LMFDB - Embedded newform 320.6.d.d.161.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [320,6,Mod(161,320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(320, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("320.161");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 320 = 2^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	320.d (of order $2$, degree $1$, 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:	$51.3228223402$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 343 x^{10} - 696 x^{9} + 44406 x^{8} + 179640 x^{7} - 2401691 x^{6} - 15554592 x^{5} + \cdots + 653157349 $ x^12 - 343x^10 - 696x^9 + 44406x^8 + 179640x^7 - 2401691x^6 - 15554592x^5 + 26901210x^4 + 434775816x^3 + 1271335685x^2 + 1475231592x + 653157349
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{30} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.1
Root			$10.0457 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	320.161
Dual form			320.6.d.d.161.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-30.0196i q^{3} -25.0000i q^{5} -45.4594 q^{7} -658.179 q^{9} +O(q^{10})$	$q-30.0196i q^{3} -25.0000i q^{5} -45.4594 q^{7} -658.179 q^{9} -142.717i q^{11} +897.074i q^{13} -750.491 q^{15} +11.4886 q^{17} +1818.86i q^{19} +1364.67i q^{21} +3875.05 q^{23} -625.000 q^{25} +12463.5i q^{27} -3463.55i q^{29} +1550.37 q^{31} -4284.30 q^{33} +1136.48i q^{35} +9610.99i q^{37} +26929.8 q^{39} -14807.5 q^{41} +6324.64i q^{43} +16454.5i q^{45} -29003.7 q^{47} -14740.4 q^{49} -344.883i q^{51} -32583.1i q^{53} -3567.91 q^{55} +54601.4 q^{57} +652.257i q^{59} +4715.33i q^{61} +29920.4 q^{63} +22426.9 q^{65} +46314.1i q^{67} -116327. i q^{69} +52668.5 q^{71} +69352.8 q^{73} +18762.3i q^{75} +6487.80i q^{77} -33288.8 q^{79} +214213. q^{81} -55352.9i q^{83} -287.214i q^{85} -103975. q^{87} +33235.3 q^{89} -40780.4i q^{91} -46541.5i q^{93} +45471.4 q^{95} -82468.3 q^{97} +93933.0i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 268 q^{7} - 428 q^{9}+O(q^{10}) $ 12 * q + 268 * q^7 - 428 * q^9	$ 12 q + 268 q^{7} - 428 q^{9} - 900 q^{15} + 2400 q^{17} + 8108 q^{23} - 7500 q^{25} + 7976 q^{31} - 20776 q^{33} + 40984 q^{39} - 56408 q^{41} + 21172 q^{47} - 5540 q^{49} + 18400 q^{55} + 39992 q^{57} + 179516 q^{63} + 44000 q^{65} + 367704 q^{71} + 58736 q^{73} + 26192 q^{79} + 411692 q^{81} + 183200 q^{87} + 87672 q^{89} + 121000 q^{95} - 172336 q^{97}+O(q^{100}) $ 12 * q + 268 * q^7 - 428 * q^9 - 900 * q^15 + 2400 * q^17 + 8108 * q^23 - 7500 * q^25 + 7976 * q^31 - 20776 * q^33 + 40984 * q^39 - 56408 * q^41 + 21172 * q^47 - 5540 * q^49 + 18400 * q^55 + 39992 * q^57 + 179516 * q^63 + 44000 * q^65 + 367704 * q^71 + 58736 * q^73 + 26192 * q^79 + 411692 * q^81 + 183200 * q^87 + 87672 * q^89 + 121000 * q^95 - 172336 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/320\mathbb{Z}\right)^\times$.

$n$	$191$	$257$	$261$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	0	0
$2$	0	0
$3$	− 30.0196i	− 1.92576i	−0.269928	−	0.962880i	$-0.587000\pi$
$3$	− 30.0196i	− 1.92576i	0.269928	−	0.962880i	$-0.413000\pi$
$4$	0	0
$5$	− 25.0000i	− 0.447214i
$5$	− 25.0000i	− 0.447214i
$6$	0	0
$7$	−45.4594	−0.350654	−0.175327	−	0.984510i	$-0.556098\pi$
$7$	−45.4594	−0.350654	−0.175327	+	0.984510i	$0.556098\pi$
$8$	0	0
$9$	−658.179	−2.70855
$10$	0	0
$11$	− 142.717i	− 0.355625i	−0.984064	−	0.177813i	$-0.943098\pi$
$11$	− 142.717i	− 0.355625i	0.984064	−	0.177813i	$-0.0569021\pi$
$12$	0	0
$13$	897.074i	1.47221i	0.676867	+	0.736105i	$0.263337\pi$
$13$	897.074i	1.47221i	−0.676867	+	0.736105i	$0.736663\pi$
$14$	0	0
$15$	−750.491	−0.861226
$16$	0	0
$17$	11.4886	0.00964148	0.00482074	−	0.999988i	$-0.498466\pi$
$17$	11.4886	0.00964148	0.00482074	+	0.999988i	$0.498466\pi$
$18$	0	0
$19$	1818.86i	1.15588i	0.816078	+	0.577942i	$0.196144\pi$
$19$	1818.86i	1.15588i	−0.816078	+	0.577942i	$0.803856\pi$
$20$	0	0
$21$	1364.67i	0.675275i
$22$	0	0
$23$	3875.05	1.52742	0.763708	−	0.645562i	$-0.223377\pi$
$23$	3875.05	1.52742	0.763708	+	0.645562i	$0.223377\pi$
$24$	0	0
$25$	−625.000	−0.200000
$26$	0	0
$27$	12463.5i	3.29027i
$28$	0	0
$29$	− 3463.55i	− 0.764763i	−0.924004	−	0.382382i	$-0.875104\pi$
$29$	− 3463.55i	− 0.764763i	0.924004	−	0.382382i	$-0.124896\pi$
$30$	0	0
$31$	1550.37	0.289755	0.144877	−	0.989450i	$-0.453721\pi$
$31$	1550.37	0.289755	0.144877	+	0.989450i	$0.453721\pi$
$32$	0	0
$33$	−4284.30	−0.684849
$34$	0	0
$35$	1136.48i	0.156817i
$36$	0	0
$37$	9610.99i	1.15415i	0.816690	+	0.577077i	$0.195807\pi$
$37$	9610.99i	1.15415i	−0.816690	+	0.577077i	$0.804193\pi$
$38$	0	0
$39$	26929.8	2.83513
$40$	0	0
$41$	−14807.5	−1.37569	−0.687847	−	0.725856i	$-0.741444\pi$
$41$	−14807.5	−1.37569	−0.687847	+	0.725856i	$0.741444\pi$
$42$	0	0
$43$	6324.64i	0.521632i	0.965388	+	0.260816i	$0.0839916\pi$
$43$	6324.64i	0.521632i	−0.965388	+	0.260816i	$0.916008\pi$
$44$	0	0
$45$	16454.5i	1.21130i
$46$	0	0
$47$	−29003.7	−1.91517	−0.957587	−	0.288145i	$-0.906962\pi$
$47$	−29003.7	−1.91517	−0.957587	+	0.288145i	$0.906962\pi$
$48$	0	0
$49$	−14740.4	−0.877042
$50$	0	0
$51$	− 344.883i	− 0.0185672i
$52$	0	0
$53$	− 32583.1i	− 1.59332i	−0.604429	−	0.796659i	$-0.706599\pi$
$53$	− 32583.1i	− 1.59332i	0.604429	−	0.796659i	$-0.293401\pi$
$54$	0	0
$55$	−3567.91	−0.159040
$56$	0	0
$57$	54601.4	2.22596
$58$	0	0
$59$	652.257i	0.0243943i	0.999926	+	0.0121972i	$0.00388257\pi$
$59$	652.257i	0.0243943i	−0.999926	+	0.0121972i	$0.996117\pi$
$60$	0	0
$61$	4715.33i	0.162251i	0.996704	+	0.0811255i	$0.0258515\pi$
$61$	4715.33i	0.162251i	−0.996704	+	0.0811255i	$0.974149\pi$
$62$	0	0
$63$	29920.4	0.949765
$64$	0	0
$65$	22426.9	0.658393
$66$	0	0
$67$	46314.1i	1.26045i	0.776411	+	0.630226i	$0.217038\pi$
$67$	46314.1i	1.26045i	−0.776411	+	0.630226i	$0.782962\pi$
$68$	0	0
$69$	− 116327.i	− 2.94144i
$70$	0	0
$71$	52668.5	1.23995	0.619976	−	0.784621i	$-0.287142\pi$
$71$	52668.5	1.23995	0.619976	+	0.784621i	$0.287142\pi$
$72$	0	0
$73$	69352.8	1.52320	0.761600	−	0.648048i	$-0.224414\pi$
$73$	69352.8	1.52320	0.761600	+	0.648048i	$0.224414\pi$
$74$	0	0
$75$	18762.3i	0.385152i
$76$	0	0
$77$	6487.80i	0.124701i
$78$	0	0
$79$	−33288.8	−0.600109	−0.300054	−	0.953922i	$-0.597005\pi$
$79$	−33288.8	−0.600109	−0.300054	+	0.953922i	$0.597005\pi$
$80$	0	0
$81$	214213.	3.62771
$82$	0	0
$83$	− 55352.9i	− 0.881953i	−0.897519	−	0.440976i	$-0.854632\pi$
$83$	− 55352.9i	− 0.881953i	0.897519	−	0.440976i	$-0.145368\pi$
$84$	0	0
$85$	− 287.214i	− 0.00431180i
$86$	0	0
$87$	−103975.	−1.47275
$88$	0	0
$89$	33235.3	0.444758	0.222379	−	0.974960i	$-0.428618\pi$
$89$	33235.3	0.444758	0.222379	+	0.974960i	$0.428618\pi$
$90$	0	0
$91$	− 40780.4i	− 0.516236i
$92$	0	0
$93$	− 46541.5i	− 0.557998i
$94$	0	0
$95$	45471.4	0.516927
$96$	0	0
$97$	−82468.3	−0.889934	−0.444967	−	0.895547i	$-0.646785\pi$
$97$	−82468.3	−0.889934	−0.444967	+	0.895547i	$0.646785\pi$
$98$	0	0
$99$	93933.0i	0.963231i
$100$	0	0
$101$	36413.2i	0.355185i	0.984104	+	0.177593i	$0.0568310\pi$
$101$	36413.2i	0.355185i	−0.984104	+	0.177593i	$0.943169\pi$
$102$	0	0
$103$	−83074.5	−0.771569	−0.385785	−	0.922589i	$-0.626069\pi$
$103$	−83074.5	−0.771569	−0.385785	+	0.922589i	$0.626069\pi$
$104$	0	0
$105$	34116.8	0.301992
$106$	0	0
$107$	30476.7i	0.257340i	0.991687	+	0.128670i	$0.0410709\pi$
$107$	30476.7i	0.257340i	−0.991687	+	0.128670i	$0.958929\pi$
$108$	0	0
$109$	200282.i	1.61464i	0.590117	+	0.807318i	$0.299082\pi$
$109$	200282.i	1.61464i	−0.590117	+	0.807318i	$0.700918\pi$
$110$	0	0
$111$	288519.	2.22263
$112$	0	0
$113$	−24491.9	−0.180437	−0.0902186	−	0.995922i	$-0.528757\pi$
$113$	−24491.9	−0.180437	−0.0902186	+	0.995922i	$0.528757\pi$
$114$	0	0
$115$	− 96876.1i	− 0.683081i
$116$	0	0
$117$	− 590435.i	− 3.98756i
$118$	0	0
$119$	−522.263	−0.00338082
$120$	0	0
$121$	140683.	0.873531
$122$	0	0
$123$	444515.i	2.64926i
$124$	0	0
$125$	15625.0i	0.0894427i
$126$	0	0
$127$	133675.	0.735431	0.367716	−	0.929938i	$-0.380140\pi$
$127$	133675.	0.735431	0.367716	+	0.929938i	$0.380140\pi$
$128$	0	0
$129$	189863.	1.00454
$130$	0	0
$131$	227954.i	1.16057i	0.814415	+	0.580283i	$0.197058\pi$
$131$	227954.i	1.16057i	−0.814415	+	0.580283i	$0.802942\pi$
$132$	0	0
$133$	− 82684.1i	− 0.405315i
$134$	0	0
$135$	311588.	1.47145
$136$	0	0
$137$	−354970.	−1.61581	−0.807905	−	0.589313i	$-0.799398\pi$
$137$	−354970.	−1.61581	−0.807905	+	0.589313i	$0.799398\pi$
$138$	0	0
$139$	414105.i	1.81792i	0.416887	+	0.908958i	$0.363121\pi$
$139$	414105.i	1.81792i	−0.416887	+	0.908958i	$0.636879\pi$
$140$	0	0
$141$	870679.i	3.68817i
$142$	0	0
$143$	128027.	0.523555
$144$	0	0
$145$	−86588.9	−0.342013
$146$	0	0
$147$	442503.i	1.68897i
$148$	0	0
$149$	− 233661.i	− 0.862224i	−0.902298	−	0.431112i	$-0.858121\pi$
$149$	− 233661.i	− 0.862224i	0.902298	−	0.431112i	$-0.141879\pi$
$150$	0	0
$151$	408202.	1.45691	0.728455	−	0.685094i	$-0.240239\pi$
$151$	408202.	1.45691	0.728455	+	0.685094i	$0.240239\pi$
$152$	0	0
$153$	−7561.53	−0.0261145
$154$	0	0
$155$	− 38759.2i	− 0.129582i
$156$	0	0
$157$	− 342377.i	− 1.10855i	−0.832334	−	0.554275i	$-0.812996\pi$
$157$	− 342377.i	− 1.10855i	0.832334	−	0.554275i	$-0.187004\pi$
$158$	0	0
$159$	−978132.	−3.06835
$160$	0	0
$161$	−176157.	−0.535594
$162$	0	0
$163$	− 374906.i	− 1.10523i	−0.833436	−	0.552616i	$-0.813630\pi$
$163$	− 374906.i	− 1.10523i	0.833436	−	0.552616i	$-0.186370\pi$
$164$	0	0
$165$	107107.i	0.306274i
$166$	0	0
$167$	−75465.0	−0.209389	−0.104695	−	0.994504i	$-0.533387\pi$
$167$	−75465.0	−0.209389	−0.104695	+	0.994504i	$0.533387\pi$
$168$	0	0
$169$	−433449.	−1.16740
$170$	0	0
$171$	− 1.19713e6i	− 3.13078i
$172$	0	0
$173$	372218.i	0.945545i	0.881185	+	0.472772i	$0.156747\pi$
$173$	372218.i	0.945545i	−0.881185	+	0.472772i	$0.843253\pi$
$174$	0	0
$175$	28412.1	0.0701307
$176$	0	0
$177$	19580.5	0.0469776
$178$	0	0
$179$	235406.i	0.549141i	0.961567	+	0.274571i	$0.0885357\pi$
$179$	235406.i	0.549141i	−0.961567	+	0.274571i	$0.911464\pi$
$180$	0	0
$181$	625543.i	1.41926i	0.704577	+	0.709628i	$0.251137\pi$
$181$	625543.i	1.41926i	−0.704577	+	0.709628i	$0.748863\pi$
$182$	0	0
$183$	141553.	0.312457
$184$	0	0
$185$	240275.	0.516154
$186$	0	0
$187$	− 1639.61i	− 0.00342875i
$188$	0	0
$189$	− 566584.i	− 1.15374i
$190$	0	0
$191$	51939.4	0.103018	0.0515090	−	0.998673i	$-0.483597\pi$
$191$	51939.4	0.103018	0.0515090	+	0.998673i	$0.483597\pi$
$192$	0	0
$193$	−355320.	−0.686635	−0.343318	−	0.939219i	$-0.611551\pi$
$193$	−355320.	−0.686635	−0.343318	+	0.939219i	$0.611551\pi$
$194$	0	0
$195$	− 673246.i	− 1.26791i
$196$	0	0
$197$	− 125802.i	− 0.230953i	−0.993310	−	0.115476i	$-0.963161\pi$
$197$	− 125802.i	− 0.230953i	0.993310	−	0.115476i	$-0.0368394\pi$
$198$	0	0
$199$	593616.	1.06261	0.531304	−	0.847181i	$-0.321702\pi$
$199$	593616.	1.06261	0.531304	+	0.847181i	$0.321702\pi$
$200$	0	0
$201$	1.39033e6	2.42733
$202$	0	0
$203$	157451.i	0.268167i
$204$	0	0
$205$	370187.i	0.615229i
$206$	0	0
$207$	−2.55047e6	−4.13709
$208$	0	0
$209$	259581.	0.411062
$210$	0	0
$211$	316256.i	0.489027i	0.969646	+	0.244513i	$0.0786282\pi$
$211$	316256.i	0.489027i	−0.969646	+	0.244513i	$0.921372\pi$
$212$	0	0
$213$	− 1.58109e6i	− 2.38785i
$214$	0	0
$215$	158116.	0.233281
$216$	0	0
$217$	−70478.7	−0.101603
$218$	0	0
$219$	− 2.08195e6i	− 2.93332i
$220$	0	0
$221$	10306.1i	0.0141943i
$222$	0	0
$223$	−154021.	−0.207405	−0.103702	−	0.994608i	$-0.533069\pi$
$223$	−154021.	−0.207405	−0.103702	+	0.994608i	$0.533069\pi$
$224$	0	0
$225$	411362.	0.541711
$226$	0	0
$227$	570137.i	0.734370i	0.930148	+	0.367185i	$0.119678\pi$
$227$	570137.i	0.734370i	−0.930148	+	0.367185i	$0.880322\pi$
$228$	0	0
$229$	− 80803.8i	− 0.101822i	−0.998703	−	0.0509111i	$-0.983787\pi$
$229$	− 80803.8i	− 0.101822i	0.998703	−	0.0509111i	$-0.0162125\pi$
$230$	0	0
$231$	194762.	0.240145
$232$	0	0
$233$	−932668.	−1.12548	−0.562739	−	0.826635i	$-0.690252\pi$
$233$	−932668.	−1.12548	−0.562739	+	0.826635i	$0.690252\pi$
$234$	0	0
$235$	725092.i	0.856492i
$236$	0	0
$237$	999317.i	1.15567i
$238$	0	0
$239$	483104.	0.547074	0.273537	−	0.961861i	$-0.411806\pi$
$239$	483104.	0.547074	0.273537	+	0.961861i	$0.411806\pi$
$240$	0	0
$241$	−194028.	−0.215190	−0.107595	−	0.994195i	$-0.534315\pi$
$241$	−194028.	−0.215190	−0.107595	+	0.994195i	$0.534315\pi$
$242$	0	0
$243$	− 3.40196e6i	− 3.69584i
$244$	0	0
$245$	368511.i	0.392225i
$246$	0	0
$247$	−1.63165e6	−1.70171
$248$	0	0
$249$	−1.66167e6	−1.69843
$250$	0	0
$251$	− 863008.i	− 0.864631i	−0.901723	−	0.432315i	$-0.857697\pi$
$251$	− 863008.i	− 0.864631i	0.901723	−	0.432315i	$-0.142303\pi$
$252$	0	0
$253$	− 553033.i	− 0.543188i
$254$	0	0
$255$	−8622.07	−0.00830350
$256$	0	0
$257$	−1.00468e6	−0.948841	−0.474420	−	0.880298i	$-0.657342\pi$
$257$	−1.00468e6	−0.948841	−0.474420	+	0.880298i	$0.657342\pi$
$258$	0	0
$259$	− 436910.i	− 0.404708i
$260$	0	0
$261$	2.27964e6i	2.07140i
$262$	0	0
$263$	1.50898e6	1.34522	0.672611	−	0.739996i	$-0.265173\pi$
$263$	1.50898e6	1.34522	0.672611	+	0.739996i	$0.265173\pi$
$264$	0	0
$265$	−814577.	−0.712553
$266$	0	0
$267$	− 997711.i	− 0.856498i
$268$	0	0
$269$	− 695284.i	− 0.585844i	−0.956136	−	0.292922i	$-0.905372\pi$
$269$	− 695284.i	− 0.585844i	0.956136	−	0.292922i	$-0.0946276\pi$
$270$	0	0
$271$	−361686.	−0.299163	−0.149582	−	0.988749i	$-0.547793\pi$
$271$	−361686.	−0.299163	−0.149582	+	0.988749i	$0.547793\pi$
$272$	0	0
$273$	−1.22421e6	−0.994147
$274$	0	0
$275$	89197.8i	0.0711251i
$276$	0	0
$277$	2.07506e6i	1.62491i	0.583020	+	0.812457i	$0.301871\pi$
$277$	2.07506e6i	1.62491i	−0.583020	+	0.812457i	$0.698129\pi$
$278$	0	0
$279$	−1.02042e6	−0.784816
$280$	0	0
$281$	−1.34117e6	−1.01325	−0.506626	−	0.862166i	$-0.669108\pi$
$281$	−1.34117e6	−1.01325	−0.506626	+	0.862166i	$0.669108\pi$
$282$	0	0
$283$	2.41343e6i	1.79130i	0.444762	+	0.895649i	$0.353288\pi$
$283$	2.41343e6i	1.79130i	−0.444762	+	0.895649i	$0.646712\pi$
$284$	0	0
$285$	− 1.36504e6i	− 0.995478i
$286$	0	0
$287$	673139.	0.482392
$288$	0	0
$289$	−1.41973e6	−0.999907
$290$	0	0
$291$	2.47567e6i	1.71380i
$292$	0	0
$293$	2.04581e6i	1.39219i	0.717952	+	0.696093i	$0.245080\pi$
$293$	2.04581e6i	1.39219i	−0.717952	+	0.696093i	$0.754920\pi$
$294$	0	0
$295$	16306.4	0.0109095
$296$	0	0
$297$	1.77875e6	1.17010
$298$	0	0
$299$	3.47620e6i	2.24868i
$300$	0	0
$301$	− 287514.i	− 0.182912i
$302$	0	0
$303$	1.09311e6	0.684002
$304$	0	0
$305$	117883.	0.0725609
$306$	0	0
$307$	454883.i	0.275457i	0.990470	+	0.137728i	$0.0439801\pi$
$307$	454883.i	0.275457i	−0.990470	+	0.137728i	$0.956020\pi$
$308$	0	0
$309$	2.49387e6i	1.48586i
$310$	0	0
$311$	−1.36777e6	−0.801884	−0.400942	−	0.916103i	$-0.631317\pi$
$311$	−1.36777e6	−0.801884	−0.400942	+	0.916103i	$0.631317\pi$
$312$	0	0
$313$	391986.	0.226157	0.113078	−	0.993586i	$-0.463929\pi$
$313$	391986.	0.226157	0.113078	+	0.993586i	$0.463929\pi$
$314$	0	0
$315$	− 748010.i	− 0.424748i
$316$	0	0
$317$	− 51502.5i	− 0.0287859i	−0.999896	−	0.0143930i	$-0.995418\pi$
$317$	− 51502.5i	− 0.0287859i	0.999896	−	0.0143930i	$-0.00458158\pi$
$318$	0	0
$319$	−494307.	−0.271969
$320$	0	0
$321$	914898.	0.495576
$322$	0	0
$323$	20896.1i	0.0111444i
$324$	0	0
$325$	− 560671.i	− 0.294442i
$326$	0	0
$327$	6.01238e6	3.10940
$328$	0	0
$329$	1.31849e6	0.671563
$330$	0	0
$331$	235665.i	0.118229i	0.998251	+	0.0591147i	$0.0188278\pi$
$331$	235665.i	0.118229i	−0.998251	+	0.0591147i	$0.981172\pi$
$332$	0	0
$333$	− 6.32575e6i	− 3.12609i
$334$	0	0
$335$	1.15785e6	0.563692
$336$	0	0
$337$	−1.60833e6	−0.771435	−0.385717	−	0.922617i	$-0.626046\pi$
$337$	−1.60833e6	−0.771435	−0.385717	+	0.922617i	$0.626046\pi$
$338$	0	0
$339$	735237.i	0.347479i
$340$	0	0
$341$	− 221263.i	− 0.103044i
$342$	0	0
$343$	1.43413e6	0.658192
$344$	0	0
$345$	−2.90819e6	−1.31545
$346$	0	0
$347$	266162.i	0.118665i	0.998238	+	0.0593324i	$0.0188972\pi$
$347$	266162.i	0.118665i	−0.998238	+	0.0593324i	$0.981103\pi$
$348$	0	0
$349$	4.43079e6i	1.94723i	0.228192	+	0.973616i	$0.426719\pi$
$349$	4.43079e6i	1.94723i	−0.228192	+	0.973616i	$0.573281\pi$
$350$	0	0
$351$	−1.11807e7	−4.84397
$352$	0	0
$353$	−2.80868e6	−1.19968	−0.599840	−	0.800120i	$-0.704769\pi$
$353$	−2.80868e6	−1.19968	−0.599840	+	0.800120i	$0.704769\pi$
$354$	0	0
$355$	− 1.31671e6i	− 0.554523i
$356$	0	0
$357$	15678.1i	0.00651065i
$358$	0	0
$359$	2.32643e6	0.952697	0.476348	−	0.879257i	$-0.341960\pi$
$359$	2.32643e6	0.952697	0.476348	+	0.879257i	$0.341960\pi$
$360$	0	0
$361$	−832139.	−0.336069
$362$	0	0
$363$	− 4.22325e6i	− 1.68221i
$364$	0	0
$365$	− 1.73382e6i	− 0.681196i
$366$	0	0
$367$	2.99596e6	1.16110	0.580552	−	0.814223i	$-0.302837\pi$
$367$	2.99596e6	1.16110	0.580552	+	0.814223i	$0.302837\pi$
$368$	0	0
$369$	9.74598e6	3.72614
$370$	0	0
$371$	1.48121e6i	0.558703i
$372$	0	0
$373$	1.15331e6i	0.429214i	0.976700	+	0.214607i	$0.0688471\pi$
$373$	1.15331e6i	0.429214i	−0.976700	+	0.214607i	$0.931153\pi$
$374$	0	0
$375$	469057.	0.172245
$376$	0	0
$377$	3.10707e6	1.12589
$378$	0	0
$379$	− 2.98305e6i	− 1.06675i	−0.845879	−	0.533375i	$-0.820923\pi$
$379$	− 2.98305e6i	− 1.06675i	0.845879	−	0.533375i	$-0.179077\pi$
$380$	0	0
$381$	− 4.01289e6i	− 1.41626i
$382$	0	0
$383$	−3.42628e6	−1.19351	−0.596755	−	0.802423i	$-0.703544\pi$
$383$	−3.42628e6	−1.19351	−0.596755	+	0.802423i	$0.703544\pi$
$384$	0	0
$385$	162195.	0.0557681
$386$	0	0
$387$	− 4.16274e6i	− 1.41287i
$388$	0	0
$389$	578513.i	0.193838i	0.995292	+	0.0969191i	$0.0308988\pi$
$389$	578513.i	0.193838i	−0.995292	+	0.0969191i	$0.969101\pi$
$390$	0	0
$391$	44518.7	0.0147265
$392$	0	0
$393$	6.84311e6	2.23497
$394$	0	0
$395$	832219.i	0.268377i
$396$	0	0
$397$	− 2.80594e6i	− 0.893514i	−0.894655	−	0.446757i	$-0.852579\pi$
$397$	− 2.80594e6i	− 0.893514i	0.894655	−	0.446757i	$-0.147421\pi$
$398$	0	0
$399$	−2.48215e6	−0.780540
$400$	0	0
$401$	−333020.	−0.103421	−0.0517106	−	0.998662i	$-0.516467\pi$
$401$	−333020.	−0.103421	−0.0517106	+	0.998662i	$0.516467\pi$
$402$	0	0
$403$	1.39079e6i	0.426580i
$404$	0	0
$405$	− 5.35532e6i	− 1.62236i
$406$	0	0
$407$	1.37165e6	0.410447
$408$	0	0
$409$	−2.51090e6	−0.742201	−0.371101	−	0.928593i	$-0.621020\pi$
$409$	−2.51090e6	−0.742201	−0.371101	+	0.928593i	$0.621020\pi$
$410$	0	0
$411$	1.06561e7i	3.11166i
$412$	0	0
$413$	− 29651.2i	− 0.00855396i
$414$	0	0
$415$	−1.38382e6	−0.394421
$416$	0	0
$417$	1.24313e7	3.50087
$418$	0	0
$419$	1.44535e6i	0.402196i	0.979571	+	0.201098i	$0.0644510\pi$
$419$	1.44535e6i	0.402196i	−0.979571	+	0.201098i	$0.935549\pi$
$420$	0	0
$421$	− 1.66158e6i	− 0.456896i	−0.973556	−	0.228448i	$-0.926635\pi$
$421$	− 1.66158e6i	− 0.456896i	0.973556	−	0.228448i	$-0.0733651\pi$
$422$	0	0
$423$	1.90896e7	5.18735
$424$	0	0
$425$	−7180.35	−0.00192830
$426$	0	0
$427$	− 214356.i	− 0.0568939i
$428$	0	0
$429$	− 3.84333e6i	− 1.00824i
$430$	0	0
$431$	−2.83017e6	−0.733870	−0.366935	−	0.930247i	$-0.619593\pi$
$431$	−2.83017e6	−0.733870	−0.366935	+	0.930247i	$0.619593\pi$
$432$	0	0
$433$	−3.79105e6	−0.971717	−0.485859	−	0.874037i	$-0.661493\pi$
$433$	−3.79105e6	−0.971717	−0.485859	+	0.874037i	$0.661493\pi$
$434$	0	0
$435$	2.59937e6i	0.658634i
$436$	0	0
$437$	7.04815e6i	1.76552i
$438$	0	0
$439$	651598.	0.161368	0.0806842	−	0.996740i	$-0.474289\pi$
$439$	651598.	0.161368	0.0806842	+	0.996740i	$0.474289\pi$
$440$	0	0
$441$	9.70185e6	2.37552
$442$	0	0
$443$	− 5.06806e6i	− 1.22697i	−0.789708	−	0.613483i	$-0.789768\pi$
$443$	− 5.06806e6i	− 1.22697i	0.789708	−	0.613483i	$-0.210232\pi$
$444$	0	0
$445$	− 830882.i	− 0.198902i
$446$	0	0
$447$	−7.01441e6	−1.66044
$448$	0	0
$449$	4.01985e6	0.941010	0.470505	−	0.882397i	$-0.344072\pi$
$449$	4.01985e6	0.941010	0.470505	+	0.882397i	$0.344072\pi$
$450$	0	0
$451$	2.11327e6i	0.489232i
$452$	0	0
$453$	− 1.22541e7i	− 2.80566i
$454$	0	0
$455$	−1.01951e6	−0.230868
$456$	0	0
$457$	−4.13558e6	−0.926288	−0.463144	−	0.886283i	$-0.653279\pi$
$457$	−4.13558e6	−0.926288	−0.463144	+	0.886283i	$0.653279\pi$
$458$	0	0
$459$	143188.i	0.0317230i
$460$	0	0
$461$	− 710960.i	− 0.155809i	−0.996961	−	0.0779046i	$-0.975177\pi$
$461$	− 710960.i	− 0.155809i	0.996961	−	0.0779046i	$-0.0248230\pi$
$462$	0	0
$463$	1.72786e6	0.374589	0.187295	−	0.982304i	$-0.440028\pi$
$463$	1.72786e6	0.374589	0.187295	+	0.982304i	$0.440028\pi$
$464$	0	0
$465$	−1.16354e6	−0.249544
$466$	0	0
$467$	2.42693e6i	0.514950i	0.966285	+	0.257475i	$0.0828905\pi$
$467$	2.42693e6i	0.514950i	−0.966285	+	0.257475i	$0.917110\pi$
$468$	0	0
$469$	− 2.10541e6i	− 0.441982i
$470$	0	0
$471$	−1.02780e7	−2.13480
$472$	0	0
$473$	902630.	0.185506
$474$	0	0
$475$	− 1.13679e6i	− 0.231177i
$476$	0	0
$477$	2.14455e7i	4.31559i
$478$	0	0
$479$	7.40523e6	1.47469	0.737343	−	0.675519i	$-0.236080\pi$
$479$	7.40523e6	1.47469	0.737343	+	0.675519i	$0.236080\pi$
$480$	0	0
$481$	−8.62177e6	−1.69916
$482$	0	0
$483$	5.28817e6i	1.03143i
$484$	0	0
$485$	2.06171e6i	0.397991i
$486$	0	0
$487$	920253.	0.175827	0.0879133	−	0.996128i	$-0.471980\pi$
$487$	920253.	0.175827	0.0879133	+	0.996128i	$0.471980\pi$
$488$	0	0
$489$	−1.12545e7	−2.12841
$490$	0	0
$491$	8.74081e6i	1.63624i	0.575045	+	0.818122i	$0.304984\pi$
$491$	8.74081e6i	1.63624i	−0.575045	+	0.818122i	$0.695016\pi$
$492$	0	0
$493$	− 39791.3i	− 0.00737345i
$494$	0	0
$495$	2.34833e6	0.430770
$496$	0	0
$497$	−2.39428e6	−0.434794
$498$	0	0
$499$	− 5.85693e6i	− 1.05298i	−0.850182	−	0.526488i	$-0.823508\pi$
$499$	− 5.85693e6i	− 1.05298i	0.850182	−	0.526488i	$-0.176492\pi$
$500$	0	0
$501$	2.26543e6i	0.403234i
$502$	0	0
$503$	−3.54886e6	−0.625416	−0.312708	−	0.949849i	$-0.601236\pi$
$503$	−3.54886e6	−0.625416	−0.312708	+	0.949849i	$0.601236\pi$
$504$	0	0
$505$	910329.	0.158844
$506$	0	0
$507$	1.30120e7i	2.24814i
$508$	0	0
$509$	2.90600e6i	0.497165i	0.968611	+	0.248582i	$0.0799647\pi$
$509$	2.90600e6i	0.497165i	−0.968611	+	0.248582i	$0.920035\pi$
$510$	0	0
$511$	−3.15273e6	−0.534115
$512$	0	0
$513$	−2.26694e7	−3.80317
$514$	0	0
$515$	2.07686e6i	0.345056i
$516$	0	0
$517$	4.13930e6i	0.681084i
$518$	0	0
$519$	1.11739e7	1.82089
$520$	0	0
$521$	5.04123e6	0.813658	0.406829	−	0.913504i	$-0.366634\pi$
$521$	5.04123e6	0.813658	0.406829	+	0.913504i	$0.366634\pi$
$522$	0	0
$523$	− 7.90605e6i	− 1.26388i	−0.775018	−	0.631939i	$-0.782259\pi$
$523$	− 7.90605e6i	− 1.26388i	0.775018	−	0.631939i	$-0.217741\pi$
$524$	0	0
$525$	− 852921.i	− 0.135055i
$526$	0	0
$527$	17811.5	0.00279366
$528$	0	0
$529$	8.57964e6	1.33300
$530$	0	0
$531$	− 429302.i	− 0.0660734i
$532$	0	0
$533$	− 1.32834e7i	− 2.02531i
$534$	0	0
$535$	761916.	0.115086
$536$	0	0
$537$	7.06679e6	1.05751
$538$	0	0
$539$	2.10371e6i	0.311898i
$540$	0	0
$541$	9.78603e6i	1.43752i	0.695259	+	0.718759i	$0.255289\pi$
$541$	9.78603e6i	1.43752i	−0.695259	+	0.718759i	$0.744711\pi$
$542$	0	0
$543$	1.87786e7	2.73315
$544$	0	0
$545$	5.00704e6	0.722087
$546$	0	0
$547$	− 2.64510e6i	− 0.377984i	−0.981979	−	0.188992i	$-0.939478\pi$
$547$	− 2.64510e6i	− 0.377984i	0.981979	−	0.188992i	$-0.0605220\pi$
$548$	0	0
$549$	− 3.10353e6i	− 0.439466i
$550$	0	0
$551$	6.29971e6	0.883978
$552$	0	0
$553$	1.51329e6	0.210430
$554$	0	0
$555$	− 7.21296e6i	− 0.993988i
$556$	0	0
$557$	− 4.84707e6i	− 0.661975i	−0.943635	−	0.330987i	$-0.892618\pi$
$557$	− 4.84707e6i	− 0.661975i	0.943635	−	0.330987i	$-0.107382\pi$
$558$	0	0
$559$	−5.67367e6	−0.767953
$560$	0	0
$561$	−49220.5	−0.00660296
$562$	0	0
$563$	7.18722e6i	0.955631i	0.878460	+	0.477815i	$0.158571\pi$
$563$	7.18722e6i	0.955631i	−0.878460	+	0.477815i	$0.841429\pi$
$564$	0	0
$565$	612297.i	0.0806940i
$566$	0	0
$567$	−9.73798e6	−1.27207
$568$	0	0
$569$	−1.30594e7	−1.69100	−0.845498	−	0.533979i	$-0.820696\pi$
$569$	−1.30594e7	−1.69100	−0.845498	+	0.533979i	$0.820696\pi$
$570$	0	0
$571$	− 2.93976e6i	− 0.377330i	−0.982041	−	0.188665i	$-0.939584\pi$
$571$	− 2.93976e6i	− 0.377330i	0.982041	−	0.188665i	$-0.0604161\pi$
$572$	0	0
$573$	− 1.55920e6i	− 0.198388i
$574$	0	0
$575$	−2.42190e6	−0.305483
$576$	0	0
$577$	4.78611e6	0.598471	0.299236	−	0.954179i	$-0.403268\pi$
$577$	4.78611e6	0.598471	0.299236	+	0.954179i	$0.403268\pi$
$578$	0	0
$579$	1.06666e7i	1.32230i
$580$	0	0
$581$	2.51631e6i	0.309260i
$582$	0	0
$583$	−4.65014e6	−0.566624
$584$	0	0
$585$	−1.47609e7	−1.78329
$586$	0	0
$587$	2.07501e6i	0.248557i	0.992247	+	0.124278i	$0.0396616\pi$
$587$	2.07501e6i	0.248557i	−0.992247	+	0.124278i	$0.960338\pi$
$588$	0	0
$589$	2.81989e6i	0.334923i
$590$	0	0
$591$	−3.77654e6	−0.444759
$592$	0	0
$593$	3.08207e6	0.359920	0.179960	−	0.983674i	$-0.442403\pi$
$593$	3.08207e6	0.359920	0.179960	+	0.983674i	$0.442403\pi$
$594$	0	0
$595$	13056.6i	0.00151195i
$596$	0	0
$597$	− 1.78201e7i	− 2.04633i
$598$	0	0
$599$	−1.34954e7	−1.53680	−0.768400	−	0.639970i	$-0.778947\pi$
$599$	−1.34954e7	−1.53680	−0.768400	+	0.639970i	$0.778947\pi$
$600$	0	0
$601$	−1.69923e6	−0.191897	−0.0959483	−	0.995386i	$-0.530588\pi$
$601$	−1.69923e6	−0.191897	−0.0959483	+	0.995386i	$0.530588\pi$
$602$	0	0
$603$	− 3.04830e7i	− 3.41401i
$604$	0	0
$605$	− 3.51707e6i	− 0.390655i
$606$	0	0
$607$	−1.47300e7	−1.62267	−0.811335	−	0.584581i	$-0.801259\pi$
$607$	−1.47300e7	−1.62267	−0.811335	+	0.584581i	$0.801259\pi$
$608$	0	0
$609$	4.72662e6	0.516426
$610$	0	0
$611$	− 2.60184e7i	− 2.81954i
$612$	0	0
$613$	5.34059e6i	0.574035i	0.957925	+	0.287017i	$0.0926638\pi$
$613$	5.34059e6i	0.574035i	−0.957925	+	0.287017i	$0.907336\pi$
$614$	0	0
$615$	1.11129e7	1.18478
$616$	0	0
$617$	−16763.1	−0.00177272	−0.000886361	−	1.00000i	$-0.500282\pi$
$617$	−16763.1	−0.00177272	−0.000886361		1.00000i	$0.500282\pi$
$618$	0	0
$619$	− 1.20604e7i	− 1.26513i	−0.774507	−	0.632566i	$-0.782002\pi$
$619$	− 1.20604e7i	− 1.26513i	0.774507	−	0.632566i	$-0.217998\pi$
$620$	0	0
$621$	4.82967e7i	5.02561i
$622$	0	0
$623$	−1.51085e6	−0.155956
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	− 7.79253e6i	− 0.791607i
$628$	0	0
$629$	110417.i	0.0111278i
$630$	0	0
$631$	−9.37030e6	−0.936871	−0.468436	−	0.883498i	$-0.655182\pi$
$631$	−9.37030e6	−0.936871	−0.468436	+	0.883498i	$0.655182\pi$
$632$	0	0
$633$	9.49389e6	0.941749
$634$	0	0
$635$	− 3.34188e6i	− 0.328895i
$636$	0	0
$637$	− 1.32233e7i	− 1.29119i
$638$	0	0
$639$	−3.46653e7	−3.35848
$640$	0	0
$641$	1.24983e7	1.20145	0.600723	−	0.799457i	$-0.294879\pi$
$641$	1.24983e7	1.20145	0.600723	+	0.799457i	$0.294879\pi$
$642$	0	0
$643$	− 8.90727e6i	− 0.849605i	−0.905286	−	0.424803i	$-0.860344\pi$
$643$	− 8.90727e6i	− 0.849605i	0.905286	−	0.424803i	$-0.139656\pi$
$644$	0	0
$645$	− 4.74658e6i	− 0.449243i
$646$	0	0
$647$	4.19736e6	0.394199	0.197100	−	0.980383i	$-0.436848\pi$
$647$	4.19736e6	0.394199	0.197100	+	0.980383i	$0.436848\pi$
$648$	0	0
$649$	93087.9	0.00867524
$650$	0	0
$651$	2.11575e6i	0.195664i
$652$	0	0
$653$	− 7.16059e6i	− 0.657152i	−0.944477	−	0.328576i	$-0.893431\pi$
$653$	− 7.16059e6i	− 0.657152i	0.944477	−	0.328576i	$-0.106569\pi$
$654$	0	0
$655$	5.69886e6	0.519021
$656$	0	0
$657$	−4.56465e7	−4.12567
$658$	0	0
$659$	− 6.13902e6i	− 0.550662i	−0.961349	−	0.275331i	$-0.911213\pi$
$659$	− 6.13902e6i	− 0.550662i	0.961349	−	0.275331i	$-0.0887875\pi$
$660$	0	0
$661$	− 3.75445e6i	− 0.334228i	−0.985938	−	0.167114i	$-0.946555\pi$
$661$	− 3.75445e6i	− 0.334228i	0.985938	−	0.167114i	$-0.0534448\pi$
$662$	0	0
$663$	309385.	0.0273348
$664$	0	0
$665$	−2.06710e6	−0.181262
$666$	0	0
$667$	− 1.34214e7i	− 1.16811i
$668$	0	0
$669$	4.62367e6i	0.399412i
$670$	0	0
$671$	672956.	0.0577006
$672$	0	0
$673$	−1.77234e7	−1.50838	−0.754188	−	0.656659i	$-0.771969\pi$
$673$	−1.77234e7	−1.50838	−0.754188	+	0.656659i	$0.771969\pi$
$674$	0	0
$675$	− 7.78970e6i	− 0.658054i
$676$	0	0
$677$	− 1.16984e7i	− 0.980969i	−0.871450	−	0.490485i	$-0.836820\pi$
$677$	− 1.16984e7i	− 0.980969i	0.871450	−	0.490485i	$-0.163180\pi$
$678$	0	0
$679$	3.74896e6	0.312059
$680$	0	0
$681$	1.71153e7	1.41422
$682$	0	0
$683$	1.76343e7i	1.44646i	0.690608	+	0.723230i	$0.257343\pi$
$683$	1.76343e7i	1.44646i	−0.690608	+	0.723230i	$0.742657\pi$
$684$	0	0
$685$	8.87425e6i	0.722612i
$686$	0	0
$687$	−2.42570e6	−0.196085
$688$	0	0
$689$	2.92294e7	2.34570
$690$	0	0
$691$	− 6.23688e6i	− 0.496903i	−0.968644	−	0.248452i	$-0.920078\pi$
$691$	− 6.23688e6i	− 0.496903i	0.968644	−	0.248452i	$-0.0799217\pi$
$692$	0	0
$693$	− 4.27014e6i	− 0.337760i
$694$	0	0
$695$	1.03526e7	0.812997
$696$	0	0
$697$	−170117.	−0.0132637
$698$	0	0
$699$	2.79983e7i	2.16740i
$700$	0	0
$701$	− 1.52056e7i	− 1.16872i	−0.811496	−	0.584358i	$-0.801346\pi$
$701$	− 1.52056e7i	− 1.16872i	0.811496	−	0.584358i	$-0.198654\pi$
$702$	0	0
$703$	−1.74810e7	−1.33407
$704$	0	0
$705$	2.17670e7	1.64940
$706$	0	0
$707$	− 1.65532e6i	− 0.124547i
$708$	0	0
$709$	2.26475e7i	1.69202i	0.533170	+	0.846008i	$0.321000\pi$
$709$	2.26475e7i	1.69202i	−0.533170	+	0.846008i	$0.679000\pi$
$710$	0	0
$711$	2.19100e7	1.62543
$712$	0	0
$713$	6.00774e6	0.442576
$714$	0	0
$715$	− 3.20068e6i	− 0.234141i
$716$	0	0
$717$	− 1.45026e7i	− 1.05353i
$718$	0	0
$719$	1.27400e7	0.919063	0.459532	−	0.888161i	$-0.348017\pi$
$719$	1.27400e7	0.919063	0.459532	+	0.888161i	$0.348017\pi$
$720$	0	0
$721$	3.77652e6	0.270553
$722$	0	0
$723$	5.82465e6i	0.414404i
$724$	0	0
$725$	2.16472e6i	0.152953i
$726$	0	0
$727$	−1.70816e7	−1.19865	−0.599325	−	0.800506i	$-0.704564\pi$
$727$	−1.70816e7	−1.19865	−0.599325	+	0.800506i	$0.704564\pi$
$728$	0	0
$729$	−5.00719e7	−3.48959
$730$	0	0
$731$	72661.0i	0.00502931i
$732$	0	0
$733$	− 9.13585e6i	− 0.628042i	−0.949416	−	0.314021i	$-0.898324\pi$
$733$	− 9.13585e6i	− 0.628042i	0.949416	−	0.314021i	$-0.101676\pi$
$734$	0	0
$735$	1.10626e7	0.755332
$736$	0	0
$737$	6.60980e6	0.448249
$738$	0	0
$739$	5.83288e6i	0.392891i	0.980515	+	0.196446i	$0.0629399\pi$
$739$	5.83288e6i	0.392891i	−0.980515	+	0.196446i	$0.937060\pi$
$740$	0	0
$741$	4.89815e7i	3.27708i
$742$	0	0
$743$	2.70851e6	0.179994	0.0899970	−	0.995942i	$-0.471314\pi$
$743$	2.70851e6	0.179994	0.0899970	+	0.995942i	$0.471314\pi$
$744$	0	0
$745$	−5.84152e6	−0.385598
$746$	0	0
$747$	3.64321e7i	2.38882i
$748$	0	0
$749$	− 1.38545e6i	− 0.0902373i
$750$	0	0
$751$	1.54490e7	0.999539	0.499769	−	0.866159i	$-0.333418\pi$
$751$	1.54490e7	0.999539	0.499769	+	0.866159i	$0.333418\pi$
$752$	0	0
$753$	−2.59072e7	−1.66507
$754$	0	0
$755$	− 1.02050e7i	− 0.651550i
$756$	0	0
$757$	2.16555e7i	1.37350i	0.726893	+	0.686750i	$0.240963\pi$
$757$	2.16555e7i	1.37350i	−0.726893	+	0.686750i	$0.759037\pi$
$758$	0	0
$759$	−1.66019e7	−1.04605
$760$	0	0
$761$	1.80099e7	1.12732	0.563662	−	0.826006i	$-0.309392\pi$
$761$	1.80099e7	1.12732	0.563662	+	0.826006i	$0.309392\pi$
$762$	0	0
$763$	− 9.10467e6i	− 0.566178i
$764$	0	0
$765$	189038.i	0.0116787i
$766$	0	0
$767$	−585123.	−0.0359136
$768$	0	0
$769$	−2.00138e7	−1.22043	−0.610215	−	0.792236i	$-0.708917\pi$
$769$	−2.00138e7	−1.22043	−0.610215	+	0.792236i	$0.708917\pi$
$770$	0	0
$771$	3.01600e7i	1.82724i
$772$	0	0
$773$	5.74507e6i	0.345817i	0.984938	+	0.172909i	$0.0553166\pi$
$773$	5.74507e6i	0.345817i	−0.984938	+	0.172909i	$0.944683\pi$
$774$	0	0
$775$	−968979.	−0.0579509
$776$	0	0
$777$	−1.31159e7	−0.779372
$778$	0	0
$779$	− 2.69327e7i	− 1.59014i
$780$	0	0
$781$	− 7.51666e6i	− 0.440958i
$782$	0	0
$783$	4.31681e7	2.51628
$784$	0	0
$785$	−8.55941e6	−0.495758
$786$	0	0
$787$	− 5.04447e6i	− 0.290321i	−0.989408	−	0.145160i	$-0.953630\pi$
$787$	− 5.04447e6i	− 0.290321i	0.989408	−	0.145160i	$-0.0463698\pi$
$788$	0	0
$789$	− 4.52990e7i	− 2.59057i
$790$	0	0
$791$	1.11339e6	0.0632709
$792$	0	0
$793$	−4.23000e6	−0.238868
$794$	0	0
$795$	2.44533e7i	1.37221i
$796$	0	0
$797$	963795.i	0.0537451i	0.999639	+	0.0268726i	$0.00855483\pi$
$797$	963795.i	0.0537451i	−0.999639	+	0.0268726i	$0.991445\pi$
$798$	0	0
$799$	−333211.	−0.0184651
$800$	0	0
$801$	−2.18748e7	−1.20465
$802$	0	0
$803$	− 9.89779e6i	− 0.541688i
$804$	0	0
$805$	4.40393e6i	0.239525i
$806$	0	0
$807$	−2.08722e7	−1.12820
$808$	0	0
$809$	−6.63331e6	−0.356335	−0.178168	−	0.984000i	$-0.557017\pi$
$809$	−6.63331e6	−0.356335	−0.178168	+	0.984000i	$0.557017\pi$
$810$	0	0
$811$	− 3.04422e7i	− 1.62526i	−0.582779	−	0.812631i	$-0.698035\pi$
$811$	− 3.04422e7i	− 1.62526i	0.582779	−	0.812631i	$-0.301965\pi$
$812$	0	0
$813$	1.08577e7i	0.576117i
$814$	0	0
$815$	−9.37264e6	−0.494274
$816$	0	0
$817$	−1.15036e7	−0.602947
$818$	0	0
$819$	2.68408e7i	1.39825i
$820$	0	0
$821$	9.68811e6i	0.501627i	0.968035	+	0.250814i	$0.0806981\pi$
$821$	9.68811e6i	0.501627i	−0.968035	+	0.250814i	$0.919302\pi$
$822$	0	0
$823$	−8.72636e6	−0.449090	−0.224545	−	0.974464i	$-0.572090\pi$
$823$	−8.72636e6	−0.449090	−0.224545	+	0.974464i	$0.572090\pi$
$824$	0	0
$825$	2.67769e6	0.136970
$826$	0	0
$827$	2.23162e7i	1.13464i	0.823498	+	0.567319i	$0.192019\pi$
$827$	2.23162e7i	1.13464i	−0.823498	+	0.567319i	$0.807981\pi$
$828$	0	0
$829$	− 3.36746e7i	− 1.70183i	−0.525302	−	0.850916i	$-0.676048\pi$
$829$	− 3.36746e7i	− 1.70183i	0.525302	−	0.850916i	$-0.323952\pi$
$830$	0	0
$831$	6.22925e7	3.12920
$832$	0	0
$833$	−169347.	−0.00845598
$834$	0	0
$835$	1.88662e6i	0.0936417i
$836$	0	0
$837$	1.93230e7i	0.953370i
$838$	0	0
$839$	8.64415e6	0.423952	0.211976	−	0.977275i	$-0.432010\pi$
$839$	8.64415e6	0.423952	0.211976	+	0.977275i	$0.432010\pi$
$840$	0	0
$841$	8.51494e6	0.415137
$842$	0	0
$843$	4.02614e7i	1.95128i
$844$	0	0
$845$	1.08362e7i	0.522079i
$846$	0	0
$847$	−6.39536e6	−0.306307
$848$	0	0
$849$	7.24502e7	3.44961
$850$	0	0
$851$	3.72430e7i	1.76287i
$852$	0	0
$853$	− 2.91285e6i	− 0.137071i	−0.997649	−	0.0685355i	$-0.978167\pi$
$853$	− 2.91285e6i	− 0.137071i	0.997649	−	0.0685355i	$-0.0218326\pi$
$854$	0	0
$855$	−2.99283e7	−1.40013
$856$	0	0
$857$	3.92591e7	1.82595	0.912973	−	0.408019i	$-0.133780\pi$
$857$	3.92591e7	1.82595	0.912973	+	0.408019i	$0.133780\pi$
$858$	0	0
$859$	− 1.53343e7i	− 0.709059i	−0.935045	−	0.354529i	$-0.884641\pi$
$859$	− 1.53343e7i	− 0.709059i	0.935045	−	0.354529i	$-0.115359\pi$
$860$	0	0
$861$	− 2.02074e7i	− 0.928972i
$862$	0	0
$863$	1.19629e7	0.546774	0.273387	−	0.961904i	$-0.411856\pi$
$863$	1.19629e7	0.546774	0.273387	+	0.961904i	$0.411856\pi$
$864$	0	0
$865$	9.30545e6	0.422861
$866$	0	0
$867$	4.26196e7i	1.92558i
$868$	0	0
$869$	4.75086e6i	0.213414i
$870$	0	0
$871$	−4.15472e7	−1.85565
$872$	0	0
$873$	5.42789e7	2.41044
$874$	0	0
$875$	− 710303.i	− 0.0313634i
$876$	0	0
$877$	− 1.64985e7i	− 0.724346i	−0.932111	−	0.362173i	$-0.882035\pi$
$877$	− 1.64985e7i	− 0.724346i	0.932111	−	0.362173i	$-0.117965\pi$
$878$	0	0
$879$	6.14146e7	2.68102
$880$	0	0
$881$	−1.28694e7	−0.558623	−0.279312	−	0.960201i	$-0.590106\pi$
$881$	−1.28694e7	−0.558623	−0.279312	+	0.960201i	$0.590106\pi$
$882$	0	0
$883$	− 2.00179e7i	− 0.864005i	−0.901872	−	0.432002i	$-0.857807\pi$
$883$	− 2.00179e7i	− 0.864005i	0.901872	−	0.432002i	$-0.142193\pi$
$884$	0	0
$885$	− 489513.i	− 0.0210090i
$886$	0	0
$887$	−3.41401e7	−1.45699	−0.728493	−	0.685053i	$-0.759779\pi$
$887$	−3.41401e7	−1.45699	−0.728493	+	0.685053i	$0.759779\pi$
$888$	0	0
$889$	−6.07680e6	−0.257882
$890$	0	0
$891$	− 3.05717e7i	− 1.29011i
$892$	0	0
$893$	− 5.27535e7i	− 2.21372i
$894$	0	0
$895$	5.88514e6	0.245583
$896$	0	0
$897$	1.04354e8	4.33041
$898$	0	0
$899$	− 5.36978e6i	− 0.221594i
$900$	0	0
$901$	− 374333.i	− 0.0153619i
$902$	0	0
$903$	−8.63107e6	−0.352245
$904$	0	0
$905$	1.56386e7	0.634710
$906$	0	0
$907$	− 2.89420e7i	− 1.16818i	−0.811688	−	0.584091i	$-0.801451\pi$
$907$	− 2.89420e7i	− 1.16818i	0.811688	−	0.584091i	$-0.198549\pi$
$908$	0	0
$909$	− 2.39664e7i	− 0.962039i
$910$	0	0
$911$	−3.26460e7	−1.30327	−0.651634	−	0.758534i	$-0.725916\pi$
$911$	−3.26460e7	−1.30327	−0.651634	+	0.758534i	$0.725916\pi$
$912$	0	0
$913$	−7.89978e6	−0.313645
$914$	0	0
$915$	− 3.53881e6i	− 0.139735i
$916$	0	0
$917$	− 1.03627e7i	− 0.406957i
$918$	0	0
$919$	2.68175e7	1.04744	0.523720	−	0.851890i	$-0.324544\pi$
$919$	2.68175e7	1.04744	0.523720	+	0.851890i	$0.324544\pi$
$920$	0	0
$921$	1.36554e7	0.530464
$922$	0	0
$923$	4.72475e7i	1.82547i
$924$	0	0
$925$	− 6.00687e6i	− 0.230831i
$926$	0	0
$927$	5.46779e7	2.08984
$928$	0	0
$929$	2.02792e6	0.0770923	0.0385462	−	0.999257i	$-0.487727\pi$
$929$	2.02792e6	0.0770923	0.0385462	+	0.999257i	$0.487727\pi$
$930$	0	0
$931$	− 2.68108e7i	− 1.01376i
$932$	0	0
$933$	4.10599e7i	1.54424i
$934$	0	0
$935$	−40990.2	−0.00153339
$936$	0	0
$937$	−3.18276e7	−1.18428	−0.592141	−	0.805834i	$-0.701717\pi$
$937$	−3.18276e7	−1.18428	−0.592141	+	0.805834i	$0.701717\pi$
$938$	0	0
$939$	− 1.17673e7i	− 0.435523i
$940$	0	0
$941$	− 4.36130e7i	− 1.60562i	−0.596237	−	0.802809i	$-0.703338\pi$
$941$	− 4.36130e7i	− 1.60562i	0.596237	−	0.802809i	$-0.296662\pi$
$942$	0	0
$943$	−5.73797e7	−2.10126
$944$	0	0
$945$	−1.41646e7	−0.515970
$946$	0	0
$947$	− 1.10000e7i	− 0.398582i	−0.979940	−	0.199291i	$-0.936136\pi$
$947$	− 1.10000e7i	− 0.398582i	0.979940	−	0.199291i	$-0.0638638\pi$
$948$	0	0
$949$	6.22146e7i	2.24247i
$950$	0	0
$951$	−1.54609e6	−0.0554348
$952$	0	0
$953$	2.76840e7	0.987406	0.493703	−	0.869630i	$-0.335643\pi$
$953$	2.76840e7	0.987406	0.493703	+	0.869630i	$0.335643\pi$
$954$	0	0
$955$	− 1.29848e6i	− 0.0460710i
$956$	0	0
$957$	1.48389e7i	0.523748i
$958$	0	0
$959$	1.61367e7	0.566589
$960$	0	0
$961$	−2.62255e7	−0.916042
$962$	0	0
$963$	− 2.00591e7i	− 0.697020i
$964$	0	0
$965$	8.88300e6i	0.307073i
$966$	0	0
$967$	3.67852e7	1.26505	0.632523	−	0.774541i	$-0.282019\pi$
$967$	3.67852e7	1.26505	0.632523	+	0.774541i	$0.282019\pi$
$968$	0	0
$969$	627292.	0.0214615
$970$	0	0
$971$	3.30736e6i	0.112573i	0.998415	+	0.0562865i	$0.0179260\pi$
$971$	3.30736e6i	0.112573i	−0.998415	+	0.0562865i	$0.982074\pi$
$972$	0	0
$973$	− 1.88250e7i	− 0.637459i
$974$	0	0
$975$	−1.68312e7	−0.567025
$976$	0	0
$977$	−4.78379e7	−1.60338	−0.801689	−	0.597741i	$-0.796065\pi$
$977$	−4.78379e7	−1.60338	−0.801689	+	0.597741i	$0.796065\pi$
$978$	0	0
$979$	− 4.74322e6i	− 0.158167i
$980$	0	0
$981$	− 1.31821e8i	− 4.37333i
$982$	0	0
$983$	1.03384e7	0.341248	0.170624	−	0.985336i	$-0.445422\pi$
$983$	1.03384e7	0.341248	0.170624	+	0.985336i	$0.445422\pi$
$984$	0	0
$985$	−3.14506e6	−0.103285
$986$	0	0
$987$	− 3.95805e7i	− 1.29327i
$988$	0	0
$989$	2.45083e7i	0.796749i
$990$	0	0
$991$	−3.54794e7	−1.14760	−0.573802	−	0.818994i	$-0.694532\pi$
$991$	−3.54794e7	−1.14760	−0.573802	+	0.818994i	$0.694532\pi$
$992$	0	0
$993$	7.07459e6	0.227682
$994$	0	0
$995$	− 1.48404e7i	− 0.475213i
$996$	0	0
$997$	3.07614e7i	0.980094i	0.871696	+	0.490047i	$0.163020\pi$
$997$	3.07614e7i	0.980094i	−0.871696	+	0.490047i	$0.836980\pi$
$998$	0	0
$999$	−1.19787e8	−3.79748

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.6.d.d.161.1	yes	12
4.3	odd	2		320.6.d.c.161.12	yes	12
8.3	odd	2		320.6.d.c.161.1	✓	12
8.5	even	2	inner	320.6.d.d.161.12	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.6.d.c.161.1	✓	12	8.3	odd	2
320.6.d.c.161.12	yes	12	4.3	odd	2
320.6.d.d.161.1	yes	12	1.1	even	1	trivial
320.6.d.d.161.12	yes	12	8.5	even	2	inner

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:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	320.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-30.0196i q^{3} -25.0000i q^{5} -45.4594 q^{7} -658.179 q^{9} +O(q^{10})\)	\(q-30.0196i q^{3} -25.0000i q^{5} -45.4594 q^{7} -658.179 q^{9} -142.717i q^{11} +897.074i q^{13} -750.491 q^{15} +11.4886 q^{17} +1818.86i q^{19} +1364.67i q^{21} +3875.05 q^{23} -625.000 q^{25} +12463.5i q^{27} -3463.55i q^{29} +1550.37 q^{31} -4284.30 q^{33} +1136.48i q^{35} +9610.99i q^{37} +26929.8 q^{39} -14807.5 q^{41} +6324.64i q^{43} +16454.5i q^{45} -29003.7 q^{47} -14740.4 q^{49} -344.883i q^{51} -32583.1i q^{53} -3567.91 q^{55} +54601.4 q^{57} +652.257i q^{59} +4715.33i q^{61} +29920.4 q^{63} +22426.9 q^{65} +46314.1i q^{67} -116327. i q^{69} +52668.5 q^{71} +69352.8 q^{73} +18762.3i q^{75} +6487.80i q^{77} -33288.8 q^{79} +214213. q^{81} -55352.9i q^{83} -287.214i q^{85} -103975. q^{87} +33235.3 q^{89} -40780.4i q^{91} -46541.5i q^{93} +45471.4 q^{95} -82468.3 q^{97} +93933.0i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 268 q^{7} - 428 q^{9}+O(q^{10}) \) 12 * q + 268 * q^7 - 428 * q^9	\( 12 q + 268 q^{7} - 428 q^{9} - 900 q^{15} + 2400 q^{17} + 8108 q^{23} - 7500 q^{25} + 7976 q^{31} - 20776 q^{33} + 40984 q^{39} - 56408 q^{41} + 21172 q^{47} - 5540 q^{49} + 18400 q^{55} + 39992 q^{57} + 179516 q^{63} + 44000 q^{65} + 367704 q^{71} + 58736 q^{73} + 26192 q^{79} + 411692 q^{81} + 183200 q^{87} + 87672 q^{89} + 121000 q^{95} - 172336 q^{97}+O(q^{100}) \) 12 * q + 268 * q^7 - 428 * q^9 - 900 * q^15 + 2400 * q^17 + 8108 * q^23 - 7500 * q^25 + 7976 * q^31 - 20776 * q^33 + 40984 * q^39 - 56408 * q^41 + 21172 * q^47 - 5540 * q^49 + 18400 * q^55 + 39992 * q^57 + 179516 * q^63 + 44000 * q^65 + 367704 * q^71 + 58736 * q^73 + 26192 * q^79 + 411692 * q^81 + 183200 * q^87 + 87672 * q^89 + 121000 * q^95 - 172336 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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