LMFDB - Embedded newform 7800.2.a.bl.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7800,2,Mod(1,7800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7800, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("7800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7800.a (trivial)

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:	yes
Analytic conductor:	$62.2833135766$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.17009$ of defining polynomial
Character	$\chi$	$=$	7800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +3.24846 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +3.24846 q^{7} +1.00000 q^{9} -0.460811 q^{11} -1.00000 q^{13} +4.26180 q^{17} +1.70928 q^{19} -3.24846 q^{21} -8.38962 q^{23} -1.00000 q^{27} +3.34017 q^{29} -7.87936 q^{31} +0.460811 q^{33} -7.46800 q^{37} +1.00000 q^{39} -8.78765 q^{41} +3.89269 q^{43} -0.751536 q^{47} +3.55252 q^{49} -4.26180 q^{51} +8.70928 q^{53} -1.70928 q^{57} -3.27513 q^{59} -8.07838 q^{61} +3.24846 q^{63} -4.95774 q^{67} +8.38962 q^{69} +8.23287 q^{71} -13.8082 q^{73} -1.49693 q^{77} -12.2062 q^{79} +1.00000 q^{81} -7.40522 q^{83} -3.34017 q^{87} -9.44521 q^{89} -3.24846 q^{91} +7.87936 q^{93} +2.49693 q^{97} -0.460811 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 + q^7 + 3 * q^9	$ 3 q - 3 q^{3} + q^{7} + 3 q^{9} - 3 q^{11} - 3 q^{13} + 5 q^{17} - 2 q^{19} - q^{21} + 4 q^{23} - 3 q^{27} - q^{29} - 11 q^{31} + 3 q^{33} + 10 q^{37} + 3 q^{39} - 16 q^{41} - 11 q^{47} + 10 q^{49} - 5 q^{51} + 19 q^{53} + 2 q^{57} - 3 q^{59} - 21 q^{61} + q^{63} + q^{67} - 4 q^{69} + 2 q^{71} + 2 q^{73} + 13 q^{77} - 12 q^{79} + 3 q^{81} - 7 q^{83} + q^{87} - 16 q^{89} - q^{91} + 11 q^{93} - 10 q^{97} - 3 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + q^7 + 3 * q^9 - 3 * q^11 - 3 * q^13 + 5 * q^17 - 2 * q^19 - q^21 + 4 * q^23 - 3 * q^27 - q^29 - 11 * q^31 + 3 * q^33 + 10 * q^37 + 3 * q^39 - 16 * q^41 - 11 * q^47 + 10 * q^49 - 5 * q^51 + 19 * q^53 + 2 * q^57 - 3 * q^59 - 21 * q^61 + q^63 + q^67 - 4 * q^69 + 2 * q^71 + 2 * q^73 + 13 * q^77 - 12 * q^79 + 3 * q^81 - 7 * q^83 + q^87 - 16 * q^89 - q^91 + 11 * q^93 - 10 * q^97 - 3 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.24846	1.22780	0.613902	−	0.789382i	$-0.289599\pi$
$7$	3.24846	1.22780	0.613902	+	0.789382i	$0.289599\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.460811	−0.138940	−0.0694699	−	0.997584i	$-0.522131\pi$
$11$	−0.460811	−0.138940	−0.0694699	+	0.997584i	$0.522131\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.26180	1.03364	0.516819	−	0.856095i	$-0.327116\pi$
$17$	4.26180	1.03364	0.516819	+	0.856095i	$0.327116\pi$
$18$	0	0
$19$	1.70928	0.392135	0.196067	−	0.980590i	$-0.437183\pi$
$19$	1.70928	0.392135	0.196067	+	0.980590i	$0.437183\pi$
$20$	0	0
$21$	−3.24846	−0.708873
$22$	0	0
$23$	−8.38962	−1.74936	−0.874678	−	0.484704i	$-0.838927\pi$
$23$	−8.38962	−1.74936	−0.874678	+	0.484704i	$0.838927\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	3.34017	0.620255	0.310127	−	0.950695i	$-0.399628\pi$
$29$	3.34017	0.620255	0.310127	+	0.950695i	$0.399628\pi$
$30$	0	0
$31$	−7.87936	−1.41518	−0.707588	−	0.706626i	$-0.750217\pi$
$31$	−7.87936	−1.41518	−0.707588	+	0.706626i	$0.750217\pi$
$32$	0	0
$33$	0.460811	0.0802169
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−7.46800	−1.22773	−0.613866	−	0.789410i	$-0.710386\pi$
$37$	−7.46800	−1.22773	−0.613866	+	0.789410i	$0.710386\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	−8.78765	−1.37240	−0.686200	−	0.727413i	$-0.740723\pi$
$41$	−8.78765	−1.37240	−0.686200	+	0.727413i	$0.740723\pi$
$42$	0	0
$43$	3.89269	0.593630	0.296815	−	0.954935i	$-0.404076\pi$
$43$	3.89269	0.593630	0.296815	+	0.954935i	$0.404076\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−0.751536	−0.109623	−0.0548114	−	0.998497i	$-0.517456\pi$
$47$	−0.751536	−0.109623	−0.0548114	+	0.998497i	$0.517456\pi$
$48$	0	0
$49$	3.55252	0.507503
$50$	0	0
$51$	−4.26180	−0.596771
$52$	0	0
$53$	8.70928	1.19631	0.598155	−	0.801380i	$-0.295901\pi$
$53$	8.70928	1.19631	0.598155	+	0.801380i	$0.295901\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−1.70928	−0.226399
$58$	0	0
$59$	−3.27513	−0.426385	−0.213193	−	0.977010i	$-0.568386\pi$
$59$	−3.27513	−0.426385	−0.213193	+	0.977010i	$0.568386\pi$
$60$	0	0
$61$	−8.07838	−1.03433	−0.517165	−	0.855886i	$-0.673013\pi$
$61$	−8.07838	−1.03433	−0.517165	+	0.855886i	$0.673013\pi$
$62$	0	0
$63$	3.24846	0.409268
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−4.95774	−0.605684	−0.302842	−	0.953041i	$-0.597935\pi$
$67$	−4.95774	−0.605684	−0.302842	+	0.953041i	$0.597935\pi$
$68$	0	0
$69$	8.38962	1.00999
$70$	0	0
$71$	8.23287	0.977061	0.488531	−	0.872547i	$-0.337533\pi$
$71$	8.23287	0.977061	0.488531	+	0.872547i	$0.337533\pi$
$72$	0	0
$73$	−13.8082	−1.61612	−0.808062	−	0.589097i	$-0.799483\pi$
$73$	−13.8082	−1.61612	−0.808062	+	0.589097i	$0.799483\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.49693	−0.170591
$78$	0	0
$79$	−12.2062	−1.37331	−0.686653	−	0.726986i	$-0.740921\pi$
$79$	−12.2062	−1.37331	−0.686653	+	0.726986i	$0.740921\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−7.40522	−0.812828	−0.406414	−	0.913689i	$-0.633221\pi$
$83$	−7.40522	−0.812828	−0.406414	+	0.913689i	$0.633221\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−3.34017	−0.358104
$88$	0	0
$89$	−9.44521	−1.00119	−0.500595	−	0.865681i	$-0.666886\pi$
$89$	−9.44521	−1.00119	−0.500595	+	0.865681i	$0.666886\pi$
$90$	0	0
$91$	−3.24846	−0.340532
$92$	0	0
$93$	7.87936	0.817052
$94$	0	0
$95$	0	0
$96$	0	0
$97$	2.49693	0.253525	0.126762	−	0.991933i	$-0.459541\pi$
$97$	2.49693	0.253525	0.126762	+	0.991933i	$0.459541\pi$
$98$	0	0
$99$	−0.460811	−0.0463133
$100$	0	0
$101$	7.72979	0.769143	0.384572	−	0.923095i	$-0.374349\pi$
$101$	7.72979	0.769143	0.384572	+	0.923095i	$0.374349\pi$
$102$	0	0
$103$	1.89269	0.186493	0.0932463	−	0.995643i	$-0.470276\pi$
$103$	1.89269	0.186493	0.0932463	+	0.995643i	$0.470276\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−7.17727	−0.693853	−0.346927	−	0.937892i	$-0.612775\pi$
$107$	−7.17727	−0.693853	−0.346927	+	0.937892i	$0.612775\pi$
$108$	0	0
$109$	−11.2267	−1.07533	−0.537663	−	0.843160i	$-0.680693\pi$
$109$	−11.2267	−1.07533	−0.537663	+	0.843160i	$0.680693\pi$
$110$	0	0
$111$	7.46800	0.708831
$112$	0	0
$113$	17.9155	1.68535	0.842673	−	0.538425i	$-0.180981\pi$
$113$	17.9155	1.68535	0.842673	+	0.538425i	$0.180981\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	13.8443	1.26910
$120$	0	0
$121$	−10.7877	−0.980696
$122$	0	0
$123$	8.78765	0.792356
$124$	0	0
$125$	0	0
$126$	0	0
$127$	11.2267	0.996211	0.498105	−	0.867117i	$-0.334029\pi$
$127$	11.2267	0.996211	0.498105	+	0.867117i	$0.334029\pi$
$128$	0	0
$129$	−3.89269	−0.342732
$130$	0	0
$131$	−13.4947	−1.17903	−0.589517	−	0.807756i	$-0.700682\pi$
$131$	−13.4947	−1.17903	−0.589517	+	0.807756i	$0.700682\pi$
$132$	0	0
$133$	5.55252	0.481465
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1.65983	−0.141809	−0.0709043	−	0.997483i	$-0.522588\pi$
$137$	−1.65983	−0.141809	−0.0709043	+	0.997483i	$0.522588\pi$
$138$	0	0
$139$	14.6803	1.24517	0.622585	−	0.782552i	$-0.286082\pi$
$139$	14.6803	1.24517	0.622585	+	0.782552i	$0.286082\pi$
$140$	0	0
$141$	0.751536	0.0632907
$142$	0	0
$143$	0.460811	0.0385350
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−3.55252	−0.293007
$148$	0	0
$149$	10.1834	0.834258	0.417129	−	0.908847i	$-0.363036\pi$
$149$	10.1834	0.834258	0.417129	+	0.908847i	$0.363036\pi$
$150$	0	0
$151$	1.34736	0.109647	0.0548233	−	0.998496i	$-0.482540\pi$
$151$	1.34736	0.109647	0.0548233	+	0.998496i	$0.482540\pi$
$152$	0	0
$153$	4.26180	0.344546
$154$	0	0
$155$	0	0
$156$	0	0
$157$	18.9649	1.51357	0.756783	−	0.653666i	$-0.226770\pi$
$157$	18.9649	1.51357	0.756783	+	0.653666i	$0.226770\pi$
$158$	0	0
$159$	−8.70928	−0.690690
$160$	0	0
$161$	−27.2534	−2.14787
$162$	0	0
$163$	−6.07611	−0.475918	−0.237959	−	0.971275i	$-0.576478\pi$
$163$	−6.07611	−0.475918	−0.237959	+	0.971275i	$0.576478\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	16.1483	1.24960	0.624798	−	0.780786i	$-0.285181\pi$
$167$	16.1483	1.24960	0.624798	+	0.780786i	$0.285181\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	1.70928	0.130712
$172$	0	0
$173$	22.3112	1.69629	0.848146	−	0.529762i	$-0.177719\pi$
$173$	22.3112	1.69629	0.848146	+	0.529762i	$0.177719\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	3.27513	0.246174
$178$	0	0
$179$	−9.52586	−0.711996	−0.355998	−	0.934487i	$-0.615859\pi$
$179$	−9.52586	−0.711996	−0.355998	+	0.934487i	$0.615859\pi$
$180$	0	0
$181$	−19.5730	−1.45485	−0.727426	−	0.686186i	$-0.759284\pi$
$181$	−19.5730	−1.45485	−0.727426	+	0.686186i	$0.759284\pi$
$182$	0	0
$183$	8.07838	0.597171
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−1.96388	−0.143613
$188$	0	0
$189$	−3.24846	−0.236291
$190$	0	0
$191$	−4.52359	−0.327316	−0.163658	−	0.986517i	$-0.552329\pi$
$191$	−4.52359	−0.327316	−0.163658	+	0.986517i	$0.552329\pi$
$192$	0	0
$193$	−4.38962	−0.315972	−0.157986	−	0.987441i	$-0.550500\pi$
$193$	−4.38962	−0.315972	−0.157986	+	0.987441i	$0.550500\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−10.6309	−0.757420	−0.378710	−	0.925515i	$-0.623632\pi$
$197$	−10.6309	−0.757420	−0.378710	+	0.925515i	$0.623632\pi$
$198$	0	0
$199$	−17.7321	−1.25699	−0.628496	−	0.777813i	$-0.716329\pi$
$199$	−17.7321	−1.25699	−0.628496	+	0.777813i	$0.716329\pi$
$200$	0	0
$201$	4.95774	0.349692
$202$	0	0
$203$	10.8504	0.761551
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−8.38962	−0.583119
$208$	0	0
$209$	−0.787653	−0.0544831
$210$	0	0
$211$	14.9132	1.02667	0.513334	−	0.858189i	$-0.328410\pi$
$211$	14.9132	1.02667	0.513334	+	0.858189i	$0.328410\pi$
$212$	0	0
$213$	−8.23287	−0.564107
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−25.5958	−1.73756
$218$	0	0
$219$	13.8082	0.933070
$220$	0	0
$221$	−4.26180	−0.286679
$222$	0	0
$223$	−20.5464	−1.37589	−0.687944	−	0.725764i	$-0.741486\pi$
$223$	−20.5464	−1.37589	−0.687944	+	0.725764i	$0.741486\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−21.6381	−1.43617	−0.718085	−	0.695955i	$-0.754981\pi$
$227$	−21.6381	−1.43617	−0.718085	+	0.695955i	$0.754981\pi$
$228$	0	0
$229$	19.3607	1.27939	0.639695	−	0.768629i	$-0.279061\pi$
$229$	19.3607	1.27939	0.639695	+	0.768629i	$0.279061\pi$
$230$	0	0
$231$	1.49693	0.0984907
$232$	0	0
$233$	24.0722	1.57702	0.788512	−	0.615019i	$-0.210852\pi$
$233$	24.0722	1.57702	0.788512	+	0.615019i	$0.210852\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	12.2062	0.792878
$238$	0	0
$239$	−14.2423	−0.921259	−0.460630	−	0.887592i	$-0.652376\pi$
$239$	−14.2423	−0.921259	−0.460630	+	0.887592i	$0.652376\pi$
$240$	0	0
$241$	−28.3896	−1.82874	−0.914368	−	0.404884i	$-0.867312\pi$
$241$	−28.3896	−1.82874	−0.914368	+	0.404884i	$0.867312\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1.70928	−0.108759
$248$	0	0
$249$	7.40522	0.469287
$250$	0	0
$251$	4.41628	0.278753	0.139377	−	0.990239i	$-0.455490\pi$
$251$	4.41628	0.278753	0.139377	+	0.990239i	$0.455490\pi$
$252$	0	0
$253$	3.86603	0.243055
$254$	0	0
$255$	0	0
$256$	0	0
$257$	8.34244	0.520387	0.260194	−	0.965556i	$-0.416214\pi$
$257$	8.34244	0.520387	0.260194	+	0.965556i	$0.416214\pi$
$258$	0	0
$259$	−24.2595	−1.50741
$260$	0	0
$261$	3.34017	0.206752
$262$	0	0
$263$	7.95055	0.490252	0.245126	−	0.969491i	$-0.421171\pi$
$263$	7.95055	0.490252	0.245126	+	0.969491i	$0.421171\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	9.44521	0.578038
$268$	0	0
$269$	−6.41855	−0.391346	−0.195673	−	0.980669i	$-0.562689\pi$
$269$	−6.41855	−0.391346	−0.195673	+	0.980669i	$0.562689\pi$
$270$	0	0
$271$	−17.9555	−1.09072	−0.545359	−	0.838203i	$-0.683607\pi$
$271$	−17.9555	−1.09072	−0.545359	+	0.838203i	$0.683607\pi$
$272$	0	0
$273$	3.24846	0.196606
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−1.86991	−0.112352	−0.0561759	−	0.998421i	$-0.517891\pi$
$277$	−1.86991	−0.112352	−0.0561759	+	0.998421i	$0.517891\pi$
$278$	0	0
$279$	−7.87936	−0.471725
$280$	0	0
$281$	6.77924	0.404416	0.202208	−	0.979343i	$-0.435188\pi$
$281$	6.77924	0.404416	0.202208	+	0.979343i	$0.435188\pi$
$282$	0	0
$283$	9.73206	0.578511	0.289256	−	0.957252i	$-0.406592\pi$
$283$	9.73206	0.578511	0.289256	+	0.957252i	$0.406592\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−28.5464	−1.68504
$288$	0	0
$289$	1.16290	0.0684058
$290$	0	0
$291$	−2.49693	−0.146373
$292$	0	0
$293$	26.9939	1.57700	0.788499	−	0.615036i	$-0.210859\pi$
$293$	26.9939	1.57700	0.788499	+	0.615036i	$0.210859\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0.460811	0.0267390
$298$	0	0
$299$	8.38962	0.485184
$300$	0	0
$301$	12.6453	0.728861
$302$	0	0
$303$	−7.72979	−0.444065
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−7.95055	−0.453762	−0.226881	−	0.973922i	$-0.572853\pi$
$307$	−7.95055	−0.453762	−0.226881	+	0.973922i	$0.572853\pi$
$308$	0	0
$309$	−1.89269	−0.107672
$310$	0	0
$311$	−16.2557	−0.921773	−0.460887	−	0.887459i	$-0.652469\pi$
$311$	−16.2557	−0.921773	−0.460887	+	0.887459i	$0.652469\pi$
$312$	0	0
$313$	11.4885	0.649369	0.324685	−	0.945822i	$-0.394742\pi$
$313$	11.4885	0.649369	0.324685	+	0.945822i	$0.394742\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	7.18568	0.403588	0.201794	−	0.979428i	$-0.435323\pi$
$317$	7.18568	0.403588	0.201794	+	0.979428i	$0.435323\pi$
$318$	0	0
$319$	−1.53919	−0.0861780
$320$	0	0
$321$	7.17727	0.400596
$322$	0	0
$323$	7.28458	0.405325
$324$	0	0
$325$	0	0
$326$	0	0
$327$	11.2267	0.620839
$328$	0	0
$329$	−2.44134	−0.134595
$330$	0	0
$331$	−9.49466	−0.521874	−0.260937	−	0.965356i	$-0.584031\pi$
$331$	−9.49466	−0.521874	−0.260937	+	0.965356i	$0.584031\pi$
$332$	0	0
$333$	−7.46800	−0.409244
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−3.10731	−0.169266	−0.0846329	−	0.996412i	$-0.526972\pi$
$337$	−3.10731	−0.169266	−0.0846329	+	0.996412i	$0.526972\pi$
$338$	0	0
$339$	−17.9155	−0.973035
$340$	0	0
$341$	3.63090	0.196624
$342$	0	0
$343$	−11.1990	−0.604690
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−6.99386	−0.375450	−0.187725	−	0.982222i	$-0.560111\pi$
$347$	−6.99386	−0.375450	−0.187725	+	0.982222i	$0.560111\pi$
$348$	0	0
$349$	5.52586	0.295792	0.147896	−	0.989003i	$-0.452750\pi$
$349$	5.52586	0.295792	0.147896	+	0.989003i	$0.452750\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	−24.3051	−1.29363	−0.646815	−	0.762647i	$-0.723899\pi$
$353$	−24.3051	−1.29363	−0.646815	+	0.762647i	$0.723899\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−13.8443	−0.732717
$358$	0	0
$359$	−30.9493	−1.63344	−0.816722	−	0.577032i	$-0.804211\pi$
$359$	−30.9493	−1.63344	−0.816722	+	0.577032i	$0.804211\pi$
$360$	0	0
$361$	−16.0784	−0.846230
$362$	0	0
$363$	10.7877	0.566205
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−28.1978	−1.47191	−0.735956	−	0.677029i	$-0.763267\pi$
$367$	−28.1978	−1.47191	−0.735956	+	0.677029i	$0.763267\pi$
$368$	0	0
$369$	−8.78765	−0.457467
$370$	0	0
$371$	28.2918	1.46884
$372$	0	0
$373$	21.6719	1.12213	0.561065	−	0.827772i	$-0.310392\pi$
$373$	21.6719	1.12213	0.561065	+	0.827772i	$0.310392\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−3.34017	−0.172028
$378$	0	0
$379$	−16.1350	−0.828800	−0.414400	−	0.910095i	$-0.636009\pi$
$379$	−16.1350	−0.828800	−0.414400	+	0.910095i	$0.636009\pi$
$380$	0	0
$381$	−11.2267	−0.575162
$382$	0	0
$383$	29.3256	1.49847	0.749235	−	0.662305i	$-0.230422\pi$
$383$	29.3256	1.49847	0.749235	+	0.662305i	$0.230422\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3.89269	0.197877
$388$	0	0
$389$	−6.39803	−0.324393	−0.162197	−	0.986758i	$-0.551858\pi$
$389$	−6.39803	−0.324393	−0.162197	+	0.986758i	$0.551858\pi$
$390$	0	0
$391$	−35.7548	−1.80820
$392$	0	0
$393$	13.4947	0.680716
$394$	0	0
$395$	0	0
$396$	0	0
$397$	29.4863	1.47987	0.739936	−	0.672677i	$-0.234856\pi$
$397$	29.4863	1.47987	0.739936	+	0.672677i	$0.234856\pi$
$398$	0	0
$399$	−5.55252	−0.277974
$400$	0	0
$401$	9.99159	0.498956	0.249478	−	0.968380i	$-0.419741\pi$
$401$	9.99159	0.498956	0.249478	+	0.968380i	$0.419741\pi$
$402$	0	0
$403$	7.87936	0.392499
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.44134	0.170581
$408$	0	0
$409$	31.8225	1.57352	0.786762	−	0.617257i	$-0.211756\pi$
$409$	31.8225	1.57352	0.786762	+	0.617257i	$0.211756\pi$
$410$	0	0
$411$	1.65983	0.0818732
$412$	0	0
$413$	−10.6391	−0.523517
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−14.6803	−0.718900
$418$	0	0
$419$	−2.37525	−0.116038	−0.0580192	−	0.998315i	$-0.518478\pi$
$419$	−2.37525	−0.116038	−0.0580192	+	0.998315i	$0.518478\pi$
$420$	0	0
$421$	14.3090	0.697377	0.348688	−	0.937239i	$-0.386627\pi$
$421$	14.3090	0.697377	0.348688	+	0.937239i	$0.386627\pi$
$422$	0	0
$423$	−0.751536	−0.0365409
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−26.2423	−1.26995
$428$	0	0
$429$	−0.460811	−0.0222482
$430$	0	0
$431$	−17.5402	−0.844883	−0.422442	−	0.906390i	$-0.638827\pi$
$431$	−17.5402	−0.844883	−0.422442	+	0.906390i	$0.638827\pi$
$432$	0	0
$433$	−35.0082	−1.68239	−0.841194	−	0.540733i	$-0.818147\pi$
$433$	−35.0082	−1.68239	−0.841194	+	0.540733i	$0.818147\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−14.3402	−0.685984
$438$	0	0
$439$	−10.7610	−0.513594	−0.256797	−	0.966465i	$-0.582667\pi$
$439$	−10.7610	−0.513594	−0.256797	+	0.966465i	$0.582667\pi$
$440$	0	0
$441$	3.55252	0.169168
$442$	0	0
$443$	−37.6658	−1.78956	−0.894778	−	0.446511i	$-0.852666\pi$
$443$	−37.6658	−1.78956	−0.894778	+	0.446511i	$0.852666\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−10.1834	−0.481659
$448$	0	0
$449$	31.9299	1.50686	0.753432	−	0.657526i	$-0.228397\pi$
$449$	31.9299	1.50686	0.753432	+	0.657526i	$0.228397\pi$
$450$	0	0
$451$	4.04945	0.190681
$452$	0	0
$453$	−1.34736	−0.0633045
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−15.2579	−0.713735	−0.356868	−	0.934155i	$-0.616155\pi$
$457$	−15.2579	−0.713735	−0.356868	+	0.934155i	$0.616155\pi$
$458$	0	0
$459$	−4.26180	−0.198924
$460$	0	0
$461$	5.73206	0.266969	0.133484	−	0.991051i	$-0.457383\pi$
$461$	5.73206	0.266969	0.133484	+	0.991051i	$0.457383\pi$
$462$	0	0
$463$	13.6959	0.636505	0.318252	−	0.948006i	$-0.396904\pi$
$463$	13.6959	0.636505	0.318252	+	0.948006i	$0.396904\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	21.6430	1.00152	0.500759	−	0.865586i	$-0.333054\pi$
$467$	21.6430	1.00152	0.500759	+	0.865586i	$0.333054\pi$
$468$	0	0
$469$	−16.1050	−0.743662
$470$	0	0
$471$	−18.9649	−0.873858
$472$	0	0
$473$	−1.79380	−0.0824788
$474$	0	0
$475$	0	0
$476$	0	0
$477$	8.70928	0.398770
$478$	0	0
$479$	−13.1990	−0.603078	−0.301539	−	0.953454i	$-0.597500\pi$
$479$	−13.1990	−0.603078	−0.301539	+	0.953454i	$0.597500\pi$
$480$	0	0
$481$	7.46800	0.340511
$482$	0	0
$483$	27.2534	1.24007
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−4.51867	−0.204760	−0.102380	−	0.994745i	$-0.532646\pi$
$487$	−4.51867	−0.204760	−0.102380	+	0.994745i	$0.532646\pi$
$488$	0	0
$489$	6.07611	0.274771
$490$	0	0
$491$	−5.15061	−0.232444	−0.116222	−	0.993223i	$-0.537078\pi$
$491$	−5.15061	−0.232444	−0.116222	+	0.993223i	$0.537078\pi$
$492$	0	0
$493$	14.2351	0.641118
$494$	0	0
$495$	0	0
$496$	0	0
$497$	26.7442	1.19964
$498$	0	0
$499$	12.3991	0.555059	0.277529	−	0.960717i	$-0.410484\pi$
$499$	12.3991	0.555059	0.277529	+	0.960717i	$0.410484\pi$
$500$	0	0
$501$	−16.1483	−0.721455
$502$	0	0
$503$	−18.6042	−0.829522	−0.414761	−	0.909930i	$-0.636135\pi$
$503$	−18.6042	−0.829522	−0.414761	+	0.909930i	$0.636135\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	−0.0761103	−0.00337353	−0.00168677	−	0.999999i	$-0.500537\pi$
$509$	−0.0761103	−0.00337353	−0.00168677	+	0.999999i	$0.500537\pi$
$510$	0	0
$511$	−44.8554	−1.98428
$512$	0	0
$513$	−1.70928	−0.0754664
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0.346316	0.0152310
$518$	0	0
$519$	−22.3112	−0.979355
$520$	0	0
$521$	−15.8432	−0.694105	−0.347053	−	0.937846i	$-0.612817\pi$
$521$	−15.8432	−0.694105	−0.347053	+	0.937846i	$0.612817\pi$
$522$	0	0
$523$	−3.97334	−0.173742	−0.0868710	−	0.996220i	$-0.527687\pi$
$523$	−3.97334	−0.173742	−0.0868710	+	0.996220i	$0.527687\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−33.5802	−1.46278
$528$	0	0
$529$	47.3857	2.06025
$530$	0	0
$531$	−3.27513	−0.142128
$532$	0	0
$533$	8.78765	0.380636
$534$	0	0
$535$	0	0
$536$	0	0
$537$	9.52586	0.411071
$538$	0	0
$539$	−1.63704	−0.0705123
$540$	0	0
$541$	−42.2739	−1.81750	−0.908749	−	0.417344i	$-0.862961\pi$
$541$	−42.2739	−1.81750	−0.908749	+	0.417344i	$0.862961\pi$
$542$	0	0
$543$	19.5730	0.839959
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−37.6781	−1.61100	−0.805499	−	0.592597i	$-0.798103\pi$
$547$	−37.6781	−1.61100	−0.805499	+	0.592597i	$0.798103\pi$
$548$	0	0
$549$	−8.07838	−0.344777
$550$	0	0
$551$	5.70928	0.243223
$552$	0	0
$553$	−39.6514	−1.68615
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−4.33630	−0.183735	−0.0918674	−	0.995771i	$-0.529284\pi$
$557$	−4.33630	−0.183735	−0.0918674	+	0.995771i	$0.529284\pi$
$558$	0	0
$559$	−3.89269	−0.164643
$560$	0	0
$561$	1.96388	0.0829152
$562$	0	0
$563$	18.0183	0.759379	0.379689	−	0.925114i	$-0.376031\pi$
$563$	18.0183	0.759379	0.379689	+	0.925114i	$0.376031\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	3.24846	0.136423
$568$	0	0
$569$	−41.9854	−1.76012	−0.880061	−	0.474861i	$-0.842498\pi$
$569$	−41.9854	−1.76012	−0.880061	+	0.474861i	$0.842498\pi$
$570$	0	0
$571$	39.6925	1.66108	0.830539	−	0.556961i	$-0.188033\pi$
$571$	39.6925	1.66108	0.830539	+	0.556961i	$0.188033\pi$
$572$	0	0
$573$	4.52359	0.188976
$574$	0	0
$575$	0	0
$576$	0	0
$577$	23.8043	0.990986	0.495493	−	0.868612i	$-0.334987\pi$
$577$	23.8043	0.990986	0.495493	+	0.868612i	$0.334987\pi$
$578$	0	0
$579$	4.38962	0.182426
$580$	0	0
$581$	−24.0556	−0.997994
$582$	0	0
$583$	−4.01333	−0.166215
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.87936	0.0775696	0.0387848	−	0.999248i	$-0.487651\pi$
$587$	1.87936	0.0775696	0.0387848	+	0.999248i	$0.487651\pi$
$588$	0	0
$589$	−13.4680	−0.554939
$590$	0	0
$591$	10.6309	0.437297
$592$	0	0
$593$	−1.10504	−0.0453785	−0.0226893	−	0.999743i	$-0.507223\pi$
$593$	−1.10504	−0.0453785	−0.0226893	+	0.999743i	$0.507223\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	17.7321	0.725725
$598$	0	0
$599$	26.4885	1.08229	0.541146	−	0.840929i	$-0.317991\pi$
$599$	26.4885	1.08229	0.541146	+	0.840929i	$0.317991\pi$
$600$	0	0
$601$	−8.43680	−0.344144	−0.172072	−	0.985084i	$-0.555046\pi$
$601$	−8.43680	−0.344144	−0.172072	+	0.985084i	$0.555046\pi$
$602$	0	0
$603$	−4.95774	−0.201895
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−31.0784	−1.26143	−0.630716	−	0.776014i	$-0.717239\pi$
$607$	−31.0784	−1.26143	−0.630716	+	0.776014i	$0.717239\pi$
$608$	0	0
$609$	−10.8504	−0.439682
$610$	0	0
$611$	0.751536	0.0304039
$612$	0	0
$613$	−4.47027	−0.180552	−0.0902762	−	0.995917i	$-0.528775\pi$
$613$	−4.47027	−0.180552	−0.0902762	+	0.995917i	$0.528775\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−9.60197	−0.386561	−0.193280	−	0.981144i	$-0.561913\pi$
$617$	−9.60197	−0.386561	−0.193280	+	0.981144i	$0.561913\pi$
$618$	0	0
$619$	15.8348	0.636456	0.318228	−	0.948014i	$-0.396912\pi$
$619$	15.8348	0.636456	0.318228	+	0.948014i	$0.396912\pi$
$620$	0	0
$621$	8.38962	0.336664
$622$	0	0
$623$	−30.6824	−1.22927
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0.787653	0.0314558
$628$	0	0
$629$	−31.8271	−1.26903
$630$	0	0
$631$	6.88655	0.274149	0.137075	−	0.990561i	$-0.456230\pi$
$631$	6.88655	0.274149	0.137075	+	0.990561i	$0.456230\pi$
$632$	0	0
$633$	−14.9132	−0.592747
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−3.55252	−0.140756
$638$	0	0
$639$	8.23287	0.325687
$640$	0	0
$641$	−1.73820	−0.0686550	−0.0343275	−	0.999411i	$-0.510929\pi$
$641$	−1.73820	−0.0686550	−0.0343275	+	0.999411i	$0.510929\pi$
$642$	0	0
$643$	−11.4536	−0.451687	−0.225843	−	0.974164i	$-0.572514\pi$
$643$	−11.4536	−0.451687	−0.225843	+	0.974164i	$0.572514\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−23.9688	−0.942311	−0.471155	−	0.882050i	$-0.656163\pi$
$647$	−23.9688	−0.942311	−0.471155	+	0.882050i	$0.656163\pi$
$648$	0	0
$649$	1.50921	0.0592419
$650$	0	0
$651$	25.5958	1.00318
$652$	0	0
$653$	−25.2967	−0.989936	−0.494968	−	0.868911i	$-0.664820\pi$
$653$	−25.2967	−0.989936	−0.494968	+	0.868911i	$0.664820\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−13.8082	−0.538708
$658$	0	0
$659$	40.4040	1.57392	0.786958	−	0.617006i	$-0.211655\pi$
$659$	40.4040	1.57392	0.786958	+	0.617006i	$0.211655\pi$
$660$	0	0
$661$	0.160631	0.00624783	0.00312391	−	0.999995i	$-0.499006\pi$
$661$	0.160631	0.00624783	0.00312391	+	0.999995i	$0.499006\pi$
$662$	0	0
$663$	4.26180	0.165514
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−28.0228	−1.08505
$668$	0	0
$669$	20.5464	0.794369
$670$	0	0
$671$	3.72261	0.143710
$672$	0	0
$673$	−4.89723	−0.188774	−0.0943871	−	0.995536i	$-0.530089\pi$
$673$	−4.89723	−0.188774	−0.0943871	+	0.995536i	$0.530089\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	14.7792	0.568012	0.284006	−	0.958822i	$-0.408336\pi$
$677$	14.7792	0.568012	0.284006	+	0.958822i	$0.408336\pi$
$678$	0	0
$679$	8.11118	0.311279
$680$	0	0
$681$	21.6381	0.829173
$682$	0	0
$683$	−22.2462	−0.851227	−0.425614	−	0.904905i	$-0.639942\pi$
$683$	−22.2462	−0.851227	−0.425614	+	0.904905i	$0.639942\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−19.3607	−0.738656
$688$	0	0
$689$	−8.70928	−0.331797
$690$	0	0
$691$	−13.6732	−0.520151	−0.260076	−	0.965588i	$-0.583748\pi$
$691$	−13.6732	−0.520151	−0.260076	+	0.965588i	$0.583748\pi$
$692$	0	0
$693$	−1.49693	−0.0568636
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−37.4512	−1.41856
$698$	0	0
$699$	−24.0722	−0.910496
$700$	0	0
$701$	−6.34632	−0.239697	−0.119849	−	0.992792i	$-0.538241\pi$
$701$	−6.34632	−0.239697	−0.119849	+	0.992792i	$0.538241\pi$
$702$	0	0
$703$	−12.7649	−0.481436
$704$	0	0
$705$	0	0
$706$	0	0
$707$	25.1100	0.944357
$708$	0	0
$709$	41.5136	1.55907	0.779537	−	0.626356i	$-0.215454\pi$
$709$	41.5136	1.55907	0.779537	+	0.626356i	$0.215454\pi$
$710$	0	0
$711$	−12.2062	−0.457768
$712$	0	0
$713$	66.1049	2.47565
$714$	0	0
$715$	0	0
$716$	0	0
$717$	14.2423	0.531889
$718$	0	0
$719$	−46.9854	−1.75226	−0.876131	−	0.482074i	$-0.839884\pi$
$719$	−46.9854	−1.75226	−0.876131	+	0.482074i	$0.839884\pi$
$720$	0	0
$721$	6.14834	0.228976
$722$	0	0
$723$	28.3896	1.05582
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−48.7754	−1.80898	−0.904489	−	0.426497i	$-0.859748\pi$
$727$	−48.7754	−1.80898	−0.904489	+	0.426497i	$0.859748\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	16.5899	0.613598
$732$	0	0
$733$	−46.6225	−1.72204	−0.861020	−	0.508570i	$-0.830174\pi$
$733$	−46.6225	−1.72204	−0.861020	+	0.508570i	$0.830174\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2.28458	0.0841536
$738$	0	0
$739$	23.6837	0.871217	0.435609	−	0.900136i	$-0.356533\pi$
$739$	23.6837	0.871217	0.435609	+	0.900136i	$0.356533\pi$
$740$	0	0
$741$	1.70928	0.0627918
$742$	0	0
$743$	12.9044	0.473417	0.236709	−	0.971581i	$-0.423931\pi$
$743$	12.9044	0.473417	0.236709	+	0.971581i	$0.423931\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−7.40522	−0.270943
$748$	0	0
$749$	−23.3151	−0.851916
$750$	0	0
$751$	−8.17501	−0.298310	−0.149155	−	0.988814i	$-0.547655\pi$
$751$	−8.17501	−0.298310	−0.149155	+	0.988814i	$0.547655\pi$
$752$	0	0
$753$	−4.41628	−0.160938
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−4.83096	−0.175584	−0.0877921	−	0.996139i	$-0.527981\pi$
$757$	−4.83096	−0.175584	−0.0877921	+	0.996139i	$0.527981\pi$
$758$	0	0
$759$	−3.86603	−0.140328
$760$	0	0
$761$	−41.4413	−1.50225	−0.751124	−	0.660162i	$-0.770488\pi$
$761$	−41.4413	−1.50225	−0.751124	+	0.660162i	$0.770488\pi$
$762$	0	0
$763$	−36.4696	−1.32029
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3.27513	0.118258
$768$	0	0
$769$	−18.1711	−0.655268	−0.327634	−	0.944805i	$-0.606251\pi$
$769$	−18.1711	−0.655268	−0.327634	+	0.944805i	$0.606251\pi$
$770$	0	0
$771$	−8.34244	−0.300446
$772$	0	0
$773$	23.6598	0.850985	0.425492	−	0.904962i	$-0.360101\pi$
$773$	23.6598	0.850985	0.425492	+	0.904962i	$0.360101\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	24.2595	0.870306
$778$	0	0
$779$	−15.0205	−0.538166
$780$	0	0
$781$	−3.79380	−0.135753
$782$	0	0
$783$	−3.34017	−0.119368
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−39.2089	−1.39764	−0.698822	−	0.715295i	$-0.746292\pi$
$787$	−39.2089	−1.39764	−0.698822	+	0.715295i	$0.746292\pi$
$788$	0	0
$789$	−7.95055	−0.283047
$790$	0	0
$791$	58.1978	2.06928
$792$	0	0
$793$	8.07838	0.286872
$794$	0	0
$795$	0	0
$796$	0	0
$797$	40.7998	1.44520	0.722601	−	0.691266i	$-0.242946\pi$
$797$	40.7998	1.44520	0.722601	+	0.691266i	$0.242946\pi$
$798$	0	0
$799$	−3.20289	−0.113310
$800$	0	0
$801$	−9.44521	−0.333730
$802$	0	0
$803$	6.36296	0.224544
$804$	0	0
$805$	0	0
$806$	0	0
$807$	6.41855	0.225944
$808$	0	0
$809$	13.0205	0.457777	0.228889	−	0.973453i	$-0.426491\pi$
$809$	13.0205	0.457777	0.228889	+	0.973453i	$0.426491\pi$
$810$	0	0
$811$	−47.2628	−1.65962	−0.829811	−	0.558044i	$-0.811552\pi$
$811$	−47.2628	−1.65962	−0.829811	+	0.558044i	$0.811552\pi$
$812$	0	0
$813$	17.9555	0.629726
$814$	0	0
$815$	0	0
$816$	0	0
$817$	6.65368	0.232783
$818$	0	0
$819$	−3.24846	−0.113511
$820$	0	0
$821$	−43.4512	−1.51646	−0.758228	−	0.651989i	$-0.773935\pi$
$821$	−43.4512	−1.51646	−0.758228	+	0.651989i	$0.773935\pi$
$822$	0	0
$823$	−36.7936	−1.28254	−0.641272	−	0.767313i	$-0.721593\pi$
$823$	−36.7936	−1.28254	−0.641272	+	0.767313i	$0.721593\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−24.5018	−0.852013	−0.426006	−	0.904720i	$-0.640080\pi$
$827$	−24.5018	−0.852013	−0.426006	+	0.904720i	$0.640080\pi$
$828$	0	0
$829$	40.4908	1.40630	0.703152	−	0.711040i	$-0.251776\pi$
$829$	40.4908	1.40630	0.703152	+	0.711040i	$0.251776\pi$
$830$	0	0
$831$	1.86991	0.0648663
$832$	0	0
$833$	15.1401	0.524574
$834$	0	0
$835$	0	0
$836$	0	0
$837$	7.87936	0.272351
$838$	0	0
$839$	−25.4245	−0.877752	−0.438876	−	0.898548i	$-0.644623\pi$
$839$	−25.4245	−0.877752	−0.438876	+	0.898548i	$0.644623\pi$
$840$	0	0
$841$	−17.8432	−0.615284
$842$	0	0
$843$	−6.77924	−0.233490
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−35.0433	−1.20410
$848$	0	0
$849$	−9.73206	−0.334003
$850$	0	0
$851$	62.6537	2.14774
$852$	0	0
$853$	21.1917	0.725588	0.362794	−	0.931869i	$-0.381823\pi$
$853$	21.1917	0.725588	0.362794	+	0.931869i	$0.381823\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	18.3135	0.625578	0.312789	−	0.949823i	$-0.398737\pi$
$857$	18.3135	0.625578	0.312789	+	0.949823i	$0.398737\pi$
$858$	0	0
$859$	16.8865	0.576162	0.288081	−	0.957606i	$-0.406983\pi$
$859$	16.8865	0.576162	0.288081	+	0.957606i	$0.406983\pi$
$860$	0	0
$861$	28.5464	0.972858
$862$	0	0
$863$	−10.0095	−0.340726	−0.170363	−	0.985381i	$-0.554494\pi$
$863$	−10.0095	−0.340726	−0.170363	+	0.985381i	$0.554494\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−1.16290	−0.0394941
$868$	0	0
$869$	5.62475	0.190807
$870$	0	0
$871$	4.95774	0.167987
$872$	0	0
$873$	2.49693	0.0845082
$874$	0	0
$875$	0	0
$876$	0	0
$877$	38.7070	1.30704	0.653521	−	0.756908i	$-0.273291\pi$
$877$	38.7070	1.30704	0.653521	+	0.756908i	$0.273291\pi$
$878$	0	0
$879$	−26.9939	−0.910480
$880$	0	0
$881$	−41.9998	−1.41501	−0.707505	−	0.706708i	$-0.750179\pi$
$881$	−41.9998	−1.41501	−0.707505	+	0.706708i	$0.750179\pi$
$882$	0	0
$883$	−26.4846	−0.891279	−0.445640	−	0.895212i	$-0.647024\pi$
$883$	−26.4846	−0.891279	−0.445640	+	0.895212i	$0.647024\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	43.9916	1.47709	0.738547	−	0.674203i	$-0.235513\pi$
$887$	43.9916	1.47709	0.738547	+	0.674203i	$0.235513\pi$
$888$	0	0
$889$	36.4696	1.22315
$890$	0	0
$891$	−0.460811	−0.0154378
$892$	0	0
$893$	−1.28458	−0.0429869
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−8.38962	−0.280121
$898$	0	0
$899$	−26.3184	−0.877769
$900$	0	0
$901$	37.1171	1.23655
$902$	0	0
$903$	−12.6453	−0.420808
$904$	0	0
$905$	0	0
$906$	0	0
$907$	29.1917	0.969293	0.484646	−	0.874710i	$-0.338948\pi$
$907$	29.1917	0.969293	0.484646	+	0.874710i	$0.338948\pi$
$908$	0	0
$909$	7.72979	0.256381
$910$	0	0
$911$	38.6453	1.28038	0.640188	−	0.768219i	$-0.278857\pi$
$911$	38.6453	1.28038	0.640188	+	0.768219i	$0.278857\pi$
$912$	0	0
$913$	3.41241	0.112934
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−43.8369	−1.44762
$918$	0	0
$919$	24.3857	0.804412	0.402206	−	0.915549i	$-0.368244\pi$
$919$	24.3857	0.804412	0.402206	+	0.915549i	$0.368244\pi$
$920$	0	0
$921$	7.95055	0.261980
$922$	0	0
$923$	−8.23287	−0.270988
$924$	0	0
$925$	0	0
$926$	0	0
$927$	1.89269	0.0621642
$928$	0	0
$929$	−7.14608	−0.234455	−0.117228	−	0.993105i	$-0.537401\pi$
$929$	−7.14608	−0.234455	−0.117228	+	0.993105i	$0.537401\pi$
$930$	0	0
$931$	6.07223	0.199009
$932$	0	0
$933$	16.2557	0.532186
$934$	0	0
$935$	0	0
$936$	0	0
$937$	9.76487	0.319004	0.159502	−	0.987198i	$-0.449011\pi$
$937$	9.76487	0.319004	0.159502	+	0.987198i	$0.449011\pi$
$938$	0	0
$939$	−11.4885	−0.374914
$940$	0	0
$941$	32.6453	1.06421	0.532103	−	0.846680i	$-0.321402\pi$
$941$	32.6453	1.06421	0.532103	+	0.846680i	$0.321402\pi$
$942$	0	0
$943$	73.7251	2.40082
$944$	0	0
$945$	0	0
$946$	0	0
$947$	30.8466	1.00238	0.501189	−	0.865338i	$-0.332896\pi$
$947$	30.8466	1.00238	0.501189	+	0.865338i	$0.332896\pi$
$948$	0	0
$949$	13.8082	0.448232
$950$	0	0
$951$	−7.18568	−0.233012
$952$	0	0
$953$	35.6596	1.15513	0.577565	−	0.816345i	$-0.304003\pi$
$953$	35.6596	1.15513	0.577565	+	0.816345i	$0.304003\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	1.53919	0.0497549
$958$	0	0
$959$	−5.39189	−0.174113
$960$	0	0
$961$	31.0843	1.00272
$962$	0	0
$963$	−7.17727	−0.231284
$964$	0	0
$965$	0	0
$966$	0	0
$967$	31.4112	1.01012	0.505058	−	0.863086i	$-0.331471\pi$
$967$	31.4112	1.01012	0.505058	+	0.863086i	$0.331471\pi$
$968$	0	0
$969$	−7.28458	−0.234014
$970$	0	0
$971$	−24.0722	−0.772515	−0.386257	−	0.922391i	$-0.626232\pi$
$971$	−24.0722	−0.772515	−0.386257	+	0.922391i	$0.626232\pi$
$972$	0	0
$973$	47.6886	1.52883
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−9.99547	−0.319783	−0.159892	−	0.987135i	$-0.551114\pi$
$977$	−9.99547	−0.319783	−0.159892	+	0.987135i	$0.551114\pi$
$978$	0	0
$979$	4.35246	0.139105
$980$	0	0
$981$	−11.2267	−0.358442
$982$	0	0
$983$	25.4112	0.810491	0.405245	−	0.914208i	$-0.367186\pi$
$983$	25.4112	0.810491	0.405245	+	0.914208i	$0.367186\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	2.44134	0.0777086
$988$	0	0
$989$	−32.6582	−1.03847
$990$	0	0
$991$	38.0638	1.20914	0.604569	−	0.796553i	$-0.293346\pi$
$991$	38.0638	1.20914	0.604569	+	0.796553i	$0.293346\pi$
$992$	0	0
$993$	9.49466	0.301304
$994$	0	0
$995$	0	0
$996$	0	0
$997$	28.5441	0.904001	0.452001	−	0.892018i	$-0.350711\pi$
$997$	28.5441	0.904001	0.452001	+	0.892018i	$0.350711\pi$
$998$	0	0
$999$	7.46800	0.236277

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7800.2.a.bl.1.3	✓	3
5.4	even	2		7800.2.a.bo.1.1	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7800.2.a.bl.1.3	✓	3	1.1	even	1	trivial
7800.2.a.bo.1.1	yes	3	5.4	even	2

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:	\( N \)	\(=\)	\( 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7800.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +3.24846 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +3.24846 q^{7} +1.00000 q^{9} -0.460811 q^{11} -1.00000 q^{13} +4.26180 q^{17} +1.70928 q^{19} -3.24846 q^{21} -8.38962 q^{23} -1.00000 q^{27} +3.34017 q^{29} -7.87936 q^{31} +0.460811 q^{33} -7.46800 q^{37} +1.00000 q^{39} -8.78765 q^{41} +3.89269 q^{43} -0.751536 q^{47} +3.55252 q^{49} -4.26180 q^{51} +8.70928 q^{53} -1.70928 q^{57} -3.27513 q^{59} -8.07838 q^{61} +3.24846 q^{63} -4.95774 q^{67} +8.38962 q^{69} +8.23287 q^{71} -13.8082 q^{73} -1.49693 q^{77} -12.2062 q^{79} +1.00000 q^{81} -7.40522 q^{83} -3.34017 q^{87} -9.44521 q^{89} -3.24846 q^{91} +7.87936 q^{93} +2.49693 q^{97} -0.460811 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 + q^7 + 3 * q^9	\( 3 q - 3 q^{3} + q^{7} + 3 q^{9} - 3 q^{11} - 3 q^{13} + 5 q^{17} - 2 q^{19} - q^{21} + 4 q^{23} - 3 q^{27} - q^{29} - 11 q^{31} + 3 q^{33} + 10 q^{37} + 3 q^{39} - 16 q^{41} - 11 q^{47} + 10 q^{49} - 5 q^{51} + 19 q^{53} + 2 q^{57} - 3 q^{59} - 21 q^{61} + q^{63} + q^{67} - 4 q^{69} + 2 q^{71} + 2 q^{73} + 13 q^{77} - 12 q^{79} + 3 q^{81} - 7 q^{83} + q^{87} - 16 q^{89} - q^{91} + 11 q^{93} - 10 q^{97} - 3 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + q^7 + 3 * q^9 - 3 * q^11 - 3 * q^13 + 5 * q^17 - 2 * q^19 - q^21 + 4 * q^23 - 3 * q^27 - q^29 - 11 * q^31 + 3 * q^33 + 10 * q^37 + 3 * q^39 - 16 * q^41 - 11 * q^47 + 10 * q^49 - 5 * q^51 + 19 * q^53 + 2 * q^57 - 3 * q^59 - 21 * q^61 + q^63 + q^67 - 4 * q^69 + 2 * q^71 + 2 * q^73 + 13 * q^77 - 12 * q^79 + 3 * q^81 - 7 * q^83 + q^87 - 16 * q^89 - q^91 + 11 * q^93 - 10 * q^97 - 3 * q^99

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