LMFDB - Embedded newform 3380.2.f.i.3041.2

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:	$8$
Coefficient field:	8.0.22581504.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 $ x^8 - 4x^7 + 5x^6 + 2x^5 - 11x^4 + 4x^3 + 20x^2 - 32*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 260)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3041.2
Root			$1.40994 + 0.109843i$ of defining polynomial
Character	$\chi$	$=$	3380.3041
Dual form			3380.2.f.i.3041.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.33225 q^{3} +1.00000i q^{5} -0.399804i q^{7} +2.43937 q^{9} +O(q^{10})$	$q-2.33225 q^{3} +1.00000i q^{5} -0.399804i q^{7} +2.43937 q^{9} +1.73205i q^{11} -2.33225i q^{15} -0.692481 q^{17} +5.37182i q^{19} +0.932442i q^{21} +0.107127 q^{23} -1.00000 q^{25} +1.30752 q^{27} -4.90348 q^{29} -7.86488i q^{31} -4.03957i q^{33} +0.399804 q^{35} +2.26469i q^{37} +7.73205i q^{41} -6.01060 q^{43} +2.43937i q^{45} +3.46410i q^{47} +6.84016 q^{49} +1.61504 q^{51} +11.7189 q^{53} -1.73205 q^{55} -12.5284i q^{57} +7.27529i q^{59} +8.68922 q^{61} -0.975272i q^{63} -1.32801i q^{67} -0.249847 q^{69} +3.87803i q^{71} -10.2251i q^{73} +2.33225 q^{75} +0.692481 q^{77} -13.1533 q^{79} -10.3676 q^{81} -14.0791i q^{83} -0.692481i q^{85} +11.4361 q^{87} +0.347088i q^{89} +18.3428i q^{93} -5.37182 q^{95} +8.85004i q^{97} +4.22512i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{3} + 8 q^{9}+O(q^{10}) $ 8 * q + 4 * q^3 + 8 * q^9	$ 8 q + 4 q^{3} + 8 q^{9} - 12 q^{17} + 12 q^{23} - 8 q^{25} + 4 q^{27} + 12 q^{35} - 20 q^{43} + 8 q^{49} + 24 q^{53} + 8 q^{61} + 48 q^{69} - 4 q^{75} + 12 q^{77} - 16 q^{79} - 16 q^{81} + 12 q^{87}+O(q^{100}) $ 8 * q + 4 * q^3 + 8 * q^9 - 12 * q^17 + 12 * q^23 - 8 * q^25 + 4 * q^27 + 12 * q^35 - 20 * q^43 + 8 * q^49 + 24 * q^53 + 8 * q^61 + 48 * q^69 - 4 * q^75 + 12 * q^77 - 16 * q^79 - 16 * q^81 + 12 * 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
$3$	−2.33225	−1.34652	−0.673262	−	0.739404i	$-0.735107\pi$
$3$	−2.33225	−1.34652	−0.673262	+	0.739404i	$0.735107\pi$
$4$	0	0
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	0	0
$7$	− 0.399804i	− 0.151112i	−0.997142	−	0.0755559i	$-0.975927\pi$
$7$	− 0.399804i	− 0.151112i	0.997142	−	0.0755559i	$-0.0240731\pi$
$8$	0	0
$9$	2.43937	0.813125
$10$	0	0
$11$	1.73205i	0.522233i	0.965307	+	0.261116i	$0.0840907\pi$
$11$	1.73205i	0.522233i	−0.965307	+	0.261116i	$0.915909\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	− 2.33225i	− 0.602183i
$16$	0	0
$17$	−0.692481	−0.167951	−0.0839757	−	0.996468i	$-0.526762\pi$
$17$	−0.692481	−0.167951	−0.0839757	+	0.996468i	$0.526762\pi$
$18$	0	0
$19$	5.37182i	1.23238i	0.787598	+	0.616190i	$0.211324\pi$
$19$	5.37182i	1.23238i	−0.787598	+	0.616190i	$0.788676\pi$
$20$	0	0
$21$	0.932442i	0.203476i
$22$	0	0
$23$	0.107127	0.0223376	0.0111688	−	0.999938i	$-0.496445\pi$
$23$	0.107127	0.0223376	0.0111688	+	0.999938i	$0.496445\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	1.30752	0.251632
$28$	0	0
$29$	−4.90348	−0.910553	−0.455276	−	0.890350i	$-0.650460\pi$
$29$	−4.90348	−0.910553	−0.455276	+	0.890350i	$0.650460\pi$
$30$	0	0
$31$	− 7.86488i	− 1.41257i	−0.707925	−	0.706287i	$-0.750369\pi$
$31$	− 7.86488i	− 1.41257i	0.707925	−	0.706287i	$-0.249631\pi$
$32$	0	0
$33$	− 4.03957i	− 0.703199i
$34$	0	0
$35$	0.399804	0.0675793
$36$	0	0
$37$	2.26469i	0.372313i	0.982520	+	0.186156i	$0.0596031\pi$
$37$	2.26469i	0.372313i	−0.982520	+	0.186156i	$0.940397\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7.73205i	1.20754i	0.797157	+	0.603772i	$0.206336\pi$
$41$	7.73205i	1.20754i	−0.797157	+	0.603772i	$0.793664\pi$
$42$	0	0
$43$	−6.01060	−0.916608	−0.458304	−	0.888795i	$-0.651543\pi$
$43$	−6.01060	−0.916608	−0.458304	+	0.888795i	$0.651543\pi$
$44$	0	0
$45$	2.43937i	0.363640i
$46$	0	0
$47$	3.46410i	0.505291i	0.967559	+	0.252646i	$0.0813007\pi$
$47$	3.46410i	0.505291i	−0.967559	+	0.252646i	$0.918699\pi$
$48$	0	0
$49$	6.84016	0.977165
$50$	0	0
$51$	1.61504	0.226150
$52$	0	0
$53$	11.7189	1.60972	0.804858	−	0.593468i	$-0.202242\pi$
$53$	11.7189	1.60972	0.804858	+	0.593468i	$0.202242\pi$
$54$	0	0
$55$	−1.73205	−0.233550
$56$	0	0
$57$	− 12.5284i	− 1.65943i
$58$	0	0
$59$	7.27529i	0.947162i	0.880750	+	0.473581i	$0.157039\pi$
$59$	7.27529i	0.947162i	−0.880750	+	0.473581i	$0.842961\pi$
$60$	0	0
$61$	8.68922	1.11254	0.556270	−	0.831001i	$-0.312232\pi$
$61$	8.68922	1.11254	0.556270	+	0.831001i	$0.312232\pi$
$62$	0	0
$63$	− 0.975272i	− 0.122873i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 1.32801i	− 0.162242i	−0.996704	−	0.0811210i	$-0.974150\pi$
$67$	− 1.32801i	− 0.162242i	0.996704	−	0.0811210i	$-0.0258500\pi$
$68$	0	0
$69$	−0.249847	−0.0300781
$70$	0	0
$71$	3.87803i	0.460238i	0.973163	+	0.230119i	$0.0739115\pi$
$71$	3.87803i	0.460238i	−0.973163	+	0.230119i	$0.926089\pi$
$72$	0	0
$73$	− 10.2251i	− 1.19676i	−0.801213	−	0.598380i	$-0.795811\pi$
$73$	− 10.2251i	− 1.19676i	0.801213	−	0.598380i	$-0.204189\pi$
$74$	0	0
$75$	2.33225	0.269305
$76$	0	0
$77$	0.692481	0.0789156
$78$	0	0
$79$	−13.1533	−1.47986	−0.739932	−	0.672681i	$-0.765142\pi$
$79$	−13.1533	−1.47986	−0.739932	+	0.672681i	$0.765142\pi$
$80$	0	0
$81$	−10.3676	−1.15195
$82$	0	0
$83$	− 14.0791i	− 1.54539i	−0.634780	−	0.772693i	$-0.718909\pi$
$83$	− 14.0791i	− 1.54539i	0.634780	−	0.772693i	$-0.281091\pi$
$84$	0	0
$85$	− 0.692481i	− 0.0751102i
$86$	0	0
$87$	11.4361	1.22608
$88$	0	0
$89$	0.347088i	0.0367913i	0.999831	+	0.0183956i	$0.00585584\pi$
$89$	0.347088i	0.0367913i	−0.999831	+	0.0183956i	$0.994144\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	18.3428i	1.90206i
$94$	0	0
$95$	−5.37182	−0.551137
$96$	0	0
$97$	8.85004i	0.898586i	0.893385	+	0.449293i	$0.148324\pi$
$97$	8.85004i	0.898586i	−0.893385	+	0.449293i	$0.851676\pi$
$98$	0	0
$99$	4.22512i	0.424640i
$100$	0	0
$101$	−4.10387	−0.408350	−0.204175	−	0.978934i	$-0.565451\pi$
$101$	−4.10387	−0.408350	−0.204175	+	0.978934i	$0.565451\pi$
$102$	0	0
$103$	−11.2325	−1.10677	−0.553384	−	0.832926i	$-0.686664\pi$
$103$	−11.2325	−1.10677	−0.553384	+	0.832926i	$0.686664\pi$
$104$	0	0
$105$	−0.932442	−0.0909970
$106$	0	0
$107$	−17.6033	−1.70178	−0.850888	−	0.525348i	$-0.823935\pi$
$107$	−17.6033	−1.70178	−0.850888	+	0.525348i	$0.823935\pi$
$108$	0	0
$109$	15.1830i	1.45427i	0.686495	+	0.727134i	$0.259148\pi$
$109$	15.1830i	1.45427i	−0.686495	+	0.727134i	$0.740852\pi$
$110$	0	0
$111$	− 5.28181i	− 0.501327i
$112$	0	0
$113$	8.90021	0.837262	0.418631	−	0.908156i	$-0.362510\pi$
$113$	8.90021	0.837262	0.418631	+	0.908156i	$0.362510\pi$
$114$	0	0
$115$	0.107127i	0.00998967i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0.276857i	0.0253794i
$120$	0	0
$121$	8.00000	0.727273
$122$	0	0
$123$	− 18.0330i	− 1.62599i
$124$	0	0
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	−12.1566	−1.07872	−0.539361	−	0.842075i	$-0.681334\pi$
$127$	−12.1566	−1.07872	−0.539361	+	0.842075i	$0.681334\pi$
$128$	0	0
$129$	14.0182	1.23423
$130$	0	0
$131$	−2.11773	−0.185027	−0.0925135	−	0.995711i	$-0.529490\pi$
$131$	−2.11773	−0.185027	−0.0925135	+	0.995711i	$0.529490\pi$
$132$	0	0
$133$	2.14768	0.186227
$134$	0	0
$135$	1.30752i	0.112533i
$136$	0	0
$137$	− 18.0330i	− 1.54067i	−0.637641	−	0.770334i	$-0.720090\pi$
$137$	− 18.0330i	− 1.54067i	0.637641	−	0.770334i	$-0.279910\pi$
$138$	0	0
$139$	−9.84016	−0.834631	−0.417316	−	0.908762i	$-0.637029\pi$
$139$	−9.84016	−0.834631	−0.417316	+	0.908762i	$0.637029\pi$
$140$	0	0
$141$	− 8.07914i	− 0.680386i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	− 4.90348i	− 0.407211i
$146$	0	0
$147$	−15.9529	−1.31578
$148$	0	0
$149$	8.88299i	0.727723i	0.931453	+	0.363861i	$0.118542\pi$
$149$	8.88299i	0.727723i	−0.931453	+	0.363861i	$0.881458\pi$
$150$	0	0
$151$	− 4.43937i	− 0.361271i	−0.983550	−	0.180636i	$-0.942185\pi$
$151$	− 4.43937i	− 0.361271i	0.983550	−	0.180636i	$-0.0578155\pi$
$152$	0	0
$153$	−1.68922	−0.136565
$154$	0	0
$155$	7.86488	0.631723
$156$	0	0
$157$	−4.16719	−0.332578	−0.166289	−	0.986077i	$-0.553178\pi$
$157$	−4.16719	−0.332578	−0.166289	+	0.986077i	$0.553178\pi$
$158$	0	0
$159$	−27.3314	−2.16752
$160$	0	0
$161$	− 0.0428299i	− 0.00337547i
$162$	0	0
$163$	− 3.69672i	− 0.289549i	−0.989465	−	0.144775i	$-0.953754\pi$
$163$	− 3.69672i	− 0.289549i	0.989465	−	0.144775i	$-0.0462458\pi$
$164$	0	0
$165$	4.03957	0.314480
$166$	0	0
$167$	− 21.4972i	− 1.66350i	−0.555151	−	0.831750i	$-0.687339\pi$
$167$	− 21.4972i	− 1.66350i	0.555151	−	0.831750i	$-0.312661\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	13.1039i	1.00208i
$172$	0	0
$173$	11.6118	0.882827	0.441414	−	0.897304i	$-0.354477\pi$
$173$	11.6118	0.882827	0.441414	+	0.897304i	$0.354477\pi$
$174$	0	0
$175$	0.399804i	0.0302224i
$176$	0	0
$177$	− 16.9678i	− 1.27538i
$178$	0	0
$179$	4.97032	0.371499	0.185749	−	0.982597i	$-0.440529\pi$
$179$	4.97032	0.371499	0.185749	+	0.982597i	$0.440529\pi$
$180$	0	0
$181$	−17.3695	−1.29107	−0.645534	−	0.763732i	$-0.723365\pi$
$181$	−17.3695	−1.29107	−0.645534	+	0.763732i	$0.723365\pi$
$182$	0	0
$183$	−20.2654	−1.49806
$184$	0	0
$185$	−2.26469	−0.166503
$186$	0	0
$187$	− 1.19941i	− 0.0877098i
$188$	0	0
$189$	− 0.522752i	− 0.0380246i
$190$	0	0
$191$	−20.5046	−1.48366	−0.741832	−	0.670586i	$-0.766043\pi$
$191$	−20.5046	−1.48366	−0.741832	+	0.670586i	$0.766043\pi$
$192$	0	0
$193$	− 26.6075i	− 1.91525i	−0.288014	−	0.957626i	$-0.592995\pi$
$193$	− 26.6075i	− 1.91525i	0.288014	−	0.957626i	$-0.407005\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0.408282i	0.0290889i	0.999894	+	0.0145445i	$0.00462981\pi$
$197$	0.408282i	0.0290889i	−0.999894	+	0.0145445i	$0.995370\pi$
$198$	0	0
$199$	−24.3996	−1.72965	−0.864823	−	0.502078i	$-0.832569\pi$
$199$	−24.3996	−1.72965	−0.864823	+	0.502078i	$0.832569\pi$
$200$	0	0
$201$	3.09724i	0.218463i
$202$	0	0
$203$	1.96043i	0.137595i
$204$	0	0
$205$	−7.73205	−0.540030
$206$	0	0
$207$	0.261323	0.0181632
$208$	0	0
$209$	−9.30426	−0.643589
$210$	0	0
$211$	1.76102	0.121233	0.0606167	−	0.998161i	$-0.480693\pi$
$211$	1.76102	0.121233	0.0606167	+	0.998161i	$0.480693\pi$
$212$	0	0
$213$	− 9.04452i	− 0.619721i
$214$	0	0
$215$	− 6.01060i	− 0.409920i
$216$	0	0
$217$	−3.14441	−0.213457
$218$	0	0
$219$	23.8475i	1.61146i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	11.3191i	0.757983i	0.925400	+	0.378991i	$0.123729\pi$
$223$	11.3191i	0.757983i	−0.925400	+	0.378991i	$0.876271\pi$
$224$	0	0
$225$	−2.43937	−0.162625
$226$	0	0
$227$	8.06077i	0.535012i	0.963556	+	0.267506i	$0.0861996\pi$
$227$	8.06077i	0.535012i	−0.963556	+	0.267506i	$0.913800\pi$
$228$	0	0
$229$	− 11.5715i	− 0.764666i	−0.924025	−	0.382333i	$-0.875121\pi$
$229$	− 11.5715i	− 0.764666i	0.924025	−	0.382333i	$-0.124879\pi$
$230$	0	0
$231$	−1.61504	−0.106262
$232$	0	0
$233$	−24.0900	−1.57819	−0.789094	−	0.614272i	$-0.789450\pi$
$233$	−24.0900	−1.57819	−0.789094	+	0.614272i	$0.789450\pi$
$234$	0	0
$235$	−3.46410	−0.225973
$236$	0	0
$237$	30.6768	1.99267
$238$	0	0
$239$	30.7089i	1.98639i	0.116459	+	0.993196i	$0.462846\pi$
$239$	30.7089i	1.98639i	−0.116459	+	0.993196i	$0.537154\pi$
$240$	0	0
$241$	− 7.92749i	− 0.510654i	−0.966855	−	0.255327i	$-0.917817\pi$
$241$	− 7.92749i	− 0.510654i	0.966855	−	0.255327i	$-0.0821832\pi$
$242$	0	0
$243$	20.2572	1.29950
$244$	0	0
$245$	6.84016i	0.437002i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	32.8360i	2.08090i
$250$	0	0
$251$	−22.6224	−1.42791	−0.713956	−	0.700191i	$-0.753098\pi$
$251$	−22.6224	−1.42791	−0.713956	+	0.700191i	$0.753098\pi$
$252$	0	0
$253$	0.185550i	0.0116654i
$254$	0	0
$255$	1.61504i	0.101138i
$256$	0	0
$257$	−25.7963	−1.60913	−0.804566	−	0.593863i	$-0.797602\pi$
$257$	−25.7963	−1.60913	−0.804566	+	0.593863i	$0.797602\pi$
$258$	0	0
$259$	0.905432	0.0562608
$260$	0	0
$261$	−11.9614	−0.740393
$262$	0	0
$263$	1.59057	0.0980788	0.0490394	−	0.998797i	$-0.484384\pi$
$263$	1.59057	0.0980788	0.0490394	+	0.998797i	$0.484384\pi$
$264$	0	0
$265$	11.7189i	0.719887i
$266$	0	0
$267$	− 0.809495i	− 0.0495403i
$268$	0	0
$269$	−27.8228	−1.69638	−0.848192	−	0.529689i	$-0.822309\pi$
$269$	−27.8228	−1.69638	−0.848192	+	0.529689i	$0.822309\pi$
$270$	0	0
$271$	− 23.5004i	− 1.42755i	−0.700376	−	0.713774i	$-0.746984\pi$
$271$	− 23.5004i	− 1.42755i	0.700376	−	0.713774i	$-0.253016\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 1.73205i	− 0.104447i
$276$	0	0
$277$	2.99674	0.180057	0.0900283	−	0.995939i	$-0.471304\pi$
$277$	2.99674	0.180057	0.0900283	+	0.995939i	$0.471304\pi$
$278$	0	0
$279$	− 19.1854i	− 1.14860i
$280$	0	0
$281$	− 24.6085i	− 1.46802i	−0.679138	−	0.734011i	$-0.737646\pi$
$281$	− 24.6085i	− 1.46802i	0.679138	−	0.734011i	$-0.262354\pi$
$282$	0	0
$283$	−8.16888	−0.485590	−0.242795	−	0.970078i	$-0.578064\pi$
$283$	−8.16888	−0.485590	−0.242795	+	0.970078i	$0.578064\pi$
$284$	0	0
$285$	12.5284	0.742118
$286$	0	0
$287$	3.09131	0.182474
$288$	0	0
$289$	−16.5205	−0.971792
$290$	0	0
$291$	− 20.6405i	− 1.20997i
$292$	0	0
$293$	7.50367i	0.438369i	0.975683	+	0.219185i	$0.0703397\pi$
$293$	7.50367i	0.438369i	−0.975683	+	0.219185i	$0.929660\pi$
$294$	0	0
$295$	−7.27529	−0.423584
$296$	0	0
$297$	2.26469i	0.131411i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	2.40306i	0.138510i
$302$	0	0
$303$	9.57123	0.549853
$304$	0	0
$305$	8.68922i	0.497543i
$306$	0	0
$307$	− 5.95293i	− 0.339752i	−0.985465	−	0.169876i	$-0.945663\pi$
$307$	− 5.95293i	− 0.339752i	0.985465	−	0.169876i	$-0.0543367\pi$
$308$	0	0
$309$	26.1969	1.49029
$310$	0	0
$311$	−17.9247	−1.01642	−0.508208	−	0.861235i	$-0.669692\pi$
$311$	−17.9247	−1.01642	−0.508208	+	0.861235i	$0.669692\pi$
$312$	0	0
$313$	−17.0073	−0.961312	−0.480656	−	0.876909i	$-0.659601\pi$
$313$	−17.0073	−0.961312	−0.480656	+	0.876909i	$0.659601\pi$
$314$	0	0
$315$	0.975272	0.0549504
$316$	0	0
$317$	− 3.09300i	− 0.173720i	−0.996221	−	0.0868601i	$-0.972317\pi$
$317$	− 3.09300i	− 0.173720i	0.996221	−	0.0868601i	$-0.0276833\pi$
$318$	0	0
$319$	− 8.49307i	− 0.475521i
$320$	0	0
$321$	41.0552	2.29148
$322$	0	0
$323$	− 3.71988i	− 0.206980i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 35.4105i	− 1.95821i
$328$	0	0
$329$	1.38496	0.0763555
$330$	0	0
$331$	− 22.5722i	− 1.24068i	−0.784333	−	0.620340i	$-0.786994\pi$
$331$	− 22.5722i	− 1.24068i	0.784333	−	0.620340i	$-0.213006\pi$
$332$	0	0
$333$	5.52442i	0.302736i
$334$	0	0
$335$	1.32801	0.0725568
$336$	0	0
$337$	18.0603	0.983808	0.491904	−	0.870649i	$-0.336301\pi$
$337$	18.0603	0.983808	0.491904	+	0.870649i	$0.336301\pi$
$338$	0	0
$339$	−20.7575	−1.12739
$340$	0	0
$341$	13.6224	0.737693
$342$	0	0
$343$	− 5.53335i	− 0.298773i
$344$	0	0
$345$	− 0.249847i	− 0.0134513i
$346$	0	0
$347$	26.4717	1.42108	0.710538	−	0.703659i	$-0.248452\pi$
$347$	26.4717	1.42108	0.710538	+	0.703659i	$0.248452\pi$
$348$	0	0
$349$	12.3772i	0.662536i	0.943537	+	0.331268i	$0.107476\pi$
$349$	12.3772i	0.662536i	−0.943537	+	0.331268i	$0.892524\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	19.2809i	1.02622i	0.858323	+	0.513110i	$0.171507\pi$
$353$	19.2809i	1.02622i	−0.858323	+	0.513110i	$0.828493\pi$
$354$	0	0
$355$	−3.87803	−0.205825
$356$	0	0
$357$	− 0.645699i	− 0.0341740i
$358$	0	0
$359$	26.5506i	1.40129i	0.713512	+	0.700643i	$0.247103\pi$
$359$	26.5506i	1.40129i	−0.713512	+	0.700643i	$0.752897\pi$
$360$	0	0
$361$	−9.85641	−0.518758
$362$	0	0
$363$	−18.6580	−0.979290
$364$	0	0
$365$	10.2251	0.535207
$366$	0	0
$367$	7.37436	0.384938	0.192469	−	0.981303i	$-0.438350\pi$
$367$	7.37436	0.384938	0.192469	+	0.981303i	$0.438350\pi$
$368$	0	0
$369$	18.8614i	0.981883i
$370$	0	0
$371$	− 4.68527i	− 0.243247i
$372$	0	0
$373$	−28.3149	−1.46609	−0.733044	−	0.680181i	$-0.761901\pi$
$373$	−28.3149	−1.46609	−0.733044	+	0.680181i	$0.761901\pi$
$374$	0	0
$375$	2.33225i	0.120437i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	10.4267i	0.535581i	0.963477	+	0.267791i	$0.0862936\pi$
$379$	10.4267i	0.535581i	−0.963477	+	0.267791i	$0.913706\pi$
$380$	0	0
$381$	28.3521	1.45252
$382$	0	0
$383$	10.8520i	0.554511i	0.960796	+	0.277256i	$0.0894249\pi$
$383$	10.8520i	0.554511i	−0.960796	+	0.277256i	$0.910575\pi$
$384$	0	0
$385$	0.692481i	0.0352921i
$386$	0	0
$387$	−14.6621	−0.745317
$388$	0	0
$389$	20.2893	1.02871	0.514353	−	0.857578i	$-0.328032\pi$
$389$	20.2893	1.02871	0.514353	+	0.857578i	$0.328032\pi$
$390$	0	0
$391$	−0.0741836	−0.00375163
$392$	0	0
$393$	4.93907	0.249143
$394$	0	0
$395$	− 13.1533i	− 0.661815i
$396$	0	0
$397$	− 25.2527i	− 1.26740i	−0.773581	−	0.633698i	$-0.781536\pi$
$397$	− 25.2527i	− 1.26740i	0.773581	−	0.633698i	$-0.218464\pi$
$398$	0	0
$399$	−5.00891	−0.250759
$400$	0	0
$401$	12.5143i	0.624933i	0.949929	+	0.312467i	$0.101155\pi$
$401$	12.5143i	0.624933i	−0.949929	+	0.312467i	$0.898845\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	− 10.3676i	− 0.515169i
$406$	0	0
$407$	−3.92256	−0.194434
$408$	0	0
$409$	− 5.99576i	− 0.296471i	−0.988952	−	0.148236i	$-0.952641\pi$
$409$	− 5.99576i	− 0.296471i	0.988952	−	0.148236i	$-0.0473594\pi$
$410$	0	0
$411$	42.0575i	2.07454i
$412$	0	0
$413$	2.90869	0.143127
$414$	0	0
$415$	14.0791	0.691118
$416$	0	0
$417$	22.9497	1.12385
$418$	0	0
$419$	−10.9703	−0.535935	−0.267968	−	0.963428i	$-0.586352\pi$
$419$	−10.9703	−0.535935	−0.267968	+	0.963428i	$0.586352\pi$
$420$	0	0
$421$	36.6085i	1.78419i	0.451848	+	0.892095i	$0.350765\pi$
$421$	36.6085i	1.78419i	−0.451848	+	0.892095i	$0.649235\pi$
$422$	0	0
$423$	8.45024i	0.410865i
$424$	0	0
$425$	0.692481	0.0335903
$426$	0	0
$427$	− 3.47399i	− 0.168118i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 12.1097i	− 0.583302i	−0.956525	−	0.291651i	$-0.905795\pi$
$431$	− 12.1097i	− 0.583302i	0.956525	−	0.291651i	$-0.0942046\pi$
$432$	0	0
$433$	9.01794	0.433375	0.216687	−	0.976241i	$-0.430475\pi$
$433$	9.01794	0.433375	0.216687	+	0.976241i	$0.430475\pi$
$434$	0	0
$435$	11.4361i	0.548320i
$436$	0	0
$437$	0.575468i	0.0275284i
$438$	0	0
$439$	12.1509	0.579933	0.289966	−	0.957037i	$-0.406356\pi$
$439$	12.1509	0.579933	0.289966	+	0.957037i	$0.406356\pi$
$440$	0	0
$441$	16.6857	0.794557
$442$	0	0
$443$	15.3116	0.727476	0.363738	−	0.931501i	$-0.381500\pi$
$443$	15.3116	0.727476	0.363738	+	0.931501i	$0.381500\pi$
$444$	0	0
$445$	−0.347088	−0.0164536
$446$	0	0
$447$	− 20.7173i	− 0.979895i
$448$	0	0
$449$	22.7394i	1.07314i	0.843856	+	0.536569i	$0.180280\pi$
$449$	22.7394i	1.07314i	−0.843856	+	0.536569i	$0.819720\pi$
$450$	0	0
$451$	−13.3923	−0.630619
$452$	0	0
$453$	10.3537i	0.486460i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 10.7450i	− 0.502632i	−0.967905	−	0.251316i	$-0.919137\pi$
$457$	− 10.7450i	− 0.502632i	0.967905	−	0.251316i	$-0.0808633\pi$
$458$	0	0
$459$	−0.905432	−0.0422620
$460$	0	0
$461$	− 11.8903i	− 0.553788i	−0.960900	−	0.276894i	$-0.910695\pi$
$461$	− 11.8903i	− 0.553788i	0.960900	−	0.276894i	$-0.0893051\pi$
$462$	0	0
$463$	− 3.39726i	− 0.157884i	−0.996879	−	0.0789420i	$-0.974846\pi$
$463$	− 3.39726i	− 0.157884i	0.996879	−	0.0789420i	$-0.0251542\pi$
$464$	0	0
$465$	−18.3428	−0.850629
$466$	0	0
$467$	−6.39426	−0.295891	−0.147946	−	0.988996i	$-0.547266\pi$
$467$	−6.39426	−0.295891	−0.147946	+	0.988996i	$0.547266\pi$
$468$	0	0
$469$	−0.530943	−0.0245167
$470$	0	0
$471$	9.71890	0.447823
$472$	0	0
$473$	− 10.4107i	− 0.478683i
$474$	0	0
$475$	− 5.37182i	− 0.246476i
$476$	0	0
$477$	28.5868	1.30890
$478$	0	0
$479$	16.3194i	0.745651i	0.927902	+	0.372825i	$0.121611\pi$
$479$	16.3194i	0.745651i	−0.927902	+	0.372825i	$0.878389\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0.0998900i	0.00454515i
$484$	0	0
$485$	−8.85004	−0.401860
$486$	0	0
$487$	11.9346i	0.540807i	0.962747	+	0.270403i	$0.0871571\pi$
$487$	11.9346i	0.540807i	−0.962747	+	0.270403i	$0.912843\pi$
$488$	0	0
$489$	8.62166i	0.389885i
$490$	0	0
$491$	34.0517	1.53673	0.768366	−	0.640011i	$-0.221070\pi$
$491$	34.0517	1.53673	0.768366	+	0.640011i	$0.221070\pi$
$492$	0	0
$493$	3.39557	0.152929
$494$	0	0
$495$	−4.22512	−0.189905
$496$	0	0
$497$	1.55045	0.0695473
$498$	0	0
$499$	12.5854i	0.563398i	0.959503	+	0.281699i	$0.0908979\pi$
$499$	12.5854i	0.563398i	−0.959503	+	0.281699i	$0.909102\pi$
$500$	0	0
$501$	50.1367i	2.23994i
$502$	0	0
$503$	−18.1887	−0.810992	−0.405496	−	0.914097i	$-0.632901\pi$
$503$	−18.1887	−0.810992	−0.405496	+	0.914097i	$0.632901\pi$
$504$	0	0
$505$	− 4.10387i	− 0.182620i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 29.0135i	− 1.28600i	−0.765865	−	0.643001i	$-0.777689\pi$
$509$	− 29.0135i	− 1.28600i	0.765865	−	0.643001i	$-0.222311\pi$
$510$	0	0
$511$	−4.08805	−0.180845
$512$	0	0
$513$	7.02375i	0.310106i
$514$	0	0
$515$	− 11.2325i	− 0.494961i
$516$	0	0
$517$	−6.00000	−0.263880
$518$	0	0
$519$	−27.0815	−1.18875
$520$	0	0
$521$	35.0240	1.53443	0.767214	−	0.641391i	$-0.221642\pi$
$521$	35.0240	1.53443	0.767214	+	0.641391i	$0.221642\pi$
$522$	0	0
$523$	−9.27741	−0.405673	−0.202836	−	0.979213i	$-0.565016\pi$
$523$	−9.27741	−0.405673	−0.202836	+	0.979213i	$0.565016\pi$
$524$	0	0
$525$	− 0.932442i	− 0.0406951i
$526$	0	0
$527$	5.44629i	0.237244i
$528$	0	0
$529$	−22.9885	−0.999501
$530$	0	0
$531$	17.7472i	0.770161i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	− 17.6033i	− 0.761057i
$536$	0	0
$537$	−11.5920	−0.500232
$538$	0	0
$539$	11.8475i	0.510308i
$540$	0	0
$541$	− 24.3814i	− 1.04824i	−0.851644	−	0.524120i	$-0.824394\pi$
$541$	− 24.3814i	− 1.04824i	0.851644	−	0.524120i	$-0.175606\pi$
$542$	0	0
$543$	40.5100	1.73845
$544$	0	0
$545$	−15.1830	−0.650369
$546$	0	0
$547$	26.5270	1.13421	0.567106	−	0.823645i	$-0.308063\pi$
$547$	26.5270	1.13421	0.567106	+	0.823645i	$0.308063\pi$
$548$	0	0
$549$	21.1963	0.904634
$550$	0	0
$551$	− 26.3406i	− 1.12215i
$552$	0	0
$553$	5.25875i	0.223625i
$554$	0	0
$555$	5.28181	0.224200
$556$	0	0
$557$	− 3.21171i	− 0.136085i	−0.997682	−	0.0680423i	$-0.978325\pi$
$557$	− 3.21171i	− 0.136085i	0.997682	−	0.0680423i	$-0.0216753\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	2.79733i	0.118103i
$562$	0	0
$563$	34.2852	1.44495	0.722474	−	0.691398i	$-0.243005\pi$
$563$	34.2852	1.44495	0.722474	+	0.691398i	$0.243005\pi$
$564$	0	0
$565$	8.90021i	0.374435i
$566$	0	0
$567$	4.14500i	0.174074i
$568$	0	0
$569$	35.6457	1.49434	0.747172	−	0.664630i	$-0.231411\pi$
$569$	35.6457	1.49434	0.747172	+	0.664630i	$0.231411\pi$
$570$	0	0
$571$	−4.53590	−0.189821	−0.0949107	−	0.995486i	$-0.530257\pi$
$571$	−4.53590	−0.189821	−0.0949107	+	0.995486i	$0.530257\pi$
$572$	0	0
$573$	47.8219	1.99779
$574$	0	0
$575$	−0.107127	−0.00446751
$576$	0	0
$577$	− 18.4475i	− 0.767981i	−0.923337	−	0.383991i	$-0.874550\pi$
$577$	− 18.4475i	− 0.767981i	0.923337	−	0.383991i	$-0.125450\pi$
$578$	0	0
$579$	62.0553i	2.57893i
$580$	0	0
$581$	−5.62890	−0.233526
$582$	0	0
$583$	20.2977i	0.840646i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 42.9914i	− 1.77444i	−0.461343	−	0.887222i	$-0.652632\pi$
$587$	− 42.9914i	− 1.77444i	0.461343	−	0.887222i	$-0.347368\pi$
$588$	0	0
$589$	42.2487	1.74083
$590$	0	0
$591$	− 0.952215i	− 0.0391689i
$592$	0	0
$593$	12.8614i	0.528153i	0.964502	+	0.264076i	$0.0850671\pi$
$593$	12.8614i	0.528153i	−0.964502	+	0.264076i	$0.914933\pi$
$594$	0	0
$595$	−0.276857	−0.0113500
$596$	0	0
$597$	56.9060	2.32901
$598$	0	0
$599$	28.3170	1.15700	0.578500	−	0.815682i	$-0.303638\pi$
$599$	28.3170	1.15700	0.578500	+	0.815682i	$0.303638\pi$
$600$	0	0
$601$	−7.13469	−0.291030	−0.145515	−	0.989356i	$-0.546484\pi$
$601$	−7.13469	−0.291030	−0.145515	+	0.989356i	$0.546484\pi$
$602$	0	0
$603$	− 3.23951i	− 0.131923i
$604$	0	0
$605$	8.00000i	0.325246i
$606$	0	0
$607$	−22.1802	−0.900266	−0.450133	−	0.892962i	$-0.648623\pi$
$607$	−22.1802	−0.900266	−0.450133	+	0.892962i	$0.648623\pi$
$608$	0	0
$609$	− 4.57221i	− 0.185275i
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 0.322622i	− 0.0130306i	−0.999979	−	0.00651530i	$-0.997926\pi$
$613$	− 0.322622i	− 0.0130306i	0.999979	−	0.00651530i	$-0.00207390\pi$
$614$	0	0
$615$	18.0330	0.727163
$616$	0	0
$617$	37.1987i	1.49756i	0.662817	+	0.748781i	$0.269361\pi$
$617$	37.1987i	1.49756i	−0.662817	+	0.748781i	$0.730639\pi$
$618$	0	0
$619$	3.94911i	0.158728i	0.996846	+	0.0793641i	$0.0252890\pi$
$619$	3.94911i	0.158728i	−0.996846	+	0.0793641i	$0.974711\pi$
$620$	0	0
$621$	0.140071	0.00562085
$622$	0	0
$623$	0.138767	0.00555959
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	21.6998	0.866607
$628$	0	0
$629$	− 1.56825i	− 0.0625304i
$630$	0	0
$631$	− 33.5815i	− 1.33686i	−0.743776	−	0.668429i	$-0.766967\pi$
$631$	− 33.5815i	− 1.33686i	0.743776	−	0.668429i	$-0.233033\pi$
$632$	0	0
$633$	−4.10713	−0.163244
$634$	0	0
$635$	− 12.1566i	− 0.482419i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	9.45997i	0.374231i
$640$	0	0
$641$	−33.1800	−1.31053	−0.655266	−	0.755398i	$-0.727443\pi$
$641$	−33.1800	−1.31053	−0.655266	+	0.755398i	$0.727443\pi$
$642$	0	0
$643$	− 31.3666i	− 1.23698i	−0.785793	−	0.618489i	$-0.787745\pi$
$643$	− 31.3666i	− 1.23698i	0.785793	−	0.618489i	$-0.212255\pi$
$644$	0	0
$645$	14.0182i	0.551966i
$646$	0	0
$647$	−19.5906	−0.770185	−0.385092	−	0.922878i	$-0.625830\pi$
$647$	−19.5906	−0.770185	−0.385092	+	0.922878i	$0.625830\pi$
$648$	0	0
$649$	−12.6012	−0.494639
$650$	0	0
$651$	7.33355	0.287424
$652$	0	0
$653$	7.11251	0.278334	0.139167	−	0.990269i	$-0.455557\pi$
$653$	7.11251	0.278334	0.139167	+	0.990269i	$0.455557\pi$
$654$	0	0
$655$	− 2.11773i	− 0.0827466i
$656$	0	0
$657$	− 24.9429i	− 0.973115i
$658$	0	0
$659$	18.5842	0.723938	0.361969	−	0.932190i	$-0.382105\pi$
$659$	18.5842	0.723938	0.361969	+	0.932190i	$0.382105\pi$
$660$	0	0
$661$	16.7908i	0.653087i	0.945182	+	0.326543i	$0.105884\pi$
$661$	16.7908i	0.653087i	−0.945182	+	0.326543i	$0.894116\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.14768i	0.0832833i
$666$	0	0
$667$	−0.525296	−0.0203395
$668$	0	0
$669$	− 26.3989i	− 1.02064i
$670$	0	0
$671$	15.0502i	0.581005i
$672$	0	0
$673$	−22.5914	−0.870834	−0.435417	−	0.900229i	$-0.643399\pi$
$673$	−22.5914	−0.870834	−0.435417	+	0.900229i	$0.643399\pi$
$674$	0	0
$675$	−1.30752	−0.0503264
$676$	0	0
$677$	5.31616	0.204317	0.102158	−	0.994768i	$-0.467425\pi$
$677$	5.31616	0.204317	0.102158	+	0.994768i	$0.467425\pi$
$678$	0	0
$679$	3.53829	0.135787
$680$	0	0
$681$	− 18.7997i	− 0.720407i
$682$	0	0
$683$	− 3.95391i	− 0.151292i	−0.997135	−	0.0756461i	$-0.975898\pi$
$683$	− 3.95391i	− 0.151292i	0.997135	−	0.0756461i	$-0.0241019\pi$
$684$	0	0
$685$	18.0330	0.689007
$686$	0	0
$687$	26.9876i	1.02964i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	− 12.8741i	− 0.489753i	−0.969554	−	0.244877i	$-0.921253\pi$
$691$	− 12.8741i	− 0.489753i	0.969554	−	0.244877i	$-0.0787474\pi$
$692$	0	0
$693$	1.68922	0.0641682
$694$	0	0
$695$	− 9.84016i	− 0.373258i
$696$	0	0
$697$	− 5.35430i	− 0.202809i
$698$	0	0
$699$	56.1838	2.12507
$700$	0	0
$701$	34.9777	1.32109	0.660544	−	0.750787i	$-0.270326\pi$
$701$	34.9777	1.32109	0.660544	+	0.750787i	$0.270326\pi$
$702$	0	0
$703$	−12.1655	−0.458830
$704$	0	0
$705$	8.07914	0.304278
$706$	0	0
$707$	1.64074i	0.0617065i
$708$	0	0
$709$	19.7665i	0.742347i	0.928564	+	0.371173i	$0.121044\pi$
$709$	19.7665i	0.742347i	−0.928564	+	0.371173i	$0.878956\pi$
$710$	0	0
$711$	−32.0859	−1.20331
$712$	0	0
$713$	− 0.842543i	− 0.0315535i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 71.6206i	− 2.67472i
$718$	0	0
$719$	44.2467	1.65012	0.825062	−	0.565042i	$-0.191140\pi$
$719$	44.2467	1.65012	0.825062	+	0.565042i	$0.191140\pi$
$720$	0	0
$721$	4.49079i	0.167246i
$722$	0	0
$723$	18.4889i	0.687608i
$724$	0	0
$725$	4.90348	0.182111
$726$	0	0
$727$	−31.4877	−1.16781	−0.583907	−	0.811821i	$-0.698477\pi$
$727$	−31.4877	−1.16781	−0.583907	+	0.811821i	$0.698477\pi$
$728$	0	0
$729$	−16.1420	−0.597853
$730$	0	0
$731$	4.16223	0.153946
$732$	0	0
$733$	42.4714i	1.56872i	0.620307	+	0.784359i	$0.287008\pi$
$733$	42.4714i	1.56872i	−0.620307	+	0.784359i	$0.712992\pi$
$734$	0	0
$735$	− 15.9529i	− 0.588433i
$736$	0	0
$737$	2.30018	0.0847281
$738$	0	0
$739$	− 13.7835i	− 0.507033i	−0.967331	−	0.253516i	$-0.918413\pi$
$739$	− 13.7835i	− 0.507033i	0.967331	−	0.253516i	$-0.0815872\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 15.9392i	− 0.584753i	−0.956303	−	0.292377i	$-0.905554\pi$
$743$	− 15.9392i	− 0.584753i	0.956303	−	0.292377i	$-0.0944461\pi$
$744$	0	0
$745$	−8.88299	−0.325447
$746$	0	0
$747$	− 34.3443i	− 1.25659i
$748$	0	0
$749$	7.03787i	0.257158i
$750$	0	0
$751$	1.51708	0.0553590	0.0276795	−	0.999617i	$-0.491188\pi$
$751$	1.51708	0.0553590	0.0276795	+	0.999617i	$0.491188\pi$
$752$	0	0
$753$	52.7610	1.92272
$754$	0	0
$755$	4.43937	0.161565
$756$	0	0
$757$	47.4829	1.72579	0.862897	−	0.505380i	$-0.168648\pi$
$757$	47.4829	1.72579	0.862897	+	0.505380i	$0.168648\pi$
$758$	0	0
$759$	− 0.432748i	− 0.0157078i
$760$	0	0
$761$	38.5445i	1.39724i	0.715495	+	0.698618i	$0.246201\pi$
$761$	38.5445i	1.39724i	−0.715495	+	0.698618i	$0.753799\pi$
$762$	0	0
$763$	6.07023	0.219757
$764$	0	0
$765$	− 1.68922i	− 0.0610739i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	39.8645i	1.43755i	0.695243	+	0.718775i	$0.255297\pi$
$769$	39.8645i	1.43755i	−0.695243	+	0.718775i	$0.744703\pi$
$770$	0	0
$771$	60.1634	2.16673
$772$	0	0
$773$	− 32.9547i	− 1.18530i	−0.805460	−	0.592650i	$-0.798082\pi$
$773$	− 32.9547i	− 1.18530i	0.805460	−	0.592650i	$-0.201918\pi$
$774$	0	0
$775$	7.86488i	0.282515i
$776$	0	0
$777$	−2.11169	−0.0757565
$778$	0	0
$779$	−41.5352	−1.48815
$780$	0	0
$781$	−6.71695	−0.240351
$782$	0	0
$783$	−6.41139	−0.229124
$784$	0	0
$785$	− 4.16719i	− 0.148733i
$786$	0	0
$787$	1.07930i	0.0384728i	0.999815	+	0.0192364i	$0.00612351\pi$
$787$	1.07930i	0.0384728i	−0.999815	+	0.0192364i	$0.993876\pi$
$788$	0	0
$789$	−3.70960	−0.132065
$790$	0	0
$791$	− 3.55834i	− 0.126520i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	− 27.3314i	− 0.969344i
$796$	0	0
$797$	19.1014	0.676607	0.338304	−	0.941037i	$-0.390147\pi$
$797$	19.1014	0.676607	0.338304	+	0.941037i	$0.390147\pi$
$798$	0	0
$799$	− 2.39883i	− 0.0848644i
$800$	0	0
$801$	0.846678i	0.0299159i
$802$	0	0
$803$	17.7104	0.624987
$804$	0	0
$805$	0.0428299	0.00150956
$806$	0	0
$807$	64.8896	2.28422
$808$	0	0
$809$	1.76340	0.0619980	0.0309990	−	0.999519i	$-0.490131\pi$
$809$	1.76340	0.0619980	0.0309990	+	0.999519i	$0.490131\pi$
$810$	0	0
$811$	52.3298i	1.83755i	0.394784	+	0.918774i	$0.370819\pi$
$811$	52.3298i	1.83755i	−0.394784	+	0.918774i	$0.629181\pi$
$812$	0	0
$813$	54.8088i	1.92223i
$814$	0	0
$815$	3.69672	0.129490
$816$	0	0
$817$	− 32.2879i	− 1.12961i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 14.2703i	− 0.498038i	−0.968499	−	0.249019i	$-0.919892\pi$
$821$	− 14.2703i	− 0.498038i	0.968499	−	0.249019i	$-0.0801082\pi$
$822$	0	0
$823$	−43.7145	−1.52379	−0.761896	−	0.647699i	$-0.775731\pi$
$823$	−43.7145	−1.52379	−0.761896	+	0.647699i	$0.775731\pi$
$824$	0	0
$825$	4.03957i	0.140640i
$826$	0	0
$827$	22.5962i	0.785748i	0.919592	+	0.392874i	$0.128519\pi$
$827$	22.5962i	0.785748i	−0.919592	+	0.392874i	$0.871481\pi$
$828$	0	0
$829$	−54.0946	−1.87878	−0.939392	−	0.342844i	$-0.888610\pi$
$829$	−54.0946	−1.87878	−0.939392	+	0.342844i	$0.888610\pi$
$830$	0	0
$831$	−6.98914	−0.242450
$832$	0	0
$833$	−4.73668	−0.164116
$834$	0	0
$835$	21.4972	0.743940
$836$	0	0
$837$	− 10.2835i	− 0.355449i
$838$	0	0
$839$	− 22.2337i	− 0.767593i	−0.923418	−	0.383796i	$-0.874616\pi$
$839$	− 22.2337i	− 0.767593i	0.923418	−	0.383796i	$-0.125384\pi$
$840$	0	0
$841$	−4.95593	−0.170894
$842$	0	0
$843$	57.3931i	1.97672i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 3.19843i	− 0.109900i
$848$	0	0
$849$	19.0518	0.653858
$850$	0	0
$851$	0.242610i	0.00831656i
$852$	0	0
$853$	16.5312i	0.566019i	0.959117	+	0.283009i	$0.0913328\pi$
$853$	16.5312i	0.566019i	−0.959117	+	0.283009i	$0.908667\pi$
$854$	0	0
$855$	−13.1039	−0.448143
$856$	0	0
$857$	27.5842	0.942259	0.471129	−	0.882064i	$-0.343846\pi$
$857$	27.5842	0.942259	0.471129	+	0.882064i	$0.343846\pi$
$858$	0	0
$859$	−32.5016	−1.10894	−0.554469	−	0.832204i	$-0.687079\pi$
$859$	−32.5016	−1.10894	−0.554469	+	0.832204i	$0.687079\pi$
$860$	0	0
$861$	−7.20969	−0.245706
$862$	0	0
$863$	29.7986i	1.01436i	0.861842	+	0.507178i	$0.169311\pi$
$863$	29.7986i	1.01436i	−0.861842	+	0.507178i	$0.830689\pi$
$864$	0	0
$865$	11.6118i	0.394812i
$866$	0	0
$867$	38.5298	1.30854
$868$	0	0
$869$	− 22.7822i	− 0.772834i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	21.5886i	0.730662i
$874$	0	0
$875$	−0.399804	−0.0135159
$876$	0	0
$877$	14.4036i	0.486376i	0.969979	+	0.243188i	$0.0781932\pi$
$877$	14.4036i	0.486376i	−0.969979	+	0.243188i	$0.921807\pi$
$878$	0	0
$879$	− 17.5004i	− 0.590274i
$880$	0	0
$881$	7.36914	0.248273	0.124136	−	0.992265i	$-0.460384\pi$
$881$	7.36914	0.248273	0.124136	+	0.992265i	$0.460384\pi$
$882$	0	0
$883$	24.3646	0.819933	0.409967	−	0.912101i	$-0.365540\pi$
$883$	24.3646	0.819933	0.409967	+	0.912101i	$0.365540\pi$
$884$	0	0
$885$	16.9678	0.570365
$886$	0	0
$887$	−24.0644	−0.808004	−0.404002	−	0.914758i	$-0.632381\pi$
$887$	−24.0644	−0.808004	−0.404002	+	0.914758i	$0.632381\pi$
$888$	0	0
$889$	4.86025i	0.163008i
$890$	0	0
$891$	− 17.9572i	− 0.601588i
$892$	0	0
$893$	−18.6085	−0.622710
$894$	0	0
$895$	4.97032i	0.166139i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	38.5653i	1.28622i
$900$	0	0
$901$	−8.11512	−0.270354
$902$	0	0
$903$	− 5.60454i	− 0.186507i
$904$	0	0
$905$	− 17.3695i	− 0.577383i
$906$	0	0
$907$	−37.5073	−1.24541	−0.622705	−	0.782457i	$-0.713966\pi$
$907$	−37.5073	−1.24541	−0.622705	+	0.782457i	$0.713966\pi$
$908$	0	0
$909$	−10.0109	−0.332039
$910$	0	0
$911$	12.6000	0.417458	0.208729	−	0.977974i	$-0.433067\pi$
$911$	12.6000	0.417458	0.208729	+	0.977974i	$0.433067\pi$
$912$	0	0
$913$	24.3858	0.807052
$914$	0	0
$915$	− 20.2654i	− 0.669954i
$916$	0	0
$917$	0.846678i	0.0279598i
$918$	0	0
$919$	−31.6664	−1.04458	−0.522288	−	0.852769i	$-0.674922\pi$
$919$	−31.6664	−1.04458	−0.522288	+	0.852769i	$0.674922\pi$
$920$	0	0
$921$	13.8837i	0.457484i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	− 2.26469i	− 0.0744625i
$926$	0	0
$927$	−27.4002	−0.899940
$928$	0	0
$929$	− 29.7378i	− 0.975666i	−0.872937	−	0.487833i	$-0.837787\pi$
$929$	− 29.7378i	− 0.975666i	0.872937	−	0.487833i	$-0.162213\pi$
$930$	0	0
$931$	36.7441i	1.20424i
$932$	0	0
$933$	41.8048	1.36863
$934$	0	0
$935$	1.19941	0.0392250
$936$	0	0
$937$	−52.3124	−1.70897	−0.854486	−	0.519474i	$-0.826128\pi$
$937$	−52.3124	−1.70897	−0.854486	+	0.519474i	$0.826128\pi$
$938$	0	0
$939$	39.6653	1.29443
$940$	0	0
$941$	29.5767i	0.964174i	0.876123	+	0.482087i	$0.160121\pi$
$941$	29.5767i	0.964174i	−0.876123	+	0.482087i	$0.839879\pi$
$942$	0	0
$943$	0.828313i	0.0269736i
$944$	0	0
$945$	0.522752	0.0170051
$946$	0	0
$947$	48.0330i	1.56086i	0.625240	+	0.780432i	$0.285001\pi$
$947$	48.0330i	1.56086i	−0.625240	+	0.780432i	$0.714999\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	7.21364i	0.233918i
$952$	0	0
$953$	−29.7824	−0.964745	−0.482373	−	0.875966i	$-0.660225\pi$
$953$	−29.7824	−0.964745	−0.482373	+	0.875966i	$0.660225\pi$
$954$	0	0
$955$	− 20.5046i	− 0.663515i
$956$	0	0
$957$	19.8079i	0.640299i
$958$	0	0
$959$	−7.20969	−0.232813
$960$	0	0
$961$	−30.8564	−0.995368
$962$	0	0
$963$	−42.9410	−1.38376
$964$	0	0
$965$	26.6075	0.856527
$966$	0	0
$967$	20.6730i	0.664798i	0.943139	+	0.332399i	$0.107858\pi$
$967$	20.6730i	0.664798i	−0.943139	+	0.332399i	$0.892142\pi$
$968$	0	0
$969$	8.67568i	0.278703i
$970$	0	0
$971$	−19.9861	−0.641386	−0.320693	−	0.947183i	$-0.603916\pi$
$971$	−19.9861	−0.641386	−0.320693	+	0.947183i	$0.603916\pi$
$972$	0	0
$973$	3.93414i	0.126123i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 45.6276i	− 1.45976i	−0.683577	−	0.729878i	$-0.739577\pi$
$977$	− 45.6276i	− 1.45976i	0.683577	−	0.729878i	$-0.260423\pi$
$978$	0	0
$979$	−0.601174	−0.0192136
$980$	0	0
$981$	37.0370i	1.18250i
$982$	0	0
$983$	− 54.1966i	− 1.72860i	−0.502975	−	0.864301i	$-0.667761\pi$
$983$	− 54.1966i	− 1.72860i	0.502975	−	0.864301i	$-0.332239\pi$
$984$	0	0
$985$	−0.408282	−0.0130090
$986$	0	0
$987$	−3.23007	−0.102814
$988$	0	0
$989$	−0.643899	−0.0204748
$990$	0	0
$991$	46.5278	1.47800	0.739002	−	0.673703i	$-0.235297\pi$
$991$	46.5278	1.47800	0.739002	+	0.673703i	$0.235297\pi$
$992$	0	0
$993$	52.6440i	1.67061i
$994$	0	0
$995$	− 24.3996i	− 0.773521i
$996$	0	0
$997$	−39.1029	−1.23840	−0.619200	−	0.785233i	$-0.712543\pi$
$997$	−39.1029	−1.23840	−0.619200	+	0.785233i	$0.712543\pi$
$998$	0	0
$999$	2.96112i	0.0936858i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3380.2.f.i.3041.2		8
13.5	odd	4		3380.2.a.q.1.1		4
13.8	odd	4		3380.2.a.p.1.1		4
13.9	even	3		260.2.x.a.101.4	✓	8
13.10	even	6		260.2.x.a.121.4	yes	8
13.12	even	2	inner	3380.2.f.i.3041.1		8
39.23	odd	6		2340.2.dj.d.901.3		8
39.35	odd	6		2340.2.dj.d.361.1		8
52.23	odd	6		1040.2.da.c.641.1		8
52.35	odd	6		1040.2.da.c.881.1		8
65.9	even	6		1300.2.y.b.101.1		8
65.22	odd	12		1300.2.ba.c.49.3		8
65.23	odd	12		1300.2.ba.c.849.3		8
65.48	odd	12		1300.2.ba.b.49.2		8
65.49	even	6		1300.2.y.b.901.1		8
65.62	odd	12		1300.2.ba.b.849.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
260.2.x.a.101.4	✓	8	13.9	even	3
260.2.x.a.121.4	yes	8	13.10	even	6
1040.2.da.c.641.1		8	52.23	odd	6
1040.2.da.c.881.1		8	52.35	odd	6
1300.2.y.b.101.1		8	65.9	even	6
1300.2.y.b.901.1		8	65.49	even	6
1300.2.ba.b.49.2		8	65.48	odd	12
1300.2.ba.b.849.2		8	65.62	odd	12
1300.2.ba.c.49.3		8	65.22	odd	12
1300.2.ba.c.849.3		8	65.23	odd	12
2340.2.dj.d.361.1		8	39.35	odd	6
2340.2.dj.d.901.3		8	39.23	odd	6
3380.2.a.p.1.1		4	13.8	odd	4
3380.2.a.q.1.1		4	13.5	odd	4
3380.2.f.i.3041.1		8	13.12	even	2	inner
3380.2.f.i.3041.2		8	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q-2.33225 q^{3} +1.00000i q^{5} -0.399804i q^{7} +2.43937 q^{9} +O(q^{10})\)	\(q-2.33225 q^{3} +1.00000i q^{5} -0.399804i q^{7} +2.43937 q^{9} +1.73205i q^{11} -2.33225i q^{15} -0.692481 q^{17} +5.37182i q^{19} +0.932442i q^{21} +0.107127 q^{23} -1.00000 q^{25} +1.30752 q^{27} -4.90348 q^{29} -7.86488i q^{31} -4.03957i q^{33} +0.399804 q^{35} +2.26469i q^{37} +7.73205i q^{41} -6.01060 q^{43} +2.43937i q^{45} +3.46410i q^{47} +6.84016 q^{49} +1.61504 q^{51} +11.7189 q^{53} -1.73205 q^{55} -12.5284i q^{57} +7.27529i q^{59} +8.68922 q^{61} -0.975272i q^{63} -1.32801i q^{67} -0.249847 q^{69} +3.87803i q^{71} -10.2251i q^{73} +2.33225 q^{75} +0.692481 q^{77} -13.1533 q^{79} -10.3676 q^{81} -14.0791i q^{83} -0.692481i q^{85} +11.4361 q^{87} +0.347088i q^{89} +18.3428i q^{93} -5.37182 q^{95} +8.85004i q^{97} +4.22512i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{3} + 8 q^{9}+O(q^{10}) \) 8 * q + 4 * q^3 + 8 * q^9	\( 8 q + 4 q^{3} + 8 q^{9} - 12 q^{17} + 12 q^{23} - 8 q^{25} + 4 q^{27} + 12 q^{35} - 20 q^{43} + 8 q^{49} + 24 q^{53} + 8 q^{61} + 48 q^{69} - 4 q^{75} + 12 q^{77} - 16 q^{79} - 16 q^{81} + 12 q^{87}+O(q^{100}) \) 8 * q + 4 * q^3 + 8 * q^9 - 12 * q^17 + 12 * q^23 - 8 * q^25 + 4 * q^27 + 12 * q^35 - 20 * q^43 + 8 * q^49 + 24 * q^53 + 8 * q^61 + 48 * q^69 - 4 * q^75 + 12 * q^77 - 16 * q^79 - 16 * q^81 + 12 * q^87

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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