LMFDB - Embedded newform 855.2.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [855,2,Mod(1,855)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(855, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("855.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 855 = 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	855.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:	$6.82720937282$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.11344.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 4x + 3 $ x^4 - 2x^3 - 4x^2 + 4*x + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 95)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.78165$ of defining polynomial
Character	$\chi$	$=$	855.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.95594 q^{2} +1.82571 q^{4} +1.00000 q^{5} -3.56331 q^{7} +0.340899 q^{8} +O(q^{10})$	\(q-1.95594 q^{2} +1.82571 q^{4} +1.00000 q^{5} -3.56331 q^{7} +0.340899 q^{8} -1.95594 q^{10} -5.56331 q^{11} +5.26647 q^{13} +6.96962 q^{14} -4.31820 q^{16} -1.40632 q^{17} +1.00000 q^{19} +1.82571 q^{20} +10.8815 q^{22} +6.96962 q^{23} +1.00000 q^{25} -10.3009 q^{26} -6.50557 q^{28} -1.40632 q^{29} +1.75489 q^{31} +7.76435 q^{32} +2.75067 q^{34} -3.56331 q^{35} +3.61504 q^{37} -1.95594 q^{38} +0.340899 q^{40} -4.34858 q^{41} -3.56331 q^{43} -10.1570 q^{44} -13.6322 q^{46} +8.26046 q^{47} +5.69716 q^{49} -1.95594 q^{50} +9.61504 q^{52} +7.61504 q^{53} -5.56331 q^{55} -1.21473 q^{56} +2.75067 q^{58} -9.47519 q^{59} +9.21473 q^{61} -3.43247 q^{62} -6.55023 q^{64} +5.26647 q^{65} -4.76090 q^{67} -2.56753 q^{68} +6.96962 q^{70} +14.0689 q^{71} +6.59368 q^{73} -7.07082 q^{74} +1.82571 q^{76} +19.8238 q^{77} +5.47519 q^{79} -4.31820 q^{80} +8.50557 q^{82} +4.15699 q^{83} -1.40632 q^{85} +6.96962 q^{86} -1.89653 q^{88} +9.23009 q^{89} -18.7660 q^{91} +12.7245 q^{92} -16.1570 q^{94} +1.00000 q^{95} +11.5116 q^{97} -11.1433 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} + 8 q^{4} + 4 q^{5} + 4 q^{7} + 12 q^{8}+O(q^{10}) $ 4 * q + 2 * q^2 + 8 * q^4 + 4 * q^5 + 4 * q^7 + 12 * q^8	$ 4 q + 2 q^{2} + 8 q^{4} + 4 q^{5} + 4 q^{7} + 12 q^{8} + 2 q^{10} - 4 q^{11} + 2 q^{13} + 8 q^{14} + 4 q^{16} - 4 q^{17} + 4 q^{19} + 8 q^{20} + 4 q^{22} + 8 q^{23} + 4 q^{25} - 4 q^{26} - 8 q^{28} - 4 q^{29} + 4 q^{31} + 6 q^{32} - 4 q^{34} + 4 q^{35} - 6 q^{37} + 2 q^{38} + 12 q^{40} - 16 q^{41} + 4 q^{43} - 24 q^{44} + 12 q^{47} + 20 q^{49} + 2 q^{50} + 18 q^{52} + 10 q^{53} - 4 q^{55} + 12 q^{56} - 4 q^{58} + 20 q^{61} - 20 q^{62} - 4 q^{64} + 2 q^{65} - 18 q^{67} - 4 q^{68} + 8 q^{70} + 20 q^{71} + 28 q^{73} - 32 q^{74} + 8 q^{76} + 40 q^{77} - 16 q^{79} + 4 q^{80} + 16 q^{82} - 4 q^{85} + 8 q^{86} - 12 q^{88} - 4 q^{89} - 36 q^{91} + 28 q^{92} - 48 q^{94} + 4 q^{95} + 30 q^{97} - 38 q^{98}+O(q^{100}) $ 4 * q + 2 * q^2 + 8 * q^4 + 4 * q^5 + 4 * q^7 + 12 * q^8 + 2 * q^10 - 4 * q^11 + 2 * q^13 + 8 * q^14 + 4 * q^16 - 4 * q^17 + 4 * q^19 + 8 * q^20 + 4 * q^22 + 8 * q^23 + 4 * q^25 - 4 * q^26 - 8 * q^28 - 4 * q^29 + 4 * q^31 + 6 * q^32 - 4 * q^34 + 4 * q^35 - 6 * q^37 + 2 * q^38 + 12 * q^40 - 16 * q^41 + 4 * q^43 - 24 * q^44 + 12 * q^47 + 20 * q^49 + 2 * q^50 + 18 * q^52 + 10 * q^53 - 4 * q^55 + 12 * q^56 - 4 * q^58 + 20 * q^61 - 20 * q^62 - 4 * q^64 + 2 * q^65 - 18 * q^67 - 4 * q^68 + 8 * q^70 + 20 * q^71 + 28 * q^73 - 32 * q^74 + 8 * q^76 + 40 * q^77 - 16 * q^79 + 4 * q^80 + 16 * q^82 - 4 * q^85 + 8 * q^86 - 12 * q^88 - 4 * q^89 - 36 * q^91 + 28 * q^92 - 48 * q^94 + 4 * q^95 + 30 * q^97 - 38 * q^98

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 $

$2$	−1.95594	−1.38306	−0.691530	−	0.722348i	$-0.743063\pi$
$2$	−1.95594	−1.38306	−0.691530	+	0.722348i	$0.743063\pi$
$3$	0	0
$3$	0	0
$4$	1.82571	0.912855
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−3.56331	−1.34680	−0.673402	−	0.739277i	$-0.735168\pi$
$7$	−3.56331	−1.34680	−0.673402	+	0.739277i	$0.735168\pi$
$8$	0.340899	0.120526
$9$	0	0
$10$	−1.95594	−0.618523
$11$	−5.56331	−1.67740	−0.838700	−	0.544594i	$-0.816684\pi$
$11$	−5.56331	−1.67740	−0.838700	+	0.544594i	$0.816684\pi$
$12$	0	0
$13$	5.26647	1.46065	0.730327	−	0.683097i	$-0.239368\pi$
$13$	5.26647	1.46065	0.730327	+	0.683097i	$0.239368\pi$
$14$	6.96962	1.86271
$15$	0	0
$16$	−4.31820	−1.07955
$17$	−1.40632	−0.341082	−0.170541	−	0.985351i	$-0.554552\pi$
$17$	−1.40632	−0.341082	−0.170541	+	0.985351i	$0.554552\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	1.82571	0.408241
$21$	0	0
$22$	10.8815	2.31995
$23$	6.96962	1.45327	0.726633	−	0.687025i	$-0.241084\pi$
$23$	6.96962	1.45327	0.726633	+	0.687025i	$0.241084\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−10.3009	−2.02017
$27$	0	0
$28$	−6.50557	−1.22944
$29$	−1.40632	−0.261146	−0.130573	−	0.991439i	$-0.541682\pi$
$29$	−1.40632	−0.261146	−0.130573	+	0.991439i	$0.541682\pi$
$30$	0	0
$31$	1.75489	0.315188	0.157594	−	0.987504i	$-0.449626\pi$
$31$	1.75489	0.315188	0.157594	+	0.987504i	$0.449626\pi$
$32$	7.76435	1.37256
$33$	0	0
$34$	2.75067	0.471737
$35$	−3.56331	−0.602309
$36$	0	0
$37$	3.61504	0.594309	0.297155	−	0.954829i	$-0.403962\pi$
$37$	3.61504	0.594309	0.297155	+	0.954829i	$0.403962\pi$
$38$	−1.95594	−0.317296
$39$	0	0
$40$	0.340899	0.0539009
$41$	−4.34858	−0.679134	−0.339567	−	0.940582i	$-0.610280\pi$
$41$	−4.34858	−0.679134	−0.339567	+	0.940582i	$0.610280\pi$
$42$	0	0
$43$	−3.56331	−0.543399	−0.271700	−	0.962382i	$-0.587586\pi$
$43$	−3.56331	−0.543399	−0.271700	+	0.962382i	$0.587586\pi$
$44$	−10.1570	−1.53122
$45$	0	0
$46$	−13.6322	−2.00996
$47$	8.26046	1.20491	0.602456	−	0.798152i	$-0.294189\pi$
$47$	8.26046	1.20491	0.602456	+	0.798152i	$0.294189\pi$
$48$	0	0
$49$	5.69716	0.813879
$50$	−1.95594	−0.276612
$51$	0	0
$52$	9.61504	1.33337
$53$	7.61504	1.04601	0.523003	−	0.852331i	$-0.324812\pi$
$53$	7.61504	1.04601	0.523003	+	0.852331i	$0.324812\pi$
$54$	0	0
$55$	−5.56331	−0.750156
$56$	−1.21473	−0.162325
$57$	0	0
$58$	2.75067	0.361181
$59$	−9.47519	−1.23356	−0.616782	−	0.787134i	$-0.711564\pi$
$59$	−9.47519	−1.23356	−0.616782	+	0.787134i	$0.711564\pi$
$60$	0	0
$61$	9.21473	1.17983	0.589913	−	0.807467i	$-0.299162\pi$
$61$	9.21473	1.17983	0.589913	+	0.807467i	$0.299162\pi$
$62$	−3.43247	−0.435924
$63$	0	0
$64$	−6.55023	−0.818779
$65$	5.26647	0.653225
$66$	0	0
$67$	−4.76090	−0.581636	−0.290818	−	0.956778i	$-0.593927\pi$
$67$	−4.76090	−0.581636	−0.290818	+	0.956778i	$0.593927\pi$
$68$	−2.56753	−0.311358
$69$	0	0
$70$	6.96962	0.833029
$71$	14.0689	1.66967	0.834834	−	0.550502i	$-0.185563\pi$
$71$	14.0689	1.66967	0.834834	+	0.550502i	$0.185563\pi$
$72$	0	0
$73$	6.59368	0.771732	0.385866	−	0.922555i	$-0.373903\pi$
$73$	6.59368	0.771732	0.385866	+	0.922555i	$0.373903\pi$
$74$	−7.07082	−0.821966
$75$	0	0
$76$	1.82571	0.209423
$77$	19.8238	2.25913
$78$	0	0
$79$	5.47519	0.616007	0.308004	−	0.951385i	$-0.400339\pi$
$79$	5.47519	0.616007	0.308004	+	0.951385i	$0.400339\pi$
$80$	−4.31820	−0.482790
$81$	0	0
$82$	8.50557	0.939283
$83$	4.15699	0.456289	0.228144	−	0.973627i	$-0.426734\pi$
$83$	4.15699	0.456289	0.228144	+	0.973627i	$0.426734\pi$
$84$	0	0
$85$	−1.40632	−0.152536
$86$	6.96962	0.751554
$87$	0	0
$88$	−1.89653	−0.202171
$89$	9.23009	0.978387	0.489194	−	0.872175i	$-0.337291\pi$
$89$	9.23009	0.978387	0.489194	+	0.872175i	$0.337291\pi$
$90$	0	0
$91$	−18.7660	−1.96721
$92$	12.7245	1.32662
$93$	0	0
$94$	−16.1570	−1.66647
$95$	1.00000	0.102598
$96$	0	0
$97$	11.5116	1.16882	0.584411	−	0.811457i	$-0.301325\pi$
$97$	11.5116	1.16882	0.584411	+	0.811457i	$0.301325\pi$
$98$	−11.1433	−1.12564
$99$	0	0
$100$	1.82571	0.182571
$101$	11.8511	1.17923	0.589616	−	0.807684i	$-0.299279\pi$
$101$	11.8511	1.17923	0.589616	+	0.807684i	$0.299279\pi$
$102$	0	0
$103$	1.35458	0.133471	0.0667354	−	0.997771i	$-0.478742\pi$
$103$	1.35458	0.133471	0.0667354	+	0.997771i	$0.478742\pi$
$104$	1.79533	0.176047
$105$	0	0
$106$	−14.8946	−1.44669
$107$	7.06287	0.682794	0.341397	−	0.939919i	$-0.389100\pi$
$107$	7.06287	0.682794	0.341397	+	0.939919i	$0.389100\pi$
$108$	0	0
$109$	10.1844	0.975484	0.487742	−	0.872988i	$-0.337821\pi$
$109$	10.1844	0.975484	0.487742	+	0.872988i	$0.337821\pi$
$110$	10.8815	1.03751
$111$	0	0
$112$	15.3871	1.45394
$113$	−1.86015	−0.174988	−0.0874940	−	0.996165i	$-0.527886\pi$
$113$	−1.86015	−0.174988	−0.0874940	+	0.996165i	$0.527886\pi$
$114$	0	0
$115$	6.96962	0.649921
$116$	−2.56753	−0.238389
$117$	0	0
$118$	18.5329	1.70609
$119$	5.01114	0.459370
$120$	0	0
$121$	19.9504	1.81367
$122$	−18.0235	−1.63177
$123$	0	0
$124$	3.20393	0.287721
$125$	1.00000	0.0894427
$126$	0	0
$127$	−4.89053	−0.433964	−0.216982	−	0.976176i	$-0.569621\pi$
$127$	−4.89053	−0.433964	−0.216982	+	0.976176i	$0.569621\pi$
$128$	−2.71684	−0.240137
$129$	0	0
$130$	−10.3009	−0.903449
$131$	−2.81263	−0.245741	−0.122870	−	0.992423i	$-0.539210\pi$
$131$	−2.81263	−0.245741	−0.122870	+	0.992423i	$0.539210\pi$
$132$	0	0
$133$	−3.56331	−0.308978
$134$	9.31204	0.804438
$135$	0	0
$136$	−0.479412	−0.0411093
$137$	−9.23009	−0.788579	−0.394290	−	0.918986i	$-0.629009\pi$
$137$	−9.23009	−0.788579	−0.394290	+	0.918986i	$0.629009\pi$
$138$	0	0
$139$	−3.67878	−0.312030	−0.156015	−	0.987755i	$-0.549865\pi$
$139$	−3.67878	−0.312030	−0.156015	+	0.987755i	$0.549865\pi$
$140$	−6.50557	−0.549821
$141$	0	0
$142$	−27.5179	−2.30925
$143$	−29.2990	−2.45010
$144$	0	0
$145$	−1.40632	−0.116788
$146$	−12.8969	−1.06735
$147$	0	0
$148$	6.60002	0.542519
$149$	−7.09925	−0.581593	−0.290797	−	0.956785i	$-0.593920\pi$
$149$	−7.09925	−0.581593	−0.290797	+	0.956785i	$0.593920\pi$
$150$	0	0
$151$	−18.3567	−1.49385	−0.746924	−	0.664910i	$-0.768470\pi$
$151$	−18.3567	−1.49385	−0.746924	+	0.664910i	$0.768470\pi$
$152$	0.340899	0.0276506
$153$	0	0
$154$	−38.7742	−3.12451
$155$	1.75489	0.140957
$156$	0	0
$157$	17.2301	1.37511	0.687555	−	0.726132i	$-0.258684\pi$
$157$	17.2301	1.37511	0.687555	+	0.726132i	$0.258684\pi$
$158$	−10.7092	−0.851975
$159$	0	0
$160$	7.76435	0.613826
$161$	−24.8349	−1.95726
$162$	0	0
$163$	10.8662	0.851103	0.425551	−	0.904934i	$-0.360080\pi$
$163$	10.8662	0.851103	0.425551	+	0.904934i	$0.360080\pi$
$164$	−7.93925	−0.619951
$165$	0	0
$166$	−8.13083	−0.631075
$167$	−2.82977	−0.218974	−0.109487	−	0.993988i	$-0.534921\pi$
$167$	−2.82977	−0.218974	−0.109487	+	0.993988i	$0.534921\pi$
$168$	0	0
$169$	14.7357	1.13351
$170$	2.75067	0.210967
$171$	0	0
$172$	−6.50557	−0.496045
$173$	9.26647	0.704516	0.352258	−	0.935903i	$-0.385414\pi$
$173$	9.26647	0.704516	0.352258	+	0.935903i	$0.385414\pi$
$174$	0	0
$175$	−3.56331	−0.269361
$176$	24.0235	1.81084
$177$	0	0
$178$	−18.0535	−1.35317
$179$	3.59067	0.268379	0.134190	−	0.990956i	$-0.457157\pi$
$179$	3.59067	0.268379	0.134190	+	0.990956i	$0.457157\pi$
$180$	0	0
$181$	−19.7630	−1.46897	−0.734487	−	0.678623i	$-0.762577\pi$
$181$	−19.7630	−1.46897	−0.734487	+	0.678623i	$0.762577\pi$
$182$	36.7053	2.72078
$183$	0	0
$184$	2.37594	0.175157
$185$	3.61504	0.265783
$186$	0	0
$187$	7.82377	0.572131
$188$	15.0812	1.09991
$189$	0	0
$190$	−1.95594	−0.141899
$191$	−8.31398	−0.601579	−0.300789	−	0.953691i	$-0.597250\pi$
$191$	−8.31398	−0.601579	−0.300789	+	0.953691i	$0.597250\pi$
$192$	0	0
$193$	−22.2514	−1.60169	−0.800847	−	0.598869i	$-0.795617\pi$
$193$	−22.2514	−1.60169	−0.800847	+	0.598869i	$0.795617\pi$
$194$	−22.5160	−1.61655
$195$	0	0
$196$	10.4014	0.742954
$197$	8.81263	0.627874	0.313937	−	0.949444i	$-0.398352\pi$
$197$	8.81263	0.627874	0.313937	+	0.949444i	$0.398352\pi$
$198$	0	0
$199$	21.0659	1.49332	0.746660	−	0.665206i	$-0.231656\pi$
$199$	21.0659	1.49332	0.746660	+	0.665206i	$0.231656\pi$
$200$	0.340899	0.0241052
$201$	0	0
$202$	−23.1801	−1.63095
$203$	5.01114	0.351713
$204$	0	0
$205$	−4.34858	−0.303718
$206$	−2.64948	−0.184598
$207$	0	0
$208$	−22.7417	−1.57685
$209$	−5.56331	−0.384822
$210$	0	0
$211$	5.34556	0.368004	0.184002	−	0.982926i	$-0.441095\pi$
$211$	5.34556	0.368004	0.184002	+	0.982926i	$0.441095\pi$
$212$	13.9029	0.954853
$213$	0	0
$214$	−13.8146	−0.944345
$215$	−3.56331	−0.243016
$216$	0	0
$217$	−6.25323	−0.424497
$218$	−19.9200	−1.34915
$219$	0	0
$220$	−10.1570	−0.684784
$221$	−7.40632	−0.498203
$222$	0	0
$223$	−3.10947	−0.208226	−0.104113	−	0.994565i	$-0.533200\pi$
$223$	−3.10947	−0.208226	−0.104113	+	0.994565i	$0.533200\pi$
$224$	−27.6668	−1.84856
$225$	0	0
$226$	3.63834	0.242019
$227$	−14.4692	−0.960354	−0.480177	−	0.877172i	$-0.659428\pi$
$227$	−14.4692	−0.960354	−0.480177	+	0.877172i	$0.659428\pi$
$228$	0	0
$229$	−5.21473	−0.344599	−0.172299	−	0.985045i	$-0.555120\pi$
$229$	−5.21473	−0.344599	−0.172299	+	0.985045i	$0.555120\pi$
$230$	−13.6322	−0.898879
$231$	0	0
$232$	−0.479412	−0.0314750
$233$	3.18737	0.208811	0.104406	−	0.994535i	$-0.466706\pi$
$233$	3.18737	0.208811	0.104406	+	0.994535i	$0.466706\pi$
$234$	0	0
$235$	8.26046	0.538853
$236$	−17.2990	−1.12607
$237$	0	0
$238$	−9.80150	−0.635337
$239$	−16.5209	−1.06865	−0.534325	−	0.845279i	$-0.679434\pi$
$239$	−16.5209	−1.06865	−0.534325	+	0.845279i	$0.679434\pi$
$240$	0	0
$241$	−12.2271	−0.787615	−0.393807	−	0.919193i	$-0.628842\pi$
$241$	−12.2271	−0.787615	−0.393807	+	0.919193i	$0.628842\pi$
$242$	−39.0218	−2.50842
$243$	0	0
$244$	16.8234	1.07701
$245$	5.69716	0.363978
$246$	0	0
$247$	5.26647	0.335097
$248$	0.598242	0.0379884
$249$	0	0
$250$	−1.95594	−0.123705
$251$	−4.52093	−0.285358	−0.142679	−	0.989769i	$-0.545572\pi$
$251$	−4.52093	−0.285358	−0.142679	+	0.989769i	$0.545572\pi$
$252$	0	0
$253$	−38.7742	−2.43771
$254$	9.56559	0.600198
$255$	0	0
$256$	18.4144	1.15090
$257$	−15.9290	−0.993625	−0.496813	−	0.867858i	$-0.665496\pi$
$257$	−15.9290	−0.993625	−0.496813	+	0.867858i	$0.665496\pi$
$258$	0	0
$259$	−12.8815	−0.800418
$260$	9.61504	0.596300
$261$	0	0
$262$	5.50135	0.339874
$263$	−0.854147	−0.0526689	−0.0263345	−	0.999653i	$-0.508383\pi$
$263$	−0.854147	−0.0526689	−0.0263345	+	0.999653i	$0.508383\pi$
$264$	0	0
$265$	7.61504	0.467788
$266$	6.96962	0.427335
$267$	0	0
$268$	−8.69202	−0.530950
$269$	−10.3913	−0.633569	−0.316784	−	0.948498i	$-0.602603\pi$
$269$	−10.3913	−0.633569	−0.316784	+	0.948498i	$0.602603\pi$
$270$	0	0
$271$	−8.19971	−0.498097	−0.249048	−	0.968491i	$-0.580118\pi$
$271$	−8.19971	−0.498097	−0.249048	+	0.968491i	$0.580118\pi$
$272$	6.07276	0.368215
$273$	0	0
$274$	18.0535	1.09065
$275$	−5.56331	−0.335480
$276$	0	0
$277$	14.6484	0.880137	0.440069	−	0.897964i	$-0.354954\pi$
$277$	14.6484	0.880137	0.440069	+	0.897964i	$0.354954\pi$
$278$	7.19549	0.431557
$279$	0	0
$280$	−1.21473	−0.0725939
$281$	31.6129	1.88587	0.942935	−	0.332977i	$-0.108053\pi$
$281$	31.6129	1.88587	0.942935	+	0.332977i	$0.108053\pi$
$282$	0	0
$283$	24.7326	1.47020	0.735101	−	0.677957i	$-0.237135\pi$
$283$	24.7326	1.47020	0.735101	+	0.677957i	$0.237135\pi$
$284$	25.6857	1.52417
$285$	0	0
$286$	57.3071	3.38864
$287$	15.4953	0.914660
$288$	0	0
$289$	−15.0223	−0.883663
$290$	2.75067	0.161525
$291$	0	0
$292$	12.0382	0.704480
$293$	−30.2857	−1.76931	−0.884655	−	0.466245i	$-0.845606\pi$
$293$	−30.2857	−1.76931	−0.884655	+	0.466245i	$0.845606\pi$
$294$	0	0
$295$	−9.47519	−0.551667
$296$	1.23237	0.0716298
$297$	0	0
$298$	13.8857	0.804379
$299$	36.7053	2.12272
$300$	0	0
$301$	12.6972	0.731852
$302$	35.9046	2.06608
$303$	0	0
$304$	−4.31820	−0.247666
$305$	9.21473	0.527634
$306$	0	0
$307$	−23.0629	−1.31627	−0.658134	−	0.752901i	$-0.728654\pi$
$307$	−23.0629	−1.31627	−0.658134	+	0.752901i	$0.728654\pi$
$308$	36.1925	2.06226
$309$	0	0
$310$	−3.43247	−0.194951
$311$	10.3152	0.584921	0.292460	−	0.956278i	$-0.405526\pi$
$311$	10.3152	0.584921	0.292460	+	0.956278i	$0.405526\pi$
$312$	0	0
$313$	−4.95038	−0.279812	−0.139906	−	0.990165i	$-0.544680\pi$
$313$	−4.95038	−0.279812	−0.139906	+	0.990165i	$0.544680\pi$
$314$	−33.7011	−1.90186
$315$	0	0
$316$	9.99612	0.562326
$317$	14.2433	0.799985	0.399992	−	0.916518i	$-0.369013\pi$
$317$	14.2433	0.799985	0.399992	+	0.916518i	$0.369013\pi$
$318$	0	0
$319$	7.82377	0.438047
$320$	−6.55023	−0.366169
$321$	0	0
$322$	48.5756	2.70702
$323$	−1.40632	−0.0782495
$324$	0	0
$325$	5.26647	0.292131
$326$	−21.2536	−1.17713
$327$	0	0
$328$	−1.48243	−0.0818534
$329$	−29.4346	−1.62278
$330$	0	0
$331$	2.04272	0.112278	0.0561390	−	0.998423i	$-0.482121\pi$
$331$	2.04272	0.112278	0.0561390	+	0.998423i	$0.482121\pi$
$332$	7.58946	0.416526
$333$	0	0
$334$	5.53487	0.302855
$335$	−4.76090	−0.260116
$336$	0	0
$337$	34.6951	1.88996	0.944980	−	0.327128i	$-0.106081\pi$
$337$	34.6951	1.88996	0.944980	+	0.327128i	$0.106081\pi$
$338$	−28.8221	−1.56772
$339$	0	0
$340$	−2.56753	−0.139244
$341$	−9.76302	−0.528697
$342$	0	0
$343$	4.64243	0.250668
$344$	−1.21473	−0.0654938
$345$	0	0
$346$	−18.1247	−0.974388
$347$	−7.35280	−0.394719	−0.197359	−	0.980331i	$-0.563237\pi$
$347$	−7.35280	−0.394719	−0.197359	+	0.980331i	$0.563237\pi$
$348$	0	0
$349$	−31.7630	−1.70024	−0.850118	−	0.526593i	$-0.823469\pi$
$349$	−31.7630	−1.70024	−0.850118	+	0.526593i	$0.823469\pi$
$350$	6.96962	0.372542
$351$	0	0
$352$	−43.1955	−2.30233
$353$	7.52179	0.400345	0.200172	−	0.979761i	$-0.435850\pi$
$353$	7.52179	0.400345	0.200172	+	0.979761i	$0.435850\pi$
$354$	0	0
$355$	14.0689	0.746698
$356$	16.8515	0.893126
$357$	0	0
$358$	−7.02314	−0.371185
$359$	30.3982	1.60436	0.802178	−	0.597085i	$-0.203674\pi$
$359$	30.3982	1.60436	0.802178	+	0.597085i	$0.203674\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	38.6553	2.03168
$363$	0	0
$364$	−34.2613	−1.79578
$365$	6.59368	0.345129
$366$	0	0
$367$	3.91577	0.204401	0.102201	−	0.994764i	$-0.467412\pi$
$367$	3.91577	0.204401	0.102201	+	0.994764i	$0.467412\pi$
$368$	−30.0962	−1.56887
$369$	0	0
$370$	−7.07082	−0.367594
$371$	−27.1347	−1.40877
$372$	0	0
$373$	26.7759	1.38641	0.693203	−	0.720743i	$-0.256199\pi$
$373$	26.7759	1.38641	0.693203	+	0.720743i	$0.256199\pi$
$374$	−15.3028	−0.791291
$375$	0	0
$376$	2.81599	0.145223
$377$	−7.40632	−0.381445
$378$	0	0
$379$	14.9504	0.767950	0.383975	−	0.923344i	$-0.374555\pi$
$379$	14.9504	0.767950	0.383975	+	0.923344i	$0.374555\pi$
$380$	1.82571	0.0936570
$381$	0	0
$382$	16.2617	0.832019
$383$	27.9910	1.43027	0.715136	−	0.698985i	$-0.246365\pi$
$383$	27.9910	1.43027	0.715136	+	0.698985i	$0.246365\pi$
$384$	0	0
$385$	19.8238	1.01031
$386$	43.5225	2.21524
$387$	0	0
$388$	21.0168	1.06697
$389$	35.2036	1.78489	0.892447	−	0.451152i	$-0.148987\pi$
$389$	35.2036	1.78489	0.892447	+	0.451152i	$0.148987\pi$
$390$	0	0
$391$	−9.80150	−0.495683
$392$	1.94216	0.0980937
$393$	0	0
$394$	−17.2370	−0.868388
$395$	5.47519	0.275487
$396$	0	0
$397$	−35.9735	−1.80546	−0.902730	−	0.430208i	$-0.858440\pi$
$397$	−35.9735	−1.80546	−0.902730	+	0.430208i	$0.858440\pi$
$398$	−41.2036	−2.06535
$399$	0	0
$400$	−4.31820	−0.215910
$401$	−23.2421	−1.16065	−0.580327	−	0.814383i	$-0.697075\pi$
$401$	−23.2421	−1.16065	−0.580327	+	0.814383i	$0.697075\pi$
$402$	0	0
$403$	9.24209	0.460381
$404$	21.6367	1.07647
$405$	0	0
$406$	−9.80150	−0.486440
$407$	−20.1116	−0.996895
$408$	0	0
$409$	−31.8926	−1.57699	−0.788495	−	0.615041i	$-0.789139\pi$
$409$	−31.8926	−1.57699	−0.788495	+	0.615041i	$0.789139\pi$
$410$	8.50557	0.420060
$411$	0	0
$412$	2.47307	0.121840
$413$	33.7630	1.66137
$414$	0	0
$415$	4.15699	0.204059
$416$	40.8907	2.00483
$417$	0	0
$418$	10.8815	0.532232
$419$	−31.8238	−1.55469	−0.777346	−	0.629073i	$-0.783435\pi$
$419$	−31.8238	−1.55469	−0.777346	+	0.629073i	$0.783435\pi$
$420$	0	0
$421$	0.348578	0.0169887	0.00849433	−	0.999964i	$-0.497296\pi$
$421$	0.348578	0.0169887	0.00849433	+	0.999964i	$0.497296\pi$
$422$	−10.4556	−0.508971
$423$	0	0
$424$	2.59596	0.126071
$425$	−1.40632	−0.0682164
$426$	0	0
$427$	−32.8349	−1.58899
$428$	12.8948	0.623292
$429$	0	0
$430$	6.96962	0.336105
$431$	−29.2764	−1.41019	−0.705097	−	0.709111i	$-0.749096\pi$
$431$	−29.2764	−1.41019	−0.705097	+	0.709111i	$0.749096\pi$
$432$	0	0
$433$	−0.883290	−0.0424482	−0.0212241	−	0.999775i	$-0.506756\pi$
$433$	−0.883290	−0.0424482	−0.0212241	+	0.999775i	$0.506756\pi$
$434$	12.2310	0.587105
$435$	0	0
$436$	18.5937	0.890476
$437$	6.96962	0.333402
$438$	0	0
$439$	−13.8584	−0.661424	−0.330712	−	0.943732i	$-0.607289\pi$
$439$	−13.8584	−0.661424	−0.330712	+	0.943732i	$0.607289\pi$
$440$	−1.89653	−0.0904134
$441$	0	0
$442$	14.4863	0.689044
$443$	13.5753	0.644982	0.322491	−	0.946572i	$-0.395480\pi$
$443$	13.5753	0.644982	0.322491	+	0.946572i	$0.395480\pi$
$444$	0	0
$445$	9.23009	0.437548
$446$	6.08195	0.287989
$447$	0	0
$448$	23.3405	1.10273
$449$	15.6334	0.737785	0.368893	−	0.929472i	$-0.379737\pi$
$449$	15.6334	0.737785	0.368893	+	0.929472i	$0.379737\pi$
$450$	0	0
$451$	24.1925	1.13918
$452$	−3.39609	−0.159739
$453$	0	0
$454$	28.3009	1.32823
$455$	−18.7660	−0.879765
$456$	0	0
$457$	5.30284	0.248057	0.124028	−	0.992279i	$-0.460419\pi$
$457$	5.30284	0.248057	0.124028	+	0.992279i	$0.460419\pi$
$458$	10.1997	0.476601
$459$	0	0
$460$	12.7245	0.593284
$461$	−0.374734	−0.0174531	−0.00872656	−	0.999962i	$-0.502778\pi$
$461$	−0.374734	−0.0174531	−0.00872656	+	0.999962i	$0.502778\pi$
$462$	0	0
$463$	6.65564	0.309314	0.154657	−	0.987968i	$-0.450573\pi$
$463$	6.65564	0.309314	0.154657	+	0.987968i	$0.450573\pi$
$464$	6.07276	0.281921
$465$	0	0
$466$	−6.23431	−0.288799
$467$	0.854147	0.0395252	0.0197626	−	0.999805i	$-0.493709\pi$
$467$	0.854147	0.0395252	0.0197626	+	0.999805i	$0.493709\pi$
$468$	0	0
$469$	16.9645	0.783349
$470$	−16.1570	−0.745266
$471$	0	0
$472$	−3.23009	−0.148677
$473$	19.8238	0.911498
$474$	0	0
$475$	1.00000	0.0458831
$476$	9.14889	0.419339
$477$	0	0
$478$	32.3140	1.47801
$479$	17.0731	0.780090	0.390045	−	0.920796i	$-0.372460\pi$
$479$	17.0731	0.780090	0.390045	+	0.920796i	$0.372460\pi$
$480$	0	0
$481$	19.0385	0.868081
$482$	23.9154	1.08932
$483$	0	0
$484$	36.4236	1.65562
$485$	11.5116	0.522713
$486$	0	0
$487$	12.8259	0.581197	0.290598	−	0.956845i	$-0.406146\pi$
$487$	12.8259	0.581197	0.290598	+	0.956845i	$0.406146\pi$
$488$	3.14129	0.142200
$489$	0	0
$490$	−11.1433	−0.503403
$491$	10.4054	0.469591	0.234796	−	0.972045i	$-0.424558\pi$
$491$	10.4054	0.469591	0.234796	+	0.972045i	$0.424558\pi$
$492$	0	0
$493$	1.97773	0.0890723
$494$	−10.3009	−0.463460
$495$	0	0
$496$	−7.57799	−0.340262
$497$	−50.1317	−2.24872
$498$	0	0
$499$	36.1612	1.61880	0.809399	−	0.587258i	$-0.199793\pi$
$499$	36.1612	1.61880	0.809399	+	0.587258i	$0.199793\pi$
$500$	1.82571	0.0816483
$501$	0	0
$502$	8.84267	0.394668
$503$	−33.2536	−1.48270	−0.741352	−	0.671117i	$-0.765815\pi$
$503$	−33.2536	−1.48270	−0.741352	+	0.671117i	$0.765815\pi$
$504$	0	0
$505$	11.8511	0.527368
$506$	75.8400	3.37150
$507$	0	0
$508$	−8.92869	−0.396146
$509$	−7.29084	−0.323161	−0.161580	−	0.986860i	$-0.551659\pi$
$509$	−7.29084	−0.323161	−0.161580	+	0.986860i	$0.551659\pi$
$510$	0	0
$511$	−23.4953	−1.03937
$512$	−30.5839	−1.35163
$513$	0	0
$514$	31.1563	1.37424
$515$	1.35458	0.0596899
$516$	0	0
$517$	−45.9555	−2.02112
$518$	25.1955	1.10703
$519$	0	0
$520$	1.79533	0.0787306
$521$	9.87849	0.432785	0.216392	−	0.976306i	$-0.430571\pi$
$521$	9.87849	0.432785	0.216392	+	0.976306i	$0.430571\pi$
$522$	0	0
$523$	−11.1925	−0.489414	−0.244707	−	0.969597i	$-0.578692\pi$
$523$	−11.1925	−0.489414	−0.244707	+	0.969597i	$0.578692\pi$
$524$	−5.13505	−0.224326
$525$	0	0
$526$	1.67066	0.0728443
$527$	−2.46794	−0.107505
$528$	0	0
$529$	25.5756	1.11198
$530$	−14.8946	−0.646979
$531$	0	0
$532$	−6.50557	−0.282052
$533$	−22.9016	−0.991980
$534$	0	0
$535$	7.06287	0.305355
$536$	−1.62299	−0.0701023
$537$	0	0
$538$	20.3248	0.876263
$539$	−31.6950	−1.36520
$540$	0	0
$541$	2.22587	0.0956974	0.0478487	−	0.998855i	$-0.484763\pi$
$541$	2.22587	0.0956974	0.0478487	+	0.998855i	$0.484763\pi$
$542$	16.0382	0.688898
$543$	0	0
$544$	−10.9191	−0.468154
$545$	10.1844	0.436250
$546$	0	0
$547$	−34.9675	−1.49510	−0.747552	−	0.664204i	$-0.768771\pi$
$547$	−34.9675	−1.49510	−0.747552	+	0.664204i	$0.768771\pi$
$548$	−16.8515	−0.719859
$549$	0	0
$550$	10.8815	0.463989
$551$	−1.40632	−0.0599111
$552$	0	0
$553$	−19.5098	−0.829641
$554$	−28.6514	−1.21728
$555$	0	0
$556$	−6.71640	−0.284839
$557$	8.88539	0.376486	0.188243	−	0.982122i	$-0.439721\pi$
$557$	8.88539	0.376486	0.188243	+	0.982122i	$0.439721\pi$
$558$	0	0
$559$	−18.7660	−0.793719
$560$	15.3871	0.650223
$561$	0	0
$562$	−61.8331	−2.60827
$563$	29.6767	1.25072	0.625362	−	0.780335i	$-0.284952\pi$
$563$	29.6767	1.25072	0.625362	+	0.780335i	$0.284952\pi$
$564$	0	0
$565$	−1.86015	−0.0782570
$566$	−48.3756	−2.03338
$567$	0	0
$568$	4.79607	0.201239
$569$	22.1152	0.927116	0.463558	−	0.886067i	$-0.346573\pi$
$569$	22.1152	0.927116	0.463558	+	0.886067i	$0.346573\pi$
$570$	0	0
$571$	−33.2656	−1.39212	−0.696060	−	0.717983i	$-0.745065\pi$
$571$	−33.2656	−1.39212	−0.696060	+	0.717983i	$0.745065\pi$
$572$	−53.4914	−2.23659
$573$	0	0
$574$	−30.3080	−1.26503
$575$	6.96962	0.290653
$576$	0	0
$577$	0.934140	0.0388887	0.0194444	−	0.999811i	$-0.493810\pi$
$577$	0.934140	0.0388887	0.0194444	+	0.999811i	$0.493810\pi$
$578$	29.3827	1.22216
$579$	0	0
$580$	−2.56753	−0.106611
$581$	−14.8126	−0.614532
$582$	0	0
$583$	−42.3648	−1.75457
$584$	2.24778	0.0930139
$585$	0	0
$586$	59.2371	2.44706
$587$	−1.57531	−0.0650200	−0.0325100	−	0.999471i	$-0.510350\pi$
$587$	−1.57531	−0.0650200	−0.0325100	+	0.999471i	$0.510350\pi$
$588$	0	0
$589$	1.75489	0.0723092
$590$	18.5329	0.762989
$591$	0	0
$592$	−15.6105	−0.641587
$593$	−26.0162	−1.06836	−0.534180	−	0.845371i	$-0.679379\pi$
$593$	−26.0162	−1.06836	−0.534180	+	0.845371i	$0.679379\pi$
$594$	0	0
$595$	5.01114	0.205437
$596$	−12.9612	−0.530911
$597$	0	0
$598$	−71.7934	−2.93585
$599$	−19.5359	−0.798217	−0.399109	−	0.916904i	$-0.630680\pi$
$599$	−19.5359	−0.798217	−0.399109	+	0.916904i	$0.630680\pi$
$600$	0	0
$601$	43.5299	1.77562	0.887811	−	0.460208i	$-0.152225\pi$
$601$	43.5299	1.77562	0.887811	+	0.460208i	$0.152225\pi$
$602$	−24.8349	−1.01220
$603$	0	0
$604$	−33.5140	−1.36367
$605$	19.9504	0.811098
$606$	0	0
$607$	−12.5847	−0.510796	−0.255398	−	0.966836i	$-0.582206\pi$
$607$	−12.5847	−0.510796	−0.255398	+	0.966836i	$0.582206\pi$
$608$	7.76435	0.314886
$609$	0	0
$610$	−18.0235	−0.729749
$611$	43.5034	1.75996
$612$	0	0
$613$	−18.2412	−0.736756	−0.368378	−	0.929676i	$-0.620087\pi$
$613$	−18.2412	−0.736756	−0.368378	+	0.929676i	$0.620087\pi$
$614$	45.1097	1.82048
$615$	0	0
$616$	6.75791	0.272284
$617$	−26.7314	−1.07617	−0.538084	−	0.842892i	$-0.680852\pi$
$617$	−26.7314	−1.07617	−0.538084	+	0.842892i	$0.680852\pi$
$618$	0	0
$619$	2.92690	0.117642	0.0588211	−	0.998269i	$-0.481266\pi$
$619$	2.92690	0.117642	0.0588211	+	0.998269i	$0.481266\pi$
$620$	3.20393	0.128673
$621$	0	0
$622$	−20.1759	−0.808980
$623$	−32.8896	−1.31770
$624$	0	0
$625$	1.00000	0.0400000
$626$	9.68267	0.386997
$627$	0	0
$628$	31.4572	1.25528
$629$	−5.08389	−0.202708
$630$	0	0
$631$	−13.2493	−0.527447	−0.263724	−	0.964598i	$-0.584951\pi$
$631$	−13.2493	−0.527447	−0.263724	+	0.964598i	$0.584951\pi$
$632$	1.86649	0.0742450
$633$	0	0
$634$	−27.8591	−1.10643
$635$	−4.89053	−0.194075
$636$	0	0
$637$	30.0039	1.18880
$638$	−15.3028	−0.605845
$639$	0	0
$640$	−2.71684	−0.107392
$641$	−6.93026	−0.273729	−0.136864	−	0.990590i	$-0.543702\pi$
$641$	−6.93026	−0.273729	−0.136864	+	0.990590i	$0.543702\pi$
$642$	0	0
$643$	14.4794	0.571012	0.285506	−	0.958377i	$-0.407838\pi$
$643$	14.4794	0.571012	0.285506	+	0.958377i	$0.407838\pi$
$644$	−45.3414	−1.78670
$645$	0	0
$646$	2.75067	0.108224
$647$	−6.35549	−0.249860	−0.124930	−	0.992166i	$-0.539871\pi$
$647$	−6.35549	−0.249860	−0.124930	+	0.992166i	$0.539871\pi$
$648$	0	0
$649$	52.7134	2.06918
$650$	−10.3009	−0.404035
$651$	0	0
$652$	19.8385	0.776934
$653$	34.4030	1.34629	0.673146	−	0.739509i	$-0.264942\pi$
$653$	34.4030	1.34629	0.673146	+	0.739509i	$0.264942\pi$
$654$	0	0
$655$	−2.81263	−0.109899
$656$	18.7780	0.733159
$657$	0	0
$658$	57.5723	2.24440
$659$	19.6214	0.764341	0.382170	−	0.924092i	$-0.375177\pi$
$659$	19.6214	0.764341	0.382170	+	0.924092i	$0.375177\pi$
$660$	0	0
$661$	39.8054	1.54825	0.774126	−	0.633032i	$-0.218190\pi$
$661$	39.8054	1.54825	0.774126	+	0.633032i	$0.218190\pi$
$662$	−3.99544	−0.155287
$663$	0	0
$664$	1.41712	0.0549947
$665$	−3.56331	−0.138179
$666$	0	0
$667$	−9.80150	−0.379515
$668$	−5.16635	−0.199892
$669$	0	0
$670$	9.31204	0.359755
$671$	−51.2644	−1.97904
$672$	0	0
$673$	−8.90374	−0.343214	−0.171607	−	0.985166i	$-0.554896\pi$
$673$	−8.90374	−0.343214	−0.171607	+	0.985166i	$0.554896\pi$
$674$	−67.8615	−2.61393
$675$	0	0
$676$	26.9030	1.03473
$677$	−22.2695	−0.855886	−0.427943	−	0.903806i	$-0.640762\pi$
$677$	−22.2695	−0.855886	−0.427943	+	0.903806i	$0.640762\pi$
$678$	0	0
$679$	−41.0193	−1.57417
$680$	−0.479412	−0.0183846
$681$	0	0
$682$	19.0959	0.731220
$683$	−15.4054	−0.589472	−0.294736	−	0.955579i	$-0.595232\pi$
$683$	−15.4054	−0.589472	−0.294736	+	0.955579i	$0.595232\pi$
$684$	0	0
$685$	−9.23009	−0.352663
$686$	−9.08033	−0.346689
$687$	0	0
$688$	15.3871	0.586627
$689$	40.1044	1.52785
$690$	0	0
$691$	−17.8773	−0.680084	−0.340042	−	0.940410i	$-0.610441\pi$
$691$	−17.8773	−0.680084	−0.340042	+	0.940410i	$0.610441\pi$
$692$	16.9179	0.643122
$693$	0	0
$694$	14.3817	0.545920
$695$	−3.67878	−0.139544
$696$	0	0
$697$	6.11548	0.231640
$698$	62.1266	2.35153
$699$	0	0
$700$	−6.50557	−0.245887
$701$	−35.3609	−1.33556	−0.667782	−	0.744357i	$-0.732756\pi$
$701$	−35.3609	−1.33556	−0.667782	+	0.744357i	$0.732756\pi$
$702$	0	0
$703$	3.61504	0.136344
$704$	36.4409	1.37342
$705$	0	0
$706$	−14.7122	−0.553701
$707$	−42.2292	−1.58819
$708$	0	0
$709$	6.90410	0.259289	0.129644	−	0.991561i	$-0.458616\pi$
$709$	6.90410	0.259289	0.129644	+	0.991561i	$0.458616\pi$
$710$	−27.5179	−1.03273
$711$	0	0
$712$	3.14653	0.117921
$713$	12.2310	0.458053
$714$	0	0
$715$	−29.2990	−1.09572
$716$	6.55552	0.244991
$717$	0	0
$718$	−59.4572	−2.21892
$719$	38.9431	1.45233	0.726167	−	0.687518i	$-0.241300\pi$
$719$	38.9431	1.45233	0.726167	+	0.687518i	$0.241300\pi$
$720$	0	0
$721$	−4.82678	−0.179759
$722$	−1.95594	−0.0727926
$723$	0	0
$724$	−36.0816	−1.34096
$725$	−1.40632	−0.0522293
$726$	0	0
$727$	−6.18347	−0.229332	−0.114666	−	0.993404i	$-0.536580\pi$
$727$	−6.18347	−0.229332	−0.114666	+	0.993404i	$0.536580\pi$
$728$	−6.39733	−0.237101
$729$	0	0
$730$	−12.8969	−0.477334
$731$	5.01114	0.185344
$732$	0	0
$733$	−0.688715	−0.0254383	−0.0127191	−	0.999919i	$-0.504049\pi$
$733$	−0.688715	−0.0254383	−0.0127191	+	0.999919i	$0.504049\pi$
$734$	−7.65902	−0.282699
$735$	0	0
$736$	54.1146	1.99469
$737$	26.4863	0.975636
$738$	0	0
$739$	−26.2532	−0.965741	−0.482870	−	0.875692i	$-0.660406\pi$
$739$	−26.2532	−0.965741	−0.482870	+	0.875692i	$0.660406\pi$
$740$	6.60002	0.242622
$741$	0	0
$742$	53.0740	1.94841
$743$	32.5688	1.19483	0.597416	−	0.801931i	$-0.296194\pi$
$743$	32.5688	1.19483	0.597416	+	0.801931i	$0.296194\pi$
$744$	0	0
$745$	−7.09925	−0.260096
$746$	−52.3722	−1.91748
$747$	0	0
$748$	14.2839	0.522273
$749$	−25.1672	−0.919589
$750$	0	0
$751$	45.4833	1.65971	0.829855	−	0.557979i	$-0.188423\pi$
$751$	45.4833	1.65971	0.829855	+	0.557979i	$0.188423\pi$
$752$	−35.6703	−1.30076
$753$	0	0
$754$	14.4863	0.527561
$755$	−18.3567	−0.668069
$756$	0	0
$757$	2.74947	0.0999311	0.0499656	−	0.998751i	$-0.484089\pi$
$757$	2.74947	0.0999311	0.0499656	+	0.998751i	$0.484089\pi$
$758$	−29.2421	−1.06212
$759$	0	0
$760$	0.340899	0.0123657
$761$	33.2978	1.20704	0.603521	−	0.797347i	$-0.293764\pi$
$761$	33.2978	1.20704	0.603521	+	0.797347i	$0.293764\pi$
$762$	0	0
$763$	−36.2900	−1.31379
$764$	−15.1789	−0.549154
$765$	0	0
$766$	−54.7488	−1.97815
$767$	−49.9008	−1.80181
$768$	0	0
$769$	−19.1540	−0.690710	−0.345355	−	0.938472i	$-0.612241\pi$
$769$	−19.1540	−0.690710	−0.345355	+	0.938472i	$0.612241\pi$
$770$	−38.7742	−1.39732
$771$	0	0
$772$	−40.6247	−1.46212
$773$	−0.569309	−0.0204766	−0.0102383	−	0.999948i	$-0.503259\pi$
$773$	−0.569309	−0.0204766	−0.0102383	+	0.999948i	$0.503259\pi$
$774$	0	0
$775$	1.75489	0.0630377
$776$	3.92429	0.140874
$777$	0	0
$778$	−68.8562	−2.46862
$779$	−4.34858	−0.155804
$780$	0	0
$781$	−78.2695	−2.80070
$782$	19.1712	0.685559
$783$	0	0
$784$	−24.6015	−0.878624
$785$	17.2301	0.614968
$786$	0	0
$787$	15.9991	0.570307	0.285153	−	0.958482i	$-0.407956\pi$
$787$	15.9991	0.570307	0.285153	+	0.958482i	$0.407956\pi$
$788$	16.0893	0.573158
$789$	0	0
$790$	−10.7092	−0.381015
$791$	6.62828	0.235675
$792$	0	0
$793$	48.5290	1.72332
$794$	70.3621	2.49706
$795$	0	0
$796$	38.4602	1.36318
$797$	35.7528	1.26643	0.633214	−	0.773976i	$-0.281735\pi$
$797$	35.7528	1.26643	0.633214	+	0.773976i	$0.281735\pi$
$798$	0	0
$799$	−11.6168	−0.410974
$800$	7.76435	0.274511
$801$	0	0
$802$	45.4602	1.60526
$803$	−36.6827	−1.29450
$804$	0	0
$805$	−24.8349	−0.875315
$806$	−18.0770	−0.636735
$807$	0	0
$808$	4.04004	0.142128
$809$	−23.2036	−0.815796	−0.407898	−	0.913028i	$-0.633738\pi$
$809$	−23.2036	−0.815796	−0.407898	+	0.913028i	$0.633738\pi$
$810$	0	0
$811$	−21.7549	−0.763918	−0.381959	−	0.924179i	$-0.624750\pi$
$811$	−21.7549	−0.763918	−0.381959	+	0.924179i	$0.624750\pi$
$812$	9.14889	0.321063
$813$	0	0
$814$	39.3371	1.37877
$815$	10.8662	0.380625
$816$	0	0
$817$	−3.56331	−0.124664
$818$	62.3802	2.18107
$819$	0	0
$820$	−7.93925	−0.277251
$821$	−52.2532	−1.82365	−0.911825	−	0.410579i	$-0.865327\pi$
$821$	−52.2532	−1.82365	−0.911825	+	0.410579i	$0.865327\pi$
$822$	0	0
$823$	−9.44783	−0.329331	−0.164665	−	0.986349i	$-0.552654\pi$
$823$	−9.44783	−0.329331	−0.164665	+	0.986349i	$0.552654\pi$
$824$	0.461775	0.0160867
$825$	0	0
$826$	−66.0385	−2.29777
$827$	50.2216	1.74638	0.873188	−	0.487383i	$-0.162048\pi$
$827$	50.2216	1.74638	0.873188	+	0.487383i	$0.162048\pi$
$828$	0	0
$829$	44.2066	1.53536	0.767680	−	0.640834i	$-0.221411\pi$
$829$	44.2066	1.53536	0.767680	+	0.640834i	$0.221411\pi$
$830$	−8.13083	−0.282225
$831$	0	0
$832$	−34.4966	−1.19595
$833$	−8.01200	−0.277599
$834$	0	0
$835$	−2.82977	−0.0979283
$836$	−10.1570	−0.351287
$837$	0	0
$838$	62.2455	2.15023
$839$	30.4033	1.04964	0.524819	−	0.851214i	$-0.324133\pi$
$839$	30.4033	1.04964	0.524819	+	0.851214i	$0.324133\pi$
$840$	0	0
$841$	−27.0223	−0.931803
$842$	−0.681799	−0.0234963
$843$	0	0
$844$	9.75945	0.335934
$845$	14.7357	0.506922
$846$	0	0
$847$	−71.0893	−2.44266
$848$	−32.8833	−1.12922
$849$	0	0
$850$	2.75067	0.0943473
$851$	25.1955	0.863690
$852$	0	0
$853$	3.93925	0.134877	0.0674386	−	0.997723i	$-0.478517\pi$
$853$	3.93925	0.134877	0.0674386	+	0.997723i	$0.478517\pi$
$854$	64.2232	2.19767
$855$	0	0
$856$	2.40773	0.0822945
$857$	−27.4388	−0.937292	−0.468646	−	0.883386i	$-0.655258\pi$
$857$	−27.4388	−0.937292	−0.468646	+	0.883386i	$0.655258\pi$
$858$	0	0
$859$	32.4517	1.10724	0.553619	−	0.832770i	$-0.313246\pi$
$859$	32.4517	1.10724	0.553619	+	0.832770i	$0.313246\pi$
$860$	−6.50557	−0.221838
$861$	0	0
$862$	57.2629	1.95038
$863$	2.10861	0.0717778	0.0358889	−	0.999356i	$-0.488574\pi$
$863$	2.10861	0.0717778	0.0358889	+	0.999356i	$0.488574\pi$
$864$	0	0
$865$	9.26647	0.315069
$866$	1.72766	0.0587084
$867$	0	0
$868$	−11.4166	−0.387504
$869$	−30.4602	−1.03329
$870$	0	0
$871$	−25.0731	−0.849569
$872$	3.47184	0.117571
$873$	0	0
$874$	−13.6322	−0.461115
$875$	−3.56331	−0.120462
$876$	0	0
$877$	−37.6613	−1.27173	−0.635866	−	0.771799i	$-0.719357\pi$
$877$	−37.6613	−1.27173	−0.635866	+	0.771799i	$0.719357\pi$
$878$	27.1062	0.914789
$879$	0	0
$880$	24.0235	0.809831
$881$	39.8818	1.34365	0.671827	−	0.740708i	$-0.265510\pi$
$881$	39.8818	1.34365	0.671827	+	0.740708i	$0.265510\pi$
$882$	0	0
$883$	36.0458	1.21304	0.606518	−	0.795070i	$-0.292566\pi$
$883$	36.0458	1.21304	0.606518	+	0.795070i	$0.292566\pi$
$884$	−13.5218	−0.454787
$885$	0	0
$886$	−26.5525	−0.892050
$887$	26.2057	0.879902	0.439951	−	0.898022i	$-0.354996\pi$
$887$	26.2057	0.879902	0.439951	+	0.898022i	$0.354996\pi$
$888$	0	0
$889$	17.4264	0.584464
$890$	−18.0535	−0.605155
$891$	0	0
$892$	−5.67700	−0.190080
$893$	8.26046	0.276426
$894$	0	0
$895$	3.59067	0.120023
$896$	9.68093	0.323417
$897$	0	0
$898$	−30.5780	−1.02040
$899$	−2.46794	−0.0823103
$900$	0	0
$901$	−10.7092	−0.356774
$902$	−47.3191	−1.57555
$903$	0	0
$904$	−0.634123	−0.0210906
$905$	−19.7630	−0.656945
$906$	0	0
$907$	−22.8036	−0.757182	−0.378591	−	0.925564i	$-0.623591\pi$
$907$	−22.8036	−0.757182	−0.378591	+	0.925564i	$0.623591\pi$
$908$	−26.4166	−0.876664
$909$	0	0
$910$	36.7053	1.21677
$911$	−4.44146	−0.147152	−0.0735761	−	0.997290i	$-0.523441\pi$
$911$	−4.44146	−0.147152	−0.0735761	+	0.997290i	$0.523441\pi$
$912$	0	0
$913$	−23.1266	−0.765379
$914$	−10.3721	−0.343077
$915$	0	0
$916$	−9.52059	−0.314569
$917$	10.0223	0.330965
$918$	0	0
$919$	37.0111	1.22088	0.610442	−	0.792061i	$-0.290992\pi$
$919$	37.0111	1.22088	0.610442	+	0.792061i	$0.290992\pi$
$920$	2.37594	0.0783324
$921$	0	0
$922$	0.732959	0.0241387
$923$	74.0932	2.43881
$924$	0	0
$925$	3.61504	0.118862
$926$	−13.0181	−0.427800
$927$	0	0
$928$	−10.9191	−0.358438
$929$	3.04185	0.0997999	0.0499000	−	0.998754i	$-0.484110\pi$
$929$	3.04185	0.0997999	0.0499000	+	0.998754i	$0.484110\pi$
$930$	0	0
$931$	5.69716	0.186717
$932$	5.81921	0.190615
$933$	0	0
$934$	−1.67066	−0.0546657
$935$	7.82377	0.255865
$936$	0	0
$937$	38.6099	1.26133	0.630666	−	0.776055i	$-0.282782\pi$
$937$	38.6099	1.26133	0.630666	+	0.776055i	$0.282782\pi$
$938$	−33.1817	−1.08342
$939$	0	0
$940$	15.0812	0.491895
$941$	−2.69716	−0.0879248	−0.0439624	−	0.999033i	$-0.513998\pi$
$941$	−2.69716	−0.0879248	−0.0439624	+	0.999033i	$0.513998\pi$
$942$	0	0
$943$	−30.3080	−0.986963
$944$	40.9158	1.33170
$945$	0	0
$946$	−38.7742	−1.26066
$947$	20.1877	0.656012	0.328006	−	0.944676i	$-0.393623\pi$
$947$	20.1877	0.656012	0.328006	+	0.944676i	$0.393623\pi$
$948$	0	0
$949$	34.7254	1.12723
$950$	−1.95594	−0.0634592
$951$	0	0
$952$	1.70829	0.0553661
$953$	−5.18559	−0.167978	−0.0839888	−	0.996467i	$-0.526766\pi$
$953$	−5.18559	−0.167978	−0.0839888	+	0.996467i	$0.526766\pi$
$954$	0	0
$955$	−8.31398	−0.269034
$956$	−30.1624	−0.975523
$957$	0	0
$958$	−33.3940	−1.07891
$959$	32.8896	1.06206
$960$	0	0
$961$	−27.9203	−0.900656
$962$	−37.2382	−1.20061
$963$	0	0
$964$	−22.3231	−0.718979
$965$	−22.2514	−0.716299
$966$	0	0
$967$	58.0054	1.86533	0.932665	−	0.360744i	$-0.117477\pi$
$967$	58.0054	1.86533	0.932665	+	0.360744i	$0.117477\pi$
$968$	6.80107	0.218595
$969$	0	0
$970$	−22.5160	−0.722944
$971$	24.3221	0.780533	0.390267	−	0.920702i	$-0.372383\pi$
$971$	24.3221	0.780533	0.390267	+	0.920702i	$0.372383\pi$
$972$	0	0
$973$	13.1086	0.420244
$974$	−25.0867	−0.803830
$975$	0	0
$976$	−39.7911	−1.27368
$977$	36.1134	1.15537	0.577685	−	0.816260i	$-0.303956\pi$
$977$	36.1134	1.15537	0.577685	+	0.816260i	$0.303956\pi$
$978$	0	0
$979$	−51.3498	−1.64115
$980$	10.4014	0.332259
$981$	0	0
$982$	−20.3525	−0.649473
$983$	−21.1603	−0.674909	−0.337455	−	0.941342i	$-0.609566\pi$
$983$	−21.1603	−0.674909	−0.337455	+	0.941342i	$0.609566\pi$
$984$	0	0
$985$	8.81263	0.280794
$986$	−3.86832	−0.123192
$987$	0	0
$988$	9.61504	0.305895
$989$	−24.8349	−0.789704
$990$	0	0
$991$	20.2652	0.643746	0.321873	−	0.946783i	$-0.395688\pi$
$991$	20.2652	0.643746	0.321873	+	0.946783i	$0.395688\pi$
$992$	13.6256	0.432614
$993$	0	0
$994$	98.0548	3.11011
$995$	21.0659	0.667833
$996$	0	0
$997$	24.2875	0.769193	0.384597	−	0.923085i	$-0.374341\pi$
$997$	24.2875	0.769193	0.384597	+	0.923085i	$0.374341\pi$
$998$	−70.7293	−2.23890
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	855.2.a.m.1.1		4
3.2	odd	2		95.2.a.b.1.4	✓	4
5.4	even	2		4275.2.a.bo.1.4		4
12.11	even	2		1520.2.a.t.1.3		4
15.2	even	4		475.2.b.e.324.6		8
15.8	even	4		475.2.b.e.324.3		8
15.14	odd	2		475.2.a.i.1.1		4
21.20	even	2		4655.2.a.y.1.4		4
24.5	odd	2		6080.2.a.cc.1.3		4
24.11	even	2		6080.2.a.ch.1.2		4
57.56	even	2		1805.2.a.p.1.1		4
60.59	even	2		7600.2.a.cf.1.2		4
285.284	even	2		9025.2.a.bf.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.2.a.b.1.4	✓	4	3.2	odd	2
475.2.a.i.1.1		4	15.14	odd	2
475.2.b.e.324.3		8	15.8	even	4
475.2.b.e.324.6		8	15.2	even	4
855.2.a.m.1.1		4	1.1	even	1	trivial
1520.2.a.t.1.3		4	12.11	even	2
1805.2.a.p.1.1		4	57.56	even	2
4275.2.a.bo.1.4		4	5.4	even	2
4655.2.a.y.1.4		4	21.20	even	2
6080.2.a.cc.1.3		4	24.5	odd	2
6080.2.a.ch.1.2		4	24.11	even	2
7600.2.a.cf.1.2		4	60.59	even	2
9025.2.a.bf.1.4		4	285.284	even	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:	\( N \)	\(=\)	\( 855 = 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	855.a (trivial)

\(f(q)\)	\(=\)	\(q-1.95594 q^{2} +1.82571 q^{4} +1.00000 q^{5} -3.56331 q^{7} +0.340899 q^{8} +O(q^{10})\)	\(q-1.95594 q^{2} +1.82571 q^{4} +1.00000 q^{5} -3.56331 q^{7} +0.340899 q^{8} -1.95594 q^{10} -5.56331 q^{11} +5.26647 q^{13} +6.96962 q^{14} -4.31820 q^{16} -1.40632 q^{17} +1.00000 q^{19} +1.82571 q^{20} +10.8815 q^{22} +6.96962 q^{23} +1.00000 q^{25} -10.3009 q^{26} -6.50557 q^{28} -1.40632 q^{29} +1.75489 q^{31} +7.76435 q^{32} +2.75067 q^{34} -3.56331 q^{35} +3.61504 q^{37} -1.95594 q^{38} +0.340899 q^{40} -4.34858 q^{41} -3.56331 q^{43} -10.1570 q^{44} -13.6322 q^{46} +8.26046 q^{47} +5.69716 q^{49} -1.95594 q^{50} +9.61504 q^{52} +7.61504 q^{53} -5.56331 q^{55} -1.21473 q^{56} +2.75067 q^{58} -9.47519 q^{59} +9.21473 q^{61} -3.43247 q^{62} -6.55023 q^{64} +5.26647 q^{65} -4.76090 q^{67} -2.56753 q^{68} +6.96962 q^{70} +14.0689 q^{71} +6.59368 q^{73} -7.07082 q^{74} +1.82571 q^{76} +19.8238 q^{77} +5.47519 q^{79} -4.31820 q^{80} +8.50557 q^{82} +4.15699 q^{83} -1.40632 q^{85} +6.96962 q^{86} -1.89653 q^{88} +9.23009 q^{89} -18.7660 q^{91} +12.7245 q^{92} -16.1570 q^{94} +1.00000 q^{95} +11.5116 q^{97} -11.1433 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 8 q^{4} + 4 q^{5} + 4 q^{7} + 12 q^{8}+O(q^{10}) \) 4 * q + 2 * q^2 + 8 * q^4 + 4 * q^5 + 4 * q^7 + 12 * q^8	\( 4 q + 2 q^{2} + 8 q^{4} + 4 q^{5} + 4 q^{7} + 12 q^{8} + 2 q^{10} - 4 q^{11} + 2 q^{13} + 8 q^{14} + 4 q^{16} - 4 q^{17} + 4 q^{19} + 8 q^{20} + 4 q^{22} + 8 q^{23} + 4 q^{25} - 4 q^{26} - 8 q^{28} - 4 q^{29} + 4 q^{31} + 6 q^{32} - 4 q^{34} + 4 q^{35} - 6 q^{37} + 2 q^{38} + 12 q^{40} - 16 q^{41} + 4 q^{43} - 24 q^{44} + 12 q^{47} + 20 q^{49} + 2 q^{50} + 18 q^{52} + 10 q^{53} - 4 q^{55} + 12 q^{56} - 4 q^{58} + 20 q^{61} - 20 q^{62} - 4 q^{64} + 2 q^{65} - 18 q^{67} - 4 q^{68} + 8 q^{70} + 20 q^{71} + 28 q^{73} - 32 q^{74} + 8 q^{76} + 40 q^{77} - 16 q^{79} + 4 q^{80} + 16 q^{82} - 4 q^{85} + 8 q^{86} - 12 q^{88} - 4 q^{89} - 36 q^{91} + 28 q^{92} - 48 q^{94} + 4 q^{95} + 30 q^{97} - 38 q^{98}+O(q^{100}) \) 4 * q + 2 * q^2 + 8 * q^4 + 4 * q^5 + 4 * q^7 + 12 * q^8 + 2 * q^10 - 4 * q^11 + 2 * q^13 + 8 * q^14 + 4 * q^16 - 4 * q^17 + 4 * q^19 + 8 * q^20 + 4 * q^22 + 8 * q^23 + 4 * q^25 - 4 * q^26 - 8 * q^28 - 4 * q^29 + 4 * q^31 + 6 * q^32 - 4 * q^34 + 4 * q^35 - 6 * q^37 + 2 * q^38 + 12 * q^40 - 16 * q^41 + 4 * q^43 - 24 * q^44 + 12 * q^47 + 20 * q^49 + 2 * q^50 + 18 * q^52 + 10 * q^53 - 4 * q^55 + 12 * q^56 - 4 * q^58 + 20 * q^61 - 20 * q^62 - 4 * q^64 + 2 * q^65 - 18 * q^67 - 4 * q^68 + 8 * q^70 + 20 * q^71 + 28 * q^73 - 32 * q^74 + 8 * q^76 + 40 * q^77 - 16 * q^79 + 4 * q^80 + 16 * q^82 - 4 * q^85 + 8 * q^86 - 12 * q^88 - 4 * q^89 - 36 * q^91 + 28 * q^92 - 48 * q^94 + 4 * q^95 + 30 * q^97 - 38 * q^98

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