LMFDB - Embedded newform 5780.2.c.g.5201.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5780,2,Mod(5201,5780)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5780, 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("5780.5201");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5780 = 2^{2} \cdot 5 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5780.c (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:	$46.1535323683$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	12.0.851059918206111744.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 33x^{10} + 360x^{8} + 1423x^{6} + 1269x^{4} + 234x^{2} + 9 $ x^12 + 33x^10 + 360x^8 + 1423x^6 + 1269x^4 + 234*x^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			5201.4
Root			$-3.18216i$ of defining polynomial
Character	$\chi$	$=$	5780.5201
Dual form			5780.2.c.g.5201.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.30278i q^{3} -1.00000i q^{5} -1.53209i q^{7} +1.30278 q^{9} +O(q^{10})$	$q-1.30278i q^{3} -1.00000i q^{5} -1.53209i q^{7} +1.30278 q^{9} +4.32780i q^{11} -2.57661 q^{13} -1.30278 q^{15} -1.57258 q^{19} -1.99597 q^{21} -3.25959i q^{23} -1.00000 q^{25} -5.60555i q^{27} +8.93335i q^{29} +5.75259i q^{31} +5.63816 q^{33} -1.53209 q^{35} +2.02288i q^{37} +3.35674i q^{39} +2.02503i q^{41} -0.0646641 q^{43} -1.30278i q^{45} -9.35283 q^{47} +4.65270 q^{49} -8.66167 q^{53} +4.32780 q^{55} +2.04872i q^{57} -0.790461 q^{59} -4.06803i q^{61} -1.99597i q^{63} +2.57661i q^{65} -8.46050 q^{67} -4.24652 q^{69} +10.8013i q^{71} +7.50609i q^{73} +1.30278i q^{75} +6.63058 q^{77} +15.4629i q^{79} -3.39445 q^{81} -0.633803 q^{83} +11.6382 q^{87} +3.53945 q^{89} +3.94759i q^{91} +7.49434 q^{93} +1.57258i q^{95} -12.1049i q^{97} +5.63816i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 6 q^{9}+O(q^{10}) $ 12 * q - 6 * q^9	$ 12 q - 6 q^{9} - 30 q^{13} + 6 q^{15} + 6 q^{19} - 12 q^{25} - 6 q^{43} - 30 q^{47} + 60 q^{49} + 24 q^{53} + 12 q^{59} - 12 q^{67} + 6 q^{77} - 84 q^{81} + 30 q^{83} + 72 q^{87} - 12 q^{89} + 72 q^{93}+O(q^{100}) $ 12 * q - 6 * q^9 - 30 * q^13 + 6 * q^15 + 6 * q^19 - 12 * q^25 - 6 * q^43 - 30 * q^47 + 60 * q^49 + 24 * q^53 + 12 * q^59 - 12 * q^67 + 6 * q^77 - 84 * q^81 + 30 * q^83 + 72 * q^87 - 12 * q^89 + 72 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5780\mathbb{Z}\right)^\times$.

$n$	$581$	$1157$	$2891$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	− 1.30278i	− 0.752158i	−0.926588	−	0.376079i	$-0.877272\pi$
$3$	− 1.30278i	− 0.752158i	0.926588	−	0.376079i	$-0.122728\pi$
$4$	0	0
$5$	− 1.00000i	− 0.447214i
$5$	− 1.00000i	− 0.447214i
$6$	0	0
$7$	− 1.53209i	− 0.579075i	−0.957167	−	0.289538i	$-0.906498\pi$
$7$	− 1.53209i	− 0.579075i	0.957167	−	0.289538i	$-0.0935015\pi$
$8$	0	0
$9$	1.30278	0.434259
$10$	0	0
$11$	4.32780i	1.30488i	0.757840	+	0.652441i	$0.226255\pi$
$11$	4.32780i	1.30488i	−0.757840	+	0.652441i	$0.773745\pi$
$12$	0	0
$13$	−2.57661	−0.714623	−0.357311	−	0.933985i	$-0.616307\pi$
$13$	−2.57661	−0.714623	−0.357311	+	0.933985i	$0.616307\pi$
$14$	0	0
$15$	−1.30278	−0.336375
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	−1.57258	−0.360774	−0.180387	−	0.983596i	$-0.557735\pi$
$19$	−1.57258	−0.360774	−0.180387	+	0.983596i	$0.557735\pi$
$20$	0	0
$21$	−1.99597	−0.435556
$22$	0	0
$23$	− 3.25959i	− 0.679672i	−0.940485	−	0.339836i	$-0.889628\pi$
$23$	− 3.25959i	− 0.679672i	0.940485	−	0.339836i	$-0.110372\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	− 5.60555i	− 1.07879i
$28$	0	0
$29$	8.93335i	1.65888i	0.558594	+	0.829441i	$0.311341\pi$
$29$	8.93335i	1.65888i	−0.558594	+	0.829441i	$0.688659\pi$
$30$	0	0
$31$	5.75259i	1.03320i	0.856228	+	0.516598i	$0.172802\pi$
$31$	5.75259i	1.03320i	−0.856228	+	0.516598i	$0.827198\pi$
$32$	0	0
$33$	5.63816	0.981477
$34$	0	0
$35$	−1.53209	−0.258970
$36$	0	0
$37$	2.02288i	0.332560i	0.986079	+	0.166280i	$0.0531755\pi$
$37$	2.02288i	0.332560i	−0.986079	+	0.166280i	$0.946824\pi$
$38$	0	0
$39$	3.35674i	0.537509i
$40$	0	0
$41$	2.02503i	0.316256i	0.987419	+	0.158128i	$0.0505459\pi$
$41$	2.02503i	0.316256i	−0.987419	+	0.158128i	$0.949454\pi$
$42$	0	0
$43$	−0.0646641	−0.00986118	−0.00493059	−	0.999988i	$-0.501569\pi$
$43$	−0.0646641	−0.00986118	−0.00493059	+	0.999988i	$0.501569\pi$
$44$	0	0
$45$	− 1.30278i	− 0.194206i
$46$	0	0
$47$	−9.35283	−1.36425	−0.682125	−	0.731235i	$-0.738944\pi$
$47$	−9.35283	−1.36425	−0.682125	+	0.731235i	$0.738944\pi$
$48$	0	0
$49$	4.65270	0.664672
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−8.66167	−1.18977	−0.594886	−	0.803810i	$-0.702803\pi$
$53$	−8.66167	−1.18977	−0.594886	+	0.803810i	$0.702803\pi$
$54$	0	0
$55$	4.32780	0.583561
$56$	0	0
$57$	2.04872i	0.271359i
$58$	0	0
$59$	−0.790461	−0.102909	−0.0514546	−	0.998675i	$-0.516386\pi$
$59$	−0.790461	−0.102909	−0.0514546	+	0.998675i	$0.516386\pi$
$60$	0	0
$61$	− 4.06803i	− 0.520858i	−0.965493	−	0.260429i	$-0.916136\pi$
$61$	− 4.06803i	− 0.520858i	0.965493	−	0.260429i	$-0.0838640\pi$
$62$	0	0
$63$	− 1.99597i	− 0.251468i
$64$	0	0
$65$	2.57661i	0.319589i
$66$	0	0
$67$	−8.46050	−1.03361	−0.516807	−	0.856102i	$-0.672880\pi$
$67$	−8.46050	−1.03361	−0.516807	+	0.856102i	$0.672880\pi$
$68$	0	0
$69$	−4.24652	−0.511221
$70$	0	0
$71$	10.8013i	1.28188i	0.767591	+	0.640940i	$0.221455\pi$
$71$	10.8013i	1.28188i	−0.767591	+	0.640940i	$0.778545\pi$
$72$	0	0
$73$	7.50609i	0.878521i	0.898360	+	0.439261i	$0.144760\pi$
$73$	7.50609i	0.878521i	−0.898360	+	0.439261i	$0.855240\pi$
$74$	0	0
$75$	1.30278i	0.150432i
$76$	0	0
$77$	6.63058	0.755625
$78$	0	0
$79$	15.4629i	1.73972i	0.493303	+	0.869858i	$0.335789\pi$
$79$	15.4629i	1.73972i	−0.493303	+	0.869858i	$0.664211\pi$
$80$	0	0
$81$	−3.39445	−0.377161
$82$	0	0
$83$	−0.633803	−0.0695689	−0.0347844	−	0.999395i	$-0.511074\pi$
$83$	−0.633803	−0.0695689	−0.0347844	+	0.999395i	$0.511074\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	11.6382	1.24774
$88$	0	0
$89$	3.53945	0.375181	0.187590	−	0.982247i	$-0.439932\pi$
$89$	3.53945	0.375181	0.187590	+	0.982247i	$0.439932\pi$
$90$	0	0
$91$	3.94759i	0.413820i
$92$	0	0
$93$	7.49434	0.777127
$94$	0	0
$95$	1.57258i	0.161343i
$96$	0	0
$97$	− 12.1049i	− 1.22907i	−0.788890	−	0.614535i	$-0.789344\pi$
$97$	− 12.1049i	− 1.22907i	0.788890	−	0.614535i	$-0.210656\pi$
$98$	0	0
$99$	5.63816i	0.566656i
$100$	0	0
$101$	5.72000	0.569162	0.284581	−	0.958652i	$-0.408146\pi$
$101$	5.72000	0.569162	0.284581	+	0.958652i	$0.408146\pi$
$102$	0	0
$103$	−8.90640	−0.877574	−0.438787	−	0.898591i	$-0.644592\pi$
$103$	−8.90640	−0.877574	−0.438787	+	0.898591i	$0.644592\pi$
$104$	0	0
$105$	1.99597i	0.194787i
$106$	0	0
$107$	2.20940i	0.213590i	0.994281	+	0.106795i	$0.0340589\pi$
$107$	2.20940i	0.213590i	−0.994281	+	0.106795i	$0.965941\pi$
$108$	0	0
$109$	12.5160i	1.19881i	0.800445	+	0.599407i	$0.204597\pi$
$109$	12.5160i	1.19881i	−0.800445	+	0.599407i	$0.795403\pi$
$110$	0	0
$111$	2.63536	0.250137
$112$	0	0
$113$	15.2659i	1.43610i	0.695993	+	0.718049i	$0.254964\pi$
$113$	15.2659i	1.43610i	−0.695993	+	0.718049i	$0.745036\pi$
$114$	0	0
$115$	−3.25959	−0.303959
$116$	0	0
$117$	−3.35674	−0.310331
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−7.72987	−0.702716
$122$	0	0
$123$	2.63816	0.237874
$124$	0	0
$125$	1.00000i	0.0894427i
$126$	0	0
$127$	−0.901981	−0.0800378	−0.0400189	−	0.999199i	$-0.512742\pi$
$127$	−0.901981	−0.0800378	−0.0400189	+	0.999199i	$0.512742\pi$
$128$	0	0
$129$	0.0842428i	0.00741716i
$130$	0	0
$131$	− 7.53434i	− 0.658278i	−0.944281	−	0.329139i	$-0.893241\pi$
$131$	− 7.53434i	− 0.658278i	0.944281	−	0.329139i	$-0.106759\pi$
$132$	0	0
$133$	2.40933i	0.208915i
$134$	0	0
$135$	−5.60555	−0.482449
$136$	0	0
$137$	−22.6289	−1.93332	−0.966659	−	0.256065i	$-0.917574\pi$
$137$	−22.6289	−1.93332	−0.966659	+	0.256065i	$0.917574\pi$
$138$	0	0
$139$	− 12.1251i	− 1.02844i	−0.857659	−	0.514218i	$-0.828082\pi$
$139$	− 12.1251i	− 1.02844i	0.857659	−	0.514218i	$-0.171918\pi$
$140$	0	0
$141$	12.1846i	1.02613i
$142$	0	0
$143$	− 11.1511i	− 0.932498i
$144$	0	0
$145$	8.93335	0.741875
$146$	0	0
$147$	− 6.06143i	− 0.499938i
$148$	0	0
$149$	−15.9678	−1.30813	−0.654067	−	0.756436i	$-0.726939\pi$
$149$	−15.9678	−1.30813	−0.654067	+	0.756436i	$0.726939\pi$
$150$	0	0
$151$	13.6491	1.11074	0.555372	−	0.831602i	$-0.312576\pi$
$151$	13.6491	1.11074	0.555372	+	0.831602i	$0.312576\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	5.75259	0.462059
$156$	0	0
$157$	5.97123	0.476556	0.238278	−	0.971197i	$-0.423417\pi$
$157$	5.97123	0.476556	0.238278	+	0.971197i	$0.423417\pi$
$158$	0	0
$159$	11.2842i	0.894896i
$160$	0	0
$161$	−4.99399	−0.393581
$162$	0	0
$163$	− 19.7254i	− 1.54501i	−0.635009	−	0.772505i	$-0.719004\pi$
$163$	− 19.7254i	− 1.54501i	0.635009	−	0.772505i	$-0.280996\pi$
$164$	0	0
$165$	− 5.63816i	− 0.438930i
$166$	0	0
$167$	20.7498i	1.60567i	0.596202	+	0.802835i	$0.296676\pi$
$167$	20.7498i	1.60567i	−0.596202	+	0.802835i	$0.703324\pi$
$168$	0	0
$169$	−6.36108	−0.489314
$170$	0	0
$171$	−2.04872	−0.156669
$172$	0	0
$173$	13.0624i	0.993117i	0.868003	+	0.496559i	$0.165403\pi$
$173$	13.0624i	0.993117i	−0.868003	+	0.496559i	$0.834597\pi$
$174$	0	0
$175$	1.53209i	0.115815i
$176$	0	0
$177$	1.02979i	0.0774040i
$178$	0	0
$179$	−8.58449	−0.641635	−0.320817	−	0.947141i	$-0.603958\pi$
$179$	−8.58449	−0.641635	−0.320817	+	0.947141i	$0.603958\pi$
$180$	0	0
$181$	− 2.20644i	− 0.164003i	−0.996632	−	0.0820016i	$-0.973869\pi$
$181$	− 2.20644i	− 0.164003i	0.996632	−	0.0820016i	$-0.0261313\pi$
$182$	0	0
$183$	−5.29973	−0.391767
$184$	0	0
$185$	2.02288	0.148725
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−8.58820	−0.624700
$190$	0	0
$191$	12.5834	0.910506	0.455253	−	0.890362i	$-0.349549\pi$
$191$	12.5834	0.910506	0.455253	+	0.890362i	$0.349549\pi$
$192$	0	0
$193$	19.9445i	1.43564i	0.696230	+	0.717819i	$0.254860\pi$
$193$	19.9445i	1.43564i	−0.696230	+	0.717819i	$0.745140\pi$
$194$	0	0
$195$	3.35674	0.240381
$196$	0	0
$197$	− 21.0779i	− 1.50174i	−0.660450	−	0.750870i	$-0.729635\pi$
$197$	− 21.0779i	− 1.50174i	0.660450	−	0.750870i	$-0.270365\pi$
$198$	0	0
$199$	17.7009i	1.25479i	0.778703	+	0.627393i	$0.215878\pi$
$199$	17.7009i	1.25479i	−0.778703	+	0.627393i	$0.784122\pi$
$200$	0	0
$201$	11.0221i	0.777441i
$202$	0	0
$203$	13.6867	0.960617
$204$	0	0
$205$	2.02503	0.141434
$206$	0	0
$207$	− 4.24652i	− 0.295153i
$208$	0	0
$209$	− 6.80581i	− 0.470767i
$210$	0	0
$211$	− 5.16250i	− 0.355401i	−0.984085	−	0.177701i	$-0.943134\pi$
$211$	− 5.16250i	− 0.355401i	0.984085	−	0.177701i	$-0.0568659\pi$
$212$	0	0
$213$	14.0717	0.964176
$214$	0	0
$215$	0.0646641i	0.00441005i
$216$	0	0
$217$	8.81348	0.598298
$218$	0	0
$219$	9.77875	0.660787
$220$	0	0
$221$	0	0
$222$	0	0
$223$	9.10309	0.609588	0.304794	−	0.952418i	$-0.401412\pi$
$223$	9.10309	0.609588	0.304794	+	0.952418i	$0.401412\pi$
$224$	0	0
$225$	−1.30278	−0.0868517
$226$	0	0
$227$	− 2.78142i	− 0.184609i	−0.995731	−	0.0923045i	$-0.970577\pi$
$227$	− 2.78142i	− 0.184609i	0.995731	−	0.0923045i	$-0.0294233\pi$
$228$	0	0
$229$	26.1712	1.72944	0.864720	−	0.502255i	$-0.167496\pi$
$229$	26.1712	1.72944	0.864720	+	0.502255i	$0.167496\pi$
$230$	0	0
$231$	− 8.63816i	− 0.568349i
$232$	0	0
$233$	− 2.61356i	− 0.171220i	−0.996329	−	0.0856101i	$-0.972716\pi$
$233$	− 2.61356i	− 0.171220i	0.996329	−	0.0856101i	$-0.0272839\pi$
$234$	0	0
$235$	9.35283i	0.610111i
$236$	0	0
$237$	20.1447	1.30854
$238$	0	0
$239$	3.23459	0.209228	0.104614	−	0.994513i	$-0.466639\pi$
$239$	3.23459	0.209228	0.104614	+	0.994513i	$0.466639\pi$
$240$	0	0
$241$	− 21.8140i	− 1.40516i	−0.711603	−	0.702582i	$-0.752030\pi$
$241$	− 21.8140i	− 1.40516i	0.711603	−	0.702582i	$-0.247970\pi$
$242$	0	0
$243$	− 12.3944i	− 0.795104i
$244$	0	0
$245$	− 4.65270i	− 0.297250i
$246$	0	0
$247$	4.05192	0.257817
$248$	0	0
$249$	0.825703i	0.0523268i
$250$	0	0
$251$	−10.7468	−0.678329	−0.339165	−	0.940727i	$-0.610144\pi$
$251$	−10.7468	−0.678329	−0.339165	+	0.940727i	$0.610144\pi$
$252$	0	0
$253$	14.1069	0.886892
$254$	0	0
$255$	0	0
$256$	0	0
$257$	28.7578	1.79386	0.896931	−	0.442170i	$-0.145791\pi$
$257$	28.7578	1.79386	0.896931	+	0.442170i	$0.145791\pi$
$258$	0	0
$259$	3.09923	0.192577
$260$	0	0
$261$	11.6382i	0.720384i
$262$	0	0
$263$	−5.64140	−0.347864	−0.173932	−	0.984758i	$-0.555647\pi$
$263$	−5.64140	−0.347864	−0.173932	+	0.984758i	$0.555647\pi$
$264$	0	0
$265$	8.66167i	0.532082i
$266$	0	0
$267$	− 4.61110i	− 0.282195i
$268$	0	0
$269$	8.86933i	0.540773i	0.962752	+	0.270386i	$0.0871514\pi$
$269$	8.86933i	0.540773i	−0.962752	+	0.270386i	$0.912849\pi$
$270$	0	0
$271$	16.1244	0.979490	0.489745	−	0.871866i	$-0.337090\pi$
$271$	16.1244	0.979490	0.489745	+	0.871866i	$0.337090\pi$
$272$	0	0
$273$	5.14283	0.311258
$274$	0	0
$275$	− 4.32780i	− 0.260976i
$276$	0	0
$277$	14.1877i	0.852454i	0.904616	+	0.426227i	$0.140158\pi$
$277$	14.1877i	0.852454i	−0.904616	+	0.426227i	$0.859842\pi$
$278$	0	0
$279$	7.49434i	0.448674i
$280$	0	0
$281$	14.3248	0.854546	0.427273	−	0.904123i	$-0.359474\pi$
$281$	14.3248	0.854546	0.427273	+	0.904123i	$0.359474\pi$
$282$	0	0
$283$	28.7084i	1.70654i	0.521472	+	0.853268i	$0.325383\pi$
$283$	28.7084i	1.70654i	−0.521472	+	0.853268i	$0.674617\pi$
$284$	0	0
$285$	2.04872	0.121355
$286$	0	0
$287$	3.10252	0.183136
$288$	0	0
$289$	0	0
$290$	0	0
$291$	−15.7700	−0.924455
$292$	0	0
$293$	−7.80999	−0.456265	−0.228132	−	0.973630i	$-0.573262\pi$
$293$	−7.80999	−0.456265	−0.228132	+	0.973630i	$0.573262\pi$
$294$	0	0
$295$	0.790461i	0.0460224i
$296$	0	0
$297$	24.2597	1.40769
$298$	0	0
$299$	8.39870i	0.485709i
$300$	0	0
$301$	0.0990711i	0.00571036i
$302$	0	0
$303$	− 7.45188i	− 0.428099i
$304$	0	0
$305$	−4.06803	−0.232935
$306$	0	0
$307$	9.53891	0.544414	0.272207	−	0.962239i	$-0.412246\pi$
$307$	9.53891	0.544414	0.272207	+	0.962239i	$0.412246\pi$
$308$	0	0
$309$	11.6030i	0.660074i
$310$	0	0
$311$	− 12.8100i	− 0.726389i	−0.931713	−	0.363194i	$-0.881686\pi$
$311$	− 12.8100i	− 0.726389i	0.931713	−	0.363194i	$-0.118314\pi$
$312$	0	0
$313$	5.36034i	0.302984i	0.988458	+	0.151492i	$0.0484078\pi$
$313$	5.36034i	0.302984i	−0.988458	+	0.151492i	$0.951592\pi$
$314$	0	0
$315$	−1.99597	−0.112460
$316$	0	0
$317$	2.28667i	0.128432i	0.997936	+	0.0642160i	$0.0204547\pi$
$317$	2.28667i	0.128432i	−0.997936	+	0.0642160i	$0.979545\pi$
$318$	0	0
$319$	−38.6618	−2.16464
$320$	0	0
$321$	2.87835	0.160654
$322$	0	0
$323$	0	0
$324$	0	0
$325$	2.57661	0.142925
$326$	0	0
$327$	16.3055	0.901697
$328$	0	0
$329$	14.3294i	0.790004i
$330$	0	0
$331$	−19.6659	−1.08094	−0.540469	−	0.841364i	$-0.681753\pi$
$331$	−19.6659	−1.08094	−0.540469	+	0.841364i	$0.681753\pi$
$332$	0	0
$333$	2.63536i	0.144417i
$334$	0	0
$335$	8.46050i	0.462246i
$336$	0	0
$337$	22.0097i	1.19895i	0.800395	+	0.599473i	$0.204623\pi$
$337$	22.0097i	1.19895i	−0.800395	+	0.599473i	$0.795377\pi$
$338$	0	0
$339$	19.8881	1.08017
$340$	0	0
$341$	−24.8961	−1.34820
$342$	0	0
$343$	− 17.8530i	− 0.963970i
$344$	0	0
$345$	4.24652i	0.228625i
$346$	0	0
$347$	− 18.6406i	− 1.00068i	−0.865829	−	0.500340i	$-0.833208\pi$
$347$	− 18.6406i	− 1.00068i	0.865829	−	0.500340i	$-0.166792\pi$
$348$	0	0
$349$	18.1409	0.971060	0.485530	−	0.874220i	$-0.338627\pi$
$349$	18.1409	0.971060	0.485530	+	0.874220i	$0.338627\pi$
$350$	0	0
$351$	14.4433i	0.770927i
$352$	0	0
$353$	12.2122	0.649988	0.324994	−	0.945716i	$-0.394638\pi$
$353$	12.2122	0.649988	0.324994	+	0.945716i	$0.394638\pi$
$354$	0	0
$355$	10.8013	0.573274
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−4.72152	−0.249192	−0.124596	−	0.992208i	$-0.539764\pi$
$359$	−4.72152	−0.249192	−0.124596	+	0.992208i	$0.539764\pi$
$360$	0	0
$361$	−16.5270	−0.869842
$362$	0	0
$363$	10.0703i	0.528553i
$364$	0	0
$365$	7.50609	0.392887
$366$	0	0
$367$	4.76821i	0.248899i	0.992226	+	0.124449i	$0.0397164\pi$
$367$	4.76821i	0.248899i	−0.992226	+	0.124449i	$0.960284\pi$
$368$	0	0
$369$	2.63816i	0.137337i
$370$	0	0
$371$	13.2704i	0.688967i
$372$	0	0
$373$	30.4398	1.57611	0.788055	−	0.615604i	$-0.211088\pi$
$373$	30.4398	1.57611	0.788055	+	0.615604i	$0.211088\pi$
$374$	0	0
$375$	1.30278	0.0672750
$376$	0	0
$377$	− 23.0178i	− 1.18548i
$378$	0	0
$379$	− 18.1687i	− 0.933263i	−0.884452	−	0.466631i	$-0.845467\pi$
$379$	− 18.1687i	− 0.933263i	0.884452	−	0.466631i	$-0.154533\pi$
$380$	0	0
$381$	1.17508i	0.0602011i
$382$	0	0
$383$	−20.1617	−1.03021	−0.515107	−	0.857126i	$-0.672248\pi$
$383$	−20.1617	−1.03021	−0.515107	+	0.857126i	$0.672248\pi$
$384$	0	0
$385$	− 6.63058i	− 0.337926i
$386$	0	0
$387$	−0.0842428	−0.00428230
$388$	0	0
$389$	25.8930	1.31283	0.656415	−	0.754400i	$-0.272072\pi$
$389$	25.8930	1.31283	0.656415	+	0.754400i	$0.272072\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−9.81556	−0.495129
$394$	0	0
$395$	15.4629	0.778024
$396$	0	0
$397$	14.7913i	0.742356i	0.928562	+	0.371178i	$0.121046\pi$
$397$	14.7913i	0.742356i	−0.928562	+	0.371178i	$0.878954\pi$
$398$	0	0
$399$	3.13881	0.157137
$400$	0	0
$401$	8.62491i	0.430707i	0.976536	+	0.215354i	$0.0690904\pi$
$401$	8.62491i	0.430707i	−0.976536	+	0.215354i	$0.930910\pi$
$402$	0	0
$403$	− 14.8222i	− 0.738346i
$404$	0	0
$405$	3.39445i	0.168672i
$406$	0	0
$407$	−8.75463	−0.433951
$408$	0	0
$409$	−20.6445	−1.02081	−0.510403	−	0.859935i	$-0.670504\pi$
$409$	−20.6445	−1.02081	−0.510403	+	0.859935i	$0.670504\pi$
$410$	0	0
$411$	29.4804i	1.45416i
$412$	0	0
$413$	1.21106i	0.0595922i
$414$	0	0
$415$	0.633803i	0.0311122i
$416$	0	0
$417$	−15.7963	−0.773547
$418$	0	0
$419$	− 15.3532i	− 0.750053i	−0.927014	−	0.375026i	$-0.877634\pi$
$419$	− 15.3532i	− 0.750053i	0.927014	−	0.375026i	$-0.122366\pi$
$420$	0	0
$421$	15.5255	0.756668	0.378334	−	0.925669i	$-0.376497\pi$
$421$	15.5255	0.756668	0.378334	+	0.925669i	$0.376497\pi$
$422$	0	0
$423$	−12.1846	−0.592437
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−6.23258	−0.301616
$428$	0	0
$429$	−14.5273	−0.701386
$430$	0	0
$431$	26.7168i	1.28691i	0.765486	+	0.643453i	$0.222499\pi$
$431$	26.7168i	1.28691i	−0.765486	+	0.643453i	$0.777501\pi$
$432$	0	0
$433$	−31.2540	−1.50197	−0.750985	−	0.660319i	$-0.770421\pi$
$433$	−31.2540	−1.50197	−0.750985	+	0.660319i	$0.770421\pi$
$434$	0	0
$435$	− 11.6382i	− 0.558007i
$436$	0	0
$437$	5.12596i	0.245208i
$438$	0	0
$439$	− 37.9934i	− 1.81332i	−0.421857	−	0.906662i	$-0.638622\pi$
$439$	− 37.9934i	− 1.81332i	0.421857	−	0.906662i	$-0.361378\pi$
$440$	0	0
$441$	6.06143	0.288639
$442$	0	0
$443$	−34.1637	−1.62317	−0.811583	−	0.584237i	$-0.801394\pi$
$443$	−34.1637	−1.62317	−0.811583	+	0.584237i	$0.801394\pi$
$444$	0	0
$445$	− 3.53945i	− 0.167786i
$446$	0	0
$447$	20.8025i	0.983924i
$448$	0	0
$449$	41.2925i	1.94871i	0.225008	+	0.974357i	$0.427759\pi$
$449$	41.2925i	1.94871i	−0.225008	+	0.974357i	$0.572241\pi$
$450$	0	0
$451$	−8.76392	−0.412677
$452$	0	0
$453$	− 17.7817i	− 0.835455i
$454$	0	0
$455$	3.94759	0.185066
$456$	0	0
$457$	34.6259	1.61973	0.809866	−	0.586614i	$-0.199540\pi$
$457$	34.6259	1.61973	0.809866	+	0.586614i	$0.199540\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	31.3380	1.45956	0.729778	−	0.683684i	$-0.239623\pi$
$461$	31.3380	1.45956	0.729778	+	0.683684i	$0.239623\pi$
$462$	0	0
$463$	−19.8760	−0.923716	−0.461858	−	0.886954i	$-0.652817\pi$
$463$	−19.8760	−0.923716	−0.461858	+	0.886954i	$0.652817\pi$
$464$	0	0
$465$	− 7.49434i	− 0.347542i
$466$	0	0
$467$	−1.67557	−0.0775359	−0.0387680	−	0.999248i	$-0.512343\pi$
$467$	−1.67557	−0.0775359	−0.0387680	+	0.999248i	$0.512343\pi$
$468$	0	0
$469$	12.9622i	0.598540i
$470$	0	0
$471$	− 7.77917i	− 0.358445i
$472$	0	0
$473$	− 0.279853i	− 0.0128677i
$474$	0	0
$475$	1.57258	0.0721548
$476$	0	0
$477$	−11.2842	−0.516668
$478$	0	0
$479$	34.2756i	1.56609i	0.621964	+	0.783046i	$0.286335\pi$
$479$	34.2756i	1.56609i	−0.621964	+	0.783046i	$0.713665\pi$
$480$	0	0
$481$	− 5.21218i	− 0.237655i
$482$	0	0
$483$	6.50604i	0.296035i
$484$	0	0
$485$	−12.1049	−0.549657
$486$	0	0
$487$	14.8657i	0.673628i	0.941571	+	0.336814i	$0.109349\pi$
$487$	14.8657i	0.673628i	−0.941571	+	0.336814i	$0.890651\pi$
$488$	0	0
$489$	−25.6977	−1.16209
$490$	0	0
$491$	4.51823	0.203905	0.101952	−	0.994789i	$-0.467491\pi$
$491$	4.51823	0.203905	0.101952	+	0.994789i	$0.467491\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	5.63816	0.253416
$496$	0	0
$497$	16.5486	0.742305
$498$	0	0
$499$	9.66238i	0.432547i	0.976333	+	0.216274i	$0.0693903\pi$
$499$	9.66238i	0.432547i	−0.976333	+	0.216274i	$0.930610\pi$
$500$	0	0
$501$	27.0324	1.20772
$502$	0	0
$503$	2.39415i	0.106750i	0.998575	+	0.0533750i	$0.0169979\pi$
$503$	2.39415i	0.106750i	−0.998575	+	0.0533750i	$0.983002\pi$
$504$	0	0
$505$	− 5.72000i	− 0.254537i
$506$	0	0
$507$	8.28706i	0.368041i
$508$	0	0
$509$	0.756082	0.0335127	0.0167564	−	0.999860i	$-0.494666\pi$
$509$	0.756082	0.0335127	0.0167564	+	0.999860i	$0.494666\pi$
$510$	0	0
$511$	11.5000	0.508730
$512$	0	0
$513$	8.81516i	0.389199i
$514$	0	0
$515$	8.90640i	0.392463i
$516$	0	0
$517$	− 40.4772i	− 1.78019i
$518$	0	0
$519$	17.0174	0.746981
$520$	0	0
$521$	26.6782i	1.16879i	0.811468	+	0.584397i	$0.198669\pi$
$521$	26.6782i	1.16879i	−0.811468	+	0.584397i	$0.801331\pi$
$522$	0	0
$523$	−41.4317	−1.81168	−0.905841	−	0.423618i	$-0.860760\pi$
$523$	−41.4317	−1.81168	−0.905841	+	0.423618i	$0.860760\pi$
$524$	0	0
$525$	1.99597	0.0871112
$526$	0	0
$527$	0	0
$528$	0	0
$529$	12.3751	0.538046
$530$	0	0
$531$	−1.02979	−0.0446892
$532$	0	0
$533$	− 5.21770i	− 0.226004i
$534$	0	0
$535$	2.20940	0.0955205
$536$	0	0
$537$	11.1837i	0.482611i
$538$	0	0
$539$	20.1360i	0.867318i
$540$	0	0
$541$	− 33.1546i	− 1.42543i	−0.701455	−	0.712713i	$-0.747466\pi$
$541$	− 33.1546i	− 1.42543i	0.701455	−	0.712713i	$-0.252534\pi$
$542$	0	0
$543$	−2.87449	−0.123356
$544$	0	0
$545$	12.5160	0.536126
$546$	0	0
$547$	− 14.9959i	− 0.641179i	−0.947218	−	0.320589i	$-0.896119\pi$
$547$	− 14.9959i	− 0.641179i	0.947218	−	0.320589i	$-0.103881\pi$
$548$	0	0
$549$	− 5.29973i	− 0.226187i
$550$	0	0
$551$	− 14.0484i	− 0.598482i
$552$	0	0
$553$	23.6906	1.00743
$554$	0	0
$555$	− 2.63536i	− 0.111865i
$556$	0	0
$557$	44.3644	1.87978	0.939889	−	0.341480i	$-0.110928\pi$
$557$	44.3644	1.87978	0.939889	+	0.341480i	$0.110928\pi$
$558$	0	0
$559$	0.166614	0.00704703
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−16.7839	−0.707355	−0.353678	−	0.935367i	$-0.615069\pi$
$563$	−16.7839	−0.707355	−0.353678	+	0.935367i	$0.615069\pi$
$564$	0	0
$565$	15.2659	0.642242
$566$	0	0
$567$	5.20060i	0.218405i
$568$	0	0
$569$	−4.40782	−0.184786	−0.0923928	−	0.995723i	$-0.529452\pi$
$569$	−4.40782	−0.184786	−0.0923928	+	0.995723i	$0.529452\pi$
$570$	0	0
$571$	− 0.746911i	− 0.0312573i	−0.999878	−	0.0156286i	$-0.995025\pi$
$571$	− 0.746911i	− 0.0312573i	0.999878	−	0.0156286i	$-0.00497495\pi$
$572$	0	0
$573$	− 16.3934i	− 0.684844i
$574$	0	0
$575$	3.25959i	0.135934i
$576$	0	0
$577$	−29.3683	−1.22262	−0.611309	−	0.791392i	$-0.709357\pi$
$577$	−29.3683	−1.22262	−0.611309	+	0.791392i	$0.709357\pi$
$578$	0	0
$579$	25.9832	1.07983
$580$	0	0
$581$	0.971042i	0.0402856i
$582$	0	0
$583$	− 37.4860i	− 1.55251i
$584$	0	0
$585$	3.35674i	0.138784i
$586$	0	0
$587$	−35.0300	−1.44584	−0.722921	−	0.690931i	$-0.757201\pi$
$587$	−35.0300	−1.44584	−0.722921	+	0.690931i	$0.757201\pi$
$588$	0	0
$589$	− 9.04640i	− 0.372750i
$590$	0	0
$591$	−27.4598	−1.12955
$592$	0	0
$593$	−9.20502	−0.378005	−0.189002	−	0.981977i	$-0.560525\pi$
$593$	−9.20502	−0.378005	−0.189002	+	0.981977i	$0.560525\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	23.0604	0.943797
$598$	0	0
$599$	12.7555	0.521175	0.260588	−	0.965450i	$-0.416084\pi$
$599$	12.7555	0.521175	0.260588	+	0.965450i	$0.416084\pi$
$600$	0	0
$601$	− 14.2167i	− 0.579912i	−0.957040	−	0.289956i	$-0.906359\pi$
$601$	− 14.2167i	− 0.579912i	0.957040	−	0.289956i	$-0.0936406\pi$
$602$	0	0
$603$	−11.0221	−0.448856
$604$	0	0
$605$	7.72987i	0.314264i
$606$	0	0
$607$	− 28.4087i	− 1.15308i	−0.817071	−	0.576538i	$-0.804403\pi$
$607$	− 28.4087i	− 1.15308i	0.817071	−	0.576538i	$-0.195597\pi$
$608$	0	0
$609$	− 17.8307i	− 0.722536i
$610$	0	0
$611$	24.0986	0.974925
$612$	0	0
$613$	−29.4996	−1.19148	−0.595738	−	0.803179i	$-0.703140\pi$
$613$	−29.4996	−1.19148	−0.595738	+	0.803179i	$0.703140\pi$
$614$	0	0
$615$	− 2.63816i	− 0.106381i
$616$	0	0
$617$	19.0514i	0.766981i	0.923545	+	0.383490i	$0.125278\pi$
$617$	19.0514i	0.766981i	−0.923545	+	0.383490i	$0.874722\pi$
$618$	0	0
$619$	1.95145i	0.0784354i	0.999231	+	0.0392177i	$0.0124866\pi$
$619$	1.95145i	0.0784354i	−0.999231	+	0.0392177i	$0.987513\pi$
$620$	0	0
$621$	−18.2718	−0.733223
$622$	0	0
$623$	− 5.42275i	− 0.217258i
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−8.86644	−0.354091
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−2.46367	−0.0980771	−0.0490385	−	0.998797i	$-0.515616\pi$
$631$	−2.46367	−0.0980771	−0.0490385	+	0.998797i	$0.515616\pi$
$632$	0	0
$633$	−6.72558	−0.267318
$634$	0	0
$635$	0.901981i	0.0357940i
$636$	0	0
$637$	−11.9882	−0.474990
$638$	0	0
$639$	14.0717i	0.556667i
$640$	0	0
$641$	− 4.49672i	− 0.177610i	−0.996049	−	0.0888050i	$-0.971695\pi$
$641$	− 4.49672i	− 0.177610i	0.996049	−	0.0888050i	$-0.0283048\pi$
$642$	0	0
$643$	9.56181i	0.377081i	0.982065	+	0.188541i	$0.0603757\pi$
$643$	9.56181i	0.377081i	−0.982065	+	0.188541i	$0.939624\pi$
$644$	0	0
$645$	0.0842428	0.00331706
$646$	0	0
$647$	28.1130	1.10524	0.552618	−	0.833435i	$-0.313629\pi$
$647$	28.1130	1.10524	0.552618	+	0.833435i	$0.313629\pi$
$648$	0	0
$649$	− 3.42096i	− 0.134284i
$650$	0	0
$651$	− 11.4820i	− 0.450015i
$652$	0	0
$653$	14.5983i	0.571278i	0.958337	+	0.285639i	$0.0922058\pi$
$653$	14.5983i	0.571278i	−0.958337	+	0.285639i	$0.907794\pi$
$654$	0	0
$655$	−7.53434	−0.294391
$656$	0	0
$657$	9.77875i	0.381505i
$658$	0	0
$659$	−43.7768	−1.70530	−0.852652	−	0.522480i	$-0.825007\pi$
$659$	−43.7768	−1.70530	−0.852652	+	0.522480i	$0.825007\pi$
$660$	0	0
$661$	−41.2896	−1.60598	−0.802990	−	0.595992i	$-0.796759\pi$
$661$	−41.2896	−1.60598	−0.802990	+	0.595992i	$0.796759\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.40933	0.0934298
$666$	0	0
$667$	29.1191	1.12750
$668$	0	0
$669$	− 11.8593i	− 0.458506i
$670$	0	0
$671$	17.6056	0.679658
$672$	0	0
$673$	− 18.7572i	− 0.723036i	−0.932365	−	0.361518i	$-0.882259\pi$
$673$	− 18.7572i	− 0.723036i	0.932365	−	0.361518i	$-0.117741\pi$
$674$	0	0
$675$	5.60555i	0.215758i
$676$	0	0
$677$	38.3496i	1.47390i	0.675950	+	0.736948i	$0.263734\pi$
$677$	38.3496i	1.47390i	−0.675950	+	0.736948i	$0.736266\pi$
$678$	0	0
$679$	−18.5458	−0.711724
$680$	0	0
$681$	−3.62356	−0.138855
$682$	0	0
$683$	− 1.82309i	− 0.0697586i	−0.999392	−	0.0348793i	$-0.988895\pi$
$683$	− 1.82309i	− 0.0697586i	0.999392	−	0.0348793i	$-0.0111047\pi$
$684$	0	0
$685$	22.6289i	0.864606i
$686$	0	0
$687$	− 34.0952i	− 1.30081i
$688$	0	0
$689$	22.3177	0.850238
$690$	0	0
$691$	− 23.7152i	− 0.902168i	−0.892482	−	0.451084i	$-0.851038\pi$
$691$	− 23.7152i	− 0.902168i	0.892482	−	0.451084i	$-0.148962\pi$
$692$	0	0
$693$	8.63816	0.328136
$694$	0	0
$695$	−12.1251	−0.459931
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−3.40489	−0.128785
$700$	0	0
$701$	−13.2451	−0.500261	−0.250130	−	0.968212i	$-0.580473\pi$
$701$	−13.2451	−0.500261	−0.250130	+	0.968212i	$0.580473\pi$
$702$	0	0
$703$	− 3.18114i	− 0.119979i
$704$	0	0
$705$	12.1846	0.458900
$706$	0	0
$707$	− 8.76355i	− 0.329587i
$708$	0	0
$709$	20.5685i	0.772466i	0.922401	+	0.386233i	$0.126224\pi$
$709$	20.5685i	0.772466i	−0.922401	+	0.386233i	$0.873776\pi$
$710$	0	0
$711$	20.1447i	0.755486i
$712$	0	0
$713$	18.7511	0.702235
$714$	0	0
$715$	−11.1511	−0.417026
$716$	0	0
$717$	− 4.21394i	− 0.157373i
$718$	0	0
$719$	− 40.3854i	− 1.50612i	−0.657950	−	0.753061i	$-0.728576\pi$
$719$	− 40.3854i	− 1.50612i	0.657950	−	0.753061i	$-0.271424\pi$
$720$	0	0
$721$	13.6454i	0.508181i
$722$	0	0
$723$	−28.4188	−1.05691
$724$	0	0
$725$	− 8.93335i	− 0.331776i
$726$	0	0
$727$	52.4431	1.94501	0.972503	−	0.232890i	$-0.0748182\pi$
$727$	52.4431	1.94501	0.972503	+	0.232890i	$0.0748182\pi$
$728$	0	0
$729$	−26.3305	−0.975205
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−11.2639	−0.416041	−0.208020	−	0.978125i	$-0.566702\pi$
$733$	−11.2639	−0.416041	−0.208020	+	0.978125i	$0.566702\pi$
$734$	0	0
$735$	−6.06143	−0.223579
$736$	0	0
$737$	− 36.6154i	− 1.34874i
$738$	0	0
$739$	−28.3290	−1.04210	−0.521049	−	0.853527i	$-0.674459\pi$
$739$	−28.3290	−1.04210	−0.521049	+	0.853527i	$0.674459\pi$
$740$	0	0
$741$	− 5.27874i	− 0.193919i
$742$	0	0
$743$	34.2519i	1.25658i	0.777980	+	0.628289i	$0.216245\pi$
$743$	34.2519i	1.25658i	−0.777980	+	0.628289i	$0.783755\pi$
$744$	0	0
$745$	15.9678i	0.585016i
$746$	0	0
$747$	−0.825703	−0.0302109
$748$	0	0
$749$	3.38499	0.123685
$750$	0	0
$751$	− 20.0518i	− 0.731702i	−0.930673	−	0.365851i	$-0.880778\pi$
$751$	− 20.0518i	− 0.731702i	0.930673	−	0.365851i	$-0.119222\pi$
$752$	0	0
$753$	14.0006i	0.510211i
$754$	0	0
$755$	− 13.6491i	− 0.496740i
$756$	0	0
$757$	31.4912	1.14457	0.572284	−	0.820055i	$-0.306057\pi$
$757$	31.4912	1.14457	0.572284	+	0.820055i	$0.306057\pi$
$758$	0	0
$759$	− 18.3781i	− 0.667082i
$760$	0	0
$761$	29.5693	1.07189	0.535943	−	0.844254i	$-0.319956\pi$
$761$	29.5693	1.07189	0.535943	+	0.844254i	$0.319956\pi$
$762$	0	0
$763$	19.1756	0.694203
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.03671	0.0735413
$768$	0	0
$769$	−51.4699	−1.85605	−0.928026	−	0.372516i	$-0.878495\pi$
$769$	−51.4699	−1.85605	−0.928026	+	0.372516i	$0.878495\pi$
$770$	0	0
$771$	− 37.4650i	− 1.34927i
$772$	0	0
$773$	−7.90891	−0.284464	−0.142232	−	0.989833i	$-0.545428\pi$
$773$	−7.90891	−0.284464	−0.142232	+	0.989833i	$0.545428\pi$
$774$	0	0
$775$	− 5.75259i	− 0.206639i
$776$	0	0
$777$	− 4.03761i	− 0.144848i
$778$	0	0
$779$	− 3.18451i	− 0.114097i
$780$	0	0
$781$	−46.7459	−1.67270
$782$	0	0
$783$	50.0764	1.78958
$784$	0	0
$785$	− 5.97123i	− 0.213122i
$786$	0	0
$787$	39.7026i	1.41525i	0.706591	+	0.707623i	$0.250232\pi$
$787$	39.7026i	1.41525i	−0.706591	+	0.707623i	$0.749768\pi$
$788$	0	0
$789$	7.34948i	0.261649i
$790$	0	0
$791$	23.3888	0.831608
$792$	0	0
$793$	10.4817i	0.372217i
$794$	0	0
$795$	11.2842	0.400210
$796$	0	0
$797$	8.87054	0.314211	0.157105	−	0.987582i	$-0.449784\pi$
$797$	8.87054	0.314211	0.157105	+	0.987582i	$0.449784\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	4.61110	0.162925
$802$	0	0
$803$	−32.4849	−1.14637
$804$	0	0
$805$	4.99399i	0.176015i
$806$	0	0
$807$	11.5547	0.406746
$808$	0	0
$809$	− 1.01405i	− 0.0356520i	−0.999841	−	0.0178260i	$-0.994326\pi$
$809$	− 1.01405i	− 0.0356520i	0.999841	−	0.0178260i	$-0.00567449\pi$
$810$	0	0
$811$	13.7227i	0.481868i	0.970542	+	0.240934i	$0.0774537\pi$
$811$	13.7227i	0.481868i	−0.970542	+	0.240934i	$0.922546\pi$
$812$	0	0
$813$	− 21.0065i	− 0.736731i
$814$	0	0
$815$	−19.7254	−0.690949
$816$	0	0
$817$	0.101689	0.00355766
$818$	0	0
$819$	5.14283i	0.179705i
$820$	0	0
$821$	− 15.3820i	− 0.536837i	−0.963302	−	0.268419i	$-0.913499\pi$
$821$	− 15.3820i	− 0.536837i	0.963302	−	0.268419i	$-0.0865010\pi$
$822$	0	0
$823$	− 39.6733i	− 1.38293i	−0.722412	−	0.691463i	$-0.756967\pi$
$823$	− 39.6733i	− 1.38293i	0.722412	−	0.691463i	$-0.243033\pi$
$824$	0	0
$825$	−5.63816	−0.196295
$826$	0	0
$827$	− 36.4651i	− 1.26802i	−0.773327	−	0.634008i	$-0.781409\pi$
$827$	− 36.4651i	− 1.26802i	0.773327	−	0.634008i	$-0.218591\pi$
$828$	0	0
$829$	−5.39734	−0.187458	−0.0937288	−	0.995598i	$-0.529879\pi$
$829$	−5.39734	−0.187458	−0.0937288	+	0.995598i	$0.529879\pi$
$830$	0	0
$831$	18.4833	0.641180
$832$	0	0
$833$	0	0
$834$	0	0
$835$	20.7498	0.718077
$836$	0	0
$837$	32.2465	1.11460
$838$	0	0
$839$	25.5770i	0.883015i	0.897258	+	0.441507i	$0.145556\pi$
$839$	25.5770i	0.883015i	−0.897258	+	0.441507i	$0.854444\pi$
$840$	0	0
$841$	−50.8048	−1.75189
$842$	0	0
$843$	− 18.6620i	− 0.642754i
$844$	0	0
$845$	6.36108i	0.218828i
$846$	0	0
$847$	11.8429i	0.406925i
$848$	0	0
$849$	37.4006	1.28358
$850$	0	0
$851$	6.59377	0.226032
$852$	0	0
$853$	30.5030i	1.04440i	0.852822	+	0.522202i	$0.174889\pi$
$853$	30.5030i	1.04440i	−0.852822	+	0.522202i	$0.825111\pi$
$854$	0	0
$855$	2.04872i	0.0700646i
$856$	0	0
$857$	37.5513i	1.28273i	0.767237	+	0.641363i	$0.221631\pi$
$857$	37.5513i	1.28273i	−0.767237	+	0.641363i	$0.778369\pi$
$858$	0	0
$859$	−1.12743	−0.0384673	−0.0192337	−	0.999815i	$-0.506123\pi$
$859$	−1.12743	−0.0384673	−0.0192337	+	0.999815i	$0.506123\pi$
$860$	0	0
$861$	− 4.04189i	− 0.137747i
$862$	0	0
$863$	47.4062	1.61373	0.806863	−	0.590739i	$-0.201164\pi$
$863$	47.4062	1.61373	0.806863	+	0.590739i	$0.201164\pi$
$864$	0	0
$865$	13.0624	0.444136
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−66.9205	−2.27012
$870$	0	0
$871$	21.7994	0.738644
$872$	0	0
$873$	− 15.7700i	− 0.533734i
$874$	0	0
$875$	1.53209	0.0517941
$876$	0	0
$877$	34.4016i	1.16166i	0.814025	+	0.580830i	$0.197272\pi$
$877$	34.4016i	1.16166i	−0.814025	+	0.580830i	$0.802728\pi$
$878$	0	0
$879$	10.1747i	0.343183i
$880$	0	0
$881$	− 46.1758i	− 1.55570i	−0.628448	−	0.777852i	$-0.716309\pi$
$881$	− 46.1758i	− 1.55570i	0.628448	−	0.777852i	$-0.283691\pi$
$882$	0	0
$883$	−36.2109	−1.21859	−0.609297	−	0.792942i	$-0.708548\pi$
$883$	−36.2109	−1.21859	−0.609297	+	0.792942i	$0.708548\pi$
$884$	0	0
$885$	1.02979	0.0346161
$886$	0	0
$887$	12.1417i	0.407678i	0.979004	+	0.203839i	$0.0653418\pi$
$887$	12.1417i	0.407678i	−0.979004	+	0.203839i	$0.934658\pi$
$888$	0	0
$889$	1.38191i	0.0463479i
$890$	0	0
$891$	− 14.6905i	− 0.492150i
$892$	0	0
$893$	14.7081	0.492186
$894$	0	0
$895$	8.58449i	0.286948i
$896$	0	0
$897$	10.9416	0.365330
$898$	0	0
$899$	−51.3900	−1.71395
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0.129067	0.00429510
$904$	0	0
$905$	−2.20644	−0.0733445
$906$	0	0
$907$	− 54.7495i	− 1.81793i	−0.416874	−	0.908964i	$-0.636874\pi$
$907$	− 54.7495i	− 1.81793i	0.416874	−	0.908964i	$-0.363126\pi$
$908$	0	0
$909$	7.45188	0.247163
$910$	0	0
$911$	− 36.0707i	− 1.19508i	−0.801841	−	0.597538i	$-0.796146\pi$
$911$	− 36.0707i	− 1.19508i	0.801841	−	0.597538i	$-0.203854\pi$
$912$	0	0
$913$	− 2.74297i	− 0.0907792i
$914$	0	0
$915$	5.29973i	0.175204i
$916$	0	0
$917$	−11.5433	−0.381193
$918$	0	0
$919$	30.8329	1.01708	0.508541	−	0.861038i	$-0.330185\pi$
$919$	30.8329	1.01708	0.508541	+	0.861038i	$0.330185\pi$
$920$	0	0
$921$	− 12.4271i	− 0.409485i
$922$	0	0
$923$	− 27.8308i	− 0.916061i
$924$	0	0
$925$	− 2.02288i	− 0.0665119i
$926$	0	0
$927$	−11.6030	−0.381094
$928$	0	0
$929$	50.6432i	1.66155i	0.556609	+	0.830775i	$0.312102\pi$
$929$	50.6432i	1.66155i	−0.556609	+	0.830775i	$0.687898\pi$
$930$	0	0
$931$	−7.31674	−0.239796
$932$	0	0
$933$	−16.6886	−0.546359
$934$	0	0
$935$	0	0
$936$	0	0
$937$	28.2721	0.923610	0.461805	−	0.886982i	$-0.347202\pi$
$937$	28.2721	0.923610	0.461805	+	0.886982i	$0.347202\pi$
$938$	0	0
$939$	6.98332	0.227892
$940$	0	0
$941$	− 37.9482i	− 1.23708i	−0.785755	−	0.618538i	$-0.787725\pi$
$941$	− 37.9482i	− 1.23708i	0.785755	−	0.618538i	$-0.212275\pi$
$942$	0	0
$943$	6.60076	0.214950
$944$	0	0
$945$	8.58820i	0.279374i
$946$	0	0
$947$	− 48.3995i	− 1.57277i	−0.617735	−	0.786386i	$-0.711950\pi$
$947$	− 48.3995i	− 1.57277i	0.617735	−	0.786386i	$-0.288050\pi$
$948$	0	0
$949$	− 19.3403i	− 0.627811i
$950$	0	0
$951$	2.97901	0.0966011
$952$	0	0
$953$	28.1684	0.912464	0.456232	−	0.889861i	$-0.349199\pi$
$953$	28.1684	0.912464	0.456232	+	0.889861i	$0.349199\pi$
$954$	0	0
$955$	− 12.5834i	− 0.407191i
$956$	0	0
$957$	50.3676i	1.62815i
$958$	0	0
$959$	34.6695i	1.11954i
$960$	0	0
$961$	−2.09233	−0.0674945
$962$	0	0
$963$	2.87835i	0.0927534i
$964$	0	0
$965$	19.9445	0.642037
$966$	0	0
$967$	−44.9945	−1.44693	−0.723463	−	0.690363i	$-0.757451\pi$
$967$	−44.9945	−1.44693	−0.723463	+	0.690363i	$0.757451\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−27.3436	−0.877498	−0.438749	−	0.898610i	$-0.644578\pi$
$971$	−27.3436	−0.877498	−0.438749	+	0.898610i	$0.644578\pi$
$972$	0	0
$973$	−18.5767	−0.595542
$974$	0	0
$975$	− 3.35674i	− 0.107502i
$976$	0	0
$977$	24.5988	0.786986	0.393493	−	0.919328i	$-0.371267\pi$
$977$	24.5988	0.786986	0.393493	+	0.919328i	$0.371267\pi$
$978$	0	0
$979$	15.3180i	0.489566i
$980$	0	0
$981$	16.3055i	0.520595i
$982$	0	0
$983$	− 25.9972i	− 0.829181i	−0.910008	−	0.414591i	$-0.863925\pi$
$983$	− 25.9972i	− 0.829181i	0.910008	−	0.414591i	$-0.136075\pi$
$984$	0	0
$985$	−21.0779	−0.671598
$986$	0	0
$987$	18.6679	0.594207
$988$	0	0
$989$	0.210779i	0.00670237i
$990$	0	0
$991$	32.0237i	1.01727i	0.860983	+	0.508633i	$0.169849\pi$
$991$	32.0237i	1.01727i	−0.860983	+	0.508633i	$0.830151\pi$
$992$	0	0
$993$	25.6203i	0.813036i
$994$	0	0
$995$	17.7009	0.561157
$996$	0	0
$997$	− 37.6854i	− 1.19351i	−0.802424	−	0.596754i	$-0.796457\pi$
$997$	− 37.6854i	− 1.19351i	0.802424	−	0.596754i	$-0.203543\pi$
$998$	0	0
$999$	11.3394	0.358762

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5780.2.c.g.5201.4		12
17.4	even	4		5780.2.a.o.1.3	yes	6
17.13	even	4		5780.2.a.l.1.4	✓	6
17.16	even	2	inner	5780.2.c.g.5201.9		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5780.2.a.l.1.4	✓	6	17.13	even	4
5780.2.a.o.1.3	yes	6	17.4	even	4
5780.2.c.g.5201.4		12	1.1	even	1	trivial
5780.2.c.g.5201.9		12	17.16	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5780 = 2^{2} \cdot 5 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5780.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.30278i q^{3} -1.00000i q^{5} -1.53209i q^{7} +1.30278 q^{9} +O(q^{10})\)	\(q-1.30278i q^{3} -1.00000i q^{5} -1.53209i q^{7} +1.30278 q^{9} +4.32780i q^{11} -2.57661 q^{13} -1.30278 q^{15} -1.57258 q^{19} -1.99597 q^{21} -3.25959i q^{23} -1.00000 q^{25} -5.60555i q^{27} +8.93335i q^{29} +5.75259i q^{31} +5.63816 q^{33} -1.53209 q^{35} +2.02288i q^{37} +3.35674i q^{39} +2.02503i q^{41} -0.0646641 q^{43} -1.30278i q^{45} -9.35283 q^{47} +4.65270 q^{49} -8.66167 q^{53} +4.32780 q^{55} +2.04872i q^{57} -0.790461 q^{59} -4.06803i q^{61} -1.99597i q^{63} +2.57661i q^{65} -8.46050 q^{67} -4.24652 q^{69} +10.8013i q^{71} +7.50609i q^{73} +1.30278i q^{75} +6.63058 q^{77} +15.4629i q^{79} -3.39445 q^{81} -0.633803 q^{83} +11.6382 q^{87} +3.53945 q^{89} +3.94759i q^{91} +7.49434 q^{93} +1.57258i q^{95} -12.1049i q^{97} +5.63816i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 6 q^{9}+O(q^{10}) \) 12 * q - 6 * q^9	\( 12 q - 6 q^{9} - 30 q^{13} + 6 q^{15} + 6 q^{19} - 12 q^{25} - 6 q^{43} - 30 q^{47} + 60 q^{49} + 24 q^{53} + 12 q^{59} - 12 q^{67} + 6 q^{77} - 84 q^{81} + 30 q^{83} + 72 q^{87} - 12 q^{89} + 72 q^{93}+O(q^{100}) \) 12 * q - 6 * q^9 - 30 * q^13 + 6 * q^15 + 6 * q^19 - 12 * q^25 - 6 * q^43 - 30 * q^47 + 60 * q^49 + 24 * q^53 + 12 * q^59 - 12 * q^67 + 6 * q^77 - 84 * q^81 + 30 * q^83 + 72 * q^87 - 12 * q^89 + 72 * q^93

Introduction

L-functions

Modular forms

Varieties

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Database

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