LMFDB - Embedded newform 9801.2.a.bb.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9801,2,Mod(1,9801)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9801, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("9801.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9801 = 3^{4} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9801.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:	$78.2613790211$
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 99)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	9801.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.381966 q^{2} -1.85410 q^{4} +1.23607 q^{5} -1.00000 q^{7} -1.47214 q^{8} +0.472136 q^{10} -6.47214 q^{13} -0.381966 q^{14} +3.14590 q^{16} -4.85410 q^{17} +1.00000 q^{19} -2.29180 q^{20} -4.61803 q^{23} -3.47214 q^{25} -2.47214 q^{26} +1.85410 q^{28} +4.85410 q^{29} +0.618034 q^{31} +4.14590 q^{32} -1.85410 q^{34} -1.23607 q^{35} -5.09017 q^{37} +0.381966 q^{38} -1.81966 q^{40} +2.52786 q^{41} +1.85410 q^{43} -1.76393 q^{46} -11.9443 q^{47} -6.00000 q^{49} -1.32624 q^{50} +12.0000 q^{52} -4.09017 q^{53} +1.47214 q^{56} +1.85410 q^{58} -1.61803 q^{59} +10.8541 q^{61} +0.236068 q^{62} -4.70820 q^{64} -8.00000 q^{65} +6.00000 q^{67} +9.00000 q^{68} -0.472136 q^{70} +2.90983 q^{71} -0.145898 q^{73} -1.94427 q^{74} -1.85410 q^{76} +10.5623 q^{79} +3.88854 q^{80} +0.965558 q^{82} +9.76393 q^{83} -6.00000 q^{85} +0.708204 q^{86} +6.76393 q^{89} +6.47214 q^{91} +8.56231 q^{92} -4.56231 q^{94} +1.23607 q^{95} +6.00000 q^{97} -2.29180 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{2} + 3 q^{4} - 2 q^{5} - 2 q^{7} + 6 q^{8} - 8 q^{10} - 4 q^{13} - 3 q^{14} + 13 q^{16} - 3 q^{17} + 2 q^{19} - 18 q^{20} - 7 q^{23} + 2 q^{25} + 4 q^{26} - 3 q^{28} + 3 q^{29} - q^{31} + 15 q^{32}+ \cdots - 18 q^{98}+O(q^{100}) $ 2 * q + 3 * q^2 + 3 * q^4 - 2 * q^5 - 2 * q^7 + 6 * q^8 - 8 * q^10 - 4 * q^13 - 3 * q^14 + 13 * q^16 - 3 * q^17 + 2 * q^19 - 18 * q^20 - 7 * q^23 + 2 * q^25 + 4 * q^26 - 3 * q^28 + 3 * q^29 - q^31 + 15 * q^32 + 3 * q^34 + 2 * q^35 + q^37 + 3 * q^38 - 26 * q^40 + 14 * q^41 - 3 * q^43 - 8 * q^46 - 6 * q^47 - 12 * q^49 + 13 * q^50 + 24 * q^52 + 3 * q^53 - 6 * q^56 - 3 * q^58 - q^59 + 15 * q^61 - 4 * q^62 + 4 * q^64 - 16 * q^65 + 12 * q^67 + 18 * q^68 + 8 * q^70 + 17 * q^71 - 7 * q^73 + 14 * q^74 + 3 * q^76 + q^79 - 28 * q^80 + 31 * q^82 + 24 * q^83 - 12 * q^85 - 12 * q^86 + 18 * q^89 + 4 * q^91 - 3 * q^92 + 11 * q^94 - 2 * q^95 + 12 * q^97 - 18 * q^98

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 $

$2$	0.381966	0.270091	0.135045	−	0.990839i	$-0.456882\pi$
$2$	0.381966	0.270091	0.135045	+	0.990839i	$0.456882\pi$
$3$	0	0
$3$	0	0
$4$	−1.85410	−0.927051
$5$	1.23607	0.552786	0.276393	−	0.961045i	$-0.410861\pi$
$5$	1.23607	0.552786	0.276393	+	0.961045i	$0.410861\pi$
$6$	0	0
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	−1.47214	−0.520479
$9$	0	0
$10$	0.472136	0.149302
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−6.47214	−1.79505	−0.897524	−	0.440966i	$-0.854636\pi$
$13$	−6.47214	−1.79505	−0.897524	+	0.440966i	$0.854636\pi$
$14$	−0.381966	−0.102085
$15$	0	0
$16$	3.14590	0.786475
$17$	−4.85410	−1.17729	−0.588646	−	0.808391i	$-0.700339\pi$
$17$	−4.85410	−1.17729	−0.588646	+	0.808391i	$0.700339\pi$
$18$	0	0
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	−2.29180	−0.512461
$21$	0	0
$22$	0	0
$23$	−4.61803	−0.962927	−0.481463	−	0.876466i	$-0.659895\pi$
$23$	−4.61803	−0.962927	−0.481463	+	0.876466i	$0.659895\pi$
$24$	0	0
$25$	−3.47214	−0.694427
$26$	−2.47214	−0.484826
$27$	0	0
$28$	1.85410	0.350392
$29$	4.85410	0.901384	0.450692	−	0.892679i	$-0.351177\pi$
$29$	4.85410	0.901384	0.450692	+	0.892679i	$0.351177\pi$
$30$	0	0
$31$	0.618034	0.111002	0.0555011	−	0.998459i	$-0.482324\pi$
$31$	0.618034	0.111002	0.0555011	+	0.998459i	$0.482324\pi$
$32$	4.14590	0.732898
$33$	0	0
$34$	−1.85410	−0.317976
$35$	−1.23607	−0.208934
$36$	0	0
$37$	−5.09017	−0.836819	−0.418409	−	0.908259i	$-0.637412\pi$
$37$	−5.09017	−0.836819	−0.418409	+	0.908259i	$0.637412\pi$
$38$	0.381966	0.0619631
$39$	0	0
$40$	−1.81966	−0.287714
$41$	2.52786	0.394786	0.197393	−	0.980324i	$-0.436752\pi$
$41$	2.52786	0.394786	0.197393	+	0.980324i	$0.436752\pi$
$42$	0	0
$43$	1.85410	0.282748	0.141374	−	0.989956i	$-0.454848\pi$
$43$	1.85410	0.282748	0.141374	+	0.989956i	$0.454848\pi$
$44$	0	0
$45$	0	0
$46$	−1.76393	−0.260078
$47$	−11.9443	−1.74225	−0.871126	−	0.491060i	$-0.836609\pi$
$47$	−11.9443	−1.74225	−0.871126	+	0.491060i	$0.836609\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	−1.32624	−0.187558
$51$	0	0
$52$	12.0000	1.66410
$53$	−4.09017	−0.561828	−0.280914	−	0.959733i	$-0.590638\pi$
$53$	−4.09017	−0.561828	−0.280914	+	0.959733i	$0.590638\pi$
$54$	0	0
$55$	0	0
$56$	1.47214	0.196722
$57$	0	0
$58$	1.85410	0.243456
$59$	−1.61803	−0.210650	−0.105325	−	0.994438i	$-0.533588\pi$
$59$	−1.61803	−0.210650	−0.105325	+	0.994438i	$0.533588\pi$
$60$	0	0
$61$	10.8541	1.38973	0.694863	−	0.719142i	$-0.255465\pi$
$61$	10.8541	1.38973	0.694863	+	0.719142i	$0.255465\pi$
$62$	0.236068	0.0299807
$63$	0	0
$64$	−4.70820	−0.588525
$65$	−8.00000	−0.992278
$66$	0	0
$67$	6.00000	0.733017	0.366508	−	0.930415i	$-0.380553\pi$
$67$	6.00000	0.733017	0.366508	+	0.930415i	$0.380553\pi$
$68$	9.00000	1.09141
$69$	0	0
$70$	−0.472136	−0.0564310
$71$	2.90983	0.345333	0.172667	−	0.984980i	$-0.444762\pi$
$71$	2.90983	0.345333	0.172667	+	0.984980i	$0.444762\pi$
$72$	0	0
$73$	−0.145898	−0.0170761	−0.00853804	−	0.999964i	$-0.502718\pi$
$73$	−0.145898	−0.0170761	−0.00853804	+	0.999964i	$0.502718\pi$
$74$	−1.94427	−0.226017
$75$	0	0
$76$	−1.85410	−0.212680
$77$	0	0
$78$	0	0
$79$	10.5623	1.18835	0.594176	−	0.804335i	$-0.297478\pi$
$79$	10.5623	1.18835	0.594176	+	0.804335i	$0.297478\pi$
$80$	3.88854	0.434752
$81$	0	0
$82$	0.965558	0.106628
$83$	9.76393	1.07173	0.535865	−	0.844303i	$-0.319985\pi$
$83$	9.76393	1.07173	0.535865	+	0.844303i	$0.319985\pi$
$84$	0	0
$85$	−6.00000	−0.650791
$86$	0.708204	0.0763676
$87$	0	0
$88$	0	0
$89$	6.76393	0.716975	0.358488	−	0.933534i	$-0.383293\pi$
$89$	6.76393	0.716975	0.358488	+	0.933534i	$0.383293\pi$
$90$	0	0
$91$	6.47214	0.678464
$92$	8.56231	0.892682
$93$	0	0
$94$	−4.56231	−0.470566
$95$	1.23607	0.126818
$96$	0	0
$97$	6.00000	0.609208	0.304604	−	0.952479i	$-0.401476\pi$
$97$	6.00000	0.609208	0.304604	+	0.952479i	$0.401476\pi$
$98$	−2.29180	−0.231506
$99$	0	0
$100$	6.43769	0.643769
$101$	8.47214	0.843009	0.421505	−	0.906826i	$-0.361502\pi$
$101$	8.47214	0.843009	0.421505	+	0.906826i	$0.361502\pi$
$102$	0	0
$103$	−10.2361	−1.00859	−0.504295	−	0.863532i	$-0.668248\pi$
$103$	−10.2361	−1.00859	−0.504295	+	0.863532i	$0.668248\pi$
$104$	9.52786	0.934284
$105$	0	0
$106$	−1.56231	−0.151745
$107$	−14.0344	−1.35676	−0.678380	−	0.734711i	$-0.737318\pi$
$107$	−14.0344	−1.35676	−0.678380	+	0.734711i	$0.737318\pi$
$108$	0	0
$109$	7.14590	0.684453	0.342226	−	0.939618i	$-0.388819\pi$
$109$	7.14590	0.684453	0.342226	+	0.939618i	$0.388819\pi$
$110$	0	0
$111$	0	0
$112$	−3.14590	−0.297259
$113$	−17.2361	−1.62143	−0.810716	−	0.585439i	$-0.800922\pi$
$113$	−17.2361	−1.62143	−0.810716	+	0.585439i	$0.800922\pi$
$114$	0	0
$115$	−5.70820	−0.532293
$116$	−9.00000	−0.835629
$117$	0	0
$118$	−0.618034	−0.0568946
$119$	4.85410	0.444975
$120$	0	0
$121$	0	0
$122$	4.14590	0.375352
$123$	0	0
$124$	−1.14590	−0.102905
$125$	−10.4721	−0.936656
$126$	0	0
$127$	−10.5623	−0.937253	−0.468627	−	0.883396i	$-0.655251\pi$
$127$	−10.5623	−0.937253	−0.468627	+	0.883396i	$0.655251\pi$
$128$	−10.0902	−0.891853
$129$	0	0
$130$	−3.05573	−0.268005
$131$	−21.2361	−1.85540	−0.927702	−	0.373322i	$-0.878219\pi$
$131$	−21.2361	−1.85540	−0.927702	+	0.373322i	$0.878219\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	2.29180	0.197981
$135$	0	0
$136$	7.14590	0.612756
$137$	8.61803	0.736288	0.368144	−	0.929769i	$-0.379993\pi$
$137$	8.61803	0.736288	0.368144	+	0.929769i	$0.379993\pi$
$138$	0	0
$139$	9.70820	0.823439	0.411720	−	0.911311i	$-0.364928\pi$
$139$	9.70820	0.823439	0.411720	+	0.911311i	$0.364928\pi$
$140$	2.29180	0.193692
$141$	0	0
$142$	1.11146	0.0932713
$143$	0	0
$144$	0	0
$145$	6.00000	0.498273
$146$	−0.0557281	−0.00461209
$147$	0	0
$148$	9.43769	0.775774
$149$	15.1803	1.24362	0.621811	−	0.783167i	$-0.286397\pi$
$149$	15.1803	1.24362	0.621811	+	0.783167i	$0.286397\pi$
$150$	0	0
$151$	4.32624	0.352064	0.176032	−	0.984384i	$-0.443674\pi$
$151$	4.32624	0.352064	0.176032	+	0.984384i	$0.443674\pi$
$152$	−1.47214	−0.119406
$153$	0	0
$154$	0	0
$155$	0.763932	0.0613605
$156$	0	0
$157$	7.14590	0.570305	0.285152	−	0.958482i	$-0.407956\pi$
$157$	7.14590	0.570305	0.285152	+	0.958482i	$0.407956\pi$
$158$	4.03444	0.320963
$159$	0	0
$160$	5.12461	0.405136
$161$	4.61803	0.363952
$162$	0	0
$163$	20.7082	1.62199	0.810996	−	0.585052i	$-0.198926\pi$
$163$	20.7082	1.62199	0.810996	+	0.585052i	$0.198926\pi$
$164$	−4.68692	−0.365987
$165$	0	0
$166$	3.72949	0.289465
$167$	5.76393	0.446026	0.223013	−	0.974815i	$-0.428411\pi$
$167$	5.76393	0.446026	0.223013	+	0.974815i	$0.428411\pi$
$168$	0	0
$169$	28.8885	2.22220
$170$	−2.29180	−0.175773
$171$	0	0
$172$	−3.43769	−0.262122
$173$	23.0902	1.75551	0.877757	−	0.479107i	$-0.159039\pi$
$173$	23.0902	1.75551	0.877757	+	0.479107i	$0.159039\pi$
$174$	0	0
$175$	3.47214	0.262469
$176$	0	0
$177$	0	0
$178$	2.58359	0.193648
$179$	5.05573	0.377883	0.188941	−	0.981988i	$-0.439494\pi$
$179$	5.05573	0.377883	0.188941	+	0.981988i	$0.439494\pi$
$180$	0	0
$181$	−9.90983	−0.736592	−0.368296	−	0.929709i	$-0.620059\pi$
$181$	−9.90983	−0.736592	−0.368296	+	0.929709i	$0.620059\pi$
$182$	2.47214	0.183247
$183$	0	0
$184$	6.79837	0.501183
$185$	−6.29180	−0.462582
$186$	0	0
$187$	0	0
$188$	22.1459	1.61516
$189$	0	0
$190$	0.472136	0.0342523
$191$	−5.94427	−0.430112	−0.215056	−	0.976602i	$-0.568993\pi$
$191$	−5.94427	−0.430112	−0.215056	+	0.976602i	$0.568993\pi$
$192$	0	0
$193$	−11.0000	−0.791797	−0.395899	−	0.918294i	$-0.629567\pi$
$193$	−11.0000	−0.791797	−0.395899	+	0.918294i	$0.629567\pi$
$194$	2.29180	0.164541
$195$	0	0
$196$	11.1246	0.794615
$197$	−8.23607	−0.586796	−0.293398	−	0.955990i	$-0.594786\pi$
$197$	−8.23607	−0.586796	−0.293398	+	0.955990i	$0.594786\pi$
$198$	0	0
$199$	−7.14590	−0.506559	−0.253280	−	0.967393i	$-0.581509\pi$
$199$	−7.14590	−0.506559	−0.253280	+	0.967393i	$0.581509\pi$
$200$	5.11146	0.361435
$201$	0	0
$202$	3.23607	0.227689
$203$	−4.85410	−0.340691
$204$	0	0
$205$	3.12461	0.218232
$206$	−3.90983	−0.272411
$207$	0	0
$208$	−20.3607	−1.41176
$209$	0	0
$210$	0	0
$211$	20.4721	1.40936	0.704680	−	0.709525i	$-0.251091\pi$
$211$	20.4721	1.40936	0.704680	+	0.709525i	$0.251091\pi$
$212$	7.58359	0.520843
$213$	0	0
$214$	−5.36068	−0.366449
$215$	2.29180	0.156299
$216$	0	0
$217$	−0.618034	−0.0419549
$218$	2.72949	0.184864
$219$	0	0
$220$	0	0
$221$	31.4164	2.11330
$222$	0	0
$223$	−15.5066	−1.03840	−0.519199	−	0.854654i	$-0.673770\pi$
$223$	−15.5066	−1.03840	−0.519199	+	0.854654i	$0.673770\pi$
$224$	−4.14590	−0.277009
$225$	0	0
$226$	−6.58359	−0.437934
$227$	16.6180	1.10298	0.551489	−	0.834182i	$-0.314060\pi$
$227$	16.6180	1.10298	0.551489	+	0.834182i	$0.314060\pi$
$228$	0	0
$229$	10.1803	0.672736	0.336368	−	0.941731i	$-0.390801\pi$
$229$	10.1803	0.672736	0.336368	+	0.941731i	$0.390801\pi$
$230$	−2.18034	−0.143767
$231$	0	0
$232$	−7.14590	−0.469151
$233$	11.1246	0.728798	0.364399	−	0.931243i	$-0.381275\pi$
$233$	11.1246	0.728798	0.364399	+	0.931243i	$0.381275\pi$
$234$	0	0
$235$	−14.7639	−0.963093
$236$	3.00000	0.195283
$237$	0	0
$238$	1.85410	0.120184
$239$	−10.4721	−0.677386	−0.338693	−	0.940897i	$-0.609985\pi$
$239$	−10.4721	−0.677386	−0.338693	+	0.940897i	$0.609985\pi$
$240$	0	0
$241$	−26.8328	−1.72845	−0.864227	−	0.503102i	$-0.832192\pi$
$241$	−26.8328	−1.72845	−0.864227	+	0.503102i	$0.832192\pi$
$242$	0	0
$243$	0	0
$244$	−20.1246	−1.28835
$245$	−7.41641	−0.473817
$246$	0	0
$247$	−6.47214	−0.411812
$248$	−0.909830	−0.0577743
$249$	0	0
$250$	−4.00000	−0.252982
$251$	−20.5623	−1.29788	−0.648941	−	0.760839i	$-0.724788\pi$
$251$	−20.5623	−1.29788	−0.648941	+	0.760839i	$0.724788\pi$
$252$	0	0
$253$	0	0
$254$	−4.03444	−0.253143
$255$	0	0
$256$	5.56231	0.347644
$257$	14.9098	0.930050	0.465025	−	0.885298i	$-0.346045\pi$
$257$	14.9098	0.930050	0.465025	+	0.885298i	$0.346045\pi$
$258$	0	0
$259$	5.09017	0.316288
$260$	14.8328	0.919892
$261$	0	0
$262$	−8.11146	−0.501127
$263$	12.7082	0.783621	0.391811	−	0.920046i	$-0.371849\pi$
$263$	12.7082	0.783621	0.391811	+	0.920046i	$0.371849\pi$
$264$	0	0
$265$	−5.05573	−0.310571
$266$	−0.381966	−0.0234198
$267$	0	0
$268$	−11.1246	−0.679544
$269$	21.2705	1.29689	0.648443	−	0.761263i	$-0.275421\pi$
$269$	21.2705	1.29689	0.648443	+	0.761263i	$0.275421\pi$
$270$	0	0
$271$	10.7639	0.653862	0.326931	−	0.945048i	$-0.393985\pi$
$271$	10.7639	0.653862	0.326931	+	0.945048i	$0.393985\pi$
$272$	−15.2705	−0.925911
$273$	0	0
$274$	3.29180	0.198865
$275$	0	0
$276$	0	0
$277$	−19.2361	−1.15578	−0.577892	−	0.816113i	$-0.696124\pi$
$277$	−19.2361	−1.15578	−0.577892	+	0.816113i	$0.696124\pi$
$278$	3.70820	0.222403
$279$	0	0
$280$	1.81966	0.108745
$281$	9.32624	0.556357	0.278178	−	0.960529i	$-0.410269\pi$
$281$	9.32624	0.556357	0.278178	+	0.960529i	$0.410269\pi$
$282$	0	0
$283$	−17.0000	−1.01055	−0.505273	−	0.862960i	$-0.668608\pi$
$283$	−17.0000	−1.01055	−0.505273	+	0.862960i	$0.668608\pi$
$284$	−5.39512	−0.320142
$285$	0	0
$286$	0	0
$287$	−2.52786	−0.149215
$288$	0	0
$289$	6.56231	0.386018
$290$	2.29180	0.134579
$291$	0	0
$292$	0.270510	0.0158304
$293$	29.1803	1.70473	0.852367	−	0.522944i	$-0.175166\pi$
$293$	29.1803	1.70473	0.852367	+	0.522944i	$0.175166\pi$
$294$	0	0
$295$	−2.00000	−0.116445
$296$	7.49342	0.435546
$297$	0	0
$298$	5.79837	0.335891
$299$	29.8885	1.72850
$300$	0	0
$301$	−1.85410	−0.106869
$302$	1.65248	0.0950893
$303$	0	0
$304$	3.14590	0.180430
$305$	13.4164	0.768221
$306$	0	0
$307$	7.85410	0.448257	0.224129	−	0.974560i	$-0.428046\pi$
$307$	7.85410	0.448257	0.224129	+	0.974560i	$0.428046\pi$
$308$	0	0
$309$	0	0
$310$	0.291796	0.0165729
$311$	−15.1803	−0.860798	−0.430399	−	0.902639i	$-0.641627\pi$
$311$	−15.1803	−0.860798	−0.430399	+	0.902639i	$0.641627\pi$
$312$	0	0
$313$	3.47214	0.196257	0.0981284	−	0.995174i	$-0.468714\pi$
$313$	3.47214	0.196257	0.0981284	+	0.995174i	$0.468714\pi$
$314$	2.72949	0.154034
$315$	0	0
$316$	−19.5836	−1.10166
$317$	−20.9443	−1.17635	−0.588174	−	0.808735i	$-0.700153\pi$
$317$	−20.9443	−1.17635	−0.588174	+	0.808735i	$0.700153\pi$
$318$	0	0
$319$	0	0
$320$	−5.81966	−0.325329
$321$	0	0
$322$	1.76393	0.0983001
$323$	−4.85410	−0.270089
$324$	0	0
$325$	22.4721	1.24653
$326$	7.90983	0.438085
$327$	0	0
$328$	−3.72136	−0.205478
$329$	11.9443	0.658509
$330$	0	0
$331$	14.4164	0.792397	0.396199	−	0.918165i	$-0.370329\pi$
$331$	14.4164	0.792397	0.396199	+	0.918165i	$0.370329\pi$
$332$	−18.1033	−0.993549
$333$	0	0
$334$	2.20163	0.120468
$335$	7.41641	0.405202
$336$	0	0
$337$	−11.4721	−0.624927	−0.312464	−	0.949930i	$-0.601154\pi$
$337$	−11.4721	−0.624927	−0.312464	+	0.949930i	$0.601154\pi$
$338$	11.0344	0.600195
$339$	0	0
$340$	11.1246	0.603317
$341$	0	0
$342$	0	0
$343$	13.0000	0.701934
$344$	−2.72949	−0.147164
$345$	0	0
$346$	8.81966	0.474148
$347$	8.88854	0.477162	0.238581	−	0.971123i	$-0.423318\pi$
$347$	8.88854	0.477162	0.238581	+	0.971123i	$0.423318\pi$
$348$	0	0
$349$	−4.14590	−0.221925	−0.110962	−	0.993825i	$-0.535393\pi$
$349$	−4.14590	−0.221925	−0.110962	+	0.993825i	$0.535393\pi$
$350$	1.32624	0.0708904
$351$	0	0
$352$	0	0
$353$	16.4164	0.873757	0.436879	−	0.899520i	$-0.356084\pi$
$353$	16.4164	0.873757	0.436879	+	0.899520i	$0.356084\pi$
$354$	0	0
$355$	3.59675	0.190896
$356$	−12.5410	−0.664673
$357$	0	0
$358$	1.93112	0.102063
$359$	22.8541	1.20619	0.603097	−	0.797668i	$-0.293933\pi$
$359$	22.8541	1.20619	0.603097	+	0.797668i	$0.293933\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	−3.78522	−0.198947
$363$	0	0
$364$	−12.0000	−0.628971
$365$	−0.180340	−0.00943942
$366$	0	0
$367$	−12.7082	−0.663363	−0.331681	−	0.943391i	$-0.607616\pi$
$367$	−12.7082	−0.663363	−0.331681	+	0.943391i	$0.607616\pi$
$368$	−14.5279	−0.757317
$369$	0	0
$370$	−2.40325	−0.124939
$371$	4.09017	0.212351
$372$	0	0
$373$	−5.65248	−0.292674	−0.146337	−	0.989235i	$-0.546748\pi$
$373$	−5.65248	−0.292674	−0.146337	+	0.989235i	$0.546748\pi$
$374$	0	0
$375$	0	0
$376$	17.5836	0.906805
$377$	−31.4164	−1.61803
$378$	0	0
$379$	15.7984	0.811508	0.405754	−	0.913982i	$-0.367009\pi$
$379$	15.7984	0.811508	0.405754	+	0.913982i	$0.367009\pi$
$380$	−2.29180	−0.117567
$381$	0	0
$382$	−2.27051	−0.116169
$383$	−19.8541	−1.01450	−0.507249	−	0.861800i	$-0.669337\pi$
$383$	−19.8541	−1.01450	−0.507249	+	0.861800i	$0.669337\pi$
$384$	0	0
$385$	0	0
$386$	−4.20163	−0.213857
$387$	0	0
$388$	−11.1246	−0.564767
$389$	22.6525	1.14853	0.574263	−	0.818671i	$-0.305289\pi$
$389$	22.6525	1.14853	0.574263	+	0.818671i	$0.305289\pi$
$390$	0	0
$391$	22.4164	1.13365
$392$	8.83282	0.446125
$393$	0	0
$394$	−3.14590	−0.158488
$395$	13.0557	0.656905
$396$	0	0
$397$	−2.12461	−0.106631	−0.0533156	−	0.998578i	$-0.516979\pi$
$397$	−2.12461	−0.106631	−0.0533156	+	0.998578i	$0.516979\pi$
$398$	−2.72949	−0.136817
$399$	0	0
$400$	−10.9230	−0.546149
$401$	−14.5623	−0.727207	−0.363603	−	0.931554i	$-0.618454\pi$
$401$	−14.5623	−0.727207	−0.363603	+	0.931554i	$0.618454\pi$
$402$	0	0
$403$	−4.00000	−0.199254
$404$	−15.7082	−0.781512
$405$	0	0
$406$	−1.85410	−0.0920175
$407$	0	0
$408$	0	0
$409$	−33.7426	−1.66847	−0.834233	−	0.551412i	$-0.814089\pi$
$409$	−33.7426	−1.66847	−0.834233	+	0.551412i	$0.814089\pi$
$410$	1.19350	0.0589425
$411$	0	0
$412$	18.9787	0.935014
$413$	1.61803	0.0796182
$414$	0	0
$415$	12.0689	0.592438
$416$	−26.8328	−1.31559
$417$	0	0
$418$	0	0
$419$	−8.09017	−0.395231	−0.197615	−	0.980280i	$-0.563320\pi$
$419$	−8.09017	−0.395231	−0.197615	+	0.980280i	$0.563320\pi$
$420$	0	0
$421$	10.2361	0.498875	0.249438	−	0.968391i	$-0.419754\pi$
$421$	10.2361	0.498875	0.249438	+	0.968391i	$0.419754\pi$
$422$	7.81966	0.380655
$423$	0	0
$424$	6.02129	0.292420
$425$	16.8541	0.817544
$426$	0	0
$427$	−10.8541	−0.525267
$428$	26.0213	1.25779
$429$	0	0
$430$	0.875388	0.0422150
$431$	34.0689	1.64104	0.820520	−	0.571618i	$-0.193684\pi$
$431$	34.0689	1.64104	0.820520	+	0.571618i	$0.193684\pi$
$432$	0	0
$433$	14.7426	0.708486	0.354243	−	0.935153i	$-0.384739\pi$
$433$	14.7426	0.708486	0.354243	+	0.935153i	$0.384739\pi$
$434$	−0.236068	−0.0113316
$435$	0	0
$436$	−13.2492	−0.634523
$437$	−4.61803	−0.220911
$438$	0	0
$439$	−0.708204	−0.0338007	−0.0169004	−	0.999857i	$-0.505380\pi$
$439$	−0.708204	−0.0338007	−0.0169004	+	0.999857i	$0.505380\pi$
$440$	0	0
$441$	0	0
$442$	12.0000	0.570782
$443$	1.23607	0.0587274	0.0293637	−	0.999569i	$-0.490652\pi$
$443$	1.23607	0.0587274	0.0293637	+	0.999569i	$0.490652\pi$
$444$	0	0
$445$	8.36068	0.396334
$446$	−5.92299	−0.280462
$447$	0	0
$448$	4.70820	0.222442
$449$	−7.47214	−0.352632	−0.176316	−	0.984334i	$-0.556418\pi$
$449$	−7.47214	−0.352632	−0.176316	+	0.984334i	$0.556418\pi$
$450$	0	0
$451$	0	0
$452$	31.9574	1.50315
$453$	0	0
$454$	6.34752	0.297904
$455$	8.00000	0.375046
$456$	0	0
$457$	−1.00000	−0.0467780	−0.0233890	−	0.999726i	$-0.507446\pi$
$457$	−1.00000	−0.0467780	−0.0233890	+	0.999726i	$0.507446\pi$
$458$	3.88854	0.181700
$459$	0	0
$460$	10.5836	0.493463
$461$	16.1459	0.751989	0.375995	−	0.926622i	$-0.377301\pi$
$461$	16.1459	0.751989	0.375995	+	0.926622i	$0.377301\pi$
$462$	0	0
$463$	17.0000	0.790057	0.395029	−	0.918669i	$-0.370735\pi$
$463$	17.0000	0.790057	0.395029	+	0.918669i	$0.370735\pi$
$464$	15.2705	0.708916
$465$	0	0
$466$	4.24922	0.196841
$467$	0.763932	0.0353506	0.0176753	−	0.999844i	$-0.494373\pi$
$467$	0.763932	0.0353506	0.0176753	+	0.999844i	$0.494373\pi$
$468$	0	0
$469$	−6.00000	−0.277054
$470$	−5.63932	−0.260122
$471$	0	0
$472$	2.38197	0.109639
$473$	0	0
$474$	0	0
$475$	−3.47214	−0.159313
$476$	−9.00000	−0.412514
$477$	0	0
$478$	−4.00000	−0.182956
$479$	11.7639	0.537508	0.268754	−	0.963209i	$-0.413388\pi$
$479$	11.7639	0.537508	0.268754	+	0.963209i	$0.413388\pi$
$480$	0	0
$481$	32.9443	1.50213
$482$	−10.2492	−0.466839
$483$	0	0
$484$	0	0
$485$	7.41641	0.336762
$486$	0	0
$487$	−12.9787	−0.588122	−0.294061	−	0.955787i	$-0.595007\pi$
$487$	−12.9787	−0.588122	−0.294061	+	0.955787i	$0.595007\pi$
$488$	−15.9787	−0.723322
$489$	0	0
$490$	−2.83282	−0.127974
$491$	−7.32624	−0.330628	−0.165314	−	0.986241i	$-0.552864\pi$
$491$	−7.32624	−0.330628	−0.165314	+	0.986241i	$0.552864\pi$
$492$	0	0
$493$	−23.5623	−1.06119
$494$	−2.47214	−0.111227
$495$	0	0
$496$	1.94427	0.0873004
$497$	−2.90983	−0.130524
$498$	0	0
$499$	38.4164	1.71975	0.859877	−	0.510501i	$-0.170540\pi$
$499$	38.4164	1.71975	0.859877	+	0.510501i	$0.170540\pi$
$500$	19.4164	0.868328
$501$	0	0
$502$	−7.85410	−0.350546
$503$	9.52786	0.424826	0.212413	−	0.977180i	$-0.431868\pi$
$503$	9.52786	0.424826	0.212413	+	0.977180i	$0.431868\pi$
$504$	0	0
$505$	10.4721	0.466004
$506$	0	0
$507$	0	0
$508$	19.5836	0.868881
$509$	27.1803	1.20475	0.602374	−	0.798214i	$-0.294222\pi$
$509$	27.1803	1.20475	0.602374	+	0.798214i	$0.294222\pi$
$510$	0	0
$511$	0.145898	0.00645415
$512$	22.3050	0.985749
$513$	0	0
$514$	5.69505	0.251198
$515$	−12.6525	−0.557535
$516$	0	0
$517$	0	0
$518$	1.94427	0.0854264
$519$	0	0
$520$	11.7771	0.516459
$521$	40.3607	1.76823	0.884117	−	0.467266i	$-0.154761\pi$
$521$	40.3607	1.76823	0.884117	+	0.467266i	$0.154761\pi$
$522$	0	0
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	39.3738	1.72005
$525$	0	0
$526$	4.85410	0.211649
$527$	−3.00000	−0.130682
$528$	0	0
$529$	−1.67376	−0.0727723
$530$	−1.93112	−0.0838823
$531$	0	0
$532$	1.85410	0.0803855
$533$	−16.3607	−0.708660
$534$	0	0
$535$	−17.3475	−0.749999
$536$	−8.83282	−0.381520
$537$	0	0
$538$	8.12461	0.350277
$539$	0	0
$540$	0	0
$541$	−11.9098	−0.512044	−0.256022	−	0.966671i	$-0.582412\pi$
$541$	−11.9098	−0.512044	−0.256022	+	0.966671i	$0.582412\pi$
$542$	4.11146	0.176602
$543$	0	0
$544$	−20.1246	−0.862836
$545$	8.83282	0.378356
$546$	0	0
$547$	−32.3262	−1.38217	−0.691085	−	0.722773i	$-0.742867\pi$
$547$	−32.3262	−1.38217	−0.691085	+	0.722773i	$0.742867\pi$
$548$	−15.9787	−0.682577
$549$	0	0
$550$	0	0
$551$	4.85410	0.206792
$552$	0	0
$553$	−10.5623	−0.449155
$554$	−7.34752	−0.312166
$555$	0	0
$556$	−18.0000	−0.763370
$557$	0.652476	0.0276463	0.0138231	−	0.999904i	$-0.495600\pi$
$557$	0.652476	0.0276463	0.0138231	+	0.999904i	$0.495600\pi$
$558$	0	0
$559$	−12.0000	−0.507546
$560$	−3.88854	−0.164321
$561$	0	0
$562$	3.56231	0.150267
$563$	20.3050	0.855752	0.427876	−	0.903838i	$-0.359262\pi$
$563$	20.3050	0.855752	0.427876	+	0.903838i	$0.359262\pi$
$564$	0	0
$565$	−21.3050	−0.896306
$566$	−6.49342	−0.272939
$567$	0	0
$568$	−4.28367	−0.179739
$569$	−38.7984	−1.62651	−0.813256	−	0.581906i	$-0.802307\pi$
$569$	−38.7984	−1.62651	−0.813256	+	0.581906i	$0.802307\pi$
$570$	0	0
$571$	−18.5279	−0.775367	−0.387683	−	0.921793i	$-0.626725\pi$
$571$	−18.5279	−0.775367	−0.387683	+	0.921793i	$0.626725\pi$
$572$	0	0
$573$	0	0
$574$	−0.965558	−0.0403016
$575$	16.0344	0.668682
$576$	0	0
$577$	35.4164	1.47440	0.737202	−	0.675672i	$-0.236147\pi$
$577$	35.4164	1.47440	0.737202	+	0.675672i	$0.236147\pi$
$578$	2.50658	0.104260
$579$	0	0
$580$	−11.1246	−0.461924
$581$	−9.76393	−0.405076
$582$	0	0
$583$	0	0
$584$	0.214782	0.00888773
$585$	0	0
$586$	11.1459	0.460433
$587$	−13.4164	−0.553754	−0.276877	−	0.960905i	$-0.589300\pi$
$587$	−13.4164	−0.553754	−0.276877	+	0.960905i	$0.589300\pi$
$588$	0	0
$589$	0.618034	0.0254656
$590$	−0.763932	−0.0314506
$591$	0	0
$592$	−16.0132	−0.658137
$593$	10.9443	0.449427	0.224714	−	0.974425i	$-0.427855\pi$
$593$	10.9443	0.449427	0.224714	+	0.974425i	$0.427855\pi$
$594$	0	0
$595$	6.00000	0.245976
$596$	−28.1459	−1.15290
$597$	0	0
$598$	11.4164	0.466852
$599$	13.3607	0.545903	0.272951	−	0.962028i	$-0.412000\pi$
$599$	13.3607	0.545903	0.272951	+	0.962028i	$0.412000\pi$
$600$	0	0
$601$	−17.4721	−0.712703	−0.356352	−	0.934352i	$-0.615979\pi$
$601$	−17.4721	−0.712703	−0.356352	+	0.934352i	$0.615979\pi$
$602$	−0.708204	−0.0288642
$603$	0	0
$604$	−8.02129	−0.326382
$605$	0	0
$606$	0	0
$607$	−3.58359	−0.145454	−0.0727268	−	0.997352i	$-0.523170\pi$
$607$	−3.58359	−0.145454	−0.0727268	+	0.997352i	$0.523170\pi$
$608$	4.14590	0.168138
$609$	0	0
$610$	5.12461	0.207489
$611$	77.3050	3.12742
$612$	0	0
$613$	19.8541	0.801900	0.400950	−	0.916100i	$-0.368680\pi$
$613$	19.8541	0.801900	0.400950	+	0.916100i	$0.368680\pi$
$614$	3.00000	0.121070
$615$	0	0
$616$	0	0
$617$	27.1591	1.09338	0.546691	−	0.837334i	$-0.315887\pi$
$617$	27.1591	1.09338	0.546691	+	0.837334i	$0.315887\pi$
$618$	0	0
$619$	27.5623	1.10782	0.553911	−	0.832576i	$-0.313135\pi$
$619$	27.5623	1.10782	0.553911	+	0.832576i	$0.313135\pi$
$620$	−1.41641	−0.0568843
$621$	0	0
$622$	−5.79837	−0.232494
$623$	−6.76393	−0.270991
$624$	0	0
$625$	4.41641	0.176656
$626$	1.32624	0.0530071
$627$	0	0
$628$	−13.2492	−0.528702
$629$	24.7082	0.985181
$630$	0	0
$631$	−10.8885	−0.433466	−0.216733	−	0.976231i	$-0.569540\pi$
$631$	−10.8885	−0.433466	−0.216733	+	0.976231i	$0.569540\pi$
$632$	−15.5492	−0.618512
$633$	0	0
$634$	−8.00000	−0.317721
$635$	−13.0557	−0.518101
$636$	0	0
$637$	38.8328	1.53861
$638$	0	0
$639$	0	0
$640$	−12.4721	−0.493004
$641$	20.4508	0.807760	0.403880	−	0.914812i	$-0.367661\pi$
$641$	20.4508	0.807760	0.403880	+	0.914812i	$0.367661\pi$
$642$	0	0
$643$	−6.85410	−0.270299	−0.135150	−	0.990825i	$-0.543152\pi$
$643$	−6.85410	−0.270299	−0.135150	+	0.990825i	$0.543152\pi$
$644$	−8.56231	−0.337402
$645$	0	0
$646$	−1.85410	−0.0729487
$647$	4.18034	0.164346	0.0821731	−	0.996618i	$-0.473814\pi$
$647$	4.18034	0.164346	0.0821731	+	0.996618i	$0.473814\pi$
$648$	0	0
$649$	0	0
$650$	8.58359	0.336676
$651$	0	0
$652$	−38.3951	−1.50367
$653$	1.36068	0.0532475	0.0266238	−	0.999646i	$-0.491524\pi$
$653$	1.36068	0.0532475	0.0266238	+	0.999646i	$0.491524\pi$
$654$	0	0
$655$	−26.2492	−1.02564
$656$	7.95240	0.310489
$657$	0	0
$658$	4.56231	0.177857
$659$	−9.97871	−0.388715	−0.194358	−	0.980931i	$-0.562262\pi$
$659$	−9.97871	−0.388715	−0.194358	+	0.980931i	$0.562262\pi$
$660$	0	0
$661$	−4.29180	−0.166932	−0.0834658	−	0.996511i	$-0.526599\pi$
$661$	−4.29180	−0.166932	−0.0834658	+	0.996511i	$0.526599\pi$
$662$	5.50658	0.214019
$663$	0	0
$664$	−14.3738	−0.557813
$665$	−1.23607	−0.0479327
$666$	0	0
$667$	−22.4164	−0.867967
$668$	−10.6869	−0.413489
$669$	0	0
$670$	2.83282	0.109441
$671$	0	0
$672$	0	0
$673$	−29.5623	−1.13954	−0.569772	−	0.821803i	$-0.692968\pi$
$673$	−29.5623	−1.13954	−0.569772	+	0.821803i	$0.692968\pi$
$674$	−4.38197	−0.168787
$675$	0	0
$676$	−53.5623	−2.06009
$677$	−6.05573	−0.232741	−0.116370	−	0.993206i	$-0.537126\pi$
$677$	−6.05573	−0.232741	−0.116370	+	0.993206i	$0.537126\pi$
$678$	0	0
$679$	−6.00000	−0.230259
$680$	8.83282	0.338723
$681$	0	0
$682$	0	0
$683$	5.52786	0.211518	0.105759	−	0.994392i	$-0.466273\pi$
$683$	5.52786	0.211518	0.105759	+	0.994392i	$0.466273\pi$
$684$	0	0
$685$	10.6525	0.407010
$686$	4.96556	0.189586
$687$	0	0
$688$	5.83282	0.222374
$689$	26.4721	1.00851
$690$	0	0
$691$	14.5967	0.555286	0.277643	−	0.960684i	$-0.410447\pi$
$691$	14.5967	0.555286	0.277643	+	0.960684i	$0.410447\pi$
$692$	−42.8115	−1.62745
$693$	0	0
$694$	3.39512	0.128877
$695$	12.0000	0.455186
$696$	0	0
$697$	−12.2705	−0.464779
$698$	−1.58359	−0.0599398
$699$	0	0
$700$	−6.43769	−0.243322
$701$	47.7214	1.80241	0.901205	−	0.433392i	$-0.142683\pi$
$701$	47.7214	1.80241	0.901205	+	0.433392i	$0.142683\pi$
$702$	0	0
$703$	−5.09017	−0.191979
$704$	0	0
$705$	0	0
$706$	6.27051	0.235994
$707$	−8.47214	−0.318627
$708$	0	0
$709$	−25.7639	−0.967585	−0.483792	−	0.875183i	$-0.660741\pi$
$709$	−25.7639	−0.967585	−0.483792	+	0.875183i	$0.660741\pi$
$710$	1.37384	0.0515591
$711$	0	0
$712$	−9.95743	−0.373170
$713$	−2.85410	−0.106887
$714$	0	0
$715$	0	0
$716$	−9.37384	−0.350317
$717$	0	0
$718$	8.72949	0.325782
$719$	19.5967	0.730835	0.365418	−	0.930844i	$-0.380926\pi$
$719$	19.5967	0.730835	0.365418	+	0.930844i	$0.380926\pi$
$720$	0	0
$721$	10.2361	0.381211
$722$	−6.87539	−0.255875
$723$	0	0
$724$	18.3738	0.682858
$725$	−16.8541	−0.625946
$726$	0	0
$727$	18.8328	0.698470	0.349235	−	0.937035i	$-0.386441\pi$
$727$	18.8328	0.698470	0.349235	+	0.937035i	$0.386441\pi$
$728$	−9.52786	−0.353126
$729$	0	0
$730$	−0.0688837	−0.00254950
$731$	−9.00000	−0.332877
$732$	0	0
$733$	48.4721	1.79036	0.895180	−	0.445706i	$-0.147047\pi$
$733$	48.4721	1.79036	0.895180	+	0.445706i	$0.147047\pi$
$734$	−4.85410	−0.179168
$735$	0	0
$736$	−19.1459	−0.705727
$737$	0	0
$738$	0	0
$739$	42.5410	1.56490	0.782448	−	0.622715i	$-0.213971\pi$
$739$	42.5410	1.56490	0.782448	+	0.622715i	$0.213971\pi$
$740$	11.6656	0.428837
$741$	0	0
$742$	1.56231	0.0573541
$743$	42.3607	1.55406	0.777031	−	0.629462i	$-0.216725\pi$
$743$	42.3607	1.55406	0.777031	+	0.629462i	$0.216725\pi$
$744$	0	0
$745$	18.7639	0.687457
$746$	−2.15905	−0.0790486
$747$	0	0
$748$	0	0
$749$	14.0344	0.512807
$750$	0	0
$751$	−31.2492	−1.14030	−0.570150	−	0.821540i	$-0.693115\pi$
$751$	−31.2492	−1.14030	−0.570150	+	0.821540i	$0.693115\pi$
$752$	−37.5755	−1.37024
$753$	0	0
$754$	−12.0000	−0.437014
$755$	5.34752	0.194616
$756$	0	0
$757$	11.0000	0.399802	0.199901	−	0.979816i	$-0.435938\pi$
$757$	11.0000	0.399802	0.199901	+	0.979816i	$0.435938\pi$
$758$	6.03444	0.219181
$759$	0	0
$760$	−1.81966	−0.0660060
$761$	−6.97871	−0.252978	−0.126489	−	0.991968i	$-0.540371\pi$
$761$	−6.97871	−0.252978	−0.126489	+	0.991968i	$0.540371\pi$
$762$	0	0
$763$	−7.14590	−0.258699
$764$	11.0213	0.398736
$765$	0	0
$766$	−7.58359	−0.274006
$767$	10.4721	0.378127
$768$	0	0
$769$	−15.7984	−0.569704	−0.284852	−	0.958572i	$-0.591944\pi$
$769$	−15.7984	−0.569704	−0.284852	+	0.958572i	$0.591944\pi$
$770$	0	0
$771$	0	0
$772$	20.3951	0.734036
$773$	−23.6525	−0.850720	−0.425360	−	0.905024i	$-0.639853\pi$
$773$	−23.6525	−0.850720	−0.425360	+	0.905024i	$0.639853\pi$
$774$	0	0
$775$	−2.14590	−0.0770829
$776$	−8.83282	−0.317080
$777$	0	0
$778$	8.65248	0.310206
$779$	2.52786	0.0905701
$780$	0	0
$781$	0	0
$782$	8.56231	0.306187
$783$	0	0
$784$	−18.8754	−0.674121
$785$	8.83282	0.315257
$786$	0	0
$787$	36.0000	1.28326	0.641631	−	0.767014i	$-0.278258\pi$
$787$	36.0000	1.28326	0.641631	+	0.767014i	$0.278258\pi$
$788$	15.2705	0.543989
$789$	0	0
$790$	4.98684	0.177424
$791$	17.2361	0.612844
$792$	0	0
$793$	−70.2492	−2.49462
$794$	−0.811529	−0.0288001
$795$	0	0
$796$	13.2492	0.469606
$797$	−24.2705	−0.859706	−0.429853	−	0.902899i	$-0.641435\pi$
$797$	−24.2705	−0.859706	−0.429853	+	0.902899i	$0.641435\pi$
$798$	0	0
$799$	57.9787	2.05114
$800$	−14.3951	−0.508944
$801$	0	0
$802$	−5.56231	−0.196412
$803$	0	0
$804$	0	0
$805$	5.70820	0.201188
$806$	−1.52786	−0.0538167
$807$	0	0
$808$	−12.4721	−0.438768
$809$	18.0902	0.636017	0.318008	−	0.948088i	$-0.396986\pi$
$809$	18.0902	0.636017	0.318008	+	0.948088i	$0.396986\pi$
$810$	0	0
$811$	8.68692	0.305039	0.152519	−	0.988300i	$-0.451261\pi$
$811$	8.68692	0.305039	0.152519	+	0.988300i	$0.451261\pi$
$812$	9.00000	0.315838
$813$	0	0
$814$	0	0
$815$	25.5967	0.896615
$816$	0	0
$817$	1.85410	0.0648668
$818$	−12.8885	−0.450637
$819$	0	0
$820$	−5.79335	−0.202313
$821$	−38.2148	−1.33371	−0.666853	−	0.745190i	$-0.732359\pi$
$821$	−38.2148	−1.33371	−0.666853	+	0.745190i	$0.732359\pi$
$822$	0	0
$823$	−42.2918	−1.47420	−0.737100	−	0.675784i	$-0.763805\pi$
$823$	−42.2918	−1.47420	−0.737100	+	0.675784i	$0.763805\pi$
$824$	15.0689	0.524949
$825$	0	0
$826$	0.618034	0.0215042
$827$	57.0344	1.98328	0.991641	−	0.129028i	$-0.0411858\pi$
$827$	57.0344	1.98328	0.991641	+	0.129028i	$0.0411858\pi$
$828$	0	0
$829$	43.5410	1.51224	0.756121	−	0.654432i	$-0.227092\pi$
$829$	43.5410	1.51224	0.756121	+	0.654432i	$0.227092\pi$
$830$	4.60990	0.160012
$831$	0	0
$832$	30.4721	1.05643
$833$	29.1246	1.00911
$834$	0	0
$835$	7.12461	0.246557
$836$	0	0
$837$	0	0
$838$	−3.09017	−0.106748
$839$	17.2918	0.596979	0.298490	−	0.954413i	$-0.403517\pi$
$839$	17.2918	0.596979	0.298490	+	0.954413i	$0.403517\pi$
$840$	0	0
$841$	−5.43769	−0.187507
$842$	3.90983	0.134742
$843$	0	0
$844$	−37.9574	−1.30655
$845$	35.7082	1.22840
$846$	0	0
$847$	0	0
$848$	−12.8673	−0.441863
$849$	0	0
$850$	6.43769	0.220811
$851$	23.5066	0.805795
$852$	0	0
$853$	−35.4377	−1.21336	−0.606682	−	0.794945i	$-0.707500\pi$
$853$	−35.4377	−1.21336	−0.606682	+	0.794945i	$0.707500\pi$
$854$	−4.14590	−0.141870
$855$	0	0
$856$	20.6606	0.706165
$857$	−25.3607	−0.866304	−0.433152	−	0.901321i	$-0.642599\pi$
$857$	−25.3607	−0.866304	−0.433152	+	0.901321i	$0.642599\pi$
$858$	0	0
$859$	−32.5623	−1.11101	−0.555506	−	0.831513i	$-0.687475\pi$
$859$	−32.5623	−1.11101	−0.555506	+	0.831513i	$0.687475\pi$
$860$	−4.24922	−0.144897
$861$	0	0
$862$	13.0132	0.443230
$863$	−14.4508	−0.491913	−0.245956	−	0.969281i	$-0.579102\pi$
$863$	−14.4508	−0.491913	−0.245956	+	0.969281i	$0.579102\pi$
$864$	0	0
$865$	28.5410	0.970424
$866$	5.63119	0.191356
$867$	0	0
$868$	1.14590	0.0388943
$869$	0	0
$870$	0	0
$871$	−38.8328	−1.31580
$872$	−10.5197	−0.356243
$873$	0	0
$874$	−1.76393	−0.0596659
$875$	10.4721	0.354023
$876$	0	0
$877$	11.5967	0.391594	0.195797	−	0.980644i	$-0.437271\pi$
$877$	11.5967	0.391594	0.195797	+	0.980644i	$0.437271\pi$
$878$	−0.270510	−0.00912926
$879$	0	0
$880$	0	0
$881$	−1.81966	−0.0613059	−0.0306530	−	0.999530i	$-0.509759\pi$
$881$	−1.81966	−0.0613059	−0.0306530	+	0.999530i	$0.509759\pi$
$882$	0	0
$883$	−35.6525	−1.19980	−0.599901	−	0.800074i	$-0.704793\pi$
$883$	−35.6525	−1.19980	−0.599901	+	0.800074i	$0.704793\pi$
$884$	−58.2492	−1.95913
$885$	0	0
$886$	0.472136	0.0158617
$887$	4.03444	0.135463	0.0677316	−	0.997704i	$-0.478424\pi$
$887$	4.03444	0.135463	0.0677316	+	0.997704i	$0.478424\pi$
$888$	0	0
$889$	10.5623	0.354248
$890$	3.19350	0.107046
$891$	0	0
$892$	28.7508	0.962647
$893$	−11.9443	−0.399700
$894$	0	0
$895$	6.24922	0.208889
$896$	10.0902	0.337089
$897$	0	0
$898$	−2.85410	−0.0952426
$899$	3.00000	0.100056
$900$	0	0
$901$	19.8541	0.661436
$902$	0	0
$903$	0	0
$904$	25.3738	0.843921
$905$	−12.2492	−0.407178
$906$	0	0
$907$	−14.5836	−0.484240	−0.242120	−	0.970246i	$-0.577843\pi$
$907$	−14.5836	−0.484240	−0.242120	+	0.970246i	$0.577843\pi$
$908$	−30.8115	−1.02252
$909$	0	0
$910$	3.05573	0.101296
$911$	19.1803	0.635473	0.317737	−	0.948179i	$-0.397077\pi$
$911$	19.1803	0.635473	0.317737	+	0.948179i	$0.397077\pi$
$912$	0	0
$913$	0	0
$914$	−0.381966	−0.0126343
$915$	0	0
$916$	−18.8754	−0.623660
$917$	21.2361	0.701277
$918$	0	0
$919$	−26.8673	−0.886269	−0.443135	−	0.896455i	$-0.646134\pi$
$919$	−26.8673	−0.886269	−0.443135	+	0.896455i	$0.646134\pi$
$920$	8.40325	0.277047
$921$	0	0
$922$	6.16718	0.203105
$923$	−18.8328	−0.619890
$924$	0	0
$925$	17.6738	0.581110
$926$	6.49342	0.213387
$927$	0	0
$928$	20.1246	0.660623
$929$	48.2705	1.58370	0.791852	−	0.610713i	$-0.209117\pi$
$929$	48.2705	1.58370	0.791852	+	0.610713i	$0.209117\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	−20.6262	−0.675632
$933$	0	0
$934$	0.291796	0.00954786
$935$	0	0
$936$	0	0
$937$	−23.0344	−0.752502	−0.376251	−	0.926518i	$-0.622787\pi$
$937$	−23.0344	−0.752502	−0.376251	+	0.926518i	$0.622787\pi$
$938$	−2.29180	−0.0748298
$939$	0	0
$940$	27.3738	0.892836
$941$	−27.1591	−0.885360	−0.442680	−	0.896680i	$-0.645972\pi$
$941$	−27.1591	−0.885360	−0.442680	+	0.896680i	$0.645972\pi$
$942$	0	0
$943$	−11.6738	−0.380150
$944$	−5.09017	−0.165671
$945$	0	0
$946$	0	0
$947$	−36.7082	−1.19286	−0.596428	−	0.802666i	$-0.703414\pi$
$947$	−36.7082	−1.19286	−0.596428	+	0.802666i	$0.703414\pi$
$948$	0	0
$949$	0.944272	0.0306524
$950$	−1.32624	−0.0430288
$951$	0	0
$952$	−7.14590	−0.231600
$953$	9.49342	0.307522	0.153761	−	0.988108i	$-0.450861\pi$
$953$	9.49342	0.307522	0.153761	+	0.988108i	$0.450861\pi$
$954$	0	0
$955$	−7.34752	−0.237760
$956$	19.4164	0.627972
$957$	0	0
$958$	4.49342	0.145176
$959$	−8.61803	−0.278291
$960$	0	0
$961$	−30.6180	−0.987679
$962$	12.5836	0.405711
$963$	0	0
$964$	49.7508	1.60236
$965$	−13.5967	−0.437695
$966$	0	0
$967$	52.4853	1.68781	0.843907	−	0.536490i	$-0.180250\pi$
$967$	52.4853	1.68781	0.843907	+	0.536490i	$0.180250\pi$
$968$	0	0
$969$	0	0
$970$	2.83282	0.0909562
$971$	−9.27051	−0.297505	−0.148752	−	0.988874i	$-0.547526\pi$
$971$	−9.27051	−0.297505	−0.148752	+	0.988874i	$0.547526\pi$
$972$	0	0
$973$	−9.70820	−0.311231
$974$	−4.95743	−0.158846
$975$	0	0
$976$	34.1459	1.09298
$977$	−15.7639	−0.504333	−0.252166	−	0.967684i	$-0.581143\pi$
$977$	−15.7639	−0.504333	−0.252166	+	0.967684i	$0.581143\pi$
$978$	0	0
$979$	0	0
$980$	13.7508	0.439252
$981$	0	0
$982$	−2.79837	−0.0892997
$983$	−9.25735	−0.295264	−0.147632	−	0.989042i	$-0.547165\pi$
$983$	−9.25735	−0.295264	−0.147632	+	0.989042i	$0.547165\pi$
$984$	0	0
$985$	−10.1803	−0.324373
$986$	−9.00000	−0.286618
$987$	0	0
$988$	12.0000	0.381771
$989$	−8.56231	−0.272265
$990$	0	0
$991$	20.5967	0.654277	0.327139	−	0.944976i	$-0.393916\pi$
$991$	20.5967	0.654277	0.327139	+	0.944976i	$0.393916\pi$
$992$	2.56231	0.0813533
$993$	0	0
$994$	−1.11146	−0.0352532
$995$	−8.83282	−0.280019
$996$	0	0
$997$	7.38197	0.233789	0.116895	−	0.993144i	$-0.462706\pi$
$997$	7.38197	0.233789	0.116895	+	0.993144i	$0.462706\pi$
$998$	14.6738	0.464490
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9801.2.a.bb.1.1		2
3.2	odd	2		9801.2.a.n.1.2		2
9.2	odd	6		1089.2.e.g.364.1		4
9.5	odd	6		1089.2.e.g.727.1		4
11.3	even	5		891.2.f.a.163.1		4
11.4	even	5		891.2.f.a.82.1		4
11.10	odd	2		9801.2.a.m.1.2		2
33.14	odd	10		891.2.f.b.163.1		4
33.26	odd	10		891.2.f.b.82.1		4
33.32	even	2		9801.2.a.bc.1.1		2
99.4	even	15		297.2.n.a.181.1		8
99.14	odd	30		99.2.m.a.97.1	yes	8
99.25	even	15		297.2.n.a.64.1		8
99.32	even	6		1089.2.e.d.727.2		4
99.47	odd	30		99.2.m.a.31.1	yes	8
99.58	even	15		297.2.n.a.262.1		8
99.59	odd	30		99.2.m.a.16.1	✓	8
99.65	even	6		1089.2.e.d.364.2		4
99.70	even	15		297.2.n.a.280.1		8
99.92	odd	30		99.2.m.a.49.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.2.m.a.16.1	✓	8	99.59	odd	30
99.2.m.a.31.1	yes	8	99.47	odd	30
99.2.m.a.49.1	yes	8	99.92	odd	30
99.2.m.a.97.1	yes	8	99.14	odd	30
297.2.n.a.64.1		8	99.25	even	15
297.2.n.a.181.1		8	99.4	even	15
297.2.n.a.262.1		8	99.58	even	15
297.2.n.a.280.1		8	99.70	even	15
891.2.f.a.82.1		4	11.4	even	5
891.2.f.a.163.1		4	11.3	even	5
891.2.f.b.82.1		4	33.26	odd	10
891.2.f.b.163.1		4	33.14	odd	10
1089.2.e.d.364.2		4	99.65	even	6
1089.2.e.d.727.2		4	99.32	even	6
1089.2.e.g.364.1		4	9.2	odd	6
1089.2.e.g.727.1		4	9.5	odd	6
9801.2.a.m.1.2		2	11.10	odd	2
9801.2.a.n.1.2		2	3.2	odd	2
9801.2.a.bb.1.1		2	1.1	even	1	trivial
9801.2.a.bc.1.1		2	33.32	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 9801 = 3^{4} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9801.a (trivial)

\(f(q)\)	\(=\)	\(q+0.381966 q^{2} -1.85410 q^{4} +1.23607 q^{5} -1.00000 q^{7} -1.47214 q^{8} +0.472136 q^{10} -6.47214 q^{13} -0.381966 q^{14} +3.14590 q^{16} -4.85410 q^{17} +1.00000 q^{19} -2.29180 q^{20} -4.61803 q^{23} -3.47214 q^{25} -2.47214 q^{26} +1.85410 q^{28} +4.85410 q^{29} +0.618034 q^{31} +4.14590 q^{32} -1.85410 q^{34} -1.23607 q^{35} -5.09017 q^{37} +0.381966 q^{38} -1.81966 q^{40} +2.52786 q^{41} +1.85410 q^{43} -1.76393 q^{46} -11.9443 q^{47} -6.00000 q^{49} -1.32624 q^{50} +12.0000 q^{52} -4.09017 q^{53} +1.47214 q^{56} +1.85410 q^{58} -1.61803 q^{59} +10.8541 q^{61} +0.236068 q^{62} -4.70820 q^{64} -8.00000 q^{65} +6.00000 q^{67} +9.00000 q^{68} -0.472136 q^{70} +2.90983 q^{71} -0.145898 q^{73} -1.94427 q^{74} -1.85410 q^{76} +10.5623 q^{79} +3.88854 q^{80} +0.965558 q^{82} +9.76393 q^{83} -6.00000 q^{85} +0.708204 q^{86} +6.76393 q^{89} +6.47214 q^{91} +8.56231 q^{92} -4.56231 q^{94} +1.23607 q^{95} +6.00000 q^{97} -2.29180 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{2} + 3 q^{4} - 2 q^{5} - 2 q^{7} + 6 q^{8} - 8 q^{10} - 4 q^{13} - 3 q^{14} + 13 q^{16} - 3 q^{17} + 2 q^{19} - 18 q^{20} - 7 q^{23} + 2 q^{25} + 4 q^{26} - 3 q^{28} + 3 q^{29} - q^{31} + 15 q^{32}+ \cdots - 18 q^{98}+O(q^{100}) \) 2 * q + 3 * q^2 + 3 * q^4 - 2 * q^5 - 2 * q^7 + 6 * q^8 - 8 * q^10 - 4 * q^13 - 3 * q^14 + 13 * q^16 - 3 * q^17 + 2 * q^19 - 18 * q^20 - 7 * q^23 + 2 * q^25 + 4 * q^26 - 3 * q^28 + 3 * q^29 - q^31 + 15 * q^32 + 3 * q^34 + 2 * q^35 + q^37 + 3 * q^38 - 26 * q^40 + 14 * q^41 - 3 * q^43 - 8 * q^46 - 6 * q^47 - 12 * q^49 + 13 * q^50 + 24 * q^52 + 3 * q^53 - 6 * q^56 - 3 * q^58 - q^59 + 15 * q^61 - 4 * q^62 + 4 * q^64 - 16 * q^65 + 12 * q^67 + 18 * q^68 + 8 * q^70 + 17 * q^71 - 7 * q^73 + 14 * q^74 + 3 * q^76 + q^79 - 28 * q^80 + 31 * q^82 + 24 * q^83 - 12 * q^85 - 12 * q^86 + 18 * q^89 + 4 * q^91 - 3 * q^92 + 11 * q^94 - 2 * q^95 + 12 * q^97 - 18 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more