LMFDB - Embedded newform 3468.2.b.a.577.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3468,2,Mod(577,3468)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3468, 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("3468.577");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3468 = 2^{2} \cdot 3 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3468.b (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:	$27.6921194210$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 204)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			577.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	3468.577
Dual form			3468.2.b.a.577.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +1.00000i q^{5} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +1.00000i q^{5} -1.00000 q^{9} -5.00000i q^{11} -5.00000 q^{13} -1.00000 q^{15} -1.00000 q^{19} +3.00000i q^{23} +4.00000 q^{25} -1.00000i q^{27} +2.00000i q^{29} +2.00000i q^{31} +5.00000 q^{33} -8.00000i q^{37} -5.00000i q^{39} +5.00000i q^{41} +9.00000 q^{43} -1.00000i q^{45} +6.00000 q^{47} +7.00000 q^{49} +6.00000 q^{53} +5.00000 q^{55} -1.00000i q^{57} -6.00000 q^{59} +4.00000i q^{61} -5.00000i q^{65} +12.0000 q^{67} -3.00000 q^{69} -12.0000i q^{71} -2.00000i q^{73} +4.00000i q^{75} -10.0000i q^{79} +1.00000 q^{81} +2.00000 q^{83} -2.00000 q^{87} +12.0000 q^{89} -2.00000 q^{93} -1.00000i q^{95} +16.0000i q^{97} +5.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^9	$ 2 q - 2 q^{9} - 10 q^{13} - 2 q^{15} - 2 q^{19} + 8 q^{25} + 10 q^{33} + 18 q^{43} + 12 q^{47} + 14 q^{49} + 12 q^{53} + 10 q^{55} - 12 q^{59} + 24 q^{67} - 6 q^{69} + 2 q^{81} + 4 q^{83} - 4 q^{87} + 24 q^{89} - 4 q^{93}+O(q^{100}) $ 2 * q - 2 * q^9 - 10 * q^13 - 2 * q^15 - 2 * q^19 + 8 * q^25 + 10 * q^33 + 18 * q^43 + 12 * q^47 + 14 * q^49 + 12 * q^53 + 10 * q^55 - 12 * q^59 + 24 * q^67 - 6 * q^69 + 2 * q^81 + 4 * q^83 - 4 * q^87 + 24 * q^89 - 4 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3468\mathbb{Z}\right)^\times$.

$n$	$1157$	$1735$	$2893$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	0	0
$5$	1.00000i	0.447214i	0.974679	+	0.223607i	$0.0717831\pi$
$5$	1.00000i	0.447214i	−0.974679	+	0.223607i	$0.928217\pi$
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	− 5.00000i	− 1.50756i	−0.657129	−	0.753778i	$-0.728229\pi$
$11$	− 5.00000i	− 1.50756i	0.657129	−	0.753778i	$-0.271771\pi$
$12$	0	0
$13$	−5.00000	−1.38675	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	−5.00000	−1.38675	−0.693375	+	0.720577i	$0.743877\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.00000i	0.625543i	0.949828	+	0.312772i	$0.101257\pi$
$23$	3.00000i	0.625543i	−0.949828	+	0.312772i	$0.898743\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	2.00000i	0.371391i	0.982607	+	0.185695i	$0.0594537\pi$
$29$	2.00000i	0.371391i	−0.982607	+	0.185695i	$0.940546\pi$
$30$	0	0
$31$	2.00000i	0.359211i	0.983739	+	0.179605i	$0.0574821\pi$
$31$	2.00000i	0.359211i	−0.983739	+	0.179605i	$0.942518\pi$
$32$	0	0
$33$	5.00000	0.870388
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 8.00000i	− 1.31519i	−0.753371	−	0.657596i	$-0.771573\pi$
$37$	− 8.00000i	− 1.31519i	0.753371	−	0.657596i	$-0.228427\pi$
$38$	0	0
$39$	− 5.00000i	− 0.800641i
$40$	0	0
$41$	5.00000i	0.780869i	0.920631	+	0.390434i	$0.127675\pi$
$41$	5.00000i	0.780869i	−0.920631	+	0.390434i	$0.872325\pi$
$42$	0	0
$43$	9.00000	1.37249	0.686244	−	0.727372i	$-0.259258\pi$
$43$	9.00000	1.37249	0.686244	+	0.727372i	$0.259258\pi$
$44$	0	0
$45$	− 1.00000i	− 0.149071i
$46$	0	0
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	0	0
$49$	7.00000	1.00000
$50$	0	0
$51$	0	0
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	5.00000	0.674200
$56$	0	0
$57$	− 1.00000i	− 0.132453i
$58$	0	0
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	4.00000i	0.512148i	0.966657	+	0.256074i	$0.0824290\pi$
$61$	4.00000i	0.512148i	−0.966657	+	0.256074i	$0.917571\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 5.00000i	− 0.620174i
$66$	0	0
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	0	0
$69$	−3.00000	−0.361158
$70$	0	0
$71$	− 12.0000i	− 1.42414i	−0.702109	−	0.712069i	$-0.747758\pi$
$71$	− 12.0000i	− 1.42414i	0.702109	−	0.712069i	$-0.252242\pi$
$72$	0	0
$73$	− 2.00000i	− 0.234082i	−0.993127	−	0.117041i	$-0.962659\pi$
$73$	− 2.00000i	− 0.234082i	0.993127	−	0.117041i	$-0.0373409\pi$
$74$	0	0
$75$	4.00000i	0.461880i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 10.0000i	− 1.12509i	−0.826767	−	0.562544i	$-0.809823\pi$
$79$	− 10.0000i	− 1.12509i	0.826767	−	0.562544i	$-0.190177\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	2.00000	0.219529	0.109764	−	0.993958i	$-0.464990\pi$
$83$	2.00000	0.219529	0.109764	+	0.993958i	$0.464990\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−2.00000	−0.214423
$88$	0	0
$89$	12.0000	1.27200	0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000	1.27200	0.635999	+	0.771690i	$0.280588\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−2.00000	−0.207390
$94$	0	0
$95$	− 1.00000i	− 0.102598i
$96$	0	0
$97$	16.0000i	1.62455i	0.583272	+	0.812277i	$0.301772\pi$
$97$	16.0000i	1.62455i	−0.583272	+	0.812277i	$0.698228\pi$
$98$	0	0
$99$	5.00000i	0.502519i
$100$	0	0
$101$	8.00000	0.796030	0.398015	−	0.917379i	$-0.369699\pi$
$101$	8.00000	0.796030	0.398015	+	0.917379i	$0.369699\pi$
$102$	0	0
$103$	7.00000	0.689730	0.344865	−	0.938652i	$-0.387925\pi$
$103$	7.00000	0.689730	0.344865	+	0.938652i	$0.387925\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 7.00000i	− 0.676716i	−0.941018	−	0.338358i	$-0.890129\pi$
$107$	− 7.00000i	− 0.676716i	0.941018	−	0.338358i	$-0.109871\pi$
$108$	0	0
$109$	4.00000i	0.383131i	0.981480	+	0.191565i	$0.0613564\pi$
$109$	4.00000i	0.383131i	−0.981480	+	0.191565i	$0.938644\pi$
$110$	0	0
$111$	8.00000	0.759326
$112$	0	0
$113$	− 17.0000i	− 1.59923i	−0.600516	−	0.799613i	$-0.705038\pi$
$113$	− 17.0000i	− 1.59923i	0.600516	−	0.799613i	$-0.294962\pi$
$114$	0	0
$115$	−3.00000	−0.279751
$116$	0	0
$117$	5.00000	0.462250
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−14.0000	−1.27273
$122$	0	0
$123$	−5.00000	−0.450835
$124$	0	0
$125$	9.00000i	0.804984i
$126$	0	0
$127$	−9.00000	−0.798621	−0.399310	−	0.916816i	$-0.630750\pi$
$127$	−9.00000	−0.798621	−0.399310	+	0.916816i	$0.630750\pi$
$128$	0	0
$129$	9.00000i	0.792406i
$130$	0	0
$131$	3.00000i	0.262111i	0.991375	+	0.131056i	$0.0418366\pi$
$131$	3.00000i	0.262111i	−0.991375	+	0.131056i	$0.958163\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	22.0000	1.87959	0.939793	−	0.341743i	$-0.111017\pi$
$137$	22.0000	1.87959	0.939793	+	0.341743i	$0.111017\pi$
$138$	0	0
$139$	14.0000i	1.18746i	0.804663	+	0.593732i	$0.202346\pi$
$139$	14.0000i	1.18746i	−0.804663	+	0.593732i	$0.797654\pi$
$140$	0	0
$141$	6.00000i	0.505291i
$142$	0	0
$143$	25.0000i	2.09061i
$144$	0	0
$145$	−2.00000	−0.166091
$146$	0	0
$147$	7.00000i	0.577350i
$148$	0	0
$149$	−22.0000	−1.80231	−0.901155	−	0.433497i	$-0.857280\pi$
$149$	−22.0000	−1.80231	−0.901155	+	0.433497i	$0.857280\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−2.00000	−0.160644
$156$	0	0
$157$	−17.0000	−1.35675	−0.678374	−	0.734717i	$-0.737315\pi$
$157$	−17.0000	−1.35675	−0.678374	+	0.734717i	$0.737315\pi$
$158$	0	0
$159$	6.00000i	0.475831i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	14.0000i	1.09656i	0.836293	+	0.548282i	$0.184718\pi$
$163$	14.0000i	1.09656i	−0.836293	+	0.548282i	$0.815282\pi$
$164$	0	0
$165$	5.00000i	0.389249i
$166$	0	0
$167$	− 23.0000i	− 1.77979i	−0.456162	−	0.889897i	$-0.650776\pi$
$167$	− 23.0000i	− 1.77979i	0.456162	−	0.889897i	$-0.349224\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	− 3.00000i	− 0.228086i	−0.993476	−	0.114043i	$-0.963620\pi$
$173$	− 3.00000i	− 0.228086i	0.993476	−	0.114043i	$-0.0363801\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 6.00000i	− 0.450988i
$178$	0	0
$179$	18.0000	1.34538	0.672692	−	0.739923i	$-0.265138\pi$
$179$	18.0000	1.34538	0.672692	+	0.739923i	$0.265138\pi$
$180$	0	0
$181$	− 14.0000i	− 1.04061i	−0.853980	−	0.520306i	$-0.825818\pi$
$181$	− 14.0000i	− 1.04061i	0.853980	−	0.520306i	$-0.174182\pi$
$182$	0	0
$183$	−4.00000	−0.295689
$184$	0	0
$185$	8.00000	0.588172
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	26.0000	1.88129	0.940647	−	0.339387i	$-0.110219\pi$
$191$	26.0000	1.88129	0.940647	+	0.339387i	$0.110219\pi$
$192$	0	0
$193$	18.0000i	1.29567i	0.761781	+	0.647834i	$0.224325\pi$
$193$	18.0000i	1.29567i	−0.761781	+	0.647834i	$0.775675\pi$
$194$	0	0
$195$	5.00000	0.358057
$196$	0	0
$197$	− 17.0000i	− 1.21120i	−0.795769	−	0.605600i	$-0.792933\pi$
$197$	− 17.0000i	− 1.21120i	0.795769	−	0.605600i	$-0.207067\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	0	0
$201$	12.0000i	0.846415i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−5.00000	−0.349215
$206$	0	0
$207$	− 3.00000i	− 0.208514i
$208$	0	0
$209$	5.00000i	0.345857i
$210$	0	0
$211$	22.0000i	1.51454i	0.653101	+	0.757271i	$0.273468\pi$
$211$	22.0000i	1.51454i	−0.653101	+	0.757271i	$0.726532\pi$
$212$	0	0
$213$	12.0000	0.822226
$214$	0	0
$215$	9.00000i	0.613795i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	2.00000	0.135147
$220$	0	0
$221$	0	0
$222$	0	0
$223$	11.0000	0.736614	0.368307	−	0.929704i	$-0.379937\pi$
$223$	11.0000	0.736614	0.368307	+	0.929704i	$0.379937\pi$
$224$	0	0
$225$	−4.00000	−0.266667
$226$	0	0
$227$	5.00000i	0.331862i	0.986137	+	0.165931i	$0.0530628\pi$
$227$	5.00000i	0.331862i	−0.986137	+	0.165931i	$0.946937\pi$
$228$	0	0
$229$	−22.0000	−1.45380	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000	−1.45380	−0.726900	+	0.686743i	$0.759040\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 13.0000i	− 0.851658i	−0.904804	−	0.425829i	$-0.859982\pi$
$233$	− 13.0000i	− 0.851658i	0.904804	−	0.425829i	$-0.140018\pi$
$234$	0	0
$235$	6.00000i	0.391397i
$236$	0	0
$237$	10.0000	0.649570
$238$	0	0
$239$	12.0000	0.776215	0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000	0.776215	0.388108	+	0.921614i	$0.373129\pi$
$240$	0	0
$241$	− 4.00000i	− 0.257663i	−0.991667	−	0.128831i	$-0.958877\pi$
$241$	− 4.00000i	− 0.257663i	0.991667	−	0.128831i	$-0.0411226\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	7.00000i	0.447214i
$246$	0	0
$247$	5.00000	0.318142
$248$	0	0
$249$	2.00000i	0.126745i
$250$	0	0
$251$	28.0000	1.76734	0.883672	−	0.468106i	$-0.155064\pi$
$251$	28.0000	1.76734	0.883672	+	0.468106i	$0.155064\pi$
$252$	0	0
$253$	15.0000	0.943042
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−8.00000	−0.499026	−0.249513	−	0.968371i	$-0.580271\pi$
$257$	−8.00000	−0.499026	−0.249513	+	0.968371i	$0.580271\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	− 2.00000i	− 0.123797i
$262$	0	0
$263$	−16.0000	−0.986602	−0.493301	−	0.869859i	$-0.664210\pi$
$263$	−16.0000	−0.986602	−0.493301	+	0.869859i	$0.664210\pi$
$264$	0	0
$265$	6.00000i	0.368577i
$266$	0	0
$267$	12.0000i	0.734388i
$268$	0	0
$269$	− 21.0000i	− 1.28039i	−0.768211	−	0.640196i	$-0.778853\pi$
$269$	− 21.0000i	− 1.28039i	0.768211	−	0.640196i	$-0.221147\pi$
$270$	0	0
$271$	25.0000	1.51864	0.759321	−	0.650716i	$-0.225531\pi$
$271$	25.0000	1.51864	0.759321	+	0.650716i	$0.225531\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 20.0000i	− 1.20605i
$276$	0	0
$277$	− 18.0000i	− 1.08152i	−0.841178	−	0.540758i	$-0.818138\pi$
$277$	− 18.0000i	− 1.08152i	0.841178	−	0.540758i	$-0.181862\pi$
$278$	0	0
$279$	− 2.00000i	− 0.119737i
$280$	0	0
$281$	−24.0000	−1.43172	−0.715860	−	0.698244i	$-0.753965\pi$
$281$	−24.0000	−1.43172	−0.715860	+	0.698244i	$0.753965\pi$
$282$	0	0
$283$	30.0000i	1.78331i	0.452711	+	0.891657i	$0.350457\pi$
$283$	30.0000i	1.78331i	−0.452711	+	0.891657i	$0.649543\pi$
$284$	0	0
$285$	1.00000	0.0592349
$286$	0	0
$287$	0	0
$288$	0	0
$289$	0	0
$290$	0	0
$291$	−16.0000	−0.937937
$292$	0	0
$293$	4.00000	0.233682	0.116841	−	0.993151i	$-0.462723\pi$
$293$	4.00000	0.233682	0.116841	+	0.993151i	$0.462723\pi$
$294$	0	0
$295$	− 6.00000i	− 0.349334i
$296$	0	0
$297$	−5.00000	−0.290129
$298$	0	0
$299$	− 15.0000i	− 0.867472i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	8.00000i	0.459588i
$304$	0	0
$305$	−4.00000	−0.229039
$306$	0	0
$307$	−12.0000	−0.684876	−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000	−0.684876	−0.342438	+	0.939540i	$0.611253\pi$
$308$	0	0
$309$	7.00000i	0.398216i
$310$	0	0
$311$	24.0000i	1.36092i	0.732787	+	0.680458i	$0.238219\pi$
$311$	24.0000i	1.36092i	−0.732787	+	0.680458i	$0.761781\pi$
$312$	0	0
$313$	− 16.0000i	− 0.904373i	−0.891923	−	0.452187i	$-0.850644\pi$
$313$	− 16.0000i	− 0.904373i	0.891923	−	0.452187i	$-0.149356\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 30.0000i	− 1.68497i	−0.538721	−	0.842484i	$-0.681092\pi$
$317$	− 30.0000i	− 1.68497i	0.538721	−	0.842484i	$-0.318908\pi$
$318$	0	0
$319$	10.0000	0.559893
$320$	0	0
$321$	7.00000	0.390702
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−20.0000	−1.10940
$326$	0	0
$327$	−4.00000	−0.221201
$328$	0	0
$329$	0	0
$330$	0	0
$331$	11.0000	0.604615	0.302307	−	0.953211i	$-0.402243\pi$
$331$	11.0000	0.604615	0.302307	+	0.953211i	$0.402243\pi$
$332$	0	0
$333$	8.00000i	0.438397i
$334$	0	0
$335$	12.0000i	0.655630i
$336$	0	0
$337$	− 2.00000i	− 0.108947i	−0.998515	−	0.0544735i	$-0.982652\pi$
$337$	− 2.00000i	− 0.108947i	0.998515	−	0.0544735i	$-0.0173480\pi$
$338$	0	0
$339$	17.0000	0.923313
$340$	0	0
$341$	10.0000	0.541530
$342$	0	0
$343$	0	0
$344$	0	0
$345$	− 3.00000i	− 0.161515i
$346$	0	0
$347$	− 4.00000i	− 0.214731i	−0.994220	−	0.107366i	$-0.965758\pi$
$347$	− 4.00000i	− 0.214731i	0.994220	−	0.107366i	$-0.0342415\pi$
$348$	0	0
$349$	7.00000	0.374701	0.187351	−	0.982293i	$-0.440010\pi$
$349$	7.00000	0.374701	0.187351	+	0.982293i	$0.440010\pi$
$350$	0	0
$351$	5.00000i	0.266880i
$352$	0	0
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	0	0
$355$	12.0000	0.636894
$356$	0	0
$357$	0	0
$358$	0	0
$359$	36.0000	1.90001	0.950004	−	0.312239i	$-0.101079\pi$
$359$	36.0000	1.90001	0.950004	+	0.312239i	$0.101079\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	− 14.0000i	− 0.734809i
$364$	0	0
$365$	2.00000	0.104685
$366$	0	0
$367$	− 12.0000i	− 0.626395i	−0.949688	−	0.313197i	$-0.898600\pi$
$367$	− 12.0000i	− 0.626395i	0.949688	−	0.313197i	$-0.101400\pi$
$368$	0	0
$369$	− 5.00000i	− 0.260290i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−6.00000	−0.310668	−0.155334	−	0.987862i	$-0.549645\pi$
$373$	−6.00000	−0.310668	−0.155334	+	0.987862i	$0.549645\pi$
$374$	0	0
$375$	−9.00000	−0.464758
$376$	0	0
$377$	− 10.0000i	− 0.515026i
$378$	0	0
$379$	− 4.00000i	− 0.205466i	−0.994709	−	0.102733i	$-0.967241\pi$
$379$	− 4.00000i	− 0.205466i	0.994709	−	0.102733i	$-0.0327588\pi$
$380$	0	0
$381$	− 9.00000i	− 0.461084i
$382$	0	0
$383$	−20.0000	−1.02195	−0.510976	−	0.859595i	$-0.670716\pi$
$383$	−20.0000	−1.02195	−0.510976	+	0.859595i	$0.670716\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−9.00000	−0.457496
$388$	0	0
$389$	8.00000	0.405616	0.202808	−	0.979219i	$-0.434993\pi$
$389$	8.00000	0.405616	0.202808	+	0.979219i	$0.434993\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−3.00000	−0.151330
$394$	0	0
$395$	10.0000	0.503155
$396$	0	0
$397$	− 4.00000i	− 0.200754i	−0.994949	−	0.100377i	$-0.967995\pi$
$397$	− 4.00000i	− 0.200754i	0.994949	−	0.100377i	$-0.0320049\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	17.0000i	0.848939i	0.905442	+	0.424470i	$0.139539\pi$
$401$	17.0000i	0.848939i	−0.905442	+	0.424470i	$0.860461\pi$
$402$	0	0
$403$	− 10.0000i	− 0.498135i
$404$	0	0
$405$	1.00000i	0.0496904i
$406$	0	0
$407$	−40.0000	−1.98273
$408$	0	0
$409$	5.00000	0.247234	0.123617	−	0.992330i	$-0.460551\pi$
$409$	5.00000	0.247234	0.123617	+	0.992330i	$0.460551\pi$
$410$	0	0
$411$	22.0000i	1.08518i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	2.00000i	0.0981761i
$416$	0	0
$417$	−14.0000	−0.685583
$418$	0	0
$419$	− 28.0000i	− 1.36789i	−0.729534	−	0.683945i	$-0.760263\pi$
$419$	− 28.0000i	− 1.36789i	0.729534	−	0.683945i	$-0.239737\pi$
$420$	0	0
$421$	−29.0000	−1.41337	−0.706687	−	0.707527i	$-0.749811\pi$
$421$	−29.0000	−1.41337	−0.706687	+	0.707527i	$0.749811\pi$
$422$	0	0
$423$	−6.00000	−0.291730
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−25.0000	−1.20701
$430$	0	0
$431$	32.0000i	1.54139i	0.637207	+	0.770693i	$0.280090\pi$
$431$	32.0000i	1.54139i	−0.637207	+	0.770693i	$0.719910\pi$
$432$	0	0
$433$	41.0000	1.97033	0.985167	−	0.171598i	$-0.0548929\pi$
$433$	41.0000	1.97033	0.985167	+	0.171598i	$0.0548929\pi$
$434$	0	0
$435$	− 2.00000i	− 0.0958927i
$436$	0	0
$437$	− 3.00000i	− 0.143509i
$438$	0	0
$439$	28.0000i	1.33637i	0.743996	+	0.668184i	$0.232928\pi$
$439$	28.0000i	1.33637i	−0.743996	+	0.668184i	$0.767072\pi$
$440$	0	0
$441$	−7.00000	−0.333333
$442$	0	0
$443$	2.00000	0.0950229	0.0475114	−	0.998871i	$-0.484871\pi$
$443$	2.00000	0.0950229	0.0475114	+	0.998871i	$0.484871\pi$
$444$	0	0
$445$	12.0000i	0.568855i
$446$	0	0
$447$	− 22.0000i	− 1.04056i
$448$	0	0
$449$	10.0000i	0.471929i	0.971762	+	0.235965i	$0.0758249\pi$
$449$	10.0000i	0.471929i	−0.971762	+	0.235965i	$0.924175\pi$
$450$	0	0
$451$	25.0000	1.17720
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−13.0000	−0.608114	−0.304057	−	0.952654i	$-0.598341\pi$
$457$	−13.0000	−0.608114	−0.304057	+	0.952654i	$0.598341\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	6.00000	0.279448	0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000	0.279448	0.139724	+	0.990190i	$0.455378\pi$
$462$	0	0
$463$	8.00000	0.371792	0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000	0.371792	0.185896	+	0.982569i	$0.440481\pi$
$464$	0	0
$465$	− 2.00000i	− 0.0927478i
$466$	0	0
$467$	−18.0000	−0.832941	−0.416470	−	0.909149i	$-0.636733\pi$
$467$	−18.0000	−0.832941	−0.416470	+	0.909149i	$0.636733\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	− 17.0000i	− 0.783319i
$472$	0	0
$473$	− 45.0000i	− 2.06910i
$474$	0	0
$475$	−4.00000	−0.183533
$476$	0	0
$477$	−6.00000	−0.274721
$478$	0	0
$479$	9.00000i	0.411220i	0.978634	+	0.205610i	$0.0659179\pi$
$479$	9.00000i	0.411220i	−0.978634	+	0.205610i	$0.934082\pi$
$480$	0	0
$481$	40.0000i	1.82384i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−16.0000	−0.726523
$486$	0	0
$487$	34.0000i	1.54069i	0.637629	+	0.770344i	$0.279915\pi$
$487$	34.0000i	1.54069i	−0.637629	+	0.770344i	$0.720085\pi$
$488$	0	0
$489$	−14.0000	−0.633102
$490$	0	0
$491$	−30.0000	−1.35388	−0.676941	−	0.736038i	$-0.736695\pi$
$491$	−30.0000	−1.35388	−0.676941	+	0.736038i	$0.736695\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−5.00000	−0.224733
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 10.0000i	− 0.447661i	−0.974628	−	0.223831i	$-0.928144\pi$
$499$	− 10.0000i	− 0.447661i	0.974628	−	0.223831i	$-0.0718563\pi$
$500$	0	0
$501$	23.0000	1.02756
$502$	0	0
$503$	− 27.0000i	− 1.20387i	−0.798545	−	0.601935i	$-0.794397\pi$
$503$	− 27.0000i	− 1.20387i	0.798545	−	0.601935i	$-0.205603\pi$
$504$	0	0
$505$	8.00000i	0.355995i
$506$	0	0
$507$	12.0000i	0.532939i
$508$	0	0
$509$	−36.0000	−1.59567	−0.797836	−	0.602875i	$-0.794022\pi$
$509$	−36.0000	−1.59567	−0.797836	+	0.602875i	$0.794022\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	1.00000i	0.0441511i
$514$	0	0
$515$	7.00000i	0.308457i
$516$	0	0
$517$	− 30.0000i	− 1.31940i
$518$	0	0
$519$	3.00000	0.131685
$520$	0	0
$521$	13.0000i	0.569540i	0.958596	+	0.284770i	$0.0919173\pi$
$521$	13.0000i	0.569540i	−0.958596	+	0.284770i	$0.908083\pi$
$522$	0	0
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	14.0000	0.608696
$530$	0	0
$531$	6.00000	0.260378
$532$	0	0
$533$	− 25.0000i	− 1.08287i
$534$	0	0
$535$	7.00000	0.302636
$536$	0	0
$537$	18.0000i	0.776757i
$538$	0	0
$539$	− 35.0000i	− 1.50756i
$540$	0	0
$541$	44.0000i	1.89171i	0.324593	+	0.945854i	$0.394773\pi$
$541$	44.0000i	1.89171i	−0.324593	+	0.945854i	$0.605227\pi$
$542$	0	0
$543$	14.0000	0.600798
$544$	0	0
$545$	−4.00000	−0.171341
$546$	0	0
$547$	− 12.0000i	− 0.513083i	−0.966533	−	0.256541i	$-0.917417\pi$
$547$	− 12.0000i	− 0.513083i	0.966533	−	0.256541i	$-0.0825830\pi$
$548$	0	0
$549$	− 4.00000i	− 0.170716i
$550$	0	0
$551$	− 2.00000i	− 0.0852029i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	8.00000i	0.339581i
$556$	0	0
$557$	14.0000	0.593199	0.296600	−	0.955002i	$-0.404147\pi$
$557$	14.0000	0.593199	0.296600	+	0.955002i	$0.404147\pi$
$558$	0	0
$559$	−45.0000	−1.90330
$560$	0	0
$561$	0	0
$562$	0	0
$563$	18.0000	0.758610	0.379305	−	0.925272i	$-0.376163\pi$
$563$	18.0000	0.758610	0.379305	+	0.925272i	$0.376163\pi$
$564$	0	0
$565$	17.0000	0.715195
$566$	0	0
$567$	0	0
$568$	0	0
$569$	24.0000	1.00613	0.503066	−	0.864248i	$-0.332205\pi$
$569$	24.0000	1.00613	0.503066	+	0.864248i	$0.332205\pi$
$570$	0	0
$571$	− 16.0000i	− 0.669579i	−0.942293	−	0.334790i	$-0.891335\pi$
$571$	− 16.0000i	− 0.669579i	0.942293	−	0.334790i	$-0.108665\pi$
$572$	0	0
$573$	26.0000i	1.08617i
$574$	0	0
$575$	12.0000i	0.500435i
$576$	0	0
$577$	−7.00000	−0.291414	−0.145707	−	0.989328i	$-0.546546\pi$
$577$	−7.00000	−0.291414	−0.145707	+	0.989328i	$0.546546\pi$
$578$	0	0
$579$	−18.0000	−0.748054
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 30.0000i	− 1.24247i
$584$	0	0
$585$	5.00000i	0.206725i
$586$	0	0
$587$	−4.00000	−0.165098	−0.0825488	−	0.996587i	$-0.526306\pi$
$587$	−4.00000	−0.165098	−0.0825488	+	0.996587i	$0.526306\pi$
$588$	0	0
$589$	− 2.00000i	− 0.0824086i
$590$	0	0
$591$	17.0000	0.699287
$592$	0	0
$593$	−22.0000	−0.903432	−0.451716	−	0.892162i	$-0.649188\pi$
$593$	−22.0000	−0.903432	−0.451716	+	0.892162i	$0.649188\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	10.0000	0.408589	0.204294	−	0.978909i	$-0.434510\pi$
$599$	10.0000	0.408589	0.204294	+	0.978909i	$0.434510\pi$
$600$	0	0
$601$	− 42.0000i	− 1.71322i	−0.515968	−	0.856608i	$-0.672568\pi$
$601$	− 42.0000i	− 1.71322i	0.515968	−	0.856608i	$-0.327432\pi$
$602$	0	0
$603$	−12.0000	−0.488678
$604$	0	0
$605$	− 14.0000i	− 0.569181i
$606$	0	0
$607$	− 6.00000i	− 0.243532i	−0.992559	−	0.121766i	$-0.961144\pi$
$607$	− 6.00000i	− 0.243532i	0.992559	−	0.121766i	$-0.0388558\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−30.0000	−1.21367
$612$	0	0
$613$	−25.0000	−1.00974	−0.504870	−	0.863195i	$-0.668460\pi$
$613$	−25.0000	−1.00974	−0.504870	+	0.863195i	$0.668460\pi$
$614$	0	0
$615$	− 5.00000i	− 0.201619i
$616$	0	0
$617$	− 18.0000i	− 0.724653i	−0.932051	−	0.362326i	$-0.881983\pi$
$617$	− 18.0000i	− 0.724653i	0.932051	−	0.362326i	$-0.118017\pi$
$618$	0	0
$619$	− 34.0000i	− 1.36658i	−0.730149	−	0.683288i	$-0.760549\pi$
$619$	− 34.0000i	− 1.36658i	0.730149	−	0.683288i	$-0.239451\pi$
$620$	0	0
$621$	3.00000	0.120386
$622$	0	0
$623$	0	0
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	−5.00000	−0.199681
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−9.00000	−0.358284	−0.179142	−	0.983823i	$-0.557332\pi$
$631$	−9.00000	−0.358284	−0.179142	+	0.983823i	$0.557332\pi$
$632$	0	0
$633$	−22.0000	−0.874421
$634$	0	0
$635$	− 9.00000i	− 0.357154i
$636$	0	0
$637$	−35.0000	−1.38675
$638$	0	0
$639$	12.0000i	0.474713i
$640$	0	0
$641$	− 7.00000i	− 0.276483i	−0.990399	−	0.138242i	$-0.955855\pi$
$641$	− 7.00000i	− 0.276483i	0.990399	−	0.138242i	$-0.0441451\pi$
$642$	0	0
$643$	8.00000i	0.315489i	0.987480	+	0.157745i	$0.0504223\pi$
$643$	8.00000i	0.315489i	−0.987480	+	0.157745i	$0.949578\pi$
$644$	0	0
$645$	−9.00000	−0.354375
$646$	0	0
$647$	−22.0000	−0.864909	−0.432455	−	0.901656i	$-0.642352\pi$
$647$	−22.0000	−0.864909	−0.432455	+	0.901656i	$0.642352\pi$
$648$	0	0
$649$	30.0000i	1.17760i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 33.0000i	− 1.29139i	−0.763596	−	0.645695i	$-0.776568\pi$
$653$	− 33.0000i	− 1.29139i	0.763596	−	0.645695i	$-0.223432\pi$
$654$	0	0
$655$	−3.00000	−0.117220
$656$	0	0
$657$	2.00000i	0.0780274i
$658$	0	0
$659$	30.0000	1.16863	0.584317	−	0.811525i	$-0.301362\pi$
$659$	30.0000	1.16863	0.584317	+	0.811525i	$0.301362\pi$
$660$	0	0
$661$	3.00000	0.116686	0.0583432	−	0.998297i	$-0.481418\pi$
$661$	3.00000	0.116686	0.0583432	+	0.998297i	$0.481418\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−6.00000	−0.232321
$668$	0	0
$669$	11.0000i	0.425285i
$670$	0	0
$671$	20.0000	0.772091
$672$	0	0
$673$	34.0000i	1.31060i	0.755367	+	0.655302i	$0.227459\pi$
$673$	34.0000i	1.31060i	−0.755367	+	0.655302i	$0.772541\pi$
$674$	0	0
$675$	− 4.00000i	− 0.153960i
$676$	0	0
$677$	− 41.0000i	− 1.57576i	−0.615830	−	0.787879i	$-0.711179\pi$
$677$	− 41.0000i	− 1.57576i	0.615830	−	0.787879i	$-0.288821\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−5.00000	−0.191600
$682$	0	0
$683$	− 5.00000i	− 0.191320i	−0.995414	−	0.0956598i	$-0.969504\pi$
$683$	− 5.00000i	− 0.191320i	0.995414	−	0.0956598i	$-0.0304961\pi$
$684$	0	0
$685$	22.0000i	0.840577i
$686$	0	0
$687$	− 22.0000i	− 0.839352i
$688$	0	0
$689$	−30.0000	−1.14291
$690$	0	0
$691$	28.0000i	1.06517i	0.846376	+	0.532585i	$0.178779\pi$
$691$	28.0000i	1.06517i	−0.846376	+	0.532585i	$0.821221\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−14.0000	−0.531050
$696$	0	0
$697$	0	0
$698$	0	0
$699$	13.0000	0.491705
$700$	0	0
$701$	40.0000	1.51078	0.755390	−	0.655276i	$-0.227448\pi$
$701$	40.0000	1.51078	0.755390	+	0.655276i	$0.227448\pi$
$702$	0	0
$703$	8.00000i	0.301726i
$704$	0	0
$705$	−6.00000	−0.225973
$706$	0	0
$707$	0	0
$708$	0	0
$709$	10.0000i	0.375558i	0.982211	+	0.187779i	$0.0601289\pi$
$709$	10.0000i	0.375558i	−0.982211	+	0.187779i	$0.939871\pi$
$710$	0	0
$711$	10.0000i	0.375029i
$712$	0	0
$713$	−6.00000	−0.224702
$714$	0	0
$715$	−25.0000	−0.934947
$716$	0	0
$717$	12.0000i	0.448148i
$718$	0	0
$719$	7.00000i	0.261056i	0.991445	+	0.130528i	$0.0416672\pi$
$719$	7.00000i	0.261056i	−0.991445	+	0.130528i	$0.958333\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	4.00000	0.148762
$724$	0	0
$725$	8.00000i	0.297113i
$726$	0	0
$727$	16.0000	0.593407	0.296704	−	0.954970i	$-0.404113\pi$
$727$	16.0000	0.593407	0.296704	+	0.954970i	$0.404113\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−10.0000	−0.369358	−0.184679	−	0.982799i	$-0.559125\pi$
$733$	−10.0000	−0.369358	−0.184679	+	0.982799i	$0.559125\pi$
$734$	0	0
$735$	−7.00000	−0.258199
$736$	0	0
$737$	− 60.0000i	− 2.21013i
$738$	0	0
$739$	23.0000	0.846069	0.423034	−	0.906114i	$-0.360965\pi$
$739$	23.0000	0.846069	0.423034	+	0.906114i	$0.360965\pi$
$740$	0	0
$741$	5.00000i	0.183680i
$742$	0	0
$743$	− 16.0000i	− 0.586983i	−0.955962	−	0.293492i	$-0.905183\pi$
$743$	− 16.0000i	− 0.586983i	0.955962	−	0.293492i	$-0.0948173\pi$
$744$	0	0
$745$	− 22.0000i	− 0.806018i
$746$	0	0
$747$	−2.00000	−0.0731762
$748$	0	0
$749$	0	0
$750$	0	0
$751$	− 22.0000i	− 0.802791i	−0.915905	−	0.401396i	$-0.868525\pi$
$751$	− 22.0000i	− 0.802791i	0.915905	−	0.401396i	$-0.131475\pi$
$752$	0	0
$753$	28.0000i	1.02038i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−41.0000	−1.49017	−0.745085	−	0.666969i	$-0.767591\pi$
$757$	−41.0000	−1.49017	−0.745085	+	0.666969i	$0.767591\pi$
$758$	0	0
$759$	15.0000i	0.544466i
$760$	0	0
$761$	−42.0000	−1.52250	−0.761249	−	0.648459i	$-0.775414\pi$
$761$	−42.0000	−1.52250	−0.761249	+	0.648459i	$0.775414\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	30.0000	1.08324
$768$	0	0
$769$	−39.0000	−1.40638	−0.703188	−	0.711004i	$-0.748241\pi$
$769$	−39.0000	−1.40638	−0.703188	+	0.711004i	$0.748241\pi$
$770$	0	0
$771$	− 8.00000i	− 0.288113i
$772$	0	0
$773$	16.0000	0.575480	0.287740	−	0.957709i	$-0.407096\pi$
$773$	16.0000	0.575480	0.287740	+	0.957709i	$0.407096\pi$
$774$	0	0
$775$	8.00000i	0.287368i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 5.00000i	− 0.179144i
$780$	0	0
$781$	−60.0000	−2.14697
$782$	0	0
$783$	2.00000	0.0714742
$784$	0	0
$785$	− 17.0000i	− 0.606756i
$786$	0	0
$787$	− 32.0000i	− 1.14068i	−0.821410	−	0.570338i	$-0.806812\pi$
$787$	− 32.0000i	− 1.14068i	0.821410	−	0.570338i	$-0.193188\pi$
$788$	0	0
$789$	− 16.0000i	− 0.569615i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	− 20.0000i	− 0.710221i
$794$	0	0
$795$	−6.00000	−0.212798
$796$	0	0
$797$	−8.00000	−0.283375	−0.141687	−	0.989911i	$-0.545253\pi$
$797$	−8.00000	−0.283375	−0.141687	+	0.989911i	$0.545253\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−12.0000	−0.423999
$802$	0	0
$803$	−10.0000	−0.352892
$804$	0	0
$805$	0	0
$806$	0	0
$807$	21.0000	0.739235
$808$	0	0
$809$	− 11.0000i	− 0.386739i	−0.981126	−	0.193370i	$-0.938058\pi$
$809$	− 11.0000i	− 0.386739i	0.981126	−	0.193370i	$-0.0619417\pi$
$810$	0	0
$811$	− 26.0000i	− 0.912983i	−0.889728	−	0.456492i	$-0.849106\pi$
$811$	− 26.0000i	− 0.912983i	0.889728	−	0.456492i	$-0.150894\pi$
$812$	0	0
$813$	25.0000i	0.876788i
$814$	0	0
$815$	−14.0000	−0.490399
$816$	0	0
$817$	−9.00000	−0.314870
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 31.0000i	− 1.08191i	−0.841052	−	0.540954i	$-0.818063\pi$
$821$	− 31.0000i	− 1.08191i	0.841052	−	0.540954i	$-0.181937\pi$
$822$	0	0
$823$	− 8.00000i	− 0.278862i	−0.990232	−	0.139431i	$-0.955473\pi$
$823$	− 8.00000i	− 0.278862i	0.990232	−	0.139431i	$-0.0445274\pi$
$824$	0	0
$825$	20.0000	0.696311
$826$	0	0
$827$	5.00000i	0.173867i	0.996214	+	0.0869335i	$0.0277068\pi$
$827$	5.00000i	0.173867i	−0.996214	+	0.0869335i	$0.972293\pi$
$828$	0	0
$829$	−10.0000	−0.347314	−0.173657	−	0.984806i	$-0.555558\pi$
$829$	−10.0000	−0.347314	−0.173657	+	0.984806i	$0.555558\pi$
$830$	0	0
$831$	18.0000	0.624413
$832$	0	0
$833$	0	0
$834$	0	0
$835$	23.0000	0.795948
$836$	0	0
$837$	2.00000	0.0691301
$838$	0	0
$839$	21.0000i	0.725001i	0.931984	+	0.362500i	$0.118077\pi$
$839$	21.0000i	0.725001i	−0.931984	+	0.362500i	$0.881923\pi$
$840$	0	0
$841$	25.0000	0.862069
$842$	0	0
$843$	− 24.0000i	− 0.826604i
$844$	0	0
$845$	12.0000i	0.412813i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−30.0000	−1.02960
$850$	0	0
$851$	24.0000	0.822709
$852$	0	0
$853$	58.0000i	1.98588i	0.118609	+	0.992941i	$0.462157\pi$
$853$	58.0000i	1.98588i	−0.118609	+	0.992941i	$0.537843\pi$
$854$	0	0
$855$	1.00000i	0.0341993i
$856$	0	0
$857$	6.00000i	0.204956i	0.994735	+	0.102478i	$0.0326771\pi$
$857$	6.00000i	0.204956i	−0.994735	+	0.102478i	$0.967323\pi$
$858$	0	0
$859$	−36.0000	−1.22830	−0.614152	−	0.789188i	$-0.710502\pi$
$859$	−36.0000	−1.22830	−0.614152	+	0.789188i	$0.710502\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	4.00000	0.136162	0.0680808	−	0.997680i	$-0.478312\pi$
$863$	4.00000	0.136162	0.0680808	+	0.997680i	$0.478312\pi$
$864$	0	0
$865$	3.00000	0.102003
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−50.0000	−1.69613
$870$	0	0
$871$	−60.0000	−2.03302
$872$	0	0
$873$	− 16.0000i	− 0.541518i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	26.0000i	0.877958i	0.898497	+	0.438979i	$0.144660\pi$
$877$	26.0000i	0.877958i	−0.898497	+	0.438979i	$0.855340\pi$
$878$	0	0
$879$	4.00000i	0.134917i
$880$	0	0
$881$	14.0000i	0.471672i	0.971793	+	0.235836i	$0.0757828\pi$
$881$	14.0000i	0.471672i	−0.971793	+	0.235836i	$0.924217\pi$
$882$	0	0
$883$	−27.0000	−0.908622	−0.454311	−	0.890843i	$-0.650115\pi$
$883$	−27.0000	−0.908622	−0.454311	+	0.890843i	$0.650115\pi$
$884$	0	0
$885$	6.00000	0.201688
$886$	0	0
$887$	27.0000i	0.906571i	0.891365	+	0.453286i	$0.149748\pi$
$887$	27.0000i	0.906571i	−0.891365	+	0.453286i	$0.850252\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	− 5.00000i	− 0.167506i
$892$	0	0
$893$	−6.00000	−0.200782
$894$	0	0
$895$	18.0000i	0.601674i
$896$	0	0
$897$	15.0000	0.500835
$898$	0	0
$899$	−4.00000	−0.133407
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	14.0000	0.465376
$906$	0	0
$907$	2.00000i	0.0664089i	0.999449	+	0.0332045i	$0.0105712\pi$
$907$	2.00000i	0.0664089i	−0.999449	+	0.0332045i	$0.989429\pi$
$908$	0	0
$909$	−8.00000	−0.265343
$910$	0	0
$911$	− 17.0000i	− 0.563235i	−0.959527	−	0.281618i	$-0.909129\pi$
$911$	− 17.0000i	− 0.563235i	0.959527	−	0.281618i	$-0.0908709\pi$
$912$	0	0
$913$	− 10.0000i	− 0.330952i
$914$	0	0
$915$	− 4.00000i	− 0.132236i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−15.0000	−0.494804	−0.247402	−	0.968913i	$-0.579577\pi$
$919$	−15.0000	−0.494804	−0.247402	+	0.968913i	$0.579577\pi$
$920$	0	0
$921$	− 12.0000i	− 0.395413i
$922$	0	0
$923$	60.0000i	1.97492i
$924$	0	0
$925$	− 32.0000i	− 1.05215i
$926$	0	0
$927$	−7.00000	−0.229910
$928$	0	0
$929$	31.0000i	1.01708i	0.861039	+	0.508539i	$0.169814\pi$
$929$	31.0000i	1.01708i	−0.861039	+	0.508539i	$0.830186\pi$
$930$	0	0
$931$	−7.00000	−0.229416
$932$	0	0
$933$	−24.0000	−0.785725
$934$	0	0
$935$	0	0
$936$	0	0
$937$	22.0000	0.718709	0.359354	−	0.933201i	$-0.382997\pi$
$937$	22.0000	0.718709	0.359354	+	0.933201i	$0.382997\pi$
$938$	0	0
$939$	16.0000	0.522140
$940$	0	0
$941$	34.0000i	1.10837i	0.832394	+	0.554184i	$0.186970\pi$
$941$	34.0000i	1.10837i	−0.832394	+	0.554184i	$0.813030\pi$
$942$	0	0
$943$	−15.0000	−0.488467
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 28.0000i	− 0.909878i	−0.890523	−	0.454939i	$-0.849661\pi$
$947$	− 28.0000i	− 0.909878i	0.890523	−	0.454939i	$-0.150339\pi$
$948$	0	0
$949$	10.0000i	0.324614i
$950$	0	0
$951$	30.0000	0.972817
$952$	0	0
$953$	36.0000	1.16615	0.583077	−	0.812417i	$-0.301849\pi$
$953$	36.0000	1.16615	0.583077	+	0.812417i	$0.301849\pi$
$954$	0	0
$955$	26.0000i	0.841340i
$956$	0	0
$957$	10.0000i	0.323254i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	27.0000	0.870968
$962$	0	0
$963$	7.00000i	0.225572i
$964$	0	0
$965$	−18.0000	−0.579441
$966$	0	0
$967$	−19.0000	−0.610999	−0.305499	−	0.952192i	$-0.598823\pi$
$967$	−19.0000	−0.610999	−0.305499	+	0.952192i	$0.598823\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	22.0000	0.706014	0.353007	−	0.935621i	$-0.385159\pi$
$971$	22.0000	0.706014	0.353007	+	0.935621i	$0.385159\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	− 20.0000i	− 0.640513i
$976$	0	0
$977$	−14.0000	−0.447900	−0.223950	−	0.974601i	$-0.571895\pi$
$977$	−14.0000	−0.447900	−0.223950	+	0.974601i	$0.571895\pi$
$978$	0	0
$979$	− 60.0000i	− 1.91761i
$980$	0	0
$981$	− 4.00000i	− 0.127710i
$982$	0	0
$983$	51.0000i	1.62665i	0.581811	+	0.813324i	$0.302344\pi$
$983$	51.0000i	1.62665i	−0.581811	+	0.813324i	$0.697656\pi$
$984$	0	0
$985$	17.0000	0.541665
$986$	0	0
$987$	0	0
$988$	0	0
$989$	27.0000i	0.858550i
$990$	0	0
$991$	20.0000i	0.635321i	0.948205	+	0.317660i	$0.102897\pi$
$991$	20.0000i	0.635321i	−0.948205	+	0.317660i	$0.897103\pi$
$992$	0	0
$993$	11.0000i	0.349074i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	2.00000i	0.0633406i	0.999498	+	0.0316703i	$0.0100827\pi$
$997$	2.00000i	0.0633406i	−0.999498	+	0.0316703i	$0.989917\pi$
$998$	0	0
$999$	−8.00000	−0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3468.2.b.a.577.2		2
17.2	even	8		3468.2.j.f.829.2		4
17.4	even	4		204.2.a.b.1.1	✓	1
17.8	even	8		3468.2.j.f.3217.2		4
17.9	even	8		3468.2.j.f.3217.1		4
17.13	even	4		3468.2.a.a.1.1		1
17.15	even	8		3468.2.j.f.829.1		4
17.16	even	2	inner	3468.2.b.a.577.1		2
51.38	odd	4		612.2.a.b.1.1		1
68.55	odd	4		816.2.a.e.1.1		1
85.4	even	4		5100.2.a.f.1.1		1
85.38	odd	4		5100.2.g.l.2449.2		2
85.72	odd	4		5100.2.g.l.2449.1		2
119.55	odd	4		9996.2.a.b.1.1		1
136.21	even	4		3264.2.a.f.1.1		1
136.123	odd	4		3264.2.a.u.1.1		1
204.191	even	4		2448.2.a.e.1.1		1
408.293	odd	4		9792.2.a.bo.1.1		1
408.395	even	4		9792.2.a.bn.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
204.2.a.b.1.1	✓	1	17.4	even	4
612.2.a.b.1.1		1	51.38	odd	4
816.2.a.e.1.1		1	68.55	odd	4
2448.2.a.e.1.1		1	204.191	even	4
3264.2.a.f.1.1		1	136.21	even	4
3264.2.a.u.1.1		1	136.123	odd	4
3468.2.a.a.1.1		1	17.13	even	4
3468.2.b.a.577.1		2	17.16	even	2	inner
3468.2.b.a.577.2		2	1.1	even	1	trivial
3468.2.j.f.829.1		4	17.15	even	8
3468.2.j.f.829.2		4	17.2	even	8
3468.2.j.f.3217.1		4	17.9	even	8
3468.2.j.f.3217.2		4	17.8	even	8
5100.2.a.f.1.1		1	85.4	even	4
5100.2.g.l.2449.1		2	85.72	odd	4
5100.2.g.l.2449.2		2	85.38	odd	4
9792.2.a.bn.1.1		1	408.395	even	4
9792.2.a.bo.1.1		1	408.293	odd	4
9996.2.a.b.1.1		1	119.55	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3468 = 2^{2} \cdot 3 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3468.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +1.00000i q^{5} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +1.00000i q^{5} -1.00000 q^{9} -5.00000i q^{11} -5.00000 q^{13} -1.00000 q^{15} -1.00000 q^{19} +3.00000i q^{23} +4.00000 q^{25} -1.00000i q^{27} +2.00000i q^{29} +2.00000i q^{31} +5.00000 q^{33} -8.00000i q^{37} -5.00000i q^{39} +5.00000i q^{41} +9.00000 q^{43} -1.00000i q^{45} +6.00000 q^{47} +7.00000 q^{49} +6.00000 q^{53} +5.00000 q^{55} -1.00000i q^{57} -6.00000 q^{59} +4.00000i q^{61} -5.00000i q^{65} +12.0000 q^{67} -3.00000 q^{69} -12.0000i q^{71} -2.00000i q^{73} +4.00000i q^{75} -10.0000i q^{79} +1.00000 q^{81} +2.00000 q^{83} -2.00000 q^{87} +12.0000 q^{89} -2.00000 q^{93} -1.00000i q^{95} +16.0000i q^{97} +5.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^9	\( 2 q - 2 q^{9} - 10 q^{13} - 2 q^{15} - 2 q^{19} + 8 q^{25} + 10 q^{33} + 18 q^{43} + 12 q^{47} + 14 q^{49} + 12 q^{53} + 10 q^{55} - 12 q^{59} + 24 q^{67} - 6 q^{69} + 2 q^{81} + 4 q^{83} - 4 q^{87} + 24 q^{89} - 4 q^{93}+O(q^{100}) \) 2 * q - 2 * q^9 - 10 * q^13 - 2 * q^15 - 2 * q^19 + 8 * q^25 + 10 * q^33 + 18 * q^43 + 12 * q^47 + 14 * q^49 + 12 * q^53 + 10 * q^55 - 12 * q^59 + 24 * q^67 - 6 * q^69 + 2 * q^81 + 4 * q^83 - 4 * q^87 + 24 * q^89 - 4 * q^93

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