LMFDB - Embedded newform 1850.2.a.bc.1.3

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:	3.3.1524.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 7x + 1 $ x^3 - x^2 - 7*x + 1
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			1.3
Root			$3.13264$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +3.13264 q^{3} +1.00000 q^{4} +3.13264 q^{6} +4.13264 q^{7} +1.00000 q^{8} +6.81342 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +3.13264 q^{3} +1.00000 q^{4} +3.13264 q^{6} +4.13264 q^{7} +1.00000 q^{8} +6.81342 q^{9} -3.68078 q^{11} +3.13264 q^{12} -0.132637 q^{13} +4.13264 q^{14} +1.00000 q^{16} -5.13264 q^{17} +6.81342 q^{18} +0.451857 q^{19} +12.9461 q^{21} -3.68078 q^{22} -4.26527 q^{23} +3.13264 q^{24} -0.132637 q^{26} +11.9461 q^{27} +4.13264 q^{28} -10.2653 q^{29} +5.58449 q^{31} +1.00000 q^{32} -11.5305 q^{33} -5.13264 q^{34} +6.81342 q^{36} +1.00000 q^{37} +0.451857 q^{38} -0.415505 q^{39} +10.3616 q^{41} +12.9461 q^{42} -5.07869 q^{43} -3.68078 q^{44} -4.26527 q^{46} +3.13264 q^{48} +10.0787 q^{49} -16.0787 q^{51} -0.132637 q^{52} -11.6268 q^{53} +11.9461 q^{54} +4.13264 q^{56} +1.41551 q^{57} -10.2653 q^{58} -13.0787 q^{59} -4.39791 q^{61} +5.58449 q^{62} +28.1574 q^{63} +1.00000 q^{64} -11.5305 q^{66} -0.228923 q^{67} -5.13264 q^{68} -13.3616 q^{69} +9.49420 q^{71} +6.81342 q^{72} +3.54814 q^{73} +1.00000 q^{74} +0.451857 q^{76} -15.2113 q^{77} -0.415505 q^{78} -0.903715 q^{79} +16.9824 q^{81} +10.3616 q^{82} -13.9461 q^{83} +12.9461 q^{84} -5.07869 q^{86} -32.1574 q^{87} -3.68078 q^{88} +0.777066 q^{89} -0.548143 q^{91} -4.26527 q^{92} +17.4942 q^{93} +3.13264 q^{96} +0.638440 q^{97} +10.0787 q^{98} -25.0787 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + q^{3} + 3 q^{4} + q^{6} + 4 q^{7} + 3 q^{8} + 6 q^{9}+O(q^{10}) $ 3 * q + 3 * q^2 + q^3 + 3 * q^4 + q^6 + 4 * q^7 + 3 * q^8 + 6 * q^9	$ 3 q + 3 q^{2} + q^{3} + 3 q^{4} + q^{6} + 4 q^{7} + 3 q^{8} + 6 q^{9} - 5 q^{11} + q^{12} + 8 q^{13} + 4 q^{14} + 3 q^{16} - 7 q^{17} + 6 q^{18} - q^{19} + 16 q^{21} - 5 q^{22} + 4 q^{23} + q^{24} + 8 q^{26} + 13 q^{27} + 4 q^{28} - 14 q^{29} + 6 q^{31} + 3 q^{32} - q^{33} - 7 q^{34} + 6 q^{36} + 3 q^{37} - q^{38} - 12 q^{39} + 19 q^{41} + 16 q^{42} + 16 q^{43} - 5 q^{44} + 4 q^{46} + q^{48} - q^{49} - 17 q^{51} + 8 q^{52} - 6 q^{53} + 13 q^{54} + 4 q^{56} + 15 q^{57} - 14 q^{58} - 8 q^{59} + 12 q^{61} + 6 q^{62} + 22 q^{63} + 3 q^{64} - q^{66} + 3 q^{67} - 7 q^{68} - 28 q^{69} + 8 q^{71} + 6 q^{72} + 13 q^{73} + 3 q^{74} - q^{76} - 6 q^{77} - 12 q^{78} + 2 q^{79} + 15 q^{81} + 19 q^{82} - 19 q^{83} + 16 q^{84} + 16 q^{86} - 34 q^{87} - 5 q^{88} + q^{89} - 4 q^{91} + 4 q^{92} + 32 q^{93} + q^{96} + 14 q^{97} - q^{98} - 44 q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 + q^3 + 3 * q^4 + q^6 + 4 * q^7 + 3 * q^8 + 6 * q^9 - 5 * q^11 + q^12 + 8 * q^13 + 4 * q^14 + 3 * q^16 - 7 * q^17 + 6 * q^18 - q^19 + 16 * q^21 - 5 * q^22 + 4 * q^23 + q^24 + 8 * q^26 + 13 * q^27 + 4 * q^28 - 14 * q^29 + 6 * q^31 + 3 * q^32 - q^33 - 7 * q^34 + 6 * q^36 + 3 * q^37 - q^38 - 12 * q^39 + 19 * q^41 + 16 * q^42 + 16 * q^43 - 5 * q^44 + 4 * q^46 + q^48 - q^49 - 17 * q^51 + 8 * q^52 - 6 * q^53 + 13 * q^54 + 4 * q^56 + 15 * q^57 - 14 * q^58 - 8 * q^59 + 12 * q^61 + 6 * q^62 + 22 * q^63 + 3 * q^64 - q^66 + 3 * q^67 - 7 * q^68 - 28 * q^69 + 8 * q^71 + 6 * q^72 + 13 * q^73 + 3 * q^74 - q^76 - 6 * q^77 - 12 * q^78 + 2 * q^79 + 15 * q^81 + 19 * q^82 - 19 * q^83 + 16 * q^84 + 16 * q^86 - 34 * q^87 - 5 * q^88 + q^89 - 4 * q^91 + 4 * q^92 + 32 * q^93 + q^96 + 14 * q^97 - q^98 - 44 * 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
$3$	3.13264	1.80863	0.904315	−	0.426867i	$-0.140383\pi$
$3$	3.13264	1.80863	0.904315	+	0.426867i	$0.140383\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	3.13264	1.27889
$7$	4.13264	1.56199	0.780995	−	0.624537i	$-0.214712\pi$
$7$	4.13264	1.56199	0.780995	+	0.624537i	$0.214712\pi$
$8$	1.00000	0.353553
$9$	6.81342	2.27114
$10$	0	0
$11$	−3.68078	−1.10980	−0.554898	−	0.831918i	$-0.687243\pi$
$11$	−3.68078	−1.10980	−0.554898	+	0.831918i	$0.687243\pi$
$12$	3.13264	0.904315
$13$	−0.132637	−0.0367870	−0.0183935	−	0.999831i	$-0.505855\pi$
$13$	−0.132637	−0.0367870	−0.0183935	+	0.999831i	$0.505855\pi$
$14$	4.13264	1.10449
$15$	0	0
$16$	1.00000	0.250000
$17$	−5.13264	−1.24485	−0.622424	−	0.782680i	$-0.713852\pi$
$17$	−5.13264	−1.24485	−0.622424	+	0.782680i	$0.713852\pi$
$18$	6.81342	1.60594
$19$	0.451857	0.103663	0.0518316	−	0.998656i	$-0.483494\pi$
$19$	0.451857	0.103663	0.0518316	+	0.998656i	$0.483494\pi$
$20$	0	0
$21$	12.9461	2.82506
$22$	−3.68078	−0.784745
$23$	−4.26527	−0.889371	−0.444686	−	0.895687i	$-0.646685\pi$
$23$	−4.26527	−0.889371	−0.444686	+	0.895687i	$0.646685\pi$
$24$	3.13264	0.639447
$25$	0	0
$26$	−0.132637	−0.0260124
$27$	11.9461	2.29902
$28$	4.13264	0.780995
$29$	−10.2653	−1.90621	−0.953107	−	0.302634i	$-0.902134\pi$
$29$	−10.2653	−1.90621	−0.953107	+	0.302634i	$0.902134\pi$
$30$	0	0
$31$	5.58449	1.00300	0.501502	−	0.865156i	$-0.332781\pi$
$31$	5.58449	1.00300	0.501502	+	0.865156i	$0.332781\pi$
$32$	1.00000	0.176777
$33$	−11.5305	−2.00721
$34$	−5.13264	−0.880240
$35$	0	0
$36$	6.81342	1.13557
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0.451857	0.0733009
$39$	−0.415505	−0.0665341
$40$	0	0
$41$	10.3616	1.61820	0.809102	−	0.587668i	$-0.199954\pi$
$41$	10.3616	1.61820	0.809102	+	0.587668i	$0.199954\pi$
$42$	12.9461	1.99762
$43$	−5.07869	−0.774493	−0.387247	−	0.921976i	$-0.626574\pi$
$43$	−5.07869	−0.774493	−0.387247	+	0.921976i	$0.626574\pi$
$44$	−3.68078	−0.554898
$45$	0	0
$46$	−4.26527	−0.628880
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	3.13264	0.452157
$49$	10.0787	1.43981
$50$	0	0
$51$	−16.0787	−2.25147
$52$	−0.132637	−0.0183935
$53$	−11.6268	−1.59707	−0.798534	−	0.601949i	$-0.794391\pi$
$53$	−11.6268	−1.59707	−0.798534	+	0.601949i	$0.794391\pi$
$54$	11.9461	1.62565
$55$	0	0
$56$	4.13264	0.552247
$57$	1.41551	0.187488
$58$	−10.2653	−1.34790
$59$	−13.0787	−1.70270	−0.851350	−	0.524597i	$-0.824216\pi$
$59$	−13.0787	−1.70270	−0.851350	+	0.524597i	$0.824216\pi$
$60$	0	0
$61$	−4.39791	−0.563095	−0.281547	−	0.959547i	$-0.590848\pi$
$61$	−4.39791	−0.563095	−0.281547	+	0.959547i	$0.590848\pi$
$62$	5.58449	0.709232
$63$	28.1574	3.54750
$64$	1.00000	0.125000
$65$	0	0
$66$	−11.5305	−1.41931
$67$	−0.228923	−0.0279674	−0.0139837	−	0.999902i	$-0.504451\pi$
$67$	−0.228923	−0.0279674	−0.0139837	+	0.999902i	$0.504451\pi$
$68$	−5.13264	−0.622424
$69$	−13.3616	−1.60854
$70$	0	0
$71$	9.49420	1.12675	0.563377	−	0.826200i	$-0.309502\pi$
$71$	9.49420	1.12675	0.563377	+	0.826200i	$0.309502\pi$
$72$	6.81342	0.802969
$73$	3.54814	0.415279	0.207639	−	0.978205i	$-0.433422\pi$
$73$	3.54814	0.415279	0.207639	+	0.978205i	$0.433422\pi$
$74$	1.00000	0.116248
$75$	0	0
$76$	0.451857	0.0518316
$77$	−15.2113	−1.73349
$78$	−0.415505	−0.0470467
$79$	−0.903715	−0.101676	−0.0508379	−	0.998707i	$-0.516189\pi$
$79$	−0.903715	−0.101676	−0.0508379	+	0.998707i	$0.516189\pi$
$80$	0	0
$81$	16.9824	1.88693
$82$	10.3616	1.14424
$83$	−13.9461	−1.53078	−0.765389	−	0.643568i	$-0.777454\pi$
$83$	−13.9461	−1.53078	−0.765389	+	0.643568i	$0.777454\pi$
$84$	12.9461	1.41253
$85$	0	0
$86$	−5.07869	−0.547650
$87$	−32.1574	−3.44763
$88$	−3.68078	−0.392372
$89$	0.777066	0.0823688	0.0411844	−	0.999152i	$-0.486887\pi$
$89$	0.777066	0.0823688	0.0411844	+	0.999152i	$0.486887\pi$
$90$	0	0
$91$	−0.548143	−0.0574610
$92$	−4.26527	−0.444686
$93$	17.4942	1.81406
$94$	0	0
$95$	0	0
$96$	3.13264	0.319723
$97$	0.638440	0.0648237	0.0324119	−	0.999475i	$-0.489681\pi$
$97$	0.638440	0.0648237	0.0324119	+	0.999475i	$0.489681\pi$
$98$	10.0787	1.01810
$99$	−25.0787	−2.52050
$100$	0	0
$101$	−5.62684	−0.559891	−0.279946	−	0.960016i	$-0.590316\pi$
$101$	−5.62684	−0.559891	−0.279946	+	0.960016i	$0.590316\pi$
$102$	−16.0787	−1.59203
$103$	10.0423	0.989501	0.494751	−	0.869035i	$-0.335259\pi$
$103$	10.0423	0.989501	0.494751	+	0.869035i	$0.335259\pi$
$104$	−0.132637	−0.0130062
$105$	0	0
$106$	−11.6268	−1.12930
$107$	19.6632	1.90091	0.950456	−	0.310859i	$-0.100617\pi$
$107$	19.6632	1.90091	0.950456	+	0.310859i	$0.100617\pi$
$108$	11.9461	1.14951
$109$	13.6268	1.30521	0.652607	−	0.757697i	$-0.273675\pi$
$109$	13.6268	1.30521	0.652607	+	0.757697i	$0.273675\pi$
$110$	0	0
$111$	3.13264	0.297337
$112$	4.13264	0.390498
$113$	7.94606	0.747502	0.373751	−	0.927529i	$-0.378071\pi$
$113$	7.94606	0.747502	0.373751	+	0.927529i	$0.378071\pi$
$114$	1.41551	0.132574
$115$	0	0
$116$	−10.2653	−0.953107
$117$	−0.903715	−0.0835484
$118$	−13.0787	−1.20399
$119$	−21.2113	−1.94444
$120$	0	0
$121$	2.54814	0.231649
$122$	−4.39791	−0.398168
$123$	32.4590	2.92673
$124$	5.58449	0.501502
$125$	0	0
$126$	28.1574	2.50846
$127$	17.4942	1.55236	0.776180	−	0.630512i	$-0.217155\pi$
$127$	17.4942	1.55236	0.776180	+	0.630512i	$0.217155\pi$
$128$	1.00000	0.0883883
$129$	−15.9097	−1.40077
$130$	0	0
$131$	8.81342	0.770032	0.385016	−	0.922910i	$-0.374196\pi$
$131$	8.81342	0.770032	0.385016	+	0.922910i	$0.374196\pi$
$132$	−11.5305	−1.00361
$133$	1.86736	0.161921
$134$	−0.228923	−0.0197759
$135$	0	0
$136$	−5.13264	−0.440120
$137$	18.8921	1.61406	0.807031	−	0.590509i	$-0.201073\pi$
$137$	18.8921	1.61406	0.807031	+	0.590509i	$0.201073\pi$
$138$	−13.3616	−1.13741
$139$	−14.8498	−1.25954	−0.629771	−	0.776781i	$-0.716851\pi$
$139$	−14.8498	−1.25954	−0.629771	+	0.776781i	$0.716851\pi$
$140$	0	0
$141$	0	0
$142$	9.49420	0.796735
$143$	0.488209	0.0408261
$144$	6.81342	0.567785
$145$	0	0
$146$	3.54814	0.293646
$147$	31.5729	2.60409
$148$	1.00000	0.0821995
$149$	−2.41551	−0.197886	−0.0989429	−	0.995093i	$-0.531546\pi$
$149$	−2.41551	−0.197886	−0.0989429	+	0.995093i	$0.531546\pi$
$150$	0	0
$151$	10.7231	0.872635	0.436318	−	0.899793i	$-0.356282\pi$
$151$	10.7231	0.872635	0.436318	+	0.899793i	$0.356282\pi$
$152$	0.451857	0.0366505
$153$	−34.9708	−2.82722
$154$	−15.2113	−1.22576
$155$	0	0
$156$	−0.415505	−0.0332670
$157$	−3.31922	−0.264903	−0.132451	−	0.991190i	$-0.542285\pi$
$157$	−3.31922	−0.264903	−0.132451	+	0.991190i	$0.542285\pi$
$158$	−0.903715	−0.0718957
$159$	−36.4227	−2.88850
$160$	0	0
$161$	−17.6268	−1.38919
$162$	16.9824	1.33426
$163$	15.5481	1.21782	0.608912	−	0.793238i	$-0.291606\pi$
$163$	15.5481	1.21782	0.608912	+	0.793238i	$0.291606\pi$
$164$	10.3616	0.809102
$165$	0	0
$166$	−13.9461	−1.08242
$167$	−3.77707	−0.292278	−0.146139	−	0.989264i	$-0.546685\pi$
$167$	−3.77707	−0.292278	−0.146139	+	0.989264i	$0.546685\pi$
$168$	12.9461	0.998810
$169$	−12.9824	−0.998647
$170$	0	0
$171$	3.07869	0.235434
$172$	−5.07869	−0.387247
$173$	20.1574	1.53254	0.766269	−	0.642520i	$-0.222111\pi$
$173$	20.1574	1.53254	0.766269	+	0.642520i	$0.222111\pi$
$174$	−32.1574	−2.43785
$175$	0	0
$176$	−3.68078	−0.277449
$177$	−40.9708	−3.07955
$178$	0.777066	0.0582435
$179$	20.2537	1.51383	0.756915	−	0.653513i	$-0.226706\pi$
$179$	20.2537	1.51383	0.756915	+	0.653513i	$0.226706\pi$
$180$	0	0
$181$	−17.1690	−1.27616	−0.638080	−	0.769970i	$-0.720271\pi$
$181$	−17.1690	−1.27616	−0.638080	+	0.769970i	$0.720271\pi$
$182$	−0.548143	−0.0406310
$183$	−13.7771	−1.01843
$184$	−4.26527	−0.314440
$185$	0	0
$186$	17.4942	1.28274
$187$	18.8921	1.38153
$188$	0	0
$189$	49.3687	3.59105
$190$	0	0
$191$	−0.192571	−0.0139339	−0.00696696	−	0.999976i	$-0.502218\pi$
$191$	−0.192571	−0.0139339	−0.00696696	+	0.999976i	$0.502218\pi$
$192$	3.13264	0.226079
$193$	−14.6692	−1.05591	−0.527955	−	0.849272i	$-0.677041\pi$
$193$	−14.6692	−1.05591	−0.527955	+	0.849272i	$0.677041\pi$
$194$	0.638440	0.0458373
$195$	0	0
$196$	10.0787	0.719907
$197$	−7.84977	−0.559273	−0.279636	−	0.960106i	$-0.590214\pi$
$197$	−7.84977	−0.559273	−0.279636	+	0.960106i	$0.590214\pi$
$198$	−25.0787	−1.78227
$199$	−11.3616	−0.805400	−0.402700	−	0.915332i	$-0.631928\pi$
$199$	−11.3616	−0.805400	−0.402700	+	0.915332i	$0.631928\pi$
$200$	0	0
$201$	−0.717132	−0.0505826
$202$	−5.62684	−0.395903
$203$	−42.4227	−2.97749
$204$	−16.0787	−1.12573
$205$	0	0
$206$	10.0423	0.699683
$207$	−29.0611	−2.01989
$208$	−0.132637	−0.00919676
$209$	−1.66319	−0.115045
$210$	0	0
$211$	−12.2289	−0.841874	−0.420937	−	0.907090i	$-0.638299\pi$
$211$	−12.2289	−0.841874	−0.420937	+	0.907090i	$0.638299\pi$
$212$	−11.6268	−0.798534
$213$	29.7419	2.03788
$214$	19.6632	1.34415
$215$	0	0
$216$	11.9461	0.812826
$217$	23.0787	1.56668
$218$	13.6268	0.922926
$219$	11.1150	0.751085
$220$	0	0
$221$	0.680780	0.0457942
$222$	3.13264	0.210249
$223$	28.1574	1.88556	0.942779	−	0.333418i	$-0.108202\pi$
$223$	28.1574	1.88556	0.942779	+	0.333418i	$0.108202\pi$
$224$	4.13264	0.276123
$225$	0	0
$226$	7.94606	0.528564
$227$	0.282868	0.0187746	0.00938729	−	0.999956i	$-0.497012\pi$
$227$	0.282868	0.0187746	0.00938729	+	0.999956i	$0.497012\pi$
$228$	1.41551	0.0937441
$229$	−8.45785	−0.558910	−0.279455	−	0.960159i	$-0.590154\pi$
$229$	−8.45785	−0.558910	−0.279455	+	0.960159i	$0.590154\pi$
$230$	0	0
$231$	−47.6516	−3.13524
$232$	−10.2653	−0.673948
$233$	−14.2477	−0.933397	−0.466698	−	0.884417i	$-0.654557\pi$
$233$	−14.2477	−0.933397	−0.466698	+	0.884417i	$0.654557\pi$
$234$	−0.903715	−0.0590777
$235$	0	0
$236$	−13.0787	−0.851350
$237$	−2.83101	−0.183894
$238$	−21.2113	−1.37493
$239$	21.8921	1.41608	0.708041	−	0.706171i	$-0.249579\pi$
$239$	21.8921	1.41608	0.708041	+	0.706171i	$0.249579\pi$
$240$	0	0
$241$	−9.13264	−0.588285	−0.294142	−	0.955762i	$-0.595034\pi$
$241$	−9.13264	−0.588285	−0.294142	+	0.955762i	$0.595034\pi$
$242$	2.54814	0.163801
$243$	17.3616	1.11374
$244$	−4.39791	−0.281547
$245$	0	0
$246$	32.4590	2.06951
$247$	−0.0599332	−0.00381346
$248$	5.58449	0.354616
$249$	−43.6879	−2.76861
$250$	0	0
$251$	−9.00000	−0.568075	−0.284037	−	0.958813i	$-0.591674\pi$
$251$	−9.00000	−0.568075	−0.284037	+	0.958813i	$0.591674\pi$
$252$	28.1574	1.77375
$253$	15.6995	0.987022
$254$	17.4942	1.09768
$255$	0	0
$256$	1.00000	0.0625000
$257$	−8.90371	−0.555398	−0.277699	−	0.960668i	$-0.589572\pi$
$257$	−8.90371	−0.555398	−0.277699	+	0.960668i	$0.589572\pi$
$258$	−15.9097	−0.990495
$259$	4.13264	0.256790
$260$	0	0
$261$	−69.9416	−4.32928
$262$	8.81342	0.544495
$263$	−14.9037	−0.919002	−0.459501	−	0.888177i	$-0.651972\pi$
$263$	−14.9037	−0.919002	−0.459501	+	0.888177i	$0.651972\pi$
$264$	−11.5305	−0.709656
$265$	0	0
$266$	1.86736	0.114495
$267$	2.43426	0.148975
$268$	−0.228923	−0.0139837
$269$	−1.55413	−0.0947570	−0.0473785	−	0.998877i	$-0.515087\pi$
$269$	−1.55413	−0.0947570	−0.0473785	+	0.998877i	$0.515087\pi$
$270$	0	0
$271$	5.68677	0.345447	0.172723	−	0.984970i	$-0.444743\pi$
$271$	5.68677	0.345447	0.172723	+	0.984970i	$0.444743\pi$
$272$	−5.13264	−0.311212
$273$	−1.71713	−0.103926
$274$	18.8921	1.14131
$275$	0	0
$276$	−13.3616	−0.804271
$277$	−12.9037	−0.775309	−0.387655	−	0.921805i	$-0.626715\pi$
$277$	−12.9037	−0.775309	−0.387655	+	0.921805i	$0.626715\pi$
$278$	−14.8498	−0.890630
$279$	38.0495	2.27796
$280$	0	0
$281$	−12.2829	−0.732734	−0.366367	−	0.930470i	$-0.619399\pi$
$281$	−12.2829	−0.732734	−0.366367	+	0.930470i	$0.619399\pi$
$282$	0	0
$283$	−26.8745	−1.59752	−0.798762	−	0.601647i	$-0.794511\pi$
$283$	−26.8745	−1.59752	−0.798762	+	0.601647i	$0.794511\pi$
$284$	9.49420	0.563377
$285$	0	0
$286$	0.488209	0.0288684
$287$	42.8206	2.52762
$288$	6.81342	0.401484
$289$	9.34397	0.549645
$290$	0	0
$291$	2.00000	0.117242
$292$	3.54814	0.207639
$293$	−16.7535	−0.978749	−0.489375	−	0.872074i	$-0.662775\pi$
$293$	−16.7535	−0.978749	−0.489375	+	0.872074i	$0.662775\pi$
$294$	31.5729	1.84137
$295$	0	0
$296$	1.00000	0.0581238
$297$	−43.9708	−2.55144
$298$	−2.41551	−0.139926
$299$	0.565735	0.0327173
$300$	0	0
$301$	−20.9884	−1.20975
$302$	10.7231	0.617046
$303$	−17.6268	−1.01264
$304$	0.451857	0.0259158
$305$	0	0
$306$	−34.9708	−1.99915
$307$	−8.47661	−0.483785	−0.241893	−	0.970303i	$-0.577768\pi$
$307$	−8.47661	−0.483785	−0.241893	+	0.970303i	$0.577768\pi$
$308$	−15.2113	−0.866746
$309$	31.4590	1.78964
$310$	0	0
$311$	32.4650	1.84092	0.920461	−	0.390835i	$-0.127814\pi$
$311$	32.4650	1.84092	0.920461	+	0.390835i	$0.127814\pi$
$312$	−0.415505	−0.0235233
$313$	28.8134	1.62863	0.814315	−	0.580423i	$-0.197113\pi$
$313$	28.8134	1.62863	0.814315	+	0.580423i	$0.197113\pi$
$314$	−3.31922	−0.187314
$315$	0	0
$316$	−0.903715	−0.0508379
$317$	−6.87335	−0.386046	−0.193023	−	0.981194i	$-0.561829\pi$
$317$	−6.87335	−0.386046	−0.193023	+	0.981194i	$0.561829\pi$
$318$	−36.4227	−2.04248
$319$	37.7842	2.11551
$320$	0	0
$321$	61.5976	3.43804
$322$	−17.6268	−0.982305
$323$	−2.31922	−0.129045
$324$	16.9824	0.943467
$325$	0	0
$326$	15.5481	0.861132
$327$	42.6879	2.36065
$328$	10.3616	0.572121
$329$	0	0
$330$	0	0
$331$	−1.65603	−0.0910238	−0.0455119	−	0.998964i	$-0.514492\pi$
$331$	−1.65603	−0.0910238	−0.0455119	+	0.998964i	$0.514492\pi$
$332$	−13.9461	−0.765389
$333$	6.81342	0.373373
$334$	−3.77707	−0.206672
$335$	0	0
$336$	12.9461	0.706265
$337$	10.2537	0.558553	0.279277	−	0.960211i	$-0.409905\pi$
$337$	10.2537	0.558553	0.279277	+	0.960211i	$0.409905\pi$
$338$	−12.9824	−0.706150
$339$	24.8921	1.35195
$340$	0	0
$341$	−20.5553	−1.11313
$342$	3.07869	0.166477
$343$	12.7231	0.686984
$344$	−5.07869	−0.273825
$345$	0	0
$346$	20.1574	1.08367
$347$	25.8805	1.38934	0.694669	−	0.719329i	$-0.255551\pi$
$347$	25.8805	1.38934	0.694669	+	0.719329i	$0.255551\pi$
$348$	−32.1574	−1.72382
$349$	12.7535	0.682678	0.341339	−	0.939940i	$-0.389120\pi$
$349$	12.7535	0.682678	0.341339	+	0.939940i	$0.389120\pi$
$350$	0	0
$351$	−1.58449	−0.0845741
$352$	−3.68078	−0.196186
$353$	8.72312	0.464285	0.232142	−	0.972682i	$-0.425426\pi$
$353$	8.72312	0.464285	0.232142	+	0.972682i	$0.425426\pi$
$354$	−40.9708	−2.17757
$355$	0	0
$356$	0.777066	0.0411844
$357$	−66.4474	−3.51677
$358$	20.2537	1.07044
$359$	14.9037	0.786588	0.393294	−	0.919413i	$-0.371335\pi$
$359$	14.9037	0.786588	0.393294	+	0.919413i	$0.371335\pi$
$360$	0	0
$361$	−18.7958	−0.989254
$362$	−17.1690	−0.902382
$363$	7.98241	0.418968
$364$	−0.548143	−0.0287305
$365$	0	0
$366$	−13.7771	−0.720139
$367$	−14.8558	−0.775464	−0.387732	−	0.921772i	$-0.626741\pi$
$367$	−14.8558	−0.775464	−0.387732	+	0.921772i	$0.626741\pi$
$368$	−4.26527	−0.222343
$369$	70.5976	3.67517
$370$	0	0
$371$	−48.0495	−2.49461
$372$	17.4942	0.907032
$373$	−1.77707	−0.0920130	−0.0460065	−	0.998941i	$-0.514649\pi$
$373$	−1.77707	−0.0920130	−0.0460065	+	0.998941i	$0.514649\pi$
$374$	18.8921	0.976888
$375$	0	0
$376$	0	0
$377$	1.36156	0.0701239
$378$	49.3687	2.53925
$379$	−26.3863	−1.35537	−0.677687	−	0.735351i	$-0.737017\pi$
$379$	−26.3863	−1.35537	−0.677687	+	0.735351i	$0.737017\pi$
$380$	0	0
$381$	54.8030	2.80764
$382$	−0.192571	−0.00985277
$383$	4.57289	0.233664	0.116832	−	0.993152i	$-0.462726\pi$
$383$	4.57289	0.233664	0.116832	+	0.993152i	$0.462726\pi$
$384$	3.13264	0.159862
$385$	0	0
$386$	−14.6692	−0.746641
$387$	−34.6033	−1.75898
$388$	0.638440	0.0324119
$389$	5.03635	0.255353	0.127677	−	0.991816i	$-0.459248\pi$
$389$	5.03635	0.255353	0.127677	+	0.991816i	$0.459248\pi$
$390$	0	0
$391$	21.8921	1.10713
$392$	10.0787	0.509051
$393$	27.6092	1.39270
$394$	−7.84977	−0.395466
$395$	0	0
$396$	−25.0787	−1.26025
$397$	34.4650	1.72975	0.864874	−	0.501988i	$-0.167398\pi$
$397$	34.4650	1.72975	0.864874	+	0.501988i	$0.167398\pi$
$398$	−11.3616	−0.569504
$399$	5.84977	0.292855
$400$	0	0
$401$	−4.47661	−0.223551	−0.111775	−	0.993733i	$-0.535654\pi$
$401$	−4.47661	−0.223551	−0.111775	+	0.993733i	$0.535654\pi$
$402$	−0.717132	−0.0357673
$403$	−0.740713	−0.0368976
$404$	−5.62684	−0.279946
$405$	0	0
$406$	−42.4227	−2.10540
$407$	−3.68078	−0.182449
$408$	−16.0787	−0.796014
$409$	−34.7243	−1.71701	−0.858503	−	0.512809i	$-0.828605\pi$
$409$	−34.7243	−1.71701	−0.858503	+	0.512809i	$0.828605\pi$
$410$	0	0
$411$	59.1821	2.91924
$412$	10.0423	0.494751
$413$	−54.0495	−2.65960
$414$	−29.0611	−1.42828
$415$	0	0
$416$	−0.132637	−0.00650309
$417$	−46.5189	−2.27804
$418$	−1.66319	−0.0813492
$419$	−7.15023	−0.349312	−0.174656	−	0.984630i	$-0.555881\pi$
$419$	−7.15023	−0.349312	−0.174656	+	0.984630i	$0.555881\pi$
$420$	0	0
$421$	26.6033	1.29656	0.648282	−	0.761401i	$-0.275488\pi$
$421$	26.6033	1.29656	0.648282	+	0.761401i	$0.275488\pi$
$422$	−12.2289	−0.595295
$423$	0	0
$424$	−11.6268	−0.564649
$425$	0	0
$426$	29.7419	1.44100
$427$	−18.1750	−0.879549
$428$	19.6632	0.950456
$429$	1.52938	0.0738393
$430$	0	0
$431$	−1.16899	−0.0563082	−0.0281541	−	0.999604i	$-0.508963\pi$
$431$	−1.16899	−0.0563082	−0.0281541	+	0.999604i	$0.508963\pi$
$432$	11.9461	0.574755
$433$	5.00000	0.240285	0.120142	−	0.992757i	$-0.461665\pi$
$433$	5.00000	0.240285	0.120142	+	0.992757i	$0.461665\pi$
$434$	23.0787	1.10781
$435$	0	0
$436$	13.6268	0.652607
$437$	−1.92730	−0.0921951
$438$	11.1150	0.531097
$439$	−32.3148	−1.54230	−0.771150	−	0.636654i	$-0.780318\pi$
$439$	−32.3148	−1.54230	−0.771150	+	0.636654i	$0.780318\pi$
$440$	0	0
$441$	68.6703	3.27002
$442$	0.680780	0.0323814
$443$	−17.4155	−0.827436	−0.413718	−	0.910405i	$-0.635770\pi$
$443$	−17.4155	−0.827436	−0.413718	+	0.910405i	$0.635770\pi$
$444$	3.13264	0.148668
$445$	0	0
$446$	28.1574	1.33329
$447$	−7.56690	−0.357902
$448$	4.13264	0.195249
$449$	−7.38032	−0.348299	−0.174149	−	0.984719i	$-0.555718\pi$
$449$	−7.38032	−0.348299	−0.174149	+	0.984719i	$0.555718\pi$
$450$	0	0
$451$	−38.1386	−1.79588
$452$	7.94606	0.373751
$453$	33.5916	1.57827
$454$	0.282868	0.0132756
$455$	0	0
$456$	1.41551	0.0662871
$457$	22.7419	1.06382	0.531910	−	0.846801i	$-0.321474\pi$
$457$	22.7419	1.06382	0.531910	+	0.846801i	$0.321474\pi$
$458$	−8.45785	−0.395209
$459$	−61.3148	−2.86193
$460$	0	0
$461$	39.7115	1.84955	0.924775	−	0.380515i	$-0.124253\pi$
$461$	39.7115	1.84955	0.924775	+	0.380515i	$0.124253\pi$
$462$	−47.6516	−2.21695
$463$	−30.6456	−1.42422	−0.712111	−	0.702067i	$-0.752261\pi$
$463$	−30.6456	−1.42422	−0.712111	+	0.702067i	$0.752261\pi$
$464$	−10.2653	−0.476553
$465$	0	0
$466$	−14.2477	−0.660011
$467$	24.7958	1.14741	0.573707	−	0.819061i	$-0.305505\pi$
$467$	24.7958	1.14741	0.573707	+	0.819061i	$0.305505\pi$
$468$	−0.903715	−0.0417742
$469$	−0.946055	−0.0436848
$470$	0	0
$471$	−10.3979	−0.479111
$472$	−13.0787	−0.601996
$473$	18.6936	0.859530
$474$	−2.83101	−0.130033
$475$	0	0
$476$	−21.2113	−0.972220
$477$	−79.2185	−3.62717
$478$	21.8921	1.00132
$479$	−15.5845	−0.712074	−0.356037	−	0.934472i	$-0.615872\pi$
$479$	−15.5845	−0.712074	−0.356037	+	0.934472i	$0.615872\pi$
$480$	0	0
$481$	−0.132637	−0.00604775
$482$	−9.13264	−0.415980
$483$	−55.2185	−2.51253
$484$	2.54814	0.115825
$485$	0	0
$486$	17.3616	0.787536
$487$	3.65720	0.165724	0.0828618	−	0.996561i	$-0.473594\pi$
$487$	3.65720	0.165724	0.0828618	+	0.996561i	$0.473594\pi$
$488$	−4.39791	−0.199084
$489$	48.7067	2.20259
$490$	0	0
$491$	33.3264	1.50400	0.751999	−	0.659164i	$-0.229090\pi$
$491$	33.3264	1.50400	0.751999	+	0.659164i	$0.229090\pi$
$492$	32.4590	1.46337
$493$	52.6879	2.37295
$494$	−0.0599332	−0.00269652
$495$	0	0
$496$	5.58449	0.250751
$497$	39.2361	1.75998
$498$	−43.6879	−1.95770
$499$	7.53654	0.337382	0.168691	−	0.985669i	$-0.446046\pi$
$499$	7.53654	0.337382	0.168691	+	0.985669i	$0.446046\pi$
$500$	0	0
$501$	−11.8322	−0.528623
$502$	−9.00000	−0.401690
$503$	−24.0000	−1.07011	−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000	−1.07011	−0.535054	+	0.844818i	$0.679709\pi$
$504$	28.1574	1.25423
$505$	0	0
$506$	15.6995	0.697930
$507$	−40.6692	−1.80618
$508$	17.4942	0.776180
$509$	24.1150	1.06888	0.534440	−	0.845206i	$-0.320522\pi$
$509$	24.1150	1.06888	0.534440	+	0.845206i	$0.320522\pi$
$510$	0	0
$511$	14.6632	0.648661
$512$	1.00000	0.0441942
$513$	5.39791	0.238324
$514$	−8.90371	−0.392726
$515$	0	0
$516$	−15.9097	−0.700386
$517$	0	0
$518$	4.13264	0.181578
$519$	63.1458	2.77179
$520$	0	0
$521$	−11.8134	−0.517555	−0.258778	−	0.965937i	$-0.583320\pi$
$521$	−11.8134	−0.517555	−0.258778	+	0.965937i	$0.583320\pi$
$522$	−69.9416	−3.06126
$523$	−33.9532	−1.48467	−0.742335	−	0.670029i	$-0.766282\pi$
$523$	−33.9532	−1.48467	−0.742335	+	0.670029i	$0.766282\pi$
$524$	8.81342	0.385016
$525$	0	0
$526$	−14.9037	−0.649833
$527$	−28.6632	−1.24859
$528$	−11.5305	−0.501803
$529$	−4.80743	−0.209019
$530$	0	0
$531$	−89.1106	−3.86707
$532$	1.86736	0.0809604
$533$	−1.37433	−0.0595289
$534$	2.43426	0.105341
$535$	0	0
$536$	−0.228923	−0.00988796
$537$	63.4474	2.73796
$538$	−1.55413	−0.0670033
$539$	−37.0975	−1.59790
$540$	0	0
$541$	−33.4590	−1.43852	−0.719258	−	0.694743i	$-0.755518\pi$
$541$	−33.4590	−1.43852	−0.719258	+	0.694743i	$0.755518\pi$
$542$	5.68677	0.244268
$543$	−53.7842	−2.30810
$544$	−5.13264	−0.220060
$545$	0	0
$546$	−1.71713	−0.0734865
$547$	−42.1514	−1.80226	−0.901132	−	0.433545i	$-0.857262\pi$
$547$	−42.1514	−1.80226	−0.901132	+	0.433545i	$0.857262\pi$
$548$	18.8921	0.807031
$549$	−29.9648	−1.27887
$550$	0	0
$551$	−4.63844	−0.197604
$552$	−13.3616	−0.568706
$553$	−3.73473	−0.158817
$554$	−12.9037	−0.548226
$555$	0	0
$556$	−14.8498	−0.629771
$557$	12.3732	0.524268	0.262134	−	0.965032i	$-0.415574\pi$
$557$	12.3732	0.524268	0.262134	+	0.965032i	$0.415574\pi$
$558$	38.0495	1.61076
$559$	0.673625	0.0284913
$560$	0	0
$561$	59.1821	2.49867
$562$	−12.2829	−0.518122
$563$	40.7782	1.71860	0.859299	−	0.511474i	$-0.170900\pi$
$563$	40.7782	1.71860	0.859299	+	0.511474i	$0.170900\pi$
$564$	0	0
$565$	0	0
$566$	−26.8745	−1.12962
$567$	70.1821	2.94737
$568$	9.49420	0.398368
$569$	−20.7419	−0.869545	−0.434772	−	0.900540i	$-0.643171\pi$
$569$	−20.7419	−0.869545	−0.434772	+	0.900540i	$0.643171\pi$
$570$	0	0
$571$	−37.1458	−1.55450	−0.777251	−	0.629190i	$-0.783387\pi$
$571$	−37.1458	−1.55450	−0.777251	+	0.629190i	$0.783387\pi$
$572$	0.488209	0.0204131
$573$	−0.603254	−0.0252013
$574$	42.8206	1.78730
$575$	0	0
$576$	6.81342	0.283892
$577$	−23.0975	−0.961560	−0.480780	−	0.876841i	$-0.659646\pi$
$577$	−23.0975	−0.961560	−0.480780	+	0.876841i	$0.659646\pi$
$578$	9.34397	0.388658
$579$	−45.9532	−1.90975
$580$	0	0
$581$	−57.6340	−2.39106
$582$	2.00000	0.0829027
$583$	42.7958	1.77242
$584$	3.54814	0.146823
$585$	0	0
$586$	−16.7535	−0.692080
$587$	11.0176	0.454745	0.227372	−	0.973808i	$-0.426987\pi$
$587$	11.0176	0.454745	0.227372	+	0.973808i	$0.426987\pi$
$588$	31.5729	1.30204
$589$	2.52339	0.103975
$590$	0	0
$591$	−24.5905	−1.01152
$592$	1.00000	0.0410997
$593$	−0.469450	−0.0192780	−0.00963900	−	0.999954i	$-0.503068\pi$
$593$	−0.469450	−0.0192780	−0.00963900	+	0.999954i	$0.503068\pi$
$594$	−43.9708	−1.80414
$595$	0	0
$596$	−2.41551	−0.0989429
$597$	−35.5916	−1.45667
$598$	0.565735	0.0231346
$599$	−24.7711	−1.01212	−0.506059	−	0.862499i	$-0.668898\pi$
$599$	−24.7711	−1.01212	−0.506059	+	0.862499i	$0.668898\pi$
$600$	0	0
$601$	−23.4051	−0.954713	−0.477356	−	0.878710i	$-0.658405\pi$
$601$	−23.4051	−0.954713	−0.477356	+	0.878710i	$0.658405\pi$
$602$	−20.9884	−0.855423
$603$	−1.55975	−0.0635178
$604$	10.7231	0.436318
$605$	0	0
$606$	−17.6268	−0.716041
$607$	0.342801	0.0139139	0.00695693	−	0.999976i	$-0.497786\pi$
$607$	0.342801	0.0139139	0.00695693	+	0.999976i	$0.497786\pi$
$608$	0.451857	0.0183252
$609$	−132.895	−5.38517
$610$	0	0
$611$	0	0
$612$	−34.9708	−1.41361
$613$	35.8921	1.44967	0.724834	−	0.688923i	$-0.241916\pi$
$613$	35.8921	1.44967	0.724834	+	0.688923i	$0.241916\pi$
$614$	−8.47661	−0.342088
$615$	0	0
$616$	−15.2113	−0.612882
$617$	7.64443	0.307753	0.153877	−	0.988090i	$-0.450824\pi$
$617$	7.64443	0.307753	0.153877	+	0.988090i	$0.450824\pi$
$618$	31.4590	1.26547
$619$	−10.4227	−0.418922	−0.209461	−	0.977817i	$-0.567171\pi$
$619$	−10.4227	−0.418922	−0.209461	+	0.977817i	$0.567171\pi$
$620$	0	0
$621$	−50.9532	−2.04468
$622$	32.4650	1.30173
$623$	3.21133	0.128659
$624$	−0.415505	−0.0166335
$625$	0	0
$626$	28.8134	1.15162
$627$	−5.21016	−0.208074
$628$	−3.31922	−0.132451
$629$	−5.13264	−0.204652
$630$	0	0
$631$	26.4227	1.05187	0.525935	−	0.850525i	$-0.323716\pi$
$631$	26.4227	1.05187	0.525935	+	0.850525i	$0.323716\pi$
$632$	−0.903715	−0.0359478
$633$	−38.3088	−1.52264
$634$	−6.87335	−0.272976
$635$	0	0
$636$	−36.4227	−1.44425
$637$	−1.33681	−0.0529664
$638$	37.7842	1.49589
$639$	64.6879	2.55902
$640$	0	0
$641$	11.8074	0.466365	0.233183	−	0.972433i	$-0.425086\pi$
$641$	11.8074	0.466365	0.233183	+	0.972433i	$0.425086\pi$
$642$	61.5976	2.43106
$643$	18.3556	0.723873	0.361937	−	0.932203i	$-0.382116\pi$
$643$	18.3556	0.723873	0.361937	+	0.932203i	$0.382116\pi$
$644$	−17.6268	−0.694595
$645$	0	0
$646$	−2.31922	−0.0912485
$647$	33.7771	1.32791	0.663957	−	0.747771i	$-0.268876\pi$
$647$	33.7771	1.32791	0.663957	+	0.747771i	$0.268876\pi$
$648$	16.9824	0.667132
$649$	48.1398	1.88965
$650$	0	0
$651$	72.2972	2.83355
$652$	15.5481	0.608912
$653$	18.0247	0.705363	0.352681	−	0.935743i	$-0.385270\pi$
$653$	18.0247	0.705363	0.352681	+	0.935743i	$0.385270\pi$
$654$	42.6879	1.66923
$655$	0	0
$656$	10.3616	0.404551
$657$	24.1750	0.943156
$658$	0	0
$659$	16.5669	0.645355	0.322677	−	0.946509i	$-0.395417\pi$
$659$	16.5669	0.645355	0.322677	+	0.946509i	$0.395417\pi$
$660$	0	0
$661$	−23.7842	−0.925099	−0.462549	−	0.886593i	$-0.653065\pi$
$661$	−23.7842	−0.925099	−0.462549	+	0.886593i	$0.653065\pi$
$662$	−1.65603	−0.0643635
$663$	2.13264	0.0828248
$664$	−13.9461	−0.541212
$665$	0	0
$666$	6.81342	0.264015
$667$	43.7842	1.69533
$668$	−3.77707	−0.146139
$669$	88.2069	3.41028
$670$	0	0
$671$	16.1877	0.624921
$672$	12.9461	0.499405
$673$	−12.0551	−0.464690	−0.232345	−	0.972633i	$-0.574640\pi$
$673$	−12.0551	−0.464690	−0.232345	+	0.972633i	$0.574640\pi$
$674$	10.2537	0.394957
$675$	0	0
$676$	−12.9824	−0.499323
$677$	−28.8805	−1.10997	−0.554984	−	0.831861i	$-0.687276\pi$
$677$	−28.8805	−1.10997	−0.554984	+	0.831861i	$0.687276\pi$
$678$	24.8921	0.955976
$679$	2.63844	0.101254
$680$	0	0
$681$	0.886122	0.0339563
$682$	−20.5553	−0.787103
$683$	17.6328	0.674701	0.337351	−	0.941379i	$-0.390469\pi$
$683$	17.6328	0.674701	0.337351	+	0.941379i	$0.390469\pi$
$684$	3.07869	0.117717
$685$	0	0
$686$	12.7231	0.485771
$687$	−26.4954	−1.01086
$688$	−5.07869	−0.193623
$689$	1.54215	0.0587514
$690$	0	0
$691$	−20.6572	−0.785837	−0.392918	−	0.919573i	$-0.628535\pi$
$691$	−20.6572	−0.785837	−0.392918	+	0.919573i	$0.628535\pi$
$692$	20.1574	0.766269
$693$	−103.641	−3.93700
$694$	25.8805	0.982411
$695$	0	0
$696$	−32.1574	−1.21892
$697$	−53.1821	−2.01442
$698$	12.7535	0.482727
$699$	−44.6328	−1.68817
$700$	0	0
$701$	49.2057	1.85847	0.929237	−	0.369484i	$-0.120466\pi$
$701$	49.2057	1.85847	0.929237	+	0.369484i	$0.120466\pi$
$702$	−1.58449	−0.0598029
$703$	0.451857	0.0170421
$704$	−3.68078	−0.138725
$705$	0	0
$706$	8.72312	0.328299
$707$	−23.2537	−0.874544
$708$	−40.9708	−1.53978
$709$	3.73473	0.140261	0.0701303	−	0.997538i	$-0.477659\pi$
$709$	3.73473	0.140261	0.0701303	+	0.997538i	$0.477659\pi$
$710$	0	0
$711$	−6.15739	−0.230920
$712$	0.777066	0.0291218
$713$	−23.8194	−0.892044
$714$	−66.4474	−2.48673
$715$	0	0
$716$	20.2537	0.756915
$717$	68.5800	2.56117
$718$	14.9037	0.556202
$719$	−47.0858	−1.75601	−0.878003	−	0.478655i	$-0.841124\pi$
$719$	−47.0858	−1.75601	−0.878003	+	0.478655i	$0.841124\pi$
$720$	0	0
$721$	41.5014	1.54559
$722$	−18.7958	−0.699508
$723$	−28.6092	−1.06399
$724$	−17.1690	−0.638080
$725$	0	0
$726$	7.98241	0.296255
$727$	27.3616	1.01478	0.507392	−	0.861715i	$-0.330610\pi$
$727$	27.3616	1.01478	0.507392	+	0.861715i	$0.330610\pi$
$728$	−0.548143	−0.0203155
$729$	3.44025	0.127417
$730$	0	0
$731$	26.0671	0.964126
$732$	−13.7771	−0.509215
$733$	47.5070	1.75471	0.877355	−	0.479842i	$-0.159306\pi$
$733$	47.5070	1.75471	0.877355	+	0.479842i	$0.159306\pi$
$734$	−14.8558	−0.548336
$735$	0	0
$736$	−4.26527	−0.157220
$737$	0.842615	0.0310381
$738$	70.5976	2.59873
$739$	19.6149	0.721544	0.360772	−	0.932654i	$-0.382513\pi$
$739$	19.6149	0.721544	0.360772	+	0.932654i	$0.382513\pi$
$740$	0	0
$741$	−0.187749	−0.00689713
$742$	−48.0495	−1.76395
$743$	22.3148	0.818650	0.409325	−	0.912389i	$-0.365764\pi$
$743$	22.3148	0.818650	0.409325	+	0.912389i	$0.365764\pi$
$744$	17.4942	0.641368
$745$	0	0
$746$	−1.77707	−0.0650630
$747$	−95.0203	−3.47661
$748$	18.8921	0.690764
$749$	81.2608	2.96921
$750$	0	0
$751$	20.7958	0.758850	0.379425	−	0.925222i	$-0.376122\pi$
$751$	20.7958	0.758850	0.379425	+	0.925222i	$0.376122\pi$
$752$	0	0
$753$	−28.1937	−1.02744
$754$	1.36156	0.0495851
$755$	0	0
$756$	49.3687	1.79552
$757$	−0.530550	−0.0192832	−0.00964158	−	0.999954i	$-0.503069\pi$
$757$	−0.530550	−0.0192832	−0.00964158	+	0.999954i	$0.503069\pi$
$758$	−26.3863	−0.958394
$759$	49.1810	1.78516
$760$	0	0
$761$	4.63245	0.167926	0.0839631	−	0.996469i	$-0.473242\pi$
$761$	4.63245	0.167926	0.0839631	+	0.996469i	$0.473242\pi$
$762$	54.8030	1.98530
$763$	56.3148	2.03873
$764$	−0.192571	−0.00696696
$765$	0	0
$766$	4.57289	0.165225
$767$	1.73473	0.0626373
$768$	3.13264	0.113039
$769$	7.78867	0.280867	0.140433	−	0.990090i	$-0.455150\pi$
$769$	7.78867	0.280867	0.140433	+	0.990090i	$0.455150\pi$
$770$	0	0
$771$	−27.8921	−1.00451
$772$	−14.6692	−0.527955
$773$	−8.04234	−0.289263	−0.144631	−	0.989486i	$-0.546200\pi$
$773$	−8.04234	−0.289263	−0.144631	+	0.989486i	$0.546200\pi$
$774$	−34.6033	−1.24379
$775$	0	0
$776$	0.638440	0.0229186
$777$	12.9461	0.464437
$778$	5.03635	0.180562
$779$	4.68195	0.167748
$780$	0	0
$781$	−34.9461	−1.25047
$782$	21.8921	0.782860
$783$	−122.630	−4.38242
$784$	10.0787	0.359953
$785$	0	0
$786$	27.6092	0.984789
$787$	25.6268	0.913498	0.456749	−	0.889596i	$-0.349014\pi$
$787$	25.6268	0.913498	0.456749	+	0.889596i	$0.349014\pi$
$788$	−7.84977	−0.279636
$789$	−46.6879	−1.66213
$790$	0	0
$791$	32.8382	1.16759
$792$	−25.0787	−0.891133
$793$	0.583328	0.0207146
$794$	34.4650	1.22312
$795$	0	0
$796$	−11.3616	−0.402700
$797$	11.4095	0.404146	0.202073	−	0.979370i	$-0.435232\pi$
$797$	11.4095	0.404146	0.202073	+	0.979370i	$0.435232\pi$
$798$	5.84977	0.207080
$799$	0	0
$800$	0	0
$801$	5.29447	0.187071
$802$	−4.47661	−0.158074
$803$	−13.0599	−0.460875
$804$	−0.717132	−0.0252913
$805$	0	0
$806$	−0.740713	−0.0260905
$807$	−4.86853	−0.171380
$808$	−5.62684	−0.197951
$809$	21.5189	0.756566	0.378283	−	0.925690i	$-0.376515\pi$
$809$	21.5189	0.756566	0.378283	+	0.925690i	$0.376515\pi$
$810$	0	0
$811$	−44.0847	−1.54802	−0.774011	−	0.633172i	$-0.781753\pi$
$811$	−44.0847	−1.54802	−0.774011	+	0.633172i	$0.781753\pi$
$812$	−42.4227	−1.48874
$813$	17.8146	0.624785
$814$	−3.68078	−0.129011
$815$	0	0
$816$	−16.0787	−0.562867
$817$	−2.29484	−0.0802864
$818$	−34.7243	−1.21411
$819$	−3.73473	−0.130502
$820$	0	0
$821$	−9.95766	−0.347525	−0.173762	−	0.984788i	$-0.555592\pi$
$821$	−9.95766	−0.347525	−0.173762	+	0.984788i	$0.555592\pi$
$822$	59.1821	2.06421
$823$	0.663187	0.0231173	0.0115586	−	0.999933i	$-0.496321\pi$
$823$	0.663187	0.0231173	0.0115586	+	0.999933i	$0.496321\pi$
$824$	10.0423	0.349842
$825$	0	0
$826$	−54.0495	−1.88062
$827$	−41.0671	−1.42804	−0.714021	−	0.700124i	$-0.753128\pi$
$827$	−41.0671	−1.42804	−0.714021	+	0.700124i	$0.753128\pi$
$828$	−29.0611	−1.00994
$829$	6.48259	0.225150	0.112575	−	0.993643i	$-0.464090\pi$
$829$	6.48259	0.225150	0.112575	+	0.993643i	$0.464090\pi$
$830$	0	0
$831$	−40.4227	−1.40225
$832$	−0.132637	−0.00459838
$833$	−51.7303	−1.79235
$834$	−46.5189	−1.61082
$835$	0	0
$836$	−1.66319	−0.0575225
$837$	66.7127	2.30593
$838$	−7.15023	−0.247001
$839$	0.771077	0.0266205	0.0133103	−	0.999911i	$-0.495763\pi$
$839$	0.771077	0.0266205	0.0133103	+	0.999911i	$0.495763\pi$
$840$	0	0
$841$	76.3759	2.63365
$842$	26.6033	0.916809
$843$	−38.4778	−1.32524
$844$	−12.2289	−0.420937
$845$	0	0
$846$	0	0
$847$	10.5305	0.361834
$848$	−11.6268	−0.399267
$849$	−84.1881	−2.88933
$850$	0	0
$851$	−4.26527	−0.146212
$852$	29.7419	1.01894
$853$	30.0975	1.03052	0.515259	−	0.857035i	$-0.327696\pi$
$853$	30.0975	1.03052	0.515259	+	0.857035i	$0.327696\pi$
$854$	−18.1750	−0.621935
$855$	0	0
$856$	19.6632	0.672074
$857$	8.22892	0.281095	0.140547	−	0.990074i	$-0.455114\pi$
$857$	8.22892	0.281095	0.140547	+	0.990074i	$0.455114\pi$
$858$	1.52938	0.0522123
$859$	14.4286	0.492299	0.246150	−	0.969232i	$-0.420835\pi$
$859$	14.4286	0.492299	0.246150	+	0.969232i	$0.420835\pi$
$860$	0	0
$861$	134.141	4.57152
$862$	−1.16899	−0.0398159
$863$	33.9049	1.15414	0.577068	−	0.816696i	$-0.304197\pi$
$863$	33.9049	1.15414	0.577068	+	0.816696i	$0.304197\pi$
$864$	11.9461	0.406413
$865$	0	0
$866$	5.00000	0.169907
$867$	29.2713	0.994104
$868$	23.0787	0.783342
$869$	3.32637	0.112840
$870$	0	0
$871$	0.0303638	0.00102884
$872$	13.6268	0.461463
$873$	4.34996	0.147224
$874$	−1.92730	−0.0651918
$875$	0	0
$876$	11.1150	0.375543
$877$	1.36872	0.0462182	0.0231091	−	0.999733i	$-0.492643\pi$
$877$	1.36872	0.0462182	0.0231091	+	0.999733i	$0.492643\pi$
$878$	−32.3148	−1.09057
$879$	−52.4826	−1.77019
$880$	0	0
$881$	47.5741	1.60281	0.801405	−	0.598122i	$-0.204086\pi$
$881$	47.5741	1.60281	0.801405	+	0.598122i	$0.204086\pi$
$882$	68.6703	2.31225
$883$	9.88051	0.332506	0.166253	−	0.986083i	$-0.446833\pi$
$883$	9.88051	0.332506	0.166253	+	0.986083i	$0.446833\pi$
$884$	0.680780	0.0228971
$885$	0	0
$886$	−17.4155	−0.585085
$887$	21.8801	0.734663	0.367331	−	0.930090i	$-0.380271\pi$
$887$	21.8801	0.734663	0.367331	+	0.930090i	$0.380271\pi$
$888$	3.13264	0.105124
$889$	72.2972	2.42477
$890$	0	0
$891$	−62.5085	−2.09411
$892$	28.1574	0.942779
$893$	0	0
$894$	−7.56690	−0.253075
$895$	0	0
$896$	4.13264	0.138062
$897$	1.77224	0.0591735
$898$	−7.38032	−0.246284
$899$	−57.3264	−1.91194
$900$	0	0
$901$	59.6763	1.98811
$902$	−38.1386	−1.26988
$903$	−65.7490	−2.18799
$904$	7.94606	0.264282
$905$	0	0
$906$	33.5916	1.11601
$907$	8.46346	0.281025	0.140512	−	0.990079i	$-0.455125\pi$
$907$	8.46346	0.281025	0.140512	+	0.990079i	$0.455125\pi$
$908$	0.282868	0.00938729
$909$	−38.3380	−1.27159
$910$	0	0
$911$	−34.1997	−1.13309	−0.566544	−	0.824032i	$-0.691720\pi$
$911$	−34.1997	−1.13309	−0.566544	+	0.824032i	$0.691720\pi$
$912$	1.41551	0.0468721
$913$	51.3324	1.69885
$914$	22.7419	0.752235
$915$	0	0
$916$	−8.45785	−0.279455
$917$	36.4227	1.20278
$918$	−61.3148	−2.02369
$919$	−0.723121	−0.0238536	−0.0119268	−	0.999929i	$-0.503797\pi$
$919$	−0.723121	−0.0238536	−0.0119268	+	0.999929i	$0.503797\pi$
$920$	0	0
$921$	−26.5541	−0.874988
$922$	39.7115	1.30783
$923$	−1.25929	−0.0414499
$924$	−47.6516	−1.56762
$925$	0	0
$926$	−30.6456	−1.00708
$927$	68.4227	2.24730
$928$	−10.2653	−0.336974
$929$	−16.7902	−0.550869	−0.275434	−	0.961320i	$-0.588822\pi$
$929$	−16.7902	−0.550869	−0.275434	+	0.961320i	$0.588822\pi$
$930$	0	0
$931$	4.55413	0.149256
$932$	−14.2477	−0.466698
$933$	101.701	3.32954
$934$	24.7958	0.811344
$935$	0	0
$936$	−0.903715	−0.0295388
$937$	7.01759	0.229255	0.114627	−	0.993409i	$-0.463433\pi$
$937$	7.01759	0.229255	0.114627	+	0.993409i	$0.463433\pi$
$938$	−0.946055	−0.0308898
$939$	90.2620	2.94559
$940$	0	0
$941$	30.7958	1.00392	0.501958	−	0.864892i	$-0.332613\pi$
$941$	30.7958	1.00392	0.501958	+	0.864892i	$0.332613\pi$
$942$	−10.3979	−0.338782
$943$	−44.1949	−1.43918
$944$	−13.0787	−0.425675
$945$	0	0
$946$	18.6936	0.607780
$947$	6.19257	0.201232	0.100616	−	0.994925i	$-0.467919\pi$
$947$	6.19257	0.201232	0.100616	+	0.994925i	$0.467919\pi$
$948$	−2.83101	−0.0919469
$949$	−0.470617	−0.0152769
$950$	0	0
$951$	−21.5317	−0.698214
$952$	−21.2113	−0.687463
$953$	−37.1750	−1.20422	−0.602108	−	0.798415i	$-0.705672\pi$
$953$	−37.1750	−1.20422	−0.602108	+	0.798415i	$0.705672\pi$
$954$	−79.2185	−2.56479
$955$	0	0
$956$	21.8921	0.708041
$957$	118.364	3.82617
$958$	−15.5845	−0.503512
$959$	78.0742	2.52115
$960$	0	0
$961$	0.186582	0.00601878
$962$	−0.132637	−0.00427640
$963$	133.974	4.31724
$964$	−9.13264	−0.294142
$965$	0	0
$966$	−55.2185	−1.77663
$967$	−34.4722	−1.10855	−0.554275	−	0.832334i	$-0.687004\pi$
$967$	−34.4722	−1.10855	−0.554275	+	0.832334i	$0.687004\pi$
$968$	2.54814	0.0819004
$969$	−7.26527	−0.233394
$970$	0	0
$971$	−54.2488	−1.74093	−0.870464	−	0.492232i	$-0.836181\pi$
$971$	−54.2488	−1.74093	−0.870464	+	0.492232i	$0.836181\pi$
$972$	17.3616	0.556872
$973$	−61.3687	−1.96739
$974$	3.65720	0.117184
$975$	0	0
$976$	−4.39791	−0.140774
$977$	−55.4706	−1.77466	−0.887331	−	0.461133i	$-0.847443\pi$
$977$	−55.4706	−1.77466	−0.887331	+	0.461133i	$0.847443\pi$
$978$	48.7067	1.55747
$979$	−2.86021	−0.0914126
$980$	0	0
$981$	92.8453	2.96432
$982$	33.3264	1.06349
$983$	−0.422660	−0.0134808	−0.00674038	−	0.999977i	$-0.502146\pi$
$983$	−0.422660	−0.0134808	−0.00674038	+	0.999977i	$0.502146\pi$
$984$	32.4590	1.03476
$985$	0	0
$986$	52.6879	1.67793
$987$	0	0
$988$	−0.0599332	−0.00190673
$989$	21.6620	0.688812
$990$	0	0
$991$	7.93445	0.252046	0.126023	−	0.992027i	$-0.459779\pi$
$991$	7.93445	0.252046	0.126023	+	0.992027i	$0.459779\pi$
$992$	5.58449	0.177308
$993$	−5.18775	−0.164628
$994$	39.2361	1.24449
$995$	0	0
$996$	−43.6879	−1.38431
$997$	48.8933	1.54847	0.774233	−	0.632901i	$-0.218136\pi$
$997$	48.8933	1.54847	0.774233	+	0.632901i	$0.218136\pi$
$998$	7.53654	0.238565
$999$	11.9461	0.377956

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1850.2.a.bc.1.3	yes	3
5.2	odd	4		1850.2.b.p.149.4		6
5.3	odd	4		1850.2.b.p.149.3		6
5.4	even	2		1850.2.a.y.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1850.2.a.y.1.1	✓	3	5.4	even	2
1850.2.a.bc.1.3	yes	3	1.1	even	1	trivial
1850.2.b.p.149.3		6	5.3	odd	4
1850.2.b.p.149.4		6	5.2	odd	4

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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +3.13264 q^{3} +1.00000 q^{4} +3.13264 q^{6} +4.13264 q^{7} +1.00000 q^{8} +6.81342 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +3.13264 q^{3} +1.00000 q^{4} +3.13264 q^{6} +4.13264 q^{7} +1.00000 q^{8} +6.81342 q^{9} -3.68078 q^{11} +3.13264 q^{12} -0.132637 q^{13} +4.13264 q^{14} +1.00000 q^{16} -5.13264 q^{17} +6.81342 q^{18} +0.451857 q^{19} +12.9461 q^{21} -3.68078 q^{22} -4.26527 q^{23} +3.13264 q^{24} -0.132637 q^{26} +11.9461 q^{27} +4.13264 q^{28} -10.2653 q^{29} +5.58449 q^{31} +1.00000 q^{32} -11.5305 q^{33} -5.13264 q^{34} +6.81342 q^{36} +1.00000 q^{37} +0.451857 q^{38} -0.415505 q^{39} +10.3616 q^{41} +12.9461 q^{42} -5.07869 q^{43} -3.68078 q^{44} -4.26527 q^{46} +3.13264 q^{48} +10.0787 q^{49} -16.0787 q^{51} -0.132637 q^{52} -11.6268 q^{53} +11.9461 q^{54} +4.13264 q^{56} +1.41551 q^{57} -10.2653 q^{58} -13.0787 q^{59} -4.39791 q^{61} +5.58449 q^{62} +28.1574 q^{63} +1.00000 q^{64} -11.5305 q^{66} -0.228923 q^{67} -5.13264 q^{68} -13.3616 q^{69} +9.49420 q^{71} +6.81342 q^{72} +3.54814 q^{73} +1.00000 q^{74} +0.451857 q^{76} -15.2113 q^{77} -0.415505 q^{78} -0.903715 q^{79} +16.9824 q^{81} +10.3616 q^{82} -13.9461 q^{83} +12.9461 q^{84} -5.07869 q^{86} -32.1574 q^{87} -3.68078 q^{88} +0.777066 q^{89} -0.548143 q^{91} -4.26527 q^{92} +17.4942 q^{93} +3.13264 q^{96} +0.638440 q^{97} +10.0787 q^{98} -25.0787 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + q^{3} + 3 q^{4} + q^{6} + 4 q^{7} + 3 q^{8} + 6 q^{9}+O(q^{10}) \) 3 * q + 3 * q^2 + q^3 + 3 * q^4 + q^6 + 4 * q^7 + 3 * q^8 + 6 * q^9	\( 3 q + 3 q^{2} + q^{3} + 3 q^{4} + q^{6} + 4 q^{7} + 3 q^{8} + 6 q^{9} - 5 q^{11} + q^{12} + 8 q^{13} + 4 q^{14} + 3 q^{16} - 7 q^{17} + 6 q^{18} - q^{19} + 16 q^{21} - 5 q^{22} + 4 q^{23} + q^{24} + 8 q^{26} + 13 q^{27} + 4 q^{28} - 14 q^{29} + 6 q^{31} + 3 q^{32} - q^{33} - 7 q^{34} + 6 q^{36} + 3 q^{37} - q^{38} - 12 q^{39} + 19 q^{41} + 16 q^{42} + 16 q^{43} - 5 q^{44} + 4 q^{46} + q^{48} - q^{49} - 17 q^{51} + 8 q^{52} - 6 q^{53} + 13 q^{54} + 4 q^{56} + 15 q^{57} - 14 q^{58} - 8 q^{59} + 12 q^{61} + 6 q^{62} + 22 q^{63} + 3 q^{64} - q^{66} + 3 q^{67} - 7 q^{68} - 28 q^{69} + 8 q^{71} + 6 q^{72} + 13 q^{73} + 3 q^{74} - q^{76} - 6 q^{77} - 12 q^{78} + 2 q^{79} + 15 q^{81} + 19 q^{82} - 19 q^{83} + 16 q^{84} + 16 q^{86} - 34 q^{87} - 5 q^{88} + q^{89} - 4 q^{91} + 4 q^{92} + 32 q^{93} + q^{96} + 14 q^{97} - q^{98} - 44 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 + q^3 + 3 * q^4 + q^6 + 4 * q^7 + 3 * q^8 + 6 * q^9 - 5 * q^11 + q^12 + 8 * q^13 + 4 * q^14 + 3 * q^16 - 7 * q^17 + 6 * q^18 - q^19 + 16 * q^21 - 5 * q^22 + 4 * q^23 + q^24 + 8 * q^26 + 13 * q^27 + 4 * q^28 - 14 * q^29 + 6 * q^31 + 3 * q^32 - q^33 - 7 * q^34 + 6 * q^36 + 3 * q^37 - q^38 - 12 * q^39 + 19 * q^41 + 16 * q^42 + 16 * q^43 - 5 * q^44 + 4 * q^46 + q^48 - q^49 - 17 * q^51 + 8 * q^52 - 6 * q^53 + 13 * q^54 + 4 * q^56 + 15 * q^57 - 14 * q^58 - 8 * q^59 + 12 * q^61 + 6 * q^62 + 22 * q^63 + 3 * q^64 - q^66 + 3 * q^67 - 7 * q^68 - 28 * q^69 + 8 * q^71 + 6 * q^72 + 13 * q^73 + 3 * q^74 - q^76 - 6 * q^77 - 12 * q^78 + 2 * q^79 + 15 * q^81 + 19 * q^82 - 19 * q^83 + 16 * q^84 + 16 * q^86 - 34 * q^87 - 5 * q^88 + q^89 - 4 * q^91 + 4 * q^92 + 32 * q^93 + q^96 + 14 * q^97 - q^98 - 44 * q^99

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