LMFDB - Embedded newform 7623.2.a.cn.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7623,2,Mod(1,7623)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7623, 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("7623.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7623 = 3^{2} \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7623.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:	$60.8699614608$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.7488.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 2x + 1 $ x^4 - 2x^3 - 4x^2 + 2*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2541)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.326909$ of defining polynomial
Character	$\chi$	$=$	7623.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.05896 q^{2} +2.23931 q^{4} +3.05896 q^{5} +1.00000 q^{7} -0.492737 q^{8} +O(q^{10})$	$q-2.05896 q^{2} +2.23931 q^{4} +3.05896 q^{5} +1.00000 q^{7} -0.492737 q^{8} -6.29827 q^{10} -3.32691 q^{13} -2.05896 q^{14} -3.46410 q^{16} +5.37651 q^{17} -8.29827 q^{19} +6.84997 q^{20} +1.74141 q^{23} +4.35723 q^{25} +6.84997 q^{26} +2.23931 q^{28} -8.16277 q^{29} -5.44999 q^{31} +8.11792 q^{32} -11.0700 q^{34} +3.05896 q^{35} +4.53242 q^{37} +17.0858 q^{38} -1.50726 q^{40} -6.84997 q^{41} +1.84997 q^{43} -3.58550 q^{46} +7.28416 q^{47} +1.00000 q^{49} -8.97136 q^{50} -7.44999 q^{52} +0.985474 q^{53} -0.492737 q^{56} +16.8068 q^{58} -4.52306 q^{59} +12.0845 q^{61} +11.2213 q^{62} -9.78626 q^{64} -10.1769 q^{65} +0.170993 q^{67} +12.0397 q^{68} -6.29827 q^{70} +10.0017 q^{71} -12.7910 q^{73} -9.33207 q^{74} -18.5824 q^{76} -14.6794 q^{79} -10.5965 q^{80} +14.1038 q^{82} -1.62993 q^{83} +16.4465 q^{85} -3.80901 q^{86} -13.9872 q^{89} -3.32691 q^{91} +3.89957 q^{92} -14.9978 q^{94} -25.3841 q^{95} +3.44483 q^{97} -2.05896 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} + 4 q^{4} + 2 q^{5} + 4 q^{7}+O(q^{10}) $ 4 * q + 2 * q^2 + 4 * q^4 + 2 * q^5 + 4 * q^7	$ 4 q + 2 q^{2} + 4 q^{4} + 2 q^{5} + 4 q^{7} - 10 q^{10} - 10 q^{13} + 2 q^{14} + 6 q^{17} - 18 q^{19} + 2 q^{23} - 8 q^{25} + 4 q^{28} + 6 q^{29} + 12 q^{32} - 2 q^{34} + 2 q^{35} - 4 q^{37} - 8 q^{40} - 20 q^{43} - 16 q^{46} + 6 q^{47} + 4 q^{49} - 24 q^{50} - 8 q^{52} + 24 q^{58} + 6 q^{59} + 10 q^{61} - 16 q^{64} - 10 q^{65} + 4 q^{67} + 28 q^{68} - 10 q^{70} + 6 q^{71} - 34 q^{73} - 36 q^{74} - 36 q^{76} - 24 q^{79} - 12 q^{80} + 28 q^{82} + 6 q^{83} + 8 q^{85} - 38 q^{86} - 18 q^{89} - 10 q^{91} - 24 q^{92} + 6 q^{94} - 18 q^{95} - 10 q^{97} + 2 q^{98}+O(q^{100}) $ 4 * q + 2 * q^2 + 4 * q^4 + 2 * q^5 + 4 * q^7 - 10 * q^10 - 10 * q^13 + 2 * q^14 + 6 * q^17 - 18 * q^19 + 2 * q^23 - 8 * q^25 + 4 * q^28 + 6 * q^29 + 12 * q^32 - 2 * q^34 + 2 * q^35 - 4 * q^37 - 8 * q^40 - 20 * q^43 - 16 * q^46 + 6 * q^47 + 4 * q^49 - 24 * q^50 - 8 * q^52 + 24 * q^58 + 6 * q^59 + 10 * q^61 - 16 * q^64 - 10 * q^65 + 4 * q^67 + 28 * q^68 - 10 * q^70 + 6 * q^71 - 34 * q^73 - 36 * q^74 - 36 * q^76 - 24 * q^79 - 12 * q^80 + 28 * q^82 + 6 * q^83 + 8 * q^85 - 38 * q^86 - 18 * q^89 - 10 * q^91 - 24 * q^92 + 6 * q^94 - 18 * q^95 - 10 * q^97 + 2 * 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 $

$2$	−2.05896	−1.45590	−0.727952	−	0.685628i	$-0.759528\pi$
$2$	−2.05896	−1.45590	−0.727952	+	0.685628i	$0.759528\pi$
$3$	0	0
$3$	0	0
$4$	2.23931	1.11966
$5$	3.05896	1.36801	0.684004	−	0.729478i	$-0.260237\pi$
$5$	3.05896	1.36801	0.684004	+	0.729478i	$0.260237\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−0.492737	−0.174209
$9$	0	0
$10$	−6.29827	−1.99169
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−3.32691	−0.922718	−0.461359	−	0.887213i	$-0.652638\pi$
$13$	−3.32691	−0.922718	−0.461359	+	0.887213i	$0.652638\pi$
$14$	−2.05896	−0.550280
$15$	0	0
$16$	−3.46410	−0.866025
$17$	5.37651	1.30399	0.651997	−	0.758221i	$-0.273931\pi$
$17$	5.37651	1.30399	0.651997	+	0.758221i	$0.273931\pi$
$18$	0	0
$19$	−8.29827	−1.90375	−0.951877	−	0.306480i	$-0.900849\pi$
$19$	−8.29827	−1.90375	−0.951877	+	0.306480i	$0.900849\pi$
$20$	6.84997	1.53170
$21$	0	0
$22$	0	0
$23$	1.74141	0.363110	0.181555	−	0.983381i	$-0.441887\pi$
$23$	1.74141	0.363110	0.181555	+	0.983381i	$0.441887\pi$
$24$	0	0
$25$	4.35723	0.871446
$26$	6.84997	1.34339
$27$	0	0
$28$	2.23931	0.423191
$29$	−8.16277	−1.51579	−0.757894	−	0.652378i	$-0.773772\pi$
$29$	−8.16277	−1.51579	−0.757894	+	0.652378i	$0.773772\pi$
$30$	0	0
$31$	−5.44999	−0.978847	−0.489424	−	0.872046i	$-0.662793\pi$
$31$	−5.44999	−0.978847	−0.489424	+	0.872046i	$0.662793\pi$
$32$	8.11792	1.43506
$33$	0	0
$34$	−11.0700	−1.89849
$35$	3.05896	0.517059
$36$	0	0
$37$	4.53242	0.745126	0.372563	−	0.928007i	$-0.378479\pi$
$37$	4.53242	0.745126	0.372563	+	0.928007i	$0.378479\pi$
$38$	17.0858	2.77168
$39$	0	0
$40$	−1.50726	−0.238319
$41$	−6.84997	−1.06979	−0.534893	−	0.844920i	$-0.679648\pi$
$41$	−6.84997	−1.06979	−0.534893	+	0.844920i	$0.679648\pi$
$42$	0	0
$43$	1.84997	0.282118	0.141059	−	0.990001i	$-0.454949\pi$
$43$	1.84997	0.282118	0.141059	+	0.990001i	$0.454949\pi$
$44$	0	0
$45$	0	0
$46$	−3.58550	−0.528653
$47$	7.28416	1.06250	0.531252	−	0.847214i	$-0.321722\pi$
$47$	7.28416	1.06250	0.531252	+	0.847214i	$0.321722\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−8.97136	−1.26874
$51$	0	0
$52$	−7.44999	−1.03313
$53$	0.985474	0.135365	0.0676827	−	0.997707i	$-0.478439\pi$
$53$	0.985474	0.135365	0.0676827	+	0.997707i	$0.478439\pi$
$54$	0	0
$55$	0	0
$56$	−0.492737	−0.0658448
$57$	0	0
$58$	16.8068	2.20684
$59$	−4.52306	−0.588852	−0.294426	−	0.955674i	$-0.595129\pi$
$59$	−4.52306	−0.588852	−0.294426	+	0.955674i	$0.595129\pi$
$60$	0	0
$61$	12.0845	1.54727	0.773633	−	0.633634i	$-0.218437\pi$
$61$	12.0845	1.54727	0.773633	+	0.633634i	$0.218437\pi$
$62$	11.2213	1.42511
$63$	0	0
$64$	−9.78626	−1.22328
$65$	−10.1769	−1.26229
$66$	0	0
$67$	0.170993	0.0208901	0.0104451	−	0.999945i	$-0.496675\pi$
$67$	0.170993	0.0208901	0.0104451	+	0.999945i	$0.496675\pi$
$68$	12.0397	1.46003
$69$	0	0
$70$	−6.29827	−0.752788
$71$	10.0017	1.18698	0.593491	−	0.804841i	$-0.297749\pi$
$71$	10.0017	1.18698	0.593491	+	0.804841i	$0.297749\pi$
$72$	0	0
$73$	−12.7910	−1.49707	−0.748537	−	0.663093i	$-0.769243\pi$
$73$	−12.7910	−1.49707	−0.748537	+	0.663093i	$0.769243\pi$
$74$	−9.33207	−1.08483
$75$	0	0
$76$	−18.5824	−2.13155
$77$	0	0
$78$	0	0
$79$	−14.6794	−1.65156	−0.825780	−	0.563992i	$-0.809265\pi$
$79$	−14.6794	−1.65156	−0.825780	+	0.563992i	$0.809265\pi$
$80$	−10.5965	−1.18473
$81$	0	0
$82$	14.1038	1.55751
$83$	−1.62993	−0.178908	−0.0894540	−	0.995991i	$-0.528512\pi$
$83$	−1.62993	−0.178908	−0.0894540	+	0.995991i	$0.528512\pi$
$84$	0	0
$85$	16.4465	1.78388
$86$	−3.80901	−0.410736
$87$	0	0
$88$	0	0
$89$	−13.9872	−1.48264	−0.741318	−	0.671154i	$-0.765799\pi$
$89$	−13.9872	−1.48264	−0.741318	+	0.671154i	$0.765799\pi$
$90$	0	0
$91$	−3.32691	−0.348755
$92$	3.89957	0.406558
$93$	0	0
$94$	−14.9978	−1.54690
$95$	−25.3841	−2.60435
$96$	0	0
$97$	3.44483	0.349769	0.174885	−	0.984589i	$-0.444045\pi$
$97$	3.44483	0.349769	0.174885	+	0.984589i	$0.444045\pi$
$98$	−2.05896	−0.207986
$99$	0	0
$100$	9.75721	0.975721
$101$	9.35554	0.930911	0.465456	−	0.885071i	$-0.345890\pi$
$101$	9.35554	0.930911	0.465456	+	0.885071i	$0.345890\pi$
$102$	0	0
$103$	12.6952	1.25089	0.625447	−	0.780267i	$-0.284917\pi$
$103$	12.6952	1.25089	0.625447	+	0.780267i	$0.284917\pi$
$104$	1.63929	0.160746
$105$	0	0
$106$	−2.02905	−0.197079
$107$	11.7589	1.13678	0.568388	−	0.822761i	$-0.307567\pi$
$107$	11.7589	1.13678	0.568388	+	0.822761i	$0.307567\pi$
$108$	0	0
$109$	−15.9038	−1.52330	−0.761652	−	0.647986i	$-0.775611\pi$
$109$	−15.9038	−1.52330	−0.761652	+	0.647986i	$0.775611\pi$
$110$	0	0
$111$	0	0
$112$	−3.46410	−0.327327
$113$	−7.30763	−0.687444	−0.343722	−	0.939071i	$-0.611688\pi$
$113$	−7.30763	−0.687444	−0.343722	+	0.939071i	$0.611688\pi$
$114$	0	0
$115$	5.32691	0.496737
$116$	−18.2790	−1.69716
$117$	0	0
$118$	9.31280	0.857313
$119$	5.37651	0.492864
$120$	0	0
$121$	0	0
$122$	−24.8816	−2.25267
$123$	0	0
$124$	−12.2042	−1.09597
$125$	−1.96620	−0.175862
$126$	0	0
$127$	4.32860	0.384101	0.192050	−	0.981385i	$-0.438486\pi$
$127$	4.32860	0.384101	0.192050	+	0.981385i	$0.438486\pi$
$128$	3.91368	0.345923
$129$	0	0
$130$	20.9538	1.83777
$131$	8.71236	0.761202	0.380601	−	0.924739i	$-0.375717\pi$
$131$	8.71236	0.761202	0.380601	+	0.924739i	$0.375717\pi$
$132$	0	0
$133$	−8.29827	−0.719552
$134$	−0.352068	−0.0304140
$135$	0	0
$136$	−2.64920	−0.227167
$137$	0.234149	0.0200047	0.0100024	−	0.999950i	$-0.496816\pi$
$137$	0.234149	0.0200047	0.0100024	+	0.999950i	$0.496816\pi$
$138$	0	0
$139$	−19.3200	−1.63870	−0.819349	−	0.573296i	$-0.805665\pi$
$139$	−19.3200	−1.63870	−0.819349	+	0.573296i	$0.805665\pi$
$140$	6.84997	0.578928
$141$	0	0
$142$	−20.5931	−1.72813
$143$	0	0
$144$	0	0
$145$	−24.9696	−2.07361
$146$	26.3362	2.17960
$147$	0	0
$148$	10.1495	0.834285
$149$	−5.20090	−0.426074	−0.213037	−	0.977044i	$-0.568336\pi$
$149$	−5.20090	−0.426074	−0.213037	+	0.977044i	$0.568336\pi$
$150$	0	0
$151$	−6.35682	−0.517310	−0.258655	−	0.965970i	$-0.583279\pi$
$151$	−6.35682	−0.517310	−0.258655	+	0.965970i	$0.583279\pi$
$152$	4.08887	0.331651
$153$	0	0
$154$	0	0
$155$	−16.6713	−1.33907
$156$	0	0
$157$	−10.0141	−0.799213	−0.399606	−	0.916687i	$-0.630853\pi$
$157$	−10.0141	−0.799213	−0.399606	+	0.916687i	$0.630853\pi$
$158$	30.2243	2.40451
$159$	0	0
$160$	24.8324	1.96317
$161$	1.74141	0.137242
$162$	0	0
$163$	−8.50337	−0.666035	−0.333018	−	0.942921i	$-0.608067\pi$
$163$	−8.50337	−0.666035	−0.333018	+	0.942921i	$0.608067\pi$
$164$	−15.3392	−1.19779
$165$	0	0
$166$	3.35596	0.260473
$167$	2.93756	0.227316	0.113658	−	0.993520i	$-0.463743\pi$
$167$	2.93756	0.227316	0.113658	+	0.993520i	$0.463743\pi$
$168$	0	0
$169$	−1.93168	−0.148591
$170$	−33.8627	−2.59715
$171$	0	0
$172$	4.14266	0.315875
$173$	−20.6841	−1.57259	−0.786293	−	0.617854i	$-0.788002\pi$
$173$	−20.6841	−1.57259	−0.786293	+	0.617854i	$0.788002\pi$
$174$	0	0
$175$	4.35723	0.329376
$176$	0	0
$177$	0	0
$178$	28.7990	2.15858
$179$	−6.14697	−0.459446	−0.229723	−	0.973256i	$-0.573782\pi$
$179$	−6.14697	−0.459446	−0.229723	+	0.973256i	$0.573782\pi$
$180$	0	0
$181$	−19.2458	−1.43053	−0.715263	−	0.698856i	$-0.753693\pi$
$181$	−19.2458	−1.43053	−0.715263	+	0.698856i	$0.753693\pi$
$182$	6.84997	0.507754
$183$	0	0
$184$	−0.858058	−0.0632569
$185$	13.8645	1.01934
$186$	0	0
$187$	0	0
$188$	16.3115	1.18964
$189$	0	0
$190$	52.2648	3.79169
$191$	−25.5978	−1.85219	−0.926097	−	0.377287i	$-0.876857\pi$
$191$	−25.5978	−1.85219	−0.926097	+	0.377287i	$0.876857\pi$
$192$	0	0
$193$	−11.1962	−0.805917	−0.402958	−	0.915218i	$-0.632018\pi$
$193$	−11.1962	−0.805917	−0.402958	+	0.915218i	$0.632018\pi$
$194$	−7.09276	−0.509230
$195$	0	0
$196$	2.23931	0.159951
$197$	−17.4687	−1.24459	−0.622297	−	0.782781i	$-0.713800\pi$
$197$	−17.4687	−1.24459	−0.622297	+	0.782781i	$0.713800\pi$
$198$	0	0
$199$	−6.71963	−0.476342	−0.238171	−	0.971223i	$-0.576548\pi$
$199$	−6.71963	−0.476342	−0.238171	+	0.971223i	$0.576548\pi$
$200$	−2.14697	−0.151814
$201$	0	0
$202$	−19.2627	−1.35532
$203$	−8.16277	−0.572914
$204$	0	0
$205$	−20.9538	−1.46348
$206$	−26.1389	−1.82118
$207$	0	0
$208$	11.5247	0.799098
$209$	0	0
$210$	0	0
$211$	20.2293	1.39264	0.696321	−	0.717730i	$-0.254819\pi$
$211$	20.2293	1.39264	0.696321	+	0.717730i	$0.254819\pi$
$212$	2.20679	0.151563
$213$	0	0
$214$	−24.2111	−1.65504
$215$	5.65898	0.385939
$216$	0	0
$217$	−5.44999	−0.369970
$218$	32.7452	2.21779
$219$	0	0
$220$	0	0
$221$	−17.8871	−1.20322
$222$	0	0
$223$	11.8705	0.794909	0.397454	−	0.917622i	$-0.369894\pi$
$223$	11.8705	0.794909	0.397454	+	0.917622i	$0.369894\pi$
$224$	8.11792	0.542401
$225$	0	0
$226$	15.0461	1.00085
$227$	13.9521	0.926033	0.463016	−	0.886350i	$-0.346767\pi$
$227$	13.9521	0.926033	0.463016	+	0.886350i	$0.346767\pi$
$228$	0	0
$229$	24.8324	1.64097	0.820485	−	0.571668i	$-0.193703\pi$
$229$	24.8324	1.64097	0.820485	+	0.571668i	$0.193703\pi$
$230$	−10.9679	−0.723201
$231$	0	0
$232$	4.02210	0.264064
$233$	−11.8265	−0.774780	−0.387390	−	0.921916i	$-0.626623\pi$
$233$	−11.8265	−0.774780	−0.387390	+	0.921916i	$0.626623\pi$
$234$	0	0
$235$	22.2820	1.45351
$236$	−10.1286	−0.659313
$237$	0	0
$238$	−11.0700	−0.717562
$239$	−8.86522	−0.573443	−0.286722	−	0.958014i	$-0.592565\pi$
$239$	−8.86522	−0.573443	−0.286722	+	0.958014i	$0.592565\pi$
$240$	0	0
$241$	−11.3922	−0.733834	−0.366917	−	0.930254i	$-0.619587\pi$
$241$	−11.3922	−0.733834	−0.366917	+	0.930254i	$0.619587\pi$
$242$	0	0
$243$	0	0
$244$	27.0611	1.73241
$245$	3.05896	0.195430
$246$	0	0
$247$	27.6076	1.75663
$248$	2.68541	0.170524
$249$	0	0
$250$	4.04833	0.256039
$251$	−9.71866	−0.613436	−0.306718	−	0.951800i	$-0.599231\pi$
$251$	−9.71866	−0.613436	−0.306718	+	0.951800i	$0.599231\pi$
$252$	0	0
$253$	0	0
$254$	−8.91241	−0.559214
$255$	0	0
$256$	11.5144	0.719651
$257$	20.0456	1.25041	0.625204	−	0.780461i	$-0.285016\pi$
$257$	20.0456	1.25041	0.625204	+	0.780461i	$0.285016\pi$
$258$	0	0
$259$	4.53242	0.281631
$260$	−22.7892	−1.41333
$261$	0	0
$262$	−17.9384	−1.10824
$263$	12.9345	0.797576	0.398788	−	0.917043i	$-0.369431\pi$
$263$	12.9345	0.797576	0.398788	+	0.917043i	$0.369431\pi$
$264$	0	0
$265$	3.01453	0.185181
$266$	17.0858	1.04760
$267$	0	0
$268$	0.382907	0.0233898
$269$	−4.44958	−0.271295	−0.135648	−	0.990757i	$-0.543312\pi$
$269$	−4.44958	−0.271295	−0.135648	+	0.990757i	$0.543312\pi$
$270$	0	0
$271$	−9.59696	−0.582974	−0.291487	−	0.956575i	$-0.594150\pi$
$271$	−9.59696	−0.582974	−0.291487	+	0.956575i	$0.594150\pi$
$272$	−18.6248	−1.12929
$273$	0	0
$274$	−0.482104	−0.0291249
$275$	0	0
$276$	0	0
$277$	2.80037	0.168258	0.0841290	−	0.996455i	$-0.473189\pi$
$277$	2.80037	0.168258	0.0841290	+	0.996455i	$0.473189\pi$
$278$	39.7790	2.38579
$279$	0	0
$280$	−1.50726	−0.0900762
$281$	15.4721	0.922988	0.461494	−	0.887143i	$-0.347314\pi$
$281$	15.4721	0.922988	0.461494	+	0.887143i	$0.347314\pi$
$282$	0	0
$283$	25.9076	1.54004	0.770022	−	0.638017i	$-0.220245\pi$
$283$	25.9076	1.54004	0.770022	+	0.638017i	$0.220245\pi$
$284$	22.3969	1.32901
$285$	0	0
$286$	0	0
$287$	−6.84997	−0.404341
$288$	0	0
$289$	11.9068	0.700401
$290$	51.4113	3.01898
$291$	0	0
$292$	−28.6431	−1.67621
$293$	23.3799	1.36587	0.682934	−	0.730480i	$-0.260704\pi$
$293$	23.3799	1.36587	0.682934	+	0.730480i	$0.260704\pi$
$294$	0	0
$295$	−13.8359	−0.805555
$296$	−2.23329	−0.129808
$297$	0	0
$298$	10.7084	0.620323
$299$	−5.79352	−0.335048
$300$	0	0
$301$	1.84997	0.106630
$302$	13.0884	0.753154
$303$	0	0
$304$	28.7461	1.64870
$305$	36.9661	2.11667
$306$	0	0
$307$	10.0742	0.574965	0.287483	−	0.957786i	$-0.407182\pi$
$307$	10.0742	0.574965	0.287483	+	0.957786i	$0.407182\pi$
$308$	0	0
$309$	0	0
$310$	34.3255	1.94956
$311$	−8.26014	−0.468390	−0.234195	−	0.972190i	$-0.575245\pi$
$311$	−8.26014	−0.468390	−0.234195	+	0.972190i	$0.575245\pi$
$312$	0	0
$313$	−7.39189	−0.417814	−0.208907	−	0.977935i	$-0.566991\pi$
$313$	−7.39189	−0.417814	−0.208907	+	0.977935i	$0.566991\pi$
$314$	20.6186	1.16358
$315$	0	0
$316$	−32.8718	−1.84918
$317$	−31.1600	−1.75012	−0.875060	−	0.484014i	$-0.839178\pi$
$317$	−31.1600	−1.75012	−0.875060	+	0.484014i	$0.839178\pi$
$318$	0	0
$319$	0	0
$320$	−29.9358	−1.67346
$321$	0	0
$322$	−3.58550	−0.199812
$323$	−44.6157	−2.48249
$324$	0	0
$325$	−14.4961	−0.804100
$326$	17.5081	0.969684
$327$	0	0
$328$	3.37523	0.186366
$329$	7.28416	0.401589
$330$	0	0
$331$	−20.0110	−1.09991	−0.549953	−	0.835195i	$-0.685355\pi$
$331$	−20.0110	−1.09991	−0.549953	+	0.835195i	$0.685355\pi$
$332$	−3.64992	−0.200316
$333$	0	0
$334$	−6.04833	−0.330950
$335$	0.523061	0.0285779
$336$	0	0
$337$	20.4973	1.11656	0.558278	−	0.829654i	$-0.311462\pi$
$337$	20.4973	1.11656	0.558278	+	0.829654i	$0.311462\pi$
$338$	3.97725	0.216334
$339$	0	0
$340$	36.8289	1.99733
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	−0.911549	−0.0491474
$345$	0	0
$346$	42.5878	2.28953
$347$	33.7959	1.81426	0.907131	−	0.420849i	$-0.138268\pi$
$347$	33.7959	1.81426	0.907131	+	0.420849i	$0.138268\pi$
$348$	0	0
$349$	−33.0030	−1.76661	−0.883304	−	0.468801i	$-0.844686\pi$
$349$	−33.0030	−1.76661	−0.883304	+	0.468801i	$0.844686\pi$
$350$	−8.97136	−0.479540
$351$	0	0
$352$	0	0
$353$	−9.49315	−0.505270	−0.252635	−	0.967562i	$-0.581297\pi$
$353$	−9.49315	−0.505270	−0.252635	+	0.967562i	$0.581297\pi$
$354$	0	0
$355$	30.5948	1.62380
$356$	−31.3216	−1.66004
$357$	0	0
$358$	12.6564	0.668910
$359$	−13.0490	−0.688702	−0.344351	−	0.938841i	$-0.611901\pi$
$359$	−13.0490	−0.688702	−0.344351	+	0.938841i	$0.611901\pi$
$360$	0	0
$361$	49.8613	2.62428
$362$	39.6262	2.08271
$363$	0	0
$364$	−7.44999	−0.390486
$365$	−39.1272	−2.04801
$366$	0	0
$367$	−1.93419	−0.100964	−0.0504819	−	0.998725i	$-0.516076\pi$
$367$	−1.93419	−0.100964	−0.0504819	+	0.998725i	$0.516076\pi$
$368$	−6.03243	−0.314462
$369$	0	0
$370$	−28.5464	−1.48406
$371$	0.985474	0.0511633
$372$	0	0
$373$	−23.8816	−1.23654	−0.618270	−	0.785966i	$-0.712166\pi$
$373$	−23.8816	−1.23654	−0.618270	+	0.785966i	$0.712166\pi$
$374$	0	0
$375$	0	0
$376$	−3.58918	−0.185098
$377$	27.1568	1.39865
$378$	0	0
$379$	−38.0262	−1.95327	−0.976637	−	0.214895i	$-0.931059\pi$
$379$	−38.0262	−1.95327	−0.976637	+	0.214895i	$0.931059\pi$
$380$	−56.8429	−2.91598
$381$	0	0
$382$	52.7049	2.69662
$383$	−31.5678	−1.61304	−0.806519	−	0.591208i	$-0.798651\pi$
$383$	−31.5678	−1.61304	−0.806519	+	0.591208i	$0.798651\pi$
$384$	0	0
$385$	0	0
$386$	23.0524	1.17334
$387$	0	0
$388$	7.71405	0.391621
$389$	−10.1308	−0.513652	−0.256826	−	0.966458i	$-0.582677\pi$
$389$	−10.1308	−0.513652	−0.256826	+	0.966458i	$0.582677\pi$
$390$	0	0
$391$	9.36271	0.473493
$392$	−0.492737	−0.0248870
$393$	0	0
$394$	35.9674	1.81201
$395$	−44.9037	−2.25935
$396$	0	0
$397$	−7.51188	−0.377010	−0.188505	−	0.982072i	$-0.560364\pi$
$397$	−7.51188	−0.377010	−0.188505	+	0.982072i	$0.560364\pi$
$398$	13.8354	0.693508
$399$	0	0
$400$	−15.0939	−0.754695
$401$	−19.2645	−0.962022	−0.481011	−	0.876715i	$-0.659730\pi$
$401$	−19.2645	−0.962022	−0.481011	+	0.876715i	$0.659730\pi$
$402$	0	0
$403$	18.1316	0.903201
$404$	20.9500	1.04230
$405$	0	0
$406$	16.8068	0.834108
$407$	0	0
$408$	0	0
$409$	−13.0605	−0.645801	−0.322900	−	0.946433i	$-0.604658\pi$
$409$	−13.0605	−0.645801	−0.322900	+	0.946433i	$0.604658\pi$
$410$	43.1430	2.13068
$411$	0	0
$412$	28.4285	1.40057
$413$	−4.52306	−0.222565
$414$	0	0
$415$	−4.98589	−0.244748
$416$	−27.0076	−1.32416
$417$	0	0
$418$	0	0
$419$	30.7589	1.50267	0.751335	−	0.659921i	$-0.229410\pi$
$419$	30.7589	1.50267	0.751335	+	0.659921i	$0.229410\pi$
$420$	0	0
$421$	16.0657	0.782993	0.391497	−	0.920180i	$-0.371957\pi$
$421$	16.0657	0.782993	0.391497	+	0.920180i	$0.371957\pi$
$422$	−41.6513	−2.02755
$423$	0	0
$424$	−0.485580	−0.0235818
$425$	23.4267	1.13636
$426$	0	0
$427$	12.0845	0.584812
$428$	26.3319	1.27280
$429$	0	0
$430$	−11.6516	−0.561891
$431$	13.0752	0.629809	0.314904	−	0.949123i	$-0.398028\pi$
$431$	13.0752	0.629809	0.314904	+	0.949123i	$0.398028\pi$
$432$	0	0
$433$	12.9757	0.623572	0.311786	−	0.950152i	$-0.399073\pi$
$433$	12.9757	0.623572	0.311786	+	0.950152i	$0.399073\pi$
$434$	11.2213	0.538640
$435$	0	0
$436$	−35.6135	−1.70558
$437$	−14.4507	−0.691271
$438$	0	0
$439$	15.9709	0.762252	0.381126	−	0.924523i	$-0.375536\pi$
$439$	15.9709	0.762252	0.381126	+	0.924523i	$0.375536\pi$
$440$	0	0
$441$	0	0
$442$	36.8289	1.75177
$443$	21.2695	1.01055	0.505273	−	0.862959i	$-0.331392\pi$
$443$	21.2695	1.01055	0.505273	+	0.862959i	$0.331392\pi$
$444$	0	0
$445$	−42.7862	−2.02826
$446$	−24.4409	−1.15731
$447$	0	0
$448$	−9.78626	−0.462357
$449$	6.63021	0.312899	0.156449	−	0.987686i	$-0.449995\pi$
$449$	6.63021	0.312899	0.156449	+	0.987686i	$0.449995\pi$
$450$	0	0
$451$	0	0
$452$	−16.3641	−0.769702
$453$	0	0
$454$	−28.7268	−1.34821
$455$	−10.1769	−0.477099
$456$	0	0
$457$	−17.1191	−0.800796	−0.400398	−	0.916341i	$-0.631128\pi$
$457$	−17.1191	−0.800796	−0.400398	+	0.916341i	$0.631128\pi$
$458$	−51.1289	−2.38910
$459$	0	0
$460$	11.9286	0.556175
$461$	19.1173	0.890380	0.445190	−	0.895436i	$-0.353136\pi$
$461$	19.1173	0.890380	0.445190	+	0.895436i	$0.353136\pi$
$462$	0	0
$463$	16.0828	0.747433	0.373717	−	0.927543i	$-0.378083\pi$
$463$	16.0828	0.747433	0.373717	+	0.927543i	$0.378083\pi$
$464$	28.2767	1.31271
$465$	0	0
$466$	24.3503	1.12800
$467$	13.4920	0.624336	0.312168	−	0.950027i	$-0.398945\pi$
$467$	13.4920	0.624336	0.312168	+	0.950027i	$0.398945\pi$
$468$	0	0
$469$	0.170993	0.00789573
$470$	−45.8776	−2.11618
$471$	0	0
$472$	2.22868	0.102583
$473$	0	0
$474$	0	0
$475$	−36.1575	−1.65902
$476$	12.0397	0.551838
$477$	0	0
$478$	18.2531	0.834878
$479$	25.0471	1.14443	0.572215	−	0.820103i	$-0.306084\pi$
$479$	25.0471	1.14443	0.572215	+	0.820103i	$0.306084\pi$
$480$	0	0
$481$	−15.0790	−0.687541
$482$	23.4560	1.06839
$483$	0	0
$484$	0	0
$485$	10.5376	0.478487
$486$	0	0
$487$	−30.7888	−1.39517	−0.697587	−	0.716500i	$-0.745743\pi$
$487$	−30.7888	−1.39517	−0.697587	+	0.716500i	$0.745743\pi$
$488$	−5.95450	−0.269548
$489$	0	0
$490$	−6.29827	−0.284527
$491$	39.1200	1.76546	0.882731	−	0.469879i	$-0.155702\pi$
$491$	39.1200	1.76546	0.882731	+	0.469879i	$0.155702\pi$
$492$	0	0
$493$	−43.8872	−1.97658
$494$	−56.8429	−2.55748
$495$	0	0
$496$	18.8793	0.847707
$497$	10.0017	0.448637
$498$	0	0
$499$	13.5522	0.606682	0.303341	−	0.952882i	$-0.401898\pi$
$499$	13.5522	0.606682	0.303341	+	0.952882i	$0.401898\pi$
$500$	−4.40294	−0.196905
$501$	0	0
$502$	20.0103	0.893105
$503$	−18.1214	−0.807995	−0.403997	−	0.914760i	$-0.632380\pi$
$503$	−18.1214	−0.807995	−0.403997	+	0.914760i	$0.632380\pi$
$504$	0	0
$505$	28.6182	1.27349
$506$	0	0
$507$	0	0
$508$	9.69309	0.430061
$509$	40.4192	1.79155	0.895774	−	0.444510i	$-0.146622\pi$
$509$	40.4192	1.79155	0.895774	+	0.444510i	$0.146622\pi$
$510$	0	0
$511$	−12.7910	−0.565841
$512$	−31.5351	−1.39367
$513$	0	0
$514$	−41.2730	−1.82047
$515$	38.8341	1.71123
$516$	0	0
$517$	0	0
$518$	−9.33207	−0.410028
$519$	0	0
$520$	5.01453	0.219902
$521$	7.74795	0.339444	0.169722	−	0.985492i	$-0.445713\pi$
$521$	7.74795	0.339444	0.169722	+	0.985492i	$0.445713\pi$
$522$	0	0
$523$	−21.6047	−0.944706	−0.472353	−	0.881409i	$-0.656595\pi$
$523$	−21.6047	−0.944706	−0.472353	+	0.881409i	$0.656595\pi$
$524$	19.5097	0.852286
$525$	0	0
$526$	−26.6316	−1.16119
$527$	−29.3019	−1.27641
$528$	0	0
$529$	−19.9675	−0.868151
$530$	−6.20679	−0.269606
$531$	0	0
$532$	−18.5824	−0.805651
$533$	22.7892	0.987111
$534$	0	0
$535$	35.9700	1.55512
$536$	−0.0842547	−0.00363925
$537$	0	0
$538$	9.16150	0.394980
$539$	0	0
$540$	0	0
$541$	−35.2507	−1.51555	−0.757773	−	0.652519i	$-0.773712\pi$
$541$	−35.2507	−1.51555	−0.757773	+	0.652519i	$0.773712\pi$
$542$	19.7598	0.848754
$543$	0	0
$544$	43.6460	1.87131
$545$	−48.6490	−2.08389
$546$	0	0
$547$	−5.44775	−0.232929	−0.116465	−	0.993195i	$-0.537156\pi$
$547$	−5.44775	−0.232929	−0.116465	+	0.993195i	$0.537156\pi$
$548$	0.524333	0.0223984
$549$	0	0
$550$	0	0
$551$	67.7369	2.88569
$552$	0	0
$553$	−14.6794	−0.624231
$554$	−5.76585	−0.244968
$555$	0	0
$556$	−43.2634	−1.83478
$557$	8.73429	0.370084	0.185042	−	0.982731i	$-0.440758\pi$
$557$	8.73429	0.370084	0.185042	+	0.982731i	$0.440758\pi$
$558$	0	0
$559$	−6.15468	−0.260315
$560$	−10.5965	−0.447786
$561$	0	0
$562$	−31.8564	−1.34378
$563$	−15.2200	−0.641448	−0.320724	−	0.947173i	$-0.603926\pi$
$563$	−15.2200	−0.641448	−0.320724	+	0.947173i	$0.603926\pi$
$564$	0	0
$565$	−22.3538	−0.940430
$566$	−53.3426	−2.24216
$567$	0	0
$568$	−4.92820	−0.206783
$569$	−0.0166300	−0.000697167 0	−0.000348583	−	1.00000i	$-0.500111\pi$
$569$	−0.0166300	−0.000697167 0	−0.000348583		1.00000i	$0.500111\pi$
$570$	0	0
$571$	−6.08632	−0.254705	−0.127352	−	0.991858i	$-0.540648\pi$
$571$	−6.08632	−0.254705	−0.127352	+	0.991858i	$0.540648\pi$
$572$	0	0
$573$	0	0
$574$	14.1038	0.588682
$575$	7.58774	0.316430
$576$	0	0
$577$	−36.6359	−1.52517	−0.762585	−	0.646888i	$-0.776070\pi$
$577$	−36.6359	−1.52517	−0.762585	+	0.646888i	$0.776070\pi$
$578$	−24.5157	−1.01972
$579$	0	0
$580$	−55.9147	−2.32173
$581$	−1.62993	−0.0676209
$582$	0	0
$583$	0	0
$584$	6.30261	0.260804
$585$	0	0
$586$	−48.1382	−1.98857
$587$	−5.52836	−0.228180	−0.114090	−	0.993470i	$-0.536395\pi$
$587$	−5.52836	−0.228180	−0.114090	+	0.993470i	$0.536395\pi$
$588$	0	0
$589$	45.2255	1.86349
$590$	28.4875	1.17281
$591$	0	0
$592$	−15.7008	−0.645298
$593$	−33.1440	−1.36106	−0.680531	−	0.732719i	$-0.738251\pi$
$593$	−33.1440	−1.36106	−0.680531	+	0.732719i	$0.738251\pi$
$594$	0	0
$595$	16.4465	0.674241
$596$	−11.6464	−0.477057
$597$	0	0
$598$	11.9286	0.487798
$599$	−18.8290	−0.769332	−0.384666	−	0.923056i	$-0.625683\pi$
$599$	−18.8290	−0.769332	−0.384666	+	0.923056i	$0.625683\pi$
$600$	0	0
$601$	7.88491	0.321632	0.160816	−	0.986984i	$-0.448587\pi$
$601$	7.88491	0.321632	0.160816	+	0.986984i	$0.448587\pi$
$602$	−3.80901	−0.155244
$603$	0	0
$604$	−14.2349	−0.579210
$605$	0	0
$606$	0	0
$607$	−1.34563	−0.0546175	−0.0273087	−	0.999627i	$-0.508694\pi$
$607$	−1.34563	−0.0546175	−0.0273087	+	0.999627i	$0.508694\pi$
$608$	−67.3647	−2.73200
$609$	0	0
$610$	−76.1117	−3.08167
$611$	−24.2337	−0.980392
$612$	0	0
$613$	−21.8217	−0.881372	−0.440686	−	0.897661i	$-0.645265\pi$
$613$	−21.8217	−0.881372	−0.440686	+	0.897661i	$0.645265\pi$
$614$	−20.7424	−0.837094
$615$	0	0
$616$	0	0
$617$	−30.1973	−1.21570	−0.607849	−	0.794053i	$-0.707967\pi$
$617$	−30.1973	−1.21570	−0.607849	+	0.794053i	$0.707967\pi$
$618$	0	0
$619$	−39.2011	−1.57562	−0.787812	−	0.615915i	$-0.788786\pi$
$619$	−39.2011	−1.57562	−0.787812	+	0.615915i	$0.788786\pi$
$620$	−37.3323	−1.49930
$621$	0	0
$622$	17.0073	0.681930
$623$	−13.9872	−0.560384
$624$	0	0
$625$	−27.8007	−1.11203
$626$	15.2196	0.608298
$627$	0	0
$628$	−22.4247	−0.894844
$629$	24.3686	0.971640
$630$	0	0
$631$	5.44272	0.216671	0.108336	−	0.994114i	$-0.465448\pi$
$631$	5.44272	0.216671	0.108336	+	0.994114i	$0.465448\pi$
$632$	7.23308	0.287717
$633$	0	0
$634$	64.1572	2.54801
$635$	13.2410	0.525453
$636$	0	0
$637$	−3.32691	−0.131817
$638$	0	0
$639$	0	0
$640$	11.9718	0.473226
$641$	−45.6066	−1.80135	−0.900677	−	0.434489i	$-0.856929\pi$
$641$	−45.6066	−1.80135	−0.900677	+	0.434489i	$0.856929\pi$
$642$	0	0
$643$	21.2599	0.838407	0.419204	−	0.907892i	$-0.362309\pi$
$643$	21.2599	0.838407	0.419204	+	0.907892i	$0.362309\pi$
$644$	3.89957	0.153664
$645$	0	0
$646$	91.8620	3.61426
$647$	18.5756	0.730283	0.365141	−	0.930952i	$-0.381021\pi$
$647$	18.5756	0.730283	0.365141	+	0.930952i	$0.381021\pi$
$648$	0	0
$649$	0	0
$650$	29.8469	1.17069
$651$	0	0
$652$	−19.0417	−0.745731
$653$	32.1882	1.25962	0.629812	−	0.776748i	$-0.283132\pi$
$653$	32.1882	1.25962	0.629812	+	0.776748i	$0.283132\pi$
$654$	0	0
$655$	26.6508	1.04133
$656$	23.7290	0.926461
$657$	0	0
$658$	−14.9978	−0.584675
$659$	40.6203	1.58234	0.791172	−	0.611594i	$-0.209471\pi$
$659$	40.6203	1.58234	0.791172	+	0.611594i	$0.209471\pi$
$660$	0	0
$661$	7.69278	0.299215	0.149607	−	0.988746i	$-0.452199\pi$
$661$	7.69278	0.299215	0.149607	+	0.988746i	$0.452199\pi$
$662$	41.2019	1.60136
$663$	0	0
$664$	0.803127	0.0311674
$665$	−25.3841	−0.984352
$666$	0	0
$667$	−14.2147	−0.550397
$668$	6.57813	0.254515
$669$	0	0
$670$	−1.07696	−0.0416066
$671$	0	0
$672$	0	0
$673$	−7.24740	−0.279367	−0.139683	−	0.990196i	$-0.544608\pi$
$673$	−7.24740	−0.279367	−0.139683	+	0.990196i	$0.544608\pi$
$674$	−42.2030	−1.62560
$675$	0	0
$676$	−4.32564	−0.166371
$677$	6.52296	0.250698	0.125349	−	0.992113i	$-0.459995\pi$
$677$	6.52296	0.250698	0.125349	+	0.992113i	$0.459995\pi$
$678$	0	0
$679$	3.44483	0.132200
$680$	−8.10381	−0.310767
$681$	0	0
$682$	0	0
$683$	−2.12170	−0.0811846	−0.0405923	−	0.999176i	$-0.512924\pi$
$683$	−2.12170	−0.0811846	−0.0405923	+	0.999176i	$0.512924\pi$
$684$	0	0
$685$	0.716253	0.0273666
$686$	−2.05896	−0.0786114
$687$	0	0
$688$	−6.40848	−0.244321
$689$	−3.27858	−0.124904
$690$	0	0
$691$	−5.66834	−0.215634	−0.107817	−	0.994171i	$-0.534386\pi$
$691$	−5.66834	−0.215634	−0.107817	+	0.994171i	$0.534386\pi$
$692$	−46.3183	−1.76076
$693$	0	0
$694$	−69.5845	−2.64139
$695$	−59.0990	−2.24175
$696$	0	0
$697$	−36.8289	−1.39499
$698$	67.9518	2.57201
$699$	0	0
$700$	9.75721	0.368788
$701$	8.22087	0.310498	0.155249	−	0.987875i	$-0.450382\pi$
$701$	8.22087	0.310498	0.155249	+	0.987875i	$0.450382\pi$
$702$	0	0
$703$	−37.6113	−1.41854
$704$	0	0
$705$	0	0
$706$	19.5460	0.735624
$707$	9.35554	0.351851
$708$	0	0
$709$	−6.38116	−0.239649	−0.119825	−	0.992795i	$-0.538233\pi$
$709$	−6.38116	−0.239649	−0.119825	+	0.992795i	$0.538233\pi$
$710$	−62.9934	−2.36410
$711$	0	0
$712$	6.89199	0.258288
$713$	−9.49068	−0.355429
$714$	0	0
$715$	0	0
$716$	−13.7650	−0.514422
$717$	0	0
$718$	26.8675	1.00268
$719$	−22.6376	−0.844241	−0.422121	−	0.906540i	$-0.638714\pi$
$719$	−22.6376	−0.844241	−0.422121	+	0.906540i	$0.638714\pi$
$720$	0	0
$721$	12.6952	0.472794
$722$	−102.662	−3.82070
$723$	0	0
$724$	−43.0973	−1.60170
$725$	−35.5671	−1.32093
$726$	0	0
$727$	24.6482	0.914153	0.457076	−	0.889427i	$-0.348897\pi$
$727$	24.6482	0.914153	0.457076	+	0.889427i	$0.348897\pi$
$728$	1.63929	0.0607562
$729$	0	0
$730$	80.5613	2.98171
$731$	9.94637	0.367880
$732$	0	0
$733$	42.4229	1.56693	0.783464	−	0.621438i	$-0.213451\pi$
$733$	42.4229	1.56693	0.783464	+	0.621438i	$0.213451\pi$
$734$	3.98241	0.146994
$735$	0	0
$736$	14.1366	0.521084
$737$	0	0
$738$	0	0
$739$	−17.2443	−0.634343	−0.317172	−	0.948368i	$-0.602733\pi$
$739$	−17.2443	−0.634343	−0.317172	+	0.948368i	$0.602733\pi$
$740$	31.0470	1.14131
$741$	0	0
$742$	−2.02905	−0.0744888
$743$	31.2944	1.14808	0.574040	−	0.818827i	$-0.305375\pi$
$743$	31.2944	1.14808	0.574040	+	0.818827i	$0.305375\pi$
$744$	0	0
$745$	−15.9093	−0.582873
$746$	49.1712	1.80028
$747$	0	0
$748$	0	0
$749$	11.7589	0.429661
$750$	0	0
$751$	−31.4340	−1.14704	−0.573521	−	0.819191i	$-0.694423\pi$
$751$	−31.4340	−1.14704	−0.573521	+	0.819191i	$0.694423\pi$
$752$	−25.2331	−0.920156
$753$	0	0
$754$	−55.9147	−2.03629
$755$	−19.4452	−0.707685
$756$	0	0
$757$	25.3641	0.921873	0.460937	−	0.887433i	$-0.347514\pi$
$757$	25.3641	0.921873	0.460937	+	0.887433i	$0.347514\pi$
$758$	78.2944	2.84378
$759$	0	0
$760$	12.5077	0.453701
$761$	10.0619	0.364744	0.182372	−	0.983230i	$-0.441622\pi$
$761$	10.0619	0.364744	0.182372	+	0.983230i	$0.441622\pi$
$762$	0	0
$763$	−15.9038	−0.575755
$764$	−57.3215	−2.07382
$765$	0	0
$766$	64.9968	2.34843
$767$	15.0478	0.543345
$768$	0	0
$769$	0.408482	0.0147303	0.00736513	−	0.999973i	$-0.497656\pi$
$769$	0.408482	0.0147303	0.00736513	+	0.999973i	$0.497656\pi$
$770$	0	0
$771$	0	0
$772$	−25.0717	−0.902350
$773$	−11.9310	−0.429129	−0.214565	−	0.976710i	$-0.568833\pi$
$773$	−11.9310	−0.429129	−0.214565	+	0.976710i	$0.568833\pi$
$774$	0	0
$775$	−23.7469	−0.853013
$776$	−1.69739	−0.0609329
$777$	0	0
$778$	20.8589	0.747827
$779$	56.8429	2.03661
$780$	0	0
$781$	0	0
$782$	−19.2774	−0.689360
$783$	0	0
$784$	−3.46410	−0.123718
$785$	−30.6328	−1.09333
$786$	0	0
$787$	−20.3396	−0.725030	−0.362515	−	0.931978i	$-0.618082\pi$
$787$	−20.3396	−0.725030	−0.362515	+	0.931978i	$0.618082\pi$
$788$	−39.1179	−1.39352
$789$	0	0
$790$	92.4548	3.28939
$791$	−7.30763	−0.259830
$792$	0	0
$793$	−40.2041	−1.42769
$794$	15.4666	0.548891
$795$	0	0
$796$	−15.0474	−0.533339
$797$	−6.82763	−0.241847	−0.120924	−	0.992662i	$-0.538586\pi$
$797$	−6.82763	−0.241847	−0.120924	+	0.992662i	$0.538586\pi$
$798$	0	0
$799$	39.1634	1.38550
$800$	35.3717	1.25058
$801$	0	0
$802$	39.6648	1.40061
$803$	0	0
$804$	0	0
$805$	5.32691	0.187749
$806$	−37.3323	−1.31497
$807$	0	0
$808$	−4.60982	−0.162173
$809$	15.3890	0.541047	0.270524	−	0.962713i	$-0.412803\pi$
$809$	15.3890	0.541047	0.270524	+	0.962713i	$0.412803\pi$
$810$	0	0
$811$	−9.42694	−0.331025	−0.165512	−	0.986208i	$-0.552928\pi$
$811$	−9.42694	−0.331025	−0.165512	+	0.986208i	$0.552928\pi$
$812$	−18.2790	−0.641467
$813$	0	0
$814$	0	0
$815$	−26.0115	−0.911142
$816$	0	0
$817$	−15.3516	−0.537083
$818$	26.8911	0.940224
$819$	0	0
$820$	−46.9221	−1.63859
$821$	32.3301	1.12833	0.564165	−	0.825662i	$-0.309198\pi$
$821$	32.3301	1.12833	0.564165	+	0.825662i	$0.309198\pi$
$822$	0	0
$823$	−18.3223	−0.638676	−0.319338	−	0.947641i	$-0.603461\pi$
$823$	−18.3223	−0.638676	−0.319338	+	0.947641i	$0.603461\pi$
$824$	−6.25539	−0.217917
$825$	0	0
$826$	9.31280	0.324034
$827$	−6.37527	−0.221690	−0.110845	−	0.993838i	$-0.535356\pi$
$827$	−6.37527	−0.221690	−0.110845	+	0.993838i	$0.535356\pi$
$828$	0	0
$829$	0.949545	0.0329791	0.0164895	−	0.999864i	$-0.494751\pi$
$829$	0.949545	0.0329791	0.0164895	+	0.999864i	$0.494751\pi$
$830$	10.2657	0.356329
$831$	0	0
$832$	32.5580	1.12875
$833$	5.37651	0.186285
$834$	0	0
$835$	8.98589	0.310970
$836$	0	0
$837$	0	0
$838$	−63.3313	−2.18774
$839$	−14.3937	−0.496925	−0.248462	−	0.968642i	$-0.579925\pi$
$839$	−14.3937	−0.496925	−0.248462	+	0.968642i	$0.579925\pi$
$840$	0	0
$841$	37.6308	1.29761
$842$	−33.0786	−1.13996
$843$	0	0
$844$	45.2998	1.55928
$845$	−5.90893	−0.203273
$846$	0	0
$847$	0	0
$848$	−3.41378	−0.117230
$849$	0	0
$850$	−48.2346	−1.65443
$851$	7.89281	0.270562
$852$	0	0
$853$	39.9349	1.36735	0.683673	−	0.729788i	$-0.260381\pi$
$853$	39.9349	1.36735	0.683673	+	0.729788i	$0.260381\pi$
$854$	−24.8816	−0.851430
$855$	0	0
$856$	−5.79405	−0.198036
$857$	−52.2332	−1.78425	−0.892127	−	0.451786i	$-0.850787\pi$
$857$	−52.2332	−1.78425	−0.892127	+	0.451786i	$0.850787\pi$
$858$	0	0
$859$	−37.1209	−1.26655	−0.633274	−	0.773928i	$-0.718289\pi$
$859$	−37.1209	−1.26655	−0.633274	+	0.773928i	$0.718289\pi$
$860$	12.6722	0.432120
$861$	0	0
$862$	−26.9213	−0.916941
$863$	34.0072	1.15762	0.578809	−	0.815463i	$-0.303518\pi$
$863$	34.0072	1.15762	0.578809	+	0.815463i	$0.303518\pi$
$864$	0	0
$865$	−63.2719	−2.15131
$866$	−26.7164	−0.907861
$867$	0	0
$868$	−12.2042	−0.414239
$869$	0	0
$870$	0	0
$871$	−0.568878	−0.0192757
$872$	7.83638	0.265373
$873$	0	0
$874$	29.7534	1.00642
$875$	−1.96620	−0.0664697
$876$	0	0
$877$	−45.3869	−1.53261	−0.766303	−	0.642479i	$-0.777906\pi$
$877$	−45.3869	−1.53261	−0.766303	+	0.642479i	$0.777906\pi$
$878$	−32.8835	−1.10977
$879$	0	0
$880$	0	0
$881$	−18.0836	−0.609251	−0.304626	−	0.952472i	$-0.598531\pi$
$881$	−18.0836	−0.609251	−0.304626	+	0.952472i	$0.598531\pi$
$882$	0	0
$883$	14.8564	0.499958	0.249979	−	0.968251i	$-0.419576\pi$
$883$	14.8564	0.499958	0.249979	+	0.968251i	$0.419576\pi$
$884$	−40.0549	−1.34719
$885$	0	0
$886$	−43.7931	−1.47126
$887$	−29.3108	−0.984162	−0.492081	−	0.870549i	$-0.663764\pi$
$887$	−29.3108	−0.984162	−0.492081	+	0.870549i	$0.663764\pi$
$888$	0	0
$889$	4.32860	0.145176
$890$	88.0950	2.95295
$891$	0	0
$892$	26.5818	0.890025
$893$	−60.4460	−2.02275
$894$	0	0
$895$	−18.8033	−0.628526
$896$	3.91368	0.130747
$897$	0	0
$898$	−13.6513	−0.455551
$899$	44.4870	1.48373
$900$	0	0
$901$	5.29841	0.176516
$902$	0	0
$903$	0	0
$904$	3.60074	0.119759
$905$	−58.8720	−1.95697
$906$	0	0
$907$	0.784709	0.0260558	0.0130279	−	0.999915i	$-0.495853\pi$
$907$	0.784709	0.0260558	0.0130279	+	0.999915i	$0.495853\pi$
$908$	31.2431	1.03684
$909$	0	0
$910$	20.9538	0.694611
$911$	−49.3824	−1.63611	−0.818056	−	0.575139i	$-0.804948\pi$
$911$	−49.3824	−1.63611	−0.818056	+	0.575139i	$0.804948\pi$
$912$	0	0
$913$	0	0
$914$	35.2474	1.16588
$915$	0	0
$916$	55.6075	1.83732
$917$	8.71236	0.287707
$918$	0	0
$919$	−1.20242	−0.0396641	−0.0198320	−	0.999803i	$-0.506313\pi$
$919$	−1.20242	−0.0396641	−0.0198320	+	0.999803i	$0.506313\pi$
$920$	−2.62477	−0.0865360
$921$	0	0
$922$	−39.3617	−1.29631
$923$	−33.2747	−1.09525
$924$	0	0
$925$	19.7488	0.649337
$926$	−33.1139	−1.08819
$927$	0	0
$928$	−66.2647	−2.17525
$929$	16.9944	0.557569	0.278785	−	0.960354i	$-0.410068\pi$
$929$	16.9944	0.557569	0.278785	+	0.960354i	$0.410068\pi$
$930$	0	0
$931$	−8.29827	−0.271965
$932$	−26.4832	−0.867487
$933$	0	0
$934$	−27.7795	−0.908973
$935$	0	0
$936$	0	0
$937$	28.4196	0.928427	0.464213	−	0.885723i	$-0.346337\pi$
$937$	28.4196	0.928427	0.464213	+	0.885723i	$0.346337\pi$
$938$	−0.352068	−0.0114954
$939$	0	0
$940$	49.8963	1.62744
$941$	−27.9160	−0.910034	−0.455017	−	0.890483i	$-0.650367\pi$
$941$	−27.9160	−0.910034	−0.455017	+	0.890483i	$0.650367\pi$
$942$	0	0
$943$	−11.9286	−0.388449
$944$	15.6683	0.509961
$945$	0	0
$946$	0	0
$947$	−34.5418	−1.12246	−0.561229	−	0.827660i	$-0.689671\pi$
$947$	−34.5418	−1.12246	−0.561229	+	0.827660i	$0.689671\pi$
$948$	0	0
$949$	42.5545	1.38138
$950$	74.4468	2.41537
$951$	0	0
$952$	−2.64920	−0.0858612
$953$	−13.8398	−0.448314	−0.224157	−	0.974553i	$-0.571963\pi$
$953$	−13.8398	−0.448314	−0.224157	+	0.974553i	$0.571963\pi$
$954$	0	0
$955$	−78.3027	−2.53382
$956$	−19.8520	−0.642060
$957$	0	0
$958$	−51.5709	−1.66618
$959$	0.234149	0.00756107
$960$	0	0
$961$	−1.29759	−0.0418577
$962$	31.0470	1.00099
$963$	0	0
$964$	−25.5106	−0.821642
$965$	−34.2486	−1.10250
$966$	0	0
$967$	7.67468	0.246801	0.123401	−	0.992357i	$-0.460620\pi$
$967$	7.67468	0.246801	0.123401	+	0.992357i	$0.460620\pi$
$968$	0	0
$969$	0	0
$970$	−21.6965	−0.696631
$971$	−6.99587	−0.224508	−0.112254	−	0.993680i	$-0.535807\pi$
$971$	−6.99587	−0.224508	−0.112254	+	0.993680i	$0.535807\pi$
$972$	0	0
$973$	−19.3200	−0.619369
$974$	63.3929	2.03124
$975$	0	0
$976$	−41.8621	−1.33997
$977$	11.0312	0.352918	0.176459	−	0.984308i	$-0.443536\pi$
$977$	11.0312	0.352918	0.176459	+	0.984308i	$0.443536\pi$
$978$	0	0
$979$	0	0
$980$	6.84997	0.218814
$981$	0	0
$982$	−80.5465	−2.57034
$983$	−13.4873	−0.430178	−0.215089	−	0.976594i	$-0.569004\pi$
$983$	−13.4873	−0.430178	−0.215089	+	0.976594i	$0.569004\pi$
$984$	0	0
$985$	−53.4361	−1.70262
$986$	90.3619	2.87771
$987$	0	0
$988$	61.8221	1.96682
$989$	3.22156	0.102440
$990$	0	0
$991$	0.759034	0.0241115	0.0120558	−	0.999927i	$-0.496162\pi$
$991$	0.759034	0.0241115	0.0120558	+	0.999927i	$0.496162\pi$
$992$	−44.2426	−1.40470
$993$	0	0
$994$	−20.5931	−0.653173
$995$	−20.5551	−0.651640
$996$	0	0
$997$	−47.7793	−1.51319	−0.756593	−	0.653886i	$-0.773138\pi$
$997$	−47.7793	−1.51319	−0.756593	+	0.653886i	$0.773138\pi$
$998$	−27.9035	−0.883271
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.cn.1.1		4
3.2	odd	2		2541.2.a.bl.1.4	✓	4
11.10	odd	2		7623.2.a.cg.1.4		4
33.32	even	2		2541.2.a.bp.1.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2541.2.a.bl.1.4	✓	4	3.2	odd	2
2541.2.a.bp.1.1	yes	4	33.32	even	2
7623.2.a.cg.1.4		4	11.10	odd	2
7623.2.a.cn.1.1		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7623.a (trivial)

\(f(q)\)	\(=\)	\(q-2.05896 q^{2} +2.23931 q^{4} +3.05896 q^{5} +1.00000 q^{7} -0.492737 q^{8} +O(q^{10})\)	\(q-2.05896 q^{2} +2.23931 q^{4} +3.05896 q^{5} +1.00000 q^{7} -0.492737 q^{8} -6.29827 q^{10} -3.32691 q^{13} -2.05896 q^{14} -3.46410 q^{16} +5.37651 q^{17} -8.29827 q^{19} +6.84997 q^{20} +1.74141 q^{23} +4.35723 q^{25} +6.84997 q^{26} +2.23931 q^{28} -8.16277 q^{29} -5.44999 q^{31} +8.11792 q^{32} -11.0700 q^{34} +3.05896 q^{35} +4.53242 q^{37} +17.0858 q^{38} -1.50726 q^{40} -6.84997 q^{41} +1.84997 q^{43} -3.58550 q^{46} +7.28416 q^{47} +1.00000 q^{49} -8.97136 q^{50} -7.44999 q^{52} +0.985474 q^{53} -0.492737 q^{56} +16.8068 q^{58} -4.52306 q^{59} +12.0845 q^{61} +11.2213 q^{62} -9.78626 q^{64} -10.1769 q^{65} +0.170993 q^{67} +12.0397 q^{68} -6.29827 q^{70} +10.0017 q^{71} -12.7910 q^{73} -9.33207 q^{74} -18.5824 q^{76} -14.6794 q^{79} -10.5965 q^{80} +14.1038 q^{82} -1.62993 q^{83} +16.4465 q^{85} -3.80901 q^{86} -13.9872 q^{89} -3.32691 q^{91} +3.89957 q^{92} -14.9978 q^{94} -25.3841 q^{95} +3.44483 q^{97} -2.05896 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 4 q^{4} + 2 q^{5} + 4 q^{7}+O(q^{10}) \) 4 * q + 2 * q^2 + 4 * q^4 + 2 * q^5 + 4 * q^7	\( 4 q + 2 q^{2} + 4 q^{4} + 2 q^{5} + 4 q^{7} - 10 q^{10} - 10 q^{13} + 2 q^{14} + 6 q^{17} - 18 q^{19} + 2 q^{23} - 8 q^{25} + 4 q^{28} + 6 q^{29} + 12 q^{32} - 2 q^{34} + 2 q^{35} - 4 q^{37} - 8 q^{40} - 20 q^{43} - 16 q^{46} + 6 q^{47} + 4 q^{49} - 24 q^{50} - 8 q^{52} + 24 q^{58} + 6 q^{59} + 10 q^{61} - 16 q^{64} - 10 q^{65} + 4 q^{67} + 28 q^{68} - 10 q^{70} + 6 q^{71} - 34 q^{73} - 36 q^{74} - 36 q^{76} - 24 q^{79} - 12 q^{80} + 28 q^{82} + 6 q^{83} + 8 q^{85} - 38 q^{86} - 18 q^{89} - 10 q^{91} - 24 q^{92} + 6 q^{94} - 18 q^{95} - 10 q^{97} + 2 q^{98}+O(q^{100}) \) 4 * q + 2 * q^2 + 4 * q^4 + 2 * q^5 + 4 * q^7 - 10 * q^10 - 10 * q^13 + 2 * q^14 + 6 * q^17 - 18 * q^19 + 2 * q^23 - 8 * q^25 + 4 * q^28 + 6 * q^29 + 12 * q^32 - 2 * q^34 + 2 * q^35 - 4 * q^37 - 8 * q^40 - 20 * q^43 - 16 * q^46 + 6 * q^47 + 4 * q^49 - 24 * q^50 - 8 * q^52 + 24 * q^58 + 6 * q^59 + 10 * q^61 - 16 * q^64 - 10 * q^65 + 4 * q^67 + 28 * q^68 - 10 * q^70 + 6 * q^71 - 34 * q^73 - 36 * q^74 - 36 * q^76 - 24 * q^79 - 12 * q^80 + 28 * q^82 + 6 * q^83 + 8 * q^85 - 38 * q^86 - 18 * q^89 - 10 * q^91 - 24 * q^92 + 6 * q^94 - 18 * q^95 - 10 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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