LMFDB - Embedded newform 3400.2.e.i.2449.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3400,2,Mod(2449,3400)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3400.2449"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3400 = 2^{3} \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3400.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,0,0,0,0,-16,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$27.1491366872$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.3534400.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 3x^{4} + 16x^{3} + x^{2} - 12x + 40 $ x^6 - 2x^5 - 3x^4 + 16x^3 + x^2 - 12x + 40
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 680)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2449.6
Root			$0.627553 - 1.14620i$ of defining polynomial
Character	$\chi$	$=$	3400.2449
Dual form			3400.2.e.i.2449.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.29240i q^{3} +1.54751i q^{7} -7.83991 q^{9} -3.54751 q^{11} +2.25511i q^{13} +1.00000i q^{17} -4.83991 q^{19} -5.09501 q^{21} +1.54751i q^{23} -15.9349i q^{27} +2.83991 q^{29} -7.87720 q^{31} -11.6798i q^{33} +0.584803i q^{37} -7.42471 q^{39} +9.09501 q^{41} +2.00000i q^{43} +6.83991i q^{47} +4.60522 q^{49} -3.29240 q^{51} -12.9145i q^{53} -15.9349i q^{57} +13.3501 q^{59} +3.74489 q^{61} -12.1323i q^{63} +10.2646i q^{67} -5.09501 q^{69} -4.78219 q^{71} -6.83991i q^{73} -5.48979i q^{77} -2.64252 q^{79} +28.9444 q^{81} +2.51021i q^{83} +9.35012i q^{87} -18.0095 q^{89} -3.48979 q^{91} -25.9349i q^{93} -11.7449i q^{97} +27.8121 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 16 q^{9} - 4 q^{11} + 2 q^{19} + 4 q^{21} - 14 q^{29} - 6 q^{31} + 14 q^{39} + 20 q^{41} - 14 q^{49} - 6 q^{51} + 42 q^{59} + 26 q^{61} + 4 q^{69} - 22 q^{71} + 36 q^{79} + 22 q^{81} - 22 q^{89} - 28 q^{91}+ \cdots + 60 q^{99}+O(q^{100}) $ 6 * q - 16 * q^9 - 4 * q^11 + 2 * q^19 + 4 * q^21 - 14 * q^29 - 6 * q^31 + 14 * q^39 + 20 * q^41 - 14 * q^49 - 6 * q^51 + 42 * q^59 + 26 * q^61 + 4 * q^69 - 22 * q^71 + 36 * q^79 + 22 * q^81 - 22 * q^89 - 28 * q^91 + 60 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3400\mathbb{Z}\right)^\times$.

$n$	$1601$	$1701$	$2177$	$2551$
$\chi(n)$	$1$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	3.29240i	1.90087i	0.310925	+	0.950434i	$0.399361\pi$
$3$	3.29240i	1.90087i	−0.310925	+	0.950434i	$0.600639\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.54751i	0.584903i	0.956280	+	0.292451i	$0.0944709\pi$
$7$	1.54751i	0.584903i	−0.956280	+	0.292451i	$0.905529\pi$
$8$	0	0
$9$	−7.83991	−2.61330
$10$	0	0
$11$	−3.54751	−1.06961	−0.534807	−	0.844974i	$-0.679616\pi$
$11$	−3.54751	−1.06961	−0.534807	+	0.844974i	$0.679616\pi$
$12$	0	0
$13$	2.25511i	0.625454i	0.949843	+	0.312727i	$0.101242\pi$
$13$	2.25511i	0.625454i	−0.949843	+	0.312727i	$0.898758\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.00000i	0.242536i
$17$	1.00000i	0.242536i
$18$	0	0
$19$	−4.83991	−1.11035	−0.555176	−	0.831733i	$-0.687349\pi$
$19$	−4.83991	−1.11035	−0.555176	+	0.831733i	$0.687349\pi$
$20$	0	0
$21$	−5.09501	−1.11182
$22$	0	0
$23$	1.54751i	0.322677i	0.986899	+	0.161339i	$0.0515812\pi$
$23$	1.54751i	0.322677i	−0.986899	+	0.161339i	$0.948419\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 15.9349i	− 3.06668i
$28$	0	0
$29$	2.83991	0.527358	0.263679	−	0.964611i	$-0.415064\pi$
$29$	2.83991	0.527358	0.263679	+	0.964611i	$0.415064\pi$
$30$	0	0
$31$	−7.87720	−1.41479	−0.707394	−	0.706820i	$-0.750129\pi$
$31$	−7.87720	−1.41479	−0.707394	+	0.706820i	$0.750129\pi$
$32$	0	0
$33$	− 11.6798i	− 2.03320i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0.584803i	0.0961410i	0.998844	+	0.0480705i	$0.0153072\pi$
$37$	0.584803i	0.0961410i	−0.998844	+	0.0480705i	$0.984693\pi$
$38$	0	0
$39$	−7.42471	−1.18891
$40$	0	0
$41$	9.09501	1.42040	0.710201	−	0.703999i	$-0.248604\pi$
$41$	9.09501	1.42040	0.710201	+	0.703999i	$0.248604\pi$
$42$	0	0
$43$	2.00000i	0.304997i	0.988304	+	0.152499i	$0.0487319\pi$
$43$	2.00000i	0.304997i	−0.988304	+	0.152499i	$0.951268\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.83991i	0.997703i	0.866687	+	0.498852i	$0.166245\pi$
$47$	6.83991i	0.997703i	−0.866687	+	0.498852i	$0.833755\pi$
$48$	0	0
$49$	4.60522	0.657889
$50$	0	0
$51$	−3.29240	−0.461028
$52$	0	0
$53$	− 12.9145i	− 1.77394i	−0.461824	−	0.886972i	$-0.652805\pi$
$53$	− 12.9145i	− 1.77394i	0.461824	−	0.886972i	$-0.347195\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 15.9349i	− 2.11063i
$58$	0	0
$59$	13.3501	1.73804	0.869019	−	0.494779i	$-0.164751\pi$
$59$	13.3501	1.73804	0.869019	+	0.494779i	$0.164751\pi$
$60$	0	0
$61$	3.74489	0.479485	0.239742	−	0.970837i	$-0.422937\pi$
$61$	3.74489	0.479485	0.239742	+	0.970837i	$0.422937\pi$
$62$	0	0
$63$	− 12.1323i	− 1.52853i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	10.2646i	1.25402i	0.779010	+	0.627011i	$0.215722\pi$
$67$	10.2646i	1.25402i	−0.779010	+	0.627011i	$0.784278\pi$
$68$	0	0
$69$	−5.09501	−0.613368
$70$	0	0
$71$	−4.78219	−0.567542	−0.283771	−	0.958892i	$-0.591586\pi$
$71$	−4.78219	−0.567542	−0.283771	+	0.958892i	$0.591586\pi$
$72$	0	0
$73$	− 6.83991i	− 0.800551i	−0.916395	−	0.400275i	$-0.868914\pi$
$73$	− 6.83991i	− 0.800551i	0.916395	−	0.400275i	$-0.131086\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 5.48979i	− 0.625620i
$78$	0	0
$79$	−2.64252	−0.297307	−0.148653	−	0.988889i	$-0.547494\pi$
$79$	−2.64252	−0.297307	−0.148653	+	0.988889i	$0.547494\pi$
$80$	0	0
$81$	28.9444	3.21605
$82$	0	0
$83$	2.51021i	0.275531i	0.990465	+	0.137766i	$0.0439921\pi$
$83$	2.51021i	0.275531i	−0.990465	+	0.137766i	$0.956008\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	9.35012i	1.00244i
$88$	0	0
$89$	−18.0095	−1.90900	−0.954502	−	0.298203i	$-0.903613\pi$
$89$	−18.0095	−1.90900	−0.954502	+	0.298203i	$0.903613\pi$
$90$	0	0
$91$	−3.48979	−0.365829
$92$	0	0
$93$	− 25.9349i	− 2.68933i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 11.7449i	− 1.19251i	−0.802794	−	0.596257i	$-0.796654\pi$
$97$	− 11.7449i	− 1.19251i	0.802794	−	0.596257i	$-0.203346\pi$
$98$	0	0
$99$	27.8121	2.79522
$100$	0	0
$101$	−10.5848	−1.05323	−0.526614	−	0.850105i	$-0.676539\pi$
$101$	−10.5848	−1.05323	−0.526614	+	0.850105i	$0.676539\pi$
$102$	0	0
$103$	− 9.60522i	− 0.946431i	−0.880947	−	0.473215i	$-0.843093\pi$
$103$	− 9.60522i	− 0.946431i	0.880947	−	0.473215i	$-0.156907\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 19.2273i	− 1.85878i	−0.369105	−	0.929388i	$-0.620336\pi$
$107$	− 19.2273i	− 1.85878i	0.369105	−	0.929388i	$-0.379664\pi$
$108$	0	0
$109$	−9.93492	−0.951593	−0.475796	−	0.879555i	$-0.657840\pi$
$109$	−9.93492	−0.951593	−0.475796	+	0.879555i	$0.657840\pi$
$110$	0	0
$111$	−1.92541	−0.182752
$112$	0	0
$113$	12.0095i	1.12976i	0.825173	+	0.564880i	$0.191078\pi$
$113$	12.0095i	1.12976i	−0.825173	+	0.564880i	$0.808922\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 17.6798i	− 1.63450i
$118$	0	0
$119$	−1.54751	−0.141860
$120$	0	0
$121$	1.58480	0.144073
$122$	0	0
$123$	29.9444i	2.70000i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 13.9349i	− 1.23652i	−0.785972	−	0.618262i	$-0.787837\pi$
$127$	− 13.9349i	− 1.23652i	0.785972	−	0.618262i	$-0.212163\pi$
$128$	0	0
$129$	−6.58480	−0.579760
$130$	0	0
$131$	−17.2273	−1.50516	−0.752579	−	0.658502i	$-0.771190\pi$
$131$	−17.2273	−1.50516	−0.752579	+	0.658502i	$0.771190\pi$
$132$	0	0
$133$	− 7.48979i	− 0.649447i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	8.51021i	0.727076i	0.931579	+	0.363538i	$0.118431\pi$
$137$	8.51021i	0.727076i	−0.931579	+	0.363538i	$0.881569\pi$
$138$	0	0
$139$	0.452493	0.0383800	0.0191900	−	0.999816i	$-0.493891\pi$
$139$	0.452493	0.0383800	0.0191900	+	0.999816i	$0.493891\pi$
$140$	0	0
$141$	−22.5197	−1.89650
$142$	0	0
$143$	− 8.00000i	− 0.668994i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	15.1622i	1.25056i
$148$	0	0
$149$	21.8698	1.79165	0.895824	−	0.444410i	$-0.146587\pi$
$149$	21.8698	1.79165	0.895824	+	0.444410i	$0.146587\pi$
$150$	0	0
$151$	10.5848	0.861379	0.430690	−	0.902500i	$-0.358270\pi$
$151$	10.5848	0.861379	0.430690	+	0.902500i	$0.358270\pi$
$152$	0	0
$153$	− 7.83991i	− 0.633819i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 11.1696i	− 0.891432i	−0.895175	−	0.445716i	$-0.852949\pi$
$157$	− 11.1696i	− 0.891432i	0.895175	−	0.445716i	$-0.147051\pi$
$158$	0	0
$159$	42.5197	3.37203
$160$	0	0
$161$	−2.39478	−0.188735
$162$	0	0
$163$	− 5.15273i	− 0.403593i	−0.979427	−	0.201796i	$-0.935322\pi$
$163$	− 5.15273i	− 0.403593i	0.979427	−	0.201796i	$-0.0646780\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	23.6221i	1.82793i	0.405790	+	0.913966i	$0.366997\pi$
$167$	23.6221i	1.82793i	−0.405790	+	0.913966i	$0.633003\pi$
$168$	0	0
$169$	7.91450	0.608808
$170$	0	0
$171$	37.9444	2.90168
$172$	0	0
$173$	19.2850i	1.46621i	0.680113	+	0.733107i	$0.261931\pi$
$173$	19.2850i	1.46621i	−0.680113	+	0.733107i	$0.738069\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	43.9540i	3.30378i
$178$	0	0
$179$	4.90499	0.366616	0.183308	−	0.983056i	$-0.441319\pi$
$179$	4.90499	0.366616	0.183308	+	0.983056i	$0.441319\pi$
$180$	0	0
$181$	18.2646	1.35760	0.678799	−	0.734324i	$-0.262501\pi$
$181$	18.2646	1.35760	0.678799	+	0.734324i	$0.262501\pi$
$182$	0	0
$183$	12.3297i	0.911438i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 3.54751i	− 0.259419i
$188$	0	0
$189$	24.6594	1.79371
$190$	0	0
$191$	−25.1696	−1.82121	−0.910604	−	0.413279i	$-0.864383\pi$
$191$	−25.1696	−1.82121	−0.910604	+	0.413279i	$0.864383\pi$
$192$	0	0
$193$	4.07459i	0.293296i	0.989189	+	0.146648i	$0.0468484\pi$
$193$	4.07459i	0.293296i	−0.989189	+	0.146648i	$0.953152\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 21.7544i	− 1.54994i	−0.632000	−	0.774969i	$-0.717766\pi$
$197$	− 21.7544i	− 1.54994i	0.632000	−	0.774969i	$-0.282234\pi$
$198$	0	0
$199$	−19.9926	−1.41724	−0.708620	−	0.705590i	$-0.750682\pi$
$199$	−19.9926	−1.41724	−0.708620	+	0.705590i	$0.750682\pi$
$200$	0	0
$201$	−33.7952	−2.38373
$202$	0	0
$203$	4.39478i	0.308453i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 12.1323i	− 0.843254i
$208$	0	0
$209$	17.1696	1.18765
$210$	0	0
$211$	−1.11189	−0.0765456	−0.0382728	−	0.999267i	$-0.512186\pi$
$211$	−1.11189	−0.0765456	−0.0382728	+	0.999267i	$0.512186\pi$
$212$	0	0
$213$	− 15.7449i	− 1.07882i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 12.1900i	− 0.827513i
$218$	0	0
$219$	22.5197	1.52174
$220$	0	0
$221$	−2.25511	−0.151695
$222$	0	0
$223$	5.78574i	0.387441i	0.981057	+	0.193721i	$0.0620556\pi$
$223$	5.78574i	0.387441i	−0.981057	+	0.193721i	$0.937944\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	15.5570i	1.03256i	0.856421	+	0.516278i	$0.172683\pi$
$227$	15.5570i	1.03256i	−0.856421	+	0.516278i	$0.827317\pi$
$228$	0	0
$229$	−7.60522	−0.502567	−0.251284	−	0.967913i	$-0.580853\pi$
$229$	−7.60522	−0.502567	−0.251284	+	0.967913i	$0.580853\pi$
$230$	0	0
$231$	18.0746	1.18922
$232$	0	0
$233$	9.93492i	0.650858i	0.945566	+	0.325429i	$0.105509\pi$
$233$	9.93492i	0.650858i	−0.945566	+	0.325429i	$0.894491\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 8.70024i	− 0.565141i
$238$	0	0
$239$	8.51021	0.550480	0.275240	−	0.961376i	$-0.411243\pi$
$239$	8.51021	0.550480	0.275240	+	0.961376i	$0.411243\pi$
$240$	0	0
$241$	−29.6243	−1.90827	−0.954133	−	0.299383i	$-0.903219\pi$
$241$	−29.6243	−1.90827	−0.954133	+	0.299383i	$0.903219\pi$
$242$	0	0
$243$	47.4919i	3.04661i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 10.9145i	− 0.694473i
$248$	0	0
$249$	−8.26462	−0.523749
$250$	0	0
$251$	−2.19003	−0.138233	−0.0691166	−	0.997609i	$-0.522018\pi$
$251$	−2.19003	−0.138233	−0.0691166	+	0.997609i	$0.522018\pi$
$252$	0	0
$253$	− 5.48979i	− 0.345140i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.67982i	0.104784i	0.998627	+	0.0523920i	$0.0166845\pi$
$257$	1.67982i	0.104784i	−0.998627	+	0.0523920i	$0.983315\pi$
$258$	0	0
$259$	−0.904987	−0.0562331
$260$	0	0
$261$	−22.2646	−1.37815
$262$	0	0
$263$	31.7639i	1.95865i	0.202299	+	0.979324i	$0.435159\pi$
$263$	31.7639i	1.95865i	−0.202299	+	0.979324i	$0.564841\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 59.2946i	− 3.62877i
$268$	0	0
$269$	3.23468	0.197222	0.0986111	−	0.995126i	$-0.468560\pi$
$269$	3.23468	0.197222	0.0986111	+	0.995126i	$0.468560\pi$
$270$	0	0
$271$	10.8304	0.657900	0.328950	−	0.944347i	$-0.393305\pi$
$271$	10.8304	0.657900	0.328950	+	0.944347i	$0.393305\pi$
$272$	0	0
$273$	− 11.4898i	− 0.695394i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 23.1696i	− 1.39213i	−0.717980	−	0.696063i	$-0.754933\pi$
$277$	− 23.1696i	− 1.39213i	0.717980	−	0.696063i	$-0.245067\pi$
$278$	0	0
$279$	61.7566	3.69727
$280$	0	0
$281$	−15.7449	−0.939262	−0.469631	−	0.882863i	$-0.655613\pi$
$281$	−15.7449	−0.939262	−0.469631	+	0.882863i	$0.655613\pi$
$282$	0	0
$283$	9.61259i	0.571409i	0.958318	+	0.285704i	$0.0922276\pi$
$283$	9.61259i	0.571409i	−0.958318	+	0.285704i	$0.907772\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	14.0746i	0.830797i
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	38.6689	2.26681
$292$	0	0
$293$	− 5.78574i	− 0.338006i	−0.985616	−	0.169003i	$-0.945945\pi$
$293$	− 5.78574i	− 0.338006i	0.985616	−	0.169003i	$-0.0540548\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	56.5292i	3.28016i
$298$	0	0
$299$	−3.48979	−0.201820
$300$	0	0
$301$	−3.09501	−0.178394
$302$	0	0
$303$	− 34.8494i	− 2.00205i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 28.5848i	− 1.63142i	−0.578460	−	0.815710i	$-0.696346\pi$
$307$	− 28.5848i	− 1.63142i	0.578460	−	0.815710i	$-0.303654\pi$
$308$	0	0
$309$	31.6243	1.79904
$310$	0	0
$311$	−25.7375	−1.45944	−0.729721	−	0.683745i	$-0.760350\pi$
$311$	−25.7375	−1.45944	−0.729721	+	0.683745i	$0.760350\pi$
$312$	0	0
$313$	17.2442i	0.974700i	0.873207	+	0.487350i	$0.162036\pi$
$313$	17.2442i	0.974700i	−0.873207	+	0.487350i	$0.837964\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 9.63898i	− 0.541379i	−0.962667	−	0.270689i	$-0.912748\pi$
$317$	− 9.63898i	− 0.541379i	0.962667	−	0.270689i	$-0.0872516\pi$
$318$	0	0
$319$	−10.0746	−0.564069
$320$	0	0
$321$	63.3041	3.53329
$322$	0	0
$323$	− 4.83991i	− 0.269300i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 32.7098i	− 1.80885i
$328$	0	0
$329$	−10.5848	−0.583559
$330$	0	0
$331$	13.4993	0.741989	0.370994	−	0.928635i	$-0.379017\pi$
$331$	13.4993	0.741989	0.370994	+	0.928635i	$0.379017\pi$
$332$	0	0
$333$	− 4.58480i	− 0.251246i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	2.72447i	0.148412i	0.997243	+	0.0742058i	$0.0236422\pi$
$337$	2.72447i	0.148412i	−0.997243	+	0.0742058i	$0.976358\pi$
$338$	0	0
$339$	−39.5401	−2.14753
$340$	0	0
$341$	27.9444	1.51328
$342$	0	0
$343$	17.9592i	0.969703i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 20.4620i	− 1.09846i	−0.835672	−	0.549229i	$-0.814921\pi$
$347$	− 20.4620i	− 1.09846i	0.835672	−	0.549229i	$-0.185079\pi$
$348$	0	0
$349$	8.70024	0.465713	0.232856	−	0.972511i	$-0.425193\pi$
$349$	8.70024	0.465713	0.232856	+	0.972511i	$0.425193\pi$
$350$	0	0
$351$	35.9349	1.91806
$352$	0	0
$353$	− 14.5292i	− 0.773313i	−0.922224	−	0.386657i	$-0.873630\pi$
$353$	− 14.5292i	− 0.773313i	0.922224	−	0.386657i	$-0.126370\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 5.09501i	− 0.269657i
$358$	0	0
$359$	4.24559	0.224074	0.112037	−	0.993704i	$-0.464263\pi$
$359$	4.24559	0.224074	0.112037	+	0.993704i	$0.464263\pi$
$360$	0	0
$361$	4.42471	0.232880
$362$	0	0
$363$	5.21781i	0.273864i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 12.6425i	− 0.659934i	−0.943992	−	0.329967i	$-0.892962\pi$
$367$	− 12.6425i	− 0.659934i	0.943992	−	0.329967i	$-0.107038\pi$
$368$	0	0
$369$	−71.3041	−3.71194
$370$	0	0
$371$	19.9853	1.03758
$372$	0	0
$373$	20.0190i	1.03655i	0.855215	+	0.518273i	$0.173425\pi$
$373$	20.0190i	1.03655i	−0.855215	+	0.518273i	$0.826575\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6.40429i	0.329838i
$378$	0	0
$379$	−12.4525	−0.639642	−0.319821	−	0.947478i	$-0.603623\pi$
$379$	−12.4525	−0.639642	−0.319821	+	0.947478i	$0.603623\pi$
$380$	0	0
$381$	45.8794	2.35047
$382$	0	0
$383$	6.95534i	0.355401i	0.984085	+	0.177701i	$0.0568659\pi$
$383$	6.95534i	0.355401i	−0.984085	+	0.177701i	$0.943134\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 15.6798i	− 0.797050i
$388$	0	0
$389$	−31.0950	−1.57658	−0.788290	−	0.615304i	$-0.789033\pi$
$389$	−31.0950	−1.57658	−0.788290	+	0.615304i	$0.789033\pi$
$390$	0	0
$391$	−1.54751	−0.0782608
$392$	0	0
$393$	− 56.7193i	− 2.86111i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	7.53063i	0.377951i	0.981982	+	0.188976i	$0.0605168\pi$
$397$	7.53063i	0.377951i	−0.981982	+	0.188976i	$0.939483\pi$
$398$	0	0
$399$	24.6594	1.23451
$400$	0	0
$401$	−6.65940	−0.332554	−0.166277	−	0.986079i	$-0.553175\pi$
$401$	−6.65940	−0.332554	−0.166277	+	0.986079i	$0.553175\pi$
$402$	0	0
$403$	− 17.7639i	− 0.884884i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 2.07459i	− 0.102834i
$408$	0	0
$409$	−5.93492	−0.293463	−0.146731	−	0.989176i	$-0.546875\pi$
$409$	−5.93492	−0.293463	−0.146731	+	0.989176i	$0.546875\pi$
$410$	0	0
$411$	−28.0190	−1.38208
$412$	0	0
$413$	20.6594i	1.01658i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	1.48979i	0.0729553i
$418$	0	0
$419$	−9.07814	−0.443496	−0.221748	−	0.975104i	$-0.571176\pi$
$419$	−9.07814	−0.443496	−0.221748	+	0.975104i	$0.571176\pi$
$420$	0	0
$421$	7.73538	0.376999	0.188500	−	0.982073i	$-0.439638\pi$
$421$	7.73538	0.376999	0.188500	+	0.982073i	$0.439638\pi$
$422$	0	0
$423$	− 53.6243i	− 2.60730i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	5.79525i	0.280452i
$428$	0	0
$429$	26.3392	1.27167
$430$	0	0
$431$	5.62210	0.270807	0.135404	−	0.990791i	$-0.456767\pi$
$431$	5.62210	0.270807	0.135404	+	0.990791i	$0.456767\pi$
$432$	0	0
$433$	6.84942i	0.329162i	0.986364	+	0.164581i	$0.0526272\pi$
$433$	6.84942i	0.329162i	−0.986364	+	0.164581i	$0.947373\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 7.48979i	− 0.358285i
$438$	0	0
$439$	−7.43207	−0.354714	−0.177357	−	0.984147i	$-0.556755\pi$
$439$	−7.43207	−0.354714	−0.177357	+	0.984147i	$0.556755\pi$
$440$	0	0
$441$	−36.1045	−1.71926
$442$	0	0
$443$	− 15.5644i	− 0.739486i	−0.929134	−	0.369743i	$-0.879446\pi$
$443$	− 15.5644i	− 0.739486i	0.929134	−	0.369743i	$-0.120554\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	72.0043i	3.40569i
$448$	0	0
$449$	−11.4152	−0.538716	−0.269358	−	0.963040i	$-0.586812\pi$
$449$	−11.4152	−0.538716	−0.269358	+	0.963040i	$0.586812\pi$
$450$	0	0
$451$	−32.2646	−1.51928
$452$	0	0
$453$	34.8494i	1.63737i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 7.63898i	− 0.357336i	−0.983909	−	0.178668i	$-0.942821\pi$
$457$	− 7.63898i	− 0.357336i	0.983909	−	0.178668i	$-0.0571788\pi$
$458$	0	0
$459$	15.9349	0.743778
$460$	0	0
$461$	−1.85081	−0.0862010	−0.0431005	−	0.999071i	$-0.513724\pi$
$461$	−1.85081	−0.0862010	−0.0431005	+	0.999071i	$0.513724\pi$
$462$	0	0
$463$	− 22.1995i	− 1.03170i	−0.856679	−	0.515850i	$-0.827476\pi$
$463$	− 22.1995i	− 1.03170i	0.856679	−	0.515850i	$-0.172524\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	10.5102i	0.486355i	0.969982	+	0.243177i	$0.0781897\pi$
$467$	10.5102i	0.486355i	−0.969982	+	0.243177i	$0.921810\pi$
$468$	0	0
$469$	−15.8846	−0.733481
$470$	0	0
$471$	36.7748	1.69449
$472$	0	0
$473$	− 7.09501i	− 0.326229i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	101.249i	4.63585i
$478$	0	0
$479$	−14.3319	−0.654839	−0.327419	−	0.944879i	$-0.606179\pi$
$479$	−14.3319	−0.654839	−0.327419	+	0.944879i	$0.606179\pi$
$480$	0	0
$481$	−1.31879	−0.0601318
$482$	0	0
$483$	− 7.88457i	− 0.358760i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	26.8663i	1.21743i	0.793390	+	0.608714i	$0.208314\pi$
$487$	26.8663i	1.21743i	−0.793390	+	0.608714i	$0.791686\pi$
$488$	0	0
$489$	16.9649	0.767177
$490$	0	0
$491$	−8.69072	−0.392207	−0.196103	−	0.980583i	$-0.562829\pi$
$491$	−8.69072	−0.392207	−0.196103	+	0.980583i	$0.562829\pi$
$492$	0	0
$493$	2.83991i	0.127903i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 7.40047i	− 0.331957i
$498$	0	0
$499$	−11.4173	−0.511111	−0.255555	−	0.966794i	$-0.582258\pi$
$499$	−11.4173	−0.511111	−0.255555	+	0.966794i	$0.582258\pi$
$500$	0	0
$501$	−77.7734	−3.47466
$502$	0	0
$503$	2.81352i	0.125449i	0.998031	+	0.0627243i	$0.0199789\pi$
$503$	2.81352i	0.125449i	−0.998031	+	0.0627243i	$0.980021\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	26.0577i	1.15726i
$508$	0	0
$509$	−9.48979	−0.420628	−0.210314	−	0.977634i	$-0.567449\pi$
$509$	−9.48979	−0.420628	−0.210314	+	0.977634i	$0.567449\pi$
$510$	0	0
$511$	10.5848	0.468244
$512$	0	0
$513$	77.1236i	3.40509i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 24.2646i	− 1.06716i
$518$	0	0
$519$	−63.4941	−2.78708
$520$	0	0
$521$	−27.8290	−1.21921	−0.609605	−	0.792705i	$-0.708672\pi$
$521$	−27.8290	−1.21921	−0.609605	+	0.792705i	$0.708672\pi$
$522$	0	0
$523$	− 19.4342i	− 0.849799i	−0.905240	−	0.424900i	$-0.860309\pi$
$523$	− 19.4342i	− 0.849799i	0.905240	−	0.424900i	$-0.139691\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 7.87720i	− 0.343136i
$528$	0	0
$529$	20.6052	0.895879
$530$	0	0
$531$	−104.664	−4.54202
$532$	0	0
$533$	20.5102i	0.888396i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	16.1492i	0.696889i
$538$	0	0
$539$	−16.3371	−0.703687
$540$	0	0
$541$	3.28504	0.141235	0.0706174	−	0.997503i	$-0.477503\pi$
$541$	3.28504	0.141235	0.0706174	+	0.997503i	$0.477503\pi$
$542$	0	0
$543$	60.1345i	2.58062i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 14.2720i	− 0.610226i	−0.952316	−	0.305113i	$-0.901306\pi$
$547$	− 14.2720i	− 0.610226i	0.952316	−	0.305113i	$-0.0986942\pi$
$548$	0	0
$549$	−29.3596	−1.25304
$550$	0	0
$551$	−13.7449	−0.585552
$552$	0	0
$553$	− 4.08932i	− 0.173895i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 8.44513i	− 0.357832i	−0.983864	−	0.178916i	$-0.942741\pi$
$557$	− 8.44513i	− 0.357832i	0.983864	−	0.178916i	$-0.0572590\pi$
$558$	0	0
$559$	−4.51021	−0.190762
$560$	0	0
$561$	11.6798	0.493122
$562$	0	0
$563$	− 7.16961i	− 0.302163i	−0.988521	−	0.151081i	$-0.951724\pi$
$563$	− 7.16961i	− 0.302163i	0.988521	−	0.151081i	$-0.0482756\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	44.7917i	1.88107i
$568$	0	0
$569$	−21.5739	−0.904425	−0.452212	−	0.891910i	$-0.649365\pi$
$569$	−21.5739	−0.904425	−0.452212	+	0.891910i	$0.649365\pi$
$570$	0	0
$571$	32.7361	1.36996	0.684982	−	0.728560i	$-0.259810\pi$
$571$	32.7361	1.36996	0.684982	+	0.728560i	$0.259810\pi$
$572$	0	0
$573$	− 82.8685i	− 3.46188i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	44.5292i	1.85378i	0.375337	+	0.926888i	$0.377527\pi$
$577$	44.5292i	1.85378i	−0.375337	+	0.926888i	$0.622473\pi$
$578$	0	0
$579$	−13.4152	−0.557517
$580$	0	0
$581$	−3.88457	−0.161159
$582$	0	0
$583$	45.8143i	1.89743i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	18.0000i	0.742940i	0.928445	+	0.371470i	$0.121146\pi$
$587$	18.0000i	0.742940i	−0.928445	+	0.371470i	$0.878854\pi$
$588$	0	0
$589$	38.1249	1.57091
$590$	0	0
$591$	71.6243	2.94623
$592$	0	0
$593$	20.9796i	0.861528i	0.902465	+	0.430764i	$0.141756\pi$
$593$	20.9796i	0.861528i	−0.902465	+	0.430764i	$0.858244\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 65.8238i	− 2.69399i
$598$	0	0
$599$	22.7002	0.927507	0.463753	−	0.885964i	$-0.346502\pi$
$599$	22.7002	0.927507	0.463753	+	0.885964i	$0.346502\pi$
$600$	0	0
$601$	−33.1140	−1.35075	−0.675375	−	0.737474i	$-0.736018\pi$
$601$	−33.1140	−1.35075	−0.675375	+	0.737474i	$0.736018\pi$
$602$	0	0
$603$	− 80.4737i	− 3.27714i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	3.58835i	0.145647i	0.997345	+	0.0728233i	$0.0232009\pi$
$607$	3.58835i	0.145647i	−0.997345	+	0.0728233i	$0.976799\pi$
$608$	0	0
$609$	−14.4694	−0.586328
$610$	0	0
$611$	−15.4247	−0.624017
$612$	0	0
$613$	− 32.7843i	− 1.32415i	−0.749439	−	0.662074i	$-0.769677\pi$
$613$	− 32.7843i	− 1.32415i	0.749439	−	0.662074i	$-0.230323\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 22.0504i	− 0.887714i	−0.896098	−	0.443857i	$-0.853610\pi$
$617$	− 22.0504i	− 0.887714i	0.896098	−	0.443857i	$-0.146390\pi$
$618$	0	0
$619$	−19.4511	−0.781806	−0.390903	−	0.920432i	$-0.627837\pi$
$619$	−19.4511	−0.781806	−0.390903	+	0.920432i	$0.627837\pi$
$620$	0	0
$621$	24.6594	0.989547
$622$	0	0
$623$	− 27.8698i	− 1.11658i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	56.5292i	2.25756i
$628$	0	0
$629$	−0.584803	−0.0233176
$630$	0	0
$631$	11.0950	0.441686	0.220843	−	0.975309i	$-0.429119\pi$
$631$	11.0950	0.441686	0.220843	+	0.975309i	$0.429119\pi$
$632$	0	0
$633$	− 3.66079i	− 0.145503i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	10.3853i	0.411479i
$638$	0	0
$639$	37.4919	1.48316
$640$	0	0
$641$	−11.1359	−0.439840	−0.219920	−	0.975518i	$-0.570580\pi$
$641$	−11.1359	−0.439840	−0.219920	+	0.975518i	$0.570580\pi$
$642$	0	0
$643$	30.7361i	1.21212i	0.795421	+	0.606058i	$0.207250\pi$
$643$	30.7361i	1.21212i	−0.795421	+	0.606058i	$0.792750\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 19.3691i	− 0.761480i	−0.924682	−	0.380740i	$-0.875669\pi$
$647$	− 19.3691i	− 0.761480i	0.924682	−	0.380740i	$-0.124331\pi$
$648$	0	0
$649$	−47.3596	−1.85903
$650$	0	0
$651$	40.1345	1.57299
$652$	0	0
$653$	17.7354i	0.694039i	0.937858	+	0.347020i	$0.112806\pi$
$653$	17.7354i	0.694039i	−0.937858	+	0.347020i	$0.887194\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	53.6243i	2.09208i
$658$	0	0
$659$	−18.7843	−0.731734	−0.365867	−	0.930667i	$-0.619228\pi$
$659$	−18.7843	−0.731734	−0.365867	+	0.930667i	$0.619228\pi$
$660$	0	0
$661$	−40.2091	−1.56395	−0.781976	−	0.623309i	$-0.785788\pi$
$661$	−40.2091	−1.56395	−0.781976	+	0.623309i	$0.785788\pi$
$662$	0	0
$663$	− 7.42471i	− 0.288352i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	4.39478i	0.170166i
$668$	0	0
$669$	−19.0490	−0.736475
$670$	0	0
$671$	−13.2850	−0.512863
$672$	0	0
$673$	1.01091i	0.0389675i	0.999810	+	0.0194838i	$0.00620227\pi$
$673$	1.01091i	0.0389675i	−0.999810	+	0.0194838i	$0.993798\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.85081i	0.0711326i	0.999367	+	0.0355663i	$0.0113235\pi$
$677$	1.85081i	0.0711326i	−0.999367	+	0.0355663i	$0.988677\pi$
$678$	0	0
$679$	18.1753	0.697504
$680$	0	0
$681$	−51.2200	−1.96275
$682$	0	0
$683$	36.6112i	1.40089i	0.713707	+	0.700444i	$0.247015\pi$
$683$	36.6112i	1.40089i	−0.713707	+	0.700444i	$0.752985\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 25.0394i	− 0.955315i
$688$	0	0
$689$	29.1236	1.10952
$690$	0	0
$691$	−29.8867	−1.13694	−0.568472	−	0.822702i	$-0.692465\pi$
$691$	−29.8867	−1.13694	−0.568472	+	0.822702i	$0.692465\pi$
$692$	0	0
$693$	43.0394i	1.63493i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	9.09501i	0.344498i
$698$	0	0
$699$	−32.7098	−1.23720
$700$	0	0
$701$	−15.2442	−0.575765	−0.287883	−	0.957666i	$-0.592951\pi$
$701$	−15.2442	−0.575765	−0.287883	+	0.957666i	$0.592951\pi$
$702$	0	0
$703$	− 2.83039i	− 0.106750i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 16.3801i	− 0.616035i
$708$	0	0
$709$	−34.5606	−1.29795	−0.648975	−	0.760810i	$-0.724802\pi$
$709$	−34.5606	−1.29795	−0.648975	+	0.760810i	$0.724802\pi$
$710$	0	0
$711$	20.7171	0.776952
$712$	0	0
$713$	− 12.1900i	− 0.456520i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	28.0190i	1.04639i
$718$	0	0
$719$	20.1418	0.751163	0.375582	−	0.926789i	$-0.377443\pi$
$719$	20.1418	0.751163	0.375582	+	0.926789i	$0.377443\pi$
$720$	0	0
$721$	14.8641	0.553570
$722$	0	0
$723$	− 97.5349i	− 3.62736i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 36.8998i	− 1.36854i	−0.729230	−	0.684268i	$-0.760122\pi$
$727$	− 36.8998i	− 1.36854i	0.729230	−	0.684268i	$-0.239878\pi$
$728$	0	0
$729$	−69.5292	−2.57516
$730$	0	0
$731$	−2.00000	−0.0739727
$732$	0	0
$733$	− 42.6784i	− 1.57636i	−0.615442	−	0.788182i	$-0.711023\pi$
$733$	− 42.6784i	− 1.57636i	0.615442	−	0.788182i	$-0.288977\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 36.4138i	− 1.34132i
$738$	0	0
$739$	−28.7098	−1.05611	−0.528053	−	0.849212i	$-0.677078\pi$
$739$	−28.7098	−1.05611	−0.528053	+	0.849212i	$0.677078\pi$
$740$	0	0
$741$	35.9349	1.32010
$742$	0	0
$743$	18.4525i	0.676956i	0.940974	+	0.338478i	$0.109912\pi$
$743$	18.4525i	0.676956i	−0.940974	+	0.338478i	$0.890088\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 19.6798i	− 0.720047i
$748$	0	0
$749$	29.7544	1.08720
$750$	0	0
$751$	−0.271981	−0.00992474	−0.00496237	−	0.999988i	$-0.501580\pi$
$751$	−0.271981	−0.00992474	−0.00496237	+	0.999988i	$0.501580\pi$
$752$	0	0
$753$	− 7.21045i	− 0.262763i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 30.7843i	− 1.11888i	−0.828872	−	0.559438i	$-0.811017\pi$
$757$	− 30.7843i	− 1.11888i	0.828872	−	0.559438i	$-0.188983\pi$
$758$	0	0
$759$	18.0746	0.656066
$760$	0	0
$761$	−39.9853	−1.44947	−0.724733	−	0.689030i	$-0.758037\pi$
$761$	−39.9853	−1.44947	−0.724733	+	0.689030i	$0.758037\pi$
$762$	0	0
$763$	− 15.3744i	− 0.556589i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	30.1059i	1.08706i
$768$	0	0
$769$	19.8195	0.714709	0.357355	−	0.933969i	$-0.383679\pi$
$769$	19.8195	0.714709	0.357355	+	0.933969i	$0.383679\pi$
$770$	0	0
$771$	−5.53063	−0.199181
$772$	0	0
$773$	16.5102i	0.593831i	0.954904	+	0.296915i	$0.0959580\pi$
$773$	16.5102i	0.593831i	−0.954904	+	0.296915i	$0.904042\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 2.97958i	− 0.106892i
$778$	0	0
$779$	−44.0190	−1.57715
$780$	0	0
$781$	16.9649	0.607050
$782$	0	0
$783$	− 45.2537i	− 1.61724i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	33.1214i	1.18065i	0.807165	+	0.590325i	$0.201001\pi$
$787$	33.1214i	1.18065i	−0.807165	+	0.590325i	$0.798999\pi$
$788$	0	0
$789$	−104.580	−3.72313
$790$	0	0
$791$	−18.5848	−0.660800
$792$	0	0
$793$	8.44513i	0.299895i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 52.5701i	− 1.86213i	−0.364856	−	0.931064i	$-0.618882\pi$
$797$	− 52.5701i	− 1.86213i	0.364856	−	0.931064i	$-0.381118\pi$
$798$	0	0
$799$	−6.83991	−0.241979
$800$	0	0
$801$	141.193	4.98881
$802$	0	0
$803$	24.2646i	0.856280i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	10.6499i	0.374894i
$808$	0	0
$809$	0.979580	0.0344402	0.0172201	−	0.999852i	$-0.494518\pi$
$809$	0.979580	0.0344402	0.0172201	+	0.999852i	$0.494518\pi$
$810$	0	0
$811$	43.7157	1.53507	0.767533	−	0.641009i	$-0.221484\pi$
$811$	43.7157	1.53507	0.767533	+	0.641009i	$0.221484\pi$
$812$	0	0
$813$	35.6580i	1.25058i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 9.67982i	− 0.338654i
$818$	0	0
$819$	27.3596	0.956023
$820$	0	0
$821$	−44.9335	−1.56819	−0.784096	−	0.620640i	$-0.786873\pi$
$821$	−44.9335	−1.56819	−0.784096	+	0.620640i	$0.786873\pi$
$822$	0	0
$823$	18.0767i	0.630116i	0.949072	+	0.315058i	$0.102024\pi$
$823$	18.0767i	0.630116i	−0.949072	+	0.315058i	$0.897976\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	30.7361i	1.06880i	0.845232	+	0.534400i	$0.179462\pi$
$827$	30.7361i	1.06880i	−0.845232	+	0.534400i	$0.820538\pi$
$828$	0	0
$829$	16.8494	0.585205	0.292602	−	0.956234i	$-0.405479\pi$
$829$	16.8494	0.585205	0.292602	+	0.956234i	$0.405479\pi$
$830$	0	0
$831$	76.2836	2.64625
$832$	0	0
$833$	4.60522i	0.159562i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	125.523i	4.33870i
$838$	0	0
$839$	41.5570	1.43471	0.717354	−	0.696709i	$-0.245353\pi$
$839$	41.5570	1.43471	0.717354	+	0.696709i	$0.245353\pi$
$840$	0	0
$841$	−20.9349	−0.721894
$842$	0	0
$843$	− 51.8385i	− 1.78541i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	2.45249i	0.0842687i
$848$	0	0
$849$	−31.6485	−1.08617
$850$	0	0
$851$	−0.904987	−0.0310225
$852$	0	0
$853$	− 30.6447i	− 1.04925i	−0.851332	−	0.524627i	$-0.824205\pi$
$853$	− 30.6447i	− 1.04925i	0.851332	−	0.524627i	$-0.175795\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	51.8794i	1.77217i	0.463527	+	0.886083i	$0.346584\pi$
$857$	51.8794i	1.77217i	−0.463527	+	0.886083i	$0.653416\pi$
$858$	0	0
$859$	−10.5007	−0.358279	−0.179140	−	0.983824i	$-0.557331\pi$
$859$	−10.5007	−0.358279	−0.179140	+	0.983824i	$0.557331\pi$
$860$	0	0
$861$	−46.3392	−1.57924
$862$	0	0
$863$	5.22517i	0.177867i	0.996038	+	0.0889334i	$0.0283458\pi$
$863$	5.22517i	0.177867i	−0.996038	+	0.0889334i	$0.971654\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 3.29240i	− 0.111816i
$868$	0	0
$869$	9.37436	0.318003
$870$	0	0
$871$	−23.1478	−0.784333
$872$	0	0
$873$	92.0789i	3.11640i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	3.38144i	0.114183i	0.998369	+	0.0570916i	$0.0181827\pi$
$877$	3.38144i	0.114183i	−0.998369	+	0.0570916i	$0.981817\pi$
$878$	0	0
$879$	19.0490	0.642506
$880$	0	0
$881$	−38.6447	−1.30197	−0.650986	−	0.759090i	$-0.725644\pi$
$881$	−38.6447	−1.30197	−0.650986	+	0.759090i	$0.725644\pi$
$882$	0	0
$883$	18.4138i	0.619674i	0.950790	+	0.309837i	$0.100274\pi$
$883$	18.4138i	0.619674i	−0.950790	+	0.309837i	$0.899726\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	30.7171i	1.03138i	0.856775	+	0.515690i	$0.172464\pi$
$887$	30.7171i	1.03138i	−0.856775	+	0.515690i	$0.827536\pi$
$888$	0	0
$889$	21.5644	0.723246
$890$	0	0
$891$	−102.681	−3.43993
$892$	0	0
$893$	− 33.1045i	− 1.10780i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 11.4898i	− 0.383633i
$898$	0	0
$899$	−22.3705	−0.746099
$900$	0	0
$901$	12.9145	0.430244
$902$	0	0
$903$	− 10.1900i	− 0.339103i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 26.5219i	− 0.880644i	−0.897840	−	0.440322i	$-0.854864\pi$
$907$	− 26.5219i	− 0.880644i	0.897840	−	0.440322i	$-0.145136\pi$
$908$	0	0
$909$	82.9839	2.75240
$910$	0	0
$911$	23.6967	0.785106	0.392553	−	0.919729i	$-0.371592\pi$
$911$	23.6967	0.785106	0.392553	+	0.919729i	$0.371592\pi$
$912$	0	0
$913$	− 8.90499i	− 0.294712i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 26.6594i	− 0.880371i
$918$	0	0
$919$	27.2295	0.898218	0.449109	−	0.893477i	$-0.351742\pi$
$919$	27.2295	0.898218	0.449109	+	0.893477i	$0.351742\pi$
$920$	0	0
$921$	94.1127	3.10112
$922$	0	0
$923$	− 10.7843i	− 0.354971i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	75.3041i	2.47331i
$928$	0	0
$929$	−1.09501	−0.0359262	−0.0179631	−	0.999839i	$-0.505718\pi$
$929$	−1.09501	−0.0359262	−0.0179631	+	0.999839i	$0.505718\pi$
$930$	0	0
$931$	−22.2889	−0.730488
$932$	0	0
$933$	− 84.7383i	− 2.77421i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	19.6798i	0.642912i	0.946925	+	0.321456i	$0.104172\pi$
$937$	19.6798i	0.642912i	−0.946925	+	0.321456i	$0.895828\pi$
$938$	0	0
$939$	−56.7748	−1.85278
$940$	0	0
$941$	42.5606	1.38743	0.693717	−	0.720247i	$-0.255972\pi$
$941$	42.5606	1.38743	0.693717	+	0.720247i	$0.255972\pi$
$942$	0	0
$943$	14.0746i	0.458332i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 5.72802i	− 0.186136i	−0.995660	−	0.0930678i	$-0.970333\pi$
$947$	− 5.72802i	− 0.186136i	0.995660	−	0.0930678i	$-0.0296673\pi$
$948$	0	0
$949$	15.4247	0.500707
$950$	0	0
$951$	31.7354	1.02909
$952$	0	0
$953$	7.85081i	0.254313i	0.991883	+	0.127156i	$0.0405850\pi$
$953$	7.85081i	0.254313i	−0.991883	+	0.127156i	$0.959415\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 33.1696i	− 1.07222i
$958$	0	0
$959$	−13.1696	−0.425269
$960$	0	0
$961$	31.0504	1.00162
$962$	0	0
$963$	150.740i	4.85754i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 12.3202i	− 0.396190i	−0.980183	−	0.198095i	$-0.936524\pi$
$967$	− 12.3202i	− 0.396190i	0.980183	−	0.198095i	$-0.0634755\pi$
$968$	0	0
$969$	15.9349	0.511903
$970$	0	0
$971$	4.69072	0.150532	0.0752662	−	0.997163i	$-0.476019\pi$
$971$	4.69072	0.150532	0.0752662	+	0.997163i	$0.476019\pi$
$972$	0	0
$973$	0.700237i	0.0224486i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 19.8290i	− 0.634386i	−0.948361	−	0.317193i	$-0.897260\pi$
$977$	− 19.8290i	− 0.634386i	0.948361	−	0.317193i	$-0.102740\pi$
$978$	0	0
$979$	63.8889	2.04190
$980$	0	0
$981$	77.8889	2.48680
$982$	0	0
$983$	2.86630i	0.0914207i	0.998955	+	0.0457104i	$0.0145551\pi$
$983$	2.86630i	0.0914207i	−0.998955	+	0.0457104i	$0.985445\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 34.8494i	− 1.10927i
$988$	0	0
$989$	−3.09501	−0.0984157
$990$	0	0
$991$	−38.3319	−1.21765	−0.608826	−	0.793304i	$-0.708359\pi$
$991$	−38.3319	−1.21765	−0.608826	+	0.793304i	$0.708359\pi$
$992$	0	0
$993$	44.4451i	1.41042i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 20.3055i	− 0.643080i	−0.946896	−	0.321540i	$-0.895799\pi$
$997$	− 20.3055i	− 0.643080i	0.946896	−	0.321540i	$-0.104201\pi$
$998$	0	0
$999$	9.31879	0.294834

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3400.2.e.i.2449.6		6
5.2	odd	4		680.2.a.h.1.3	✓	3
5.3	odd	4		3400.2.a.k.1.1		3
5.4	even	2	inner	3400.2.e.i.2449.1		6
15.2	even	4		6120.2.a.bq.1.1		3
20.3	even	4		6800.2.a.bs.1.3		3
20.7	even	4		1360.2.a.p.1.1		3
40.27	even	4		5440.2.a.bu.1.3		3
40.37	odd	4		5440.2.a.bn.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
680.2.a.h.1.3	✓	3	5.2	odd	4
1360.2.a.p.1.1		3	20.7	even	4
3400.2.a.k.1.1		3	5.3	odd	4
3400.2.e.i.2449.1		6	5.4	even	2	inner
3400.2.e.i.2449.6		6	1.1	even	1	trivial
5440.2.a.bn.1.1		3	40.37	odd	4
5440.2.a.bu.1.3		3	40.27	even	4
6120.2.a.bq.1.1		3	15.2	even	4
6800.2.a.bs.1.3		3	20.3	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3400 = 2^{3} \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3400.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.29240i q^{3} +1.54751i q^{7} -7.83991 q^{9} -3.54751 q^{11} +2.25511i q^{13} +1.00000i q^{17} -4.83991 q^{19} -5.09501 q^{21} +1.54751i q^{23} -15.9349i q^{27} +2.83991 q^{29} -7.87720 q^{31} -11.6798i q^{33} +0.584803i q^{37} -7.42471 q^{39} +9.09501 q^{41} +2.00000i q^{43} +6.83991i q^{47} +4.60522 q^{49} -3.29240 q^{51} -12.9145i q^{53} -15.9349i q^{57} +13.3501 q^{59} +3.74489 q^{61} -12.1323i q^{63} +10.2646i q^{67} -5.09501 q^{69} -4.78219 q^{71} -6.83991i q^{73} -5.48979i q^{77} -2.64252 q^{79} +28.9444 q^{81} +2.51021i q^{83} +9.35012i q^{87} -18.0095 q^{89} -3.48979 q^{91} -25.9349i q^{93} -11.7449i q^{97} +27.8121 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 16 q^{9} - 4 q^{11} + 2 q^{19} + 4 q^{21} - 14 q^{29} - 6 q^{31} + 14 q^{39} + 20 q^{41} - 14 q^{49} - 6 q^{51} + 42 q^{59} + 26 q^{61} + 4 q^{69} - 22 q^{71} + 36 q^{79} + 22 q^{81} - 22 q^{89} - 28 q^{91}+ \cdots + 60 q^{99}+O(q^{100}) \) 6 * q - 16 * q^9 - 4 * q^11 + 2 * q^19 + 4 * q^21 - 14 * q^29 - 6 * q^31 + 14 * q^39 + 20 * q^41 - 14 * q^49 - 6 * q^51 + 42 * q^59 + 26 * q^61 + 4 * q^69 - 22 * q^71 + 36 * q^79 + 22 * q^81 - 22 * q^89 - 28 * q^91 + 60 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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