LMFDB - Embedded newform 4851.2.a.bg.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4851,2,Mod(1,4851)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4851, 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("4851.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4851 = 3^{2} \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4851.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:	$38.7354300205$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1617)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	4851.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.61803 q^{2} +4.85410 q^{4} -2.23607 q^{5} +7.47214 q^{8} +O(q^{10})$	$q+2.61803 q^{2} +4.85410 q^{4} -2.23607 q^{5} +7.47214 q^{8} -5.85410 q^{10} +1.00000 q^{11} -3.47214 q^{13} +9.85410 q^{16} +6.00000 q^{17} +4.23607 q^{19} -10.8541 q^{20} +2.61803 q^{22} +2.47214 q^{23} -9.09017 q^{26} +10.2361 q^{29} -8.47214 q^{31} +10.8541 q^{32} +15.7082 q^{34} -0.527864 q^{37} +11.0902 q^{38} -16.7082 q^{40} +6.00000 q^{41} -0.472136 q^{43} +4.85410 q^{44} +6.47214 q^{46} +11.4721 q^{47} -16.8541 q^{52} +12.9443 q^{53} -2.23607 q^{55} +26.7984 q^{58} -11.9443 q^{59} +6.94427 q^{61} -22.1803 q^{62} +8.70820 q^{64} +7.76393 q^{65} +0.708204 q^{67} +29.1246 q^{68} -4.47214 q^{71} +1.00000 q^{73} -1.38197 q^{74} +20.5623 q^{76} -2.47214 q^{79} -22.0344 q^{80} +15.7082 q^{82} +12.4721 q^{83} -13.4164 q^{85} -1.23607 q^{86} +7.47214 q^{88} -13.4164 q^{89} +12.0000 q^{92} +30.0344 q^{94} -9.47214 q^{95} +6.94427 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{2} + 3 q^{4} + 6 q^{8}+O(q^{10}) $ 2 * q + 3 * q^2 + 3 * q^4 + 6 * q^8	$ 2 q + 3 q^{2} + 3 q^{4} + 6 q^{8} - 5 q^{10} + 2 q^{11} + 2 q^{13} + 13 q^{16} + 12 q^{17} + 4 q^{19} - 15 q^{20} + 3 q^{22} - 4 q^{23} - 7 q^{26} + 16 q^{29} - 8 q^{31} + 15 q^{32} + 18 q^{34} - 10 q^{37} + 11 q^{38} - 20 q^{40} + 12 q^{41} + 8 q^{43} + 3 q^{44} + 4 q^{46} + 14 q^{47} - 27 q^{52} + 8 q^{53} + 29 q^{58} - 6 q^{59} - 4 q^{61} - 22 q^{62} + 4 q^{64} + 20 q^{65} - 12 q^{67} + 18 q^{68} + 2 q^{73} - 5 q^{74} + 21 q^{76} + 4 q^{79} - 15 q^{80} + 18 q^{82} + 16 q^{83} + 2 q^{86} + 6 q^{88} + 24 q^{92} + 31 q^{94} - 10 q^{95} - 4 q^{97}+O(q^{100}) $ 2 * q + 3 * q^2 + 3 * q^4 + 6 * q^8 - 5 * q^10 + 2 * q^11 + 2 * q^13 + 13 * q^16 + 12 * q^17 + 4 * q^19 - 15 * q^20 + 3 * q^22 - 4 * q^23 - 7 * q^26 + 16 * q^29 - 8 * q^31 + 15 * q^32 + 18 * q^34 - 10 * q^37 + 11 * q^38 - 20 * q^40 + 12 * q^41 + 8 * q^43 + 3 * q^44 + 4 * q^46 + 14 * q^47 - 27 * q^52 + 8 * q^53 + 29 * q^58 - 6 * q^59 - 4 * q^61 - 22 * q^62 + 4 * q^64 + 20 * q^65 - 12 * q^67 + 18 * q^68 + 2 * q^73 - 5 * q^74 + 21 * q^76 + 4 * q^79 - 15 * q^80 + 18 * q^82 + 16 * q^83 + 2 * q^86 + 6 * q^88 + 24 * q^92 + 31 * q^94 - 10 * q^95 - 4 * q^97

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 $

$2$	2.61803	1.85123	0.925615	−	0.378467i	$-0.123549\pi$
$2$	2.61803	1.85123	0.925615	+	0.378467i	$0.123549\pi$
$3$	0	0
$3$	0	0
$4$	4.85410	2.42705
$5$	−2.23607	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$5$	−2.23607	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	7.47214	2.64180
$9$	0	0
$10$	−5.85410	−1.85123
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−3.47214	−0.962997	−0.481499	−	0.876447i	$-0.659907\pi$
$13$	−3.47214	−0.962997	−0.481499	+	0.876447i	$0.659907\pi$
$14$	0	0
$15$	0	0
$16$	9.85410	2.46353
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	0	0
$19$	4.23607	0.971821	0.485910	−	0.874009i	$-0.338488\pi$
$19$	4.23607	0.971821	0.485910	+	0.874009i	$0.338488\pi$
$20$	−10.8541	−2.42705
$21$	0	0
$22$	2.61803	0.558167
$23$	2.47214	0.515476	0.257738	−	0.966215i	$-0.417023\pi$
$23$	2.47214	0.515476	0.257738	+	0.966215i	$0.417023\pi$
$24$	0	0
$25$	0	0
$26$	−9.09017	−1.78273
$27$	0	0
$28$	0	0
$29$	10.2361	1.90079	0.950395	−	0.311045i	$-0.100679\pi$
$29$	10.2361	1.90079	0.950395	+	0.311045i	$0.100679\pi$
$30$	0	0
$31$	−8.47214	−1.52164	−0.760820	−	0.648963i	$-0.775203\pi$
$31$	−8.47214	−1.52164	−0.760820	+	0.648963i	$0.775203\pi$
$32$	10.8541	1.91875
$33$	0	0
$34$	15.7082	2.69393
$35$	0	0
$36$	0	0
$37$	−0.527864	−0.0867803	−0.0433902	−	0.999058i	$-0.513816\pi$
$37$	−0.527864	−0.0867803	−0.0433902	+	0.999058i	$0.513816\pi$
$38$	11.0902	1.79906
$39$	0	0
$40$	−16.7082	−2.64180
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	−0.472136	−0.0720001	−0.0360000	−	0.999352i	$-0.511462\pi$
$43$	−0.472136	−0.0720001	−0.0360000	+	0.999352i	$0.511462\pi$
$44$	4.85410	0.731783
$45$	0	0
$46$	6.47214	0.954264
$47$	11.4721	1.67338	0.836692	−	0.547674i	$-0.184487\pi$
$47$	11.4721	1.67338	0.836692	+	0.547674i	$0.184487\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	−16.8541	−2.33724
$53$	12.9443	1.77803	0.889016	−	0.457876i	$-0.151390\pi$
$53$	12.9443	1.77803	0.889016	+	0.457876i	$0.151390\pi$
$54$	0	0
$55$	−2.23607	−0.301511
$56$	0	0
$57$	0	0
$58$	26.7984	3.51880
$59$	−11.9443	−1.55501	−0.777506	−	0.628876i	$-0.783515\pi$
$59$	−11.9443	−1.55501	−0.777506	+	0.628876i	$0.783515\pi$
$60$	0	0
$61$	6.94427	0.889123	0.444561	−	0.895748i	$-0.353360\pi$
$61$	6.94427	0.889123	0.444561	+	0.895748i	$0.353360\pi$
$62$	−22.1803	−2.81691
$63$	0	0
$64$	8.70820	1.08853
$65$	7.76393	0.962997
$66$	0	0
$67$	0.708204	0.0865209	0.0432604	−	0.999064i	$-0.486225\pi$
$67$	0.708204	0.0865209	0.0432604	+	0.999064i	$0.486225\pi$
$68$	29.1246	3.53188
$69$	0	0
$70$	0	0
$71$	−4.47214	−0.530745	−0.265372	−	0.964146i	$-0.585495\pi$
$71$	−4.47214	−0.530745	−0.265372	+	0.964146i	$0.585495\pi$
$72$	0	0
$73$	1.00000	0.117041	0.0585206	−	0.998286i	$-0.481362\pi$
$73$	1.00000	0.117041	0.0585206	+	0.998286i	$0.481362\pi$
$74$	−1.38197	−0.160650
$75$	0	0
$76$	20.5623	2.35866
$77$	0	0
$78$	0	0
$79$	−2.47214	−0.278137	−0.139069	−	0.990283i	$-0.544411\pi$
$79$	−2.47214	−0.278137	−0.139069	+	0.990283i	$0.544411\pi$
$80$	−22.0344	−2.46353
$81$	0	0
$82$	15.7082	1.73468
$83$	12.4721	1.36899	0.684497	−	0.729015i	$-0.260022\pi$
$83$	12.4721	1.36899	0.684497	+	0.729015i	$0.260022\pi$
$84$	0	0
$85$	−13.4164	−1.45521
$86$	−1.23607	−0.133289
$87$	0	0
$88$	7.47214	0.796532
$89$	−13.4164	−1.42214	−0.711068	−	0.703123i	$-0.751788\pi$
$89$	−13.4164	−1.42214	−0.711068	+	0.703123i	$0.751788\pi$
$90$	0	0
$91$	0	0
$92$	12.0000	1.25109
$93$	0	0
$94$	30.0344	3.09782
$95$	−9.47214	−0.971821
$96$	0	0
$97$	6.94427	0.705084	0.352542	−	0.935796i	$-0.385317\pi$
$97$	6.94427	0.705084	0.352542	+	0.935796i	$0.385317\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	3.52786	0.351036	0.175518	−	0.984476i	$-0.443840\pi$
$101$	3.52786	0.351036	0.175518	+	0.984476i	$0.443840\pi$
$102$	0	0
$103$	7.41641	0.730760	0.365380	−	0.930858i	$-0.380939\pi$
$103$	7.41641	0.730760	0.365380	+	0.930858i	$0.380939\pi$
$104$	−25.9443	−2.54405
$105$	0	0
$106$	33.8885	3.29155
$107$	−19.9443	−1.92809	−0.964043	−	0.265747i	$-0.914381\pi$
$107$	−19.9443	−1.92809	−0.964043	+	0.265747i	$0.914381\pi$
$108$	0	0
$109$	6.00000	0.574696	0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	−5.85410	−0.558167
$111$	0	0
$112$	0	0
$113$	−6.47214	−0.608847	−0.304424	−	0.952537i	$-0.598464\pi$
$113$	−6.47214	−0.608847	−0.304424	+	0.952537i	$0.598464\pi$
$114$	0	0
$115$	−5.52786	−0.515476
$116$	49.6869	4.61331
$117$	0	0
$118$	−31.2705	−2.87868
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	18.1803	1.64597
$123$	0	0
$124$	−41.1246	−3.69310
$125$	11.1803	1.00000
$126$	0	0
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	1.09017	0.0963583
$129$	0	0
$130$	20.3262	1.78273
$131$	−4.47214	−0.390732	−0.195366	−	0.980730i	$-0.562590\pi$
$131$	−4.47214	−0.390732	−0.195366	+	0.980730i	$0.562590\pi$
$132$	0	0
$133$	0	0
$134$	1.85410	0.160170
$135$	0	0
$136$	44.8328	3.84438
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	0	0
$141$	0	0
$142$	−11.7082	−0.982531
$143$	−3.47214	−0.290355
$144$	0	0
$145$	−22.8885	−1.90079
$146$	2.61803	0.216670
$147$	0	0
$148$	−2.56231	−0.210620
$149$	−15.6525	−1.28230	−0.641150	−	0.767415i	$-0.721543\pi$
$149$	−15.6525	−1.28230	−0.641150	+	0.767415i	$0.721543\pi$
$150$	0	0
$151$	−6.94427	−0.565117	−0.282558	−	0.959250i	$-0.591183\pi$
$151$	−6.94427	−0.565117	−0.282558	+	0.959250i	$0.591183\pi$
$152$	31.6525	2.56735
$153$	0	0
$154$	0	0
$155$	18.9443	1.52164
$156$	0	0
$157$	−3.41641	−0.272659	−0.136330	−	0.990664i	$-0.543531\pi$
$157$	−3.41641	−0.272659	−0.136330	+	0.990664i	$0.543531\pi$
$158$	−6.47214	−0.514895
$159$	0	0
$160$	−24.2705	−1.91875
$161$	0	0
$162$	0	0
$163$	−18.1246	−1.41963	−0.709815	−	0.704389i	$-0.751221\pi$
$163$	−18.1246	−1.41963	−0.709815	+	0.704389i	$0.751221\pi$
$164$	29.1246	2.27425
$165$	0	0
$166$	32.6525	2.53432
$167$	18.4721	1.42942	0.714708	−	0.699423i	$-0.246559\pi$
$167$	18.4721	1.42942	0.714708	+	0.699423i	$0.246559\pi$
$168$	0	0
$169$	−0.944272	−0.0726363
$170$	−35.1246	−2.69393
$171$	0	0
$172$	−2.29180	−0.174748
$173$	5.52786	0.420276	0.210138	−	0.977672i	$-0.432609\pi$
$173$	5.52786	0.420276	0.210138	+	0.977672i	$0.432609\pi$
$174$	0	0
$175$	0	0
$176$	9.85410	0.742781
$177$	0	0
$178$	−35.1246	−2.63270
$179$	5.52786	0.413172	0.206586	−	0.978428i	$-0.433765\pi$
$179$	5.52786	0.413172	0.206586	+	0.978428i	$0.433765\pi$
$180$	0	0
$181$	−0.472136	−0.0350936	−0.0175468	−	0.999846i	$-0.505586\pi$
$181$	−0.472136	−0.0350936	−0.0175468	+	0.999846i	$0.505586\pi$
$182$	0	0
$183$	0	0
$184$	18.4721	1.36178
$185$	1.18034	0.0867803
$186$	0	0
$187$	6.00000	0.438763
$188$	55.6869	4.06139
$189$	0	0
$190$	−24.7984	−1.79906
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	0	0
$193$	25.8885	1.86350	0.931749	−	0.363103i	$-0.118283\pi$
$193$	25.8885	1.86350	0.931749	+	0.363103i	$0.118283\pi$
$194$	18.1803	1.30527
$195$	0	0
$196$	0	0
$197$	8.47214	0.603615	0.301807	−	0.953369i	$-0.402410\pi$
$197$	8.47214	0.603615	0.301807	+	0.953369i	$0.402410\pi$
$198$	0	0
$199$	−24.4721	−1.73478	−0.867392	−	0.497626i	$-0.834205\pi$
$199$	−24.4721	−1.73478	−0.867392	+	0.497626i	$0.834205\pi$
$200$	0	0
$201$	0	0
$202$	9.23607	0.649847
$203$	0	0
$204$	0	0
$205$	−13.4164	−0.937043
$206$	19.4164	1.35281
$207$	0	0
$208$	−34.2148	−2.37237
$209$	4.23607	0.293015
$210$	0	0
$211$	18.9443	1.30418	0.652089	−	0.758143i	$-0.273893\pi$
$211$	18.9443	1.30418	0.652089	+	0.758143i	$0.273893\pi$
$212$	62.8328	4.31538
$213$	0	0
$214$	−52.2148	−3.56933
$215$	1.05573	0.0720001
$216$	0	0
$217$	0	0
$218$	15.7082	1.06389
$219$	0	0
$220$	−10.8541	−0.731783
$221$	−20.8328	−1.40137
$222$	0	0
$223$	−18.0000	−1.20537	−0.602685	−	0.797980i	$-0.705902\pi$
$223$	−18.0000	−1.20537	−0.602685	+	0.797980i	$0.705902\pi$
$224$	0	0
$225$	0	0
$226$	−16.9443	−1.12712
$227$	26.8328	1.78096	0.890478	−	0.455026i	$-0.150370\pi$
$227$	26.8328	1.78096	0.890478	+	0.455026i	$0.150370\pi$
$228$	0	0
$229$	−15.4164	−1.01874	−0.509372	−	0.860546i	$-0.670122\pi$
$229$	−15.4164	−1.01874	−0.509372	+	0.860546i	$0.670122\pi$
$230$	−14.4721	−0.954264
$231$	0	0
$232$	76.4853	5.02151
$233$	−13.4164	−0.878938	−0.439469	−	0.898258i	$-0.644833\pi$
$233$	−13.4164	−0.878938	−0.439469	+	0.898258i	$0.644833\pi$
$234$	0	0
$235$	−25.6525	−1.67338
$236$	−57.9787	−3.77409
$237$	0	0
$238$	0	0
$239$	−17.3607	−1.12297	−0.561485	−	0.827487i	$-0.689770\pi$
$239$	−17.3607	−1.12297	−0.561485	+	0.827487i	$0.689770\pi$
$240$	0	0
$241$	19.0000	1.22390	0.611949	−	0.790897i	$-0.290386\pi$
$241$	19.0000	1.22390	0.611949	+	0.790897i	$0.290386\pi$
$242$	2.61803	0.168294
$243$	0	0
$244$	33.7082	2.15795
$245$	0	0
$246$	0	0
$247$	−14.7082	−0.935861
$248$	−63.3050	−4.01987
$249$	0	0
$250$	29.2705	1.85123
$251$	3.94427	0.248960	0.124480	−	0.992222i	$-0.460274\pi$
$251$	3.94427	0.248960	0.124480	+	0.992222i	$0.460274\pi$
$252$	0	0
$253$	2.47214	0.155422
$254$	41.8885	2.62832
$255$	0	0
$256$	−14.5623	−0.910144
$257$	−7.76393	−0.484301	−0.242150	−	0.970239i	$-0.577853\pi$
$257$	−7.76393	−0.484301	−0.242150	+	0.970239i	$0.577853\pi$
$258$	0	0
$259$	0	0
$260$	37.6869	2.33724
$261$	0	0
$262$	−11.7082	−0.723335
$263$	4.41641	0.272327	0.136164	−	0.990686i	$-0.456523\pi$
$263$	4.41641	0.272327	0.136164	+	0.990686i	$0.456523\pi$
$264$	0	0
$265$	−28.9443	−1.77803
$266$	0	0
$267$	0	0
$268$	3.43769	0.209991
$269$	−16.4721	−1.00432	−0.502162	−	0.864774i	$-0.667462\pi$
$269$	−16.4721	−1.00432	−0.502162	+	0.864774i	$0.667462\pi$
$270$	0	0
$271$	−5.29180	−0.321454	−0.160727	−	0.986999i	$-0.551384\pi$
$271$	−5.29180	−0.321454	−0.160727	+	0.986999i	$0.551384\pi$
$272$	59.1246	3.58496
$273$	0	0
$274$	−15.7082	−0.948967
$275$	0	0
$276$	0	0
$277$	−18.9443	−1.13825	−0.569125	−	0.822251i	$-0.692718\pi$
$277$	−18.9443	−1.13825	−0.569125	+	0.822251i	$0.692718\pi$
$278$	−20.9443	−1.25615
$279$	0	0
$280$	0	0
$281$	−5.18034	−0.309033	−0.154517	−	0.987990i	$-0.549382\pi$
$281$	−5.18034	−0.309033	−0.154517	+	0.987990i	$0.549382\pi$
$282$	0	0
$283$	−4.70820	−0.279874	−0.139937	−	0.990160i	$-0.544690\pi$
$283$	−4.70820	−0.279874	−0.139937	+	0.990160i	$0.544690\pi$
$284$	−21.7082	−1.28814
$285$	0	0
$286$	−9.09017	−0.537513
$287$	0	0
$288$	0	0
$289$	19.0000	1.11765
$290$	−59.9230	−3.51880
$291$	0	0
$292$	4.85410	0.284065
$293$	−9.88854	−0.577695	−0.288847	−	0.957375i	$-0.593272\pi$
$293$	−9.88854	−0.577695	−0.288847	+	0.957375i	$0.593272\pi$
$294$	0	0
$295$	26.7082	1.55501
$296$	−3.94427	−0.229256
$297$	0	0
$298$	−40.9787	−2.37383
$299$	−8.58359	−0.496402
$300$	0	0
$301$	0	0
$302$	−18.1803	−1.04616
$303$	0	0
$304$	41.7426	2.39410
$305$	−15.5279	−0.889123
$306$	0	0
$307$	−16.0000	−0.913168	−0.456584	−	0.889680i	$-0.650927\pi$
$307$	−16.0000	−0.913168	−0.456584	+	0.889680i	$0.650927\pi$
$308$	0	0
$309$	0	0
$310$	49.5967	2.81691
$311$	−12.9443	−0.734002	−0.367001	−	0.930220i	$-0.619615\pi$
$311$	−12.9443	−0.734002	−0.367001	+	0.930220i	$0.619615\pi$
$312$	0	0
$313$	−26.3607	−1.48999	−0.744997	−	0.667068i	$-0.767549\pi$
$313$	−26.3607	−1.48999	−0.744997	+	0.667068i	$0.767549\pi$
$314$	−8.94427	−0.504754
$315$	0	0
$316$	−12.0000	−0.675053
$317$	−26.9443	−1.51334	−0.756671	−	0.653796i	$-0.773175\pi$
$317$	−26.9443	−1.51334	−0.756671	+	0.653796i	$0.773175\pi$
$318$	0	0
$319$	10.2361	0.573110
$320$	−19.4721	−1.08853
$321$	0	0
$322$	0	0
$323$	25.4164	1.41421
$324$	0	0
$325$	0	0
$326$	−47.4508	−2.62806
$327$	0	0
$328$	44.8328	2.47548
$329$	0	0
$330$	0	0
$331$	−16.0000	−0.879440	−0.439720	−	0.898135i	$-0.644922\pi$
$331$	−16.0000	−0.879440	−0.439720	+	0.898135i	$0.644922\pi$
$332$	60.5410	3.32262
$333$	0	0
$334$	48.3607	2.64618
$335$	−1.58359	−0.0865209
$336$	0	0
$337$	11.4164	0.621891	0.310946	−	0.950428i	$-0.399354\pi$
$337$	11.4164	0.621891	0.310946	+	0.950428i	$0.399354\pi$
$338$	−2.47214	−0.134466
$339$	0	0
$340$	−65.1246	−3.53188
$341$	−8.47214	−0.458792
$342$	0	0
$343$	0	0
$344$	−3.52786	−0.190210
$345$	0	0
$346$	14.4721	0.778027
$347$	−0.944272	−0.0506912	−0.0253456	−	0.999679i	$-0.508069\pi$
$347$	−0.944272	−0.0506912	−0.0253456	+	0.999679i	$0.508069\pi$
$348$	0	0
$349$	2.41641	0.129347	0.0646737	−	0.997906i	$-0.479399\pi$
$349$	2.41641	0.129347	0.0646737	+	0.997906i	$0.479399\pi$
$350$	0	0
$351$	0	0
$352$	10.8541	0.578526
$353$	−4.23607	−0.225463	−0.112732	−	0.993625i	$-0.535960\pi$
$353$	−4.23607	−0.225463	−0.112732	+	0.993625i	$0.535960\pi$
$354$	0	0
$355$	10.0000	0.530745
$356$	−65.1246	−3.45160
$357$	0	0
$358$	14.4721	0.764876
$359$	20.9443	1.10540	0.552698	−	0.833381i	$-0.313598\pi$
$359$	20.9443	1.10540	0.552698	+	0.833381i	$0.313598\pi$
$360$	0	0
$361$	−1.05573	−0.0555646
$362$	−1.23607	−0.0649663
$363$	0	0
$364$	0	0
$365$	−2.23607	−0.117041
$366$	0	0
$367$	25.4164	1.32673	0.663363	−	0.748298i	$-0.269129\pi$
$367$	25.4164	1.32673	0.663363	+	0.748298i	$0.269129\pi$
$368$	24.3607	1.26989
$369$	0	0
$370$	3.09017	0.160650
$371$	0	0
$372$	0	0
$373$	−16.9443	−0.877341	−0.438671	−	0.898648i	$-0.644550\pi$
$373$	−16.9443	−0.877341	−0.438671	+	0.898648i	$0.644550\pi$
$374$	15.7082	0.812252
$375$	0	0
$376$	85.7214	4.42074
$377$	−35.5410	−1.83046
$378$	0	0
$379$	−25.6525	−1.31768	−0.658840	−	0.752283i	$-0.728952\pi$
$379$	−25.6525	−1.31768	−0.658840	+	0.752283i	$0.728952\pi$
$380$	−45.9787	−2.35866
$381$	0	0
$382$	−31.4164	−1.60740
$383$	8.94427	0.457031	0.228515	−	0.973540i	$-0.426613\pi$
$383$	8.94427	0.457031	0.228515	+	0.973540i	$0.426613\pi$
$384$	0	0
$385$	0	0
$386$	67.7771	3.44976
$387$	0	0
$388$	33.7082	1.71127
$389$	−20.8328	−1.05627	−0.528133	−	0.849162i	$-0.677108\pi$
$389$	−20.8328	−1.05627	−0.528133	+	0.849162i	$0.677108\pi$
$390$	0	0
$391$	14.8328	0.750128
$392$	0	0
$393$	0	0
$394$	22.1803	1.11743
$395$	5.52786	0.278137
$396$	0	0
$397$	−9.41641	−0.472596	−0.236298	−	0.971681i	$-0.575934\pi$
$397$	−9.41641	−0.472596	−0.236298	+	0.971681i	$0.575934\pi$
$398$	−64.0689	−3.21148
$399$	0	0
$400$	0	0
$401$	8.47214	0.423078	0.211539	−	0.977370i	$-0.432152\pi$
$401$	8.47214	0.423078	0.211539	+	0.977370i	$0.432152\pi$
$402$	0	0
$403$	29.4164	1.46534
$404$	17.1246	0.851981
$405$	0	0
$406$	0	0
$407$	−0.527864	−0.0261652
$408$	0	0
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	−35.1246	−1.73468
$411$	0	0
$412$	36.0000	1.77359
$413$	0	0
$414$	0	0
$415$	−27.8885	−1.36899
$416$	−37.6869	−1.84775
$417$	0	0
$418$	11.0902	0.542438
$419$	3.00000	0.146560	0.0732798	−	0.997311i	$-0.476653\pi$
$419$	3.00000	0.146560	0.0732798	+	0.997311i	$0.476653\pi$
$420$	0	0
$421$	29.4721	1.43638	0.718192	−	0.695845i	$-0.244970\pi$
$421$	29.4721	1.43638	0.718192	+	0.695845i	$0.244970\pi$
$422$	49.5967	2.41433
$423$	0	0
$424$	96.7214	4.69720
$425$	0	0
$426$	0	0
$427$	0	0
$428$	−96.8115	−4.67956
$429$	0	0
$430$	2.76393	0.133289
$431$	17.3607	0.836235	0.418117	−	0.908393i	$-0.362690\pi$
$431$	17.3607	0.836235	0.418117	+	0.908393i	$0.362690\pi$
$432$	0	0
$433$	12.5836	0.604729	0.302364	−	0.953192i	$-0.402224\pi$
$433$	12.5836	0.604729	0.302364	+	0.953192i	$0.402224\pi$
$434$	0	0
$435$	0	0
$436$	29.1246	1.39482
$437$	10.4721	0.500950
$438$	0	0
$439$	−8.12461	−0.387767	−0.193883	−	0.981025i	$-0.562108\pi$
$439$	−8.12461	−0.387767	−0.193883	+	0.981025i	$0.562108\pi$
$440$	−16.7082	−0.796532
$441$	0	0
$442$	−54.5410	−2.59425
$443$	−9.05573	−0.430251	−0.215125	−	0.976586i	$-0.569016\pi$
$443$	−9.05573	−0.430251	−0.215125	+	0.976586i	$0.569016\pi$
$444$	0	0
$445$	30.0000	1.42214
$446$	−47.1246	−2.23142
$447$	0	0
$448$	0	0
$449$	25.4164	1.19947	0.599737	−	0.800197i	$-0.295272\pi$
$449$	25.4164	1.19947	0.599737	+	0.800197i	$0.295272\pi$
$450$	0	0
$451$	6.00000	0.282529
$452$	−31.4164	−1.47770
$453$	0	0
$454$	70.2492	3.29696
$455$	0	0
$456$	0	0
$457$	−11.4164	−0.534037	−0.267019	−	0.963691i	$-0.586038\pi$
$457$	−11.4164	−0.534037	−0.267019	+	0.963691i	$0.586038\pi$
$458$	−40.3607	−1.88593
$459$	0	0
$460$	−26.8328	−1.25109
$461$	16.4721	0.767184	0.383592	−	0.923503i	$-0.374687\pi$
$461$	16.4721	0.767184	0.383592	+	0.923503i	$0.374687\pi$
$462$	0	0
$463$	−2.70820	−0.125861	−0.0629305	−	0.998018i	$-0.520045\pi$
$463$	−2.70820	−0.125861	−0.0629305	+	0.998018i	$0.520045\pi$
$464$	100.867	4.68264
$465$	0	0
$466$	−35.1246	−1.62712
$467$	23.8328	1.10285	0.551426	−	0.834224i	$-0.314084\pi$
$467$	23.8328	1.10285	0.551426	+	0.834224i	$0.314084\pi$
$468$	0	0
$469$	0	0
$470$	−67.1591	−3.09782
$471$	0	0
$472$	−89.2492	−4.10803
$473$	−0.472136	−0.0217088
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	−45.4508	−2.07887
$479$	21.8885	1.00011	0.500057	−	0.865993i	$-0.333313\pi$
$479$	21.8885	1.00011	0.500057	+	0.865993i	$0.333313\pi$
$480$	0	0
$481$	1.83282	0.0835692
$482$	49.7426	2.26572
$483$	0	0
$484$	4.85410	0.220641
$485$	−15.5279	−0.705084
$486$	0	0
$487$	−21.8885	−0.991865	−0.495932	−	0.868361i	$-0.665174\pi$
$487$	−21.8885	−0.991865	−0.495932	+	0.868361i	$0.665174\pi$
$488$	51.8885	2.34888
$489$	0	0
$490$	0	0
$491$	10.8885	0.491393	0.245697	−	0.969347i	$-0.420983\pi$
$491$	10.8885	0.491393	0.245697	+	0.969347i	$0.420983\pi$
$492$	0	0
$493$	61.4164	2.76606
$494$	−38.5066	−1.73249
$495$	0	0
$496$	−83.4853	−3.74860
$497$	0	0
$498$	0	0
$499$	22.1246	0.990434	0.495217	−	0.868769i	$-0.335089\pi$
$499$	22.1246	0.990434	0.495217	+	0.868769i	$0.335089\pi$
$500$	54.2705	2.42705
$501$	0	0
$502$	10.3262	0.460883
$503$	10.5836	0.471899	0.235950	−	0.971765i	$-0.424180\pi$
$503$	10.5836	0.471899	0.235950	+	0.971765i	$0.424180\pi$
$504$	0	0
$505$	−7.88854	−0.351036
$506$	6.47214	0.287722
$507$	0	0
$508$	77.6656	3.44586
$509$	35.5279	1.57474	0.787372	−	0.616478i	$-0.211441\pi$
$509$	35.5279	1.57474	0.787372	+	0.616478i	$0.211441\pi$
$510$	0	0
$511$	0	0
$512$	−40.3050	−1.78124
$513$	0	0
$514$	−20.3262	−0.896552
$515$	−16.5836	−0.730760
$516$	0	0
$517$	11.4721	0.504544
$518$	0	0
$519$	0	0
$520$	58.0132	2.54405
$521$	−23.7639	−1.04112	−0.520558	−	0.853826i	$-0.674276\pi$
$521$	−23.7639	−1.04112	−0.520558	+	0.853826i	$0.674276\pi$
$522$	0	0
$523$	−3.29180	−0.143940	−0.0719701	−	0.997407i	$-0.522929\pi$
$523$	−3.29180	−0.143940	−0.0719701	+	0.997407i	$0.522929\pi$
$524$	−21.7082	−0.948327
$525$	0	0
$526$	11.5623	0.504140
$527$	−50.8328	−2.21431
$528$	0	0
$529$	−16.8885	−0.734285
$530$	−75.7771	−3.29155
$531$	0	0
$532$	0	0
$533$	−20.8328	−0.902369
$534$	0	0
$535$	44.5967	1.92809
$536$	5.29180	0.228571
$537$	0	0
$538$	−43.1246	−1.85923
$539$	0	0
$540$	0	0
$541$	21.4164	0.920763	0.460382	−	0.887721i	$-0.347713\pi$
$541$	21.4164	0.920763	0.460382	+	0.887721i	$0.347713\pi$
$542$	−13.8541	−0.595085
$543$	0	0
$544$	65.1246	2.79219
$545$	−13.4164	−0.574696
$546$	0	0
$547$	−13.4164	−0.573644	−0.286822	−	0.957984i	$-0.592599\pi$
$547$	−13.4164	−0.573644	−0.286822	+	0.957984i	$0.592599\pi$
$548$	−29.1246	−1.24414
$549$	0	0
$550$	0	0
$551$	43.3607	1.84723
$552$	0	0
$553$	0	0
$554$	−49.5967	−2.10716
$555$	0	0
$556$	−38.8328	−1.64688
$557$	8.12461	0.344251	0.172125	−	0.985075i	$-0.444937\pi$
$557$	8.12461	0.344251	0.172125	+	0.985075i	$0.444937\pi$
$558$	0	0
$559$	1.63932	0.0693359
$560$	0	0
$561$	0	0
$562$	−13.5623	−0.572091
$563$	−4.47214	−0.188478	−0.0942390	−	0.995550i	$-0.530042\pi$
$563$	−4.47214	−0.188478	−0.0942390	+	0.995550i	$0.530042\pi$
$564$	0	0
$565$	14.4721	0.608847
$566$	−12.3262	−0.518110
$567$	0	0
$568$	−33.4164	−1.40212
$569$	12.4721	0.522859	0.261430	−	0.965223i	$-0.415806\pi$
$569$	12.4721	0.522859	0.261430	+	0.965223i	$0.415806\pi$
$570$	0	0
$571$	−10.5836	−0.442910	−0.221455	−	0.975171i	$-0.571081\pi$
$571$	−10.5836	−0.442910	−0.221455	+	0.975171i	$0.571081\pi$
$572$	−16.8541	−0.704705
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−34.9443	−1.45475	−0.727375	−	0.686241i	$-0.759260\pi$
$577$	−34.9443	−1.45475	−0.727375	+	0.686241i	$0.759260\pi$
$578$	49.7426	2.06902
$579$	0	0
$580$	−111.103	−4.61331
$581$	0	0
$582$	0	0
$583$	12.9443	0.536097
$584$	7.47214	0.309199
$585$	0	0
$586$	−25.8885	−1.06945
$587$	14.0557	0.580142	0.290071	−	0.957005i	$-0.406321\pi$
$587$	14.0557	0.580142	0.290071	+	0.957005i	$0.406321\pi$
$588$	0	0
$589$	−35.8885	−1.47876
$590$	69.9230	2.87868
$591$	0	0
$592$	−5.20163	−0.213786
$593$	−37.4164	−1.53651	−0.768254	−	0.640145i	$-0.778874\pi$
$593$	−37.4164	−1.53651	−0.768254	+	0.640145i	$0.778874\pi$
$594$	0	0
$595$	0	0
$596$	−75.9787	−3.11221
$597$	0	0
$598$	−22.4721	−0.918954
$599$	−25.3050	−1.03393	−0.516966	−	0.856006i	$-0.672939\pi$
$599$	−25.3050	−1.03393	−0.516966	+	0.856006i	$0.672939\pi$
$600$	0	0
$601$	−5.83282	−0.237926	−0.118963	−	0.992899i	$-0.537957\pi$
$601$	−5.83282	−0.237926	−0.118963	+	0.992899i	$0.537957\pi$
$602$	0	0
$603$	0	0
$604$	−33.7082	−1.37157
$605$	−2.23607	−0.0909091
$606$	0	0
$607$	22.7082	0.921698	0.460849	−	0.887479i	$-0.347545\pi$
$607$	22.7082	0.921698	0.460849	+	0.887479i	$0.347545\pi$
$608$	45.9787	1.86468
$609$	0	0
$610$	−40.6525	−1.64597
$611$	−39.8328	−1.61146
$612$	0	0
$613$	19.4164	0.784221	0.392111	−	0.919918i	$-0.371745\pi$
$613$	19.4164	0.784221	0.392111	+	0.919918i	$0.371745\pi$
$614$	−41.8885	−1.69048
$615$	0	0
$616$	0	0
$617$	40.2492	1.62037	0.810186	−	0.586172i	$-0.199366\pi$
$617$	40.2492	1.62037	0.810186	+	0.586172i	$0.199366\pi$
$618$	0	0
$619$	−27.4164	−1.10196	−0.550979	−	0.834519i	$-0.685746\pi$
$619$	−27.4164	−1.10196	−0.550979	+	0.834519i	$0.685746\pi$
$620$	91.9574	3.69310
$621$	0	0
$622$	−33.8885	−1.35881
$623$	0	0
$624$	0	0
$625$	−25.0000	−1.00000
$626$	−69.0132	−2.75832
$627$	0	0
$628$	−16.5836	−0.661757
$629$	−3.16718	−0.126284
$630$	0	0
$631$	−4.00000	−0.159237	−0.0796187	−	0.996825i	$-0.525370\pi$
$631$	−4.00000	−0.159237	−0.0796187	+	0.996825i	$0.525370\pi$
$632$	−18.4721	−0.734782
$633$	0	0
$634$	−70.5410	−2.80154
$635$	−35.7771	−1.41977
$636$	0	0
$637$	0	0
$638$	26.7984	1.06096
$639$	0	0
$640$	−2.43769	−0.0963583
$641$	−31.5279	−1.24528	−0.622638	−	0.782510i	$-0.713939\pi$
$641$	−31.5279	−1.24528	−0.622638	+	0.782510i	$0.713939\pi$
$642$	0	0
$643$	−5.41641	−0.213602	−0.106801	−	0.994280i	$-0.534061\pi$
$643$	−5.41641	−0.213602	−0.106801	+	0.994280i	$0.534061\pi$
$644$	0	0
$645$	0	0
$646$	66.5410	2.61802
$647$	−4.41641	−0.173627	−0.0868135	−	0.996225i	$-0.527668\pi$
$647$	−4.41641	−0.173627	−0.0868135	+	0.996225i	$0.527668\pi$
$648$	0	0
$649$	−11.9443	−0.468854
$650$	0	0
$651$	0	0
$652$	−87.9787	−3.44551
$653$	−5.88854	−0.230437	−0.115218	−	0.993340i	$-0.536757\pi$
$653$	−5.88854	−0.230437	−0.115218	+	0.993340i	$0.536757\pi$
$654$	0	0
$655$	10.0000	0.390732
$656$	59.1246	2.30843
$657$	0	0
$658$	0	0
$659$	−32.8885	−1.28116	−0.640578	−	0.767893i	$-0.721305\pi$
$659$	−32.8885	−1.28116	−0.640578	+	0.767893i	$0.721305\pi$
$660$	0	0
$661$	4.00000	0.155582	0.0777910	−	0.996970i	$-0.475213\pi$
$661$	4.00000	0.155582	0.0777910	+	0.996970i	$0.475213\pi$
$662$	−41.8885	−1.62804
$663$	0	0
$664$	93.1935	3.61661
$665$	0	0
$666$	0	0
$667$	25.3050	0.979812
$668$	89.6656	3.46927
$669$	0	0
$670$	−4.14590	−0.160170
$671$	6.94427	0.268081
$672$	0	0
$673$	−18.5836	−0.716345	−0.358172	−	0.933655i	$-0.616600\pi$
$673$	−18.5836	−0.716345	−0.358172	+	0.933655i	$0.616600\pi$
$674$	29.8885	1.15126
$675$	0	0
$676$	−4.58359	−0.176292
$677$	43.4164	1.66863	0.834314	−	0.551289i	$-0.185864\pi$
$677$	43.4164	1.66863	0.834314	+	0.551289i	$0.185864\pi$
$678$	0	0
$679$	0	0
$680$	−100.249	−3.84438
$681$	0	0
$682$	−22.1803	−0.849329
$683$	−9.52786	−0.364574	−0.182287	−	0.983245i	$-0.558350\pi$
$683$	−9.52786	−0.364574	−0.182287	+	0.983245i	$0.558350\pi$
$684$	0	0
$685$	13.4164	0.512615
$686$	0	0
$687$	0	0
$688$	−4.65248	−0.177374
$689$	−44.9443	−1.71224
$690$	0	0
$691$	−6.58359	−0.250452	−0.125226	−	0.992128i	$-0.539966\pi$
$691$	−6.58359	−0.250452	−0.125226	+	0.992128i	$0.539966\pi$
$692$	26.8328	1.02003
$693$	0	0
$694$	−2.47214	−0.0938410
$695$	17.8885	0.678551
$696$	0	0
$697$	36.0000	1.36360
$698$	6.32624	0.239452
$699$	0	0
$700$	0	0
$701$	13.4164	0.506731	0.253365	−	0.967371i	$-0.418463\pi$
$701$	13.4164	0.506731	0.253365	+	0.967371i	$0.418463\pi$
$702$	0	0
$703$	−2.23607	−0.0843349
$704$	8.70820	0.328203
$705$	0	0
$706$	−11.0902	−0.417384
$707$	0	0
$708$	0	0
$709$	22.4164	0.841866	0.420933	−	0.907092i	$-0.361703\pi$
$709$	22.4164	0.841866	0.420933	+	0.907092i	$0.361703\pi$
$710$	26.1803	0.982531
$711$	0	0
$712$	−100.249	−3.75700
$713$	−20.9443	−0.784369
$714$	0	0
$715$	7.76393	0.290355
$716$	26.8328	1.00279
$717$	0	0
$718$	54.8328	2.04634
$719$	−4.63932	−0.173017	−0.0865087	−	0.996251i	$-0.527571\pi$
$719$	−4.63932	−0.173017	−0.0865087	+	0.996251i	$0.527571\pi$
$720$	0	0
$721$	0	0
$722$	−2.76393	−0.102863
$723$	0	0
$724$	−2.29180	−0.0851739
$725$	0	0
$726$	0	0
$727$	−4.58359	−0.169996	−0.0849980	−	0.996381i	$-0.527088\pi$
$727$	−4.58359	−0.169996	−0.0849980	+	0.996381i	$0.527088\pi$
$728$	0	0
$729$	0	0
$730$	−5.85410	−0.216670
$731$	−2.83282	−0.104775
$732$	0	0
$733$	−29.0557	−1.07320	−0.536599	−	0.843837i	$-0.680291\pi$
$733$	−29.0557	−1.07320	−0.536599	+	0.843837i	$0.680291\pi$
$734$	66.5410	2.45607
$735$	0	0
$736$	26.8328	0.989071
$737$	0.708204	0.0260870
$738$	0	0
$739$	43.4164	1.59710	0.798549	−	0.601930i	$-0.205601\pi$
$739$	43.4164	1.59710	0.798549	+	0.601930i	$0.205601\pi$
$740$	5.72949	0.210620
$741$	0	0
$742$	0	0
$743$	−12.3050	−0.451425	−0.225712	−	0.974194i	$-0.572471\pi$
$743$	−12.3050	−0.451425	−0.225712	+	0.974194i	$0.572471\pi$
$744$	0	0
$745$	35.0000	1.28230
$746$	−44.3607	−1.62416
$747$	0	0
$748$	29.1246	1.06490
$749$	0	0
$750$	0	0
$751$	12.1246	0.442433	0.221217	−	0.975225i	$-0.428997\pi$
$751$	12.1246	0.442433	0.221217	+	0.975225i	$0.428997\pi$
$752$	113.048	4.12242
$753$	0	0
$754$	−93.0476	−3.38859
$755$	15.5279	0.565117
$756$	0	0
$757$	−11.5836	−0.421013	−0.210506	−	0.977592i	$-0.567511\pi$
$757$	−11.5836	−0.421013	−0.210506	+	0.977592i	$0.567511\pi$
$758$	−67.1591	−2.43933
$759$	0	0
$760$	−70.7771	−2.56735
$761$	13.5279	0.490385	0.245192	−	0.969474i	$-0.421149\pi$
$761$	13.5279	0.490385	0.245192	+	0.969474i	$0.421149\pi$
$762$	0	0
$763$	0	0
$764$	−58.2492	−2.10738
$765$	0	0
$766$	23.4164	0.846069
$767$	41.4721	1.49747
$768$	0	0
$769$	−1.00000	−0.0360609	−0.0180305	−	0.999837i	$-0.505740\pi$
$769$	−1.00000	−0.0360609	−0.0180305	+	0.999837i	$0.505740\pi$
$770$	0	0
$771$	0	0
$772$	125.666	4.52281
$773$	−20.3475	−0.731850	−0.365925	−	0.930644i	$-0.619247\pi$
$773$	−20.3475	−0.731850	−0.365925	+	0.930644i	$0.619247\pi$
$774$	0	0
$775$	0	0
$776$	51.8885	1.86269
$777$	0	0
$778$	−54.5410	−1.95539
$779$	25.4164	0.910637
$780$	0	0
$781$	−4.47214	−0.160026
$782$	38.8328	1.38866
$783$	0	0
$784$	0	0
$785$	7.63932	0.272659
$786$	0	0
$787$	−38.1246	−1.35900	−0.679498	−	0.733678i	$-0.737802\pi$
$787$	−38.1246	−1.35900	−0.679498	+	0.733678i	$0.737802\pi$
$788$	41.1246	1.46500
$789$	0	0
$790$	14.4721	0.514895
$791$	0	0
$792$	0	0
$793$	−24.1115	−0.856223
$794$	−24.6525	−0.874884
$795$	0	0
$796$	−118.790	−4.21041
$797$	−45.7639	−1.62104	−0.810521	−	0.585710i	$-0.800816\pi$
$797$	−45.7639	−1.62104	−0.810521	+	0.585710i	$0.800816\pi$
$798$	0	0
$799$	68.8328	2.43513
$800$	0	0
$801$	0	0
$802$	22.1803	0.783215
$803$	1.00000	0.0352892
$804$	0	0
$805$	0	0
$806$	77.0132	2.71267
$807$	0	0
$808$	26.3607	0.927365
$809$	14.8197	0.521032	0.260516	−	0.965470i	$-0.416107\pi$
$809$	14.8197	0.521032	0.260516	+	0.965470i	$0.416107\pi$
$810$	0	0
$811$	27.5410	0.967096	0.483548	−	0.875318i	$-0.339348\pi$
$811$	27.5410	0.967096	0.483548	+	0.875318i	$0.339348\pi$
$812$	0	0
$813$	0	0
$814$	−1.38197	−0.0484379
$815$	40.5279	1.41963
$816$	0	0
$817$	−2.00000	−0.0699711
$818$	−36.6525	−1.28152
$819$	0	0
$820$	−65.1246	−2.27425
$821$	33.5410	1.17059	0.585295	−	0.810821i	$-0.300979\pi$
$821$	33.5410	1.17059	0.585295	+	0.810821i	$0.300979\pi$
$822$	0	0
$823$	−45.5410	−1.58746	−0.793730	−	0.608270i	$-0.791864\pi$
$823$	−45.5410	−1.58746	−0.793730	+	0.608270i	$0.791864\pi$
$824$	55.4164	1.93052
$825$	0	0
$826$	0	0
$827$	−0.167184	−0.00581357	−0.00290678	−	0.999996i	$-0.500925\pi$
$827$	−0.167184	−0.00581357	−0.00290678	+	0.999996i	$0.500925\pi$
$828$	0	0
$829$	34.8328	1.20979	0.604897	−	0.796304i	$-0.293214\pi$
$829$	34.8328	1.20979	0.604897	+	0.796304i	$0.293214\pi$
$830$	−73.0132	−2.53432
$831$	0	0
$832$	−30.2361	−1.04825
$833$	0	0
$834$	0	0
$835$	−41.3050	−1.42942
$836$	20.5623	0.711162
$837$	0	0
$838$	7.85410	0.271315
$839$	−20.5279	−0.708701	−0.354350	−	0.935113i	$-0.615298\pi$
$839$	−20.5279	−0.708701	−0.354350	+	0.935113i	$0.615298\pi$
$840$	0	0
$841$	75.7771	2.61300
$842$	77.1591	2.65908
$843$	0	0
$844$	91.9574	3.16531
$845$	2.11146	0.0726363
$846$	0	0
$847$	0	0
$848$	127.554	4.38023
$849$	0	0
$850$	0	0
$851$	−1.30495	−0.0447332
$852$	0	0
$853$	54.7214	1.87362	0.936812	−	0.349834i	$-0.113762\pi$
$853$	54.7214	1.87362	0.936812	+	0.349834i	$0.113762\pi$
$854$	0	0
$855$	0	0
$856$	−149.026	−5.09361
$857$	−50.7214	−1.73261	−0.866304	−	0.499517i	$-0.833511\pi$
$857$	−50.7214	−1.73261	−0.866304	+	0.499517i	$0.833511\pi$
$858$	0	0
$859$	−15.4164	−0.526001	−0.263001	−	0.964796i	$-0.584712\pi$
$859$	−15.4164	−0.526001	−0.263001	+	0.964796i	$0.584712\pi$
$860$	5.12461	0.174748
$861$	0	0
$862$	45.4508	1.54806
$863$	−35.3050	−1.20179	−0.600897	−	0.799326i	$-0.705190\pi$
$863$	−35.3050	−1.20179	−0.600897	+	0.799326i	$0.705190\pi$
$864$	0	0
$865$	−12.3607	−0.420276
$866$	32.9443	1.11949
$867$	0	0
$868$	0	0
$869$	−2.47214	−0.0838615
$870$	0	0
$871$	−2.45898	−0.0833194
$872$	44.8328	1.51823
$873$	0	0
$874$	27.4164	0.927374
$875$	0	0
$876$	0	0
$877$	−21.8885	−0.739124	−0.369562	−	0.929206i	$-0.620492\pi$
$877$	−21.8885	−0.739124	−0.369562	+	0.929206i	$0.620492\pi$
$878$	−21.2705	−0.717845
$879$	0	0
$880$	−22.0344	−0.742781
$881$	−22.5967	−0.761304	−0.380652	−	0.924718i	$-0.624300\pi$
$881$	−22.5967	−0.761304	−0.380652	+	0.924718i	$0.624300\pi$
$882$	0	0
$883$	4.23607	0.142555	0.0712775	−	0.997457i	$-0.477292\pi$
$883$	4.23607	0.142555	0.0712775	+	0.997457i	$0.477292\pi$
$884$	−101.125	−3.40119
$885$	0	0
$886$	−23.7082	−0.796493
$887$	−28.3607	−0.952258	−0.476129	−	0.879375i	$-0.657961\pi$
$887$	−28.3607	−0.952258	−0.476129	+	0.879375i	$0.657961\pi$
$888$	0	0
$889$	0	0
$890$	78.5410	2.63270
$891$	0	0
$892$	−87.3738	−2.92549
$893$	48.5967	1.62623
$894$	0	0
$895$	−12.3607	−0.413172
$896$	0	0
$897$	0	0
$898$	66.5410	2.22050
$899$	−86.7214	−2.89232
$900$	0	0
$901$	77.6656	2.58742
$902$	15.7082	0.523026
$903$	0	0
$904$	−48.3607	−1.60845
$905$	1.05573	0.0350936
$906$	0	0
$907$	−14.1115	−0.468563	−0.234282	−	0.972169i	$-0.575274\pi$
$907$	−14.1115	−0.468563	−0.234282	+	0.972169i	$0.575274\pi$
$908$	130.249	4.32247
$909$	0	0
$910$	0	0
$911$	31.4164	1.04087	0.520436	−	0.853901i	$-0.325769\pi$
$911$	31.4164	1.04087	0.520436	+	0.853901i	$0.325769\pi$
$912$	0	0
$913$	12.4721	0.412767
$914$	−29.8885	−0.988625
$915$	0	0
$916$	−74.8328	−2.47255
$917$	0	0
$918$	0	0
$919$	1.88854	0.0622973	0.0311487	−	0.999515i	$-0.490083\pi$
$919$	1.88854	0.0622973	0.0311487	+	0.999515i	$0.490083\pi$
$920$	−41.3050	−1.36178
$921$	0	0
$922$	43.1246	1.42023
$923$	15.5279	0.511106
$924$	0	0
$925$	0	0
$926$	−7.09017	−0.232997
$927$	0	0
$928$	111.103	3.64715
$929$	52.4853	1.72199	0.860993	−	0.508616i	$-0.169843\pi$
$929$	52.4853	1.72199	0.860993	+	0.508616i	$0.169843\pi$
$930$	0	0
$931$	0	0
$932$	−65.1246	−2.13323
$933$	0	0
$934$	62.3951	2.04163
$935$	−13.4164	−0.438763
$936$	0	0
$937$	−42.7214	−1.39565	−0.697823	−	0.716270i	$-0.745848\pi$
$937$	−42.7214	−1.39565	−0.697823	+	0.716270i	$0.745848\pi$
$938$	0	0
$939$	0	0
$940$	−124.520	−4.06139
$941$	51.1935	1.66886	0.834430	−	0.551114i	$-0.185797\pi$
$941$	51.1935	1.66886	0.834430	+	0.551114i	$0.185797\pi$
$942$	0	0
$943$	14.8328	0.483023
$944$	−117.700	−3.83081
$945$	0	0
$946$	−1.23607	−0.0401880
$947$	−0.111456	−0.00362184	−0.00181092	−	0.999998i	$-0.500576\pi$
$947$	−0.111456	−0.00362184	−0.00181092	+	0.999998i	$0.500576\pi$
$948$	0	0
$949$	−3.47214	−0.112710
$950$	0	0
$951$	0	0
$952$	0	0
$953$	28.4853	0.922729	0.461365	−	0.887211i	$-0.347360\pi$
$953$	28.4853	0.922729	0.461365	+	0.887211i	$0.347360\pi$
$954$	0	0
$955$	26.8328	0.868290
$956$	−84.2705	−2.72550
$957$	0	0
$958$	57.3050	1.85144
$959$	0	0
$960$	0	0
$961$	40.7771	1.31539
$962$	4.79837	0.154706
$963$	0	0
$964$	92.2279	2.97046
$965$	−57.8885	−1.86350
$966$	0	0
$967$	46.2492	1.48727	0.743637	−	0.668583i	$-0.233099\pi$
$967$	46.2492	1.48727	0.743637	+	0.668583i	$0.233099\pi$
$968$	7.47214	0.240164
$969$	0	0
$970$	−40.6525	−1.30527
$971$	24.8885	0.798711	0.399356	−	0.916796i	$-0.369234\pi$
$971$	24.8885	0.798711	0.399356	+	0.916796i	$0.369234\pi$
$972$	0	0
$973$	0	0
$974$	−57.3050	−1.83617
$975$	0	0
$976$	68.4296	2.19038
$977$	13.5279	0.432795	0.216397	−	0.976305i	$-0.430569\pi$
$977$	13.5279	0.432795	0.216397	+	0.976305i	$0.430569\pi$
$978$	0	0
$979$	−13.4164	−0.428790
$980$	0	0
$981$	0	0
$982$	28.5066	0.909681
$983$	38.8328	1.23857	0.619287	−	0.785165i	$-0.287422\pi$
$983$	38.8328	1.23857	0.619287	+	0.785165i	$0.287422\pi$
$984$	0	0
$985$	−18.9443	−0.603615
$986$	160.790	5.12060
$987$	0	0
$988$	−71.3951	−2.27138
$989$	−1.16718	−0.0371143
$990$	0	0
$991$	13.2918	0.422228	0.211114	−	0.977461i	$-0.432291\pi$
$991$	13.2918	0.422228	0.211114	+	0.977461i	$0.432291\pi$
$992$	−91.9574	−2.91965
$993$	0	0
$994$	0	0
$995$	54.7214	1.73478
$996$	0	0
$997$	−42.0000	−1.33015	−0.665077	−	0.746775i	$-0.731601\pi$
$997$	−42.0000	−1.33015	−0.665077	+	0.746775i	$0.731601\pi$
$998$	57.9230	1.83352
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4851.2.a.bg.1.2		2
3.2	odd	2		1617.2.a.l.1.1	yes	2
7.6	odd	2		4851.2.a.bh.1.2		2
21.20	even	2		1617.2.a.k.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1617.2.a.k.1.1	✓	2	21.20	even	2
1617.2.a.l.1.1	yes	2	3.2	odd	2
4851.2.a.bg.1.2		2	1.1	even	1	trivial
4851.2.a.bh.1.2		2	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4851 = 3^{2} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4851.a (trivial)

\(f(q)\)	\(=\)	\(q+2.61803 q^{2} +4.85410 q^{4} -2.23607 q^{5} +7.47214 q^{8} +O(q^{10})\)	\(q+2.61803 q^{2} +4.85410 q^{4} -2.23607 q^{5} +7.47214 q^{8} -5.85410 q^{10} +1.00000 q^{11} -3.47214 q^{13} +9.85410 q^{16} +6.00000 q^{17} +4.23607 q^{19} -10.8541 q^{20} +2.61803 q^{22} +2.47214 q^{23} -9.09017 q^{26} +10.2361 q^{29} -8.47214 q^{31} +10.8541 q^{32} +15.7082 q^{34} -0.527864 q^{37} +11.0902 q^{38} -16.7082 q^{40} +6.00000 q^{41} -0.472136 q^{43} +4.85410 q^{44} +6.47214 q^{46} +11.4721 q^{47} -16.8541 q^{52} +12.9443 q^{53} -2.23607 q^{55} +26.7984 q^{58} -11.9443 q^{59} +6.94427 q^{61} -22.1803 q^{62} +8.70820 q^{64} +7.76393 q^{65} +0.708204 q^{67} +29.1246 q^{68} -4.47214 q^{71} +1.00000 q^{73} -1.38197 q^{74} +20.5623 q^{76} -2.47214 q^{79} -22.0344 q^{80} +15.7082 q^{82} +12.4721 q^{83} -13.4164 q^{85} -1.23607 q^{86} +7.47214 q^{88} -13.4164 q^{89} +12.0000 q^{92} +30.0344 q^{94} -9.47214 q^{95} +6.94427 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{2} + 3 q^{4} + 6 q^{8}+O(q^{10}) \) 2 * q + 3 * q^2 + 3 * q^4 + 6 * q^8	\( 2 q + 3 q^{2} + 3 q^{4} + 6 q^{8} - 5 q^{10} + 2 q^{11} + 2 q^{13} + 13 q^{16} + 12 q^{17} + 4 q^{19} - 15 q^{20} + 3 q^{22} - 4 q^{23} - 7 q^{26} + 16 q^{29} - 8 q^{31} + 15 q^{32} + 18 q^{34} - 10 q^{37} + 11 q^{38} - 20 q^{40} + 12 q^{41} + 8 q^{43} + 3 q^{44} + 4 q^{46} + 14 q^{47} - 27 q^{52} + 8 q^{53} + 29 q^{58} - 6 q^{59} - 4 q^{61} - 22 q^{62} + 4 q^{64} + 20 q^{65} - 12 q^{67} + 18 q^{68} + 2 q^{73} - 5 q^{74} + 21 q^{76} + 4 q^{79} - 15 q^{80} + 18 q^{82} + 16 q^{83} + 2 q^{86} + 6 q^{88} + 24 q^{92} + 31 q^{94} - 10 q^{95} - 4 q^{97}+O(q^{100}) \) 2 * q + 3 * q^2 + 3 * q^4 + 6 * q^8 - 5 * q^10 + 2 * q^11 + 2 * q^13 + 13 * q^16 + 12 * q^17 + 4 * q^19 - 15 * q^20 + 3 * q^22 - 4 * q^23 - 7 * q^26 + 16 * q^29 - 8 * q^31 + 15 * q^32 + 18 * q^34 - 10 * q^37 + 11 * q^38 - 20 * q^40 + 12 * q^41 + 8 * q^43 + 3 * q^44 + 4 * q^46 + 14 * q^47 - 27 * q^52 + 8 * q^53 + 29 * q^58 - 6 * q^59 - 4 * q^61 - 22 * q^62 + 4 * q^64 + 20 * q^65 - 12 * q^67 + 18 * q^68 + 2 * q^73 - 5 * q^74 + 21 * q^76 + 4 * q^79 - 15 * q^80 + 18 * q^82 + 16 * q^83 + 2 * q^86 + 6 * q^88 + 24 * q^92 + 31 * q^94 - 10 * q^95 - 4 * q^97

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