LMFDB - Embedded newform 3380.2.f.g.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:	$6$
Coefficient field:	6.0.153664.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 5x^{4} + 6x^{2} + 1 $ x^6 + 5x^4 + 6x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3041.1
Root			$-1.80194i$ of defining polynomial
Character	$\chi$	$=$	3380.3041
Dual form			3380.2.f.g.3041.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.80194 q^{3} -1.00000i q^{5} +1.80194i q^{7} +0.246980 q^{9} +O(q^{10})$	$q-1.80194 q^{3} -1.00000i q^{5} +1.80194i q^{7} +0.246980 q^{9} -1.55496i q^{11} +1.80194i q^{15} -0.911854 q^{17} +1.69202i q^{19} -3.24698i q^{21} -0.246980 q^{23} -1.00000 q^{25} +4.96077 q^{27} -5.29590 q^{29} -1.08815i q^{31} +2.80194i q^{33} +1.80194 q^{35} -3.00000i q^{37} +6.96077i q^{41} -2.26875 q^{43} -0.246980i q^{45} -8.52111i q^{47} +3.75302 q^{49} +1.64310 q^{51} +6.92692 q^{53} -1.55496 q^{55} -3.04892i q^{57} +14.7017i q^{59} -9.55496 q^{61} +0.445042i q^{63} +5.21983i q^{67} +0.445042 q^{69} -5.74094i q^{71} -14.5918i q^{73} +1.80194 q^{75} +2.80194 q^{77} +7.75063 q^{79} -9.67994 q^{81} +4.94869i q^{83} +0.911854i q^{85} +9.54288 q^{87} +11.9095i q^{89} +1.96077i q^{93} +1.69202 q^{95} -9.17390i q^{97} -0.384043i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{3} - 8 q^{9}+O(q^{10}) $ 6 * q - 2 * q^3 - 8 * q^9	$ 6 q - 2 q^{3} - 8 q^{9} + 2 q^{17} + 8 q^{23} - 6 q^{25} + 4 q^{27} - 4 q^{29} + 2 q^{35} + 2 q^{43} + 32 q^{49} + 18 q^{51} - 16 q^{53} - 10 q^{55} - 58 q^{61} + 2 q^{69} + 2 q^{75} + 8 q^{77} - 26 q^{79} - 10 q^{81} + 20 q^{87}+O(q^{100}) $ 6 * q - 2 * q^3 - 8 * q^9 + 2 * q^17 + 8 * q^23 - 6 * q^25 + 4 * q^27 - 4 * q^29 + 2 * q^35 + 2 * q^43 + 32 * q^49 + 18 * q^51 - 16 * q^53 - 10 * q^55 - 58 * q^61 + 2 * q^69 + 2 * q^75 + 8 * q^77 - 26 * q^79 - 10 * q^81 + 20 * q^87

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
$3$	−1.80194	−1.04035	−0.520175	−	0.854060i	$-0.674133\pi$
$3$	−1.80194	−1.04035	−0.520175	+	0.854060i	$0.674133\pi$
$4$	0	0
$5$	− 1.00000i	− 0.447214i
$5$	− 1.00000i	− 0.447214i
$6$	0	0
$7$	1.80194i	0.681068i	0.940232	+	0.340534i	$0.110608\pi$
$7$	1.80194i	0.681068i	−0.940232	+	0.340534i	$0.889392\pi$
$8$	0	0
$9$	0.246980	0.0823265
$10$	0	0
$11$	− 1.55496i	− 0.468838i	−0.972136	−	0.234419i	$-0.924681\pi$
$11$	− 1.55496i	− 0.468838i	0.972136	−	0.234419i	$-0.0753187\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	1.80194i	0.465258i
$16$	0	0
$17$	−0.911854	−0.221157	−0.110579	−	0.993867i	$-0.535270\pi$
$17$	−0.911854	−0.221157	−0.110579	+	0.993867i	$0.535270\pi$
$18$	0	0
$19$	1.69202i	0.388176i	0.980984	+	0.194088i	$0.0621748\pi$
$19$	1.69202i	0.388176i	−0.980984	+	0.194088i	$0.937825\pi$
$20$	0	0
$21$	− 3.24698i	− 0.708549i
$22$	0	0
$23$	−0.246980	−0.0514988	−0.0257494	−	0.999668i	$-0.508197\pi$
$23$	−0.246980	−0.0514988	−0.0257494	+	0.999668i	$0.508197\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	4.96077	0.954701
$28$	0	0
$29$	−5.29590	−0.983423	−0.491712	−	0.870758i	$-0.663629\pi$
$29$	−5.29590	−0.983423	−0.491712	+	0.870758i	$0.663629\pi$
$30$	0	0
$31$	− 1.08815i	− 0.195437i	−0.995214	−	0.0977184i	$-0.968846\pi$
$31$	− 1.08815i	− 0.195437i	0.995214	−	0.0977184i	$-0.0311544\pi$
$32$	0	0
$33$	2.80194i	0.487755i
$34$	0	0
$35$	1.80194	0.304583
$36$	0	0
$37$	− 3.00000i	− 0.493197i	−0.969118	−	0.246598i	$-0.920687\pi$
$37$	− 3.00000i	− 0.493197i	0.969118	−	0.246598i	$-0.0793129\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	6.96077i	1.08709i	0.839380	+	0.543545i	$0.182918\pi$
$41$	6.96077i	1.08709i	−0.839380	+	0.543545i	$0.817082\pi$
$42$	0	0
$43$	−2.26875	−0.345981	−0.172991	−	0.984923i	$-0.555343\pi$
$43$	−2.26875	−0.345981	−0.172991	+	0.984923i	$0.555343\pi$
$44$	0	0
$45$	− 0.246980i	− 0.0368175i
$46$	0	0
$47$	− 8.52111i	− 1.24293i	−0.783441	−	0.621466i	$-0.786538\pi$
$47$	− 8.52111i	− 1.24293i	0.783441	−	0.621466i	$-0.213462\pi$
$48$	0	0
$49$	3.75302	0.536146
$50$	0	0
$51$	1.64310	0.230081
$52$	0	0
$53$	6.92692	0.951486	0.475743	−	0.879584i	$-0.342179\pi$
$53$	6.92692	0.951486	0.475743	+	0.879584i	$0.342179\pi$
$54$	0	0
$55$	−1.55496	−0.209671
$56$	0	0
$57$	− 3.04892i	− 0.403839i
$58$	0	0
$59$	14.7017i	1.91400i	0.290089	+	0.957000i	$0.406315\pi$
$59$	14.7017i	1.91400i	−0.290089	+	0.957000i	$0.593685\pi$
$60$	0	0
$61$	−9.55496	−1.22339	−0.611694	−	0.791095i	$-0.709511\pi$
$61$	−9.55496	−1.22339	−0.611694	+	0.791095i	$0.709511\pi$
$62$	0	0
$63$	0.445042i	0.0560700i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	5.21983i	0.637704i	0.947805	+	0.318852i	$0.103297\pi$
$67$	5.21983i	0.637704i	−0.947805	+	0.318852i	$0.896703\pi$
$68$	0	0
$69$	0.445042	0.0535767
$70$	0	0
$71$	− 5.74094i	− 0.681324i	−0.940186	−	0.340662i	$-0.889349\pi$
$71$	− 5.74094i	− 0.681324i	0.940186	−	0.340662i	$-0.110651\pi$
$72$	0	0
$73$	− 14.5918i	− 1.70784i	−0.520404	−	0.853920i	$-0.674219\pi$
$73$	− 14.5918i	− 1.70784i	0.520404	−	0.853920i	$-0.325781\pi$
$74$	0	0
$75$	1.80194	0.208070
$76$	0	0
$77$	2.80194	0.319310
$78$	0	0
$79$	7.75063	0.872014	0.436007	−	0.899943i	$-0.356392\pi$
$79$	7.75063	0.872014	0.436007	+	0.899943i	$0.356392\pi$
$80$	0	0
$81$	−9.67994	−1.07555
$82$	0	0
$83$	4.94869i	0.543189i	0.962412	+	0.271595i	$0.0875510\pi$
$83$	4.94869i	0.543189i	−0.962412	+	0.271595i	$0.912449\pi$
$84$	0	0
$85$	0.911854i	0.0989045i
$86$	0	0
$87$	9.54288	1.02310
$88$	0	0
$89$	11.9095i	1.26240i	0.775620	+	0.631200i	$0.217437\pi$
$89$	11.9095i	1.26240i	−0.775620	+	0.631200i	$0.782563\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	1.96077i	0.203323i
$94$	0	0
$95$	1.69202	0.173598
$96$	0	0
$97$	− 9.17390i	− 0.931468i	−0.884925	−	0.465734i	$-0.845790\pi$
$97$	− 9.17390i	− 0.931468i	0.884925	−	0.465734i	$-0.154210\pi$
$98$	0	0
$99$	− 0.384043i	− 0.0385978i
$100$	0	0
$101$	2.18060	0.216978	0.108489	−	0.994098i	$-0.465399\pi$
$101$	2.18060	0.216978	0.108489	+	0.994098i	$0.465399\pi$
$102$	0	0
$103$	9.58211	0.944153	0.472076	−	0.881558i	$-0.343505\pi$
$103$	9.58211	0.944153	0.472076	+	0.881558i	$0.343505\pi$
$104$	0	0
$105$	−3.24698	−0.316873
$106$	0	0
$107$	6.03684	0.583603	0.291801	−	0.956479i	$-0.405745\pi$
$107$	6.03684	0.583603	0.291801	+	0.956479i	$0.405745\pi$
$108$	0	0
$109$	− 10.6703i	− 1.02202i	−0.859573	−	0.511012i	$-0.829271\pi$
$109$	− 10.6703i	− 1.02202i	0.859573	−	0.511012i	$-0.170729\pi$
$110$	0	0
$111$	5.40581i	0.513097i
$112$	0	0
$113$	−17.6136	−1.65694	−0.828472	−	0.560030i	$-0.810790\pi$
$113$	−17.6136	−1.65694	−0.828472	+	0.560030i	$0.810790\pi$
$114$	0	0
$115$	0.246980i	0.0230310i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 1.64310i	− 0.150623i
$120$	0	0
$121$	8.58211	0.780191
$122$	0	0
$123$	− 12.5429i	− 1.13095i
$124$	0	0
$125$	1.00000i	0.0894427i
$126$	0	0
$127$	21.4644	1.90466	0.952330	−	0.305071i	$-0.0986802\pi$
$127$	21.4644	1.90466	0.952330	+	0.305071i	$0.0986802\pi$
$128$	0	0
$129$	4.08815	0.359941
$130$	0	0
$131$	7.94869	0.694480	0.347240	−	0.937776i	$-0.387119\pi$
$131$	7.94869	0.694480	0.347240	+	0.937776i	$0.387119\pi$
$132$	0	0
$133$	−3.04892	−0.264375
$134$	0	0
$135$	− 4.96077i	− 0.426955i
$136$	0	0
$137$	1.39075i	0.118820i	0.998234	+	0.0594098i	$0.0189219\pi$
$137$	1.39075i	0.118820i	−0.998234	+	0.0594098i	$0.981078\pi$
$138$	0	0
$139$	20.1008	1.70493	0.852465	−	0.522785i	$-0.175107\pi$
$139$	20.1008	1.70493	0.852465	+	0.522785i	$0.175107\pi$
$140$	0	0
$141$	15.3545i	1.29308i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	5.29590i	0.439800i
$146$	0	0
$147$	−6.76271	−0.557779
$148$	0	0
$149$	7.34050i	0.601357i	0.953725	+	0.300679i	$0.0972132\pi$
$149$	7.34050i	0.601357i	−0.953725	+	0.300679i	$0.902787\pi$
$150$	0	0
$151$	− 8.46011i	− 0.688474i	−0.938883	−	0.344237i	$-0.888138\pi$
$151$	− 8.46011i	− 0.688474i	0.938883	−	0.344237i	$-0.111862\pi$
$152$	0	0
$153$	−0.225209	−0.0182071
$154$	0	0
$155$	−1.08815	−0.0874020
$156$	0	0
$157$	−17.7681	−1.41805	−0.709024	−	0.705185i	$-0.750864\pi$
$157$	−17.7681	−1.41805	−0.709024	+	0.705185i	$0.750864\pi$
$158$	0	0
$159$	−12.4819	−0.989877
$160$	0	0
$161$	− 0.445042i	− 0.0350742i
$162$	0	0
$163$	− 11.6286i	− 0.910825i	−0.890281	−	0.455412i	$-0.849492\pi$
$163$	− 11.6286i	− 0.910825i	0.890281	−	0.455412i	$-0.150508\pi$
$164$	0	0
$165$	2.80194	0.218131
$166$	0	0
$167$	− 9.20237i	− 0.712101i	−0.934467	−	0.356051i	$-0.884123\pi$
$167$	− 9.20237i	− 0.712101i	0.934467	−	0.356051i	$-0.115877\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	0.417895i	0.0319572i
$172$	0	0
$173$	15.4601	1.17541	0.587705	−	0.809075i	$-0.300032\pi$
$173$	15.4601	1.17541	0.587705	+	0.809075i	$0.300032\pi$
$174$	0	0
$175$	− 1.80194i	− 0.136214i
$176$	0	0
$177$	− 26.4916i	− 1.99123i
$178$	0	0
$179$	21.0519	1.57349	0.786746	−	0.617276i	$-0.211764\pi$
$179$	21.0519	1.57349	0.786746	+	0.617276i	$0.211764\pi$
$180$	0	0
$181$	14.9608	1.11203	0.556013	−	0.831174i	$-0.312331\pi$
$181$	14.9608	1.11203	0.556013	+	0.831174i	$0.312331\pi$
$182$	0	0
$183$	17.2174	1.27275
$184$	0	0
$185$	−3.00000	−0.220564
$186$	0	0
$187$	1.41789i	0.103687i
$188$	0	0
$189$	8.93900i	0.650217i
$190$	0	0
$191$	3.15452	0.228253	0.114127	−	0.993466i	$-0.463593\pi$
$191$	3.15452	0.228253	0.114127	+	0.993466i	$0.463593\pi$
$192$	0	0
$193$	4.04892i	0.291447i	0.989325	+	0.145724i	$0.0465511\pi$
$193$	4.04892i	0.291447i	−0.989325	+	0.145724i	$0.953449\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	5.20775i	0.371037i	0.982641	+	0.185518i	$0.0593964\pi$
$197$	5.20775i	0.371037i	−0.982641	+	0.185518i	$0.940604\pi$
$198$	0	0
$199$	22.5013	1.59507	0.797536	−	0.603272i	$-0.206136\pi$
$199$	22.5013	1.59507	0.797536	+	0.603272i	$0.206136\pi$
$200$	0	0
$201$	− 9.40581i	− 0.663435i
$202$	0	0
$203$	− 9.54288i	− 0.669779i
$204$	0	0
$205$	6.96077	0.486161
$206$	0	0
$207$	−0.0609989	−0.00423972
$208$	0	0
$209$	2.63102	0.181992
$210$	0	0
$211$	−0.0435405	−0.00299745	−0.00149873	−	0.999999i	$-0.500477\pi$
$211$	−0.0435405	−0.00299745	−0.00149873	+	0.999999i	$0.500477\pi$
$212$	0	0
$213$	10.3448i	0.708815i
$214$	0	0
$215$	2.26875i	0.154727i
$216$	0	0
$217$	1.96077	0.133106
$218$	0	0
$219$	26.2935i	1.77675i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 16.0911i	− 1.07754i	−0.842452	−	0.538771i	$-0.818889\pi$
$223$	− 16.0911i	− 1.07754i	0.842452	−	0.538771i	$-0.181111\pi$
$224$	0	0
$225$	−0.246980	−0.0164653
$226$	0	0
$227$	− 1.08575i	− 0.0720640i	−0.999351	−	0.0360320i	$-0.988528\pi$
$227$	− 1.08575i	− 0.0720640i	0.999351	−	0.0360320i	$-0.0114718\pi$
$228$	0	0
$229$	25.2620i	1.66936i	0.550733	+	0.834681i	$0.314348\pi$
$229$	25.2620i	1.66936i	−0.550733	+	0.834681i	$0.685652\pi$
$230$	0	0
$231$	−5.04892	−0.332194
$232$	0	0
$233$	23.9879	1.57150	0.785750	−	0.618544i	$-0.212277\pi$
$233$	23.9879	1.57150	0.785750	+	0.618544i	$0.212277\pi$
$234$	0	0
$235$	−8.52111	−0.555856
$236$	0	0
$237$	−13.9661	−0.907199
$238$	0	0
$239$	3.88769i	0.251474i	0.992064	+	0.125737i	$0.0401295\pi$
$239$	3.88769i	0.251474i	−0.992064	+	0.125737i	$0.959870\pi$
$240$	0	0
$241$	− 24.5894i	− 1.58394i	−0.610558	−	0.791971i	$-0.709055\pi$
$241$	− 24.5894i	− 1.58394i	0.610558	−	0.791971i	$-0.290945\pi$
$242$	0	0
$243$	2.56033	0.164246
$244$	0	0
$245$	− 3.75302i	− 0.239772i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	− 8.91723i	− 0.565107i
$250$	0	0
$251$	14.5090	0.915802	0.457901	−	0.889003i	$-0.348601\pi$
$251$	14.5090	0.915802	0.457901	+	0.889003i	$0.348601\pi$
$252$	0	0
$253$	0.384043i	0.0241446i
$254$	0	0
$255$	− 1.64310i	− 0.102895i
$256$	0	0
$257$	5.19700	0.324180	0.162090	−	0.986776i	$-0.448177\pi$
$257$	5.19700	0.324180	0.162090	+	0.986776i	$0.448177\pi$
$258$	0	0
$259$	5.40581	0.335901
$260$	0	0
$261$	−1.30798	−0.0809618
$262$	0	0
$263$	7.86294	0.484849	0.242425	−	0.970170i	$-0.422057\pi$
$263$	7.86294	0.484849	0.242425	+	0.970170i	$0.422057\pi$
$264$	0	0
$265$	− 6.92692i	− 0.425517i
$266$	0	0
$267$	− 21.4601i	− 1.31334i
$268$	0	0
$269$	6.81700	0.415640	0.207820	−	0.978167i	$-0.433363\pi$
$269$	6.81700	0.415640	0.207820	+	0.978167i	$0.433363\pi$
$270$	0	0
$271$	− 16.2228i	− 0.985466i	−0.870181	−	0.492733i	$-0.835998\pi$
$271$	− 16.2228i	− 0.985466i	0.870181	−	0.492733i	$-0.164002\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.55496i	0.0937675i
$276$	0	0
$277$	−2.15644	−0.129568	−0.0647840	−	0.997899i	$-0.520636\pi$
$277$	−2.15644	−0.129568	−0.0647840	+	0.997899i	$0.520636\pi$
$278$	0	0
$279$	− 0.268750i	− 0.0160896i
$280$	0	0
$281$	− 10.7235i	− 0.639709i	−0.947467	−	0.319855i	$-0.896366\pi$
$281$	− 10.7235i	− 0.639709i	0.947467	−	0.319855i	$-0.103634\pi$
$282$	0	0
$283$	5.50902	0.327478	0.163739	−	0.986504i	$-0.447645\pi$
$283$	5.50902	0.327478	0.163739	+	0.986504i	$0.447645\pi$
$284$	0	0
$285$	−3.04892	−0.180602
$286$	0	0
$287$	−12.5429	−0.740383
$288$	0	0
$289$	−16.1685	−0.951090
$290$	0	0
$291$	16.5308i	0.969052i
$292$	0	0
$293$	− 1.80864i	− 0.105662i	−0.998603	−	0.0528310i	$-0.983176\pi$
$293$	− 1.80864i	− 0.105662i	0.998603	−	0.0528310i	$-0.0168245\pi$
$294$	0	0
$295$	14.7017	0.855967
$296$	0	0
$297$	− 7.71379i	− 0.447600i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	− 4.08815i	− 0.235637i
$302$	0	0
$303$	−3.92931	−0.225733
$304$	0	0
$305$	9.55496i	0.547115i
$306$	0	0
$307$	− 10.9414i	− 0.624458i	−0.950007	−	0.312229i	$-0.898924\pi$
$307$	− 10.9414i	− 0.624458i	0.950007	−	0.312229i	$-0.101076\pi$
$308$	0	0
$309$	−17.2664	−0.982249
$310$	0	0
$311$	2.95646	0.167645	0.0838227	−	0.996481i	$-0.473287\pi$
$311$	2.95646	0.167645	0.0838227	+	0.996481i	$0.473287\pi$
$312$	0	0
$313$	−1.17523	−0.0664278	−0.0332139	−	0.999448i	$-0.510574\pi$
$313$	−1.17523	−0.0664278	−0.0332139	+	0.999448i	$0.510574\pi$
$314$	0	0
$315$	0.445042	0.0250753
$316$	0	0
$317$	− 17.4620i	− 0.980765i	−0.871507	−	0.490383i	$-0.836857\pi$
$317$	− 17.4620i	− 0.980765i	0.871507	−	0.490383i	$-0.163143\pi$
$318$	0	0
$319$	8.23490i	0.461066i
$320$	0	0
$321$	−10.8780	−0.607151
$322$	0	0
$323$	− 1.54288i	− 0.0858479i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	19.2271i	1.06326i
$328$	0	0
$329$	15.3545	0.846521
$330$	0	0
$331$	− 11.7289i	− 0.644676i	−0.946625	−	0.322338i	$-0.895531\pi$
$331$	− 11.7289i	− 0.644676i	0.946625	−	0.322338i	$-0.104469\pi$
$332$	0	0
$333$	− 0.740939i	− 0.0406032i
$334$	0	0
$335$	5.21983	0.285190
$336$	0	0
$337$	28.1879	1.53549	0.767746	−	0.640754i	$-0.221378\pi$
$337$	28.1879	1.53549	0.767746	+	0.640754i	$0.221378\pi$
$338$	0	0
$339$	31.7385	1.72380
$340$	0	0
$341$	−1.69202	−0.0916281
$342$	0	0
$343$	19.3763i	1.04622i
$344$	0	0
$345$	− 0.445042i	− 0.0239602i
$346$	0	0
$347$	4.92394	0.264331	0.132165	−	0.991228i	$-0.457807\pi$
$347$	4.92394	0.264331	0.132165	+	0.991228i	$0.457807\pi$
$348$	0	0
$349$	14.3220i	0.766638i	0.923616	+	0.383319i	$0.125219\pi$
$349$	14.3220i	0.766638i	−0.923616	+	0.383319i	$0.874781\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	4.57242i	0.243365i	0.992569	+	0.121683i	$0.0388290\pi$
$353$	4.57242i	0.243365i	−0.992569	+	0.121683i	$0.961171\pi$
$354$	0	0
$355$	−5.74094	−0.304697
$356$	0	0
$357$	2.96077i	0.156701i
$358$	0	0
$359$	29.0315i	1.53222i	0.642708	+	0.766111i	$0.277811\pi$
$359$	29.0315i	1.53222i	−0.642708	+	0.766111i	$0.722189\pi$
$360$	0	0
$361$	16.1371	0.849319
$362$	0	0
$363$	−15.4644	−0.811672
$364$	0	0
$365$	−14.5918	−0.763769
$366$	0	0
$367$	27.8297	1.45270	0.726349	−	0.687326i	$-0.241216\pi$
$367$	27.8297	1.45270	0.726349	+	0.687326i	$0.241216\pi$
$368$	0	0
$369$	1.71917i	0.0894963i
$370$	0	0
$371$	12.4819i	0.648027i
$372$	0	0
$373$	−16.1588	−0.836673	−0.418336	−	0.908292i	$-0.637387\pi$
$373$	−16.1588	−0.836673	−0.418336	+	0.908292i	$0.637387\pi$
$374$	0	0
$375$	− 1.80194i	− 0.0930517i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 4.05861i	− 0.208477i	−0.994552	−	0.104238i	$-0.966760\pi$
$379$	− 4.05861i	− 0.208477i	0.994552	−	0.104238i	$-0.0332405\pi$
$380$	0	0
$381$	−38.6775	−1.98151
$382$	0	0
$383$	− 12.9269i	− 0.660535i	−0.943887	−	0.330267i	$-0.892861\pi$
$383$	− 12.9269i	− 0.660535i	0.943887	−	0.330267i	$-0.107139\pi$
$384$	0	0
$385$	− 2.80194i	− 0.142800i
$386$	0	0
$387$	−0.560335	−0.0284834
$388$	0	0
$389$	−10.2784	−0.521138	−0.260569	−	0.965455i	$-0.583910\pi$
$389$	−10.2784	−0.521138	−0.260569	+	0.965455i	$0.583910\pi$
$390$	0	0
$391$	0.225209	0.0113893
$392$	0	0
$393$	−14.3230	−0.722502
$394$	0	0
$395$	− 7.75063i	− 0.389976i
$396$	0	0
$397$	− 15.0694i	− 0.756309i	−0.925742	−	0.378155i	$-0.876559\pi$
$397$	− 15.0694i	− 0.756309i	0.925742	−	0.378155i	$-0.123441\pi$
$398$	0	0
$399$	5.49396	0.275042
$400$	0	0
$401$	− 1.25129i	− 0.0624865i	−0.999512	−	0.0312433i	$-0.990053\pi$
$401$	− 1.25129i	− 0.0624865i	0.999512	−	0.0312433i	$-0.00994666\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	9.67994i	0.481000i
$406$	0	0
$407$	−4.66487	−0.231229
$408$	0	0
$409$	13.4849i	0.666783i	0.942788	+	0.333392i	$0.108193\pi$
$409$	13.4849i	0.666783i	−0.942788	+	0.333392i	$0.891807\pi$
$410$	0	0
$411$	− 2.50604i	− 0.123614i
$412$	0	0
$413$	−26.4916	−1.30356
$414$	0	0
$415$	4.94869	0.242922
$416$	0	0
$417$	−36.2204	−1.77372
$418$	0	0
$419$	−8.84846	−0.432276	−0.216138	−	0.976363i	$-0.569346\pi$
$419$	−8.84846	−0.432276	−0.216138	+	0.976363i	$0.569346\pi$
$420$	0	0
$421$	4.39373i	0.214137i	0.994252	+	0.107069i	$0.0341465\pi$
$421$	4.39373i	0.214137i	−0.994252	+	0.107069i	$0.965854\pi$
$422$	0	0
$423$	− 2.10454i	− 0.102326i
$424$	0	0
$425$	0.911854	0.0442314
$426$	0	0
$427$	− 17.2174i	− 0.833210i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 17.7235i	− 0.853710i	−0.904320	−	0.426855i	$-0.859621\pi$
$431$	− 17.7235i	− 0.853710i	0.904320	−	0.426855i	$-0.140379\pi$
$432$	0	0
$433$	7.81402	0.375518	0.187759	−	0.982215i	$-0.439878\pi$
$433$	7.81402	0.375518	0.187759	+	0.982215i	$0.439878\pi$
$434$	0	0
$435$	− 9.54288i	− 0.457546i
$436$	0	0
$437$	− 0.417895i	− 0.0199906i
$438$	0	0
$439$	5.64178	0.269267	0.134634	−	0.990895i	$-0.457014\pi$
$439$	5.64178	0.269267	0.134634	+	0.990895i	$0.457014\pi$
$440$	0	0
$441$	0.926919	0.0441390
$442$	0	0
$443$	26.3491	1.25188	0.625942	−	0.779869i	$-0.284715\pi$
$443$	26.3491	1.25188	0.625942	+	0.779869i	$0.284715\pi$
$444$	0	0
$445$	11.9095	0.564563
$446$	0	0
$447$	− 13.2271i	− 0.625622i
$448$	0	0
$449$	39.0941i	1.84497i	0.386039	+	0.922483i	$0.373843\pi$
$449$	39.0941i	1.84497i	−0.386039	+	0.922483i	$0.626157\pi$
$450$	0	0
$451$	10.8237	0.509669
$452$	0	0
$453$	15.2446i	0.716253i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	19.6025i	0.916968i	0.888703	+	0.458484i	$0.151607\pi$
$457$	19.6025i	0.916968i	−0.888703	+	0.458484i	$0.848393\pi$
$458$	0	0
$459$	−4.52350	−0.211139
$460$	0	0
$461$	40.5435i	1.88830i	0.329520	+	0.944149i	$0.393113\pi$
$461$	40.5435i	1.88830i	−0.329520	+	0.944149i	$0.606887\pi$
$462$	0	0
$463$	31.4959i	1.46374i	0.681446	+	0.731869i	$0.261352\pi$
$463$	31.4959i	1.46374i	−0.681446	+	0.731869i	$0.738648\pi$
$464$	0	0
$465$	1.96077	0.0909286
$466$	0	0
$467$	−35.6189	−1.64825	−0.824124	−	0.566409i	$-0.808332\pi$
$467$	−35.6189	−1.64825	−0.824124	+	0.566409i	$0.808332\pi$
$468$	0	0
$469$	−9.40581	−0.434320
$470$	0	0
$471$	32.0170	1.47526
$472$	0	0
$473$	3.52781i	0.162209i
$474$	0	0
$475$	− 1.69202i	− 0.0776353i
$476$	0	0
$477$	1.71081	0.0783325
$478$	0	0
$479$	0.698726i	0.0319256i	0.999873	+	0.0159628i	$0.00508134\pi$
$479$	0.698726i	0.0319256i	−0.999873	+	0.0159628i	$0.994919\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0.801938i	0.0364894i
$484$	0	0
$485$	−9.17390	−0.416565
$486$	0	0
$487$	− 9.49263i	− 0.430152i	−0.976597	−	0.215076i	$-0.931000\pi$
$487$	− 9.49263i	− 0.430152i	0.976597	−	0.215076i	$-0.0690000\pi$
$488$	0	0
$489$	20.9541i	0.947576i
$490$	0	0
$491$	−23.0261	−1.03915	−0.519576	−	0.854424i	$-0.673910\pi$
$491$	−23.0261	−1.03915	−0.519576	+	0.854424i	$0.673910\pi$
$492$	0	0
$493$	4.82908	0.217491
$494$	0	0
$495$	−0.384043	−0.0172614
$496$	0	0
$497$	10.3448	0.464028
$498$	0	0
$499$	15.5109i	0.694365i	0.937798	+	0.347183i	$0.112862\pi$
$499$	15.5109i	0.694365i	−0.937798	+	0.347183i	$0.887138\pi$
$500$	0	0
$501$	16.5821i	0.740834i
$502$	0	0
$503$	−27.8538	−1.24194	−0.620971	−	0.783834i	$-0.713261\pi$
$503$	−27.8538	−1.24194	−0.620971	+	0.783834i	$0.713261\pi$
$504$	0	0
$505$	− 2.18060i	− 0.0970356i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 20.6474i	− 0.915181i	−0.889163	−	0.457590i	$-0.848713\pi$
$509$	− 20.6474i	− 0.915181i	0.889163	−	0.457590i	$-0.151287\pi$
$510$	0	0
$511$	26.2935	1.16316
$512$	0	0
$513$	8.39373i	0.370592i
$514$	0	0
$515$	− 9.58211i	− 0.422238i
$516$	0	0
$517$	−13.2500	−0.582733
$518$	0	0
$519$	−27.8582	−1.22284
$520$	0	0
$521$	27.5700	1.20786	0.603932	−	0.797036i	$-0.293600\pi$
$521$	27.5700	1.20786	0.603932	+	0.797036i	$0.293600\pi$
$522$	0	0
$523$	14.2427	0.622788	0.311394	−	0.950281i	$-0.399204\pi$
$523$	14.2427	0.622788	0.311394	+	0.950281i	$0.399204\pi$
$524$	0	0
$525$	3.24698i	0.141710i
$526$	0	0
$527$	0.992230i	0.0432222i
$528$	0	0
$529$	−22.9390	−0.997348
$530$	0	0
$531$	3.63102i	0.157573i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	− 6.03684i	− 0.260995i
$536$	0	0
$537$	−37.9342	−1.63698
$538$	0	0
$539$	− 5.83579i	− 0.251365i
$540$	0	0
$541$	26.9667i	1.15939i	0.814833	+	0.579695i	$0.196828\pi$
$541$	26.9667i	1.15939i	−0.814833	+	0.579695i	$0.803172\pi$
$542$	0	0
$543$	−26.9584	−1.15689
$544$	0	0
$545$	−10.6703	−0.457063
$546$	0	0
$547$	−32.7294	−1.39941	−0.699705	−	0.714432i	$-0.746685\pi$
$547$	−32.7294	−1.39941	−0.699705	+	0.714432i	$0.746685\pi$
$548$	0	0
$549$	−2.35988	−0.100717
$550$	0	0
$551$	− 8.96077i	− 0.381742i
$552$	0	0
$553$	13.9661i	0.593901i
$554$	0	0
$555$	5.40581	0.229464
$556$	0	0
$557$	− 41.7603i	− 1.76944i	−0.466122	−	0.884721i	$-0.654349\pi$
$557$	− 41.7603i	− 1.76944i	0.466122	−	0.884721i	$-0.345651\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	− 2.55496i	− 0.107870i
$562$	0	0
$563$	−24.8834	−1.04871	−0.524355	−	0.851500i	$-0.675693\pi$
$563$	−24.8834	−1.04871	−0.524355	+	0.851500i	$0.675693\pi$
$564$	0	0
$565$	17.6136i	0.741008i
$566$	0	0
$567$	− 17.4426i	− 0.732522i
$568$	0	0
$569$	−28.2271	−1.18334	−0.591671	−	0.806179i	$-0.701532\pi$
$569$	−28.2271	−1.18334	−0.591671	+	0.806179i	$0.701532\pi$
$570$	0	0
$571$	−22.4316	−0.938735	−0.469367	−	0.883003i	$-0.655518\pi$
$571$	−22.4316	−0.938735	−0.469367	+	0.883003i	$0.655518\pi$
$572$	0	0
$573$	−5.68425	−0.237463
$574$	0	0
$575$	0.246980	0.0102998
$576$	0	0
$577$	− 36.8437i	− 1.53382i	−0.641753	−	0.766911i	$-0.721793\pi$
$577$	− 36.8437i	− 1.53382i	0.641753	−	0.766911i	$-0.278207\pi$
$578$	0	0
$579$	− 7.29590i	− 0.303207i
$580$	0	0
$581$	−8.91723	−0.369949
$582$	0	0
$583$	− 10.7711i	− 0.446092i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	44.4480i	1.83457i	0.398236	+	0.917283i	$0.369623\pi$
$587$	44.4480i	1.83457i	−0.398236	+	0.917283i	$0.630377\pi$
$588$	0	0
$589$	1.84117	0.0758639
$590$	0	0
$591$	− 9.38404i	− 0.386008i
$592$	0	0
$593$	− 20.6276i	− 0.847073i	−0.905879	−	0.423536i	$-0.860789\pi$
$593$	− 20.6276i	− 0.847073i	0.905879	−	0.423536i	$-0.139211\pi$
$594$	0	0
$595$	−1.64310	−0.0673607
$596$	0	0
$597$	−40.5459	−1.65943
$598$	0	0
$599$	31.2674	1.27755	0.638776	−	0.769393i	$-0.279441\pi$
$599$	31.2674	1.27755	0.638776	+	0.769393i	$0.279441\pi$
$600$	0	0
$601$	−7.94438	−0.324058	−0.162029	−	0.986786i	$-0.551804\pi$
$601$	−7.94438	−0.324058	−0.162029	+	0.986786i	$0.551804\pi$
$602$	0	0
$603$	1.28919i	0.0525000i
$604$	0	0
$605$	− 8.58211i	− 0.348912i
$606$	0	0
$607$	−3.97690	−0.161417	−0.0807087	−	0.996738i	$-0.525718\pi$
$607$	−3.97690	−0.161417	−0.0807087	+	0.996738i	$0.525718\pi$
$608$	0	0
$609$	17.1957i	0.696804i
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 36.0062i	− 1.45428i	−0.686490	−	0.727139i	$-0.740850\pi$
$613$	− 36.0062i	− 1.45428i	0.686490	−	0.727139i	$-0.259150\pi$
$614$	0	0
$615$	−12.5429	−0.505778
$616$	0	0
$617$	3.14808i	0.126737i	0.997990	+	0.0633685i	$0.0201843\pi$
$617$	3.14808i	0.126737i	−0.997990	+	0.0633685i	$0.979816\pi$
$618$	0	0
$619$	24.9594i	1.00320i	0.865098	+	0.501602i	$0.167256\pi$
$619$	24.9594i	1.00320i	−0.865098	+	0.501602i	$0.832744\pi$
$620$	0	0
$621$	−1.22521	−0.0491660
$622$	0	0
$623$	−21.4601	−0.859781
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−4.74094	−0.189335
$628$	0	0
$629$	2.73556i	0.109074i
$630$	0	0
$631$	− 1.34960i	− 0.0537267i	−0.999639	−	0.0268634i	$-0.991448\pi$
$631$	− 1.34960i	− 0.0537267i	0.999639	−	0.0268634i	$-0.00855190\pi$
$632$	0	0
$633$	0.0784573	0.00311840
$634$	0	0
$635$	− 21.4644i	− 0.851789i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	− 1.41789i	− 0.0560911i
$640$	0	0
$641$	−30.7808	−1.21577	−0.607883	−	0.794026i	$-0.707981\pi$
$641$	−30.7808	−1.21577	−0.607883	+	0.794026i	$0.707981\pi$
$642$	0	0
$643$	0.528402i	0.0208381i	0.999946	+	0.0104191i	$0.00331655\pi$
$643$	0.528402i	0.0208381i	−0.999946	+	0.0104191i	$0.996683\pi$
$644$	0	0
$645$	− 4.08815i	− 0.160971i
$646$	0	0
$647$	12.7114	0.499737	0.249868	−	0.968280i	$-0.419613\pi$
$647$	12.7114	0.499737	0.249868	+	0.968280i	$0.419613\pi$
$648$	0	0
$649$	22.8605	0.897355
$650$	0	0
$651$	−3.53319	−0.138477
$652$	0	0
$653$	16.2241	0.634900	0.317450	−	0.948275i	$-0.397173\pi$
$653$	16.2241	0.634900	0.317450	+	0.948275i	$0.397173\pi$
$654$	0	0
$655$	− 7.94869i	− 0.310581i
$656$	0	0
$657$	− 3.60388i	− 0.140601i
$658$	0	0
$659$	−36.5478	−1.42370	−0.711850	−	0.702332i	$-0.752142\pi$
$659$	−36.5478	−1.42370	−0.711850	+	0.702332i	$0.752142\pi$
$660$	0	0
$661$	7.94810i	0.309145i	0.987981	+	0.154573i	$0.0494001\pi$
$661$	7.94810i	0.309145i	−0.987981	+	0.154573i	$0.950600\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	3.04892i	0.118232i
$666$	0	0
$667$	1.30798	0.0506451
$668$	0	0
$669$	28.9952i	1.12102i
$670$	0	0
$671$	14.8576i	0.573570i
$672$	0	0
$673$	10.0771	0.388445	0.194222	−	0.980958i	$-0.437782\pi$
$673$	10.0771	0.388445	0.194222	+	0.980958i	$0.437782\pi$
$674$	0	0
$675$	−4.96077	−0.190940
$676$	0	0
$677$	−37.3236	−1.43446	−0.717232	−	0.696835i	$-0.754591\pi$
$677$	−37.3236	−1.43446	−0.717232	+	0.696835i	$0.754591\pi$
$678$	0	0
$679$	16.5308	0.634394
$680$	0	0
$681$	1.95646i	0.0749717i
$682$	0	0
$683$	− 10.8237i	− 0.414158i	−0.978324	−	0.207079i	$-0.933604\pi$
$683$	− 10.8237i	− 0.414158i	0.978324	−	0.207079i	$-0.0663957\pi$
$684$	0	0
$685$	1.39075	0.0531377
$686$	0	0
$687$	− 45.5206i	− 1.73672i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	− 24.4838i	− 0.931408i	−0.884941	−	0.465704i	$-0.845801\pi$
$691$	− 24.4838i	− 0.931408i	0.884941	−	0.465704i	$-0.154199\pi$
$692$	0	0
$693$	0.692021	0.0262877
$694$	0	0
$695$	− 20.1008i	− 0.762468i
$696$	0	0
$697$	− 6.34721i	− 0.240418i
$698$	0	0
$699$	−43.2247	−1.63491
$700$	0	0
$701$	−30.9554	−1.16917	−0.584585	−	0.811333i	$-0.698742\pi$
$701$	−30.9554	−1.16917	−0.584585	+	0.811333i	$0.698742\pi$
$702$	0	0
$703$	5.07606	0.191447
$704$	0	0
$705$	15.3545	0.578284
$706$	0	0
$707$	3.92931i	0.147777i
$708$	0	0
$709$	− 21.6396i	− 0.812694i	−0.913719	−	0.406347i	$-0.866802\pi$
$709$	− 21.6396i	− 0.812694i	0.913719	−	0.406347i	$-0.133198\pi$
$710$	0	0
$711$	1.91425	0.0717899
$712$	0	0
$713$	0.268750i	0.0100648i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 7.00538i	− 0.261621i
$718$	0	0
$719$	11.2556	0.419763	0.209882	−	0.977727i	$-0.432692\pi$
$719$	11.2556	0.419763	0.209882	+	0.977727i	$0.432692\pi$
$720$	0	0
$721$	17.2664i	0.643033i
$722$	0	0
$723$	44.3086i	1.64785i
$724$	0	0
$725$	5.29590	0.196685
$726$	0	0
$727$	−18.6485	−0.691634	−0.345817	−	0.938302i	$-0.612398\pi$
$727$	−18.6485	−0.691634	−0.345817	+	0.938302i	$0.612398\pi$
$728$	0	0
$729$	24.4263	0.904676
$730$	0	0
$731$	2.06877	0.0765162
$732$	0	0
$733$	5.01639i	0.185285i	0.995699	+	0.0926424i	$0.0295313\pi$
$733$	5.01639i	0.185285i	−0.995699	+	0.0926424i	$0.970469\pi$
$734$	0	0
$735$	6.76271i	0.249446i
$736$	0	0
$737$	8.11662	0.298980
$738$	0	0
$739$	26.4276i	0.972154i	0.873916	+	0.486077i	$0.161573\pi$
$739$	26.4276i	0.972154i	−0.873916	+	0.486077i	$0.838427\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	2.25129i	0.0825919i	0.999147	+	0.0412959i	$0.0131486\pi$
$743$	2.25129i	0.0825919i	−0.999147	+	0.0412959i	$0.986851\pi$
$744$	0	0
$745$	7.34050	0.268935
$746$	0	0
$747$	1.22223i	0.0447189i
$748$	0	0
$749$	10.8780i	0.397474i
$750$	0	0
$751$	−29.7375	−1.08514	−0.542568	−	0.840012i	$-0.682548\pi$
$751$	−29.7375	−1.08514	−0.542568	+	0.840012i	$0.682548\pi$
$752$	0	0
$753$	−26.1444	−0.952753
$754$	0	0
$755$	−8.46011	−0.307895
$756$	0	0
$757$	24.9764	0.907784	0.453892	−	0.891057i	$-0.350035\pi$
$757$	24.9764	0.907784	0.453892	+	0.891057i	$0.350035\pi$
$758$	0	0
$759$	− 0.692021i	− 0.0251188i
$760$	0	0
$761$	− 7.37090i	− 0.267195i	−0.991036	−	0.133597i	$-0.957347\pi$
$761$	− 7.37090i	− 0.267195i	0.991036	−	0.133597i	$-0.0426529\pi$
$762$	0	0
$763$	19.2271	0.696069
$764$	0	0
$765$	0.225209i	0.00814246i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	− 55.2780i	− 1.99337i	−0.0813304	−	0.996687i	$-0.525917\pi$
$769$	− 55.2780i	− 1.99337i	0.0813304	−	0.996687i	$-0.474083\pi$
$770$	0	0
$771$	−9.36467	−0.337260
$772$	0	0
$773$	54.5120i	1.96066i	0.197364	+	0.980330i	$0.436762\pi$
$773$	54.5120i	1.96066i	−0.197364	+	0.980330i	$0.563238\pi$
$774$	0	0
$775$	1.08815i	0.0390874i
$776$	0	0
$777$	−9.74094	−0.349454
$778$	0	0
$779$	−11.7778	−0.421983
$780$	0	0
$781$	−8.92692	−0.319430
$782$	0	0
$783$	−26.2717	−0.938875
$784$	0	0
$785$	17.7681i	0.634170i
$786$	0	0
$787$	− 54.7493i	− 1.95160i	−0.218661	−	0.975801i	$-0.570169\pi$
$787$	− 54.7493i	− 1.95160i	0.218661	−	0.975801i	$-0.429831\pi$
$788$	0	0
$789$	−14.1685	−0.504413
$790$	0	0
$791$	− 31.7385i	− 1.12849i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	12.4819i	0.442687i
$796$	0	0
$797$	−33.9982	−1.20428	−0.602139	−	0.798391i	$-0.705685\pi$
$797$	−33.9982	−1.20428	−0.602139	+	0.798391i	$0.705685\pi$
$798$	0	0
$799$	7.77000i	0.274883i
$800$	0	0
$801$	2.94139i	0.103929i
$802$	0	0
$803$	−22.6896	−0.800700
$804$	0	0
$805$	−0.445042	−0.0156857
$806$	0	0
$807$	−12.2838	−0.432411
$808$	0	0
$809$	−11.3532	−0.399156	−0.199578	−	0.979882i	$-0.563957\pi$
$809$	−11.3532	−0.399156	−0.199578	+	0.979882i	$0.563957\pi$
$810$	0	0
$811$	32.6082i	1.14503i	0.819895	+	0.572514i	$0.194032\pi$
$811$	32.6082i	1.14503i	−0.819895	+	0.572514i	$0.805968\pi$
$812$	0	0
$813$	29.2325i	1.02523i
$814$	0	0
$815$	−11.6286	−0.407333
$816$	0	0
$817$	− 3.83877i	− 0.134302i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 13.6377i	− 0.475960i	−0.971270	−	0.237980i	$-0.923515\pi$
$821$	− 13.6377i	− 0.475960i	0.971270	−	0.237980i	$-0.0764853\pi$
$822$	0	0
$823$	33.4077	1.16452	0.582260	−	0.813003i	$-0.302169\pi$
$823$	33.4077	1.16452	0.582260	+	0.813003i	$0.302169\pi$
$824$	0	0
$825$	− 2.80194i	− 0.0975510i
$826$	0	0
$827$	13.9801i	0.486137i	0.970009	+	0.243069i	$0.0781540\pi$
$827$	13.9801i	0.486137i	−0.970009	+	0.243069i	$0.921846\pi$
$828$	0	0
$829$	−36.8528	−1.27995	−0.639975	−	0.768396i	$-0.721055\pi$
$829$	−36.8528	−1.27995	−0.639975	+	0.768396i	$0.721055\pi$
$830$	0	0
$831$	3.88577	0.134796
$832$	0	0
$833$	−3.42221	−0.118572
$834$	0	0
$835$	−9.20237	−0.318461
$836$	0	0
$837$	− 5.39804i	− 0.186584i
$838$	0	0
$839$	− 3.58775i	− 0.123863i	−0.998080	−	0.0619314i	$-0.980274\pi$
$839$	− 3.58775i	− 0.123863i	0.998080	−	0.0619314i	$-0.0197260\pi$
$840$	0	0
$841$	−0.953476	−0.0328785
$842$	0	0
$843$	19.3230i	0.665521i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	15.4644i	0.531364i
$848$	0	0
$849$	−9.92692	−0.340691
$850$	0	0
$851$	0.740939i	0.0253991i
$852$	0	0
$853$	47.2723i	1.61857i	0.587414	+	0.809287i	$0.300146\pi$
$853$	47.2723i	1.61857i	−0.587414	+	0.809287i	$0.699854\pi$
$854$	0	0
$855$	0.417895	0.0142917
$856$	0	0
$857$	20.5435	0.701751	0.350876	−	0.936422i	$-0.385884\pi$
$857$	20.5435	0.701751	0.350876	+	0.936422i	$0.385884\pi$
$858$	0	0
$859$	20.0597	0.684427	0.342214	−	0.939622i	$-0.388823\pi$
$859$	20.0597	0.684427	0.342214	+	0.939622i	$0.388823\pi$
$860$	0	0
$861$	22.6015	0.770256
$862$	0	0
$863$	− 43.7103i	− 1.48792i	−0.668226	−	0.743959i	$-0.732946\pi$
$863$	− 43.7103i	− 1.48792i	0.668226	−	0.743959i	$-0.267054\pi$
$864$	0	0
$865$	− 15.4601i	− 0.525659i
$866$	0	0
$867$	29.1347	0.989465
$868$	0	0
$869$	− 12.0519i	− 0.408833i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	− 2.26577i	− 0.0766846i
$874$	0	0
$875$	−1.80194	−0.0609166
$876$	0	0
$877$	− 23.6655i	− 0.799126i	−0.916706	−	0.399563i	$-0.869162\pi$
$877$	− 23.6655i	− 0.799126i	0.916706	−	0.399563i	$-0.130838\pi$
$878$	0	0
$879$	3.25906i	0.109925i
$880$	0	0
$881$	33.5743	1.13115	0.565574	−	0.824698i	$-0.308655\pi$
$881$	33.5743	1.13115	0.565574	+	0.824698i	$0.308655\pi$
$882$	0	0
$883$	8.10513	0.272759	0.136380	−	0.990657i	$-0.456453\pi$
$883$	8.10513	0.272759	0.136380	+	0.990657i	$0.456453\pi$
$884$	0	0
$885$	−26.4916	−0.890504
$886$	0	0
$887$	42.9687	1.44275	0.721373	−	0.692547i	$-0.243511\pi$
$887$	42.9687	1.44275	0.721373	+	0.692547i	$0.243511\pi$
$888$	0	0
$889$	38.6775i	1.29720i
$890$	0	0
$891$	15.0519i	0.504258i
$892$	0	0
$893$	14.4179	0.482476
$894$	0	0
$895$	− 21.0519i	− 0.703687i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	5.76271i	0.192197i
$900$	0	0
$901$	−6.31634	−0.210428
$902$	0	0
$903$	7.36658i	0.245145i
$904$	0	0
$905$	− 14.9608i	− 0.497313i
$906$	0	0
$907$	26.9536	0.894979	0.447490	−	0.894289i	$-0.352318\pi$
$907$	26.9536	0.894979	0.447490	+	0.894289i	$0.352318\pi$
$908$	0	0
$909$	0.538565	0.0178631
$910$	0	0
$911$	−30.1909	−1.00027	−0.500134	−	0.865948i	$-0.666716\pi$
$911$	−30.1909	−1.00027	−0.500134	+	0.865948i	$0.666716\pi$
$912$	0	0
$913$	7.69501	0.254668
$914$	0	0
$915$	− 17.2174i	− 0.569191i
$916$	0	0
$917$	14.3230i	0.472989i
$918$	0	0
$919$	−28.9336	−0.954432	−0.477216	−	0.878786i	$-0.658354\pi$
$919$	−28.9336	−0.954432	−0.477216	+	0.878786i	$0.658354\pi$
$920$	0	0
$921$	19.7157i	0.649655i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	3.00000i	0.0986394i
$926$	0	0
$927$	2.36658	0.0777288
$928$	0	0
$929$	− 9.34854i	− 0.306715i	−0.988171	−	0.153358i	$-0.950991\pi$
$929$	− 9.34854i	− 0.306715i	0.988171	−	0.153358i	$-0.0490087\pi$
$930$	0	0
$931$	6.35019i	0.208119i
$932$	0	0
$933$	−5.32736	−0.174410
$934$	0	0
$935$	1.41789	0.0463701
$936$	0	0
$937$	−40.7808	−1.33225	−0.666125	−	0.745840i	$-0.732048\pi$
$937$	−40.7808	−1.33225	−0.666125	+	0.745840i	$0.732048\pi$
$938$	0	0
$939$	2.11769	0.0691081
$940$	0	0
$941$	− 37.4161i	− 1.21973i	−0.792505	−	0.609865i	$-0.791224\pi$
$941$	− 37.4161i	− 1.21973i	0.792505	−	0.609865i	$-0.208776\pi$
$942$	0	0
$943$	− 1.71917i	− 0.0559838i
$944$	0	0
$945$	8.93900	0.290786
$946$	0	0
$947$	12.4612i	0.404934i	0.979289	+	0.202467i	$0.0648958\pi$
$947$	12.4612i	0.404934i	−0.979289	+	0.202467i	$0.935104\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	31.4655i	1.02034i
$952$	0	0
$953$	6.66978	0.216055	0.108028	−	0.994148i	$-0.465546\pi$
$953$	6.66978	0.216055	0.108028	+	0.994148i	$0.465546\pi$
$954$	0	0
$955$	− 3.15452i	− 0.102078i
$956$	0	0
$957$	− 14.8388i	− 0.479669i
$958$	0	0
$959$	−2.50604	−0.0809243
$960$	0	0
$961$	29.8159	0.961804
$962$	0	0
$963$	1.49098	0.0480460
$964$	0	0
$965$	4.04892	0.130339
$966$	0	0
$967$	45.9584i	1.47792i	0.673748	+	0.738961i	$0.264683\pi$
$967$	45.9584i	1.47792i	−0.673748	+	0.738961i	$0.735317\pi$
$968$	0	0
$969$	2.78017i	0.0893118i
$970$	0	0
$971$	45.3991	1.45693	0.728463	−	0.685085i	$-0.240235\pi$
$971$	45.3991	1.45693	0.728463	+	0.685085i	$0.240235\pi$
$972$	0	0
$973$	36.2204i	1.16117i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 12.2174i	− 0.390871i	−0.980717	−	0.195435i	$-0.937388\pi$
$977$	− 12.2174i	− 0.390871i	0.980717	−	0.195435i	$-0.0626120\pi$
$978$	0	0
$979$	18.5187	0.591861
$980$	0	0
$981$	− 2.63533i	− 0.0841398i
$982$	0	0
$983$	52.9081i	1.68751i	0.536731	+	0.843754i	$0.319659\pi$
$983$	52.9081i	1.68751i	−0.536731	+	0.843754i	$0.680341\pi$
$984$	0	0
$985$	5.20775	0.165933
$986$	0	0
$987$	−27.6679	−0.880678
$988$	0	0
$989$	0.560335	0.0178176
$990$	0	0
$991$	−11.9011	−0.378051	−0.189025	−	0.981972i	$-0.560533\pi$
$991$	−11.9011	−0.378051	−0.189025	+	0.981972i	$0.560533\pi$
$992$	0	0
$993$	21.1347i	0.670688i
$994$	0	0
$995$	− 22.5013i	− 0.713338i
$996$	0	0
$997$	32.5937	1.03225	0.516127	−	0.856512i	$-0.327373\pi$
$997$	32.5937	1.03225	0.516127	+	0.856512i	$0.327373\pi$
$998$	0	0
$999$	− 14.8823i	− 0.470856i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3380.2.f.g.3041.1		6
13.5	odd	4		3380.2.a.k.1.1	✓	3
13.8	odd	4		3380.2.a.l.1.1	yes	3
13.12	even	2	inner	3380.2.f.g.3041.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3380.2.a.k.1.1	✓	3	13.5	odd	4
3380.2.a.l.1.1	yes	3	13.8	odd	4
3380.2.f.g.3041.1		6	1.1	even	1	trivial
3380.2.f.g.3041.2		6	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)

\(f(q)\)	\(=\)	\(q-1.80194 q^{3} -1.00000i q^{5} +1.80194i q^{7} +0.246980 q^{9} +O(q^{10})\)	\(q-1.80194 q^{3} -1.00000i q^{5} +1.80194i q^{7} +0.246980 q^{9} -1.55496i q^{11} +1.80194i q^{15} -0.911854 q^{17} +1.69202i q^{19} -3.24698i q^{21} -0.246980 q^{23} -1.00000 q^{25} +4.96077 q^{27} -5.29590 q^{29} -1.08815i q^{31} +2.80194i q^{33} +1.80194 q^{35} -3.00000i q^{37} +6.96077i q^{41} -2.26875 q^{43} -0.246980i q^{45} -8.52111i q^{47} +3.75302 q^{49} +1.64310 q^{51} +6.92692 q^{53} -1.55496 q^{55} -3.04892i q^{57} +14.7017i q^{59} -9.55496 q^{61} +0.445042i q^{63} +5.21983i q^{67} +0.445042 q^{69} -5.74094i q^{71} -14.5918i q^{73} +1.80194 q^{75} +2.80194 q^{77} +7.75063 q^{79} -9.67994 q^{81} +4.94869i q^{83} +0.911854i q^{85} +9.54288 q^{87} +11.9095i q^{89} +1.96077i q^{93} +1.69202 q^{95} -9.17390i q^{97} -0.384043i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{3} - 8 q^{9}+O(q^{10}) \) 6 * q - 2 * q^3 - 8 * q^9	\( 6 q - 2 q^{3} - 8 q^{9} + 2 q^{17} + 8 q^{23} - 6 q^{25} + 4 q^{27} - 4 q^{29} + 2 q^{35} + 2 q^{43} + 32 q^{49} + 18 q^{51} - 16 q^{53} - 10 q^{55} - 58 q^{61} + 2 q^{69} + 2 q^{75} + 8 q^{77} - 26 q^{79} - 10 q^{81} + 20 q^{87}+O(q^{100}) \) 6 * q - 2 * q^3 - 8 * q^9 + 2 * q^17 + 8 * q^23 - 6 * q^25 + 4 * q^27 - 4 * q^29 + 2 * q^35 + 2 * q^43 + 32 * q^49 + 18 * q^51 - 16 * q^53 - 10 * q^55 - 58 * q^61 + 2 * q^69 + 2 * q^75 + 8 * q^77 - 26 * q^79 - 10 * q^81 + 20 * q^87

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more