LMFDB - Embedded newform 2475.2.a.bd.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,2,Mod(1,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, 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("2475.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2475.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:	$19.7629745003$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.568.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x - 2 $ x^3 - x^2 - 6*x - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 825)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.76156$ of defining polynomial
Character	$\chi$	$=$	2475.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.76156 q^{2} +5.62620 q^{4} +1.86464 q^{7} +10.0140 q^{8} +O(q^{10})$	$q+2.76156 q^{2} +5.62620 q^{4} +1.86464 q^{7} +10.0140 q^{8} -1.00000 q^{11} +4.62620 q^{13} +5.14931 q^{14} +16.4017 q^{16} -2.49084 q^{17} -5.38776 q^{19} -2.76156 q^{22} -7.14931 q^{23} +12.7755 q^{26} +10.4908 q^{28} +3.52311 q^{29} +8.62620 q^{31} +25.2663 q^{32} -6.87859 q^{34} -8.87859 q^{37} -14.8786 q^{38} +0.761557 q^{41} -7.40171 q^{43} -5.62620 q^{44} -19.7432 q^{46} +0.373802 q^{47} -3.52311 q^{49} +26.0279 q^{52} +5.45856 q^{53} +18.6724 q^{56} +9.72928 q^{58} -5.14931 q^{59} +4.42003 q^{61} +23.8217 q^{62} +36.9711 q^{64} +11.9431 q^{67} -14.0140 q^{68} -11.6262 q^{71} +6.77551 q^{73} -24.5187 q^{74} -30.3126 q^{76} -1.86464 q^{77} -6.01395 q^{79} +2.10308 q^{82} -14.5693 q^{83} -20.4402 q^{86} -10.0140 q^{88} -9.04623 q^{89} +8.62620 q^{91} -40.2234 q^{92} +1.03228 q^{94} +16.3169 q^{97} -9.72928 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{2} + 8 q^{4} + 3 q^{7} + 6 q^{8}+O(q^{10}) $ 3 * q + 2 * q^2 + 8 * q^4 + 3 * q^7 + 6 * q^8	$ 3 q + 2 q^{2} + 8 q^{4} + 3 q^{7} + 6 q^{8} - 3 q^{11} + 5 q^{13} - 6 q^{14} + 10 q^{16} + 4 q^{17} - q^{19} - 2 q^{22} + 8 q^{26} + 20 q^{28} - 2 q^{29} + 17 q^{31} + 34 q^{32} + 6 q^{34} - 18 q^{38} - 4 q^{41} + 17 q^{43} - 8 q^{44} - 30 q^{46} + 10 q^{47} + 2 q^{49} + 30 q^{52} + 6 q^{53} + 22 q^{56} + 24 q^{58} + 6 q^{59} - 3 q^{61} + 16 q^{62} + 34 q^{64} + 7 q^{67} - 18 q^{68} - 26 q^{71} - 10 q^{73} - 14 q^{74} - 24 q^{76} - 3 q^{77} + 6 q^{79} + 10 q^{82} - 6 q^{83} - 28 q^{86} - 6 q^{88} - 2 q^{89} + 17 q^{91} - 26 q^{92} + 2 q^{94} + 29 q^{97} - 24 q^{98}+O(q^{100}) $ 3 * q + 2 * q^2 + 8 * q^4 + 3 * q^7 + 6 * q^8 - 3 * q^11 + 5 * q^13 - 6 * q^14 + 10 * q^16 + 4 * q^17 - q^19 - 2 * q^22 + 8 * q^26 + 20 * q^28 - 2 * q^29 + 17 * q^31 + 34 * q^32 + 6 * q^34 - 18 * q^38 - 4 * q^41 + 17 * q^43 - 8 * q^44 - 30 * q^46 + 10 * q^47 + 2 * q^49 + 30 * q^52 + 6 * q^53 + 22 * q^56 + 24 * q^58 + 6 * q^59 - 3 * q^61 + 16 * q^62 + 34 * q^64 + 7 * q^67 - 18 * q^68 - 26 * q^71 - 10 * q^73 - 14 * q^74 - 24 * q^76 - 3 * q^77 + 6 * q^79 + 10 * q^82 - 6 * q^83 - 28 * q^86 - 6 * q^88 - 2 * q^89 + 17 * q^91 - 26 * q^92 + 2 * q^94 + 29 * q^97 - 24 * 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 $

$2$	2.76156	1.95272	0.976358	−	0.216160i	$-0.0693534\pi$
$2$	2.76156	1.95272	0.976358	+	0.216160i	$0.0693534\pi$
$3$	0	0
$3$	0	0
$4$	5.62620	2.81310
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.86464	0.704768	0.352384	−	0.935855i	$-0.385371\pi$
$7$	1.86464	0.704768	0.352384	+	0.935855i	$0.385371\pi$
$8$	10.0140	3.54047
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	4.62620	1.28308	0.641538	−	0.767091i	$-0.278297\pi$
$13$	4.62620	1.28308	0.641538	+	0.767091i	$0.278297\pi$
$14$	5.14931	1.37621
$15$	0	0
$16$	16.4017	4.10043
$17$	−2.49084	−0.604117	−0.302059	−	0.953289i	$-0.597674\pi$
$17$	−2.49084	−0.604117	−0.302059	+	0.953289i	$0.597674\pi$
$18$	0	0
$19$	−5.38776	−1.23604	−0.618018	−	0.786164i	$-0.712064\pi$
$19$	−5.38776	−1.23604	−0.618018	+	0.786164i	$0.712064\pi$
$20$	0	0
$21$	0	0
$22$	−2.76156	−0.588766
$23$	−7.14931	−1.49073	−0.745367	−	0.666654i	$-0.767726\pi$
$23$	−7.14931	−1.49073	−0.745367	+	0.666654i	$0.767726\pi$
$24$	0	0
$25$	0	0
$26$	12.7755	2.50548
$27$	0	0
$28$	10.4908	1.98258
$29$	3.52311	0.654226	0.327113	−	0.944985i	$-0.393924\pi$
$29$	3.52311	0.654226	0.327113	+	0.944985i	$0.393924\pi$
$30$	0	0
$31$	8.62620	1.54931	0.774655	−	0.632384i	$-0.217923\pi$
$31$	8.62620	1.54931	0.774655	+	0.632384i	$0.217923\pi$
$32$	25.2663	4.46650
$33$	0	0
$34$	−6.87859	−1.17967
$35$	0	0
$36$	0	0
$37$	−8.87859	−1.45963	−0.729816	−	0.683644i	$-0.760394\pi$
$37$	−8.87859	−1.45963	−0.729816	+	0.683644i	$0.760394\pi$
$38$	−14.8786	−2.41363
$39$	0	0
$40$	0	0
$41$	0.761557	0.118935	0.0594676	−	0.998230i	$-0.481060\pi$
$41$	0.761557	0.118935	0.0594676	+	0.998230i	$0.481060\pi$
$42$	0	0
$43$	−7.40171	−1.12875	−0.564375	−	0.825519i	$-0.690883\pi$
$43$	−7.40171	−1.12875	−0.564375	+	0.825519i	$0.690883\pi$
$44$	−5.62620	−0.848181
$45$	0	0
$46$	−19.7432	−2.91098
$47$	0.373802	0.0545246	0.0272623	−	0.999628i	$-0.491321\pi$
$47$	0.373802	0.0545246	0.0272623	+	0.999628i	$0.491321\pi$
$48$	0	0
$49$	−3.52311	−0.503302
$50$	0	0
$51$	0	0
$52$	26.0279	3.60942
$53$	5.45856	0.749791	0.374896	−	0.927067i	$-0.377679\pi$
$53$	5.45856	0.749791	0.374896	+	0.927067i	$0.377679\pi$
$54$	0	0
$55$	0	0
$56$	18.6724	2.49521
$57$	0	0
$58$	9.72928	1.27752
$59$	−5.14931	−0.670383	−0.335192	−	0.942150i	$-0.608801\pi$
$59$	−5.14931	−0.670383	−0.335192	+	0.942150i	$0.608801\pi$
$60$	0	0
$61$	4.42003	0.565927	0.282963	−	0.959131i	$-0.408682\pi$
$61$	4.42003	0.565927	0.282963	+	0.959131i	$0.408682\pi$
$62$	23.8217	3.02536
$63$	0	0
$64$	36.9711	4.62138
$65$	0	0
$66$	0	0
$67$	11.9431	1.45909	0.729544	−	0.683934i	$-0.239732\pi$
$67$	11.9431	1.45909	0.729544	+	0.683934i	$0.239732\pi$
$68$	−14.0140	−1.69944
$69$	0	0
$70$	0	0
$71$	−11.6262	−1.37978	−0.689888	−	0.723916i	$-0.742340\pi$
$71$	−11.6262	−1.37978	−0.689888	+	0.723916i	$0.742340\pi$
$72$	0	0
$73$	6.77551	0.793014	0.396507	−	0.918032i	$-0.370222\pi$
$73$	6.77551	0.793014	0.396507	+	0.918032i	$0.370222\pi$
$74$	−24.5187	−2.85025
$75$	0	0
$76$	−30.3126	−3.47709
$77$	−1.86464	−0.212496
$78$	0	0
$79$	−6.01395	−0.676623	−0.338311	−	0.941034i	$-0.609856\pi$
$79$	−6.01395	−0.676623	−0.338311	+	0.941034i	$0.609856\pi$
$80$	0	0
$81$	0	0
$82$	2.10308	0.232247
$83$	−14.5693	−1.59919	−0.799597	−	0.600538i	$-0.794953\pi$
$83$	−14.5693	−1.59919	−0.799597	+	0.600538i	$0.794953\pi$
$84$	0	0
$85$	0	0
$86$	−20.4402	−2.20413
$87$	0	0
$88$	−10.0140	−1.06749
$89$	−9.04623	−0.958898	−0.479449	−	0.877570i	$-0.659164\pi$
$89$	−9.04623	−0.958898	−0.479449	+	0.877570i	$0.659164\pi$
$90$	0	0
$91$	8.62620	0.904271
$92$	−40.2234	−4.19358
$93$	0	0
$94$	1.03228	0.106471
$95$	0	0
$96$	0	0
$97$	16.3169	1.65673	0.828367	−	0.560185i	$-0.189270\pi$
$97$	16.3169	1.65673	0.828367	+	0.560185i	$0.189270\pi$
$98$	−9.72928	−0.982806
$99$	0	0
$100$	0	0
$101$	1.03228	0.102715	0.0513576	−	0.998680i	$-0.483645\pi$
$101$	1.03228	0.102715	0.0513576	+	0.998680i	$0.483645\pi$
$102$	0	0
$103$	3.04623	0.300154	0.150077	−	0.988674i	$-0.452048\pi$
$103$	3.04623	0.300154	0.150077	+	0.988674i	$0.452048\pi$
$104$	46.3265	4.54269
$105$	0	0
$106$	15.0741	1.46413
$107$	8.50479	0.822189	0.411095	−	0.911593i	$-0.365147\pi$
$107$	8.50479	0.822189	0.411095	+	0.911593i	$0.365147\pi$
$108$	0	0
$109$	−6.14931	−0.588997	−0.294499	−	0.955652i	$-0.595153\pi$
$109$	−6.14931	−0.588997	−0.294499	+	0.955652i	$0.595153\pi$
$110$	0	0
$111$	0	0
$112$	30.5833	2.88985
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	0	0
$116$	19.8217	1.84040
$117$	0	0
$118$	−14.2201	−1.30907
$119$	−4.64452	−0.425762
$120$	0	0
$121$	1.00000	0.0909091
$122$	12.2062	1.10509
$123$	0	0
$124$	48.5327	4.35837
$125$	0	0
$126$	0	0
$127$	−9.26635	−0.822256	−0.411128	−	0.911578i	$-0.634865\pi$
$127$	−9.26635	−0.822256	−0.411128	+	0.911578i	$0.634865\pi$
$128$	51.5650	4.55774
$129$	0	0
$130$	0	0
$131$	−4.06455	−0.355121	−0.177561	−	0.984110i	$-0.556821\pi$
$131$	−4.06455	−0.355121	−0.177561	+	0.984110i	$0.556821\pi$
$132$	0	0
$133$	−10.0462	−0.871119
$134$	32.9817	2.84918
$135$	0	0
$136$	−24.9431	−2.13886
$137$	−5.93545	−0.507100	−0.253550	−	0.967322i	$-0.581598\pi$
$137$	−5.93545	−0.507100	−0.253550	+	0.967322i	$0.581598\pi$
$138$	0	0
$139$	6.71096	0.569216	0.284608	−	0.958644i	$-0.408137\pi$
$139$	6.71096	0.569216	0.284608	+	0.958644i	$0.408137\pi$
$140$	0	0
$141$	0	0
$142$	−32.1064	−2.69431
$143$	−4.62620	−0.386862
$144$	0	0
$145$	0	0
$146$	18.7110	1.54853
$147$	0	0
$148$	−49.9527	−4.10609
$149$	−5.74324	−0.470504	−0.235252	−	0.971934i	$-0.575592\pi$
$149$	−5.74324	−0.470504	−0.235252	+	0.971934i	$0.575592\pi$
$150$	0	0
$151$	10.5616	0.859495	0.429747	−	0.902949i	$-0.358603\pi$
$151$	10.5616	0.859495	0.429747	+	0.902949i	$0.358603\pi$
$152$	−53.9527	−4.37614
$153$	0	0
$154$	−5.14931	−0.414943
$155$	0	0
$156$	0	0
$157$	3.10308	0.247653	0.123827	−	0.992304i	$-0.460483\pi$
$157$	3.10308	0.247653	0.123827	+	0.992304i	$0.460483\pi$
$158$	−16.6079	−1.32125
$159$	0	0
$160$	0	0
$161$	−13.3309	−1.05062
$162$	0	0
$163$	15.6724	1.22756	0.613780	−	0.789477i	$-0.289648\pi$
$163$	15.6724	1.22756	0.613780	+	0.789477i	$0.289648\pi$
$164$	4.28467	0.334577
$165$	0	0
$166$	−40.2341	−3.12277
$167$	8.98168	0.695023	0.347512	−	0.937676i	$-0.387027\pi$
$167$	8.98168	0.695023	0.347512	+	0.937676i	$0.387027\pi$
$168$	0	0
$169$	8.40171	0.646285
$170$	0	0
$171$	0	0
$172$	−41.6435	−3.17529
$173$	11.5092	0.875025	0.437513	−	0.899212i	$-0.355860\pi$
$173$	11.5092	0.875025	0.437513	+	0.899212i	$0.355860\pi$
$174$	0	0
$175$	0	0
$176$	−16.4017	−1.23633
$177$	0	0
$178$	−24.9817	−1.87246
$179$	−10.3738	−0.775374	−0.387687	−	0.921791i	$-0.626726\pi$
$179$	−10.3738	−0.775374	−0.387687	+	0.921791i	$0.626726\pi$
$180$	0	0
$181$	−2.66473	−0.198068	−0.0990339	−	0.995084i	$-0.531575\pi$
$181$	−2.66473	−0.198068	−0.0990339	+	0.995084i	$0.531575\pi$
$182$	23.8217	1.76578
$183$	0	0
$184$	−71.5929	−5.27790
$185$	0	0
$186$	0	0
$187$	2.49084	0.182148
$188$	2.10308	0.153383
$189$	0	0
$190$	0	0
$191$	−14.6724	−1.06166	−0.530830	−	0.847478i	$-0.678120\pi$
$191$	−14.6724	−1.06166	−0.530830	+	0.847478i	$0.678120\pi$
$192$	0	0
$193$	5.10308	0.367328	0.183664	−	0.982989i	$-0.441204\pi$
$193$	5.10308	0.367328	0.183664	+	0.982989i	$0.441204\pi$
$194$	45.0602	3.23513
$195$	0	0
$196$	−19.8217	−1.41584
$197$	−3.74324	−0.266694	−0.133347	−	0.991069i	$-0.542573\pi$
$197$	−3.74324	−0.266694	−0.133347	+	0.991069i	$0.542573\pi$
$198$	0	0
$199$	−8.08476	−0.573114	−0.286557	−	0.958063i	$-0.592511\pi$
$199$	−8.08476	−0.573114	−0.286557	+	0.958063i	$0.592511\pi$
$200$	0	0
$201$	0	0
$202$	2.85069	0.200574
$203$	6.56934	0.461077
$204$	0	0
$205$	0	0
$206$	8.41233	0.586115
$207$	0	0
$208$	75.8776	5.26116
$209$	5.38776	0.372679
$210$	0	0
$211$	−15.9431	−1.09757	−0.548786	−	0.835963i	$-0.684910\pi$
$211$	−15.9431	−1.09757	−0.548786	+	0.835963i	$0.684910\pi$
$212$	30.7110	2.10924
$213$	0	0
$214$	23.4865	1.60550
$215$	0	0
$216$	0	0
$217$	16.0848	1.09190
$218$	−16.9817	−1.15014
$219$	0	0
$220$	0	0
$221$	−11.5231	−0.775129
$222$	0	0
$223$	−22.1772	−1.48510	−0.742548	−	0.669793i	$-0.766383\pi$
$223$	−22.1772	−1.48510	−0.742548	+	0.669793i	$0.766383\pi$
$224$	47.1127	3.14785
$225$	0	0
$226$	16.5693	1.10218
$227$	−5.93545	−0.393950	−0.196975	−	0.980409i	$-0.563112\pi$
$227$	−5.93545	−0.393950	−0.196975	+	0.980409i	$0.563112\pi$
$228$	0	0
$229$	−8.31695	−0.549599	−0.274800	−	0.961502i	$-0.588612\pi$
$229$	−8.31695	−0.549599	−0.274800	+	0.961502i	$0.588612\pi$
$230$	0	0
$231$	0	0
$232$	35.2803	2.31627
$233$	−28.5187	−1.86833	−0.934163	−	0.356848i	$-0.883852\pi$
$233$	−28.5187	−1.86833	−0.934163	+	0.356848i	$0.883852\pi$
$234$	0	0
$235$	0	0
$236$	−28.9711	−1.88585
$237$	0	0
$238$	−12.8261	−0.831393
$239$	−24.0925	−1.55841	−0.779206	−	0.626768i	$-0.784377\pi$
$239$	−24.0925	−1.55841	−0.779206	+	0.626768i	$0.784377\pi$
$240$	0	0
$241$	−5.16763	−0.332877	−0.166438	−	0.986052i	$-0.553227\pi$
$241$	−5.16763	−0.332877	−0.166438	+	0.986052i	$0.553227\pi$
$242$	2.76156	0.177520
$243$	0	0
$244$	24.8680	1.59201
$245$	0	0
$246$	0	0
$247$	−24.9248	−1.58593
$248$	86.3823	5.48528
$249$	0	0
$250$	0	0
$251$	9.25240	0.584006	0.292003	−	0.956417i	$-0.405678\pi$
$251$	9.25240	0.584006	0.292003	+	0.956417i	$0.405678\pi$
$252$	0	0
$253$	7.14931	0.449473
$254$	−25.5896	−1.60563
$255$	0	0
$256$	68.4575	4.27860
$257$	25.2158	1.57292	0.786458	−	0.617644i	$-0.211913\pi$
$257$	25.2158	1.57292	0.786458	+	0.617644i	$0.211913\pi$
$258$	0	0
$259$	−16.5554	−1.02870
$260$	0	0
$261$	0	0
$262$	−11.2245	−0.693451
$263$	−5.45856	−0.336589	−0.168295	−	0.985737i	$-0.553826\pi$
$263$	−5.45856	−0.336589	−0.168295	+	0.985737i	$0.553826\pi$
$264$	0	0
$265$	0	0
$266$	−27.7432	−1.70105
$267$	0	0
$268$	67.1945	4.10456
$269$	−17.0741	−1.04103	−0.520514	−	0.853853i	$-0.674260\pi$
$269$	−17.0741	−1.04103	−0.520514	+	0.853853i	$0.674260\pi$
$270$	0	0
$271$	9.77988	0.594085	0.297043	−	0.954864i	$-0.404000\pi$
$271$	9.77988	0.594085	0.297043	+	0.954864i	$0.404000\pi$
$272$	−40.8540	−2.47714
$273$	0	0
$274$	−16.3911	−0.990221
$275$	0	0
$276$	0	0
$277$	5.91524	0.355412	0.177706	−	0.984084i	$-0.443132\pi$
$277$	5.91524	0.355412	0.177706	+	0.984084i	$0.443132\pi$
$278$	18.5327	1.11152
$279$	0	0
$280$	0	0
$281$	6.76156	0.403361	0.201680	−	0.979451i	$-0.435360\pi$
$281$	6.76156	0.403361	0.201680	+	0.979451i	$0.435360\pi$
$282$	0	0
$283$	8.81841	0.524200	0.262100	−	0.965041i	$-0.415585\pi$
$283$	8.81841	0.524200	0.262100	+	0.965041i	$0.415585\pi$
$284$	−65.4113	−3.88145
$285$	0	0
$286$	−12.7755	−0.755432
$287$	1.42003	0.0838218
$288$	0	0
$289$	−10.7957	−0.635042
$290$	0	0
$291$	0	0
$292$	38.1204	2.23083
$293$	30.4908	1.78129	0.890647	−	0.454696i	$-0.150252\pi$
$293$	30.4908	1.78129	0.890647	+	0.454696i	$0.150252\pi$
$294$	0	0
$295$	0	0
$296$	−88.9098	−5.16778
$297$	0	0
$298$	−15.8603	−0.918761
$299$	−33.0741	−1.91273
$300$	0	0
$301$	−13.8015	−0.795507
$302$	29.1666	1.67835
$303$	0	0
$304$	−88.3684	−5.06827
$305$	0	0
$306$	0	0
$307$	0.767815	0.0438215	0.0219107	−	0.999760i	$-0.493025\pi$
$307$	0.767815	0.0438215	0.0219107	+	0.999760i	$0.493025\pi$
$308$	−10.4908	−0.597771
$309$	0	0
$310$	0	0
$311$	33.8496	1.91944	0.959719	−	0.280963i	$-0.0906538\pi$
$311$	33.8496	1.91944	0.959719	+	0.280963i	$0.0906538\pi$
$312$	0	0
$313$	−8.11078	−0.458448	−0.229224	−	0.973374i	$-0.573619\pi$
$313$	−8.11078	−0.458448	−0.229224	+	0.973374i	$0.573619\pi$
$314$	8.56934	0.483596
$315$	0	0
$316$	−33.8357	−1.90341
$317$	−8.06455	−0.452950	−0.226475	−	0.974017i	$-0.572720\pi$
$317$	−8.06455	−0.452950	−0.226475	+	0.974017i	$0.572720\pi$
$318$	0	0
$319$	−3.52311	−0.197257
$320$	0	0
$321$	0	0
$322$	−36.8140	−2.05157
$323$	13.4200	0.746710
$324$	0	0
$325$	0	0
$326$	43.2803	2.39707
$327$	0	0
$328$	7.62620	0.421086
$329$	0.697006	0.0384272
$330$	0	0
$331$	22.4769	1.23544	0.617721	−	0.786398i	$-0.288056\pi$
$331$	22.4769	1.23544	0.617721	+	0.786398i	$0.288056\pi$
$332$	−81.9700	−4.49869
$333$	0	0
$334$	24.8034	1.35718
$335$	0	0
$336$	0	0
$337$	6.32757	0.344685	0.172342	−	0.985037i	$-0.444866\pi$
$337$	6.32757	0.344685	0.172342	+	0.985037i	$0.444866\pi$
$338$	23.2018	1.26201
$339$	0	0
$340$	0	0
$341$	−8.62620	−0.467135
$342$	0	0
$343$	−19.6218	−1.05948
$344$	−74.1204	−3.99630
$345$	0	0
$346$	31.7832	1.70868
$347$	0.178261	0.00956954	0.00478477	−	0.999989i	$-0.498477\pi$
$347$	0.178261	0.00956954	0.00478477	+	0.999989i	$0.498477\pi$
$348$	0	0
$349$	−19.7293	−1.05608	−0.528042	−	0.849218i	$-0.677074\pi$
$349$	−19.7293	−1.05608	−0.528042	+	0.849218i	$0.677074\pi$
$350$	0	0
$351$	0	0
$352$	−25.2663	−1.34670
$353$	19.5231	1.03911	0.519555	−	0.854437i	$-0.326098\pi$
$353$	19.5231	1.03911	0.519555	+	0.854437i	$0.326098\pi$
$354$	0	0
$355$	0	0
$356$	−50.8959	−2.69748
$357$	0	0
$358$	−28.6478	−1.51409
$359$	−35.6926	−1.88379	−0.941893	−	0.335914i	$-0.890955\pi$
$359$	−35.6926	−1.88379	−0.941893	+	0.335914i	$0.890955\pi$
$360$	0	0
$361$	10.0279	0.527785
$362$	−7.35881	−0.386770
$363$	0	0
$364$	48.5327	2.54380
$365$	0	0
$366$	0	0
$367$	20.1127	1.04987	0.524936	−	0.851141i	$-0.324089\pi$
$367$	20.1127	1.04987	0.524936	+	0.851141i	$0.324089\pi$
$368$	−117.261	−6.11265
$369$	0	0
$370$	0	0
$371$	10.1783	0.528429
$372$	0	0
$373$	−2.56165	−0.132637	−0.0663185	−	0.997799i	$-0.521125\pi$
$373$	−2.56165	−0.132637	−0.0663185	+	0.997799i	$0.521125\pi$
$374$	6.87859	0.355684
$375$	0	0
$376$	3.74324	0.193043
$377$	16.2986	0.839422
$378$	0	0
$379$	23.2601	1.19479	0.597395	−	0.801947i	$-0.296202\pi$
$379$	23.2601	1.19479	0.597395	+	0.801947i	$0.296202\pi$
$380$	0	0
$381$	0	0
$382$	−40.5187	−2.07312
$383$	−25.7938	−1.31800	−0.659002	−	0.752142i	$-0.729021\pi$
$383$	−25.7938	−1.31800	−0.659002	+	0.752142i	$0.729021\pi$
$384$	0	0
$385$	0	0
$386$	14.0925	0.717287
$387$	0	0
$388$	91.8024	4.66056
$389$	−2.33527	−0.118403	−0.0592014	−	0.998246i	$-0.518855\pi$
$389$	−2.33527	−0.118403	−0.0592014	+	0.998246i	$0.518855\pi$
$390$	0	0
$391$	17.8078	0.900578
$392$	−35.2803	−1.78192
$393$	0	0
$394$	−10.3372	−0.520778
$395$	0	0
$396$	0	0
$397$	17.1955	0.863019	0.431510	−	0.902108i	$-0.357981\pi$
$397$	17.1955	0.863019	0.431510	+	0.902108i	$0.357981\pi$
$398$	−22.3265	−1.11913
$399$	0	0
$400$	0	0
$401$	26.5693	1.32681	0.663405	−	0.748261i	$-0.269111\pi$
$401$	26.5693	1.32681	0.663405	+	0.748261i	$0.269111\pi$
$402$	0	0
$403$	39.9065	1.98788
$404$	5.80779	0.288948
$405$	0	0
$406$	18.1416	0.900353
$407$	8.87859	0.440096
$408$	0	0
$409$	−20.4200	−1.00971	−0.504853	−	0.863205i	$-0.668453\pi$
$409$	−20.4200	−1.00971	−0.504853	+	0.863205i	$0.668453\pi$
$410$	0	0
$411$	0	0
$412$	17.1387	0.844362
$413$	−9.60162	−0.472465
$414$	0	0
$415$	0	0
$416$	116.887	5.73086
$417$	0	0
$418$	14.8786	0.727736
$419$	13.5896	0.663893	0.331947	−	0.943298i	$-0.392295\pi$
$419$	13.5896	0.663893	0.331947	+	0.943298i	$0.392295\pi$
$420$	0	0
$421$	−39.6175	−1.93084	−0.965418	−	0.260705i	$-0.916045\pi$
$421$	−39.6175	−1.93084	−0.965418	+	0.260705i	$0.916045\pi$
$422$	−44.0279	−2.14324
$423$	0	0
$424$	54.6618	2.65461
$425$	0	0
$426$	0	0
$427$	8.24177	0.398847
$428$	47.8496	2.31290
$429$	0	0
$430$	0	0
$431$	6.56934	0.316434	0.158217	−	0.987404i	$-0.449425\pi$
$431$	6.56934	0.316434	0.158217	+	0.987404i	$0.449425\pi$
$432$	0	0
$433$	−36.4113	−1.74982	−0.874908	−	0.484290i	$-0.839078\pi$
$433$	−36.4113	−1.74982	−0.874908	+	0.484290i	$0.839078\pi$
$434$	44.4190	2.13218
$435$	0	0
$436$	−34.5972	−1.65691
$437$	38.5187	1.84260
$438$	0	0
$439$	25.3878	1.21169	0.605846	−	0.795582i	$-0.292835\pi$
$439$	25.3878	1.21169	0.605846	+	0.795582i	$0.292835\pi$
$440$	0	0
$441$	0	0
$442$	−31.8217	−1.51361
$443$	20.9065	0.993298	0.496649	−	0.867952i	$-0.334564\pi$
$443$	20.9065	0.993298	0.496649	+	0.867952i	$0.334564\pi$
$444$	0	0
$445$	0	0
$446$	−61.2437	−2.89997
$447$	0	0
$448$	68.9377	3.25700
$449$	34.9450	1.64916	0.824579	−	0.565747i	$-0.191412\pi$
$449$	34.9450	1.64916	0.824579	+	0.565747i	$0.191412\pi$
$450$	0	0
$451$	−0.761557	−0.0358603
$452$	33.7572	1.58780
$453$	0	0
$454$	−16.3911	−0.769272
$455$	0	0
$456$	0	0
$457$	7.31695	0.342272	0.171136	−	0.985247i	$-0.445256\pi$
$457$	7.31695	0.342272	0.171136	+	0.985247i	$0.445256\pi$
$458$	−22.9677	−1.07321
$459$	0	0
$460$	0	0
$461$	14.0279	0.653345	0.326672	−	0.945138i	$-0.394073\pi$
$461$	14.0279	0.653345	0.326672	+	0.945138i	$0.394073\pi$
$462$	0	0
$463$	−4.33527	−0.201477	−0.100739	−	0.994913i	$-0.532121\pi$
$463$	−4.33527	−0.201477	−0.100739	+	0.994913i	$0.532121\pi$
$464$	57.7851	2.68261
$465$	0	0
$466$	−78.7561	−3.64831
$467$	3.70138	0.171279	0.0856396	−	0.996326i	$-0.472707\pi$
$467$	3.70138	0.171279	0.0856396	+	0.996326i	$0.472707\pi$
$468$	0	0
$469$	22.2697	1.02832
$470$	0	0
$471$	0	0
$472$	−51.5650	−2.37347
$473$	7.40171	0.340331
$474$	0	0
$475$	0	0
$476$	−26.1310	−1.19771
$477$	0	0
$478$	−66.5327	−3.04313
$479$	22.8680	1.04486	0.522432	−	0.852681i	$-0.325025\pi$
$479$	22.8680	1.04486	0.522432	+	0.852681i	$0.325025\pi$
$480$	0	0
$481$	−41.0741	−1.87282
$482$	−14.2707	−0.650013
$483$	0	0
$484$	5.62620	0.255736
$485$	0	0
$486$	0	0
$487$	5.42962	0.246039	0.123020	−	0.992404i	$-0.460742\pi$
$487$	5.42962	0.246039	0.123020	+	0.992404i	$0.460742\pi$
$488$	44.2620	2.00365
$489$	0	0
$490$	0	0
$491$	−21.2803	−0.960367	−0.480183	−	0.877168i	$-0.659430\pi$
$491$	−21.2803	−0.960367	−0.480183	+	0.877168i	$0.659430\pi$
$492$	0	0
$493$	−8.77551	−0.395229
$494$	−68.8313	−3.09687
$495$	0	0
$496$	141.484	6.35284
$497$	−21.6787	−0.972422
$498$	0	0
$499$	−16.0077	−0.716603	−0.358301	−	0.933606i	$-0.616644\pi$
$499$	−16.0077	−0.716603	−0.358301	+	0.933606i	$0.616644\pi$
$500$	0	0
$501$	0	0
$502$	25.5510	1.14040
$503$	12.0925	0.539176	0.269588	−	0.962976i	$-0.413112\pi$
$503$	12.0925	0.539176	0.269588	+	0.962976i	$0.413112\pi$
$504$	0	0
$505$	0	0
$506$	19.7432	0.877694
$507$	0	0
$508$	−52.1343	−2.31309
$509$	−14.7110	−0.652052	−0.326026	−	0.945361i	$-0.605710\pi$
$509$	−14.7110	−0.652052	−0.326026	+	0.945361i	$0.605710\pi$
$510$	0	0
$511$	12.6339	0.558891
$512$	85.9194	3.79714
$513$	0	0
$514$	69.6347	3.07146
$515$	0	0
$516$	0	0
$517$	−0.373802	−0.0164398
$518$	−45.7187	−2.00876
$519$	0	0
$520$	0	0
$521$	17.2803	0.757064	0.378532	−	0.925588i	$-0.376429\pi$
$521$	17.2803	0.757064	0.378532	+	0.925588i	$0.376429\pi$
$522$	0	0
$523$	42.7326	1.86857	0.934283	−	0.356532i	$-0.116041\pi$
$523$	42.7326	1.86857	0.934283	+	0.356532i	$0.116041\pi$
$524$	−22.8680	−0.998992
$525$	0	0
$526$	−15.0741	−0.657264
$527$	−21.4865	−0.935965
$528$	0	0
$529$	28.1127	1.22229
$530$	0	0
$531$	0	0
$532$	−56.5221	−2.45054
$533$	3.52311	0.152603
$534$	0	0
$535$	0	0
$536$	119.598	5.16585
$537$	0	0
$538$	−47.1512	−2.03283
$539$	3.52311	0.151751
$540$	0	0
$541$	8.69075	0.373644	0.186822	−	0.982394i	$-0.440181\pi$
$541$	8.69075	0.373644	0.186822	+	0.982394i	$0.440181\pi$
$542$	27.0077	1.16008
$543$	0	0
$544$	−62.9344	−2.69829
$545$	0	0
$546$	0	0
$547$	−44.1064	−1.88585	−0.942927	−	0.333000i	$-0.891939\pi$
$547$	−44.1064	−1.88585	−0.942927	+	0.333000i	$0.891939\pi$
$548$	−33.3940	−1.42652
$549$	0	0
$550$	0	0
$551$	−18.9817	−0.808647
$552$	0	0
$553$	−11.2139	−0.476862
$554$	16.3353	0.694019
$555$	0	0
$556$	37.7572	1.60126
$557$	7.69264	0.325948	0.162974	−	0.986630i	$-0.447891\pi$
$557$	7.69264	0.325948	0.162974	+	0.986630i	$0.447891\pi$
$558$	0	0
$559$	−34.2418	−1.44827
$560$	0	0
$561$	0	0
$562$	18.6724	0.787649
$563$	−9.25240	−0.389942	−0.194971	−	0.980809i	$-0.562461\pi$
$563$	−9.25240	−0.389942	−0.194971	+	0.980809i	$0.562461\pi$
$564$	0	0
$565$	0	0
$566$	24.3525	1.02361
$567$	0	0
$568$	−116.424	−4.88505
$569$	−4.96772	−0.208258	−0.104129	−	0.994564i	$-0.533205\pi$
$569$	−4.96772	−0.208258	−0.104129	+	0.994564i	$0.533205\pi$
$570$	0	0
$571$	6.02021	0.251938	0.125969	−	0.992034i	$-0.459796\pi$
$571$	6.02021	0.251938	0.125969	+	0.992034i	$0.459796\pi$
$572$	−26.0279	−1.08828
$573$	0	0
$574$	3.92150	0.163680
$575$	0	0
$576$	0	0
$577$	8.25240	0.343552	0.171776	−	0.985136i	$-0.445050\pi$
$577$	8.25240	0.343552	0.171776	+	0.985136i	$0.445050\pi$
$578$	−29.8130	−1.24006
$579$	0	0
$580$	0	0
$581$	−27.1666	−1.12706
$582$	0	0
$583$	−5.45856	−0.226071
$584$	67.8496	2.80764
$585$	0	0
$586$	84.2022	3.47836
$587$	11.4846	0.474019	0.237010	−	0.971507i	$-0.423833\pi$
$587$	11.4846	0.474019	0.237010	+	0.971507i	$0.423833\pi$
$588$	0	0
$589$	−46.4758	−1.91500
$590$	0	0
$591$	0	0
$592$	−145.624	−5.98511
$593$	39.0375	1.60308	0.801539	−	0.597943i	$-0.204015\pi$
$593$	39.0375	1.60308	0.801539	+	0.597943i	$0.204015\pi$
$594$	0	0
$595$	0	0
$596$	−32.3126	−1.32357
$597$	0	0
$598$	−91.3361	−3.73501
$599$	30.1589	1.23226	0.616130	−	0.787645i	$-0.288700\pi$
$599$	30.1589	1.23226	0.616130	+	0.787645i	$0.288700\pi$
$600$	0	0
$601$	19.1310	0.780369	0.390185	−	0.920737i	$-0.372411\pi$
$601$	19.1310	0.780369	0.390185	+	0.920737i	$0.372411\pi$
$602$	−38.1137	−1.55340
$603$	0	0
$604$	59.4219	2.41784
$605$	0	0
$606$	0	0
$607$	4.20617	0.170723	0.0853615	−	0.996350i	$-0.472795\pi$
$607$	4.20617	0.170723	0.0853615	+	0.996350i	$0.472795\pi$
$608$	−136.129	−5.52076
$609$	0	0
$610$	0	0
$611$	1.72928	0.0699593
$612$	0	0
$613$	5.45856	0.220469	0.110235	−	0.993906i	$-0.464840\pi$
$613$	5.45856	0.220469	0.110235	+	0.993906i	$0.464840\pi$
$614$	2.12036	0.0855709
$615$	0	0
$616$	−18.6724	−0.752334
$617$	−30.1974	−1.21570	−0.607851	−	0.794051i	$-0.707968\pi$
$617$	−30.1974	−1.21570	−0.607851	+	0.794051i	$0.707968\pi$
$618$	0	0
$619$	23.1955	0.932308	0.466154	−	0.884704i	$-0.345639\pi$
$619$	23.1955	0.932308	0.466154	+	0.884704i	$0.345639\pi$
$620$	0	0
$621$	0	0
$622$	93.4777	3.74812
$623$	−16.8680	−0.675801
$624$	0	0
$625$	0	0
$626$	−22.3984	−0.895219
$627$	0	0
$628$	17.4586	0.696673
$629$	22.1151	0.881789
$630$	0	0
$631$	−3.19554	−0.127212	−0.0636062	−	0.997975i	$-0.520260\pi$
$631$	−3.19554	−0.127212	−0.0636062	+	0.997975i	$0.520260\pi$
$632$	−60.2234	−2.39556
$633$	0	0
$634$	−22.2707	−0.884483
$635$	0	0
$636$	0	0
$637$	−16.2986	−0.645775
$638$	−9.72928	−0.385186
$639$	0	0
$640$	0	0
$641$	−28.1974	−1.11373	−0.556866	−	0.830603i	$-0.687996\pi$
$641$	−28.1974	−1.11373	−0.556866	+	0.830603i	$0.687996\pi$
$642$	0	0
$643$	17.9634	0.708406	0.354203	−	0.935169i	$-0.384752\pi$
$643$	17.9634	0.708406	0.354203	+	0.935169i	$0.384752\pi$
$644$	−75.0023	−2.95550
$645$	0	0
$646$	37.0602	1.45811
$647$	40.9990	1.61184	0.805918	−	0.592028i	$-0.201672\pi$
$647$	40.9990	1.61184	0.805918	+	0.592028i	$0.201672\pi$
$648$	0	0
$649$	5.14931	0.202128
$650$	0	0
$651$	0	0
$652$	88.1762	3.45325
$653$	−11.6926	−0.457568	−0.228784	−	0.973477i	$-0.573475\pi$
$653$	−11.6926	−0.457568	−0.228784	+	0.973477i	$0.573475\pi$
$654$	0	0
$655$	0	0
$656$	12.4908	0.487685
$657$	0	0
$658$	1.92482	0.0750374
$659$	1.72928	0.0673633	0.0336816	−	0.999433i	$-0.489277\pi$
$659$	1.72928	0.0673633	0.0336816	+	0.999433i	$0.489277\pi$
$660$	0	0
$661$	0.232185	0.00903096	0.00451548	−	0.999990i	$-0.498563\pi$
$661$	0.232185	0.00903096	0.00451548	+	0.999990i	$0.498563\pi$
$662$	62.0712	2.41247
$663$	0	0
$664$	−145.897	−5.66189
$665$	0	0
$666$	0	0
$667$	−25.1878	−0.975277
$668$	50.5327	1.95517
$669$	0	0
$670$	0	0
$671$	−4.42003	−0.170633
$672$	0	0
$673$	33.5144	1.29188	0.645942	−	0.763386i	$-0.276465\pi$
$673$	33.5144	1.29188	0.645942	+	0.763386i	$0.276465\pi$
$674$	17.4740	0.673072
$675$	0	0
$676$	47.2697	1.81806
$677$	13.0183	0.500335	0.250167	−	0.968203i	$-0.419514\pi$
$677$	13.0183	0.500335	0.250167	+	0.968203i	$0.419514\pi$
$678$	0	0
$679$	30.4252	1.16761
$680$	0	0
$681$	0	0
$682$	−23.8217	−0.912182
$683$	4.54333	0.173846	0.0869228	−	0.996215i	$-0.472297\pi$
$683$	4.54333	0.173846	0.0869228	+	0.996215i	$0.472297\pi$
$684$	0	0
$685$	0	0
$686$	−54.1868	−2.06886
$687$	0	0
$688$	−121.401	−4.62836
$689$	25.2524	0.962040
$690$	0	0
$691$	−19.8988	−0.756986	−0.378493	−	0.925604i	$-0.623558\pi$
$691$	−19.8988	−0.756986	−0.378493	+	0.925604i	$0.623558\pi$
$692$	64.7528	2.46153
$693$	0	0
$694$	0.492277	0.0186866
$695$	0	0
$696$	0	0
$697$	−1.89692	−0.0718508
$698$	−54.4835	−2.06223
$699$	0	0
$700$	0	0
$701$	15.8444	0.598436	0.299218	−	0.954185i	$-0.403274\pi$
$701$	15.8444	0.598436	0.299218	+	0.954185i	$0.403274\pi$
$702$	0	0
$703$	47.8357	1.80416
$704$	−36.9711	−1.39340
$705$	0	0
$706$	53.9142	2.02909
$707$	1.92482	0.0723904
$708$	0	0
$709$	8.82174	0.331307	0.165654	−	0.986184i	$-0.447027\pi$
$709$	8.82174	0.331307	0.165654	+	0.986184i	$0.447027\pi$
$710$	0	0
$711$	0	0
$712$	−90.5885	−3.39495
$713$	−61.6714	−2.30961
$714$	0	0
$715$	0	0
$716$	−58.3651	−2.18120
$717$	0	0
$718$	−98.5673	−3.67850
$719$	−36.7668	−1.37117	−0.685585	−	0.727993i	$-0.740453\pi$
$719$	−36.7668	−1.37117	−0.685585	+	0.727993i	$0.740453\pi$
$720$	0	0
$721$	5.68012	0.211539
$722$	27.6926	1.03061
$723$	0	0
$724$	−14.9923	−0.557185
$725$	0	0
$726$	0	0
$727$	−5.27261	−0.195550	−0.0977751	−	0.995209i	$-0.531173\pi$
$727$	−5.27261	−0.195550	−0.0977751	+	0.995209i	$0.531173\pi$
$728$	86.3823	3.20154
$729$	0	0
$730$	0	0
$731$	18.4365	0.681897
$732$	0	0
$733$	37.9508	1.40175	0.700873	−	0.713286i	$-0.252794\pi$
$733$	37.9508	1.40175	0.700873	+	0.713286i	$0.252794\pi$
$734$	55.5423	2.05010
$735$	0	0
$736$	−180.637	−6.65837
$737$	−11.9431	−0.439931
$738$	0	0
$739$	37.2943	1.37189	0.685946	−	0.727653i	$-0.259389\pi$
$739$	37.2943	1.37189	0.685946	+	0.727653i	$0.259389\pi$
$740$	0	0
$741$	0	0
$742$	28.1078	1.03187
$743$	−22.5693	−0.827989	−0.413994	−	0.910279i	$-0.635867\pi$
$743$	−22.5693	−0.827989	−0.413994	+	0.910279i	$0.635867\pi$
$744$	0	0
$745$	0	0
$746$	−7.07414	−0.259002
$747$	0	0
$748$	14.0140	0.512401
$749$	15.8584	0.579453
$750$	0	0
$751$	−9.55102	−0.348522	−0.174261	−	0.984700i	$-0.555754\pi$
$751$	−9.55102	−0.348522	−0.174261	+	0.984700i	$0.555754\pi$
$752$	6.13099	0.223574
$753$	0	0
$754$	45.0096	1.63915
$755$	0	0
$756$	0	0
$757$	30.3188	1.10196	0.550978	−	0.834519i	$-0.314255\pi$
$757$	30.3188	1.10196	0.550978	+	0.834519i	$0.314255\pi$
$758$	64.2341	2.33309
$759$	0	0
$760$	0	0
$761$	1.43066	0.0518613	0.0259306	−	0.999664i	$-0.491745\pi$
$761$	1.43066	0.0518613	0.0259306	+	0.999664i	$0.491745\pi$
$762$	0	0
$763$	−11.4663	−0.415106
$764$	−82.5500	−2.98655
$765$	0	0
$766$	−71.2311	−2.57369
$767$	−23.8217	−0.860153
$768$	0	0
$769$	−35.4017	−1.27662	−0.638309	−	0.769780i	$-0.720366\pi$
$769$	−35.4017	−1.27662	−0.638309	+	0.769780i	$0.720366\pi$
$770$	0	0
$771$	0	0
$772$	28.7110	1.03333
$773$	13.4094	0.482303	0.241151	−	0.970488i	$-0.422475\pi$
$773$	13.4094	0.482303	0.241151	+	0.970488i	$0.422475\pi$
$774$	0	0
$775$	0	0
$776$	163.397	5.86562
$777$	0	0
$778$	−6.44898	−0.231207
$779$	−4.10308	−0.147008
$780$	0	0
$781$	11.6262	0.416018
$782$	49.1772	1.75857
$783$	0	0
$784$	−57.7851	−2.06375
$785$	0	0
$786$	0	0
$787$	−37.3232	−1.33043	−0.665214	−	0.746653i	$-0.731660\pi$
$787$	−37.3232	−1.33043	−0.665214	+	0.746653i	$0.731660\pi$
$788$	−21.0602	−0.750238
$789$	0	0
$790$	0	0
$791$	11.1878	0.397794
$792$	0	0
$793$	20.4479	0.726128
$794$	47.4865	1.68523
$795$	0	0
$796$	−45.4865	−1.61223
$797$	8.05581	0.285352	0.142676	−	0.989769i	$-0.454429\pi$
$797$	8.05581	0.285352	0.142676	+	0.989769i	$0.454429\pi$
$798$	0	0
$799$	−0.931080	−0.0329393
$800$	0	0
$801$	0	0
$802$	73.3728	2.59088
$803$	−6.77551	−0.239103
$804$	0	0
$805$	0	0
$806$	110.204	3.88177
$807$	0	0
$808$	10.3372	0.363660
$809$	42.3771	1.48990	0.744950	−	0.667120i	$-0.232473\pi$
$809$	42.3771	1.48990	0.744950	+	0.667120i	$0.232473\pi$
$810$	0	0
$811$	−21.3878	−0.751026	−0.375513	−	0.926817i	$-0.622533\pi$
$811$	−21.3878	−0.751026	−0.375513	+	0.926817i	$0.622533\pi$
$812$	36.9604	1.29706
$813$	0	0
$814$	24.5187	0.859382
$815$	0	0
$816$	0	0
$817$	39.8786	1.39518
$818$	−56.3911	−1.97167
$819$	0	0
$820$	0	0
$821$	47.8776	1.67094	0.835469	−	0.549537i	$-0.185196\pi$
$821$	47.8776	1.67094	0.835469	+	0.549537i	$0.185196\pi$
$822$	0	0
$823$	16.3555	0.570116	0.285058	−	0.958510i	$-0.407987\pi$
$823$	16.3555	0.570116	0.285058	+	0.958510i	$0.407987\pi$
$824$	30.5048	1.06268
$825$	0	0
$826$	−26.5154	−0.922589
$827$	44.7110	1.55475	0.777376	−	0.629036i	$-0.216550\pi$
$827$	44.7110	1.55475	0.777376	+	0.629036i	$0.216550\pi$
$828$	0	0
$829$	−38.5972	−1.34054	−0.670269	−	0.742118i	$-0.733821\pi$
$829$	−38.5972	−1.34054	−0.670269	+	0.742118i	$0.733821\pi$
$830$	0	0
$831$	0	0
$832$	171.035	5.92959
$833$	8.77551	0.304053
$834$	0	0
$835$	0	0
$836$	30.3126	1.04838
$837$	0	0
$838$	37.5283	1.29639
$839$	0.710960	0.0245451	0.0122725	−	0.999925i	$-0.496093\pi$
$839$	0.710960	0.0245451	0.0122725	+	0.999925i	$0.496093\pi$
$840$	0	0
$841$	−16.5877	−0.571988
$842$	−109.406	−3.77038
$843$	0	0
$844$	−89.6993	−3.08758
$845$	0	0
$846$	0	0
$847$	1.86464	0.0640698
$848$	89.5298	3.07446
$849$	0	0
$850$	0	0
$851$	63.4758	2.17592
$852$	0	0
$853$	52.7187	1.80505	0.902526	−	0.430635i	$-0.141710\pi$
$853$	52.7187	1.80505	0.902526	+	0.430635i	$0.141710\pi$
$854$	22.7601	0.778835
$855$	0	0
$856$	85.1666	2.91093
$857$	−16.7124	−0.570885	−0.285442	−	0.958396i	$-0.592141\pi$
$857$	−16.7124	−0.570885	−0.285442	+	0.958396i	$0.592141\pi$
$858$	0	0
$859$	−41.9142	−1.43009	−0.715047	−	0.699076i	$-0.753595\pi$
$859$	−41.9142	−1.43009	−0.715047	+	0.699076i	$0.753595\pi$
$860$	0	0
$861$	0	0
$862$	18.1416	0.617906
$863$	41.4219	1.41002	0.705009	−	0.709198i	$-0.250943\pi$
$863$	41.4219	1.41002	0.705009	+	0.709198i	$0.250943\pi$
$864$	0	0
$865$	0	0
$866$	−100.552	−3.41689
$867$	0	0
$868$	90.4961	3.07164
$869$	6.01395	0.204009
$870$	0	0
$871$	55.2514	1.87212
$872$	−61.5789	−2.08533
$873$	0	0
$874$	106.372	3.59808
$875$	0	0
$876$	0	0
$877$	−12.0848	−0.408073	−0.204037	−	0.978963i	$-0.565406\pi$
$877$	−12.0848	−0.408073	−0.204037	+	0.978963i	$0.565406\pi$
$878$	70.1097	2.36609
$879$	0	0
$880$	0	0
$881$	31.8130	1.07181	0.535904	−	0.844279i	$-0.319971\pi$
$881$	31.8130	1.07181	0.535904	+	0.844279i	$0.319971\pi$
$882$	0	0
$883$	−20.9894	−0.706349	−0.353174	−	0.935558i	$-0.614898\pi$
$883$	−20.9894	−0.706349	−0.353174	+	0.935558i	$0.614898\pi$
$884$	−64.8313	−2.18051
$885$	0	0
$886$	57.7345	1.93963
$887$	−49.2032	−1.65208	−0.826042	−	0.563609i	$-0.809412\pi$
$887$	−49.2032	−1.65208	−0.826042	+	0.563609i	$0.809412\pi$
$888$	0	0
$889$	−17.2784	−0.579499
$890$	0	0
$891$	0	0
$892$	−124.773	−4.17772
$893$	−2.01395	−0.0673944
$894$	0	0
$895$	0	0
$896$	96.1502	3.21215
$897$	0	0
$898$	96.5027	3.22034
$899$	30.3911	1.01360
$900$	0	0
$901$	−13.5964	−0.452962
$902$	−2.10308	−0.0700250
$903$	0	0
$904$	60.0837	1.99835
$905$	0	0
$906$	0	0
$907$	−24.8526	−0.825216	−0.412608	−	0.910909i	$-0.635382\pi$
$907$	−24.8526	−0.825216	−0.412608	+	0.910909i	$0.635382\pi$
$908$	−33.3940	−1.10822
$909$	0	0
$910$	0	0
$911$	−45.4758	−1.50668	−0.753341	−	0.657630i	$-0.771559\pi$
$911$	−45.4758	−1.50668	−0.753341	+	0.657630i	$0.771559\pi$
$912$	0	0
$913$	14.5693	0.482175
$914$	20.2062	0.668361
$915$	0	0
$916$	−46.7928	−1.54608
$917$	−7.57893	−0.250278
$918$	0	0
$919$	−15.7355	−0.519068	−0.259534	−	0.965734i	$-0.583569\pi$
$919$	−15.7355	−0.519068	−0.259534	+	0.965734i	$0.583569\pi$
$920$	0	0
$921$	0	0
$922$	38.7389	1.27580
$923$	−53.7851	−1.77036
$924$	0	0
$925$	0	0
$926$	−11.9721	−0.393427
$927$	0	0
$928$	89.0162	2.92210
$929$	2.06455	0.0677357	0.0338679	−	0.999426i	$-0.489217\pi$
$929$	2.06455	0.0677357	0.0338679	+	0.999426i	$0.489217\pi$
$930$	0	0
$931$	18.9817	0.622099
$932$	−160.452	−5.25578
$933$	0	0
$934$	10.2216	0.334460
$935$	0	0
$936$	0	0
$937$	40.1493	1.31162	0.655810	−	0.754926i	$-0.272327\pi$
$937$	40.1493	1.31162	0.655810	+	0.754926i	$0.272327\pi$
$938$	61.4990	2.00801
$939$	0	0
$940$	0	0
$941$	−26.4050	−0.860780	−0.430390	−	0.902643i	$-0.641624\pi$
$941$	−26.4050	−0.860780	−0.430390	+	0.902643i	$0.641624\pi$
$942$	0	0
$943$	−5.44461	−0.177301
$944$	−84.4575	−2.74886
$945$	0	0
$946$	20.4402	0.664570
$947$	−8.67243	−0.281816	−0.140908	−	0.990023i	$-0.545002\pi$
$947$	−8.67243	−0.281816	−0.140908	+	0.990023i	$0.545002\pi$
$948$	0	0
$949$	31.3449	1.01750
$950$	0	0
$951$	0	0
$952$	−46.5100	−1.50740
$953$	26.8540	0.869887	0.434943	−	0.900458i	$-0.356768\pi$
$953$	26.8540	0.869887	0.434943	+	0.900458i	$0.356768\pi$
$954$	0	0
$955$	0	0
$956$	−135.549	−4.38397
$957$	0	0
$958$	63.1512	2.04032
$959$	−11.0675	−0.357388
$960$	0	0
$961$	43.4113	1.40036
$962$	−113.429	−3.65708
$963$	0	0
$964$	−29.0741	−0.936415
$965$	0	0
$966$	0	0
$967$	−55.8496	−1.79600	−0.898002	−	0.439992i	$-0.854981\pi$
$967$	−55.8496	−1.79600	−0.898002	+	0.439992i	$0.854981\pi$
$968$	10.0140	0.321861
$969$	0	0
$970$	0	0
$971$	−30.0664	−0.964878	−0.482439	−	0.875930i	$-0.660249\pi$
$971$	−30.0664	−0.964878	−0.482439	+	0.875930i	$0.660249\pi$
$972$	0	0
$973$	12.5135	0.401165
$974$	14.9942	0.480445
$975$	0	0
$976$	72.4961	2.32054
$977$	−15.7014	−0.502331	−0.251166	−	0.967944i	$-0.580814\pi$
$977$	−15.7014	−0.502331	−0.251166	+	0.967944i	$0.580814\pi$
$978$	0	0
$979$	9.04623	0.289119
$980$	0	0
$981$	0	0
$982$	−58.7668	−1.87532
$983$	51.9894	1.65820	0.829102	−	0.559098i	$-0.188852\pi$
$983$	51.9894	1.65820	0.829102	+	0.559098i	$0.188852\pi$
$984$	0	0
$985$	0	0
$986$	−24.2341	−0.771770
$987$	0	0
$988$	−140.232	−4.46137
$989$	52.9171	1.68267
$990$	0	0
$991$	27.6445	0.878157	0.439079	−	0.898449i	$-0.355305\pi$
$991$	27.6445	0.878157	0.439079	+	0.898449i	$0.355305\pi$
$992$	217.953	6.92000
$993$	0	0
$994$	−59.8669	−1.89886
$995$	0	0
$996$	0	0
$997$	40.5048	1.28280	0.641400	−	0.767207i	$-0.278354\pi$
$997$	40.5048	1.28280	0.641400	+	0.767207i	$0.278354\pi$
$998$	−44.2062	−1.39932
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.2.a.bd.1.3		3
3.2	odd	2		825.2.a.i.1.1	✓	3
5.2	odd	4		2475.2.c.q.199.6		6
5.3	odd	4		2475.2.c.q.199.1		6
5.4	even	2		2475.2.a.z.1.1		3
15.2	even	4		825.2.c.f.199.1		6
15.8	even	4		825.2.c.f.199.6		6
15.14	odd	2		825.2.a.m.1.3	yes	3
33.32	even	2		9075.2.a.cj.1.3		3
165.164	even	2		9075.2.a.cd.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.2.a.i.1.1	✓	3	3.2	odd	2
825.2.a.m.1.3	yes	3	15.14	odd	2
825.2.c.f.199.1		6	15.2	even	4
825.2.c.f.199.6		6	15.8	even	4
2475.2.a.z.1.1		3	5.4	even	2
2475.2.a.bd.1.3		3	1.1	even	1	trivial
2475.2.c.q.199.1		6	5.3	odd	4
2475.2.c.q.199.6		6	5.2	odd	4
9075.2.a.cd.1.1		3	165.164	even	2
9075.2.a.cj.1.3		3	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2475.a (trivial)

\(f(q)\)	\(=\)	\(q+2.76156 q^{2} +5.62620 q^{4} +1.86464 q^{7} +10.0140 q^{8} +O(q^{10})\)	\(q+2.76156 q^{2} +5.62620 q^{4} +1.86464 q^{7} +10.0140 q^{8} -1.00000 q^{11} +4.62620 q^{13} +5.14931 q^{14} +16.4017 q^{16} -2.49084 q^{17} -5.38776 q^{19} -2.76156 q^{22} -7.14931 q^{23} +12.7755 q^{26} +10.4908 q^{28} +3.52311 q^{29} +8.62620 q^{31} +25.2663 q^{32} -6.87859 q^{34} -8.87859 q^{37} -14.8786 q^{38} +0.761557 q^{41} -7.40171 q^{43} -5.62620 q^{44} -19.7432 q^{46} +0.373802 q^{47} -3.52311 q^{49} +26.0279 q^{52} +5.45856 q^{53} +18.6724 q^{56} +9.72928 q^{58} -5.14931 q^{59} +4.42003 q^{61} +23.8217 q^{62} +36.9711 q^{64} +11.9431 q^{67} -14.0140 q^{68} -11.6262 q^{71} +6.77551 q^{73} -24.5187 q^{74} -30.3126 q^{76} -1.86464 q^{77} -6.01395 q^{79} +2.10308 q^{82} -14.5693 q^{83} -20.4402 q^{86} -10.0140 q^{88} -9.04623 q^{89} +8.62620 q^{91} -40.2234 q^{92} +1.03228 q^{94} +16.3169 q^{97} -9.72928 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{2} + 8 q^{4} + 3 q^{7} + 6 q^{8}+O(q^{10}) \) 3 * q + 2 * q^2 + 8 * q^4 + 3 * q^7 + 6 * q^8	\( 3 q + 2 q^{2} + 8 q^{4} + 3 q^{7} + 6 q^{8} - 3 q^{11} + 5 q^{13} - 6 q^{14} + 10 q^{16} + 4 q^{17} - q^{19} - 2 q^{22} + 8 q^{26} + 20 q^{28} - 2 q^{29} + 17 q^{31} + 34 q^{32} + 6 q^{34} - 18 q^{38} - 4 q^{41} + 17 q^{43} - 8 q^{44} - 30 q^{46} + 10 q^{47} + 2 q^{49} + 30 q^{52} + 6 q^{53} + 22 q^{56} + 24 q^{58} + 6 q^{59} - 3 q^{61} + 16 q^{62} + 34 q^{64} + 7 q^{67} - 18 q^{68} - 26 q^{71} - 10 q^{73} - 14 q^{74} - 24 q^{76} - 3 q^{77} + 6 q^{79} + 10 q^{82} - 6 q^{83} - 28 q^{86} - 6 q^{88} - 2 q^{89} + 17 q^{91} - 26 q^{92} + 2 q^{94} + 29 q^{97} - 24 q^{98}+O(q^{100}) \) 3 * q + 2 * q^2 + 8 * q^4 + 3 * q^7 + 6 * q^8 - 3 * q^11 + 5 * q^13 - 6 * q^14 + 10 * q^16 + 4 * q^17 - q^19 - 2 * q^22 + 8 * q^26 + 20 * q^28 - 2 * q^29 + 17 * q^31 + 34 * q^32 + 6 * q^34 - 18 * q^38 - 4 * q^41 + 17 * q^43 - 8 * q^44 - 30 * q^46 + 10 * q^47 + 2 * q^49 + 30 * q^52 + 6 * q^53 + 22 * q^56 + 24 * q^58 + 6 * q^59 - 3 * q^61 + 16 * q^62 + 34 * q^64 + 7 * q^67 - 18 * q^68 - 26 * q^71 - 10 * q^73 - 14 * q^74 - 24 * q^76 - 3 * q^77 + 6 * q^79 + 10 * q^82 - 6 * q^83 - 28 * q^86 - 6 * q^88 - 2 * q^89 + 17 * q^91 - 26 * q^92 + 2 * q^94 + 29 * q^97 - 24 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more