$
$318$	29.8216	1.67231
$319$	6.82430	0.382087
$320$	0	0
$321$	38.9119	2.17185
$322$	−19.8134	−1.10416
$323$	20.6653	1.14985
$324$	1.90240	0.105689
$325$	0	0
$326$	−17.0813	−0.946043
$327$	11.1182	0.614835
$328$	−3.98348	−0.219951
$329$	−47.9973	−2.64618
$330$	0	0
$331$	−0.808833	−0.0444575	−0.0222287	−	0.999753i	$-0.507076\pi$
$331$	−0.808833	−0.0444575	−0.0222287	+	0.999753i	$0.507076\pi$
$332$	11.4611	0.629007
$333$	−37.2790	−2.04287
$334$	−2.94829	−0.161323
$335$	0	0
$336$	−12.8955	−0.703508
$337$	24.0398	1.30953	0.654766	−	0.755832i	$-0.272767\pi$
$337$	24.0398	1.30953	0.654766	+	0.755832i	$0.272767\pi$
$338$	−11.1156	−0.604610
$339$	4.56169	0.247757
$340$	0	0
$341$	−7.70445	−0.417219
$342$	−22.9571	−1.24138
$343$	29.2356	1.57857
$344$	−6.25152	−0.337059
$345$	0	0
$346$	−5.36147	−0.288235
$347$	−21.4905	−1.15367	−0.576835	−	0.816860i	$-0.695713\pi$
$347$	−21.4905	−1.15367	−0.576835	+	0.816860i	$0.695713\pi$
$348$	15.8339	0.848786
$349$	−30.2476	−1.61912	−0.809558	−	0.587040i	$-0.800293\pi$
$349$	−30.2476	−1.61912	−0.809558	+	0.587040i	$0.800293\pi$
$350$	0	0
$351$	−8.32205	−0.444198
$352$	−1.22864	−0.0654868
$353$	24.0131	1.27809	0.639043	−	0.769171i	$-0.279331\pi$
$353$	24.0131	1.27809	0.639043	+	0.769171i	$0.279331\pi$
$354$	19.5095	1.03692
$355$	0	0
$356$	−1.00000	−0.0529999
$357$	59.5108	3.14965
$358$	11.5470	0.610281
$359$	−6.63025	−0.349931	−0.174966	−	0.984575i	$-0.555981\pi$
$359$	−6.63025	−0.349931	−0.174966	+	0.984575i	$0.555981\pi$
$360$	0	0
$361$	1.05262	0.0554009
$362$	−25.3052	−1.33001
$363$	27.0546	1.42000
$364$	6.20967	0.325475
$365$	0	0
$366$	−39.8491	−2.08294
$367$	9.60858	0.501564	0.250782	−	0.968044i	$-0.419312\pi$
$367$	9.60858	0.501564	0.250782	+	0.968044i	$0.419312\pi$
$368$	−4.38002	−0.228324
$369$	−20.4218	−1.06312
$370$	0	0
$371$	−47.3216	−2.45682
$372$	−17.8761	−0.926831
$373$	23.0875	1.19543	0.597714	−	0.801710i	$-0.296076\pi$
$373$	23.0875	1.19543	0.597714	+	0.801710i	$0.296076\pi$
$374$	5.66999	0.293188
$375$	0	0
$376$	−10.6104	−0.547191
$377$	−7.62460	−0.392687
$378$	−27.4239	−1.41053
$379$	−30.1248	−1.54741	−0.773704	−	0.633547i	$-0.781598\pi$
$379$	−30.1248	−1.54741	−0.773704	+	0.633547i	$0.781598\pi$
$380$	0	0
$381$	−25.1605	−1.28901
$382$	15.7942	0.808100
$383$	4.58478	0.234271	0.117136	−	0.993116i	$-0.462629\pi$
$383$	4.58478	0.234271	0.117136	+	0.993116i	$0.462629\pi$
$384$	−2.85072	−0.145475
$385$	0	0
$386$	−7.69294	−0.391560
$387$	−32.0492	−1.62915
$388$	−15.1680	−0.770036
$389$	−1.43129	−0.0725691	−0.0362846	−	0.999341i	$-0.511552\pi$
$389$	−1.43129	−0.0725691	−0.0362846	+	0.999341i	$0.511552\pi$
$390$	0	0
$391$	20.2131	1.02222
$392$	13.4629	0.679980
$393$	−15.1594	−0.764693
$394$	0.115358	0.00581166
$395$	0	0
$396$	−6.29878	−0.316526
$397$	8.96970	0.450176	0.225088	−	0.974338i	$-0.427733\pi$
$397$	8.96970	0.450176	0.225088	+	0.974338i	$0.427733\pi$
$398$	0.584508	0.0292987
$399$	57.7463	2.89093
$400$	0	0
$401$	6.18849	0.309039	0.154519	−	0.987990i	$-0.450617\pi$
$401$	6.18849	0.309039	0.154519	+	0.987990i	$0.450617\pi$
$402$	−1.87294	−0.0934136
$403$	8.60798	0.428794
$404$	−4.96914	−0.247224
$405$	0	0
$406$	−25.1256	−1.24696
$407$	8.93423	0.442854
$408$	13.1556	0.651301
$409$	−15.5684	−0.769807	−0.384904	−	0.922957i	$-0.625765\pi$
$409$	−15.5684	−0.769807	−0.384904	+	0.922957i	$0.625765\pi$
$410$	0	0
$411$	−2.78942	−0.137592
$412$	−12.8655	−0.633837
$413$	−30.9582	−1.52335
$414$	−22.4547	−1.10359
$415$	0	0
$416$	1.37273	0.0673035
$417$	−12.0750	−0.591315
$418$	5.50187	0.269105
$419$	26.7703	1.30782	0.653908	−	0.756574i	$-0.273128\pi$
$419$	26.7703	1.30782	0.653908	+	0.756574i	$0.273128\pi$
$420$	0	0
$421$	−30.2450	−1.47405	−0.737025	−	0.675865i	$-0.763770\pi$
$421$	−30.2450	−1.47405	−0.737025	+	0.675865i	$0.763770\pi$
$422$	−15.0506	−0.732651
$423$	−54.3957	−2.64481
$424$	−10.4611	−0.508034
$425$	0	0
$426$	3.12489	0.151401
$427$	63.2334	3.06008
$428$	−13.6498	−0.659790
$429$	4.80800	0.232132
$430$	0	0
$431$	32.0590	1.54423	0.772114	−	0.635484i	$-0.219199\pi$
$431$	32.0590	1.54423	0.772114	+	0.635484i	$0.219199\pi$
$432$	−6.06241	−0.291678
$433$	−8.68039	−0.417153	−0.208576	−	0.978006i	$-0.566883\pi$
$433$	−8.68039	−0.417153	−0.208576	+	0.978006i	$0.566883\pi$
$434$	28.3662	1.36162
$435$	0	0
$436$	−3.90012	−0.186782
$437$	19.6138	0.938256
$438$	−27.3240	−1.30559
$439$	11.4514	0.546547	0.273273	−	0.961936i	$-0.411894\pi$
$439$	11.4514	0.546547	0.273273	+	0.961936i	$0.411894\pi$
$440$	0	0
$441$	69.0193	3.28663
$442$	−6.33493	−0.301322
$443$	0.605849	0.0287848	0.0143924	−	0.999896i	$-0.495419\pi$
$443$	0.605849	0.0287848	0.0143924	+	0.999896i	$0.495419\pi$
$444$	20.7294	0.983775
$445$	0	0
$446$	−11.2778	−0.534020
$447$	−24.2126	−1.14522
$448$	4.52360	0.213720
$449$	7.76655	0.366526	0.183263	−	0.983064i	$-0.441334\pi$
$449$	7.76655	0.366526	0.183263	+	0.983064i	$0.441334\pi$
$450$	0	0
$451$	4.89426	0.230462
$452$	−1.60019	−0.0752665
$453$	−25.9252	−1.21807
$454$	−19.6413	−0.921810
$455$	0	0
$456$	12.7656	0.597803
$457$	−24.3876	−1.14080	−0.570402	−	0.821366i	$-0.693213\pi$
$457$	−24.3876	−1.14080	−0.570402	+	0.821366i	$0.693213\pi$
$458$	−13.2889	−0.620948
$459$	27.9771	1.30586
$460$	0	0
$461$	7.95362	0.370437	0.185218	−	0.982697i	$-0.440701\pi$
$461$	7.95362	0.370437	0.185218	+	0.982697i	$0.440701\pi$
$462$	15.8440	0.737128
$463$	15.0594	0.699868	0.349934	−	0.936774i	$-0.386204\pi$
$463$	15.0594	0.699868	0.349934	+	0.936774i	$0.386204\pi$
$464$	−5.55434	−0.257854
$465$	0	0
$466$	−19.0035	−0.880319
$467$	39.4937	1.82755	0.913775	−	0.406222i	$-0.133154\pi$
$467$	39.4937	1.82755	0.913775	+	0.406222i	$0.133154\pi$
$468$	7.03746	0.325307
$469$	2.97202	0.137235
$470$	0	0
$471$	−47.0945	−2.17000
$472$	−6.84371	−0.315007
$473$	7.68087	0.353167
$474$	−45.6069	−2.09480
$475$	0	0
$476$	−20.8757	−0.956835
$477$	−53.6299	−2.45555
$478$	1.66519	0.0761641
$479$	15.7045	0.717556	0.358778	−	0.933423i	$-0.383193\pi$
$479$	15.7045	0.717556	0.358778	+	0.933423i	$0.383193\pi$
$480$	0	0
$481$	−9.98198	−0.455139
$482$	5.69530	0.259414
$483$	56.4827	2.57005
$484$	−9.49044	−0.431384
$485$	0	0
$486$	12.7640	0.578988
$487$	−40.3721	−1.82943	−0.914717	−	0.404096i	$-0.867586\pi$
$487$	−40.3721	−1.82943	−0.914717	+	0.404096i	$0.867586\pi$
$488$	13.9786	0.632781
$489$	48.6939	2.20202
$490$	0	0
$491$	−17.2348	−0.777795	−0.388898	−	0.921281i	$-0.627144\pi$
$491$	−17.2348	−0.777795	−0.388898	+	0.921281i	$0.627144\pi$
$492$	11.3558	0.511958
$493$	25.6324	1.15443
$494$	−6.14710	−0.276571
$495$	0	0
$496$	6.27071	0.281563
$497$	−4.95865	−0.222426
$498$	−32.6723	−1.46408
$499$	37.6760	1.68661	0.843305	−	0.537435i	$-0.180607\pi$
$499$	37.6760	1.68661	0.843305	+	0.537435i	$0.180607\pi$
$500$	0	0
$501$	8.40476	0.375497
$502$	−5.89313	−0.263023
$503$	−19.5597	−0.872126	−0.436063	−	0.899916i	$-0.643627\pi$
$503$	−19.5597	−0.872126	−0.436063	+	0.899916i	$0.643627\pi$
$504$	23.1908	1.03300
$505$	0	0
$506$	5.38147	0.239236
$507$	31.6876	1.40729
$508$	8.82600	0.391590
$509$	34.9703	1.55003	0.775015	−	0.631943i	$-0.217742\pi$
$509$	34.9703	1.55003	0.775015	+	0.631943i	$0.217742\pi$
$510$	0	0
$511$	43.3583	1.91806
$512$	1.00000	0.0441942
$513$	27.1476	1.19860
$514$	0.854848	0.0377057
$515$	0	0
$516$	17.8214	0.784541
$517$	13.0364	0.573341
$518$	−32.8940	−1.44528
$519$	15.2841	0.670896
$520$	0	0
$521$	20.0609	0.878885	0.439443	−	0.898271i	$-0.355176\pi$
$521$	20.0609	0.878885	0.439443	+	0.898271i	$0.355176\pi$
$522$	−28.4750	−1.24632
$523$	10.1567	0.444124	0.222062	−	0.975033i	$-0.428721\pi$
$523$	10.1567	0.444124	0.222062	+	0.975033i	$0.428721\pi$
$524$	5.31775	0.232307
$525$	0	0
$526$	18.3357	0.799476
$527$	−28.9384	−1.26057
$528$	3.50252	0.152427
$529$	−3.81541	−0.165887
$530$	0	0
$531$	−35.0851	−1.52257
$532$	−20.2567	−0.878240
$533$	−5.46823	−0.236855
$534$	2.85072	0.123363
$535$	0	0
$536$	0.657004	0.0283782
$537$	−32.9174	−1.42049
$538$	−12.3327	−0.531702
$539$	−16.5411	−0.712475
$540$	0	0
$541$	9.93773	0.427256	0.213628	−	0.976915i	$-0.431472\pi$
$541$	9.93773	0.427256	0.213628	+	0.976915i	$0.431472\pi$
$542$	21.1708	0.909363
$543$	72.1380	3.09574
$544$	−4.61484	−0.197860
$545$	0	0
$546$	−17.7020	−0.757577
$547$	2.50925	0.107288	0.0536438	−	0.998560i	$-0.482916\pi$
$547$	2.50925	0.107288	0.0536438	+	0.998560i	$0.482916\pi$
$548$	0.978495	0.0417992
$549$	71.6629	3.05850
$550$	0	0
$551$	24.8724	1.05960
$552$	12.4862	0.531449
$553$	72.3702	3.07749
$554$	−19.7804	−0.840391
$555$	0	0
$556$	4.23577	0.179637
$557$	−22.8456	−0.967999	−0.484000	−	0.875068i	$-0.660816\pi$
$557$	−22.8456	−0.967999	−0.484000	+	0.875068i	$0.660816\pi$
$558$	32.1476	1.36092
$559$	−8.58164	−0.362965
$560$	0	0
$561$	−16.1636	−0.682426
$562$	−12.0499	−0.508295
$563$	46.4571	1.95793	0.978967	−	0.204020i	$-0.0654008\pi$
$563$	46.4571	1.95793	0.978967	+	0.204020i	$0.0654008\pi$
$564$	30.2474	1.27365
$565$	0	0
$566$	−14.5109	−0.609938
$567$	8.60567	0.361404
$568$	−1.09617	−0.0459945
$569$	−18.3145	−0.767782	−0.383891	−	0.923378i	$-0.625416\pi$
$569$	−18.3145	−0.767782	−0.383891	+	0.923378i	$0.625416\pi$
$570$	0	0
$571$	0.758516	0.0317429	0.0158715	−	0.999874i	$-0.494948\pi$
$571$	0.758516	0.0317429	0.0158715	+	0.999874i	$0.494948\pi$
$572$	−1.68659	−0.0705199
$573$	−45.0248	−1.88094
$574$	−18.0196	−0.752125
$575$	0	0
$576$	5.12662	0.213609
$577$	41.3238	1.72033	0.860167	−	0.510013i	$-0.170359\pi$
$577$	41.3238	1.72033	0.860167	+	0.510013i	$0.170359\pi$
$578$	4.29679	0.178723
$579$	21.9304	0.911398
$580$	0	0
$581$	51.8452	2.15090
$582$	43.2396	1.79234
$583$	12.8529	0.532312
$584$	9.58493	0.396627
$585$	0	0
$586$	14.8147	0.611990
$587$	8.87893	0.366473	0.183236	−	0.983069i	$-0.441343\pi$
$587$	8.87893	0.366473	0.183236	+	0.983069i	$0.441343\pi$
$588$	−38.3790	−1.58272
$589$	−28.0803	−1.15703
$590$	0	0
$591$	−0.328854	−0.0135272
$592$	−7.27164	−0.298863
$593$	−46.1189	−1.89387	−0.946937	−	0.321419i	$-0.895840\pi$
$593$	−46.1189	−1.89387	−0.946937	+	0.321419i	$0.895840\pi$
$594$	7.44853	0.305617
$595$	0	0
$596$	8.49351	0.347908
$597$	−1.66627	−0.0681959
$598$	−6.01258	−0.245873
$599$	−0.242231	−0.00989730	−0.00494865	−	0.999988i	$-0.501575\pi$
$599$	−0.242231	−0.00989730	−0.00494865	+	0.999988i	$0.501575\pi$
$600$	0	0
$601$	13.3805	0.545803	0.272902	−	0.962042i	$-0.412017\pi$
$601$	13.3805	0.545803	0.272902	+	0.962042i	$0.412017\pi$
$602$	−28.2793	−1.15258
$603$	3.36821	0.137164
$604$	9.09426	0.370040
$605$	0	0
$606$	14.1656	0.575440
$607$	−8.09568	−0.328593	−0.164297	−	0.986411i	$-0.552535\pi$
$607$	−8.09568	−0.328593	−0.164297	+	0.986411i	$0.552535\pi$
$608$	−4.47801	−0.181607
$609$	71.6262	2.90244
$610$	0	0
$611$	−14.5652	−0.589247
$612$	−23.6586	−0.956341
$613$	−38.8721	−1.57003	−0.785015	−	0.619477i	$-0.787345\pi$
$613$	−38.8721	−1.57003	−0.785015	+	0.619477i	$0.787345\pi$
$614$	33.2294	1.34103
$615$	0	0
$616$	−5.55787	−0.223933
$617$	22.6884	0.913401	0.456700	−	0.889621i	$-0.349031\pi$
$617$	22.6884	0.913401	0.456700	+	0.889621i	$0.349031\pi$
$618$	36.6759	1.47532
$619$	−7.03326	−0.282691	−0.141345	−	0.989960i	$-0.545143\pi$
$619$	−7.03326	−0.282691	−0.141345	+	0.989960i	$0.545143\pi$
$620$	0	0
$621$	26.5535	1.06556
$622$	13.6843	0.548691
$623$	−4.52360	−0.181234
$624$	−3.91327	−0.156656
$625$	0	0
$626$	8.07789	0.322857
$627$	−15.6843	−0.626371
$628$	16.5202	0.659227
$629$	33.5575	1.33802
$630$	0	0
$631$	16.9874	0.676256	0.338128	−	0.941100i	$-0.390206\pi$
$631$	16.9874	0.676256	0.338128	+	0.941100i	$0.390206\pi$
$632$	15.9984	0.636381
$633$	42.9051	1.70532
$634$	−30.2354	−1.20080
$635$	0	0
$636$	29.8216	1.18250
$637$	18.4809	0.732241
$638$	6.82430	0.270176
$639$	−5.61968	−0.222311
$640$	0	0
$641$	37.5611	1.48357	0.741787	−	0.670635i	$-0.233978\pi$
$641$	37.5611	1.48357	0.741787	+	0.670635i	$0.233978\pi$
$642$	38.9119	1.53573
$643$	−18.0731	−0.712732	−0.356366	−	0.934347i	$-0.615984\pi$
$643$	−18.0731	−0.712732	−0.356366	+	0.934347i	$0.615984\pi$
$644$	−19.8134	−0.780759
$645$	0	0
$646$	20.6653	0.813067
$647$	−43.8207	−1.72277	−0.861386	−	0.507952i	$-0.830403\pi$
$647$	−43.8207	−1.72277	−0.861386	+	0.507952i	$0.830403\pi$
$648$	1.90240	0.0747332
$649$	8.40847	0.330061
$650$	0	0
$651$	−80.8641	−3.16931
$652$	−17.0813	−0.668954
$653$	12.0487	0.471504	0.235752	−	0.971813i	$-0.424245\pi$
$653$	12.0487	0.471504	0.235752	+	0.971813i	$0.424245\pi$
$654$	11.1182	0.434754
$655$	0	0
$656$	−3.98348	−0.155529
$657$	49.1383	1.91707
$658$	−47.9973	−1.87113
$659$	−40.0336	−1.55949	−0.779744	−	0.626098i	$-0.784651\pi$
$659$	−40.0336	−1.55949	−0.779744	+	0.626098i	$0.784651\pi$
$660$	0	0
$661$	−24.2278	−0.942351	−0.471175	−	0.882040i	$-0.656170\pi$
$661$	−24.2278	−0.942351	−0.471175	+	0.882040i	$0.656170\pi$
$662$	−0.808833	−0.0314362
$663$	18.0591	0.701358
$664$	11.4611	0.444775
$665$	0	0
$666$	−37.2790	−1.44453
$667$	24.3282	0.941990
$668$	−2.94829	−0.114073
$669$	32.1499	1.24299
$670$	0	0
$671$	−17.1747	−0.663020
$672$	−12.8955	−0.497455
$673$	−33.6487	−1.29706	−0.648532	−	0.761188i	$-0.724617\pi$
$673$	−33.6487	−1.29706	−0.648532	+	0.761188i	$0.724617\pi$
$674$	24.0398	0.925979
$675$	0	0
$676$	−11.1156	−0.427524
$677$	−36.8340	−1.41565	−0.707823	−	0.706389i	$-0.750323\pi$
$677$	−36.8340	−1.41565	−0.707823	+	0.706389i	$0.750323\pi$
$678$	4.56169	0.175191
$679$	−68.6137	−2.63315
$680$	0	0
$681$	55.9918	2.14561
$682$	−7.70445	−0.295019
$683$	44.2202	1.69204	0.846020	−	0.533151i	$-0.178992\pi$
$683$	44.2202	1.69204	0.846020	+	0.533151i	$0.178992\pi$
$684$	−22.9571	−0.877786
$685$	0	0
$686$	29.2356	1.11622
$687$	37.8829	1.44532
$688$	−6.25152	−0.238337
$689$	−14.3602	−0.547080
$690$	0	0
$691$	15.6482	0.595286	0.297643	−	0.954677i	$-0.403800\pi$
$691$	15.6482	0.595286	0.297643	+	0.954677i	$0.403800\pi$
$692$	−5.36147	−0.203813
$693$	−28.4931	−1.08236
$694$	−21.4905	−0.815768
$695$	0	0
$696$	15.8339	0.600182
$697$	18.3831	0.696310
$698$	−30.2476	−1.14489
$699$	54.1737	2.04904
$700$	0	0
$701$	−24.8434	−0.938324	−0.469162	−	0.883112i	$-0.655444\pi$
$701$	−24.8434	−0.938324	−0.469162	+	0.883112i	$0.655444\pi$
$702$	−8.32205	−0.314095
$703$	32.5625	1.22812
$704$	−1.22864	−0.0463061
$705$	0	0
$706$	24.0131	0.903743
$707$	−22.4784	−0.845386
$708$	19.5095	0.733213
$709$	−25.7323	−0.966397	−0.483198	−	0.875511i	$-0.660525\pi$
$709$	−25.7323	−0.966397	−0.483198	+	0.875511i	$0.660525\pi$
$710$	0	0
$711$	82.0176	3.07590
$712$	−1.00000	−0.0374766
$713$	−27.4659	−1.02860
$714$	59.5108	2.22714
$715$	0	0
$716$	11.5470	0.431533
$717$	−4.74700	−0.177280
$718$	−6.63025	−0.247439
$719$	−30.9572	−1.15451	−0.577254	−	0.816564i	$-0.695876\pi$
$719$	−30.9572	−1.15451	−0.577254	+	0.816564i	$0.695876\pi$
$720$	0	0
$721$	−58.1982	−2.16742
$722$	1.05262	0.0391744
$723$	−16.2357	−0.603813
$724$	−25.3052	−0.940459
$725$	0	0
$726$	27.0546	1.00409
$727$	0.0138657	0.000514250 0	0.000257125	−	1.00000i	$-0.499918\pi$
$727$	0.0138657	0.000514250 0	0.000257125		1.00000i	$0.499918\pi$
$728$	6.20967	0.230146
$729$	−42.0939	−1.55903
$730$	0	0
$731$	28.8498	1.06705
$732$	−39.8491	−1.47286
$733$	11.7817	0.435168	0.217584	−	0.976042i	$-0.430182\pi$
$733$	11.7817	0.435168	0.217584	+	0.976042i	$0.430182\pi$
$734$	9.60858	0.354659
$735$	0	0
$736$	−4.38002	−0.161450
$737$	−0.807222	−0.0297344
$738$	−20.4218	−0.751736
$739$	12.7734	0.469877	0.234939	−	0.972010i	$-0.424511\pi$
$739$	12.7734	0.469877	0.234939	+	0.972010i	$0.424511\pi$
$740$	0	0
$741$	17.5237	0.643748
$742$	−47.3216	−1.73723
$743$	−34.0456	−1.24901	−0.624507	−	0.781019i	$-0.714700\pi$
$743$	−34.0456	−1.24901	−0.624507	+	0.781019i	$0.714700\pi$
$744$	−17.8761	−0.655368
$745$	0	0
$746$	23.0875	0.845295
$747$	58.7565	2.14979
$748$	5.66999	0.207315
$749$	−61.7463	−2.25616
$750$	0	0
$751$	−18.0609	−0.659053	−0.329526	−	0.944146i	$-0.606889\pi$
$751$	−18.0609	−0.659053	−0.329526	+	0.944146i	$0.606889\pi$
$752$	−10.6104	−0.386923
$753$	16.7997	0.612215
$754$	−7.62460	−0.277672
$755$	0	0
$756$	−27.4239	−0.997398
$757$	−28.5767	−1.03864	−0.519319	−	0.854581i	$-0.673814\pi$
$757$	−28.5767	−1.03864	−0.519319	+	0.854581i	$0.673814\pi$
$758$	−30.1248	−1.09418
$759$	−15.3411	−0.556846
$760$	0	0
$761$	11.0520	0.400636	0.200318	−	0.979731i	$-0.435802\pi$
$761$	11.0520	0.400636	0.200318	+	0.979731i	$0.435802\pi$
$762$	−25.1605	−0.911468
$763$	−17.6425	−0.638703
$764$	15.7942	0.571413
$765$	0	0
$766$	4.58478	0.165655
$767$	−9.39456	−0.339218
$768$	−2.85072	−0.102867
$769$	−10.9106	−0.393445	−0.196722	−	0.980459i	$-0.563030\pi$
$769$	−10.9106	−0.393445	−0.196722	+	0.980459i	$0.563030\pi$
$770$	0	0
$771$	−2.43694	−0.0877641
$772$	−7.69294	−0.276875
$773$	27.6095	0.993046	0.496523	−	0.868024i	$-0.334610\pi$
$773$	27.6095	0.993046	0.496523	+	0.868024i	$0.334610\pi$
$774$	−32.0492	−1.15198
$775$	0	0
$776$	−15.1680	−0.544498
$777$	93.7716	3.36404
$778$	−1.43129	−0.0513141
$779$	17.8381	0.639115
$780$	0	0
$781$	1.34681	0.0481925
$782$	20.2131	0.722820
$783$	33.6727	1.20337
$784$	13.4629	0.480818
$785$	0	0
$786$	−15.1594	−0.540720
$787$	54.2969	1.93547	0.967737	−	0.251962i	$-0.0810759\pi$
$787$	54.2969	1.93547	0.967737	+	0.251962i	$0.0810759\pi$
$788$	0.115358	0.00410946
$789$	−52.2701	−1.86086
$790$	0	0
$791$	−7.23860	−0.257375
$792$	−6.29878	−0.223817
$793$	19.1888	0.681414
$794$	8.96970	0.318323
$795$	0	0
$796$	0.584508	0.0207173
$797$	31.7349	1.12411	0.562053	−	0.827101i	$-0.310012\pi$
$797$	31.7349	1.12411	0.562053	+	0.827101i	$0.310012\pi$
$798$	57.7463	2.04420
$799$	48.9655	1.73227
$800$	0	0
$801$	−5.12662	−0.181140
$802$	6.18849	0.218523
$803$	−11.7764	−0.415581
$804$	−1.87294	−0.0660534
$805$	0	0
$806$	8.60798	0.303203
$807$	35.1572	1.23759
$808$	−4.96914	−0.174814
$809$	0.435169	0.0152997	0.00764986	−	0.999971i	$-0.497565\pi$
$809$	0.435169	0.0152997	0.00764986	+	0.999971i	$0.497565\pi$
$810$	0	0
$811$	8.86615	0.311333	0.155666	−	0.987810i	$-0.450248\pi$
$811$	8.86615	0.311333	0.155666	+	0.987810i	$0.450248\pi$
$812$	−25.1256	−0.881736
$813$	−60.3520	−2.11664
$814$	8.93423	0.313145
$815$	0	0
$816$	13.1556	0.460540
$817$	27.9944	0.979400
$818$	−15.5684	−0.544336
$819$	31.8346	1.11239
$820$	0	0
$821$	−6.17004	−0.215336	−0.107668	−	0.994187i	$-0.534338\pi$
$821$	−6.17004	−0.215336	−0.107668	+	0.994187i	$0.534338\pi$
$822$	−2.78942	−0.0972922
$823$	12.4662	0.434544	0.217272	−	0.976111i	$-0.430284\pi$
$823$	12.4662	0.434544	0.217272	+	0.976111i	$0.430284\pi$
$824$	−12.8655	−0.448190
$825$	0	0
$826$	−30.9582	−1.07717
$827$	−35.5408	−1.23587	−0.617937	−	0.786228i	$-0.712031\pi$
$827$	−35.5408	−1.23587	−0.617937	+	0.786228i	$0.712031\pi$
$828$	−22.4547	−0.780356
$829$	38.5645	1.33940	0.669700	−	0.742632i	$-0.266423\pi$
$829$	38.5645	1.33940	0.669700	+	0.742632i	$0.266423\pi$
$830$	0	0
$831$	56.3886	1.95610
$832$	1.37273	0.0475908
$833$	−62.1292	−2.15265
$834$	−12.0750	−0.418123
$835$	0	0
$836$	5.50187	0.190286
$837$	−38.0156	−1.31401
$838$	26.7703	0.924765
$839$	10.5761	0.365128	0.182564	−	0.983194i	$-0.441560\pi$
$839$	10.5761	0.365128	0.182564	+	0.983194i	$0.441560\pi$
$840$	0	0
$841$	1.85075	0.0638189
$842$	−30.2450	−1.04231
$843$	34.3509	1.18311
$844$	−15.0506	−0.518063
$845$	0	0
$846$	−54.3957	−1.87016
$847$	−42.9309	−1.47512
$848$	−10.4611	−0.359234
$849$	41.3665	1.41970
$850$	0	0
$851$	31.8499	1.09180
$852$	3.12489	0.107057
$853$	−0.122918	−0.00420863	−0.00210431	−	0.999998i	$-0.500670\pi$
$853$	−0.122918	−0.00420863	−0.00210431	+	0.999998i	$0.500670\pi$
$854$	63.2334	2.16380
$855$	0	0
$856$	−13.6498	−0.466542
$857$	−34.3154	−1.17219	−0.586096	−	0.810242i	$-0.699336\pi$
$857$	−34.3154	−1.17219	−0.586096	+	0.810242i	$0.699336\pi$
$858$	4.80800	0.164142
$859$	12.4774	0.425725	0.212863	−	0.977082i	$-0.431721\pi$
$859$	12.4774	0.425725	0.212863	+	0.977082i	$0.431721\pi$
$860$	0	0
$861$	51.3690	1.75065
$862$	32.0590	1.09193
$863$	−23.0720	−0.785381	−0.392691	−	0.919671i	$-0.628456\pi$
$863$	−23.0720	−0.785381	−0.392691	+	0.919671i	$0.628456\pi$
$864$	−6.06241	−0.206248
$865$	0	0
$866$	−8.68039	−0.294972
$867$	−12.2489	−0.415996
$868$	28.3662	0.962810
$869$	−19.6563	−0.666793
$870$	0	0
$871$	0.901888	0.0305593
$872$	−3.90012	−0.132075
$873$	−77.7604	−2.63179
$874$	19.6138	0.663447
$875$	0	0
$876$	−27.3240	−0.923192
$877$	−29.9035	−1.00977	−0.504885	−	0.863186i	$-0.668465\pi$
$877$	−29.9035	−1.00977	−0.504885	+	0.863186i	$0.668465\pi$
$878$	11.4514	0.386467
$879$	−42.2327	−1.42447
$880$	0	0
$881$	19.5416	0.658373	0.329186	−	0.944265i	$-0.393226\pi$
$881$	19.5416	0.658373	0.329186	+	0.944265i	$0.393226\pi$
$882$	69.0193	2.32400
$883$	−47.3292	−1.59276	−0.796378	−	0.604799i	$-0.793253\pi$
$883$	−47.3292	−1.59276	−0.796378	+	0.604799i	$0.793253\pi$
$884$	−6.33493	−0.213067
$885$	0	0
$886$	0.605849	0.0203539
$887$	49.1245	1.64944	0.824719	−	0.565543i	$-0.191333\pi$
$887$	49.1245	1.64944	0.824719	+	0.565543i	$0.191333\pi$
$888$	20.7294	0.695634
$889$	39.9252	1.33905
$890$	0	0
$891$	−2.33736	−0.0783046
$892$	−11.2778	−0.377609
$893$	47.5137	1.58998
$894$	−24.2126	−0.809792
$895$	0	0
$896$	4.52360	0.151123
$897$	17.1402	0.572295
$898$	7.76655	0.259173
$899$	−34.8297	−1.16164
$900$	0	0
$901$	48.2762	1.60831
$902$	4.89426	0.162961
$903$	80.6166	2.68275
$904$	−1.60019	−0.0532215
$905$	0	0
$906$	−25.9252	−0.861308
$907$	5.68576	0.188793	0.0943963	−	0.995535i	$-0.469908\pi$
$907$	5.68576	0.188793	0.0943963	+	0.995535i	$0.469908\pi$
$908$	−19.6413	−0.651818
$909$	−25.4749	−0.844949
$910$	0	0
$911$	2.89022	0.0957572	0.0478786	−	0.998853i	$-0.484754\pi$
$911$	2.89022	0.0957572	0.0478786	+	0.998853i	$0.484754\pi$
$912$	12.7656	0.422711
$913$	−14.0815	−0.466030
$914$	−24.3876	−0.806670
$915$	0	0
$916$	−13.2889	−0.439076
$917$	24.0554	0.794378
$918$	27.9771	0.923382
$919$	−41.2095	−1.35938	−0.679688	−	0.733501i	$-0.737885\pi$
$919$	−41.2095	−1.35938	−0.679688	+	0.733501i	$0.737885\pi$
$920$	0	0
$921$	−94.7279	−3.12139
$922$	7.95362	0.261938
$923$	−1.50475	−0.0495295
$924$	15.8440	0.521228
$925$	0	0
$926$	15.0594	0.494882
$927$	−65.9565	−2.16630
$928$	−5.55434	−0.182330
$929$	−7.26885	−0.238483	−0.119242	−	0.992865i	$-0.538046\pi$
$929$	−7.26885	−0.238483	−0.119242	+	0.992865i	$0.538046\pi$
$930$	0	0
$931$	−60.2871	−1.97583
$932$	−19.0035	−0.622480
$933$	−39.0102	−1.27714
$934$	39.4937	1.29227
$935$	0	0
$936$	7.03746	0.230027
$937$	2.12881	0.0695453	0.0347726	−	0.999395i	$-0.488929\pi$
$937$	2.12881	0.0695453	0.0347726	+	0.999395i	$0.488929\pi$
$938$	2.97202	0.0970399
$939$	−23.0278	−0.751485
$940$	0	0
$941$	−4.37139	−0.142503	−0.0712517	−	0.997458i	$-0.522699\pi$
$941$	−4.37139	−0.142503	−0.0712517	+	0.997458i	$0.522699\pi$
$942$	−47.0945	−1.53442
$943$	17.4477	0.568176
$944$	−6.84371	−0.222744
$945$	0	0
$946$	7.68087	0.249727
$947$	8.73145	0.283734	0.141867	−	0.989886i	$-0.454689\pi$
$947$	8.73145	0.283734	0.141867	+	0.989886i	$0.454689\pi$
$948$	−45.6069	−1.48124
$949$	13.1575	0.427111
$950$	0	0
$951$	86.1929	2.79500
$952$	−20.8757	−0.676585
$953$	10.6958	0.346470	0.173235	−	0.984881i	$-0.444578\pi$
$953$	10.6958	0.346470	0.173235	+	0.984881i	$0.444578\pi$
$954$	−53.6299	−1.73633
$955$	0	0
$956$	1.66519	0.0538561
$957$	−19.4542	−0.628864
$958$	15.7045	0.507389
$959$	4.42632	0.142933
$960$	0	0
$961$	8.32181	0.268445
$962$	−9.98198	−0.321832
$963$	−69.9776	−2.25500
$964$	5.69530	0.183433
$965$	0	0
$966$	56.4827	1.81730
$967$	25.2596	0.812293	0.406146	−	0.913808i	$-0.366872\pi$
$967$	25.2596	0.812293	0.406146	+	0.913808i	$0.366872\pi$
$968$	−9.49044	−0.305034
$969$	−58.9112	−1.89250
$970$	0	0
$971$	−46.0863	−1.47898	−0.739490	−	0.673167i	$-0.764933\pi$
$971$	−46.0863	−1.47898	−0.739490	+	0.673167i	$0.764933\pi$
$972$	12.7640	0.409407
$973$	19.1609	0.614270
$974$	−40.3721	−1.29360
$975$	0	0
$976$	13.9786	0.447444
$977$	−10.3767	−0.331980	−0.165990	−	0.986127i	$-0.553082\pi$
$977$	−10.3767	−0.331980	−0.165990	+	0.986127i	$0.553082\pi$
$978$	48.6939	1.55706
$979$	1.22864	0.0392675
$980$	0	0
$981$	−19.9944	−0.638373
$982$	−17.2348	−0.549984
$983$	0.254970	0.00813227	0.00406614	−	0.999992i	$-0.498706\pi$
$983$	0.254970	0.00813227	0.00406614	+	0.999992i	$0.498706\pi$
$984$	11.3558	0.362009
$985$	0	0
$986$	25.6324	0.816303
$987$	136.827	4.35525
$988$	−6.14710	−0.195565
$989$	27.3818	0.870690
$990$	0	0
$991$	−16.0684	−0.510428	−0.255214	−	0.966885i	$-0.582146\pi$
$991$	−16.0684	−0.510428	−0.255214	+	0.966885i	$0.582146\pi$
$992$	6.27071	0.199095
$993$	2.30576	0.0731710
$994$	−4.95865	−0.157279
$995$	0	0
$996$	−32.6723	−1.03526
$997$	−50.0379	−1.58472	−0.792358	−	0.610057i	$-0.791147\pi$
$997$	−50.0379	−1.58472	−0.792358	+	0.610057i	$0.791147\pi$
$998$	37.6760	1.19261
$999$	44.0837	1.39475

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4450.2.a.bj.1.1		8
5.2	odd	4		890.2.b.a.179.16	yes	16
5.3	odd	4		890.2.b.a.179.1	✓	16
5.4	even	2		4450.2.a.bi.1.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
890.2.b.a.179.1	✓	16	5.3	odd	4
890.2.b.a.179.16	yes	16	5.2	odd	4
4450.2.a.bi.1.8		8	5.4	even	2
4450.2.a.bj.1.1		8	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:	\( N \)	\(=\)	\( 4450 = 2 \cdot 5^{2} \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4450.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.85072 q^{3} +1.00000 q^{4} -2.85072 q^{6} +4.52360 q^{7} +1.00000 q^{8} +5.12662 q^{9} -1.22864 q^{11} -2.85072 q^{12} +1.37273 q^{13} +4.52360 q^{14} +1.00000 q^{16} -4.61484 q^{17} +5.12662 q^{18} -4.47801 q^{19} -12.8955 q^{21} -1.22864 q^{22} -4.38002 q^{23} -2.85072 q^{24} +1.37273 q^{26} -6.06241 q^{27} +4.52360 q^{28} -5.55434 q^{29} +6.27071 q^{31} +1.00000 q^{32} +3.50252 q^{33} -4.61484 q^{34} +5.12662 q^{36} -7.27164 q^{37} -4.47801 q^{38} -3.91327 q^{39} -3.98348 q^{41} -12.8955 q^{42} -6.25152 q^{43} -1.22864 q^{44} -4.38002 q^{46} -10.6104 q^{47} -2.85072 q^{48} +13.4629 q^{49} +13.1556 q^{51} +1.37273 q^{52} -10.4611 q^{53} -6.06241 q^{54} +4.52360 q^{56} +12.7656 q^{57} -5.55434 q^{58} -6.84371 q^{59} +13.9786 q^{61} +6.27071 q^{62} +23.1908 q^{63} +1.00000 q^{64} +3.50252 q^{66} +0.657004 q^{67} -4.61484 q^{68} +12.4862 q^{69} -1.09617 q^{71} +5.12662 q^{72} +9.58493 q^{73} -7.27164 q^{74} -4.47801 q^{76} -5.55787 q^{77} -3.91327 q^{78} +15.9984 q^{79} +1.90240 q^{81} -3.98348 q^{82} +11.4611 q^{83} -12.8955 q^{84} -6.25152 q^{86} +15.8339 q^{87} -1.22864 q^{88} -1.00000 q^{89} +6.20967 q^{91} -4.38002 q^{92} -17.8761 q^{93} -10.6104 q^{94} -2.85072 q^{96} -15.1680 q^{97} +13.4629 q^{98} -6.29878 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} - 4 q^{3} + 8 q^{4} - 4 q^{6} - 2 q^{7} + 8 q^{8} - 4 q^{11} - 4 q^{12} - 4 q^{13} - 2 q^{14} + 8 q^{16} - 14 q^{17} - 8 q^{19} - 10 q^{21} - 4 q^{22} - 24 q^{23} - 4 q^{24} - 4 q^{26} - 10 q^{27}+ \cdots + 12 q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 - 4 * q^3 + 8 * q^4 - 4 * q^6 - 2 * q^7 + 8 * q^8 - 4 * q^11 - 4 * q^12 - 4 * q^13 - 2 * q^14 + 8 * q^16 - 14 * q^17 - 8 * q^19 - 10 * q^21 - 4 * q^22 - 24 * q^23 - 4 * q^24 - 4 * q^26 - 10 * q^27 - 2 * q^28 - 10 * q^29 + 8 * q^32 - 10 * q^33 - 14 * q^34 - 14 * q^37 - 8 * q^38 - 10 * q^39 - 10 * q^42 - 4 * q^43 - 4 * q^44 - 24 * q^46 - 28 * q^47 - 4 * q^48 - 4 * q^49 + 4 * q^51 - 4 * q^52 - 42 * q^53 - 10 * q^54 - 2 * q^56 + 20 * q^57 - 10 * q^58 - 4 * q^59 - 2 * q^61 + 6 * q^63 + 8 * q^64 - 10 * q^66 - 18 * q^67 - 14 * q^68 - 18 * q^69 + 20 * q^71 - 18 * q^73 - 14 * q^74 - 8 * q^76 - 34 * q^77 - 10 * q^78 + 2 * q^79 - 4 * q^81 - 20 * q^83 - 10 * q^84 - 4 * q^86 + 30 * q^87 - 4 * q^88 - 8 * q^89 + 20 * q^91 - 24 * q^92 - 46 * q^93 - 28 * q^94 - 4 * q^96 - 16 * q^97 - 4 * q^98 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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