LMFDB - Embedded newform 6900.2.a.v.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6900,2,Mod(1,6900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6900, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6900.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6900.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:	$55.0967773947$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.30278$ of defining polynomial
Character	$\chi$	$=$	6900.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +4.30278 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +4.30278 q^{7} +1.00000 q^{9} +2.30278 q^{11} -0.302776 q^{13} +3.00000 q^{17} +6.60555 q^{19} +4.30278 q^{21} -1.00000 q^{23} +1.00000 q^{27} +1.39445 q^{29} -9.51388 q^{31} +2.30278 q^{33} +3.60555 q^{37} -0.302776 q^{39} -1.60555 q^{41} -1.00000 q^{43} -3.00000 q^{47} +11.5139 q^{49} +3.00000 q^{51} +12.9083 q^{53} +6.60555 q^{57} +11.5139 q^{59} +2.90833 q^{61} +4.30278 q^{63} -10.6972 q^{67} -1.00000 q^{69} +10.1194 q^{71} -10.2111 q^{73} +9.90833 q^{77} -4.21110 q^{79} +1.00000 q^{81} +13.6056 q^{83} +1.39445 q^{87} -4.81665 q^{89} -1.30278 q^{91} -9.51388 q^{93} -18.7250 q^{97} +2.30278 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 5 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 5 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} + 5 q^{7} + 2 q^{9} + q^{11} + 3 q^{13} + 6 q^{17} + 6 q^{19} + 5 q^{21} - 2 q^{23} + 2 q^{27} + 10 q^{29} - q^{31} + q^{33} + 3 q^{39} + 4 q^{41} - 2 q^{43} - 6 q^{47} + 5 q^{49} + 6 q^{51} + 15 q^{53} + 6 q^{57} + 5 q^{59} - 5 q^{61} + 5 q^{63} - 25 q^{67} - 2 q^{69} - 5 q^{71} - 6 q^{73} + 9 q^{77} + 6 q^{79} + 2 q^{81} + 20 q^{83} + 10 q^{87} + 12 q^{89} + q^{91} - q^{93} - 5 q^{97} + q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 5 * q^7 + 2 * q^9 + q^11 + 3 * q^13 + 6 * q^17 + 6 * q^19 + 5 * q^21 - 2 * q^23 + 2 * q^27 + 10 * q^29 - q^31 + q^33 + 3 * q^39 + 4 * q^41 - 2 * q^43 - 6 * q^47 + 5 * q^49 + 6 * q^51 + 15 * q^53 + 6 * q^57 + 5 * q^59 - 5 * q^61 + 5 * q^63 - 25 * q^67 - 2 * q^69 - 5 * q^71 - 6 * q^73 + 9 * q^77 + 6 * q^79 + 2 * q^81 + 20 * q^83 + 10 * q^87 + 12 * q^89 + q^91 - q^93 - 5 * q^97 + 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
$7$	4.30278	1.62630	0.813148	−	0.582057i	$-0.197752\pi$
$7$	4.30278	1.62630	0.813148	+	0.582057i	$0.197752\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.30278	0.694313	0.347156	−	0.937807i	$-0.387147\pi$
$11$	2.30278	0.694313	0.347156	+	0.937807i	$0.387147\pi$
$12$	0	0
$13$	−0.302776	−0.0839749	−0.0419874	−	0.999118i	$-0.513369\pi$
$13$	−0.302776	−0.0839749	−0.0419874	+	0.999118i	$0.513369\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	0	0
$19$	6.60555	1.51542	0.757709	−	0.652593i	$-0.226319\pi$
$19$	6.60555	1.51542	0.757709	+	0.652593i	$0.226319\pi$
$20$	0	0
$21$	4.30278	0.938943
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	1.39445	0.258943	0.129471	−	0.991583i	$-0.458672\pi$
$29$	1.39445	0.258943	0.129471	+	0.991583i	$0.458672\pi$
$30$	0	0
$31$	−9.51388	−1.70874	−0.854371	−	0.519663i	$-0.826058\pi$
$31$	−9.51388	−1.70874	−0.854371	+	0.519663i	$0.826058\pi$
$32$	0	0
$33$	2.30278	0.400862
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3.60555	0.592749	0.296374	−	0.955072i	$-0.404222\pi$
$37$	3.60555	0.592749	0.296374	+	0.955072i	$0.404222\pi$
$38$	0	0
$39$	−0.302776	−0.0484829
$40$	0	0
$41$	−1.60555	−0.250745	−0.125372	−	0.992110i	$-0.540013\pi$
$41$	−1.60555	−0.250745	−0.125372	+	0.992110i	$0.540013\pi$
$42$	0	0
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3.00000	−0.437595	−0.218797	−	0.975770i	$-0.570213\pi$
$47$	−3.00000	−0.437595	−0.218797	+	0.975770i	$0.570213\pi$
$48$	0	0
$49$	11.5139	1.64484
$50$	0	0
$51$	3.00000	0.420084
$52$	0	0
$53$	12.9083	1.77310	0.886548	−	0.462638i	$-0.153097\pi$
$53$	12.9083	1.77310	0.886548	+	0.462638i	$0.153097\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	6.60555	0.874927
$58$	0	0
$59$	11.5139	1.49898	0.749490	−	0.662016i	$-0.230299\pi$
$59$	11.5139	1.49898	0.749490	+	0.662016i	$0.230299\pi$
$60$	0	0
$61$	2.90833	0.372373	0.186187	−	0.982514i	$-0.440387\pi$
$61$	2.90833	0.372373	0.186187	+	0.982514i	$0.440387\pi$
$62$	0	0
$63$	4.30278	0.542099
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−10.6972	−1.30687	−0.653437	−	0.756981i	$-0.726674\pi$
$67$	−10.6972	−1.30687	−0.653437	+	0.756981i	$0.726674\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	10.1194	1.20096	0.600478	−	0.799642i	$-0.294977\pi$
$71$	10.1194	1.20096	0.600478	+	0.799642i	$0.294977\pi$
$72$	0	0
$73$	−10.2111	−1.19512	−0.597560	−	0.801825i	$-0.703863\pi$
$73$	−10.2111	−1.19512	−0.597560	+	0.801825i	$0.703863\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	9.90833	1.12916
$78$	0	0
$79$	−4.21110	−0.473786	−0.236893	−	0.971536i	$-0.576129\pi$
$79$	−4.21110	−0.473786	−0.236893	+	0.971536i	$0.576129\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	13.6056	1.49340	0.746702	−	0.665159i	$-0.231636\pi$
$83$	13.6056	1.49340	0.746702	+	0.665159i	$0.231636\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1.39445	0.149501
$88$	0	0
$89$	−4.81665	−0.510564	−0.255282	−	0.966867i	$-0.582168\pi$
$89$	−4.81665	−0.510564	−0.255282	+	0.966867i	$0.582168\pi$
$90$	0	0
$91$	−1.30278	−0.136568
$92$	0	0
$93$	−9.51388	−0.986543
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−18.7250	−1.90123	−0.950617	−	0.310367i	$-0.899548\pi$
$97$	−18.7250	−1.90123	−0.950617	+	0.310367i	$0.899548\pi$
$98$	0	0
$99$	2.30278	0.231438
$100$	0	0
$101$	−9.90833	−0.985915	−0.492958	−	0.870053i	$-0.664084\pi$
$101$	−9.90833	−0.985915	−0.492958	+	0.870053i	$0.664084\pi$
$102$	0	0
$103$	3.39445	0.334465	0.167232	−	0.985917i	$-0.446517\pi$
$103$	3.39445	0.334465	0.167232	+	0.985917i	$0.446517\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−10.8167	−1.04569	−0.522843	−	0.852429i	$-0.675128\pi$
$107$	−10.8167	−1.04569	−0.522843	+	0.852429i	$0.675128\pi$
$108$	0	0
$109$	−7.00000	−0.670478	−0.335239	−	0.942133i	$-0.608817\pi$
$109$	−7.00000	−0.670478	−0.335239	+	0.942133i	$0.608817\pi$
$110$	0	0
$111$	3.60555	0.342224
$112$	0	0
$113$	−17.7250	−1.66743	−0.833713	−	0.552198i	$-0.813789\pi$
$113$	−17.7250	−1.66743	−0.833713	+	0.552198i	$0.813789\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−0.302776	−0.0279916
$118$	0	0
$119$	12.9083	1.18330
$120$	0	0
$121$	−5.69722	−0.517929
$122$	0	0
$123$	−1.60555	−0.144768
$124$	0	0
$125$	0	0
$126$	0	0
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	0	0
$129$	−1.00000	−0.0880451
$130$	0	0
$131$	−0.211103	−0.0184441	−0.00922206	−	0.999957i	$-0.502936\pi$
$131$	−0.211103	−0.0184441	−0.00922206	+	0.999957i	$0.502936\pi$
$132$	0	0
$133$	28.4222	2.46452
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−3.21110	−0.274343	−0.137172	−	0.990547i	$-0.543801\pi$
$137$	−3.21110	−0.274343	−0.137172	+	0.990547i	$0.543801\pi$
$138$	0	0
$139$	13.5139	1.14623	0.573116	−	0.819474i	$-0.305734\pi$
$139$	13.5139	1.14623	0.573116	+	0.819474i	$0.305734\pi$
$140$	0	0
$141$	−3.00000	−0.252646
$142$	0	0
$143$	−0.697224	−0.0583048
$144$	0	0
$145$	0	0
$146$	0	0
$147$	11.5139	0.949649
$148$	0	0
$149$	−12.2111	−1.00037	−0.500186	−	0.865918i	$-0.666735\pi$
$149$	−12.2111	−1.00037	−0.500186	+	0.865918i	$0.666735\pi$
$150$	0	0
$151$	0.816654	0.0664583	0.0332292	−	0.999448i	$-0.489421\pi$
$151$	0.816654	0.0664583	0.0332292	+	0.999448i	$0.489421\pi$
$152$	0	0
$153$	3.00000	0.242536
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−4.00000	−0.319235	−0.159617	−	0.987179i	$-0.551026\pi$
$157$	−4.00000	−0.319235	−0.159617	+	0.987179i	$0.551026\pi$
$158$	0	0
$159$	12.9083	1.02370
$160$	0	0
$161$	−4.30278	−0.339106
$162$	0	0
$163$	18.8167	1.47383	0.736917	−	0.675983i	$-0.236281\pi$
$163$	18.8167	1.47383	0.736917	+	0.675983i	$0.236281\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2.51388	0.194530	0.0972649	−	0.995259i	$-0.468991\pi$
$167$	2.51388	0.194530	0.0972649	+	0.995259i	$0.468991\pi$
$168$	0	0
$169$	−12.9083	−0.992948
$170$	0	0
$171$	6.60555	0.505139
$172$	0	0
$173$	23.5139	1.78773	0.893864	−	0.448339i	$-0.147984\pi$
$173$	23.5139	1.78773	0.893864	+	0.448339i	$0.147984\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	11.5139	0.865436
$178$	0	0
$179$	−19.3305	−1.44483	−0.722416	−	0.691459i	$-0.756968\pi$
$179$	−19.3305	−1.44483	−0.722416	+	0.691459i	$0.756968\pi$
$180$	0	0
$181$	−21.0278	−1.56298	−0.781490	−	0.623917i	$-0.785540\pi$
$181$	−21.0278	−1.56298	−0.781490	+	0.623917i	$0.785540\pi$
$182$	0	0
$183$	2.90833	0.214990
$184$	0	0
$185$	0	0
$186$	0	0
$187$	6.90833	0.505187
$188$	0	0
$189$	4.30278	0.312981
$190$	0	0
$191$	16.8167	1.21681	0.608405	−	0.793627i	$-0.291810\pi$
$191$	16.8167	1.21681	0.608405	+	0.793627i	$0.291810\pi$
$192$	0	0
$193$	11.0000	0.791797	0.395899	−	0.918294i	$-0.370433\pi$
$193$	11.0000	0.791797	0.395899	+	0.918294i	$0.370433\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	19.8167	1.41188	0.705939	−	0.708273i	$-0.250525\pi$
$197$	19.8167	1.41188	0.705939	+	0.708273i	$0.250525\pi$
$198$	0	0
$199$	6.18335	0.438326	0.219163	−	0.975688i	$-0.429667\pi$
$199$	6.18335	0.438326	0.219163	+	0.975688i	$0.429667\pi$
$200$	0	0
$201$	−10.6972	−0.754524
$202$	0	0
$203$	6.00000	0.421117
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	15.2111	1.05217
$210$	0	0
$211$	−11.3944	−0.784426	−0.392213	−	0.919874i	$-0.628290\pi$
$211$	−11.3944	−0.784426	−0.392213	+	0.919874i	$0.628290\pi$
$212$	0	0
$213$	10.1194	0.693372
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−40.9361	−2.77892
$218$	0	0
$219$	−10.2111	−0.690002
$220$	0	0
$221$	−0.908327	−0.0611007
$222$	0	0
$223$	14.6972	0.984199	0.492099	−	0.870539i	$-0.336230\pi$
$223$	14.6972	0.984199	0.492099	+	0.870539i	$0.336230\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	12.2111	0.810479	0.405240	−	0.914210i	$-0.367188\pi$
$227$	12.2111	0.810479	0.405240	+	0.914210i	$0.367188\pi$
$228$	0	0
$229$	−14.8167	−0.979112	−0.489556	−	0.871972i	$-0.662841\pi$
$229$	−14.8167	−0.979112	−0.489556	+	0.871972i	$0.662841\pi$
$230$	0	0
$231$	9.90833	0.651920
$232$	0	0
$233$	11.0917	0.726640	0.363320	−	0.931664i	$-0.381643\pi$
$233$	11.0917	0.726640	0.363320	+	0.931664i	$0.381643\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−4.21110	−0.273541
$238$	0	0
$239$	−4.60555	−0.297908	−0.148954	−	0.988844i	$-0.547591\pi$
$239$	−4.60555	−0.297908	−0.148954	+	0.988844i	$0.547591\pi$
$240$	0	0
$241$	−6.51388	−0.419596	−0.209798	−	0.977745i	$-0.567281\pi$
$241$	−6.51388	−0.419596	−0.209798	+	0.977745i	$0.567281\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−2.00000	−0.127257
$248$	0	0
$249$	13.6056	0.862217
$250$	0	0
$251$	−22.1194	−1.39617	−0.698083	−	0.716017i	$-0.745963\pi$
$251$	−22.1194	−1.39617	−0.698083	+	0.716017i	$0.745963\pi$
$252$	0	0
$253$	−2.30278	−0.144774
$254$	0	0
$255$	0	0
$256$	0	0
$257$	31.1194	1.94118	0.970588	−	0.240745i	$-0.0773918\pi$
$257$	31.1194	1.94118	0.970588	+	0.240745i	$0.0773918\pi$
$258$	0	0
$259$	15.5139	0.963985
$260$	0	0
$261$	1.39445	0.0863142
$262$	0	0
$263$	3.90833	0.240998	0.120499	−	0.992713i	$-0.461551\pi$
$263$	3.90833	0.240998	0.120499	+	0.992713i	$0.461551\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−4.81665	−0.294774
$268$	0	0
$269$	26.2389	1.59981	0.799906	−	0.600126i	$-0.204883\pi$
$269$	26.2389	1.59981	0.799906	+	0.600126i	$0.204883\pi$
$270$	0	0
$271$	−3.78890	−0.230159	−0.115080	−	0.993356i	$-0.536712\pi$
$271$	−3.78890	−0.230159	−0.115080	+	0.993356i	$0.536712\pi$
$272$	0	0
$273$	−1.30278	−0.0788476
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−0.0916731	−0.00550810	−0.00275405	−	0.999996i	$-0.500877\pi$
$277$	−0.0916731	−0.00550810	−0.00275405	+	0.999996i	$0.500877\pi$
$278$	0	0
$279$	−9.51388	−0.569581
$280$	0	0
$281$	−7.18335	−0.428523	−0.214261	−	0.976776i	$-0.568734\pi$
$281$	−7.18335	−0.428523	−0.214261	+	0.976776i	$0.568734\pi$
$282$	0	0
$283$	−10.6333	−0.632085	−0.316042	−	0.948745i	$-0.602354\pi$
$283$	−10.6333	−0.632085	−0.316042	+	0.948745i	$0.602354\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−6.90833	−0.407786
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	−18.7250	−1.09768
$292$	0	0
$293$	−1.60555	−0.0937973	−0.0468987	−	0.998900i	$-0.514934\pi$
$293$	−1.60555	−0.0937973	−0.0468987	+	0.998900i	$0.514934\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	2.30278	0.133621
$298$	0	0
$299$	0.302776	0.0175100
$300$	0	0
$301$	−4.30278	−0.248008
$302$	0	0
$303$	−9.90833	−0.569219
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−31.2111	−1.78131	−0.890656	−	0.454678i	$-0.849754\pi$
$307$	−31.2111	−1.78131	−0.890656	+	0.454678i	$0.849754\pi$
$308$	0	0
$309$	3.39445	0.193103
$310$	0	0
$311$	−0.211103	−0.0119705	−0.00598526	−	0.999982i	$-0.501905\pi$
$311$	−0.211103	−0.0119705	−0.00598526	+	0.999982i	$0.501905\pi$
$312$	0	0
$313$	7.78890	0.440255	0.220127	−	0.975471i	$-0.429353\pi$
$313$	7.78890	0.440255	0.220127	+	0.975471i	$0.429353\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−0.908327	−0.0510167	−0.0255084	−	0.999675i	$-0.508120\pi$
$317$	−0.908327	−0.0510167	−0.0255084	+	0.999675i	$0.508120\pi$
$318$	0	0
$319$	3.21110	0.179787
$320$	0	0
$321$	−10.8167	−0.603727
$322$	0	0
$323$	19.8167	1.10263
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−7.00000	−0.387101
$328$	0	0
$329$	−12.9083	−0.711659
$330$	0	0
$331$	−15.3028	−0.841117	−0.420558	−	0.907266i	$-0.638166\pi$
$331$	−15.3028	−0.841117	−0.420558	+	0.907266i	$0.638166\pi$
$332$	0	0
$333$	3.60555	0.197583
$334$	0	0
$335$	0	0
$336$	0	0
$337$	26.6333	1.45081	0.725404	−	0.688323i	$-0.241653\pi$
$337$	26.6333	1.45081	0.725404	+	0.688323i	$0.241653\pi$
$338$	0	0
$339$	−17.7250	−0.962689
$340$	0	0
$341$	−21.9083	−1.18640
$342$	0	0
$343$	19.4222	1.04870
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−9.63331	−0.517143	−0.258572	−	0.965992i	$-0.583252\pi$
$347$	−9.63331	−0.517143	−0.258572	+	0.965992i	$0.583252\pi$
$348$	0	0
$349$	−5.18335	−0.277458	−0.138729	−	0.990330i	$-0.544302\pi$
$349$	−5.18335	−0.277458	−0.138729	+	0.990330i	$0.544302\pi$
$350$	0	0
$351$	−0.302776	−0.0161610
$352$	0	0
$353$	−21.6972	−1.15483	−0.577413	−	0.816452i	$-0.695938\pi$
$353$	−21.6972	−1.15483	−0.577413	+	0.816452i	$0.695938\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	12.9083	0.683181
$358$	0	0
$359$	11.7889	0.622194	0.311097	−	0.950378i	$-0.399304\pi$
$359$	11.7889	0.622194	0.311097	+	0.950378i	$0.399304\pi$
$360$	0	0
$361$	24.6333	1.29649
$362$	0	0
$363$	−5.69722	−0.299027
$364$	0	0
$365$	0	0
$366$	0	0
$367$	1.78890	0.0933797	0.0466898	−	0.998909i	$-0.485133\pi$
$367$	1.78890	0.0933797	0.0466898	+	0.998909i	$0.485133\pi$
$368$	0	0
$369$	−1.60555	−0.0835817
$370$	0	0
$371$	55.5416	2.88358
$372$	0	0
$373$	24.3944	1.26310	0.631548	−	0.775337i	$-0.282420\pi$
$373$	24.3944	1.26310	0.631548	+	0.775337i	$0.282420\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−0.422205	−0.0217447
$378$	0	0
$379$	17.2111	0.884075	0.442037	−	0.896997i	$-0.354256\pi$
$379$	17.2111	0.884075	0.442037	+	0.896997i	$0.354256\pi$
$380$	0	0
$381$	2.00000	0.102463
$382$	0	0
$383$	−19.3305	−0.987744	−0.493872	−	0.869535i	$-0.664419\pi$
$383$	−19.3305	−0.987744	−0.493872	+	0.869535i	$0.664419\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1.00000	−0.0508329
$388$	0	0
$389$	29.5139	1.49641	0.748207	−	0.663466i	$-0.230915\pi$
$389$	29.5139	1.49641	0.748207	+	0.663466i	$0.230915\pi$
$390$	0	0
$391$	−3.00000	−0.151717
$392$	0	0
$393$	−0.211103	−0.0106487
$394$	0	0
$395$	0	0
$396$	0	0
$397$	10.7889	0.541479	0.270740	−	0.962653i	$-0.412732\pi$
$397$	10.7889	0.541479	0.270740	+	0.962653i	$0.412732\pi$
$398$	0	0
$399$	28.4222	1.42289
$400$	0	0
$401$	−21.6972	−1.08351	−0.541754	−	0.840537i	$-0.682240\pi$
$401$	−21.6972	−1.08351	−0.541754	+	0.840537i	$0.682240\pi$
$402$	0	0
$403$	2.88057	0.143491
$404$	0	0
$405$	0	0
$406$	0	0
$407$	8.30278	0.411553
$408$	0	0
$409$	−23.8167	−1.17766	−0.588829	−	0.808258i	$-0.700411\pi$
$409$	−23.8167	−1.17766	−0.588829	+	0.808258i	$0.700411\pi$
$410$	0	0
$411$	−3.21110	−0.158392
$412$	0	0
$413$	49.5416	2.43778
$414$	0	0
$415$	0	0
$416$	0	0
$417$	13.5139	0.661777
$418$	0	0
$419$	14.0917	0.688423	0.344212	−	0.938892i	$-0.388146\pi$
$419$	14.0917	0.688423	0.344212	+	0.938892i	$0.388146\pi$
$420$	0	0
$421$	−15.3028	−0.745812	−0.372906	−	0.927869i	$-0.621639\pi$
$421$	−15.3028	−0.745812	−0.372906	+	0.927869i	$0.621639\pi$
$422$	0	0
$423$	−3.00000	−0.145865
$424$	0	0
$425$	0	0
$426$	0	0
$427$	12.5139	0.605589
$428$	0	0
$429$	−0.697224	−0.0336623
$430$	0	0
$431$	−33.4222	−1.60989	−0.804945	−	0.593349i	$-0.797806\pi$
$431$	−33.4222	−1.60989	−0.804945	+	0.593349i	$0.797806\pi$
$432$	0	0
$433$	3.60555	0.173272	0.0866359	−	0.996240i	$-0.472388\pi$
$433$	3.60555	0.173272	0.0866359	+	0.996240i	$0.472388\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−6.60555	−0.315986
$438$	0	0
$439$	11.4222	0.545152	0.272576	−	0.962134i	$-0.412124\pi$
$439$	11.4222	0.545152	0.272576	+	0.962134i	$0.412124\pi$
$440$	0	0
$441$	11.5139	0.548280
$442$	0	0
$443$	−27.2111	−1.29284	−0.646419	−	0.762982i	$-0.723734\pi$
$443$	−27.2111	−1.29284	−0.646419	+	0.762982i	$0.723734\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−12.2111	−0.577565
$448$	0	0
$449$	22.5416	1.06381	0.531903	−	0.846805i	$-0.321477\pi$
$449$	22.5416	1.06381	0.531903	+	0.846805i	$0.321477\pi$
$450$	0	0
$451$	−3.69722	−0.174095
$452$	0	0
$453$	0.816654	0.0383697
$454$	0	0
$455$	0	0
$456$	0	0
$457$	32.4222	1.51665	0.758323	−	0.651879i	$-0.226019\pi$
$457$	32.4222	1.51665	0.758323	+	0.651879i	$0.226019\pi$
$458$	0	0
$459$	3.00000	0.140028
$460$	0	0
$461$	−20.7250	−0.965259	−0.482629	−	0.875825i	$-0.660318\pi$
$461$	−20.7250	−0.965259	−0.482629	+	0.875825i	$0.660318\pi$
$462$	0	0
$463$	9.33053	0.433627	0.216813	−	0.976213i	$-0.430434\pi$
$463$	9.33053	0.433627	0.216813	+	0.976213i	$0.430434\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	23.4500	1.08513	0.542567	−	0.840012i	$-0.317452\pi$
$467$	23.4500	1.08513	0.542567	+	0.840012i	$0.317452\pi$
$468$	0	0
$469$	−46.0278	−2.12536
$470$	0	0
$471$	−4.00000	−0.184310
$472$	0	0
$473$	−2.30278	−0.105882
$474$	0	0
$475$	0	0
$476$	0	0
$477$	12.9083	0.591032
$478$	0	0
$479$	22.6056	1.03287	0.516437	−	0.856325i	$-0.327258\pi$
$479$	22.6056	1.03287	0.516437	+	0.856325i	$0.327258\pi$
$480$	0	0
$481$	−1.09167	−0.0497760
$482$	0	0
$483$	−4.30278	−0.195783
$484$	0	0
$485$	0	0
$486$	0	0
$487$	30.6056	1.38687	0.693435	−	0.720519i	$-0.256096\pi$
$487$	30.6056	1.38687	0.693435	+	0.720519i	$0.256096\pi$
$488$	0	0
$489$	18.8167	0.850918
$490$	0	0
$491$	−26.3028	−1.18703	−0.593514	−	0.804824i	$-0.702260\pi$
$491$	−26.3028	−1.18703	−0.593514	+	0.804824i	$0.702260\pi$
$492$	0	0
$493$	4.18335	0.188408
$494$	0	0
$495$	0	0
$496$	0	0
$497$	43.5416	1.95311
$498$	0	0
$499$	−7.00000	−0.313363	−0.156682	−	0.987649i	$-0.550080\pi$
$499$	−7.00000	−0.313363	−0.156682	+	0.987649i	$0.550080\pi$
$500$	0	0
$501$	2.51388	0.112312
$502$	0	0
$503$	−4.39445	−0.195939	−0.0979694	−	0.995189i	$-0.531235\pi$
$503$	−4.39445	−0.195939	−0.0979694	+	0.995189i	$0.531235\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−12.9083	−0.573279
$508$	0	0
$509$	−23.9361	−1.06095	−0.530474	−	0.847701i	$-0.677986\pi$
$509$	−23.9361	−1.06095	−0.530474	+	0.847701i	$0.677986\pi$
$510$	0	0
$511$	−43.9361	−1.94362
$512$	0	0
$513$	6.60555	0.291642
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−6.90833	−0.303828
$518$	0	0
$519$	23.5139	1.03214
$520$	0	0
$521$	−10.6056	−0.464638	−0.232319	−	0.972640i	$-0.574631\pi$
$521$	−10.6056	−0.464638	−0.232319	+	0.972640i	$0.574631\pi$
$522$	0	0
$523$	16.0278	0.700845	0.350422	−	0.936592i	$-0.386038\pi$
$523$	16.0278	0.700845	0.350422	+	0.936592i	$0.386038\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−28.5416	−1.24329
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	11.5139	0.499660
$532$	0	0
$533$	0.486122	0.0210563
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−19.3305	−0.834174
$538$	0	0
$539$	26.5139	1.14203
$540$	0	0
$541$	43.4500	1.86806	0.934030	−	0.357195i	$-0.116267\pi$
$541$	43.4500	1.86806	0.934030	+	0.357195i	$0.116267\pi$
$542$	0	0
$543$	−21.0278	−0.902387
$544$	0	0
$545$	0	0
$546$	0	0
$547$	5.42221	0.231837	0.115918	−	0.993259i	$-0.463019\pi$
$547$	5.42221	0.231837	0.115918	+	0.993259i	$0.463019\pi$
$548$	0	0
$549$	2.90833	0.124124
$550$	0	0
$551$	9.21110	0.392406
$552$	0	0
$553$	−18.1194	−0.770517
$554$	0	0
$555$	0	0
$556$	0	0
$557$	10.6056	0.449372	0.224686	−	0.974431i	$-0.427864\pi$
$557$	10.6056	0.449372	0.224686	+	0.974431i	$0.427864\pi$
$558$	0	0
$559$	0.302776	0.0128060
$560$	0	0
$561$	6.90833	0.291670
$562$	0	0
$563$	−27.9083	−1.17620	−0.588098	−	0.808790i	$-0.700123\pi$
$563$	−27.9083	−1.17620	−0.588098	+	0.808790i	$0.700123\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	4.30278	0.180700
$568$	0	0
$569$	15.0000	0.628833	0.314416	−	0.949285i	$-0.398191\pi$
$569$	15.0000	0.628833	0.314416	+	0.949285i	$0.398191\pi$
$570$	0	0
$571$	−13.0000	−0.544033	−0.272017	−	0.962293i	$-0.587691\pi$
$571$	−13.0000	−0.544033	−0.272017	+	0.962293i	$0.587691\pi$
$572$	0	0
$573$	16.8167	0.702526
$574$	0	0
$575$	0	0
$576$	0	0
$577$	14.9083	0.620642	0.310321	−	0.950632i	$-0.399564\pi$
$577$	14.9083	0.620642	0.310321	+	0.950632i	$0.399564\pi$
$578$	0	0
$579$	11.0000	0.457144
$580$	0	0
$581$	58.5416	2.42872
$582$	0	0
$583$	29.7250	1.23108
$584$	0	0
$585$	0	0
$586$	0	0
$587$	5.30278	0.218869	0.109434	−	0.993994i	$-0.465096\pi$
$587$	5.30278	0.218869	0.109434	+	0.993994i	$0.465096\pi$
$588$	0	0
$589$	−62.8444	−2.58946
$590$	0	0
$591$	19.8167	0.815148
$592$	0	0
$593$	−3.00000	−0.123195	−0.0615976	−	0.998101i	$-0.519620\pi$
$593$	−3.00000	−0.123195	−0.0615976	+	0.998101i	$0.519620\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	6.18335	0.253068
$598$	0	0
$599$	47.2389	1.93013	0.965064	−	0.262015i	$-0.0843871\pi$
$599$	47.2389	1.93013	0.965064	+	0.262015i	$0.0843871\pi$
$600$	0	0
$601$	4.78890	0.195343	0.0976716	−	0.995219i	$-0.468861\pi$
$601$	4.78890	0.195343	0.0976716	+	0.995219i	$0.468861\pi$
$602$	0	0
$603$	−10.6972	−0.435625
$604$	0	0
$605$	0	0
$606$	0	0
$607$	11.9083	0.483344	0.241672	−	0.970358i	$-0.422304\pi$
$607$	11.9083	0.483344	0.241672	+	0.970358i	$0.422304\pi$
$608$	0	0
$609$	6.00000	0.243132
$610$	0	0
$611$	0.908327	0.0367470
$612$	0	0
$613$	−16.1472	−0.652179	−0.326089	−	0.945339i	$-0.605731\pi$
$613$	−16.1472	−0.652179	−0.326089	+	0.945339i	$0.605731\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	38.7250	1.55901	0.779505	−	0.626397i	$-0.215471\pi$
$617$	38.7250	1.55901	0.779505	+	0.626397i	$0.215471\pi$
$618$	0	0
$619$	−6.72498	−0.270300	−0.135150	−	0.990825i	$-0.543152\pi$
$619$	−6.72498	−0.270300	−0.135150	+	0.990825i	$0.543152\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	−20.7250	−0.830329
$624$	0	0
$625$	0	0
$626$	0	0
$627$	15.2111	0.607473
$628$	0	0
$629$	10.8167	0.431288
$630$	0	0
$631$	−9.30278	−0.370338	−0.185169	−	0.982707i	$-0.559283\pi$
$631$	−9.30278	−0.370338	−0.185169	+	0.982707i	$0.559283\pi$
$632$	0	0
$633$	−11.3944	−0.452889
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−3.48612	−0.138125
$638$	0	0
$639$	10.1194	0.400318
$640$	0	0
$641$	49.3305	1.94844	0.974219	−	0.225603i	$-0.0724351\pi$
$641$	49.3305	1.94844	0.974219	+	0.225603i	$0.0724351\pi$
$642$	0	0
$643$	−15.0917	−0.595157	−0.297579	−	0.954697i	$-0.596179\pi$
$643$	−15.0917	−0.595157	−0.297579	+	0.954697i	$0.596179\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−18.8444	−0.740850	−0.370425	−	0.928862i	$-0.620788\pi$
$647$	−18.8444	−0.740850	−0.370425	+	0.928862i	$0.620788\pi$
$648$	0	0
$649$	26.5139	1.04076
$650$	0	0
$651$	−40.9361	−1.60441
$652$	0	0
$653$	−45.8444	−1.79403	−0.897015	−	0.442000i	$-0.854269\pi$
$653$	−45.8444	−1.79403	−0.897015	+	0.442000i	$0.854269\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−10.2111	−0.398373
$658$	0	0
$659$	31.1194	1.21224	0.606120	−	0.795373i	$-0.292725\pi$
$659$	31.1194	1.21224	0.606120	+	0.795373i	$0.292725\pi$
$660$	0	0
$661$	3.88057	0.150937	0.0754684	−	0.997148i	$-0.475955\pi$
$661$	3.88057	0.150937	0.0754684	+	0.997148i	$0.475955\pi$
$662$	0	0
$663$	−0.908327	−0.0352765
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1.39445	−0.0539933
$668$	0	0
$669$	14.6972	0.568228
$670$	0	0
$671$	6.69722	0.258543
$672$	0	0
$673$	8.00000	0.308377	0.154189	−	0.988041i	$-0.450724\pi$
$673$	8.00000	0.308377	0.154189	+	0.988041i	$0.450724\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−33.4861	−1.28698	−0.643488	−	0.765456i	$-0.722513\pi$
$677$	−33.4861	−1.28698	−0.643488	+	0.765456i	$0.722513\pi$
$678$	0	0
$679$	−80.5694	−3.09197
$680$	0	0
$681$	12.2111	0.467930
$682$	0	0
$683$	−10.8167	−0.413888	−0.206944	−	0.978353i	$-0.566352\pi$
$683$	−10.8167	−0.413888	−0.206944	+	0.978353i	$0.566352\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−14.8167	−0.565291
$688$	0	0
$689$	−3.90833	−0.148895
$690$	0	0
$691$	−37.0000	−1.40755	−0.703773	−	0.710425i	$-0.748503\pi$
$691$	−37.0000	−1.40755	−0.703773	+	0.710425i	$0.748503\pi$
$692$	0	0
$693$	9.90833	0.376386
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−4.81665	−0.182444
$698$	0	0
$699$	11.0917	0.419526
$700$	0	0
$701$	14.0278	0.529821	0.264911	−	0.964273i	$-0.414658\pi$
$701$	14.0278	0.529821	0.264911	+	0.964273i	$0.414658\pi$
$702$	0	0
$703$	23.8167	0.898262
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−42.6333	−1.60339
$708$	0	0
$709$	−29.8167	−1.11979	−0.559894	−	0.828564i	$-0.689158\pi$
$709$	−29.8167	−1.11979	−0.559894	+	0.828564i	$0.689158\pi$
$710$	0	0
$711$	−4.21110	−0.157929
$712$	0	0
$713$	9.51388	0.356298
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−4.60555	−0.171997
$718$	0	0
$719$	−1.60555	−0.0598770	−0.0299385	−	0.999552i	$-0.509531\pi$
$719$	−1.60555	−0.0598770	−0.0299385	+	0.999552i	$0.509531\pi$
$720$	0	0
$721$	14.6056	0.543939
$722$	0	0
$723$	−6.51388	−0.242254
$724$	0	0
$725$	0	0
$726$	0	0
$727$	20.4222	0.757418	0.378709	−	0.925516i	$-0.376368\pi$
$727$	20.4222	0.757418	0.378709	+	0.925516i	$0.376368\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−3.00000	−0.110959
$732$	0	0
$733$	22.0278	0.813614	0.406807	−	0.913514i	$-0.366642\pi$
$733$	22.0278	0.813614	0.406807	+	0.913514i	$0.366642\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−24.6333	−0.907380
$738$	0	0
$739$	−51.5139	−1.89497	−0.947484	−	0.319802i	$-0.896384\pi$
$739$	−51.5139	−1.89497	−0.947484	+	0.319802i	$0.896384\pi$
$740$	0	0
$741$	−2.00000	−0.0734718
$742$	0	0
$743$	15.9083	0.583620	0.291810	−	0.956476i	$-0.405743\pi$
$743$	15.9083	0.583620	0.291810	+	0.956476i	$0.405743\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	13.6056	0.497801
$748$	0	0
$749$	−46.5416	−1.70059
$750$	0	0
$751$	−42.7250	−1.55906	−0.779528	−	0.626367i	$-0.784541\pi$
$751$	−42.7250	−1.55906	−0.779528	+	0.626367i	$0.784541\pi$
$752$	0	0
$753$	−22.1194	−0.806077
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−13.6972	−0.497834	−0.248917	−	0.968525i	$-0.580075\pi$
$757$	−13.6972	−0.497834	−0.248917	+	0.968525i	$0.580075\pi$
$758$	0	0
$759$	−2.30278	−0.0835855
$760$	0	0
$761$	−27.8444	−1.00936	−0.504680	−	0.863307i	$-0.668389\pi$
$761$	−27.8444	−1.00936	−0.504680	+	0.863307i	$0.668389\pi$
$762$	0	0
$763$	−30.1194	−1.09040
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−3.48612	−0.125877
$768$	0	0
$769$	27.9638	1.00840	0.504201	−	0.863586i	$-0.331787\pi$
$769$	27.9638	1.00840	0.504201	+	0.863586i	$0.331787\pi$
$770$	0	0
$771$	31.1194	1.12074
$772$	0	0
$773$	12.0000	0.431610	0.215805	−	0.976436i	$-0.430762\pi$
$773$	12.0000	0.431610	0.215805	+	0.976436i	$0.430762\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	15.5139	0.556557
$778$	0	0
$779$	−10.6056	−0.379983
$780$	0	0
$781$	23.3028	0.833839
$782$	0	0
$783$	1.39445	0.0498335
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−42.0278	−1.49813	−0.749064	−	0.662498i	$-0.769496\pi$
$787$	−42.0278	−1.49813	−0.749064	+	0.662498i	$0.769496\pi$
$788$	0	0
$789$	3.90833	0.139140
$790$	0	0
$791$	−76.2666	−2.71173
$792$	0	0
$793$	−0.880571	−0.0312700
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−33.8444	−1.19883	−0.599415	−	0.800438i	$-0.704600\pi$
$797$	−33.8444	−1.19883	−0.599415	+	0.800438i	$0.704600\pi$
$798$	0	0
$799$	−9.00000	−0.318397
$800$	0	0
$801$	−4.81665	−0.170188
$802$	0	0
$803$	−23.5139	−0.829787
$804$	0	0
$805$	0	0
$806$	0	0
$807$	26.2389	0.923652
$808$	0	0
$809$	−26.0278	−0.915087	−0.457544	−	0.889187i	$-0.651271\pi$
$809$	−26.0278	−0.915087	−0.457544	+	0.889187i	$0.651271\pi$
$810$	0	0
$811$	−16.2111	−0.569249	−0.284624	−	0.958639i	$-0.591869\pi$
$811$	−16.2111	−0.569249	−0.284624	+	0.958639i	$0.591869\pi$
$812$	0	0
$813$	−3.78890	−0.132882
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−6.60555	−0.231099
$818$	0	0
$819$	−1.30278	−0.0455227
$820$	0	0
$821$	−12.6333	−0.440905	−0.220453	−	0.975398i	$-0.570753\pi$
$821$	−12.6333	−0.440905	−0.220453	+	0.975398i	$0.570753\pi$
$822$	0	0
$823$	11.2750	0.393022	0.196511	−	0.980502i	$-0.437039\pi$
$823$	11.2750	0.393022	0.196511	+	0.980502i	$0.437039\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	34.2666	1.19157	0.595783	−	0.803145i	$-0.296842\pi$
$827$	34.2666	1.19157	0.595783	+	0.803145i	$0.296842\pi$
$828$	0	0
$829$	4.23886	0.147222	0.0736108	−	0.997287i	$-0.476548\pi$
$829$	4.23886	0.147222	0.0736108	+	0.997287i	$0.476548\pi$
$830$	0	0
$831$	−0.0916731	−0.00318010
$832$	0	0
$833$	34.5416	1.19680
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−9.51388	−0.328848
$838$	0	0
$839$	47.7250	1.64765	0.823825	−	0.566845i	$-0.191836\pi$
$839$	47.7250	1.64765	0.823825	+	0.566845i	$0.191836\pi$
$840$	0	0
$841$	−27.0555	−0.932949
$842$	0	0
$843$	−7.18335	−0.247408
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−24.5139	−0.842307
$848$	0	0
$849$	−10.6333	−0.364934
$850$	0	0
$851$	−3.60555	−0.123597
$852$	0	0
$853$	7.57779	0.259459	0.129729	−	0.991549i	$-0.458589\pi$
$853$	7.57779	0.259459	0.129729	+	0.991549i	$0.458589\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−0.908327	−0.0310279	−0.0155139	−	0.999880i	$-0.504938\pi$
$857$	−0.908327	−0.0310279	−0.0155139	+	0.999880i	$0.504938\pi$
$858$	0	0
$859$	35.3583	1.20641	0.603205	−	0.797586i	$-0.293890\pi$
$859$	35.3583	1.20641	0.603205	+	0.797586i	$0.293890\pi$
$860$	0	0
$861$	−6.90833	−0.235435
$862$	0	0
$863$	15.8444	0.539350	0.269675	−	0.962951i	$-0.413084\pi$
$863$	15.8444	0.539350	0.269675	+	0.962951i	$0.413084\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−8.00000	−0.271694
$868$	0	0
$869$	−9.69722	−0.328956
$870$	0	0
$871$	3.23886	0.109745
$872$	0	0
$873$	−18.7250	−0.633745
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−14.1194	−0.476779	−0.238390	−	0.971170i	$-0.576620\pi$
$877$	−14.1194	−0.476779	−0.238390	+	0.971170i	$0.576620\pi$
$878$	0	0
$879$	−1.60555	−0.0541539
$880$	0	0
$881$	32.9361	1.10964	0.554822	−	0.831969i	$-0.312786\pi$
$881$	32.9361	1.10964	0.554822	+	0.831969i	$0.312786\pi$
$882$	0	0
$883$	41.6333	1.40107	0.700536	−	0.713617i	$-0.252944\pi$
$883$	41.6333	1.40107	0.700536	+	0.713617i	$0.252944\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−26.7250	−0.897337	−0.448669	−	0.893698i	$-0.648102\pi$
$887$	−26.7250	−0.897337	−0.448669	+	0.893698i	$0.648102\pi$
$888$	0	0
$889$	8.60555	0.288621
$890$	0	0
$891$	2.30278	0.0771459
$892$	0	0
$893$	−19.8167	−0.663139
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0.302776	0.0101094
$898$	0	0
$899$	−13.2666	−0.442466
$900$	0	0
$901$	38.7250	1.29012
$902$	0	0
$903$	−4.30278	−0.143187
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−41.8167	−1.38850	−0.694250	−	0.719734i	$-0.744264\pi$
$907$	−41.8167	−1.38850	−0.694250	+	0.719734i	$0.744264\pi$
$908$	0	0
$909$	−9.90833	−0.328638
$910$	0	0
$911$	1.60555	0.0531943	0.0265971	−	0.999646i	$-0.491533\pi$
$911$	1.60555	0.0531943	0.0265971	+	0.999646i	$0.491533\pi$
$912$	0	0
$913$	31.3305	1.03689
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−0.908327	−0.0299956
$918$	0	0
$919$	−7.76114	−0.256016	−0.128008	−	0.991773i	$-0.540858\pi$
$919$	−7.76114	−0.256016	−0.128008	+	0.991773i	$0.540858\pi$
$920$	0	0
$921$	−31.2111	−1.02844
$922$	0	0
$923$	−3.06392	−0.100850
$924$	0	0
$925$	0	0
$926$	0	0
$927$	3.39445	0.111488
$928$	0	0
$929$	−3.27502	−0.107450	−0.0537249	−	0.998556i	$-0.517109\pi$
$929$	−3.27502	−0.107450	−0.0537249	+	0.998556i	$0.517109\pi$
$930$	0	0
$931$	76.0555	2.49262
$932$	0	0
$933$	−0.211103	−0.00691119
$934$	0	0
$935$	0	0
$936$	0	0
$937$	53.0833	1.73415	0.867077	−	0.498173i	$-0.165996\pi$
$937$	53.0833	1.73415	0.867077	+	0.498173i	$0.165996\pi$
$938$	0	0
$939$	7.78890	0.254181
$940$	0	0
$941$	−20.9361	−0.682497	−0.341248	−	0.939973i	$-0.610850\pi$
$941$	−20.9361	−0.682497	−0.341248	+	0.939973i	$0.610850\pi$
$942$	0	0
$943$	1.60555	0.0522839
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−10.8806	−0.353571	−0.176786	−	0.984249i	$-0.556570\pi$
$947$	−10.8806	−0.353571	−0.176786	+	0.984249i	$0.556570\pi$
$948$	0	0
$949$	3.09167	0.100360
$950$	0	0
$951$	−0.908327	−0.0294545
$952$	0	0
$953$	13.6056	0.440727	0.220364	−	0.975418i	$-0.429276\pi$
$953$	13.6056	0.440727	0.220364	+	0.975418i	$0.429276\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	3.21110	0.103800
$958$	0	0
$959$	−13.8167	−0.446163
$960$	0	0
$961$	59.5139	1.91980
$962$	0	0
$963$	−10.8167	−0.348562
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−27.7889	−0.893631	−0.446815	−	0.894626i	$-0.647442\pi$
$967$	−27.7889	−0.893631	−0.446815	+	0.894626i	$0.647442\pi$
$968$	0	0
$969$	19.8167	0.636603
$970$	0	0
$971$	−53.8722	−1.72884	−0.864420	−	0.502770i	$-0.832314\pi$
$971$	−53.8722	−1.72884	−0.864420	+	0.502770i	$0.832314\pi$
$972$	0	0
$973$	58.1472	1.86411
$974$	0	0
$975$	0	0
$976$	0	0
$977$	22.1833	0.709708	0.354854	−	0.934922i	$-0.384531\pi$
$977$	22.1833	0.709708	0.354854	+	0.934922i	$0.384531\pi$
$978$	0	0
$979$	−11.0917	−0.354491
$980$	0	0
$981$	−7.00000	−0.223493
$982$	0	0
$983$	36.9083	1.17719	0.588596	−	0.808427i	$-0.299681\pi$
$983$	36.9083	1.17719	0.588596	+	0.808427i	$0.299681\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−12.9083	−0.410877
$988$	0	0
$989$	1.00000	0.0317982
$990$	0	0
$991$	−20.6056	−0.654557	−0.327278	−	0.944928i	$-0.606132\pi$
$991$	−20.6056	−0.654557	−0.327278	+	0.944928i	$0.606132\pi$
$992$	0	0
$993$	−15.3028	−0.485619
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−48.7250	−1.54314	−0.771568	−	0.636147i	$-0.780527\pi$
$997$	−48.7250	−1.54314	−0.771568	+	0.636147i	$0.780527\pi$
$998$	0	0
$999$	3.60555	0.114075

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6900.2.a.v.1.2	yes	2
5.2	odd	4		6900.2.f.m.6349.2		4
5.3	odd	4		6900.2.f.m.6349.3		4
5.4	even	2		6900.2.a.k.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6900.2.a.k.1.1	✓	2	5.4	even	2
6900.2.a.v.1.2	yes	2	1.1	even	1	trivial
6900.2.f.m.6349.2		4	5.2	odd	4
6900.2.f.m.6349.3		4	5.3	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6900.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +4.30278 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +4.30278 q^{7} +1.00000 q^{9} +2.30278 q^{11} -0.302776 q^{13} +3.00000 q^{17} +6.60555 q^{19} +4.30278 q^{21} -1.00000 q^{23} +1.00000 q^{27} +1.39445 q^{29} -9.51388 q^{31} +2.30278 q^{33} +3.60555 q^{37} -0.302776 q^{39} -1.60555 q^{41} -1.00000 q^{43} -3.00000 q^{47} +11.5139 q^{49} +3.00000 q^{51} +12.9083 q^{53} +6.60555 q^{57} +11.5139 q^{59} +2.90833 q^{61} +4.30278 q^{63} -10.6972 q^{67} -1.00000 q^{69} +10.1194 q^{71} -10.2111 q^{73} +9.90833 q^{77} -4.21110 q^{79} +1.00000 q^{81} +13.6056 q^{83} +1.39445 q^{87} -4.81665 q^{89} -1.30278 q^{91} -9.51388 q^{93} -18.7250 q^{97} +2.30278 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 5 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 5 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} + 5 q^{7} + 2 q^{9} + q^{11} + 3 q^{13} + 6 q^{17} + 6 q^{19} + 5 q^{21} - 2 q^{23} + 2 q^{27} + 10 q^{29} - q^{31} + q^{33} + 3 q^{39} + 4 q^{41} - 2 q^{43} - 6 q^{47} + 5 q^{49} + 6 q^{51} + 15 q^{53} + 6 q^{57} + 5 q^{59} - 5 q^{61} + 5 q^{63} - 25 q^{67} - 2 q^{69} - 5 q^{71} - 6 q^{73} + 9 q^{77} + 6 q^{79} + 2 q^{81} + 20 q^{83} + 10 q^{87} + 12 q^{89} + q^{91} - q^{93} - 5 q^{97} + q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 5 * q^7 + 2 * q^9 + q^11 + 3 * q^13 + 6 * q^17 + 6 * q^19 + 5 * q^21 - 2 * q^23 + 2 * q^27 + 10 * q^29 - q^31 + q^33 + 3 * q^39 + 4 * q^41 - 2 * q^43 - 6 * q^47 + 5 * q^49 + 6 * q^51 + 15 * q^53 + 6 * q^57 + 5 * q^59 - 5 * q^61 + 5 * q^63 - 25 * q^67 - 2 * q^69 - 5 * q^71 - 6 * q^73 + 9 * q^77 + 6 * q^79 + 2 * q^81 + 20 * q^83 + 10 * q^87 + 12 * q^89 + q^91 - q^93 - 5 * q^97 + q^99

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