LMFDB - Embedded newform 3800.2.d.o.3649.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3800,2,Mod(3649,3800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3800.3649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3800 = 2^{3} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3800.d (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$30.3431527681$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.419904.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 6x^{4} + 9x^{2} + 1 $ x^6 + 6x^4 + 9x^2 + 1
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			3649.5
Root			$1.53209i$ of defining polynomial
Character	$\chi$	$=$	3800.3649
Dual form			3800.2.d.o.3649.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.53209i q^{3} -2.22668i q^{7} +0.652704 q^{9} -3.22668 q^{11} -1.57398i q^{13} +3.53209i q^{17} -1.00000 q^{19} +3.41147 q^{21} -4.47565i q^{23} +5.59627i q^{27} -1.92127 q^{29} -3.81521 q^{31} -4.94356i q^{33} +11.3550i q^{37} +2.41147 q^{39} +3.47565 q^{41} +1.69459i q^{43} +1.57398i q^{47} +2.04189 q^{49} -5.41147 q^{51} +7.12836i q^{53} -1.53209i q^{57} +7.88713 q^{59} -2.79561 q^{61} -1.45336i q^{63} +3.22668i q^{67} +6.85710 q^{69} -4.38919 q^{71} +6.41147i q^{73} +7.18479i q^{77} +8.59627 q^{79} -6.61587 q^{81} +14.6236i q^{83} -2.94356i q^{87} -6.10607 q^{89} -3.50475 q^{91} -5.84524i q^{93} +15.7023i q^{97} -2.10607 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{9} - 6 q^{11} - 6 q^{19} + 6 q^{29} - 30 q^{31} - 6 q^{39} - 18 q^{41} + 6 q^{49} - 12 q^{51} - 12 q^{59} - 18 q^{61} + 42 q^{69} - 18 q^{71} + 24 q^{79} - 18 q^{81} - 12 q^{89} - 54 q^{91} + 12 q^{99}+O(q^{100}) $ 6 * q + 6 * q^9 - 6 * q^11 - 6 * q^19 + 6 * q^29 - 30 * q^31 - 6 * q^39 - 18 * q^41 + 6 * q^49 - 12 * q^51 - 12 * q^59 - 18 * q^61 + 42 * q^69 - 18 * q^71 + 24 * q^79 - 18 * q^81 - 12 * q^89 - 54 * q^91 + 12 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times$.

$n$	$401$	$951$	$1901$	$1977$
$\chi(n)$	$1$	$1$	$1$	$-1$

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 $

$2$	0	0
$2$	0	0
$3$	1.53209i	0.884552i	0.896879	+	0.442276i	$0.145829\pi$
$3$	1.53209i	0.884552i	−0.896879	+	0.442276i	$0.854171\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.22668i	− 0.841607i	−0.907152	−	0.420803i	$-0.861748\pi$
$7$	− 2.22668i	− 0.841607i	0.907152	−	0.420803i	$-0.138252\pi$
$8$	0	0
$9$	0.652704	0.217568
$10$	0	0
$11$	−3.22668	−0.972881	−0.486441	−	0.873714i	$-0.661705\pi$
$11$	−3.22668	−0.972881	−0.486441	+	0.873714i	$0.661705\pi$
$12$	0	0
$13$	− 1.57398i	− 0.436543i	−0.975888	−	0.218271i	$-0.929958\pi$
$13$	− 1.57398i	− 0.436543i	0.975888	−	0.218271i	$-0.0700418\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.53209i	0.856657i	0.903623	+	0.428329i	$0.140897\pi$
$17$	3.53209i	0.856657i	−0.903623	+	0.428329i	$0.859103\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	3.41147	0.744445
$22$	0	0
$23$	− 4.47565i	− 0.933238i	−0.884458	−	0.466619i	$-0.845472\pi$
$23$	− 4.47565i	− 0.933238i	0.884458	−	0.466619i	$-0.154528\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.59627i	1.07700i
$28$	0	0
$29$	−1.92127	−0.356772	−0.178386	−	0.983961i	$-0.557088\pi$
$29$	−1.92127	−0.356772	−0.178386	+	0.983961i	$0.557088\pi$
$30$	0	0
$31$	−3.81521	−0.685231	−0.342616	−	0.939476i	$-0.611313\pi$
$31$	−3.81521	−0.685231	−0.342616	+	0.939476i	$0.611313\pi$
$32$	0	0
$33$	− 4.94356i	− 0.860564i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	11.3550i	1.86676i	0.358894	+	0.933378i	$0.383154\pi$
$37$	11.3550i	1.86676i	−0.358894	+	0.933378i	$0.616846\pi$
$38$	0	0
$39$	2.41147	0.386145
$40$	0	0
$41$	3.47565	0.542806	0.271403	−	0.962466i	$-0.412512\pi$
$41$	3.47565	0.542806	0.271403	+	0.962466i	$0.412512\pi$
$42$	0	0
$43$	1.69459i	0.258423i	0.991617	+	0.129211i	$0.0412446\pi$
$43$	1.69459i	0.258423i	−0.991617	+	0.129211i	$0.958755\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.57398i	0.229588i	0.993389	+	0.114794i	$0.0366208\pi$
$47$	1.57398i	0.229588i	−0.993389	+	0.114794i	$0.963379\pi$
$48$	0	0
$49$	2.04189	0.291698
$50$	0	0
$51$	−5.41147	−0.757758
$52$	0	0
$53$	7.12836i	0.979155i	0.871960	+	0.489577i	$0.162849\pi$
$53$	7.12836i	0.979155i	−0.871960	+	0.489577i	$0.837151\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 1.53209i	− 0.202930i
$58$	0	0
$59$	7.88713	1.02682	0.513408	−	0.858145i	$-0.328383\pi$
$59$	7.88713	1.02682	0.513408	+	0.858145i	$0.328383\pi$
$60$	0	0
$61$	−2.79561	−0.357941	−0.178970	−	0.983854i	$-0.557277\pi$
$61$	−2.79561	−0.357941	−0.178970	+	0.983854i	$0.557277\pi$
$62$	0	0
$63$	− 1.45336i	− 0.183107i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	3.22668i	0.394202i	0.980383	+	0.197101i	$0.0631527\pi$
$67$	3.22668i	0.394202i	−0.980383	+	0.197101i	$0.936847\pi$
$68$	0	0
$69$	6.85710	0.825497
$70$	0	0
$71$	−4.38919	−0.520900	−0.260450	−	0.965487i	$-0.583871\pi$
$71$	−4.38919	−0.520900	−0.260450	+	0.965487i	$0.583871\pi$
$72$	0	0
$73$	6.41147i	0.750406i	0.926943	+	0.375203i	$0.122427\pi$
$73$	6.41147i	0.750406i	−0.926943	+	0.375203i	$0.877573\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	7.18479i	0.818783i
$78$	0	0
$79$	8.59627	0.967156	0.483578	−	0.875301i	$-0.339337\pi$
$79$	8.59627	0.967156	0.483578	+	0.875301i	$0.339337\pi$
$80$	0	0
$81$	−6.61587	−0.735096
$82$	0	0
$83$	14.6236i	1.60515i	0.596552	+	0.802575i	$0.296537\pi$
$83$	14.6236i	1.60515i	−0.596552	+	0.802575i	$0.703463\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 2.94356i	− 0.315583i
$88$	0	0
$89$	−6.10607	−0.647242	−0.323621	−	0.946187i	$-0.604900\pi$
$89$	−6.10607	−0.647242	−0.323621	+	0.946187i	$0.604900\pi$
$90$	0	0
$91$	−3.50475	−0.367397
$92$	0	0
$93$	− 5.84524i	− 0.606123i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	15.7023i	1.59433i	0.603761	+	0.797165i	$0.293668\pi$
$97$	15.7023i	1.59433i	−0.603761	+	0.797165i	$0.706332\pi$
$98$	0	0
$99$	−2.10607	−0.211668
$100$	0	0
$101$	4.35504	0.433342	0.216671	−	0.976245i	$-0.430480\pi$
$101$	4.35504	0.433342	0.216671	+	0.976245i	$0.430480\pi$
$102$	0	0
$103$	− 4.27126i	− 0.420860i	−0.977609	−	0.210430i	$-0.932514\pi$
$103$	− 4.27126i	− 0.420860i	0.977609	−	0.210430i	$-0.0674863\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0.106067i	0.0102539i	0.999987	+	0.00512693i	$0.00163196\pi$
$107$	0.106067i	0.0102539i	−0.999987	+	0.00512693i	$0.998368\pi$
$108$	0	0
$109$	14.4115	1.38037	0.690184	−	0.723634i	$-0.257529\pi$
$109$	14.4115	1.38037	0.690184	+	0.723634i	$0.257529\pi$
$110$	0	0
$111$	−17.3969	−1.65124
$112$	0	0
$113$	16.5895i	1.56061i	0.625402	+	0.780303i	$0.284935\pi$
$113$	16.5895i	1.56061i	−0.625402	+	0.780303i	$0.715065\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 1.02734i	− 0.0949777i
$118$	0	0
$119$	7.86484	0.720968
$120$	0	0
$121$	−0.588526	−0.0535024
$122$	0	0
$123$	5.32501i	0.480140i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 4.02229i	− 0.356920i	−0.983947	−	0.178460i	$-0.942888\pi$
$127$	− 4.02229i	− 0.356920i	0.983947	−	0.178460i	$-0.0571116\pi$
$128$	0	0
$129$	−2.59627	−0.228589
$130$	0	0
$131$	−4.32770	−0.378113	−0.189056	−	0.981966i	$-0.560543\pi$
$131$	−4.32770	−0.378113	−0.189056	+	0.981966i	$0.560543\pi$
$132$	0	0
$133$	2.22668i	0.193078i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	11.0223i	0.941698i	0.882214	+	0.470849i	$0.156052\pi$
$137$	11.0223i	0.941698i	−0.882214	+	0.470849i	$0.843948\pi$
$138$	0	0
$139$	17.2567	1.46370	0.731848	−	0.681468i	$-0.238658\pi$
$139$	17.2567	1.46370	0.731848	+	0.681468i	$0.238658\pi$
$140$	0	0
$141$	−2.41147	−0.203083
$142$	0	0
$143$	5.07873i	0.424704i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	3.12836i	0.258022i
$148$	0	0
$149$	−12.3824	−1.01440	−0.507202	−	0.861827i	$-0.669320\pi$
$149$	−12.3824	−1.01440	−0.507202	+	0.861827i	$0.669320\pi$
$150$	0	0
$151$	20.8016	1.69281	0.846405	−	0.532540i	$-0.178762\pi$
$151$	20.8016	1.69281	0.846405	+	0.532540i	$0.178762\pi$
$152$	0	0
$153$	2.30541i	0.186381i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 5.25402i	− 0.419317i	−0.977775	−	0.209658i	$-0.932765\pi$
$157$	− 5.25402i	− 0.419317i	0.977775	−	0.209658i	$-0.0672352\pi$
$158$	0	0
$159$	−10.9213	−0.866113
$160$	0	0
$161$	−9.96585	−0.785419
$162$	0	0
$163$	6.63816i	0.519940i	0.965617	+	0.259970i	$0.0837128\pi$
$163$	6.63816i	0.519940i	−0.965617	+	0.259970i	$0.916287\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 16.3824i	− 1.26771i	−0.773453	−	0.633853i	$-0.781472\pi$
$167$	− 16.3824i	− 1.26771i	0.773453	−	0.633853i	$-0.218528\pi$
$168$	0	0
$169$	10.5226	0.809430
$170$	0	0
$171$	−0.652704	−0.0499135
$172$	0	0
$173$	4.01455i	0.305220i	0.988286	+	0.152610i	$0.0487679\pi$
$173$	4.01455i	0.305220i	−0.988286	+	0.152610i	$0.951232\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	12.0838i	0.908272i
$178$	0	0
$179$	−22.2472	−1.66283	−0.831417	−	0.555648i	$-0.812470\pi$
$179$	−22.2472	−1.66283	−0.831417	+	0.555648i	$0.812470\pi$
$180$	0	0
$181$	−7.29591	−0.542301	−0.271150	−	0.962537i	$-0.587404\pi$
$181$	−7.29591	−0.542301	−0.271150	+	0.962537i	$0.587404\pi$
$182$	0	0
$183$	− 4.28312i	− 0.316617i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 11.3969i	− 0.833426i
$188$	0	0
$189$	12.4611	0.906412
$190$	0	0
$191$	−9.55169	−0.691136	−0.345568	−	0.938394i	$-0.612314\pi$
$191$	−9.55169	−0.691136	−0.345568	+	0.938394i	$0.612314\pi$
$192$	0	0
$193$	1.23442i	0.0888557i	0.999013	+	0.0444278i	$0.0141465\pi$
$193$	1.23442i	0.0888557i	−0.999013	+	0.0444278i	$0.985854\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 11.5672i	− 0.824127i	−0.911155	−	0.412063i	$-0.864808\pi$
$197$	− 11.5672i	− 0.824127i	0.911155	−	0.412063i	$-0.135192\pi$
$198$	0	0
$199$	−12.0104	−0.851397	−0.425698	−	0.904865i	$-0.639972\pi$
$199$	−12.0104	−0.851397	−0.425698	+	0.904865i	$0.639972\pi$
$200$	0	0
$201$	−4.94356	−0.348692
$202$	0	0
$203$	4.27807i	0.300261i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 2.92127i	− 0.203043i
$208$	0	0
$209$	3.22668	0.223194
$210$	0	0
$211$	−11.3696	−0.782715	−0.391357	−	0.920239i	$-0.627994\pi$
$211$	−11.3696	−0.782715	−0.391357	+	0.920239i	$0.627994\pi$
$212$	0	0
$213$	− 6.72462i	− 0.460764i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	8.49525i	0.576695i
$218$	0	0
$219$	−9.82295	−0.663773
$220$	0	0
$221$	5.55943	0.373968
$222$	0	0
$223$	− 13.7493i	− 0.920720i	−0.887732	−	0.460360i	$-0.847720\pi$
$223$	− 13.7493i	− 0.920720i	0.887732	−	0.460360i	$-0.152280\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	7.06149i	0.468688i	0.972154	+	0.234344i	$0.0752941\pi$
$227$	7.06149i	0.468688i	−0.972154	+	0.234344i	$0.924706\pi$
$228$	0	0
$229$	7.05644	0.466302	0.233151	−	0.972440i	$-0.425096\pi$
$229$	7.05644	0.466302	0.233151	+	0.972440i	$0.425096\pi$
$230$	0	0
$231$	−11.0077	−0.724256
$232$	0	0
$233$	− 11.9659i	− 0.783909i	−0.919985	−	0.391955i	$-0.871799\pi$
$233$	− 11.9659i	− 0.783909i	0.919985	−	0.391955i	$-0.128201\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	13.1702i	0.855499i
$238$	0	0
$239$	−2.75103	−0.177949	−0.0889747	−	0.996034i	$-0.528359\pi$
$239$	−2.75103	−0.177949	−0.0889747	+	0.996034i	$0.528359\pi$
$240$	0	0
$241$	−7.95811	−0.512627	−0.256313	−	0.966594i	$-0.582508\pi$
$241$	−7.95811	−0.512627	−0.256313	+	0.966594i	$0.582508\pi$
$242$	0	0
$243$	6.65270i	0.426771i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.57398i	0.100150i
$248$	0	0
$249$	−22.4047	−1.41984
$250$	0	0
$251$	−13.6827	−0.863646	−0.431823	−	0.901958i	$-0.642130\pi$
$251$	−13.6827	−0.863646	−0.431823	+	0.901958i	$0.642130\pi$
$252$	0	0
$253$	14.4415i	0.907930i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	10.5517i	0.658196i	0.944296	+	0.329098i	$0.106745\pi$
$257$	10.5517i	0.658196i	−0.944296	+	0.329098i	$0.893255\pi$
$258$	0	0
$259$	25.2841	1.57107
$260$	0	0
$261$	−1.25402	−0.0776221
$262$	0	0
$263$	9.40642i	0.580025i	0.957023	+	0.290012i	$0.0936594\pi$
$263$	9.40642i	0.580025i	−0.957023	+	0.290012i	$0.906341\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 9.35504i	− 0.572519i
$268$	0	0
$269$	−9.02734	−0.550407	−0.275203	−	0.961386i	$-0.588745\pi$
$269$	−9.02734	−0.550407	−0.275203	+	0.961386i	$0.588745\pi$
$270$	0	0
$271$	−22.3354	−1.35678	−0.678391	−	0.734701i	$-0.737322\pi$
$271$	−22.3354	−1.35678	−0.678391	+	0.734701i	$0.737322\pi$
$272$	0	0
$273$	− 5.36959i	− 0.324982i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	17.2959i	1.03921i	0.854406	+	0.519605i	$0.173921\pi$
$277$	17.2959i	1.03921i	−0.854406	+	0.519605i	$0.826079\pi$
$278$	0	0
$279$	−2.49020	−0.149084
$280$	0	0
$281$	−10.4534	−0.623595	−0.311798	−	0.950149i	$-0.600931\pi$
$281$	−10.4534	−0.623595	−0.311798	+	0.950149i	$0.600931\pi$
$282$	0	0
$283$	23.3628i	1.38877i	0.719602	+	0.694386i	$0.244324\pi$
$283$	23.3628i	1.38877i	−0.719602	+	0.694386i	$0.755676\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 7.73917i	− 0.456829i
$288$	0	0
$289$	4.52435	0.266138
$290$	0	0
$291$	−24.0574	−1.41027
$292$	0	0
$293$	27.9641i	1.63368i	0.576864	+	0.816840i	$0.304276\pi$
$293$	27.9641i	1.63368i	−0.576864	+	0.816840i	$0.695724\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 18.0574i	− 1.04779i
$298$	0	0
$299$	−7.04458	−0.407398
$300$	0	0
$301$	3.77332	0.217490
$302$	0	0
$303$	6.67230i	0.383314i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 22.8452i	− 1.30385i	−0.758285	−	0.651923i	$-0.773962\pi$
$307$	− 22.8452i	− 1.30385i	0.758285	−	0.651923i	$-0.226038\pi$
$308$	0	0
$309$	6.54395	0.372272
$310$	0	0
$311$	−11.8699	−0.673080	−0.336540	−	0.941669i	$-0.609257\pi$
$311$	−11.8699	−0.673080	−0.336540	+	0.941669i	$0.609257\pi$
$312$	0	0
$313$	− 25.5672i	− 1.44514i	−0.691297	−	0.722571i	$-0.742960\pi$
$313$	− 25.5672i	− 1.44514i	0.691297	−	0.722571i	$-0.257040\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	17.4020i	0.977392i	0.872454	+	0.488696i	$0.162527\pi$
$317$	17.4020i	0.977392i	−0.872454	+	0.488696i	$0.837473\pi$
$318$	0	0
$319$	6.19934	0.347096
$320$	0	0
$321$	−0.162504	−0.00907008
$322$	0	0
$323$	− 3.53209i	− 0.196531i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	22.0797i	1.22101i
$328$	0	0
$329$	3.50475	0.193223
$330$	0	0
$331$	−6.77063	−0.372147	−0.186074	−	0.982536i	$-0.559576\pi$
$331$	−6.77063	−0.372147	−0.186074	+	0.982536i	$0.559576\pi$
$332$	0	0
$333$	7.41147i	0.406146i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	4.11112i	0.223947i	0.993711	+	0.111973i	$0.0357172\pi$
$337$	4.11112i	0.223947i	−0.993711	+	0.111973i	$0.964283\pi$
$338$	0	0
$339$	−25.4165	−1.38044
$340$	0	0
$341$	12.3105	0.666649
$342$	0	0
$343$	− 20.1334i	− 1.08710i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 11.0247i	− 0.591834i	−0.955214	−	0.295917i	$-0.904375\pi$
$347$	− 11.0247i	− 0.591834i	0.955214	−	0.295917i	$-0.0956252\pi$
$348$	0	0
$349$	22.8307	1.22210	0.611049	−	0.791592i	$-0.290748\pi$
$349$	22.8307	1.22210	0.611049	+	0.791592i	$0.290748\pi$
$350$	0	0
$351$	8.80840	0.470158
$352$	0	0
$353$	− 14.6732i	− 0.780978i	−0.920608	−	0.390489i	$-0.872306\pi$
$353$	− 14.6732i	− 0.780978i	0.920608	−	0.390489i	$-0.127694\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	12.0496i	0.637734i
$358$	0	0
$359$	−3.60307	−0.190163	−0.0950815	−	0.995469i	$-0.530311\pi$
$359$	−3.60307	−0.190163	−0.0950815	+	0.995469i	$0.530311\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	− 0.901674i	− 0.0473256i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 19.8007i	− 1.03359i	−0.856110	−	0.516793i	$-0.827126\pi$
$367$	− 19.8007i	− 1.03359i	0.856110	−	0.516793i	$-0.172874\pi$
$368$	0	0
$369$	2.26857	0.118097
$370$	0	0
$371$	15.8726	0.824063
$372$	0	0
$373$	17.6631i	0.914562i	0.889322	+	0.457281i	$0.151177\pi$
$373$	17.6631i	0.914562i	−0.889322	+	0.457281i	$0.848823\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.02404i	0.155746i
$378$	0	0
$379$	−6.05644	−0.311098	−0.155549	−	0.987828i	$-0.549715\pi$
$379$	−6.05644	−0.311098	−0.155549	+	0.987828i	$0.549715\pi$
$380$	0	0
$381$	6.16250	0.315715
$382$	0	0
$383$	20.8895i	1.06740i	0.845673	+	0.533702i	$0.179199\pi$
$383$	20.8895i	1.06740i	−0.845673	+	0.533702i	$0.820801\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1.10607i	0.0562245i
$388$	0	0
$389$	2.30541	0.116889	0.0584444	−	0.998291i	$-0.481386\pi$
$389$	2.30541	0.116889	0.0584444	+	0.998291i	$0.481386\pi$
$390$	0	0
$391$	15.8084	0.799465
$392$	0	0
$393$	− 6.63041i	− 0.334460i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 22.2354i	− 1.11596i	−0.829854	−	0.557980i	$-0.811576\pi$
$397$	− 22.2354i	− 1.11596i	0.829854	−	0.557980i	$-0.188424\pi$
$398$	0	0
$399$	−3.41147	−0.170787
$400$	0	0
$401$	−14.1584	−0.707036	−0.353518	−	0.935428i	$-0.615015\pi$
$401$	−14.1584	−0.707036	−0.353518	+	0.935428i	$0.615015\pi$
$402$	0	0
$403$	6.00505i	0.299133i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 36.6391i	− 1.81613i
$408$	0	0
$409$	4.22432	0.208879	0.104440	−	0.994531i	$-0.466695\pi$
$409$	4.22432	0.208879	0.104440	+	0.994531i	$0.466695\pi$
$410$	0	0
$411$	−16.8871	−0.832980
$412$	0	0
$413$	− 17.5621i	− 0.864175i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	26.4388i	1.29471i
$418$	0	0
$419$	11.9632	0.584439	0.292219	−	0.956351i	$-0.405606\pi$
$419$	11.9632	0.584439	0.292219	+	0.956351i	$0.405606\pi$
$420$	0	0
$421$	−10.2395	−0.499041	−0.249521	−	0.968369i	$-0.580273\pi$
$421$	−10.2395	−0.499041	−0.249521	+	0.968369i	$0.580273\pi$
$422$	0	0
$423$	1.02734i	0.0499510i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	6.22493i	0.301245i
$428$	0	0
$429$	−7.78106	−0.375673
$430$	0	0
$431$	22.2517	1.07182	0.535912	−	0.844274i	$-0.319968\pi$
$431$	22.2517	1.07182	0.535912	+	0.844274i	$0.319968\pi$
$432$	0	0
$433$	− 24.9581i	− 1.19941i	−0.800221	−	0.599705i	$-0.795285\pi$
$433$	− 24.9581i	− 1.19941i	0.800221	−	0.599705i	$-0.204715\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4.47565i	0.214099i
$438$	0	0
$439$	−16.0942	−0.768135	−0.384067	−	0.923305i	$-0.625477\pi$
$439$	−16.0942	−0.768135	−0.384067	+	0.923305i	$0.625477\pi$
$440$	0	0
$441$	1.33275	0.0634642
$442$	0	0
$443$	− 17.7270i	− 0.842235i	−0.907006	−	0.421117i	$-0.861638\pi$
$443$	− 17.7270i	− 0.842235i	0.907006	−	0.421117i	$-0.138362\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 18.9709i	− 0.897293i
$448$	0	0
$449$	37.6049	1.77469	0.887343	−	0.461109i	$-0.152548\pi$
$449$	37.6049	1.77469	0.887343	+	0.461109i	$0.152548\pi$
$450$	0	0
$451$	−11.2148	−0.528085
$452$	0	0
$453$	31.8699i	1.49738i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 20.1949i	− 0.944677i	−0.881417	−	0.472339i	$-0.843410\pi$
$457$	− 20.1949i	− 0.944677i	0.881417	−	0.472339i	$-0.156590\pi$
$458$	0	0
$459$	−19.7665	−0.922622
$460$	0	0
$461$	6.81521	0.317416	0.158708	−	0.987326i	$-0.449267\pi$
$461$	6.81521	0.317416	0.158708	+	0.987326i	$0.449267\pi$
$462$	0	0
$463$	33.7083i	1.56656i	0.621670	+	0.783279i	$0.286454\pi$
$463$	33.7083i	1.56656i	−0.621670	+	0.783279i	$0.713546\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	27.5648i	1.27555i	0.770224	+	0.637774i	$0.220144\pi$
$467$	27.5648i	1.27555i	−0.770224	+	0.637774i	$0.779856\pi$
$468$	0	0
$469$	7.18479	0.331763
$470$	0	0
$471$	8.04963	0.370907
$472$	0	0
$473$	− 5.46791i	− 0.251415i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	4.65270i	0.213033i
$478$	0	0
$479$	−26.1438	−1.19454	−0.597271	−	0.802039i	$-0.703748\pi$
$479$	−26.1438	−1.19454	−0.597271	+	0.802039i	$0.703748\pi$
$480$	0	0
$481$	17.8726	0.814919
$482$	0	0
$483$	− 15.2686i	− 0.694744i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 5.63310i	− 0.255260i	−0.991822	−	0.127630i	$-0.959263\pi$
$487$	− 5.63310i	− 0.255260i	0.991822	−	0.127630i	$-0.0407370\pi$
$488$	0	0
$489$	−10.1702	−0.459914
$490$	0	0
$491$	14.6682	0.661966	0.330983	−	0.943637i	$-0.392620\pi$
$491$	14.6682	0.661966	0.330983	+	0.943637i	$0.392620\pi$
$492$	0	0
$493$	− 6.78611i	− 0.305631i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	9.77332i	0.438393i
$498$	0	0
$499$	−16.7793	−0.751145	−0.375572	−	0.926793i	$-0.622554\pi$
$499$	−16.7793	−0.751145	−0.375572	+	0.926793i	$0.622554\pi$
$500$	0	0
$501$	25.0993	1.12135
$502$	0	0
$503$	− 3.12330i	− 0.139261i	−0.997573	−	0.0696306i	$-0.977818\pi$
$503$	− 3.12330i	− 0.139261i	0.997573	−	0.0696306i	$-0.0221821\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	16.1215i	0.715983i
$508$	0	0
$509$	40.1566	1.77991	0.889956	−	0.456047i	$-0.150735\pi$
$509$	40.1566	1.77991	0.889956	+	0.456047i	$0.150735\pi$
$510$	0	0
$511$	14.2763	0.631547
$512$	0	0
$513$	− 5.59627i	− 0.247081i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 5.07873i	− 0.223362i
$518$	0	0
$519$	−6.15064	−0.269983
$520$	0	0
$521$	−8.75465	−0.383548	−0.191774	−	0.981439i	$-0.561424\pi$
$521$	−8.75465	−0.383548	−0.191774	+	0.981439i	$0.561424\pi$
$522$	0	0
$523$	26.9522i	1.17854i	0.807937	+	0.589270i	$0.200584\pi$
$523$	26.9522i	1.17854i	−0.807937	+	0.589270i	$0.799416\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 13.4757i	− 0.587009i
$528$	0	0
$529$	2.96854	0.129067
$530$	0	0
$531$	5.14796	0.223402
$532$	0	0
$533$	− 5.47060i	− 0.236958i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 34.0847i	− 1.47086i
$538$	0	0
$539$	−6.58853	−0.283788
$540$	0	0
$541$	8.48751	0.364907	0.182453	−	0.983215i	$-0.441596\pi$
$541$	8.48751	0.364907	0.182453	+	0.983215i	$0.441596\pi$
$542$	0	0
$543$	− 11.1780i	− 0.479693i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 3.62267i	− 0.154894i	−0.996996	−	0.0774472i	$-0.975323\pi$
$547$	− 3.62267i	− 0.154894i	0.996996	−	0.0774472i	$-0.0246769\pi$
$548$	0	0
$549$	−1.82470	−0.0778764
$550$	0	0
$551$	1.92127	0.0818490
$552$	0	0
$553$	− 19.1411i	− 0.813964i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 2.01455i	− 0.0853591i	−0.999089	−	0.0426796i	$-0.986411\pi$
$557$	− 2.01455i	− 0.0853591i	0.999089	−	0.0426796i	$-0.0135895\pi$
$558$	0	0
$559$	2.66725	0.112813
$560$	0	0
$561$	17.4611	0.737208
$562$	0	0
$563$	16.5185i	0.696171i	0.937463	+	0.348085i	$0.113168\pi$
$563$	16.5185i	0.696171i	−0.937463	+	0.348085i	$0.886832\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	14.7314i	0.618662i
$568$	0	0
$569$	8.18304	0.343051	0.171525	−	0.985180i	$-0.445130\pi$
$569$	8.18304	0.343051	0.171525	+	0.985180i	$0.445130\pi$
$570$	0	0
$571$	−36.9796	−1.54755	−0.773774	−	0.633462i	$-0.781633\pi$
$571$	−36.9796	−1.54755	−0.773774	+	0.633462i	$0.781633\pi$
$572$	0	0
$573$	− 14.6340i	− 0.611346i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 18.7347i	− 0.779937i	−0.920828	−	0.389968i	$-0.872486\pi$
$577$	− 18.7347i	− 0.779937i	0.920828	−	0.389968i	$-0.127514\pi$
$578$	0	0
$579$	−1.89124	−0.0785975
$580$	0	0
$581$	32.5621	1.35090
$582$	0	0
$583$	− 23.0009i	− 0.952601i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	29.2968i	1.20921i	0.796525	+	0.604605i	$0.206669\pi$
$587$	29.2968i	1.20921i	−0.796525	+	0.604605i	$0.793331\pi$
$588$	0	0
$589$	3.81521	0.157203
$590$	0	0
$591$	17.7219	0.728983
$592$	0	0
$593$	− 22.4260i	− 0.920926i	−0.887679	−	0.460463i	$-0.847683\pi$
$593$	− 22.4260i	− 0.920926i	0.887679	−	0.460463i	$-0.152317\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 18.4010i	− 0.753105i
$598$	0	0
$599$	33.1002	1.35244	0.676219	−	0.736701i	$-0.263617\pi$
$599$	33.1002	1.35244	0.676219	+	0.736701i	$0.263617\pi$
$600$	0	0
$601$	43.5613	1.77690	0.888451	−	0.458971i	$-0.151782\pi$
$601$	43.5613	1.77690	0.888451	+	0.458971i	$0.151782\pi$
$602$	0	0
$603$	2.10607i	0.0857657i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 2.03178i	− 0.0824676i	−0.999150	−	0.0412338i	$-0.986871\pi$
$607$	− 2.03178i	− 0.0824676i	0.999150	−	0.0412338i	$-0.0131288\pi$
$608$	0	0
$609$	−6.55438	−0.265597
$610$	0	0
$611$	2.47741	0.100225
$612$	0	0
$613$	− 25.1266i	− 1.01485i	−0.861695	−	0.507427i	$-0.830597\pi$
$613$	− 25.1266i	− 1.01485i	0.861695	−	0.507427i	$-0.169403\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	24.4979i	0.986250i	0.869958	+	0.493125i	$0.164145\pi$
$617$	24.4979i	0.986250i	−0.869958	+	0.493125i	$0.835855\pi$
$618$	0	0
$619$	−9.05138	−0.363806	−0.181903	−	0.983316i	$-0.558226\pi$
$619$	−9.05138	−0.363806	−0.181903	+	0.983316i	$0.558226\pi$
$620$	0	0
$621$	25.0469	1.00510
$622$	0	0
$623$	13.5963i	0.544723i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	4.94356i	0.197427i
$628$	0	0
$629$	−40.1070	−1.59917
$630$	0	0
$631$	23.7324	0.944770	0.472385	−	0.881392i	$-0.343393\pi$
$631$	23.7324	0.944770	0.472385	+	0.881392i	$0.343393\pi$
$632$	0	0
$633$	− 17.4192i	− 0.692352i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 3.21389i	− 0.127339i
$638$	0	0
$639$	−2.86484	−0.113331
$640$	0	0
$641$	10.6067	0.418939	0.209470	−	0.977815i	$-0.432826\pi$
$641$	10.6067	0.418939	0.209470	+	0.977815i	$0.432826\pi$
$642$	0	0
$643$	− 31.1634i	− 1.22897i	−0.788930	−	0.614483i	$-0.789365\pi$
$643$	− 31.1634i	− 1.22897i	0.788930	−	0.614483i	$-0.210635\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	12.9154i	0.507757i	0.967236	+	0.253878i	$0.0817063\pi$
$647$	12.9154i	0.507757i	−0.967236	+	0.253878i	$0.918294\pi$
$648$	0	0
$649$	−25.4492	−0.998970
$650$	0	0
$651$	−13.0155	−0.510117
$652$	0	0
$653$	10.8060i	0.422873i	0.977392	+	0.211436i	$0.0678141\pi$
$653$	10.8060i	0.422873i	−0.977392	+	0.211436i	$0.932186\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	4.18479i	0.163264i
$658$	0	0
$659$	28.3583	1.10468	0.552342	−	0.833618i	$-0.313734\pi$
$659$	28.3583	1.10468	0.552342	+	0.833618i	$0.313734\pi$
$660$	0	0
$661$	−28.9691	−1.12677	−0.563385	−	0.826195i	$-0.690501\pi$
$661$	−28.9691	−1.12677	−0.563385	+	0.826195i	$0.690501\pi$
$662$	0	0
$663$	8.51754i	0.330794i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	8.59896i	0.332953i
$668$	0	0
$669$	21.0651	0.814424
$670$	0	0
$671$	9.02053	0.348234
$672$	0	0
$673$	− 15.6973i	− 0.605086i	−0.953136	−	0.302543i	$-0.902164\pi$
$673$	− 15.6973i	− 0.605086i	0.953136	−	0.302543i	$-0.0978355\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 3.25578i	− 0.125130i	−0.998041	−	0.0625648i	$-0.980072\pi$
$677$	− 3.25578i	− 0.125130i	0.998041	−	0.0625648i	$-0.0199280\pi$
$678$	0	0
$679$	34.9641	1.34180
$680$	0	0
$681$	−10.8188	−0.414578
$682$	0	0
$683$	33.1834i	1.26973i	0.772625	+	0.634863i	$0.218943\pi$
$683$	33.1834i	1.26973i	−0.772625	+	0.634863i	$0.781057\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	10.8111i	0.412469i
$688$	0	0
$689$	11.2199	0.427443
$690$	0	0
$691$	41.1242	1.56444	0.782220	−	0.623002i	$-0.214087\pi$
$691$	41.1242	1.56444	0.782220	+	0.623002i	$0.214087\pi$
$692$	0	0
$693$	4.68954i	0.178141i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	12.2763i	0.464998i
$698$	0	0
$699$	18.3327	0.693408
$700$	0	0
$701$	44.9198	1.69660	0.848300	−	0.529517i	$-0.177627\pi$
$701$	44.9198	1.69660	0.848300	+	0.529517i	$0.177627\pi$
$702$	0	0
$703$	− 11.3550i	− 0.428263i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 9.69728i	− 0.364704i
$708$	0	0
$709$	34.9240	1.31160	0.655798	−	0.754936i	$-0.272332\pi$
$709$	34.9240	1.31160	0.655798	+	0.754936i	$0.272332\pi$
$710$	0	0
$711$	5.61081	0.210422
$712$	0	0
$713$	17.0755i	0.639484i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 4.21482i	− 0.157405i
$718$	0	0
$719$	−29.0077	−1.08181	−0.540903	−	0.841085i	$-0.681917\pi$
$719$	−29.0077	−1.08181	−0.540903	+	0.841085i	$0.681917\pi$
$720$	0	0
$721$	−9.51073	−0.354198
$722$	0	0
$723$	− 12.1925i	− 0.453445i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	8.27126i	0.306764i	0.988167	+	0.153382i	$0.0490165\pi$
$727$	8.27126i	0.306764i	−0.988167	+	0.153382i	$0.950983\pi$
$728$	0	0
$729$	−30.0401	−1.11260
$730$	0	0
$731$	−5.98545	−0.221380
$732$	0	0
$733$	12.6696i	0.467963i	0.972241	+	0.233981i	$0.0751755\pi$
$733$	12.6696i	0.467963i	−0.972241	+	0.233981i	$0.924824\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 10.4115i	− 0.383512i
$738$	0	0
$739$	−17.9733	−0.661157	−0.330579	−	0.943778i	$-0.607244\pi$
$739$	−17.9733	−0.661157	−0.330579	+	0.943778i	$0.607244\pi$
$740$	0	0
$741$	−2.41147	−0.0885877
$742$	0	0
$743$	5.47977i	0.201033i	0.994935	+	0.100517i	$0.0320496\pi$
$743$	5.47977i	0.201033i	−0.994935	+	0.100517i	$0.967950\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	9.54488i	0.349229i
$748$	0	0
$749$	0.236177	0.00862972
$750$	0	0
$751$	32.5803	1.18887	0.594436	−	0.804143i	$-0.297375\pi$
$751$	32.5803	1.18887	0.594436	+	0.804143i	$0.297375\pi$
$752$	0	0
$753$	− 20.9632i	− 0.763940i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	4.97359i	0.180768i	0.995907	+	0.0903841i	$0.0288095\pi$
$757$	4.97359i	0.180768i	−0.995907	+	0.0903841i	$0.971191\pi$
$758$	0	0
$759$	−22.1257	−0.803111
$760$	0	0
$761$	24.5262	0.889075	0.444537	−	0.895760i	$-0.353368\pi$
$761$	24.5262	0.889075	0.444537	+	0.895760i	$0.353368\pi$
$762$	0	0
$763$	− 32.0898i	− 1.16173i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 12.4142i	− 0.448249i
$768$	0	0
$769$	24.4688	0.882369	0.441185	−	0.897416i	$-0.354558\pi$
$769$	24.4688	0.882369	0.441185	+	0.897416i	$0.354558\pi$
$770$	0	0
$771$	−16.1661	−0.582209
$772$	0	0
$773$	− 4.43645i	− 0.159568i	−0.996812	−	0.0797840i	$-0.974577\pi$
$773$	− 4.43645i	− 0.159568i	0.996812	−	0.0797840i	$-0.0254231\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	38.7374i	1.38970i
$778$	0	0
$779$	−3.47565	−0.124528
$780$	0	0
$781$	14.1625	0.506774
$782$	0	0
$783$	− 10.7520i	− 0.384244i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	1.73143i	0.0617188i	0.999524	+	0.0308594i	$0.00982441\pi$
$787$	1.73143i	0.0617188i	−0.999524	+	0.0308594i	$0.990176\pi$
$788$	0	0
$789$	−14.4115	−0.513062
$790$	0	0
$791$	36.9394	1.31342
$792$	0	0
$793$	4.40022i	0.156257i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 48.3979i	− 1.71434i	−0.515033	−	0.857170i	$-0.672221\pi$
$797$	− 48.3979i	− 1.71434i	0.515033	−	0.857170i	$-0.327779\pi$
$798$	0	0
$799$	−5.55943	−0.196678
$800$	0	0
$801$	−3.98545	−0.140819
$802$	0	0
$803$	− 20.6878i	− 0.730056i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 13.8307i	− 0.486863i
$808$	0	0
$809$	−45.7861	−1.60975	−0.804877	−	0.593442i	$-0.797769\pi$
$809$	−45.7861	−1.60975	−0.804877	+	0.593442i	$0.797769\pi$
$810$	0	0
$811$	−40.0779	−1.40733	−0.703663	−	0.710534i	$-0.748453\pi$
$811$	−40.0779	−1.40733	−0.703663	+	0.710534i	$0.748453\pi$
$812$	0	0
$813$	− 34.2199i	− 1.20014i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 1.69459i	− 0.0592863i
$818$	0	0
$819$	−2.28756	−0.0799339
$820$	0	0
$821$	−51.9208	−1.81205	−0.906024	−	0.423227i	$-0.860897\pi$
$821$	−51.9208	−1.81205	−0.906024	+	0.423227i	$0.860897\pi$
$822$	0	0
$823$	18.0779i	0.630156i	0.949066	+	0.315078i	$0.102031\pi$
$823$	18.0779i	0.630156i	−0.949066	+	0.315078i	$0.897969\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 46.5022i	− 1.61704i	−0.588469	−	0.808519i	$-0.700269\pi$
$827$	− 46.5022i	− 1.61704i	0.588469	−	0.808519i	$-0.299731\pi$
$828$	0	0
$829$	44.6946	1.55231	0.776154	−	0.630544i	$-0.217168\pi$
$829$	44.6946	1.55231	0.776154	+	0.630544i	$0.217168\pi$
$830$	0	0
$831$	−26.4989	−0.919236
$832$	0	0
$833$	7.21213i	0.249886i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 21.3509i	− 0.737996i
$838$	0	0
$839$	6.37134	0.219963	0.109982	−	0.993934i	$-0.464921\pi$
$839$	6.37134	0.219963	0.109982	+	0.993934i	$0.464921\pi$
$840$	0	0
$841$	−25.3087	−0.872714
$842$	0	0
$843$	− 16.0155i	− 0.551602i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1.31046i	0.0450279i
$848$	0	0
$849$	−35.7939	−1.22844
$850$	0	0
$851$	50.8212	1.74213
$852$	0	0
$853$	37.9813i	1.30046i	0.759739	+	0.650228i	$0.225327\pi$
$853$	37.9813i	1.30046i	−0.759739	+	0.650228i	$0.774673\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	47.2104i	1.61268i	0.591455	+	0.806338i	$0.298554\pi$
$857$	47.2104i	1.61268i	−0.591455	+	0.806338i	$0.701446\pi$
$858$	0	0
$859$	−15.6587	−0.534268	−0.267134	−	0.963659i	$-0.586077\pi$
$859$	−15.6587	−0.534268	−0.267134	+	0.963659i	$0.586077\pi$
$860$	0	0
$861$	11.8571	0.404089
$862$	0	0
$863$	− 54.9026i	− 1.86891i	−0.356086	−	0.934453i	$-0.615889\pi$
$863$	− 54.9026i	− 1.86891i	0.356086	−	0.934453i	$-0.384111\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	6.93170i	0.235413i
$868$	0	0
$869$	−27.7374	−0.940927
$870$	0	0
$871$	5.07873	0.172086
$872$	0	0
$873$	10.2490i	0.346875i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 7.74329i	− 0.261472i	−0.991417	−	0.130736i	$-0.958266\pi$
$877$	− 7.74329i	− 0.261472i	0.991417	−	0.130736i	$-0.0417341\pi$
$878$	0	0
$879$	−42.8435	−1.44507
$880$	0	0
$881$	−48.6005	−1.63739	−0.818696	−	0.574227i	$-0.805303\pi$
$881$	−48.6005	−1.63739	−0.818696	+	0.574227i	$0.805303\pi$
$882$	0	0
$883$	− 31.1625i	− 1.04870i	−0.851502	−	0.524351i	$-0.824308\pi$
$883$	− 31.1625i	− 1.04870i	0.851502	−	0.524351i	$-0.175692\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 4.52259i	− 0.151854i	−0.997113	−	0.0759269i	$-0.975808\pi$
$887$	− 4.52259i	− 0.151854i	0.997113	−	0.0759269i	$-0.0241916\pi$
$888$	0	0
$889$	−8.95636	−0.300387
$890$	0	0
$891$	21.3473	0.715161
$892$	0	0
$893$	− 1.57398i	− 0.0526712i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 10.7929i	− 0.360365i
$898$	0	0
$899$	7.33006	0.244471
$900$	0	0
$901$	−25.1780	−0.838800
$902$	0	0
$903$	5.78106i	0.192382i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 27.9777i	− 0.928985i	−0.885577	−	0.464492i	$-0.846237\pi$
$907$	− 27.9777i	− 0.928985i	0.885577	−	0.464492i	$-0.153763\pi$
$908$	0	0
$909$	2.84255	0.0942814
$910$	0	0
$911$	35.5895	1.17913	0.589566	−	0.807720i	$-0.299299\pi$
$911$	35.5895	1.17913	0.589566	+	0.807720i	$0.299299\pi$
$912$	0	0
$913$	− 47.1857i	− 1.56162i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	9.63640i	0.318222i
$918$	0	0
$919$	−20.8854	−0.688945	−0.344472	−	0.938796i	$-0.611942\pi$
$919$	−20.8854	−0.688945	−0.344472	+	0.938796i	$0.611942\pi$
$920$	0	0
$921$	35.0009	1.15332
$922$	0	0
$923$	6.90848i	0.227395i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 2.78787i	− 0.0915655i
$928$	0	0
$929$	−45.5885	−1.49571	−0.747856	−	0.663862i	$-0.768916\pi$
$929$	−45.5885	−1.49571	−0.747856	+	0.663862i	$0.768916\pi$
$930$	0	0
$931$	−2.04189	−0.0669202
$932$	0	0
$933$	− 18.1857i	− 0.595374i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 34.7638i	− 1.13568i	−0.823137	−	0.567842i	$-0.807778\pi$
$937$	− 34.7638i	− 1.13568i	0.823137	−	0.567842i	$-0.192222\pi$
$938$	0	0
$939$	39.1712	1.27830
$940$	0	0
$941$	5.80066	0.189096	0.0945480	−	0.995520i	$-0.469859\pi$
$941$	5.80066	0.189096	0.0945480	+	0.995520i	$0.469859\pi$
$942$	0	0
$943$	− 15.5558i	− 0.506567i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 18.4466i	− 0.599433i	−0.954028	−	0.299716i	$-0.903108\pi$
$947$	− 18.4466i	− 0.599433i	0.954028	−	0.299716i	$-0.0968920\pi$
$948$	0	0
$949$	10.0915	0.327585
$950$	0	0
$951$	−26.6614	−0.864554
$952$	0	0
$953$	− 15.6568i	− 0.507174i	−0.967313	−	0.253587i	$-0.918390\pi$
$953$	− 15.6568i	− 0.507174i	0.967313	−	0.253587i	$-0.0816105\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	9.49794i	0.307025i
$958$	0	0
$959$	24.5431	0.792539
$960$	0	0
$961$	−16.4442	−0.530458
$962$	0	0
$963$	0.0692302i	0.00223091i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	13.9923i	0.449961i	0.974363	+	0.224980i	$0.0722318\pi$
$967$	13.9923i	0.449961i	−0.974363	+	0.224980i	$0.927768\pi$
$968$	0	0
$969$	5.41147	0.173842
$970$	0	0
$971$	13.0487	0.418753	0.209376	−	0.977835i	$-0.432857\pi$
$971$	13.0487	0.418753	0.209376	+	0.977835i	$0.432857\pi$
$972$	0	0
$973$	− 38.4252i	− 1.23186i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	17.3327i	0.554524i	0.960794	+	0.277262i	$0.0894270\pi$
$977$	17.3327i	0.554524i	−0.960794	+	0.277262i	$0.910573\pi$
$978$	0	0
$979$	19.7023	0.629689
$980$	0	0
$981$	9.40642	0.300324
$982$	0	0
$983$	− 25.3114i	− 0.807308i	−0.914912	−	0.403654i	$-0.867740\pi$
$983$	− 25.3114i	− 0.807308i	0.914912	−	0.403654i	$-0.132260\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	5.36959i	0.170916i
$988$	0	0
$989$	7.58441	0.241170
$990$	0	0
$991$	37.1762	1.18094	0.590471	−	0.807059i	$-0.298942\pi$
$991$	37.1762	1.18094	0.590471	+	0.807059i	$0.298942\pi$
$992$	0	0
$993$	− 10.3732i	− 0.329184i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 26.3865i	− 0.835669i	−0.908523	−	0.417834i	$-0.862789\pi$
$997$	− 26.3865i	− 0.835669i	0.908523	−	0.417834i	$-0.137211\pi$
$998$	0	0
$999$	−63.5458	−2.01050

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3800.2.d.o.3649.5		6
5.2	odd	4		3800.2.a.t.1.3	yes	3
5.3	odd	4		3800.2.a.s.1.1	✓	3
5.4	even	2	inner	3800.2.d.o.3649.2		6
20.3	even	4		7600.2.a.br.1.3		3
20.7	even	4		7600.2.a.bs.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3800.2.a.s.1.1	✓	3	5.3	odd	4
3800.2.a.t.1.3	yes	3	5.2	odd	4
3800.2.d.o.3649.2		6	5.4	even	2	inner
3800.2.d.o.3649.5		6	1.1	even	1	trivial
7600.2.a.br.1.3		3	20.3	even	4
7600.2.a.bs.1.1		3	20.7	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3800.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.53209i q^{3} -2.22668i q^{7} +0.652704 q^{9} -3.22668 q^{11} -1.57398i q^{13} +3.53209i q^{17} -1.00000 q^{19} +3.41147 q^{21} -4.47565i q^{23} +5.59627i q^{27} -1.92127 q^{29} -3.81521 q^{31} -4.94356i q^{33} +11.3550i q^{37} +2.41147 q^{39} +3.47565 q^{41} +1.69459i q^{43} +1.57398i q^{47} +2.04189 q^{49} -5.41147 q^{51} +7.12836i q^{53} -1.53209i q^{57} +7.88713 q^{59} -2.79561 q^{61} -1.45336i q^{63} +3.22668i q^{67} +6.85710 q^{69} -4.38919 q^{71} +6.41147i q^{73} +7.18479i q^{77} +8.59627 q^{79} -6.61587 q^{81} +14.6236i q^{83} -2.94356i q^{87} -6.10607 q^{89} -3.50475 q^{91} -5.84524i q^{93} +15.7023i q^{97} -2.10607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{9} - 6 q^{11} - 6 q^{19} + 6 q^{29} - 30 q^{31} - 6 q^{39} - 18 q^{41} + 6 q^{49} - 12 q^{51} - 12 q^{59} - 18 q^{61} + 42 q^{69} - 18 q^{71} + 24 q^{79} - 18 q^{81} - 12 q^{89} - 54 q^{91} + 12 q^{99}+O(q^{100}) \) 6 * q + 6 * q^9 - 6 * q^11 - 6 * q^19 + 6 * q^29 - 30 * q^31 - 6 * q^39 - 18 * q^41 + 6 * q^49 - 12 * q^51 - 12 * q^59 - 18 * q^61 + 42 * q^69 - 18 * q^71 + 24 * q^79 - 18 * q^81 - 12 * q^89 - 54 * q^91 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more