LMFDB - Embedded newform 374.2.a.d.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [374,2,Mod(1,374)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(374, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("374.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$374 = 2 \cdot 11 \cdot 17$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	374.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$2.98640503560$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.55585.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - x^{3} - 10x^{2} + 9x + 16$ x^4 - x^3 - 10x^2 + 9x + 16
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.61078$ of defining polynomial
Character	$\chi$	$=$	374.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +2.61078 q^{3} +1.00000 q^{4} +2.81619 q^{5} -2.61078 q^{6} -3.15249 q^{7} -1.00000 q^{8} +3.81619 q^{9} -2.81619 q^{10} +1.00000 q^{11} +2.61078 q^{12} +5.28338 q^{13} +3.15249 q^{14} +7.35246 q^{15} +1.00000 q^{16} -1.00000 q^{17} -3.81619 q^{18} -6.37406 q^{19} +2.81619 q^{20} -8.23046 q^{21} -1.00000 q^{22} -3.22157 q^{23} -2.61078 q^{24} +2.93092 q^{25} -5.28338 q^{26} +2.13089 q^{27} -3.15249 q^{28} +4.67260 q^{29} -7.35246 q^{30} -2.67260 q^{31} -1.00000 q^{32} +2.61078 q^{33} +1.00000 q^{34} -8.87800 q^{35} +3.81619 q^{36} +2.84751 q^{37} +6.37406 q^{38} +13.7938 q^{39} -2.81619 q^{40} -1.36516 q^{41} +8.23046 q^{42} +12.7104 q^{43} +1.00000 q^{44} +10.7471 q^{45} +3.22157 q^{46} -11.0780 q^{47} +2.61078 q^{48} +2.93818 q^{49} -2.93092 q^{50} -2.61078 q^{51} +5.28338 q^{52} -7.89417 q^{53} -2.13089 q^{54} +2.81619 q^{55} +3.15249 q^{56} -16.6413 q^{57} -4.67260 q^{58} -1.08341 q^{59} +7.35246 q^{60} -1.22157 q^{61} +2.67260 q^{62} -12.0305 q^{63} +1.00000 q^{64} +14.8790 q^{65} -2.61078 q^{66} -1.22157 q^{67} -1.00000 q^{68} -8.41081 q^{69} +8.87800 q^{70} +6.93439 q^{71} -3.81619 q^{72} -7.65100 q^{73} -2.84751 q^{74} +7.65200 q^{75} -6.37406 q^{76} -3.15249 q^{77} -13.7938 q^{78} +16.5481 q^{79} +2.81619 q^{80} -5.88527 q^{81} +1.36516 q^{82} -13.4143 q^{83} -8.23046 q^{84} -2.81619 q^{85} -12.7104 q^{86} +12.1991 q^{87} -1.00000 q^{88} +7.41428 q^{89} -10.7471 q^{90} -16.6558 q^{91} -3.22157 q^{92} -6.97758 q^{93} +11.0780 q^{94} -17.9505 q^{95} -2.61078 q^{96} -2.54897 q^{97} -2.93818 q^{98} +3.81619 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 4 q^{2} + q^{3} + 4 q^{4} + 5 q^{5} - q^{6} + q^{7} - 4 q^{8} + 9 q^{9} - 5 q^{10} + 4 q^{11} + q^{12} - 3 q^{13} - q^{14} + 4 q^{16} - 4 q^{17} - 9 q^{18} + 7 q^{19} + 5 q^{20} + 8 q^{21} - 4 q^{22}+ \cdots + 9 q^{99}+O(q^{100})$ 4 * q - 4 * q^2 + q^3 + 4 * q^4 + 5 * q^5 - q^6 + q^7 - 4 * q^8 + 9 * q^9 - 5 * q^10 + 4 * q^11 + q^12 - 3 * q^13 - q^14 + 4 * q^16 - 4 * q^17 - 9 * q^18 + 7 * q^19 + 5 * q^20 + 8 * q^21 - 4 * q^22 + 6 * q^23 - q^24 + 17 * q^25 + 3 * q^26 - 2 * q^27 + q^28 + 4 * q^29 + 4 * q^31 - 4 * q^32 + q^33 + 4 * q^34 - 24 * q^35 + 9 * q^36 + 25 * q^37 - 7 * q^38 + 7 * q^39 - 5 * q^40 + 5 * q^41 - 8 * q^42 + 11 * q^43 + 4 * q^44 + 42 * q^45 - 6 * q^46 - 17 * q^47 + q^48 + 17 * q^49 - 17 * q^50 - q^51 - 3 * q^52 + 2 * q^53 + 2 * q^54 + 5 * q^55 - q^56 - 32 * q^57 - 4 * q^58 + 4 * q^59 + 14 * q^61 - 4 * q^62 - 23 * q^63 + 4 * q^64 - 16 * q^65 - q^66 + 14 * q^67 - 4 * q^68 - 40 * q^69 + 24 * q^70 - 8 * q^71 - 9 * q^72 - 19 * q^73 - 25 * q^74 - 45 * q^75 + 7 * q^76 + q^77 - 7 * q^78 + 17 * q^79 + 5 * q^80 - 12 * q^81 - 5 * q^82 - 19 * q^83 + 8 * q^84 - 5 * q^85 - 11 * q^86 - 12 * q^87 - 4 * q^88 - 5 * q^89 - 42 * q^90 - 32 * q^91 + 6 * q^92 + 14 * q^93 + 17 * q^94 - 14 * q^95 - q^96 - 6 * q^97 - 17 * q^98 + 9 * 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
$3$	2.61078	1.50734	0.753668	−	0.657255i	$-0.228283\pi$
$3$	2.61078	1.50734	0.753668	+	0.657255i	$0.228283\pi$
$4$	1.00000	0.500000
$5$	2.81619	1.25944	0.629719	−	0.776823i	$-0.283170\pi$
$5$	2.81619	1.25944	0.629719	+	0.776823i	$0.283170\pi$
$6$	−2.61078	−1.06585
$7$	−3.15249	−1.19153	−0.595764	−	0.803159i	$-0.703151\pi$
$7$	−3.15249	−1.19153	−0.595764	+	0.803159i	$0.703151\pi$
$8$	−1.00000	−0.353553
$9$	3.81619	1.27206
$10$	−2.81619	−0.890557
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	2.61078	0.753668
$13$	5.28338	1.46535	0.732673	−	0.680581i	$-0.238272\pi$
$13$	5.28338	1.46535	0.732673	+	0.680581i	$0.238272\pi$
$14$	3.15249	0.842538
$15$	7.35246	1.89840
$16$	1.00000	0.250000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	−3.81619	−0.899484
$19$	−6.37406	−1.46231	−0.731154	−	0.682212i	$-0.761018\pi$
$19$	−6.37406	−1.46231	−0.731154	+	0.682212i	$0.761018\pi$
$20$	2.81619	0.629719
$21$	−8.23046	−1.79603
$22$	−1.00000	−0.213201
$23$	−3.22157	−0.671743	−0.335872	−	0.941908i	$-0.609031\pi$
$23$	−3.22157	−0.671743	−0.335872	+	0.941908i	$0.609031\pi$
$24$	−2.61078	−0.532924
$25$	2.93092	0.586184
$26$	−5.28338	−1.03616
$27$	2.13089	0.410091
$28$	−3.15249	−0.595764
$29$	4.67260	0.867680	0.433840	−	0.900990i	$-0.357158\pi$
$29$	4.67260	0.867680	0.433840	+	0.900990i	$0.357158\pi$
$30$	−7.35246	−1.34237
$31$	−2.67260	−0.480013	−0.240006	−	0.970771i	$-0.577150\pi$
$31$	−2.67260	−0.480013	−0.240006	+	0.970771i	$0.577150\pi$
$32$	−1.00000	−0.176777
$33$	2.61078	0.454479
$34$	1.00000	0.171499
$35$	−8.87800	−1.50066
$36$	3.81619	0.636032
$37$	2.84751	0.468128	0.234064	−	0.972221i	$-0.424797\pi$
$37$	2.84751	0.468128	0.234064	+	0.972221i	$0.424797\pi$
$38$	6.37406	1.03401
$39$	13.7938	2.20877
$40$	−2.81619	−0.445279
$41$	−1.36516	−0.213202	−0.106601	−	0.994302i	$-0.533997\pi$
$41$	−1.36516	−0.213202	−0.106601	+	0.994302i	$0.533997\pi$
$42$	8.23046	1.26999
$43$	12.7104	1.93831	0.969155	−	0.246450i	$-0.0792642\pi$
$43$	12.7104	1.93831	0.969155	+	0.246450i	$0.0792642\pi$
$44$	1.00000	0.150756
$45$	10.7471	1.60208
$46$	3.22157	0.474994
$47$	−11.0780	−1.61589	−0.807944	−	0.589259i	$-0.799420\pi$
$47$	−11.0780	−1.61589	−0.807944	+	0.589259i	$0.799420\pi$
$48$	2.61078	0.376834
$49$	2.93818	0.419741
$50$	−2.93092	−0.414495
$51$	−2.61078	−0.365583
$52$	5.28338	0.732673
$53$	−7.89417	−1.08435	−0.542173	−	0.840267i	$-0.682398\pi$
$53$	−7.89417	−1.08435	−0.542173	+	0.840267i	$0.682398\pi$
$54$	−2.13089	−0.289978
$55$	2.81619	0.379735
$56$	3.15249	0.421269
$57$	−16.6413	−2.20419
$58$	−4.67260	−0.613542
$59$	−1.08341	−0.141048	−0.0705240	−	0.997510i	$-0.522467\pi$
$59$	−1.08341	−0.141048	−0.0705240	+	0.997510i	$0.522467\pi$
$60$	7.35246	0.949198
$61$	−1.22157	−0.156406	−0.0782028	−	0.996937i	$-0.524918\pi$
$61$	−1.22157	−0.156406	−0.0782028	+	0.996937i	$0.524918\pi$
$62$	2.67260	0.339420
$63$	−12.0305	−1.51570
$64$	1.00000	0.125000
$65$	14.8790	1.84551
$66$	−2.61078	−0.321365
$67$	−1.22157	−0.149238	−0.0746191	−	0.997212i	$-0.523774\pi$
$67$	−1.22157	−0.149238	−0.0746191	+	0.997212i	$0.523774\pi$
$68$	−1.00000	−0.121268
$69$	−8.41081	−1.01254
$70$	8.87800	1.06112
$71$	6.93439	0.822960	0.411480	−	0.911419i	$-0.365012\pi$
$71$	6.93439	0.822960	0.411480	+	0.911419i	$0.365012\pi$
$72$	−3.81619	−0.449742
$73$	−7.65100	−0.895482	−0.447741	−	0.894163i	$-0.647771\pi$
$73$	−7.65100	−0.895482	−0.447741	+	0.894163i	$0.647771\pi$
$74$	−2.84751	−0.331016
$75$	7.65200	0.883577
$76$	−6.37406	−0.731154
$77$	−3.15249	−0.359259
$78$	−13.7938	−1.56184
$79$	16.5481	1.86181	0.930906	−	0.365260i	$-0.119020\pi$
$79$	16.5481	1.86181	0.930906	+	0.365260i	$0.119020\pi$
$80$	2.81619	0.314860
$81$	−5.88527	−0.653919
$82$	1.36516	0.150756
$83$	−13.4143	−1.47241	−0.736204	−	0.676760i	$-0.763384\pi$
$83$	−13.4143	−1.47241	−0.736204	+	0.676760i	$0.763384\pi$
$84$	−8.23046	−0.898017
$85$	−2.81619	−0.305459
$86$	−12.7104	−1.37059
$87$	12.1991	1.30789
$88$	−1.00000	−0.106600
$89$	7.41428	0.785912	0.392956	−	0.919557i	$-0.371453\pi$
$89$	7.41428	0.785912	0.392956	+	0.919557i	$0.371453\pi$
$90$	−10.7471	−1.13285
$91$	−16.6558	−1.74600
$92$	−3.22157	−0.335872
$93$	−6.97758	−0.723541
$94$	11.0780	1.14261
$95$	−17.9505	−1.84169
$96$	−2.61078	−0.266462
$97$	−2.54897	−0.258808	−0.129404	−	0.991592i	$-0.541306\pi$
$97$	−2.54897	−0.258808	−0.129404	+	0.991592i	$0.541306\pi$
$98$	−2.93818	−0.296801
$99$	3.81619	0.383541
$100$	2.93092	0.293092
$101$	−8.81619	−0.877244	−0.438622	−	0.898672i	$-0.644533\pi$
$101$	−8.81619	−0.877244	−0.438622	+	0.898672i	$0.644533\pi$
$102$	2.61078	0.258506
$103$	16.0315	1.57963	0.789815	−	0.613345i	$-0.210177\pi$
$103$	16.0315	1.57963	0.789815	+	0.613345i	$0.210177\pi$
$104$	−5.28338	−0.518078
$105$	−23.1785	−2.26199
$106$	7.89417	0.766749
$107$	−13.8942	−1.34320	−0.671600	−	0.740914i	$-0.734393\pi$
$107$	−13.8942	−1.34320	−0.671600	+	0.740914i	$0.734393\pi$
$108$	2.13089	0.205045
$109$	0.548968	0.0525816	0.0262908	−	0.999654i	$-0.491630\pi$
$109$	0.548968	0.0525816	0.0262908	+	0.999654i	$0.491630\pi$
$110$	−2.81619	−0.268513
$111$	7.43424	0.705626
$112$	−3.15249	−0.297882
$113$	−13.5265	−1.27247	−0.636235	−	0.771495i	$-0.719509\pi$
$113$	−13.5265	−1.27247	−0.636235	+	0.771495i	$0.719509\pi$
$114$	16.6413	1.55860
$115$	−9.07254	−0.846019
$116$	4.67260	0.433840
$117$	20.1624	1.86401
$118$	1.08341	0.0997360
$119$	3.15249	0.288988
$120$	−7.35246	−0.671185
$121$	1.00000	0.0909091
$122$	1.22157	0.110595
$123$	−3.56413	−0.321367
$124$	−2.67260	−0.240006
$125$	−5.82692	−0.521175
$126$	12.0305	1.07176
$127$	17.2216	1.52817	0.764083	−	0.645117i	$-0.223192\pi$
$127$	17.2216	1.52817	0.764083	+	0.645117i	$0.223192\pi$
$128$	−1.00000	−0.0883883
$129$	33.1840	2.92169
$130$	−14.8790	−1.30497
$131$	−15.7883	−1.37943	−0.689716	−	0.724080i	$-0.742265\pi$
$131$	−15.7883	−1.37943	−0.689716	+	0.724080i	$0.742265\pi$
$132$	2.61078	0.227240
$133$	20.0941	1.74238
$134$	1.22157	0.105527
$135$	6.00100	0.516484
$136$	1.00000	0.0857493
$137$	6.50495	0.555755	0.277878	−	0.960616i	$-0.410369\pi$
$137$	6.50495	0.555755	0.277878	+	0.960616i	$0.410369\pi$
$138$	8.41081	0.715976
$139$	−12.4863	−1.05908	−0.529538	−	0.848286i	$-0.677635\pi$
$139$	−12.4863	−1.05908	−0.529538	+	0.848286i	$0.677635\pi$
$140$	−8.87800	−0.750328
$141$	−28.9222	−2.43569
$142$	−6.93439	−0.581921
$143$	5.28338	0.441819
$144$	3.81619	0.318016
$145$	13.1589	1.09279
$146$	7.65100	0.633202
$147$	7.67096	0.632690
$148$	2.84751	0.234064
$149$	16.4618	1.34860	0.674300	−	0.738458i	$-0.264446\pi$
$149$	16.4618	1.34860	0.674300	+	0.738458i	$0.264446\pi$
$150$	−7.65200	−0.624783
$151$	−5.89417	−0.479660	−0.239830	−	0.970815i	$-0.577092\pi$
$151$	−5.89417	−0.479660	−0.239830	+	0.970815i	$0.577092\pi$
$152$	6.37406	0.517004
$153$	−3.81619	−0.308521
$154$	3.15249	0.254035
$155$	−7.52654	−0.604547
$156$	13.7938	1.10439
$157$	−20.3805	−1.62654	−0.813270	−	0.581886i	$-0.802315\pi$
$157$	−20.3805	−1.62654	−0.813270	+	0.581886i	$0.802315\pi$
$158$	−16.5481	−1.31650
$159$	−20.6100	−1.63448
$160$	−2.81619	−0.222639
$161$	10.1560	0.800401
$162$	5.88527	0.462390
$163$	8.16139	0.639249	0.319625	−	0.947544i	$-0.396443\pi$
$163$	8.16139	0.639249	0.319625	+	0.947544i	$0.396443\pi$
$164$	−1.36516	−0.106601
$165$	7.35246	0.572388
$166$	13.4143	1.04115
$167$	−16.3427	−1.26464	−0.632319	−	0.774708i	$-0.717897\pi$
$167$	−16.3427	−1.26464	−0.632319	+	0.774708i	$0.717897\pi$
$168$	8.23046	0.634994
$169$	14.9141	1.14724
$170$	2.81619	0.215992
$171$	−24.3246	−1.86015
$172$	12.7104	0.969155
$173$	18.4431	1.40221	0.701103	−	0.713060i	$-0.252691\pi$
$173$	18.4431	1.40221	0.701103	+	0.713060i	$0.252691\pi$
$174$	−12.1991	−0.924815
$175$	−9.23970	−0.698456
$176$	1.00000	0.0753778
$177$	−2.82855	−0.212607
$178$	−7.41428	−0.555723
$179$	−16.7481	−1.25181	−0.625906	−	0.779898i	$-0.715271\pi$
$179$	−16.7481	−1.25181	−0.625906	+	0.779898i	$0.715271\pi$
$180$	10.7471	0.801042
$181$	12.9784	0.964677	0.482339	−	0.875985i	$-0.339787\pi$
$181$	12.9784	0.964677	0.482339	+	0.875985i	$0.339787\pi$
$182$	16.6558	1.23461
$183$	−3.18925	−0.235756
$184$	3.22157	0.237497
$185$	8.01913	0.589578
$186$	6.97758	0.511621
$187$	−1.00000	−0.0731272
$188$	−11.0780	−0.807944
$189$	−6.71762	−0.488635
$190$	17.9505	1.30227
$191$	22.8099	1.65047	0.825234	−	0.564791i	$-0.191043\pi$
$191$	22.8099	1.65047	0.825234	+	0.564791i	$0.191043\pi$
$192$	2.61078	0.188417
$193$	−3.12743	−0.225117	−0.112559	−	0.993645i	$-0.535905\pi$
$193$	−3.12743	−0.225117	−0.112559	+	0.993645i	$0.535905\pi$
$194$	2.54897	0.183005
$195$	38.8459	2.78181
$196$	2.93818	0.209870
$197$	22.3050	1.58916	0.794582	−	0.607157i	$-0.207690\pi$
$197$	22.3050	1.58916	0.794582	+	0.607157i	$0.207690\pi$
$198$	−3.81619	−0.271205
$199$	5.58919	0.396207	0.198104	−	0.980181i	$-0.436522\pi$
$199$	5.58919	0.396207	0.198104	+	0.980181i	$0.436522\pi$
$200$	−2.93092	−0.207247
$201$	−3.18925	−0.224952
$202$	8.81619	0.620305
$203$	−14.7303	−1.03387
$204$	−2.61078	−0.182791
$205$	−3.84454	−0.268514
$206$	−16.0315	−1.11697
$207$	−12.2941	−0.854500
$208$	5.28338	0.366337
$209$	−6.37406	−0.440903
$210$	23.1785	1.59947
$211$	7.26476	0.500127	0.250063	−	0.968230i	$-0.419549\pi$
$211$	7.26476	0.500127	0.250063	+	0.968230i	$0.419549\pi$
$212$	−7.89417	−0.542173
$213$	18.1042	1.24048
$214$	13.8942	0.949786
$215$	35.7948	2.44118
$216$	−2.13089	−0.144989
$217$	8.42534	0.571949
$218$	−0.548968	−0.0371808
$219$	−19.9751	−1.34979
$220$	2.81619	0.189867
$221$	−5.28338	−0.355399
$222$	−7.43424	−0.498953
$223$	−1.62594	−0.108881	−0.0544407	−	0.998517i	$-0.517338\pi$
$223$	−1.62594	−0.108881	−0.0544407	+	0.998517i	$0.517338\pi$
$224$	3.15249	0.210635
$225$	11.1850	0.745664
$226$	13.5265	0.899772
$227$	5.02077	0.333240	0.166620	−	0.986021i	$-0.446715\pi$
$227$	5.02077	0.333240	0.166620	+	0.986021i	$0.446715\pi$
$228$	−16.6413	−1.10210
$229$	9.40291	0.621362	0.310681	−	0.950514i	$-0.399443\pi$
$229$	9.40291	0.621362	0.310681	+	0.950514i	$0.399443\pi$
$230$	9.07254	0.598226
$231$	−8.23046	−0.541525
$232$	−4.67260	−0.306771
$233$	−8.47263	−0.555060	−0.277530	−	0.960717i	$-0.589516\pi$
$233$	−8.47263	−0.555060	−0.277530	+	0.960717i	$0.589516\pi$
$234$	−20.1624	−1.31806
$235$	−31.1977	−2.03511
$236$	−1.08341	−0.0705240
$237$	43.2036	2.80638
$238$	−3.15249	−0.204345
$239$	24.8863	1.60976	0.804879	−	0.593438i	$-0.202230\pi$
$239$	24.8863	1.60976	0.804879	+	0.593438i	$0.202230\pi$
$240$	7.35246	0.474599
$241$	8.44313	0.543870	0.271935	−	0.962316i	$-0.412336\pi$
$241$	8.44313	0.543870	0.271935	+	0.962316i	$0.412336\pi$
$242$	−1.00000	−0.0642824
$243$	−21.7578	−1.39577
$244$	−1.22157	−0.0782028
$245$	8.27448	0.528637
$246$	3.56413	0.227241
$247$	−33.6766	−2.14279
$248$	2.67260	0.169710
$249$	−35.0218	−2.21941
$250$	5.82692	0.368526
$251$	22.3805	1.41264	0.706322	−	0.707891i	$-0.250353\pi$
$251$	22.3805	1.41264	0.706322	+	0.707891i	$0.250353\pi$
$252$	−12.0305	−0.757850
$253$	−3.22157	−0.202538
$254$	−17.2216	−1.08058
$255$	−7.35246	−0.460429
$256$	1.00000	0.0625000
$257$	14.8172	0.924271	0.462135	−	0.886809i	$-0.347083\pi$
$257$	14.8172	0.924271	0.462135	+	0.886809i	$0.347083\pi$
$258$	−33.1840	−2.06594
$259$	−8.97675	−0.557788
$260$	14.8790	0.922757
$261$	17.8315	1.10374
$262$	15.7883	0.975406
$263$	−26.0933	−1.60898	−0.804491	−	0.593965i	$-0.797562\pi$
$263$	−26.0933	−1.60898	−0.804491	+	0.593965i	$0.797562\pi$
$264$	−2.61078	−0.160683
$265$	−22.2315	−1.36567
$266$	−20.0941	−1.23205
$267$	19.3571	1.18463
$268$	−1.22157	−0.0746191
$269$	17.1525	1.04581	0.522903	−	0.852392i	$-0.324849\pi$
$269$	17.1525	1.04581	0.522903	+	0.852392i	$0.324849\pi$
$270$	−6.00100	−0.365209
$271$	−17.1589	−1.04233	−0.521165	−	0.853456i	$-0.674502\pi$
$271$	−17.1589	−1.04233	−0.521165	+	0.853456i	$0.674502\pi$
$272$	−1.00000	−0.0606339
$273$	−43.4847	−2.63181
$274$	−6.50495	−0.392978
$275$	2.93092	0.176741
$276$	−8.41081	−0.506271
$277$	−8.62448	−0.518195	−0.259097	−	0.965851i	$-0.583425\pi$
$277$	−8.62448	−0.518195	−0.259097	+	0.965851i	$0.583425\pi$
$278$	12.4863	0.748880
$279$	−10.1991	−0.610607
$280$	8.87800	0.530562
$281$	−4.99210	−0.297804	−0.148902	−	0.988852i	$-0.547574\pi$
$281$	−4.99210	−0.297804	−0.148902	+	0.988852i	$0.547574\pi$
$282$	28.9222	1.72229
$283$	12.5236	0.744449	0.372225	−	0.928143i	$-0.378595\pi$
$283$	12.5236	0.744449	0.372225	+	0.928143i	$0.378595\pi$
$284$	6.93439	0.411480
$285$	−46.8650	−2.77604
$286$	−5.28338	−0.312413
$287$	4.30364	0.254036
$288$	−3.81619	−0.224871
$289$	1.00000	0.0588235
$290$	−13.1589	−0.772719
$291$	−6.65480	−0.390111
$292$	−7.65100	−0.447741
$293$	−0.453496	−0.0264935	−0.0132468	−	0.999912i	$-0.504217\pi$
$293$	−0.453496	−0.0264935	−0.0132468	+	0.999912i	$0.504217\pi$
$294$	−7.67096	−0.447380
$295$	−3.05109	−0.177641
$296$	−2.84751	−0.165508
$297$	2.13089	0.123647
$298$	−16.4618	−0.953604
$299$	−17.0208	−0.984336
$300$	7.65200	0.441789
$301$	−40.0692	−2.30955
$302$	5.89417	0.339171
$303$	−23.0172	−1.32230
$304$	−6.37406	−0.365577
$305$	−3.44016	−0.196983
$306$	3.81619	0.218157
$307$	−18.3236	−1.04578	−0.522892	−	0.852399i	$-0.675147\pi$
$307$	−18.3236	−1.04578	−0.522892	+	0.852399i	$0.675147\pi$
$308$	−3.15249	−0.179630
$309$	41.8548	2.38103
$310$	7.52654	0.427479
$311$	23.7079	1.34435	0.672176	−	0.740392i	$-0.265360\pi$
$311$	23.7079	1.34435	0.672176	+	0.740392i	$0.265360\pi$
$312$	−13.7938	−0.780918
$313$	29.4013	1.66186	0.830929	−	0.556379i	$-0.187810\pi$
$313$	29.4013	1.66186	0.830929	+	0.556379i	$0.187810\pi$
$314$	20.3805	1.15014
$315$	−33.8801	−1.90893
$316$	16.5481	0.930906
$317$	7.45103	0.418492	0.209246	−	0.977863i	$-0.432899\pi$
$317$	7.45103	0.418492	0.209246	+	0.977863i	$0.432899\pi$
$318$	20.6100	1.15575
$319$	4.67260	0.261615
$320$	2.81619	0.157430
$321$	−36.2747	−2.02465
$322$	−10.1560	−0.565969
$323$	6.37406	0.354662
$324$	−5.88527	−0.326959
$325$	15.4852	0.858963
$326$	−8.16139	−0.452017
$327$	1.43324	0.0792581
$328$	1.36516	0.0753782
$329$	34.9232	1.92538
$330$	−7.35246	−0.404740
$331$	35.0531	1.92669	0.963346	−	0.268262i	$-0.0864493\pi$
$331$	35.0531	1.92669	0.963346	+	0.268262i	$0.0864493\pi$
$332$	−13.4143	−0.736204
$333$	10.8666	0.595488
$334$	16.3427	0.894235
$335$	−3.44016	−0.187956
$336$	−8.23046	−0.449009
$337$	18.7104	1.01922	0.509609	−	0.860406i	$-0.329790\pi$
$337$	18.7104	1.01922	0.509609	+	0.860406i	$0.329790\pi$
$338$	−14.9141	−0.811221
$339$	−35.3149	−1.91804
$340$	−2.81619	−0.152729
$341$	−2.67260	−0.144729
$342$	24.3246	1.31532
$343$	12.8048	0.691396
$344$	−12.7104	−0.685296
$345$	−23.6864	−1.27523
$346$	−18.4431	−0.991509
$347$	1.89417	0.101684	0.0508421	−	0.998707i	$-0.483809\pi$
$347$	1.89417	0.101684	0.0508421	+	0.998707i	$0.483809\pi$
$348$	12.1991	0.653943
$349$	−21.2648	−1.13828	−0.569138	−	0.822242i	$-0.692723\pi$
$349$	−21.2648	−1.13828	−0.569138	+	0.822242i	$0.692723\pi$
$350$	9.23970	0.493883
$351$	11.2583	0.600925
$352$	−1.00000	−0.0533002
$353$	−12.5290	−0.666852	−0.333426	−	0.942776i	$-0.608205\pi$
$353$	−12.5290	−0.666852	−0.333426	+	0.942776i	$0.608205\pi$
$354$	2.82855	0.150336
$355$	19.5285	1.03647
$356$	7.41428	0.392956
$357$	8.23046	0.435602
$358$	16.7481	0.885165
$359$	2.77843	0.146640	0.0733201	−	0.997308i	$-0.476641\pi$
$359$	2.77843	0.146640	0.0733201	+	0.997308i	$0.476641\pi$
$360$	−10.7471	−0.566423
$361$	21.6286	1.13835
$362$	−12.9784	−0.682130
$363$	2.61078	0.137031
$364$	−16.6558	−0.873001
$365$	−21.5467	−1.12780
$366$	3.18925	0.166705
$367$	11.9568	0.624140	0.312070	−	0.950059i	$-0.398978\pi$
$367$	11.9568	0.624140	0.312070	+	0.950059i	$0.398978\pi$
$368$	−3.22157	−0.167936
$369$	−5.20970	−0.271206
$370$	−8.01913	−0.416895
$371$	24.8863	1.29203
$372$	−6.97758	−0.361770
$373$	−19.5524	−1.01239	−0.506193	−	0.862420i	$-0.668948\pi$
$373$	−19.5524	−1.01239	−0.506193	+	0.862420i	$0.668948\pi$
$374$	1.00000	0.0517088
$375$	−15.2128	−0.785586
$376$	11.0780	0.571303
$377$	24.6871	1.27145
$378$	6.71762	0.345517
$379$	6.36813	0.327109	0.163554	−	0.986534i	$-0.447704\pi$
$379$	6.36813	0.327109	0.163554	+	0.986534i	$0.447704\pi$
$380$	−17.9505	−0.920844
$381$	44.9618	2.30346
$382$	−22.8099	−1.16706
$383$	−5.55687	−0.283943	−0.141971	−	0.989871i	$-0.545344\pi$
$383$	−5.55687	−0.283943	−0.141971	+	0.989871i	$0.545344\pi$
$384$	−2.61078	−0.133231
$385$	−8.87800	−0.452465
$386$	3.12743	0.159182
$387$	48.5051	2.46565
$388$	−2.54897	−0.129404
$389$	6.07551	0.308041	0.154020	−	0.988068i	$-0.450778\pi$
$389$	6.07551	0.308041	0.154020	+	0.988068i	$0.450778\pi$
$390$	−38.8459	−1.96704
$391$	3.22157	0.162922
$392$	−2.93818	−0.148401
$393$	−41.2199	−2.07927
$394$	−22.3050	−1.12371
$395$	46.6027	2.34484
$396$	3.81619	0.191771
$397$	−16.8027	−0.843301	−0.421651	−	0.906758i	$-0.638549\pi$
$397$	−16.8027	−0.843301	−0.421651	+	0.906758i	$0.638549\pi$
$398$	−5.58919	−0.280161
$399$	52.4614	2.62636
$400$	2.93092	0.146546
$401$	−17.4639	−0.872106	−0.436053	−	0.899921i	$-0.643624\pi$
$401$	−17.4639	−0.872106	−0.436053	+	0.899921i	$0.643624\pi$
$402$	3.18925	0.159065
$403$	−14.1204	−0.703385
$404$	−8.81619	−0.438622
$405$	−16.5740	−0.823570
$406$	14.7303	0.731053
$407$	2.84751	0.141146
$408$	2.61078	0.129253
$409$	8.71579	0.430968	0.215484	−	0.976507i	$-0.430867\pi$
$409$	8.71579	0.430968	0.215484	+	0.976507i	$0.430867\pi$
$410$	3.84454	0.189868
$411$	16.9830	0.837710
$412$	16.0315	0.789815
$413$	3.41544	0.168063
$414$	12.2941	0.604222
$415$	−37.7771	−1.85441
$416$	−5.28338	−0.259039
$417$	−32.5991	−1.59638
$418$	6.37406	0.311765
$419$	38.6741	1.88935	0.944677	−	0.328003i	$-0.106375\pi$
$419$	38.6741	1.88935	0.944677	+	0.328003i	$0.106375\pi$
$420$	−23.1785	−1.13100
$421$	33.8493	1.64971	0.824857	−	0.565341i	$-0.191255\pi$
$421$	33.8493	1.64971	0.824857	+	0.565341i	$0.191255\pi$
$422$	−7.26476	−0.353643
$423$	−42.2757	−2.05551
$424$	7.89417	0.383375
$425$	−2.93092	−0.142171
$426$	−18.1042	−0.877150
$427$	3.85097	0.186362
$428$	−13.8942	−0.671600
$429$	13.7938	0.665969
$430$	−35.7948	−1.72618
$431$	23.0790	1.11168	0.555838	−	0.831291i	$-0.312398\pi$
$431$	23.0790	1.11168	0.555838	+	0.831291i	$0.312398\pi$
$432$	2.13089	0.102523
$433$	29.2145	1.40396	0.701979	−	0.712198i	$-0.252300\pi$
$433$	29.2145	1.40396	0.701979	+	0.712198i	$0.252300\pi$
$434$	−8.42534	−0.404429
$435$	34.3551	1.64720
$436$	0.548968	0.0262908
$437$	20.5344	0.982296
$438$	19.9751	0.954448
$439$	20.8772	0.996413	0.498207	−	0.867058i	$-0.333992\pi$
$439$	20.8772	0.996413	0.498207	+	0.867058i	$0.333992\pi$
$440$	−2.81619	−0.134257
$441$	11.2127	0.533937
$442$	5.28338	0.251305
$443$	−10.2569	−0.487318	−0.243659	−	0.969861i	$-0.578348\pi$
$443$	−10.2569	−0.487318	−0.243659	+	0.969861i	$0.578348\pi$
$444$	7.43424	0.352813
$445$	20.8800	0.989807
$446$	1.62594	0.0769907
$447$	42.9781	2.03279
$448$	−3.15249	−0.148941
$449$	−23.0963	−1.08998	−0.544990	−	0.838442i	$-0.683467\pi$
$449$	−23.0963	−1.08998	−0.544990	+	0.838442i	$0.683467\pi$
$450$	−11.1850	−0.527264
$451$	−1.36516	−0.0642827
$452$	−13.5265	−0.636235
$453$	−15.3884	−0.723010
$454$	−5.02077	−0.235636
$455$	−46.9059	−2.19898
$456$	16.6413	0.779299
$457$	7.75601	0.362811	0.181405	−	0.983408i	$-0.441935\pi$
$457$	7.75601	0.362811	0.181405	+	0.983408i	$0.441935\pi$
$458$	−9.40291	−0.439369
$459$	−2.13089	−0.0994616
$460$	−9.07254	−0.423009
$461$	3.90093	0.181685	0.0908423	−	0.995865i	$-0.471044\pi$
$461$	3.90093	0.181685	0.0908423	+	0.995865i	$0.471044\pi$
$462$	8.23046	0.382916
$463$	−28.1179	−1.30675	−0.653374	−	0.757035i	$-0.726647\pi$
$463$	−28.1179	−1.30675	−0.653374	+	0.757035i	$0.726647\pi$
$464$	4.67260	0.216920
$465$	−19.6502	−0.911255
$466$	8.47263	0.392487
$467$	−24.5473	−1.13591	−0.567957	−	0.823058i	$-0.692266\pi$
$467$	−24.5473	−1.13591	−0.567957	+	0.823058i	$0.692266\pi$
$468$	20.1624	0.932007
$469$	3.85097	0.177821
$470$	31.1977	1.43904
$471$	−53.2090	−2.45174
$472$	1.08341	0.0498680
$473$	12.7104	0.584423
$474$	−43.2036	−1.98441
$475$	−18.6819	−0.857182
$476$	3.15249	0.144494
$477$	−30.1256	−1.37936
$478$	−24.8863	−1.13827
$479$	−14.1050	−0.644474	−0.322237	−	0.946659i	$-0.604435\pi$
$479$	−14.1050	−0.644474	−0.322237	+	0.946659i	$0.604435\pi$
$480$	−7.35246	−0.335592
$481$	15.0445	0.685970
$482$	−8.44313	−0.384574
$483$	26.5150	1.20647
$484$	1.00000	0.0454545
$485$	−7.17838	−0.325953
$486$	21.7578	0.986956
$487$	−27.3343	−1.23864	−0.619318	−	0.785140i	$-0.712591\pi$
$487$	−27.3343	−1.23864	−0.619318	+	0.785140i	$0.712591\pi$
$488$	1.22157	0.0552977
$489$	21.3076	0.963564
$490$	−8.27448	−0.373803
$491$	2.54353	0.114788	0.0573940	−	0.998352i	$-0.481721\pi$
$491$	2.54353	0.114788	0.0573940	+	0.998352i	$0.481721\pi$
$492$	−3.56413	−0.160683
$493$	−4.67260	−0.210443
$494$	33.6766	1.51518
$495$	10.7471	0.483047
$496$	−2.67260	−0.120003
$497$	−21.8606	−0.980581
$498$	35.0218	1.56936
$499$	−29.6639	−1.32794	−0.663969	−	0.747760i	$-0.731129\pi$
$499$	−29.6639	−1.32794	−0.663969	+	0.747760i	$0.731129\pi$
$500$	−5.82692	−0.260588
$501$	−42.6673	−1.90624
$502$	−22.3805	−0.998890
$503$	−2.27909	−0.101620	−0.0508098	−	0.998708i	$-0.516180\pi$
$503$	−2.27909	−0.101620	−0.0508098	+	0.998708i	$0.516180\pi$
$504$	12.0305	0.535881
$505$	−24.8281	−1.10483
$506$	3.22157	0.143216
$507$	38.9375	1.72928
$508$	17.2216	0.764083
$509$	−7.84605	−0.347770	−0.173885	−	0.984766i	$-0.555632\pi$
$509$	−7.84605	−0.347770	−0.173885	+	0.984766i	$0.555632\pi$
$510$	7.35246	0.325572
$511$	24.1197	1.06699
$512$	−1.00000	−0.0441942
$513$	−13.5824	−0.599679
$514$	−14.8172	−0.653558
$515$	45.1477	1.98945
$516$	33.1840	1.46084
$517$	−11.0780	−0.487209
$518$	8.97675	0.394416
$519$	48.1510	2.11360
$520$	−14.8790	−0.652487
$521$	15.3953	0.674481	0.337240	−	0.941419i	$-0.390506\pi$
$521$	15.3953	0.674481	0.337240	+	0.941419i	$0.390506\pi$
$522$	−17.8315	−0.780464
$523$	7.15349	0.312800	0.156400	−	0.987694i	$-0.450011\pi$
$523$	7.15349	0.312800	0.156400	+	0.987694i	$0.450011\pi$
$524$	−15.7883	−0.689716
$525$	−24.1229	−1.05281
$526$	26.0933	1.13772
$527$	2.67260	0.116420
$528$	2.61078	0.113620
$529$	−12.6215	−0.548761
$530$	22.2315	0.965673
$531$	−4.13450	−0.179422
$532$	20.0941	0.871191
$533$	−7.21265	−0.312414
$534$	−19.3571	−0.837662
$535$	−39.1286	−1.69168
$536$	1.22157	0.0527636
$537$	−43.7257	−1.88690
$538$	−17.1525	−0.739496
$539$	2.93818	0.126557
$540$	6.00100	0.258242
$541$	11.6146	0.499350	0.249675	−	0.968330i	$-0.419676\pi$
$541$	11.6146	0.499350	0.249675	+	0.968330i	$0.419676\pi$
$542$	17.1589	0.737039
$543$	33.8838	1.45409
$544$	1.00000	0.0428746
$545$	1.54600	0.0662233
$546$	43.4847	1.86097
$547$	12.5490	0.536555	0.268278	−	0.963342i	$-0.413546\pi$
$547$	12.5490	0.536555	0.268278	+	0.963342i	$0.413546\pi$
$548$	6.50495	0.277878
$549$	−4.66173	−0.198958
$550$	−2.93092	−0.124975
$551$	−29.7834	−1.26882
$552$	8.41081	0.357988
$553$	−52.1678	−2.21840
$554$	8.62448	0.366419
$555$	20.9362	0.888693
$556$	−12.4863	−0.529538
$557$	−36.7555	−1.55738	−0.778691	−	0.627407i	$-0.784116\pi$
$557$	−36.7555	−1.55738	−0.778691	+	0.627407i	$0.784116\pi$
$558$	10.1991	0.431764
$559$	67.1537	2.84030
$560$	−8.87800	−0.375164
$561$	−2.61078	−0.110227
$562$	4.99210	0.210579
$563$	23.1535	0.975803	0.487902	−	0.872899i	$-0.337762\pi$
$563$	23.1535	0.975803	0.487902	+	0.872899i	$0.337762\pi$
$564$	−28.9222	−1.21784
$565$	−38.0933	−1.60260
$566$	−12.5236	−0.526405
$567$	18.5532	0.779163
$568$	−6.93439	−0.290960
$569$	−38.4609	−1.61237	−0.806183	−	0.591666i	$-0.798470\pi$
$569$	−38.4609	−1.61237	−0.806183	+	0.591666i	$0.798470\pi$
$570$	46.8650	1.96296
$571$	21.8460	0.914229	0.457114	−	0.889408i	$-0.348883\pi$
$571$	21.8460	0.914229	0.457114	+	0.889408i	$0.348883\pi$
$572$	5.28338	0.220909
$573$	59.5518	2.48781
$574$	−4.30364	−0.179631
$575$	−9.44216	−0.393765
$576$	3.81619	0.159008
$577$	−25.5811	−1.06495	−0.532477	−	0.846444i	$-0.678739\pi$
$577$	−25.5811	−1.06495	−0.532477	+	0.846444i	$0.678739\pi$
$578$	−1.00000	−0.0415945
$579$	−8.16504	−0.339328
$580$	13.1589	0.546395
$581$	42.2884	1.75442
$582$	6.65480	0.275850
$583$	−7.89417	−0.326943
$584$	7.65100	0.316601
$585$	56.7811	2.34761
$586$	0.453496	0.0187338
$587$	−23.8510	−0.984435	−0.492218	−	0.870472i	$-0.663814\pi$
$587$	−23.8510	−0.984435	−0.492218	+	0.870472i	$0.663814\pi$
$588$	7.67096	0.316345
$589$	17.0353	0.701927
$590$	3.05109	0.125611
$591$	58.2335	2.39540
$592$	2.84751	0.117032
$593$	−1.32740	−0.0545098	−0.0272549	−	0.999629i	$-0.508677\pi$
$593$	−1.32740	−0.0545098	−0.0272549	+	0.999629i	$0.508677\pi$
$594$	−2.13089	−0.0874316
$595$	8.87800	0.363963
$596$	16.4618	0.674300
$597$	14.5922	0.597217
$598$	17.0208	0.696031
$599$	24.8863	1.01683	0.508413	−	0.861114i	$-0.330232\pi$
$599$	24.8863	1.01683	0.508413	+	0.861114i	$0.330232\pi$
$600$	−7.65200	−0.312392
$601$	−7.47692	−0.304990	−0.152495	−	0.988304i	$-0.548731\pi$
$601$	−7.47692	−0.304990	−0.152495	+	0.988304i	$0.548731\pi$
$602$	40.0692	1.63310
$603$	−4.66173	−0.189840
$604$	−5.89417	−0.239830
$605$	2.81619	0.114494
$606$	23.0172	0.935008
$607$	16.0556	0.651675	0.325837	−	0.945426i	$-0.394354\pi$
$607$	16.0556	0.651675	0.325837	+	0.945426i	$0.394354\pi$
$608$	6.37406	0.258502
$609$	−38.4577	−1.55838
$610$	3.44016	0.139288
$611$	−58.5292	−2.36784
$612$	−3.81619	−0.154260
$613$	−19.0644	−0.770006	−0.385003	−	0.922915i	$-0.625800\pi$
$613$	−19.0644	−0.770006	−0.385003	+	0.922915i	$0.625800\pi$
$614$	18.3236	0.739480
$615$	−10.0373	−0.404742
$616$	3.15249	0.127017
$617$	−12.3950	−0.499004	−0.249502	−	0.968374i	$-0.580267\pi$
$617$	−12.3950	−0.499004	−0.249502	+	0.968374i	$0.580267\pi$
$618$	−41.8548	−1.68365
$619$	−20.4423	−0.821646	−0.410823	−	0.911715i	$-0.634759\pi$
$619$	−20.4423	−0.821646	−0.410823	+	0.911715i	$0.634759\pi$
$620$	−7.52654	−0.302273
$621$	−6.86481	−0.275476
$622$	−23.7079	−0.950600
$623$	−23.3734	−0.936436
$624$	13.7938	0.552193
$625$	−31.0643	−1.24257
$626$	−29.4013	−1.17511
$627$	−16.6413	−0.664589
$628$	−20.3805	−0.813270
$629$	−2.84751	−0.113538
$630$	33.8801	1.34982
$631$	41.3394	1.64570	0.822849	−	0.568261i	$-0.192383\pi$
$631$	41.3394	1.64570	0.822849	+	0.568261i	$0.192383\pi$
$632$	−16.5481	−0.658250
$633$	18.9667	0.753859
$634$	−7.45103	−0.295918
$635$	48.4992	1.92463
$636$	−20.6100	−0.817238
$637$	15.5236	0.615066
$638$	−4.67260	−0.184990
$639$	26.4629	1.04686
$640$	−2.81619	−0.111320
$641$	6.85887	0.270909	0.135455	−	0.990784i	$-0.456751\pi$
$641$	6.85887	0.270909	0.135455	+	0.990784i	$0.456751\pi$
$642$	36.2747	1.43165
$643$	−23.1784	−0.914066	−0.457033	−	0.889450i	$-0.651088\pi$
$643$	−23.1784	−0.914066	−0.457033	+	0.889450i	$0.651088\pi$
$644$	10.1560	0.400201
$645$	93.4524	3.67968
$646$	−6.37406	−0.250784
$647$	−30.6846	−1.20634	−0.603169	−	0.797614i	$-0.706095\pi$
$647$	−30.6846	−1.20634	−0.603169	+	0.797614i	$0.706095\pi$
$648$	5.88527	0.231195
$649$	−1.08341	−0.0425276
$650$	−15.4852	−0.607379
$651$	21.9967	0.862120
$652$	8.16139	0.319625
$653$	0.578126	0.0226238	0.0113119	−	0.999936i	$-0.496399\pi$
$653$	0.578126	0.0226238	0.0113119	+	0.999936i	$0.496399\pi$
$654$	−1.43324	−0.0560440
$655$	−44.4629	−1.73731
$656$	−1.36516	−0.0533004
$657$	−29.1977	−1.13911
$658$	−34.9232	−1.36145
$659$	−17.5007	−0.681729	−0.340864	−	0.940112i	$-0.610720\pi$
$659$	−17.5007	−0.681729	−0.340864	+	0.940112i	$0.610720\pi$
$660$	7.35246	0.286194
$661$	−23.0099	−0.894981	−0.447491	−	0.894289i	$-0.647682\pi$
$661$	−23.0099	−0.894981	−0.447491	+	0.894289i	$0.647682\pi$
$662$	−35.0531	−1.36238
$663$	−13.7938	−0.535705
$664$	13.4143	0.520575
$665$	56.5889	2.19442
$666$	−10.8666	−0.421074
$667$	−15.0531	−0.582858
$668$	−16.3427	−0.632319
$669$	−4.24499	−0.164121
$670$	3.44016	0.132905
$671$	−1.22157	−0.0471581
$672$	8.23046	0.317497
$673$	10.1004	0.389342	0.194671	−	0.980869i	$-0.437636\pi$
$673$	10.1004	0.389342	0.194671	+	0.980869i	$0.437636\pi$
$674$	−18.7104	−0.720696
$675$	6.24548	0.240389
$676$	14.9141	0.573620
$677$	0.911661	0.0350380	0.0175190	−	0.999847i	$-0.494423\pi$
$677$	0.911661	0.0350380	0.0175190	+	0.999847i	$0.494423\pi$
$678$	35.3149	1.35626
$679$	8.03559	0.308378
$680$	2.81619	0.107996
$681$	13.1081	0.502305
$682$	2.67260	0.102339
$683$	−39.1390	−1.49761	−0.748805	−	0.662790i	$-0.769372\pi$
$683$	−39.1390	−1.49761	−0.748805	+	0.662790i	$0.769372\pi$
$684$	−24.3246	−0.930074
$685$	18.3192	0.699939
$686$	−12.8048	−0.488891
$687$	24.5490	0.936602
$688$	12.7104	0.484578
$689$	−41.7079	−1.58894
$690$	23.6864	0.901727
$691$	30.5877	1.16361	0.581806	−	0.813328i	$-0.302346\pi$
$691$	30.5877	1.16361	0.581806	+	0.813328i	$0.302346\pi$
$692$	18.4431	0.701103
$693$	−12.0305	−0.457001
$694$	−1.89417	−0.0719015
$695$	−35.1639	−1.33384
$696$	−12.1991	−0.462407
$697$	1.36516	0.0517090
$698$	21.2648	0.804883
$699$	−22.1202	−0.836663
$700$	−9.23970	−0.349228
$701$	−21.7321	−0.820811	−0.410405	−	0.911903i	$-0.634613\pi$
$701$	−21.7321	−0.820811	−0.410405	+	0.911903i	$0.634613\pi$
$702$	−11.2583	−0.424918
$703$	−18.1502	−0.684548
$704$	1.00000	0.0376889
$705$	−81.4504	−3.06760
$706$	12.5290	0.471535
$707$	27.7929	1.04526
$708$	−2.82855	−0.106303
$709$	32.4819	1.21988	0.609941	−	0.792446i	$-0.291193\pi$
$709$	32.4819	1.21988	0.609941	+	0.792446i	$0.291193\pi$
$710$	−19.5285	−0.732893
$711$	63.1508	2.36834
$712$	−7.41428	−0.277862
$713$	8.60995	0.322445
$714$	−8.23046	−0.308017
$715$	14.8790	0.556443
$716$	−16.7481	−0.625906
$717$	64.9726	2.42645
$718$	−2.77843	−0.103690
$719$	−7.94228	−0.296197	−0.148099	−	0.988973i	$-0.547315\pi$
$719$	−7.94228	−0.296197	−0.148099	+	0.988973i	$0.547315\pi$
$720$	10.7471	0.400521
$721$	−50.5391	−1.88217
$722$	−21.6286	−0.804932
$723$	22.0432	0.819795
$724$	12.9784	0.482339
$725$	13.6950	0.508620
$726$	−2.61078	−0.0968953
$727$	11.5079	0.426805	0.213403	−	0.976964i	$-0.431545\pi$
$727$	11.5079	0.426805	0.213403	+	0.976964i	$0.431545\pi$
$728$	16.6558	0.617305
$729$	−39.1492	−1.44997
$730$	21.5467	0.797478
$731$	−12.7104	−0.470109
$732$	−3.18925	−0.117878
$733$	−7.91905	−0.292497	−0.146248	−	0.989248i	$-0.546720\pi$
$733$	−7.91905	−0.292497	−0.146248	+	0.989248i	$0.546720\pi$
$734$	−11.9568	−0.441334
$735$	21.6029	0.796834
$736$	3.22157	0.118749
$737$	−1.22157	−0.0449970
$738$	5.20970	0.191772
$739$	36.2432	1.33323	0.666613	−	0.745404i	$-0.267743\pi$
$739$	36.2432	1.33323	0.666613	+	0.745404i	$0.267743\pi$
$740$	8.01913	0.294789
$741$	−87.9222	−3.22990
$742$	−24.8863	−0.913604
$743$	−4.67969	−0.171681	−0.0858406	−	0.996309i	$-0.527358\pi$
$743$	−4.67969	−0.171681	−0.0858406	+	0.996309i	$0.527358\pi$
$744$	6.97758	0.255810
$745$	46.3594	1.69848
$746$	19.5524	0.715865
$747$	−51.1914	−1.87300
$748$	−1.00000	−0.0365636
$749$	43.8012	1.60046
$750$	15.2128	0.555493
$751$	−22.0020	−0.802864	−0.401432	−	0.915889i	$-0.631487\pi$
$751$	−22.0020	−0.802864	−0.401432	+	0.915889i	$0.631487\pi$
$752$	−11.0780	−0.403972
$753$	58.4306	2.12933
$754$	−24.6871	−0.899052
$755$	−16.5991	−0.604103
$756$	−6.71762	−0.244317
$757$	12.7013	0.461635	0.230818	−	0.972997i	$-0.425860\pi$
$757$	12.7013	0.461635	0.230818	+	0.972997i	$0.425860\pi$
$758$	−6.36813	−0.231301
$759$	−8.41081	−0.305293
$760$	17.9505	0.651135
$761$	−19.7451	−0.715761	−0.357880	−	0.933767i	$-0.616500\pi$
$761$	−19.7451	−0.715761	−0.357880	+	0.933767i	$0.616500\pi$
$762$	−44.9618	−1.62879
$763$	−1.73061	−0.0626525
$764$	22.8099	0.825234
$765$	−10.7471	−0.388563
$766$	5.55687	0.200778
$767$	−5.72407	−0.206684
$768$	2.61078	0.0942085
$769$	−11.9423	−0.430650	−0.215325	−	0.976542i	$-0.569081\pi$
$769$	−11.9423	−0.430650	−0.215325	+	0.976542i	$0.569081\pi$
$770$	8.87800	0.319941
$771$	38.6845	1.39319
$772$	−3.12743	−0.112559
$773$	31.5961	1.13643	0.568217	−	0.822879i	$-0.307634\pi$
$773$	31.5961	1.13643	0.568217	+	0.822879i	$0.307634\pi$
$774$	−48.5051	−1.74348
$775$	−7.83318	−0.281376
$776$	2.54897	0.0915026
$777$	−23.4363	−0.840774
$778$	−6.07551	−0.217818
$779$	8.70159	0.311767
$780$	38.8459	1.39090
$781$	6.93439	0.248132
$782$	−3.22157	−0.115203
$783$	9.95681	0.355827
$784$	2.93818	0.104935
$785$	−57.3953	−2.04853
$786$	41.2199	1.47027
$787$	20.3983	0.727120	0.363560	−	0.931571i	$-0.381561\pi$
$787$	20.3983	0.727120	0.363560	+	0.931571i	$0.381561\pi$
$788$	22.3050	0.794582
$789$	−68.1240	−2.42528
$790$	−46.6027	−1.65805
$791$	42.6423	1.51618
$792$	−3.81619	−0.135602
$793$	−6.45400	−0.229188
$794$	16.8027	0.596304
$795$	−58.0415	−2.05852
$796$	5.58919	0.198104
$797$	−7.91956	−0.280525	−0.140263	−	0.990114i	$-0.544795\pi$
$797$	−7.91956	−0.280525	−0.140263	+	0.990114i	$0.544795\pi$
$798$	−52.4614	−1.85711
$799$	11.0780	0.391911
$800$	−2.93092	−0.103624
$801$	28.2943	0.999729
$802$	17.4639	0.616672
$803$	−7.65100	−0.269998
$804$	−3.18925	−0.112476
$805$	28.6011	1.00806
$806$	14.1204	0.497368
$807$	44.7814	1.57638
$808$	8.81619	0.310152
$809$	34.4847	1.21242	0.606208	−	0.795306i	$-0.292690\pi$
$809$	34.4847	1.21242	0.606208	+	0.795306i	$0.292690\pi$
$810$	16.5740	0.582352
$811$	−36.5846	−1.28466	−0.642329	−	0.766429i	$-0.722032\pi$
$811$	−36.5846	−1.28466	−0.642329	+	0.766429i	$0.722032\pi$
$812$	−14.7303	−0.516933
$813$	−44.7982	−1.57114
$814$	−2.84751	−0.0998052
$815$	22.9840	0.805095
$816$	−2.61078	−0.0913957
$817$	−81.0165	−2.83441
$818$	−8.71579	−0.304740
$819$	−63.5617	−2.22103
$820$	−3.84454	−0.134257
$821$	49.3099	1.72093	0.860464	−	0.509510i	$-0.170173\pi$
$821$	49.3099	1.72093	0.860464	+	0.509510i	$0.170173\pi$
$822$	−16.9830	−0.592351
$823$	−50.7728	−1.76983	−0.884915	−	0.465753i	$-0.845784\pi$
$823$	−50.7728	−1.76983	−0.884915	+	0.465753i	$0.845784\pi$
$824$	−16.0315	−0.558483
$825$	7.65200	0.266409
$826$	−3.41544	−0.118838
$827$	−22.0323	−0.766139	−0.383069	−	0.923720i	$-0.625133\pi$
$827$	−22.0323	−0.766139	−0.383069	+	0.923720i	$0.625133\pi$
$828$	−12.2941	−0.427250
$829$	−39.8088	−1.38262	−0.691308	−	0.722560i	$-0.742965\pi$
$829$	−39.8088	−1.38262	−0.691308	+	0.722560i	$0.742965\pi$
$830$	37.7771	1.31126
$831$	−22.5166	−0.781094
$832$	5.28338	0.183168
$833$	−2.93818	−0.101802
$834$	32.5991	1.12881
$835$	−46.0242	−1.59273
$836$	−6.37406	−0.220451
$837$	−5.69502	−0.196849
$838$	−38.6741	−1.33597
$839$	2.23639	0.0772088	0.0386044	−	0.999255i	$-0.487709\pi$
$839$	2.23639	0.0772088	0.0386044	+	0.999255i	$0.487709\pi$
$840$	23.1785	0.799736
$841$	−7.16682	−0.247132
$842$	−33.8493	−1.16652
$843$	−13.0333	−0.448890
$844$	7.26476	0.250063
$845$	42.0010	1.44488
$846$	42.2757	1.45347
$847$	−3.15249	−0.108321
$848$	−7.89417	−0.271087
$849$	32.6963	1.12214
$850$	2.93092	0.100530
$851$	−9.17345	−0.314462
$852$	18.1042	0.620239
$853$	30.2061	1.03424	0.517118	−	0.855914i	$-0.327005\pi$
$853$	30.2061	1.03424	0.517118	+	0.855914i	$0.327005\pi$
$854$	−3.85097	−0.131778
$855$	−68.5027	−2.34274
$856$	13.8942	0.474893
$857$	8.71301	0.297631	0.148815	−	0.988865i	$-0.452454\pi$
$857$	8.71301	0.297631	0.148815	+	0.988865i	$0.452454\pi$
$858$	−13.7938	−0.470911
$859$	16.6855	0.569301	0.284650	−	0.958631i	$-0.408122\pi$
$859$	16.6855	0.569301	0.284650	+	0.958631i	$0.408122\pi$
$860$	35.7948	1.22059
$861$	11.2359	0.382918
$862$	−23.0790	−0.786073
$863$	−12.2072	−0.415539	−0.207770	−	0.978178i	$-0.566620\pi$
$863$	−12.2072	−0.415539	−0.207770	+	0.978178i	$0.566620\pi$
$864$	−2.13089	−0.0724945
$865$	51.9394	1.76599
$866$	−29.2145	−0.992748
$867$	2.61078	0.0886668
$868$	8.42534	0.285975
$869$	16.5481	0.561357
$870$	−34.3551	−1.16475
$871$	−6.45400	−0.218686
$872$	−0.548968	−0.0185904
$873$	−9.72734	−0.329221
$874$	−20.5344	−0.694588
$875$	18.3693	0.620995
$876$	−19.9751	−0.674896
$877$	−7.93736	−0.268025	−0.134013	−	0.990980i	$-0.542786\pi$
$877$	−7.93736	−0.268025	−0.134013	+	0.990980i	$0.542786\pi$
$878$	−20.8772	−0.704571
$879$	−1.18398	−0.0399347
$880$	2.81619	0.0949337
$881$	37.0294	1.24755	0.623775	−	0.781604i	$-0.285598\pi$
$881$	37.0294	1.24755	0.623775	+	0.781604i	$0.285598\pi$
$882$	−11.2127	−0.377550
$883$	6.89119	0.231907	0.115954	−	0.993255i	$-0.463008\pi$
$883$	6.89119	0.231907	0.115954	+	0.993255i	$0.463008\pi$
$884$	−5.28338	−0.177699
$885$	−7.96573	−0.267765
$886$	10.2569	0.344586
$887$	−1.80729	−0.0606829	−0.0303415	−	0.999540i	$-0.509659\pi$
$887$	−1.80729	−0.0606829	−0.0303415	+	0.999540i	$0.509659\pi$
$888$	−7.43424	−0.249477
$889$	−54.2908	−1.82085
$890$	−20.8800	−0.699899
$891$	−5.88527	−0.197164
$892$	−1.62594	−0.0544407
$893$	70.6116	2.36293
$894$	−42.9781	−1.43740
$895$	−47.1658	−1.57658
$896$	3.15249	0.105317
$897$	−44.4375	−1.48373
$898$	23.0963	0.770733
$899$	−12.4880	−0.416497
$900$	11.1850	0.372832
$901$	7.89417	0.262993
$902$	1.36516	0.0454548
$903$	−104.612	−3.48127
$904$	13.5265	0.449886
$905$	36.5496	1.21495
$906$	15.3884	0.511245
$907$	38.1874	1.26799	0.633997	−	0.773336i	$-0.281413\pi$
$907$	38.1874	1.26799	0.633997	+	0.773336i	$0.281413\pi$
$908$	5.02077	0.166620
$909$	−33.6442	−1.11591
$910$	46.9059	1.55492
$911$	−23.5569	−0.780474	−0.390237	−	0.920714i	$-0.627607\pi$
$911$	−23.5569	−0.780474	−0.390237	+	0.920714i	$0.627607\pi$
$912$	−16.6413	−0.551048
$913$	−13.4143	−0.443948
$914$	−7.75601	−0.256546
$915$	−8.98152	−0.296920
$916$	9.40291	0.310681
$917$	49.7725	1.64363
$918$	2.13089	0.0703300
$919$	30.9667	1.02150	0.510749	−	0.859730i	$-0.329368\pi$
$919$	30.9667	1.02150	0.510749	+	0.859730i	$0.329368\pi$
$920$	9.07254	0.299113
$921$	−47.8390	−1.57635
$922$	−3.90093	−0.128470
$923$	36.6370	1.20592
$924$	−8.23046	−0.270762
$925$	8.34583	0.274409
$926$	28.1179	0.924011
$927$	61.1792	2.00939
$928$	−4.67260	−0.153386
$929$	−15.8170	−0.518939	−0.259469	−	0.965751i	$-0.583548\pi$
$929$	−15.8170	−0.518939	−0.259469	+	0.965751i	$0.583548\pi$
$930$	19.6502	0.644355
$931$	−18.7282	−0.613790
$932$	−8.47263	−0.277530
$933$	61.8962	2.02639
$934$	24.5473	0.803213
$935$	−2.81619	−0.0920992
$936$	−20.1624	−0.659028
$937$	−51.0983	−1.66931	−0.834654	−	0.550775i	$-0.814332\pi$
$937$	−51.0983	−1.66931	−0.834654	+	0.550775i	$0.814332\pi$
$938$	−3.85097	−0.125739
$939$	76.7603	2.50498
$940$	−31.1977	−1.01756
$941$	−39.8384	−1.29870	−0.649348	−	0.760491i	$-0.724958\pi$
$941$	−39.8384	−1.29870	−0.649348	+	0.760491i	$0.724958\pi$
$942$	53.2090	1.73364
$943$	4.39794	0.143217
$944$	−1.08341	−0.0352620
$945$	−18.9181	−0.615405
$946$	−12.7104	−0.413249
$947$	15.4552	0.502226	0.251113	−	0.967958i	$-0.419203\pi$
$947$	15.4552	0.502226	0.251113	+	0.967958i	$0.419203\pi$
$948$	43.2036	1.40319
$949$	−40.4232	−1.31219
$950$	18.6819	0.606120
$951$	19.4530	0.630808
$952$	−3.15249	−0.102173
$953$	−34.2601	−1.10979	−0.554897	−	0.831919i	$-0.687242\pi$
$953$	−34.2601	−1.10979	−0.554897	+	0.831919i	$0.687242\pi$
$954$	30.1256	0.975353
$955$	64.2371	2.07866
$956$	24.8863	0.804879
$957$	12.1991	0.394342
$958$	14.1050	0.455712
$959$	−20.5068	−0.662198
$960$	7.35246	0.237300
$961$	−23.8572	−0.769588
$962$	−15.0445	−0.485054
$963$	−53.0228	−1.70863
$964$	8.44313	0.271935
$965$	−8.80744	−0.283521
$966$	−26.5150	−0.853106
$967$	−25.5717	−0.822331	−0.411166	−	0.911561i	$-0.634878\pi$
$967$	−25.5717	−0.822331	−0.411166	+	0.911561i	$0.634878\pi$
$968$	−1.00000	−0.0321412
$969$	16.6413	0.534595
$970$	7.17838	0.230484
$971$	−34.9278	−1.12089	−0.560443	−	0.828193i	$-0.689369\pi$
$971$	−34.9278	−1.12089	−0.560443	+	0.828193i	$0.689369\pi$
$972$	−21.7578	−0.697883
$973$	39.3630	1.26192
$974$	27.3343	0.875848
$975$	40.4285	1.29475
$976$	−1.22157	−0.0391014
$977$	−5.46505	−0.174842	−0.0874212	−	0.996171i	$-0.527863\pi$
$977$	−5.46505	−0.174842	−0.0874212	+	0.996171i	$0.527863\pi$
$978$	−21.3076	−0.681342
$979$	7.41428	0.236961
$980$	8.27448	0.264319
$981$	2.09497	0.0668871
$982$	−2.54353	−0.0811674
$983$	−23.0871	−0.736363	−0.368182	−	0.929754i	$-0.620020\pi$
$983$	−23.0871	−0.736363	−0.368182	+	0.929754i	$0.620020\pi$
$984$	3.56413	0.113620
$985$	62.8150	2.00145
$986$	4.67260	0.148806
$987$	91.1769	2.90219
$988$	−33.6766	−1.07139
$989$	−40.9473	−1.30205
$990$	−10.7471	−0.341566
$991$	−34.6871	−1.10187	−0.550936	−	0.834547i	$-0.685729\pi$
$991$	−34.6871	−1.10187	−0.550936	+	0.834547i	$0.685729\pi$
$992$	2.67260	0.0848551
$993$	91.5160	2.90417
$994$	21.8606	0.693375
$995$	15.7402	0.498998
$996$	−35.0218	−1.10971
$997$	−16.0274	−0.507593	−0.253796	−	0.967258i	$-0.581679\pi$
$997$	−16.0274	−0.507593	−0.253796	+	0.967258i	$0.581679\pi$
$998$	29.6639	0.938993
$999$	6.06774	0.191975

Display

a_p

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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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	374.2.a.d.1.4	✓	4
3.2	odd	2		3366.2.a.bg.1.2		4
4.3	odd	2		2992.2.a.w.1.1		4
5.4	even	2		9350.2.a.cl.1.1		4
11.10	odd	2		4114.2.a.bc.1.4		4
17.16	even	2		6358.2.a.t.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
374.2.a.d.1.4	✓	4	1.1	even	1	trivial
2992.2.a.w.1.1		4	4.3	odd	2
3366.2.a.bg.1.2		4	3.2	odd	2
4114.2.a.bc.1.4		4	11.10	odd	2
6358.2.a.t.1.1		4	17.16	even	2
9350.2.a.cl.1.1		4	5.4	even	2

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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	374.2.a.d.1.4

Level	$374$
Weight	$2$
Character	374.1
Self dual	yes
Analytic conductor	$2.986$
Analytic rank	$0$
Dimension	$4$
CM	no
Inner twists	$1$