LMFDB - Embedded newform 7248.2.a.o.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7248,2,Mod(1,7248)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7248, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("7248.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7248 = 2^{4} \cdot 3 \cdot 151 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7248.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:	$57.8755713850$
Analytic rank:	$0$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 453)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	7248.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.61803 q^{5} +3.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.61803 q^{5} +3.00000 q^{7} +1.00000 q^{9} +0.763932 q^{11} -1.00000 q^{13} +1.61803 q^{15} -2.23607 q^{17} +4.00000 q^{19} -3.00000 q^{21} +7.47214 q^{23} -2.38197 q^{25} -1.00000 q^{27} +7.85410 q^{29} -3.09017 q^{31} -0.763932 q^{33} -4.85410 q^{35} -4.85410 q^{37} +1.00000 q^{39} +1.14590 q^{41} +0.527864 q^{43} -1.61803 q^{45} -4.47214 q^{47} +2.00000 q^{49} +2.23607 q^{51} +7.94427 q^{53} -1.23607 q^{55} -4.00000 q^{57} +3.61803 q^{59} +12.5623 q^{61} +3.00000 q^{63} +1.61803 q^{65} +10.8541 q^{67} -7.47214 q^{69} -12.0902 q^{71} -14.7082 q^{73} +2.38197 q^{75} +2.29180 q^{77} +8.79837 q^{79} +1.00000 q^{81} +6.38197 q^{83} +3.61803 q^{85} -7.85410 q^{87} -3.00000 q^{89} -3.00000 q^{91} +3.09017 q^{93} -6.47214 q^{95} -13.4164 q^{97} +0.763932 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - q^{5} + 6 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - q^5 + 6 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} - q^{5} + 6 q^{7} + 2 q^{9} + 6 q^{11} - 2 q^{13} + q^{15} + 8 q^{19} - 6 q^{21} + 6 q^{23} - 7 q^{25} - 2 q^{27} + 9 q^{29} + 5 q^{31} - 6 q^{33} - 3 q^{35} - 3 q^{37} + 2 q^{39} + 9 q^{41} + 10 q^{43} - q^{45} + 4 q^{49} - 2 q^{53} + 2 q^{55} - 8 q^{57} + 5 q^{59} + 5 q^{61} + 6 q^{63} + q^{65} + 15 q^{67} - 6 q^{69} - 13 q^{71} - 16 q^{73} + 7 q^{75} + 18 q^{77} - 7 q^{79} + 2 q^{81} + 15 q^{83} + 5 q^{85} - 9 q^{87} - 6 q^{89} - 6 q^{91} - 5 q^{93} - 4 q^{95} + 6 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - q^5 + 6 * q^7 + 2 * q^9 + 6 * q^11 - 2 * q^13 + q^15 + 8 * q^19 - 6 * q^21 + 6 * q^23 - 7 * q^25 - 2 * q^27 + 9 * q^29 + 5 * q^31 - 6 * q^33 - 3 * q^35 - 3 * q^37 + 2 * q^39 + 9 * q^41 + 10 * q^43 - q^45 + 4 * q^49 - 2 * q^53 + 2 * q^55 - 8 * q^57 + 5 * q^59 + 5 * q^61 + 6 * q^63 + q^65 + 15 * q^67 - 6 * q^69 - 13 * q^71 - 16 * q^73 + 7 * q^75 + 18 * q^77 - 7 * q^79 + 2 * q^81 + 15 * q^83 + 5 * q^85 - 9 * q^87 - 6 * q^89 - 6 * q^91 - 5 * q^93 - 4 * q^95 + 6 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−1.61803	−0.723607	−0.361803	−	0.932254i	$-0.617839\pi$
$5$	−1.61803	−0.723607	−0.361803	+	0.932254i	$0.617839\pi$
$6$	0	0
$7$	3.00000	1.13389	0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000	1.13389	0.566947	+	0.823754i	$0.308125\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0.763932	0.230334	0.115167	−	0.993346i	$-0.463260\pi$
$11$	0.763932	0.230334	0.115167	+	0.993346i	$0.463260\pi$
$12$	0	0
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	0	0
$15$	1.61803	0.417775
$16$	0	0
$17$	−2.23607	−0.542326	−0.271163	−	0.962533i	$-0.587408\pi$
$17$	−2.23607	−0.542326	−0.271163	+	0.962533i	$0.587408\pi$
$18$	0	0
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	−3.00000	−0.654654
$22$	0	0
$23$	7.47214	1.55805	0.779024	−	0.626994i	$-0.215715\pi$
$23$	7.47214	1.55805	0.779024	+	0.626994i	$0.215715\pi$
$24$	0	0
$25$	−2.38197	−0.476393
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	7.85410	1.45847	0.729235	−	0.684263i	$-0.239876\pi$
$29$	7.85410	1.45847	0.729235	+	0.684263i	$0.239876\pi$
$30$	0	0
$31$	−3.09017	−0.555011	−0.277505	−	0.960724i	$-0.589508\pi$
$31$	−3.09017	−0.555011	−0.277505	+	0.960724i	$0.589508\pi$
$32$	0	0
$33$	−0.763932	−0.132983
$34$	0	0
$35$	−4.85410	−0.820493
$36$	0	0
$37$	−4.85410	−0.798009	−0.399005	−	0.916949i	$-0.630644\pi$
$37$	−4.85410	−0.798009	−0.399005	+	0.916949i	$0.630644\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	1.14590	0.178959	0.0894796	−	0.995989i	$-0.471480\pi$
$41$	1.14590	0.178959	0.0894796	+	0.995989i	$0.471480\pi$
$42$	0	0
$43$	0.527864	0.0804985	0.0402493	−	0.999190i	$-0.487185\pi$
$43$	0.527864	0.0804985	0.0402493	+	0.999190i	$0.487185\pi$
$44$	0	0
$45$	−1.61803	−0.241202
$46$	0	0
$47$	−4.47214	−0.652328	−0.326164	−	0.945313i	$-0.605756\pi$
$47$	−4.47214	−0.652328	−0.326164	+	0.945313i	$0.605756\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	0	0
$51$	2.23607	0.313112
$52$	0	0
$53$	7.94427	1.09123	0.545615	−	0.838036i	$-0.316296\pi$
$53$	7.94427	1.09123	0.545615	+	0.838036i	$0.316296\pi$
$54$	0	0
$55$	−1.23607	−0.166671
$56$	0	0
$57$	−4.00000	−0.529813
$58$	0	0
$59$	3.61803	0.471028	0.235514	−	0.971871i	$-0.424323\pi$
$59$	3.61803	0.471028	0.235514	+	0.971871i	$0.424323\pi$
$60$	0	0
$61$	12.5623	1.60844	0.804219	−	0.594333i	$-0.202584\pi$
$61$	12.5623	1.60844	0.804219	+	0.594333i	$0.202584\pi$
$62$	0	0
$63$	3.00000	0.377964
$64$	0	0
$65$	1.61803	0.200692
$66$	0	0
$67$	10.8541	1.32604	0.663020	−	0.748602i	$-0.269275\pi$
$67$	10.8541	1.32604	0.663020	+	0.748602i	$0.269275\pi$
$68$	0	0
$69$	−7.47214	−0.899539
$70$	0	0
$71$	−12.0902	−1.43484	−0.717420	−	0.696641i	$-0.754677\pi$
$71$	−12.0902	−1.43484	−0.717420	+	0.696641i	$0.754677\pi$
$72$	0	0
$73$	−14.7082	−1.72147	−0.860733	−	0.509057i	$-0.829994\pi$
$73$	−14.7082	−1.72147	−0.860733	+	0.509057i	$0.829994\pi$
$74$	0	0
$75$	2.38197	0.275046
$76$	0	0
$77$	2.29180	0.261174
$78$	0	0
$79$	8.79837	0.989894	0.494947	−	0.868923i	$-0.335187\pi$
$79$	8.79837	0.989894	0.494947	+	0.868923i	$0.335187\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	6.38197	0.700512	0.350256	−	0.936654i	$-0.386095\pi$
$83$	6.38197	0.700512	0.350256	+	0.936654i	$0.386095\pi$
$84$	0	0
$85$	3.61803	0.392431
$86$	0	0
$87$	−7.85410	−0.842048
$88$	0	0
$89$	−3.00000	−0.317999	−0.159000	−	0.987279i	$-0.550827\pi$
$89$	−3.00000	−0.317999	−0.159000	+	0.987279i	$0.550827\pi$
$90$	0	0
$91$	−3.00000	−0.314485
$92$	0	0
$93$	3.09017	0.320436
$94$	0	0
$95$	−6.47214	−0.664027
$96$	0	0
$97$	−13.4164	−1.36223	−0.681115	−	0.732177i	$-0.738505\pi$
$97$	−13.4164	−1.36223	−0.681115	+	0.732177i	$0.738505\pi$
$98$	0	0
$99$	0.763932	0.0767781
$100$	0	0
$101$	1.90983	0.190035	0.0950176	−	0.995476i	$-0.469709\pi$
$101$	1.90983	0.190035	0.0950176	+	0.995476i	$0.469709\pi$
$102$	0	0
$103$	−7.00000	−0.689730	−0.344865	−	0.938652i	$-0.612075\pi$
$103$	−7.00000	−0.689730	−0.344865	+	0.938652i	$0.612075\pi$
$104$	0	0
$105$	4.85410	0.473712
$106$	0	0
$107$	9.94427	0.961349	0.480675	−	0.876899i	$-0.340392\pi$
$107$	9.94427	0.961349	0.480675	+	0.876899i	$0.340392\pi$
$108$	0	0
$109$	−12.0344	−1.15269	−0.576345	−	0.817206i	$-0.695522\pi$
$109$	−12.0344	−1.15269	−0.576345	+	0.817206i	$0.695522\pi$
$110$	0	0
$111$	4.85410	0.460731
$112$	0	0
$113$	1.23607	0.116279	0.0581397	−	0.998308i	$-0.481483\pi$
$113$	1.23607	0.116279	0.0581397	+	0.998308i	$0.481483\pi$
$114$	0	0
$115$	−12.0902	−1.12741
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	−6.70820	−0.614940
$120$	0	0
$121$	−10.4164	−0.946946
$122$	0	0
$123$	−1.14590	−0.103322
$124$	0	0
$125$	11.9443	1.06833
$126$	0	0
$127$	2.85410	0.253261	0.126630	−	0.991950i	$-0.459584\pi$
$127$	2.85410	0.253261	0.126630	+	0.991950i	$0.459584\pi$
$128$	0	0
$129$	−0.527864	−0.0464758
$130$	0	0
$131$	−3.90983	−0.341603	−0.170802	−	0.985305i	$-0.554636\pi$
$131$	−3.90983	−0.341603	−0.170802	+	0.985305i	$0.554636\pi$
$132$	0	0
$133$	12.0000	1.04053
$134$	0	0
$135$	1.61803	0.139258
$136$	0	0
$137$	15.1803	1.29694	0.648472	−	0.761239i	$-0.275408\pi$
$137$	15.1803	1.29694	0.648472	+	0.761239i	$0.275408\pi$
$138$	0	0
$139$	−3.14590	−0.266832	−0.133416	−	0.991060i	$-0.542595\pi$
$139$	−3.14590	−0.266832	−0.133416	+	0.991060i	$0.542595\pi$
$140$	0	0
$141$	4.47214	0.376622
$142$	0	0
$143$	−0.763932	−0.0638832
$144$	0	0
$145$	−12.7082	−1.05536
$146$	0	0
$147$	−2.00000	−0.164957
$148$	0	0
$149$	2.32624	0.190573	0.0952864	−	0.995450i	$-0.469623\pi$
$149$	2.32624	0.190573	0.0952864	+	0.995450i	$0.469623\pi$
$150$	0	0
$151$	−1.00000	−0.0813788
$151$	−1.00000	−0.0813788
$152$	0	0
$153$	−2.23607	−0.180775
$154$	0	0
$155$	5.00000	0.401610
$156$	0	0
$157$	3.56231	0.284303	0.142151	−	0.989845i	$-0.454598\pi$
$157$	3.56231	0.284303	0.142151	+	0.989845i	$0.454598\pi$
$158$	0	0
$159$	−7.94427	−0.630022
$160$	0	0
$161$	22.4164	1.76666
$162$	0	0
$163$	14.0000	1.09656	0.548282	−	0.836293i	$-0.315282\pi$
$163$	14.0000	1.09656	0.548282	+	0.836293i	$0.315282\pi$
$164$	0	0
$165$	1.23607	0.0962278
$166$	0	0
$167$	6.38197	0.493851	0.246926	−	0.969034i	$-0.420580\pi$
$167$	6.38197	0.493851	0.246926	+	0.969034i	$0.420580\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	4.00000	0.305888
$172$	0	0
$173$	13.2361	1.00632	0.503160	−	0.864193i	$-0.332171\pi$
$173$	13.2361	1.00632	0.503160	+	0.864193i	$0.332171\pi$
$174$	0	0
$175$	−7.14590	−0.540179
$176$	0	0
$177$	−3.61803	−0.271948
$178$	0	0
$179$	4.41641	0.330098	0.165049	−	0.986285i	$-0.447222\pi$
$179$	4.41641	0.330098	0.165049	+	0.986285i	$0.447222\pi$
$180$	0	0
$181$	−9.29180	−0.690654	−0.345327	−	0.938482i	$-0.612232\pi$
$181$	−9.29180	−0.690654	−0.345327	+	0.938482i	$0.612232\pi$
$182$	0	0
$183$	−12.5623	−0.928632
$184$	0	0
$185$	7.85410	0.577445
$186$	0	0
$187$	−1.70820	−0.124916
$188$	0	0
$189$	−3.00000	−0.218218
$190$	0	0
$191$	19.3607	1.40089	0.700445	−	0.713707i	$-0.252985\pi$
$191$	19.3607	1.40089	0.700445	+	0.713707i	$0.252985\pi$
$192$	0	0
$193$	25.8885	1.86350	0.931749	−	0.363103i	$-0.118283\pi$
$193$	25.8885	1.86350	0.931749	+	0.363103i	$0.118283\pi$
$194$	0	0
$195$	−1.61803	−0.115870
$196$	0	0
$197$	−20.0344	−1.42739	−0.713697	−	0.700454i	$-0.752981\pi$
$197$	−20.0344	−1.42739	−0.713697	+	0.700454i	$0.752981\pi$
$198$	0	0
$199$	−0.236068	−0.0167344	−0.00836721	−	0.999965i	$-0.502663\pi$
$199$	−0.236068	−0.0167344	−0.00836721	+	0.999965i	$0.502663\pi$
$200$	0	0
$201$	−10.8541	−0.765589
$202$	0	0
$203$	23.5623	1.65375
$204$	0	0
$205$	−1.85410	−0.129496
$206$	0	0
$207$	7.47214	0.519349
$208$	0	0
$209$	3.05573	0.211369
$210$	0	0
$211$	−14.5623	−1.00251	−0.501255	−	0.865299i	$-0.667128\pi$
$211$	−14.5623	−1.00251	−0.501255	+	0.865299i	$0.667128\pi$
$212$	0	0
$213$	12.0902	0.828405
$214$	0	0
$215$	−0.854102	−0.0582493
$216$	0	0
$217$	−9.27051	−0.629323
$218$	0	0
$219$	14.7082	0.993888
$220$	0	0
$221$	2.23607	0.150414
$222$	0	0
$223$	−0.673762	−0.0451184	−0.0225592	−	0.999746i	$-0.507181\pi$
$223$	−0.673762	−0.0451184	−0.0225592	+	0.999746i	$0.507181\pi$
$224$	0	0
$225$	−2.38197	−0.158798
$226$	0	0
$227$	13.9098	0.923228	0.461614	−	0.887081i	$-0.347271\pi$
$227$	13.9098	0.923228	0.461614	+	0.887081i	$0.347271\pi$
$228$	0	0
$229$	−22.2361	−1.46940	−0.734700	−	0.678392i	$-0.762677\pi$
$229$	−22.2361	−1.46940	−0.734700	+	0.678392i	$0.762677\pi$
$230$	0	0
$231$	−2.29180	−0.150789
$232$	0	0
$233$	−2.23607	−0.146490	−0.0732448	−	0.997314i	$-0.523335\pi$
$233$	−2.23607	−0.146490	−0.0732448	+	0.997314i	$0.523335\pi$
$234$	0	0
$235$	7.23607	0.472029
$236$	0	0
$237$	−8.79837	−0.571516
$238$	0	0
$239$	−2.94427	−0.190449	−0.0952246	−	0.995456i	$-0.530357\pi$
$239$	−2.94427	−0.190449	−0.0952246	+	0.995456i	$0.530357\pi$
$240$	0	0
$241$	9.94427	0.640567	0.320283	−	0.947322i	$-0.396222\pi$
$241$	9.94427	0.640567	0.320283	+	0.947322i	$0.396222\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−3.23607	−0.206745
$246$	0	0
$247$	−4.00000	−0.254514
$248$	0	0
$249$	−6.38197	−0.404441
$250$	0	0
$251$	16.5279	1.04323	0.521615	−	0.853181i	$-0.325330\pi$
$251$	16.5279	1.04323	0.521615	+	0.853181i	$0.325330\pi$
$252$	0	0
$253$	5.70820	0.358872
$254$	0	0
$255$	−3.61803	−0.226570
$256$	0	0
$257$	−10.4164	−0.649758	−0.324879	−	0.945756i	$-0.605324\pi$
$257$	−10.4164	−0.649758	−0.324879	+	0.945756i	$0.605324\pi$
$258$	0	0
$259$	−14.5623	−0.904858
$260$	0	0
$261$	7.85410	0.486157
$262$	0	0
$263$	−10.0344	−0.618750	−0.309375	−	0.950940i	$-0.600120\pi$
$263$	−10.0344	−0.618750	−0.309375	+	0.950940i	$0.600120\pi$
$264$	0	0
$265$	−12.8541	−0.789621
$266$	0	0
$267$	3.00000	0.183597
$268$	0	0
$269$	−9.03444	−0.550840	−0.275420	−	0.961324i	$-0.588817\pi$
$269$	−9.03444	−0.550840	−0.275420	+	0.961324i	$0.588817\pi$
$270$	0	0
$271$	16.9443	1.02929	0.514646	−	0.857403i	$-0.327924\pi$
$271$	16.9443	1.02929	0.514646	+	0.857403i	$0.327924\pi$
$272$	0	0
$273$	3.00000	0.181568
$274$	0	0
$275$	−1.81966	−0.109730
$276$	0	0
$277$	17.7639	1.06733	0.533666	−	0.845696i	$-0.320814\pi$
$277$	17.7639	1.06733	0.533666	+	0.845696i	$0.320814\pi$
$278$	0	0
$279$	−3.09017	−0.185004
$280$	0	0
$281$	20.1803	1.20386	0.601929	−	0.798550i	$-0.294399\pi$
$281$	20.1803	1.20386	0.601929	+	0.798550i	$0.294399\pi$
$282$	0	0
$283$	21.9443	1.30445	0.652226	−	0.758025i	$-0.273835\pi$
$283$	21.9443	1.30445	0.652226	+	0.758025i	$0.273835\pi$
$284$	0	0
$285$	6.47214	0.383376
$286$	0	0
$287$	3.43769	0.202921
$288$	0	0
$289$	−12.0000	−0.705882
$290$	0	0
$291$	13.4164	0.786484
$292$	0	0
$293$	−11.6525	−0.680745	−0.340372	−	0.940291i	$-0.610553\pi$
$293$	−11.6525	−0.680745	−0.340372	+	0.940291i	$0.610553\pi$
$294$	0	0
$295$	−5.85410	−0.340839
$296$	0	0
$297$	−0.763932	−0.0443278
$298$	0	0
$299$	−7.47214	−0.432125
$300$	0	0
$301$	1.58359	0.0912767
$302$	0	0
$303$	−1.90983	−0.109717
$304$	0	0
$305$	−20.3262	−1.16388
$306$	0	0
$307$	−17.0344	−0.972207	−0.486103	−	0.873901i	$-0.661582\pi$
$307$	−17.0344	−0.972207	−0.486103	+	0.873901i	$0.661582\pi$
$308$	0	0
$309$	7.00000	0.398216
$310$	0	0
$311$	−3.00000	−0.170114	−0.0850572	−	0.996376i	$-0.527107\pi$
$311$	−3.00000	−0.170114	−0.0850572	+	0.996376i	$0.527107\pi$
$312$	0	0
$313$	13.5066	0.763437	0.381718	−	0.924279i	$-0.375332\pi$
$313$	13.5066	0.763437	0.381718	+	0.924279i	$0.375332\pi$
$314$	0	0
$315$	−4.85410	−0.273498
$316$	0	0
$317$	12.0000	0.673987	0.336994	−	0.941507i	$-0.390590\pi$
$317$	12.0000	0.673987	0.336994	+	0.941507i	$0.390590\pi$
$318$	0	0
$319$	6.00000	0.335936
$320$	0	0
$321$	−9.94427	−0.555035
$322$	0	0
$323$	−8.94427	−0.497673
$324$	0	0
$325$	2.38197	0.132128
$326$	0	0
$327$	12.0344	0.665506
$328$	0	0
$329$	−13.4164	−0.739671
$330$	0	0
$331$	−19.1246	−1.05118	−0.525592	−	0.850737i	$-0.676156\pi$
$331$	−19.1246	−1.05118	−0.525592	+	0.850737i	$0.676156\pi$
$332$	0	0
$333$	−4.85410	−0.266003
$334$	0	0
$335$	−17.5623	−0.959531
$336$	0	0
$337$	17.7984	0.969539	0.484770	−	0.874642i	$-0.338903\pi$
$337$	17.7984	0.969539	0.484770	+	0.874642i	$0.338903\pi$
$338$	0	0
$339$	−1.23607	−0.0671340
$340$	0	0
$341$	−2.36068	−0.127838
$342$	0	0
$343$	−15.0000	−0.809924
$344$	0	0
$345$	12.0902	0.650913
$346$	0	0
$347$	29.7639	1.59781	0.798906	−	0.601456i	$-0.205413\pi$
$347$	29.7639	1.59781	0.798906	+	0.601456i	$0.205413\pi$
$348$	0	0
$349$	30.1803	1.61552	0.807758	−	0.589514i	$-0.200681\pi$
$349$	30.1803	1.61552	0.807758	+	0.589514i	$0.200681\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	20.3607	1.08369	0.541845	−	0.840479i	$-0.317726\pi$
$353$	20.3607	1.08369	0.541845	+	0.840479i	$0.317726\pi$
$354$	0	0
$355$	19.5623	1.03826
$356$	0	0
$357$	6.70820	0.355036
$358$	0	0
$359$	18.0344	0.951821	0.475911	−	0.879494i	$-0.342119\pi$
$359$	18.0344	0.951821	0.475911	+	0.879494i	$0.342119\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	10.4164	0.546720
$364$	0	0
$365$	23.7984	1.24566
$366$	0	0
$367$	4.32624	0.225828	0.112914	−	0.993605i	$-0.463982\pi$
$367$	4.32624	0.225828	0.112914	+	0.993605i	$0.463982\pi$
$368$	0	0
$369$	1.14590	0.0596531
$370$	0	0
$371$	23.8328	1.23734
$372$	0	0
$373$	17.1459	0.887781	0.443890	−	0.896081i	$-0.353598\pi$
$373$	17.1459	0.887781	0.443890	+	0.896081i	$0.353598\pi$
$374$	0	0
$375$	−11.9443	−0.616800
$376$	0	0
$377$	−7.85410	−0.404507
$378$	0	0
$379$	34.1246	1.75286	0.876432	−	0.481526i	$-0.159917\pi$
$379$	34.1246	1.75286	0.876432	+	0.481526i	$0.159917\pi$
$380$	0	0
$381$	−2.85410	−0.146220
$382$	0	0
$383$	3.76393	0.192328	0.0961640	−	0.995366i	$-0.469343\pi$
$383$	3.76393	0.192328	0.0961640	+	0.995366i	$0.469343\pi$
$384$	0	0
$385$	−3.70820	−0.188988
$386$	0	0
$387$	0.527864	0.0268328
$388$	0	0
$389$	15.9787	0.810153	0.405076	−	0.914283i	$-0.367245\pi$
$389$	15.9787	0.810153	0.405076	+	0.914283i	$0.367245\pi$
$390$	0	0
$391$	−16.7082	−0.844970
$392$	0	0
$393$	3.90983	0.197225
$394$	0	0
$395$	−14.2361	−0.716294
$396$	0	0
$397$	−3.00000	−0.150566	−0.0752828	−	0.997162i	$-0.523986\pi$
$397$	−3.00000	−0.150566	−0.0752828	+	0.997162i	$0.523986\pi$
$398$	0	0
$399$	−12.0000	−0.600751
$400$	0	0
$401$	−1.47214	−0.0735150	−0.0367575	−	0.999324i	$-0.511703\pi$
$401$	−1.47214	−0.0735150	−0.0367575	+	0.999324i	$0.511703\pi$
$402$	0	0
$403$	3.09017	0.153932
$404$	0	0
$405$	−1.61803	−0.0804008
$406$	0	0
$407$	−3.70820	−0.183809
$408$	0	0
$409$	24.4164	1.20731	0.603657	−	0.797244i	$-0.293710\pi$
$409$	24.4164	1.20731	0.603657	+	0.797244i	$0.293710\pi$
$410$	0	0
$411$	−15.1803	−0.748791
$412$	0	0
$413$	10.8541	0.534095
$414$	0	0
$415$	−10.3262	−0.506895
$416$	0	0
$417$	3.14590	0.154055
$418$	0	0
$419$	−24.5279	−1.19826	−0.599132	−	0.800650i	$-0.704488\pi$
$419$	−24.5279	−1.19826	−0.599132	+	0.800650i	$0.704488\pi$
$420$	0	0
$421$	−6.27051	−0.305606	−0.152803	−	0.988257i	$-0.548830\pi$
$421$	−6.27051	−0.305606	−0.152803	+	0.988257i	$0.548830\pi$
$422$	0	0
$423$	−4.47214	−0.217443
$424$	0	0
$425$	5.32624	0.258360
$426$	0	0
$427$	37.6869	1.82380
$428$	0	0
$429$	0.763932	0.0368830
$430$	0	0
$431$	30.2148	1.45539	0.727697	−	0.685898i	$-0.240591\pi$
$431$	30.2148	1.45539	0.727697	+	0.685898i	$0.240591\pi$
$432$	0	0
$433$	35.3951	1.70098	0.850490	−	0.525990i	$-0.176305\pi$
$433$	35.3951	1.70098	0.850490	+	0.525990i	$0.176305\pi$
$434$	0	0
$435$	12.7082	0.609312
$436$	0	0
$437$	29.8885	1.42976
$438$	0	0
$439$	−23.4721	−1.12026	−0.560132	−	0.828403i	$-0.689250\pi$
$439$	−23.4721	−1.12026	−0.560132	+	0.828403i	$0.689250\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	0	0
$443$	26.7639	1.27159	0.635796	−	0.771857i	$-0.280672\pi$
$443$	26.7639	1.27159	0.635796	+	0.771857i	$0.280672\pi$
$444$	0	0
$445$	4.85410	0.230107
$446$	0	0
$447$	−2.32624	−0.110027
$448$	0	0
$449$	31.9443	1.50754	0.753772	−	0.657136i	$-0.228232\pi$
$449$	31.9443	1.50754	0.753772	+	0.657136i	$0.228232\pi$
$450$	0	0
$451$	0.875388	0.0412204
$452$	0	0
$453$	1.00000	0.0469841
$454$	0	0
$455$	4.85410	0.227564
$456$	0	0
$457$	−19.4164	−0.908261	−0.454131	−	0.890935i	$-0.650050\pi$
$457$	−19.4164	−0.908261	−0.454131	+	0.890935i	$0.650050\pi$
$458$	0	0
$459$	2.23607	0.104371
$460$	0	0
$461$	42.0344	1.95774	0.978870	−	0.204486i	$-0.0655522\pi$
$461$	42.0344	1.95774	0.978870	+	0.204486i	$0.0655522\pi$
$462$	0	0
$463$	−4.65248	−0.216219	−0.108109	−	0.994139i	$-0.534480\pi$
$463$	−4.65248	−0.216219	−0.108109	+	0.994139i	$0.534480\pi$
$464$	0	0
$465$	−5.00000	−0.231869
$466$	0	0
$467$	−41.8328	−1.93579	−0.967896	−	0.251351i	$-0.919125\pi$
$467$	−41.8328	−1.93579	−0.967896	+	0.251351i	$0.919125\pi$
$468$	0	0
$469$	32.5623	1.50359
$470$	0	0
$471$	−3.56231	−0.164142
$472$	0	0
$473$	0.403252	0.0185416
$474$	0	0
$475$	−9.52786	−0.437168
$476$	0	0
$477$	7.94427	0.363743
$478$	0	0
$479$	9.43769	0.431219	0.215610	−	0.976480i	$-0.430826\pi$
$479$	9.43769	0.431219	0.215610	+	0.976480i	$0.430826\pi$
$480$	0	0
$481$	4.85410	0.221328
$482$	0	0
$483$	−22.4164	−1.01998
$484$	0	0
$485$	21.7082	0.985719
$486$	0	0
$487$	10.8197	0.490286	0.245143	−	0.969487i	$-0.421165\pi$
$487$	10.8197	0.490286	0.245143	+	0.969487i	$0.421165\pi$
$488$	0	0
$489$	−14.0000	−0.633102
$490$	0	0
$491$	−19.0902	−0.861527	−0.430764	−	0.902465i	$-0.641756\pi$
$491$	−19.0902	−0.861527	−0.430764	+	0.902465i	$0.641756\pi$
$492$	0	0
$493$	−17.5623	−0.790966
$494$	0	0
$495$	−1.23607	−0.0555571
$496$	0	0
$497$	−36.2705	−1.62695
$498$	0	0
$499$	38.1246	1.70669	0.853346	−	0.521345i	$-0.174570\pi$
$499$	38.1246	1.70669	0.853346	+	0.521345i	$0.174570\pi$
$500$	0	0
$501$	−6.38197	−0.285125
$502$	0	0
$503$	5.67376	0.252981	0.126490	−	0.991968i	$-0.459629\pi$
$503$	5.67376	0.252981	0.126490	+	0.991968i	$0.459629\pi$
$504$	0	0
$505$	−3.09017	−0.137511
$506$	0	0
$507$	12.0000	0.532939
$508$	0	0
$509$	−22.3607	−0.991120	−0.495560	−	0.868574i	$-0.665037\pi$
$509$	−22.3607	−0.991120	−0.495560	+	0.868574i	$0.665037\pi$
$510$	0	0
$511$	−44.1246	−1.95196
$512$	0	0
$513$	−4.00000	−0.176604
$514$	0	0
$515$	11.3262	0.499094
$516$	0	0
$517$	−3.41641	−0.150253
$518$	0	0
$519$	−13.2361	−0.580999
$520$	0	0
$521$	19.0344	0.833914	0.416957	−	0.908926i	$-0.363097\pi$
$521$	19.0344	0.833914	0.416957	+	0.908926i	$0.363097\pi$
$522$	0	0
$523$	−0.798374	−0.0349105	−0.0174552	−	0.999848i	$-0.505556\pi$
$523$	−0.798374	−0.0349105	−0.0174552	+	0.999848i	$0.505556\pi$
$524$	0	0
$525$	7.14590	0.311873
$526$	0	0
$527$	6.90983	0.300997
$528$	0	0
$529$	32.8328	1.42751
$530$	0	0
$531$	3.61803	0.157009
$532$	0	0
$533$	−1.14590	−0.0496344
$534$	0	0
$535$	−16.0902	−0.695639
$536$	0	0
$537$	−4.41641	−0.190582
$538$	0	0
$539$	1.52786	0.0658098
$540$	0	0
$541$	13.8328	0.594719	0.297360	−	0.954766i	$-0.403894\pi$
$541$	13.8328	0.594719	0.297360	+	0.954766i	$0.403894\pi$
$542$	0	0
$543$	9.29180	0.398749
$544$	0	0
$545$	19.4721	0.834095
$546$	0	0
$547$	8.12461	0.347383	0.173692	−	0.984800i	$-0.444430\pi$
$547$	8.12461	0.347383	0.173692	+	0.984800i	$0.444430\pi$
$548$	0	0
$549$	12.5623	0.536146
$550$	0	0
$551$	31.4164	1.33838
$552$	0	0
$553$	26.3951	1.12243
$554$	0	0
$555$	−7.85410	−0.333388
$556$	0	0
$557$	−46.0902	−1.95290	−0.976452	−	0.215737i	$-0.930785\pi$
$557$	−46.0902	−1.95290	−0.976452	+	0.215737i	$0.930785\pi$
$558$	0	0
$559$	−0.527864	−0.0223263
$560$	0	0
$561$	1.70820	0.0721204
$562$	0	0
$563$	33.6525	1.41828	0.709141	−	0.705066i	$-0.249083\pi$
$563$	33.6525	1.41828	0.709141	+	0.705066i	$0.249083\pi$
$564$	0	0
$565$	−2.00000	−0.0841406
$566$	0	0
$567$	3.00000	0.125988
$568$	0	0
$569$	1.34752	0.0564912	0.0282456	−	0.999601i	$-0.491008\pi$
$569$	1.34752	0.0564912	0.0282456	+	0.999601i	$0.491008\pi$
$570$	0	0
$571$	−20.3607	−0.852068	−0.426034	−	0.904707i	$-0.640090\pi$
$571$	−20.3607	−0.852068	−0.426034	+	0.904707i	$0.640090\pi$
$572$	0	0
$573$	−19.3607	−0.808804
$574$	0	0
$575$	−17.7984	−0.742243
$576$	0	0
$577$	−9.90983	−0.412552	−0.206276	−	0.978494i	$-0.566134\pi$
$577$	−9.90983	−0.412552	−0.206276	+	0.978494i	$0.566134\pi$
$578$	0	0
$579$	−25.8885	−1.07589
$580$	0	0
$581$	19.1459	0.794306
$582$	0	0
$583$	6.06888	0.251347
$584$	0	0
$585$	1.61803	0.0668975
$586$	0	0
$587$	−25.7426	−1.06251	−0.531256	−	0.847211i	$-0.678280\pi$
$587$	−25.7426	−1.06251	−0.531256	+	0.847211i	$0.678280\pi$
$588$	0	0
$589$	−12.3607	−0.509313
$590$	0	0
$591$	20.0344	0.824107
$592$	0	0
$593$	30.9787	1.27214	0.636072	−	0.771630i	$-0.280558\pi$
$593$	30.9787	1.27214	0.636072	+	0.771630i	$0.280558\pi$
$594$	0	0
$595$	10.8541	0.444975
$596$	0	0
$597$	0.236068	0.00966162
$598$	0	0
$599$	12.8197	0.523797	0.261899	−	0.965095i	$-0.415651\pi$
$599$	12.8197	0.523797	0.261899	+	0.965095i	$0.415651\pi$
$600$	0	0
$601$	−35.2148	−1.43644	−0.718220	−	0.695816i	$-0.755043\pi$
$601$	−35.2148	−1.43644	−0.718220	+	0.695816i	$0.755043\pi$
$602$	0	0
$603$	10.8541	0.442013
$604$	0	0
$605$	16.8541	0.685217
$606$	0	0
$607$	34.2705	1.39100	0.695499	−	0.718528i	$-0.255184\pi$
$607$	34.2705	1.39100	0.695499	+	0.718528i	$0.255184\pi$
$608$	0	0
$609$	−23.5623	−0.954793
$610$	0	0
$611$	4.47214	0.180923
$612$	0	0
$613$	−2.29180	−0.0925648	−0.0462824	−	0.998928i	$-0.514737\pi$
$613$	−2.29180	−0.0925648	−0.0462824	+	0.998928i	$0.514737\pi$
$614$	0	0
$615$	1.85410	0.0747646
$616$	0	0
$617$	20.2918	0.816917	0.408458	−	0.912777i	$-0.366066\pi$
$617$	20.2918	0.816917	0.408458	+	0.912777i	$0.366066\pi$
$618$	0	0
$619$	−10.1803	−0.409182	−0.204591	−	0.978848i	$-0.565586\pi$
$619$	−10.1803	−0.409182	−0.204591	+	0.978848i	$0.565586\pi$
$620$	0	0
$621$	−7.47214	−0.299846
$622$	0	0
$623$	−9.00000	−0.360577
$624$	0	0
$625$	−7.41641	−0.296656
$626$	0	0
$627$	−3.05573	−0.122034
$628$	0	0
$629$	10.8541	0.432781
$630$	0	0
$631$	46.9787	1.87019	0.935097	−	0.354393i	$-0.115313\pi$
$631$	46.9787	1.87019	0.935097	+	0.354393i	$0.115313\pi$
$632$	0	0
$633$	14.5623	0.578800
$634$	0	0
$635$	−4.61803	−0.183261
$636$	0	0
$637$	−2.00000	−0.0792429
$638$	0	0
$639$	−12.0902	−0.478280
$640$	0	0
$641$	−23.5066	−0.928454	−0.464227	−	0.885716i	$-0.653668\pi$
$641$	−23.5066	−0.928454	−0.464227	+	0.885716i	$0.653668\pi$
$642$	0	0
$643$	−10.4508	−0.412141	−0.206071	−	0.978537i	$-0.566068\pi$
$643$	−10.4508	−0.412141	−0.206071	+	0.978537i	$0.566068\pi$
$644$	0	0
$645$	0.854102	0.0336302
$646$	0	0
$647$	2.88854	0.113560	0.0567802	−	0.998387i	$-0.481917\pi$
$647$	2.88854	0.113560	0.0567802	+	0.998387i	$0.481917\pi$
$648$	0	0
$649$	2.76393	0.108494
$650$	0	0
$651$	9.27051	0.363340
$652$	0	0
$653$	−18.0000	−0.704394	−0.352197	−	0.935926i	$-0.614565\pi$
$653$	−18.0000	−0.704394	−0.352197	+	0.935926i	$0.614565\pi$
$654$	0	0
$655$	6.32624	0.247187
$656$	0	0
$657$	−14.7082	−0.573822
$658$	0	0
$659$	−23.6180	−0.920028	−0.460014	−	0.887912i	$-0.652156\pi$
$659$	−23.6180	−0.920028	−0.460014	+	0.887912i	$0.652156\pi$
$660$	0	0
$661$	21.7984	0.847858	0.423929	−	0.905695i	$-0.360651\pi$
$661$	21.7984	0.847858	0.423929	+	0.905695i	$0.360651\pi$
$662$	0	0
$663$	−2.23607	−0.0868417
$664$	0	0
$665$	−19.4164	−0.752936
$666$	0	0
$667$	58.6869	2.27237
$668$	0	0
$669$	0.673762	0.0260491
$670$	0	0
$671$	9.59675	0.370478
$672$	0	0
$673$	−18.9443	−0.730248	−0.365124	−	0.930959i	$-0.618973\pi$
$673$	−18.9443	−0.730248	−0.365124	+	0.930959i	$0.618973\pi$
$674$	0	0
$675$	2.38197	0.0916819
$676$	0	0
$677$	−2.40325	−0.0923645	−0.0461822	−	0.998933i	$-0.514705\pi$
$677$	−2.40325	−0.0923645	−0.0461822	+	0.998933i	$0.514705\pi$
$678$	0	0
$679$	−40.2492	−1.54462
$680$	0	0
$681$	−13.9098	−0.533026
$682$	0	0
$683$	−10.0557	−0.384772	−0.192386	−	0.981319i	$-0.561623\pi$
$683$	−10.0557	−0.384772	−0.192386	+	0.981319i	$0.561623\pi$
$684$	0	0
$685$	−24.5623	−0.938477
$686$	0	0
$687$	22.2361	0.848359
$688$	0	0
$689$	−7.94427	−0.302653
$690$	0	0
$691$	37.0689	1.41017	0.705083	−	0.709124i	$-0.250910\pi$
$691$	37.0689	1.41017	0.705083	+	0.709124i	$0.250910\pi$
$692$	0	0
$693$	2.29180	0.0870581
$694$	0	0
$695$	5.09017	0.193081
$696$	0	0
$697$	−2.56231	−0.0970543
$698$	0	0
$699$	2.23607	0.0845759
$700$	0	0
$701$	41.1803	1.55536	0.777680	−	0.628660i	$-0.216396\pi$
$701$	41.1803	1.55536	0.777680	+	0.628660i	$0.216396\pi$
$702$	0	0
$703$	−19.4164	−0.732304
$704$	0	0
$705$	−7.23607	−0.272526
$706$	0	0
$707$	5.72949	0.215480
$708$	0	0
$709$	−35.6525	−1.33896	−0.669478	−	0.742832i	$-0.733482\pi$
$709$	−35.6525	−1.33896	−0.669478	+	0.742832i	$0.733482\pi$
$710$	0	0
$711$	8.79837	0.329965
$712$	0	0
$713$	−23.0902	−0.864734
$714$	0	0
$715$	1.23607	0.0462263
$716$	0	0
$717$	2.94427	0.109956
$718$	0	0
$719$	−19.7426	−0.736276	−0.368138	−	0.929771i	$-0.620005\pi$
$719$	−19.7426	−0.736276	−0.368138	+	0.929771i	$0.620005\pi$
$720$	0	0
$721$	−21.0000	−0.782081
$722$	0	0
$723$	−9.94427	−0.369831
$724$	0	0
$725$	−18.7082	−0.694805
$726$	0	0
$727$	−2.29180	−0.0849980	−0.0424990	−	0.999097i	$-0.513532\pi$
$727$	−2.29180	−0.0849980	−0.0424990	+	0.999097i	$0.513532\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−1.18034	−0.0436564
$732$	0	0
$733$	−34.8673	−1.28785	−0.643926	−	0.765088i	$-0.722695\pi$
$733$	−34.8673	−1.28785	−0.643926	+	0.765088i	$0.722695\pi$
$734$	0	0
$735$	3.23607	0.119364
$736$	0	0
$737$	8.29180	0.305432
$738$	0	0
$739$	22.6869	0.834552	0.417276	−	0.908780i	$-0.362985\pi$
$739$	22.6869	0.834552	0.417276	+	0.908780i	$0.362985\pi$
$740$	0	0
$741$	4.00000	0.146944
$742$	0	0
$743$	29.3607	1.07714	0.538569	−	0.842581i	$-0.318965\pi$
$743$	29.3607	1.07714	0.538569	+	0.842581i	$0.318965\pi$
$744$	0	0
$745$	−3.76393	−0.137900
$746$	0	0
$747$	6.38197	0.233504
$748$	0	0
$749$	29.8328	1.09007
$750$	0	0
$751$	−42.7082	−1.55844	−0.779222	−	0.626748i	$-0.784386\pi$
$751$	−42.7082	−1.55844	−0.779222	+	0.626748i	$0.784386\pi$
$752$	0	0
$753$	−16.5279	−0.602309
$754$	0	0
$755$	1.61803	0.0588863
$756$	0	0
$757$	−46.9574	−1.70670	−0.853348	−	0.521341i	$-0.825432\pi$
$757$	−46.9574	−1.70670	−0.853348	+	0.521341i	$0.825432\pi$
$758$	0	0
$759$	−5.70820	−0.207195
$760$	0	0
$761$	38.6525	1.40115	0.700576	−	0.713578i	$-0.252927\pi$
$761$	38.6525	1.40115	0.700576	+	0.713578i	$0.252927\pi$
$762$	0	0
$763$	−36.1033	−1.30703
$764$	0	0
$765$	3.61803	0.130810
$766$	0	0
$767$	−3.61803	−0.130640
$768$	0	0
$769$	15.6869	0.565685	0.282842	−	0.959166i	$-0.408723\pi$
$769$	15.6869	0.565685	0.282842	+	0.959166i	$0.408723\pi$
$770$	0	0
$771$	10.4164	0.375138
$772$	0	0
$773$	−50.4853	−1.81583	−0.907915	−	0.419155i	$-0.862327\pi$
$773$	−50.4853	−1.81583	−0.907915	+	0.419155i	$0.862327\pi$
$774$	0	0
$775$	7.36068	0.264403
$776$	0	0
$777$	14.5623	0.522420
$778$	0	0
$779$	4.58359	0.164224
$780$	0	0
$781$	−9.23607	−0.330492
$782$	0	0
$783$	−7.85410	−0.280683
$784$	0	0
$785$	−5.76393	−0.205724
$786$	0	0
$787$	26.7082	0.952045	0.476022	−	0.879433i	$-0.342078\pi$
$787$	26.7082	0.952045	0.476022	+	0.879433i	$0.342078\pi$
$788$	0	0
$789$	10.0344	0.357236
$790$	0	0
$791$	3.70820	0.131849
$792$	0	0
$793$	−12.5623	−0.446101
$794$	0	0
$795$	12.8541	0.455888
$796$	0	0
$797$	15.2016	0.538469	0.269235	−	0.963075i	$-0.413229\pi$
$797$	15.2016	0.538469	0.269235	+	0.963075i	$0.413229\pi$
$798$	0	0
$799$	10.0000	0.353775
$800$	0	0
$801$	−3.00000	−0.106000
$802$	0	0
$803$	−11.2361	−0.396512
$804$	0	0
$805$	−36.2705	−1.27837
$806$	0	0
$807$	9.03444	0.318027
$808$	0	0
$809$	−1.20163	−0.0422469	−0.0211235	−	0.999777i	$-0.506724\pi$
$809$	−1.20163	−0.0422469	−0.0211235	+	0.999777i	$0.506724\pi$
$810$	0	0
$811$	−12.1246	−0.425753	−0.212876	−	0.977079i	$-0.568283\pi$
$811$	−12.1246	−0.425753	−0.212876	+	0.977079i	$0.568283\pi$
$812$	0	0
$813$	−16.9443	−0.594262
$814$	0	0
$815$	−22.6525	−0.793482
$816$	0	0
$817$	2.11146	0.0738705
$818$	0	0
$819$	−3.00000	−0.104828
$820$	0	0
$821$	−1.47214	−0.0513779	−0.0256889	−	0.999670i	$-0.508178\pi$
$821$	−1.47214	−0.0513779	−0.0256889	+	0.999670i	$0.508178\pi$
$822$	0	0
$823$	32.0344	1.11665	0.558325	−	0.829622i	$-0.311444\pi$
$823$	32.0344	1.11665	0.558325	+	0.829622i	$0.311444\pi$
$824$	0	0
$825$	1.81966	0.0633524
$826$	0	0
$827$	10.9098	0.379372	0.189686	−	0.981845i	$-0.439253\pi$
$827$	10.9098	0.379372	0.189686	+	0.981845i	$0.439253\pi$
$828$	0	0
$829$	24.8328	0.862479	0.431240	−	0.902237i	$-0.358076\pi$
$829$	24.8328	0.862479	0.431240	+	0.902237i	$0.358076\pi$
$830$	0	0
$831$	−17.7639	−0.616224
$832$	0	0
$833$	−4.47214	−0.154950
$834$	0	0
$835$	−10.3262	−0.357354
$836$	0	0
$837$	3.09017	0.106812
$838$	0	0
$839$	−3.96556	−0.136906	−0.0684531	−	0.997654i	$-0.521806\pi$
$839$	−3.96556	−0.136906	−0.0684531	+	0.997654i	$0.521806\pi$
$840$	0	0
$841$	32.6869	1.12714
$842$	0	0
$843$	−20.1803	−0.695048
$844$	0	0
$845$	19.4164	0.667945
$846$	0	0
$847$	−31.2492	−1.07374
$848$	0	0
$849$	−21.9443	−0.753125
$850$	0	0
$851$	−36.2705	−1.24334
$852$	0	0
$853$	−41.5279	−1.42189	−0.710943	−	0.703249i	$-0.751732\pi$
$853$	−41.5279	−1.42189	−0.710943	+	0.703249i	$0.751732\pi$
$854$	0	0
$855$	−6.47214	−0.221342
$856$	0	0
$857$	7.58359	0.259051	0.129525	−	0.991576i	$-0.458655\pi$
$857$	7.58359	0.259051	0.129525	+	0.991576i	$0.458655\pi$
$858$	0	0
$859$	−13.9787	−0.476948	−0.238474	−	0.971149i	$-0.576647\pi$
$859$	−13.9787	−0.476948	−0.238474	+	0.971149i	$0.576647\pi$
$860$	0	0
$861$	−3.43769	−0.117156
$862$	0	0
$863$	−6.00000	−0.204242	−0.102121	−	0.994772i	$-0.532563\pi$
$863$	−6.00000	−0.204242	−0.102121	+	0.994772i	$0.532563\pi$
$864$	0	0
$865$	−21.4164	−0.728180
$866$	0	0
$867$	12.0000	0.407541
$868$	0	0
$869$	6.72136	0.228007
$870$	0	0
$871$	−10.8541	−0.367777
$872$	0	0
$873$	−13.4164	−0.454077
$874$	0	0
$875$	35.8328	1.21137
$876$	0	0
$877$	15.8885	0.536518	0.268259	−	0.963347i	$-0.413552\pi$
$877$	15.8885	0.536518	0.268259	+	0.963347i	$0.413552\pi$
$878$	0	0
$879$	11.6525	0.393028
$880$	0	0
$881$	51.1033	1.72171	0.860857	−	0.508846i	$-0.169928\pi$
$881$	51.1033	1.72171	0.860857	+	0.508846i	$0.169928\pi$
$882$	0	0
$883$	18.5410	0.623955	0.311977	−	0.950089i	$-0.399009\pi$
$883$	18.5410	0.623955	0.311977	+	0.950089i	$0.399009\pi$
$884$	0	0
$885$	5.85410	0.196783
$886$	0	0
$887$	14.8885	0.499908	0.249954	−	0.968258i	$-0.419584\pi$
$887$	14.8885	0.499908	0.249954	+	0.968258i	$0.419584\pi$
$888$	0	0
$889$	8.56231	0.287171
$890$	0	0
$891$	0.763932	0.0255927
$892$	0	0
$893$	−17.8885	−0.598617
$894$	0	0
$895$	−7.14590	−0.238861
$896$	0	0
$897$	7.47214	0.249487
$898$	0	0
$899$	−24.2705	−0.809467
$900$	0	0
$901$	−17.7639	−0.591802
$902$	0	0
$903$	−1.58359	−0.0526986
$904$	0	0
$905$	15.0344	0.499762
$906$	0	0
$907$	−17.5410	−0.582440	−0.291220	−	0.956656i	$-0.594061\pi$
$907$	−17.5410	−0.582440	−0.291220	+	0.956656i	$0.594061\pi$
$908$	0	0
$909$	1.90983	0.0633451
$910$	0	0
$911$	−56.5967	−1.87513	−0.937567	−	0.347805i	$-0.886927\pi$
$911$	−56.5967	−1.87513	−0.937567	+	0.347805i	$0.886927\pi$
$912$	0	0
$913$	4.87539	0.161352
$914$	0	0
$915$	20.3262	0.671965
$916$	0	0
$917$	−11.7295	−0.387342
$918$	0	0
$919$	−21.4377	−0.707164	−0.353582	−	0.935403i	$-0.615037\pi$
$919$	−21.4377	−0.707164	−0.353582	+	0.935403i	$0.615037\pi$
$920$	0	0
$921$	17.0344	0.561304
$922$	0	0
$923$	12.0902	0.397953
$924$	0	0
$925$	11.5623	0.380166
$926$	0	0
$927$	−7.00000	−0.229910
$928$	0	0
$929$	−24.7984	−0.813608	−0.406804	−	0.913515i	$-0.633357\pi$
$929$	−24.7984	−0.813608	−0.406804	+	0.913515i	$0.633357\pi$
$930$	0	0
$931$	8.00000	0.262189
$932$	0	0
$933$	3.00000	0.0982156
$934$	0	0
$935$	2.76393	0.0903902
$936$	0	0
$937$	29.0000	0.947389	0.473694	−	0.880689i	$-0.342920\pi$
$937$	29.0000	0.947389	0.473694	+	0.880689i	$0.342920\pi$
$938$	0	0
$939$	−13.5066	−0.440771
$940$	0	0
$941$	−21.4721	−0.699972	−0.349986	−	0.936755i	$-0.613814\pi$
$941$	−21.4721	−0.699972	−0.349986	+	0.936755i	$0.613814\pi$
$942$	0	0
$943$	8.56231	0.278827
$944$	0	0
$945$	4.85410	0.157904
$946$	0	0
$947$	54.7426	1.77890	0.889448	−	0.457035i	$-0.151089\pi$
$947$	54.7426	1.77890	0.889448	+	0.457035i	$0.151089\pi$
$948$	0	0
$949$	14.7082	0.477449
$950$	0	0
$951$	−12.0000	−0.389127
$952$	0	0
$953$	−42.1033	−1.36386	−0.681930	−	0.731417i	$-0.738859\pi$
$953$	−42.1033	−1.36386	−0.681930	+	0.731417i	$0.738859\pi$
$954$	0	0
$955$	−31.3262	−1.01369
$956$	0	0
$957$	−6.00000	−0.193952
$958$	0	0
$959$	45.5410	1.47060
$960$	0	0
$961$	−21.4508	−0.691963
$962$	0	0
$963$	9.94427	0.320450
$964$	0	0
$965$	−41.8885	−1.34844
$966$	0	0
$967$	−23.7082	−0.762404	−0.381202	−	0.924492i	$-0.624490\pi$
$967$	−23.7082	−0.762404	−0.381202	+	0.924492i	$0.624490\pi$
$968$	0	0
$969$	8.94427	0.287331
$970$	0	0
$971$	20.2361	0.649406	0.324703	−	0.945816i	$-0.394736\pi$
$971$	20.2361	0.649406	0.324703	+	0.945816i	$0.394736\pi$
$972$	0	0
$973$	−9.43769	−0.302559
$974$	0	0
$975$	−2.38197	−0.0762840
$976$	0	0
$977$	−23.5967	−0.754927	−0.377463	−	0.926024i	$-0.623204\pi$
$977$	−23.5967	−0.754927	−0.377463	+	0.926024i	$0.623204\pi$
$978$	0	0
$979$	−2.29180	−0.0732461
$980$	0	0
$981$	−12.0344	−0.384230
$982$	0	0
$983$	−38.5967	−1.23105	−0.615523	−	0.788119i	$-0.711055\pi$
$983$	−38.5967	−1.23105	−0.615523	+	0.788119i	$0.711055\pi$
$984$	0	0
$985$	32.4164	1.03287
$986$	0	0
$987$	13.4164	0.427049
$988$	0	0
$989$	3.94427	0.125421
$990$	0	0
$991$	−13.7082	−0.435455	−0.217728	−	0.976010i	$-0.569864\pi$
$991$	−13.7082	−0.435455	−0.217728	+	0.976010i	$0.569864\pi$
$992$	0	0
$993$	19.1246	0.606901
$994$	0	0
$995$	0.381966	0.0121091
$996$	0	0
$997$	44.6869	1.41525	0.707624	−	0.706589i	$-0.249767\pi$
$997$	44.6869	1.41525	0.707624	+	0.706589i	$0.249767\pi$
$998$	0	0
$999$	4.85410	0.153577

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7248.2.a.o.1.1		2
4.3	odd	2		453.2.a.b.1.2	✓	2
12.11	even	2		1359.2.a.c.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
453.2.a.b.1.2	✓	2	4.3	odd	2
1359.2.a.c.1.1		2	12.11	even	2
7248.2.a.o.1.1		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 7248 = 2^{4} \cdot 3 \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7248.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.61803 q^{5} +3.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.61803 q^{5} +3.00000 q^{7} +1.00000 q^{9} +0.763932 q^{11} -1.00000 q^{13} +1.61803 q^{15} -2.23607 q^{17} +4.00000 q^{19} -3.00000 q^{21} +7.47214 q^{23} -2.38197 q^{25} -1.00000 q^{27} +7.85410 q^{29} -3.09017 q^{31} -0.763932 q^{33} -4.85410 q^{35} -4.85410 q^{37} +1.00000 q^{39} +1.14590 q^{41} +0.527864 q^{43} -1.61803 q^{45} -4.47214 q^{47} +2.00000 q^{49} +2.23607 q^{51} +7.94427 q^{53} -1.23607 q^{55} -4.00000 q^{57} +3.61803 q^{59} +12.5623 q^{61} +3.00000 q^{63} +1.61803 q^{65} +10.8541 q^{67} -7.47214 q^{69} -12.0902 q^{71} -14.7082 q^{73} +2.38197 q^{75} +2.29180 q^{77} +8.79837 q^{79} +1.00000 q^{81} +6.38197 q^{83} +3.61803 q^{85} -7.85410 q^{87} -3.00000 q^{89} -3.00000 q^{91} +3.09017 q^{93} -6.47214 q^{95} -13.4164 q^{97} +0.763932 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - q^{5} + 6 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - q^5 + 6 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} - q^{5} + 6 q^{7} + 2 q^{9} + 6 q^{11} - 2 q^{13} + q^{15} + 8 q^{19} - 6 q^{21} + 6 q^{23} - 7 q^{25} - 2 q^{27} + 9 q^{29} + 5 q^{31} - 6 q^{33} - 3 q^{35} - 3 q^{37} + 2 q^{39} + 9 q^{41} + 10 q^{43} - q^{45} + 4 q^{49} - 2 q^{53} + 2 q^{55} - 8 q^{57} + 5 q^{59} + 5 q^{61} + 6 q^{63} + q^{65} + 15 q^{67} - 6 q^{69} - 13 q^{71} - 16 q^{73} + 7 q^{75} + 18 q^{77} - 7 q^{79} + 2 q^{81} + 15 q^{83} + 5 q^{85} - 9 q^{87} - 6 q^{89} - 6 q^{91} - 5 q^{93} - 4 q^{95} + 6 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - q^5 + 6 * q^7 + 2 * q^9 + 6 * q^11 - 2 * q^13 + q^15 + 8 * q^19 - 6 * q^21 + 6 * q^23 - 7 * q^25 - 2 * q^27 + 9 * q^29 + 5 * q^31 - 6 * q^33 - 3 * q^35 - 3 * q^37 + 2 * q^39 + 9 * q^41 + 10 * q^43 - q^45 + 4 * q^49 - 2 * q^53 + 2 * q^55 - 8 * q^57 + 5 * q^59 + 5 * q^61 + 6 * q^63 + q^65 + 15 * q^67 - 6 * q^69 - 13 * q^71 - 16 * q^73 + 7 * q^75 + 18 * q^77 - 7 * q^79 + 2 * q^81 + 15 * q^83 + 5 * q^85 - 9 * q^87 - 6 * q^89 - 6 * q^91 - 5 * q^93 - 4 * q^95 + 6 * q^99

Introduction

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