LMFDB - Embedded newform 1850.2.a.z.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1850,2,Mod(1,1850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1850, 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("1850.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1850 = 2 \cdot 5^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1850.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:	$14.7723243739$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.892.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 8x + 10 $ x^3 - x^2 - 8*x + 10
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 370)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.91729$ of defining polynomial
Character	$\chi$	$=$	1850.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -0.406728 q^{3} +1.00000 q^{4} +0.406728 q^{6} -2.91729 q^{7} -1.00000 q^{8} -2.83457 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -0.406728 q^{3} +1.00000 q^{4} +0.406728 q^{6} -2.91729 q^{7} -1.00000 q^{8} -2.83457 q^{9} +6.51056 q^{11} -0.406728 q^{12} +0.813457 q^{13} +2.91729 q^{14} +1.00000 q^{16} +2.51056 q^{17} +2.83457 q^{18} +0.406728 q^{19} +1.18654 q^{21} -6.51056 q^{22} -5.02112 q^{23} +0.406728 q^{24} -0.813457 q^{26} +2.37309 q^{27} -2.91729 q^{28} -5.32401 q^{29} -8.75186 q^{31} -1.00000 q^{32} -2.64803 q^{33} -2.51056 q^{34} -2.83457 q^{36} +1.00000 q^{37} -0.406728 q^{38} -0.330856 q^{39} +6.34513 q^{41} -1.18654 q^{42} +7.32401 q^{43} +6.51056 q^{44} +5.02112 q^{46} +5.42784 q^{47} -0.406728 q^{48} +1.51056 q^{49} -1.02112 q^{51} +0.813457 q^{52} +2.34513 q^{53} -2.37309 q^{54} +2.91729 q^{56} -0.165428 q^{57} +5.32401 q^{58} -1.42784 q^{59} -1.32401 q^{61} +8.75186 q^{62} +8.26926 q^{63} +1.00000 q^{64} +2.64803 q^{66} +5.42784 q^{67} +2.51056 q^{68} +2.04223 q^{69} +14.6480 q^{71} +2.83457 q^{72} +11.0211 q^{73} -1.00000 q^{74} +0.406728 q^{76} -18.9932 q^{77} +0.330856 q^{78} -1.75870 q^{79} +7.53851 q^{81} -6.34513 q^{82} +7.05476 q^{83} +1.18654 q^{84} -7.32401 q^{86} +2.16543 q^{87} -6.51056 q^{88} +6.00000 q^{89} -2.37309 q^{91} -5.02112 q^{92} +3.55963 q^{93} -5.42784 q^{94} +0.406728 q^{96} -2.34513 q^{97} -1.51056 q^{98} -18.4546 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{4} + q^{7} - 3 q^{8} + 11 q^{9}+O(q^{10}) $ 3 * q - 3 * q^2 + 3 * q^4 + q^7 - 3 * q^8 + 11 * q^9	$ 3 q - 3 q^{2} + 3 q^{4} + q^{7} - 3 q^{8} + 11 q^{9} + 11 q^{11} - q^{14} + 3 q^{16} - q^{17} - 11 q^{18} + 6 q^{21} - 11 q^{22} + 2 q^{23} + 12 q^{27} + q^{28} - 5 q^{29} + 3 q^{31} - 3 q^{32} + 14 q^{33} + q^{34} + 11 q^{36} + 3 q^{37} - 40 q^{39} - 9 q^{41} - 6 q^{42} + 11 q^{43} + 11 q^{44} - 2 q^{46} - 2 q^{47} - 4 q^{49} + 14 q^{51} - 21 q^{53} - 12 q^{54} - q^{56} - 20 q^{57} + 5 q^{58} + 14 q^{59} + 7 q^{61} - 3 q^{62} + 37 q^{63} + 3 q^{64} - 14 q^{66} - 2 q^{67} - q^{68} - 28 q^{69} + 22 q^{71} - 11 q^{72} + 16 q^{73} - 3 q^{74} - 7 q^{77} + 40 q^{78} - 26 q^{79} + 47 q^{81} + 9 q^{82} - 2 q^{83} + 6 q^{84} - 11 q^{86} + 26 q^{87} - 11 q^{88} + 18 q^{89} - 12 q^{91} + 2 q^{92} + 18 q^{93} + 2 q^{94} + 21 q^{97} + 4 q^{98} + 19 q^{99}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^4 + q^7 - 3 * q^8 + 11 * q^9 + 11 * q^11 - q^14 + 3 * q^16 - q^17 - 11 * q^18 + 6 * q^21 - 11 * q^22 + 2 * q^23 + 12 * q^27 + q^28 - 5 * q^29 + 3 * q^31 - 3 * q^32 + 14 * q^33 + q^34 + 11 * q^36 + 3 * q^37 - 40 * q^39 - 9 * q^41 - 6 * q^42 + 11 * q^43 + 11 * q^44 - 2 * q^46 - 2 * q^47 - 4 * q^49 + 14 * q^51 - 21 * q^53 - 12 * q^54 - q^56 - 20 * q^57 + 5 * q^58 + 14 * q^59 + 7 * q^61 - 3 * q^62 + 37 * q^63 + 3 * q^64 - 14 * q^66 - 2 * q^67 - q^68 - 28 * q^69 + 22 * q^71 - 11 * q^72 + 16 * q^73 - 3 * q^74 - 7 * q^77 + 40 * q^78 - 26 * q^79 + 47 * q^81 + 9 * q^82 - 2 * q^83 + 6 * q^84 - 11 * q^86 + 26 * q^87 - 11 * q^88 + 18 * q^89 - 12 * q^91 + 2 * q^92 + 18 * q^93 + 2 * q^94 + 21 * q^97 + 4 * q^98 + 19 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−0.406728	−0.234825	−0.117412	−	0.993083i	$-0.537460\pi$
$3$	−0.406728	−0.234825	−0.117412	+	0.993083i	$0.537460\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0.406728	0.166046
$7$	−2.91729	−1.10263	−0.551315	−	0.834297i	$-0.685874\pi$
$7$	−2.91729	−1.10263	−0.551315	+	0.834297i	$0.685874\pi$
$8$	−1.00000	−0.353553
$9$	−2.83457	−0.944857
$10$	0	0
$11$	6.51056	1.96301	0.981503	−	0.191444i	$-0.0613171\pi$
$11$	6.51056	1.96301	0.981503	+	0.191444i	$0.0613171\pi$
$12$	−0.406728	−0.117412
$13$	0.813457	0.225612	0.112806	−	0.993617i	$-0.464016\pi$
$13$	0.813457	0.225612	0.112806	+	0.993617i	$0.464016\pi$
$14$	2.91729	0.779677
$15$	0	0
$16$	1.00000	0.250000
$17$	2.51056	0.608900	0.304450	−	0.952528i	$-0.401527\pi$
$17$	2.51056	0.608900	0.304450	+	0.952528i	$0.401527\pi$
$18$	2.83457	0.668115
$19$	0.406728	0.0933099	0.0466550	−	0.998911i	$-0.485144\pi$
$19$	0.406728	0.0933099	0.0466550	+	0.998911i	$0.485144\pi$
$20$	0	0
$21$	1.18654	0.258925
$22$	−6.51056	−1.38806
$23$	−5.02112	−1.04697	−0.523487	−	0.852033i	$-0.675369\pi$
$23$	−5.02112	−1.04697	−0.523487	+	0.852033i	$0.675369\pi$
$24$	0.406728	0.0830231
$25$	0	0
$26$	−0.813457	−0.159532
$27$	2.37309	0.456701
$28$	−2.91729	−0.551315
$29$	−5.32401	−0.988645	−0.494322	−	0.869279i	$-0.664584\pi$
$29$	−5.32401	−0.988645	−0.494322	+	0.869279i	$0.664584\pi$
$30$	0	0
$31$	−8.75186	−1.57188	−0.785940	−	0.618303i	$-0.787821\pi$
$31$	−8.75186	−1.57188	−0.785940	+	0.618303i	$0.787821\pi$
$32$	−1.00000	−0.176777
$33$	−2.64803	−0.460963
$34$	−2.51056	−0.430557
$35$	0	0
$36$	−2.83457	−0.472429
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	−0.406728	−0.0659801
$39$	−0.330856	−0.0529794
$40$	0	0
$41$	6.34513	0.990943	0.495471	−	0.868624i	$-0.334995\pi$
$41$	6.34513	0.990943	0.495471	+	0.868624i	$0.334995\pi$
$42$	−1.18654	−0.183088
$43$	7.32401	1.11690	0.558451	−	0.829538i	$-0.311396\pi$
$43$	7.32401	1.11690	0.558451	+	0.829538i	$0.311396\pi$
$44$	6.51056	0.981503
$45$	0	0
$46$	5.02112	0.740323
$47$	5.42784	0.791732	0.395866	−	0.918308i	$-0.370444\pi$
$47$	5.42784	0.791732	0.395866	+	0.918308i	$0.370444\pi$
$48$	−0.406728	−0.0587062
$49$	1.51056	0.215794
$50$	0	0
$51$	−1.02112	−0.142985
$52$	0.813457	0.112806
$53$	2.34513	0.322128	0.161064	−	0.986944i	$-0.448507\pi$
$53$	2.34513	0.322128	0.161064	+	0.986944i	$0.448507\pi$
$54$	−2.37309	−0.322936
$55$	0	0
$56$	2.91729	0.389839
$57$	−0.165428	−0.0219115
$58$	5.32401	0.699077
$59$	−1.42784	−0.185889	−0.0929447	−	0.995671i	$-0.529628\pi$
$59$	−1.42784	−0.185889	−0.0929447	+	0.995671i	$0.529628\pi$
$60$	0	0
$61$	−1.32401	−0.169523	−0.0847613	−	0.996401i	$-0.527013\pi$
$61$	−1.32401	−0.169523	−0.0847613	+	0.996401i	$0.527013\pi$
$62$	8.75186	1.11149
$63$	8.26926	1.04183
$64$	1.00000	0.125000
$65$	0	0
$66$	2.64803	0.325950
$67$	5.42784	0.663117	0.331558	−	0.943435i	$-0.392426\pi$
$67$	5.42784	0.663117	0.331558	+	0.943435i	$0.392426\pi$
$68$	2.51056	0.304450
$69$	2.04223	0.245856
$70$	0	0
$71$	14.6480	1.73840	0.869201	−	0.494460i	$-0.164634\pi$
$71$	14.6480	1.73840	0.869201	+	0.494460i	$0.164634\pi$
$72$	2.83457	0.334058
$73$	11.0211	1.28992	0.644962	−	0.764215i	$-0.276873\pi$
$73$	11.0211	1.28992	0.644962	+	0.764215i	$0.276873\pi$
$74$	−1.00000	−0.116248
$75$	0	0
$76$	0.406728	0.0466550
$77$	−18.9932	−2.16447
$78$	0.330856	0.0374621
$79$	−1.75870	−0.197869	−0.0989346	−	0.995094i	$-0.531543\pi$
$79$	−1.75870	−0.197869	−0.0989346	+	0.995094i	$0.531543\pi$
$80$	0	0
$81$	7.53851	0.837613
$82$	−6.34513	−0.700702
$83$	7.05476	0.774360	0.387180	−	0.922004i	$-0.373449\pi$
$83$	7.05476	0.774360	0.387180	+	0.922004i	$0.373449\pi$
$84$	1.18654	0.129462
$85$	0	0
$86$	−7.32401	−0.789769
$87$	2.16543	0.232158
$88$	−6.51056	−0.694028
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	−2.37309	−0.248767
$92$	−5.02112	−0.523487
$93$	3.55963	0.369116
$94$	−5.42784	−0.559839
$95$	0	0
$96$	0.406728	0.0415115
$97$	−2.34513	−0.238112	−0.119056	−	0.992888i	$-0.537987\pi$
$97$	−2.34513	−0.238112	−0.119056	+	0.992888i	$0.537987\pi$
$98$	−1.51056	−0.152589
$99$	−18.4546	−1.85476
$100$	0	0
$101$	−8.20766	−0.816693	−0.408346	−	0.912827i	$-0.633894\pi$
$101$	−8.20766	−0.816693	−0.408346	+	0.912827i	$0.633894\pi$
$102$	1.02112	0.101105
$103$	8.81346	0.868416	0.434208	−	0.900813i	$-0.357028\pi$
$103$	8.81346	0.868416	0.434208	+	0.900813i	$0.357028\pi$
$104$	−0.813457	−0.0797660
$105$	0	0
$106$	−2.34513	−0.227779
$107$	−15.2624	−1.47547	−0.737737	−	0.675089i	$-0.764105\pi$
$107$	−15.2624	−1.47547	−0.737737	+	0.675089i	$0.764105\pi$
$108$	2.37309	0.228350
$109$	4.30290	0.412143	0.206072	−	0.978537i	$-0.433932\pi$
$109$	4.30290	0.412143	0.206072	+	0.978537i	$0.433932\pi$
$110$	0	0
$111$	−0.406728	−0.0386050
$112$	−2.91729	−0.275658
$113$	9.15859	0.861567	0.430784	−	0.902455i	$-0.358237\pi$
$113$	9.15859	0.861567	0.430784	+	0.902455i	$0.358237\pi$
$114$	0.165428	0.0154938
$115$	0	0
$116$	−5.32401	−0.494322
$117$	−2.30580	−0.213171
$118$	1.42784	0.131444
$119$	−7.32401	−0.671391
$120$	0	0
$121$	31.3874	2.85340
$122$	1.32401	0.119871
$123$	−2.58074	−0.232698
$124$	−8.75186	−0.785940
$125$	0	0
$126$	−8.26926	−0.736684
$127$	8.61439	0.764403	0.382202	−	0.924079i	$-0.375166\pi$
$127$	8.61439	0.764403	0.382202	+	0.924079i	$0.375166\pi$
$128$	−1.00000	−0.0883883
$129$	−2.97888	−0.262276
$130$	0	0
$131$	13.4278	1.17320	0.586598	−	0.809878i	$-0.300467\pi$
$131$	13.4278	1.17320	0.586598	+	0.809878i	$0.300467\pi$
$132$	−2.64803	−0.230481
$133$	−1.18654	−0.102886
$134$	−5.42784	−0.468894
$135$	0	0
$136$	−2.51056	−0.215279
$137$	14.6903	1.25507	0.627537	−	0.778587i	$-0.284063\pi$
$137$	14.6903	1.25507	0.627537	+	0.778587i	$0.284063\pi$
$138$	−2.04223	−0.173846
$139$	17.3662	1.47299	0.736493	−	0.676445i	$-0.236481\pi$
$139$	17.3662	1.47299	0.736493	+	0.676445i	$0.236481\pi$
$140$	0	0
$141$	−2.20766	−0.185918
$142$	−14.6480	−1.22924
$143$	5.29606	0.442879
$144$	−2.83457	−0.236214
$145$	0	0
$146$	−11.0211	−0.912114
$147$	−0.614387	−0.0506738
$148$	1.00000	0.0821995
$149$	−0.207658	−0.0170120	−0.00850601	−	0.999964i	$-0.502708\pi$
$149$	−0.207658	−0.0170120	−0.00850601	+	0.999964i	$0.502708\pi$
$150$	0	0
$151$	0.813457	0.0661982	0.0330991	−	0.999452i	$-0.489462\pi$
$151$	0.813457	0.0661982	0.0330991	+	0.999452i	$0.489462\pi$
$152$	−0.406728	−0.0329900
$153$	−7.11636	−0.575323
$154$	18.9932	1.53051
$155$	0	0
$156$	−0.330856	−0.0264897
$157$	−5.32401	−0.424903	−0.212451	−	0.977172i	$-0.568145\pi$
$157$	−5.32401	−0.424903	−0.212451	+	0.977172i	$0.568145\pi$
$158$	1.75870	0.139915
$159$	−0.953831	−0.0756437
$160$	0	0
$161$	14.6480	1.15443
$162$	−7.53851	−0.592282
$163$	−15.2008	−1.19062	−0.595310	−	0.803496i	$-0.702971\pi$
$163$	−15.2008	−1.19062	−0.595310	+	0.803496i	$0.702971\pi$
$164$	6.34513	0.495471
$165$	0	0
$166$	−7.05476	−0.547555
$167$	12.4826	0.965933	0.482966	−	0.875639i	$-0.339559\pi$
$167$	12.4826	0.965933	0.482966	+	0.875639i	$0.339559\pi$
$168$	−1.18654	−0.0915438
$169$	−12.3383	−0.949099
$170$	0	0
$171$	−1.15290	−0.0881645
$172$	7.32401	0.558451
$173$	−23.0354	−1.75135	−0.875674	−	0.482903i	$-0.839583\pi$
$173$	−23.0354	−1.75135	−0.875674	+	0.482903i	$0.839583\pi$
$174$	−2.16543	−0.164161
$175$	0	0
$176$	6.51056	0.490752
$177$	0.580745	0.0436514
$178$	−6.00000	−0.449719
$179$	24.2835	1.81504	0.907518	−	0.420013i	$-0.137974\pi$
$179$	24.2835	1.81504	0.907518	+	0.420013i	$0.137974\pi$
$180$	0	0
$181$	2.16543	0.160955	0.0804775	−	0.996756i	$-0.474355\pi$
$181$	2.16543	0.160955	0.0804775	+	0.996756i	$0.474355\pi$
$182$	2.37309	0.175905
$183$	0.538514	0.0398081
$184$	5.02112	0.370162
$185$	0	0
$186$	−3.55963	−0.261005
$187$	16.3451	1.19527
$188$	5.42784	0.395866
$189$	−6.92297	−0.503572
$190$	0	0
$191$	−19.3326	−1.39886	−0.699429	−	0.714702i	$-0.746562\pi$
$191$	−19.3326	−1.39886	−0.699429	+	0.714702i	$0.746562\pi$
$192$	−0.406728	−0.0293531
$193$	20.0422	1.44267	0.721336	−	0.692586i	$-0.243529\pi$
$193$	20.0422	1.44267	0.721336	+	0.692586i	$0.243529\pi$
$194$	2.34513	0.168370
$195$	0	0
$196$	1.51056	0.107897
$197$	−15.6269	−1.11337	−0.556686	−	0.830723i	$-0.687927\pi$
$197$	−15.6269	−1.11337	−0.556686	+	0.830723i	$0.687927\pi$
$198$	18.4546	1.31151
$199$	20.2835	1.43786	0.718931	−	0.695082i	$-0.244632\pi$
$199$	20.2835	1.43786	0.718931	+	0.695082i	$0.244632\pi$
$200$	0	0
$201$	−2.20766	−0.155716
$202$	8.20766	0.577489
$203$	15.5317	1.09011
$204$	−1.02112	−0.0714924
$205$	0	0
$206$	−8.81346	−0.614063
$207$	14.2327	0.989242
$208$	0.813457	0.0564031
$209$	2.64803	0.183168
$210$	0	0
$211$	18.7182	1.28862	0.644308	−	0.764766i	$-0.277146\pi$
$211$	18.7182	1.28862	0.644308	+	0.764766i	$0.277146\pi$
$212$	2.34513	0.161064
$213$	−5.95777	−0.408220
$214$	15.2624	1.04332
$215$	0	0
$216$	−2.37309	−0.161468
$217$	25.5317	1.73320
$218$	−4.30290	−0.291429
$219$	−4.48260	−0.302906
$220$	0	0
$221$	2.04223	0.137375
$222$	0.406728	0.0272978
$223$	15.7307	1.05341	0.526704	−	0.850049i	$-0.323428\pi$
$223$	15.7307	1.05341	0.526704	+	0.850049i	$0.323428\pi$
$224$	2.91729	0.194919
$225$	0	0
$226$	−9.15859	−0.609220
$227$	−12.8836	−0.855117	−0.427559	−	0.903988i	$-0.640626\pi$
$227$	−12.8836	−0.855117	−0.427559	+	0.903988i	$0.640626\pi$
$228$	−0.165428	−0.0109557
$229$	5.25383	0.347183	0.173591	−	0.984818i	$-0.444463\pi$
$229$	5.25383	0.347183	0.173591	+	0.984818i	$0.444463\pi$
$230$	0	0
$231$	7.72506	0.508271
$232$	5.32401	0.349539
$233$	−28.3172	−1.85512	−0.927560	−	0.373675i	$-0.878098\pi$
$233$	−28.3172	−1.85512	−0.927560	+	0.373675i	$0.878098\pi$
$234$	2.30580	0.150735
$235$	0	0
$236$	−1.42784	−0.0929447
$237$	0.715313	0.0464646
$238$	7.32401	0.474745
$239$	14.3788	0.930085	0.465043	−	0.885288i	$-0.346039\pi$
$239$	14.3788	0.930085	0.465043	+	0.885288i	$0.346039\pi$
$240$	0	0
$241$	18.2749	1.17719	0.588596	−	0.808427i	$-0.299681\pi$
$241$	18.2749	1.17719	0.588596	+	0.808427i	$0.299681\pi$
$242$	−31.3874	−2.01766
$243$	−10.1854	−0.653393
$244$	−1.32401	−0.0847613
$245$	0	0
$246$	2.58074	0.164542
$247$	0.330856	0.0210519
$248$	8.75186	0.555744
$249$	−2.86937	−0.181839
$250$	0	0
$251$	−1.22019	−0.0770174	−0.0385087	−	0.999258i	$-0.512261\pi$
$251$	−1.22019	−0.0770174	−0.0385087	+	0.999258i	$0.512261\pi$
$252$	8.26926	0.520914
$253$	−32.6903	−2.05522
$254$	−8.61439	−0.540515
$255$	0	0
$256$	1.00000	0.0625000
$257$	−27.1306	−1.69236	−0.846181	−	0.532895i	$-0.821104\pi$
$257$	−27.1306	−1.69236	−0.846181	+	0.532895i	$0.821104\pi$
$258$	2.97888	0.185457
$259$	−2.91729	−0.181271
$260$	0	0
$261$	15.0913	0.934128
$262$	−13.4278	−0.829575
$263$	20.2133	1.24641	0.623204	−	0.782059i	$-0.285831\pi$
$263$	20.2133	1.24641	0.623204	+	0.782059i	$0.285831\pi$
$264$	2.64803	0.162975
$265$	0	0
$266$	1.18654	0.0727516
$267$	−2.44037	−0.149348
$268$	5.42784	0.331558
$269$	−0.207658	−0.0126611	−0.00633057	−	0.999980i	$-0.502015\pi$
$269$	−0.207658	−0.0126611	−0.00633057	+	0.999980i	$0.502015\pi$
$270$	0	0
$271$	−30.6480	−1.86174	−0.930868	−	0.365357i	$-0.880947\pi$
$271$	−30.6480	−1.86174	−0.930868	+	0.365357i	$0.880947\pi$
$272$	2.51056	0.152225
$273$	0.965202	0.0584167
$274$	−14.6903	−0.887471
$275$	0	0
$276$	2.04223	0.122928
$277$	16.8135	1.01022	0.505111	−	0.863054i	$-0.331451\pi$
$277$	16.8135	1.01022	0.505111	+	0.863054i	$0.331451\pi$
$278$	−17.3662	−1.04156
$279$	24.8078	1.48520
$280$	0	0
$281$	−14.2749	−0.851572	−0.425786	−	0.904824i	$-0.640002\pi$
$281$	−14.2749	−0.851572	−0.425786	+	0.904824i	$0.640002\pi$
$282$	2.20766	0.131464
$283$	13.2288	0.786369	0.393184	−	0.919460i	$-0.371373\pi$
$283$	13.2288	0.786369	0.393184	+	0.919460i	$0.371373\pi$
$284$	14.6480	0.869201
$285$	0	0
$286$	−5.29606	−0.313162
$287$	−18.5106	−1.09264
$288$	2.83457	0.167029
$289$	−10.6971	−0.629241
$290$	0	0
$291$	0.953831	0.0559146
$292$	11.0211	0.644962
$293$	−2.34513	−0.137004	−0.0685020	−	0.997651i	$-0.521822\pi$
$293$	−2.34513	−0.137004	−0.0685020	+	0.997651i	$0.521822\pi$
$294$	0.614387	0.0358318
$295$	0	0
$296$	−1.00000	−0.0581238
$297$	15.4501	0.896507
$298$	0.207658	0.0120293
$299$	−4.08446	−0.236210
$300$	0	0
$301$	−21.3662	−1.23153
$302$	−0.813457	−0.0468092
$303$	3.33829	0.191780
$304$	0.406728	0.0233275
$305$	0	0
$306$	7.11636	0.406815
$307$	23.2624	1.32766	0.663828	−	0.747885i	$-0.268931\pi$
$307$	23.2624	1.32766	0.663828	+	0.747885i	$0.268931\pi$
$308$	−18.9932	−1.08224
$309$	−3.58468	−0.203926
$310$	0	0
$311$	−31.3999	−1.78052	−0.890262	−	0.455449i	$-0.849479\pi$
$311$	−31.3999	−1.78052	−0.890262	+	0.455449i	$0.849479\pi$
$312$	0.330856	0.0187310
$313$	8.04223	0.454574	0.227287	−	0.973828i	$-0.427014\pi$
$313$	8.04223	0.454574	0.227287	+	0.973828i	$0.427014\pi$
$314$	5.32401	0.300452
$315$	0	0
$316$	−1.75870	−0.0989346
$317$	12.3029	0.691000	0.345500	−	0.938419i	$-0.387709\pi$
$317$	12.3029	0.691000	0.345500	+	0.938419i	$0.387709\pi$
$318$	0.953831	0.0534882
$319$	−34.6623	−1.94072
$320$	0	0
$321$	6.20766	0.346478
$322$	−14.6480	−0.816303
$323$	1.02112	0.0568164
$324$	7.53851	0.418806
$325$	0	0
$326$	15.2008	0.841895
$327$	−1.75011	−0.0967814
$328$	−6.34513	−0.350351
$329$	−15.8346	−0.872988
$330$	0	0
$331$	−6.77981	−0.372652	−0.186326	−	0.982488i	$-0.559658\pi$
$331$	−6.77981	−0.372652	−0.186326	+	0.982488i	$0.559658\pi$
$332$	7.05476	0.387180
$333$	−2.83457	−0.155334
$334$	−12.4826	−0.683018
$335$	0	0
$336$	1.18654	0.0647312
$337$	−10.6903	−0.582336	−0.291168	−	0.956672i	$-0.594044\pi$
$337$	−10.6903	−0.582336	−0.291168	+	0.956672i	$0.594044\pi$
$338$	12.3383	0.671114
$339$	−3.72506	−0.202317
$340$	0	0
$341$	−56.9795	−3.08561
$342$	1.15290	0.0623417
$343$	16.0143	0.864689
$344$	−7.32401	−0.394884
$345$	0	0
$346$	23.0354	1.23839
$347$	−5.62691	−0.302069	−0.151034	−	0.988529i	$-0.548260\pi$
$347$	−5.62691	−0.302069	−0.151034	+	0.988529i	$0.548260\pi$
$348$	2.16543	0.116079
$349$	15.1306	0.809924	0.404962	−	0.914334i	$-0.367285\pi$
$349$	15.1306	0.809924	0.404962	+	0.914334i	$0.367285\pi$
$350$	0	0
$351$	1.93040	0.103037
$352$	−6.51056	−0.347014
$353$	−35.7816	−1.90446	−0.952230	−	0.305381i	$-0.901216\pi$
$353$	−35.7816	−1.90446	−0.952230	+	0.305381i	$0.901216\pi$
$354$	−0.580745	−0.0308662
$355$	0	0
$356$	6.00000	0.317999
$357$	2.97888	0.157659
$358$	−24.2835	−1.28342
$359$	4.48260	0.236583	0.118291	−	0.992979i	$-0.462258\pi$
$359$	4.48260	0.236583	0.118291	+	0.992979i	$0.462258\pi$
$360$	0	0
$361$	−18.8346	−0.991293
$362$	−2.16543	−0.113812
$363$	−12.7661	−0.670048
$364$	−2.37309	−0.124384
$365$	0	0
$366$	−0.538514	−0.0281486
$367$	21.5653	1.12570	0.562850	−	0.826559i	$-0.309705\pi$
$367$	21.5653	1.12570	0.562850	+	0.826559i	$0.309705\pi$
$368$	−5.02112	−0.261744
$369$	−17.9857	−0.936300
$370$	0	0
$371$	−6.84141	−0.355188
$372$	3.55963	0.184558
$373$	−16.9230	−0.876238	−0.438119	−	0.898917i	$-0.644355\pi$
$373$	−16.9230	−0.876238	−0.438119	+	0.898917i	$0.644355\pi$
$374$	−16.3451	−0.845187
$375$	0	0
$376$	−5.42784	−0.279920
$377$	−4.33086	−0.223050
$378$	6.92297	0.356079
$379$	31.2710	1.60628	0.803142	−	0.595788i	$-0.203160\pi$
$379$	31.2710	1.60628	0.803142	+	0.595788i	$0.203160\pi$
$380$	0	0
$381$	−3.50372	−0.179501
$382$	19.3326	0.989142
$383$	−1.62691	−0.0831314	−0.0415657	−	0.999136i	$-0.513235\pi$
$383$	−1.62691	−0.0831314	−0.0415657	+	0.999136i	$0.513235\pi$
$384$	0.406728	0.0207558
$385$	0	0
$386$	−20.0422	−1.02012
$387$	−20.7604	−1.05531
$388$	−2.34513	−0.119056
$389$	−31.6412	−1.60427	−0.802136	−	0.597141i	$-0.796303\pi$
$389$	−31.6412	−1.60427	−0.802136	+	0.597141i	$0.796303\pi$
$390$	0	0
$391$	−12.6058	−0.637503
$392$	−1.51056	−0.0762947
$393$	−5.46149	−0.275496
$394$	15.6269	0.787273
$395$	0	0
$396$	−18.4546	−0.927381
$397$	39.2961	1.97221	0.986106	−	0.166116i	$-0.0531225\pi$
$397$	39.2961	1.97221	0.986106	+	0.166116i	$0.0531225\pi$
$398$	−20.2835	−1.01672
$399$	0.482601	0.0241603
$400$	0	0
$401$	−9.66914	−0.482854	−0.241427	−	0.970419i	$-0.577615\pi$
$401$	−9.66914	−0.482854	−0.241427	+	0.970419i	$0.577615\pi$
$402$	2.20766	0.110108
$403$	−7.11926	−0.354636
$404$	−8.20766	−0.408346
$405$	0	0
$406$	−15.5317	−0.770824
$407$	6.51056	0.322716
$408$	1.02112	0.0505527
$409$	−26.2749	−1.29921	−0.649606	−	0.760271i	$-0.725066\pi$
$409$	−26.2749	−1.29921	−0.649606	+	0.760271i	$0.725066\pi$
$410$	0	0
$411$	−5.97495	−0.294722
$412$	8.81346	0.434208
$413$	4.16543	0.204967
$414$	−14.2327	−0.699500
$415$	0	0
$416$	−0.813457	−0.0398830
$417$	−7.06335	−0.345894
$418$	−2.64803	−0.129519
$419$	−30.1940	−1.47507	−0.737536	−	0.675308i	$-0.764011\pi$
$419$	−30.1940	−1.47507	−0.737536	+	0.675308i	$0.764011\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	−18.7182	−0.911188
$423$	−15.3856	−0.748074
$424$	−2.34513	−0.113890
$425$	0	0
$426$	5.95777	0.288655
$427$	3.86253	0.186921
$428$	−15.2624	−0.737737
$429$	−2.15406	−0.103999
$430$	0	0
$431$	25.5653	1.23144	0.615719	−	0.787966i	$-0.288866\pi$
$431$	25.5653	1.23144	0.615719	+	0.787966i	$0.288866\pi$
$432$	2.37309	0.114175
$433$	−32.0422	−1.53985	−0.769926	−	0.638134i	$-0.779707\pi$
$433$	−32.0422	−1.53985	−0.769926	+	0.638134i	$0.779707\pi$
$434$	−25.5317	−1.22556
$435$	0	0
$436$	4.30290	0.206072
$437$	−2.04223	−0.0976931
$438$	4.48260	0.214187
$439$	3.93840	0.187970	0.0939848	−	0.995574i	$-0.470039\pi$
$439$	3.93840	0.187970	0.0939848	+	0.995574i	$0.470039\pi$
$440$	0	0
$441$	−4.28178	−0.203894
$442$	−2.04223	−0.0971390
$443$	0.0758724	0.00360481	0.00180240	−	0.999998i	$-0.499426\pi$
$443$	0.0758724	0.00360481	0.00180240	+	0.999998i	$0.499426\pi$
$444$	−0.406728	−0.0193025
$445$	0	0
$446$	−15.7307	−0.744872
$447$	0.0844605	0.00399485
$448$	−2.91729	−0.137829
$449$	19.2961	0.910637	0.455319	−	0.890329i	$-0.349525\pi$
$449$	19.2961	0.910637	0.455319	+	0.890329i	$0.349525\pi$
$450$	0	0
$451$	41.3103	1.94523
$452$	9.15859	0.430784
$453$	−0.330856	−0.0155450
$454$	12.8836	0.604659
$455$	0	0
$456$	0.165428	0.00774688
$457$	−3.00684	−0.140654	−0.0703271	−	0.997524i	$-0.522404\pi$
$457$	−3.00684	−0.140654	−0.0703271	+	0.997524i	$0.522404\pi$
$458$	−5.25383	−0.245495
$459$	5.95777	0.278085
$460$	0	0
$461$	−0.633755	−0.0295169	−0.0147585	−	0.999891i	$-0.504698\pi$
$461$	−0.633755	−0.0295169	−0.0147585	+	0.999891i	$0.504698\pi$
$462$	−7.72506	−0.359402
$463$	−18.0422	−0.838494	−0.419247	−	0.907872i	$-0.637706\pi$
$463$	−18.0422	−0.838494	−0.419247	+	0.907872i	$0.637706\pi$
$464$	−5.32401	−0.247161
$465$	0	0
$466$	28.3172	1.31177
$467$	23.4758	1.08633	0.543164	−	0.839626i	$-0.317226\pi$
$467$	23.4758	1.08633	0.543164	+	0.839626i	$0.317226\pi$
$468$	−2.30580	−0.106586
$469$	−15.8346	−0.731173
$470$	0	0
$471$	2.16543	0.0997777
$472$	1.42784	0.0657218
$473$	47.6834	2.19249
$474$	−0.715313	−0.0328554
$475$	0	0
$476$	−7.32401	−0.335696
$477$	−6.64744	−0.304365
$478$	−14.3788	−0.657670
$479$	−8.19907	−0.374625	−0.187313	−	0.982300i	$-0.559978\pi$
$479$	−8.19907	−0.374625	−0.187313	+	0.982300i	$0.559978\pi$
$480$	0	0
$481$	0.813457	0.0370904
$482$	−18.2749	−0.832401
$483$	−5.95777	−0.271088
$484$	31.3874	1.42670
$485$	0	0
$486$	10.1854	0.462019
$487$	−0.953831	−0.0432222	−0.0216111	−	0.999766i	$-0.506880\pi$
$487$	−0.953831	−0.0432222	−0.0216111	+	0.999766i	$0.506880\pi$
$488$	1.32401	0.0599353
$489$	6.18260	0.279587
$490$	0	0
$491$	19.3383	0.872725	0.436362	−	0.899771i	$-0.356267\pi$
$491$	19.3383	0.872725	0.436362	+	0.899771i	$0.356267\pi$
$492$	−2.58074	−0.116349
$493$	−13.3662	−0.601985
$494$	−0.330856	−0.0148859
$495$	0	0
$496$	−8.75186	−0.392970
$497$	−42.7325	−1.91681
$498$	2.86937	0.128580
$499$	−5.63550	−0.252280	−0.126140	−	0.992012i	$-0.540259\pi$
$499$	−5.63550	−0.252280	−0.126140	+	0.992012i	$0.540259\pi$
$500$	0	0
$501$	−5.07703	−0.226825
$502$	1.22019	0.0544595
$503$	33.6269	1.49935	0.749675	−	0.661806i	$-0.230210\pi$
$503$	33.6269	1.49935	0.749675	+	0.661806i	$0.230210\pi$
$504$	−8.26926	−0.368342
$505$	0	0
$506$	32.6903	1.45326
$507$	5.01833	0.222872
$508$	8.61439	0.382202
$509$	−18.6903	−0.828431	−0.414216	−	0.910179i	$-0.635944\pi$
$509$	−18.6903	−0.828431	−0.414216	+	0.910179i	$0.635944\pi$
$510$	0	0
$511$	−32.1517	−1.42231
$512$	−1.00000	−0.0441942
$513$	0.965202	0.0426147
$514$	27.1306	1.19668
$515$	0	0
$516$	−2.97888	−0.131138
$517$	35.3383	1.55418
$518$	2.91729	0.128178
$519$	9.36915	0.411260
$520$	0	0
$521$	25.1586	1.10222	0.551109	−	0.834433i	$-0.314205\pi$
$521$	25.1586	1.10222	0.551109	+	0.834433i	$0.314205\pi$
$522$	−15.0913	−0.660528
$523$	7.25383	0.317188	0.158594	−	0.987344i	$-0.449304\pi$
$523$	7.25383	0.317188	0.158594	+	0.987344i	$0.449304\pi$
$524$	13.4278	0.586598
$525$	0	0
$526$	−20.2133	−0.881344
$527$	−21.9720	−0.957117
$528$	−2.64803	−0.115241
$529$	2.21160	0.0961564
$530$	0	0
$531$	4.04733	0.175639
$532$	−1.18654	−0.0514432
$533$	5.16149	0.223569
$534$	2.44037	0.105605
$535$	0	0
$536$	−5.42784	−0.234447
$537$	−9.87680	−0.426215
$538$	0.207658	0.00895278
$539$	9.83457	0.423605
$540$	0	0
$541$	16.0422	0.689709	0.344855	−	0.938656i	$-0.387928\pi$
$541$	16.0422	0.689709	0.344855	+	0.938656i	$0.387928\pi$
$542$	30.6480	1.31645
$543$	−0.880741	−0.0377962
$544$	−2.51056	−0.107639
$545$	0	0
$546$	−0.965202	−0.0413068
$547$	−16.0143	−0.684721	−0.342360	−	0.939569i	$-0.611226\pi$
$547$	−16.0143	−0.684721	−0.342360	+	0.939569i	$0.611226\pi$
$548$	14.6903	0.627537
$549$	3.75301	0.160175
$550$	0	0
$551$	−2.16543	−0.0922503
$552$	−2.04223	−0.0869231
$553$	5.13063	0.218177
$554$	−16.8135	−0.714335
$555$	0	0
$556$	17.3662	0.736493
$557$	10.8557	0.459970	0.229985	−	0.973194i	$-0.426132\pi$
$557$	10.8557	0.459970	0.229985	+	0.973194i	$0.426132\pi$
$558$	−24.8078	−1.05020
$559$	5.95777	0.251987
$560$	0	0
$561$	−6.64803	−0.280680
$562$	14.2749	0.602152
$563$	−31.0776	−1.30977	−0.654883	−	0.755731i	$-0.727282\pi$
$563$	−31.0776	−1.30977	−0.654883	+	0.755731i	$0.727282\pi$
$564$	−2.20766	−0.0929592
$565$	0	0
$566$	−13.2288	−0.556047
$567$	−21.9920	−0.923577
$568$	−14.6480	−0.614618
$569$	30.0845	1.26121	0.630603	−	0.776105i	$-0.282808\pi$
$569$	30.0845	1.26121	0.630603	+	0.776105i	$0.282808\pi$
$570$	0	0
$571$	5.55673	0.232542	0.116271	−	0.993218i	$-0.462906\pi$
$571$	5.55673	0.232542	0.116271	+	0.993218i	$0.462906\pi$
$572$	5.29606	0.221439
$573$	7.86312	0.328487
$574$	18.5106	0.772616
$575$	0	0
$576$	−2.83457	−0.118107
$577$	22.2499	0.926275	0.463137	−	0.886286i	$-0.346724\pi$
$577$	22.2499	0.926275	0.463137	+	0.886286i	$0.346724\pi$
$578$	10.6971	0.444941
$579$	−8.15174	−0.338775
$580$	0	0
$581$	−20.5807	−0.853833
$582$	−0.953831	−0.0395376
$583$	15.2681	0.632340
$584$	−11.0211	−0.456057
$585$	0	0
$586$	2.34513	0.0968764
$587$	−45.3103	−1.87016	−0.935079	−	0.354440i	$-0.884672\pi$
$587$	−45.3103	−1.87016	−0.935079	+	0.354440i	$0.884672\pi$
$588$	−0.614387	−0.0253369
$589$	−3.55963	−0.146672
$590$	0	0
$591$	6.35591	0.261447
$592$	1.00000	0.0410997
$593$	0.232712	0.00955635	0.00477817	−	0.999989i	$-0.498479\pi$
$593$	0.232712	0.00955635	0.00477817	+	0.999989i	$0.498479\pi$
$594$	−15.4501	−0.633926
$595$	0	0
$596$	−0.207658	−0.00850601
$597$	−8.24989	−0.337645
$598$	4.08446	0.167026
$599$	43.4056	1.77350	0.886752	−	0.462246i	$-0.152956\pi$
$599$	43.4056	1.77350	0.886752	+	0.462246i	$0.152956\pi$
$600$	0	0
$601$	35.5317	1.44937	0.724684	−	0.689082i	$-0.241986\pi$
$601$	35.5317	1.44937	0.724684	+	0.689082i	$0.241986\pi$
$602$	21.3662	0.870823
$603$	−15.3856	−0.626551
$604$	0.813457	0.0330991
$605$	0	0
$606$	−3.33829	−0.135609
$607$	−26.7325	−1.08504	−0.542519	−	0.840043i	$-0.682529\pi$
$607$	−26.7325	−1.08504	−0.542519	+	0.840043i	$0.682529\pi$
$608$	−0.406728	−0.0164950
$609$	−6.31717	−0.255985
$610$	0	0
$611$	4.41532	0.178625
$612$	−7.11636	−0.287662
$613$	−48.0565	−1.94098	−0.970492	−	0.241134i	$-0.922481\pi$
$613$	−48.0565	−1.94098	−0.970492	+	0.241134i	$0.922481\pi$
$614$	−23.2624	−0.938795
$615$	0	0
$616$	18.9932	0.765256
$617$	−12.9789	−0.522510	−0.261255	−	0.965270i	$-0.584136\pi$
$617$	−12.9789	−0.522510	−0.261255	+	0.965270i	$0.584136\pi$
$618$	3.58468	0.144197
$619$	25.3662	1.01956	0.509778	−	0.860306i	$-0.329728\pi$
$619$	25.3662	1.01956	0.509778	+	0.860306i	$0.329728\pi$
$620$	0	0
$621$	−11.9155	−0.478154
$622$	31.3999	1.25902
$623$	−17.5037	−0.701272
$624$	−0.330856	−0.0132448
$625$	0	0
$626$	−8.04223	−0.321432
$627$	−1.07703	−0.0430124
$628$	−5.32401	−0.212451
$629$	2.51056	0.100102
$630$	0	0
$631$	−31.6075	−1.25828	−0.629138	−	0.777293i	$-0.716592\pi$
$631$	−31.6075	−1.25828	−0.629138	+	0.777293i	$0.716592\pi$
$632$	1.75870	0.0699573
$633$	−7.61323	−0.302599
$634$	−12.3029	−0.488611
$635$	0	0
$636$	−0.953831	−0.0378219
$637$	1.22877	0.0486858
$638$	34.6623	1.37229
$639$	−41.5209	−1.64254
$640$	0	0
$641$	−23.8625	−0.942513	−0.471257	−	0.881996i	$-0.656199\pi$
$641$	−23.8625	−0.942513	−0.471257	+	0.881996i	$0.656199\pi$
$642$	−6.20766	−0.244997
$643$	−32.8277	−1.29460	−0.647300	−	0.762236i	$-0.724102\pi$
$643$	−32.8277	−1.29460	−0.647300	+	0.762236i	$0.724102\pi$
$644$	14.6480	0.577213
$645$	0	0
$646$	−1.02112	−0.0401752
$647$	13.9578	0.548737	0.274368	−	0.961625i	$-0.411531\pi$
$647$	13.9578	0.548737	0.274368	+	0.961625i	$0.411531\pi$
$648$	−7.53851	−0.296141
$649$	−9.29606	−0.364902
$650$	0	0
$651$	−10.3845	−0.406999
$652$	−15.2008	−0.595310
$653$	−28.5921	−1.11890	−0.559448	−	0.828865i	$-0.688987\pi$
$653$	−28.5921	−1.11890	−0.559448	+	0.828865i	$0.688987\pi$
$654$	1.75011	0.0684348
$655$	0	0
$656$	6.34513	0.247736
$657$	−31.2401	−1.21879
$658$	15.8346	0.617296
$659$	−5.62691	−0.219193	−0.109597	−	0.993976i	$-0.534956\pi$
$659$	−5.62691	−0.219193	−0.109597	+	0.993976i	$0.534956\pi$
$660$	0	0
$661$	13.3240	0.518244	0.259122	−	0.965845i	$-0.416567\pi$
$661$	13.3240	0.518244	0.259122	+	0.965845i	$0.416567\pi$
$662$	6.77981	0.263505
$663$	−0.830633	−0.0322591
$664$	−7.05476	−0.273778
$665$	0	0
$666$	2.83457	0.109837
$667$	26.7325	1.03509
$668$	12.4826	0.482966
$669$	−6.39814	−0.247366
$670$	0	0
$671$	−8.62007	−0.332774
$672$	−1.18654	−0.0457719
$673$	−29.6691	−1.14366	−0.571831	−	0.820372i	$-0.693767\pi$
$673$	−29.6691	−1.14366	−0.571831	+	0.820372i	$0.693767\pi$
$674$	10.6903	0.411773
$675$	0	0
$676$	−12.3383	−0.474550
$677$	1.25383	0.0481885	0.0240943	−	0.999710i	$-0.492330\pi$
$677$	1.25383	0.0481885	0.0240943	+	0.999710i	$0.492330\pi$
$678$	3.72506	0.143060
$679$	6.84141	0.262549
$680$	0	0
$681$	5.24014	0.200803
$682$	56.9795	2.18186
$683$	4.76045	0.182153	0.0910767	−	0.995844i	$-0.470969\pi$
$683$	4.76045	0.182153	0.0910767	+	0.995844i	$0.470969\pi$
$684$	−1.15290	−0.0440823
$685$	0	0
$686$	−16.0143	−0.611428
$687$	−2.13688	−0.0815271
$688$	7.32401	0.279225
$689$	1.90766	0.0726761
$690$	0	0
$691$	−20.2219	−0.769279	−0.384639	−	0.923067i	$-0.625674\pi$
$691$	−20.2219	−0.769279	−0.384639	+	0.923067i	$0.625674\pi$
$692$	−23.0354	−0.875674
$693$	53.8375	2.04512
$694$	5.62691	0.213595
$695$	0	0
$696$	−2.16543	−0.0820803
$697$	15.9298	0.603385
$698$	−15.1306	−0.572703
$699$	11.5174	0.435628
$700$	0	0
$701$	−22.0845	−0.834119	−0.417059	−	0.908879i	$-0.636939\pi$
$701$	−22.0845	−0.834119	−0.417059	+	0.908879i	$0.636939\pi$
$702$	−1.93040	−0.0728584
$703$	0.406728	0.0153401
$704$	6.51056	0.245376
$705$	0	0
$706$	35.7816	1.34666
$707$	23.9441	0.900510
$708$	0.580745	0.0218257
$709$	5.59896	0.210273	0.105137	−	0.994458i	$-0.466472\pi$
$709$	5.59896	0.210273	0.105137	+	0.994458i	$0.466472\pi$
$710$	0	0
$711$	4.98516	0.186958
$712$	−6.00000	−0.224860
$713$	43.9441	1.64572
$714$	−2.97888	−0.111482
$715$	0	0
$716$	24.2835	0.907518
$717$	−5.84826	−0.218407
$718$	−4.48260	−0.167289
$719$	40.2921	1.50264	0.751321	−	0.659937i	$-0.229417\pi$
$719$	40.2921	1.50264	0.751321	+	0.659937i	$0.229417\pi$
$720$	0	0
$721$	−25.7114	−0.957542
$722$	18.8346	0.700950
$723$	−7.43294	−0.276434
$724$	2.16543	0.0804775
$725$	0	0
$726$	12.7661	0.473796
$727$	−11.2538	−0.417381	−0.208691	−	0.977982i	$-0.566920\pi$
$727$	−11.2538	−0.417381	−0.208691	+	0.977982i	$0.566920\pi$
$728$	2.37309	0.0879524
$729$	−18.4729	−0.684180
$730$	0	0
$731$	18.3874	0.680081
$732$	0.538514	0.0199041
$733$	42.2892	1.56199	0.780994	−	0.624539i	$-0.214713\pi$
$733$	42.2892	1.56199	0.780994	+	0.624539i	$0.214713\pi$
$734$	−21.5653	−0.795990
$735$	0	0
$736$	5.02112	0.185081
$737$	35.3383	1.30170
$738$	17.9857	0.662064
$739$	−29.3103	−1.07820	−0.539099	−	0.842242i	$-0.681235\pi$
$739$	−29.3103	−1.07820	−0.539099	+	0.842242i	$0.681235\pi$
$740$	0	0
$741$	−0.134569	−0.00494350
$742$	6.84141	0.251156
$743$	4.39245	0.161144	0.0805718	−	0.996749i	$-0.474325\pi$
$743$	4.39245	0.161144	0.0805718	+	0.996749i	$0.474325\pi$
$744$	−3.55963	−0.130502
$745$	0	0
$746$	16.9230	0.619594
$747$	−19.9972	−0.731660
$748$	16.3451	0.597637
$749$	44.5248	1.62690
$750$	0	0
$751$	2.58074	0.0941727	0.0470864	−	0.998891i	$-0.485006\pi$
$751$	2.58074	0.0941727	0.0470864	+	0.998891i	$0.485006\pi$
$752$	5.42784	0.197933
$753$	0.496284	0.0180856
$754$	4.33086	0.157720
$755$	0	0
$756$	−6.92297	−0.251786
$757$	−27.3805	−0.995162	−0.497581	−	0.867418i	$-0.665778\pi$
$757$	−27.3805	−0.995162	−0.497581	+	0.867418i	$0.665778\pi$
$758$	−31.2710	−1.13581
$759$	13.2961	0.482616
$760$	0	0
$761$	−42.9509	−1.55697	−0.778485	−	0.627663i	$-0.784011\pi$
$761$	−42.9509	−1.55697	−0.778485	+	0.627663i	$0.784011\pi$
$762$	3.50372	0.126926
$763$	−12.5528	−0.454441
$764$	−19.3326	−0.699429
$765$	0	0
$766$	1.62691	0.0587828
$767$	−1.16149	−0.0419389
$768$	−0.406728	−0.0146765
$769$	−13.0633	−0.471076	−0.235538	−	0.971865i	$-0.575685\pi$
$769$	−13.0633	−0.471076	−0.235538	+	0.971865i	$0.575685\pi$
$770$	0	0
$771$	11.0348	0.397409
$772$	20.0422	0.721336
$773$	15.3662	0.552685	0.276343	−	0.961059i	$-0.410878\pi$
$773$	15.3662	0.552685	0.276343	+	0.961059i	$0.410878\pi$
$774$	20.7604	0.746219
$775$	0	0
$776$	2.34513	0.0841852
$777$	1.18654	0.0425670
$778$	31.6412	1.13439
$779$	2.58074	0.0924648
$780$	0	0
$781$	95.3668	3.41249
$782$	12.6058	0.450782
$783$	−12.6343	−0.451515
$784$	1.51056	0.0539485
$785$	0	0
$786$	5.46149	0.194805
$787$	−10.9316	−0.389668	−0.194834	−	0.980836i	$-0.562417\pi$
$787$	−10.9316	−0.389668	−0.194834	+	0.980836i	$0.562417\pi$
$788$	−15.6269	−0.556686
$789$	−8.22134	−0.292688
$790$	0	0
$791$	−26.7182	−0.949990
$792$	18.4546	0.655757
$793$	−1.07703	−0.0382464
$794$	−39.2961	−1.39456
$795$	0	0
$796$	20.2835	0.718931
$797$	24.2921	0.860471	0.430235	−	0.902717i	$-0.358431\pi$
$797$	24.2921	0.860471	0.430235	+	0.902717i	$0.358431\pi$
$798$	−0.482601	−0.0170839
$799$	13.6269	0.482086
$800$	0	0
$801$	−17.0074	−0.600928
$802$	9.66914	0.341429
$803$	71.7536	2.53213
$804$	−2.20766	−0.0778581
$805$	0	0
$806$	7.11926	0.250765
$807$	0.0844605	0.00297315
$808$	8.20766	0.288744
$809$	−40.0422	−1.40781	−0.703905	−	0.710294i	$-0.748562\pi$
$809$	−40.0422	−1.40781	−0.703905	+	0.710294i	$0.748562\pi$
$810$	0	0
$811$	−40.6343	−1.42686	−0.713432	−	0.700724i	$-0.752860\pi$
$811$	−40.6343	−1.42686	−0.713432	+	0.700724i	$0.752860\pi$
$812$	15.5317	0.545055
$813$	12.4654	0.437182
$814$	−6.51056	−0.228195
$815$	0	0
$816$	−1.02112	−0.0357462
$817$	2.97888	0.104218
$818$	26.2749	0.918682
$819$	6.72668	0.235049
$820$	0	0
$821$	−41.4787	−1.44762	−0.723808	−	0.690002i	$-0.757610\pi$
$821$	−41.4787	−1.44762	−0.723808	+	0.690002i	$0.757610\pi$
$822$	5.97495	0.208400
$823$	5.27610	0.183913	0.0919566	−	0.995763i	$-0.470688\pi$
$823$	5.27610	0.183913	0.0919566	+	0.995763i	$0.470688\pi$
$824$	−8.81346	−0.307031
$825$	0	0
$826$	−4.16543	−0.144934
$827$	−1.76438	−0.0613537	−0.0306768	−	0.999529i	$-0.509766\pi$
$827$	−1.76438	−0.0613537	−0.0306768	+	0.999529i	$0.509766\pi$
$828$	14.2327	0.494621
$829$	16.3029	0.566223	0.283112	−	0.959087i	$-0.408633\pi$
$829$	16.3029	0.566223	0.283112	+	0.959087i	$0.408633\pi$
$830$	0	0
$831$	−6.83851	−0.237225
$832$	0.813457	0.0282015
$833$	3.79234	0.131397
$834$	7.06335	0.244584
$835$	0	0
$836$	2.64803	0.0915840
$837$	−20.7689	−0.717879
$838$	30.1940	1.04303
$839$	14.3731	0.496214	0.248107	−	0.968733i	$-0.420192\pi$
$839$	14.3731	0.496214	0.248107	+	0.968733i	$0.420192\pi$
$840$	0	0
$841$	−0.654870	−0.0225817
$842$	−22.0000	−0.758170
$843$	5.80602	0.199970
$844$	18.7182	0.644308
$845$	0	0
$846$	15.3856	0.528968
$847$	−91.5659	−3.14624
$848$	2.34513	0.0805321
$849$	−5.38052	−0.184659
$850$	0	0
$851$	−5.02112	−0.172122
$852$	−5.95777	−0.204110
$853$	−43.2961	−1.48243	−0.741214	−	0.671268i	$-0.765750\pi$
$853$	−43.2961	−1.48243	−0.741214	+	0.671268i	$0.765750\pi$
$854$	−3.86253	−0.132173
$855$	0	0
$856$	15.2624	0.521659
$857$	49.0776	1.67646	0.838230	−	0.545317i	$-0.183591\pi$
$857$	49.0776	1.67646	0.838230	+	0.545317i	$0.183591\pi$
$858$	2.15406	0.0735383
$859$	15.5374	0.530128	0.265064	−	0.964231i	$-0.414607\pi$
$859$	15.5374	0.530128	0.265064	+	0.964231i	$0.414607\pi$
$860$	0	0
$861$	7.52877	0.256580
$862$	−25.5653	−0.870758
$863$	36.0901	1.22852	0.614261	−	0.789103i	$-0.289454\pi$
$863$	36.0901	1.22852	0.614261	+	0.789103i	$0.289454\pi$
$864$	−2.37309	−0.0807340
$865$	0	0
$866$	32.0422	1.08884
$867$	4.35081	0.147761
$868$	25.5317	0.866601
$869$	−11.4501	−0.388419
$870$	0	0
$871$	4.41532	0.149607
$872$	−4.30290	−0.145715
$873$	6.64744	0.224982
$874$	2.04223	0.0690795
$875$	0	0
$876$	−4.48260	−0.151453
$877$	−29.3240	−0.990202	−0.495101	−	0.868836i	$-0.664869\pi$
$877$	−29.3240	−0.990202	−0.495101	+	0.868836i	$0.664869\pi$
$878$	−3.93840	−0.132915
$879$	0.953831	0.0321719
$880$	0	0
$881$	23.8066	0.802065	0.401033	−	0.916064i	$-0.368651\pi$
$881$	23.8066	0.802065	0.401033	+	0.916064i	$0.368651\pi$
$882$	4.28178	0.144175
$883$	3.94699	0.132827	0.0664134	−	0.997792i	$-0.478844\pi$
$883$	3.94699	0.132827	0.0664134	+	0.997792i	$0.478844\pi$
$884$	2.04223	0.0686876
$885$	0	0
$886$	−0.0758724	−0.00254898
$887$	−1.96346	−0.0659264	−0.0329632	−	0.999457i	$-0.510494\pi$
$887$	−1.96346	−0.0659264	−0.0329632	+	0.999457i	$0.510494\pi$
$888$	0.406728	0.0136489
$889$	−25.1306	−0.842854
$890$	0	0
$891$	49.0799	1.64424
$892$	15.7307	0.526704
$893$	2.20766	0.0738765
$894$	−0.0844605	−0.00282478
$895$	0	0
$896$	2.91729	0.0974597
$897$	1.66127	0.0554681
$898$	−19.2961	−0.643918
$899$	46.5950	1.55403
$900$	0	0
$901$	5.88758	0.196144
$902$	−41.3103	−1.37548
$903$	8.69026	0.289194
$904$	−9.15859	−0.304610
$905$	0	0
$906$	0.330856	0.0109920
$907$	−5.89049	−0.195590	−0.0977952	−	0.995207i	$-0.531179\pi$
$907$	−5.89049	−0.195590	−0.0977952	+	0.995207i	$0.531179\pi$
$908$	−12.8836	−0.427559
$909$	23.2652	0.771658
$910$	0	0
$911$	−27.9527	−0.926113	−0.463057	−	0.886329i	$-0.653247\pi$
$911$	−27.9527	−0.926113	−0.463057	+	0.886329i	$0.653247\pi$
$912$	−0.165428	−0.00547787
$913$	45.9304	1.52007
$914$	3.00684	0.0994575
$915$	0	0
$916$	5.25383	0.173591
$917$	−39.1729	−1.29360
$918$	−5.95777	−0.196636
$919$	17.7587	0.585805	0.292903	−	0.956142i	$-0.405379\pi$
$919$	17.7587	0.585805	0.292903	+	0.956142i	$0.405379\pi$
$920$	0	0
$921$	−9.46149	−0.311767
$922$	0.633755	0.0208716
$923$	11.9155	0.392205
$924$	7.72506	0.254136
$925$	0	0
$926$	18.0422	0.592904
$927$	−24.9824	−0.820529
$928$	5.32401	0.174769
$929$	34.2892	1.12499	0.562496	−	0.826800i	$-0.309841\pi$
$929$	34.2892	1.12499	0.562496	+	0.826800i	$0.309841\pi$
$930$	0	0
$931$	0.614387	0.0201357
$932$	−28.3172	−0.927560
$933$	12.7712	0.418111
$934$	−23.4758	−0.768150
$935$	0	0
$936$	2.30580	0.0753675
$937$	14.0845	0.460119	0.230060	−	0.973177i	$-0.426108\pi$
$937$	14.0845	0.460119	0.230060	+	0.973177i	$0.426108\pi$
$938$	15.8346	0.517017
$939$	−3.27100	−0.106745
$940$	0	0
$941$	14.2499	0.464533	0.232267	−	0.972652i	$-0.425386\pi$
$941$	14.2499	0.464533	0.232267	+	0.972652i	$0.425386\pi$
$942$	−2.16543	−0.0705535
$943$	−31.8596	−1.03749
$944$	−1.42784	−0.0464723
$945$	0	0
$946$	−47.6834	−1.55032
$947$	29.8203	0.969029	0.484515	−	0.874783i	$-0.338996\pi$
$947$	29.8203	0.969029	0.484515	+	0.874783i	$0.338996\pi$
$948$	0.715313	0.0232323
$949$	8.96520	0.291023
$950$	0	0
$951$	−5.00394	−0.162264
$952$	7.32401	0.237373
$953$	−8.64803	−0.280137	−0.140069	−	0.990142i	$-0.544732\pi$
$953$	−8.64803	−0.280137	−0.140069	+	0.990142i	$0.544732\pi$
$954$	6.64744	0.215219
$955$	0	0
$956$	14.3788	0.465043
$957$	14.0981	0.455728
$958$	8.19907	0.264900
$959$	−42.8557	−1.38388
$960$	0	0
$961$	45.5950	1.47081
$962$	−0.813457	−0.0262269
$963$	43.2624	1.39411
$964$	18.2749	0.588596
$965$	0	0
$966$	5.95777	0.191688
$967$	45.0325	1.44815	0.724074	−	0.689723i	$-0.242268\pi$
$967$	45.0325	1.44815	0.724074	+	0.689723i	$0.242268\pi$
$968$	−31.3874	−1.00883
$969$	−0.415317	−0.0133419
$970$	0	0
$971$	26.3029	0.844100	0.422050	−	0.906573i	$-0.361311\pi$
$971$	26.3029	0.844100	0.422050	+	0.906573i	$0.361311\pi$
$972$	−10.1854	−0.326696
$973$	−50.6623	−1.62416
$974$	0.953831	0.0305627
$975$	0	0
$976$	−1.32401	−0.0423807
$977$	−25.6549	−0.820772	−0.410386	−	0.911912i	$-0.634606\pi$
$977$	−25.6549	−0.820772	−0.410386	+	0.911912i	$0.634606\pi$
$978$	−6.18260	−0.197698
$979$	39.0633	1.24847
$980$	0	0
$981$	−12.1969	−0.389416
$982$	−19.3383	−0.617110
$983$	55.3611	1.76575	0.882873	−	0.469611i	$-0.155606\pi$
$983$	55.3611	1.76575	0.882873	+	0.469611i	$0.155606\pi$
$984$	2.58074	0.0822711
$985$	0	0
$986$	13.3662	0.425668
$987$	6.44037	0.204999
$988$	0.330856	0.0105259
$989$	−36.7747	−1.16937
$990$	0	0
$991$	−3.24814	−0.103181	−0.0515903	−	0.998668i	$-0.516429\pi$
$991$	−3.24814	−0.103181	−0.0515903	+	0.998668i	$0.516429\pi$
$992$	8.75186	0.277872
$993$	2.75754	0.0875080
$994$	42.7325	1.35539
$995$	0	0
$996$	−2.86937	−0.0909195
$997$	12.6480	0.400567	0.200284	−	0.979738i	$-0.435814\pi$
$997$	12.6480	0.400567	0.200284	+	0.979738i	$0.435814\pi$
$998$	5.63550	0.178389
$999$	2.37309	0.0750811

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1850.2.a.z.1.2		3
5.2	odd	4		1850.2.b.o.149.2		6
5.3	odd	4		1850.2.b.o.149.5		6
5.4	even	2		370.2.a.g.1.2	✓	3
15.14	odd	2		3330.2.a.bg.1.3		3
20.19	odd	2		2960.2.a.u.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
370.2.a.g.1.2	✓	3	5.4	even	2
1850.2.a.z.1.2		3	1.1	even	1	trivial
1850.2.b.o.149.2		6	5.2	odd	4
1850.2.b.o.149.5		6	5.3	odd	4
2960.2.a.u.1.2		3	20.19	odd	2
3330.2.a.bg.1.3		3	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1850.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -0.406728 q^{3} +1.00000 q^{4} +0.406728 q^{6} -2.91729 q^{7} -1.00000 q^{8} -2.83457 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -0.406728 q^{3} +1.00000 q^{4} +0.406728 q^{6} -2.91729 q^{7} -1.00000 q^{8} -2.83457 q^{9} +6.51056 q^{11} -0.406728 q^{12} +0.813457 q^{13} +2.91729 q^{14} +1.00000 q^{16} +2.51056 q^{17} +2.83457 q^{18} +0.406728 q^{19} +1.18654 q^{21} -6.51056 q^{22} -5.02112 q^{23} +0.406728 q^{24} -0.813457 q^{26} +2.37309 q^{27} -2.91729 q^{28} -5.32401 q^{29} -8.75186 q^{31} -1.00000 q^{32} -2.64803 q^{33} -2.51056 q^{34} -2.83457 q^{36} +1.00000 q^{37} -0.406728 q^{38} -0.330856 q^{39} +6.34513 q^{41} -1.18654 q^{42} +7.32401 q^{43} +6.51056 q^{44} +5.02112 q^{46} +5.42784 q^{47} -0.406728 q^{48} +1.51056 q^{49} -1.02112 q^{51} +0.813457 q^{52} +2.34513 q^{53} -2.37309 q^{54} +2.91729 q^{56} -0.165428 q^{57} +5.32401 q^{58} -1.42784 q^{59} -1.32401 q^{61} +8.75186 q^{62} +8.26926 q^{63} +1.00000 q^{64} +2.64803 q^{66} +5.42784 q^{67} +2.51056 q^{68} +2.04223 q^{69} +14.6480 q^{71} +2.83457 q^{72} +11.0211 q^{73} -1.00000 q^{74} +0.406728 q^{76} -18.9932 q^{77} +0.330856 q^{78} -1.75870 q^{79} +7.53851 q^{81} -6.34513 q^{82} +7.05476 q^{83} +1.18654 q^{84} -7.32401 q^{86} +2.16543 q^{87} -6.51056 q^{88} +6.00000 q^{89} -2.37309 q^{91} -5.02112 q^{92} +3.55963 q^{93} -5.42784 q^{94} +0.406728 q^{96} -2.34513 q^{97} -1.51056 q^{98} -18.4546 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{4} + q^{7} - 3 q^{8} + 11 q^{9}+O(q^{10}) \) 3 * q - 3 * q^2 + 3 * q^4 + q^7 - 3 * q^8 + 11 * q^9	\( 3 q - 3 q^{2} + 3 q^{4} + q^{7} - 3 q^{8} + 11 q^{9} + 11 q^{11} - q^{14} + 3 q^{16} - q^{17} - 11 q^{18} + 6 q^{21} - 11 q^{22} + 2 q^{23} + 12 q^{27} + q^{28} - 5 q^{29} + 3 q^{31} - 3 q^{32} + 14 q^{33} + q^{34} + 11 q^{36} + 3 q^{37} - 40 q^{39} - 9 q^{41} - 6 q^{42} + 11 q^{43} + 11 q^{44} - 2 q^{46} - 2 q^{47} - 4 q^{49} + 14 q^{51} - 21 q^{53} - 12 q^{54} - q^{56} - 20 q^{57} + 5 q^{58} + 14 q^{59} + 7 q^{61} - 3 q^{62} + 37 q^{63} + 3 q^{64} - 14 q^{66} - 2 q^{67} - q^{68} - 28 q^{69} + 22 q^{71} - 11 q^{72} + 16 q^{73} - 3 q^{74} - 7 q^{77} + 40 q^{78} - 26 q^{79} + 47 q^{81} + 9 q^{82} - 2 q^{83} + 6 q^{84} - 11 q^{86} + 26 q^{87} - 11 q^{88} + 18 q^{89} - 12 q^{91} + 2 q^{92} + 18 q^{93} + 2 q^{94} + 21 q^{97} + 4 q^{98} + 19 q^{99}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^4 + q^7 - 3 * q^8 + 11 * q^9 + 11 * q^11 - q^14 + 3 * q^16 - q^17 - 11 * q^18 + 6 * q^21 - 11 * q^22 + 2 * q^23 + 12 * q^27 + q^28 - 5 * q^29 + 3 * q^31 - 3 * q^32 + 14 * q^33 + q^34 + 11 * q^36 + 3 * q^37 - 40 * q^39 - 9 * q^41 - 6 * q^42 + 11 * q^43 + 11 * q^44 - 2 * q^46 - 2 * q^47 - 4 * q^49 + 14 * q^51 - 21 * q^53 - 12 * q^54 - q^56 - 20 * q^57 + 5 * q^58 + 14 * q^59 + 7 * q^61 - 3 * q^62 + 37 * q^63 + 3 * q^64 - 14 * q^66 - 2 * q^67 - q^68 - 28 * q^69 + 22 * q^71 - 11 * q^72 + 16 * q^73 - 3 * q^74 - 7 * q^77 + 40 * q^78 - 26 * q^79 + 47 * q^81 + 9 * q^82 - 2 * q^83 + 6 * q^84 - 11 * q^86 + 26 * q^87 - 11 * q^88 + 18 * q^89 - 12 * q^91 + 2 * q^92 + 18 * q^93 + 2 * q^94 + 21 * q^97 + 4 * q^98 + 19 * q^99

Introduction

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