LMFDB - Embedded newform 1425.2.c.p.799.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1425,2,Mod(799,1425)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1425, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1425.799"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1425 = 3 \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1425.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,-12,0,0,0,0,-6,0,-6,0,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)

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

Self dual:	no
Analytic conductor:	$11.3786822880$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.24681024.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 12x^{4} + 36x^{2} + 9 $ x^6 + 12x^4 + 36x^2 + 9
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			799.1
Root			$-2.66908i$ of defining polynomial
Character	$\chi$	$=$	1425.799
Dual form			1425.2.c.p.799.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.66908i q^{2} -1.00000i q^{3} -5.12398 q^{4} -2.66908 q^{6} +0.454904i q^{7} +8.33816i q^{8} -1.00000 q^{9} -1.45490 q^{11} +5.12398i q^{12} +3.33816i q^{13} +1.21417 q^{14} +12.0072 q^{16} +2.66908i q^{18} -1.00000 q^{19} +0.454904 q^{21} +3.88325i q^{22} +6.79306i q^{23} +8.33816 q^{24} +8.90981 q^{26} +1.00000i q^{27} -2.33092i q^{28} -3.00000 q^{29} +8.79306 q^{31} -15.3719i q^{32} +1.45490i q^{33} +5.12398 q^{36} +10.2480i q^{37} +2.66908i q^{38} +3.33816 q^{39} +3.97345 q^{41} -1.21417i q^{42} -8.00000i q^{43} +7.45490 q^{44} +18.1312 q^{46} +2.42835i q^{47} -12.0072i q^{48} +6.79306 q^{49} -17.1047i q^{52} -13.6763i q^{53} +2.66908 q^{54} -3.79306 q^{56} +1.00000i q^{57} +8.00724i q^{58} -7.54510 q^{59} -3.90981 q^{61} -23.4694i q^{62} -0.454904i q^{63} -17.0145 q^{64} +3.88325 q^{66} +14.1312i q^{67} +6.79306 q^{69} -9.13122 q^{71} -8.33816i q^{72} +11.6763i q^{73} +27.3526 q^{74} +5.12398 q^{76} -0.661842i q^{77} -8.90981i q^{78} -8.97345 q^{79} +1.00000 q^{81} -10.6054i q^{82} +4.54510i q^{83} -2.33092 q^{84} -21.3526 q^{86} +3.00000i q^{87} -12.1312i q^{88} +0.0901918 q^{89} -1.51854 q^{91} -34.8075i q^{92} -8.79306i q^{93} +6.48146 q^{94} -15.3719 q^{96} +12.0145i q^{97} -18.1312i q^{98} +1.45490 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 12 q^{4} - 6 q^{9} - 6 q^{11} - 6 q^{14} + 24 q^{16} - 6 q^{19} + 18 q^{24} + 48 q^{26} - 18 q^{29} + 18 q^{31} + 12 q^{36} - 12 q^{39} + 42 q^{44} + 42 q^{46} + 6 q^{49} + 12 q^{56} - 48 q^{59} - 18 q^{61}+ \cdots + 6 q^{99}+O(q^{100}) $ 6 * q - 12 * q^4 - 6 * q^9 - 6 * q^11 - 6 * q^14 + 24 * q^16 - 6 * q^19 + 18 * q^24 + 48 * q^26 - 18 * q^29 + 18 * q^31 + 12 * q^36 - 12 * q^39 + 42 * q^44 + 42 * q^46 + 6 * q^49 + 12 * q^56 - 48 * q^59 - 18 * q^61 - 6 * q^64 - 6 * q^66 + 6 * q^69 + 12 * q^71 + 36 * q^74 + 12 * q^76 - 30 * q^79 + 6 * q^81 - 30 * q^84 + 6 * q^89 + 12 * q^91 + 60 * q^94 - 36 * q^96 + 6 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1425\mathbb{Z}\right)^\times$.

$n$	$476$	$1027$	$1351$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$	− 2.66908i	− 1.88732i	−0.330911	−	0.943662i	$-0.607356\pi$
$2$	− 2.66908i	− 1.88732i	0.330911	−	0.943662i	$-0.392644\pi$
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	−5.12398	−2.56199
$5$	0	0
$5$	0	0
$6$	−2.66908	−1.08965
$7$	0.454904i	0.171938i	0.996298	+	0.0859688i	$0.0273985\pi$
$7$	0.454904i	0.171938i	−0.996298	+	0.0859688i	$0.972601\pi$
$8$	8.33816i	2.94798i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−1.45490	−0.438670	−0.219335	−	0.975650i	$-0.570389\pi$
$11$	−1.45490	−0.438670	−0.219335	+	0.975650i	$0.570389\pi$
$12$	5.12398i	1.47917i
$13$	3.33816i	0.925838i	0.886400	+	0.462919i	$0.153198\pi$
$13$	3.33816i	0.925838i	−0.886400	+	0.462919i	$0.846802\pi$
$14$	1.21417	0.324502
$15$	0	0
$16$	12.0072	3.00181
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	2.66908i	0.629108i
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0.454904	0.0992682
$22$	3.88325i	0.827913i
$23$	6.79306i	1.41645i	0.705986	+	0.708226i	$0.250504\pi$
$23$	6.79306i	1.41645i	−0.705986	+	0.708226i	$0.749496\pi$
$24$	8.33816	1.70202
$25$	0	0
$26$	8.90981	1.74736
$27$	1.00000i	0.192450i
$28$	− 2.33092i	− 0.440503i
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	0	0
$31$	8.79306	1.57928	0.789640	−	0.613570i	$-0.210267\pi$
$31$	8.79306	1.57928	0.789640	+	0.613570i	$0.210267\pi$
$32$	− 15.3719i	− 2.71740i
$33$	1.45490i	0.253266i
$34$	0	0
$35$	0	0
$36$	5.12398	0.853997
$37$	10.2480i	1.68476i	0.538888	+	0.842378i	$0.318845\pi$
$37$	10.2480i	1.68476i	−0.538888	+	0.842378i	$0.681155\pi$
$38$	2.66908i	0.432982i
$39$	3.33816	0.534533
$40$	0	0
$41$	3.97345	0.620548	0.310274	−	0.950647i	$-0.399579\pi$
$41$	3.97345	0.620548	0.310274	+	0.950647i	$0.399579\pi$
$42$	− 1.21417i	− 0.187351i
$43$	− 8.00000i	− 1.21999i	−0.792406	−	0.609994i	$-0.791172\pi$
$43$	− 8.00000i	− 1.21999i	0.792406	−	0.609994i	$-0.208828\pi$
$44$	7.45490	1.12387
$45$	0	0
$46$	18.1312	2.67330
$47$	2.42835i	0.354211i	0.984192	+	0.177106i	$0.0566734\pi$
$47$	2.42835i	0.354211i	−0.984192	+	0.177106i	$0.943327\pi$
$48$	− 12.0072i	− 1.73310i
$49$	6.79306	0.970437
$50$	0	0
$51$	0	0
$52$	− 17.1047i	− 2.37199i
$53$	− 13.6763i	− 1.87859i	−0.343115	−	0.939293i	$-0.611482\pi$
$53$	− 13.6763i	− 1.87859i	0.343115	−	0.939293i	$-0.388518\pi$
$54$	2.66908	0.363216
$55$	0	0
$56$	−3.79306	−0.506869
$57$	1.00000i	0.132453i
$58$	8.00724i	1.05140i
$59$	−7.54510	−0.982288	−0.491144	−	0.871078i	$-0.663421\pi$
$59$	−7.54510	−0.982288	−0.491144	+	0.871078i	$0.663421\pi$
$60$	0	0
$61$	−3.90981	−0.500600	−0.250300	−	0.968168i	$-0.580529\pi$
$61$	−3.90981	−0.500600	−0.250300	+	0.968168i	$0.580529\pi$
$62$	− 23.4694i	− 2.98061i
$63$	− 0.454904i	− 0.0573125i
$64$	−17.0145	−2.12681
$65$	0	0
$66$	3.88325	0.477996
$67$	14.1312i	1.72640i	0.504859	+	0.863202i	$0.331544\pi$
$67$	14.1312i	1.72640i	−0.504859	+	0.863202i	$0.668456\pi$
$68$	0	0
$69$	6.79306	0.817789
$70$	0	0
$71$	−9.13122	−1.08368	−0.541838	−	0.840483i	$-0.682271\pi$
$71$	−9.13122	−1.08368	−0.541838	+	0.840483i	$0.682271\pi$
$72$	− 8.33816i	− 0.982661i
$73$	11.6763i	1.36661i	0.730133	+	0.683305i	$0.239458\pi$
$73$	11.6763i	1.36661i	−0.730133	+	0.683305i	$0.760542\pi$
$74$	27.3526	3.17968
$75$	0	0
$76$	5.12398	0.587761
$77$	− 0.661842i	− 0.0754239i
$78$	− 8.90981i	− 1.00884i
$79$	−8.97345	−1.00959	−0.504796	−	0.863239i	$-0.668432\pi$
$79$	−8.97345	−1.00959	−0.504796	+	0.863239i	$0.668432\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 10.6054i	− 1.17118i
$83$	4.54510i	0.498889i	0.968389	+	0.249445i	$0.0802480\pi$
$83$	4.54510i	0.498889i	−0.968389	+	0.249445i	$0.919752\pi$
$84$	−2.33092	−0.254324
$85$	0	0
$86$	−21.3526	−2.30251
$87$	3.00000i	0.321634i
$88$	− 12.1312i	− 1.29319i
$89$	0.0901918	0.00956031	0.00478016	−	0.999989i	$-0.498478\pi$
$89$	0.0901918	0.00956031	0.00478016	+	0.999989i	$0.498478\pi$
$90$	0	0
$91$	−1.51854	−0.159186
$92$	− 34.8075i	− 3.62894i
$93$	− 8.79306i	− 0.911798i
$94$	6.48146	0.668511
$95$	0	0
$96$	−15.3719	−1.56889
$97$	12.0145i	1.21989i	0.792446	+	0.609943i	$0.208807\pi$
$97$	12.0145i	1.21989i	−0.792446	+	0.609943i	$0.791193\pi$
$98$	− 18.1312i	− 1.83153i
$99$	1.45490	0.146223
$100$	0	0
$101$	5.51854	0.549115	0.274558	−	0.961571i	$-0.411469\pi$
$101$	5.51854	0.549115	0.274558	+	0.961571i	$0.411469\pi$
$102$	0	0
$103$	2.54510i	0.250776i	0.992108	+	0.125388i	$0.0400175\pi$
$103$	2.54510i	0.250776i	−0.992108	+	0.125388i	$0.959982\pi$
$104$	−27.8341	−2.72936
$105$	0	0
$106$	−36.5032	−3.54550
$107$	− 4.63529i	− 0.448110i	−0.974576	−	0.224055i	$-0.928070\pi$
$107$	− 4.63529i	− 0.448110i	0.974576	−	0.224055i	$-0.0719296\pi$
$108$	− 5.12398i	− 0.493056i
$109$	6.24797	0.598447	0.299223	−	0.954183i	$-0.403272\pi$
$109$	6.24797	0.598447	0.299223	+	0.954183i	$0.403272\pi$
$110$	0	0
$111$	10.2480	0.972694
$112$	5.46214i	0.516124i
$113$	5.20694i	0.489827i	0.969545	+	0.244914i	$0.0787596\pi$
$113$	5.20694i	0.489827i	−0.969545	+	0.244914i	$0.921240\pi$
$114$	2.66908	0.249982
$115$	0	0
$116$	15.3719	1.42725
$117$	− 3.33816i	− 0.308613i
$118$	20.1385i	1.85390i
$119$	0	0
$120$	0	0
$121$	−8.88325	−0.807569
$122$	10.4356i	0.944794i
$123$	− 3.97345i	− 0.358274i
$124$	−45.0555	−4.04610
$125$	0	0
$126$	−1.21417	−0.108167
$127$	− 10.6127i	− 0.941723i	−0.882207	−	0.470861i	$-0.843943\pi$
$127$	− 10.6127i	− 0.941723i	0.882207	−	0.470861i	$-0.156057\pi$
$128$	14.6691i	1.29658i
$129$	−8.00000	−0.704361
$130$	0	0
$131$	21.7029	1.89619	0.948094	−	0.317989i	$-0.103008\pi$
$131$	21.7029	1.89619	0.948094	+	0.317989i	$0.103008\pi$
$132$	− 7.45490i	− 0.648866i
$133$	− 0.454904i	− 0.0394452i
$134$	37.7173	3.25828
$135$	0	0
$136$	0	0
$137$	22.3155i	1.90655i	0.302110	+	0.953273i	$0.402309\pi$
$137$	22.3155i	1.90655i	−0.302110	+	0.953273i	$0.597691\pi$
$138$	− 18.1312i	− 1.54343i
$139$	10.4018	0.882269	0.441134	−	0.897441i	$-0.354576\pi$
$139$	10.4018	0.882269	0.441134	+	0.897441i	$0.354576\pi$
$140$	0	0
$141$	2.42835	0.204504
$142$	24.3719i	2.04525i
$143$	− 4.85670i	− 0.406138i
$144$	−12.0072	−1.00060
$145$	0	0
$146$	31.1650	2.57923
$147$	− 6.79306i	− 0.560282i
$148$	− 52.5104i	− 4.31633i
$149$	−5.15777	−0.422541	−0.211271	−	0.977428i	$-0.567760\pi$
$149$	−5.15777	−0.422541	−0.211271	+	0.977428i	$0.567760\pi$
$150$	0	0
$151$	−8.49593	−0.691389	−0.345695	−	0.938347i	$-0.612357\pi$
$151$	−8.49593	−0.691389	−0.345695	+	0.938347i	$0.612357\pi$
$152$	− 8.33816i	− 0.676314i
$153$	0	0
$154$	−1.76651	−0.142349
$155$	0	0
$156$	−17.1047	−1.36947
$157$	22.6498i	1.80765i	0.427905	+	0.903824i	$0.359252\pi$
$157$	22.6498i	1.80765i	−0.427905	+	0.903824i	$0.640748\pi$
$158$	23.9508i	1.90543i
$159$	−13.6763	−1.08460
$160$	0	0
$161$	−3.09019	−0.243541
$162$	− 2.66908i	− 0.209703i
$163$	− 3.72548i	− 0.291802i	−0.989299	−	0.145901i	$-0.953392\pi$
$163$	− 3.72548i	− 0.291802i	0.989299	−	0.145901i	$-0.0466081\pi$
$164$	−20.3599	−1.58984
$165$	0	0
$166$	12.1312	0.941565
$167$	12.2214i	0.945721i	0.881137	+	0.472861i	$0.156779\pi$
$167$	12.2214i	0.945721i	−0.881137	+	0.472861i	$0.843221\pi$
$168$	3.79306i	0.292641i
$169$	1.85670	0.142823
$170$	0	0
$171$	1.00000	0.0764719
$172$	40.9919i	3.12560i
$173$	− 24.1722i	− 1.83778i	−0.394511	−	0.918891i	$-0.629086\pi$
$173$	− 24.1722i	− 1.83778i	0.394511	−	0.918891i	$-0.370914\pi$
$174$	8.00724	0.607027
$175$	0	0
$176$	−17.4694	−1.31680
$177$	7.54510i	0.567124i
$178$	− 0.240729i	− 0.0180434i
$179$	6.22141	0.465010	0.232505	−	0.972595i	$-0.425308\pi$
$179$	6.22141	0.465010	0.232505	+	0.972595i	$0.425308\pi$
$180$	0	0
$181$	−20.4959	−1.52345	−0.761725	−	0.647900i	$-0.775647\pi$
$181$	−20.4959	−1.52345	−0.761725	+	0.647900i	$0.775647\pi$
$182$	4.05311i	0.300436i
$183$	3.90981i	0.289021i
$184$	−56.6416	−4.17568
$185$	0	0
$186$	−23.4694	−1.72086
$187$	0	0
$188$	− 12.4428i	− 0.907486i
$189$	−0.454904	−0.0330894
$190$	0	0
$191$	0.793062	0.0573840	0.0286920	−	0.999588i	$-0.490866\pi$
$191$	0.793062	0.0573840	0.0286920	+	0.999588i	$0.490866\pi$
$192$	17.0145i	1.22791i
$193$	15.8196i	1.13872i	0.822088	+	0.569360i	$0.192809\pi$
$193$	15.8196i	1.13872i	−0.822088	+	0.569360i	$0.807191\pi$
$194$	32.0676	2.30232
$195$	0	0
$196$	−34.8075	−2.48625
$197$	3.75203i	0.267321i	0.991027	+	0.133661i	$0.0426732\pi$
$197$	3.75203i	0.267321i	−0.991027	+	0.133661i	$0.957327\pi$
$198$	− 3.88325i	− 0.275971i
$199$	−6.45490	−0.457576	−0.228788	−	0.973476i	$-0.573476\pi$
$199$	−6.45490	−0.457576	−0.228788	+	0.973476i	$0.573476\pi$
$200$	0	0
$201$	14.1312	0.996739
$202$	− 14.7294i	− 1.03636i
$203$	− 1.36471i	− 0.0957840i
$204$	0	0
$205$	0	0
$206$	6.79306	0.473295
$207$	− 6.79306i	− 0.472150i
$208$	40.0821i	2.77919i
$209$	1.45490	0.100638
$210$	0	0
$211$	23.9653	1.64984	0.824920	−	0.565249i	$-0.191220\pi$
$211$	23.9653	1.64984	0.824920	+	0.565249i	$0.191220\pi$
$212$	70.0772i	4.81292i
$213$	9.13122i	0.625661i
$214$	−12.3719	−0.845729
$215$	0	0
$216$	−8.33816	−0.567340
$217$	4.00000i	0.271538i
$218$	− 16.6763i	− 1.12946i
$219$	11.6763	0.789012
$220$	0	0
$221$	0	0
$222$	− 27.3526i	− 1.83579i
$223$	3.20694i	0.214752i	0.994218	+	0.107376i	$0.0342449\pi$
$223$	3.20694i	0.214752i	−0.994218	+	0.107376i	$0.965755\pi$
$224$	6.99276	0.467224
$225$	0	0
$226$	13.8977	0.924463
$227$	− 9.13122i	− 0.606060i	−0.952981	−	0.303030i	$-0.902002\pi$
$227$	− 9.13122i	− 0.606060i	0.952981	−	0.303030i	$-0.0979983\pi$
$228$	− 5.12398i	− 0.339344i
$229$	−16.7400	−1.10621	−0.553104	−	0.833112i	$-0.686557\pi$
$229$	−16.7400	−1.10621	−0.553104	+	0.833112i	$0.686557\pi$
$230$	0	0
$231$	−0.661842	−0.0435460
$232$	− 25.0145i	− 1.64228i
$233$	9.57165i	0.627060i	0.949578	+	0.313530i	$0.101512\pi$
$233$	9.57165i	0.627060i	−0.949578	+	0.313530i	$0.898488\pi$
$234$	−8.90981	−0.582452
$235$	0	0
$236$	38.6609	2.51661
$237$	8.97345i	0.582888i
$238$	0	0
$239$	−17.5185	−1.13318	−0.566590	−	0.824000i	$-0.691738\pi$
$239$	−17.5185	−1.13318	−0.566590	+	0.824000i	$0.691738\pi$
$240$	0	0
$241$	16.9098	1.08926	0.544628	−	0.838678i	$-0.316671\pi$
$241$	16.9098	1.08926	0.544628	+	0.838678i	$0.316671\pi$
$242$	23.7101i	1.52414i
$243$	− 1.00000i	− 0.0641500i
$244$	20.0338	1.28253
$245$	0	0
$246$	−10.6054	−0.676178
$247$	− 3.33816i	− 0.212402i
$248$	73.3179i	4.65569i
$249$	4.54510	0.288034
$250$	0	0
$251$	−11.6392	−0.734662	−0.367331	−	0.930090i	$-0.619728\pi$
$251$	−11.6392	−0.734662	−0.367331	+	0.930090i	$0.619728\pi$
$252$	2.33092i	0.146834i
$253$	− 9.88325i	− 0.621355i
$254$	−28.3261	−1.77734
$255$	0	0
$256$	5.12398	0.320249
$257$	− 4.76651i	− 0.297327i	−0.988888	−	0.148663i	$-0.952503\pi$
$257$	− 4.76651i	− 0.297327i	0.988888	−	0.148663i	$-0.0474971\pi$
$258$	21.3526i	1.32936i
$259$	−4.66184	−0.289673
$260$	0	0
$261$	3.00000	0.185695
$262$	− 57.9267i	− 3.57872i
$263$	− 14.8606i	− 0.916347i	−0.888863	−	0.458173i	$-0.848504\pi$
$263$	− 14.8606i	− 0.916347i	0.888863	−	0.458173i	$-0.151496\pi$
$264$	−12.1312	−0.746625
$265$	0	0
$266$	−1.21417	−0.0744458
$267$	− 0.0901918i	− 0.00551965i
$268$	− 72.4081i	− 4.42303i
$269$	21.5330	1.31289	0.656446	−	0.754373i	$-0.272059\pi$
$269$	21.5330	1.31289	0.656446	+	0.754373i	$0.272059\pi$
$270$	0	0
$271$	−1.13122	−0.0687167	−0.0343584	−	0.999410i	$-0.510939\pi$
$271$	−1.13122	−0.0687167	−0.0343584	+	0.999410i	$0.510939\pi$
$272$	0	0
$273$	1.51854i	0.0919063i
$274$	59.5620	3.59827
$275$	0	0
$276$	−34.8075	−2.09517
$277$	− 13.2624i	− 0.796863i	−0.917198	−	0.398431i	$-0.869555\pi$
$277$	− 13.2624i	− 0.796863i	0.917198	−	0.398431i	$-0.130445\pi$
$278$	− 27.7632i	− 1.66513i
$279$	−8.79306	−0.526427
$280$	0	0
$281$	14.1988	0.847030	0.423515	−	0.905889i	$-0.360796\pi$
$281$	14.1988	0.847030	0.423515	+	0.905889i	$0.360796\pi$
$282$	− 6.48146i	− 0.385965i
$283$	19.5330i	1.16112i	0.814218	+	0.580559i	$0.197166\pi$
$283$	19.5330i	1.16112i	−0.814218	+	0.580559i	$0.802834\pi$
$284$	46.7882	2.77637
$285$	0	0
$286$	−12.9629	−0.766513
$287$	1.80754i	0.106696i
$288$	15.3719i	0.905801i
$289$	17.0000	1.00000
$290$	0	0
$291$	12.0145	0.704301
$292$	− 59.8292i	− 3.50124i
$293$	24.9734i	1.45896i	0.684000	+	0.729482i	$0.260239\pi$
$293$	24.9734i	1.45896i	−0.684000	+	0.729482i	$0.739761\pi$
$294$	−18.1312	−1.05743
$295$	0	0
$296$	−85.4492	−4.96663
$297$	− 1.45490i	− 0.0844221i
$298$	13.7665i	0.797472i
$299$	−22.6763	−1.31141
$300$	0	0
$301$	3.63923	0.209762
$302$	22.6763i	1.30488i
$303$	− 5.51854i	− 0.317032i
$304$	−12.0072	−0.688662
$305$	0	0
$306$	0	0
$307$	11.4018i	0.650735i	0.945588	+	0.325367i	$0.105488\pi$
$307$	11.4018i	0.650735i	−0.945588	+	0.325367i	$0.894512\pi$
$308$	3.39127i	0.193235i
$309$	2.54510	0.144785
$310$	0	0
$311$	1.32368	0.0750592	0.0375296	−	0.999296i	$-0.488051\pi$
$311$	1.32368	0.0750592	0.0375296	+	0.999296i	$0.488051\pi$
$312$	27.8341i	1.57580i
$313$	− 20.6127i	− 1.16510i	−0.812796	−	0.582549i	$-0.802056\pi$
$313$	− 20.6127i	− 1.16510i	0.812796	−	0.582549i	$-0.197944\pi$
$314$	60.4540	3.41162
$315$	0	0
$316$	45.9798	2.58657
$317$	− 18.1722i	− 1.02066i	−0.859980	−	0.510328i	$-0.829524\pi$
$317$	− 18.1722i	− 1.02066i	0.859980	−	0.510328i	$-0.170476\pi$
$318$	36.5032i	2.04700i
$319$	4.36471	0.244377
$320$	0	0
$321$	−4.63529	−0.258717
$322$	8.24797i	0.459641i
$323$	0	0
$324$	−5.12398	−0.284666
$325$	0	0
$326$	−9.94360	−0.550725
$327$	− 6.24797i	− 0.345513i
$328$	33.1312i	1.82937i
$329$	−1.10467	−0.0609022
$330$	0	0
$331$	−19.5225	−1.07305	−0.536526	−	0.843883i	$-0.680264\pi$
$331$	−19.5225	−1.07305	−0.536526	+	0.843883i	$0.680264\pi$
$332$	− 23.2890i	− 1.27815i
$333$	− 10.2480i	− 0.561585i
$334$	32.6199	1.78488
$335$	0	0
$336$	5.46214	0.297984
$337$	0.413875i	0.0225452i	0.999936	+	0.0112726i	$0.00358826\pi$
$337$	0.413875i	0.0225452i	−0.999936	+	0.0112726i	$0.996412\pi$
$338$	− 4.95568i	− 0.269553i
$339$	5.20694	0.282802
$340$	0	0
$341$	−12.7931	−0.692783
$342$	− 2.66908i	− 0.144327i
$343$	6.27452i	0.338792i
$344$	66.7053	3.59651
$345$	0	0
$346$	−64.5176	−3.46849
$347$	− 30.9243i	− 1.66010i	−0.557687	−	0.830051i	$-0.688311\pi$
$347$	− 30.9243i	− 1.66010i	0.557687	−	0.830051i	$-0.311689\pi$
$348$	− 15.3719i	− 0.824023i
$349$	14.7665	0.790433	0.395217	−	0.918588i	$-0.370670\pi$
$349$	14.7665	0.790433	0.395217	+	0.918588i	$0.370670\pi$
$350$	0	0
$351$	−3.33816	−0.178178
$352$	22.3647i	1.19204i
$353$	27.5330i	1.46543i	0.680533	+	0.732717i	$0.261748\pi$
$353$	27.5330i	1.46543i	−0.680533	+	0.732717i	$0.738252\pi$
$354$	20.1385	1.07035
$355$	0	0
$356$	−0.462141	−0.0244934
$357$	0	0
$358$	− 16.6054i	− 0.877625i
$359$	3.83409	0.202356	0.101178	−	0.994868i	$-0.467739\pi$
$359$	3.83409	0.202356	0.101178	+	0.994868i	$0.467739\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	54.7053i	2.87524i
$363$	8.88325i	0.466250i
$364$	7.78098	0.407834
$365$	0	0
$366$	10.4356	0.545477
$367$	21.2254i	1.10795i	0.832532	+	0.553977i	$0.186891\pi$
$367$	21.2254i	1.10795i	−0.832532	+	0.553977i	$0.813109\pi$
$368$	81.5659i	4.25192i
$369$	−3.97345	−0.206849
$370$	0	0
$371$	6.22141	0.323000
$372$	45.0555i	2.33602i
$373$	22.2624i	1.15271i	0.817201	+	0.576353i	$0.195525\pi$
$373$	22.2624i	1.15271i	−0.817201	+	0.576353i	$0.804475\pi$
$374$	0	0
$375$	0	0
$376$	−20.2480	−1.04421
$377$	− 10.0145i	− 0.515772i
$378$	1.21417i	0.0624504i
$379$	3.33816	0.171470	0.0857348	−	0.996318i	$-0.472676\pi$
$379$	3.33816	0.171470	0.0857348	+	0.996318i	$0.472676\pi$
$380$	0	0
$381$	−10.6127	−0.543704
$382$	− 2.11675i	− 0.108302i
$383$	− 34.5370i	− 1.76476i	−0.470541	−	0.882378i	$-0.655941\pi$
$383$	− 34.5370i	− 1.76476i	0.470541	−	0.882378i	$-0.344059\pi$
$384$	14.6691	0.748578
$385$	0	0
$386$	42.2238	2.14914
$387$	8.00000i	0.406663i
$388$	− 61.5620i	− 3.12534i
$389$	−19.1047	−0.968645	−0.484323	−	0.874890i	$-0.660934\pi$
$389$	−19.1047	−0.968645	−0.484323	+	0.874890i	$0.660934\pi$
$390$	0	0
$391$	0	0
$392$	56.6416i	2.86083i
$393$	− 21.7029i	− 1.09476i
$394$	10.0145	0.504522
$395$	0	0
$396$	−7.45490	−0.374623
$397$	13.8301i	0.694115i	0.937844	+	0.347058i	$0.112819\pi$
$397$	13.8301i	0.694115i	−0.937844	+	0.347058i	$0.887181\pi$
$398$	17.2286i	0.863594i
$399$	−0.454904	−0.0227737
$400$	0	0
$401$	6.79306	0.339229	0.169615	−	0.985510i	$-0.445748\pi$
$401$	6.79306	0.339229	0.169615	+	0.985510i	$0.445748\pi$
$402$	− 37.7173i	− 1.88117i
$403$	29.3526i	1.46216i
$404$	−28.2769	−1.40683
$405$	0	0
$406$	−3.64252	−0.180775
$407$	− 14.9098i	− 0.739052i
$408$	0	0
$409$	12.5104	0.618600	0.309300	−	0.950965i	$-0.399905\pi$
$409$	12.5104	0.618600	0.309300	+	0.950965i	$0.399905\pi$
$410$	0	0
$411$	22.3155	1.10074
$412$	− 13.0410i	− 0.642485i
$413$	− 3.43229i	− 0.168892i
$414$	−18.1312	−0.891101
$415$	0	0
$416$	51.3140	2.51588
$417$	− 10.4018i	− 0.509378i
$418$	− 3.88325i	− 0.189936i
$419$	2.06758	0.101008	0.0505040	−	0.998724i	$-0.483917\pi$
$419$	2.06758	0.101008	0.0505040	+	0.998724i	$0.483917\pi$
$420$	0	0
$421$	−3.51854	−0.171483	−0.0857416	−	0.996317i	$-0.527326\pi$
$421$	−3.51854	−0.171483	−0.0857416	+	0.996317i	$0.527326\pi$
$422$	− 63.9653i	− 3.11378i
$423$	− 2.42835i	− 0.118070i
$424$	114.035	5.53804
$425$	0	0
$426$	24.3719	1.18082
$427$	− 1.77859i	− 0.0860719i
$428$	23.7511i	1.14805i
$429$	−4.85670	−0.234484
$430$	0	0
$431$	31.4468	1.51474	0.757369	−	0.652987i	$-0.226485\pi$
$431$	31.4468	1.51474	0.757369	+	0.652987i	$0.226485\pi$
$432$	12.0072i	0.577698i
$433$	20.9774i	1.00811i	0.863672	+	0.504055i	$0.168159\pi$
$433$	20.9774i	1.00811i	−0.863672	+	0.504055i	$0.831841\pi$
$434$	10.6763	0.512480
$435$	0	0
$436$	−32.0145	−1.53322
$437$	− 6.79306i	− 0.324956i
$438$	− 31.1650i	− 1.48912i
$439$	−3.75598	−0.179263	−0.0896315	−	0.995975i	$-0.528569\pi$
$439$	−3.75598	−0.179263	−0.0896315	+	0.995975i	$0.528569\pi$
$440$	0	0
$441$	−6.79306	−0.323479
$442$	0	0
$443$	20.0184i	0.951104i	0.879688	+	0.475552i	$0.157752\pi$
$443$	20.0184i	0.951104i	−0.879688	+	0.475552i	$0.842248\pi$
$444$	−52.5104	−2.49203
$445$	0	0
$446$	8.55957	0.405307
$447$	5.15777i	0.243954i
$448$	− 7.73995i	− 0.365678i
$449$	−22.4057	−1.05739	−0.528696	−	0.848811i	$-0.677319\pi$
$449$	−22.4057	−1.05739	−0.528696	+	0.848811i	$0.677319\pi$
$450$	0	0
$451$	−5.78098	−0.272216
$452$	− 26.6803i	− 1.25493i
$453$	8.49593i	0.399174i
$454$	−24.3719	−1.14383
$455$	0	0
$456$	−8.33816	−0.390470
$457$	34.2890i	1.60397i	0.597343	+	0.801986i	$0.296223\pi$
$457$	34.2890i	1.60397i	−0.597343	+	0.801986i	$0.703777\pi$
$458$	44.6803i	2.08777i
$459$	0	0
$460$	0	0
$461$	14.5104	0.675817	0.337909	−	0.941179i	$-0.390281\pi$
$461$	14.5104	0.675817	0.337909	+	0.941179i	$0.390281\pi$
$462$	1.76651i	0.0821854i
$463$	− 33.8486i	− 1.57308i	−0.617542	−	0.786538i	$-0.711871\pi$
$463$	− 33.8486i	− 1.57308i	0.617542	−	0.786538i	$-0.288129\pi$
$464$	−36.0217	−1.67227
$465$	0	0
$466$	25.5475	1.18346
$467$	− 9.22141i	− 0.426716i	−0.976974	−	0.213358i	$-0.931560\pi$
$467$	− 9.22141i	− 0.426716i	0.976974	−	0.213358i	$-0.0684401\pi$
$468$	17.1047i	0.790663i
$469$	−6.42835	−0.296834
$470$	0	0
$471$	22.6498	1.04365
$472$	− 62.9122i	− 2.89577i
$473$	11.6392i	0.535172i
$474$	23.9508	1.10010
$475$	0	0
$476$	0	0
$477$	13.6763i	0.626196i
$478$	46.7584i	2.13868i
$479$	−27.3035	−1.24753	−0.623764	−	0.781613i	$-0.714397\pi$
$479$	−27.3035	−1.24753	−0.623764	+	0.781613i	$0.714397\pi$
$480$	0	0
$481$	−34.2093	−1.55981
$482$	− 45.1336i	− 2.05578i
$483$	3.09019i	0.140609i
$484$	45.5176	2.06898
$485$	0	0
$486$	−2.66908	−0.121072
$487$	14.4815i	0.656218i	0.944640	+	0.328109i	$0.106411\pi$
$487$	14.4815i	0.656218i	−0.944640	+	0.328109i	$0.893589\pi$
$488$	− 32.6006i	− 1.47576i
$489$	−3.72548	−0.168472
$490$	0	0
$491$	−38.2093	−1.72436	−0.862182	−	0.506599i	$-0.830902\pi$
$491$	−38.2093	−1.72436	−0.862182	+	0.506599i	$0.830902\pi$
$492$	20.3599i	0.917894i
$493$	0	0
$494$	−8.90981	−0.400871
$495$	0	0
$496$	105.580	4.74070
$497$	− 4.15383i	− 0.186325i
$498$	− 12.1312i	− 0.543613i
$499$	−35.5740	−1.59251	−0.796256	−	0.604959i	$-0.793189\pi$
$499$	−35.5740	−1.59251	−0.796256	+	0.604959i	$0.793189\pi$
$500$	0	0
$501$	12.2214	0.546012
$502$	31.0660i	1.38654i
$503$	6.84223i	0.305080i	0.988297	+	0.152540i	$0.0487452\pi$
$503$	6.84223i	0.305080i	−0.988297	+	0.152540i	$0.951255\pi$
$504$	3.79306	0.168956
$505$	0	0
$506$	−26.3792	−1.17270
$507$	− 1.85670i	− 0.0824589i
$508$	54.3792i	2.41269i
$509$	−25.1352	−1.11410	−0.557048	−	0.830480i	$-0.688066\pi$
$509$	−25.1352	−1.11410	−0.557048	+	0.830480i	$0.688066\pi$
$510$	0	0
$511$	−5.31160	−0.234972
$512$	15.6618i	0.692162i
$513$	− 1.00000i	− 0.0441511i
$514$	−12.7222	−0.561152
$515$	0	0
$516$	40.9919	1.80457
$517$	− 3.53302i	− 0.155382i
$518$	12.4428i	0.546706i
$519$	−24.1722	−1.06104
$520$	0	0
$521$	10.5861	0.463787	0.231893	−	0.972741i	$-0.425508\pi$
$521$	10.5861	0.463787	0.231893	+	0.972741i	$0.425508\pi$
$522$	− 8.00724i	− 0.350467i
$523$	33.9614i	1.48503i	0.669831	+	0.742513i	$0.266366\pi$
$523$	33.9614i	1.48503i	−0.669831	+	0.742513i	$0.733634\pi$
$524$	−111.205	−4.85802
$525$	0	0
$526$	−39.6642	−1.72944
$527$	0	0
$528$	17.4694i	0.760257i
$529$	−23.1457	−1.00633
$530$	0	0
$531$	7.54510	0.327429
$532$	2.33092i	0.101058i
$533$	13.2640i	0.574527i
$534$	−0.240729	−0.0104174
$535$	0	0
$536$	−117.828	−5.08941
$537$	− 6.22141i	− 0.268474i
$538$	− 57.4733i	− 2.47785i
$539$	−9.88325	−0.425702
$540$	0	0
$541$	−33.7318	−1.45024	−0.725122	−	0.688620i	$-0.758217\pi$
$541$	−33.7318	−1.45024	−0.725122	+	0.688620i	$0.758217\pi$
$542$	3.01932i	0.129691i
$543$	20.4959i	0.879565i
$544$	0	0
$545$	0	0
$546$	4.05311	0.173457
$547$	− 20.7255i	− 0.886158i	−0.896483	−	0.443079i	$-0.853886\pi$
$547$	− 20.7255i	− 0.886158i	0.896483	−	0.443079i	$-0.146114\pi$
$548$	− 114.344i	− 4.88455i
$549$	3.90981	0.166867
$550$	0	0
$551$	3.00000	0.127804
$552$	56.6416i	2.41083i
$553$	− 4.08206i	− 0.173587i
$554$	−35.3985	−1.50394
$555$	0	0
$556$	−53.2986	−2.26037
$557$	− 13.3237i	− 0.564543i	−0.959335	−	0.282271i	$-0.908912\pi$
$557$	− 13.3237i	− 0.564543i	0.959335	−	0.282271i	$-0.0910878\pi$
$558$	23.4694i	0.993538i
$559$	26.7053	1.12951
$560$	0	0
$561$	0	0
$562$	− 37.8977i	− 1.59862i
$563$	− 11.7786i	− 0.496408i	−0.968708	−	0.248204i	$-0.920160\pi$
$563$	− 11.7786i	− 0.496408i	0.968708	−	0.248204i	$-0.0798404\pi$
$564$	−12.4428	−0.523937
$565$	0	0
$566$	52.1352	2.19140
$567$	0.454904i	0.0191042i
$568$	− 76.1376i	− 3.19466i
$569$	−43.2278	−1.81220	−0.906101	−	0.423062i	$-0.860955\pi$
$569$	−43.2278	−1.81220	−0.906101	+	0.423062i	$0.860955\pi$
$570$	0	0
$571$	−3.41782	−0.143031	−0.0715157	−	0.997439i	$-0.522784\pi$
$571$	−3.41782	−0.143031	−0.0715157	+	0.997439i	$0.522784\pi$
$572$	24.8856i	1.04052i
$573$	− 0.793062i	− 0.0331307i
$574$	4.82446	0.201369
$575$	0	0
$576$	17.0145	0.708936
$577$	− 32.4081i	− 1.34917i	−0.738198	−	0.674584i	$-0.764323\pi$
$577$	− 32.4081i	− 1.34917i	0.738198	−	0.674584i	$-0.235677\pi$
$578$	− 45.3743i	− 1.88732i
$579$	15.8196	0.657441
$580$	0	0
$581$	−2.06758	−0.0857778
$582$	− 32.0676i	− 1.32924i
$583$	19.8977i	0.824080i
$584$	−97.3590	−4.02874
$585$	0	0
$586$	66.6561	2.75354
$587$	− 7.93636i	− 0.327569i	−0.986496	−	0.163784i	$-0.947630\pi$
$587$	− 7.93636i	− 0.327569i	0.986496	−	0.163784i	$-0.0523701\pi$
$588$	34.8075i	1.43544i
$589$	−8.79306	−0.362312
$590$	0	0
$591$	3.75203	0.154338
$592$	123.050i	5.05731i
$593$	− 1.98553i	− 0.0815358i	−0.999169	−	0.0407679i	$-0.987020\pi$
$593$	− 1.98553i	− 0.0815358i	0.999169	−	0.0407679i	$-0.0129804\pi$
$594$	−3.88325	−0.159332
$595$	0	0
$596$	26.4283	1.08255
$597$	6.45490i	0.264182i
$598$	60.5249i	2.47505i
$599$	21.3526	0.872445	0.436222	−	0.899839i	$-0.356316\pi$
$599$	21.3526	0.872445	0.436222	+	0.899839i	$0.356316\pi$
$600$	0	0
$601$	−18.2480	−0.744350	−0.372175	−	0.928163i	$-0.621388\pi$
$601$	−18.2480	−0.744350	−0.372175	+	0.928163i	$0.621388\pi$
$602$	− 9.71340i	− 0.395889i
$603$	− 14.1312i	− 0.575468i
$604$	43.5330	1.77133
$605$	0	0
$606$	−14.7294	−0.598342
$607$	− 23.6353i	− 0.959327i	−0.877453	−	0.479663i	$-0.840759\pi$
$607$	− 23.6353i	− 0.959327i	0.877453	−	0.479663i	$-0.159241\pi$
$608$	15.3719i	0.623415i
$609$	−1.36471	−0.0553009
$610$	0	0
$611$	−8.10622	−0.327942
$612$	0	0
$613$	36.9122i	1.49087i	0.666578	+	0.745435i	$0.267758\pi$
$613$	36.9122i	1.49087i	−0.666578	+	0.745435i	$0.732242\pi$
$614$	30.4323	1.22815
$615$	0	0
$616$	5.51854	0.222348
$617$	14.6908i	0.591429i	0.955276	+	0.295714i	$0.0955577\pi$
$617$	14.6908i	0.591429i	−0.955276	+	0.295714i	$0.904442\pi$
$618$	− 6.79306i	− 0.273257i
$619$	24.3487	0.978656	0.489328	−	0.872100i	$-0.337242\pi$
$619$	24.3487	0.978656	0.489328	+	0.872100i	$0.337242\pi$
$620$	0	0
$621$	−6.79306	−0.272596
$622$	− 3.53302i	− 0.141661i
$623$	0.0410286i	0.00164378i
$624$	40.0821	1.60457
$625$	0	0
$626$	−55.0169	−2.19892
$627$	− 1.45490i	− 0.0581033i
$628$	− 116.057i	− 4.63118i
$629$	0	0
$630$	0	0
$631$	21.3237	0.848882	0.424441	−	0.905455i	$-0.360471\pi$
$631$	21.3237	0.848882	0.424441	+	0.905455i	$0.360471\pi$
$632$	− 74.8220i	− 2.97626i
$633$	− 23.9653i	− 0.952536i
$634$	−48.5032	−1.92631
$635$	0	0
$636$	70.0772	2.77874
$637$	22.6763i	0.898468i
$638$	− 11.6498i	− 0.461219i
$639$	9.13122	0.361225
$640$	0	0
$641$	−26.9919	−1.06611	−0.533057	−	0.846079i	$-0.678957\pi$
$641$	−26.9919	−1.06611	−0.533057	+	0.846079i	$0.678957\pi$
$642$	12.3719i	0.488282i
$643$	− 40.1683i	− 1.58408i	−0.610467	−	0.792042i	$-0.709018\pi$
$643$	− 40.1683i	− 1.58408i	0.610467	−	0.792042i	$-0.290982\pi$
$644$	15.8341	0.623951
$645$	0	0
$646$	0	0
$647$	− 17.2890i	− 0.679701i	−0.940480	−	0.339850i	$-0.889624\pi$
$647$	− 17.2890i	− 0.679701i	0.940480	−	0.339850i	$-0.110376\pi$
$648$	8.33816i	0.327554i
$649$	10.9774	0.430900
$650$	0	0
$651$	4.00000	0.156772
$652$	19.0893i	0.747594i
$653$	21.6537i	0.847375i	0.905808	+	0.423688i	$0.139265\pi$
$653$	21.6537i	0.847375i	−0.905808	+	0.423688i	$0.860735\pi$
$654$	−16.6763	−0.652096
$655$	0	0
$656$	47.7101	1.86277
$657$	− 11.6763i	− 0.455536i
$658$	2.94844i	0.114942i
$659$	11.6392	0.453400	0.226700	−	0.973965i	$-0.427206\pi$
$659$	11.6392	0.453400	0.226700	+	0.973965i	$0.427206\pi$
$660$	0	0
$661$	−4.96292	−0.193035	−0.0965175	−	0.995331i	$-0.530770\pi$
$661$	−4.96292	−0.193035	−0.0965175	+	0.995331i	$0.530770\pi$
$662$	52.1071i	2.02520i
$663$	0	0
$664$	−37.8977	−1.47072
$665$	0	0
$666$	−27.3526	−1.05989
$667$	− 20.3792i	− 0.789085i
$668$	− 62.6223i	− 2.42293i
$669$	3.20694	0.123987
$670$	0	0
$671$	5.68840	0.219598
$672$	− 6.99276i	− 0.269752i
$673$	− 13.1578i	− 0.507195i	−0.967310	−	0.253597i	$-0.918386\pi$
$673$	− 13.1578i	− 0.507195i	0.967310	−	0.253597i	$-0.0816139\pi$
$674$	1.10467	0.0425502
$675$	0	0
$676$	−9.51370	−0.365912
$677$	− 13.1352i	− 0.504825i	−0.967620	−	0.252413i	$-0.918776\pi$
$677$	− 13.1352i	− 0.504825i	0.967620	−	0.252413i	$-0.0812241\pi$
$678$	− 13.8977i	− 0.533739i
$679$	−5.46543	−0.209744
$680$	0	0
$681$	−9.13122	−0.349909
$682$	34.1457i	1.30751i
$683$	− 46.5370i	− 1.78069i	−0.455289	−	0.890344i	$-0.650464\pi$
$683$	− 46.5370i	− 1.78069i	0.455289	−	0.890344i	$-0.349536\pi$
$684$	−5.12398	−0.195920
$685$	0	0
$686$	16.7472	0.639411
$687$	16.7400i	0.638669i
$688$	− 96.0579i	− 3.66217i
$689$	45.6537	1.73927
$690$	0	0
$691$	−4.26244	−0.162151	−0.0810754	−	0.996708i	$-0.525835\pi$
$691$	−4.26244	−0.162151	−0.0810754	+	0.996708i	$0.525835\pi$
$692$	123.858i	4.70838i
$693$	0.661842i	0.0251413i
$694$	−82.5394	−3.13315
$695$	0	0
$696$	−25.0145	−0.948171
$697$	0	0
$698$	− 39.4130i	− 1.49180i
$699$	9.57165	0.362033
$700$	0	0
$701$	−36.4428	−1.37643	−0.688213	−	0.725508i	$-0.741605\pi$
$701$	−36.4428	−1.37643	−0.688213	+	0.725508i	$0.741605\pi$
$702$	8.90981i	0.336279i
$703$	− 10.2480i	− 0.386509i
$704$	24.7544	0.932968
$705$	0	0
$706$	73.4878	2.76575
$707$	2.51041i	0.0944136i
$708$	− 38.6609i	− 1.45297i
$709$	5.23349	0.196548	0.0982740	−	0.995159i	$-0.468668\pi$
$709$	5.23349	0.196548	0.0982740	+	0.995159i	$0.468668\pi$
$710$	0	0
$711$	8.97345	0.336531
$712$	0.752034i	0.0281837i
$713$	59.7318i	2.23697i
$714$	0	0
$715$	0	0
$716$	−31.8784	−1.19135
$717$	17.5185i	0.654242i
$718$	− 10.2335i	− 0.381911i
$719$	−50.2809	−1.87516	−0.937580	−	0.347770i	$-0.886939\pi$
$719$	−50.2809	−1.87516	−0.937580	+	0.347770i	$0.886939\pi$
$720$	0	0
$721$	−1.15777	−0.0431178
$722$	− 2.66908i	− 0.0993328i
$723$	− 16.9098i	− 0.628883i
$724$	105.021	3.90307
$725$	0	0
$726$	23.7101	0.879965
$727$	− 2.81567i	− 0.104427i	−0.998636	−	0.0522137i	$-0.983372\pi$
$727$	− 2.81567i	− 0.104427i	0.998636	−	0.0522137i	$-0.0166277\pi$
$728$	− 12.6618i	− 0.469279i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	0	0
$732$	− 20.0338i	− 0.740470i
$733$	− 9.23349i	− 0.341047i	−0.985354	−	0.170523i	$-0.945454\pi$
$733$	− 9.23349i	− 0.341047i	0.985354	−	0.170523i	$-0.0545458\pi$
$734$	56.6522	2.09107
$735$	0	0
$736$	104.423	3.84907
$737$	− 20.5596i	− 0.757322i
$738$	10.6054i	0.390392i
$739$	11.1843	0.411422	0.205711	−	0.978613i	$-0.434049\pi$
$739$	11.1843	0.411422	0.205711	+	0.978613i	$0.434049\pi$
$740$	0	0
$741$	−3.33816	−0.122630
$742$	− 16.6054i	− 0.609605i
$743$	23.1602i	0.849664i	0.905272	+	0.424832i	$0.139667\pi$
$743$	23.1602i	0.849664i	−0.905272	+	0.424832i	$0.860333\pi$
$744$	73.3179	2.68797
$745$	0	0
$746$	59.4202	2.17553
$747$	− 4.54510i	− 0.166296i
$748$	0	0
$749$	2.10861	0.0770470
$750$	0	0
$751$	11.7520	0.428838	0.214419	−	0.976742i	$-0.431214\pi$
$751$	11.7520	0.428838	0.214419	+	0.976742i	$0.431214\pi$
$752$	29.1578i	1.06327i
$753$	11.6392i	0.424157i
$754$	−26.7294	−0.973428
$755$	0	0
$756$	2.33092	0.0847748
$757$	17.6232i	0.640526i	0.947329	+	0.320263i	$0.103771\pi$
$757$	17.6232i	0.640526i	−0.947329	+	0.320263i	$0.896229\pi$
$758$	− 8.90981i	− 0.323619i
$759$	−9.88325	−0.358739
$760$	0	0
$761$	−22.5556	−0.817641	−0.408820	−	0.912615i	$-0.634060\pi$
$761$	−22.5556	−0.817641	−0.408820	+	0.912615i	$0.634060\pi$
$762$	28.3261i	1.02615i
$763$	2.84223i	0.102895i
$764$	−4.06364	−0.147017
$765$	0	0
$766$	−92.1819	−3.33067
$767$	− 25.1867i	− 0.909440i
$768$	− 5.12398i	− 0.184896i
$769$	26.8486	0.968184	0.484092	−	0.875017i	$-0.339150\pi$
$769$	26.8486	0.968184	0.484092	+	0.875017i	$0.339150\pi$
$770$	0	0
$771$	−4.76651	−0.171662
$772$	− 81.0594i	− 2.91739i
$773$	33.8751i	1.21840i	0.793015	+	0.609202i	$0.208510\pi$
$773$	33.8751i	1.21840i	−0.793015	+	0.609202i	$0.791490\pi$
$774$	21.3526	0.767505
$775$	0	0
$776$	−100.179	−3.59620
$777$	4.66184i	0.167243i
$778$	50.9919i	1.82815i
$779$	−3.97345	−0.142363
$780$	0	0
$781$	13.2850	0.475376
$782$	0	0
$783$	− 3.00000i	− 0.107211i
$784$	81.5659	2.91307
$785$	0	0
$786$	−57.9267	−2.06618
$787$	− 30.0781i	− 1.07217i	−0.844164	−	0.536084i	$-0.819903\pi$
$787$	− 30.0781i	− 1.07217i	0.844164	−	0.536084i	$-0.180097\pi$
$788$	− 19.2254i	− 0.684875i
$789$	−14.8606	−0.529053
$790$	0	0
$791$	−2.36866	−0.0842198
$792$	12.1312i	0.431064i
$793$	− 13.0516i	− 0.463474i
$794$	36.9138	1.31002
$795$	0	0
$796$	33.0748	1.17231
$797$	− 29.3792i	− 1.04066i	−0.853964	−	0.520332i	$-0.825808\pi$
$797$	− 29.3792i	− 1.04066i	0.853964	−	0.520332i	$-0.174192\pi$
$798$	1.21417i	0.0429813i
$799$	0	0
$800$	0	0
$801$	−0.0901918	−0.00318677
$802$	− 18.1312i	− 0.640236i
$803$	− 16.9879i	− 0.599491i
$804$	−72.4081	−2.55364
$805$	0	0
$806$	78.3445	2.75957
$807$	− 21.5330i	− 0.757998i
$808$	46.0145i	1.61878i
$809$	45.7358	1.60798	0.803992	−	0.594640i	$-0.202705\pi$
$809$	45.7358	1.60798	0.803992	+	0.594640i	$0.202705\pi$
$810$	0	0
$811$	6.10227	0.214280	0.107140	−	0.994244i	$-0.465831\pi$
$811$	6.10227	0.214280	0.107140	+	0.994244i	$0.465831\pi$
$812$	6.99276i	0.245398i
$813$	1.13122i	0.0396736i
$814$	−39.7955	−1.39483
$815$	0	0
$816$	0	0
$817$	8.00000i	0.279885i
$818$	− 33.3913i	− 1.16750i
$819$	1.51854	0.0530621
$820$	0	0
$821$	−11.2175	−0.391492	−0.195746	−	0.980655i	$-0.562713\pi$
$821$	−11.2175	−0.391492	−0.195746	+	0.980655i	$0.562713\pi$
$822$	− 59.5620i	− 2.07746i
$823$	− 24.6353i	− 0.858732i	−0.903131	−	0.429366i	$-0.858737\pi$
$823$	− 24.6353i	− 0.858732i	0.903131	−	0.429366i	$-0.141263\pi$
$824$	−21.2214	−0.739283
$825$	0	0
$826$	−9.16107	−0.318754
$827$	− 32.6311i	− 1.13469i	−0.823479	−	0.567347i	$-0.807970\pi$
$827$	− 32.6311i	− 1.13469i	0.823479	−	0.567347i	$-0.192030\pi$
$828$	34.8075i	1.20965i
$829$	33.3382	1.15788	0.578941	−	0.815369i	$-0.303466\pi$
$829$	33.3382	1.15788	0.578941	+	0.815369i	$0.303466\pi$
$830$	0	0
$831$	−13.2624	−0.460069
$832$	− 56.7970i	− 1.96908i
$833$	0	0
$834$	−27.7632	−0.961362
$835$	0	0
$836$	−7.45490	−0.257833
$837$	8.79306i	0.303933i
$838$	− 5.51854i	− 0.190635i
$839$	−51.1312	−1.76525	−0.882623	−	0.470082i	$-0.844224\pi$
$839$	−51.1312	−1.76525	−0.882623	+	0.470082i	$0.844224\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	9.39127i	0.323644i
$843$	− 14.1988i	− 0.489033i
$844$	−122.798	−4.22688
$845$	0	0
$846$	−6.48146	−0.222837
$847$	− 4.04103i	− 0.138851i
$848$	− 164.215i	− 5.63916i
$849$	19.5330	0.670371
$850$	0	0
$851$	−69.6151	−2.38637
$852$	− 46.7882i	− 1.60294i
$853$	− 40.9122i	− 1.40081i	−0.713747	−	0.700404i	$-0.753003\pi$
$853$	− 40.9122i	− 1.40081i	0.713747	−	0.700404i	$-0.246997\pi$
$854$	−4.74719	−0.162446
$855$	0	0
$856$	38.6498	1.32102
$857$	17.8220i	0.608788i	0.952546	+	0.304394i	$0.0984540\pi$
$857$	17.8220i	0.608788i	−0.952546	+	0.304394i	$0.901546\pi$
$858$	12.9629i	0.442547i
$859$	−14.1231	−0.481873	−0.240937	−	0.970541i	$-0.577455\pi$
$859$	−14.1231	−0.481873	−0.240937	+	0.970541i	$0.577455\pi$
$860$	0	0
$861$	1.80754	0.0616007
$862$	− 83.9339i	− 2.85880i
$863$	32.4307i	1.10396i	0.833859	+	0.551978i	$0.186127\pi$
$863$	32.4307i	1.10396i	−0.833859	+	0.551978i	$0.813873\pi$
$864$	15.3719	0.522964
$865$	0	0
$866$	55.9903	1.90263
$867$	− 17.0000i	− 0.577350i
$868$	− 20.4959i	− 0.695677i
$869$	13.0555	0.442878
$870$	0	0
$871$	−47.1722	−1.59837
$872$	52.0965i	1.76421i
$873$	− 12.0145i	− 0.406628i
$874$	−18.1312	−0.613298
$875$	0	0
$876$	−59.8292	−2.02144
$877$	20.9243i	0.706563i	0.935517	+	0.353281i	$0.114934\pi$
$877$	20.9243i	0.706563i	−0.935517	+	0.353281i	$0.885066\pi$
$878$	10.0250i	0.338327i
$879$	24.9734	0.842333
$880$	0	0
$881$	−31.3671	−1.05678	−0.528392	−	0.849000i	$-0.677205\pi$
$881$	−31.3671	−1.05678	−0.528392	+	0.849000i	$0.677205\pi$
$882$	18.1312i	0.610510i
$883$	− 0.635288i	− 0.0213791i	−0.999943	−	0.0106896i	$-0.996597\pi$
$883$	− 0.635288i	− 0.0213791i	0.999943	−	0.0106896i	$-0.00340266\pi$
$884$	0	0
$885$	0	0
$886$	53.4307	1.79504
$887$	− 55.0660i	− 1.84894i	−0.381259	−	0.924468i	$-0.624509\pi$
$887$	− 55.0660i	− 1.84894i	0.381259	−	0.924468i	$-0.375491\pi$
$888$	85.4492i	2.86749i
$889$	4.82775	0.161918
$890$	0	0
$891$	−1.45490	−0.0487411
$892$	− 16.4323i	− 0.550194i
$893$	− 2.42835i	− 0.0812616i
$894$	13.7665	0.460421
$895$	0	0
$896$	−6.67302	−0.222930
$897$	22.6763i	0.757140i
$898$	59.8027i	1.99564i
$899$	−26.3792	−0.879795
$900$	0	0
$901$	0	0
$902$	15.4299i	0.513759i
$903$	− 3.63923i	− 0.121106i
$904$	−43.4163	−1.44400
$905$	0	0
$906$	22.6763	0.753370
$907$	− 8.49593i	− 0.282103i	−0.990002	−	0.141051i	$-0.954952\pi$
$907$	− 8.49593i	− 0.282103i	0.990002	−	0.141051i	$-0.0450483\pi$
$908$	46.7882i	1.55272i
$909$	−5.51854	−0.183038
$910$	0	0
$911$	58.9798	1.95409	0.977044	−	0.213039i	$-0.0683361\pi$
$911$	58.9798	1.95409	0.977044	+	0.213039i	$0.0683361\pi$
$912$	12.0072i	0.397599i
$913$	− 6.61268i	− 0.218848i
$914$	91.5200	3.02721
$915$	0	0
$916$	85.7752	2.83409
$917$	9.87272i	0.326026i
$918$	0	0
$919$	−8.22141	−0.271199	−0.135600	−	0.990764i	$-0.543296\pi$
$919$	−8.22141	−0.271199	−0.135600	+	0.990764i	$0.543296\pi$
$920$	0	0
$921$	11.4018	0.375702
$922$	− 38.7294i	− 1.27549i
$923$	− 30.4815i	− 1.00331i
$924$	3.39127	0.111564
$925$	0	0
$926$	−90.3445	−2.96890
$927$	− 2.54510i	− 0.0835919i
$928$	46.1158i	1.51383i
$929$	−1.76651	−0.0579573	−0.0289786	−	0.999580i	$-0.509225\pi$
$929$	−1.76651	−0.0579573	−0.0289786	+	0.999580i	$0.509225\pi$
$930$	0	0
$931$	−6.79306	−0.222634
$932$	− 49.0450i	− 1.60652i
$933$	− 1.32368i	− 0.0433355i
$934$	−24.6127	−0.805351
$935$	0	0
$936$	27.8341	0.909786
$937$	− 6.20694i	− 0.202772i	−0.994847	−	0.101386i	$-0.967672\pi$
$937$	− 6.20694i	− 0.202772i	0.994847	−	0.101386i	$-0.0323277\pi$
$938$	17.1578i	0.560221i
$939$	−20.6127	−0.672669
$940$	0	0
$941$	15.8833	0.517779	0.258890	−	0.965907i	$-0.416643\pi$
$941$	15.8833	0.517779	0.258890	+	0.965907i	$0.416643\pi$
$942$	− 60.4540i	− 1.96970i
$943$	26.9919i	0.878976i
$944$	−90.5958	−2.94864
$945$	0	0
$946$	31.0660	1.01004
$947$	16.1352i	0.524322i	0.965024	+	0.262161i	$0.0844352\pi$
$947$	16.1352i	0.524322i	−0.965024	+	0.262161i	$0.915565\pi$
$948$	− 45.9798i	− 1.49335i
$949$	−38.9774	−1.26526
$950$	0	0
$951$	−18.1722	−0.589276
$952$	0	0
$953$	− 41.1110i	− 1.33172i	−0.746078	−	0.665858i	$-0.768066\pi$
$953$	− 41.1110i	− 1.33172i	0.746078	−	0.665858i	$-0.231934\pi$
$954$	36.5032	1.18183
$955$	0	0
$956$	89.7647	2.90320
$957$	− 4.36471i	− 0.141091i
$958$	72.8751i	2.35449i
$959$	−10.1514	−0.327807
$960$	0	0
$961$	46.3179	1.49413
$962$	91.3074i	2.94387i
$963$	4.63529i	0.149370i
$964$	−86.6456	−2.79067
$965$	0	0
$966$	8.24797	0.265374
$967$	30.9798i	0.996243i	0.867107	+	0.498121i	$0.165977\pi$
$967$	30.9798i	0.996243i	−0.867107	+	0.498121i	$0.834023\pi$
$968$	− 74.0700i	− 2.38070i
$969$	0	0
$970$	0	0
$971$	51.2133	1.64351	0.821756	−	0.569839i	$-0.192995\pi$
$971$	51.2133	1.64351	0.821756	+	0.569839i	$0.192995\pi$
$972$	5.12398i	0.164352i
$973$	4.73182i	0.151695i
$974$	38.6522	1.23850
$975$	0	0
$976$	−46.9460	−1.50270
$977$	20.2093i	0.646554i	0.946304	+	0.323277i	$0.104785\pi$
$977$	20.2093i	0.646554i	−0.946304	+	0.323277i	$0.895215\pi$
$978$	9.94360i	0.317961i
$979$	−0.131220	−0.00419382
$980$	0	0
$981$	−6.24797	−0.199482
$982$	101.984i	3.25443i
$983$	− 41.1191i	− 1.31150i	−0.754979	−	0.655748i	$-0.772353\pi$
$983$	− 41.1191i	− 1.31150i	0.754979	−	0.655748i	$-0.227647\pi$
$984$	33.1312	1.05618
$985$	0	0
$986$	0	0
$987$	1.10467i	0.0351619i
$988$	17.1047i	0.544172i
$989$	54.3445	1.72805
$990$	0	0
$991$	27.4549	0.872134	0.436067	−	0.899914i	$-0.356371\pi$
$991$	27.4549	0.872134	0.436067	+	0.899914i	$0.356371\pi$
$992$	− 135.167i	− 4.29154i
$993$	19.5225i	0.619527i
$994$	−11.0869	−0.351655
$995$	0	0
$996$	−23.2890	−0.737940
$997$	13.1086i	0.415154i	0.978219	+	0.207577i	$0.0665577\pi$
$997$	13.1086i	0.415154i	−0.978219	+	0.207577i	$0.933442\pi$
$998$	94.9499i	3.00559i
$999$	−10.2480	−0.324231

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1425.2.c.p.799.1		6
5.2	odd	4		1425.2.a.u.1.3	✓	3
5.3	odd	4		1425.2.a.v.1.1	yes	3
5.4	even	2	inner	1425.2.c.p.799.6		6
15.2	even	4		4275.2.a.be.1.1		3
15.8	even	4		4275.2.a.bh.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1425.2.a.u.1.3	✓	3	5.2	odd	4
1425.2.a.v.1.1	yes	3	5.3	odd	4
1425.2.c.p.799.1		6	1.1	even	1	trivial
1425.2.c.p.799.6		6	5.4	even	2	inner
4275.2.a.be.1.1		3	15.2	even	4
4275.2.a.bh.1.3		3	15.8	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1425 = 3 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1425.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.66908i q^{2} -1.00000i q^{3} -5.12398 q^{4} -2.66908 q^{6} +0.454904i q^{7} +8.33816i q^{8} -1.00000 q^{9} -1.45490 q^{11} +5.12398i q^{12} +3.33816i q^{13} +1.21417 q^{14} +12.0072 q^{16} +2.66908i q^{18} -1.00000 q^{19} +0.454904 q^{21} +3.88325i q^{22} +6.79306i q^{23} +8.33816 q^{24} +8.90981 q^{26} +1.00000i q^{27} -2.33092i q^{28} -3.00000 q^{29} +8.79306 q^{31} -15.3719i q^{32} +1.45490i q^{33} +5.12398 q^{36} +10.2480i q^{37} +2.66908i q^{38} +3.33816 q^{39} +3.97345 q^{41} -1.21417i q^{42} -8.00000i q^{43} +7.45490 q^{44} +18.1312 q^{46} +2.42835i q^{47} -12.0072i q^{48} +6.79306 q^{49} -17.1047i q^{52} -13.6763i q^{53} +2.66908 q^{54} -3.79306 q^{56} +1.00000i q^{57} +8.00724i q^{58} -7.54510 q^{59} -3.90981 q^{61} -23.4694i q^{62} -0.454904i q^{63} -17.0145 q^{64} +3.88325 q^{66} +14.1312i q^{67} +6.79306 q^{69} -9.13122 q^{71} -8.33816i q^{72} +11.6763i q^{73} +27.3526 q^{74} +5.12398 q^{76} -0.661842i q^{77} -8.90981i q^{78} -8.97345 q^{79} +1.00000 q^{81} -10.6054i q^{82} +4.54510i q^{83} -2.33092 q^{84} -21.3526 q^{86} +3.00000i q^{87} -12.1312i q^{88} +0.0901918 q^{89} -1.51854 q^{91} -34.8075i q^{92} -8.79306i q^{93} +6.48146 q^{94} -15.3719 q^{96} +12.0145i q^{97} -18.1312i q^{98} +1.45490 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 12 q^{4} - 6 q^{9} - 6 q^{11} - 6 q^{14} + 24 q^{16} - 6 q^{19} + 18 q^{24} + 48 q^{26} - 18 q^{29} + 18 q^{31} + 12 q^{36} - 12 q^{39} + 42 q^{44} + 42 q^{46} + 6 q^{49} + 12 q^{56} - 48 q^{59} - 18 q^{61}+ \cdots + 6 q^{99}+O(q^{100}) \) 6 * q - 12 * q^4 - 6 * q^9 - 6 * q^11 - 6 * q^14 + 24 * q^16 - 6 * q^19 + 18 * q^24 + 48 * q^26 - 18 * q^29 + 18 * q^31 + 12 * q^36 - 12 * q^39 + 42 * q^44 + 42 * q^46 + 6 * q^49 + 12 * q^56 - 48 * q^59 - 18 * q^61 - 6 * q^64 - 6 * q^66 + 6 * q^69 + 12 * q^71 + 36 * q^74 + 12 * q^76 - 30 * q^79 + 6 * q^81 - 30 * q^84 + 6 * q^89 + 12 * q^91 + 60 * q^94 - 36 * q^96 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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