LMFDB - Embedded newform 1575.2.a.s.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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 175)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ 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)

$f(q)$	$=$	$q-0.618034 q^{2} -1.61803 q^{4} -1.00000 q^{7} +2.23607 q^{8} +O(q^{10})$	$q-0.618034 q^{2} -1.61803 q^{4} -1.00000 q^{7} +2.23607 q^{8} +0.236068 q^{11} -1.23607 q^{13} +0.618034 q^{14} +1.85410 q^{16} +2.47214 q^{17} -4.47214 q^{19} -0.145898 q^{22} +6.23607 q^{23} +0.763932 q^{26} +1.61803 q^{28} -5.00000 q^{29} +3.70820 q^{31} -5.61803 q^{32} -1.52786 q^{34} -3.00000 q^{37} +2.76393 q^{38} -4.76393 q^{41} -1.76393 q^{43} -0.381966 q^{44} -3.85410 q^{46} -2.00000 q^{47} +1.00000 q^{49} +2.00000 q^{52} +8.47214 q^{53} -2.23607 q^{56} +3.09017 q^{58} -11.7082 q^{59} -9.70820 q^{61} -2.29180 q^{62} -0.236068 q^{64} +4.23607 q^{67} -4.00000 q^{68} -8.70820 q^{71} +8.76393 q^{73} +1.85410 q^{74} +7.23607 q^{76} -0.236068 q^{77} -11.1803 q^{79} +2.94427 q^{82} -7.70820 q^{83} +1.09017 q^{86} +0.527864 q^{88} -17.2361 q^{89} +1.23607 q^{91} -10.0902 q^{92} +1.23607 q^{94} -5.23607 q^{97} -0.618034 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - q^{4} - 2 q^{7}+O(q^{10}) $ 2 * q + q^2 - q^4 - 2 * q^7	$ 2 q + q^{2} - q^{4} - 2 q^{7} - 4 q^{11} + 2 q^{13} - q^{14} - 3 q^{16} - 4 q^{17} - 7 q^{22} + 8 q^{23} + 6 q^{26} + q^{28} - 10 q^{29} - 6 q^{31} - 9 q^{32} - 12 q^{34} - 6 q^{37} + 10 q^{38} - 14 q^{41} - 8 q^{43} - 3 q^{44} - q^{46} - 4 q^{47} + 2 q^{49} + 4 q^{52} + 8 q^{53} - 5 q^{58} - 10 q^{59} - 6 q^{61} - 18 q^{62} + 4 q^{64} + 4 q^{67} - 8 q^{68} - 4 q^{71} + 22 q^{73} - 3 q^{74} + 10 q^{76} + 4 q^{77} - 12 q^{82} - 2 q^{83} - 9 q^{86} + 10 q^{88} - 30 q^{89} - 2 q^{91} - 9 q^{92} - 2 q^{94} - 6 q^{97} + q^{98}+O(q^{100}) $ 2 * q + q^2 - q^4 - 2 * q^7 - 4 * q^11 + 2 * q^13 - q^14 - 3 * q^16 - 4 * q^17 - 7 * q^22 + 8 * q^23 + 6 * q^26 + q^28 - 10 * q^29 - 6 * q^31 - 9 * q^32 - 12 * q^34 - 6 * q^37 + 10 * q^38 - 14 * q^41 - 8 * q^43 - 3 * q^44 - q^46 - 4 * q^47 + 2 * q^49 + 4 * q^52 + 8 * q^53 - 5 * q^58 - 10 * q^59 - 6 * q^61 - 18 * q^62 + 4 * q^64 + 4 * q^67 - 8 * q^68 - 4 * q^71 + 22 * q^73 - 3 * q^74 + 10 * q^76 + 4 * q^77 - 12 * q^82 - 2 * q^83 - 9 * q^86 + 10 * q^88 - 30 * q^89 - 2 * q^91 - 9 * q^92 - 2 * q^94 - 6 * 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 $

$2$	−0.618034	−0.437016	−0.218508	−	0.975835i	$-0.570119\pi$
$2$	−0.618034	−0.437016	−0.218508	+	0.975835i	$0.570119\pi$
$3$	0	0
$3$	0	0
$4$	−1.61803	−0.809017
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	2.23607	0.790569
$9$	0	0
$10$	0	0
$11$	0.236068	0.0711772	0.0355886	−	0.999367i	$-0.488669\pi$
$11$	0.236068	0.0711772	0.0355886	+	0.999367i	$0.488669\pi$
$12$	0	0
$13$	−1.23607	−0.342824	−0.171412	−	0.985199i	$-0.554833\pi$
$13$	−1.23607	−0.342824	−0.171412	+	0.985199i	$0.554833\pi$
$14$	0.618034	0.165177
$15$	0	0
$16$	1.85410	0.463525
$17$	2.47214	0.599581	0.299791	−	0.954005i	$-0.403083\pi$
$17$	2.47214	0.599581	0.299791	+	0.954005i	$0.403083\pi$
$18$	0	0
$19$	−4.47214	−1.02598	−0.512989	−	0.858395i	$-0.671462\pi$
$19$	−4.47214	−1.02598	−0.512989	+	0.858395i	$0.671462\pi$
$20$	0	0
$21$	0	0
$22$	−0.145898	−0.0311056
$23$	6.23607	1.30031	0.650155	−	0.759802i	$-0.274704\pi$
$23$	6.23607	1.30031	0.650155	+	0.759802i	$0.274704\pi$
$24$	0	0
$25$	0	0
$26$	0.763932	0.149819
$27$	0	0
$28$	1.61803	0.305780
$29$	−5.00000	−0.928477	−0.464238	−	0.885710i	$-0.653672\pi$
$29$	−5.00000	−0.928477	−0.464238	+	0.885710i	$0.653672\pi$
$30$	0	0
$31$	3.70820	0.666013	0.333007	−	0.942925i	$-0.391937\pi$
$31$	3.70820	0.666013	0.333007	+	0.942925i	$0.391937\pi$
$32$	−5.61803	−0.993137
$33$	0	0
$34$	−1.52786	−0.262027
$35$	0	0
$36$	0	0
$37$	−3.00000	−0.493197	−0.246598	−	0.969118i	$-0.579313\pi$
$37$	−3.00000	−0.493197	−0.246598	+	0.969118i	$0.579313\pi$
$38$	2.76393	0.448369
$39$	0	0
$40$	0	0
$41$	−4.76393	−0.744001	−0.372001	−	0.928232i	$-0.621328\pi$
$41$	−4.76393	−0.744001	−0.372001	+	0.928232i	$0.621328\pi$
$42$	0	0
$43$	−1.76393	−0.268997	−0.134499	−	0.990914i	$-0.542942\pi$
$43$	−1.76393	−0.268997	−0.134499	+	0.990914i	$0.542942\pi$
$44$	−0.381966	−0.0575835
$45$	0	0
$46$	−3.85410	−0.568256
$47$	−2.00000	−0.291730	−0.145865	−	0.989305i	$-0.546597\pi$
$47$	−2.00000	−0.291730	−0.145865	+	0.989305i	$0.546597\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	2.00000	0.277350
$53$	8.47214	1.16374	0.581869	−	0.813283i	$-0.302322\pi$
$53$	8.47214	1.16374	0.581869	+	0.813283i	$0.302322\pi$
$54$	0	0
$55$	0	0
$56$	−2.23607	−0.298807
$57$	0	0
$58$	3.09017	0.405759
$59$	−11.7082	−1.52428	−0.762139	−	0.647413i	$-0.775851\pi$
$59$	−11.7082	−1.52428	−0.762139	+	0.647413i	$0.775851\pi$
$60$	0	0
$61$	−9.70820	−1.24301	−0.621504	−	0.783411i	$-0.713478\pi$
$61$	−9.70820	−1.24301	−0.621504	+	0.783411i	$0.713478\pi$
$62$	−2.29180	−0.291058
$63$	0	0
$64$	−0.236068	−0.0295085
$65$	0	0
$66$	0	0
$67$	4.23607	0.517518	0.258759	−	0.965942i	$-0.416686\pi$
$67$	4.23607	0.517518	0.258759	+	0.965942i	$0.416686\pi$
$68$	−4.00000	−0.485071
$69$	0	0
$70$	0	0
$71$	−8.70820	−1.03347	−0.516737	−	0.856144i	$-0.672853\pi$
$71$	−8.70820	−1.03347	−0.516737	+	0.856144i	$0.672853\pi$
$72$	0	0
$73$	8.76393	1.02574	0.512870	−	0.858466i	$-0.328582\pi$
$73$	8.76393	1.02574	0.512870	+	0.858466i	$0.328582\pi$
$74$	1.85410	0.215535
$75$	0	0
$76$	7.23607	0.830034
$77$	−0.236068	−0.0269024
$78$	0	0
$79$	−11.1803	−1.25789	−0.628943	−	0.777451i	$-0.716512\pi$
$79$	−11.1803	−1.25789	−0.628943	+	0.777451i	$0.716512\pi$
$80$	0	0
$81$	0	0
$82$	2.94427	0.325140
$83$	−7.70820	−0.846085	−0.423043	−	0.906110i	$-0.639038\pi$
$83$	−7.70820	−0.846085	−0.423043	+	0.906110i	$0.639038\pi$
$84$	0	0
$85$	0	0
$86$	1.09017	0.117556
$87$	0	0
$88$	0.527864	0.0562705
$89$	−17.2361	−1.82702	−0.913510	−	0.406817i	$-0.866639\pi$
$89$	−17.2361	−1.82702	−0.913510	+	0.406817i	$0.866639\pi$
$90$	0	0
$91$	1.23607	0.129575
$92$	−10.0902	−1.05197
$93$	0	0
$94$	1.23607	0.127491
$95$	0	0
$96$	0	0
$97$	−5.23607	−0.531642	−0.265821	−	0.964022i	$-0.585643\pi$
$97$	−5.23607	−0.531642	−0.265821	+	0.964022i	$0.585643\pi$
$98$	−0.618034	−0.0624309
$99$	0	0
$100$	0	0
$101$	−4.76393	−0.474029	−0.237014	−	0.971506i	$-0.576169\pi$
$101$	−4.76393	−0.474029	−0.237014	+	0.971506i	$0.576169\pi$
$102$	0	0
$103$	−8.47214	−0.834784	−0.417392	−	0.908726i	$-0.637056\pi$
$103$	−8.47214	−0.834784	−0.417392	+	0.908726i	$0.637056\pi$
$104$	−2.76393	−0.271026
$105$	0	0
$106$	−5.23607	−0.508572
$107$	8.00000	0.773389	0.386695	−	0.922208i	$-0.373617\pi$
$107$	8.00000	0.773389	0.386695	+	0.922208i	$0.373617\pi$
$108$	0	0
$109$	8.41641	0.806146	0.403073	−	0.915168i	$-0.367942\pi$
$109$	8.41641	0.806146	0.403073	+	0.915168i	$0.367942\pi$
$110$	0	0
$111$	0	0
$112$	−1.85410	−0.175196
$113$	−14.4164	−1.35618	−0.678091	−	0.734978i	$-0.737192\pi$
$113$	−14.4164	−1.35618	−0.678091	+	0.734978i	$0.737192\pi$
$114$	0	0
$115$	0	0
$116$	8.09017	0.751153
$117$	0	0
$118$	7.23607	0.666134
$119$	−2.47214	−0.226620
$120$	0	0
$121$	−10.9443	−0.994934
$122$	6.00000	0.543214
$123$	0	0
$124$	−6.00000	−0.538816
$125$	0	0
$126$	0	0
$127$	−13.6525	−1.21146	−0.605731	−	0.795670i	$-0.707119\pi$
$127$	−13.6525	−1.21146	−0.605731	+	0.795670i	$0.707119\pi$
$128$	11.3820	1.00603
$129$	0	0
$130$	0	0
$131$	16.9443	1.48043	0.740214	−	0.672371i	$-0.234724\pi$
$131$	16.9443	1.48043	0.740214	+	0.672371i	$0.234724\pi$
$132$	0	0
$133$	4.47214	0.387783
$134$	−2.61803	−0.226164
$135$	0	0
$136$	5.52786	0.474010
$137$	−10.9443	−0.935032	−0.467516	−	0.883985i	$-0.654851\pi$
$137$	−10.9443	−0.935032	−0.467516	+	0.883985i	$0.654851\pi$
$138$	0	0
$139$	10.6525	0.903531	0.451766	−	0.892137i	$-0.350794\pi$
$139$	10.6525	0.903531	0.451766	+	0.892137i	$0.350794\pi$
$140$	0	0
$141$	0	0
$142$	5.38197	0.451645
$143$	−0.291796	−0.0244012
$144$	0	0
$145$	0	0
$146$	−5.41641	−0.448265
$147$	0	0
$148$	4.85410	0.399005
$149$	−3.94427	−0.323127	−0.161564	−	0.986862i	$-0.551654\pi$
$149$	−3.94427	−0.323127	−0.161564	+	0.986862i	$0.551654\pi$
$150$	0	0
$151$	−20.2361	−1.64679	−0.823394	−	0.567470i	$-0.807922\pi$
$151$	−20.2361	−1.64679	−0.823394	+	0.567470i	$0.807922\pi$
$152$	−10.0000	−0.811107
$153$	0	0
$154$	0.145898	0.0117568
$155$	0	0
$156$	0	0
$157$	−0.763932	−0.0609684	−0.0304842	−	0.999535i	$-0.509705\pi$
$157$	−0.763932	−0.0609684	−0.0304842	+	0.999535i	$0.509705\pi$
$158$	6.90983	0.549717
$159$	0	0
$160$	0	0
$161$	−6.23607	−0.491471
$162$	0	0
$163$	1.52786	0.119672	0.0598358	−	0.998208i	$-0.480942\pi$
$163$	1.52786	0.119672	0.0598358	+	0.998208i	$0.480942\pi$
$164$	7.70820	0.601910
$165$	0	0
$166$	4.76393	0.369753
$167$	5.23607	0.405179	0.202590	−	0.979264i	$-0.435064\pi$
$167$	5.23607	0.405179	0.202590	+	0.979264i	$0.435064\pi$
$168$	0	0
$169$	−11.4721	−0.882472
$170$	0	0
$171$	0	0
$172$	2.85410	0.217623
$173$	−11.5279	−0.876447	−0.438224	−	0.898866i	$-0.644392\pi$
$173$	−11.5279	−0.876447	−0.438224	+	0.898866i	$0.644392\pi$
$174$	0	0
$175$	0	0
$176$	0.437694	0.0329924
$177$	0	0
$178$	10.6525	0.798437
$179$	23.4164	1.75022	0.875112	−	0.483920i	$-0.160787\pi$
$179$	23.4164	1.75022	0.875112	+	0.483920i	$0.160787\pi$
$180$	0	0
$181$	8.18034	0.608040	0.304020	−	0.952666i	$-0.401671\pi$
$181$	8.18034	0.608040	0.304020	+	0.952666i	$0.401671\pi$
$182$	−0.763932	−0.0566264
$183$	0	0
$184$	13.9443	1.02799
$185$	0	0
$186$	0	0
$187$	0.583592	0.0426765
$188$	3.23607	0.236015
$189$	0	0
$190$	0	0
$191$	−6.47214	−0.468307	−0.234154	−	0.972200i	$-0.575232\pi$
$191$	−6.47214	−0.468307	−0.234154	+	0.972200i	$0.575232\pi$
$192$	0	0
$193$	−12.4164	−0.893753	−0.446876	−	0.894596i	$-0.647464\pi$
$193$	−12.4164	−0.893753	−0.446876	+	0.894596i	$0.647464\pi$
$194$	3.23607	0.232336
$195$	0	0
$196$	−1.61803	−0.115574
$197$	−1.47214	−0.104885	−0.0524427	−	0.998624i	$-0.516701\pi$
$197$	−1.47214	−0.104885	−0.0524427	+	0.998624i	$0.516701\pi$
$198$	0	0
$199$	7.23607	0.512951	0.256476	−	0.966551i	$-0.417439\pi$
$199$	7.23607	0.512951	0.256476	+	0.966551i	$0.417439\pi$
$200$	0	0
$201$	0	0
$202$	2.94427	0.207158
$203$	5.00000	0.350931
$204$	0	0
$205$	0	0
$206$	5.23607	0.364814
$207$	0	0
$208$	−2.29180	−0.158907
$209$	−1.05573	−0.0730262
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	−13.7082	−0.941483
$213$	0	0
$214$	−4.94427	−0.337983
$215$	0	0
$216$	0	0
$217$	−3.70820	−0.251729
$218$	−5.20163	−0.352299
$219$	0	0
$220$	0	0
$221$	−3.05573	−0.205551
$222$	0	0
$223$	−20.1803	−1.35138	−0.675688	−	0.737188i	$-0.736153\pi$
$223$	−20.1803	−1.35138	−0.675688	+	0.737188i	$0.736153\pi$
$224$	5.61803	0.375371
$225$	0	0
$226$	8.90983	0.592673
$227$	21.4164	1.42146	0.710728	−	0.703466i	$-0.248365\pi$
$227$	21.4164	1.42146	0.710728	+	0.703466i	$0.248365\pi$
$228$	0	0
$229$	−4.47214	−0.295527	−0.147764	−	0.989023i	$-0.547207\pi$
$229$	−4.47214	−0.295527	−0.147764	+	0.989023i	$0.547207\pi$
$230$	0	0
$231$	0	0
$232$	−11.1803	−0.734025
$233$	7.94427	0.520447	0.260223	−	0.965548i	$-0.416204\pi$
$233$	7.94427	0.520447	0.260223	+	0.965548i	$0.416204\pi$
$234$	0	0
$235$	0	0
$236$	18.9443	1.23317
$237$	0	0
$238$	1.52786	0.0990367
$239$	5.52786	0.357568	0.178784	−	0.983888i	$-0.442784\pi$
$239$	5.52786	0.357568	0.178784	+	0.983888i	$0.442784\pi$
$240$	0	0
$241$	−3.52786	−0.227250	−0.113625	−	0.993524i	$-0.536246\pi$
$241$	−3.52786	−0.227250	−0.113625	+	0.993524i	$0.536246\pi$
$242$	6.76393	0.434802
$243$	0	0
$244$	15.7082	1.00561
$245$	0	0
$246$	0	0
$247$	5.52786	0.351730
$248$	8.29180	0.526530
$249$	0	0
$250$	0	0
$251$	−6.47214	−0.408518	−0.204259	−	0.978917i	$-0.565478\pi$
$251$	−6.47214	−0.408518	−0.204259	+	0.978917i	$0.565478\pi$
$252$	0	0
$253$	1.47214	0.0925524
$254$	8.43769	0.529428
$255$	0	0
$256$	−6.56231	−0.410144
$257$	−12.6525	−0.789240	−0.394620	−	0.918844i	$-0.629124\pi$
$257$	−12.6525	−0.789240	−0.394620	+	0.918844i	$0.629124\pi$
$258$	0	0
$259$	3.00000	0.186411
$260$	0	0
$261$	0	0
$262$	−10.4721	−0.646971
$263$	16.2361	1.00116	0.500579	−	0.865691i	$-0.333120\pi$
$263$	16.2361	1.00116	0.500579	+	0.865691i	$0.333120\pi$
$264$	0	0
$265$	0	0
$266$	−2.76393	−0.169468
$267$	0	0
$268$	−6.85410	−0.418681
$269$	11.7082	0.713862	0.356931	−	0.934131i	$-0.383823\pi$
$269$	11.7082	0.713862	0.356931	+	0.934131i	$0.383823\pi$
$270$	0	0
$271$	23.7082	1.44017	0.720085	−	0.693885i	$-0.244103\pi$
$271$	23.7082	1.44017	0.720085	+	0.693885i	$0.244103\pi$
$272$	4.58359	0.277921
$273$	0	0
$274$	6.76393	0.408624
$275$	0	0
$276$	0	0
$277$	19.8885	1.19499	0.597493	−	0.801874i	$-0.296163\pi$
$277$	19.8885	1.19499	0.597493	+	0.801874i	$0.296163\pi$
$278$	−6.58359	−0.394858
$279$	0	0
$280$	0	0
$281$	15.3607	0.916341	0.458171	−	0.888864i	$-0.348505\pi$
$281$	15.3607	0.916341	0.458171	+	0.888864i	$0.348505\pi$
$282$	0	0
$283$	−17.4164	−1.03530	−0.517649	−	0.855593i	$-0.673193\pi$
$283$	−17.4164	−1.03530	−0.517649	+	0.855593i	$0.673193\pi$
$284$	14.0902	0.836098
$285$	0	0
$286$	0.180340	0.0106637
$287$	4.76393	0.281206
$288$	0	0
$289$	−10.8885	−0.640503
$290$	0	0
$291$	0	0
$292$	−14.1803	−0.829842
$293$	−31.1246	−1.81832	−0.909160	−	0.416448i	$-0.863275\pi$
$293$	−31.1246	−1.81832	−0.909160	+	0.416448i	$0.863275\pi$
$294$	0	0
$295$	0	0
$296$	−6.70820	−0.389906
$297$	0	0
$298$	2.43769	0.141212
$299$	−7.70820	−0.445777
$300$	0	0
$301$	1.76393	0.101671
$302$	12.5066	0.719673
$303$	0	0
$304$	−8.29180	−0.475567
$305$	0	0
$306$	0	0
$307$	−4.58359	−0.261599	−0.130800	−	0.991409i	$-0.541754\pi$
$307$	−4.58359	−0.261599	−0.130800	+	0.991409i	$0.541754\pi$
$308$	0.381966	0.0217645
$309$	0	0
$310$	0	0
$311$	−24.3607	−1.38137	−0.690684	−	0.723157i	$-0.742690\pi$
$311$	−24.3607	−1.38137	−0.690684	+	0.723157i	$0.742690\pi$
$312$	0	0
$313$	−19.5279	−1.10378	−0.551890	−	0.833917i	$-0.686093\pi$
$313$	−19.5279	−1.10378	−0.551890	+	0.833917i	$0.686093\pi$
$314$	0.472136	0.0266442
$315$	0	0
$316$	18.0902	1.01765
$317$	25.3607	1.42440	0.712199	−	0.701978i	$-0.247699\pi$
$317$	25.3607	1.42440	0.712199	+	0.701978i	$0.247699\pi$
$318$	0	0
$319$	−1.18034	−0.0660863
$320$	0	0
$321$	0	0
$322$	3.85410	0.214781
$323$	−11.0557	−0.615157
$324$	0	0
$325$	0	0
$326$	−0.944272	−0.0522984
$327$	0	0
$328$	−10.6525	−0.588185
$329$	2.00000	0.110264
$330$	0	0
$331$	−24.7082	−1.35809	−0.679043	−	0.734099i	$-0.737605\pi$
$331$	−24.7082	−1.35809	−0.679043	+	0.734099i	$0.737605\pi$
$332$	12.4721	0.684497
$333$	0	0
$334$	−3.23607	−0.177070
$335$	0	0
$336$	0	0
$337$	16.4721	0.897294	0.448647	−	0.893709i	$-0.351906\pi$
$337$	16.4721	0.897294	0.448647	+	0.893709i	$0.351906\pi$
$338$	7.09017	0.385654
$339$	0	0
$340$	0	0
$341$	0.875388	0.0474049
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−3.94427	−0.212661
$345$	0	0
$346$	7.12461	0.383022
$347$	20.2361	1.08633	0.543165	−	0.839626i	$-0.317226\pi$
$347$	20.2361	1.08633	0.543165	+	0.839626i	$0.317226\pi$
$348$	0	0
$349$	4.47214	0.239388	0.119694	−	0.992811i	$-0.461809\pi$
$349$	4.47214	0.239388	0.119694	+	0.992811i	$0.461809\pi$
$350$	0	0
$351$	0	0
$352$	−1.32624	−0.0706887
$353$	−2.18034	−0.116048	−0.0580239	−	0.998315i	$-0.518480\pi$
$353$	−2.18034	−0.116048	−0.0580239	+	0.998315i	$0.518480\pi$
$354$	0	0
$355$	0	0
$356$	27.8885	1.47809
$357$	0	0
$358$	−14.4721	−0.764876
$359$	30.1246	1.58992	0.794958	−	0.606664i	$-0.207493\pi$
$359$	30.1246	1.58992	0.794958	+	0.606664i	$0.207493\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−5.05573	−0.265723
$363$	0	0
$364$	−2.00000	−0.104828
$365$	0	0
$366$	0	0
$367$	37.1246	1.93789	0.968944	−	0.247278i	$-0.0795362\pi$
$367$	37.1246	1.93789	0.968944	+	0.247278i	$0.0795362\pi$
$368$	11.5623	0.602727
$369$	0	0
$370$	0	0
$371$	−8.47214	−0.439851
$372$	0	0
$373$	37.8328	1.95891	0.979454	−	0.201665i	$-0.0646354\pi$
$373$	37.8328	1.95891	0.979454	+	0.201665i	$0.0646354\pi$
$374$	−0.360680	−0.0186503
$375$	0	0
$376$	−4.47214	−0.230633
$377$	6.18034	0.318304
$378$	0	0
$379$	−11.1803	−0.574295	−0.287148	−	0.957886i	$-0.592707\pi$
$379$	−11.1803	−0.574295	−0.287148	+	0.957886i	$0.592707\pi$
$380$	0	0
$381$	0	0
$382$	4.00000	0.204658
$383$	−33.2361	−1.69828	−0.849142	−	0.528165i	$-0.822880\pi$
$383$	−33.2361	−1.69828	−0.849142	+	0.528165i	$0.822880\pi$
$384$	0	0
$385$	0	0
$386$	7.67376	0.390584
$387$	0	0
$388$	8.47214	0.430108
$389$	−2.88854	−0.146455	−0.0732275	−	0.997315i	$-0.523330\pi$
$389$	−2.88854	−0.146455	−0.0732275	+	0.997315i	$0.523330\pi$
$390$	0	0
$391$	15.4164	0.779641
$392$	2.23607	0.112938
$393$	0	0
$394$	0.909830	0.0458366
$395$	0	0
$396$	0	0
$397$	−9.05573	−0.454494	−0.227247	−	0.973837i	$-0.572972\pi$
$397$	−9.05573	−0.454494	−0.227247	+	0.973837i	$0.572972\pi$
$398$	−4.47214	−0.224168
$399$	0	0
$400$	0	0
$401$	−2.52786	−0.126236	−0.0631178	−	0.998006i	$-0.520104\pi$
$401$	−2.52786	−0.126236	−0.0631178	+	0.998006i	$0.520104\pi$
$402$	0	0
$403$	−4.58359	−0.228325
$404$	7.70820	0.383497
$405$	0	0
$406$	−3.09017	−0.153363
$407$	−0.708204	−0.0351044
$408$	0	0
$409$	24.4721	1.21007	0.605035	−	0.796199i	$-0.293159\pi$
$409$	24.4721	1.21007	0.605035	+	0.796199i	$0.293159\pi$
$410$	0	0
$411$	0	0
$412$	13.7082	0.675355
$413$	11.7082	0.576123
$414$	0	0
$415$	0	0
$416$	6.94427	0.340471
$417$	0	0
$418$	0.652476	0.0319136
$419$	26.1803	1.27899	0.639497	−	0.768794i	$-0.279143\pi$
$419$	26.1803	1.27899	0.639497	+	0.768794i	$0.279143\pi$
$420$	0	0
$421$	−13.0000	−0.633581	−0.316791	−	0.948495i	$-0.602605\pi$
$421$	−13.0000	−0.633581	−0.316791	+	0.948495i	$0.602605\pi$
$422$	−7.41641	−0.361025
$423$	0	0
$424$	18.9443	0.920015
$425$	0	0
$426$	0	0
$427$	9.70820	0.469813
$428$	−12.9443	−0.625685
$429$	0	0
$430$	0	0
$431$	−17.5279	−0.844288	−0.422144	−	0.906529i	$-0.638722\pi$
$431$	−17.5279	−0.844288	−0.422144	+	0.906529i	$0.638722\pi$
$432$	0	0
$433$	28.3607	1.36293	0.681464	−	0.731852i	$-0.261344\pi$
$433$	28.3607	1.36293	0.681464	+	0.731852i	$0.261344\pi$
$434$	2.29180	0.110010
$435$	0	0
$436$	−13.6180	−0.652186
$437$	−27.8885	−1.33409
$438$	0	0
$439$	−8.29180	−0.395746	−0.197873	−	0.980228i	$-0.563403\pi$
$439$	−8.29180	−0.395746	−0.197873	+	0.980228i	$0.563403\pi$
$440$	0	0
$441$	0	0
$442$	1.88854	0.0898289
$443$	−19.4164	−0.922501	−0.461251	−	0.887270i	$-0.652599\pi$
$443$	−19.4164	−0.922501	−0.461251	+	0.887270i	$0.652599\pi$
$444$	0	0
$445$	0	0
$446$	12.4721	0.590573
$447$	0	0
$448$	0.236068	0.0111532
$449$	−20.5279	−0.968770	−0.484385	−	0.874855i	$-0.660957\pi$
$449$	−20.5279	−0.968770	−0.484385	+	0.874855i	$0.660957\pi$
$450$	0	0
$451$	−1.12461	−0.0529559
$452$	23.3262	1.09717
$453$	0	0
$454$	−13.2361	−0.621199
$455$	0	0
$456$	0	0
$457$	12.5279	0.586029	0.293014	−	0.956108i	$-0.405342\pi$
$457$	12.5279	0.586029	0.293014	+	0.956108i	$0.405342\pi$
$458$	2.76393	0.129150
$459$	0	0
$460$	0	0
$461$	14.1803	0.660444	0.330222	−	0.943903i	$-0.392876\pi$
$461$	14.1803	0.660444	0.330222	+	0.943903i	$0.392876\pi$
$462$	0	0
$463$	13.8885	0.645455	0.322728	−	0.946492i	$-0.395400\pi$
$463$	13.8885	0.645455	0.322728	+	0.946492i	$0.395400\pi$
$464$	−9.27051	−0.430373
$465$	0	0
$466$	−4.90983	−0.227443
$467$	6.94427	0.321343	0.160671	−	0.987008i	$-0.448634\pi$
$467$	6.94427	0.321343	0.160671	+	0.987008i	$0.448634\pi$
$468$	0	0
$469$	−4.23607	−0.195603
$470$	0	0
$471$	0	0
$472$	−26.1803	−1.20505
$473$	−0.416408	−0.0191465
$474$	0	0
$475$	0	0
$476$	4.00000	0.183340
$477$	0	0
$478$	−3.41641	−0.156263
$479$	26.1803	1.19621	0.598105	−	0.801418i	$-0.295921\pi$
$479$	26.1803	1.19621	0.598105	+	0.801418i	$0.295921\pi$
$480$	0	0
$481$	3.70820	0.169080
$482$	2.18034	0.0993118
$483$	0	0
$484$	17.7082	0.804918
$485$	0	0
$486$	0	0
$487$	−5.76393	−0.261189	−0.130594	−	0.991436i	$-0.541689\pi$
$487$	−5.76393	−0.261189	−0.130594	+	0.991436i	$0.541689\pi$
$488$	−21.7082	−0.982684
$489$	0	0
$490$	0	0
$491$	5.76393	0.260123	0.130061	−	0.991506i	$-0.458483\pi$
$491$	5.76393	0.260123	0.130061	+	0.991506i	$0.458483\pi$
$492$	0	0
$493$	−12.3607	−0.556697
$494$	−3.41641	−0.153711
$495$	0	0
$496$	6.87539	0.308714
$497$	8.70820	0.390616
$498$	0	0
$499$	11.0557	0.494922	0.247461	−	0.968898i	$-0.420404\pi$
$499$	11.0557	0.494922	0.247461	+	0.968898i	$0.420404\pi$
$500$	0	0
$501$	0	0
$502$	4.00000	0.178529
$503$	−8.11146	−0.361672	−0.180836	−	0.983513i	$-0.557880\pi$
$503$	−8.11146	−0.361672	−0.180836	+	0.983513i	$0.557880\pi$
$504$	0	0
$505$	0	0
$506$	−0.909830	−0.0404469
$507$	0	0
$508$	22.0902	0.980093
$509$	−40.6525	−1.80189	−0.900945	−	0.433934i	$-0.857125\pi$
$509$	−40.6525	−1.80189	−0.900945	+	0.433934i	$0.857125\pi$
$510$	0	0
$511$	−8.76393	−0.387694
$512$	−18.7082	−0.826794
$513$	0	0
$514$	7.81966	0.344910
$515$	0	0
$516$	0	0
$517$	−0.472136	−0.0207645
$518$	−1.85410	−0.0814646
$519$	0	0
$520$	0	0
$521$	−22.0000	−0.963837	−0.481919	−	0.876216i	$-0.660060\pi$
$521$	−22.0000	−0.963837	−0.481919	+	0.876216i	$0.660060\pi$
$522$	0	0
$523$	−16.3607	−0.715403	−0.357701	−	0.933836i	$-0.616439\pi$
$523$	−16.3607	−0.715403	−0.357701	+	0.933836i	$0.616439\pi$
$524$	−27.4164	−1.19769
$525$	0	0
$526$	−10.0344	−0.437522
$527$	9.16718	0.399329
$528$	0	0
$529$	15.8885	0.690806
$530$	0	0
$531$	0	0
$532$	−7.23607	−0.313723
$533$	5.88854	0.255061
$534$	0	0
$535$	0	0
$536$	9.47214	0.409134
$537$	0	0
$538$	−7.23607	−0.311969
$539$	0.236068	0.0101682
$540$	0	0
$541$	15.9443	0.685498	0.342749	−	0.939427i	$-0.388642\pi$
$541$	15.9443	0.685498	0.342749	+	0.939427i	$0.388642\pi$
$542$	−14.6525	−0.629378
$543$	0	0
$544$	−13.8885	−0.595466
$545$	0	0
$546$	0	0
$547$	9.76393	0.417476	0.208738	−	0.977972i	$-0.433064\pi$
$547$	9.76393	0.417476	0.208738	+	0.977972i	$0.433064\pi$
$548$	17.7082	0.756457
$549$	0	0
$550$	0	0
$551$	22.3607	0.952597
$552$	0	0
$553$	11.1803	0.475436
$554$	−12.2918	−0.522228
$555$	0	0
$556$	−17.2361	−0.730972
$557$	−9.11146	−0.386065	−0.193032	−	0.981192i	$-0.561832\pi$
$557$	−9.11146	−0.386065	−0.193032	+	0.981192i	$0.561832\pi$
$558$	0	0
$559$	2.18034	0.0922186
$560$	0	0
$561$	0	0
$562$	−9.49342	−0.400456
$563$	17.4164	0.734014	0.367007	−	0.930218i	$-0.380382\pi$
$563$	17.4164	0.734014	0.367007	+	0.930218i	$0.380382\pi$
$564$	0	0
$565$	0	0
$566$	10.7639	0.452442
$567$	0	0
$568$	−19.4721	−0.817033
$569$	3.94427	0.165352	0.0826762	−	0.996576i	$-0.473653\pi$
$569$	3.94427	0.165352	0.0826762	+	0.996576i	$0.473653\pi$
$570$	0	0
$571$	36.5967	1.53153	0.765763	−	0.643123i	$-0.222362\pi$
$571$	36.5967	1.53153	0.765763	+	0.643123i	$0.222362\pi$
$572$	0.472136	0.0197410
$573$	0	0
$574$	−2.94427	−0.122892
$575$	0	0
$576$	0	0
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	6.72949	0.279910
$579$	0	0
$580$	0	0
$581$	7.70820	0.319790
$582$	0	0
$583$	2.00000	0.0828315
$584$	19.5967	0.810919
$585$	0	0
$586$	19.2361	0.794635
$587$	−24.7639	−1.02212	−0.511058	−	0.859546i	$-0.670746\pi$
$587$	−24.7639	−1.02212	−0.511058	+	0.859546i	$0.670746\pi$
$588$	0	0
$589$	−16.5836	−0.683315
$590$	0	0
$591$	0	0
$592$	−5.56231	−0.228609
$593$	−37.3050	−1.53193	−0.765965	−	0.642882i	$-0.777739\pi$
$593$	−37.3050	−1.53193	−0.765965	+	0.642882i	$0.777739\pi$
$594$	0	0
$595$	0	0
$596$	6.38197	0.261416
$597$	0	0
$598$	4.76393	0.194812
$599$	11.1803	0.456816	0.228408	−	0.973565i	$-0.426648\pi$
$599$	11.1803	0.456816	0.228408	+	0.973565i	$0.426648\pi$
$600$	0	0
$601$	−36.9443	−1.50699	−0.753494	−	0.657455i	$-0.771633\pi$
$601$	−36.9443	−1.50699	−0.753494	+	0.657455i	$0.771633\pi$
$602$	−1.09017	−0.0444320
$603$	0	0
$604$	32.7426	1.33228
$605$	0	0
$606$	0	0
$607$	7.12461	0.289179	0.144590	−	0.989492i	$-0.453814\pi$
$607$	7.12461	0.289179	0.144590	+	0.989492i	$0.453814\pi$
$608$	25.1246	1.01894
$609$	0	0
$610$	0	0
$611$	2.47214	0.100012
$612$	0	0
$613$	44.4164	1.79396	0.896981	−	0.442069i	$-0.145755\pi$
$613$	44.4164	1.79396	0.896981	+	0.442069i	$0.145755\pi$
$614$	2.83282	0.114323
$615$	0	0
$616$	−0.527864	−0.0212682
$617$	−5.94427	−0.239307	−0.119654	−	0.992816i	$-0.538178\pi$
$617$	−5.94427	−0.239307	−0.119654	+	0.992816i	$0.538178\pi$
$618$	0	0
$619$	−11.7082	−0.470592	−0.235296	−	0.971924i	$-0.575606\pi$
$619$	−11.7082	−0.470592	−0.235296	+	0.971924i	$0.575606\pi$
$620$	0	0
$621$	0	0
$622$	15.0557	0.603680
$623$	17.2361	0.690548
$624$	0	0
$625$	0	0
$626$	12.0689	0.482370
$627$	0	0
$628$	1.23607	0.0493245
$629$	−7.41641	−0.295712
$630$	0	0
$631$	27.6525	1.10083	0.550414	−	0.834892i	$-0.314470\pi$
$631$	27.6525	1.10083	0.550414	+	0.834892i	$0.314470\pi$
$632$	−25.0000	−0.994447
$633$	0	0
$634$	−15.6738	−0.622485
$635$	0	0
$636$	0	0
$637$	−1.23607	−0.0489748
$638$	0.729490	0.0288808
$639$	0	0
$640$	0	0
$641$	−43.8328	−1.73129	−0.865646	−	0.500656i	$-0.833092\pi$
$641$	−43.8328	−1.73129	−0.865646	+	0.500656i	$0.833092\pi$
$642$	0	0
$643$	−18.4721	−0.728470	−0.364235	−	0.931307i	$-0.618669\pi$
$643$	−18.4721	−0.728470	−0.364235	+	0.931307i	$0.618669\pi$
$644$	10.0902	0.397608
$645$	0	0
$646$	6.83282	0.268834
$647$	−19.8885	−0.781899	−0.390950	−	0.920412i	$-0.627853\pi$
$647$	−19.8885	−0.781899	−0.390950	+	0.920412i	$0.627853\pi$
$648$	0	0
$649$	−2.76393	−0.108494
$650$	0	0
$651$	0	0
$652$	−2.47214	−0.0968163
$653$	25.0557	0.980506	0.490253	−	0.871580i	$-0.336904\pi$
$653$	25.0557	0.980506	0.490253	+	0.871580i	$0.336904\pi$
$654$	0	0
$655$	0	0
$656$	−8.83282	−0.344864
$657$	0	0
$658$	−1.23607	−0.0481869
$659$	−17.8885	−0.696839	−0.348419	−	0.937339i	$-0.613281\pi$
$659$	−17.8885	−0.696839	−0.348419	+	0.937339i	$0.613281\pi$
$660$	0	0
$661$	−42.7214	−1.66167	−0.830834	−	0.556520i	$-0.812136\pi$
$661$	−42.7214	−1.66167	−0.830834	+	0.556520i	$0.812136\pi$
$662$	15.2705	0.593505
$663$	0	0
$664$	−17.2361	−0.668889
$665$	0	0
$666$	0	0
$667$	−31.1803	−1.20731
$668$	−8.47214	−0.327797
$669$	0	0
$670$	0	0
$671$	−2.29180	−0.0884738
$672$	0	0
$673$	−19.5279	−0.752744	−0.376372	−	0.926469i	$-0.622829\pi$
$673$	−19.5279	−0.752744	−0.376372	+	0.926469i	$0.622829\pi$
$674$	−10.1803	−0.392132
$675$	0	0
$676$	18.5623	0.713935
$677$	−14.3607	−0.551926	−0.275963	−	0.961168i	$-0.588997\pi$
$677$	−14.3607	−0.551926	−0.275963	+	0.961168i	$0.588997\pi$
$678$	0	0
$679$	5.23607	0.200942
$680$	0	0
$681$	0	0
$682$	−0.541020	−0.0207167
$683$	14.1246	0.540463	0.270232	−	0.962795i	$-0.412900\pi$
$683$	14.1246	0.540463	0.270232	+	0.962795i	$0.412900\pi$
$684$	0	0
$685$	0	0
$686$	0.618034	0.0235966
$687$	0	0
$688$	−3.27051	−0.124687
$689$	−10.4721	−0.398957
$690$	0	0
$691$	−4.18034	−0.159028	−0.0795138	−	0.996834i	$-0.525337\pi$
$691$	−4.18034	−0.159028	−0.0795138	+	0.996834i	$0.525337\pi$
$692$	18.6525	0.709061
$693$	0	0
$694$	−12.5066	−0.474743
$695$	0	0
$696$	0	0
$697$	−11.7771	−0.446089
$698$	−2.76393	−0.104616
$699$	0	0
$700$	0	0
$701$	29.0557	1.09742	0.548710	−	0.836013i	$-0.315119\pi$
$701$	29.0557	1.09742	0.548710	+	0.836013i	$0.315119\pi$
$702$	0	0
$703$	13.4164	0.506009
$704$	−0.0557281	−0.00210033
$705$	0	0
$706$	1.34752	0.0507147
$707$	4.76393	0.179166
$708$	0	0
$709$	−12.1115	−0.454855	−0.227428	−	0.973795i	$-0.573032\pi$
$709$	−12.1115	−0.454855	−0.227428	+	0.973795i	$0.573032\pi$
$710$	0	0
$711$	0	0
$712$	−38.5410	−1.44439
$713$	23.1246	0.866024
$714$	0	0
$715$	0	0
$716$	−37.8885	−1.41596
$717$	0	0
$718$	−18.6180	−0.694819
$719$	−16.1803	−0.603425	−0.301712	−	0.953399i	$-0.597558\pi$
$719$	−16.1803	−0.603425	−0.301712	+	0.953399i	$0.597558\pi$
$720$	0	0
$721$	8.47214	0.315519
$722$	−0.618034	−0.0230008
$723$	0	0
$724$	−13.2361	−0.491915
$725$	0	0
$726$	0	0
$727$	3.05573	0.113331	0.0566653	−	0.998393i	$-0.481953\pi$
$727$	3.05573	0.113331	0.0566653	+	0.998393i	$0.481953\pi$
$728$	2.76393	0.102438
$729$	0	0
$730$	0	0
$731$	−4.36068	−0.161286
$732$	0	0
$733$	−4.00000	−0.147743	−0.0738717	−	0.997268i	$-0.523536\pi$
$733$	−4.00000	−0.147743	−0.0738717	+	0.997268i	$0.523536\pi$
$734$	−22.9443	−0.846889
$735$	0	0
$736$	−35.0344	−1.29139
$737$	1.00000	0.0368355
$738$	0	0
$739$	25.6525	0.943642	0.471821	−	0.881694i	$-0.343597\pi$
$739$	25.6525	0.943642	0.471821	+	0.881694i	$0.343597\pi$
$740$	0	0
$741$	0	0
$742$	5.23607	0.192222
$743$	−10.4721	−0.384185	−0.192093	−	0.981377i	$-0.561527\pi$
$743$	−10.4721	−0.384185	−0.192093	+	0.981377i	$0.561527\pi$
$744$	0	0
$745$	0	0
$746$	−23.3820	−0.856075
$747$	0	0
$748$	−0.944272	−0.0345260
$749$	−8.00000	−0.292314
$750$	0	0
$751$	3.05573	0.111505	0.0557526	−	0.998445i	$-0.482244\pi$
$751$	3.05573	0.111505	0.0557526	+	0.998445i	$0.482244\pi$
$752$	−3.70820	−0.135224
$753$	0	0
$754$	−3.81966	−0.139104
$755$	0	0
$756$	0	0
$757$	−19.5836	−0.711778	−0.355889	−	0.934528i	$-0.615822\pi$
$757$	−19.5836	−0.711778	−0.355889	+	0.934528i	$0.615822\pi$
$758$	6.90983	0.250976
$759$	0	0
$760$	0	0
$761$	−27.7771	−1.00692	−0.503459	−	0.864019i	$-0.667940\pi$
$761$	−27.7771	−1.00692	−0.503459	+	0.864019i	$0.667940\pi$
$762$	0	0
$763$	−8.41641	−0.304694
$764$	10.4721	0.378869
$765$	0	0
$766$	20.5410	0.742177
$767$	14.4721	0.522559
$768$	0	0
$769$	−43.0132	−1.55109	−0.775547	−	0.631290i	$-0.782526\pi$
$769$	−43.0132	−1.55109	−0.775547	+	0.631290i	$0.782526\pi$
$770$	0	0
$771$	0	0
$772$	20.0902	0.723061
$773$	50.1803	1.80486	0.902431	−	0.430835i	$-0.141781\pi$
$773$	50.1803	1.80486	0.902431	+	0.430835i	$0.141781\pi$
$774$	0	0
$775$	0	0
$776$	−11.7082	−0.420300
$777$	0	0
$778$	1.78522	0.0640032
$779$	21.3050	0.763329
$780$	0	0
$781$	−2.05573	−0.0735597
$782$	−9.52786	−0.340716
$783$	0	0
$784$	1.85410	0.0662179
$785$	0	0
$786$	0	0
$787$	−40.7639	−1.45308	−0.726539	−	0.687126i	$-0.758872\pi$
$787$	−40.7639	−1.45308	−0.726539	+	0.687126i	$0.758872\pi$
$788$	2.38197	0.0848540
$789$	0	0
$790$	0	0
$791$	14.4164	0.512588
$792$	0	0
$793$	12.0000	0.426132
$794$	5.59675	0.198621
$795$	0	0
$796$	−11.7082	−0.414986
$797$	−35.4164	−1.25451	−0.627257	−	0.778813i	$-0.715822\pi$
$797$	−35.4164	−1.25451	−0.627257	+	0.778813i	$0.715822\pi$
$798$	0	0
$799$	−4.94427	−0.174916
$800$	0	0
$801$	0	0
$802$	1.56231	0.0551669
$803$	2.06888	0.0730093
$804$	0	0
$805$	0	0
$806$	2.83282	0.0997817
$807$	0	0
$808$	−10.6525	−0.374753
$809$	−29.4721	−1.03619	−0.518093	−	0.855325i	$-0.673358\pi$
$809$	−29.4721	−1.03619	−0.518093	+	0.855325i	$0.673358\pi$
$810$	0	0
$811$	−42.7214	−1.50015	−0.750075	−	0.661353i	$-0.769983\pi$
$811$	−42.7214	−1.50015	−0.750075	+	0.661353i	$0.769983\pi$
$812$	−8.09017	−0.283909
$813$	0	0
$814$	0.437694	0.0153412
$815$	0	0
$816$	0	0
$817$	7.88854	0.275985
$818$	−15.1246	−0.528820
$819$	0	0
$820$	0	0
$821$	−28.8328	−1.00627	−0.503136	−	0.864207i	$-0.667821\pi$
$821$	−28.8328	−1.00627	−0.503136	+	0.864207i	$0.667821\pi$
$822$	0	0
$823$	31.6525	1.10334	0.551668	−	0.834064i	$-0.313992\pi$
$823$	31.6525	1.10334	0.551668	+	0.834064i	$0.313992\pi$
$824$	−18.9443	−0.659955
$825$	0	0
$826$	−7.23607	−0.251775
$827$	41.5410	1.44452	0.722261	−	0.691620i	$-0.243103\pi$
$827$	41.5410	1.44452	0.722261	+	0.691620i	$0.243103\pi$
$828$	0	0
$829$	−7.63932	−0.265325	−0.132662	−	0.991161i	$-0.542353\pi$
$829$	−7.63932	−0.265325	−0.132662	+	0.991161i	$0.542353\pi$
$830$	0	0
$831$	0	0
$832$	0.291796	0.0101162
$833$	2.47214	0.0856544
$834$	0	0
$835$	0	0
$836$	1.70820	0.0590795
$837$	0	0
$838$	−16.1803	−0.558941
$839$	30.6525	1.05824	0.529120	−	0.848547i	$-0.322522\pi$
$839$	30.6525	1.05824	0.529120	+	0.848547i	$0.322522\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	8.03444	0.276885
$843$	0	0
$844$	−19.4164	−0.668340
$845$	0	0
$846$	0	0
$847$	10.9443	0.376050
$848$	15.7082	0.539422
$849$	0	0
$850$	0	0
$851$	−18.7082	−0.641309
$852$	0	0
$853$	−27.4164	−0.938720	−0.469360	−	0.883007i	$-0.655515\pi$
$853$	−27.4164	−0.938720	−0.469360	+	0.883007i	$0.655515\pi$
$854$	−6.00000	−0.205316
$855$	0	0
$856$	17.8885	0.611418
$857$	−15.8197	−0.540389	−0.270195	−	0.962806i	$-0.587088\pi$
$857$	−15.8197	−0.540389	−0.270195	+	0.962806i	$0.587088\pi$
$858$	0	0
$859$	22.3607	0.762937	0.381468	−	0.924382i	$-0.375419\pi$
$859$	22.3607	0.762937	0.381468	+	0.924382i	$0.375419\pi$
$860$	0	0
$861$	0	0
$862$	10.8328	0.368967
$863$	18.3475	0.624557	0.312278	−	0.949991i	$-0.398908\pi$
$863$	18.3475	0.624557	0.312278	+	0.949991i	$0.398908\pi$
$864$	0	0
$865$	0	0
$866$	−17.5279	−0.595621
$867$	0	0
$868$	6.00000	0.203653
$869$	−2.63932	−0.0895328
$870$	0	0
$871$	−5.23607	−0.177417
$872$	18.8197	0.637314
$873$	0	0
$874$	17.2361	0.583019
$875$	0	0
$876$	0	0
$877$	−30.3607	−1.02521	−0.512604	−	0.858625i	$-0.671319\pi$
$877$	−30.3607	−1.02521	−0.512604	+	0.858625i	$0.671319\pi$
$878$	5.12461	0.172947
$879$	0	0
$880$	0	0
$881$	−5.81966	−0.196069	−0.0980347	−	0.995183i	$-0.531256\pi$
$881$	−5.81966	−0.196069	−0.0980347	+	0.995183i	$0.531256\pi$
$882$	0	0
$883$	1.40325	0.0472232	0.0236116	−	0.999721i	$-0.492483\pi$
$883$	1.40325	0.0472232	0.0236116	+	0.999721i	$0.492483\pi$
$884$	4.94427	0.166294
$885$	0	0
$886$	12.0000	0.403148
$887$	−21.3475	−0.716780	−0.358390	−	0.933572i	$-0.616674\pi$
$887$	−21.3475	−0.716780	−0.358390	+	0.933572i	$0.616674\pi$
$888$	0	0
$889$	13.6525	0.457889
$890$	0	0
$891$	0	0
$892$	32.6525	1.09329
$893$	8.94427	0.299309
$894$	0	0
$895$	0	0
$896$	−11.3820	−0.380245
$897$	0	0
$898$	12.6869	0.423368
$899$	−18.5410	−0.618378
$900$	0	0
$901$	20.9443	0.697755
$902$	0.695048	0.0231426
$903$	0	0
$904$	−32.2361	−1.07216
$905$	0	0
$906$	0	0
$907$	−34.8328	−1.15660	−0.578302	−	0.815823i	$-0.696285\pi$
$907$	−34.8328	−1.15660	−0.578302	+	0.815823i	$0.696285\pi$
$908$	−34.6525	−1.14998
$909$	0	0
$910$	0	0
$911$	−0.819660	−0.0271566	−0.0135783	−	0.999908i	$-0.504322\pi$
$911$	−0.819660	−0.0271566	−0.0135783	+	0.999908i	$0.504322\pi$
$912$	0	0
$913$	−1.81966	−0.0602220
$914$	−7.74265	−0.256104
$915$	0	0
$916$	7.23607	0.239086
$917$	−16.9443	−0.559549
$918$	0	0
$919$	−27.7639	−0.915848	−0.457924	−	0.888991i	$-0.651407\pi$
$919$	−27.7639	−0.915848	−0.457924	+	0.888991i	$0.651407\pi$
$920$	0	0
$921$	0	0
$922$	−8.76393	−0.288625
$923$	10.7639	0.354299
$924$	0	0
$925$	0	0
$926$	−8.58359	−0.282074
$927$	0	0
$928$	28.0902	0.922105
$929$	−38.2918	−1.25631	−0.628157	−	0.778087i	$-0.716190\pi$
$929$	−38.2918	−1.25631	−0.628157	+	0.778087i	$0.716190\pi$
$930$	0	0
$931$	−4.47214	−0.146568
$932$	−12.8541	−0.421050
$933$	0	0
$934$	−4.29180	−0.140432
$935$	0	0
$936$	0	0
$937$	−35.2361	−1.15111	−0.575556	−	0.817762i	$-0.695214\pi$
$937$	−35.2361	−1.15111	−0.575556	+	0.817762i	$0.695214\pi$
$938$	2.61803	0.0854818
$939$	0	0
$940$	0	0
$941$	5.23607	0.170691	0.0853455	−	0.996351i	$-0.472801\pi$
$941$	5.23607	0.170691	0.0853455	+	0.996351i	$0.472801\pi$
$942$	0	0
$943$	−29.7082	−0.967432
$944$	−21.7082	−0.706542
$945$	0	0
$946$	0.257354	0.00836731
$947$	34.8328	1.13191	0.565957	−	0.824435i	$-0.308507\pi$
$947$	34.8328	1.13191	0.565957	+	0.824435i	$0.308507\pi$
$948$	0	0
$949$	−10.8328	−0.351648
$950$	0	0
$951$	0	0
$952$	−5.52786	−0.179159
$953$	3.47214	0.112474	0.0562368	−	0.998417i	$-0.482090\pi$
$953$	3.47214	0.112474	0.0562368	+	0.998417i	$0.482090\pi$
$954$	0	0
$955$	0	0
$956$	−8.94427	−0.289278
$957$	0	0
$958$	−16.1803	−0.522763
$959$	10.9443	0.353409
$960$	0	0
$961$	−17.2492	−0.556427
$962$	−2.29180	−0.0738905
$963$	0	0
$964$	5.70820	0.183849
$965$	0	0
$966$	0	0
$967$	14.1115	0.453794	0.226897	−	0.973919i	$-0.427142\pi$
$967$	14.1115	0.453794	0.226897	+	0.973919i	$0.427142\pi$
$968$	−24.4721	−0.786564
$969$	0	0
$970$	0	0
$971$	18.0000	0.577647	0.288824	−	0.957382i	$-0.406736\pi$
$971$	18.0000	0.577647	0.288824	+	0.957382i	$0.406736\pi$
$972$	0	0
$973$	−10.6525	−0.341503
$974$	3.56231	0.114144
$975$	0	0
$976$	−18.0000	−0.576166
$977$	−11.4721	−0.367026	−0.183513	−	0.983017i	$-0.558747\pi$
$977$	−11.4721	−0.367026	−0.183513	+	0.983017i	$0.558747\pi$
$978$	0	0
$979$	−4.06888	−0.130042
$980$	0	0
$981$	0	0
$982$	−3.56231	−0.113678
$983$	−34.5410	−1.10169	−0.550844	−	0.834608i	$-0.685694\pi$
$983$	−34.5410	−1.10169	−0.550844	+	0.834608i	$0.685694\pi$
$984$	0	0
$985$	0	0
$986$	7.63932	0.243286
$987$	0	0
$988$	−8.94427	−0.284555
$989$	−11.0000	−0.349780
$990$	0	0
$991$	13.1803	0.418687	0.209344	−	0.977842i	$-0.432867\pi$
$991$	13.1803	0.418687	0.209344	+	0.977842i	$0.432867\pi$
$992$	−20.8328	−0.661443
$993$	0	0
$994$	−5.38197	−0.170706
$995$	0	0
$996$	0	0
$997$	45.4164	1.43835	0.719176	−	0.694828i	$-0.244519\pi$
$997$	45.4164	1.43835	0.719176	+	0.694828i	$0.244519\pi$
$998$	−6.83282	−0.216289
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.2.a.s.1.1		2
3.2	odd	2		175.2.a.d.1.2	✓	2
5.2	odd	4		1575.2.d.k.1324.2		4
5.3	odd	4		1575.2.d.k.1324.3		4
5.4	even	2		1575.2.a.n.1.2		2
12.11	even	2		2800.2.a.bh.1.1		2
15.2	even	4		175.2.b.c.99.3		4
15.8	even	4		175.2.b.c.99.2		4
15.14	odd	2		175.2.a.e.1.1	yes	2
21.20	even	2		1225.2.a.n.1.2		2
60.23	odd	4		2800.2.g.s.449.1		4
60.47	odd	4		2800.2.g.s.449.4		4
60.59	even	2		2800.2.a.bp.1.2		2
105.62	odd	4		1225.2.b.k.99.3		4
105.83	odd	4		1225.2.b.k.99.2		4
105.104	even	2		1225.2.a.u.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
175.2.a.d.1.2	✓	2	3.2	odd	2
175.2.a.e.1.1	yes	2	15.14	odd	2
175.2.b.c.99.2		4	15.8	even	4
175.2.b.c.99.3		4	15.2	even	4
1225.2.a.n.1.2		2	21.20	even	2
1225.2.a.u.1.1		2	105.104	even	2
1225.2.b.k.99.2		4	105.83	odd	4
1225.2.b.k.99.3		4	105.62	odd	4
1575.2.a.n.1.2		2	5.4	even	2
1575.2.a.s.1.1		2	1.1	even	1	trivial
1575.2.d.k.1324.2		4	5.2	odd	4
1575.2.d.k.1324.3		4	5.3	odd	4
2800.2.a.bh.1.1		2	12.11	even	2
2800.2.a.bp.1.2		2	60.59	even	2
2800.2.g.s.449.1		4	60.23	odd	4
2800.2.g.s.449.4		4	60.47	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:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1575.a (trivial)

\(f(q)\)	\(=\)	\(q-0.618034 q^{2} -1.61803 q^{4} -1.00000 q^{7} +2.23607 q^{8} +O(q^{10})\)	\(q-0.618034 q^{2} -1.61803 q^{4} -1.00000 q^{7} +2.23607 q^{8} +0.236068 q^{11} -1.23607 q^{13} +0.618034 q^{14} +1.85410 q^{16} +2.47214 q^{17} -4.47214 q^{19} -0.145898 q^{22} +6.23607 q^{23} +0.763932 q^{26} +1.61803 q^{28} -5.00000 q^{29} +3.70820 q^{31} -5.61803 q^{32} -1.52786 q^{34} -3.00000 q^{37} +2.76393 q^{38} -4.76393 q^{41} -1.76393 q^{43} -0.381966 q^{44} -3.85410 q^{46} -2.00000 q^{47} +1.00000 q^{49} +2.00000 q^{52} +8.47214 q^{53} -2.23607 q^{56} +3.09017 q^{58} -11.7082 q^{59} -9.70820 q^{61} -2.29180 q^{62} -0.236068 q^{64} +4.23607 q^{67} -4.00000 q^{68} -8.70820 q^{71} +8.76393 q^{73} +1.85410 q^{74} +7.23607 q^{76} -0.236068 q^{77} -11.1803 q^{79} +2.94427 q^{82} -7.70820 q^{83} +1.09017 q^{86} +0.527864 q^{88} -17.2361 q^{89} +1.23607 q^{91} -10.0902 q^{92} +1.23607 q^{94} -5.23607 q^{97} -0.618034 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - q^{4} - 2 q^{7}+O(q^{10}) \) 2 * q + q^2 - q^4 - 2 * q^7	\( 2 q + q^{2} - q^{4} - 2 q^{7} - 4 q^{11} + 2 q^{13} - q^{14} - 3 q^{16} - 4 q^{17} - 7 q^{22} + 8 q^{23} + 6 q^{26} + q^{28} - 10 q^{29} - 6 q^{31} - 9 q^{32} - 12 q^{34} - 6 q^{37} + 10 q^{38} - 14 q^{41} - 8 q^{43} - 3 q^{44} - q^{46} - 4 q^{47} + 2 q^{49} + 4 q^{52} + 8 q^{53} - 5 q^{58} - 10 q^{59} - 6 q^{61} - 18 q^{62} + 4 q^{64} + 4 q^{67} - 8 q^{68} - 4 q^{71} + 22 q^{73} - 3 q^{74} + 10 q^{76} + 4 q^{77} - 12 q^{82} - 2 q^{83} - 9 q^{86} + 10 q^{88} - 30 q^{89} - 2 q^{91} - 9 q^{92} - 2 q^{94} - 6 q^{97} + q^{98}+O(q^{100}) \) 2 * q + q^2 - q^4 - 2 * q^7 - 4 * q^11 + 2 * q^13 - q^14 - 3 * q^16 - 4 * q^17 - 7 * q^22 + 8 * q^23 + 6 * q^26 + q^28 - 10 * q^29 - 6 * q^31 - 9 * q^32 - 12 * q^34 - 6 * q^37 + 10 * q^38 - 14 * q^41 - 8 * q^43 - 3 * q^44 - q^46 - 4 * q^47 + 2 * q^49 + 4 * q^52 + 8 * q^53 - 5 * q^58 - 10 * q^59 - 6 * q^61 - 18 * q^62 + 4 * q^64 + 4 * q^67 - 8 * q^68 - 4 * q^71 + 22 * q^73 - 3 * q^74 + 10 * q^76 + 4 * q^77 - 12 * q^82 - 2 * q^83 - 9 * q^86 + 10 * q^88 - 30 * q^89 - 2 * q^91 - 9 * q^92 - 2 * q^94 - 6 * q^97 + q^98

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