LMFDB - Embedded newform 5175.2.a.bj.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5175,2,Mod(1,5175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5175, 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("5175.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5175 = 3^{2} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5175.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:	$41.3225830460$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 345)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	5175.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{2} +3.82843 q^{7} -2.82843 q^{8} +O(q^{10})$	$q+1.41421 q^{2} +3.82843 q^{7} -2.82843 q^{8} +5.41421 q^{11} -0.585786 q^{13} +5.41421 q^{14} -4.00000 q^{16} +8.07107 q^{17} +2.24264 q^{19} +7.65685 q^{22} -1.00000 q^{23} -0.828427 q^{26} +6.41421 q^{29} -9.82843 q^{31} +11.4142 q^{34} -3.00000 q^{37} +3.17157 q^{38} +7.58579 q^{41} -6.00000 q^{43} -1.41421 q^{46} -11.4142 q^{47} +7.65685 q^{49} -1.24264 q^{53} -10.8284 q^{56} +9.07107 q^{58} -12.8995 q^{59} +2.58579 q^{61} -13.8995 q^{62} +8.00000 q^{64} -5.48528 q^{67} +10.8995 q^{71} +1.75736 q^{73} -4.24264 q^{74} +20.7279 q^{77} +7.31371 q^{79} +10.7279 q^{82} -0.0710678 q^{83} -8.48528 q^{86} -15.3137 q^{88} +18.1421 q^{89} -2.24264 q^{91} -16.1421 q^{94} +6.82843 q^{97} +10.8284 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{7}+O(q^{10}) $ 2 * q + 2 * q^7	$ 2 q + 2 q^{7} + 8 q^{11} - 4 q^{13} + 8 q^{14} - 8 q^{16} + 2 q^{17} - 4 q^{19} + 4 q^{22} - 2 q^{23} + 4 q^{26} + 10 q^{29} - 14 q^{31} + 20 q^{34} - 6 q^{37} + 12 q^{38} + 18 q^{41} - 12 q^{43} - 20 q^{47} + 4 q^{49} + 6 q^{53} - 16 q^{56} + 4 q^{58} - 6 q^{59} + 8 q^{61} - 8 q^{62} + 16 q^{64} + 6 q^{67} + 2 q^{71} + 12 q^{73} + 16 q^{77} - 8 q^{79} - 4 q^{82} + 14 q^{83} - 8 q^{88} + 8 q^{89} + 4 q^{91} - 4 q^{94} + 8 q^{97} + 16 q^{98}+O(q^{100}) $ 2 * q + 2 * q^7 + 8 * q^11 - 4 * q^13 + 8 * q^14 - 8 * q^16 + 2 * q^17 - 4 * q^19 + 4 * q^22 - 2 * q^23 + 4 * q^26 + 10 * q^29 - 14 * q^31 + 20 * q^34 - 6 * q^37 + 12 * q^38 + 18 * q^41 - 12 * q^43 - 20 * q^47 + 4 * q^49 + 6 * q^53 - 16 * q^56 + 4 * q^58 - 6 * q^59 + 8 * q^61 - 8 * q^62 + 16 * q^64 + 6 * q^67 + 2 * q^71 + 12 * q^73 + 16 * q^77 - 8 * q^79 - 4 * q^82 + 14 * q^83 - 8 * q^88 + 8 * q^89 + 4 * q^91 - 4 * q^94 + 8 * q^97 + 16 * 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 $

$2$	1.41421	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$2$	1.41421	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.82843	1.44701	0.723505	−	0.690319i	$-0.242530\pi$
$7$	3.82843	1.44701	0.723505	+	0.690319i	$0.242530\pi$
$8$	−2.82843	−1.00000
$9$	0	0
$10$	0	0
$11$	5.41421	1.63245	0.816223	−	0.577736i	$-0.196064\pi$
$11$	5.41421	1.63245	0.816223	+	0.577736i	$0.196064\pi$
$12$	0	0
$13$	−0.585786	−0.162468	−0.0812340	−	0.996695i	$-0.525886\pi$
$13$	−0.585786	−0.162468	−0.0812340	+	0.996695i	$0.525886\pi$
$14$	5.41421	1.44701
$15$	0	0
$16$	−4.00000	−1.00000
$17$	8.07107	1.95752	0.978761	−	0.205006i	$-0.0657214\pi$
$17$	8.07107	1.95752	0.978761	+	0.205006i	$0.0657214\pi$
$18$	0	0
$19$	2.24264	0.514497	0.257249	−	0.966345i	$-0.417184\pi$
$19$	2.24264	0.514497	0.257249	+	0.966345i	$0.417184\pi$
$20$	0	0
$21$	0	0
$22$	7.65685	1.63245
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	0	0
$26$	−0.828427	−0.162468
$27$	0	0
$28$	0	0
$29$	6.41421	1.19109	0.595545	−	0.803322i	$-0.296936\pi$
$29$	6.41421	1.19109	0.595545	+	0.803322i	$0.296936\pi$
$30$	0	0
$31$	−9.82843	−1.76524	−0.882619	−	0.470089i	$-0.844222\pi$
$31$	−9.82843	−1.76524	−0.882619	+	0.470089i	$0.844222\pi$
$32$	0	0
$33$	0	0
$34$	11.4142	1.95752
$35$	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	$0.579313\pi$
$38$	3.17157	0.514497
$39$	0	0
$40$	0	0
$41$	7.58579	1.18470	0.592350	−	0.805680i	$-0.298200\pi$
$41$	7.58579	1.18470	0.592350	+	0.805680i	$0.298200\pi$
$42$	0	0
$43$	−6.00000	−0.914991	−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000	−0.914991	−0.457496	+	0.889212i	$0.651253\pi$
$44$	0	0
$45$	0	0
$46$	−1.41421	−0.208514
$47$	−11.4142	−1.66493	−0.832467	−	0.554075i	$-0.813072\pi$
$47$	−11.4142	−1.66493	−0.832467	+	0.554075i	$0.813072\pi$
$48$	0	0
$49$	7.65685	1.09384
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−1.24264	−0.170690	−0.0853449	−	0.996351i	$-0.527199\pi$
$53$	−1.24264	−0.170690	−0.0853449	+	0.996351i	$0.527199\pi$
$54$	0	0
$55$	0	0
$56$	−10.8284	−1.44701
$57$	0	0
$58$	9.07107	1.19109
$59$	−12.8995	−1.67937	−0.839686	−	0.543073i	$-0.817261\pi$
$59$	−12.8995	−1.67937	−0.839686	+	0.543073i	$0.817261\pi$
$60$	0	0
$61$	2.58579	0.331076	0.165538	−	0.986203i	$-0.447064\pi$
$61$	2.58579	0.331076	0.165538	+	0.986203i	$0.447064\pi$
$62$	−13.8995	−1.76524
$63$	0	0
$64$	8.00000	1.00000
$65$	0	0
$66$	0	0
$67$	−5.48528	−0.670134	−0.335067	−	0.942194i	$-0.608759\pi$
$67$	−5.48528	−0.670134	−0.335067	+	0.942194i	$0.608759\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	10.8995	1.29353	0.646766	−	0.762688i	$-0.276121\pi$
$71$	10.8995	1.29353	0.646766	+	0.762688i	$0.276121\pi$
$72$	0	0
$73$	1.75736	0.205683	0.102842	−	0.994698i	$-0.467206\pi$
$73$	1.75736	0.205683	0.102842	+	0.994698i	$0.467206\pi$
$74$	−4.24264	−0.493197
$75$	0	0
$76$	0	0
$77$	20.7279	2.36217
$78$	0	0
$79$	7.31371	0.822856	0.411428	−	0.911442i	$-0.365030\pi$
$79$	7.31371	0.822856	0.411428	+	0.911442i	$0.365030\pi$
$80$	0	0
$81$	0	0
$82$	10.7279	1.18470
$83$	−0.0710678	−0.00780071	−0.00390035	−	0.999992i	$-0.501242\pi$
$83$	−0.0710678	−0.00780071	−0.00390035	+	0.999992i	$0.501242\pi$
$84$	0	0
$85$	0	0
$86$	−8.48528	−0.914991
$87$	0	0
$88$	−15.3137	−1.63245
$89$	18.1421	1.92306	0.961531	−	0.274696i	$-0.0885771\pi$
$89$	18.1421	1.92306	0.961531	+	0.274696i	$0.0885771\pi$
$90$	0	0
$91$	−2.24264	−0.235093
$92$	0	0
$93$	0	0
$94$	−16.1421	−1.66493
$95$	0	0
$96$	0	0
$97$	6.82843	0.693322	0.346661	−	0.937991i	$-0.387315\pi$
$97$	6.82843	0.693322	0.346661	+	0.937991i	$0.387315\pi$
$98$	10.8284	1.09384
$99$	0	0
$100$	0	0
$101$	18.5563	1.84643	0.923213	−	0.384289i	$-0.125553\pi$
$101$	18.5563	1.84643	0.923213	+	0.384289i	$0.125553\pi$
$102$	0	0
$103$	4.82843	0.475759	0.237880	−	0.971295i	$-0.423548\pi$
$103$	4.82843	0.475759	0.237880	+	0.971295i	$0.423548\pi$
$104$	1.65685	0.162468
$105$	0	0
$106$	−1.75736	−0.170690
$107$	−1.58579	−0.153304	−0.0766519	−	0.997058i	$-0.524423\pi$
$107$	−1.58579	−0.153304	−0.0766519	+	0.997058i	$0.524423\pi$
$108$	0	0
$109$	5.75736	0.551455	0.275728	−	0.961236i	$-0.411081\pi$
$109$	5.75736	0.551455	0.275728	+	0.961236i	$0.411081\pi$
$110$	0	0
$111$	0	0
$112$	−15.3137	−1.44701
$113$	−7.58579	−0.713611	−0.356805	−	0.934179i	$-0.616134\pi$
$113$	−7.58579	−0.713611	−0.356805	+	0.934179i	$0.616134\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	−18.2426	−1.67937
$119$	30.8995	2.83255
$120$	0	0
$121$	18.3137	1.66488
$122$	3.65685	0.331076
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−8.24264	−0.731416	−0.365708	−	0.930730i	$-0.619173\pi$
$127$	−8.24264	−0.731416	−0.365708	+	0.930730i	$0.619173\pi$
$128$	11.3137	1.00000
$129$	0	0
$130$	0	0
$131$	2.34315	0.204722	0.102361	−	0.994747i	$-0.467360\pi$
$131$	2.34315	0.204722	0.102361	+	0.994747i	$0.467360\pi$
$132$	0	0
$133$	8.58579	0.744482
$134$	−7.75736	−0.670134
$135$	0	0
$136$	−22.8284	−1.95752
$137$	−2.82843	−0.241649	−0.120824	−	0.992674i	$-0.538554\pi$
$137$	−2.82843	−0.241649	−0.120824	+	0.992674i	$0.538554\pi$
$138$	0	0
$139$	−11.0000	−0.933008	−0.466504	−	0.884519i	$-0.654487\pi$
$139$	−11.0000	−0.933008	−0.466504	+	0.884519i	$0.654487\pi$
$140$	0	0
$141$	0	0
$142$	15.4142	1.29353
$143$	−3.17157	−0.265220
$144$	0	0
$145$	0	0
$146$	2.48528	0.205683
$147$	0	0
$148$	0	0
$149$	5.07107	0.415438	0.207719	−	0.978189i	$-0.433396\pi$
$149$	5.07107	0.415438	0.207719	+	0.978189i	$0.433396\pi$
$150$	0	0
$151$	13.3137	1.08345	0.541727	−	0.840554i	$-0.317771\pi$
$151$	13.3137	1.08345	0.541727	+	0.840554i	$0.317771\pi$
$152$	−6.34315	−0.514497
$153$	0	0
$154$	29.3137	2.36217
$155$	0	0
$156$	0	0
$157$	−15.4853	−1.23586	−0.617930	−	0.786233i	$-0.712028\pi$
$157$	−15.4853	−1.23586	−0.617930	+	0.786233i	$0.712028\pi$
$158$	10.3431	0.822856
$159$	0	0
$160$	0	0
$161$	−3.82843	−0.301722
$162$	0	0
$163$	2.00000	0.156652	0.0783260	−	0.996928i	$-0.475042\pi$
$163$	2.00000	0.156652	0.0783260	+	0.996928i	$0.475042\pi$
$164$	0	0
$165$	0	0
$166$	−0.100505	−0.00780071
$167$	18.7279	1.44921	0.724605	−	0.689164i	$-0.242022\pi$
$167$	18.7279	1.44921	0.724605	+	0.689164i	$0.242022\pi$
$168$	0	0
$169$	−12.6569	−0.973604
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2.34315	0.178146	0.0890730	−	0.996025i	$-0.471610\pi$
$173$	2.34315	0.178146	0.0890730	+	0.996025i	$0.471610\pi$
$174$	0	0
$175$	0	0
$176$	−21.6569	−1.63245
$177$	0	0
$178$	25.6569	1.92306
$179$	−3.65685	−0.273326	−0.136663	−	0.990618i	$-0.543638\pi$
$179$	−3.65685	−0.273326	−0.136663	+	0.990618i	$0.543638\pi$
$180$	0	0
$181$	−1.17157	−0.0870823	−0.0435412	−	0.999052i	$-0.513864\pi$
$181$	−1.17157	−0.0870823	−0.0435412	+	0.999052i	$0.513864\pi$
$182$	−3.17157	−0.235093
$183$	0	0
$184$	2.82843	0.208514
$185$	0	0
$186$	0	0
$187$	43.6985	3.19555
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1.75736	−0.127158	−0.0635790	−	0.997977i	$-0.520251\pi$
$191$	−1.75736	−0.127158	−0.0635790	+	0.997977i	$0.520251\pi$
$192$	0	0
$193$	−8.48528	−0.610784	−0.305392	−	0.952227i	$-0.598787\pi$
$193$	−8.48528	−0.610784	−0.305392	+	0.952227i	$0.598787\pi$
$194$	9.65685	0.693322
$195$	0	0
$196$	0	0
$197$	11.1716	0.795942	0.397971	−	0.917398i	$-0.369715\pi$
$197$	11.1716	0.795942	0.397971	+	0.917398i	$0.369715\pi$
$198$	0	0
$199$	−6.48528	−0.459729	−0.229865	−	0.973223i	$-0.573828\pi$
$199$	−6.48528	−0.459729	−0.229865	+	0.973223i	$0.573828\pi$
$200$	0	0
$201$	0	0
$202$	26.2426	1.84643
$203$	24.5563	1.72352
$204$	0	0
$205$	0	0
$206$	6.82843	0.475759
$207$	0	0
$208$	2.34315	0.162468
$209$	12.1421	0.839889
$210$	0	0
$211$	−10.6569	−0.733648	−0.366824	−	0.930290i	$-0.619555\pi$
$211$	−10.6569	−0.733648	−0.366824	+	0.930290i	$0.619555\pi$
$212$	0	0
$213$	0	0
$214$	−2.24264	−0.153304
$215$	0	0
$216$	0	0
$217$	−37.6274	−2.55432
$218$	8.14214	0.551455
$219$	0	0
$220$	0	0
$221$	−4.72792	−0.318034
$222$	0	0
$223$	14.0000	0.937509	0.468755	−	0.883328i	$-0.344703\pi$
$223$	14.0000	0.937509	0.468755	+	0.883328i	$0.344703\pi$
$224$	0	0
$225$	0	0
$226$	−10.7279	−0.713611
$227$	6.34315	0.421009	0.210505	−	0.977593i	$-0.432489\pi$
$227$	6.34315	0.421009	0.210505	+	0.977593i	$0.432489\pi$
$228$	0	0
$229$	−15.7990	−1.04403	−0.522013	−	0.852937i	$-0.674819\pi$
$229$	−15.7990	−1.04403	−0.522013	+	0.852937i	$0.674819\pi$
$230$	0	0
$231$	0	0
$232$	−18.1421	−1.19109
$233$	−1.65685	−0.108544	−0.0542721	−	0.998526i	$-0.517284\pi$
$233$	−1.65685	−0.108544	−0.0542721	+	0.998526i	$0.517284\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	43.6985	2.83255
$239$	4.41421	0.285532	0.142766	−	0.989756i	$-0.454400\pi$
$239$	4.41421	0.285532	0.142766	+	0.989756i	$0.454400\pi$
$240$	0	0
$241$	20.5858	1.32605	0.663024	−	0.748599i	$-0.269273\pi$
$241$	20.5858	1.32605	0.663024	+	0.748599i	$0.269273\pi$
$242$	25.8995	1.66488
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1.31371	−0.0835893
$248$	27.7990	1.76524
$249$	0	0
$250$	0	0
$251$	−24.6274	−1.55447	−0.777234	−	0.629211i	$-0.783378\pi$
$251$	−24.6274	−1.55447	−0.777234	+	0.629211i	$0.783378\pi$
$252$	0	0
$253$	−5.41421	−0.340389
$254$	−11.6569	−0.731416
$255$	0	0
$256$	0	0
$257$	−18.0416	−1.12541	−0.562703	−	0.826659i	$-0.690239\pi$
$257$	−18.0416	−1.12541	−0.562703	+	0.826659i	$0.690239\pi$
$258$	0	0
$259$	−11.4853	−0.713661
$260$	0	0
$261$	0	0
$262$	3.31371	0.204722
$263$	−21.7279	−1.33980	−0.669901	−	0.742451i	$-0.733663\pi$
$263$	−21.7279	−1.33980	−0.669901	+	0.742451i	$0.733663\pi$
$264$	0	0
$265$	0	0
$266$	12.1421	0.744482
$267$	0	0
$268$	0	0
$269$	−4.55635	−0.277806	−0.138903	−	0.990306i	$-0.544358\pi$
$269$	−4.55635	−0.277806	−0.138903	+	0.990306i	$0.544358\pi$
$270$	0	0
$271$	−23.4853	−1.42663	−0.713315	−	0.700844i	$-0.752807\pi$
$271$	−23.4853	−1.42663	−0.713315	+	0.700844i	$0.752807\pi$
$272$	−32.2843	−1.95752
$273$	0	0
$274$	−4.00000	−0.241649
$275$	0	0
$276$	0	0
$277$	−2.68629	−0.161404	−0.0807018	−	0.996738i	$-0.525716\pi$
$277$	−2.68629	−0.161404	−0.0807018	+	0.996738i	$0.525716\pi$
$278$	−15.5563	−0.933008
$279$	0	0
$280$	0	0
$281$	−14.5858	−0.870115	−0.435058	−	0.900403i	$-0.643272\pi$
$281$	−14.5858	−0.870115	−0.435058	+	0.900403i	$0.643272\pi$
$282$	0	0
$283$	17.3431	1.03094	0.515472	−	0.856907i	$-0.327617\pi$
$283$	17.3431	1.03094	0.515472	+	0.856907i	$0.327617\pi$
$284$	0	0
$285$	0	0
$286$	−4.48528	−0.265220
$287$	29.0416	1.71427
$288$	0	0
$289$	48.1421	2.83189
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−22.8995	−1.33780	−0.668901	−	0.743351i	$-0.733235\pi$
$293$	−22.8995	−1.33780	−0.668901	+	0.743351i	$0.733235\pi$
$294$	0	0
$295$	0	0
$296$	8.48528	0.493197
$297$	0	0
$298$	7.17157	0.415438
$299$	0.585786	0.0338769
$300$	0	0
$301$	−22.9706	−1.32400
$302$	18.8284	1.08345
$303$	0	0
$304$	−8.97056	−0.514497
$305$	0	0
$306$	0	0
$307$	15.2132	0.868263	0.434132	−	0.900849i	$-0.357055\pi$
$307$	15.2132	0.868263	0.434132	+	0.900849i	$0.357055\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−6.34315	−0.359687	−0.179843	−	0.983695i	$-0.557559\pi$
$311$	−6.34315	−0.359687	−0.179843	+	0.983695i	$0.557559\pi$
$312$	0	0
$313$	7.97056	0.450523	0.225261	−	0.974298i	$-0.427676\pi$
$313$	7.97056	0.450523	0.225261	+	0.974298i	$0.427676\pi$
$314$	−21.8995	−1.23586
$315$	0	0
$316$	0	0
$317$	1.55635	0.0874133	0.0437066	−	0.999044i	$-0.486083\pi$
$317$	1.55635	0.0874133	0.0437066	+	0.999044i	$0.486083\pi$
$318$	0	0
$319$	34.7279	1.94439
$320$	0	0
$321$	0	0
$322$	−5.41421	−0.301722
$323$	18.1005	1.00714
$324$	0	0
$325$	0	0
$326$	2.82843	0.156652
$327$	0	0
$328$	−21.4558	−1.18470
$329$	−43.6985	−2.40918
$330$	0	0
$331$	1.00000	0.0549650	0.0274825	−	0.999622i	$-0.491251\pi$
$331$	1.00000	0.0549650	0.0274825	+	0.999622i	$0.491251\pi$
$332$	0	0
$333$	0	0
$334$	26.4853	1.44921
$335$	0	0
$336$	0	0
$337$	2.97056	0.161817	0.0809084	−	0.996722i	$-0.474218\pi$
$337$	2.97056	0.161817	0.0809084	+	0.996722i	$0.474218\pi$
$338$	−17.8995	−0.973604
$339$	0	0
$340$	0	0
$341$	−53.2132	−2.88166
$342$	0	0
$343$	2.51472	0.135782
$344$	16.9706	0.914991
$345$	0	0
$346$	3.31371	0.178146
$347$	18.8284	1.01076	0.505381	−	0.862896i	$-0.331352\pi$
$347$	18.8284	1.01076	0.505381	+	0.862896i	$0.331352\pi$
$348$	0	0
$349$	30.1127	1.61190	0.805948	−	0.591986i	$-0.201656\pi$
$349$	30.1127	1.61190	0.805948	+	0.591986i	$0.201656\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	30.5269	1.62478	0.812392	−	0.583112i	$-0.198165\pi$
$353$	30.5269	1.62478	0.812392	+	0.583112i	$0.198165\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	−5.17157	−0.273326
$359$	25.0711	1.32320	0.661600	−	0.749857i	$-0.269878\pi$
$359$	25.0711	1.32320	0.661600	+	0.749857i	$0.269878\pi$
$360$	0	0
$361$	−13.9706	−0.735293
$362$	−1.65685	−0.0870823
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−19.9706	−1.04245	−0.521227	−	0.853418i	$-0.674526\pi$
$367$	−19.9706	−1.04245	−0.521227	+	0.853418i	$0.674526\pi$
$368$	4.00000	0.208514
$369$	0	0
$370$	0	0
$371$	−4.75736	−0.246990
$372$	0	0
$373$	−20.9706	−1.08581	−0.542907	−	0.839793i	$-0.682677\pi$
$373$	−20.9706	−1.08581	−0.542907	+	0.839793i	$0.682677\pi$
$374$	61.7990	3.19555
$375$	0	0
$376$	32.2843	1.66493
$377$	−3.75736	−0.193514
$378$	0	0
$379$	−4.97056	−0.255321	−0.127660	−	0.991818i	$-0.540747\pi$
$379$	−4.97056	−0.255321	−0.127660	+	0.991818i	$0.540747\pi$
$380$	0	0
$381$	0	0
$382$	−2.48528	−0.127158
$383$	11.5858	0.592006	0.296003	−	0.955187i	$-0.404346\pi$
$383$	11.5858	0.592006	0.296003	+	0.955187i	$0.404346\pi$
$384$	0	0
$385$	0	0
$386$	−12.0000	−0.610784
$387$	0	0
$388$	0	0
$389$	2.48528	0.126009	0.0630044	−	0.998013i	$-0.479932\pi$
$389$	2.48528	0.126009	0.0630044	+	0.998013i	$0.479932\pi$
$390$	0	0
$391$	−8.07107	−0.408171
$392$	−21.6569	−1.09384
$393$	0	0
$394$	15.7990	0.795942
$395$	0	0
$396$	0	0
$397$	−7.31371	−0.367065	−0.183532	−	0.983014i	$-0.558753\pi$
$397$	−7.31371	−0.367065	−0.183532	+	0.983014i	$0.558753\pi$
$398$	−9.17157	−0.459729
$399$	0	0
$400$	0	0
$401$	3.85786	0.192653	0.0963263	−	0.995350i	$-0.469291\pi$
$401$	3.85786	0.192653	0.0963263	+	0.995350i	$0.469291\pi$
$402$	0	0
$403$	5.75736	0.286794
$404$	0	0
$405$	0	0
$406$	34.7279	1.72352
$407$	−16.2426	−0.805118
$408$	0	0
$409$	−15.4853	−0.765698	−0.382849	−	0.923811i	$-0.625057\pi$
$409$	−15.4853	−0.765698	−0.382849	+	0.923811i	$0.625057\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−49.3848	−2.43007
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	17.1716	0.839889
$419$	−24.0416	−1.17451	−0.587255	−	0.809402i	$-0.699792\pi$
$419$	−24.0416	−1.17451	−0.587255	+	0.809402i	$0.699792\pi$
$420$	0	0
$421$	−10.2426	−0.499196	−0.249598	−	0.968350i	$-0.580298\pi$
$421$	−10.2426	−0.499196	−0.249598	+	0.968350i	$0.580298\pi$
$422$	−15.0711	−0.733648
$423$	0	0
$424$	3.51472	0.170690
$425$	0	0
$426$	0	0
$427$	9.89949	0.479070
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−11.7990	−0.568337	−0.284169	−	0.958774i	$-0.591718\pi$
$431$	−11.7990	−0.568337	−0.284169	+	0.958774i	$0.591718\pi$
$432$	0	0
$433$	−3.82843	−0.183982	−0.0919912	−	0.995760i	$-0.529323\pi$
$433$	−3.82843	−0.183982	−0.0919912	+	0.995760i	$0.529323\pi$
$434$	−53.2132	−2.55432
$435$	0	0
$436$	0	0
$437$	−2.24264	−0.107280
$438$	0	0
$439$	−1.51472	−0.0722936	−0.0361468	−	0.999346i	$-0.511508\pi$
$439$	−1.51472	−0.0722936	−0.0361468	+	0.999346i	$0.511508\pi$
$440$	0	0
$441$	0	0
$442$	−6.68629	−0.318034
$443$	−34.5269	−1.64042	−0.820212	−	0.572060i	$-0.806144\pi$
$443$	−34.5269	−1.64042	−0.820212	+	0.572060i	$0.806144\pi$
$444$	0	0
$445$	0	0
$446$	19.7990	0.937509
$447$	0	0
$448$	30.6274	1.44701
$449$	4.75736	0.224514	0.112257	−	0.993679i	$-0.464192\pi$
$449$	4.75736	0.224514	0.112257	+	0.993679i	$0.464192\pi$
$450$	0	0
$451$	41.0711	1.93396
$452$	0	0
$453$	0	0
$454$	8.97056	0.421009
$455$	0	0
$456$	0	0
$457$	31.0000	1.45012	0.725059	−	0.688686i	$-0.241812\pi$
$457$	31.0000	1.45012	0.725059	+	0.688686i	$0.241812\pi$
$458$	−22.3431	−1.04403
$459$	0	0
$460$	0	0
$461$	26.4853	1.23354	0.616771	−	0.787142i	$-0.288440\pi$
$461$	26.4853	1.23354	0.616771	+	0.787142i	$0.288440\pi$
$462$	0	0
$463$	−2.92893	−0.136119	−0.0680595	−	0.997681i	$-0.521681\pi$
$463$	−2.92893	−0.136119	−0.0680595	+	0.997681i	$0.521681\pi$
$464$	−25.6569	−1.19109
$465$	0	0
$466$	−2.34315	−0.108544
$467$	6.41421	0.296814	0.148407	−	0.988926i	$-0.452585\pi$
$467$	6.41421	0.296814	0.148407	+	0.988926i	$0.452585\pi$
$468$	0	0
$469$	−21.0000	−0.969690
$470$	0	0
$471$	0	0
$472$	36.4853	1.67937
$473$	−32.4853	−1.49367
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	6.24264	0.285532
$479$	−6.72792	−0.307407	−0.153703	−	0.988117i	$-0.549120\pi$
$479$	−6.72792	−0.307407	−0.153703	+	0.988117i	$0.549120\pi$
$480$	0	0
$481$	1.75736	0.0801287
$482$	29.1127	1.32605
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	26.3848	1.19561	0.597804	−	0.801642i	$-0.296040\pi$
$487$	26.3848	1.19561	0.597804	+	0.801642i	$0.296040\pi$
$488$	−7.31371	−0.331076
$489$	0	0
$490$	0	0
$491$	12.0711	0.544760	0.272380	−	0.962190i	$-0.412189\pi$
$491$	12.0711	0.544760	0.272380	+	0.962190i	$0.412189\pi$
$492$	0	0
$493$	51.7696	2.33158
$494$	−1.85786	−0.0835893
$495$	0	0
$496$	39.3137	1.76524
$497$	41.7279	1.87175
$498$	0	0
$499$	−13.0000	−0.581960	−0.290980	−	0.956729i	$-0.593981\pi$
$499$	−13.0000	−0.581960	−0.290980	+	0.956729i	$0.593981\pi$
$500$	0	0
$501$	0	0
$502$	−34.8284	−1.55447
$503$	−39.7279	−1.77138	−0.885690	−	0.464277i	$-0.846314\pi$
$503$	−39.7279	−1.77138	−0.885690	+	0.464277i	$0.846314\pi$
$504$	0	0
$505$	0	0
$506$	−7.65685	−0.340389
$507$	0	0
$508$	0	0
$509$	−7.79899	−0.345684	−0.172842	−	0.984950i	$-0.555295\pi$
$509$	−7.79899	−0.345684	−0.172842	+	0.984950i	$0.555295\pi$
$510$	0	0
$511$	6.72792	0.297626
$512$	−22.6274	−1.00000
$513$	0	0
$514$	−25.5147	−1.12541
$515$	0	0
$516$	0	0
$517$	−61.7990	−2.71792
$518$	−16.2426	−0.713661
$519$	0	0
$520$	0	0
$521$	31.5563	1.38251	0.691254	−	0.722612i	$-0.257058\pi$
$521$	31.5563	1.38251	0.691254	+	0.722612i	$0.257058\pi$
$522$	0	0
$523$	−12.0000	−0.524723	−0.262362	−	0.964970i	$-0.584501\pi$
$523$	−12.0000	−0.524723	−0.262362	+	0.964970i	$0.584501\pi$
$524$	0	0
$525$	0	0
$526$	−30.7279	−1.33980
$527$	−79.3259	−3.45549
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−4.44365	−0.192476
$534$	0	0
$535$	0	0
$536$	15.5147	0.670134
$537$	0	0
$538$	−6.44365	−0.277806
$539$	41.4558	1.78563
$540$	0	0
$541$	5.85786	0.251849	0.125925	−	0.992040i	$-0.459810\pi$
$541$	5.85786	0.251849	0.125925	+	0.992040i	$0.459810\pi$
$542$	−33.2132	−1.42663
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−12.9706	−0.554581	−0.277291	−	0.960786i	$-0.589436\pi$
$547$	−12.9706	−0.554581	−0.277291	+	0.960786i	$0.589436\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	14.3848	0.612812
$552$	0	0
$553$	28.0000	1.19068
$554$	−3.79899	−0.161404
$555$	0	0
$556$	0	0
$557$	−6.89949	−0.292341	−0.146170	−	0.989259i	$-0.546695\pi$
$557$	−6.89949	−0.292341	−0.146170	+	0.989259i	$0.546695\pi$
$558$	0	0
$559$	3.51472	0.148657
$560$	0	0
$561$	0	0
$562$	−20.6274	−0.870115
$563$	20.8995	0.880809	0.440404	−	0.897800i	$-0.354835\pi$
$563$	20.8995	0.880809	0.440404	+	0.897800i	$0.354835\pi$
$564$	0	0
$565$	0	0
$566$	24.5269	1.03094
$567$	0	0
$568$	−30.8284	−1.29353
$569$	−12.1421	−0.509025	−0.254512	−	0.967070i	$-0.581915\pi$
$569$	−12.1421	−0.509025	−0.254512	+	0.967070i	$0.581915\pi$
$570$	0	0
$571$	28.7279	1.20223	0.601113	−	0.799164i	$-0.294724\pi$
$571$	28.7279	1.20223	0.601113	+	0.799164i	$0.294724\pi$
$572$	0	0
$573$	0	0
$574$	41.0711	1.71427
$575$	0	0
$576$	0	0
$577$	3.02944	0.126117	0.0630586	−	0.998010i	$-0.479915\pi$
$577$	3.02944	0.126117	0.0630586	+	0.998010i	$0.479915\pi$
$578$	68.0833	2.83189
$579$	0	0
$580$	0	0
$581$	−0.272078	−0.0112877
$582$	0	0
$583$	−6.72792	−0.278642
$584$	−4.97056	−0.205683
$585$	0	0
$586$	−32.3848	−1.33780
$587$	−4.62742	−0.190994	−0.0954970	−	0.995430i	$-0.530444\pi$
$587$	−4.62742	−0.190994	−0.0954970	+	0.995430i	$0.530444\pi$
$588$	0	0
$589$	−22.0416	−0.908210
$590$	0	0
$591$	0	0
$592$	12.0000	0.493197
$593$	−33.5563	−1.37799	−0.688997	−	0.724764i	$-0.741949\pi$
$593$	−33.5563	−1.37799	−0.688997	+	0.724764i	$0.741949\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0.828427	0.0338769
$599$	−38.8284	−1.58649	−0.793243	−	0.608905i	$-0.791609\pi$
$599$	−38.8284	−1.58649	−0.793243	+	0.608905i	$0.791609\pi$
$600$	0	0
$601$	4.31371	0.175960	0.0879799	−	0.996122i	$-0.471959\pi$
$601$	4.31371	0.175960	0.0879799	+	0.996122i	$0.471959\pi$
$602$	−32.4853	−1.32400
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	1.61522	0.0655599	0.0327800	−	0.999463i	$-0.489564\pi$
$607$	1.61522	0.0655599	0.0327800	+	0.999463i	$0.489564\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	6.68629	0.270498
$612$	0	0
$613$	−12.6274	−0.510017	−0.255008	−	0.966939i	$-0.582078\pi$
$613$	−12.6274	−0.510017	−0.255008	+	0.966939i	$0.582078\pi$
$614$	21.5147	0.868263
$615$	0	0
$616$	−58.6274	−2.36217
$617$	−14.8995	−0.599831	−0.299916	−	0.953966i	$-0.596959\pi$
$617$	−14.8995	−0.599831	−0.299916	+	0.953966i	$0.596959\pi$
$618$	0	0
$619$	8.28427	0.332973	0.166486	−	0.986044i	$-0.446758\pi$
$619$	8.28427	0.332973	0.166486	+	0.986044i	$0.446758\pi$
$620$	0	0
$621$	0	0
$622$	−8.97056	−0.359687
$623$	69.4558	2.78269
$624$	0	0
$625$	0	0
$626$	11.2721	0.450523
$627$	0	0
$628$	0	0
$629$	−24.2132	−0.965444
$630$	0	0
$631$	−9.21320	−0.366772	−0.183386	−	0.983041i	$-0.558706\pi$
$631$	−9.21320	−0.366772	−0.183386	+	0.983041i	$0.558706\pi$
$632$	−20.6863	−0.822856
$633$	0	0
$634$	2.20101	0.0874133
$635$	0	0
$636$	0	0
$637$	−4.48528	−0.177713
$638$	49.1127	1.94439
$639$	0	0
$640$	0	0
$641$	−19.0711	−0.753262	−0.376631	−	0.926363i	$-0.622918\pi$
$641$	−19.0711	−0.753262	−0.376631	+	0.926363i	$0.622918\pi$
$642$	0	0
$643$	−33.0000	−1.30139	−0.650696	−	0.759338i	$-0.725523\pi$
$643$	−33.0000	−1.30139	−0.650696	+	0.759338i	$0.725523\pi$
$644$	0	0
$645$	0	0
$646$	25.5980	1.00714
$647$	−21.2132	−0.833977	−0.416989	−	0.908912i	$-0.636914\pi$
$647$	−21.2132	−0.833977	−0.416989	+	0.908912i	$0.636914\pi$
$648$	0	0
$649$	−69.8406	−2.74148
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−5.41421	−0.211875	−0.105937	−	0.994373i	$-0.533784\pi$
$653$	−5.41421	−0.211875	−0.105937	+	0.994373i	$0.533784\pi$
$654$	0	0
$655$	0	0
$656$	−30.3431	−1.18470
$657$	0	0
$658$	−61.7990	−2.40918
$659$	34.3848	1.33944	0.669720	−	0.742613i	$-0.266414\pi$
$659$	34.3848	1.33944	0.669720	+	0.742613i	$0.266414\pi$
$660$	0	0
$661$	−39.4558	−1.53465	−0.767327	−	0.641256i	$-0.778414\pi$
$661$	−39.4558	−1.53465	−0.767327	+	0.641256i	$0.778414\pi$
$662$	1.41421	0.0549650
$663$	0	0
$664$	0.201010	0.00780071
$665$	0	0
$666$	0	0
$667$	−6.41421	−0.248359
$668$	0	0
$669$	0	0
$670$	0	0
$671$	14.0000	0.540464
$672$	0	0
$673$	17.6985	0.682226	0.341113	−	0.940022i	$-0.389196\pi$
$673$	17.6985	0.682226	0.341113	+	0.940022i	$0.389196\pi$
$674$	4.20101	0.161817
$675$	0	0
$676$	0	0
$677$	−10.7574	−0.413439	−0.206719	−	0.978400i	$-0.566279\pi$
$677$	−10.7574	−0.413439	−0.206719	+	0.978400i	$0.566279\pi$
$678$	0	0
$679$	26.1421	1.00324
$680$	0	0
$681$	0	0
$682$	−75.2548	−2.88166
$683$	44.1838	1.69064	0.845322	−	0.534257i	$-0.179408\pi$
$683$	44.1838	1.69064	0.845322	+	0.534257i	$0.179408\pi$
$684$	0	0
$685$	0	0
$686$	3.55635	0.135782
$687$	0	0
$688$	24.0000	0.914991
$689$	0.727922	0.0277316
$690$	0	0
$691$	−4.34315	−0.165221	−0.0826105	−	0.996582i	$-0.526326\pi$
$691$	−4.34315	−0.165221	−0.0826105	+	0.996582i	$0.526326\pi$
$692$	0	0
$693$	0	0
$694$	26.6274	1.01076
$695$	0	0
$696$	0	0
$697$	61.2254	2.31908
$698$	42.5858	1.61190
$699$	0	0
$700$	0	0
$701$	−0.384776	−0.0145328	−0.00726640	−	0.999974i	$-0.502313\pi$
$701$	−0.384776	−0.0145328	−0.00726640	+	0.999974i	$0.502313\pi$
$702$	0	0
$703$	−6.72792	−0.253748
$704$	43.3137	1.63245
$705$	0	0
$706$	43.1716	1.62478
$707$	71.0416	2.67180
$708$	0	0
$709$	−8.72792	−0.327784	−0.163892	−	0.986478i	$-0.552405\pi$
$709$	−8.72792	−0.327784	−0.163892	+	0.986478i	$0.552405\pi$
$710$	0	0
$711$	0	0
$712$	−51.3137	−1.92306
$713$	9.82843	0.368077
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	35.4558	1.32320
$719$	6.21320	0.231713	0.115857	−	0.993266i	$-0.463039\pi$
$719$	6.21320	0.231713	0.115857	+	0.993266i	$0.463039\pi$
$720$	0	0
$721$	18.4853	0.688428
$722$	−19.7574	−0.735293
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	23.0000	0.853023	0.426511	−	0.904482i	$-0.359742\pi$
$727$	23.0000	0.853023	0.426511	+	0.904482i	$0.359742\pi$
$728$	6.34315	0.235093
$729$	0	0
$730$	0	0
$731$	−48.4264	−1.79112
$732$	0	0
$733$	0.514719	0.0190116	0.00950578	−	0.999955i	$-0.496974\pi$
$733$	0.514719	0.0190116	0.00950578	+	0.999955i	$0.496974\pi$
$734$	−28.2426	−1.04245
$735$	0	0
$736$	0	0
$737$	−29.6985	−1.09396
$738$	0	0
$739$	−8.37258	−0.307990	−0.153995	−	0.988072i	$-0.549214\pi$
$739$	−8.37258	−0.307990	−0.153995	+	0.988072i	$0.549214\pi$
$740$	0	0
$741$	0	0
$742$	−6.72792	−0.246990
$743$	−14.3431	−0.526199	−0.263099	−	0.964769i	$-0.584745\pi$
$743$	−14.3431	−0.526199	−0.263099	+	0.964769i	$0.584745\pi$
$744$	0	0
$745$	0	0
$746$	−29.6569	−1.08581
$747$	0	0
$748$	0	0
$749$	−6.07107	−0.221832
$750$	0	0
$751$	−34.7279	−1.26724	−0.633620	−	0.773644i	$-0.718432\pi$
$751$	−34.7279	−1.26724	−0.633620	+	0.773644i	$0.718432\pi$
$752$	45.6569	1.66493
$753$	0	0
$754$	−5.31371	−0.193514
$755$	0	0
$756$	0	0
$757$	−27.0000	−0.981332	−0.490666	−	0.871348i	$-0.663246\pi$
$757$	−27.0000	−0.981332	−0.490666	+	0.871348i	$0.663246\pi$
$758$	−7.02944	−0.255321
$759$	0	0
$760$	0	0
$761$	34.0711	1.23508	0.617538	−	0.786541i	$-0.288130\pi$
$761$	34.0711	1.23508	0.617538	+	0.786541i	$0.288130\pi$
$762$	0	0
$763$	22.0416	0.797961
$764$	0	0
$765$	0	0
$766$	16.3848	0.592006
$767$	7.55635	0.272844
$768$	0	0
$769$	−28.0416	−1.01121	−0.505604	−	0.862766i	$-0.668730\pi$
$769$	−28.0416	−1.01121	−0.505604	+	0.862766i	$0.668730\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−8.82843	−0.317536	−0.158768	−	0.987316i	$-0.550752\pi$
$773$	−8.82843	−0.317536	−0.158768	+	0.987316i	$0.550752\pi$
$774$	0	0
$775$	0	0
$776$	−19.3137	−0.693322
$777$	0	0
$778$	3.51472	0.126009
$779$	17.0122	0.609525
$780$	0	0
$781$	59.0122	2.11162
$782$	−11.4142	−0.408171
$783$	0	0
$784$	−30.6274	−1.09384
$785$	0	0
$786$	0	0
$787$	41.6274	1.48386	0.741929	−	0.670479i	$-0.233911\pi$
$787$	41.6274	1.48386	0.741929	+	0.670479i	$0.233911\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−29.0416	−1.03260
$792$	0	0
$793$	−1.51472	−0.0537892
$794$	−10.3431	−0.367065
$795$	0	0
$796$	0	0
$797$	25.2426	0.894140	0.447070	−	0.894499i	$-0.352467\pi$
$797$	25.2426	0.894140	0.447070	+	0.894499i	$0.352467\pi$
$798$	0	0
$799$	−92.1249	−3.25914
$800$	0	0
$801$	0	0
$802$	5.45584	0.192653
$803$	9.51472	0.335767
$804$	0	0
$805$	0	0
$806$	8.14214	0.286794
$807$	0	0
$808$	−52.4853	−1.84643
$809$	−47.8701	−1.68302	−0.841511	−	0.540240i	$-0.818333\pi$
$809$	−47.8701	−1.68302	−0.841511	+	0.540240i	$0.818333\pi$
$810$	0	0
$811$	−17.2843	−0.606933	−0.303466	−	0.952842i	$-0.598144\pi$
$811$	−17.2843	−0.606933	−0.303466	+	0.952842i	$0.598144\pi$
$812$	0	0
$813$	0	0
$814$	−22.9706	−0.805118
$815$	0	0
$816$	0	0
$817$	−13.4558	−0.470760
$818$	−21.8995	−0.765698
$819$	0	0
$820$	0	0
$821$	4.20101	0.146616	0.0733081	−	0.997309i	$-0.476644\pi$
$821$	4.20101	0.146616	0.0733081	+	0.997309i	$0.476644\pi$
$822$	0	0
$823$	−34.3431	−1.19713	−0.598563	−	0.801075i	$-0.704262\pi$
$823$	−34.3431	−1.19713	−0.598563	+	0.801075i	$0.704262\pi$
$824$	−13.6569	−0.475759
$825$	0	0
$826$	−69.8406	−2.43007
$827$	32.0122	1.11317	0.556587	−	0.830790i	$-0.312111\pi$
$827$	32.0122	1.11317	0.556587	+	0.830790i	$0.312111\pi$
$828$	0	0
$829$	6.51472	0.226266	0.113133	−	0.993580i	$-0.463911\pi$
$829$	6.51472	0.226266	0.113133	+	0.993580i	$0.463911\pi$
$830$	0	0
$831$	0	0
$832$	−4.68629	−0.162468
$833$	61.7990	2.14121
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	−34.0000	−1.17451
$839$	−40.6274	−1.40261	−0.701307	−	0.712859i	$-0.747400\pi$
$839$	−40.6274	−1.40261	−0.701307	+	0.712859i	$0.747400\pi$
$840$	0	0
$841$	12.1421	0.418694
$842$	−14.4853	−0.499196
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	70.1127	2.40910
$848$	4.97056	0.170690
$849$	0	0
$850$	0	0
$851$	3.00000	0.102839
$852$	0	0
$853$	−19.4558	−0.666155	−0.333078	−	0.942899i	$-0.608087\pi$
$853$	−19.4558	−0.666155	−0.333078	+	0.942899i	$0.608087\pi$
$854$	14.0000	0.479070
$855$	0	0
$856$	4.48528	0.153304
$857$	−57.5980	−1.96751	−0.983755	−	0.179518i	$-0.942546\pi$
$857$	−57.5980	−1.96751	−0.983755	+	0.179518i	$0.942546\pi$
$858$	0	0
$859$	−57.0000	−1.94481	−0.972407	−	0.233289i	$-0.925051\pi$
$859$	−57.0000	−1.94481	−0.972407	+	0.233289i	$0.925051\pi$
$860$	0	0
$861$	0	0
$862$	−16.6863	−0.568337
$863$	16.8284	0.572846	0.286423	−	0.958103i	$-0.407534\pi$
$863$	16.8284	0.572846	0.286423	+	0.958103i	$0.407534\pi$
$864$	0	0
$865$	0	0
$866$	−5.41421	−0.183982
$867$	0	0
$868$	0	0
$869$	39.5980	1.34327
$870$	0	0
$871$	3.21320	0.108875
$872$	−16.2843	−0.551455
$873$	0	0
$874$	−3.17157	−0.107280
$875$	0	0
$876$	0	0
$877$	18.4853	0.624204	0.312102	−	0.950049i	$-0.398967\pi$
$877$	18.4853	0.624204	0.312102	+	0.950049i	$0.398967\pi$
$878$	−2.14214	−0.0722936
$879$	0	0
$880$	0	0
$881$	−16.5858	−0.558789	−0.279395	−	0.960176i	$-0.590134\pi$
$881$	−16.5858	−0.558789	−0.279395	+	0.960176i	$0.590134\pi$
$882$	0	0
$883$	−17.8995	−0.602366	−0.301183	−	0.953566i	$-0.597381\pi$
$883$	−17.8995	−0.602366	−0.301183	+	0.953566i	$0.597381\pi$
$884$	0	0
$885$	0	0
$886$	−48.8284	−1.64042
$887$	−17.7990	−0.597632	−0.298816	−	0.954311i	$-0.596592\pi$
$887$	−17.7990	−0.597632	−0.298816	+	0.954311i	$0.596592\pi$
$888$	0	0
$889$	−31.5563	−1.05837
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−25.5980	−0.856604
$894$	0	0
$895$	0	0
$896$	43.3137	1.44701
$897$	0	0
$898$	6.72792	0.224514
$899$	−63.0416	−2.10256
$900$	0	0
$901$	−10.0294	−0.334129
$902$	58.0833	1.93396
$903$	0	0
$904$	21.4558	0.713611
$905$	0	0
$906$	0	0
$907$	−2.85786	−0.0948938	−0.0474469	−	0.998874i	$-0.515108\pi$
$907$	−2.85786	−0.0948938	−0.0474469	+	0.998874i	$0.515108\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−5.65685	−0.187420	−0.0937100	−	0.995600i	$-0.529873\pi$
$911$	−5.65685	−0.187420	−0.0937100	+	0.995600i	$0.529873\pi$
$912$	0	0
$913$	−0.384776	−0.0127342
$914$	43.8406	1.45012
$915$	0	0
$916$	0	0
$917$	8.97056	0.296234
$918$	0	0
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	0	0
$921$	0	0
$922$	37.4558	1.23354
$923$	−6.38478	−0.210157
$924$	0	0
$925$	0	0
$926$	−4.14214	−0.136119
$927$	0	0
$928$	0	0
$929$	11.8701	0.389444	0.194722	−	0.980858i	$-0.437620\pi$
$929$	11.8701	0.389444	0.194722	+	0.980858i	$0.437620\pi$
$930$	0	0
$931$	17.1716	0.562776
$932$	0	0
$933$	0	0
$934$	9.07107	0.296814
$935$	0	0
$936$	0	0
$937$	20.2010	0.659938	0.329969	−	0.943992i	$-0.392962\pi$
$937$	20.2010	0.659938	0.329969	+	0.943992i	$0.392962\pi$
$938$	−29.6985	−0.969690
$939$	0	0
$940$	0	0
$941$	4.58579	0.149492	0.0747462	−	0.997203i	$-0.476185\pi$
$941$	4.58579	0.149492	0.0747462	+	0.997203i	$0.476185\pi$
$942$	0	0
$943$	−7.58579	−0.247027
$944$	51.5980	1.67937
$945$	0	0
$946$	−45.9411	−1.49367
$947$	35.7990	1.16331	0.581655	−	0.813435i	$-0.302405\pi$
$947$	35.7990	1.16331	0.581655	+	0.813435i	$0.302405\pi$
$948$	0	0
$949$	−1.02944	−0.0334169
$950$	0	0
$951$	0	0
$952$	−87.3970	−2.83255
$953$	52.6274	1.70477	0.852385	−	0.522915i	$-0.175156\pi$
$953$	52.6274	1.70477	0.852385	+	0.522915i	$0.175156\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	−9.51472	−0.307407
$959$	−10.8284	−0.349668
$960$	0	0
$961$	65.5980	2.11606
$962$	2.48528	0.0801287
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−31.5563	−1.01478	−0.507392	−	0.861715i	$-0.669390\pi$
$967$	−31.5563	−1.01478	−0.507392	+	0.861715i	$0.669390\pi$
$968$	−51.7990	−1.66488
$969$	0	0
$970$	0	0
$971$	54.4264	1.74663	0.873313	−	0.487159i	$-0.161967\pi$
$971$	54.4264	1.74663	0.873313	+	0.487159i	$0.161967\pi$
$972$	0	0
$973$	−42.1127	−1.35007
$974$	37.3137	1.19561
$975$	0	0
$976$	−10.3431	−0.331076
$977$	−18.4142	−0.589123	−0.294561	−	0.955633i	$-0.595174\pi$
$977$	−18.4142	−0.589123	−0.294561	+	0.955633i	$0.595174\pi$
$978$	0	0
$979$	98.2254	3.13930
$980$	0	0
$981$	0	0
$982$	17.0711	0.544760
$983$	−58.0122	−1.85030	−0.925151	−	0.379600i	$-0.876062\pi$
$983$	−58.0122	−1.85030	−0.925151	+	0.379600i	$0.876062\pi$
$984$	0	0
$985$	0	0
$986$	73.2132	2.33158
$987$	0	0
$988$	0	0
$989$	6.00000	0.190789
$990$	0	0
$991$	12.6569	0.402058	0.201029	−	0.979585i	$-0.435571\pi$
$991$	12.6569	0.402058	0.201029	+	0.979585i	$0.435571\pi$
$992$	0	0
$993$	0	0
$994$	59.0122	1.87175
$995$	0	0
$996$	0	0
$997$	5.85786	0.185520	0.0927602	−	0.995688i	$-0.470431\pi$
$997$	5.85786	0.185520	0.0927602	+	0.995688i	$0.470431\pi$
$998$	−18.3848	−0.581960
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5175.2.a.bj.1.2		2
3.2	odd	2		1725.2.a.z.1.1		2
5.4	even	2		1035.2.a.j.1.1		2
15.2	even	4		1725.2.b.s.1174.1		4
15.8	even	4		1725.2.b.s.1174.4		4
15.14	odd	2		345.2.a.h.1.2	✓	2
60.59	even	2		5520.2.a.bm.1.2		2
345.344	even	2		7935.2.a.q.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
345.2.a.h.1.2	✓	2	15.14	odd	2
1035.2.a.j.1.1		2	5.4	even	2
1725.2.a.z.1.1		2	3.2	odd	2
1725.2.b.s.1174.1		4	15.2	even	4
1725.2.b.s.1174.4		4	15.8	even	4
5175.2.a.bj.1.2		2	1.1	even	1	trivial
5520.2.a.bm.1.2		2	60.59	even	2
7935.2.a.q.1.2		2	345.344	even	2

Overview	Random
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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:	\( N \)	\(=\)	\( 5175 = 3^{2} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5175.a (trivial)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} +3.82843 q^{7} -2.82843 q^{8} +O(q^{10})\)	\(q+1.41421 q^{2} +3.82843 q^{7} -2.82843 q^{8} +5.41421 q^{11} -0.585786 q^{13} +5.41421 q^{14} -4.00000 q^{16} +8.07107 q^{17} +2.24264 q^{19} +7.65685 q^{22} -1.00000 q^{23} -0.828427 q^{26} +6.41421 q^{29} -9.82843 q^{31} +11.4142 q^{34} -3.00000 q^{37} +3.17157 q^{38} +7.58579 q^{41} -6.00000 q^{43} -1.41421 q^{46} -11.4142 q^{47} +7.65685 q^{49} -1.24264 q^{53} -10.8284 q^{56} +9.07107 q^{58} -12.8995 q^{59} +2.58579 q^{61} -13.8995 q^{62} +8.00000 q^{64} -5.48528 q^{67} +10.8995 q^{71} +1.75736 q^{73} -4.24264 q^{74} +20.7279 q^{77} +7.31371 q^{79} +10.7279 q^{82} -0.0710678 q^{83} -8.48528 q^{86} -15.3137 q^{88} +18.1421 q^{89} -2.24264 q^{91} -16.1421 q^{94} +6.82843 q^{97} +10.8284 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{7}+O(q^{10}) \) 2 * q + 2 * q^7	\( 2 q + 2 q^{7} + 8 q^{11} - 4 q^{13} + 8 q^{14} - 8 q^{16} + 2 q^{17} - 4 q^{19} + 4 q^{22} - 2 q^{23} + 4 q^{26} + 10 q^{29} - 14 q^{31} + 20 q^{34} - 6 q^{37} + 12 q^{38} + 18 q^{41} - 12 q^{43} - 20 q^{47} + 4 q^{49} + 6 q^{53} - 16 q^{56} + 4 q^{58} - 6 q^{59} + 8 q^{61} - 8 q^{62} + 16 q^{64} + 6 q^{67} + 2 q^{71} + 12 q^{73} + 16 q^{77} - 8 q^{79} - 4 q^{82} + 14 q^{83} - 8 q^{88} + 8 q^{89} + 4 q^{91} - 4 q^{94} + 8 q^{97} + 16 q^{98}+O(q^{100}) \) 2 * q + 2 * q^7 + 8 * q^11 - 4 * q^13 + 8 * q^14 - 8 * q^16 + 2 * q^17 - 4 * q^19 + 4 * q^22 - 2 * q^23 + 4 * q^26 + 10 * q^29 - 14 * q^31 + 20 * q^34 - 6 * q^37 + 12 * q^38 + 18 * q^41 - 12 * q^43 - 20 * q^47 + 4 * q^49 + 6 * q^53 - 16 * q^56 + 4 * q^58 - 6 * q^59 + 8 * q^61 - 8 * q^62 + 16 * q^64 + 6 * q^67 + 2 * q^71 + 12 * q^73 + 16 * q^77 - 8 * q^79 - 4 * q^82 + 14 * q^83 - 8 * q^88 + 8 * q^89 + 4 * q^91 - 4 * q^94 + 8 * q^97 + 16 * q^98

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