LMFDB - Embedded newform 9225.2.a.bx.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9225,2,Mod(1,9225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9225, 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("9225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9225 = 3^{2} \cdot 5^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9225.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:	$73.6619958646$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 123)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.34292$ of defining polynomial
Character	$\chi$	$=$	9225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.34292 q^{2} +3.48929 q^{4} +3.83221 q^{7} +3.48929 q^{8} +O(q^{10})$	$q+2.34292 q^{2} +3.48929 q^{4} +3.83221 q^{7} +3.48929 q^{8} +3.34292 q^{11} -3.14637 q^{13} +8.97858 q^{14} +1.19656 q^{16} +3.63565 q^{17} +1.14637 q^{19} +7.83221 q^{22} -2.85363 q^{23} -7.37169 q^{26} +13.3717 q^{28} +8.02877 q^{29} +9.86098 q^{31} -4.17513 q^{32} +8.51806 q^{34} -8.19656 q^{37} +2.68585 q^{38} -1.00000 q^{41} -11.7606 q^{43} +11.6644 q^{44} -6.68585 q^{46} +8.32150 q^{47} +7.68585 q^{49} -10.9786 q^{52} +1.60688 q^{53} +13.3717 q^{56} +18.8108 q^{58} +11.6644 q^{59} -4.19656 q^{61} +23.1035 q^{62} -12.1751 q^{64} -6.10038 q^{67} +12.6858 q^{68} +8.65708 q^{71} +10.1537 q^{73} -19.2039 q^{74} +4.00000 q^{76} +12.8108 q^{77} -3.60688 q^{79} -2.34292 q^{82} -10.1249 q^{83} -27.5542 q^{86} +11.6644 q^{88} +3.37169 q^{89} -12.0575 q^{91} -9.95715 q^{92} +19.4966 q^{94} +7.53948 q^{97} +18.0073 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} + 3 q^{4} - 2 q^{7} + 3 q^{8}+O(q^{10}) $ 3 * q + q^2 + 3 * q^4 - 2 * q^7 + 3 * q^8	$ 3 q + q^{2} + 3 q^{4} - 2 q^{7} + 3 q^{8} + 4 q^{11} - 8 q^{13} + 12 q^{14} - q^{16} + 2 q^{17} + 2 q^{19} + 10 q^{22} - 10 q^{23} + 2 q^{26} + 16 q^{28} + 6 q^{29} - 2 q^{31} + 7 q^{32} - 20 q^{37} - 4 q^{38} - 3 q^{41} - 10 q^{43} + 8 q^{44} - 8 q^{46} + 4 q^{47} + 11 q^{49} - 18 q^{52} + 14 q^{53} + 16 q^{56} + 28 q^{58} + 8 q^{59} - 8 q^{61} + 38 q^{62} - 17 q^{64} - 12 q^{67} + 26 q^{68} + 32 q^{71} - 4 q^{73} - 20 q^{74} + 12 q^{76} + 10 q^{77} - 20 q^{79} - q^{82} - 14 q^{83} - 6 q^{86} + 8 q^{88} - 14 q^{89} + 18 q^{94} + 12 q^{97} + 21 q^{98}+O(q^{100}) $ 3 * q + q^2 + 3 * q^4 - 2 * q^7 + 3 * q^8 + 4 * q^11 - 8 * q^13 + 12 * q^14 - q^16 + 2 * q^17 + 2 * q^19 + 10 * q^22 - 10 * q^23 + 2 * q^26 + 16 * q^28 + 6 * q^29 - 2 * q^31 + 7 * q^32 - 20 * q^37 - 4 * q^38 - 3 * q^41 - 10 * q^43 + 8 * q^44 - 8 * q^46 + 4 * q^47 + 11 * q^49 - 18 * q^52 + 14 * q^53 + 16 * q^56 + 28 * q^58 + 8 * q^59 - 8 * q^61 + 38 * q^62 - 17 * q^64 - 12 * q^67 + 26 * q^68 + 32 * q^71 - 4 * q^73 - 20 * q^74 + 12 * q^76 + 10 * q^77 - 20 * q^79 - q^82 - 14 * q^83 - 6 * q^86 + 8 * q^88 - 14 * q^89 + 18 * q^94 + 12 * q^97 + 21 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	2.34292	1.65670	0.828348	−	0.560213i	$-0.189281\pi$
$2$	2.34292	1.65670	0.828348	+	0.560213i	$0.189281\pi$
$3$	0	0
$3$	0	0
$4$	3.48929	1.74464
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.83221	1.44844	0.724220	−	0.689569i	$-0.242200\pi$
$7$	3.83221	1.44844	0.724220	+	0.689569i	$0.242200\pi$
$8$	3.48929	1.23365
$9$	0	0
$10$	0	0
$11$	3.34292	1.00793	0.503965	−	0.863724i	$-0.331874\pi$
$11$	3.34292	1.00793	0.503965	+	0.863724i	$0.331874\pi$
$12$	0	0
$13$	−3.14637	−0.872645	−0.436322	−	0.899790i	$-0.643719\pi$
$13$	−3.14637	−0.872645	−0.436322	+	0.899790i	$0.643719\pi$
$14$	8.97858	2.39963
$15$	0	0
$16$	1.19656	0.299139
$17$	3.63565	0.881776	0.440888	−	0.897562i	$-0.354664\pi$
$17$	3.63565	0.881776	0.440888	+	0.897562i	$0.354664\pi$
$18$	0	0
$19$	1.14637	0.262994	0.131497	−	0.991317i	$-0.458022\pi$
$19$	1.14637	0.262994	0.131497	+	0.991317i	$0.458022\pi$
$20$	0	0
$21$	0	0
$22$	7.83221	1.66983
$23$	−2.85363	−0.595024	−0.297512	−	0.954718i	$-0.596157\pi$
$23$	−2.85363	−0.595024	−0.297512	+	0.954718i	$0.596157\pi$
$24$	0	0
$25$	0	0
$26$	−7.37169	−1.44571
$27$	0	0
$28$	13.3717	2.52701
$29$	8.02877	1.49091	0.745453	−	0.666559i	$-0.232233\pi$
$29$	8.02877	1.49091	0.745453	+	0.666559i	$0.232233\pi$
$30$	0	0
$31$	9.86098	1.77108	0.885542	−	0.464559i	$-0.153787\pi$
$31$	9.86098	1.77108	0.885542	+	0.464559i	$0.153787\pi$
$32$	−4.17513	−0.738067
$33$	0	0
$34$	8.51806	1.46083
$35$	0	0
$36$	0	0
$37$	−8.19656	−1.34751	−0.673753	−	0.738957i	$-0.735319\pi$
$37$	−8.19656	−1.34751	−0.673753	+	0.738957i	$0.735319\pi$
$38$	2.68585	0.435702
$39$	0	0
$40$	0	0
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	−11.7606	−1.79347	−0.896737	−	0.442564i	$-0.854069\pi$
$43$	−11.7606	−1.79347	−0.896737	+	0.442564i	$0.854069\pi$
$44$	11.6644	1.75848
$45$	0	0
$46$	−6.68585	−0.985774
$47$	8.32150	1.21382	0.606908	−	0.794772i	$-0.292410\pi$
$47$	8.32150	1.21382	0.606908	+	0.794772i	$0.292410\pi$
$48$	0	0
$49$	7.68585	1.09798
$50$	0	0
$51$	0	0
$52$	−10.9786	−1.52245
$53$	1.60688	0.220723	0.110361	−	0.993892i	$-0.464799\pi$
$53$	1.60688	0.220723	0.110361	+	0.993892i	$0.464799\pi$
$54$	0	0
$55$	0	0
$56$	13.3717	1.78687
$57$	0	0
$58$	18.8108	2.46998
$59$	11.6644	1.51858	0.759289	−	0.650753i	$-0.225547\pi$
$59$	11.6644	1.51858	0.759289	+	0.650753i	$0.225547\pi$
$60$	0	0
$61$	−4.19656	−0.537314	−0.268657	−	0.963236i	$-0.586580\pi$
$61$	−4.19656	−0.537314	−0.268657	+	0.963236i	$0.586580\pi$
$62$	23.1035	2.93415
$63$	0	0
$64$	−12.1751	−1.52189
$65$	0	0
$66$	0	0
$67$	−6.10038	−0.745281	−0.372640	−	0.927976i	$-0.621547\pi$
$67$	−6.10038	−0.745281	−0.372640	+	0.927976i	$0.621547\pi$
$68$	12.6858	1.53838
$69$	0	0
$70$	0	0
$71$	8.65708	1.02741	0.513703	−	0.857968i	$-0.328273\pi$
$71$	8.65708	1.02741	0.513703	+	0.857968i	$0.328273\pi$
$72$	0	0
$73$	10.1537	1.18840	0.594201	−	0.804317i	$-0.297468\pi$
$73$	10.1537	1.18840	0.594201	+	0.804317i	$0.297468\pi$
$74$	−19.2039	−2.23241
$75$	0	0
$76$	4.00000	0.458831
$77$	12.8108	1.45992
$78$	0	0
$79$	−3.60688	−0.405806	−0.202903	−	0.979199i	$-0.565038\pi$
$79$	−3.60688	−0.405806	−0.202903	+	0.979199i	$0.565038\pi$
$80$	0	0
$81$	0	0
$82$	−2.34292	−0.258733
$83$	−10.1249	−1.11136	−0.555678	−	0.831397i	$-0.687541\pi$
$83$	−10.1249	−1.11136	−0.555678	+	0.831397i	$0.687541\pi$
$84$	0	0
$85$	0	0
$86$	−27.5542	−2.97124
$87$	0	0
$88$	11.6644	1.24343
$89$	3.37169	0.357399	0.178699	−	0.983904i	$-0.442811\pi$
$89$	3.37169	0.357399	0.178699	+	0.983904i	$0.442811\pi$
$90$	0	0
$91$	−12.0575	−1.26397
$92$	−9.95715	−1.03811
$93$	0	0
$94$	19.4966	2.01092
$95$	0	0
$96$	0	0
$97$	7.53948	0.765518	0.382759	−	0.923848i	$-0.374974\pi$
$97$	7.53948	0.765518	0.382759	+	0.923848i	$0.374974\pi$
$98$	18.0073	1.81902
$99$	0	0
$100$	0	0
$101$	−5.00735	−0.498250	−0.249125	−	0.968471i	$-0.580143\pi$
$101$	−5.00735	−0.498250	−0.249125	+	0.968471i	$0.580143\pi$
$102$	0	0
$103$	−1.21798	−0.120011	−0.0600056	−	0.998198i	$-0.519112\pi$
$103$	−1.21798	−0.120011	−0.0600056	+	0.998198i	$0.519112\pi$
$104$	−10.9786	−1.07654
$105$	0	0
$106$	3.76481	0.365670
$107$	13.7648	1.33069	0.665347	−	0.746534i	$-0.268284\pi$
$107$	13.7648	1.33069	0.665347	+	0.746534i	$0.268284\pi$
$108$	0	0
$109$	3.14637	0.301367	0.150684	−	0.988582i	$-0.451853\pi$
$109$	3.14637	0.301367	0.150684	+	0.988582i	$0.451853\pi$
$110$	0	0
$111$	0	0
$112$	4.58546	0.433285
$113$	−7.73183	−0.727349	−0.363675	−	0.931526i	$-0.618478\pi$
$113$	−7.73183	−0.727349	−0.363675	+	0.931526i	$0.618478\pi$
$114$	0	0
$115$	0	0
$116$	28.0147	2.60110
$117$	0	0
$118$	27.3288	2.51582
$119$	13.9326	1.27720
$120$	0	0
$121$	0.175135	0.0159213
$122$	−9.83221	−0.890167
$123$	0	0
$124$	34.4078	3.08991
$125$	0	0
$126$	0	0
$127$	12.0575	1.06993	0.534967	−	0.844873i	$-0.320324\pi$
$127$	12.0575	1.06993	0.534967	+	0.844873i	$0.320324\pi$
$128$	−20.1751	−1.78325
$129$	0	0
$130$	0	0
$131$	−12.2253	−1.06813	−0.534066	−	0.845443i	$-0.679337\pi$
$131$	−12.2253	−1.06813	−0.534066	+	0.845443i	$0.679337\pi$
$132$	0	0
$133$	4.39312	0.380931
$134$	−14.2927	−1.23470
$135$	0	0
$136$	12.6858	1.08780
$137$	7.69319	0.657274	0.328637	−	0.944456i	$-0.393411\pi$
$137$	7.69319	0.657274	0.328637	+	0.944456i	$0.393411\pi$
$138$	0	0
$139$	−18.0147	−1.52799	−0.763993	−	0.645224i	$-0.776764\pi$
$139$	−18.0147	−1.52799	−0.763993	+	0.645224i	$0.776764\pi$
$140$	0	0
$141$	0	0
$142$	20.2829	1.70210
$143$	−10.5181	−0.879564
$144$	0	0
$145$	0	0
$146$	23.7894	1.96882
$147$	0	0
$148$	−28.6002	−2.35092
$149$	−12.5426	−1.02753	−0.513766	−	0.857931i	$-0.671750\pi$
$149$	−12.5426	−1.02753	−0.513766	+	0.857931i	$0.671750\pi$
$150$	0	0
$151$	2.62831	0.213889	0.106944	−	0.994265i	$-0.465893\pi$
$151$	2.62831	0.213889	0.106944	+	0.994265i	$0.465893\pi$
$152$	4.00000	0.324443
$153$	0	0
$154$	30.0147	2.41865
$155$	0	0
$156$	0	0
$157$	−20.1579	−1.60878	−0.804389	−	0.594103i	$-0.797507\pi$
$157$	−20.1579	−1.60878	−0.804389	+	0.594103i	$0.797507\pi$
$158$	−8.45065	−0.672298
$159$	0	0
$160$	0	0
$161$	−10.9357	−0.861856
$162$	0	0
$163$	−18.2541	−1.42977	−0.714886	−	0.699241i	$-0.753521\pi$
$163$	−18.2541	−1.42977	−0.714886	+	0.699241i	$0.753521\pi$
$164$	−3.48929	−0.272468
$165$	0	0
$166$	−23.7220	−1.84118
$167$	1.31415	0.101692	0.0508461	−	0.998706i	$-0.483808\pi$
$167$	1.31415	0.101692	0.0508461	+	0.998706i	$0.483808\pi$
$168$	0	0
$169$	−3.10038	−0.238491
$170$	0	0
$171$	0	0
$172$	−41.0361	−3.12897
$173$	1.18921	0.0904141	0.0452070	−	0.998978i	$-0.485605\pi$
$173$	1.18921	0.0904141	0.0452070	+	0.998978i	$0.485605\pi$
$174$	0	0
$175$	0	0
$176$	4.00000	0.301511
$177$	0	0
$178$	7.89962	0.592101
$179$	19.0073	1.42068	0.710338	−	0.703861i	$-0.248542\pi$
$179$	19.0073	1.42068	0.710338	+	0.703861i	$0.248542\pi$
$180$	0	0
$181$	10.3931	0.772514	0.386257	−	0.922391i	$-0.373768\pi$
$181$	10.3931	0.772514	0.386257	+	0.922391i	$0.373768\pi$
$182$	−28.2499	−2.09402
$183$	0	0
$184$	−9.95715	−0.734051
$185$	0	0
$186$	0	0
$187$	12.1537	0.888767
$188$	29.0361	2.11768
$189$	0	0
$190$	0	0
$191$	6.87819	0.497689	0.248844	−	0.968544i	$-0.419949\pi$
$191$	6.87819	0.497689	0.248844	+	0.968544i	$0.419949\pi$
$192$	0	0
$193$	19.5970	1.41062	0.705312	−	0.708897i	$-0.250807\pi$
$193$	19.5970	1.41062	0.705312	+	0.708897i	$0.250807\pi$
$194$	17.6644	1.26823
$195$	0	0
$196$	26.8181	1.91558
$197$	−24.7679	−1.76464	−0.882321	−	0.470647i	$-0.844020\pi$
$197$	−24.7679	−1.76464	−0.882321	+	0.470647i	$0.844020\pi$
$198$	0	0
$199$	12.8108	0.908133	0.454066	−	0.890968i	$-0.349973\pi$
$199$	12.8108	0.908133	0.454066	+	0.890968i	$0.349973\pi$
$200$	0	0
$201$	0	0
$202$	−11.7318	−0.825448
$203$	30.7679	2.15949
$204$	0	0
$205$	0	0
$206$	−2.85363	−0.198822
$207$	0	0
$208$	−3.76481	−0.261042
$209$	3.83221	0.265080
$210$	0	0
$211$	3.66442	0.252269	0.126135	−	0.992013i	$-0.459743\pi$
$211$	3.66442	0.252269	0.126135	+	0.992013i	$0.459743\pi$
$212$	5.60688	0.385082
$213$	0	0
$214$	32.2499	2.20456
$215$	0	0
$216$	0	0
$217$	37.7894	2.56531
$218$	7.37169	0.499274
$219$	0	0
$220$	0	0
$221$	−11.4391	−0.769477
$222$	0	0
$223$	−9.17092	−0.614130	−0.307065	−	0.951688i	$-0.599347\pi$
$223$	−9.17092	−0.614130	−0.307065	+	0.951688i	$0.599347\pi$
$224$	−16.0000	−1.06904
$225$	0	0
$226$	−18.1151	−1.20500
$227$	0.263962	0.0175198	0.00875988	−	0.999962i	$-0.497212\pi$
$227$	0.263962	0.0175198	0.00875988	+	0.999962i	$0.497212\pi$
$228$	0	0
$229$	7.20390	0.476047	0.238024	−	0.971259i	$-0.423500\pi$
$229$	7.20390	0.476047	0.238024	+	0.971259i	$0.423500\pi$
$230$	0	0
$231$	0	0
$232$	28.0147	1.83925
$233$	16.1579	1.05854	0.529270	−	0.848453i	$-0.322466\pi$
$233$	16.1579	1.05854	0.529270	+	0.848453i	$0.322466\pi$
$234$	0	0
$235$	0	0
$236$	40.7005	2.64938
$237$	0	0
$238$	32.6430	2.11593
$239$	15.6644	1.01325	0.506624	−	0.862167i	$-0.330893\pi$
$239$	15.6644	1.01325	0.506624	+	0.862167i	$0.330893\pi$
$240$	0	0
$241$	10.5468	0.679381	0.339690	−	0.940537i	$-0.389678\pi$
$241$	10.5468	0.679381	0.339690	+	0.940537i	$0.389678\pi$
$242$	0.410327	0.0263768
$243$	0	0
$244$	−14.6430	−0.937422
$245$	0	0
$246$	0	0
$247$	−3.60688	−0.229501
$248$	34.4078	2.18490
$249$	0	0
$250$	0	0
$251$	−21.4292	−1.35260	−0.676301	−	0.736626i	$-0.736418\pi$
$251$	−21.4292	−1.35260	−0.676301	+	0.736626i	$0.736418\pi$
$252$	0	0
$253$	−9.53948	−0.599742
$254$	28.2499	1.77256
$255$	0	0
$256$	−22.9185	−1.43241
$257$	−7.10773	−0.443368	−0.221684	−	0.975119i	$-0.571155\pi$
$257$	−7.10773	−0.443368	−0.221684	+	0.975119i	$0.571155\pi$
$258$	0	0
$259$	−31.4109	−1.95178
$260$	0	0
$261$	0	0
$262$	−28.6430	−1.76957
$263$	−9.97123	−0.614852	−0.307426	−	0.951572i	$-0.599468\pi$
$263$	−9.97123	−0.614852	−0.307426	+	0.951572i	$0.599468\pi$
$264$	0	0
$265$	0	0
$266$	10.2927	0.631088
$267$	0	0
$268$	−21.2860	−1.30025
$269$	0.685846	0.0418168	0.0209084	−	0.999781i	$-0.493344\pi$
$269$	0.685846	0.0418168	0.0209084	+	0.999781i	$0.493344\pi$
$270$	0	0
$271$	13.8034	0.838499	0.419250	−	0.907871i	$-0.362293\pi$
$271$	13.8034	0.838499	0.419250	+	0.907871i	$0.362293\pi$
$272$	4.35027	0.263774
$273$	0	0
$274$	18.0246	1.08890
$275$	0	0
$276$	0	0
$277$	−22.2113	−1.33454	−0.667272	−	0.744814i	$-0.732538\pi$
$277$	−22.2113	−1.33454	−0.667272	+	0.744814i	$0.732538\pi$
$278$	−42.2070	−2.53141
$279$	0	0
$280$	0	0
$281$	−4.61423	−0.275262	−0.137631	−	0.990484i	$-0.543949\pi$
$281$	−4.61423	−0.275262	−0.137631	+	0.990484i	$0.543949\pi$
$282$	0	0
$283$	−9.13229	−0.542858	−0.271429	−	0.962458i	$-0.587496\pi$
$283$	−9.13229	−0.542858	−0.271429	+	0.962458i	$0.587496\pi$
$284$	30.2070	1.79246
$285$	0	0
$286$	−24.6430	−1.45717
$287$	−3.83221	−0.226208
$288$	0	0
$289$	−3.78202	−0.222472
$290$	0	0
$291$	0	0
$292$	35.4292	2.07334
$293$	23.4433	1.36957	0.684786	−	0.728744i	$-0.259896\pi$
$293$	23.4433	1.36957	0.684786	+	0.728744i	$0.259896\pi$
$294$	0	0
$295$	0	0
$296$	−28.6002	−1.66235
$297$	0	0
$298$	−29.3864	−1.70231
$299$	8.97858	0.519245
$300$	0	0
$301$	−45.0691	−2.59774
$302$	6.15792	0.354349
$303$	0	0
$304$	1.37169	0.0786720
$305$	0	0
$306$	0	0
$307$	−18.2541	−1.04182	−0.520908	−	0.853613i	$-0.674407\pi$
$307$	−18.2541	−1.04182	−0.520908	+	0.853613i	$0.674407\pi$
$308$	44.7005	2.54705
$309$	0	0
$310$	0	0
$311$	−14.8782	−0.843665	−0.421832	−	0.906674i	$-0.638613\pi$
$311$	−14.8782	−0.843665	−0.421832	+	0.906674i	$0.638613\pi$
$312$	0	0
$313$	26.4752	1.49647	0.748234	−	0.663435i	$-0.230902\pi$
$313$	26.4752	1.49647	0.748234	+	0.663435i	$0.230902\pi$
$314$	−47.2285	−2.66526
$315$	0	0
$316$	−12.5855	−0.707988
$317$	29.6503	1.66533	0.832665	−	0.553777i	$-0.186814\pi$
$317$	29.6503	1.66533	0.832665	+	0.553777i	$0.186814\pi$
$318$	0	0
$319$	26.8396	1.50273
$320$	0	0
$321$	0	0
$322$	−25.6216	−1.42783
$323$	4.16779	0.231902
$324$	0	0
$325$	0	0
$326$	−42.7679	−2.36870
$327$	0	0
$328$	−3.48929	−0.192664
$329$	31.8898	1.75814
$330$	0	0
$331$	2.60375	0.143115	0.0715575	−	0.997436i	$-0.477203\pi$
$331$	2.60375	0.143115	0.0715575	+	0.997436i	$0.477203\pi$
$332$	−35.3288	−1.93892
$333$	0	0
$334$	3.07896	0.168473
$335$	0	0
$336$	0	0
$337$	1.51071	0.0822937	0.0411468	−	0.999153i	$-0.486899\pi$
$337$	1.51071	0.0822937	0.0411468	+	0.999153i	$0.486899\pi$
$338$	−7.26396	−0.395107
$339$	0	0
$340$	0	0
$341$	32.9645	1.78513
$342$	0	0
$343$	2.62831	0.141915
$344$	−41.0361	−2.21252
$345$	0	0
$346$	2.78623	0.149789
$347$	−14.2787	−0.766518	−0.383259	−	0.923641i	$-0.625198\pi$
$347$	−14.2787	−0.766518	−0.383259	+	0.923641i	$0.625198\pi$
$348$	0	0
$349$	−16.2541	−0.870062	−0.435031	−	0.900416i	$-0.643263\pi$
$349$	−16.2541	−0.870062	−0.435031	+	0.900416i	$0.643263\pi$
$350$	0	0
$351$	0	0
$352$	−13.9572	−0.743919
$353$	11.1709	0.594568	0.297284	−	0.954789i	$-0.403919\pi$
$353$	11.1709	0.594568	0.297284	+	0.954789i	$0.403919\pi$
$354$	0	0
$355$	0	0
$356$	11.7648	0.623534
$357$	0	0
$358$	44.5328	2.35363
$359$	16.2253	0.856340	0.428170	−	0.903698i	$-0.359158\pi$
$359$	16.2253	0.856340	0.428170	+	0.903698i	$0.359158\pi$
$360$	0	0
$361$	−17.6858	−0.930834
$362$	24.3503	1.27982
$363$	0	0
$364$	−42.0722	−2.20518
$365$	0	0
$366$	0	0
$367$	−0.431750	−0.0225372	−0.0112686	−	0.999937i	$-0.503587\pi$
$367$	−0.431750	−0.0225372	−0.0112686	+	0.999937i	$0.503587\pi$
$368$	−3.41454	−0.177995
$369$	0	0
$370$	0	0
$371$	6.15792	0.319703
$372$	0	0
$373$	3.66863	0.189955	0.0949773	−	0.995479i	$-0.469722\pi$
$373$	3.66863	0.189955	0.0949773	+	0.995479i	$0.469722\pi$
$374$	28.4752	1.47242
$375$	0	0
$376$	29.0361	1.49742
$377$	−25.2614	−1.30103
$378$	0	0
$379$	−9.89962	−0.508509	−0.254255	−	0.967137i	$-0.581830\pi$
$379$	−9.89962	−0.508509	−0.254255	+	0.967137i	$0.581830\pi$
$380$	0	0
$381$	0	0
$382$	16.1151	0.824519
$383$	−31.3148	−1.60011	−0.800055	−	0.599927i	$-0.795196\pi$
$383$	−31.3148	−1.60011	−0.800055	+	0.599927i	$0.795196\pi$
$384$	0	0
$385$	0	0
$386$	45.9143	2.33698
$387$	0	0
$388$	26.3074	1.33556
$389$	−7.39625	−0.375005	−0.187502	−	0.982264i	$-0.560039\pi$
$389$	−7.39625	−0.375005	−0.187502	+	0.982264i	$0.560039\pi$
$390$	0	0
$391$	−10.3748	−0.524678
$392$	26.8181	1.35452
$393$	0	0
$394$	−58.0294	−2.92348
$395$	0	0
$396$	0	0
$397$	−11.9326	−0.598880	−0.299440	−	0.954115i	$-0.596800\pi$
$397$	−11.9326	−0.598880	−0.299440	+	0.954115i	$0.596800\pi$
$398$	30.0147	1.50450
$399$	0	0
$400$	0	0
$401$	15.7648	0.787257	0.393628	−	0.919270i	$-0.371220\pi$
$401$	15.7648	0.787257	0.393628	+	0.919270i	$0.371220\pi$
$402$	0	0
$403$	−31.0263	−1.54553
$404$	−17.4721	−0.869268
$405$	0	0
$406$	72.0869	3.57761
$407$	−27.4005	−1.35819
$408$	0	0
$409$	−7.55356	−0.373499	−0.186750	−	0.982408i	$-0.559795\pi$
$409$	−7.55356	−0.373499	−0.186750	+	0.982408i	$0.559795\pi$
$410$	0	0
$411$	0	0
$412$	−4.24989	−0.209377
$413$	44.7005	2.19957
$414$	0	0
$415$	0	0
$416$	13.1365	0.644070
$417$	0	0
$418$	8.97858	0.439157
$419$	−9.59702	−0.468845	−0.234423	−	0.972135i	$-0.575320\pi$
$419$	−9.59702	−0.468845	−0.234423	+	0.972135i	$0.575320\pi$
$420$	0	0
$421$	12.3503	0.601915	0.300958	−	0.953638i	$-0.402694\pi$
$421$	12.3503	0.601915	0.300958	+	0.953638i	$0.402694\pi$
$422$	8.58546	0.417934
$423$	0	0
$424$	5.60688	0.272294
$425$	0	0
$426$	0	0
$427$	−16.0821	−0.778267
$428$	48.0294	2.32159
$429$	0	0
$430$	0	0
$431$	−5.14637	−0.247892	−0.123946	−	0.992289i	$-0.539555\pi$
$431$	−5.14637	−0.247892	−0.123946	+	0.992289i	$0.539555\pi$
$432$	0	0
$433$	−17.0894	−0.821266	−0.410633	−	0.911801i	$-0.634692\pi$
$433$	−17.0894	−0.821266	−0.410633	+	0.911801i	$0.634692\pi$
$434$	88.5376	4.24994
$435$	0	0
$436$	10.9786	0.525778
$437$	−3.27131	−0.156488
$438$	0	0
$439$	−22.3503	−1.06672	−0.533360	−	0.845888i	$-0.679071\pi$
$439$	−22.3503	−1.06672	−0.533360	+	0.845888i	$0.679071\pi$
$440$	0	0
$441$	0	0
$442$	−26.8009	−1.27479
$443$	23.4145	1.11246	0.556229	−	0.831029i	$-0.312248\pi$
$443$	23.4145	1.11246	0.556229	+	0.831029i	$0.312248\pi$
$444$	0	0
$445$	0	0
$446$	−21.4868	−1.01743
$447$	0	0
$448$	−46.6577	−2.20437
$449$	33.2713	1.57017	0.785085	−	0.619388i	$-0.212619\pi$
$449$	33.2713	1.57017	0.785085	+	0.619388i	$0.212619\pi$
$450$	0	0
$451$	−3.34292	−0.157412
$452$	−26.9786	−1.26897
$453$	0	0
$454$	0.618442	0.0290249
$455$	0	0
$456$	0	0
$457$	26.3931	1.23462	0.617309	−	0.786721i	$-0.288223\pi$
$457$	26.3931	1.23462	0.617309	+	0.786721i	$0.288223\pi$
$458$	16.8782	0.788666
$459$	0	0
$460$	0	0
$461$	−2.20077	−0.102500	−0.0512500	−	0.998686i	$-0.516321\pi$
$461$	−2.20077	−0.102500	−0.0512500	+	0.998686i	$0.516321\pi$
$462$	0	0
$463$	−24.5328	−1.14013	−0.570067	−	0.821598i	$-0.693083\pi$
$463$	−24.5328	−1.14013	−0.570067	+	0.821598i	$0.693083\pi$
$464$	9.60688	0.445988
$465$	0	0
$466$	37.8568	1.75368
$467$	−22.9357	−1.06134	−0.530670	−	0.847579i	$-0.678059\pi$
$467$	−22.9357	−1.06134	−0.530670	+	0.847579i	$0.678059\pi$
$468$	0	0
$469$	−23.3780	−1.07949
$470$	0	0
$471$	0	0
$472$	40.7005	1.87339
$473$	−39.3148	−1.80770
$474$	0	0
$475$	0	0
$476$	48.6148	2.22826
$477$	0	0
$478$	36.7005	1.67864
$479$	−41.6075	−1.90110	−0.950548	−	0.310579i	$-0.899477\pi$
$479$	−41.6075	−1.90110	−0.950548	+	0.310579i	$0.899477\pi$
$480$	0	0
$481$	25.7894	1.17589
$482$	24.7104	1.12553
$483$	0	0
$484$	0.611096	0.0277771
$485$	0	0
$486$	0	0
$487$	−40.4036	−1.83086	−0.915431	−	0.402475i	$-0.868150\pi$
$487$	−40.4036	−1.83086	−0.915431	+	0.402475i	$0.868150\pi$
$488$	−14.6430	−0.662857
$489$	0	0
$490$	0	0
$491$	16.6184	0.749980	0.374990	−	0.927029i	$-0.377646\pi$
$491$	16.6184	0.749980	0.374990	+	0.927029i	$0.377646\pi$
$492$	0	0
$493$	29.1898	1.31464
$494$	−8.45065	−0.380213
$495$	0	0
$496$	11.7992	0.529801
$497$	33.1758	1.48814
$498$	0	0
$499$	−0.921039	−0.0412314	−0.0206157	−	0.999787i	$-0.506563\pi$
$499$	−0.921039	−0.0412314	−0.0206157	+	0.999787i	$0.506563\pi$
$500$	0	0
$501$	0	0
$502$	−50.2070	−2.24085
$503$	18.9217	0.843675	0.421837	−	0.906671i	$-0.361385\pi$
$503$	18.9217	0.843675	0.421837	+	0.906671i	$0.361385\pi$
$504$	0	0
$505$	0	0
$506$	−22.3503	−0.993591
$507$	0	0
$508$	42.0722	1.86665
$509$	−27.8855	−1.23600	−0.618002	−	0.786176i	$-0.712058\pi$
$509$	−27.8855	−1.23600	−0.618002	+	0.786176i	$0.712058\pi$
$510$	0	0
$511$	38.9112	1.72133
$512$	−13.3461	−0.589818
$513$	0	0
$514$	−16.6529	−0.734526
$515$	0	0
$516$	0	0
$517$	27.8181	1.22344
$518$	−73.5934	−3.23351
$519$	0	0
$520$	0	0
$521$	−9.93681	−0.435339	−0.217670	−	0.976022i	$-0.569846\pi$
$521$	−9.93681	−0.435339	−0.217670	+	0.976022i	$0.569846\pi$
$522$	0	0
$523$	35.9718	1.57294	0.786470	−	0.617629i	$-0.211907\pi$
$523$	35.9718	1.57294	0.786470	+	0.617629i	$0.211907\pi$
$524$	−42.6577	−1.86351
$525$	0	0
$526$	−23.3618	−1.01862
$527$	35.8511	1.56170
$528$	0	0
$529$	−14.8568	−0.645947
$530$	0	0
$531$	0	0
$532$	15.3288	0.664590
$533$	3.14637	0.136284
$534$	0	0
$535$	0	0
$536$	−21.2860	−0.919415
$537$	0	0
$538$	1.60688	0.0692777
$539$	25.6932	1.10668
$540$	0	0
$541$	9.79923	0.421302	0.210651	−	0.977561i	$-0.432442\pi$
$541$	9.79923	0.421302	0.210651	+	0.977561i	$0.432442\pi$
$542$	32.3404	1.38914
$543$	0	0
$544$	−15.1793	−0.650809
$545$	0	0
$546$	0	0
$547$	33.8223	1.44614	0.723070	−	0.690775i	$-0.242731\pi$
$547$	33.8223	1.44614	0.723070	+	0.690775i	$0.242731\pi$
$548$	26.8438	1.14671
$549$	0	0
$550$	0	0
$551$	9.20390	0.392099
$552$	0	0
$553$	−13.8223	−0.587786
$554$	−52.0393	−2.21094
$555$	0	0
$556$	−62.8585	−2.66579
$557$	−20.9217	−0.886479	−0.443239	−	0.896403i	$-0.646171\pi$
$557$	−20.9217	−0.886479	−0.443239	+	0.896403i	$0.646171\pi$
$558$	0	0
$559$	37.0031	1.56507
$560$	0	0
$561$	0	0
$562$	−10.8108	−0.456026
$563$	−28.6289	−1.20657	−0.603283	−	0.797527i	$-0.706141\pi$
$563$	−28.6289	−1.20657	−0.603283	+	0.797527i	$0.706141\pi$
$564$	0	0
$565$	0	0
$566$	−21.3963	−0.899352
$567$	0	0
$568$	30.2070	1.26746
$569$	20.9603	0.878701	0.439351	−	0.898316i	$-0.355209\pi$
$569$	20.9603	0.878701	0.439351	+	0.898316i	$0.355209\pi$
$570$	0	0
$571$	−13.2860	−0.556002	−0.278001	−	0.960581i	$-0.589672\pi$
$571$	−13.2860	−0.556002	−0.278001	+	0.960581i	$0.589672\pi$
$572$	−36.7005	−1.53453
$573$	0	0
$574$	−8.97858	−0.374759
$575$	0	0
$576$	0	0
$577$	−1.35700	−0.0564926	−0.0282463	−	0.999601i	$-0.508992\pi$
$577$	−1.35700	−0.0564926	−0.0282463	+	0.999601i	$0.508992\pi$
$578$	−8.86098	−0.368568
$579$	0	0
$580$	0	0
$581$	−38.8009	−1.60973
$582$	0	0
$583$	5.37169	0.222473
$584$	35.4292	1.46607
$585$	0	0
$586$	54.9259	2.26897
$587$	29.9431	1.23588	0.617942	−	0.786224i	$-0.287967\pi$
$587$	29.9431	1.23588	0.617942	+	0.786224i	$0.287967\pi$
$588$	0	0
$589$	11.3043	0.465785
$590$	0	0
$591$	0	0
$592$	−9.80765	−0.403092
$593$	−6.01408	−0.246969	−0.123484	−	0.992347i	$-0.539407\pi$
$593$	−6.01408	−0.246969	−0.123484	+	0.992347i	$0.539407\pi$
$594$	0	0
$595$	0	0
$596$	−43.7648	−1.79268
$597$	0	0
$598$	21.0361	0.860231
$599$	−30.0477	−1.22771	−0.613857	−	0.789417i	$-0.710383\pi$
$599$	−30.0477	−1.22771	−0.613857	+	0.789417i	$0.710383\pi$
$600$	0	0
$601$	−38.1642	−1.55675	−0.778375	−	0.627800i	$-0.783956\pi$
$601$	−38.1642	−1.55675	−0.778375	+	0.627800i	$0.783956\pi$
$602$	−105.593	−4.30367
$603$	0	0
$604$	9.17092	0.373160
$605$	0	0
$606$	0	0
$607$	−35.7220	−1.44991	−0.724955	−	0.688796i	$-0.758139\pi$
$607$	−35.7220	−1.44991	−0.724955	+	0.688796i	$0.758139\pi$
$608$	−4.78623	−0.194107
$609$	0	0
$610$	0	0
$611$	−26.1825	−1.05923
$612$	0	0
$613$	8.68585	0.350818	0.175409	−	0.984496i	$-0.443875\pi$
$613$	8.68585	0.350818	0.175409	+	0.984496i	$0.443875\pi$
$614$	−42.7679	−1.72597
$615$	0	0
$616$	44.7005	1.80104
$617$	25.4391	1.02414	0.512070	−	0.858944i	$-0.328879\pi$
$617$	25.4391	1.02414	0.512070	+	0.858944i	$0.328879\pi$
$618$	0	0
$619$	4.28852	0.172370	0.0861851	−	0.996279i	$-0.472532\pi$
$619$	4.28852	0.172370	0.0861851	+	0.996279i	$0.472532\pi$
$620$	0	0
$621$	0	0
$622$	−34.8585	−1.39770
$623$	12.9210	0.517670
$624$	0	0
$625$	0	0
$626$	62.0294	2.47919
$627$	0	0
$628$	−70.3368	−2.80674
$629$	−29.7998	−1.18820
$630$	0	0
$631$	−3.17513	−0.126400	−0.0632001	−	0.998001i	$-0.520131\pi$
$631$	−3.17513	−0.126400	−0.0632001	+	0.998001i	$0.520131\pi$
$632$	−12.5855	−0.500623
$633$	0	0
$634$	69.4685	2.75895
$635$	0	0
$636$	0	0
$637$	−24.1825	−0.958145
$638$	62.8830	2.48956
$639$	0	0
$640$	0	0
$641$	21.0565	0.831680	0.415840	−	0.909438i	$-0.363488\pi$
$641$	21.0565	0.831680	0.415840	+	0.909438i	$0.363488\pi$
$642$	0	0
$643$	18.2070	0.718016	0.359008	−	0.933335i	$-0.383115\pi$
$643$	18.2070	0.718016	0.359008	+	0.933335i	$0.383115\pi$
$644$	−38.1579	−1.50363
$645$	0	0
$646$	9.76481	0.384191
$647$	−19.6890	−0.774054	−0.387027	−	0.922068i	$-0.626498\pi$
$647$	−19.6890	−0.774054	−0.387027	+	0.922068i	$0.626498\pi$
$648$	0	0
$649$	38.9933	1.53062
$650$	0	0
$651$	0	0
$652$	−63.6938	−2.49444
$653$	−2.80031	−0.109584	−0.0547922	−	0.998498i	$-0.517450\pi$
$653$	−2.80031	−0.109584	−0.0547922	+	0.998498i	$0.517450\pi$
$654$	0	0
$655$	0	0
$656$	−1.19656	−0.0467177
$657$	0	0
$658$	74.7152	2.91270
$659$	7.07896	0.275757	0.137879	−	0.990449i	$-0.455972\pi$
$659$	7.07896	0.275757	0.137879	+	0.990449i	$0.455972\pi$
$660$	0	0
$661$	−15.6791	−0.609847	−0.304923	−	0.952377i	$-0.598631\pi$
$661$	−15.6791	−0.609847	−0.304923	+	0.952377i	$0.598631\pi$
$662$	6.10038	0.237098
$663$	0	0
$664$	−35.3288	−1.37103
$665$	0	0
$666$	0	0
$667$	−22.9112	−0.887124
$668$	4.58546	0.177417
$669$	0	0
$670$	0	0
$671$	−14.0288	−0.541575
$672$	0	0
$673$	12.2070	0.470547	0.235273	−	0.971929i	$-0.424401\pi$
$673$	12.2070	0.470547	0.235273	+	0.971929i	$0.424401\pi$
$674$	3.53948	0.136336
$675$	0	0
$676$	−10.8181	−0.416082
$677$	−39.7121	−1.52626	−0.763130	−	0.646245i	$-0.776338\pi$
$677$	−39.7121	−1.52626	−0.763130	+	0.646245i	$0.776338\pi$
$678$	0	0
$679$	28.8929	1.10881
$680$	0	0
$681$	0	0
$682$	77.2333	2.95742
$683$	9.22846	0.353117	0.176559	−	0.984290i	$-0.443503\pi$
$683$	9.22846	0.353117	0.176559	+	0.984290i	$0.443503\pi$
$684$	0	0
$685$	0	0
$686$	6.15792	0.235111
$687$	0	0
$688$	−14.0722	−0.536499
$689$	−5.05585	−0.192612
$690$	0	0
$691$	−4.75325	−0.180822	−0.0904111	−	0.995905i	$-0.528818\pi$
$691$	−4.75325	−0.180822	−0.0904111	+	0.995905i	$0.528818\pi$
$692$	4.14950	0.157740
$693$	0	0
$694$	−33.4538	−1.26989
$695$	0	0
$696$	0	0
$697$	−3.63565	−0.137710
$698$	−38.0821	−1.44143
$699$	0	0
$700$	0	0
$701$	21.0790	0.796141	0.398071	−	0.917355i	$-0.369680\pi$
$701$	21.0790	0.796141	0.398071	+	0.917355i	$0.369680\pi$
$702$	0	0
$703$	−9.39625	−0.354386
$704$	−40.7005	−1.53396
$705$	0	0
$706$	26.1726	0.985019
$707$	−19.1892	−0.721685
$708$	0	0
$709$	42.7497	1.60550	0.802749	−	0.596318i	$-0.203370\pi$
$709$	42.7497	1.60550	0.802749	+	0.596318i	$0.203370\pi$
$710$	0	0
$711$	0	0
$712$	11.7648	0.440905
$713$	−28.1396	−1.05384
$714$	0	0
$715$	0	0
$716$	66.3221	2.47857
$717$	0	0
$718$	38.0147	1.41870
$719$	7.00735	0.261330	0.130665	−	0.991427i	$-0.458289\pi$
$719$	7.00735	0.261330	0.130665	+	0.991427i	$0.458289\pi$
$720$	0	0
$721$	−4.66756	−0.173829
$722$	−41.4366	−1.54211
$723$	0	0
$724$	36.2646	1.34776
$725$	0	0
$726$	0	0
$727$	33.3963	1.23860	0.619299	−	0.785155i	$-0.287417\pi$
$727$	33.3963	1.23860	0.619299	+	0.785155i	$0.287417\pi$
$728$	−42.0722	−1.55930
$729$	0	0
$730$	0	0
$731$	−42.7575	−1.58144
$732$	0	0
$733$	−13.1176	−0.484509	−0.242255	−	0.970213i	$-0.577887\pi$
$733$	−13.1176	−0.484509	−0.242255	+	0.970213i	$0.577887\pi$
$734$	−1.01156	−0.0373373
$735$	0	0
$736$	11.9143	0.439167
$737$	−20.3931	−0.751190
$738$	0	0
$739$	−47.2902	−1.73960	−0.869799	−	0.493406i	$-0.835752\pi$
$739$	−47.2902	−1.73960	−0.869799	+	0.493406i	$0.835752\pi$
$740$	0	0
$741$	0	0
$742$	14.4275	0.529652
$743$	−18.5510	−0.680572	−0.340286	−	0.940322i	$-0.610524\pi$
$743$	−18.5510	−0.680572	−0.340286	+	0.940322i	$0.610524\pi$
$744$	0	0
$745$	0	0
$746$	8.59533	0.314697
$747$	0	0
$748$	42.4078	1.55058
$749$	52.7497	1.92743
$750$	0	0
$751$	−13.6791	−0.499158	−0.249579	−	0.968354i	$-0.580292\pi$
$751$	−13.6791	−0.499158	−0.249579	+	0.968354i	$0.580292\pi$
$752$	9.95715	0.363100
$753$	0	0
$754$	−59.1856	−2.15541
$755$	0	0
$756$	0	0
$757$	23.0031	0.836063	0.418032	−	0.908432i	$-0.362720\pi$
$757$	23.0031	0.836063	0.418032	+	0.908432i	$0.362720\pi$
$758$	−23.1940	−0.842445
$759$	0	0
$760$	0	0
$761$	−20.4851	−0.742583	−0.371292	−	0.928516i	$-0.621085\pi$
$761$	−20.4851	−0.742583	−0.371292	+	0.928516i	$0.621085\pi$
$762$	0	0
$763$	12.0575	0.436512
$764$	24.0000	0.868290
$765$	0	0
$766$	−73.3681	−2.65090
$767$	−36.7005	−1.32518
$768$	0	0
$769$	−17.5107	−0.631452	−0.315726	−	0.948850i	$-0.602248\pi$
$769$	−17.5107	−0.631452	−0.315726	+	0.948850i	$0.602248\pi$
$770$	0	0
$771$	0	0
$772$	68.3797	2.46104
$773$	−31.8083	−1.14406	−0.572032	−	0.820231i	$-0.693845\pi$
$773$	−31.8083	−1.14406	−0.572032	+	0.820231i	$0.693845\pi$
$774$	0	0
$775$	0	0
$776$	26.3074	0.944381
$777$	0	0
$778$	−17.3288	−0.621269
$779$	−1.14637	−0.0410728
$780$	0	0
$781$	28.9399	1.03555
$782$	−24.3074	−0.869232
$783$	0	0
$784$	9.19656	0.328448
$785$	0	0
$786$	0	0
$787$	−14.5040	−0.517011	−0.258506	−	0.966010i	$-0.583230\pi$
$787$	−14.5040	−0.517011	−0.258506	+	0.966010i	$0.583230\pi$
$788$	−86.4225	−3.07867
$789$	0	0
$790$	0	0
$791$	−29.6300	−1.05352
$792$	0	0
$793$	13.2039	0.468884
$794$	−27.9572	−0.992162
$795$	0	0
$796$	44.7005	1.58437
$797$	50.3650	1.78402	0.892009	−	0.452017i	$-0.149295\pi$
$797$	50.3650	1.78402	0.892009	+	0.452017i	$0.149295\pi$
$798$	0	0
$799$	30.2541	1.07031
$800$	0	0
$801$	0	0
$802$	36.9357	1.30425
$803$	33.9431	1.19783
$804$	0	0
$805$	0	0
$806$	−72.6921	−2.56047
$807$	0	0
$808$	−17.4721	−0.614666
$809$	−35.2369	−1.23886	−0.619431	−	0.785051i	$-0.712637\pi$
$809$	−35.2369	−1.23886	−0.619431	+	0.785051i	$0.712637\pi$
$810$	0	0
$811$	−13.8610	−0.486725	−0.243362	−	0.969935i	$-0.578250\pi$
$811$	−13.8610	−0.486725	−0.243362	+	0.969935i	$0.578250\pi$
$812$	107.358	3.76754
$813$	0	0
$814$	−64.1972	−2.25011
$815$	0	0
$816$	0	0
$817$	−13.4819	−0.471673
$818$	−17.6974	−0.618775
$819$	0	0
$820$	0	0
$821$	−7.20390	−0.251418	−0.125709	−	0.992067i	$-0.540121\pi$
$821$	−7.20390	−0.251418	−0.125709	+	0.992067i	$0.540121\pi$
$822$	0	0
$823$	9.81392	0.342092	0.171046	−	0.985263i	$-0.445285\pi$
$823$	9.81392	0.342092	0.171046	+	0.985263i	$0.445285\pi$
$824$	−4.24989	−0.148052
$825$	0	0
$826$	104.730	3.64402
$827$	24.8788	0.865121	0.432560	−	0.901605i	$-0.357610\pi$
$827$	24.8788	0.865121	0.432560	+	0.901605i	$0.357610\pi$
$828$	0	0
$829$	13.3759	0.464564	0.232282	−	0.972648i	$-0.425381\pi$
$829$	13.3759	0.464564	0.232282	+	0.972648i	$0.425381\pi$
$830$	0	0
$831$	0	0
$832$	38.3074	1.32807
$833$	27.9431	0.968170
$834$	0	0
$835$	0	0
$836$	13.3717	0.462470
$837$	0	0
$838$	−22.4851	−0.776734
$839$	12.9210	0.446084	0.223042	−	0.974809i	$-0.428401\pi$
$839$	12.9210	0.446084	0.223042	+	0.974809i	$0.428401\pi$
$840$	0	0
$841$	35.4611	1.22280
$842$	28.9357	0.997191
$843$	0	0
$844$	12.7862	0.440120
$845$	0	0
$846$	0	0
$847$	0.671153	0.0230611
$848$	1.92273	0.0660268
$849$	0	0
$850$	0	0
$851$	23.3900	0.801798
$852$	0	0
$853$	−16.4851	−0.564438	−0.282219	−	0.959350i	$-0.591071\pi$
$853$	−16.4851	−0.564438	−0.282219	+	0.959350i	$0.591071\pi$
$854$	−37.6791	−1.28935
$855$	0	0
$856$	48.0294	1.64161
$857$	−11.5149	−0.393342	−0.196671	−	0.980470i	$-0.563013\pi$
$857$	−11.5149	−0.393342	−0.196671	+	0.980470i	$0.563013\pi$
$858$	0	0
$859$	−28.7392	−0.980568	−0.490284	−	0.871563i	$-0.663107\pi$
$859$	−28.7392	−0.980568	−0.490284	+	0.871563i	$0.663107\pi$
$860$	0	0
$861$	0	0
$862$	−12.0575	−0.410681
$863$	51.1611	1.74154	0.870771	−	0.491688i	$-0.163620\pi$
$863$	51.1611	1.74154	0.870771	+	0.491688i	$0.163620\pi$
$864$	0	0
$865$	0	0
$866$	−40.0393	−1.36059
$867$	0	0
$868$	131.858	4.47555
$869$	−12.0575	−0.409024
$870$	0	0
$871$	19.1940	0.650365
$872$	10.9786	0.371782
$873$	0	0
$874$	−7.66442	−0.259253
$875$	0	0
$876$	0	0
$877$	−34.3179	−1.15883	−0.579417	−	0.815031i	$-0.696720\pi$
$877$	−34.3179	−1.15883	−0.579417	+	0.815031i	$0.696720\pi$
$878$	−52.3650	−1.76723
$879$	0	0
$880$	0	0
$881$	38.9786	1.31322	0.656611	−	0.754230i	$-0.271989\pi$
$881$	38.9786	1.31322	0.656611	+	0.754230i	$0.271989\pi$
$882$	0	0
$883$	28.2253	0.949858	0.474929	−	0.880024i	$-0.342474\pi$
$883$	28.2253	0.949858	0.474929	+	0.880024i	$0.342474\pi$
$884$	−39.9143	−1.34246
$885$	0	0
$886$	54.8585	1.84301
$887$	−30.1438	−1.01213	−0.506066	−	0.862495i	$-0.668901\pi$
$887$	−30.1438	−1.01213	−0.506066	+	0.862495i	$0.668901\pi$
$888$	0	0
$889$	46.2070	1.54973
$890$	0	0
$891$	0	0
$892$	−32.0000	−1.07144
$893$	9.53948	0.319227
$894$	0	0
$895$	0	0
$896$	−77.3154	−2.58293
$897$	0	0
$898$	77.9521	2.60130
$899$	79.1715	2.64052
$900$	0	0
$901$	5.84208	0.194628
$902$	−7.83221	−0.260784
$903$	0	0
$904$	−26.9786	−0.897294
$905$	0	0
$906$	0	0
$907$	−41.3435	−1.37279	−0.686395	−	0.727229i	$-0.740808\pi$
$907$	−41.3435	−1.37279	−0.686395	+	0.727229i	$0.740808\pi$
$908$	0.921039	0.0305657
$909$	0	0
$910$	0	0
$911$	1.23206	0.0408199	0.0204099	−	0.999792i	$-0.493503\pi$
$911$	1.23206	0.0408199	0.0204099	+	0.999792i	$0.493503\pi$
$912$	0	0
$913$	−33.8469	−1.12017
$914$	61.8370	2.04539
$915$	0	0
$916$	25.1365	0.830533
$917$	−46.8500	−1.54712
$918$	0	0
$919$	−21.4047	−0.706075	−0.353037	−	0.935609i	$-0.614851\pi$
$919$	−21.4047	−0.706075	−0.353037	+	0.935609i	$0.614851\pi$
$920$	0	0
$921$	0	0
$922$	−5.15623	−0.169811
$923$	−27.2383	−0.896560
$924$	0	0
$925$	0	0
$926$	−57.4783	−1.88886
$927$	0	0
$928$	−33.5212	−1.10039
$929$	−32.8150	−1.07663	−0.538313	−	0.842745i	$-0.680938\pi$
$929$	−32.8150	−1.07663	−0.538313	+	0.842745i	$0.680938\pi$
$930$	0	0
$931$	8.81079	0.288762
$932$	56.3797	1.84678
$933$	0	0
$934$	−53.7367	−1.75832
$935$	0	0
$936$	0	0
$937$	−27.4783	−0.897678	−0.448839	−	0.893613i	$-0.648162\pi$
$937$	−27.4783	−0.897678	−0.448839	+	0.893613i	$0.648162\pi$
$938$	−54.7728	−1.78839
$939$	0	0
$940$	0	0
$941$	−38.1151	−1.24252	−0.621258	−	0.783606i	$-0.713378\pi$
$941$	−38.1151	−1.24252	−0.621258	+	0.783606i	$0.713378\pi$
$942$	0	0
$943$	2.85363	0.0929271
$944$	13.9572	0.454267
$945$	0	0
$946$	−92.1115	−2.99480
$947$	−9.85050	−0.320098	−0.160049	−	0.987109i	$-0.551165\pi$
$947$	−9.85050	−0.320098	−0.160049	+	0.987109i	$0.551165\pi$
$948$	0	0
$949$	−31.9473	−1.03705
$950$	0	0
$951$	0	0
$952$	48.6148	1.57562
$953$	3.92417	0.127116	0.0635582	−	0.997978i	$-0.479755\pi$
$953$	3.92417	0.127116	0.0635582	+	0.997978i	$0.479755\pi$
$954$	0	0
$955$	0	0
$956$	54.6577	1.76776
$957$	0	0
$958$	−97.4832	−3.14954
$959$	29.4819	0.952022
$960$	0	0
$961$	66.2389	2.13674
$962$	60.4225	1.94810
$963$	0	0
$964$	36.8009	1.18528
$965$	0	0
$966$	0	0
$967$	40.3074	1.29620	0.648100	−	0.761556i	$-0.275564\pi$
$967$	40.3074	1.29620	0.648100	+	0.761556i	$0.275564\pi$
$968$	0.611096	0.0196414
$969$	0	0
$970$	0	0
$971$	−37.8280	−1.21396	−0.606979	−	0.794718i	$-0.707619\pi$
$971$	−37.8280	−1.21396	−0.606979	+	0.794718i	$0.707619\pi$
$972$	0	0
$973$	−69.0361	−2.21320
$974$	−94.6625	−3.03318
$975$	0	0
$976$	−5.02142	−0.160732
$977$	8.89227	0.284489	0.142244	−	0.989832i	$-0.454568\pi$
$977$	8.89227	0.284489	0.142244	+	0.989832i	$0.454568\pi$
$978$	0	0
$979$	11.2713	0.360233
$980$	0	0
$981$	0	0
$982$	38.9357	1.24249
$983$	−30.6247	−0.976777	−0.488388	−	0.872626i	$-0.662415\pi$
$983$	−30.6247	−0.976777	−0.488388	+	0.872626i	$0.662415\pi$
$984$	0	0
$985$	0	0
$986$	68.3895	2.17797
$987$	0	0
$988$	−12.5855	−0.400397
$989$	33.5604	1.06716
$990$	0	0
$991$	−53.1512	−1.68840	−0.844202	−	0.536026i	$-0.819925\pi$
$991$	−53.1512	−1.68840	−0.844202	+	0.536026i	$0.819925\pi$
$992$	−41.1709	−1.30718
$993$	0	0
$994$	77.7282	2.46539
$995$	0	0
$996$	0	0
$997$	44.5181	1.40990	0.704951	−	0.709256i	$-0.250969\pi$
$997$	44.5181	1.40990	0.704951	+	0.709256i	$0.250969\pi$
$998$	−2.15792	−0.0683078
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9225.2.a.bx.1.3		3
3.2	odd	2		3075.2.a.t.1.1		3
5.4	even	2		369.2.a.e.1.1		3
15.14	odd	2		123.2.a.d.1.3	✓	3
20.19	odd	2		5904.2.a.bd.1.2		3
60.59	even	2		1968.2.a.w.1.2		3
105.104	even	2		6027.2.a.s.1.3		3
120.29	odd	2		7872.2.a.bx.1.2		3
120.59	even	2		7872.2.a.bs.1.2		3
615.614	odd	2		5043.2.a.n.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
123.2.a.d.1.3	✓	3	15.14	odd	2
369.2.a.e.1.1		3	5.4	even	2
1968.2.a.w.1.2		3	60.59	even	2
3075.2.a.t.1.1		3	3.2	odd	2
5043.2.a.n.1.3		3	615.614	odd	2
5904.2.a.bd.1.2		3	20.19	odd	2
6027.2.a.s.1.3		3	105.104	even	2
7872.2.a.bs.1.2		3	120.59	even	2
7872.2.a.bx.1.2		3	120.29	odd	2
9225.2.a.bx.1.3		3	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:	\( N \)	\(=\)	\( 9225 = 3^{2} \cdot 5^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9225.a (trivial)

\(f(q)\)	\(=\)	\(q+2.34292 q^{2} +3.48929 q^{4} +3.83221 q^{7} +3.48929 q^{8} +O(q^{10})\)	\(q+2.34292 q^{2} +3.48929 q^{4} +3.83221 q^{7} +3.48929 q^{8} +3.34292 q^{11} -3.14637 q^{13} +8.97858 q^{14} +1.19656 q^{16} +3.63565 q^{17} +1.14637 q^{19} +7.83221 q^{22} -2.85363 q^{23} -7.37169 q^{26} +13.3717 q^{28} +8.02877 q^{29} +9.86098 q^{31} -4.17513 q^{32} +8.51806 q^{34} -8.19656 q^{37} +2.68585 q^{38} -1.00000 q^{41} -11.7606 q^{43} +11.6644 q^{44} -6.68585 q^{46} +8.32150 q^{47} +7.68585 q^{49} -10.9786 q^{52} +1.60688 q^{53} +13.3717 q^{56} +18.8108 q^{58} +11.6644 q^{59} -4.19656 q^{61} +23.1035 q^{62} -12.1751 q^{64} -6.10038 q^{67} +12.6858 q^{68} +8.65708 q^{71} +10.1537 q^{73} -19.2039 q^{74} +4.00000 q^{76} +12.8108 q^{77} -3.60688 q^{79} -2.34292 q^{82} -10.1249 q^{83} -27.5542 q^{86} +11.6644 q^{88} +3.37169 q^{89} -12.0575 q^{91} -9.95715 q^{92} +19.4966 q^{94} +7.53948 q^{97} +18.0073 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} + 3 q^{4} - 2 q^{7} + 3 q^{8}+O(q^{10}) \) 3 * q + q^2 + 3 * q^4 - 2 * q^7 + 3 * q^8	\( 3 q + q^{2} + 3 q^{4} - 2 q^{7} + 3 q^{8} + 4 q^{11} - 8 q^{13} + 12 q^{14} - q^{16} + 2 q^{17} + 2 q^{19} + 10 q^{22} - 10 q^{23} + 2 q^{26} + 16 q^{28} + 6 q^{29} - 2 q^{31} + 7 q^{32} - 20 q^{37} - 4 q^{38} - 3 q^{41} - 10 q^{43} + 8 q^{44} - 8 q^{46} + 4 q^{47} + 11 q^{49} - 18 q^{52} + 14 q^{53} + 16 q^{56} + 28 q^{58} + 8 q^{59} - 8 q^{61} + 38 q^{62} - 17 q^{64} - 12 q^{67} + 26 q^{68} + 32 q^{71} - 4 q^{73} - 20 q^{74} + 12 q^{76} + 10 q^{77} - 20 q^{79} - q^{82} - 14 q^{83} - 6 q^{86} + 8 q^{88} - 14 q^{89} + 18 q^{94} + 12 q^{97} + 21 q^{98}+O(q^{100}) \) 3 * q + q^2 + 3 * q^4 - 2 * q^7 + 3 * q^8 + 4 * q^11 - 8 * q^13 + 12 * q^14 - q^16 + 2 * q^17 + 2 * q^19 + 10 * q^22 - 10 * q^23 + 2 * q^26 + 16 * q^28 + 6 * q^29 - 2 * q^31 + 7 * q^32 - 20 * q^37 - 4 * q^38 - 3 * q^41 - 10 * q^43 + 8 * q^44 - 8 * q^46 + 4 * q^47 + 11 * q^49 - 18 * q^52 + 14 * q^53 + 16 * q^56 + 28 * q^58 + 8 * q^59 - 8 * q^61 + 38 * q^62 - 17 * q^64 - 12 * q^67 + 26 * q^68 + 32 * q^71 - 4 * q^73 - 20 * q^74 + 12 * q^76 + 10 * q^77 - 20 * q^79 - q^82 - 14 * q^83 - 6 * q^86 + 8 * q^88 - 14 * q^89 + 18 * q^94 + 12 * q^97 + 21 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more