LMFDB - Embedded newform 945.2.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [945,2,Mod(1,945)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(945, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("945.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 945 = 3^{3} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	945.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:	$7.54586299101$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.30278$ of defining polynomial
Character	$\chi$	$=$	945.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.30278 q^{2} -0.302776 q^{4} +1.00000 q^{5} +1.00000 q^{7} +3.00000 q^{8} +O(q^{10})$	\(q-1.30278 q^{2} -0.302776 q^{4} +1.00000 q^{5} +1.00000 q^{7} +3.00000 q^{8} -1.30278 q^{10} +3.00000 q^{11} -2.30278 q^{13} -1.30278 q^{14} -3.30278 q^{16} +1.30278 q^{17} +0.302776 q^{19} -0.302776 q^{20} -3.90833 q^{22} +4.30278 q^{23} +1.00000 q^{25} +3.00000 q^{26} -0.302776 q^{28} +1.69722 q^{29} -9.60555 q^{31} -1.69722 q^{32} -1.69722 q^{34} +1.00000 q^{35} -3.60555 q^{37} -0.394449 q^{38} +3.00000 q^{40} +3.90833 q^{41} +10.2111 q^{43} -0.908327 q^{44} -5.60555 q^{46} +0.394449 q^{47} +1.00000 q^{49} -1.30278 q^{50} +0.697224 q^{52} +12.5139 q^{53} +3.00000 q^{55} +3.00000 q^{56} -2.21110 q^{58} -8.21110 q^{59} +14.1194 q^{61} +12.5139 q^{62} +8.81665 q^{64} -2.30278 q^{65} -4.51388 q^{67} -0.394449 q^{68} -1.30278 q^{70} +12.9083 q^{71} +4.21110 q^{73} +4.69722 q^{74} -0.0916731 q^{76} +3.00000 q^{77} -6.09167 q^{79} -3.30278 q^{80} -5.09167 q^{82} +2.21110 q^{83} +1.30278 q^{85} -13.3028 q^{86} +9.00000 q^{88} -10.8167 q^{89} -2.30278 q^{91} -1.30278 q^{92} -0.513878 q^{94} +0.302776 q^{95} +8.51388 q^{97} -1.30278 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} + 3 q^{4} + 2 q^{5} + 2 q^{7} + 6 q^{8}+O(q^{10}) $ 2 * q + q^2 + 3 * q^4 + 2 * q^5 + 2 * q^7 + 6 * q^8	$ 2 q + q^{2} + 3 q^{4} + 2 q^{5} + 2 q^{7} + 6 q^{8} + q^{10} + 6 q^{11} - q^{13} + q^{14} - 3 q^{16} - q^{17} - 3 q^{19} + 3 q^{20} + 3 q^{22} + 5 q^{23} + 2 q^{25} + 6 q^{26} + 3 q^{28} + 7 q^{29} - 12 q^{31} - 7 q^{32} - 7 q^{34} + 2 q^{35} - 8 q^{38} + 6 q^{40} - 3 q^{41} + 6 q^{43} + 9 q^{44} - 4 q^{46} + 8 q^{47} + 2 q^{49} + q^{50} + 5 q^{52} + 7 q^{53} + 6 q^{55} + 6 q^{56} + 10 q^{58} - 2 q^{59} + 3 q^{61} + 7 q^{62} - 4 q^{64} - q^{65} + 9 q^{67} - 8 q^{68} + q^{70} + 15 q^{71} - 6 q^{73} + 13 q^{74} - 11 q^{76} + 6 q^{77} - 23 q^{79} - 3 q^{80} - 21 q^{82} - 10 q^{83} - q^{85} - 23 q^{86} + 18 q^{88} - q^{91} + q^{92} + 17 q^{94} - 3 q^{95} - q^{97} + q^{98}+O(q^{100}) $ 2 * q + q^2 + 3 * q^4 + 2 * q^5 + 2 * q^7 + 6 * q^8 + q^10 + 6 * q^11 - q^13 + q^14 - 3 * q^16 - q^17 - 3 * q^19 + 3 * q^20 + 3 * q^22 + 5 * q^23 + 2 * q^25 + 6 * q^26 + 3 * q^28 + 7 * q^29 - 12 * q^31 - 7 * q^32 - 7 * q^34 + 2 * q^35 - 8 * q^38 + 6 * q^40 - 3 * q^41 + 6 * q^43 + 9 * q^44 - 4 * q^46 + 8 * q^47 + 2 * q^49 + q^50 + 5 * q^52 + 7 * q^53 + 6 * q^55 + 6 * q^56 + 10 * q^58 - 2 * q^59 + 3 * q^61 + 7 * q^62 - 4 * q^64 - q^65 + 9 * q^67 - 8 * q^68 + q^70 + 15 * q^71 - 6 * q^73 + 13 * q^74 - 11 * q^76 + 6 * q^77 - 23 * q^79 - 3 * q^80 - 21 * q^82 - 10 * q^83 - q^85 - 23 * q^86 + 18 * q^88 - q^91 + q^92 + 17 * q^94 - 3 * q^95 - q^97 + q^98

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 $

$2$	−1.30278	−0.921201	−0.460601	−	0.887607i	$-0.652366\pi$
$2$	−1.30278	−0.921201	−0.460601	+	0.887607i	$0.652366\pi$
$3$	0	0
$3$	0	0
$4$	−0.302776	−0.151388
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	3.00000	1.06066
$9$	0	0
$10$	−1.30278	−0.411974
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	0	0
$13$	−2.30278	−0.638675	−0.319338	−	0.947641i	$-0.603460\pi$
$13$	−2.30278	−0.638675	−0.319338	+	0.947641i	$0.603460\pi$
$14$	−1.30278	−0.348181
$15$	0	0
$16$	−3.30278	−0.825694
$17$	1.30278	0.315970	0.157985	−	0.987442i	$-0.449500\pi$
$17$	1.30278	0.315970	0.157985	+	0.987442i	$0.449500\pi$
$18$	0	0
$19$	0.302776	0.0694615	0.0347307	−	0.999397i	$-0.488943\pi$
$19$	0.302776	0.0694615	0.0347307	+	0.999397i	$0.488943\pi$
$20$	−0.302776	−0.0677027
$21$	0	0
$22$	−3.90833	−0.833258
$23$	4.30278	0.897191	0.448595	−	0.893735i	$-0.351924\pi$
$23$	4.30278	0.897191	0.448595	+	0.893735i	$0.351924\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	3.00000	0.588348
$27$	0	0
$28$	−0.302776	−0.0572192
$29$	1.69722	0.315167	0.157583	−	0.987506i	$-0.449630\pi$
$29$	1.69722	0.315167	0.157583	+	0.987506i	$0.449630\pi$
$30$	0	0
$31$	−9.60555	−1.72521	−0.862604	−	0.505880i	$-0.831168\pi$
$31$	−9.60555	−1.72521	−0.862604	+	0.505880i	$0.831168\pi$
$32$	−1.69722	−0.300030
$33$	0	0
$34$	−1.69722	−0.291072
$35$	1.00000	0.169031
$36$	0	0
$37$	−3.60555	−0.592749	−0.296374	−	0.955072i	$-0.595778\pi$
$37$	−3.60555	−0.592749	−0.296374	+	0.955072i	$0.595778\pi$
$38$	−0.394449	−0.0639880
$39$	0	0
$40$	3.00000	0.474342
$41$	3.90833	0.610378	0.305189	−	0.952292i	$-0.401280\pi$
$41$	3.90833	0.610378	0.305189	+	0.952292i	$0.401280\pi$
$42$	0	0
$43$	10.2111	1.55718	0.778589	−	0.627534i	$-0.215936\pi$
$43$	10.2111	1.55718	0.778589	+	0.627534i	$0.215936\pi$
$44$	−0.908327	−0.136935
$45$	0	0
$46$	−5.60555	−0.826493
$47$	0.394449	0.0575363	0.0287681	−	0.999586i	$-0.490842\pi$
$47$	0.394449	0.0575363	0.0287681	+	0.999586i	$0.490842\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−1.30278	−0.184240
$51$	0	0
$52$	0.697224	0.0966876
$53$	12.5139	1.71891	0.859457	−	0.511209i	$-0.170802\pi$
$53$	12.5139	1.71891	0.859457	+	0.511209i	$0.170802\pi$
$54$	0	0
$55$	3.00000	0.404520
$56$	3.00000	0.400892
$57$	0	0
$58$	−2.21110	−0.290332
$59$	−8.21110	−1.06899	−0.534497	−	0.845170i	$-0.679499\pi$
$59$	−8.21110	−1.06899	−0.534497	+	0.845170i	$0.679499\pi$
$60$	0	0
$61$	14.1194	1.80781	0.903904	−	0.427736i	$-0.140689\pi$
$61$	14.1194	1.80781	0.903904	+	0.427736i	$0.140689\pi$
$62$	12.5139	1.58926
$63$	0	0
$64$	8.81665	1.10208
$65$	−2.30278	−0.285624
$66$	0	0
$67$	−4.51388	−0.551458	−0.275729	−	0.961235i	$-0.588919\pi$
$67$	−4.51388	−0.551458	−0.275729	+	0.961235i	$0.588919\pi$
$68$	−0.394449	−0.0478339
$69$	0	0
$70$	−1.30278	−0.155711
$71$	12.9083	1.53194	0.765968	−	0.642878i	$-0.222260\pi$
$71$	12.9083	1.53194	0.765968	+	0.642878i	$0.222260\pi$
$72$	0	0
$73$	4.21110	0.492872	0.246436	−	0.969159i	$-0.420740\pi$
$73$	4.21110	0.492872	0.246436	+	0.969159i	$0.420740\pi$
$74$	4.69722	0.546041
$75$	0	0
$76$	−0.0916731	−0.0105156
$77$	3.00000	0.341882
$78$	0	0
$79$	−6.09167	−0.685367	−0.342683	−	0.939451i	$-0.611336\pi$
$79$	−6.09167	−0.685367	−0.342683	+	0.939451i	$0.611336\pi$
$80$	−3.30278	−0.369262
$81$	0	0
$82$	−5.09167	−0.562281
$83$	2.21110	0.242700	0.121350	−	0.992610i	$-0.461278\pi$
$83$	2.21110	0.242700	0.121350	+	0.992610i	$0.461278\pi$
$84$	0	0
$85$	1.30278	0.141306
$86$	−13.3028	−1.43448
$87$	0	0
$88$	9.00000	0.959403
$89$	−10.8167	−1.14656	−0.573282	−	0.819358i	$-0.694330\pi$
$89$	−10.8167	−1.14656	−0.573282	+	0.819358i	$0.694330\pi$
$90$	0	0
$91$	−2.30278	−0.241396
$92$	−1.30278	−0.135824
$93$	0	0
$94$	−0.513878	−0.0530025
$95$	0.302776	0.0310641
$96$	0	0
$97$	8.51388	0.864453	0.432227	−	0.901765i	$-0.357728\pi$
$97$	8.51388	0.864453	0.432227	+	0.901765i	$0.357728\pi$
$98$	−1.30278	−0.131600
$99$	0	0
$100$	−0.302776	−0.0302776
$101$	5.21110	0.518524	0.259262	−	0.965807i	$-0.416521\pi$
$101$	5.21110	0.518524	0.259262	+	0.965807i	$0.416521\pi$
$102$	0	0
$103$	16.7250	1.64796	0.823981	−	0.566618i	$-0.191748\pi$
$103$	16.7250	1.64796	0.823981	+	0.566618i	$0.191748\pi$
$104$	−6.90833	−0.677417
$105$	0	0
$106$	−16.3028	−1.58347
$107$	15.0000	1.45010	0.725052	−	0.688694i	$-0.241816\pi$
$107$	15.0000	1.45010	0.725052	+	0.688694i	$0.241816\pi$
$108$	0	0
$109$	−5.30278	−0.507914	−0.253957	−	0.967216i	$-0.581732\pi$
$109$	−5.30278	−0.507914	−0.253957	+	0.967216i	$0.581732\pi$
$110$	−3.90833	−0.372644
$111$	0	0
$112$	−3.30278	−0.312083
$113$	5.09167	0.478984	0.239492	−	0.970898i	$-0.423019\pi$
$113$	5.09167	0.478984	0.239492	+	0.970898i	$0.423019\pi$
$114$	0	0
$115$	4.30278	0.401236
$116$	−0.513878	−0.0477124
$117$	0	0
$118$	10.6972	0.984759
$119$	1.30278	0.119425
$120$	0	0
$121$	−2.00000	−0.181818
$122$	−18.3944	−1.66536
$123$	0	0
$124$	2.90833	0.261175
$125$	1.00000	0.0894427
$126$	0	0
$127$	13.7250	1.21790	0.608948	−	0.793210i	$-0.291592\pi$
$127$	13.7250	1.21790	0.608948	+	0.793210i	$0.291592\pi$
$128$	−8.09167	−0.715210
$129$	0	0
$130$	3.00000	0.263117
$131$	−4.69722	−0.410398	−0.205199	−	0.978720i	$-0.565784\pi$
$131$	−4.69722	−0.410398	−0.205199	+	0.978720i	$0.565784\pi$
$132$	0	0
$133$	0.302776	0.0262540
$134$	5.88057	0.508004
$135$	0	0
$136$	3.90833	0.335136
$137$	−10.4222	−0.890429	−0.445215	−	0.895424i	$-0.646873\pi$
$137$	−10.4222	−0.890429	−0.445215	+	0.895424i	$0.646873\pi$
$138$	0	0
$139$	1.60555	0.136181	0.0680905	−	0.997679i	$-0.478309\pi$
$139$	1.60555	0.136181	0.0680905	+	0.997679i	$0.478309\pi$
$140$	−0.302776	−0.0255892
$141$	0	0
$142$	−16.8167	−1.41122
$143$	−6.90833	−0.577703
$144$	0	0
$145$	1.69722	0.140947
$146$	−5.48612	−0.454035
$147$	0	0
$148$	1.09167	0.0897350
$149$	−12.5139	−1.02518	−0.512588	−	0.858634i	$-0.671313\pi$
$149$	−12.5139	−1.02518	−0.512588	+	0.858634i	$0.671313\pi$
$150$	0	0
$151$	−1.00000	−0.0813788	−0.0406894	−	0.999172i	$-0.512955\pi$
$151$	−1.00000	−0.0813788	−0.0406894	+	0.999172i	$0.512955\pi$
$152$	0.908327	0.0736750
$153$	0	0
$154$	−3.90833	−0.314942
$155$	−9.60555	−0.771536
$156$	0	0
$157$	−0.605551	−0.0483283	−0.0241641	−	0.999708i	$-0.507692\pi$
$157$	−0.605551	−0.0483283	−0.0241641	+	0.999708i	$0.507692\pi$
$158$	7.93608	0.631361
$159$	0	0
$160$	−1.69722	−0.134177
$161$	4.30278	0.339106
$162$	0	0
$163$	8.78890	0.688400	0.344200	−	0.938896i	$-0.388150\pi$
$163$	8.78890	0.688400	0.344200	+	0.938896i	$0.388150\pi$
$164$	−1.18335	−0.0924038
$165$	0	0
$166$	−2.88057	−0.223576
$167$	12.3944	0.959111	0.479556	−	0.877511i	$-0.340798\pi$
$167$	12.3944	0.959111	0.479556	+	0.877511i	$0.340798\pi$
$168$	0	0
$169$	−7.69722	−0.592094
$170$	−1.69722	−0.130171
$171$	0	0
$172$	−3.09167	−0.235738
$173$	4.81665	0.366203	0.183102	−	0.983094i	$-0.441386\pi$
$173$	4.81665	0.366203	0.183102	+	0.983094i	$0.441386\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	−9.90833	−0.746868
$177$	0	0
$178$	14.0917	1.05622
$179$	−4.81665	−0.360014	−0.180007	−	0.983665i	$-0.557612\pi$
$179$	−4.81665	−0.360014	−0.180007	+	0.983665i	$0.557612\pi$
$180$	0	0
$181$	−11.4222	−0.849006	−0.424503	−	0.905427i	$-0.639551\pi$
$181$	−11.4222	−0.849006	−0.424503	+	0.905427i	$0.639551\pi$
$182$	3.00000	0.222375
$183$	0	0
$184$	12.9083	0.951614
$185$	−3.60555	−0.265085
$186$	0	0
$187$	3.90833	0.285805
$188$	−0.119429	−0.00871029
$189$	0	0
$190$	−0.394449	−0.0286163
$191$	−10.3028	−0.745483	−0.372741	−	0.927935i	$-0.621582\pi$
$191$	−10.3028	−0.745483	−0.372741	+	0.927935i	$0.621582\pi$
$192$	0	0
$193$	−19.1194	−1.37625	−0.688123	−	0.725594i	$-0.741565\pi$
$193$	−19.1194	−1.37625	−0.688123	+	0.725594i	$0.741565\pi$
$194$	−11.0917	−0.796336
$195$	0	0
$196$	−0.302776	−0.0216268
$197$	−1.18335	−0.0843099	−0.0421550	−	0.999111i	$-0.513422\pi$
$197$	−1.18335	−0.0843099	−0.0421550	+	0.999111i	$0.513422\pi$
$198$	0	0
$199$	−9.72498	−0.689386	−0.344693	−	0.938716i	$-0.612017\pi$
$199$	−9.72498	−0.689386	−0.344693	+	0.938716i	$0.612017\pi$
$200$	3.00000	0.212132
$201$	0	0
$202$	−6.78890	−0.477665
$203$	1.69722	0.119122
$204$	0	0
$205$	3.90833	0.272969
$206$	−21.7889	−1.51810
$207$	0	0
$208$	7.60555	0.527350
$209$	0.908327	0.0628303
$210$	0	0
$211$	−18.6056	−1.28086	−0.640429	−	0.768017i	$-0.721244\pi$
$211$	−18.6056	−1.28086	−0.640429	+	0.768017i	$0.721244\pi$
$212$	−3.78890	−0.260223
$213$	0	0
$214$	−19.5416	−1.33584
$215$	10.2111	0.696391
$216$	0	0
$217$	−9.60555	−0.652067
$218$	6.90833	0.467891
$219$	0	0
$220$	−0.908327	−0.0612394
$221$	−3.00000	−0.201802
$222$	0	0
$223$	5.78890	0.387653	0.193827	−	0.981036i	$-0.437910\pi$
$223$	5.78890	0.387653	0.193827	+	0.981036i	$0.437910\pi$
$224$	−1.69722	−0.113401
$225$	0	0
$226$	−6.63331	−0.441241
$227$	−24.5139	−1.62704	−0.813522	−	0.581535i	$-0.802452\pi$
$227$	−24.5139	−1.62704	−0.813522	+	0.581535i	$0.802452\pi$
$228$	0	0
$229$	16.2111	1.07126	0.535630	−	0.844453i	$-0.320074\pi$
$229$	16.2111	1.07126	0.535630	+	0.844453i	$0.320074\pi$
$230$	−5.60555	−0.369619
$231$	0	0
$232$	5.09167	0.334285
$233$	0.908327	0.0595065	0.0297532	−	0.999557i	$-0.490528\pi$
$233$	0.908327	0.0595065	0.0297532	+	0.999557i	$0.490528\pi$
$234$	0	0
$235$	0.394449	0.0257310
$236$	2.48612	0.161833
$237$	0	0
$238$	−1.69722	−0.110015
$239$	20.2111	1.30735	0.653674	−	0.756776i	$-0.273227\pi$
$239$	20.2111	1.30735	0.653674	+	0.756776i	$0.273227\pi$
$240$	0	0
$241$	−25.1194	−1.61808	−0.809042	−	0.587750i	$-0.800014\pi$
$241$	−25.1194	−1.61808	−0.809042	+	0.587750i	$0.800014\pi$
$242$	2.60555	0.167491
$243$	0	0
$244$	−4.27502	−0.273680
$245$	1.00000	0.0638877
$246$	0	0
$247$	−0.697224	−0.0443633
$248$	−28.8167	−1.82986
$249$	0	0
$250$	−1.30278	−0.0823948
$251$	25.8167	1.62953	0.814766	−	0.579789i	$-0.196865\pi$
$251$	25.8167	1.62953	0.814766	+	0.579789i	$0.196865\pi$
$252$	0	0
$253$	12.9083	0.811540
$254$	−17.8806	−1.12193
$255$	0	0
$256$	−7.09167	−0.443230
$257$	−12.0000	−0.748539	−0.374270	−	0.927320i	$-0.622107\pi$
$257$	−12.0000	−0.748539	−0.374270	+	0.927320i	$0.622107\pi$
$258$	0	0
$259$	−3.60555	−0.224038
$260$	0.697224	0.0432400
$261$	0	0
$262$	6.11943	0.378060
$263$	30.5139	1.88157	0.940783	−	0.339009i	$-0.110092\pi$
$263$	30.5139	1.88157	0.940783	+	0.339009i	$0.110092\pi$
$264$	0	0
$265$	12.5139	0.768721
$266$	−0.394449	−0.0241852
$267$	0	0
$268$	1.36669	0.0834840
$269$	−19.4222	−1.18419	−0.592096	−	0.805867i	$-0.701700\pi$
$269$	−19.4222	−1.18419	−0.592096	+	0.805867i	$0.701700\pi$
$270$	0	0
$271$	−13.1194	−0.796949	−0.398474	−	0.917180i	$-0.630460\pi$
$271$	−13.1194	−0.796949	−0.398474	+	0.917180i	$0.630460\pi$
$272$	−4.30278	−0.260894
$273$	0	0
$274$	13.5778	0.820265
$275$	3.00000	0.180907
$276$	0	0
$277$	−13.9083	−0.835670	−0.417835	−	0.908523i	$-0.637211\pi$
$277$	−13.9083	−0.835670	−0.417835	+	0.908523i	$0.637211\pi$
$278$	−2.09167	−0.125450
$279$	0	0
$280$	3.00000	0.179284
$281$	11.7250	0.699454	0.349727	−	0.936852i	$-0.386274\pi$
$281$	11.7250	0.699454	0.349727	+	0.936852i	$0.386274\pi$
$282$	0	0
$283$	−7.90833	−0.470101	−0.235051	−	0.971983i	$-0.575526\pi$
$283$	−7.90833	−0.470101	−0.235051	+	0.971983i	$0.575526\pi$
$284$	−3.90833	−0.231917
$285$	0	0
$286$	9.00000	0.532181
$287$	3.90833	0.230701
$288$	0	0
$289$	−15.3028	−0.900163
$290$	−2.21110	−0.129840
$291$	0	0
$292$	−1.27502	−0.0746149
$293$	−30.2389	−1.76657	−0.883287	−	0.468834i	$-0.844674\pi$
$293$	−30.2389	−1.76657	−0.883287	+	0.468834i	$0.844674\pi$
$294$	0	0
$295$	−8.21110	−0.478069
$296$	−10.8167	−0.628705
$297$	0	0
$298$	16.3028	0.944394
$299$	−9.90833	−0.573013
$300$	0	0
$301$	10.2111	0.588558
$302$	1.30278	0.0749663
$303$	0	0
$304$	−1.00000	−0.0573539
$305$	14.1194	0.808476
$306$	0	0
$307$	−24.8444	−1.41795	−0.708973	−	0.705236i	$-0.750841\pi$
$307$	−24.8444	−1.41795	−0.708973	+	0.705236i	$0.750841\pi$
$308$	−0.908327	−0.0517567
$309$	0	0
$310$	12.5139	0.710741
$311$	24.9083	1.41242	0.706211	−	0.708002i	$-0.250403\pi$
$311$	24.9083	1.41242	0.706211	+	0.708002i	$0.250403\pi$
$312$	0	0
$313$	6.81665	0.385300	0.192650	−	0.981268i	$-0.438292\pi$
$313$	6.81665	0.385300	0.192650	+	0.981268i	$0.438292\pi$
$314$	0.788897	0.0445201
$315$	0	0
$316$	1.84441	0.103756
$317$	−33.2389	−1.86688	−0.933440	−	0.358733i	$-0.883209\pi$
$317$	−33.2389	−1.86688	−0.933440	+	0.358733i	$0.883209\pi$
$318$	0	0
$319$	5.09167	0.285079
$320$	8.81665	0.492866
$321$	0	0
$322$	−5.60555	−0.312385
$323$	0.394449	0.0219477
$324$	0	0
$325$	−2.30278	−0.127735
$326$	−11.4500	−0.634155
$327$	0	0
$328$	11.7250	0.647404
$329$	0.394449	0.0217467
$330$	0	0
$331$	9.69722	0.533008	0.266504	−	0.963834i	$-0.414132\pi$
$331$	9.69722	0.533008	0.266504	+	0.963834i	$0.414132\pi$
$332$	−0.669468	−0.0367418
$333$	0	0
$334$	−16.1472	−0.883535
$335$	−4.51388	−0.246620
$336$	0	0
$337$	12.3028	0.670175	0.335087	−	0.942187i	$-0.391234\pi$
$337$	12.3028	0.670175	0.335087	+	0.942187i	$0.391234\pi$
$338$	10.0278	0.545438
$339$	0	0
$340$	−0.394449	−0.0213920
$341$	−28.8167	−1.56051
$342$	0	0
$343$	1.00000	0.0539949
$344$	30.6333	1.65164
$345$	0	0
$346$	−6.27502	−0.337347
$347$	26.6056	1.42826	0.714130	−	0.700013i	$-0.246822\pi$
$347$	26.6056	1.42826	0.714130	+	0.700013i	$0.246822\pi$
$348$	0	0
$349$	−16.6333	−0.890361	−0.445180	−	0.895441i	$-0.646860\pi$
$349$	−16.6333	−0.890361	−0.445180	+	0.895441i	$0.646860\pi$
$350$	−1.30278	−0.0696363
$351$	0	0
$352$	−5.09167	−0.271387
$353$	−8.72498	−0.464384	−0.232192	−	0.972670i	$-0.574590\pi$
$353$	−8.72498	−0.464384	−0.232192	+	0.972670i	$0.574590\pi$
$354$	0	0
$355$	12.9083	0.685103
$356$	3.27502	0.173576
$357$	0	0
$358$	6.27502	0.331645
$359$	22.8167	1.20422	0.602108	−	0.798414i	$-0.294327\pi$
$359$	22.8167	1.20422	0.602108	+	0.798414i	$0.294327\pi$
$360$	0	0
$361$	−18.9083	−0.995175
$362$	14.8806	0.782105
$363$	0	0
$364$	0.697224	0.0365445
$365$	4.21110	0.220419
$366$	0	0
$367$	−2.69722	−0.140794	−0.0703970	−	0.997519i	$-0.522427\pi$
$367$	−2.69722	−0.140794	−0.0703970	+	0.997519i	$0.522427\pi$
$368$	−14.2111	−0.740805
$369$	0	0
$370$	4.69722	0.244197
$371$	12.5139	0.649688
$372$	0	0
$373$	8.90833	0.461256	0.230628	−	0.973042i	$-0.425922\pi$
$373$	8.90833	0.461256	0.230628	+	0.973042i	$0.425922\pi$
$374$	−5.09167	−0.263284
$375$	0	0
$376$	1.18335	0.0610264
$377$	−3.90833	−0.201289
$378$	0	0
$379$	17.6333	0.905762	0.452881	−	0.891571i	$-0.350396\pi$
$379$	17.6333	0.905762	0.452881	+	0.891571i	$0.350396\pi$
$380$	−0.0916731	−0.00470273
$381$	0	0
$382$	13.4222	0.686740
$383$	−14.2111	−0.726153	−0.363077	−	0.931759i	$-0.618274\pi$
$383$	−14.2111	−0.726153	−0.363077	+	0.931759i	$0.618274\pi$
$384$	0	0
$385$	3.00000	0.152894
$386$	24.9083	1.26780
$387$	0	0
$388$	−2.57779	−0.130868
$389$	23.7250	1.20290	0.601452	−	0.798909i	$-0.294589\pi$
$389$	23.7250	1.20290	0.601452	+	0.798909i	$0.294589\pi$
$390$	0	0
$391$	5.60555	0.283485
$392$	3.00000	0.151523
$393$	0	0
$394$	1.54163	0.0776664
$395$	−6.09167	−0.306505
$396$	0	0
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	12.6695	0.635063
$399$	0	0
$400$	−3.30278	−0.165139
$401$	26.7250	1.33458	0.667291	−	0.744797i	$-0.267454\pi$
$401$	26.7250	1.33458	0.667291	+	0.744797i	$0.267454\pi$
$402$	0	0
$403$	22.1194	1.10185
$404$	−1.57779	−0.0784982
$405$	0	0
$406$	−2.21110	−0.109735
$407$	−10.8167	−0.536162
$408$	0	0
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	−5.09167	−0.251460
$411$	0	0
$412$	−5.06392	−0.249481
$413$	−8.21110	−0.404042
$414$	0	0
$415$	2.21110	0.108539
$416$	3.90833	0.191621
$417$	0	0
$418$	−1.18335	−0.0578794
$419$	34.8167	1.70090	0.850452	−	0.526052i	$-0.176328\pi$
$419$	34.8167	1.70090	0.850452	+	0.526052i	$0.176328\pi$
$420$	0	0
$421$	6.30278	0.307178	0.153589	−	0.988135i	$-0.450917\pi$
$421$	6.30278	0.307178	0.153589	+	0.988135i	$0.450917\pi$
$422$	24.2389	1.17993
$423$	0	0
$424$	37.5416	1.82318
$425$	1.30278	0.0631939
$426$	0	0
$427$	14.1194	0.683287
$428$	−4.54163	−0.219528
$429$	0	0
$430$	−13.3028	−0.641517
$431$	−10.9361	−0.526773	−0.263386	−	0.964690i	$-0.584839\pi$
$431$	−10.9361	−0.526773	−0.263386	+	0.964690i	$0.584839\pi$
$432$	0	0
$433$	−24.3305	−1.16925	−0.584625	−	0.811303i	$-0.698758\pi$
$433$	−24.3305	−1.16925	−0.584625	+	0.811303i	$0.698758\pi$
$434$	12.5139	0.600685
$435$	0	0
$436$	1.60555	0.0768920
$437$	1.30278	0.0623202
$438$	0	0
$439$	−33.6056	−1.60391	−0.801953	−	0.597387i	$-0.796205\pi$
$439$	−33.6056	−1.60391	−0.801953	+	0.597387i	$0.796205\pi$
$440$	9.00000	0.429058
$441$	0	0
$442$	3.90833	0.185900
$443$	−32.7250	−1.55481	−0.777405	−	0.629000i	$-0.783465\pi$
$443$	−32.7250	−1.55481	−0.777405	+	0.629000i	$0.783465\pi$
$444$	0	0
$445$	−10.8167	−0.512759
$446$	−7.54163	−0.357107
$447$	0	0
$448$	8.81665	0.416548
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	0	0
$451$	11.7250	0.552108
$452$	−1.54163	−0.0725124
$453$	0	0
$454$	31.9361	1.49883
$455$	−2.30278	−0.107956
$456$	0	0
$457$	18.6972	0.874619	0.437310	−	0.899311i	$-0.355931\pi$
$457$	18.6972	0.874619	0.437310	+	0.899311i	$0.355931\pi$
$458$	−21.1194	−0.986846
$459$	0	0
$460$	−1.30278	−0.0607422
$461$	−8.48612	−0.395238	−0.197619	−	0.980279i	$-0.563321\pi$
$461$	−8.48612	−0.395238	−0.197619	+	0.980279i	$0.563321\pi$
$462$	0	0
$463$	−35.9361	−1.67009	−0.835046	−	0.550181i	$-0.814559\pi$
$463$	−35.9361	−1.67009	−0.835046	+	0.550181i	$0.814559\pi$
$464$	−5.60555	−0.260231
$465$	0	0
$466$	−1.18335	−0.0548175
$467$	−5.21110	−0.241141	−0.120571	−	0.992705i	$-0.538472\pi$
$467$	−5.21110	−0.241141	−0.120571	+	0.992705i	$0.538472\pi$
$468$	0	0
$469$	−4.51388	−0.208432
$470$	−0.513878	−0.0237034
$471$	0	0
$472$	−24.6333	−1.13384
$473$	30.6333	1.40852
$474$	0	0
$475$	0.302776	0.0138923
$476$	−0.394449	−0.0180795
$477$	0	0
$478$	−26.3305	−1.20433
$479$	34.9361	1.59627	0.798135	−	0.602478i	$-0.205820\pi$
$479$	34.9361	1.59627	0.798135	+	0.602478i	$0.205820\pi$
$480$	0	0
$481$	8.30278	0.378574
$482$	32.7250	1.49058
$483$	0	0
$484$	0.605551	0.0275251
$485$	8.51388	0.386595
$486$	0	0
$487$	38.0000	1.72194	0.860972	−	0.508652i	$-0.169856\pi$
$487$	38.0000	1.72194	0.860972	+	0.508652i	$0.169856\pi$
$488$	42.3583	1.91747
$489$	0	0
$490$	−1.30278	−0.0588534
$491$	11.7250	0.529141	0.264570	−	0.964366i	$-0.414770\pi$
$491$	11.7250	0.529141	0.264570	+	0.964366i	$0.414770\pi$
$492$	0	0
$493$	2.21110	0.0995831
$494$	0.908327	0.0408676
$495$	0	0
$496$	31.7250	1.42449
$497$	12.9083	0.579018
$498$	0	0
$499$	2.63331	0.117883	0.0589415	−	0.998261i	$-0.481227\pi$
$499$	2.63331	0.117883	0.0589415	+	0.998261i	$0.481227\pi$
$500$	−0.302776	−0.0135405
$501$	0	0
$502$	−33.6333	−1.50113
$503$	−6.11943	−0.272852	−0.136426	−	0.990650i	$-0.543562\pi$
$503$	−6.11943	−0.272852	−0.136426	+	0.990650i	$0.543562\pi$
$504$	0	0
$505$	5.21110	0.231891
$506$	−16.8167	−0.747591
$507$	0	0
$508$	−4.15559	−0.184374
$509$	27.2389	1.20734	0.603671	−	0.797234i	$-0.293704\pi$
$509$	27.2389	1.20734	0.603671	+	0.797234i	$0.293704\pi$
$510$	0	0
$511$	4.21110	0.186288
$512$	25.4222	1.12351
$513$	0	0
$514$	15.6333	0.689556
$515$	16.7250	0.736991
$516$	0	0
$517$	1.18335	0.0520435
$518$	4.69722	0.206384
$519$	0	0
$520$	−6.90833	−0.302950
$521$	−36.3944	−1.59447	−0.797235	−	0.603669i	$-0.793705\pi$
$521$	−36.3944	−1.59447	−0.797235	+	0.603669i	$0.793705\pi$
$522$	0	0
$523$	−18.7250	−0.818786	−0.409393	−	0.912358i	$-0.634260\pi$
$523$	−18.7250	−0.818786	−0.409393	+	0.912358i	$0.634260\pi$
$524$	1.42221	0.0621293
$525$	0	0
$526$	−39.7527	−1.73330
$527$	−12.5139	−0.545113
$528$	0	0
$529$	−4.48612	−0.195049
$530$	−16.3028	−0.708147
$531$	0	0
$532$	−0.0916731	−0.00397453
$533$	−9.00000	−0.389833
$534$	0	0
$535$	15.0000	0.648507
$536$	−13.5416	−0.584910
$537$	0	0
$538$	25.3028	1.09088
$539$	3.00000	0.129219
$540$	0	0
$541$	2.51388	0.108080	0.0540400	−	0.998539i	$-0.482790\pi$
$541$	2.51388	0.108080	0.0540400	+	0.998539i	$0.482790\pi$
$542$	17.0917	0.734150
$543$	0	0
$544$	−2.21110	−0.0948002
$545$	−5.30278	−0.227146
$546$	0	0
$547$	23.7889	1.01714	0.508570	−	0.861021i	$-0.330174\pi$
$547$	23.7889	1.01714	0.508570	+	0.861021i	$0.330174\pi$
$548$	3.15559	0.134800
$549$	0	0
$550$	−3.90833	−0.166652
$551$	0.513878	0.0218919
$552$	0	0
$553$	−6.09167	−0.259044
$554$	18.1194	0.769821
$555$	0	0
$556$	−0.486122	−0.0206162
$557$	−33.5139	−1.42003	−0.710014	−	0.704187i	$-0.751312\pi$
$557$	−33.5139	−1.42003	−0.710014	+	0.704187i	$0.751312\pi$
$558$	0	0
$559$	−23.5139	−0.994531
$560$	−3.30278	−0.139568
$561$	0	0
$562$	−15.2750	−0.644338
$563$	−10.9361	−0.460901	−0.230450	−	0.973084i	$-0.574020\pi$
$563$	−10.9361	−0.460901	−0.230450	+	0.973084i	$0.574020\pi$
$564$	0	0
$565$	5.09167	0.214208
$566$	10.3028	0.433058
$567$	0	0
$568$	38.7250	1.62486
$569$	12.6333	0.529616	0.264808	−	0.964301i	$-0.414691\pi$
$569$	12.6333	0.529616	0.264808	+	0.964301i	$0.414691\pi$
$570$	0	0
$571$	−26.9361	−1.12724	−0.563620	−	0.826034i	$-0.690592\pi$
$571$	−26.9361	−1.12724	−0.563620	+	0.826034i	$0.690592\pi$
$572$	2.09167	0.0874572
$573$	0	0
$574$	−5.09167	−0.212522
$575$	4.30278	0.179438
$576$	0	0
$577$	32.3944	1.34860	0.674299	−	0.738458i	$-0.264446\pi$
$577$	32.3944	1.34860	0.674299	+	0.738458i	$0.264446\pi$
$578$	19.9361	0.829232
$579$	0	0
$580$	−0.513878	−0.0213376
$581$	2.21110	0.0917320
$582$	0	0
$583$	37.5416	1.55482
$584$	12.6333	0.522770
$585$	0	0
$586$	39.3944	1.62737
$587$	−12.9083	−0.532784	−0.266392	−	0.963865i	$-0.585832\pi$
$587$	−12.9083	−0.532784	−0.266392	+	0.963865i	$0.585832\pi$
$588$	0	0
$589$	−2.90833	−0.119836
$590$	10.6972	0.440398
$591$	0	0
$592$	11.9083	0.489429
$593$	−13.1833	−0.541375	−0.270688	−	0.962667i	$-0.587251\pi$
$593$	−13.1833	−0.541375	−0.270688	+	0.962667i	$0.587251\pi$
$594$	0	0
$595$	1.30278	0.0534086
$596$	3.78890	0.155199
$597$	0	0
$598$	12.9083	0.527861
$599$	7.57779	0.309620	0.154810	−	0.987944i	$-0.450523\pi$
$599$	7.57779	0.309620	0.154810	+	0.987944i	$0.450523\pi$
$600$	0	0
$601$	19.8806	0.810945	0.405473	−	0.914107i	$-0.367107\pi$
$601$	19.8806	0.810945	0.405473	+	0.914107i	$0.367107\pi$
$602$	−13.3028	−0.542181
$603$	0	0
$604$	0.302776	0.0123198
$605$	−2.00000	−0.0813116
$606$	0	0
$607$	−28.6333	−1.16219	−0.581095	−	0.813836i	$-0.697376\pi$
$607$	−28.6333	−1.16219	−0.581095	+	0.813836i	$0.697376\pi$
$608$	−0.513878	−0.0208405
$609$	0	0
$610$	−18.3944	−0.744769
$611$	−0.908327	−0.0367470
$612$	0	0
$613$	23.2389	0.938609	0.469304	−	0.883036i	$-0.344505\pi$
$613$	23.2389	0.938609	0.469304	+	0.883036i	$0.344505\pi$
$614$	32.3667	1.30621
$615$	0	0
$616$	9.00000	0.362620
$617$	32.0917	1.29196	0.645981	−	0.763353i	$-0.276448\pi$
$617$	32.0917	1.29196	0.645981	+	0.763353i	$0.276448\pi$
$618$	0	0
$619$	−20.0278	−0.804983	−0.402492	−	0.915424i	$-0.631856\pi$
$619$	−20.0278	−0.804983	−0.402492	+	0.915424i	$0.631856\pi$
$620$	2.90833	0.116801
$621$	0	0
$622$	−32.4500	−1.30112
$623$	−10.8167	−0.433360
$624$	0	0
$625$	1.00000	0.0400000
$626$	−8.88057	−0.354939
$627$	0	0
$628$	0.183346	0.00731631
$629$	−4.69722	−0.187291
$630$	0	0
$631$	−45.2111	−1.79983	−0.899913	−	0.436070i	$-0.856370\pi$
$631$	−45.2111	−1.79983	−0.899913	+	0.436070i	$0.856370\pi$
$632$	−18.2750	−0.726941
$633$	0	0
$634$	43.3028	1.71977
$635$	13.7250	0.544659
$636$	0	0
$637$	−2.30278	−0.0912393
$638$	−6.63331	−0.262615
$639$	0	0
$640$	−8.09167	−0.319851
$641$	−26.7250	−1.05557	−0.527787	−	0.849377i	$-0.676978\pi$
$641$	−26.7250	−1.05557	−0.527787	+	0.849377i	$0.676978\pi$
$642$	0	0
$643$	−31.5139	−1.24279	−0.621393	−	0.783499i	$-0.713433\pi$
$643$	−31.5139	−1.24279	−0.621393	+	0.783499i	$0.713433\pi$
$644$	−1.30278	−0.0513366
$645$	0	0
$646$	−0.513878	−0.0202183
$647$	30.0000	1.17942	0.589711	−	0.807614i	$-0.299242\pi$
$647$	30.0000	1.17942	0.589711	+	0.807614i	$0.299242\pi$
$648$	0	0
$649$	−24.6333	−0.966942
$650$	3.00000	0.117670
$651$	0	0
$652$	−2.66106	−0.104215
$653$	30.1194	1.17866	0.589332	−	0.807891i	$-0.299391\pi$
$653$	30.1194	1.17866	0.589332	+	0.807891i	$0.299391\pi$
$654$	0	0
$655$	−4.69722	−0.183536
$656$	−12.9083	−0.503985
$657$	0	0
$658$	−0.513878	−0.0200331
$659$	−43.8167	−1.70685	−0.853427	−	0.521212i	$-0.825480\pi$
$659$	−43.8167	−1.70685	−0.853427	+	0.521212i	$0.825480\pi$
$660$	0	0
$661$	−24.7250	−0.961690	−0.480845	−	0.876806i	$-0.659670\pi$
$661$	−24.7250	−0.961690	−0.480845	+	0.876806i	$0.659670\pi$
$662$	−12.6333	−0.491007
$663$	0	0
$664$	6.63331	0.257422
$665$	0.302776	0.0117411
$666$	0	0
$667$	7.30278	0.282765
$668$	−3.75274	−0.145198
$669$	0	0
$670$	5.88057	0.227186
$671$	42.3583	1.63522
$672$	0	0
$673$	−28.0000	−1.07932	−0.539660	−	0.841883i	$-0.681447\pi$
$673$	−28.0000	−1.07932	−0.539660	+	0.841883i	$0.681447\pi$
$674$	−16.0278	−0.617366
$675$	0	0
$676$	2.33053	0.0896358
$677$	−32.2111	−1.23797	−0.618987	−	0.785402i	$-0.712456\pi$
$677$	−32.2111	−1.23797	−0.618987	+	0.785402i	$0.712456\pi$
$678$	0	0
$679$	8.51388	0.326733
$680$	3.90833	0.149877
$681$	0	0
$682$	37.5416	1.43754
$683$	30.3583	1.16163	0.580814	−	0.814036i	$-0.302734\pi$
$683$	30.3583	1.16163	0.580814	+	0.814036i	$0.302734\pi$
$684$	0	0
$685$	−10.4222	−0.398212
$686$	−1.30278	−0.0497402
$687$	0	0
$688$	−33.7250	−1.28575
$689$	−28.8167	−1.09783
$690$	0	0
$691$	21.4222	0.814939	0.407470	−	0.913219i	$-0.366411\pi$
$691$	21.4222	0.814939	0.407470	+	0.913219i	$0.366411\pi$
$692$	−1.45837	−0.0554387
$693$	0	0
$694$	−34.6611	−1.31572
$695$	1.60555	0.0609020
$696$	0	0
$697$	5.09167	0.192861
$698$	21.6695	0.820201
$699$	0	0
$700$	−0.302776	−0.0114438
$701$	−10.0278	−0.378743	−0.189372	−	0.981905i	$-0.560645\pi$
$701$	−10.0278	−0.378743	−0.189372	+	0.981905i	$0.560645\pi$
$702$	0	0
$703$	−1.09167	−0.0411732
$704$	26.4500	0.996870
$705$	0	0
$706$	11.3667	0.427791
$707$	5.21110	0.195984
$708$	0	0
$709$	−8.97224	−0.336960	−0.168480	−	0.985705i	$-0.553886\pi$
$709$	−8.97224	−0.336960	−0.168480	+	0.985705i	$0.553886\pi$
$710$	−16.8167	−0.631118
$711$	0	0
$712$	−32.4500	−1.21611
$713$	−41.3305	−1.54784
$714$	0	0
$715$	−6.90833	−0.258357
$716$	1.45837	0.0545017
$717$	0	0
$718$	−29.7250	−1.10933
$719$	−20.3305	−0.758201	−0.379100	−	0.925356i	$-0.623767\pi$
$719$	−20.3305	−0.758201	−0.379100	+	0.925356i	$0.623767\pi$
$720$	0	0
$721$	16.7250	0.622871
$722$	24.6333	0.916757
$723$	0	0
$724$	3.45837	0.128529
$725$	1.69722	0.0630333
$726$	0	0
$727$	−37.5139	−1.39131	−0.695656	−	0.718375i	$-0.744886\pi$
$727$	−37.5139	−1.39131	−0.695656	+	0.718375i	$0.744886\pi$
$728$	−6.90833	−0.256040
$729$	0	0
$730$	−5.48612	−0.203050
$731$	13.3028	0.492021
$732$	0	0
$733$	−41.5416	−1.53438	−0.767188	−	0.641423i	$-0.778344\pi$
$733$	−41.5416	−1.53438	−0.767188	+	0.641423i	$0.778344\pi$
$734$	3.51388	0.129700
$735$	0	0
$736$	−7.30278	−0.269184
$737$	−13.5416	−0.498813
$738$	0	0
$739$	−41.8167	−1.53825	−0.769125	−	0.639098i	$-0.779308\pi$
$739$	−41.8167	−1.53825	−0.769125	+	0.639098i	$0.779308\pi$
$740$	1.09167	0.0401307
$741$	0	0
$742$	−16.3028	−0.598494
$743$	−19.5416	−0.716913	−0.358457	−	0.933546i	$-0.616697\pi$
$743$	−19.5416	−0.716913	−0.358457	+	0.933546i	$0.616697\pi$
$744$	0	0
$745$	−12.5139	−0.458473
$746$	−11.6056	−0.424909
$747$	0	0
$748$	−1.18335	−0.0432674
$749$	15.0000	0.548088
$750$	0	0
$751$	14.0000	0.510867	0.255434	−	0.966827i	$-0.417782\pi$
$751$	14.0000	0.510867	0.255434	+	0.966827i	$0.417782\pi$
$752$	−1.30278	−0.0475073
$753$	0	0
$754$	5.09167	0.185428
$755$	−1.00000	−0.0363937
$756$	0	0
$757$	39.5416	1.43717	0.718583	−	0.695442i	$-0.244791\pi$
$757$	39.5416	1.43717	0.718583	+	0.695442i	$0.244791\pi$
$758$	−22.9722	−0.834389
$759$	0	0
$760$	0.908327	0.0329485
$761$	−40.5416	−1.46963	−0.734817	−	0.678266i	$-0.762732\pi$
$761$	−40.5416	−1.46963	−0.734817	+	0.678266i	$0.762732\pi$
$762$	0	0
$763$	−5.30278	−0.191973
$764$	3.11943	0.112857
$765$	0	0
$766$	18.5139	0.668934
$767$	18.9083	0.682740
$768$	0	0
$769$	−5.18335	−0.186916	−0.0934581	−	0.995623i	$-0.529792\pi$
$769$	−5.18335	−0.186916	−0.0934581	+	0.995623i	$0.529792\pi$
$770$	−3.90833	−0.140846
$771$	0	0
$772$	5.78890	0.208347
$773$	−52.5416	−1.88979	−0.944896	−	0.327372i	$-0.893837\pi$
$773$	−52.5416	−1.88979	−0.944896	+	0.327372i	$0.893837\pi$
$774$	0	0
$775$	−9.60555	−0.345042
$776$	25.5416	0.916891
$777$	0	0
$778$	−30.9083	−1.10812
$779$	1.18335	0.0423978
$780$	0	0
$781$	38.7250	1.38569
$782$	−7.30278	−0.261147
$783$	0	0
$784$	−3.30278	−0.117956
$785$	−0.605551	−0.0216131
$786$	0	0
$787$	33.0278	1.17731	0.588656	−	0.808384i	$-0.299657\pi$
$787$	33.0278	1.17731	0.588656	+	0.808384i	$0.299657\pi$
$788$	0.358288	0.0127635
$789$	0	0
$790$	7.93608	0.282353
$791$	5.09167	0.181039
$792$	0	0
$793$	−32.5139	−1.15460
$794$	−2.60555	−0.0924676
$795$	0	0
$796$	2.94449	0.104365
$797$	−32.0917	−1.13675	−0.568373	−	0.822771i	$-0.692427\pi$
$797$	−32.0917	−1.13675	−0.568373	+	0.822771i	$0.692427\pi$
$798$	0	0
$799$	0.513878	0.0181797
$800$	−1.69722	−0.0600059
$801$	0	0
$802$	−34.8167	−1.22942
$803$	12.6333	0.445820
$804$	0	0
$805$	4.30278	0.151653
$806$	−28.8167	−1.01502
$807$	0	0
$808$	15.6333	0.549978
$809$	−40.8167	−1.43504	−0.717519	−	0.696539i	$-0.754722\pi$
$809$	−40.8167	−1.43504	−0.717519	+	0.696539i	$0.754722\pi$
$810$	0	0
$811$	−6.48612	−0.227759	−0.113879	−	0.993495i	$-0.536328\pi$
$811$	−6.48612	−0.227759	−0.113879	+	0.993495i	$0.536328\pi$
$812$	−0.513878	−0.0180336
$813$	0	0
$814$	14.0917	0.493913
$815$	8.78890	0.307862
$816$	0	0
$817$	3.09167	0.108164
$818$	−18.2389	−0.637707
$819$	0	0
$820$	−1.18335	−0.0413242
$821$	−10.8167	−0.377504	−0.188752	−	0.982025i	$-0.560444\pi$
$821$	−10.8167	−0.377504	−0.188752	+	0.982025i	$0.560444\pi$
$822$	0	0
$823$	−10.6333	−0.370654	−0.185327	−	0.982677i	$-0.559334\pi$
$823$	−10.6333	−0.370654	−0.185327	+	0.982677i	$0.559334\pi$
$824$	50.1749	1.74793
$825$	0	0
$826$	10.6972	0.372204
$827$	−42.6333	−1.48251	−0.741253	−	0.671226i	$-0.765768\pi$
$827$	−42.6333	−1.48251	−0.741253	+	0.671226i	$0.765768\pi$
$828$	0	0
$829$	49.0555	1.70377	0.851884	−	0.523730i	$-0.175460\pi$
$829$	49.0555	1.70377	0.851884	+	0.523730i	$0.175460\pi$
$830$	−2.88057	−0.0999861
$831$	0	0
$832$	−20.3028	−0.703872
$833$	1.30278	0.0451385
$834$	0	0
$835$	12.3944	0.428928
$836$	−0.275019	−0.00951174
$837$	0	0
$838$	−45.3583	−1.56688
$839$	−21.9083	−0.756359	−0.378180	−	0.925732i	$-0.623450\pi$
$839$	−21.9083	−0.756359	−0.378180	+	0.925732i	$0.623450\pi$
$840$	0	0
$841$	−26.1194	−0.900670
$842$	−8.21110	−0.282973
$843$	0	0
$844$	5.63331	0.193906
$845$	−7.69722	−0.264793
$846$	0	0
$847$	−2.00000	−0.0687208
$848$	−41.3305	−1.41930
$849$	0	0
$850$	−1.69722	−0.0582143
$851$	−15.5139	−0.531809
$852$	0	0
$853$	−27.8444	−0.953374	−0.476687	−	0.879073i	$-0.658163\pi$
$853$	−27.8444	−0.953374	−0.476687	+	0.879073i	$0.658163\pi$
$854$	−18.3944	−0.629445
$855$	0	0
$856$	45.0000	1.53807
$857$	52.2666	1.78539	0.892697	−	0.450658i	$-0.148811\pi$
$857$	52.2666	1.78539	0.892697	+	0.450658i	$0.148811\pi$
$858$	0	0
$859$	6.18335	0.210973	0.105487	−	0.994421i	$-0.466360\pi$
$859$	6.18335	0.210973	0.105487	+	0.994421i	$0.466360\pi$
$860$	−3.09167	−0.105425
$861$	0	0
$862$	14.2473	0.485264
$863$	55.2666	1.88130	0.940649	−	0.339382i	$-0.110218\pi$
$863$	55.2666	1.88130	0.940649	+	0.339382i	$0.110218\pi$
$864$	0	0
$865$	4.81665	0.163771
$866$	31.6972	1.07712
$867$	0	0
$868$	2.90833	0.0987150
$869$	−18.2750	−0.619938
$870$	0	0
$871$	10.3944	0.352202
$872$	−15.9083	−0.538724
$873$	0	0
$874$	−1.69722	−0.0574095
$875$	1.00000	0.0338062
$876$	0	0
$877$	20.6333	0.696737	0.348369	−	0.937358i	$-0.386736\pi$
$877$	20.6333	0.696737	0.348369	+	0.937358i	$0.386736\pi$
$878$	43.7805	1.47752
$879$	0	0
$880$	−9.90833	−0.334010
$881$	−2.36669	−0.0797359	−0.0398679	−	0.999205i	$-0.512694\pi$
$881$	−2.36669	−0.0797359	−0.0398679	+	0.999205i	$0.512694\pi$
$882$	0	0
$883$	30.0278	1.01051	0.505257	−	0.862969i	$-0.331398\pi$
$883$	30.0278	1.01051	0.505257	+	0.862969i	$0.331398\pi$
$884$	0.908327	0.0305503
$885$	0	0
$886$	42.6333	1.43229
$887$	−15.3944	−0.516895	−0.258448	−	0.966025i	$-0.583211\pi$
$887$	−15.3944	−0.516895	−0.258448	+	0.966025i	$0.583211\pi$
$888$	0	0
$889$	13.7250	0.460321
$890$	14.0917	0.472354
$891$	0	0
$892$	−1.75274	−0.0586860
$893$	0.119429	0.00399655
$894$	0	0
$895$	−4.81665	−0.161003
$896$	−8.09167	−0.270324
$897$	0	0
$898$	23.4500	0.782535
$899$	−16.3028	−0.543728
$900$	0	0
$901$	16.3028	0.543124
$902$	−15.2750	−0.508603
$903$	0	0
$904$	15.2750	0.508040
$905$	−11.4222	−0.379687
$906$	0	0
$907$	37.5694	1.24747	0.623736	−	0.781635i	$-0.285614\pi$
$907$	37.5694	1.24747	0.623736	+	0.781635i	$0.285614\pi$
$908$	7.42221	0.246315
$909$	0	0
$910$	3.00000	0.0994490
$911$	15.6333	0.517955	0.258977	−	0.965883i	$-0.416615\pi$
$911$	15.6333	0.517955	0.258977	+	0.965883i	$0.416615\pi$
$912$	0	0
$913$	6.63331	0.219530
$914$	−24.3583	−0.805701
$915$	0	0
$916$	−4.90833	−0.162176
$917$	−4.69722	−0.155116
$918$	0	0
$919$	20.5139	0.676690	0.338345	−	0.941022i	$-0.390133\pi$
$919$	20.5139	0.676690	0.338345	+	0.941022i	$0.390133\pi$
$920$	12.9083	0.425575
$921$	0	0
$922$	11.0555	0.364094
$923$	−29.7250	−0.978410
$924$	0	0
$925$	−3.60555	−0.118550
$926$	46.8167	1.53849
$927$	0	0
$928$	−2.88057	−0.0945594
$929$	−50.5694	−1.65913	−0.829564	−	0.558412i	$-0.811411\pi$
$929$	−50.5694	−1.65913	−0.829564	+	0.558412i	$0.811411\pi$
$930$	0	0
$931$	0.302776	0.00992307
$932$	−0.275019	−0.00900856
$933$	0	0
$934$	6.78890	0.222140
$935$	3.90833	0.127816
$936$	0	0
$937$	15.5778	0.508904	0.254452	−	0.967085i	$-0.418105\pi$
$937$	15.5778	0.508904	0.254452	+	0.967085i	$0.418105\pi$
$938$	5.88057	0.192007
$939$	0	0
$940$	−0.119429	−0.00389536
$941$	52.2666	1.70384	0.851921	−	0.523670i	$-0.175437\pi$
$941$	52.2666	1.70384	0.851921	+	0.523670i	$0.175437\pi$
$942$	0	0
$943$	16.8167	0.547626
$944$	27.1194	0.882662
$945$	0	0
$946$	−39.9083	−1.29753
$947$	−38.3305	−1.24557	−0.622787	−	0.782391i	$-0.714000\pi$
$947$	−38.3305	−1.24557	−0.622787	+	0.782391i	$0.714000\pi$
$948$	0	0
$949$	−9.69722	−0.314785
$950$	−0.394449	−0.0127976
$951$	0	0
$952$	3.90833	0.126670
$953$	−22.5778	−0.731367	−0.365683	−	0.930739i	$-0.619165\pi$
$953$	−22.5778	−0.731367	−0.365683	+	0.930739i	$0.619165\pi$
$954$	0	0
$955$	−10.3028	−0.333390
$956$	−6.11943	−0.197916
$957$	0	0
$958$	−45.5139	−1.47049
$959$	−10.4222	−0.336551
$960$	0	0
$961$	61.2666	1.97634
$962$	−10.8167	−0.348743
$963$	0	0
$964$	7.60555	0.244958
$965$	−19.1194	−0.615476
$966$	0	0
$967$	31.8444	1.02405	0.512024	−	0.858971i	$-0.328896\pi$
$967$	31.8444	1.02405	0.512024	+	0.858971i	$0.328896\pi$
$968$	−6.00000	−0.192847
$969$	0	0
$970$	−11.0917	−0.356132
$971$	−34.4222	−1.10466	−0.552331	−	0.833625i	$-0.686261\pi$
$971$	−34.4222	−1.10466	−0.552331	+	0.833625i	$0.686261\pi$
$972$	0	0
$973$	1.60555	0.0514716
$974$	−49.5055	−1.58626
$975$	0	0
$976$	−46.6333	−1.49270
$977$	−28.4222	−0.909307	−0.454653	−	0.890668i	$-0.650237\pi$
$977$	−28.4222	−0.909307	−0.454653	+	0.890668i	$0.650237\pi$
$978$	0	0
$979$	−32.4500	−1.03711
$980$	−0.302776	−0.00967181
$981$	0	0
$982$	−15.2750	−0.487445
$983$	2.48612	0.0792950	0.0396475	−	0.999214i	$-0.487377\pi$
$983$	2.48612	0.0792950	0.0396475	+	0.999214i	$0.487377\pi$
$984$	0	0
$985$	−1.18335	−0.0377045
$986$	−2.88057	−0.0917361
$987$	0	0
$988$	0.211103	0.00671607
$989$	43.9361	1.39709
$990$	0	0
$991$	13.0555	0.414722	0.207361	−	0.978264i	$-0.433513\pi$
$991$	13.0555	0.414722	0.207361	+	0.978264i	$0.433513\pi$
$992$	16.3028	0.517614
$993$	0	0
$994$	−16.8167	−0.533392
$995$	−9.72498	−0.308303
$996$	0	0
$997$	6.81665	0.215886	0.107943	−	0.994157i	$-0.465574\pi$
$997$	6.81665	0.215886	0.107943	+	0.994157i	$0.465574\pi$
$998$	−3.43061	−0.108594
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	945.2.a.j.1.1	yes	2
3.2	odd	2		945.2.a.f.1.2	✓	2
5.4	even	2		4725.2.a.z.1.2		2
7.6	odd	2		6615.2.a.u.1.1		2
15.14	odd	2		4725.2.a.bf.1.1		2
21.20	even	2		6615.2.a.q.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
945.2.a.f.1.2	✓	2	3.2	odd	2
945.2.a.j.1.1	yes	2	1.1	even	1	trivial
4725.2.a.z.1.2		2	5.4	even	2
4725.2.a.bf.1.1		2	15.14	odd	2
6615.2.a.q.1.2		2	21.20	even	2
6615.2.a.u.1.1		2	7.6	odd	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:	\( N \)	\(=\)	\( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	945.a (trivial)

\(f(q)\)	\(=\)	\(q-1.30278 q^{2} -0.302776 q^{4} +1.00000 q^{5} +1.00000 q^{7} +3.00000 q^{8} +O(q^{10})\)	\(q-1.30278 q^{2} -0.302776 q^{4} +1.00000 q^{5} +1.00000 q^{7} +3.00000 q^{8} -1.30278 q^{10} +3.00000 q^{11} -2.30278 q^{13} -1.30278 q^{14} -3.30278 q^{16} +1.30278 q^{17} +0.302776 q^{19} -0.302776 q^{20} -3.90833 q^{22} +4.30278 q^{23} +1.00000 q^{25} +3.00000 q^{26} -0.302776 q^{28} +1.69722 q^{29} -9.60555 q^{31} -1.69722 q^{32} -1.69722 q^{34} +1.00000 q^{35} -3.60555 q^{37} -0.394449 q^{38} +3.00000 q^{40} +3.90833 q^{41} +10.2111 q^{43} -0.908327 q^{44} -5.60555 q^{46} +0.394449 q^{47} +1.00000 q^{49} -1.30278 q^{50} +0.697224 q^{52} +12.5139 q^{53} +3.00000 q^{55} +3.00000 q^{56} -2.21110 q^{58} -8.21110 q^{59} +14.1194 q^{61} +12.5139 q^{62} +8.81665 q^{64} -2.30278 q^{65} -4.51388 q^{67} -0.394449 q^{68} -1.30278 q^{70} +12.9083 q^{71} +4.21110 q^{73} +4.69722 q^{74} -0.0916731 q^{76} +3.00000 q^{77} -6.09167 q^{79} -3.30278 q^{80} -5.09167 q^{82} +2.21110 q^{83} +1.30278 q^{85} -13.3028 q^{86} +9.00000 q^{88} -10.8167 q^{89} -2.30278 q^{91} -1.30278 q^{92} -0.513878 q^{94} +0.302776 q^{95} +8.51388 q^{97} -1.30278 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + 3 q^{4} + 2 q^{5} + 2 q^{7} + 6 q^{8}+O(q^{10}) \) 2 * q + q^2 + 3 * q^4 + 2 * q^5 + 2 * q^7 + 6 * q^8	\( 2 q + q^{2} + 3 q^{4} + 2 q^{5} + 2 q^{7} + 6 q^{8} + q^{10} + 6 q^{11} - q^{13} + q^{14} - 3 q^{16} - q^{17} - 3 q^{19} + 3 q^{20} + 3 q^{22} + 5 q^{23} + 2 q^{25} + 6 q^{26} + 3 q^{28} + 7 q^{29} - 12 q^{31} - 7 q^{32} - 7 q^{34} + 2 q^{35} - 8 q^{38} + 6 q^{40} - 3 q^{41} + 6 q^{43} + 9 q^{44} - 4 q^{46} + 8 q^{47} + 2 q^{49} + q^{50} + 5 q^{52} + 7 q^{53} + 6 q^{55} + 6 q^{56} + 10 q^{58} - 2 q^{59} + 3 q^{61} + 7 q^{62} - 4 q^{64} - q^{65} + 9 q^{67} - 8 q^{68} + q^{70} + 15 q^{71} - 6 q^{73} + 13 q^{74} - 11 q^{76} + 6 q^{77} - 23 q^{79} - 3 q^{80} - 21 q^{82} - 10 q^{83} - q^{85} - 23 q^{86} + 18 q^{88} - q^{91} + q^{92} + 17 q^{94} - 3 q^{95} - q^{97} + q^{98}+O(q^{100}) \) 2 * q + q^2 + 3 * q^4 + 2 * q^5 + 2 * q^7 + 6 * q^8 + q^10 + 6 * q^11 - q^13 + q^14 - 3 * q^16 - q^17 - 3 * q^19 + 3 * q^20 + 3 * q^22 + 5 * q^23 + 2 * q^25 + 6 * q^26 + 3 * q^28 + 7 * q^29 - 12 * q^31 - 7 * q^32 - 7 * q^34 + 2 * q^35 - 8 * q^38 + 6 * q^40 - 3 * q^41 + 6 * q^43 + 9 * q^44 - 4 * q^46 + 8 * q^47 + 2 * q^49 + q^50 + 5 * q^52 + 7 * q^53 + 6 * q^55 + 6 * q^56 + 10 * q^58 - 2 * q^59 + 3 * q^61 + 7 * q^62 - 4 * q^64 - q^65 + 9 * q^67 - 8 * q^68 + q^70 + 15 * q^71 - 6 * q^73 + 13 * q^74 - 11 * q^76 + 6 * q^77 - 23 * q^79 - 3 * q^80 - 21 * q^82 - 10 * q^83 - q^85 - 23 * q^86 + 18 * q^88 - q^91 + q^92 + 17 * q^94 - 3 * q^95 - q^97 + q^98

Introduction

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