LMFDB - Embedded newform 7098.2.a.ca.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7098,2,Mod(1,7098)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7098, 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("7098.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7098.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:	$56.6778153547$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{57}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 14 $ x^2 - x - 14
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 546)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.27492$ of defining polynomial
Character	$\chi$	$=$	7098.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.27492 q^{5} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -3.27492 q^{10} -4.00000 q^{11} +1.00000 q^{12} +1.00000 q^{14} -3.27492 q^{15} +1.00000 q^{16} +3.00000 q^{17} +1.00000 q^{18} -8.54983 q^{19} -3.27492 q^{20} +1.00000 q^{21} -4.00000 q^{22} +2.27492 q^{23} +1.00000 q^{24} +5.72508 q^{25} +1.00000 q^{27} +1.00000 q^{28} +0.725083 q^{29} -3.27492 q^{30} +6.27492 q^{31} +1.00000 q^{32} -4.00000 q^{33} +3.00000 q^{34} -3.27492 q^{35} +1.00000 q^{36} -7.27492 q^{37} -8.54983 q^{38} -3.27492 q^{40} +0.725083 q^{41} +1.00000 q^{42} +10.8248 q^{43} -4.00000 q^{44} -3.27492 q^{45} +2.27492 q^{46} +8.54983 q^{47} +1.00000 q^{48} +1.00000 q^{49} +5.72508 q^{50} +3.00000 q^{51} +11.5498 q^{53} +1.00000 q^{54} +13.0997 q^{55} +1.00000 q^{56} -8.54983 q^{57} +0.725083 q^{58} +10.8248 q^{59} -3.27492 q^{60} -9.00000 q^{61} +6.27492 q^{62} +1.00000 q^{63} +1.00000 q^{64} -4.00000 q^{66} -10.2749 q^{67} +3.00000 q^{68} +2.27492 q^{69} -3.27492 q^{70} -2.27492 q^{71} +1.00000 q^{72} +13.2749 q^{73} -7.27492 q^{74} +5.72508 q^{75} -8.54983 q^{76} -4.00000 q^{77} -8.00000 q^{79} -3.27492 q^{80} +1.00000 q^{81} +0.725083 q^{82} +10.8248 q^{83} +1.00000 q^{84} -9.82475 q^{85} +10.8248 q^{86} +0.725083 q^{87} -4.00000 q^{88} +8.27492 q^{89} -3.27492 q^{90} +2.27492 q^{92} +6.27492 q^{93} +8.54983 q^{94} +28.0000 q^{95} +1.00000 q^{96} +14.5498 q^{97} +1.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + q^{5} + 2 q^{6} + 2 q^{7} + 2 q^{8} + 2 q^{9} + q^{10} - 8 q^{11} + 2 q^{12} + 2 q^{14} + q^{15} + 2 q^{16} + 6 q^{17} + 2 q^{18} - 2 q^{19} + q^{20} + 2 q^{21} - 8 q^{22}+ \cdots - 8 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + q^5 + 2 * q^6 + 2 * q^7 + 2 * q^8 + 2 * q^9 + q^10 - 8 * q^11 + 2 * q^12 + 2 * q^14 + q^15 + 2 * q^16 + 6 * q^17 + 2 * q^18 - 2 * q^19 + q^20 + 2 * q^21 - 8 * q^22 - 3 * q^23 + 2 * q^24 + 19 * q^25 + 2 * q^27 + 2 * q^28 + 9 * q^29 + q^30 + 5 * q^31 + 2 * q^32 - 8 * q^33 + 6 * q^34 + q^35 + 2 * q^36 - 7 * q^37 - 2 * q^38 + q^40 + 9 * q^41 + 2 * q^42 - q^43 - 8 * q^44 + q^45 - 3 * q^46 + 2 * q^47 + 2 * q^48 + 2 * q^49 + 19 * q^50 + 6 * q^51 + 8 * q^53 + 2 * q^54 - 4 * q^55 + 2 * q^56 - 2 * q^57 + 9 * q^58 - q^59 + q^60 - 18 * q^61 + 5 * q^62 + 2 * q^63 + 2 * q^64 - 8 * q^66 - 13 * q^67 + 6 * q^68 - 3 * q^69 + q^70 + 3 * q^71 + 2 * q^72 + 19 * q^73 - 7 * q^74 + 19 * q^75 - 2 * q^76 - 8 * q^77 - 16 * q^79 + q^80 + 2 * q^81 + 9 * q^82 - q^83 + 2 * q^84 + 3 * q^85 - q^86 + 9 * q^87 - 8 * q^88 + 9 * q^89 + q^90 - 3 * q^92 + 5 * q^93 + 2 * q^94 + 56 * q^95 + 2 * q^96 + 14 * q^97 + 2 * q^98 - 8 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	−3.27492	−1.46459	−0.732294	−	0.680989i	$-0.761550\pi$
$5$	−3.27492	−1.46459	−0.732294	+	0.680989i	$0.761550\pi$
$6$	1.00000	0.408248
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−3.27492	−1.03562
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	1.00000	0.288675
$13$	0	0
$13$	0	0
$14$	1.00000	0.267261
$15$	−3.27492	−0.845580
$16$	1.00000	0.250000
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	1.00000	0.235702
$19$	−8.54983	−1.96147	−0.980733	−	0.195352i	$-0.937415\pi$
$19$	−8.54983	−1.96147	−0.980733	+	0.195352i	$0.937415\pi$
$20$	−3.27492	−0.732294
$21$	1.00000	0.218218
$22$	−4.00000	−0.852803
$23$	2.27492	0.474353	0.237177	−	0.971467i	$-0.423778\pi$
$23$	2.27492	0.474353	0.237177	+	0.971467i	$0.423778\pi$
$24$	1.00000	0.204124
$25$	5.72508	1.14502
$26$	0	0
$27$	1.00000	0.192450
$28$	1.00000	0.188982
$29$	0.725083	0.134644	0.0673222	−	0.997731i	$-0.478554\pi$
$29$	0.725083	0.134644	0.0673222	+	0.997731i	$0.478554\pi$
$30$	−3.27492	−0.597915
$31$	6.27492	1.12701	0.563504	−	0.826113i	$-0.309453\pi$
$31$	6.27492	1.12701	0.563504	+	0.826113i	$0.309453\pi$
$32$	1.00000	0.176777
$33$	−4.00000	−0.696311
$34$	3.00000	0.514496
$35$	−3.27492	−0.553562
$36$	1.00000	0.166667
$37$	−7.27492	−1.19599	−0.597995	−	0.801500i	$-0.704036\pi$
$37$	−7.27492	−1.19599	−0.597995	+	0.801500i	$0.704036\pi$
$38$	−8.54983	−1.38697
$39$	0	0
$40$	−3.27492	−0.517810
$41$	0.725083	0.113239	0.0566195	−	0.998396i	$-0.481968\pi$
$41$	0.725083	0.113239	0.0566195	+	0.998396i	$0.481968\pi$
$42$	1.00000	0.154303
$43$	10.8248	1.65076	0.825380	−	0.564578i	$-0.190961\pi$
$43$	10.8248	1.65076	0.825380	+	0.564578i	$0.190961\pi$
$44$	−4.00000	−0.603023
$45$	−3.27492	−0.488196
$46$	2.27492	0.335418
$47$	8.54983	1.24712	0.623561	−	0.781775i	$-0.285685\pi$
$47$	8.54983	1.24712	0.623561	+	0.781775i	$0.285685\pi$
$48$	1.00000	0.144338
$49$	1.00000	0.142857
$50$	5.72508	0.809649
$51$	3.00000	0.420084
$52$	0	0
$53$	11.5498	1.58649	0.793246	−	0.608901i	$-0.208390\pi$
$53$	11.5498	1.58649	0.793246	+	0.608901i	$0.208390\pi$
$54$	1.00000	0.136083
$55$	13.0997	1.76636
$56$	1.00000	0.133631
$57$	−8.54983	−1.13245
$58$	0.725083	0.0952080
$59$	10.8248	1.40926	0.704631	−	0.709574i	$-0.251112\pi$
$59$	10.8248	1.40926	0.704631	+	0.709574i	$0.251112\pi$
$60$	−3.27492	−0.422790
$61$	−9.00000	−1.15233	−0.576166	−	0.817333i	$-0.695452\pi$
$61$	−9.00000	−1.15233	−0.576166	+	0.817333i	$0.695452\pi$
$62$	6.27492	0.796915
$63$	1.00000	0.125988
$64$	1.00000	0.125000
$65$	0	0
$66$	−4.00000	−0.492366
$67$	−10.2749	−1.25528	−0.627640	−	0.778503i	$-0.715979\pi$
$67$	−10.2749	−1.25528	−0.627640	+	0.778503i	$0.715979\pi$
$68$	3.00000	0.363803
$69$	2.27492	0.273868
$70$	−3.27492	−0.391427
$71$	−2.27492	−0.269983	−0.134992	−	0.990847i	$-0.543101\pi$
$71$	−2.27492	−0.269983	−0.134992	+	0.990847i	$0.543101\pi$
$72$	1.00000	0.117851
$73$	13.2749	1.55371	0.776856	−	0.629679i	$-0.216813\pi$
$73$	13.2749	1.55371	0.776856	+	0.629679i	$0.216813\pi$
$74$	−7.27492	−0.845692
$75$	5.72508	0.661076
$76$	−8.54983	−0.980733
$77$	−4.00000	−0.455842
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	−3.27492	−0.366147
$81$	1.00000	0.111111
$82$	0.725083	0.0800720
$83$	10.8248	1.18817	0.594085	−	0.804402i	$-0.297514\pi$
$83$	10.8248	1.18817	0.594085	+	0.804402i	$0.297514\pi$
$84$	1.00000	0.109109
$85$	−9.82475	−1.06564
$86$	10.8248	1.16726
$87$	0.725083	0.0777370
$88$	−4.00000	−0.426401
$89$	8.27492	0.877139	0.438570	−	0.898697i	$-0.355485\pi$
$89$	8.27492	0.877139	0.438570	+	0.898697i	$0.355485\pi$
$90$	−3.27492	−0.345207
$91$	0	0
$92$	2.27492	0.237177
$93$	6.27492	0.650679
$94$	8.54983	0.881848
$95$	28.0000	2.87274
$96$	1.00000	0.102062
$97$	14.5498	1.47731	0.738656	−	0.674083i	$-0.235461\pi$
$97$	14.5498	1.47731	0.738656	+	0.674083i	$0.235461\pi$
$98$	1.00000	0.101015
$99$	−4.00000	−0.402015
$100$	5.72508	0.572508
$101$	9.27492	0.922889	0.461444	−	0.887169i	$-0.347331\pi$
$101$	9.27492	0.922889	0.461444	+	0.887169i	$0.347331\pi$
$102$	3.00000	0.297044
$103$	−2.27492	−0.224154	−0.112077	−	0.993700i	$-0.535750\pi$
$103$	−2.27492	−0.224154	−0.112077	+	0.993700i	$0.535750\pi$
$104$	0	0
$105$	−3.27492	−0.319599
$106$	11.5498	1.12182
$107$	−12.5498	−1.21324	−0.606619	−	0.794993i	$-0.707475\pi$
$107$	−12.5498	−1.21324	−0.606619	+	0.794993i	$0.707475\pi$
$108$	1.00000	0.0962250
$109$	10.5498	1.01049	0.505245	−	0.862976i	$-0.331402\pi$
$109$	10.5498	1.01049	0.505245	+	0.862976i	$0.331402\pi$
$110$	13.0997	1.24900
$111$	−7.27492	−0.690505
$112$	1.00000	0.0944911
$113$	−7.82475	−0.736091	−0.368045	−	0.929808i	$-0.619973\pi$
$113$	−7.82475	−0.736091	−0.368045	+	0.929808i	$0.619973\pi$
$114$	−8.54983	−0.800765
$115$	−7.45017	−0.694732
$116$	0.725083	0.0673222
$117$	0	0
$118$	10.8248	0.996499
$119$	3.00000	0.275010
$120$	−3.27492	−0.298958
$121$	5.00000	0.454545
$122$	−9.00000	−0.814822
$123$	0.725083	0.0653785
$124$	6.27492	0.563504
$125$	−2.37459	−0.212389
$126$	1.00000	0.0890871
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	1.00000	0.0883883
$129$	10.8248	0.953066
$130$	0	0
$131$	−10.2749	−0.897724	−0.448862	−	0.893601i	$-0.648170\pi$
$131$	−10.2749	−0.897724	−0.448862	+	0.893601i	$0.648170\pi$
$132$	−4.00000	−0.348155
$133$	−8.54983	−0.741365
$134$	−10.2749	−0.887618
$135$	−3.27492	−0.281860
$136$	3.00000	0.257248
$137$	0.175248	0.0149725	0.00748624	−	0.999972i	$-0.497617\pi$
$137$	0.175248	0.0149725	0.00748624	+	0.999972i	$0.497617\pi$
$138$	2.27492	0.193654
$139$	21.0997	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$139$	21.0997	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$140$	−3.27492	−0.276781
$141$	8.54983	0.720026
$142$	−2.27492	−0.190907
$143$	0	0
$144$	1.00000	0.0833333
$145$	−2.37459	−0.197199
$146$	13.2749	1.09864
$147$	1.00000	0.0824786
$148$	−7.27492	−0.597995
$149$	−1.00000	−0.0819232	−0.0409616	−	0.999161i	$-0.513042\pi$
$149$	−1.00000	−0.0819232	−0.0409616	+	0.999161i	$0.513042\pi$
$150$	5.72508	0.467451
$151$	−17.0997	−1.39155	−0.695776	−	0.718259i	$-0.744939\pi$
$151$	−17.0997	−1.39155	−0.695776	+	0.718259i	$0.744939\pi$
$152$	−8.54983	−0.693483
$153$	3.00000	0.242536
$154$	−4.00000	−0.322329
$155$	−20.5498	−1.65060
$156$	0	0
$157$	−11.2749	−0.899836	−0.449918	−	0.893070i	$-0.648547\pi$
$157$	−11.2749	−0.899836	−0.449918	+	0.893070i	$0.648547\pi$
$158$	−8.00000	−0.636446
$159$	11.5498	0.915961
$160$	−3.27492	−0.258905
$161$	2.27492	0.179289
$162$	1.00000	0.0785674
$163$	−5.72508	−0.448423	−0.224212	−	0.974540i	$-0.571981\pi$
$163$	−5.72508	−0.448423	−0.224212	+	0.974540i	$0.571981\pi$
$164$	0.725083	0.0566195
$165$	13.0997	1.01981
$166$	10.8248	0.840164
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	1.00000	0.0771517
$169$	0	0
$170$	−9.82475	−0.753524
$171$	−8.54983	−0.653822
$172$	10.8248	0.825380
$173$	10.5498	0.802089	0.401045	−	0.916059i	$-0.368647\pi$
$173$	10.5498	0.802089	0.401045	+	0.916059i	$0.368647\pi$
$174$	0.725083	0.0549684
$175$	5.72508	0.432776
$176$	−4.00000	−0.301511
$177$	10.8248	0.813638
$178$	8.27492	0.620231
$179$	16.5498	1.23699	0.618496	−	0.785788i	$-0.287742\pi$
$179$	16.5498	1.23699	0.618496	+	0.785788i	$0.287742\pi$
$180$	−3.27492	−0.244098
$181$	5.82475	0.432950	0.216475	−	0.976288i	$-0.430544\pi$
$181$	5.82475	0.432950	0.216475	+	0.976288i	$0.430544\pi$
$182$	0	0
$183$	−9.00000	−0.665299
$184$	2.27492	0.167709
$185$	23.8248	1.75163
$186$	6.27492	0.460099
$187$	−12.0000	−0.877527
$188$	8.54983	0.623561
$189$	1.00000	0.0727393
$190$	28.0000	2.03133
$191$	14.2749	1.03290	0.516448	−	0.856318i	$-0.327254\pi$
$191$	14.2749	1.03290	0.516448	+	0.856318i	$0.327254\pi$
$192$	1.00000	0.0721688
$193$	−2.17525	−0.156578	−0.0782889	−	0.996931i	$-0.524946\pi$
$193$	−2.17525	−0.156578	−0.0782889	+	0.996931i	$0.524946\pi$
$194$	14.5498	1.04462
$195$	0	0
$196$	1.00000	0.0714286
$197$	0.824752	0.0587611	0.0293806	−	0.999568i	$-0.490647\pi$
$197$	0.824752	0.0587611	0.0293806	+	0.999568i	$0.490647\pi$
$198$	−4.00000	−0.284268
$199$	−10.8248	−0.767346	−0.383673	−	0.923469i	$-0.625341\pi$
$199$	−10.8248	−0.767346	−0.383673	+	0.923469i	$0.625341\pi$
$200$	5.72508	0.404824
$201$	−10.2749	−0.724737
$202$	9.27492	0.652581
$203$	0.725083	0.0508908
$204$	3.00000	0.210042
$205$	−2.37459	−0.165848
$206$	−2.27492	−0.158501
$207$	2.27492	0.158118
$208$	0	0
$209$	34.1993	2.36562
$210$	−3.27492	−0.225991
$211$	−9.09967	−0.626447	−0.313224	−	0.949679i	$-0.601409\pi$
$211$	−9.09967	−0.626447	−0.313224	+	0.949679i	$0.601409\pi$
$212$	11.5498	0.793246
$213$	−2.27492	−0.155875
$214$	−12.5498	−0.857889
$215$	−35.4502	−2.41768
$216$	1.00000	0.0680414
$217$	6.27492	0.425969
$218$	10.5498	0.714525
$219$	13.2749	0.897036
$220$	13.0997	0.883179
$221$	0	0
$222$	−7.27492	−0.488260
$223$	18.2749	1.22378	0.611889	−	0.790943i	$-0.290410\pi$
$223$	18.2749	1.22378	0.611889	+	0.790943i	$0.290410\pi$
$224$	1.00000	0.0668153
$225$	5.72508	0.381672
$226$	−7.82475	−0.520495
$227$	9.09967	0.603966	0.301983	−	0.953313i	$-0.402351\pi$
$227$	9.09967	0.603966	0.301983	+	0.953313i	$0.402351\pi$
$228$	−8.54983	−0.566227
$229$	20.2749	1.33980	0.669902	−	0.742449i	$-0.266336\pi$
$229$	20.2749	1.33980	0.669902	+	0.742449i	$0.266336\pi$
$230$	−7.45017	−0.491249
$231$	−4.00000	−0.263181
$232$	0.725083	0.0476040
$233$	22.5498	1.47729	0.738644	−	0.674095i	$-0.235466\pi$
$233$	22.5498	1.47729	0.738644	+	0.674095i	$0.235466\pi$
$234$	0	0
$235$	−28.0000	−1.82652
$236$	10.8248	0.704631
$237$	−8.00000	−0.519656
$238$	3.00000	0.194461
$239$	2.27492	0.147152	0.0735761	−	0.997290i	$-0.476559\pi$
$239$	2.27492	0.147152	0.0735761	+	0.997290i	$0.476559\pi$
$240$	−3.27492	−0.211395
$241$	−23.2749	−1.49927	−0.749635	−	0.661852i	$-0.769771\pi$
$241$	−23.2749	−1.49927	−0.749635	+	0.661852i	$0.769771\pi$
$242$	5.00000	0.321412
$243$	1.00000	0.0641500
$244$	−9.00000	−0.576166
$245$	−3.27492	−0.209227
$246$	0.725083	0.0462296
$247$	0	0
$248$	6.27492	0.398458
$249$	10.8248	0.685991
$250$	−2.37459	−0.150182
$251$	5.72508	0.361364	0.180682	−	0.983542i	$-0.442170\pi$
$251$	5.72508	0.361364	0.180682	+	0.983542i	$0.442170\pi$
$252$	1.00000	0.0629941
$253$	−9.09967	−0.572091
$254$	8.00000	0.501965
$255$	−9.82475	−0.615250
$256$	1.00000	0.0625000
$257$	15.0000	0.935674	0.467837	−	0.883815i	$-0.345033\pi$
$257$	15.0000	0.935674	0.467837	+	0.883815i	$0.345033\pi$
$258$	10.8248	0.673920
$259$	−7.27492	−0.452041
$260$	0	0
$261$	0.725083	0.0448815
$262$	−10.2749	−0.634787
$263$	−9.09967	−0.561110	−0.280555	−	0.959838i	$-0.590518\pi$
$263$	−9.09967	−0.561110	−0.280555	+	0.959838i	$0.590518\pi$
$264$	−4.00000	−0.246183
$265$	−37.8248	−2.32356
$266$	−8.54983	−0.524224
$267$	8.27492	0.506417
$268$	−10.2749	−0.627640
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	−3.27492	−0.199305
$271$	17.7251	1.07672	0.538361	−	0.842714i	$-0.319044\pi$
$271$	17.7251	1.07672	0.538361	+	0.842714i	$0.319044\pi$
$272$	3.00000	0.181902
$273$	0	0
$274$	0.175248	0.0105871
$275$	−22.9003	−1.38094
$276$	2.27492	0.136934
$277$	−7.82475	−0.470144	−0.235072	−	0.971978i	$-0.575533\pi$
$277$	−7.82475	−0.470144	−0.235072	+	0.971978i	$0.575533\pi$
$278$	21.0997	1.26547
$279$	6.27492	0.375669
$280$	−3.27492	−0.195714
$281$	1.82475	0.108856	0.0544278	−	0.998518i	$-0.482667\pi$
$281$	1.82475	0.108856	0.0544278	+	0.998518i	$0.482667\pi$
$282$	8.54983	0.509135
$283$	−24.5498	−1.45934	−0.729668	−	0.683801i	$-0.760325\pi$
$283$	−24.5498	−1.45934	−0.729668	+	0.683801i	$0.760325\pi$
$284$	−2.27492	−0.134992
$285$	28.0000	1.65858
$286$	0	0
$287$	0.725083	0.0428003
$288$	1.00000	0.0589256
$289$	−8.00000	−0.470588
$290$	−2.37459	−0.139440
$291$	14.5498	0.852926
$292$	13.2749	0.776856
$293$	−10.7251	−0.626566	−0.313283	−	0.949660i	$-0.601429\pi$
$293$	−10.7251	−0.626566	−0.313283	+	0.949660i	$0.601429\pi$
$294$	1.00000	0.0583212
$295$	−35.4502	−2.06399
$296$	−7.27492	−0.422846
$297$	−4.00000	−0.232104
$298$	−1.00000	−0.0579284
$299$	0	0
$300$	5.72508	0.330538
$301$	10.8248	0.623928
$302$	−17.0997	−0.983975
$303$	9.27492	0.532830
$304$	−8.54983	−0.490367
$305$	29.4743	1.68769
$306$	3.00000	0.171499
$307$	−16.5498	−0.944549	−0.472274	−	0.881452i	$-0.656567\pi$
$307$	−16.5498	−0.944549	−0.472274	+	0.881452i	$0.656567\pi$
$308$	−4.00000	−0.227921
$309$	−2.27492	−0.129416
$310$	−20.5498	−1.16715
$311$	12.5498	0.711636	0.355818	−	0.934555i	$-0.384202\pi$
$311$	12.5498	0.711636	0.355818	+	0.934555i	$0.384202\pi$
$312$	0	0
$313$	−27.6495	−1.56284	−0.781421	−	0.624004i	$-0.785505\pi$
$313$	−27.6495	−1.56284	−0.781421	+	0.624004i	$0.785505\pi$
$314$	−11.2749	−0.636280
$315$	−3.27492	−0.184521
$316$	−8.00000	−0.450035
$317$	7.54983	0.424041	0.212020	−	0.977265i	$-0.431996\pi$
$317$	7.54983	0.424041	0.212020	+	0.977265i	$0.431996\pi$
$318$	11.5498	0.647683
$319$	−2.90033	−0.162387
$320$	−3.27492	−0.183073
$321$	−12.5498	−0.700463
$322$	2.27492	0.126776
$323$	−25.6495	−1.42718
$324$	1.00000	0.0555556
$325$	0	0
$326$	−5.72508	−0.317083
$327$	10.5498	0.583407
$328$	0.725083	0.0400360
$329$	8.54983	0.471368
$330$	13.0997	0.721113
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	10.8248	0.594085
$333$	−7.27492	−0.398663
$334$	0	0
$335$	33.6495	1.83847
$336$	1.00000	0.0545545
$337$	−7.27492	−0.396290	−0.198145	−	0.980173i	$-0.563492\pi$
$337$	−7.27492	−0.396290	−0.198145	+	0.980173i	$0.563492\pi$
$338$	0	0
$339$	−7.82475	−0.424982
$340$	−9.82475	−0.532822
$341$	−25.0997	−1.35922
$342$	−8.54983	−0.462322
$343$	1.00000	0.0539949
$344$	10.8248	0.583631
$345$	−7.45017	−0.401103
$346$	10.5498	0.567163
$347$	15.4502	0.829408	0.414704	−	0.909956i	$-0.363885\pi$
$347$	15.4502	0.829408	0.414704	+	0.909956i	$0.363885\pi$
$348$	0.725083	0.0388685
$349$	17.9244	0.959472	0.479736	−	0.877413i	$-0.340732\pi$
$349$	17.9244	0.959472	0.479736	+	0.877413i	$0.340732\pi$
$350$	5.72508	0.306019
$351$	0	0
$352$	−4.00000	−0.213201
$353$	−4.45017	−0.236858	−0.118429	−	0.992962i	$-0.537786\pi$
$353$	−4.45017	−0.236858	−0.118429	+	0.992962i	$0.537786\pi$
$354$	10.8248	0.575329
$355$	7.45017	0.395414
$356$	8.27492	0.438570
$357$	3.00000	0.158777
$358$	16.5498	0.874686
$359$	29.0997	1.53582	0.767911	−	0.640557i	$-0.221296\pi$
$359$	29.0997	1.53582	0.767911	+	0.640557i	$0.221296\pi$
$360$	−3.27492	−0.172603
$361$	54.0997	2.84735
$362$	5.82475	0.306142
$363$	5.00000	0.262432
$364$	0	0
$365$	−43.4743	−2.27555
$366$	−9.00000	−0.470438
$367$	5.72508	0.298847	0.149423	−	0.988773i	$-0.452258\pi$
$367$	5.72508	0.298847	0.149423	+	0.988773i	$0.452258\pi$
$368$	2.27492	0.118588
$369$	0.725083	0.0377463
$370$	23.8248	1.23859
$371$	11.5498	0.599638
$372$	6.27492	0.325339
$373$	4.17525	0.216186	0.108093	−	0.994141i	$-0.465526\pi$
$373$	4.17525	0.216186	0.108093	+	0.994141i	$0.465526\pi$
$374$	−12.0000	−0.620505
$375$	−2.37459	−0.122623
$376$	8.54983	0.440924
$377$	0	0
$378$	1.00000	0.0514344
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	28.0000	1.43637
$381$	8.00000	0.409852
$382$	14.2749	0.730368
$383$	−38.1993	−1.95189	−0.975947	−	0.218006i	$-0.930045\pi$
$383$	−38.1993	−1.95189	−0.975947	+	0.218006i	$0.930045\pi$
$384$	1.00000	0.0510310
$385$	13.0997	0.667621
$386$	−2.17525	−0.110717
$387$	10.8248	0.550253
$388$	14.5498	0.738656
$389$	20.6495	1.04697	0.523486	−	0.852034i	$-0.324631\pi$
$389$	20.6495	1.04697	0.523486	+	0.852034i	$0.324631\pi$
$390$	0	0
$391$	6.82475	0.345143
$392$	1.00000	0.0505076
$393$	−10.2749	−0.518301
$394$	0.824752	0.0415504
$395$	26.1993	1.31823
$396$	−4.00000	−0.201008
$397$	15.7251	0.789219	0.394610	−	0.918849i	$-0.370880\pi$
$397$	15.7251	0.789219	0.394610	+	0.918849i	$0.370880\pi$
$398$	−10.8248	−0.542596
$399$	−8.54983	−0.428027
$400$	5.72508	0.286254
$401$	5.27492	0.263417	0.131708	−	0.991289i	$-0.457954\pi$
$401$	5.27492	0.263417	0.131708	+	0.991289i	$0.457954\pi$
$402$	−10.2749	−0.512466
$403$	0	0
$404$	9.27492	0.461444
$405$	−3.27492	−0.162732
$406$	0.725083	0.0359853
$407$	29.0997	1.44242
$408$	3.00000	0.148522
$409$	22.9244	1.13354	0.566770	−	0.823876i	$-0.308193\pi$
$409$	22.9244	1.13354	0.566770	+	0.823876i	$0.308193\pi$
$410$	−2.37459	−0.117272
$411$	0.175248	0.00864436
$412$	−2.27492	−0.112077
$413$	10.8248	0.532651
$414$	2.27492	0.111806
$415$	−35.4502	−1.74018
$416$	0	0
$417$	21.0997	1.03326
$418$	34.1993	1.67274
$419$	15.3746	0.751098	0.375549	−	0.926803i	$-0.377454\pi$
$419$	15.3746	0.751098	0.375549	+	0.926803i	$0.377454\pi$
$420$	−3.27492	−0.159800
$421$	0.725083	0.0353384	0.0176692	−	0.999844i	$-0.494375\pi$
$421$	0.725083	0.0353384	0.0176692	+	0.999844i	$0.494375\pi$
$422$	−9.09967	−0.442965
$423$	8.54983	0.415707
$424$	11.5498	0.560910
$425$	17.1752	0.833122
$426$	−2.27492	−0.110220
$427$	−9.00000	−0.435541
$428$	−12.5498	−0.606619
$429$	0	0
$430$	−35.4502	−1.70956
$431$	−5.72508	−0.275768	−0.137884	−	0.990448i	$-0.544030\pi$
$431$	−5.72508	−0.275768	−0.137884	+	0.990448i	$0.544030\pi$
$432$	1.00000	0.0481125
$433$	−2.72508	−0.130959	−0.0654796	−	0.997854i	$-0.520858\pi$
$433$	−2.72508	−0.130959	−0.0654796	+	0.997854i	$0.520858\pi$
$434$	6.27492	0.301206
$435$	−2.37459	−0.113853
$436$	10.5498	0.505245
$437$	−19.4502	−0.930428
$438$	13.2749	0.634300
$439$	−17.0997	−0.816123	−0.408061	−	0.912955i	$-0.633795\pi$
$439$	−17.0997	−0.816123	−0.408061	+	0.912955i	$0.633795\pi$
$440$	13.0997	0.624502
$441$	1.00000	0.0476190
$442$	0	0
$443$	19.4502	0.924105	0.462053	−	0.886853i	$-0.347113\pi$
$443$	19.4502	0.924105	0.462053	+	0.886853i	$0.347113\pi$
$444$	−7.27492	−0.345252
$445$	−27.0997	−1.28465
$446$	18.2749	0.865342
$447$	−1.00000	−0.0472984
$448$	1.00000	0.0472456
$449$	11.0997	0.523826	0.261913	−	0.965092i	$-0.415647\pi$
$449$	11.0997	0.523826	0.261913	+	0.965092i	$0.415647\pi$
$450$	5.72508	0.269883
$451$	−2.90033	−0.136571
$452$	−7.82475	−0.368045
$453$	−17.0997	−0.803413
$454$	9.09967	0.427069
$455$	0	0
$456$	−8.54983	−0.400383
$457$	−17.0000	−0.795226	−0.397613	−	0.917553i	$-0.630161\pi$
$457$	−17.0000	−0.795226	−0.397613	+	0.917553i	$0.630161\pi$
$458$	20.2749	0.947385
$459$	3.00000	0.140028
$460$	−7.45017	−0.347366
$461$	18.3746	0.855790	0.427895	−	0.903829i	$-0.359255\pi$
$461$	18.3746	0.855790	0.427895	+	0.903829i	$0.359255\pi$
$462$	−4.00000	−0.186097
$463$	4.54983	0.211449	0.105724	−	0.994395i	$-0.466284\pi$
$463$	4.54983	0.211449	0.105724	+	0.994395i	$0.466284\pi$
$464$	0.725083	0.0336611
$465$	−20.5498	−0.952976
$466$	22.5498	1.04460
$467$	−30.8248	−1.42640	−0.713200	−	0.700961i	$-0.752755\pi$
$467$	−30.8248	−1.42640	−0.713200	+	0.700961i	$0.752755\pi$
$468$	0	0
$469$	−10.2749	−0.474452
$470$	−28.0000	−1.29154
$471$	−11.2749	−0.519521
$472$	10.8248	0.498250
$473$	−43.2990	−1.99089
$474$	−8.00000	−0.367452
$475$	−48.9485	−2.24591
$476$	3.00000	0.137505
$477$	11.5498	0.528831
$478$	2.27492	0.104052
$479$	20.0000	0.913823	0.456912	−	0.889512i	$-0.348956\pi$
$479$	20.0000	0.913823	0.456912	+	0.889512i	$0.348956\pi$
$480$	−3.27492	−0.149479
$481$	0	0
$482$	−23.2749	−1.06014
$483$	2.27492	0.103512
$484$	5.00000	0.227273
$485$	−47.6495	−2.16365
$486$	1.00000	0.0453609
$487$	29.0997	1.31863	0.659316	−	0.751866i	$-0.270846\pi$
$487$	29.0997	1.31863	0.659316	+	0.751866i	$0.270846\pi$
$488$	−9.00000	−0.407411
$489$	−5.72508	−0.258897
$490$	−3.27492	−0.147946
$491$	−5.09967	−0.230145	−0.115072	−	0.993357i	$-0.536710\pi$
$491$	−5.09967	−0.230145	−0.115072	+	0.993357i	$0.536710\pi$
$492$	0.725083	0.0326893
$493$	2.17525	0.0979683
$494$	0	0
$495$	13.0997	0.588786
$496$	6.27492	0.281752
$497$	−2.27492	−0.102044
$498$	10.8248	0.485069
$499$	−9.17525	−0.410741	−0.205370	−	0.978684i	$-0.565840\pi$
$499$	−9.17525	−0.410741	−0.205370	+	0.978684i	$0.565840\pi$
$500$	−2.37459	−0.106195
$501$	0	0
$502$	5.72508	0.255523
$503$	9.09967	0.405734	0.202867	−	0.979206i	$-0.434974\pi$
$503$	9.09967	0.405734	0.202867	+	0.979206i	$0.434974\pi$
$504$	1.00000	0.0445435
$505$	−30.3746	−1.35165
$506$	−9.09967	−0.404530
$507$	0	0
$508$	8.00000	0.354943
$509$	−35.8248	−1.58790	−0.793952	−	0.607980i	$-0.791980\pi$
$509$	−35.8248	−1.58790	−0.793952	+	0.607980i	$0.791980\pi$
$510$	−9.82475	−0.435047
$511$	13.2749	0.587248
$512$	1.00000	0.0441942
$513$	−8.54983	−0.377484
$514$	15.0000	0.661622
$515$	7.45017	0.328294
$516$	10.8248	0.476533
$517$	−34.1993	−1.50409
$518$	−7.27492	−0.319642
$519$	10.5498	0.463086
$520$	0	0
$521$	1.82475	0.0799438	0.0399719	−	0.999201i	$-0.487273\pi$
$521$	1.82475	0.0799438	0.0399719	+	0.999201i	$0.487273\pi$
$522$	0.725083	0.0317360
$523$	34.1993	1.49543	0.747716	−	0.664018i	$-0.231150\pi$
$523$	34.1993	1.49543	0.747716	+	0.664018i	$0.231150\pi$
$524$	−10.2749	−0.448862
$525$	5.72508	0.249863
$526$	−9.09967	−0.396764
$527$	18.8248	0.820019
$528$	−4.00000	−0.174078
$529$	−17.8248	−0.774989
$530$	−37.8248	−1.64300
$531$	10.8248	0.469754
$532$	−8.54983	−0.370682
$533$	0	0
$534$	8.27492	0.358091
$535$	41.0997	1.77689
$536$	−10.2749	−0.443809
$537$	16.5498	0.714178
$538$	−18.0000	−0.776035
$539$	−4.00000	−0.172292
$540$	−3.27492	−0.140930
$541$	−12.3746	−0.532025	−0.266013	−	0.963970i	$-0.585706\pi$
$541$	−12.3746	−0.532025	−0.266013	+	0.963970i	$0.585706\pi$
$542$	17.7251	0.761357
$543$	5.82475	0.249964
$544$	3.00000	0.128624
$545$	−34.5498	−1.47995
$546$	0	0
$547$	33.0997	1.41524	0.707620	−	0.706593i	$-0.249769\pi$
$547$	33.0997	1.41524	0.707620	+	0.706593i	$0.249769\pi$
$548$	0.175248	0.00748624
$549$	−9.00000	−0.384111
$550$	−22.9003	−0.976473
$551$	−6.19934	−0.264101
$552$	2.27492	0.0968269
$553$	−8.00000	−0.340195
$554$	−7.82475	−0.332442
$555$	23.8248	1.01130
$556$	21.0997	0.894825
$557$	25.1993	1.06773	0.533865	−	0.845570i	$-0.320739\pi$
$557$	25.1993	1.06773	0.533865	+	0.845570i	$0.320739\pi$
$558$	6.27492	0.265638
$559$	0	0
$560$	−3.27492	−0.138391
$561$	−12.0000	−0.506640
$562$	1.82475	0.0769725
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	8.54983	0.360013
$565$	25.6254	1.07807
$566$	−24.5498	−1.03191
$567$	1.00000	0.0419961
$568$	−2.27492	−0.0954534
$569$	35.0997	1.47145	0.735727	−	0.677278i	$-0.236840\pi$
$569$	35.0997	1.47145	0.735727	+	0.677278i	$0.236840\pi$
$570$	28.0000	1.17279
$571$	−22.2749	−0.932176	−0.466088	−	0.884738i	$-0.654337\pi$
$571$	−22.2749	−0.932176	−0.466088	+	0.884738i	$0.654337\pi$
$572$	0	0
$573$	14.2749	0.596343
$574$	0.725083	0.0302644
$575$	13.0241	0.543142
$576$	1.00000	0.0416667
$577$	−28.3746	−1.18125	−0.590625	−	0.806946i	$-0.701119\pi$
$577$	−28.3746	−1.18125	−0.590625	+	0.806946i	$0.701119\pi$
$578$	−8.00000	−0.332756
$579$	−2.17525	−0.0904002
$580$	−2.37459	−0.0985993
$581$	10.8248	0.449086
$582$	14.5498	0.603110
$583$	−46.1993	−1.91338
$584$	13.2749	0.549320
$585$	0	0
$586$	−10.7251	−0.443049
$587$	−43.9244	−1.81295	−0.906477	−	0.422254i	$-0.861239\pi$
$587$	−43.9244	−1.81295	−0.906477	+	0.422254i	$0.861239\pi$
$588$	1.00000	0.0412393
$589$	−53.6495	−2.21059
$590$	−35.4502	−1.45946
$591$	0.824752	0.0339257
$592$	−7.27492	−0.298997
$593$	19.5498	0.802815	0.401408	−	0.915899i	$-0.368521\pi$
$593$	19.5498	0.802815	0.401408	+	0.915899i	$0.368521\pi$
$594$	−4.00000	−0.164122
$595$	−9.82475	−0.402776
$596$	−1.00000	−0.0409616
$597$	−10.8248	−0.443028
$598$	0	0
$599$	−18.2749	−0.746693	−0.373346	−	0.927692i	$-0.621790\pi$
$599$	−18.2749	−0.746693	−0.373346	+	0.927692i	$0.621790\pi$
$600$	5.72508	0.233726
$601$	−15.2749	−0.623077	−0.311538	−	0.950234i	$-0.600844\pi$
$601$	−15.2749	−0.623077	−0.311538	+	0.950234i	$0.600844\pi$
$602$	10.8248	0.441184
$603$	−10.2749	−0.418427
$604$	−17.0997	−0.695776
$605$	−16.3746	−0.665722
$606$	9.27492	0.376768
$607$	22.2749	0.904111	0.452055	−	0.891990i	$-0.350691\pi$
$607$	22.2749	0.904111	0.452055	+	0.891990i	$0.350691\pi$
$608$	−8.54983	−0.346742
$609$	0.725083	0.0293818
$610$	29.4743	1.19338
$611$	0	0
$612$	3.00000	0.121268
$613$	−8.37459	−0.338246	−0.169123	−	0.985595i	$-0.554094\pi$
$613$	−8.37459	−0.338246	−0.169123	+	0.985595i	$0.554094\pi$
$614$	−16.5498	−0.667897
$615$	−2.37459	−0.0957526
$616$	−4.00000	−0.161165
$617$	−12.9244	−0.520318	−0.260159	−	0.965566i	$-0.583775\pi$
$617$	−12.9244	−0.520318	−0.260159	+	0.965566i	$0.583775\pi$
$618$	−2.27492	−0.0915106
$619$	29.6495	1.19171	0.595857	−	0.803090i	$-0.296812\pi$
$619$	29.6495	1.19171	0.595857	+	0.803090i	$0.296812\pi$
$620$	−20.5498	−0.825301
$621$	2.27492	0.0912893
$622$	12.5498	0.503203
$623$	8.27492	0.331528
$624$	0	0
$625$	−20.8488	−0.833954
$626$	−27.6495	−1.10510
$627$	34.1993	1.36579
$628$	−11.2749	−0.449918
$629$	−21.8248	−0.870210
$630$	−3.27492	−0.130476
$631$	1.09967	0.0437771	0.0218886	−	0.999760i	$-0.493032\pi$
$631$	1.09967	0.0437771	0.0218886	+	0.999760i	$0.493032\pi$
$632$	−8.00000	−0.318223
$633$	−9.09967	−0.361679
$634$	7.54983	0.299842
$635$	−26.1993	−1.03969
$636$	11.5498	0.457981
$637$	0	0
$638$	−2.90033	−0.114825
$639$	−2.27492	−0.0899943
$640$	−3.27492	−0.129452
$641$	−3.82475	−0.151069	−0.0755343	−	0.997143i	$-0.524066\pi$
$641$	−3.82475	−0.151069	−0.0755343	+	0.997143i	$0.524066\pi$
$642$	−12.5498	−0.495302
$643$	42.7492	1.68586	0.842931	−	0.538021i	$-0.180828\pi$
$643$	42.7492	1.68586	0.842931	+	0.538021i	$0.180828\pi$
$644$	2.27492	0.0896443
$645$	−35.4502	−1.39585
$646$	−25.6495	−1.00917
$647$	24.5498	0.965154	0.482577	−	0.875854i	$-0.339701\pi$
$647$	24.5498	0.965154	0.482577	+	0.875854i	$0.339701\pi$
$648$	1.00000	0.0392837
$649$	−43.2990	−1.69963
$650$	0	0
$651$	6.27492	0.245933
$652$	−5.72508	−0.224212
$653$	−34.4743	−1.34908	−0.674541	−	0.738237i	$-0.735658\pi$
$653$	−34.4743	−1.34908	−0.674541	+	0.738237i	$0.735658\pi$
$654$	10.5498	0.412531
$655$	33.6495	1.31479
$656$	0.725083	0.0283097
$657$	13.2749	0.517904
$658$	8.54983	0.333307
$659$	−28.5498	−1.11214	−0.556072	−	0.831134i	$-0.687692\pi$
$659$	−28.5498	−1.11214	−0.556072	+	0.831134i	$0.687692\pi$
$660$	13.0997	0.509904
$661$	40.0997	1.55970	0.779848	−	0.625969i	$-0.215296\pi$
$661$	40.0997	1.55970	0.779848	+	0.625969i	$0.215296\pi$
$662$	4.00000	0.155464
$663$	0	0
$664$	10.8248	0.420082
$665$	28.0000	1.08579
$666$	−7.27492	−0.281897
$667$	1.64950	0.0638690
$668$	0	0
$669$	18.2749	0.706549
$670$	33.6495	1.29999
$671$	36.0000	1.38976
$672$	1.00000	0.0385758
$673$	12.0997	0.466408	0.233204	−	0.972428i	$-0.425079\pi$
$673$	12.0997	0.466408	0.233204	+	0.972428i	$0.425079\pi$
$674$	−7.27492	−0.280219
$675$	5.72508	0.220359
$676$	0	0
$677$	11.6495	0.447727	0.223863	−	0.974621i	$-0.428133\pi$
$677$	11.6495	0.447727	0.223863	+	0.974621i	$0.428133\pi$
$678$	−7.82475	−0.300508
$679$	14.5498	0.558371
$680$	−9.82475	−0.376762
$681$	9.09967	0.348700
$682$	−25.0997	−0.961116
$683$	−49.6495	−1.89979	−0.949893	−	0.312576i	$-0.898808\pi$
$683$	−49.6495	−1.89979	−0.949893	+	0.312576i	$0.898808\pi$
$684$	−8.54983	−0.326911
$685$	−0.573924	−0.0219285
$686$	1.00000	0.0381802
$687$	20.2749	0.773536
$688$	10.8248	0.412690
$689$	0	0
$690$	−7.45017	−0.283623
$691$	8.00000	0.304334	0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000	0.304334	0.152167	+	0.988355i	$0.451375\pi$
$692$	10.5498	0.401045
$693$	−4.00000	−0.151947
$694$	15.4502	0.586480
$695$	−69.0997	−2.62110
$696$	0.725083	0.0274842
$697$	2.17525	0.0823934
$698$	17.9244	0.678449
$699$	22.5498	0.852913
$700$	5.72508	0.216388
$701$	−37.9244	−1.43239	−0.716193	−	0.697902i	$-0.754117\pi$
$701$	−37.9244	−1.43239	−0.716193	+	0.697902i	$0.754117\pi$
$702$	0	0
$703$	62.1993	2.34589
$704$	−4.00000	−0.150756
$705$	−28.0000	−1.05454
$706$	−4.45017	−0.167484
$707$	9.27492	0.348819
$708$	10.8248	0.406819
$709$	−9.62541	−0.361490	−0.180745	−	0.983530i	$-0.557851\pi$
$709$	−9.62541	−0.361490	−0.180745	+	0.983530i	$0.557851\pi$
$710$	7.45017	0.279600
$711$	−8.00000	−0.300023
$712$	8.27492	0.310116
$713$	14.2749	0.534600
$714$	3.00000	0.112272
$715$	0	0
$716$	16.5498	0.618496
$717$	2.27492	0.0849583
$718$	29.0997	1.08599
$719$	−8.00000	−0.298350	−0.149175	−	0.988811i	$-0.547662\pi$
$719$	−8.00000	−0.298350	−0.149175	+	0.988811i	$0.547662\pi$
$720$	−3.27492	−0.122049
$721$	−2.27492	−0.0847223
$722$	54.0997	2.01338
$723$	−23.2749	−0.865603
$724$	5.82475	0.216475
$725$	4.15116	0.154170
$726$	5.00000	0.185567
$727$	−2.82475	−0.104764	−0.0523821	−	0.998627i	$-0.516681\pi$
$727$	−2.82475	−0.104764	−0.0523821	+	0.998627i	$0.516681\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−43.4743	−1.60905
$731$	32.4743	1.20110
$732$	−9.00000	−0.332650
$733$	12.6495	0.467220	0.233610	−	0.972330i	$-0.424946\pi$
$733$	12.6495	0.467220	0.233610	+	0.972330i	$0.424946\pi$
$734$	5.72508	0.211317
$735$	−3.27492	−0.120797
$736$	2.27492	0.0838546
$737$	41.0997	1.51393
$738$	0.725083	0.0266907
$739$	−41.7251	−1.53488	−0.767441	−	0.641120i	$-0.778470\pi$
$739$	−41.7251	−1.53488	−0.767441	+	0.641120i	$0.778470\pi$
$740$	23.8248	0.875815
$741$	0	0
$742$	11.5498	0.424008
$743$	−26.8248	−0.984105	−0.492052	−	0.870566i	$-0.663753\pi$
$743$	−26.8248	−0.984105	−0.492052	+	0.870566i	$0.663753\pi$
$744$	6.27492	0.230050
$745$	3.27492	0.119984
$746$	4.17525	0.152867
$747$	10.8248	0.396057
$748$	−12.0000	−0.438763
$749$	−12.5498	−0.458561
$750$	−2.37459	−0.0867076
$751$	−3.45017	−0.125898	−0.0629492	−	0.998017i	$-0.520051\pi$
$751$	−3.45017	−0.125898	−0.0629492	+	0.998017i	$0.520051\pi$
$752$	8.54983	0.311780
$753$	5.72508	0.208634
$754$	0	0
$755$	56.0000	2.03805
$756$	1.00000	0.0363696
$757$	−26.0000	−0.944986	−0.472493	−	0.881334i	$-0.656646\pi$
$757$	−26.0000	−0.944986	−0.472493	+	0.881334i	$0.656646\pi$
$758$	−20.0000	−0.726433
$759$	−9.09967	−0.330297
$760$	28.0000	1.01567
$761$	−40.1993	−1.45722	−0.728612	−	0.684926i	$-0.759834\pi$
$761$	−40.1993	−1.45722	−0.728612	+	0.684926i	$0.759834\pi$
$762$	8.00000	0.289809
$763$	10.5498	0.381930
$764$	14.2749	0.516448
$765$	−9.82475	−0.355215
$766$	−38.1993	−1.38020
$767$	0	0
$768$	1.00000	0.0360844
$769$	−3.64950	−0.131604	−0.0658022	−	0.997833i	$-0.520961\pi$
$769$	−3.64950	−0.131604	−0.0658022	+	0.997833i	$0.520961\pi$
$770$	13.0997	0.472079
$771$	15.0000	0.540212
$772$	−2.17525	−0.0782889
$773$	−51.0997	−1.83793	−0.918964	−	0.394342i	$-0.870972\pi$
$773$	−51.0997	−1.83793	−0.918964	+	0.394342i	$0.870972\pi$
$774$	10.8248	0.389088
$775$	35.9244	1.29044
$776$	14.5498	0.522309
$777$	−7.27492	−0.260986
$778$	20.6495	0.740321
$779$	−6.19934	−0.222114
$780$	0	0
$781$	9.09967	0.325612
$782$	6.82475	0.244053
$783$	0.725083	0.0259123
$784$	1.00000	0.0357143
$785$	36.9244	1.31789
$786$	−10.2749	−0.366494
$787$	−9.09967	−0.324368	−0.162184	−	0.986761i	$-0.551854\pi$
$787$	−9.09967	−0.324368	−0.162184	+	0.986761i	$0.551854\pi$
$788$	0.824752	0.0293806
$789$	−9.09967	−0.323957
$790$	26.1993	0.932131
$791$	−7.82475	−0.278216
$792$	−4.00000	−0.142134
$793$	0	0
$794$	15.7251	0.558062
$795$	−37.8248	−1.34151
$796$	−10.8248	−0.383673
$797$	1.45017	0.0513675	0.0256837	−	0.999670i	$-0.491824\pi$
$797$	1.45017	0.0513675	0.0256837	+	0.999670i	$0.491824\pi$
$798$	−8.54983	−0.302661
$799$	25.6495	0.907414
$800$	5.72508	0.202412
$801$	8.27492	0.292380
$802$	5.27492	0.186264
$803$	−53.0997	−1.87385
$804$	−10.2749	−0.362368
$805$	−7.45017	−0.262584
$806$	0	0
$807$	−18.0000	−0.633630
$808$	9.27492	0.326290
$809$	17.2749	0.607354	0.303677	−	0.952775i	$-0.401786\pi$
$809$	17.2749	0.607354	0.303677	+	0.952775i	$0.401786\pi$
$810$	−3.27492	−0.115069
$811$	−25.0997	−0.881369	−0.440684	−	0.897662i	$-0.645264\pi$
$811$	−25.0997	−0.881369	−0.440684	+	0.897662i	$0.645264\pi$
$812$	0.725083	0.0254454
$813$	17.7251	0.621646
$814$	29.0997	1.01994
$815$	18.7492	0.656755
$816$	3.00000	0.105021
$817$	−92.5498	−3.23791
$818$	22.9244	0.801534
$819$	0	0
$820$	−2.37459	−0.0829241
$821$	7.72508	0.269607	0.134804	−	0.990872i	$-0.456960\pi$
$821$	7.72508	0.269607	0.134804	+	0.990872i	$0.456960\pi$
$822$	0.175248	0.00611249
$823$	−37.0997	−1.29321	−0.646607	−	0.762824i	$-0.723812\pi$
$823$	−37.0997	−1.29321	−0.646607	+	0.762824i	$0.723812\pi$
$824$	−2.27492	−0.0792505
$825$	−22.9003	−0.797287
$826$	10.8248	0.376641
$827$	−25.0997	−0.872801	−0.436401	−	0.899753i	$-0.643747\pi$
$827$	−25.0997	−0.872801	−0.436401	+	0.899753i	$0.643747\pi$
$828$	2.27492	0.0790588
$829$	−22.1752	−0.770178	−0.385089	−	0.922879i	$-0.625829\pi$
$829$	−22.1752	−0.770178	−0.385089	+	0.922879i	$0.625829\pi$
$830$	−35.4502	−1.23049
$831$	−7.82475	−0.271438
$832$	0	0
$833$	3.00000	0.103944
$834$	21.0997	0.730622
$835$	0	0
$836$	34.1993	1.18281
$837$	6.27492	0.216893
$838$	15.3746	0.531106
$839$	54.7492	1.89015	0.945076	−	0.326852i	$-0.105988\pi$
$839$	54.7492	1.89015	0.945076	+	0.326852i	$0.105988\pi$
$840$	−3.27492	−0.112995
$841$	−28.4743	−0.981871
$842$	0.725083	0.0249880
$843$	1.82475	0.0628478
$844$	−9.09967	−0.313224
$845$	0	0
$846$	8.54983	0.293949
$847$	5.00000	0.171802
$848$	11.5498	0.396623
$849$	−24.5498	−0.842548
$850$	17.1752	0.589106
$851$	−16.5498	−0.567321
$852$	−2.27492	−0.0779374
$853$	−29.5498	−1.01177	−0.505884	−	0.862602i	$-0.668833\pi$
$853$	−29.5498	−1.01177	−0.505884	+	0.862602i	$0.668833\pi$
$854$	−9.00000	−0.307974
$855$	28.0000	0.957580
$856$	−12.5498	−0.428945
$857$	−14.1752	−0.484217	−0.242109	−	0.970249i	$-0.577839\pi$
$857$	−14.1752	−0.484217	−0.242109	+	0.970249i	$0.577839\pi$
$858$	0	0
$859$	17.6495	0.602193	0.301097	−	0.953594i	$-0.402647\pi$
$859$	17.6495	0.602193	0.301097	+	0.953594i	$0.402647\pi$
$860$	−35.4502	−1.20884
$861$	0.725083	0.0247108
$862$	−5.72508	−0.194997
$863$	−55.2990	−1.88240	−0.941200	−	0.337850i	$-0.890300\pi$
$863$	−55.2990	−1.88240	−0.941200	+	0.337850i	$0.890300\pi$
$864$	1.00000	0.0340207
$865$	−34.5498	−1.17473
$866$	−2.72508	−0.0926021
$867$	−8.00000	−0.271694
$868$	6.27492	0.212985
$869$	32.0000	1.08553
$870$	−2.37459	−0.0805060
$871$	0	0
$872$	10.5498	0.357262
$873$	14.5498	0.492437
$874$	−19.4502	−0.657912
$875$	−2.37459	−0.0802757
$876$	13.2749	0.448518
$877$	−11.2749	−0.380727	−0.190363	−	0.981714i	$-0.560967\pi$
$877$	−11.2749	−0.380727	−0.190363	+	0.981714i	$0.560967\pi$
$878$	−17.0997	−0.577086
$879$	−10.7251	−0.361748
$880$	13.0997	0.441590
$881$	−13.5498	−0.456506	−0.228253	−	0.973602i	$-0.573301\pi$
$881$	−13.5498	−0.456506	−0.228253	+	0.973602i	$0.573301\pi$
$882$	1.00000	0.0336718
$883$	−25.0241	−0.842128	−0.421064	−	0.907031i	$-0.638343\pi$
$883$	−25.0241	−0.842128	−0.421064	+	0.907031i	$0.638343\pi$
$884$	0	0
$885$	−35.4502	−1.19164
$886$	19.4502	0.653441
$887$	31.4502	1.05599	0.527997	−	0.849246i	$-0.322943\pi$
$887$	31.4502	1.05599	0.527997	+	0.849246i	$0.322943\pi$
$888$	−7.27492	−0.244130
$889$	8.00000	0.268311
$890$	−27.0997	−0.908383
$891$	−4.00000	−0.134005
$892$	18.2749	0.611889
$893$	−73.0997	−2.44619
$894$	−1.00000	−0.0334450
$895$	−54.1993	−1.81168
$896$	1.00000	0.0334077
$897$	0	0
$898$	11.0997	0.370401
$899$	4.54983	0.151745
$900$	5.72508	0.190836
$901$	34.6495	1.15434
$902$	−2.90033	−0.0965705
$903$	10.8248	0.360225
$904$	−7.82475	−0.260247
$905$	−19.0756	−0.634094
$906$	−17.0997	−0.568098
$907$	−17.1752	−0.570295	−0.285147	−	0.958484i	$-0.592043\pi$
$907$	−17.1752	−0.570295	−0.285147	+	0.958484i	$0.592043\pi$
$908$	9.09967	0.301983
$909$	9.27492	0.307630
$910$	0	0
$911$	5.09967	0.168960	0.0844798	−	0.996425i	$-0.473077\pi$
$911$	5.09967	0.168960	0.0844798	+	0.996425i	$0.473077\pi$
$912$	−8.54983	−0.283113
$913$	−43.2990	−1.43299
$914$	−17.0000	−0.562310
$915$	29.4743	0.974389
$916$	20.2749	0.669902
$917$	−10.2749	−0.339308
$918$	3.00000	0.0990148
$919$	−5.64950	−0.186360	−0.0931800	−	0.995649i	$-0.529703\pi$
$919$	−5.64950	−0.186360	−0.0931800	+	0.995649i	$0.529703\pi$
$920$	−7.45017	−0.245625
$921$	−16.5498	−0.545336
$922$	18.3746	0.605135
$923$	0	0
$924$	−4.00000	−0.131590
$925$	−41.6495	−1.36943
$926$	4.54983	0.149517
$927$	−2.27492	−0.0747181
$928$	0.725083	0.0238020
$929$	38.4502	1.26151	0.630755	−	0.775982i	$-0.282745\pi$
$929$	38.4502	1.26151	0.630755	+	0.775982i	$0.282745\pi$
$930$	−20.5498	−0.673856
$931$	−8.54983	−0.280210
$932$	22.5498	0.738644
$933$	12.5498	0.410863
$934$	−30.8248	−1.00862
$935$	39.2990	1.28521
$936$	0	0
$937$	41.8248	1.36636	0.683178	−	0.730252i	$-0.260598\pi$
$937$	41.8248	1.36636	0.683178	+	0.730252i	$0.260598\pi$
$938$	−10.2749	−0.335488
$939$	−27.6495	−0.902307
$940$	−28.0000	−0.913259
$941$	−7.64950	−0.249367	−0.124683	−	0.992197i	$-0.539791\pi$
$941$	−7.64950	−0.249367	−0.124683	+	0.992197i	$0.539791\pi$
$942$	−11.2749	−0.367357
$943$	1.64950	0.0537152
$944$	10.8248	0.352316
$945$	−3.27492	−0.106533
$946$	−43.2990	−1.40777
$947$	21.0997	0.685647	0.342824	−	0.939400i	$-0.388617\pi$
$947$	21.0997	0.685647	0.342824	+	0.939400i	$0.388617\pi$
$948$	−8.00000	−0.259828
$949$	0	0
$950$	−48.9485	−1.58810
$951$	7.54983	0.244820
$952$	3.00000	0.0972306
$953$	19.0997	0.618699	0.309349	−	0.950948i	$-0.399889\pi$
$953$	19.0997	0.618699	0.309349	+	0.950948i	$0.399889\pi$
$954$	11.5498	0.373940
$955$	−46.7492	−1.51277
$956$	2.27492	0.0735761
$957$	−2.90033	−0.0937544
$958$	20.0000	0.646171
$959$	0.175248	0.00565906
$960$	−3.27492	−0.105697
$961$	8.37459	0.270148
$962$	0	0
$963$	−12.5498	−0.404413
$964$	−23.2749	−0.749635
$965$	7.12376	0.229322
$966$	2.27492	0.0731943
$967$	−3.45017	−0.110950	−0.0554749	−	0.998460i	$-0.517667\pi$
$967$	−3.45017	−0.110950	−0.0554749	+	0.998460i	$0.517667\pi$
$968$	5.00000	0.160706
$969$	−25.6495	−0.823981
$970$	−47.6495	−1.52993
$971$	−49.5739	−1.59090	−0.795451	−	0.606017i	$-0.792766\pi$
$971$	−49.5739	−1.59090	−0.795451	+	0.606017i	$0.792766\pi$
$972$	1.00000	0.0320750
$973$	21.0997	0.676424
$974$	29.0997	0.932414
$975$	0	0
$976$	−9.00000	−0.288083
$977$	62.3746	1.99554	0.997770	−	0.0667475i	$-0.0212622\pi$
$977$	62.3746	1.99554	0.997770	+	0.0667475i	$0.0212622\pi$
$978$	−5.72508	−0.183068
$979$	−33.0997	−1.05787
$980$	−3.27492	−0.104613
$981$	10.5498	0.336830
$982$	−5.09967	−0.162737
$983$	−51.8488	−1.65372	−0.826861	−	0.562407i	$-0.809875\pi$
$983$	−51.8488	−1.65372	−0.826861	+	0.562407i	$0.809875\pi$
$984$	0.725083	0.0231148
$985$	−2.70099	−0.0860608
$986$	2.17525	0.0692740
$987$	8.54983	0.272144
$988$	0	0
$989$	24.6254	0.783043
$990$	13.0997	0.416335
$991$	38.7492	1.23091	0.615454	−	0.788173i	$-0.288973\pi$
$991$	38.7492	1.23091	0.615454	+	0.788173i	$0.288973\pi$
$992$	6.27492	0.199229
$993$	4.00000	0.126936
$994$	−2.27492	−0.0721560
$995$	35.4502	1.12385
$996$	10.8248	0.342995
$997$	−34.0997	−1.07995	−0.539974	−	0.841682i	$-0.681566\pi$
$997$	−34.0997	−1.07995	−0.539974	+	0.841682i	$0.681566\pi$
$998$	−9.17525	−0.290437
$999$	−7.27492	−0.230168

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7098.2.a.ca.1.1		2
13.3	even	3		546.2.l.j.295.1	yes	4
13.9	even	3		546.2.l.j.211.1	✓	4
13.12	even	2		7098.2.a.bm.1.2		2
39.29	odd	6		1638.2.r.x.1387.2		4
39.35	odd	6		1638.2.r.x.757.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.2.l.j.211.1	✓	4	13.9	even	3
546.2.l.j.295.1	yes	4	13.3	even	3
1638.2.r.x.757.2		4	39.35	odd	6
1638.2.r.x.1387.2		4	39.29	odd	6
7098.2.a.bm.1.2		2	13.12	even	2
7098.2.a.ca.1.1		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7098.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.27492 q^{5} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -3.27492 q^{10} -4.00000 q^{11} +1.00000 q^{12} +1.00000 q^{14} -3.27492 q^{15} +1.00000 q^{16} +3.00000 q^{17} +1.00000 q^{18} -8.54983 q^{19} -3.27492 q^{20} +1.00000 q^{21} -4.00000 q^{22} +2.27492 q^{23} +1.00000 q^{24} +5.72508 q^{25} +1.00000 q^{27} +1.00000 q^{28} +0.725083 q^{29} -3.27492 q^{30} +6.27492 q^{31} +1.00000 q^{32} -4.00000 q^{33} +3.00000 q^{34} -3.27492 q^{35} +1.00000 q^{36} -7.27492 q^{37} -8.54983 q^{38} -3.27492 q^{40} +0.725083 q^{41} +1.00000 q^{42} +10.8248 q^{43} -4.00000 q^{44} -3.27492 q^{45} +2.27492 q^{46} +8.54983 q^{47} +1.00000 q^{48} +1.00000 q^{49} +5.72508 q^{50} +3.00000 q^{51} +11.5498 q^{53} +1.00000 q^{54} +13.0997 q^{55} +1.00000 q^{56} -8.54983 q^{57} +0.725083 q^{58} +10.8248 q^{59} -3.27492 q^{60} -9.00000 q^{61} +6.27492 q^{62} +1.00000 q^{63} +1.00000 q^{64} -4.00000 q^{66} -10.2749 q^{67} +3.00000 q^{68} +2.27492 q^{69} -3.27492 q^{70} -2.27492 q^{71} +1.00000 q^{72} +13.2749 q^{73} -7.27492 q^{74} +5.72508 q^{75} -8.54983 q^{76} -4.00000 q^{77} -8.00000 q^{79} -3.27492 q^{80} +1.00000 q^{81} +0.725083 q^{82} +10.8248 q^{83} +1.00000 q^{84} -9.82475 q^{85} +10.8248 q^{86} +0.725083 q^{87} -4.00000 q^{88} +8.27492 q^{89} -3.27492 q^{90} +2.27492 q^{92} +6.27492 q^{93} +8.54983 q^{94} +28.0000 q^{95} +1.00000 q^{96} +14.5498 q^{97} +1.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + q^{5} + 2 q^{6} + 2 q^{7} + 2 q^{8} + 2 q^{9} + q^{10} - 8 q^{11} + 2 q^{12} + 2 q^{14} + q^{15} + 2 q^{16} + 6 q^{17} + 2 q^{18} - 2 q^{19} + q^{20} + 2 q^{21} - 8 q^{22}+ \cdots - 8 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + q^5 + 2 * q^6 + 2 * q^7 + 2 * q^8 + 2 * q^9 + q^10 - 8 * q^11 + 2 * q^12 + 2 * q^14 + q^15 + 2 * q^16 + 6 * q^17 + 2 * q^18 - 2 * q^19 + q^20 + 2 * q^21 - 8 * q^22 - 3 * q^23 + 2 * q^24 + 19 * q^25 + 2 * q^27 + 2 * q^28 + 9 * q^29 + q^30 + 5 * q^31 + 2 * q^32 - 8 * q^33 + 6 * q^34 + q^35 + 2 * q^36 - 7 * q^37 - 2 * q^38 + q^40 + 9 * q^41 + 2 * q^42 - q^43 - 8 * q^44 + q^45 - 3 * q^46 + 2 * q^47 + 2 * q^48 + 2 * q^49 + 19 * q^50 + 6 * q^51 + 8 * q^53 + 2 * q^54 - 4 * q^55 + 2 * q^56 - 2 * q^57 + 9 * q^58 - q^59 + q^60 - 18 * q^61 + 5 * q^62 + 2 * q^63 + 2 * q^64 - 8 * q^66 - 13 * q^67 + 6 * q^68 - 3 * q^69 + q^70 + 3 * q^71 + 2 * q^72 + 19 * q^73 - 7 * q^74 + 19 * q^75 - 2 * q^76 - 8 * q^77 - 16 * q^79 + q^80 + 2 * q^81 + 9 * q^82 - q^83 + 2 * q^84 + 3 * q^85 - q^86 + 9 * q^87 - 8 * q^88 + 9 * q^89 + q^90 - 3 * q^92 + 5 * q^93 + 2 * q^94 + 56 * q^95 + 2 * q^96 + 14 * q^97 + 2 * q^98 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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