LMFDB - Embedded newform 1575.2.a.o.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1575,2,Mod(1,1575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1575, 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("1575.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1575 = 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1575.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:	$12.5764383184$
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:	no (minimal twist has level 525)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.30278$ of defining polynomial
Character	$\chi$	$=$	1575.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.30278 q^{2} +3.30278 q^{4} +1.00000 q^{7} -3.00000 q^{8} +O(q^{10})$	$q-2.30278 q^{2} +3.30278 q^{4} +1.00000 q^{7} -3.00000 q^{8} +3.00000 q^{11} -2.60555 q^{13} -2.30278 q^{14} +0.302776 q^{16} -4.60555 q^{17} +6.60555 q^{19} -6.90833 q^{22} +6.21110 q^{23} +6.00000 q^{26} +3.30278 q^{28} +7.60555 q^{29} -7.21110 q^{31} +5.30278 q^{32} +10.6056 q^{34} -4.21110 q^{37} -15.2111 q^{38} +9.60555 q^{43} +9.90833 q^{44} -14.3028 q^{46} -10.6056 q^{47} +1.00000 q^{49} -8.60555 q^{52} -3.21110 q^{53} -3.00000 q^{56} -17.5139 q^{58} +10.6056 q^{59} -1.21110 q^{61} +16.6056 q^{62} -12.8167 q^{64} +15.6056 q^{67} -15.2111 q^{68} +3.00000 q^{71} +0.605551 q^{73} +9.69722 q^{74} +21.8167 q^{76} +3.00000 q^{77} -14.8167 q^{79} +3.21110 q^{83} -22.1194 q^{86} -9.00000 q^{88} -7.81665 q^{89} -2.60555 q^{91} +20.5139 q^{92} +24.4222 q^{94} -0.788897 q^{97} -2.30278 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} + 3 q^{4} + 2 q^{7} - 6 q^{8}+O(q^{10}) $ 2 * q - q^2 + 3 * q^4 + 2 * q^7 - 6 * q^8	$ 2 q - q^{2} + 3 q^{4} + 2 q^{7} - 6 q^{8} + 6 q^{11} + 2 q^{13} - q^{14} - 3 q^{16} - 2 q^{17} + 6 q^{19} - 3 q^{22} - 2 q^{23} + 12 q^{26} + 3 q^{28} + 8 q^{29} + 7 q^{32} + 14 q^{34} + 6 q^{37} - 16 q^{38} + 12 q^{43} + 9 q^{44} - 25 q^{46} - 14 q^{47} + 2 q^{49} - 10 q^{52} + 8 q^{53} - 6 q^{56} - 17 q^{58} + 14 q^{59} + 12 q^{61} + 26 q^{62} - 4 q^{64} + 24 q^{67} - 16 q^{68} + 6 q^{71} - 6 q^{73} + 23 q^{74} + 22 q^{76} + 6 q^{77} - 8 q^{79} - 8 q^{83} - 19 q^{86} - 18 q^{88} + 6 q^{89} + 2 q^{91} + 23 q^{92} + 20 q^{94} - 16 q^{97} - q^{98}+O(q^{100}) $ 2 * q - q^2 + 3 * q^4 + 2 * q^7 - 6 * q^8 + 6 * q^11 + 2 * q^13 - q^14 - 3 * q^16 - 2 * q^17 + 6 * q^19 - 3 * q^22 - 2 * q^23 + 12 * q^26 + 3 * q^28 + 8 * q^29 + 7 * q^32 + 14 * q^34 + 6 * q^37 - 16 * q^38 + 12 * q^43 + 9 * q^44 - 25 * q^46 - 14 * q^47 + 2 * q^49 - 10 * q^52 + 8 * q^53 - 6 * q^56 - 17 * q^58 + 14 * q^59 + 12 * q^61 + 26 * q^62 - 4 * q^64 + 24 * q^67 - 16 * q^68 + 6 * q^71 - 6 * q^73 + 23 * q^74 + 22 * q^76 + 6 * q^77 - 8 * q^79 - 8 * q^83 - 19 * q^86 - 18 * q^88 + 6 * q^89 + 2 * q^91 + 23 * q^92 + 20 * q^94 - 16 * 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 $

$2$	−2.30278	−1.62831	−0.814154	−	0.580649i	$-0.802799\pi$
$2$	−2.30278	−1.62831	−0.814154	+	0.580649i	$0.802799\pi$
$3$	0	0
$3$	0	0
$4$	3.30278	1.65139
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−3.00000	−1.06066
$9$	0	0
$10$	0	0
$11$	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
$13$	−2.60555	−0.722650	−0.361325	−	0.932440i	$-0.617675\pi$
$13$	−2.60555	−0.722650	−0.361325	+	0.932440i	$0.617675\pi$
$14$	−2.30278	−0.615443
$15$	0	0
$16$	0.302776	0.0756939
$17$	−4.60555	−1.11701	−0.558505	−	0.829501i	$-0.688625\pi$
$17$	−4.60555	−1.11701	−0.558505	+	0.829501i	$0.688625\pi$
$18$	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
$21$	0	0
$22$	−6.90833	−1.47286
$23$	6.21110	1.29510	0.647552	−	0.762021i	$-0.275793\pi$
$23$	6.21110	1.29510	0.647552	+	0.762021i	$0.275793\pi$
$24$	0	0
$25$	0	0
$26$	6.00000	1.17670
$27$	0	0
$28$	3.30278	0.624166
$29$	7.60555	1.41232	0.706158	−	0.708055i	$-0.250427\pi$
$29$	7.60555	1.41232	0.706158	+	0.708055i	$0.250427\pi$
$30$	0	0
$31$	−7.21110	−1.29515	−0.647576	−	0.762001i	$-0.724217\pi$
$31$	−7.21110	−1.29515	−0.647576	+	0.762001i	$0.724217\pi$
$32$	5.30278	0.937407
$33$	0	0
$34$	10.6056	1.81884
$35$	0	0
$36$	0	0
$37$	−4.21110	−0.692301	−0.346150	−	0.938179i	$-0.612511\pi$
$37$	−4.21110	−0.692301	−0.346150	+	0.938179i	$0.612511\pi$
$38$	−15.2111	−2.46757
$39$	0	0
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	9.60555	1.46483	0.732416	−	0.680857i	$-0.238392\pi$
$43$	9.60555	1.46483	0.732416	+	0.680857i	$0.238392\pi$
$44$	9.90833	1.49374
$45$	0	0
$46$	−14.3028	−2.10883
$47$	−10.6056	−1.54698	−0.773489	−	0.633809i	$-0.781490\pi$
$47$	−10.6056	−1.54698	−0.773489	+	0.633809i	$0.781490\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	−8.60555	−1.19338
$53$	−3.21110	−0.441079	−0.220539	−	0.975378i	$-0.570782\pi$
$53$	−3.21110	−0.441079	−0.220539	+	0.975378i	$0.570782\pi$
$54$	0	0
$55$	0	0
$56$	−3.00000	−0.400892
$57$	0	0
$58$	−17.5139	−2.29968
$59$	10.6056	1.38073	0.690363	−	0.723464i	$-0.257451\pi$
$59$	10.6056	1.38073	0.690363	+	0.723464i	$0.257451\pi$
$60$	0	0
$61$	−1.21110	−0.155066	−0.0775329	−	0.996990i	$-0.524704\pi$
$61$	−1.21110	−0.155066	−0.0775329	+	0.996990i	$0.524704\pi$
$62$	16.6056	2.10891
$63$	0	0
$64$	−12.8167	−1.60208
$65$	0	0
$66$	0	0
$67$	15.6056	1.90652	0.953261	−	0.302149i	$-0.0977040\pi$
$67$	15.6056	1.90652	0.953261	+	0.302149i	$0.0977040\pi$
$68$	−15.2111	−1.84462
$69$	0	0
$70$	0	0
$71$	3.00000	0.356034	0.178017	−	0.984027i	$-0.443032\pi$
$71$	3.00000	0.356034	0.178017	+	0.984027i	$0.443032\pi$
$72$	0	0
$73$	0.605551	0.0708744	0.0354372	−	0.999372i	$-0.488718\pi$
$73$	0.605551	0.0708744	0.0354372	+	0.999372i	$0.488718\pi$
$74$	9.69722	1.12728
$75$	0	0
$76$	21.8167	2.50254
$77$	3.00000	0.341882
$78$	0	0
$79$	−14.8167	−1.66700	−0.833502	−	0.552517i	$-0.813668\pi$
$79$	−14.8167	−1.66700	−0.833502	+	0.552517i	$0.813668\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	3.21110	0.352464	0.176232	−	0.984349i	$-0.443609\pi$
$83$	3.21110	0.352464	0.176232	+	0.984349i	$0.443609\pi$
$84$	0	0
$85$	0	0
$86$	−22.1194	−2.38520
$87$	0	0
$88$	−9.00000	−0.959403
$89$	−7.81665	−0.828564	−0.414282	−	0.910149i	$-0.635967\pi$
$89$	−7.81665	−0.828564	−0.414282	+	0.910149i	$0.635967\pi$
$90$	0	0
$91$	−2.60555	−0.273136
$92$	20.5139	2.13872
$93$	0	0
$94$	24.4222	2.51896
$95$	0	0
$96$	0	0
$97$	−0.788897	−0.0801004	−0.0400502	−	0.999198i	$-0.512752\pi$
$97$	−0.788897	−0.0801004	−0.0400502	+	0.999198i	$0.512752\pi$
$98$	−2.30278	−0.232615
$99$	0	0
$100$	0	0
$101$	−16.6056	−1.65231	−0.826157	−	0.563440i	$-0.809478\pi$
$101$	−16.6056	−1.65231	−0.826157	+	0.563440i	$0.809478\pi$
$102$	0	0
$103$	3.81665	0.376066	0.188033	−	0.982163i	$-0.439789\pi$
$103$	3.81665	0.376066	0.188033	+	0.982163i	$0.439789\pi$
$104$	7.81665	0.766486
$105$	0	0
$106$	7.39445	0.718212
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	2.21110	0.211785	0.105893	−	0.994378i	$-0.466230\pi$
$109$	2.21110	0.211785	0.105893	+	0.994378i	$0.466230\pi$
$110$	0	0
$111$	0	0
$112$	0.302776	0.0286096
$113$	10.8167	1.01755	0.508773	−	0.860901i	$-0.330099\pi$
$113$	10.8167	1.01755	0.508773	+	0.860901i	$0.330099\pi$
$114$	0	0
$115$	0	0
$116$	25.1194	2.33228
$117$	0	0
$118$	−24.4222	−2.24825
$119$	−4.60555	−0.422190
$120$	0	0
$121$	−2.00000	−0.181818
$122$	2.78890	0.252495
$123$	0	0
$124$	−23.8167	−2.13880
$125$	0	0
$126$	0	0
$127$	12.8167	1.13729	0.568647	−	0.822582i	$-0.307467\pi$
$127$	12.8167	1.13729	0.568647	+	0.822582i	$0.307467\pi$
$128$	18.9083	1.67128
$129$	0	0
$130$	0	0
$131$	7.39445	0.646056	0.323028	−	0.946389i	$-0.395299\pi$
$131$	7.39445	0.646056	0.323028	+	0.946389i	$0.395299\pi$
$132$	0	0
$133$	6.60555	0.572774
$134$	−35.9361	−3.10440
$135$	0	0
$136$	13.8167	1.18477
$137$	3.21110	0.274343	0.137172	−	0.990547i	$-0.456199\pi$
$137$	3.21110	0.274343	0.137172	+	0.990547i	$0.456199\pi$
$138$	0	0
$139$	19.0278	1.61391	0.806957	−	0.590611i	$-0.201113\pi$
$139$	19.0278	1.61391	0.806957	+	0.590611i	$0.201113\pi$
$140$	0	0
$141$	0	0
$142$	−6.90833	−0.579734
$143$	−7.81665	−0.653661
$144$	0	0
$145$	0	0
$146$	−1.39445	−0.115405
$147$	0	0
$148$	−13.9083	−1.14326
$149$	16.3944	1.34309	0.671543	−	0.740966i	$-0.265632\pi$
$149$	16.3944	1.34309	0.671543	+	0.740966i	$0.265632\pi$
$150$	0	0
$151$	6.81665	0.554731	0.277366	−	0.960764i	$-0.410539\pi$
$151$	6.81665	0.554731	0.277366	+	0.960764i	$0.410539\pi$
$152$	−19.8167	−1.60734
$153$	0	0
$154$	−6.90833	−0.556689
$155$	0	0
$156$	0	0
$157$	14.4222	1.15102	0.575509	−	0.817796i	$-0.304804\pi$
$157$	14.4222	1.15102	0.575509	+	0.817796i	$0.304804\pi$
$158$	34.1194	2.71440
$159$	0	0
$160$	0	0
$161$	6.21110	0.489503
$162$	0	0
$163$	17.2111	1.34808	0.674039	−	0.738696i	$-0.264558\pi$
$163$	17.2111	1.34808	0.674039	+	0.738696i	$0.264558\pi$
$164$	0	0
$165$	0	0
$166$	−7.39445	−0.573921
$167$	17.0278	1.31765	0.658824	−	0.752297i	$-0.271054\pi$
$167$	17.0278	1.31765	0.658824	+	0.752297i	$0.271054\pi$
$168$	0	0
$169$	−6.21110	−0.477777
$170$	0	0
$171$	0	0
$172$	31.7250	2.41901
$173$	7.81665	0.594289	0.297145	−	0.954832i	$-0.403966\pi$
$173$	7.81665	0.594289	0.297145	+	0.954832i	$0.403966\pi$
$174$	0	0
$175$	0	0
$176$	0.908327	0.0684677
$177$	0	0
$178$	18.0000	1.34916
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	12.6056	0.936963	0.468482	−	0.883473i	$-0.344801\pi$
$181$	12.6056	0.936963	0.468482	+	0.883473i	$0.344801\pi$
$182$	6.00000	0.444750
$183$	0	0
$184$	−18.6333	−1.37367
$185$	0	0
$186$	0	0
$187$	−13.8167	−1.01037
$188$	−35.0278	−2.55466
$189$	0	0
$190$	0	0
$191$	2.78890	0.201798	0.100899	−	0.994897i	$-0.467828\pi$
$191$	2.78890	0.201798	0.100899	+	0.994897i	$0.467828\pi$
$192$	0	0
$193$	8.21110	0.591048	0.295524	−	0.955335i	$-0.404506\pi$
$193$	8.21110	0.591048	0.295524	+	0.955335i	$0.404506\pi$
$194$	1.81665	0.130428
$195$	0	0
$196$	3.30278	0.235913
$197$	−16.8167	−1.19814	−0.599068	−	0.800698i	$-0.704462\pi$
$197$	−16.8167	−1.19814	−0.599068	+	0.800698i	$0.704462\pi$
$198$	0	0
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	0	0
$201$	0	0
$202$	38.2389	2.69048
$203$	7.60555	0.533805
$204$	0	0
$205$	0	0
$206$	−8.78890	−0.612352
$207$	0	0
$208$	−0.788897	−0.0547002
$209$	19.8167	1.37075
$210$	0	0
$211$	26.4222	1.81898	0.909490	−	0.415726i	$-0.136473\pi$
$211$	26.4222	1.81898	0.909490	+	0.415726i	$0.136473\pi$
$212$	−10.6056	−0.728392
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−7.21110	−0.489522
$218$	−5.09167	−0.344852
$219$	0	0
$220$	0	0
$221$	12.0000	0.807207
$222$	0	0
$223$	15.3944	1.03089	0.515444	−	0.856923i	$-0.327627\pi$
$223$	15.3944	1.03089	0.515444	+	0.856923i	$0.327627\pi$
$224$	5.30278	0.354307
$225$	0	0
$226$	−24.9083	−1.65688
$227$	−22.6056	−1.50038	−0.750192	−	0.661221i	$-0.770039\pi$
$227$	−22.6056	−1.50038	−0.750192	+	0.661221i	$0.770039\pi$
$228$	0	0
$229$	−7.21110	−0.476523	−0.238262	−	0.971201i	$-0.576578\pi$
$229$	−7.21110	−0.476523	−0.238262	+	0.971201i	$0.576578\pi$
$230$	0	0
$231$	0	0
$232$	−22.8167	−1.49799
$233$	1.18335	0.0775236	0.0387618	−	0.999248i	$-0.487659\pi$
$233$	1.18335	0.0775236	0.0387618	+	0.999248i	$0.487659\pi$
$234$	0	0
$235$	0	0
$236$	35.0278	2.28011
$237$	0	0
$238$	10.6056	0.687456
$239$	−14.7889	−0.956614	−0.478307	−	0.878193i	$-0.658749\pi$
$239$	−14.7889	−0.956614	−0.478307	+	0.878193i	$0.658749\pi$
$240$	0	0
$241$	9.39445	0.605150	0.302575	−	0.953126i	$-0.402154\pi$
$241$	9.39445	0.605150	0.302575	+	0.953126i	$0.402154\pi$
$242$	4.60555	0.296056
$243$	0	0
$244$	−4.00000	−0.256074
$245$	0	0
$246$	0	0
$247$	−17.2111	−1.09512
$248$	21.6333	1.37372
$249$	0	0
$250$	0	0
$251$	13.8167	0.872099	0.436050	−	0.899923i	$-0.356377\pi$
$251$	13.8167	0.872099	0.436050	+	0.899923i	$0.356377\pi$
$252$	0	0
$253$	18.6333	1.17147
$254$	−29.5139	−1.85187
$255$	0	0
$256$	−17.9083	−1.11927
$257$	21.6333	1.34945	0.674724	−	0.738070i	$-0.264263\pi$
$257$	21.6333	1.34945	0.674724	+	0.738070i	$0.264263\pi$
$258$	0	0
$259$	−4.21110	−0.261665
$260$	0	0
$261$	0	0
$262$	−17.0278	−1.05198
$263$	5.78890	0.356959	0.178479	−	0.983944i	$-0.442882\pi$
$263$	5.78890	0.356959	0.178479	+	0.983944i	$0.442882\pi$
$264$	0	0
$265$	0	0
$266$	−15.2111	−0.932653
$267$	0	0
$268$	51.5416	3.14841
$269$	3.21110	0.195784	0.0978922	−	0.995197i	$-0.468790\pi$
$269$	3.21110	0.195784	0.0978922	+	0.995197i	$0.468790\pi$
$270$	0	0
$271$	−26.6056	−1.61617	−0.808086	−	0.589064i	$-0.799496\pi$
$271$	−26.6056	−1.61617	−0.808086	+	0.589064i	$0.799496\pi$
$272$	−1.39445	−0.0845509
$273$	0	0
$274$	−7.39445	−0.446715
$275$	0	0
$276$	0	0
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	−43.8167	−2.62795
$279$	0	0
$280$	0	0
$281$	1.18335	0.0705925	0.0352963	−	0.999377i	$-0.488763\pi$
$281$	1.18335	0.0705925	0.0352963	+	0.999377i	$0.488763\pi$
$282$	0	0
$283$	−13.6333	−0.810416	−0.405208	−	0.914225i	$-0.632801\pi$
$283$	−13.6333	−0.810416	−0.405208	+	0.914225i	$0.632801\pi$
$284$	9.90833	0.587951
$285$	0	0
$286$	18.0000	1.06436
$287$	0	0
$288$	0	0
$289$	4.21110	0.247712
$290$	0	0
$291$	0	0
$292$	2.00000	0.117041
$293$	−10.6056	−0.619583	−0.309791	−	0.950805i	$-0.600259\pi$
$293$	−10.6056	−0.619583	−0.309791	+	0.950805i	$0.600259\pi$
$294$	0	0
$295$	0	0
$296$	12.6333	0.734296
$297$	0	0
$298$	−37.7527	−2.18696
$299$	−16.1833	−0.935907
$300$	0	0
$301$	9.60555	0.553655
$302$	−15.6972	−0.903274
$303$	0	0
$304$	2.00000	0.114708
$305$	0	0
$306$	0	0
$307$	−8.60555	−0.491145	−0.245572	−	0.969378i	$-0.578976\pi$
$307$	−8.60555	−0.491145	−0.245572	+	0.969378i	$0.578976\pi$
$308$	9.90833	0.564579
$309$	0	0
$310$	0	0
$311$	−13.8167	−0.783471	−0.391735	−	0.920078i	$-0.628125\pi$
$311$	−13.8167	−0.783471	−0.391735	+	0.920078i	$0.628125\pi$
$312$	0	0
$313$	−23.8167	−1.34620	−0.673098	−	0.739553i	$-0.735037\pi$
$313$	−23.8167	−1.34620	−0.673098	+	0.739553i	$0.735037\pi$
$314$	−33.2111	−1.87421
$315$	0	0
$316$	−48.9361	−2.75287
$317$	16.3944	0.920804	0.460402	−	0.887711i	$-0.347705\pi$
$317$	16.3944	0.920804	0.460402	+	0.887711i	$0.347705\pi$
$318$	0	0
$319$	22.8167	1.27749
$320$	0	0
$321$	0	0
$322$	−14.3028	−0.797063
$323$	−30.4222	−1.69274
$324$	0	0
$325$	0	0
$326$	−39.6333	−2.19509
$327$	0	0
$328$	0	0
$329$	−10.6056	−0.584703
$330$	0	0
$331$	−21.2389	−1.16739	−0.583697	−	0.811972i	$-0.698394\pi$
$331$	−21.2389	−1.16739	−0.583697	+	0.811972i	$0.698394\pi$
$332$	10.6056	0.582055
$333$	0	0
$334$	−39.2111	−2.14554
$335$	0	0
$336$	0	0
$337$	−7.21110	−0.392814	−0.196407	−	0.980522i	$-0.562927\pi$
$337$	−7.21110	−0.392814	−0.196407	+	0.980522i	$0.562927\pi$
$338$	14.3028	0.777968
$339$	0	0
$340$	0	0
$341$	−21.6333	−1.17151
$342$	0	0
$343$	1.00000	0.0539949
$344$	−28.8167	−1.55369
$345$	0	0
$346$	−18.0000	−0.967686
$347$	−30.2111	−1.62182	−0.810908	−	0.585173i	$-0.801027\pi$
$347$	−30.2111	−1.62182	−0.810908	+	0.585173i	$0.801027\pi$
$348$	0	0
$349$	31.4500	1.68348	0.841739	−	0.539885i	$-0.181532\pi$
$349$	31.4500	1.68348	0.841739	+	0.539885i	$0.181532\pi$
$350$	0	0
$351$	0	0
$352$	15.9083	0.847917
$353$	30.0000	1.59674	0.798369	−	0.602168i	$-0.205696\pi$
$353$	30.0000	1.59674	0.798369	+	0.602168i	$0.205696\pi$
$354$	0	0
$355$	0	0
$356$	−25.8167	−1.36828
$357$	0	0
$358$	0	0
$359$	−24.6333	−1.30010	−0.650048	−	0.759893i	$-0.725251\pi$
$359$	−24.6333	−1.30010	−0.650048	+	0.759893i	$0.725251\pi$
$360$	0	0
$361$	24.6333	1.29649
$362$	−29.0278	−1.52567
$363$	0	0
$364$	−8.60555	−0.451053
$365$	0	0
$366$	0	0
$367$	14.4222	0.752833	0.376416	−	0.926451i	$-0.377156\pi$
$367$	14.4222	0.752833	0.376416	+	0.926451i	$0.377156\pi$
$368$	1.88057	0.0980315
$369$	0	0
$370$	0	0
$371$	−3.21110	−0.166712
$372$	0	0
$373$	−1.00000	−0.0517780	−0.0258890	−	0.999665i	$-0.508242\pi$
$373$	−1.00000	−0.0517780	−0.0258890	+	0.999665i	$0.508242\pi$
$374$	31.8167	1.64520
$375$	0	0
$376$	31.8167	1.64082
$377$	−19.8167	−1.02061
$378$	0	0
$379$	−14.8167	−0.761080	−0.380540	−	0.924764i	$-0.624262\pi$
$379$	−14.8167	−0.761080	−0.380540	+	0.924764i	$0.624262\pi$
$380$	0	0
$381$	0	0
$382$	−6.42221	−0.328589
$383$	26.2389	1.34074	0.670372	−	0.742026i	$-0.266135\pi$
$383$	26.2389	1.34074	0.670372	+	0.742026i	$0.266135\pi$
$384$	0	0
$385$	0	0
$386$	−18.9083	−0.962408
$387$	0	0
$388$	−2.60555	−0.132277
$389$	28.8167	1.46106	0.730531	−	0.682879i	$-0.239273\pi$
$389$	28.8167	1.46106	0.730531	+	0.682879i	$0.239273\pi$
$390$	0	0
$391$	−28.6056	−1.44664
$392$	−3.00000	−0.151523
$393$	0	0
$394$	38.7250	1.95094
$395$	0	0
$396$	0	0
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	−18.4222	−0.923422
$399$	0	0
$400$	0	0
$401$	16.8167	0.839784	0.419892	−	0.907574i	$-0.362068\pi$
$401$	16.8167	0.839784	0.419892	+	0.907574i	$0.362068\pi$
$402$	0	0
$403$	18.7889	0.935942
$404$	−54.8444	−2.72861
$405$	0	0
$406$	−17.5139	−0.869199
$407$	−12.6333	−0.626210
$408$	0	0
$409$	−11.8167	−0.584296	−0.292148	−	0.956373i	$-0.594370\pi$
$409$	−11.8167	−0.584296	−0.292148	+	0.956373i	$0.594370\pi$
$410$	0	0
$411$	0	0
$412$	12.6056	0.621031
$413$	10.6056	0.521865
$414$	0	0
$415$	0	0
$416$	−13.8167	−0.677417
$417$	0	0
$418$	−45.6333	−2.23200
$419$	−6.00000	−0.293119	−0.146560	−	0.989202i	$-0.546820\pi$
$419$	−6.00000	−0.293119	−0.146560	+	0.989202i	$0.546820\pi$
$420$	0	0
$421$	−10.2111	−0.497659	−0.248829	−	0.968547i	$-0.580046\pi$
$421$	−10.2111	−0.497659	−0.248829	+	0.968547i	$0.580046\pi$
$422$	−60.8444	−2.96186
$423$	0	0
$424$	9.63331	0.467835
$425$	0	0
$426$	0	0
$427$	−1.21110	−0.0586094
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−5.57779	−0.268673	−0.134336	−	0.990936i	$-0.542890\pi$
$431$	−5.57779	−0.268673	−0.134336	+	0.990936i	$0.542890\pi$
$432$	0	0
$433$	−37.2111	−1.78825	−0.894126	−	0.447816i	$-0.852202\pi$
$433$	−37.2111	−1.78825	−0.894126	+	0.447816i	$0.852202\pi$
$434$	16.6056	0.797092
$435$	0	0
$436$	7.30278	0.349740
$437$	41.0278	1.96262
$438$	0	0
$439$	6.60555	0.315266	0.157633	−	0.987498i	$-0.449614\pi$
$439$	6.60555	0.315266	0.157633	+	0.987498i	$0.449614\pi$
$440$	0	0
$441$	0	0
$442$	−27.6333	−1.31438
$443$	−15.6333	−0.742761	−0.371380	−	0.928481i	$-0.621115\pi$
$443$	−15.6333	−0.742761	−0.371380	+	0.928481i	$0.621115\pi$
$444$	0	0
$445$	0	0
$446$	−35.4500	−1.67860
$447$	0	0
$448$	−12.8167	−0.605530
$449$	34.8167	1.64310	0.821550	−	0.570137i	$-0.193110\pi$
$449$	34.8167	1.64310	0.821550	+	0.570137i	$0.193110\pi$
$450$	0	0
$451$	0	0
$452$	35.7250	1.68036
$453$	0	0
$454$	52.0555	2.44309
$455$	0	0
$456$	0	0
$457$	−3.78890	−0.177237	−0.0886186	−	0.996066i	$-0.528245\pi$
$457$	−3.78890	−0.177237	−0.0886186	+	0.996066i	$0.528245\pi$
$458$	16.6056	0.775926
$459$	0	0
$460$	0	0
$461$	26.2389	1.22207	0.611033	−	0.791605i	$-0.290754\pi$
$461$	26.2389	1.22207	0.611033	+	0.791605i	$0.290754\pi$
$462$	0	0
$463$	17.2111	0.799868	0.399934	−	0.916544i	$-0.369033\pi$
$463$	17.2111	0.799868	0.399934	+	0.916544i	$0.369033\pi$
$464$	2.30278	0.106904
$465$	0	0
$466$	−2.72498	−0.126232
$467$	0.422205	0.0195373	0.00976866	−	0.999952i	$-0.496890\pi$
$467$	0.422205	0.0195373	0.00976866	+	0.999952i	$0.496890\pi$
$468$	0	0
$469$	15.6056	0.720597
$470$	0	0
$471$	0	0
$472$	−31.8167	−1.46448
$473$	28.8167	1.32499
$474$	0	0
$475$	0	0
$476$	−15.2111	−0.697200
$477$	0	0
$478$	34.0555	1.55766
$479$	−34.6056	−1.58117	−0.790584	−	0.612354i	$-0.790223\pi$
$479$	−34.6056	−1.58117	−0.790584	+	0.612354i	$0.790223\pi$
$480$	0	0
$481$	10.9722	0.500291
$482$	−21.6333	−0.985370
$483$	0	0
$484$	−6.60555	−0.300252
$485$	0	0
$486$	0	0
$487$	−20.8167	−0.943293	−0.471646	−	0.881788i	$-0.656340\pi$
$487$	−20.8167	−0.943293	−0.471646	+	0.881788i	$0.656340\pi$
$488$	3.63331	0.164472
$489$	0	0
$490$	0	0
$491$	3.00000	0.135388	0.0676941	−	0.997706i	$-0.478436\pi$
$491$	3.00000	0.135388	0.0676941	+	0.997706i	$0.478436\pi$
$492$	0	0
$493$	−35.0278	−1.57757
$494$	39.6333	1.78319
$495$	0	0
$496$	−2.18335	−0.0980351
$497$	3.00000	0.134568
$498$	0	0
$499$	−28.0000	−1.25345	−0.626726	−	0.779240i	$-0.715605\pi$
$499$	−28.0000	−1.25345	−0.626726	+	0.779240i	$0.715605\pi$
$500$	0	0
$501$	0	0
$502$	−31.8167	−1.42005
$503$	8.78890	0.391878	0.195939	−	0.980616i	$-0.437225\pi$
$503$	8.78890	0.391878	0.195939	+	0.980616i	$0.437225\pi$
$504$	0	0
$505$	0	0
$506$	−42.9083	−1.90751
$507$	0	0
$508$	42.3305	1.87811
$509$	−1.39445	−0.0618079	−0.0309039	−	0.999522i	$-0.509839\pi$
$509$	−1.39445	−0.0618079	−0.0309039	+	0.999522i	$0.509839\pi$
$510$	0	0
$511$	0.605551	0.0267880
$512$	3.42221	0.151242
$513$	0	0
$514$	−49.8167	−2.19732
$515$	0	0
$516$	0	0
$517$	−31.8167	−1.39929
$518$	9.69722	0.426072
$519$	0	0
$520$	0	0
$521$	36.4222	1.59569	0.797843	−	0.602865i	$-0.205974\pi$
$521$	36.4222	1.59569	0.797843	+	0.602865i	$0.205974\pi$
$522$	0	0
$523$	−16.0000	−0.699631	−0.349816	−	0.936819i	$-0.613756\pi$
$523$	−16.0000	−0.699631	−0.349816	+	0.936819i	$0.613756\pi$
$524$	24.4222	1.06689
$525$	0	0
$526$	−13.3305	−0.581239
$527$	33.2111	1.44670
$528$	0	0
$529$	15.5778	0.677295
$530$	0	0
$531$	0	0
$532$	21.8167	0.945872
$533$	0	0
$534$	0	0
$535$	0	0
$536$	−46.8167	−2.02217
$537$	0	0
$538$	−7.39445	−0.318797
$539$	3.00000	0.129219
$540$	0	0
$541$	−31.4222	−1.35095	−0.675473	−	0.737385i	$-0.736061\pi$
$541$	−31.4222	−1.35095	−0.675473	+	0.737385i	$0.736061\pi$
$542$	61.2666	2.63163
$543$	0	0
$544$	−24.4222	−1.04709
$545$	0	0
$546$	0	0
$547$	−29.6056	−1.26584	−0.632921	−	0.774216i	$-0.718144\pi$
$547$	−29.6056	−1.26584	−0.632921	+	0.774216i	$0.718144\pi$
$548$	10.6056	0.453047
$549$	0	0
$550$	0	0
$551$	50.2389	2.14025
$552$	0	0
$553$	−14.8167	−0.630068
$554$	23.0278	0.978356
$555$	0	0
$556$	62.8444	2.66520
$557$	−11.2389	−0.476206	−0.238103	−	0.971240i	$-0.576526\pi$
$557$	−11.2389	−0.476206	−0.238103	+	0.971240i	$0.576526\pi$
$558$	0	0
$559$	−25.0278	−1.05856
$560$	0	0
$561$	0	0
$562$	−2.72498	−0.114946
$563$	−15.2111	−0.641072	−0.320536	−	0.947236i	$-0.603863\pi$
$563$	−15.2111	−0.641072	−0.320536	+	0.947236i	$0.603863\pi$
$564$	0	0
$565$	0	0
$566$	31.3944	1.31961
$567$	0	0
$568$	−9.00000	−0.377632
$569$	1.18335	0.0496085	0.0248042	−	0.999692i	$-0.492104\pi$
$569$	1.18335	0.0496085	0.0248042	+	0.999692i	$0.492104\pi$
$570$	0	0
$571$	−5.60555	−0.234585	−0.117293	−	0.993097i	$-0.537422\pi$
$571$	−5.60555	−0.234585	−0.117293	+	0.993097i	$0.537422\pi$
$572$	−25.8167	−1.07945
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	30.6056	1.27413	0.637063	−	0.770812i	$-0.280149\pi$
$577$	30.6056	1.27413	0.637063	+	0.770812i	$0.280149\pi$
$578$	−9.69722	−0.403351
$579$	0	0
$580$	0	0
$581$	3.21110	0.133219
$582$	0	0
$583$	−9.63331	−0.398971
$584$	−1.81665	−0.0751737
$585$	0	0
$586$	24.4222	1.00887
$587$	−37.8167	−1.56086	−0.780430	−	0.625243i	$-0.785000\pi$
$587$	−37.8167	−1.56086	−0.780430	+	0.625243i	$0.785000\pi$
$588$	0	0
$589$	−47.6333	−1.96270
$590$	0	0
$591$	0	0
$592$	−1.27502	−0.0524030
$593$	−27.6333	−1.13476	−0.567382	−	0.823455i	$-0.692044\pi$
$593$	−27.6333	−1.13476	−0.567382	+	0.823455i	$0.692044\pi$
$594$	0	0
$595$	0	0
$596$	54.1472	2.21796
$597$	0	0
$598$	37.2666	1.52395
$599$	−5.78890	−0.236528	−0.118264	−	0.992982i	$-0.537733\pi$
$599$	−5.78890	−0.236528	−0.118264	+	0.992982i	$0.537733\pi$
$600$	0	0
$601$	17.2111	0.702056	0.351028	−	0.936365i	$-0.385832\pi$
$601$	17.2111	0.702056	0.351028	+	0.936365i	$0.385832\pi$
$602$	−22.1194	−0.901521
$603$	0	0
$604$	22.5139	0.916077
$605$	0	0
$606$	0	0
$607$	−23.8167	−0.966688	−0.483344	−	0.875430i	$-0.660578\pi$
$607$	−23.8167	−0.966688	−0.483344	+	0.875430i	$0.660578\pi$
$608$	35.0278	1.42056
$609$	0	0
$610$	0	0
$611$	27.6333	1.11792
$612$	0	0
$613$	−49.4222	−1.99614	−0.998072	−	0.0620663i	$-0.980231\pi$
$613$	−49.4222	−1.99614	−0.998072	+	0.0620663i	$0.980231\pi$
$614$	19.8167	0.799735
$615$	0	0
$616$	−9.00000	−0.362620
$617$	−22.8167	−0.918564	−0.459282	−	0.888291i	$-0.651893\pi$
$617$	−22.8167	−0.918564	−0.459282	+	0.888291i	$0.651893\pi$
$618$	0	0
$619$	−2.60555	−0.104726	−0.0523630	−	0.998628i	$-0.516675\pi$
$619$	−2.60555	−0.104726	−0.0523630	+	0.998628i	$0.516675\pi$
$620$	0	0
$621$	0	0
$622$	31.8167	1.27573
$623$	−7.81665	−0.313168
$624$	0	0
$625$	0	0
$626$	54.8444	2.19202
$627$	0	0
$628$	47.6333	1.90078
$629$	19.3944	0.773307
$630$	0	0
$631$	22.0278	0.876911	0.438456	−	0.898753i	$-0.355526\pi$
$631$	22.0278	0.876911	0.438456	+	0.898753i	$0.355526\pi$
$632$	44.4500	1.76812
$633$	0	0
$634$	−37.7527	−1.49935
$635$	0	0
$636$	0	0
$637$	−2.60555	−0.103236
$638$	−52.5416	−2.08014
$639$	0	0
$640$	0	0
$641$	−22.8167	−0.901204	−0.450602	−	0.892725i	$-0.648791\pi$
$641$	−22.8167	−0.901204	−0.450602	+	0.892725i	$0.648791\pi$
$642$	0	0
$643$	38.4222	1.51522	0.757612	−	0.652705i	$-0.226366\pi$
$643$	38.4222	1.51522	0.757612	+	0.652705i	$0.226366\pi$
$644$	20.5139	0.808360
$645$	0	0
$646$	70.0555	2.75630
$647$	−6.00000	−0.235884	−0.117942	−	0.993020i	$-0.537630\pi$
$647$	−6.00000	−0.235884	−0.117942	+	0.993020i	$0.537630\pi$
$648$	0	0
$649$	31.8167	1.24891
$650$	0	0
$651$	0	0
$652$	56.8444	2.22620
$653$	3.21110	0.125660	0.0628301	−	0.998024i	$-0.479987\pi$
$653$	3.21110	0.125660	0.0628301	+	0.998024i	$0.479987\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	24.4222	0.952077
$659$	12.0000	0.467454	0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000	0.467454	0.233727	+	0.972302i	$0.424908\pi$
$660$	0	0
$661$	−20.1833	−0.785041	−0.392521	−	0.919743i	$-0.628397\pi$
$661$	−20.1833	−0.785041	−0.392521	+	0.919743i	$0.628397\pi$
$662$	48.9083	1.90088
$663$	0	0
$664$	−9.63331	−0.373845
$665$	0	0
$666$	0	0
$667$	47.2389	1.82910
$668$	56.2389	2.17595
$669$	0	0
$670$	0	0
$671$	−3.63331	−0.140262
$672$	0	0
$673$	−10.0000	−0.385472	−0.192736	−	0.981251i	$-0.561736\pi$
$673$	−10.0000	−0.385472	−0.192736	+	0.981251i	$0.561736\pi$
$674$	16.6056	0.639622
$675$	0	0
$676$	−20.5139	−0.788995
$677$	8.78890	0.337785	0.168892	−	0.985634i	$-0.445981\pi$
$677$	8.78890	0.337785	0.168892	+	0.985634i	$0.445981\pi$
$678$	0	0
$679$	−0.788897	−0.0302751
$680$	0	0
$681$	0	0
$682$	49.8167	1.90758
$683$	18.6333	0.712984	0.356492	−	0.934298i	$-0.383973\pi$
$683$	18.6333	0.712984	0.356492	+	0.934298i	$0.383973\pi$
$684$	0	0
$685$	0	0
$686$	−2.30278	−0.0879204
$687$	0	0
$688$	2.90833	0.110879
$689$	8.36669	0.318746
$690$	0	0
$691$	41.2111	1.56774	0.783872	−	0.620922i	$-0.213242\pi$
$691$	41.2111	1.56774	0.783872	+	0.620922i	$0.213242\pi$
$692$	25.8167	0.981402
$693$	0	0
$694$	69.5694	2.64082
$695$	0	0
$696$	0	0
$697$	0	0
$698$	−72.4222	−2.74122
$699$	0	0
$700$	0	0
$701$	20.7889	0.785186	0.392593	−	0.919712i	$-0.371578\pi$
$701$	20.7889	0.785186	0.392593	+	0.919712i	$0.371578\pi$
$702$	0	0
$703$	−27.8167	−1.04912
$704$	−38.4500	−1.44914
$705$	0	0
$706$	−69.0833	−2.59998
$707$	−16.6056	−0.624516
$708$	0	0
$709$	−34.8444	−1.30861	−0.654305	−	0.756231i	$-0.727039\pi$
$709$	−34.8444	−1.30861	−0.654305	+	0.756231i	$0.727039\pi$
$710$	0	0
$711$	0	0
$712$	23.4500	0.878824
$713$	−44.7889	−1.67736
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	56.7250	2.11696
$719$	−26.7889	−0.999057	−0.499529	−	0.866297i	$-0.666493\pi$
$719$	−26.7889	−0.999057	−0.499529	+	0.866297i	$0.666493\pi$
$720$	0	0
$721$	3.81665	0.142140
$722$	−56.7250	−2.11109
$723$	0	0
$724$	41.6333	1.54729
$725$	0	0
$726$	0	0
$727$	−23.3944	−0.867652	−0.433826	−	0.900997i	$-0.642837\pi$
$727$	−23.3944	−0.867652	−0.433826	+	0.900997i	$0.642837\pi$
$728$	7.81665	0.289704
$729$	0	0
$730$	0	0
$731$	−44.2389	−1.63623
$732$	0	0
$733$	20.0000	0.738717	0.369358	−	0.929287i	$-0.379577\pi$
$733$	20.0000	0.738717	0.369358	+	0.929287i	$0.379577\pi$
$734$	−33.2111	−1.22584
$735$	0	0
$736$	32.9361	1.21404
$737$	46.8167	1.72451
$738$	0	0
$739$	3.18335	0.117101	0.0585506	−	0.998284i	$-0.481352\pi$
$739$	3.18335	0.117101	0.0585506	+	0.998284i	$0.481352\pi$
$740$	0	0
$741$	0	0
$742$	7.39445	0.271459
$743$	27.6333	1.01377	0.506884	−	0.862014i	$-0.330797\pi$
$743$	27.6333	1.01377	0.506884	+	0.862014i	$0.330797\pi$
$744$	0	0
$745$	0	0
$746$	2.30278	0.0843106
$747$	0	0
$748$	−45.6333	−1.66852
$749$	0	0
$750$	0	0
$751$	−0.366692	−0.0133808	−0.00669040	−	0.999978i	$-0.502130\pi$
$751$	−0.366692	−0.0133808	−0.00669040	+	0.999978i	$0.502130\pi$
$752$	−3.21110	−0.117097
$753$	0	0
$754$	45.6333	1.66187
$755$	0	0
$756$	0	0
$757$	41.0000	1.49017	0.745085	−	0.666969i	$-0.232409\pi$
$757$	41.0000	1.49017	0.745085	+	0.666969i	$0.232409\pi$
$758$	34.1194	1.23927
$759$	0	0
$760$	0	0
$761$	27.6333	1.00171	0.500853	−	0.865532i	$-0.333020\pi$
$761$	27.6333	1.00171	0.500853	+	0.865532i	$0.333020\pi$
$762$	0	0
$763$	2.21110	0.0800473
$764$	9.21110	0.333246
$765$	0	0
$766$	−60.4222	−2.18314
$767$	−27.6333	−0.997781
$768$	0	0
$769$	−31.6333	−1.14073	−0.570363	−	0.821393i	$-0.693198\pi$
$769$	−31.6333	−1.14073	−0.570363	+	0.821393i	$0.693198\pi$
$770$	0	0
$771$	0	0
$772$	27.1194	0.976050
$773$	−6.00000	−0.215805	−0.107903	−	0.994161i	$-0.534413\pi$
$773$	−6.00000	−0.215805	−0.107903	+	0.994161i	$0.534413\pi$
$774$	0	0
$775$	0	0
$776$	2.36669	0.0849593
$777$	0	0
$778$	−66.3583	−2.37906
$779$	0	0
$780$	0	0
$781$	9.00000	0.322045
$782$	65.8722	2.35558
$783$	0	0
$784$	0.302776	0.0108134
$785$	0	0
$786$	0	0
$787$	28.2389	1.00661	0.503303	−	0.864110i	$-0.332118\pi$
$787$	28.2389	1.00661	0.503303	+	0.864110i	$0.332118\pi$
$788$	−55.5416	−1.97859
$789$	0	0
$790$	0	0
$791$	10.8167	0.384596
$792$	0	0
$793$	3.15559	0.112058
$794$	−4.60555	−0.163445
$795$	0	0
$796$	26.4222	0.936510
$797$	29.4500	1.04317	0.521586	−	0.853199i	$-0.325341\pi$
$797$	29.4500	1.04317	0.521586	+	0.853199i	$0.325341\pi$
$798$	0	0
$799$	48.8444	1.72799
$800$	0	0
$801$	0	0
$802$	−38.7250	−1.36743
$803$	1.81665	0.0641083
$804$	0	0
$805$	0	0
$806$	−43.2666	−1.52400
$807$	0	0
$808$	49.8167	1.75254
$809$	−32.4500	−1.14088	−0.570440	−	0.821339i	$-0.693227\pi$
$809$	−32.4500	−1.14088	−0.570440	+	0.821339i	$0.693227\pi$
$810$	0	0
$811$	14.8444	0.521258	0.260629	−	0.965439i	$-0.416070\pi$
$811$	14.8444	0.521258	0.260629	+	0.965439i	$0.416070\pi$
$812$	25.1194	0.881519
$813$	0	0
$814$	29.0917	1.01966
$815$	0	0
$816$	0	0
$817$	63.4500	2.21983
$818$	27.2111	0.951414
$819$	0	0
$820$	0	0
$821$	−49.2666	−1.71942	−0.859708	−	0.510785i	$-0.829355\pi$
$821$	−49.2666	−1.71942	−0.859708	+	0.510785i	$0.829355\pi$
$822$	0	0
$823$	24.8167	0.865054	0.432527	−	0.901621i	$-0.357622\pi$
$823$	24.8167	0.865054	0.432527	+	0.901621i	$0.357622\pi$
$824$	−11.4500	−0.398878
$825$	0	0
$826$	−24.4222	−0.849757
$827$	30.6333	1.06522	0.532612	−	0.846359i	$-0.321210\pi$
$827$	30.6333	1.06522	0.532612	+	0.846359i	$0.321210\pi$
$828$	0	0
$829$	−0.238859	−0.00829591	−0.00414796	−	0.999991i	$-0.501320\pi$
$829$	−0.238859	−0.00829591	−0.00414796	+	0.999991i	$0.501320\pi$
$830$	0	0
$831$	0	0
$832$	33.3944	1.15774
$833$	−4.60555	−0.159573
$834$	0	0
$835$	0	0
$836$	65.4500	2.26363
$837$	0	0
$838$	13.8167	0.477288
$839$	22.1833	0.765854	0.382927	−	0.923779i	$-0.374916\pi$
$839$	22.1833	0.765854	0.382927	+	0.923779i	$0.374916\pi$
$840$	0	0
$841$	28.8444	0.994635
$842$	23.5139	0.810342
$843$	0	0
$844$	87.2666	3.00384
$845$	0	0
$846$	0	0
$847$	−2.00000	−0.0687208
$848$	−0.972244	−0.0333870
$849$	0	0
$850$	0	0
$851$	−26.1556	−0.896602
$852$	0	0
$853$	−6.78890	−0.232447	−0.116224	−	0.993223i	$-0.537079\pi$
$853$	−6.78890	−0.232447	−0.116224	+	0.993223i	$0.537079\pi$
$854$	2.78890	0.0954341
$855$	0	0
$856$	0	0
$857$	−27.6333	−0.943936	−0.471968	−	0.881616i	$-0.656456\pi$
$857$	−27.6333	−0.943936	−0.471968	+	0.881616i	$0.656456\pi$
$858$	0	0
$859$	−49.6333	−1.69347	−0.846733	−	0.532018i	$-0.821434\pi$
$859$	−49.6333	−1.69347	−0.846733	+	0.532018i	$0.821434\pi$
$860$	0	0
$861$	0	0
$862$	12.8444	0.437482
$863$	−5.36669	−0.182684	−0.0913422	−	0.995820i	$-0.529116\pi$
$863$	−5.36669	−0.182684	−0.0913422	+	0.995820i	$0.529116\pi$
$864$	0	0
$865$	0	0
$866$	85.6888	2.91182
$867$	0	0
$868$	−23.8167	−0.808390
$869$	−44.4500	−1.50786
$870$	0	0
$871$	−40.6611	−1.37775
$872$	−6.63331	−0.224632
$873$	0	0
$874$	−94.4777	−3.19576
$875$	0	0
$876$	0	0
$877$	−22.0000	−0.742887	−0.371444	−	0.928456i	$-0.621137\pi$
$877$	−22.0000	−0.742887	−0.371444	+	0.928456i	$0.621137\pi$
$878$	−15.2111	−0.513350
$879$	0	0
$880$	0	0
$881$	−33.6333	−1.13313	−0.566567	−	0.824015i	$-0.691729\pi$
$881$	−33.6333	−1.13313	−0.566567	+	0.824015i	$0.691729\pi$
$882$	0	0
$883$	15.6056	0.525169	0.262584	−	0.964909i	$-0.415425\pi$
$883$	15.6056	0.525169	0.262584	+	0.964909i	$0.415425\pi$
$884$	39.6333	1.33301
$885$	0	0
$886$	36.0000	1.20944
$887$	14.7889	0.496563	0.248281	−	0.968688i	$-0.420134\pi$
$887$	14.7889	0.496563	0.248281	+	0.968688i	$0.420134\pi$
$888$	0	0
$889$	12.8167	0.429857
$890$	0	0
$891$	0	0
$892$	50.8444	1.70240
$893$	−70.0555	−2.34432
$894$	0	0
$895$	0	0
$896$	18.9083	0.631683
$897$	0	0
$898$	−80.1749	−2.67547
$899$	−54.8444	−1.82916
$900$	0	0
$901$	14.7889	0.492690
$902$	0	0
$903$	0	0
$904$	−32.4500	−1.07927
$905$	0	0
$906$	0	0
$907$	−13.2111	−0.438667	−0.219334	−	0.975650i	$-0.570388\pi$
$907$	−13.2111	−0.438667	−0.219334	+	0.975650i	$0.570388\pi$
$908$	−74.6611	−2.47771
$909$	0	0
$910$	0	0
$911$	−15.0000	−0.496972	−0.248486	−	0.968635i	$-0.579933\pi$
$911$	−15.0000	−0.496972	−0.248486	+	0.968635i	$0.579933\pi$
$912$	0	0
$913$	9.63331	0.318816
$914$	8.72498	0.288597
$915$	0	0
$916$	−23.8167	−0.786924
$917$	7.39445	0.244186
$918$	0	0
$919$	−17.6056	−0.580754	−0.290377	−	0.956912i	$-0.593781\pi$
$919$	−17.6056	−0.580754	−0.290377	+	0.956912i	$0.593781\pi$
$920$	0	0
$921$	0	0
$922$	−60.4222	−1.98990
$923$	−7.81665	−0.257288
$924$	0	0
$925$	0	0
$926$	−39.6333	−1.30243
$927$	0	0
$928$	40.3305	1.32391
$929$	20.2389	0.664015	0.332008	−	0.943277i	$-0.392274\pi$
$929$	20.2389	0.664015	0.332008	+	0.943277i	$0.392274\pi$
$930$	0	0
$931$	6.60555	0.216488
$932$	3.90833	0.128022
$933$	0	0
$934$	−0.972244	−0.0318128
$935$	0	0
$936$	0	0
$937$	−56.4777	−1.84505	−0.922523	−	0.385941i	$-0.873877\pi$
$937$	−56.4777	−1.84505	−0.922523	+	0.385941i	$0.873877\pi$
$938$	−35.9361	−1.17335
$939$	0	0
$940$	0	0
$941$	−6.00000	−0.195594	−0.0977972	−	0.995206i	$-0.531180\pi$
$941$	−6.00000	−0.195594	−0.0977972	+	0.995206i	$0.531180\pi$
$942$	0	0
$943$	0	0
$944$	3.21110	0.104512
$945$	0	0
$946$	−66.3583	−2.15749
$947$	0.844410	0.0274396	0.0137198	−	0.999906i	$-0.495633\pi$
$947$	0.844410	0.0274396	0.0137198	+	0.999906i	$0.495633\pi$
$948$	0	0
$949$	−1.57779	−0.0512174
$950$	0	0
$951$	0	0
$952$	13.8167	0.447800
$953$	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
$956$	−48.8444	−1.57974
$957$	0	0
$958$	79.6888	2.57463
$959$	3.21110	0.103692
$960$	0	0
$961$	21.0000	0.677419
$962$	−25.2666	−0.814628
$963$	0	0
$964$	31.0278	0.999337
$965$	0	0
$966$	0	0
$967$	−40.8444	−1.31347	−0.656734	−	0.754122i	$-0.728063\pi$
$967$	−40.8444	−1.31347	−0.656734	+	0.754122i	$0.728063\pi$
$968$	6.00000	0.192847
$969$	0	0
$970$	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
$973$	19.0278	0.610002
$974$	47.9361	1.53597
$975$	0	0
$976$	−0.366692	−0.0117375
$977$	2.02776	0.0648737	0.0324368	−	0.999474i	$-0.489673\pi$
$977$	2.02776	0.0648737	0.0324368	+	0.999474i	$0.489673\pi$
$978$	0	0
$979$	−23.4500	−0.749464
$980$	0	0
$981$	0	0
$982$	−6.90833	−0.220454
$983$	−40.0555	−1.27757	−0.638786	−	0.769384i	$-0.720563\pi$
$983$	−40.0555	−1.27757	−0.638786	+	0.769384i	$0.720563\pi$
$984$	0	0
$985$	0	0
$986$	80.6611	2.56877
$987$	0	0
$988$	−56.8444	−1.80846
$989$	59.6611	1.89711
$990$	0	0
$991$	46.0278	1.46212	0.731060	−	0.682313i	$-0.239026\pi$
$991$	46.0278	1.46212	0.731060	+	0.682313i	$0.239026\pi$
$992$	−38.2389	−1.21408
$993$	0	0
$994$	−6.90833	−0.219119
$995$	0	0
$996$	0	0
$997$	−15.4500	−0.489305	−0.244653	−	0.969611i	$-0.578674\pi$
$997$	−15.4500	−0.489305	−0.244653	+	0.969611i	$0.578674\pi$
$998$	64.4777	2.04101
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.2.a.o.1.1		2
3.2	odd	2		525.2.a.h.1.2	yes	2
5.2	odd	4		1575.2.d.g.1324.1		4
5.3	odd	4		1575.2.d.g.1324.4		4
5.4	even	2		1575.2.a.t.1.2		2
12.11	even	2		8400.2.a.cw.1.1		2
15.2	even	4		525.2.d.d.274.4		4
15.8	even	4		525.2.d.d.274.1		4
15.14	odd	2		525.2.a.f.1.1	✓	2
21.20	even	2		3675.2.a.bb.1.2		2
60.59	even	2		8400.2.a.df.1.2		2
105.104	even	2		3675.2.a.w.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.2.a.f.1.1	✓	2	15.14	odd	2
525.2.a.h.1.2	yes	2	3.2	odd	2
525.2.d.d.274.1		4	15.8	even	4
525.2.d.d.274.4		4	15.2	even	4
1575.2.a.o.1.1		2	1.1	even	1	trivial
1575.2.a.t.1.2		2	5.4	even	2
1575.2.d.g.1324.1		4	5.2	odd	4
1575.2.d.g.1324.4		4	5.3	odd	4
3675.2.a.w.1.1		2	105.104	even	2
3675.2.a.bb.1.2		2	21.20	even	2
8400.2.a.cw.1.1		2	12.11	even	2
8400.2.a.df.1.2		2	60.59	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:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1575.a (trivial)

\(f(q)\)	\(=\)	\(q-2.30278 q^{2} +3.30278 q^{4} +1.00000 q^{7} -3.00000 q^{8} +O(q^{10})\)	\(q-2.30278 q^{2} +3.30278 q^{4} +1.00000 q^{7} -3.00000 q^{8} +3.00000 q^{11} -2.60555 q^{13} -2.30278 q^{14} +0.302776 q^{16} -4.60555 q^{17} +6.60555 q^{19} -6.90833 q^{22} +6.21110 q^{23} +6.00000 q^{26} +3.30278 q^{28} +7.60555 q^{29} -7.21110 q^{31} +5.30278 q^{32} +10.6056 q^{34} -4.21110 q^{37} -15.2111 q^{38} +9.60555 q^{43} +9.90833 q^{44} -14.3028 q^{46} -10.6056 q^{47} +1.00000 q^{49} -8.60555 q^{52} -3.21110 q^{53} -3.00000 q^{56} -17.5139 q^{58} +10.6056 q^{59} -1.21110 q^{61} +16.6056 q^{62} -12.8167 q^{64} +15.6056 q^{67} -15.2111 q^{68} +3.00000 q^{71} +0.605551 q^{73} +9.69722 q^{74} +21.8167 q^{76} +3.00000 q^{77} -14.8167 q^{79} +3.21110 q^{83} -22.1194 q^{86} -9.00000 q^{88} -7.81665 q^{89} -2.60555 q^{91} +20.5139 q^{92} +24.4222 q^{94} -0.788897 q^{97} -2.30278 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + 3 q^{4} + 2 q^{7} - 6 q^{8}+O(q^{10}) \) 2 * q - q^2 + 3 * q^4 + 2 * q^7 - 6 * q^8	\( 2 q - q^{2} + 3 q^{4} + 2 q^{7} - 6 q^{8} + 6 q^{11} + 2 q^{13} - q^{14} - 3 q^{16} - 2 q^{17} + 6 q^{19} - 3 q^{22} - 2 q^{23} + 12 q^{26} + 3 q^{28} + 8 q^{29} + 7 q^{32} + 14 q^{34} + 6 q^{37} - 16 q^{38} + 12 q^{43} + 9 q^{44} - 25 q^{46} - 14 q^{47} + 2 q^{49} - 10 q^{52} + 8 q^{53} - 6 q^{56} - 17 q^{58} + 14 q^{59} + 12 q^{61} + 26 q^{62} - 4 q^{64} + 24 q^{67} - 16 q^{68} + 6 q^{71} - 6 q^{73} + 23 q^{74} + 22 q^{76} + 6 q^{77} - 8 q^{79} - 8 q^{83} - 19 q^{86} - 18 q^{88} + 6 q^{89} + 2 q^{91} + 23 q^{92} + 20 q^{94} - 16 q^{97} - q^{98}+O(q^{100}) \) 2 * q - q^2 + 3 * q^4 + 2 * q^7 - 6 * q^8 + 6 * q^11 + 2 * q^13 - q^14 - 3 * q^16 - 2 * q^17 + 6 * q^19 - 3 * q^22 - 2 * q^23 + 12 * q^26 + 3 * q^28 + 8 * q^29 + 7 * q^32 + 14 * q^34 + 6 * q^37 - 16 * q^38 + 12 * q^43 + 9 * q^44 - 25 * q^46 - 14 * q^47 + 2 * q^49 - 10 * q^52 + 8 * q^53 - 6 * q^56 - 17 * q^58 + 14 * q^59 + 12 * q^61 + 26 * q^62 - 4 * q^64 + 24 * q^67 - 16 * q^68 + 6 * q^71 - 6 * q^73 + 23 * q^74 + 22 * q^76 + 6 * q^77 - 8 * q^79 - 8 * q^83 - 19 * q^86 - 18 * q^88 + 6 * q^89 + 2 * q^91 + 23 * q^92 + 20 * q^94 - 16 * q^97 - q^98

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