LMFDB - Embedded newform 3380.2.f.j.3041.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3380,2,Mod(3041,3380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3380, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3380.3041");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3380 = 2^{2} \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3380.f (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:	$26.9894358832$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} + 39x^{16} + 605x^{14} + 4764x^{12} + 20080x^{10} + 43783x^{8} + 43791x^{6} + 14071x^{4} + 378x^{2} + 1 $ x^18 + 39x^16 + 605x^14 + 4764x^12 + 20080x^10 + 43783x^8 + 43791x^6 + 14071x^4 + 378x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3041.1
Root			$-3.40622i$ of defining polynomial
Character	$\chi$	$=$	3380.3041
Dual form			3380.2.f.j.3041.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.40622 q^{3} -1.00000i q^{5} +2.33790i q^{7} +8.60234 q^{9} +O(q^{10})$	$q-3.40622 q^{3} -1.00000i q^{5} +2.33790i q^{7} +8.60234 q^{9} -6.16565i q^{11} +3.40622i q^{15} -3.07520 q^{17} +0.0902681i q^{19} -7.96342i q^{21} -4.81954 q^{23} -1.00000 q^{25} -19.0828 q^{27} +5.63201 q^{29} -8.34078i q^{31} +21.0016i q^{33} +2.33790 q^{35} +0.794425i q^{37} +3.95907i q^{41} +8.75108 q^{43} -8.60234i q^{45} -3.67871i q^{47} +1.53420 q^{49} +10.4748 q^{51} +8.08456 q^{53} -6.16565 q^{55} -0.307473i q^{57} +0.379577i q^{59} -4.96367 q^{61} +20.1114i q^{63} -0.376774i q^{67} +16.4164 q^{69} +9.04130i q^{71} +5.48133i q^{73} +3.40622 q^{75} +14.4147 q^{77} -4.79283 q^{79} +39.1932 q^{81} -11.6189i q^{83} +3.07520i q^{85} -19.1839 q^{87} -11.7118i q^{89} +28.4105i q^{93} +0.0902681 q^{95} -2.43658i q^{97} -53.0390i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 2 q^{3} + 24 q^{9}+O(q^{10}) $ 18 * q - 2 * q^3 + 24 * q^9	$ 18 q - 2 q^{3} + 24 q^{9} - 26 q^{17} - 24 q^{23} - 18 q^{25} - 8 q^{27} + 32 q^{29} + 2 q^{35} - 2 q^{43} - 40 q^{49} - 22 q^{51} + 60 q^{53} - 14 q^{55} + 42 q^{61} - 30 q^{69} + 2 q^{75} - 92 q^{77} + 62 q^{79} + 82 q^{81} + 56 q^{87} + 8 q^{95}+O(q^{100}) $ 18 * q - 2 * q^3 + 24 * q^9 - 26 * q^17 - 24 * q^23 - 18 * q^25 - 8 * q^27 + 32 * q^29 + 2 * q^35 - 2 * q^43 - 40 * q^49 - 22 * q^51 + 60 * q^53 - 14 * q^55 + 42 * q^61 - 30 * q^69 + 2 * q^75 - 92 * q^77 + 62 * q^79 + 82 * q^81 + 56 * q^87 + 8 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3380\mathbb{Z}\right)^\times$.

$n$	$677$	$1691$	$1861$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	−3.40622	−1.96658	−0.983291	−	0.182040i	$-0.941730\pi$
$3$	−3.40622	−1.96658	−0.983291	+	0.182040i	$0.941730\pi$
$4$	0	0
$5$	− 1.00000i	− 0.447214i
$5$	− 1.00000i	− 0.447214i
$6$	0	0
$7$	2.33790i	0.883645i	0.897103	+	0.441822i	$0.145668\pi$
$7$	2.33790i	0.883645i	−0.897103	+	0.441822i	$0.854332\pi$
$8$	0	0
$9$	8.60234	2.86745
$10$	0	0
$11$	− 6.16565i	− 1.85901i	−0.368806	−	0.929507i	$-0.620233\pi$
$11$	− 6.16565i	− 1.85901i	0.368806	−	0.929507i	$-0.379767\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	3.40622i	0.879482i
$16$	0	0
$17$	−3.07520	−0.745844	−0.372922	−	0.927863i	$-0.621644\pi$
$17$	−3.07520	−0.745844	−0.372922	+	0.927863i	$0.621644\pi$
$18$	0	0
$19$	0.0902681i	0.0207089i	0.999946	+	0.0103545i	$0.00329599\pi$
$19$	0.0902681i	0.0207089i	−0.999946	+	0.0103545i	$0.996704\pi$
$20$	0	0
$21$	− 7.96342i	− 1.73776i
$22$	0	0
$23$	−4.81954	−1.00494	−0.502472	−	0.864594i	$-0.667576\pi$
$23$	−4.81954	−1.00494	−0.502472	+	0.864594i	$0.667576\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−19.0828	−3.67249
$28$	0	0
$29$	5.63201	1.04584	0.522919	−	0.852382i	$-0.324843\pi$
$29$	5.63201	1.04584	0.522919	+	0.852382i	$0.324843\pi$
$30$	0	0
$31$	− 8.34078i	− 1.49805i	−0.662543	−	0.749024i	$-0.730523\pi$
$31$	− 8.34078i	− 1.49805i	0.662543	−	0.749024i	$-0.269477\pi$
$32$	0	0
$33$	21.0016i	3.65590i
$34$	0	0
$35$	2.33790	0.395178
$36$	0	0
$37$	0.794425i	0.130603i	0.997866	+	0.0653014i	$0.0208009\pi$
$37$	0.794425i	0.130603i	−0.997866	+	0.0653014i	$0.979199\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	3.95907i	0.618302i	0.951013	+	0.309151i	$0.100045\pi$
$41$	3.95907i	0.618302i	−0.951013	+	0.309151i	$0.899955\pi$
$42$	0	0
$43$	8.75108	1.33453	0.667264	−	0.744822i	$-0.267465\pi$
$43$	8.75108	1.33453	0.667264	+	0.744822i	$0.267465\pi$
$44$	0	0
$45$	− 8.60234i	− 1.28236i
$46$	0	0
$47$	− 3.67871i	− 0.536595i	−0.963336	−	0.268298i	$-0.913539\pi$
$47$	− 3.67871i	− 0.536595i	0.963336	−	0.268298i	$-0.0864611\pi$
$48$	0	0
$49$	1.53420	0.219172
$50$	0	0
$51$	10.4748	1.46676
$52$	0	0
$53$	8.08456	1.11050	0.555250	−	0.831683i	$-0.312623\pi$
$53$	8.08456	1.11050	0.555250	+	0.831683i	$0.312623\pi$
$54$	0	0
$55$	−6.16565	−0.831376
$56$	0	0
$57$	− 0.307473i	− 0.0407258i
$58$	0	0
$59$	0.379577i	0.0494167i	0.999695	+	0.0247084i	$0.00786572\pi$
$59$	0.379577i	0.0494167i	−0.999695	+	0.0247084i	$0.992134\pi$
$60$	0	0
$61$	−4.96367	−0.635533	−0.317766	−	0.948169i	$-0.602933\pi$
$61$	−4.96367	−0.635533	−0.317766	+	0.948169i	$0.602933\pi$
$62$	0	0
$63$	20.1114i	2.53380i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 0.376774i	− 0.0460302i	−0.999735	−	0.0230151i	$-0.992673\pi$
$67$	− 0.376774i	− 0.0460302i	0.999735	−	0.0230151i	$-0.00732659\pi$
$68$	0	0
$69$	16.4164	1.97630
$70$	0	0
$71$	9.04130i	1.07300i	0.843899	+	0.536502i	$0.180255\pi$
$71$	9.04130i	1.07300i	−0.843899	+	0.536502i	$0.819745\pi$
$72$	0	0
$73$	5.48133i	0.641541i	0.947157	+	0.320770i	$0.103942\pi$
$73$	5.48133i	0.641541i	−0.947157	+	0.320770i	$0.896058\pi$
$74$	0	0
$75$	3.40622	0.393316
$76$	0	0
$77$	14.4147	1.64271
$78$	0	0
$79$	−4.79283	−0.539235	−0.269617	−	0.962967i	$-0.586897\pi$
$79$	−4.79283	−0.539235	−0.269617	+	0.962967i	$0.586897\pi$
$80$	0	0
$81$	39.1932	4.35480
$82$	0	0
$83$	− 11.6189i	− 1.27534i	−0.770310	−	0.637670i	$-0.779898\pi$
$83$	− 11.6189i	− 1.27534i	0.770310	−	0.637670i	$-0.220102\pi$
$84$	0	0
$85$	3.07520i	0.333552i
$86$	0	0
$87$	−19.1839	−2.05673
$88$	0	0
$89$	− 11.7118i	− 1.24145i	−0.784029	−	0.620724i	$-0.786839\pi$
$89$	− 11.7118i	− 1.24145i	0.784029	−	0.620724i	$-0.213161\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	28.4105i	2.94603i
$94$	0	0
$95$	0.0902681	0.00926131
$96$	0	0
$97$	− 2.43658i	− 0.247397i	−0.992320	−	0.123698i	$-0.960524\pi$
$97$	− 2.43658i	− 0.247397i	0.992320	−	0.123698i	$-0.0394755\pi$
$98$	0	0
$99$	− 53.0390i	− 5.33062i
$100$	0	0
$101$	5.52834	0.550090	0.275045	−	0.961431i	$-0.411307\pi$
$101$	5.52834	0.550090	0.275045	+	0.961431i	$0.411307\pi$
$102$	0	0
$103$	−8.46999	−0.834573	−0.417287	−	0.908775i	$-0.637019\pi$
$103$	−8.46999	−0.834573	−0.417287	+	0.908775i	$0.637019\pi$
$104$	0	0
$105$	−7.96342	−0.777150
$106$	0	0
$107$	11.3891	1.10103	0.550514	−	0.834826i	$-0.314432\pi$
$107$	11.3891	1.10103	0.550514	+	0.834826i	$0.314432\pi$
$108$	0	0
$109$	9.04624i	0.866473i	0.901280	+	0.433236i	$0.142628\pi$
$109$	9.04624i	0.866473i	−0.901280	+	0.433236i	$0.857372\pi$
$110$	0	0
$111$	− 2.70599i	− 0.256841i
$112$	0	0
$113$	−2.99157	−0.281423	−0.140712	−	0.990051i	$-0.544939\pi$
$113$	−2.99157	−0.281423	−0.140712	+	0.990051i	$0.544939\pi$
$114$	0	0
$115$	4.81954i	0.449425i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 7.18951i	− 0.659061i
$120$	0	0
$121$	−27.0152	−2.45593
$122$	0	0
$123$	− 13.4855i	− 1.21594i
$124$	0	0
$125$	1.00000i	0.0894427i
$126$	0	0
$127$	−10.6776	−0.947482	−0.473741	−	0.880664i	$-0.657097\pi$
$127$	−10.6776	−0.947482	−0.473741	+	0.880664i	$0.657097\pi$
$128$	0	0
$129$	−29.8081	−2.62446
$130$	0	0
$131$	−20.6072	−1.80046	−0.900231	−	0.435412i	$-0.856603\pi$
$131$	−20.6072	−1.80046	−0.900231	+	0.435412i	$0.856603\pi$
$132$	0	0
$133$	−0.211038	−0.0182993
$134$	0	0
$135$	19.0828i	1.64239i
$136$	0	0
$137$	− 0.0850150i	− 0.00726332i	−0.999993	−	0.00363166i	$-0.998844\pi$
$137$	− 0.0850150i	− 0.00726332i	0.999993	−	0.00363166i	$-0.00115600\pi$
$138$	0	0
$139$	11.0101	0.933864	0.466932	−	0.884293i	$-0.345359\pi$
$139$	11.0101	0.933864	0.466932	+	0.884293i	$0.345359\pi$
$140$	0	0
$141$	12.5305i	1.05526i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	− 5.63201i	− 0.467713i
$146$	0	0
$147$	−5.22584	−0.431020
$148$	0	0
$149$	− 4.41512i	− 0.361701i	−0.983511	−	0.180850i	$-0.942115\pi$
$149$	− 4.41512i	− 0.361701i	0.983511	−	0.180850i	$-0.0578850\pi$
$150$	0	0
$151$	15.7555i	1.28216i	0.767473	+	0.641081i	$0.221514\pi$
$151$	15.7555i	1.28216i	−0.767473	+	0.641081i	$0.778486\pi$
$152$	0	0
$153$	−26.4539	−2.13867
$154$	0	0
$155$	−8.34078	−0.669947
$156$	0	0
$157$	3.40120	0.271445	0.135722	−	0.990747i	$-0.456664\pi$
$157$	3.40120	0.271445	0.135722	+	0.990747i	$0.456664\pi$
$158$	0	0
$159$	−27.5378	−2.18389
$160$	0	0
$161$	− 11.2676i	− 0.888013i
$162$	0	0
$163$	14.4332i	1.13050i	0.824920	+	0.565249i	$0.191220\pi$
$163$	14.4332i	1.13050i	−0.824920	+	0.565249i	$0.808780\pi$
$164$	0	0
$165$	21.0016	1.63497
$166$	0	0
$167$	− 18.9815i	− 1.46883i	−0.678698	−	0.734417i	$-0.737456\pi$
$167$	− 18.9815i	− 1.46883i	0.678698	−	0.734417i	$-0.262544\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	0.776517i	0.0593817i
$172$	0	0
$173$	−11.0970	−0.843691	−0.421845	−	0.906668i	$-0.638617\pi$
$173$	−11.0970	−0.843691	−0.421845	+	0.906668i	$0.638617\pi$
$174$	0	0
$175$	− 2.33790i	− 0.176729i
$176$	0	0
$177$	− 1.29292i	− 0.0971821i
$178$	0	0
$179$	−5.27165	−0.394021	−0.197011	−	0.980401i	$-0.563123\pi$
$179$	−5.27165	−0.394021	−0.197011	+	0.980401i	$0.563123\pi$
$180$	0	0
$181$	−16.1744	−1.20223	−0.601117	−	0.799161i	$-0.705277\pi$
$181$	−16.1744	−1.20223	−0.601117	+	0.799161i	$0.705277\pi$
$182$	0	0
$183$	16.9074	1.24983
$184$	0	0
$185$	0.794425	0.0584073
$186$	0	0
$187$	18.9606i	1.38653i
$188$	0	0
$189$	− 44.6138i	− 3.24517i
$190$	0	0
$191$	−6.75365	−0.488677	−0.244338	−	0.969690i	$-0.578571\pi$
$191$	−6.75365	−0.488677	−0.244338	+	0.969690i	$0.578571\pi$
$192$	0	0
$193$	0.481519i	0.0346605i	0.999850	+	0.0173302i	$0.00551666\pi$
$193$	0.481519i	0.0346605i	−0.999850	+	0.0173302i	$0.994483\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 21.8881i	− 1.55946i	−0.626113	−	0.779732i	$-0.715355\pi$
$197$	− 21.8881i	− 1.55946i	0.626113	−	0.779732i	$-0.284645\pi$
$198$	0	0
$199$	−23.5567	−1.66989	−0.834947	−	0.550331i	$-0.814502\pi$
$199$	−23.5567	−1.66989	−0.834947	+	0.550331i	$0.814502\pi$
$200$	0	0
$201$	1.28337i	0.0905223i
$202$	0	0
$203$	13.1671i	0.924149i
$204$	0	0
$205$	3.95907	0.276513
$206$	0	0
$207$	−41.4593	−2.88162
$208$	0	0
$209$	0.556561	0.0384981
$210$	0	0
$211$	−11.0661	−0.761824	−0.380912	−	0.924611i	$-0.624390\pi$
$211$	−11.0661	−0.761824	−0.380912	+	0.924611i	$0.624390\pi$
$212$	0	0
$213$	− 30.7967i	− 2.11015i
$214$	0	0
$215$	− 8.75108i	− 0.596819i
$216$	0	0
$217$	19.4999	1.32374
$218$	0	0
$219$	− 18.6706i	− 1.26164i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 24.3465i	− 1.63036i	−0.579205	−	0.815182i	$-0.696637\pi$
$223$	− 24.3465i	− 1.63036i	0.579205	−	0.815182i	$-0.303363\pi$
$224$	0	0
$225$	−8.60234	−0.573489
$226$	0	0
$227$	− 0.100791i	− 0.00668971i	−0.999994	−	0.00334485i	$-0.998935\pi$
$227$	− 0.100791i	− 0.00668971i	0.999994	−	0.00334485i	$-0.00106470\pi$
$228$	0	0
$229$	− 5.60506i	− 0.370393i	−0.982702	−	0.185196i	$-0.940708\pi$
$229$	− 5.60506i	− 0.370393i	0.982702	−	0.185196i	$-0.0592921\pi$
$230$	0	0
$231$	−49.0996	−3.23052
$232$	0	0
$233$	−13.2785	−0.869902	−0.434951	−	0.900454i	$-0.643234\pi$
$233$	−13.2785	−0.869902	−0.434951	+	0.900454i	$0.643234\pi$
$234$	0	0
$235$	−3.67871	−0.239973
$236$	0	0
$237$	16.3254	1.06045
$238$	0	0
$239$	12.9817i	0.839715i	0.907590	+	0.419858i	$0.137920\pi$
$239$	12.9817i	0.839715i	−0.907590	+	0.419858i	$0.862080\pi$
$240$	0	0
$241$	1.49222i	0.0961223i	0.998844	+	0.0480612i	$0.0153042\pi$
$241$	1.49222i	0.0961223i	−0.998844	+	0.0480612i	$0.984696\pi$
$242$	0	0
$243$	−76.2524	−4.89159
$244$	0	0
$245$	− 1.53420i	− 0.0980167i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	39.5765i	2.50806i
$250$	0	0
$251$	11.2640	0.710975	0.355487	−	0.934681i	$-0.384315\pi$
$251$	11.2640	0.710975	0.355487	+	0.934681i	$0.384315\pi$
$252$	0	0
$253$	29.7156i	1.86820i
$254$	0	0
$255$	− 10.4748i	− 0.655957i
$256$	0	0
$257$	−15.0148	−0.936600	−0.468300	−	0.883569i	$-0.655133\pi$
$257$	−15.0148	−0.936600	−0.468300	+	0.883569i	$0.655133\pi$
$258$	0	0
$259$	−1.85729	−0.115406
$260$	0	0
$261$	48.4484	2.99888
$262$	0	0
$263$	11.4305	0.704837	0.352418	−	0.935843i	$-0.385359\pi$
$263$	11.4305	0.704837	0.352418	+	0.935843i	$0.385359\pi$
$264$	0	0
$265$	− 8.08456i	− 0.496631i
$266$	0	0
$267$	39.8930i	2.44141i
$268$	0	0
$269$	25.3906	1.54809	0.774047	−	0.633129i	$-0.218230\pi$
$269$	25.3906	1.54809	0.774047	+	0.633129i	$0.218230\pi$
$270$	0	0
$271$	10.6327i	0.645892i	0.946417	+	0.322946i	$0.104673\pi$
$271$	10.6327i	0.645892i	−0.946417	+	0.322946i	$0.895327\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6.16565i	0.371803i
$276$	0	0
$277$	−25.9901	−1.56159	−0.780797	−	0.624785i	$-0.785187\pi$
$277$	−25.9901	−1.56159	−0.780797	+	0.624785i	$0.785187\pi$
$278$	0	0
$279$	− 71.7502i	− 4.29557i
$280$	0	0
$281$	− 7.65920i	− 0.456909i	−0.973555	−	0.228455i	$-0.926633\pi$
$281$	− 7.65920i	− 0.456909i	0.973555	−	0.228455i	$-0.0733673\pi$
$282$	0	0
$283$	−19.5025	−1.15930	−0.579651	−	0.814865i	$-0.696811\pi$
$283$	−19.5025	−1.15930	−0.579651	+	0.814865i	$0.696811\pi$
$284$	0	0
$285$	−0.307473	−0.0182131
$286$	0	0
$287$	−9.25592	−0.546360
$288$	0	0
$289$	−7.54317	−0.443716
$290$	0	0
$291$	8.29952i	0.486526i
$292$	0	0
$293$	− 12.7879i	− 0.747078i	−0.927614	−	0.373539i	$-0.878144\pi$
$293$	− 12.7879i	− 0.747078i	0.927614	−	0.373539i	$-0.121856\pi$
$294$	0	0
$295$	0.379577	0.0220998
$296$	0	0
$297$	117.658i	6.82720i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	20.4592i	1.17925i
$302$	0	0
$303$	−18.8307	−1.08180
$304$	0	0
$305$	4.96367i	0.284219i
$306$	0	0
$307$	32.5882i	1.85990i	0.367680	+	0.929952i	$0.380152\pi$
$307$	32.5882i	1.85990i	−0.367680	+	0.929952i	$0.619848\pi$
$308$	0	0
$309$	28.8507	1.64126
$310$	0	0
$311$	−4.51961	−0.256284	−0.128142	−	0.991756i	$-0.540901\pi$
$311$	−4.51961	−0.256284	−0.128142	+	0.991756i	$0.540901\pi$
$312$	0	0
$313$	−2.40976	−0.136208	−0.0681038	−	0.997678i	$-0.521695\pi$
$313$	−2.40976	−0.136208	−0.0681038	+	0.997678i	$0.521695\pi$
$314$	0	0
$315$	20.1114	1.13315
$316$	0	0
$317$	− 11.6751i	− 0.655739i	−0.944723	−	0.327870i	$-0.893669\pi$
$317$	− 11.6751i	− 0.655739i	0.944723	−	0.327870i	$-0.106331\pi$
$318$	0	0
$319$	− 34.7250i	− 1.94423i
$320$	0	0
$321$	−38.7939	−2.16526
$322$	0	0
$323$	− 0.277592i	− 0.0154456i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 30.8135i	− 1.70399i
$328$	0	0
$329$	8.60048	0.474160
$330$	0	0
$331$	− 22.3827i	− 1.23026i	−0.788425	−	0.615131i	$-0.789103\pi$
$331$	− 22.3827i	− 1.23026i	0.788425	−	0.615131i	$-0.210897\pi$
$332$	0	0
$333$	6.83392i	0.374496i
$334$	0	0
$335$	−0.376774	−0.0205854
$336$	0	0
$337$	−12.6686	−0.690102	−0.345051	−	0.938584i	$-0.612138\pi$
$337$	−12.6686	−0.690102	−0.345051	+	0.938584i	$0.612138\pi$
$338$	0	0
$339$	10.1900	0.553442
$340$	0	0
$341$	−51.4263	−2.78489
$342$	0	0
$343$	19.9522i	1.07731i
$344$	0	0
$345$	− 16.4164i	− 0.883830i
$346$	0	0
$347$	7.56502	0.406111	0.203056	−	0.979167i	$-0.434913\pi$
$347$	7.56502	0.406111	0.203056	+	0.979167i	$0.434913\pi$
$348$	0	0
$349$	− 18.3617i	− 0.982879i	−0.870912	−	0.491439i	$-0.836471\pi$
$349$	− 18.3617i	− 0.982879i	0.870912	−	0.491439i	$-0.163529\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1.27380i	0.0677976i	0.999425	+	0.0338988i	$0.0107924\pi$
$353$	1.27380i	0.0677976i	−0.999425	+	0.0338988i	$0.989208\pi$
$354$	0	0
$355$	9.04130	0.479862
$356$	0	0
$357$	24.4891i	1.29610i
$358$	0	0
$359$	− 11.0008i	− 0.580598i	−0.956936	−	0.290299i	$-0.906245\pi$
$359$	− 11.0008i	− 0.580598i	0.956936	−	0.290299i	$-0.0937548\pi$
$360$	0	0
$361$	18.9919	0.999571
$362$	0	0
$363$	92.0198	4.82979
$364$	0	0
$365$	5.48133	0.286906
$366$	0	0
$367$	5.95825	0.311018	0.155509	−	0.987834i	$-0.450298\pi$
$367$	5.95825	0.311018	0.155509	+	0.987834i	$0.450298\pi$
$368$	0	0
$369$	34.0572i	1.77295i
$370$	0	0
$371$	18.9009i	0.981288i
$372$	0	0
$373$	−21.7186	−1.12455	−0.562274	−	0.826951i	$-0.690073\pi$
$373$	−21.7186	−1.12455	−0.562274	+	0.826951i	$0.690073\pi$
$374$	0	0
$375$	− 3.40622i	− 0.175896i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 29.5331i	− 1.51701i	−0.651665	−	0.758507i	$-0.725929\pi$
$379$	− 29.5331i	− 1.51701i	0.651665	−	0.758507i	$-0.274071\pi$
$380$	0	0
$381$	36.3702	1.86330
$382$	0	0
$383$	26.8889i	1.37396i	0.726676	+	0.686980i	$0.241064\pi$
$383$	26.8889i	1.37396i	−0.726676	+	0.686980i	$0.758936\pi$
$384$	0	0
$385$	− 14.4147i	− 0.734641i
$386$	0	0
$387$	75.2798	3.82669
$388$	0	0
$389$	3.11775	0.158076	0.0790380	−	0.996872i	$-0.474815\pi$
$389$	3.11775	0.158076	0.0790380	+	0.996872i	$0.474815\pi$
$390$	0	0
$391$	14.8210	0.749532
$392$	0	0
$393$	70.1928	3.54076
$394$	0	0
$395$	4.79283i	0.241153i
$396$	0	0
$397$	− 21.2851i	− 1.06827i	−0.845400	−	0.534133i	$-0.820638\pi$
$397$	− 21.2851i	− 1.06827i	0.845400	−	0.534133i	$-0.179362\pi$
$398$	0	0
$399$	0.718842	0.0359871
$400$	0	0
$401$	7.30134i	0.364611i	0.983242	+	0.182306i	$0.0583560\pi$
$401$	7.30134i	0.364611i	−0.983242	+	0.182306i	$0.941644\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	− 39.1932i	− 1.94753i
$406$	0	0
$407$	4.89815	0.242792
$408$	0	0
$409$	4.26732i	0.211005i	0.994419	+	0.105503i	$0.0336451\pi$
$409$	4.26732i	0.211005i	−0.994419	+	0.105503i	$0.966355\pi$
$410$	0	0
$411$	0.289580i	0.0142839i
$412$	0	0
$413$	−0.887415	−0.0436668
$414$	0	0
$415$	−11.6189	−0.570349
$416$	0	0
$417$	−37.5028	−1.83652
$418$	0	0
$419$	−8.14300	−0.397811	−0.198906	−	0.980019i	$-0.563739\pi$
$419$	−8.14300	−0.397811	−0.198906	+	0.980019i	$0.563739\pi$
$420$	0	0
$421$	− 7.36448i	− 0.358923i	−0.983765	−	0.179461i	$-0.942564\pi$
$421$	− 7.36448i	− 0.358923i	0.983765	−	0.179461i	$-0.0574355\pi$
$422$	0	0
$423$	− 31.6455i	− 1.53866i
$424$	0	0
$425$	3.07520	0.149169
$426$	0	0
$427$	− 11.6046i	− 0.561585i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	20.1661i	0.971369i	0.874134	+	0.485684i	$0.161430\pi$
$431$	20.1661i	0.971369i	−0.874134	+	0.485684i	$0.838570\pi$
$432$	0	0
$433$	−11.0657	−0.531783	−0.265892	−	0.964003i	$-0.585666\pi$
$433$	−11.0657	−0.531783	−0.265892	+	0.964003i	$0.585666\pi$
$434$	0	0
$435$	19.1839i	0.919796i
$436$	0	0
$437$	− 0.435051i	− 0.0208113i
$438$	0	0
$439$	−34.5470	−1.64884	−0.824419	−	0.565980i	$-0.808498\pi$
$439$	−34.5470	−1.64884	−0.824419	+	0.565980i	$0.808498\pi$
$440$	0	0
$441$	13.1977	0.628464
$442$	0	0
$443$	−1.22275	−0.0580947	−0.0290473	−	0.999578i	$-0.509247\pi$
$443$	−1.22275	−0.0580947	−0.0290473	+	0.999578i	$0.509247\pi$
$444$	0	0
$445$	−11.7118	−0.555193
$446$	0	0
$447$	15.0389i	0.711315i
$448$	0	0
$449$	18.2491i	0.861228i	0.902536	+	0.430614i	$0.141703\pi$
$449$	18.2491i	0.861228i	−0.902536	+	0.430614i	$0.858297\pi$
$450$	0	0
$451$	24.4102	1.14943
$452$	0	0
$453$	− 53.6666i	− 2.52148i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	23.2180i	1.08609i	0.839703	+	0.543046i	$0.182729\pi$
$457$	23.2180i	1.08609i	−0.839703	+	0.543046i	$0.817271\pi$
$458$	0	0
$459$	58.6833	2.73910
$460$	0	0
$461$	− 29.6014i	− 1.37868i	−0.724440	−	0.689338i	$-0.757901\pi$
$461$	− 29.6014i	− 1.37868i	0.724440	−	0.689338i	$-0.242099\pi$
$462$	0	0
$463$	− 30.4362i	− 1.41449i	−0.706969	−	0.707244i	$-0.749938\pi$
$463$	− 30.4362i	− 1.41449i	0.706969	−	0.707244i	$-0.250062\pi$
$464$	0	0
$465$	28.4105	1.31751
$466$	0	0
$467$	−15.6000	−0.721880	−0.360940	−	0.932589i	$-0.617544\pi$
$467$	−15.6000	−0.721880	−0.360940	+	0.932589i	$0.617544\pi$
$468$	0	0
$469$	0.880861	0.0406744
$470$	0	0
$471$	−11.5852	−0.533819
$472$	0	0
$473$	− 53.9561i	− 2.48090i
$474$	0	0
$475$	− 0.0902681i	− 0.00414178i
$476$	0	0
$477$	69.5462	3.18430
$478$	0	0
$479$	− 41.5259i	− 1.89737i	−0.316227	−	0.948684i	$-0.602416\pi$
$479$	− 41.5259i	− 1.89737i	0.316227	−	0.948684i	$-0.397584\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	38.3800i	1.74635i
$484$	0	0
$485$	−2.43658	−0.110639
$486$	0	0
$487$	9.84163i	0.445967i	0.974822	+	0.222983i	$0.0715796\pi$
$487$	9.84163i	0.445967i	−0.974822	+	0.222983i	$0.928420\pi$
$488$	0	0
$489$	− 49.1628i	− 2.22322i
$490$	0	0
$491$	−29.1322	−1.31472	−0.657359	−	0.753577i	$-0.728327\pi$
$491$	−29.1322	−1.31472	−0.657359	+	0.753577i	$0.728327\pi$
$492$	0	0
$493$	−17.3195	−0.780032
$494$	0	0
$495$	−53.0390	−2.38393
$496$	0	0
$497$	−21.1377	−0.948155
$498$	0	0
$499$	− 33.1347i	− 1.48331i	−0.670781	−	0.741656i	$-0.734041\pi$
$499$	− 33.1347i	− 1.48331i	0.670781	−	0.741656i	$-0.265959\pi$
$500$	0	0
$501$	64.6553i	2.88858i
$502$	0	0
$503$	24.3323	1.08492	0.542462	−	0.840080i	$-0.317492\pi$
$503$	24.3323	1.08492	0.542462	+	0.840080i	$0.317492\pi$
$504$	0	0
$505$	− 5.52834i	− 0.246008i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	16.9484i	0.751223i	0.926777	+	0.375611i	$0.122567\pi$
$509$	16.9484i	0.751223i	−0.926777	+	0.375611i	$0.877433\pi$
$510$	0	0
$511$	−12.8148	−0.566894
$512$	0	0
$513$	− 1.72257i	− 0.0760532i
$514$	0	0
$515$	8.46999i	0.373232i
$516$	0	0
$517$	−22.6817	−0.997538
$518$	0	0
$519$	37.7989	1.65919
$520$	0	0
$521$	−16.0211	−0.701896	−0.350948	−	0.936395i	$-0.614141\pi$
$521$	−16.0211	−0.701896	−0.350948	+	0.936395i	$0.614141\pi$
$522$	0	0
$523$	−5.05699	−0.221127	−0.110563	−	0.993869i	$-0.535266\pi$
$523$	−5.05699	−0.221127	−0.110563	+	0.993869i	$0.535266\pi$
$524$	0	0
$525$	7.96342i	0.347552i
$526$	0	0
$527$	25.6495i	1.11731i
$528$	0	0
$529$	0.227981	0.00991221
$530$	0	0
$531$	3.26525i	0.141700i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	− 11.3891i	− 0.492395i
$536$	0	0
$537$	17.9564	0.774876
$538$	0	0
$539$	− 9.45936i	− 0.407444i
$540$	0	0
$541$	44.1770i	1.89932i	0.313286	+	0.949659i	$0.398570\pi$
$541$	44.1770i	1.89932i	−0.313286	+	0.949659i	$0.601430\pi$
$542$	0	0
$543$	55.0936	2.36429
$544$	0	0
$545$	9.04624	0.387498
$546$	0	0
$547$	16.3624	0.699605	0.349803	−	0.936823i	$-0.386249\pi$
$547$	16.3624	0.699605	0.349803	+	0.936823i	$0.386249\pi$
$548$	0	0
$549$	−42.6992	−1.82236
$550$	0	0
$551$	0.508391i	0.0216582i
$552$	0	0
$553$	− 11.2052i	− 0.476492i
$554$	0	0
$555$	−2.70599	−0.114863
$556$	0	0
$557$	− 29.8749i	− 1.26584i	−0.774218	−	0.632919i	$-0.781857\pi$
$557$	− 29.8749i	− 1.26584i	0.774218	−	0.632919i	$-0.218143\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	− 64.5839i	− 2.72673i
$562$	0	0
$563$	−16.9922	−0.716135	−0.358067	−	0.933696i	$-0.616564\pi$
$563$	−16.9922	−0.716135	−0.358067	+	0.933696i	$0.616564\pi$
$564$	0	0
$565$	2.99157i	0.125856i
$566$	0	0
$567$	91.6300i	3.84810i
$568$	0	0
$569$	−16.8326	−0.705660	−0.352830	−	0.935687i	$-0.614781\pi$
$569$	−16.8326	−0.705660	−0.352830	+	0.935687i	$0.614781\pi$
$570$	0	0
$571$	−1.09953	−0.0460138	−0.0230069	−	0.999735i	$-0.507324\pi$
$571$	−1.09953	−0.0460138	−0.0230069	+	0.999735i	$0.507324\pi$
$572$	0	0
$573$	23.0044	0.961023
$574$	0	0
$575$	4.81954	0.200989
$576$	0	0
$577$	− 21.1153i	− 0.879042i	−0.898232	−	0.439521i	$-0.855148\pi$
$577$	− 21.1153i	− 0.879042i	0.898232	−	0.439521i	$-0.144852\pi$
$578$	0	0
$579$	− 1.64016i	− 0.0681627i
$580$	0	0
$581$	27.1639	1.12695
$582$	0	0
$583$	− 49.8466i	− 2.06443i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	32.2938i	1.33291i	0.745547	+	0.666453i	$0.232188\pi$
$587$	32.2938i	1.33291i	−0.745547	+	0.666453i	$0.767812\pi$
$588$	0	0
$589$	0.752906	0.0310229
$590$	0	0
$591$	74.5558i	3.06681i
$592$	0	0
$593$	12.5827i	0.516711i	0.966050	+	0.258356i	$0.0831806\pi$
$593$	12.5827i	0.516711i	−0.966050	+	0.258356i	$0.916819\pi$
$594$	0	0
$595$	−7.18951	−0.294741
$596$	0	0
$597$	80.2395	3.28398
$598$	0	0
$599$	9.46110	0.386570	0.193285	−	0.981143i	$-0.438086\pi$
$599$	9.46110	0.386570	0.193285	+	0.981143i	$0.438086\pi$
$600$	0	0
$601$	8.02016	0.327149	0.163574	−	0.986531i	$-0.447698\pi$
$601$	8.02016	0.327149	0.163574	+	0.986531i	$0.447698\pi$
$602$	0	0
$603$	− 3.24114i	− 0.131989i
$604$	0	0
$605$	27.0152i	1.09833i
$606$	0	0
$607$	−26.5832	−1.07898	−0.539490	−	0.841992i	$-0.681383\pi$
$607$	−26.5832	−1.07898	−0.539490	+	0.841992i	$0.681383\pi$
$608$	0	0
$609$	− 44.8500i	− 1.81742i
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 44.8547i	− 1.81166i	−0.423639	−	0.905831i	$-0.639247\pi$
$613$	− 44.8547i	− 1.81166i	0.423639	−	0.905831i	$-0.360753\pi$
$614$	0	0
$615$	−13.4855	−0.543786
$616$	0	0
$617$	20.9169i	0.842082i	0.907042	+	0.421041i	$0.138335\pi$
$617$	20.9169i	0.842082i	−0.907042	+	0.421041i	$0.861665\pi$
$618$	0	0
$619$	9.55179i	0.383919i	0.981403	+	0.191959i	$0.0614842\pi$
$619$	9.55179i	0.383919i	−0.981403	+	0.191959i	$0.938516\pi$
$620$	0	0
$621$	91.9704	3.69064
$622$	0	0
$623$	27.3811	1.09700
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−1.89577	−0.0757098
$628$	0	0
$629$	− 2.44301i	− 0.0974093i
$630$	0	0
$631$	− 4.38539i	− 0.174580i	−0.996183	−	0.0872899i	$-0.972179\pi$
$631$	− 4.38539i	− 0.174580i	0.996183	−	0.0872899i	$-0.0278206\pi$
$632$	0	0
$633$	37.6937	1.49819
$634$	0	0
$635$	10.6776i	0.423727i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	77.7763i	3.07678i
$640$	0	0
$641$	19.1325	0.755690	0.377845	−	0.925869i	$-0.376665\pi$
$641$	19.1325	0.755690	0.377845	+	0.925869i	$0.376665\pi$
$642$	0	0
$643$	17.7043i	0.698190i	0.937087	+	0.349095i	$0.113511\pi$
$643$	17.7043i	0.698190i	−0.937087	+	0.349095i	$0.886489\pi$
$644$	0	0
$645$	29.8081i	1.17369i
$646$	0	0
$647$	30.0594	1.18176	0.590879	−	0.806761i	$-0.298781\pi$
$647$	30.0594	1.18176	0.590879	+	0.806761i	$0.298781\pi$
$648$	0	0
$649$	2.34034	0.0918663
$650$	0	0
$651$	−66.4211	−2.60325
$652$	0	0
$653$	−19.3677	−0.757915	−0.378958	−	0.925414i	$-0.623717\pi$
$653$	−19.3677	−0.757915	−0.378958	+	0.925414i	$0.623717\pi$
$654$	0	0
$655$	20.6072i	0.805191i
$656$	0	0
$657$	47.1522i	1.83958i
$658$	0	0
$659$	8.47410	0.330104	0.165052	−	0.986285i	$-0.447221\pi$
$659$	8.47410	0.330104	0.165052	+	0.986285i	$0.447221\pi$
$660$	0	0
$661$	− 14.9713i	− 0.582318i	−0.956675	−	0.291159i	$-0.905959\pi$
$661$	− 14.9713i	− 0.582318i	0.956675	−	0.291159i	$-0.0940408\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0.211038i	0.00818371i
$666$	0	0
$667$	−27.1437	−1.05101
$668$	0	0
$669$	82.9296i	3.20625i
$670$	0	0
$671$	30.6043i	1.18146i
$672$	0	0
$673$	−35.3418	−1.36233	−0.681163	−	0.732132i	$-0.738525\pi$
$673$	−35.3418	−1.36233	−0.681163	+	0.732132i	$0.738525\pi$
$674$	0	0
$675$	19.0828	0.734497
$676$	0	0
$677$	−31.1737	−1.19810	−0.599051	−	0.800711i	$-0.704455\pi$
$677$	−31.1737	−1.19810	−0.599051	+	0.800711i	$0.704455\pi$
$678$	0	0
$679$	5.69648	0.218611
$680$	0	0
$681$	0.343315i	0.0131559i
$682$	0	0
$683$	48.5452i	1.85753i	0.370667	+	0.928766i	$0.379129\pi$
$683$	48.5452i	1.85753i	−0.370667	+	0.928766i	$0.620871\pi$
$684$	0	0
$685$	−0.0850150	−0.00324826
$686$	0	0
$687$	19.0921i	0.728408i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	17.9719i	0.683684i	0.939757	+	0.341842i	$0.111051\pi$
$691$	17.9719i	0.683684i	−0.939757	+	0.341842i	$0.888949\pi$
$692$	0	0
$693$	124.000	4.71037
$694$	0	0
$695$	− 11.0101i	− 0.417637i
$696$	0	0
$697$	− 12.1749i	− 0.461157i
$698$	0	0
$699$	45.2294	1.71073
$700$	0	0
$701$	−13.0567	−0.493147	−0.246573	−	0.969124i	$-0.579305\pi$
$701$	−13.0567	−0.493147	−0.246573	+	0.969124i	$0.579305\pi$
$702$	0	0
$703$	−0.0717113	−0.00270464
$704$	0	0
$705$	12.5305	0.471926
$706$	0	0
$707$	12.9247i	0.486084i
$708$	0	0
$709$	− 14.9427i	− 0.561186i	−0.959827	−	0.280593i	$-0.909469\pi$
$709$	− 14.9427i	− 0.561186i	0.959827	−	0.280593i	$-0.0905312\pi$
$710$	0	0
$711$	−41.2295	−1.54623
$712$	0	0
$713$	40.1987i	1.50545i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 44.2185i	− 1.65137i
$718$	0	0
$719$	−10.7997	−0.402761	−0.201381	−	0.979513i	$-0.564543\pi$
$719$	−10.7997	−0.402761	−0.201381	+	0.979513i	$0.564543\pi$
$720$	0	0
$721$	− 19.8020i	− 0.737466i
$722$	0	0
$723$	− 5.08283i	− 0.189032i
$724$	0	0
$725$	−5.63201	−0.209168
$726$	0	0
$727$	25.1935	0.934376	0.467188	−	0.884158i	$-0.345267\pi$
$727$	25.1935	0.934376	0.467188	+	0.884158i	$0.345267\pi$
$728$	0	0
$729$	142.153	5.26491
$730$	0	0
$731$	−26.9113	−0.995350
$732$	0	0
$733$	32.1714i	1.18828i	0.804363	+	0.594139i	$0.202507\pi$
$733$	32.1714i	1.18828i	−0.804363	+	0.594139i	$0.797493\pi$
$734$	0	0
$735$	5.22584i	0.192758i
$736$	0	0
$737$	−2.32306	−0.0855708
$738$	0	0
$739$	44.1036i	1.62238i	0.584785	+	0.811188i	$0.301179\pi$
$739$	44.1036i	1.62238i	−0.584785	+	0.811188i	$0.698821\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	44.4544i	1.63087i	0.578846	+	0.815437i	$0.303503\pi$
$743$	44.4544i	1.63087i	−0.578846	+	0.815437i	$0.696497\pi$
$744$	0	0
$745$	−4.41512	−0.161758
$746$	0	0
$747$	− 99.9497i	− 3.65697i
$748$	0	0
$749$	26.6267i	0.972918i
$750$	0	0
$751$	−37.6126	−1.37250	−0.686252	−	0.727364i	$-0.740745\pi$
$751$	−37.6126	−1.37250	−0.686252	+	0.727364i	$0.740745\pi$
$752$	0	0
$753$	−38.3675	−1.39819
$754$	0	0
$755$	15.7555	0.573400
$756$	0	0
$757$	35.3294	1.28407	0.642034	−	0.766676i	$-0.278091\pi$
$757$	35.3294	1.28407	0.642034	+	0.766676i	$0.278091\pi$
$758$	0	0
$759$	− 101.218i	− 3.67398i
$760$	0	0
$761$	27.3170i	0.990240i	0.868825	+	0.495120i	$0.164876\pi$
$761$	27.3170i	0.990240i	−0.868825	+	0.495120i	$0.835124\pi$
$762$	0	0
$763$	−21.1492	−0.765654
$764$	0	0
$765$	26.4539i	0.956442i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	23.1958i	0.836460i	0.908341	+	0.418230i	$0.137349\pi$
$769$	23.1958i	0.836460i	−0.908341	+	0.418230i	$0.862651\pi$
$770$	0	0
$771$	51.1439	1.84190
$772$	0	0
$773$	− 6.52005i	− 0.234510i	−0.993102	−	0.117255i	$-0.962591\pi$
$773$	− 6.52005i	− 0.234510i	0.993102	−	0.117255i	$-0.0374095\pi$
$774$	0	0
$775$	8.34078i	0.299610i
$776$	0	0
$777$	6.32634	0.226956
$778$	0	0
$779$	−0.357377	−0.0128044
$780$	0	0
$781$	55.7455	1.99473
$782$	0	0
$783$	−107.475	−3.84083
$784$	0	0
$785$	− 3.40120i	− 0.121394i
$786$	0	0
$787$	− 40.7311i	− 1.45191i	−0.687744	−	0.725954i	$-0.741399\pi$
$787$	− 40.7311i	− 1.45191i	0.687744	−	0.725954i	$-0.258601\pi$
$788$	0	0
$789$	−38.9349	−1.38612
$790$	0	0
$791$	− 6.99401i	− 0.248678i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	27.5378i	0.976665i
$796$	0	0
$797$	−4.48208	−0.158763	−0.0793816	−	0.996844i	$-0.525295\pi$
$797$	−4.48208	−0.158763	−0.0793816	+	0.996844i	$0.525295\pi$
$798$	0	0
$799$	11.3128i	0.400217i
$800$	0	0
$801$	− 100.749i	− 3.55979i
$802$	0	0
$803$	33.7959	1.19263
$804$	0	0
$805$	−11.2676	−0.397132
$806$	0	0
$807$	−86.4860	−3.04445
$808$	0	0
$809$	32.7884	1.15278	0.576390	−	0.817175i	$-0.304461\pi$
$809$	32.7884	1.15278	0.576390	+	0.817175i	$0.304461\pi$
$810$	0	0
$811$	29.5808i	1.03872i	0.854555	+	0.519361i	$0.173830\pi$
$811$	29.5808i	1.03872i	−0.854555	+	0.519361i	$0.826170\pi$
$812$	0	0
$813$	− 36.2174i	− 1.27020i
$814$	0	0
$815$	14.4332	0.505574
$816$	0	0
$817$	0.789943i	0.0276366i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 47.1035i	− 1.64392i	−0.569543	−	0.821962i	$-0.692880\pi$
$821$	− 47.1035i	− 1.64392i	0.569543	−	0.821962i	$-0.307120\pi$
$822$	0	0
$823$	16.4099	0.572012	0.286006	−	0.958228i	$-0.407672\pi$
$823$	16.4099	0.572012	0.286006	+	0.958228i	$0.407672\pi$
$824$	0	0
$825$	− 21.0016i	− 0.731180i
$826$	0	0
$827$	− 7.42042i	− 0.258033i	−0.991642	−	0.129017i	$-0.958818\pi$
$827$	− 7.42042i	− 0.258033i	0.991642	−	0.129017i	$-0.0411821\pi$
$828$	0	0
$829$	48.0555	1.66904	0.834518	−	0.550980i	$-0.185746\pi$
$829$	48.0555	1.66904	0.834518	+	0.550980i	$0.185746\pi$
$830$	0	0
$831$	88.5280	3.07100
$832$	0	0
$833$	−4.71798	−0.163468
$834$	0	0
$835$	−18.9815	−0.656883
$836$	0	0
$837$	159.165i	5.50156i
$838$	0	0
$839$	12.5460i	0.433136i	0.976268	+	0.216568i	$0.0694862\pi$
$839$	12.5460i	0.433136i	−0.976268	+	0.216568i	$0.930514\pi$
$840$	0	0
$841$	2.71952	0.0937766
$842$	0	0
$843$	26.0889i	0.898550i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 63.1590i	− 2.17017i
$848$	0	0
$849$	66.4298	2.27986
$850$	0	0
$851$	− 3.82877i	− 0.131248i
$852$	0	0
$853$	8.26494i	0.282986i	0.989939	+	0.141493i	$0.0451903\pi$
$853$	8.26494i	0.282986i	−0.989939	+	0.141493i	$0.954810\pi$
$854$	0	0
$855$	0.776517	0.0265563
$856$	0	0
$857$	−15.0212	−0.513115	−0.256558	−	0.966529i	$-0.582588\pi$
$857$	−15.0212	−0.513115	−0.256558	+	0.966529i	$0.582588\pi$
$858$	0	0
$859$	22.3863	0.763810	0.381905	−	0.924202i	$-0.375268\pi$
$859$	22.3863	0.763810	0.381905	+	0.924202i	$0.375268\pi$
$860$	0	0
$861$	31.5277	1.07446
$862$	0	0
$863$	− 4.17657i	− 0.142172i	−0.997470	−	0.0710860i	$-0.977354\pi$
$863$	− 4.17657i	− 0.142172i	0.997470	−	0.0710860i	$-0.0226465\pi$
$864$	0	0
$865$	11.0970i	0.377310i
$866$	0	0
$867$	25.6937	0.872604
$868$	0	0
$869$	29.5509i	1.00244i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	− 20.9603i	− 0.709397i
$874$	0	0
$875$	−2.33790	−0.0790356
$876$	0	0
$877$	39.2315i	1.32475i	0.749171	+	0.662377i	$0.230452\pi$
$877$	39.2315i	1.32475i	−0.749171	+	0.662377i	$0.769548\pi$
$878$	0	0
$879$	43.5585i	1.46919i
$880$	0	0
$881$	−8.53832	−0.287663	−0.143832	−	0.989602i	$-0.545942\pi$
$881$	−8.53832	−0.287663	−0.143832	+	0.989602i	$0.545942\pi$
$882$	0	0
$883$	−44.7515	−1.50601	−0.753004	−	0.658016i	$-0.771396\pi$
$883$	−44.7515	−1.50601	−0.753004	+	0.658016i	$0.771396\pi$
$884$	0	0
$885$	−1.29292	−0.0434611
$886$	0	0
$887$	37.1035	1.24581	0.622906	−	0.782296i	$-0.285952\pi$
$887$	37.1035	1.24581	0.622906	+	0.782296i	$0.285952\pi$
$888$	0	0
$889$	− 24.9632i	− 0.837237i
$890$	0	0
$891$	− 241.652i	− 8.09563i
$892$	0	0
$893$	0.332070	0.0111123
$894$	0	0
$895$	5.27165i	0.176212i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 46.9753i	− 1.56671i
$900$	0	0
$901$	−24.8616	−0.828260
$902$	0	0
$903$	− 69.6885i	− 2.31909i
$904$	0	0
$905$	16.1744i	0.537655i
$906$	0	0
$907$	37.6342	1.24962	0.624812	−	0.780776i	$-0.285176\pi$
$907$	37.6342	1.24962	0.624812	+	0.780776i	$0.285176\pi$
$908$	0	0
$909$	47.5566	1.57735
$910$	0	0
$911$	39.8183	1.31924	0.659619	−	0.751600i	$-0.270718\pi$
$911$	39.8183	1.31924	0.659619	+	0.751600i	$0.270718\pi$
$912$	0	0
$913$	−71.6380	−2.37087
$914$	0	0
$915$	− 16.9074i	− 0.558940i
$916$	0	0
$917$	− 48.1777i	− 1.59097i
$918$	0	0
$919$	−34.1947	−1.12798	−0.563989	−	0.825782i	$-0.690734\pi$
$919$	−34.1947	−1.12798	−0.563989	+	0.825782i	$0.690734\pi$
$920$	0	0
$921$	− 111.002i	− 3.65766i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	− 0.794425i	− 0.0261205i
$926$	0	0
$927$	−72.8617	−2.39309
$928$	0	0
$929$	30.5866i	1.00351i	0.865009	+	0.501757i	$0.167313\pi$
$929$	30.5866i	1.00351i	−0.865009	+	0.501757i	$0.832687\pi$
$930$	0	0
$931$	0.138490i	0.00453881i
$932$	0	0
$933$	15.3948	0.504003
$934$	0	0
$935$	18.9606	0.620077
$936$	0	0
$937$	−39.0385	−1.27533	−0.637666	−	0.770313i	$-0.720100\pi$
$937$	−39.0385	−1.27533	−0.637666	+	0.770313i	$0.720100\pi$
$938$	0	0
$939$	8.20816	0.267863
$940$	0	0
$941$	6.92575i	0.225773i	0.993608	+	0.112886i	$0.0360096\pi$
$941$	6.92575i	0.225773i	−0.993608	+	0.112886i	$0.963990\pi$
$942$	0	0
$943$	− 19.0809i	− 0.621359i
$944$	0	0
$945$	−44.6138	−1.45129
$946$	0	0
$947$	15.2145i	0.494405i	0.968964	+	0.247202i	$0.0795113\pi$
$947$	15.2145i	0.494405i	−0.968964	+	0.247202i	$0.920489\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	39.7680i	1.28957i
$952$	0	0
$953$	5.06168	0.163964	0.0819819	−	0.996634i	$-0.473875\pi$
$953$	5.06168	0.163964	0.0819819	+	0.996634i	$0.473875\pi$
$954$	0	0
$955$	6.75365i	0.218543i
$956$	0	0
$957$	118.281i	3.82348i
$958$	0	0
$959$	0.198757	0.00641820
$960$	0	0
$961$	−38.5686	−1.24415
$962$	0	0
$963$	97.9732	3.15714
$964$	0	0
$965$	0.481519	0.0155006
$966$	0	0
$967$	− 48.3231i	− 1.55397i	−0.629522	−	0.776983i	$-0.716749\pi$
$967$	− 48.3231i	− 1.55397i	0.629522	−	0.776983i	$-0.283251\pi$
$968$	0	0
$969$	0.945539i	0.0303751i
$970$	0	0
$971$	45.8463	1.47128	0.735638	−	0.677374i	$-0.236882\pi$
$971$	45.8463	1.47128	0.735638	+	0.677374i	$0.236882\pi$
$972$	0	0
$973$	25.7405i	0.825204i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 43.2717i	− 1.38438i	−0.721714	−	0.692192i	$-0.756645\pi$
$977$	− 43.2717i	− 1.38438i	0.721714	−	0.692192i	$-0.243355\pi$
$978$	0	0
$979$	−72.2108	−2.30787
$980$	0	0
$981$	77.8188i	2.48456i
$982$	0	0
$983$	0.695243i	0.0221748i	0.999939	+	0.0110874i	$0.00352930\pi$
$983$	0.695243i	0.0221748i	−0.999939	+	0.0110874i	$0.996471\pi$
$984$	0	0
$985$	−21.8881	−0.697414
$986$	0	0
$987$	−29.2951	−0.932474
$988$	0	0
$989$	−42.1762	−1.34113
$990$	0	0
$991$	21.1973	0.673355	0.336677	−	0.941620i	$-0.390697\pi$
$991$	21.1973	0.673355	0.336677	+	0.941620i	$0.390697\pi$
$992$	0	0
$993$	76.2403i	2.41941i
$994$	0	0
$995$	23.5567i	0.746799i
$996$	0	0
$997$	−36.7860	−1.16503	−0.582513	−	0.812822i	$-0.697930\pi$
$997$	−36.7860	−1.16503	−0.582513	+	0.812822i	$0.697930\pi$
$998$	0	0
$999$	− 15.1599i	− 0.479637i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3380.2.f.j.3041.1		18
13.5	odd	4		3380.2.a.r.1.1	✓	9
13.8	odd	4		3380.2.a.s.1.1	yes	9
13.12	even	2	inner	3380.2.f.j.3041.2		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3380.2.a.r.1.1	✓	9	13.5	odd	4
3380.2.a.s.1.1	yes	9	13.8	odd	4
3380.2.f.j.3041.1		18	1.1	even	1	trivial
3380.2.f.j.3041.2		18	13.12	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:	\( N \)	\(=\)	\( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3380.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.40622 q^{3} -1.00000i q^{5} +2.33790i q^{7} +8.60234 q^{9} +O(q^{10})\)	\(q-3.40622 q^{3} -1.00000i q^{5} +2.33790i q^{7} +8.60234 q^{9} -6.16565i q^{11} +3.40622i q^{15} -3.07520 q^{17} +0.0902681i q^{19} -7.96342i q^{21} -4.81954 q^{23} -1.00000 q^{25} -19.0828 q^{27} +5.63201 q^{29} -8.34078i q^{31} +21.0016i q^{33} +2.33790 q^{35} +0.794425i q^{37} +3.95907i q^{41} +8.75108 q^{43} -8.60234i q^{45} -3.67871i q^{47} +1.53420 q^{49} +10.4748 q^{51} +8.08456 q^{53} -6.16565 q^{55} -0.307473i q^{57} +0.379577i q^{59} -4.96367 q^{61} +20.1114i q^{63} -0.376774i q^{67} +16.4164 q^{69} +9.04130i q^{71} +5.48133i q^{73} +3.40622 q^{75} +14.4147 q^{77} -4.79283 q^{79} +39.1932 q^{81} -11.6189i q^{83} +3.07520i q^{85} -19.1839 q^{87} -11.7118i q^{89} +28.4105i q^{93} +0.0902681 q^{95} -2.43658i q^{97} -53.0390i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 2 q^{3} + 24 q^{9}+O(q^{10}) \) 18 * q - 2 * q^3 + 24 * q^9	\( 18 q - 2 q^{3} + 24 q^{9} - 26 q^{17} - 24 q^{23} - 18 q^{25} - 8 q^{27} + 32 q^{29} + 2 q^{35} - 2 q^{43} - 40 q^{49} - 22 q^{51} + 60 q^{53} - 14 q^{55} + 42 q^{61} - 30 q^{69} + 2 q^{75} - 92 q^{77} + 62 q^{79} + 82 q^{81} + 56 q^{87} + 8 q^{95}+O(q^{100}) \) 18 * q - 2 * q^3 + 24 * q^9 - 26 * q^17 - 24 * q^23 - 18 * q^25 - 8 * q^27 + 32 * q^29 + 2 * q^35 - 2 * q^43 - 40 * q^49 - 22 * q^51 + 60 * q^53 - 14 * q^55 + 42 * q^61 - 30 * q^69 + 2 * q^75 - 92 * q^77 + 62 * q^79 + 82 * q^81 + 56 * q^87 + 8 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more