LMFDB - Embedded newform 8700.2.g.u.349.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8700,2,Mod(349,8700)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8700, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8700.349");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8700 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8700.g (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:	$69.4698497585$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.4227136.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 9x^{4} + 22x^{2} + 9 $ x^6 + 9x^4 + 22x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			349.6
Root			$1.91223i$ of defining polynomial
Character	$\chi$	$=$	8700.349
Dual form			8700.2.g.u.349.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +2.91223i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +2.91223i q^{7} -1.00000 q^{9} -0.656620 q^{11} +1.56885i q^{13} -3.16784i q^{17} +4.00000 q^{19} -2.91223 q^{21} -5.13770i q^{23} -1.00000i q^{27} -1.00000 q^{29} +3.31324 q^{31} -0.656620i q^{33} -8.45094i q^{37} -1.56885 q^{39} +2.13770 q^{41} -1.82446i q^{43} +11.2479i q^{47} -1.48108 q^{49} +3.16784 q^{51} +10.2754i q^{53} +4.00000i q^{57} +13.6489 q^{59} +4.62648 q^{61} -2.91223i q^{63} -7.67906i q^{67} +5.13770 q^{69} +2.51122 q^{71} -11.7642i q^{73} -1.91223i q^{77} +12.5112 q^{79} +1.00000 q^{81} +11.0224i q^{83} -1.00000i q^{87} +6.88209 q^{89} -4.56885 q^{91} +3.31324i q^{93} -8.33568i q^{97} +0.656620 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{9}+O(q^{10}) $ 6 * q - 6 * q^9	$ 6 q - 6 q^{9} + 24 q^{19} - 4 q^{21} - 6 q^{29} + 12 q^{31} + 8 q^{39} - 22 q^{41} + 22 q^{49} - 4 q^{51} + 28 q^{59} + 12 q^{61} - 4 q^{69} - 4 q^{71} + 56 q^{79} + 6 q^{81} + 16 q^{89} - 10 q^{91}+O(q^{100}) $ 6 * q - 6 * q^9 + 24 * q^19 - 4 * q^21 - 6 * q^29 + 12 * q^31 + 8 * q^39 - 22 * q^41 + 22 * q^49 - 4 * q^51 + 28 * q^59 + 12 * q^61 - 4 * q^69 - 4 * q^71 + 56 * q^79 + 6 * q^81 + 16 * q^89 - 10 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/8700\mathbb{Z}\right)^\times$.

$n$	$901$	$4177$	$4351$	$5801$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.91223i	1.10072i	0.834928	+	0.550360i	$0.185509\pi$
$7$	2.91223i	1.10072i	−0.834928	+	0.550360i	$0.814491\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−0.656620	−0.197979	−0.0989893	−	0.995089i	$-0.531561\pi$
$11$	−0.656620	−0.197979	−0.0989893	+	0.995089i	$0.531561\pi$
$12$	0	0
$13$	1.56885i	0.435121i	0.976047	+	0.217560i	$0.0698099\pi$
$13$	1.56885i	0.435121i	−0.976047	+	0.217560i	$0.930190\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 3.16784i	− 0.768314i	−0.923268	−	0.384157i	$-0.874492\pi$
$17$	− 3.16784i	− 0.768314i	0.923268	−	0.384157i	$-0.125508\pi$
$18$	0	0
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	−2.91223	−0.635501
$22$	0	0
$23$	− 5.13770i	− 1.07128i	−0.844445	−	0.535642i	$-0.820070\pi$
$23$	− 5.13770i	− 1.07128i	0.844445	−	0.535642i	$-0.179930\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	3.31324	0.595076	0.297538	−	0.954710i	$-0.403835\pi$
$31$	3.31324	0.595076	0.297538	+	0.954710i	$0.403835\pi$
$32$	0	0
$33$	− 0.656620i	− 0.114303i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 8.45094i	− 1.38933i	−0.719335	−	0.694663i	$-0.755554\pi$
$37$	− 8.45094i	− 1.38933i	0.719335	−	0.694663i	$-0.244446\pi$
$38$	0	0
$39$	−1.56885	−0.251217
$40$	0	0
$41$	2.13770	0.333853	0.166926	−	0.985969i	$-0.446616\pi$
$41$	2.13770	0.333853	0.166926	+	0.985969i	$0.446616\pi$
$42$	0	0
$43$	− 1.82446i	− 0.278227i	−0.990276	−	0.139114i	$-0.955575\pi$
$43$	− 1.82446i	− 0.278227i	0.990276	−	0.139114i	$-0.0444253\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	11.2479i	1.64068i	0.571879	+	0.820338i	$0.306215\pi$
$47$	11.2479i	1.64068i	−0.571879	+	0.820338i	$0.693785\pi$
$48$	0	0
$49$	−1.48108	−0.211583
$50$	0	0
$51$	3.16784	0.443586
$52$	0	0
$53$	10.2754i	1.41143i	0.708494	+	0.705717i	$0.249375\pi$
$53$	10.2754i	1.41143i	−0.708494	+	0.705717i	$0.750625\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	4.00000i	0.529813i
$58$	0	0
$59$	13.6489	1.77694	0.888469	−	0.458937i	$-0.151770\pi$
$59$	13.6489	1.77694	0.888469	+	0.458937i	$0.151770\pi$
$60$	0	0
$61$	4.62648	0.592360	0.296180	−	0.955132i	$-0.404287\pi$
$61$	4.62648	0.592360	0.296180	+	0.955132i	$0.404287\pi$
$62$	0	0
$63$	− 2.91223i	− 0.366906i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 7.67906i	− 0.938146i	−0.883159	−	0.469073i	$-0.844588\pi$
$67$	− 7.67906i	− 0.938146i	0.883159	−	0.469073i	$-0.155412\pi$
$68$	0	0
$69$	5.13770	0.618506
$70$	0	0
$71$	2.51122	0.298027	0.149013	−	0.988835i	$-0.452390\pi$
$71$	2.51122	0.298027	0.149013	+	0.988835i	$0.452390\pi$
$72$	0	0
$73$	− 11.7642i	− 1.37689i	−0.725287	−	0.688447i	$-0.758293\pi$
$73$	− 11.7642i	− 1.37689i	0.725287	−	0.688447i	$-0.241707\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 1.91223i	− 0.217919i
$78$	0	0
$79$	12.5112	1.40762	0.703811	−	0.710387i	$-0.251480\pi$
$79$	12.5112	1.40762	0.703811	+	0.710387i	$0.251480\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	11.0224i	1.20987i	0.796275	+	0.604935i	$0.206801\pi$
$83$	11.0224i	1.20987i	−0.796275	+	0.604935i	$0.793199\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 1.00000i	− 0.107211i
$88$	0	0
$89$	6.88209	0.729500	0.364750	−	0.931105i	$-0.381154\pi$
$89$	6.88209	0.729500	0.364750	+	0.931105i	$0.381154\pi$
$90$	0	0
$91$	−4.56885	−0.478946
$92$	0	0
$93$	3.31324i	0.343567i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 8.33568i	− 0.846360i	−0.906046	−	0.423180i	$-0.860914\pi$
$97$	− 8.33568i	− 0.846360i	0.906046	−	0.423180i	$-0.139086\pi$
$98$	0	0
$99$	0.656620	0.0659928
$100$	0	0
$101$	1.62913	0.162104	0.0810521	−	0.996710i	$-0.474172\pi$
$101$	1.62913	0.162104	0.0810521	+	0.996710i	$0.474172\pi$
$102$	0	0
$103$	− 17.1601i	− 1.69084i	−0.534103	−	0.845419i	$-0.679351\pi$
$103$	− 17.1601i	− 1.69084i	0.534103	−	0.845419i	$-0.320649\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	3.93972i	0.380867i	0.981700	+	0.190434i	$0.0609894\pi$
$107$	3.93972i	0.380867i	−0.981700	+	0.190434i	$0.939011\pi$
$108$	0	0
$109$	14.5913	1.39759	0.698796	−	0.715321i	$-0.253720\pi$
$109$	14.5913	1.39759	0.698796	+	0.715321i	$0.253720\pi$
$110$	0	0
$111$	8.45094	0.802128
$112$	0	0
$113$	5.24791i	0.493681i	0.969056	+	0.246841i	$0.0793924\pi$
$113$	5.24791i	0.493681i	−0.969056	+	0.246841i	$0.920608\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 1.56885i	− 0.145040i
$118$	0	0
$119$	9.22547	0.845697
$120$	0	0
$121$	−10.5688	−0.960805
$122$	0	0
$123$	2.13770i	0.192750i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	8.68676i	0.770825i	0.922744	+	0.385413i	$0.125941\pi$
$127$	8.68676i	0.770825i	−0.922744	+	0.385413i	$0.874059\pi$
$128$	0	0
$129$	1.82446	0.160635
$130$	0	0
$131$	−8.85195	−0.773399	−0.386699	−	0.922206i	$-0.626385\pi$
$131$	−8.85195	−0.773399	−0.386699	+	0.922206i	$0.626385\pi$
$132$	0	0
$133$	11.6489i	1.01009i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	16.7642i	1.43226i	0.697967	+	0.716130i	$0.254088\pi$
$137$	16.7642i	1.43226i	−0.697967	+	0.716130i	$0.745912\pi$
$138$	0	0
$139$	7.26331	0.616066	0.308033	−	0.951376i	$-0.400329\pi$
$139$	7.26331	0.616066	0.308033	+	0.951376i	$0.400329\pi$
$140$	0	0
$141$	−11.2479	−0.947244
$142$	0	0
$143$	− 1.03014i	− 0.0861445i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 1.48108i	− 0.122157i
$148$	0	0
$149$	−20.4355	−1.67414	−0.837072	−	0.547093i	$-0.815735\pi$
$149$	−20.4355	−1.67414	−0.837072	+	0.547093i	$0.815735\pi$
$150$	0	0
$151$	−8.76418	−0.713219	−0.356609	−	0.934254i	$-0.616067\pi$
$151$	−8.76418	−0.713219	−0.356609	+	0.934254i	$0.616067\pi$
$152$	0	0
$153$	3.16784i	0.256105i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 6.27540i	− 0.500831i	−0.968138	−	0.250416i	$-0.919433\pi$
$157$	− 6.27540i	− 0.500831i	0.968138	−	0.250416i	$-0.0805673\pi$
$158$	0	0
$159$	−10.2754	−0.814892
$160$	0	0
$161$	14.9622	1.17918
$162$	0	0
$163$	2.11526i	0.165680i	0.996563	+	0.0828401i	$0.0263991\pi$
$163$	2.11526i	0.165680i	−0.996563	+	0.0828401i	$0.973601\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	8.51122i	0.658618i	0.944222	+	0.329309i	$0.106816\pi$
$167$	8.51122i	0.658618i	−0.944222	+	0.329309i	$0.893184\pi$
$168$	0	0
$169$	10.5387	0.810670
$170$	0	0
$171$	−4.00000	−0.305888
$172$	0	0
$173$	− 3.70919i	− 0.282005i	−0.990009	−	0.141002i	$-0.954967\pi$
$173$	− 3.70919i	− 0.282005i	0.990009	−	0.141002i	$-0.0450325\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	13.6489i	1.02592i
$178$	0	0
$179$	−0.450940	−0.0337048	−0.0168524	−	0.999858i	$-0.505365\pi$
$179$	−0.450940	−0.0337048	−0.0168524	+	0.999858i	$0.505365\pi$
$180$	0	0
$181$	−20.4054	−1.51672	−0.758360	−	0.651835i	$-0.773999\pi$
$181$	−20.4054	−1.51672	−0.758360	+	0.651835i	$0.773999\pi$
$182$	0	0
$183$	4.62648i	0.341999i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	2.08007i	0.152110i
$188$	0	0
$189$	2.91223	0.211834
$190$	0	0
$191$	5.25296	0.380091	0.190046	−	0.981775i	$-0.439136\pi$
$191$	5.25296	0.380091	0.190046	+	0.981775i	$0.439136\pi$
$192$	0	0
$193$	6.80202i	0.489620i	0.969571	+	0.244810i	$0.0787256\pi$
$193$	6.80202i	0.489620i	−0.969571	+	0.244810i	$0.921274\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	18.3357i	1.30636i	0.757201	+	0.653181i	$0.226566\pi$
$197$	18.3357i	1.30636i	−0.757201	+	0.653181i	$0.773434\pi$
$198$	0	0
$199$	4.54641	0.322287	0.161143	−	0.986931i	$-0.448482\pi$
$199$	4.54641	0.322287	0.161143	+	0.986931i	$0.448482\pi$
$200$	0	0
$201$	7.67906	0.541639
$202$	0	0
$203$	− 2.91223i	− 0.204398i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	5.13770i	0.357095i
$208$	0	0
$209$	−2.62648	−0.181678
$210$	0	0
$211$	8.33568	0.573852	0.286926	−	0.957953i	$-0.407367\pi$
$211$	8.33568	0.573852	0.286926	+	0.957953i	$0.407367\pi$
$212$	0	0
$213$	2.51122i	0.172066i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	9.64892i	0.655011i
$218$	0	0
$219$	11.7642	0.794950
$220$	0	0
$221$	4.96986	0.334309
$222$	0	0
$223$	24.7015i	1.65413i	0.562103	+	0.827067i	$0.309992\pi$
$223$	24.7015i	1.65413i	−0.562103	+	0.827067i	$0.690008\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 20.5508i	− 1.36400i	−0.731350	−	0.682002i	$-0.761109\pi$
$227$	− 20.5508i	− 1.36400i	0.731350	−	0.682002i	$-0.238891\pi$
$228$	0	0
$229$	−18.4958	−1.22224	−0.611119	−	0.791539i	$-0.709280\pi$
$229$	−18.4958	−1.22224	−0.611119	+	0.791539i	$0.709280\pi$
$230$	0	0
$231$	1.91223	0.125815
$232$	0	0
$233$	6.97757i	0.457115i	0.973530	+	0.228558i	$0.0734010\pi$
$233$	6.97757i	0.457115i	−0.973530	+	0.228558i	$0.926599\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	12.5112i	0.812691i
$238$	0	0
$239$	4.68676	0.303161	0.151581	−	0.988445i	$-0.451564\pi$
$239$	4.68676	0.303161	0.151581	+	0.988445i	$0.451564\pi$
$240$	0	0
$241$	26.2650	1.69188	0.845940	−	0.533278i	$-0.179040\pi$
$241$	26.2650	1.69188	0.845940	+	0.533278i	$0.179040\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	6.27540i	0.399294i
$248$	0	0
$249$	−11.0224	−0.698518
$250$	0	0
$251$	−12.7668	−0.805835	−0.402917	−	0.915236i	$-0.632004\pi$
$251$	−12.7668	−0.805835	−0.402917	+	0.915236i	$0.632004\pi$
$252$	0	0
$253$	3.37352i	0.212091i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	9.03784i	0.563765i	0.959449	+	0.281883i	$0.0909589\pi$
$257$	9.03784i	0.563765i	−0.959449	+	0.281883i	$0.909041\pi$
$258$	0	0
$259$	24.6111	1.52926
$260$	0	0
$261$	1.00000	0.0618984
$262$	0	0
$263$	− 18.5284i	− 1.14251i	−0.820773	−	0.571254i	$-0.806457\pi$
$263$	− 18.5284i	− 1.14251i	0.820773	−	0.571254i	$-0.193543\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	6.88209i	0.421177i
$268$	0	0
$269$	19.1076	1.16501	0.582504	−	0.812828i	$-0.302073\pi$
$269$	19.1076	1.16501	0.582504	+	0.812828i	$0.302073\pi$
$270$	0	0
$271$	−24.8865	−1.51175	−0.755873	−	0.654719i	$-0.772787\pi$
$271$	−24.8865	−1.51175	−0.755873	+	0.654719i	$0.772787\pi$
$272$	0	0
$273$	− 4.56885i	− 0.276519i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	14.0697i	0.845367i	0.906277	+	0.422684i	$0.138912\pi$
$277$	14.0697i	0.845367i	−0.906277	+	0.422684i	$0.861088\pi$
$278$	0	0
$279$	−3.31324	−0.198359
$280$	0	0
$281$	−26.7712	−1.59704	−0.798518	−	0.601971i	$-0.794382\pi$
$281$	−26.7712	−1.59704	−0.798518	+	0.601971i	$0.794382\pi$
$282$	0	0
$283$	− 0.884736i	− 0.0525921i	−0.999654	−	0.0262960i	$-0.991629\pi$
$283$	− 0.884736i	− 0.0525921i	0.999654	−	0.0262960i	$-0.00837125\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	6.22547i	0.367478i
$288$	0	0
$289$	6.96480	0.409694
$290$	0	0
$291$	8.33568	0.488646
$292$	0	0
$293$	− 17.1575i	− 1.00235i	−0.865346	−	0.501176i	$-0.832901\pi$
$293$	− 17.1575i	− 1.00235i	0.865346	−	0.501176i	$-0.167099\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0.656620i	0.0381010i
$298$	0	0
$299$	8.06028	0.466138
$300$	0	0
$301$	5.31324	0.306250
$302$	0	0
$303$	1.62913i	0.0935909i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 23.5284i	− 1.34283i	−0.741080	−	0.671417i	$-0.765686\pi$
$307$	− 23.5284i	− 1.34283i	0.741080	−	0.671417i	$-0.234314\pi$
$308$	0	0
$309$	17.1601	0.976206
$310$	0	0
$311$	11.6988	0.663381	0.331690	−	0.943388i	$-0.392381\pi$
$311$	11.6988	0.663381	0.331690	+	0.943388i	$0.392381\pi$
$312$	0	0
$313$	− 2.38560i	− 0.134842i	−0.997725	−	0.0674212i	$-0.978523\pi$
$313$	− 2.38560i	− 0.134842i	0.997725	−	0.0674212i	$-0.0214771\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	28.2556i	1.58699i	0.608575	+	0.793497i	$0.291742\pi$
$317$	28.2556i	1.58699i	−0.608575	+	0.793497i	$0.708258\pi$
$318$	0	0
$319$	0.656620	0.0367637
$320$	0	0
$321$	−3.93972	−0.219894
$322$	0	0
$323$	− 12.6714i	− 0.705053i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	14.5913i	0.806900i
$328$	0	0
$329$	−32.7565	−1.80592
$330$	0	0
$331$	−1.03784	−0.0570450	−0.0285225	−	0.999593i	$-0.509080\pi$
$331$	−1.03784	−0.0570450	−0.0285225	+	0.999593i	$0.509080\pi$
$332$	0	0
$333$	8.45094i	0.463109i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 10.3960i	− 0.566304i	−0.959075	−	0.283152i	$-0.908620\pi$
$337$	− 10.3960i	− 0.566304i	0.959075	−	0.283152i	$-0.0913800\pi$
$338$	0	0
$339$	−5.24791	−0.285027
$340$	0	0
$341$	−2.17554	−0.117812
$342$	0	0
$343$	16.0724i	0.867826i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	6.78662i	0.364325i	0.983268	+	0.182162i	$0.0583096\pi$
$347$	6.78662i	0.364325i	−0.983268	+	0.182162i	$0.941690\pi$
$348$	0	0
$349$	−0.648917	−0.0347357	−0.0173679	−	0.999849i	$-0.505529\pi$
$349$	−0.648917	−0.0347357	−0.0173679	+	0.999849i	$0.505529\pi$
$350$	0	0
$351$	1.56885	0.0837390
$352$	0	0
$353$	7.31324i	0.389245i	0.980878	+	0.194622i	$0.0623481\pi$
$353$	7.31324i	0.389245i	−0.980878	+	0.194622i	$0.937652\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	9.22547i	0.488264i
$358$	0	0
$359$	19.2754	1.01732	0.508658	−	0.860968i	$-0.330142\pi$
$359$	19.2754	1.01732	0.508658	+	0.860968i	$0.330142\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	− 10.5688i	− 0.554721i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	14.6111i	0.762692i	0.924432	+	0.381346i	$0.124539\pi$
$367$	14.6111i	0.762692i	−0.924432	+	0.381346i	$0.875461\pi$
$368$	0	0
$369$	−2.13770	−0.111284
$370$	0	0
$371$	−29.9243	−1.55359
$372$	0	0
$373$	24.2754i	1.25693i	0.777837	+	0.628466i	$0.216317\pi$
$373$	24.2754i	1.25693i	−0.777837	+	0.628466i	$0.783683\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 1.56885i	− 0.0807999i
$378$	0	0
$379$	26.6111	1.36692	0.683460	−	0.729988i	$-0.260475\pi$
$379$	26.6111	1.36692	0.683460	+	0.729988i	$0.260475\pi$
$380$	0	0
$381$	−8.68676	−0.445036
$382$	0	0
$383$	− 2.96216i	− 0.151359i	−0.997132	−	0.0756796i	$-0.975887\pi$
$383$	− 2.96216i	− 0.151359i	0.997132	−	0.0756796i	$-0.0241126\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1.82446i	0.0927424i
$388$	0	0
$389$	26.5310	1.34518	0.672588	−	0.740017i	$-0.265183\pi$
$389$	26.5310	1.34518	0.672588	+	0.740017i	$0.265183\pi$
$390$	0	0
$391$	−16.2754	−0.823082
$392$	0	0
$393$	− 8.85195i	− 0.446522i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	36.8486i	1.84938i	0.380721	+	0.924690i	$0.375676\pi$
$397$	36.8486i	1.84938i	−0.380721	+	0.924690i	$0.624324\pi$
$398$	0	0
$399$	−11.6489	−0.583175
$400$	0	0
$401$	−26.1755	−1.30714	−0.653572	−	0.756864i	$-0.726730\pi$
$401$	−26.1755	−1.30714	−0.653572	+	0.756864i	$0.726730\pi$
$402$	0	0
$403$	5.19798i	0.258930i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	5.54906i	0.275057i
$408$	0	0
$409$	−10.0999	−0.499406	−0.249703	−	0.968323i	$-0.580333\pi$
$409$	−10.0999	−0.499406	−0.249703	+	0.968323i	$0.580333\pi$
$410$	0	0
$411$	−16.7642	−0.826916
$412$	0	0
$413$	39.7488i	1.95591i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	7.26331i	0.355686i
$418$	0	0
$419$	−13.4184	−0.655531	−0.327766	−	0.944759i	$-0.606296\pi$
$419$	−13.4184	−0.655531	−0.327766	+	0.944759i	$0.606296\pi$
$420$	0	0
$421$	21.6885	1.05703	0.528516	−	0.848923i	$-0.322749\pi$
$421$	21.6885	1.05703	0.528516	+	0.848923i	$0.322749\pi$
$422$	0	0
$423$	− 11.2479i	− 0.546892i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	13.4734i	0.652022i
$428$	0	0
$429$	1.03014	0.0497356
$430$	0	0
$431$	−8.39595	−0.404419	−0.202209	−	0.979342i	$-0.564812\pi$
$431$	−8.39595	−0.404419	−0.202209	+	0.979342i	$0.564812\pi$
$432$	0	0
$433$	16.1755i	0.777347i	0.921376	+	0.388673i	$0.127067\pi$
$433$	16.1755i	0.777347i	−0.921376	+	0.388673i	$0.872933\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 20.5508i	− 0.983078i
$438$	0	0
$439$	31.3882	1.49808	0.749040	−	0.662525i	$-0.230515\pi$
$439$	31.3882	1.49808	0.749040	+	0.662525i	$0.230515\pi$
$440$	0	0
$441$	1.48108	0.0705276
$442$	0	0
$443$	− 4.25561i	− 0.202190i	−0.994877	−	0.101095i	$-0.967765\pi$
$443$	− 4.25561i	− 0.202190i	0.994877	−	0.101095i	$-0.0322346\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 20.4355i	− 0.966568i
$448$	0	0
$449$	22.0499	1.04060	0.520300	−	0.853983i	$-0.325820\pi$
$449$	22.0499	1.04060	0.520300	+	0.853983i	$0.325820\pi$
$450$	0	0
$451$	−1.40366	−0.0660956
$452$	0	0
$453$	− 8.76418i	− 0.411777i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 25.8141i	− 1.20753i	−0.797161	−	0.603767i	$-0.793666\pi$
$457$	− 25.8141i	− 1.20753i	0.797161	−	0.603767i	$-0.206334\pi$
$458$	0	0
$459$	−3.16784	−0.147862
$460$	0	0
$461$	−6.48878	−0.302213	−0.151106	−	0.988518i	$-0.548284\pi$
$461$	−6.48878	−0.302213	−0.151106	+	0.988518i	$0.548284\pi$
$462$	0	0
$463$	− 23.1421i	− 1.07550i	−0.843103	−	0.537752i	$-0.819274\pi$
$463$	− 23.1421i	− 1.07550i	0.843103	−	0.537752i	$-0.180726\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 1.12055i	− 0.0518531i	−0.999664	−	0.0259265i	$-0.991746\pi$
$467$	− 1.12055i	− 0.0518531i	0.999664	−	0.0259265i	$-0.00825360\pi$
$468$	0	0
$469$	22.3632	1.03264
$470$	0	0
$471$	6.27540	0.289155
$472$	0	0
$473$	1.19798i	0.0550830i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 10.2754i	− 0.470478i
$478$	0	0
$479$	3.90188	0.178281	0.0891407	−	0.996019i	$-0.471588\pi$
$479$	3.90188	0.178281	0.0891407	+	0.996019i	$0.471588\pi$
$480$	0	0
$481$	13.2583	0.604524
$482$	0	0
$483$	14.9622i	0.680802i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 40.3203i	− 1.82709i	−0.406742	−	0.913543i	$-0.633335\pi$
$487$	− 40.3203i	− 1.82709i	0.406742	−	0.913543i	$-0.366665\pi$
$488$	0	0
$489$	−2.11526	−0.0956556
$490$	0	0
$491$	33.8262	1.52655	0.763277	−	0.646071i	$-0.223589\pi$
$491$	33.8262	1.52655	0.763277	+	0.646071i	$0.223589\pi$
$492$	0	0
$493$	3.16784i	0.142672i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	7.31324i	0.328044i
$498$	0	0
$499$	30.0801	1.34657	0.673284	−	0.739384i	$-0.264883\pi$
$499$	30.0801	1.34657	0.673284	+	0.739384i	$0.264883\pi$
$500$	0	0
$501$	−8.51122	−0.380253
$502$	0	0
$503$	20.4553i	0.912058i	0.889965	+	0.456029i	$0.150729\pi$
$503$	20.4553i	0.912058i	−0.889965	+	0.456029i	$0.849271\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	10.5387i	0.468041i
$508$	0	0
$509$	3.59393	0.159298	0.0796491	−	0.996823i	$-0.474620\pi$
$509$	3.59393	0.159298	0.0796491	+	0.996823i	$0.474620\pi$
$510$	0	0
$511$	34.2600	1.51557
$512$	0	0
$513$	− 4.00000i	− 0.176604i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 7.38560i	− 0.324819i
$518$	0	0
$519$	3.70919	0.162816
$520$	0	0
$521$	−0.681469	−0.0298557	−0.0149278	−	0.999889i	$-0.504752\pi$
$521$	−0.681469	−0.0298557	−0.0149278	+	0.999889i	$0.504752\pi$
$522$	0	0
$523$	1.56885i	0.0686010i	0.999412	+	0.0343005i	$0.0109203\pi$
$523$	1.56885i	0.0686010i	−0.999412	+	0.0343005i	$0.989080\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 10.4958i	− 0.457205i
$528$	0	0
$529$	−3.39595	−0.147650
$530$	0	0
$531$	−13.6489	−0.592313
$532$	0	0
$533$	3.35373i	0.145266i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 0.450940i	− 0.0194595i
$538$	0	0
$539$	0.972507	0.0418888
$540$	0	0
$541$	−8.50593	−0.365698	−0.182849	−	0.983141i	$-0.558532\pi$
$541$	−8.50593	−0.365698	−0.182849	+	0.983141i	$0.558532\pi$
$542$	0	0
$543$	− 20.4054i	− 0.875679i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 36.2901i	− 1.55165i	−0.630946	−	0.775827i	$-0.717333\pi$
$547$	− 36.2901i	− 1.55165i	0.630946	−	0.775827i	$-0.282667\pi$
$548$	0	0
$549$	−4.62648	−0.197453
$550$	0	0
$551$	−4.00000	−0.170406
$552$	0	0
$553$	36.4355i	1.54940i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	30.8918i	1.30893i	0.756094	+	0.654463i	$0.227105\pi$
$557$	30.8918i	1.30893i	−0.756094	+	0.654463i	$0.772895\pi$
$558$	0	0
$559$	2.86230	0.121062
$560$	0	0
$561$	−2.08007	−0.0878205
$562$	0	0
$563$	35.8339i	1.51022i	0.655599	+	0.755109i	$0.272416\pi$
$563$	35.8339i	1.51022i	−0.655599	+	0.755109i	$0.727584\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	2.91223i	0.122302i
$568$	0	0
$569$	28.1947	1.18198	0.590991	−	0.806678i	$-0.298737\pi$
$569$	28.1947	1.18198	0.590991	+	0.806678i	$0.298737\pi$
$570$	0	0
$571$	17.0396	0.713084	0.356542	−	0.934279i	$-0.383956\pi$
$571$	17.0396	0.713084	0.356542	+	0.934279i	$0.383956\pi$
$572$	0	0
$573$	5.25296i	0.219446i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 23.4131i	− 0.974700i	−0.873207	−	0.487350i	$-0.837964\pi$
$577$	− 23.4131i	− 0.974700i	0.873207	−	0.487350i	$-0.162036\pi$
$578$	0	0
$579$	−6.80202	−0.282682
$580$	0	0
$581$	−32.0999	−1.33173
$582$	0	0
$583$	− 6.74704i	− 0.279434i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 36.7866i	− 1.51835i	−0.650889	−	0.759173i	$-0.725604\pi$
$587$	− 36.7866i	− 1.51835i	0.650889	−	0.759173i	$-0.274396\pi$
$588$	0	0
$589$	13.2530	0.546079
$590$	0	0
$591$	−18.3357	−0.754229
$592$	0	0
$593$	29.8091i	1.22411i	0.790815	+	0.612056i	$0.209657\pi$
$593$	29.8091i	1.22411i	−0.790815	+	0.612056i	$0.790343\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	4.54641i	0.186072i
$598$	0	0
$599$	−28.9518	−1.18294	−0.591469	−	0.806327i	$-0.701452\pi$
$599$	−28.9518	−1.18294	−0.591469	+	0.806327i	$0.701452\pi$
$600$	0	0
$601$	−10.3753	−0.423215	−0.211608	−	0.977355i	$-0.567870\pi$
$601$	−10.3753	−0.423215	−0.211608	+	0.977355i	$0.567870\pi$
$602$	0	0
$603$	7.67906i	0.312715i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 25.1773i	− 1.02191i	−0.859606	−	0.510957i	$-0.829291\pi$
$607$	− 25.1773i	− 1.02191i	0.859606	−	0.510957i	$-0.170709\pi$
$608$	0	0
$609$	2.91223	0.118009
$610$	0	0
$611$	−17.6463	−0.713892
$612$	0	0
$613$	36.0895i	1.45764i	0.684705	+	0.728821i	$0.259931\pi$
$613$	36.0895i	1.45764i	−0.684705	+	0.728821i	$0.740069\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	34.7642i	1.39955i	0.714362	+	0.699777i	$0.246717\pi$
$617$	34.7642i	1.39955i	−0.714362	+	0.699777i	$0.753283\pi$
$618$	0	0
$619$	−27.1223	−1.09014	−0.545068	−	0.838392i	$-0.683496\pi$
$619$	−27.1223	−1.09014	−0.545068	+	0.838392i	$0.683496\pi$
$620$	0	0
$621$	−5.13770	−0.206169
$622$	0	0
$623$	20.0422i	0.802975i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 2.62648i	− 0.104892i
$628$	0	0
$629$	−26.7712	−1.06744
$630$	0	0
$631$	−13.4080	−0.533766	−0.266883	−	0.963729i	$-0.585994\pi$
$631$	−13.4080	−0.533766	−0.266883	+	0.963729i	$0.585994\pi$
$632$	0	0
$633$	8.33568i	0.331313i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 2.32359i	− 0.0920640i
$638$	0	0
$639$	−2.51122	−0.0993422
$640$	0	0
$641$	6.75209	0.266692	0.133346	−	0.991070i	$-0.457428\pi$
$641$	6.75209	0.266692	0.133346	+	0.991070i	$0.457428\pi$
$642$	0	0
$643$	39.9993i	1.57742i	0.614766	+	0.788710i	$0.289251\pi$
$643$	39.9993i	1.57742i	−0.614766	+	0.788710i	$0.710749\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 7.87945i	− 0.309773i	−0.987932	−	0.154886i	$-0.950499\pi$
$647$	− 7.87945i	− 0.309773i	0.987932	−	0.154886i	$-0.0495012\pi$
$648$	0	0
$649$	−8.96216	−0.351795
$650$	0	0
$651$	−9.64892	−0.378171
$652$	0	0
$653$	− 40.5105i	− 1.58530i	−0.609677	−	0.792650i	$-0.708701\pi$
$653$	− 40.5105i	− 1.58530i	0.609677	−	0.792650i	$-0.291299\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	11.7642i	0.458964i
$658$	0	0
$659$	18.0051	0.701377	0.350689	−	0.936492i	$-0.385948\pi$
$659$	18.0051	0.701377	0.350689	+	0.936492i	$0.385948\pi$
$660$	0	0
$661$	3.96480	0.154213	0.0771065	−	0.997023i	$-0.475432\pi$
$661$	3.96480	0.154213	0.0771065	+	0.997023i	$0.475432\pi$
$662$	0	0
$663$	4.96986i	0.193013i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	5.13770i	0.198933i
$668$	0	0
$669$	−24.7015	−0.955015
$670$	0	0
$671$	−3.03784	−0.117275
$672$	0	0
$673$	0.897498i	0.0345960i	0.999850	+	0.0172980i	$0.00550640\pi$
$673$	0.897498i	0.0345960i	−0.999850	+	0.0172980i	$0.994494\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	32.0094i	1.23022i	0.788440	+	0.615111i	$0.210889\pi$
$677$	32.0094i	1.23022i	−0.788440	+	0.615111i	$0.789111\pi$
$678$	0	0
$679$	24.2754	0.931604
$680$	0	0
$681$	20.5508	0.787508
$682$	0	0
$683$	− 5.25296i	− 0.200999i	−0.994937	−	0.100500i	$-0.967956\pi$
$683$	− 5.25296i	− 0.200999i	0.994937	−	0.100500i	$-0.0320441\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 18.4958i	− 0.705659i
$688$	0	0
$689$	−16.1206	−0.614144
$690$	0	0
$691$	−20.2901	−0.771873	−0.385936	−	0.922525i	$-0.626122\pi$
$691$	−20.2901	−0.771873	−0.385936	+	0.922525i	$0.626122\pi$
$692$	0	0
$693$	1.91223i	0.0726396i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 6.77188i	− 0.256503i
$698$	0	0
$699$	−6.97757	−0.263916
$700$	0	0
$701$	22.8416	0.862715	0.431358	−	0.902181i	$-0.358035\pi$
$701$	22.8416	0.862715	0.431358	+	0.902181i	$0.358035\pi$
$702$	0	0
$703$	− 33.8038i	− 1.27493i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	4.74439i	0.178431i
$708$	0	0
$709$	−23.8262	−0.894812	−0.447406	−	0.894331i	$-0.647652\pi$
$709$	−23.8262	−0.894812	−0.447406	+	0.894331i	$0.647652\pi$
$710$	0	0
$711$	−12.5112	−0.469207
$712$	0	0
$713$	− 17.0224i	− 0.637495i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	4.68676i	0.175030i
$718$	0	0
$719$	25.1980	0.939726	0.469863	−	0.882739i	$-0.344303\pi$
$719$	25.1980	0.939726	0.469863	+	0.882739i	$0.344303\pi$
$720$	0	0
$721$	49.9742	1.86114
$722$	0	0
$723$	26.2650i	0.976808i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	30.3598i	1.12598i	0.826462	+	0.562992i	$0.190350\pi$
$727$	30.3598i	1.12598i	−0.826462	+	0.562992i	$0.809650\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−5.77959	−0.213766
$732$	0	0
$733$	− 15.3977i	− 0.568727i	−0.958717	−	0.284363i	$-0.908218\pi$
$733$	− 15.3977i	− 0.568727i	0.958717	−	0.284363i	$-0.0917822\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	5.04222i	0.185733i
$738$	0	0
$739$	11.1980	0.411924	0.205962	−	0.978560i	$-0.433968\pi$
$739$	11.1980	0.411924	0.205962	+	0.978560i	$0.433968\pi$
$740$	0	0
$741$	−6.27540	−0.230533
$742$	0	0
$743$	33.3029i	1.22176i	0.791721	+	0.610882i	$0.209185\pi$
$743$	33.3029i	1.22176i	−0.791721	+	0.610882i	$0.790815\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 11.0224i	− 0.403290i
$748$	0	0
$749$	−11.4734	−0.419228
$750$	0	0
$751$	20.1051	0.733647	0.366824	−	0.930291i	$-0.380445\pi$
$751$	20.1051	0.733647	0.366824	+	0.930291i	$0.380445\pi$
$752$	0	0
$753$	− 12.7668i	− 0.465249i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	22.0396i	0.801042i	0.916288	+	0.400521i	$0.131171\pi$
$757$	22.0396i	0.801042i	−0.916288	+	0.400521i	$0.868829\pi$
$758$	0	0
$759$	−3.37352	−0.122451
$760$	0	0
$761$	21.3735	0.774789	0.387395	−	0.921914i	$-0.373375\pi$
$761$	21.3735	0.774789	0.387395	+	0.921914i	$0.373375\pi$
$762$	0	0
$763$	42.4932i	1.53836i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	21.4131i	0.773182i
$768$	0	0
$769$	12.7712	0.460542	0.230271	−	0.973127i	$-0.426039\pi$
$769$	12.7712	0.460542	0.230271	+	0.973127i	$0.426039\pi$
$770$	0	0
$771$	−9.03784	−0.325490
$772$	0	0
$773$	− 44.4751i	− 1.59966i	−0.600228	−	0.799829i	$-0.704923\pi$
$773$	− 44.4751i	− 1.59966i	0.600228	−	0.799829i	$-0.295077\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	24.6111i	0.882917i
$778$	0	0
$779$	8.55080	0.306364
$780$	0	0
$781$	−1.64892	−0.0590029
$782$	0	0
$783$	1.00000i	0.0357371i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	20.4580i	0.729248i	0.931155	+	0.364624i	$0.118802\pi$
$787$	20.4580i	0.729248i	−0.931155	+	0.364624i	$0.881198\pi$
$788$	0	0
$789$	18.5284	0.659627
$790$	0	0
$791$	−15.2831	−0.543405
$792$	0	0
$793$	7.25825i	0.257748i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	4.83987i	0.171437i	0.996319	+	0.0857184i	$0.0273185\pi$
$797$	4.83987i	0.171437i	−0.996319	+	0.0857184i	$0.972681\pi$
$798$	0	0
$799$	35.6315	1.26055
$800$	0	0
$801$	−6.88209	−0.243167
$802$	0	0
$803$	7.72460i	0.272595i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	19.1076i	0.672618i
$808$	0	0
$809$	30.0543	1.05665	0.528327	−	0.849041i	$-0.322820\pi$
$809$	30.0543	1.05665	0.528327	+	0.849041i	$0.322820\pi$
$810$	0	0
$811$	37.6034	1.32043	0.660216	−	0.751075i	$-0.270465\pi$
$811$	37.6034	1.32043	0.660216	+	0.751075i	$0.270465\pi$
$812$	0	0
$813$	− 24.8865i	− 0.872807i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 7.29783i	− 0.255319i
$818$	0	0
$819$	4.56885	0.159649
$820$	0	0
$821$	6.45094	0.225139	0.112570	−	0.993644i	$-0.464092\pi$
$821$	6.45094	0.225139	0.112570	+	0.993644i	$0.464092\pi$
$822$	0	0
$823$	5.19269i	0.181006i	0.995896	+	0.0905028i	$0.0288474\pi$
$823$	5.19269i	0.181006i	−0.995896	+	0.0905028i	$0.971153\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 27.8262i	− 0.967612i	−0.875175	−	0.483806i	$-0.839254\pi$
$827$	− 27.8262i	− 0.967612i	0.875175	−	0.483806i	$-0.160746\pi$
$828$	0	0
$829$	51.2067	1.77848	0.889242	−	0.457437i	$-0.151233\pi$
$829$	51.2067	1.77848	0.889242	+	0.457437i	$0.151233\pi$
$830$	0	0
$831$	−14.0697	−0.488073
$832$	0	0
$833$	4.69182i	0.162562i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 3.31324i	− 0.114522i
$838$	0	0
$839$	−38.6412	−1.33404	−0.667021	−	0.745039i	$-0.732431\pi$
$839$	−38.6412	−1.33404	−0.667021	+	0.745039i	$0.732431\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	− 26.7712i	− 0.922049i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 30.7789i	− 1.05758i
$848$	0	0
$849$	0.884736	0.0303641
$850$	0	0
$851$	−43.4184	−1.48836
$852$	0	0
$853$	− 19.7488i	− 0.676185i	−0.941113	−	0.338093i	$-0.890218\pi$
$853$	− 19.7488i	− 0.676185i	0.941113	−	0.338093i	$-0.109782\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 31.8038i	− 1.08640i	−0.839605	−	0.543198i	$-0.817213\pi$
$857$	− 31.8038i	− 1.08640i	0.839605	−	0.543198i	$-0.182787\pi$
$858$	0	0
$859$	−46.6955	−1.59323	−0.796615	−	0.604487i	$-0.793378\pi$
$859$	−46.6955	−1.59323	−0.796615	+	0.604487i	$0.793378\pi$
$860$	0	0
$861$	−6.22547	−0.212163
$862$	0	0
$863$	− 15.9397i	− 0.542594i	−0.962496	−	0.271297i	$-0.912547\pi$
$863$	− 15.9397i	− 0.542594i	0.962496	−	0.271297i	$-0.0874526\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	6.96480i	0.236537i
$868$	0	0
$869$	−8.21512	−0.278679
$870$	0	0
$871$	12.0473	0.408207
$872$	0	0
$873$	8.33568i	0.282120i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 28.1773i	− 0.951479i	−0.879586	−	0.475740i	$-0.842180\pi$
$877$	− 28.1773i	− 0.951479i	0.879586	−	0.475740i	$-0.157820\pi$
$878$	0	0
$879$	17.1575	0.578708
$880$	0	0
$881$	22.9518	0.773266	0.386633	−	0.922234i	$-0.373638\pi$
$881$	22.9518	0.773266	0.386633	+	0.922234i	$0.373638\pi$
$882$	0	0
$883$	40.9916i	1.37948i	0.724058	+	0.689739i	$0.242275\pi$
$883$	40.9916i	1.37948i	−0.724058	+	0.689739i	$0.757725\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	14.9967i	0.503539i	0.967787	+	0.251770i	$0.0810125\pi$
$887$	14.9967i	0.503539i	−0.967787	+	0.251770i	$0.918987\pi$
$888$	0	0
$889$	−25.2978	−0.848462
$890$	0	0
$891$	−0.656620	−0.0219976
$892$	0	0
$893$	44.9916i	1.50559i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	8.06028i	0.269125i
$898$	0	0
$899$	−3.31324	−0.110503
$900$	0	0
$901$	32.5508	1.08442
$902$	0	0
$903$	5.31324i	0.176814i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	4.68147i	0.155446i	0.996975	+	0.0777228i	$0.0247649\pi$
$907$	4.68147i	0.155446i	−0.996975	+	0.0777228i	$0.975235\pi$
$908$	0	0
$909$	−1.62913	−0.0540347
$910$	0	0
$911$	−12.0955	−0.400741	−0.200370	−	0.979720i	$-0.564215\pi$
$911$	−12.0955	−0.400741	−0.200370	+	0.979720i	$0.564215\pi$
$912$	0	0
$913$	− 7.23756i	− 0.239528i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 25.7789i	− 0.851295i
$918$	0	0
$919$	26.3608	0.869561	0.434781	−	0.900536i	$-0.356826\pi$
$919$	26.3608	0.869561	0.434781	+	0.900536i	$0.356826\pi$
$920$	0	0
$921$	23.5284	0.775286
$922$	0	0
$923$	3.93972i	0.129678i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	17.1601i	0.563613i
$928$	0	0
$929$	44.9261	1.47398	0.736988	−	0.675906i	$-0.236247\pi$
$929$	44.9261	1.47398	0.736988	+	0.675906i	$0.236247\pi$
$930$	0	0
$931$	−5.92432	−0.194162
$932$	0	0
$933$	11.6988i	0.383003i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 8.50684i	− 0.277906i	−0.990299	−	0.138953i	$-0.955626\pi$
$937$	− 8.50684i	− 0.277906i	0.990299	−	0.138953i	$-0.0443737\pi$
$938$	0	0
$939$	2.38560	0.0778513
$940$	0	0
$941$	−50.4751	−1.64544	−0.822721	−	0.568446i	$-0.807545\pi$
$941$	−50.4751	−1.64544	−0.822721	+	0.568446i	$0.807545\pi$
$942$	0	0
$943$	− 10.9829i	− 0.357651i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	22.9320i	0.745190i	0.927994	+	0.372595i	$0.121532\pi$
$947$	22.9320i	0.745190i	−0.927994	+	0.372595i	$0.878468\pi$
$948$	0	0
$949$	18.4562	0.599115
$950$	0	0
$951$	−28.2556	−0.916251
$952$	0	0
$953$	− 23.6643i	− 0.766563i	−0.923632	−	0.383281i	$-0.874794\pi$
$953$	− 23.6643i	− 0.766563i	0.923632	−	0.383281i	$-0.125206\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0.656620i	0.0212255i
$958$	0	0
$959$	−48.8211	−1.57652
$960$	0	0
$961$	−20.0224	−0.645885
$962$	0	0
$963$	− 3.93972i	− 0.126956i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 50.0000i	− 1.60789i	−0.594703	−	0.803946i	$-0.702730\pi$
$967$	− 50.0000i	− 1.60789i	0.594703	−	0.803946i	$-0.297270\pi$
$968$	0	0
$969$	12.6714	0.407062
$970$	0	0
$971$	−11.3960	−0.365714	−0.182857	−	0.983140i	$-0.558534\pi$
$971$	−11.3960	−0.365714	−0.182857	+	0.983140i	$0.558534\pi$
$972$	0	0
$973$	21.1524i	0.678116i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 12.8262i	− 0.410346i	−0.978726	−	0.205173i	$-0.934224\pi$
$977$	− 12.8262i	− 0.410346i	0.978726	−	0.205173i	$-0.0657758\pi$
$978$	0	0
$979$	−4.51892	−0.144425
$980$	0	0
$981$	−14.5913	−0.465864
$982$	0	0
$983$	− 30.2978i	− 0.966351i	−0.875524	−	0.483175i	$-0.839483\pi$
$983$	− 30.2978i	− 0.966351i	0.875524	−	0.483175i	$-0.160517\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 32.7565i	− 1.04265i
$988$	0	0
$989$	−9.37352	−0.298061
$990$	0	0
$991$	−5.84954	−0.185817	−0.0929084	−	0.995675i	$-0.529616\pi$
$991$	−5.84954	−0.185817	−0.0929084	+	0.995675i	$0.529616\pi$
$992$	0	0
$993$	− 1.03784i	− 0.0329349i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	44.2754i	1.40222i	0.713055	+	0.701108i	$0.247311\pi$
$997$	44.2754i	1.40222i	−0.713055	+	0.701108i	$0.752689\pi$
$998$	0	0
$999$	−8.45094	−0.267376

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8700.2.g.u.349.6		6
5.2	odd	4		8700.2.a.bc.1.1	yes	3
5.3	odd	4		8700.2.a.bb.1.3	✓	3
5.4	even	2	inner	8700.2.g.u.349.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8700.2.a.bb.1.3	✓	3	5.3	odd	4
8700.2.a.bc.1.1	yes	3	5.2	odd	4
8700.2.g.u.349.1		6	5.4	even	2	inner
8700.2.g.u.349.6		6	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:	\( N \)	\(=\)	\( 8700 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8700.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +2.91223i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +2.91223i q^{7} -1.00000 q^{9} -0.656620 q^{11} +1.56885i q^{13} -3.16784i q^{17} +4.00000 q^{19} -2.91223 q^{21} -5.13770i q^{23} -1.00000i q^{27} -1.00000 q^{29} +3.31324 q^{31} -0.656620i q^{33} -8.45094i q^{37} -1.56885 q^{39} +2.13770 q^{41} -1.82446i q^{43} +11.2479i q^{47} -1.48108 q^{49} +3.16784 q^{51} +10.2754i q^{53} +4.00000i q^{57} +13.6489 q^{59} +4.62648 q^{61} -2.91223i q^{63} -7.67906i q^{67} +5.13770 q^{69} +2.51122 q^{71} -11.7642i q^{73} -1.91223i q^{77} +12.5112 q^{79} +1.00000 q^{81} +11.0224i q^{83} -1.00000i q^{87} +6.88209 q^{89} -4.56885 q^{91} +3.31324i q^{93} -8.33568i q^{97} +0.656620 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{9}+O(q^{10}) \) 6 * q - 6 * q^9	\( 6 q - 6 q^{9} + 24 q^{19} - 4 q^{21} - 6 q^{29} + 12 q^{31} + 8 q^{39} - 22 q^{41} + 22 q^{49} - 4 q^{51} + 28 q^{59} + 12 q^{61} - 4 q^{69} - 4 q^{71} + 56 q^{79} + 6 q^{81} + 16 q^{89} - 10 q^{91}+O(q^{100}) \) 6 * q - 6 * q^9 + 24 * q^19 - 4 * q^21 - 6 * q^29 + 12 * q^31 + 8 * q^39 - 22 * q^41 + 22 * q^49 - 4 * q^51 + 28 * q^59 + 12 * q^61 - 4 * q^69 - 4 * q^71 + 56 * q^79 + 6 * q^81 + 16 * q^89 - 10 * q^91

Introduction

L-functions

Modular forms

Varieties

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Database

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