LMFDB - Embedded newform 7728.2.a.bf.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7728,2,Mod(1,7728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7728, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("7728.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7728.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:	$61.7083906820$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1932)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.30278$ of defining polynomial
Character	$\chi$	$=$	7728.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -0.697224 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -0.697224 q^{5} +1.00000 q^{7} +1.00000 q^{9} -5.60555 q^{11} +2.30278 q^{13} -0.697224 q^{15} -0.394449 q^{17} +0.394449 q^{19} +1.00000 q^{21} +1.00000 q^{23} -4.51388 q^{25} +1.00000 q^{27} -5.60555 q^{29} +3.60555 q^{31} -5.60555 q^{33} -0.697224 q^{35} +5.60555 q^{37} +2.30278 q^{39} +3.60555 q^{41} -4.30278 q^{43} -0.697224 q^{45} +4.60555 q^{47} +1.00000 q^{49} -0.394449 q^{51} +5.90833 q^{53} +3.90833 q^{55} +0.394449 q^{57} -3.90833 q^{59} +4.90833 q^{61} +1.00000 q^{63} -1.60555 q^{65} -13.3028 q^{67} +1.00000 q^{69} -12.9083 q^{71} +11.0000 q^{73} -4.51388 q^{75} -5.60555 q^{77} -13.4222 q^{79} +1.00000 q^{81} -16.2111 q^{83} +0.275019 q^{85} -5.60555 q^{87} -15.5139 q^{89} +2.30278 q^{91} +3.60555 q^{93} -0.275019 q^{95} +3.78890 q^{97} -5.60555 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 5 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 5 * q^5 + 2 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} - 5 q^{5} + 2 q^{7} + 2 q^{9} - 4 q^{11} + q^{13} - 5 q^{15} - 8 q^{17} + 8 q^{19} + 2 q^{21} + 2 q^{23} + 9 q^{25} + 2 q^{27} - 4 q^{29} - 4 q^{33} - 5 q^{35} + 4 q^{37} + q^{39} - 5 q^{43} - 5 q^{45} + 2 q^{47} + 2 q^{49} - 8 q^{51} + q^{53} - 3 q^{55} + 8 q^{57} + 3 q^{59} - q^{61} + 2 q^{63} + 4 q^{65} - 23 q^{67} + 2 q^{69} - 15 q^{71} + 22 q^{73} + 9 q^{75} - 4 q^{77} + 2 q^{79} + 2 q^{81} - 18 q^{83} + 33 q^{85} - 4 q^{87} - 13 q^{89} + q^{91} - 33 q^{95} + 22 q^{97} - 4 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 5 * q^5 + 2 * q^7 + 2 * q^9 - 4 * q^11 + q^13 - 5 * q^15 - 8 * q^17 + 8 * q^19 + 2 * q^21 + 2 * q^23 + 9 * q^25 + 2 * q^27 - 4 * q^29 - 4 * q^33 - 5 * q^35 + 4 * q^37 + q^39 - 5 * q^43 - 5 * q^45 + 2 * q^47 + 2 * q^49 - 8 * q^51 + q^53 - 3 * q^55 + 8 * q^57 + 3 * q^59 - q^61 + 2 * q^63 + 4 * q^65 - 23 * q^67 + 2 * q^69 - 15 * q^71 + 22 * q^73 + 9 * q^75 - 4 * q^77 + 2 * q^79 + 2 * q^81 - 18 * q^83 + 33 * q^85 - 4 * q^87 - 13 * q^89 + q^91 - 33 * q^95 + 22 * q^97 - 4 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−0.697224	−0.311808	−0.155904	−	0.987772i	$-0.549829\pi$
$5$	−0.697224	−0.311808	−0.155904	+	0.987772i	$0.549829\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.60555	−1.69014	−0.845069	−	0.534658i	$-0.820441\pi$
$11$	−5.60555	−1.69014	−0.845069	+	0.534658i	$0.820441\pi$
$12$	0	0
$13$	2.30278	0.638675	0.319338	−	0.947641i	$-0.396540\pi$
$13$	2.30278	0.638675	0.319338	+	0.947641i	$0.396540\pi$
$14$	0	0
$15$	−0.697224	−0.180023
$16$	0	0
$17$	−0.394449	−0.0956679	−0.0478339	−	0.998855i	$-0.515232\pi$
$17$	−0.394449	−0.0956679	−0.0478339	+	0.998855i	$0.515232\pi$
$18$	0	0
$19$	0.394449	0.0904927	0.0452464	−	0.998976i	$-0.485593\pi$
$19$	0.394449	0.0904927	0.0452464	+	0.998976i	$0.485593\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−4.51388	−0.902776
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−5.60555	−1.04092	−0.520462	−	0.853885i	$-0.674240\pi$
$29$	−5.60555	−1.04092	−0.520462	+	0.853885i	$0.674240\pi$
$30$	0	0
$31$	3.60555	0.647576	0.323788	−	0.946130i	$-0.395044\pi$
$31$	3.60555	0.647576	0.323788	+	0.946130i	$0.395044\pi$
$32$	0	0
$33$	−5.60555	−0.975801
$34$	0	0
$35$	−0.697224	−0.117852
$36$	0	0
$37$	5.60555	0.921547	0.460773	−	0.887518i	$-0.347572\pi$
$37$	5.60555	0.921547	0.460773	+	0.887518i	$0.347572\pi$
$38$	0	0
$39$	2.30278	0.368739
$40$	0	0
$41$	3.60555	0.563093	0.281546	−	0.959548i	$-0.409153\pi$
$41$	3.60555	0.563093	0.281546	+	0.959548i	$0.409153\pi$
$42$	0	0
$43$	−4.30278	−0.656167	−0.328084	−	0.944649i	$-0.606403\pi$
$43$	−4.30278	−0.656167	−0.328084	+	0.944649i	$0.606403\pi$
$44$	0	0
$45$	−0.697224	−0.103936
$46$	0	0
$47$	4.60555	0.671789	0.335894	−	0.941900i	$-0.390961\pi$
$47$	4.60555	0.671789	0.335894	+	0.941900i	$0.390961\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−0.394449	−0.0552339
$52$	0	0
$53$	5.90833	0.811571	0.405786	−	0.913968i	$-0.366998\pi$
$53$	5.90833	0.811571	0.405786	+	0.913968i	$0.366998\pi$
$54$	0	0
$55$	3.90833	0.526999
$56$	0	0
$57$	0.394449	0.0522460
$58$	0	0
$59$	−3.90833	−0.508821	−0.254410	−	0.967096i	$-0.581881\pi$
$59$	−3.90833	−0.508821	−0.254410	+	0.967096i	$0.581881\pi$
$60$	0	0
$61$	4.90833	0.628447	0.314223	−	0.949349i	$-0.398256\pi$
$61$	4.90833	0.628447	0.314223	+	0.949349i	$0.398256\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	−1.60555	−0.199144
$66$	0	0
$67$	−13.3028	−1.62519	−0.812596	−	0.582827i	$-0.801947\pi$
$67$	−13.3028	−1.62519	−0.812596	+	0.582827i	$0.801947\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−12.9083	−1.53194	−0.765968	−	0.642878i	$-0.777740\pi$
$71$	−12.9083	−1.53194	−0.765968	+	0.642878i	$0.777740\pi$
$72$	0	0
$73$	11.0000	1.28745	0.643726	−	0.765256i	$-0.277388\pi$
$73$	11.0000	1.28745	0.643726	+	0.765256i	$0.277388\pi$
$74$	0	0
$75$	−4.51388	−0.521218
$76$	0	0
$77$	−5.60555	−0.638812
$78$	0	0
$79$	−13.4222	−1.51012	−0.755058	−	0.655658i	$-0.772391\pi$
$79$	−13.4222	−1.51012	−0.755058	+	0.655658i	$0.772391\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−16.2111	−1.77940	−0.889700	−	0.456546i	$-0.849086\pi$
$83$	−16.2111	−1.77940	−0.889700	+	0.456546i	$0.849086\pi$
$84$	0	0
$85$	0.275019	0.0298300
$86$	0	0
$87$	−5.60555	−0.600978
$88$	0	0
$89$	−15.5139	−1.64447	−0.822234	−	0.569150i	$-0.807272\pi$
$89$	−15.5139	−1.64447	−0.822234	+	0.569150i	$0.807272\pi$
$90$	0	0
$91$	2.30278	0.241396
$92$	0	0
$93$	3.60555	0.373878
$94$	0	0
$95$	−0.275019	−0.0282164
$96$	0	0
$97$	3.78890	0.384704	0.192352	−	0.981326i	$-0.438388\pi$
$97$	3.78890	0.384704	0.192352	+	0.981326i	$0.438388\pi$
$98$	0	0
$99$	−5.60555	−0.563379
$100$	0	0
$101$	−12.5139	−1.24518	−0.622589	−	0.782549i	$-0.713919\pi$
$101$	−12.5139	−1.24518	−0.622589	+	0.782549i	$0.713919\pi$
$102$	0	0
$103$	−2.78890	−0.274798	−0.137399	−	0.990516i	$-0.543874\pi$
$103$	−2.78890	−0.274798	−0.137399	+	0.990516i	$0.543874\pi$
$104$	0	0
$105$	−0.697224	−0.0680421
$106$	0	0
$107$	−0.302776	−0.0292704	−0.0146352	−	0.999893i	$-0.504659\pi$
$107$	−0.302776	−0.0292704	−0.0146352	+	0.999893i	$0.504659\pi$
$108$	0	0
$109$	−4.51388	−0.432351	−0.216176	−	0.976355i	$-0.569358\pi$
$109$	−4.51388	−0.432351	−0.216176	+	0.976355i	$0.569358\pi$
$110$	0	0
$111$	5.60555	0.532055
$112$	0	0
$113$	5.30278	0.498843	0.249422	−	0.968395i	$-0.419760\pi$
$113$	5.30278	0.498843	0.249422	+	0.968395i	$0.419760\pi$
$114$	0	0
$115$	−0.697224	−0.0650165
$116$	0	0
$117$	2.30278	0.212892
$118$	0	0
$119$	−0.394449	−0.0361591
$120$	0	0
$121$	20.4222	1.85656
$122$	0	0
$123$	3.60555	0.325102
$124$	0	0
$125$	6.63331	0.593301
$126$	0	0
$127$	−18.1194	−1.60784	−0.803920	−	0.594738i	$-0.797256\pi$
$127$	−18.1194	−1.60784	−0.803920	+	0.594738i	$0.797256\pi$
$128$	0	0
$129$	−4.30278	−0.378838
$130$	0	0
$131$	6.81665	0.595574	0.297787	−	0.954632i	$-0.403752\pi$
$131$	6.81665	0.595574	0.297787	+	0.954632i	$0.403752\pi$
$132$	0	0
$133$	0.394449	0.0342030
$134$	0	0
$135$	−0.697224	−0.0600075
$136$	0	0
$137$	−5.60555	−0.478915	−0.239457	−	0.970907i	$-0.576970\pi$
$137$	−5.60555	−0.478915	−0.239457	+	0.970907i	$0.576970\pi$
$138$	0	0
$139$	15.1194	1.28241	0.641207	−	0.767368i	$-0.278434\pi$
$139$	15.1194	1.28241	0.641207	+	0.767368i	$0.278434\pi$
$140$	0	0
$141$	4.60555	0.387857
$142$	0	0
$143$	−12.9083	−1.07945
$144$	0	0
$145$	3.90833	0.324569
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	−15.3944	−1.26116	−0.630581	−	0.776123i	$-0.717183\pi$
$149$	−15.3944	−1.26116	−0.630581	+	0.776123i	$0.717183\pi$
$150$	0	0
$151$	−8.60555	−0.700310	−0.350155	−	0.936692i	$-0.613871\pi$
$151$	−8.60555	−0.700310	−0.350155	+	0.936692i	$0.613871\pi$
$152$	0	0
$153$	−0.394449	−0.0318893
$154$	0	0
$155$	−2.51388	−0.201920
$156$	0	0
$157$	11.8167	0.943072	0.471536	−	0.881847i	$-0.343700\pi$
$157$	11.8167	0.943072	0.471536	+	0.881847i	$0.343700\pi$
$158$	0	0
$159$	5.90833	0.468561
$160$	0	0
$161$	1.00000	0.0788110
$162$	0	0
$163$	9.11943	0.714289	0.357144	−	0.934049i	$-0.383750\pi$
$163$	9.11943	0.714289	0.357144	+	0.934049i	$0.383750\pi$
$164$	0	0
$165$	3.90833	0.304263
$166$	0	0
$167$	14.0278	1.08550	0.542750	−	0.839894i	$-0.317383\pi$
$167$	14.0278	1.08550	0.542750	+	0.839894i	$0.317383\pi$
$168$	0	0
$169$	−7.69722	−0.592094
$170$	0	0
$171$	0.394449	0.0301642
$172$	0	0
$173$	−6.39445	−0.486161	−0.243080	−	0.970006i	$-0.578158\pi$
$173$	−6.39445	−0.486161	−0.243080	+	0.970006i	$0.578158\pi$
$174$	0	0
$175$	−4.51388	−0.341217
$176$	0	0
$177$	−3.90833	−0.293768
$178$	0	0
$179$	−24.9361	−1.86381	−0.931905	−	0.362702i	$-0.881854\pi$
$179$	−24.9361	−1.86381	−0.931905	+	0.362702i	$0.881854\pi$
$180$	0	0
$181$	−19.0000	−1.41226	−0.706129	−	0.708083i	$-0.749560\pi$
$181$	−19.0000	−1.41226	−0.706129	+	0.708083i	$0.749560\pi$
$182$	0	0
$183$	4.90833	0.362834
$184$	0	0
$185$	−3.90833	−0.287346
$186$	0	0
$187$	2.21110	0.161692
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	8.60555	0.622676	0.311338	−	0.950299i	$-0.399223\pi$
$191$	8.60555	0.622676	0.311338	+	0.950299i	$0.399223\pi$
$192$	0	0
$193$	21.0278	1.51361	0.756806	−	0.653640i	$-0.226759\pi$
$193$	21.0278	1.51361	0.756806	+	0.653640i	$0.226759\pi$
$194$	0	0
$195$	−1.60555	−0.114976
$196$	0	0
$197$	−25.9361	−1.84787	−0.923935	−	0.382550i	$-0.875046\pi$
$197$	−25.9361	−1.84787	−0.923935	+	0.382550i	$0.875046\pi$
$198$	0	0
$199$	1.90833	0.135278	0.0676389	−	0.997710i	$-0.478453\pi$
$199$	1.90833	0.135278	0.0676389	+	0.997710i	$0.478453\pi$
$200$	0	0
$201$	−13.3028	−0.938305
$202$	0	0
$203$	−5.60555	−0.393433
$204$	0	0
$205$	−2.51388	−0.175577
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	−2.21110	−0.152945
$210$	0	0
$211$	−25.6056	−1.76276	−0.881379	−	0.472409i	$-0.843385\pi$
$211$	−25.6056	−1.76276	−0.881379	+	0.472409i	$0.843385\pi$
$212$	0	0
$213$	−12.9083	−0.884464
$214$	0	0
$215$	3.00000	0.204598
$216$	0	0
$217$	3.60555	0.244761
$218$	0	0
$219$	11.0000	0.743311
$220$	0	0
$221$	−0.908327	−0.0611007
$222$	0	0
$223$	−17.7250	−1.18695	−0.593476	−	0.804852i	$-0.702245\pi$
$223$	−17.7250	−1.18695	−0.593476	+	0.804852i	$0.702245\pi$
$224$	0	0
$225$	−4.51388	−0.300925
$226$	0	0
$227$	−11.9083	−0.790383	−0.395192	−	0.918599i	$-0.629322\pi$
$227$	−11.9083	−0.790383	−0.395192	+	0.918599i	$0.629322\pi$
$228$	0	0
$229$	3.11943	0.206138	0.103069	−	0.994674i	$-0.467134\pi$
$229$	3.11943	0.206138	0.103069	+	0.994674i	$0.467134\pi$
$230$	0	0
$231$	−5.60555	−0.368818
$232$	0	0
$233$	7.11943	0.466409	0.233205	−	0.972428i	$-0.425079\pi$
$233$	7.11943	0.466409	0.233205	+	0.972428i	$0.425079\pi$
$234$	0	0
$235$	−3.21110	−0.209469
$236$	0	0
$237$	−13.4222	−0.871866
$238$	0	0
$239$	−5.90833	−0.382178	−0.191089	−	0.981573i	$-0.561202\pi$
$239$	−5.90833	−0.382178	−0.191089	+	0.981573i	$0.561202\pi$
$240$	0	0
$241$	8.21110	0.528924	0.264462	−	0.964396i	$-0.414806\pi$
$241$	8.21110	0.528924	0.264462	+	0.964396i	$0.414806\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−0.697224	−0.0445440
$246$	0	0
$247$	0.908327	0.0577955
$248$	0	0
$249$	−16.2111	−1.02734
$250$	0	0
$251$	−4.18335	−0.264050	−0.132025	−	0.991246i	$-0.542148\pi$
$251$	−4.18335	−0.264050	−0.132025	+	0.991246i	$0.542148\pi$
$252$	0	0
$253$	−5.60555	−0.352418
$254$	0	0
$255$	0.275019	0.0172224
$256$	0	0
$257$	25.8167	1.61040	0.805199	−	0.593004i	$-0.202058\pi$
$257$	25.8167	1.61040	0.805199	+	0.593004i	$0.202058\pi$
$258$	0	0
$259$	5.60555	0.348312
$260$	0	0
$261$	−5.60555	−0.346975
$262$	0	0
$263$	9.42221	0.580998	0.290499	−	0.956875i	$-0.406179\pi$
$263$	9.42221	0.580998	0.290499	+	0.956875i	$0.406179\pi$
$264$	0	0
$265$	−4.11943	−0.253055
$266$	0	0
$267$	−15.5139	−0.949434
$268$	0	0
$269$	12.1194	0.738935	0.369467	−	0.929244i	$-0.379540\pi$
$269$	12.1194	0.738935	0.369467	+	0.929244i	$0.379540\pi$
$270$	0	0
$271$	0.577795	0.0350985	0.0175493	−	0.999846i	$-0.494414\pi$
$271$	0.577795	0.0350985	0.0175493	+	0.999846i	$0.494414\pi$
$272$	0	0
$273$	2.30278	0.139370
$274$	0	0
$275$	25.3028	1.52581
$276$	0	0
$277$	24.5416	1.47456	0.737282	−	0.675585i	$-0.236109\pi$
$277$	24.5416	1.47456	0.737282	+	0.675585i	$0.236109\pi$
$278$	0	0
$279$	3.60555	0.215859
$280$	0	0
$281$	6.60555	0.394054	0.197027	−	0.980398i	$-0.436871\pi$
$281$	6.60555	0.394054	0.197027	+	0.980398i	$0.436871\pi$
$282$	0	0
$283$	−25.3028	−1.50409	−0.752047	−	0.659110i	$-0.770933\pi$
$283$	−25.3028	−1.50409	−0.752047	+	0.659110i	$0.770933\pi$
$284$	0	0
$285$	−0.275019	−0.0162907
$286$	0	0
$287$	3.60555	0.212829
$288$	0	0
$289$	−16.8444	−0.990848
$290$	0	0
$291$	3.78890	0.222109
$292$	0	0
$293$	−18.7889	−1.09766	−0.548830	−	0.835934i	$-0.684926\pi$
$293$	−18.7889	−1.09766	−0.548830	+	0.835934i	$0.684926\pi$
$294$	0	0
$295$	2.72498	0.158655
$296$	0	0
$297$	−5.60555	−0.325267
$298$	0	0
$299$	2.30278	0.133173
$300$	0	0
$301$	−4.30278	−0.248008
$302$	0	0
$303$	−12.5139	−0.718904
$304$	0	0
$305$	−3.42221	−0.195955
$306$	0	0
$307$	25.6056	1.46139	0.730693	−	0.682706i	$-0.239197\pi$
$307$	25.6056	1.46139	0.730693	+	0.682706i	$0.239197\pi$
$308$	0	0
$309$	−2.78890	−0.158655
$310$	0	0
$311$	−10.1194	−0.573820	−0.286910	−	0.957958i	$-0.592628\pi$
$311$	−10.1194	−0.573820	−0.286910	+	0.957958i	$0.592628\pi$
$312$	0	0
$313$	−30.4222	−1.71956	−0.859782	−	0.510661i	$-0.829401\pi$
$313$	−30.4222	−1.71956	−0.859782	+	0.510661i	$0.829401\pi$
$314$	0	0
$315$	−0.697224	−0.0392841
$316$	0	0
$317$	−12.6972	−0.713147	−0.356574	−	0.934267i	$-0.616055\pi$
$317$	−12.6972	−0.713147	−0.356574	+	0.934267i	$0.616055\pi$
$318$	0	0
$319$	31.4222	1.75931
$320$	0	0
$321$	−0.302776	−0.0168993
$322$	0	0
$323$	−0.155590	−0.00865725
$324$	0	0
$325$	−10.3944	−0.576580
$326$	0	0
$327$	−4.51388	−0.249618
$328$	0	0
$329$	4.60555	0.253912
$330$	0	0
$331$	2.00000	0.109930	0.0549650	−	0.998488i	$-0.482495\pi$
$331$	2.00000	0.109930	0.0549650	+	0.998488i	$0.482495\pi$
$332$	0	0
$333$	5.60555	0.307182
$334$	0	0
$335$	9.27502	0.506748
$336$	0	0
$337$	16.7250	0.911068	0.455534	−	0.890218i	$-0.349448\pi$
$337$	16.7250	0.911068	0.455534	+	0.890218i	$0.349448\pi$
$338$	0	0
$339$	5.30278	0.288007
$340$	0	0
$341$	−20.2111	−1.09449
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−0.697224	−0.0375373
$346$	0	0
$347$	−0.211103	−0.0113326	−0.00566629	−	0.999984i	$-0.501804\pi$
$347$	−0.211103	−0.0113326	−0.00566629	+	0.999984i	$0.501804\pi$
$348$	0	0
$349$	0.697224	0.0373216	0.0186608	−	0.999826i	$-0.494060\pi$
$349$	0.697224	0.0373216	0.0186608	+	0.999826i	$0.494060\pi$
$350$	0	0
$351$	2.30278	0.122913
$352$	0	0
$353$	15.6056	0.830600	0.415300	−	0.909685i	$-0.363677\pi$
$353$	15.6056	0.830600	0.415300	+	0.909685i	$0.363677\pi$
$354$	0	0
$355$	9.00000	0.477670
$356$	0	0
$357$	−0.394449	−0.0208764
$358$	0	0
$359$	−9.90833	−0.522941	−0.261471	−	0.965211i	$-0.584207\pi$
$359$	−9.90833	−0.522941	−0.261471	+	0.965211i	$0.584207\pi$
$360$	0	0
$361$	−18.8444	−0.991811
$362$	0	0
$363$	20.4222	1.07189
$364$	0	0
$365$	−7.66947	−0.401438
$366$	0	0
$367$	−25.5139	−1.33181	−0.665907	−	0.746035i	$-0.731955\pi$
$367$	−25.5139	−1.33181	−0.665907	+	0.746035i	$0.731955\pi$
$368$	0	0
$369$	3.60555	0.187698
$370$	0	0
$371$	5.90833	0.306745
$372$	0	0
$373$	11.1833	0.579052	0.289526	−	0.957170i	$-0.406502\pi$
$373$	11.1833	0.579052	0.289526	+	0.957170i	$0.406502\pi$
$374$	0	0
$375$	6.63331	0.342543
$376$	0	0
$377$	−12.9083	−0.664813
$378$	0	0
$379$	10.4222	0.535353	0.267676	−	0.963509i	$-0.413744\pi$
$379$	10.4222	0.535353	0.267676	+	0.963509i	$0.413744\pi$
$380$	0	0
$381$	−18.1194	−0.928286
$382$	0	0
$383$	26.6333	1.36090	0.680449	−	0.732795i	$-0.261785\pi$
$383$	26.6333	1.36090	0.680449	+	0.732795i	$0.261785\pi$
$384$	0	0
$385$	3.90833	0.199187
$386$	0	0
$387$	−4.30278	−0.218722
$388$	0	0
$389$	−12.6333	−0.640534	−0.320267	−	0.947327i	$-0.603773\pi$
$389$	−12.6333	−0.640534	−0.320267	+	0.947327i	$0.603773\pi$
$390$	0	0
$391$	−0.394449	−0.0199481
$392$	0	0
$393$	6.81665	0.343855
$394$	0	0
$395$	9.35829	0.470867
$396$	0	0
$397$	−8.60555	−0.431900	−0.215950	−	0.976404i	$-0.569285\pi$
$397$	−8.60555	−0.431900	−0.215950	+	0.976404i	$0.569285\pi$
$398$	0	0
$399$	0.394449	0.0197471
$400$	0	0
$401$	−26.6333	−1.33000	−0.665002	−	0.746842i	$-0.731569\pi$
$401$	−26.6333	−1.33000	−0.665002	+	0.746842i	$0.731569\pi$
$402$	0	0
$403$	8.30278	0.413591
$404$	0	0
$405$	−0.697224	−0.0346454
$406$	0	0
$407$	−31.4222	−1.55754
$408$	0	0
$409$	8.39445	0.415079	0.207539	−	0.978227i	$-0.433454\pi$
$409$	8.39445	0.415079	0.207539	+	0.978227i	$0.433454\pi$
$410$	0	0
$411$	−5.60555	−0.276501
$412$	0	0
$413$	−3.90833	−0.192316
$414$	0	0
$415$	11.3028	0.554831
$416$	0	0
$417$	15.1194	0.740402
$418$	0	0
$419$	11.9083	0.581760	0.290880	−	0.956760i	$-0.406052\pi$
$419$	11.9083	0.581760	0.290880	+	0.956760i	$0.406052\pi$
$420$	0	0
$421$	−7.09167	−0.345627	−0.172813	−	0.984955i	$-0.555286\pi$
$421$	−7.09167	−0.345627	−0.172813	+	0.984955i	$0.555286\pi$
$422$	0	0
$423$	4.60555	0.223930
$424$	0	0
$425$	1.78049	0.0863666
$426$	0	0
$427$	4.90833	0.237531
$428$	0	0
$429$	−12.9083	−0.623220
$430$	0	0
$431$	4.30278	0.207257	0.103629	−	0.994616i	$-0.466955\pi$
$431$	4.30278	0.207257	0.103629	+	0.994616i	$0.466955\pi$
$432$	0	0
$433$	23.8167	1.14456	0.572278	−	0.820060i	$-0.306060\pi$
$433$	23.8167	1.14456	0.572278	+	0.820060i	$0.306060\pi$
$434$	0	0
$435$	3.90833	0.187390
$436$	0	0
$437$	0.394449	0.0188690
$438$	0	0
$439$	27.6056	1.31754	0.658771	−	0.752344i	$-0.271077\pi$
$439$	27.6056	1.31754	0.658771	+	0.752344i	$0.271077\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	6.78890	0.322550	0.161275	−	0.986909i	$-0.448439\pi$
$443$	6.78890	0.322550	0.161275	+	0.986909i	$0.448439\pi$
$444$	0	0
$445$	10.8167	0.512759
$446$	0	0
$447$	−15.3944	−0.728132
$448$	0	0
$449$	−7.11943	−0.335987	−0.167993	−	0.985788i	$-0.553729\pi$
$449$	−7.11943	−0.335987	−0.167993	+	0.985788i	$0.553729\pi$
$450$	0	0
$451$	−20.2111	−0.951704
$452$	0	0
$453$	−8.60555	−0.404324
$454$	0	0
$455$	−1.60555	−0.0752694
$456$	0	0
$457$	−20.6972	−0.968175	−0.484088	−	0.875020i	$-0.660848\pi$
$457$	−20.6972	−0.968175	−0.484088	+	0.875020i	$0.660848\pi$
$458$	0	0
$459$	−0.394449	−0.0184113
$460$	0	0
$461$	−26.0917	−1.21521	−0.607605	−	0.794239i	$-0.707870\pi$
$461$	−26.0917	−1.21521	−0.607605	+	0.794239i	$0.707870\pi$
$462$	0	0
$463$	−17.0000	−0.790057	−0.395029	−	0.918669i	$-0.629265\pi$
$463$	−17.0000	−0.790057	−0.395029	+	0.918669i	$0.629265\pi$
$464$	0	0
$465$	−2.51388	−0.116578
$466$	0	0
$467$	23.8444	1.10339	0.551694	−	0.834047i	$-0.313982\pi$
$467$	23.8444	1.10339	0.551694	+	0.834047i	$0.313982\pi$
$468$	0	0
$469$	−13.3028	−0.614265
$470$	0	0
$471$	11.8167	0.544483
$472$	0	0
$473$	24.1194	1.10901
$474$	0	0
$475$	−1.78049	−0.0816946
$476$	0	0
$477$	5.90833	0.270524
$478$	0	0
$479$	27.2389	1.24458	0.622288	−	0.782789i	$-0.286203\pi$
$479$	27.2389	1.24458	0.622288	+	0.782789i	$0.286203\pi$
$480$	0	0
$481$	12.9083	0.588569
$482$	0	0
$483$	1.00000	0.0455016
$484$	0	0
$485$	−2.64171	−0.119954
$486$	0	0
$487$	−4.21110	−0.190823	−0.0954116	−	0.995438i	$-0.530417\pi$
$487$	−4.21110	−0.190823	−0.0954116	+	0.995438i	$0.530417\pi$
$488$	0	0
$489$	9.11943	0.412395
$490$	0	0
$491$	−34.9083	−1.57539	−0.787695	−	0.616065i	$-0.788726\pi$
$491$	−34.9083	−1.57539	−0.787695	+	0.616065i	$0.788726\pi$
$492$	0	0
$493$	2.21110	0.0995831
$494$	0	0
$495$	3.90833	0.175666
$496$	0	0
$497$	−12.9083	−0.579018
$498$	0	0
$499$	−12.8806	−0.576614	−0.288307	−	0.957538i	$-0.593092\pi$
$499$	−12.8806	−0.576614	−0.288307	+	0.957538i	$0.593092\pi$
$500$	0	0
$501$	14.0278	0.626714
$502$	0	0
$503$	−21.1194	−0.941669	−0.470834	−	0.882222i	$-0.656047\pi$
$503$	−21.1194	−0.941669	−0.470834	+	0.882222i	$0.656047\pi$
$504$	0	0
$505$	8.72498	0.388257
$506$	0	0
$507$	−7.69722	−0.341846
$508$	0	0
$509$	−19.3944	−0.859644	−0.429822	−	0.902914i	$-0.641424\pi$
$509$	−19.3944	−0.859644	−0.429822	+	0.902914i	$0.641424\pi$
$510$	0	0
$511$	11.0000	0.486611
$512$	0	0
$513$	0.394449	0.0174153
$514$	0	0
$515$	1.94449	0.0856843
$516$	0	0
$517$	−25.8167	−1.13542
$518$	0	0
$519$	−6.39445	−0.280685
$520$	0	0
$521$	4.78890	0.209805	0.104903	−	0.994482i	$-0.466547\pi$
$521$	4.78890	0.209805	0.104903	+	0.994482i	$0.466547\pi$
$522$	0	0
$523$	−31.2111	−1.36477	−0.682383	−	0.730995i	$-0.739056\pi$
$523$	−31.2111	−1.36477	−0.682383	+	0.730995i	$0.739056\pi$
$524$	0	0
$525$	−4.51388	−0.197002
$526$	0	0
$527$	−1.42221	−0.0619522
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−3.90833	−0.169607
$532$	0	0
$533$	8.30278	0.359633
$534$	0	0
$535$	0.211103	0.00912676
$536$	0	0
$537$	−24.9361	−1.07607
$538$	0	0
$539$	−5.60555	−0.241448
$540$	0	0
$541$	27.0278	1.16201	0.581007	−	0.813899i	$-0.302659\pi$
$541$	27.0278	1.16201	0.581007	+	0.813899i	$0.302659\pi$
$542$	0	0
$543$	−19.0000	−0.815368
$544$	0	0
$545$	3.14719	0.134811
$546$	0	0
$547$	28.3305	1.21133	0.605663	−	0.795721i	$-0.292908\pi$
$547$	28.3305	1.21133	0.605663	+	0.795721i	$0.292908\pi$
$548$	0	0
$549$	4.90833	0.209482
$550$	0	0
$551$	−2.21110	−0.0941961
$552$	0	0
$553$	−13.4222	−0.570770
$554$	0	0
$555$	−3.90833	−0.165899
$556$	0	0
$557$	16.1833	0.685710	0.342855	−	0.939388i	$-0.388606\pi$
$557$	16.1833	0.685710	0.342855	+	0.939388i	$0.388606\pi$
$558$	0	0
$559$	−9.90833	−0.419078
$560$	0	0
$561$	2.21110	0.0933528
$562$	0	0
$563$	38.5416	1.62434	0.812168	−	0.583423i	$-0.198287\pi$
$563$	38.5416	1.62434	0.812168	+	0.583423i	$0.198287\pi$
$564$	0	0
$565$	−3.69722	−0.155543
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−14.7889	−0.619983	−0.309991	−	0.950739i	$-0.600326\pi$
$569$	−14.7889	−0.619983	−0.309991	+	0.950739i	$0.600326\pi$
$570$	0	0
$571$	29.7889	1.24663	0.623313	−	0.781972i	$-0.285786\pi$
$571$	29.7889	1.24663	0.623313	+	0.781972i	$0.285786\pi$
$572$	0	0
$573$	8.60555	0.359502
$574$	0	0
$575$	−4.51388	−0.188242
$576$	0	0
$577$	14.0000	0.582828	0.291414	−	0.956597i	$-0.405874\pi$
$577$	14.0000	0.582828	0.291414	+	0.956597i	$0.405874\pi$
$578$	0	0
$579$	21.0278	0.873884
$580$	0	0
$581$	−16.2111	−0.672550
$582$	0	0
$583$	−33.1194	−1.37167
$584$	0	0
$585$	−1.60555	−0.0663814
$586$	0	0
$587$	35.5139	1.46581	0.732907	−	0.680329i	$-0.238163\pi$
$587$	35.5139	1.46581	0.732907	+	0.680329i	$0.238163\pi$
$588$	0	0
$589$	1.42221	0.0586009
$590$	0	0
$591$	−25.9361	−1.06687
$592$	0	0
$593$	21.3944	0.878565	0.439282	−	0.898349i	$-0.355233\pi$
$593$	21.3944	0.878565	0.439282	+	0.898349i	$0.355233\pi$
$594$	0	0
$595$	0.275019	0.0112747
$596$	0	0
$597$	1.90833	0.0781026
$598$	0	0
$599$	−26.5416	−1.08446	−0.542231	−	0.840230i	$-0.682420\pi$
$599$	−26.5416	−1.08446	−0.542231	+	0.840230i	$0.682420\pi$
$600$	0	0
$601$	−27.5416	−1.12345	−0.561723	−	0.827325i	$-0.689861\pi$
$601$	−27.5416	−1.12345	−0.561723	+	0.827325i	$0.689861\pi$
$602$	0	0
$603$	−13.3028	−0.541731
$604$	0	0
$605$	−14.2389	−0.578892
$606$	0	0
$607$	20.5416	0.833759	0.416880	−	0.908962i	$-0.363124\pi$
$607$	20.5416	0.833759	0.416880	+	0.908962i	$0.363124\pi$
$608$	0	0
$609$	−5.60555	−0.227148
$610$	0	0
$611$	10.6056	0.429055
$612$	0	0
$613$	−28.6333	−1.15649	−0.578244	−	0.815864i	$-0.696262\pi$
$613$	−28.6333	−1.15649	−0.578244	+	0.815864i	$0.696262\pi$
$614$	0	0
$615$	−2.51388	−0.101369
$616$	0	0
$617$	20.3028	0.817359	0.408679	−	0.912678i	$-0.365989\pi$
$617$	20.3028	0.817359	0.408679	+	0.912678i	$0.365989\pi$
$618$	0	0
$619$	−40.3028	−1.61991	−0.809953	−	0.586495i	$-0.800507\pi$
$619$	−40.3028	−1.61991	−0.809953	+	0.586495i	$0.800507\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	−15.5139	−0.621550
$624$	0	0
$625$	17.9445	0.717779
$626$	0	0
$627$	−2.21110	−0.0883029
$628$	0	0
$629$	−2.21110	−0.0881624
$630$	0	0
$631$	−16.3944	−0.652653	−0.326326	−	0.945257i	$-0.605811\pi$
$631$	−16.3944	−0.652653	−0.326326	+	0.945257i	$0.605811\pi$
$632$	0	0
$633$	−25.6056	−1.01773
$634$	0	0
$635$	12.6333	0.501338
$636$	0	0
$637$	2.30278	0.0912393
$638$	0	0
$639$	−12.9083	−0.510646
$640$	0	0
$641$	−30.5416	−1.20632	−0.603161	−	0.797619i	$-0.706092\pi$
$641$	−30.5416	−1.20632	−0.603161	+	0.797619i	$0.706092\pi$
$642$	0	0
$643$	−19.0917	−0.752902	−0.376451	−	0.926437i	$-0.622856\pi$
$643$	−19.0917	−0.752902	−0.376451	+	0.926437i	$0.622856\pi$
$644$	0	0
$645$	3.00000	0.118125
$646$	0	0
$647$	16.5139	0.649228	0.324614	−	0.945847i	$-0.394766\pi$
$647$	16.5139	0.649228	0.324614	+	0.945847i	$0.394766\pi$
$648$	0	0
$649$	21.9083	0.859977
$650$	0	0
$651$	3.60555	0.141313
$652$	0	0
$653$	17.9083	0.700807	0.350403	−	0.936599i	$-0.386044\pi$
$653$	17.9083	0.700807	0.350403	+	0.936599i	$0.386044\pi$
$654$	0	0
$655$	−4.75274	−0.185705
$656$	0	0
$657$	11.0000	0.429151
$658$	0	0
$659$	13.1833	0.513550	0.256775	−	0.966471i	$-0.417340\pi$
$659$	13.1833	0.513550	0.256775	+	0.966471i	$0.417340\pi$
$660$	0	0
$661$	−5.42221	−0.210899	−0.105450	−	0.994425i	$-0.533628\pi$
$661$	−5.42221	−0.210899	−0.105450	+	0.994425i	$0.533628\pi$
$662$	0	0
$663$	−0.908327	−0.0352765
$664$	0	0
$665$	−0.275019	−0.0106648
$666$	0	0
$667$	−5.60555	−0.217048
$668$	0	0
$669$	−17.7250	−0.685287
$670$	0	0
$671$	−27.5139	−1.06216
$672$	0	0
$673$	5.00000	0.192736	0.0963679	−	0.995346i	$-0.469277\pi$
$673$	5.00000	0.192736	0.0963679	+	0.995346i	$0.469277\pi$
$674$	0	0
$675$	−4.51388	−0.173739
$676$	0	0
$677$	−30.1194	−1.15758	−0.578792	−	0.815475i	$-0.696476\pi$
$677$	−30.1194	−1.15758	−0.578792	+	0.815475i	$0.696476\pi$
$678$	0	0
$679$	3.78890	0.145405
$680$	0	0
$681$	−11.9083	−0.456328
$682$	0	0
$683$	30.6333	1.17215	0.586075	−	0.810256i	$-0.300672\pi$
$683$	30.6333	1.17215	0.586075	+	0.810256i	$0.300672\pi$
$684$	0	0
$685$	3.90833	0.149329
$686$	0	0
$687$	3.11943	0.119014
$688$	0	0
$689$	13.6056	0.518330
$690$	0	0
$691$	3.09167	0.117613	0.0588064	−	0.998269i	$-0.481271\pi$
$691$	3.09167	0.117613	0.0588064	+	0.998269i	$0.481271\pi$
$692$	0	0
$693$	−5.60555	−0.212937
$694$	0	0
$695$	−10.5416	−0.399867
$696$	0	0
$697$	−1.42221	−0.0538699
$698$	0	0
$699$	7.11943	0.269282
$700$	0	0
$701$	18.9361	0.715206	0.357603	−	0.933874i	$-0.383594\pi$
$701$	18.9361	0.715206	0.357603	+	0.933874i	$0.383594\pi$
$702$	0	0
$703$	2.21110	0.0833933
$704$	0	0
$705$	−3.21110	−0.120937
$706$	0	0
$707$	−12.5139	−0.470633
$708$	0	0
$709$	−20.1194	−0.755601	−0.377801	−	0.925887i	$-0.623319\pi$
$709$	−20.1194	−0.755601	−0.377801	+	0.925887i	$0.623319\pi$
$710$	0	0
$711$	−13.4222	−0.503372
$712$	0	0
$713$	3.60555	0.135029
$714$	0	0
$715$	9.00000	0.336581
$716$	0	0
$717$	−5.90833	−0.220651
$718$	0	0
$719$	−41.8444	−1.56053	−0.780267	−	0.625447i	$-0.784917\pi$
$719$	−41.8444	−1.56053	−0.780267	+	0.625447i	$0.784917\pi$
$720$	0	0
$721$	−2.78890	−0.103864
$722$	0	0
$723$	8.21110	0.305374
$724$	0	0
$725$	25.3028	0.939721
$726$	0	0
$727$	6.39445	0.237157	0.118578	−	0.992945i	$-0.462166\pi$
$727$	6.39445	0.237157	0.118578	+	0.992945i	$0.462166\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	1.69722	0.0627741
$732$	0	0
$733$	−37.0278	−1.36765	−0.683826	−	0.729645i	$-0.739685\pi$
$733$	−37.0278	−1.36765	−0.683826	+	0.729645i	$0.739685\pi$
$734$	0	0
$735$	−0.697224	−0.0257175
$736$	0	0
$737$	74.5694	2.74680
$738$	0	0
$739$	36.8167	1.35432	0.677161	−	0.735835i	$-0.263210\pi$
$739$	36.8167	1.35432	0.677161	+	0.735835i	$0.263210\pi$
$740$	0	0
$741$	0.908327	0.0333682
$742$	0	0
$743$	−27.4861	−1.00837	−0.504184	−	0.863596i	$-0.668207\pi$
$743$	−27.4861	−1.00837	−0.504184	+	0.863596i	$0.668207\pi$
$744$	0	0
$745$	10.7334	0.393241
$746$	0	0
$747$	−16.2111	−0.593133
$748$	0	0
$749$	−0.302776	−0.0110632
$750$	0	0
$751$	42.9361	1.56676	0.783380	−	0.621543i	$-0.213494\pi$
$751$	42.9361	1.56676	0.783380	+	0.621543i	$0.213494\pi$
$752$	0	0
$753$	−4.18335	−0.152450
$754$	0	0
$755$	6.00000	0.218362
$756$	0	0
$757$	20.8167	0.756594	0.378297	−	0.925684i	$-0.376510\pi$
$757$	20.8167	0.756594	0.378297	+	0.925684i	$0.376510\pi$
$758$	0	0
$759$	−5.60555	−0.203469
$760$	0	0
$761$	−15.6333	−0.566707	−0.283353	−	0.959016i	$-0.591447\pi$
$761$	−15.6333	−0.566707	−0.283353	+	0.959016i	$0.591447\pi$
$762$	0	0
$763$	−4.51388	−0.163413
$764$	0	0
$765$	0.275019	0.00994334
$766$	0	0
$767$	−9.00000	−0.324971
$768$	0	0
$769$	11.8167	0.426119	0.213060	−	0.977039i	$-0.431657\pi$
$769$	11.8167	0.426119	0.213060	+	0.977039i	$0.431657\pi$
$770$	0	0
$771$	25.8167	0.929764
$772$	0	0
$773$	−2.21110	−0.0795278	−0.0397639	−	0.999209i	$-0.512661\pi$
$773$	−2.21110	−0.0795278	−0.0397639	+	0.999209i	$0.512661\pi$
$774$	0	0
$775$	−16.2750	−0.584616
$776$	0	0
$777$	5.60555	0.201098
$778$	0	0
$779$	1.42221	0.0509558
$780$	0	0
$781$	72.3583	2.58918
$782$	0	0
$783$	−5.60555	−0.200326
$784$	0	0
$785$	−8.23886	−0.294057
$786$	0	0
$787$	31.1194	1.10929	0.554644	−	0.832088i	$-0.312854\pi$
$787$	31.1194	1.10929	0.554644	+	0.832088i	$0.312854\pi$
$788$	0	0
$789$	9.42221	0.335439
$790$	0	0
$791$	5.30278	0.188545
$792$	0	0
$793$	11.3028	0.401373
$794$	0	0
$795$	−4.11943	−0.146101
$796$	0	0
$797$	−49.8167	−1.76460	−0.882298	−	0.470691i	$-0.844005\pi$
$797$	−49.8167	−1.76460	−0.882298	+	0.470691i	$0.844005\pi$
$798$	0	0
$799$	−1.81665	−0.0642686
$800$	0	0
$801$	−15.5139	−0.548156
$802$	0	0
$803$	−61.6611	−2.17597
$804$	0	0
$805$	−0.697224	−0.0245739
$806$	0	0
$807$	12.1194	0.426624
$808$	0	0
$809$	25.5139	0.897020	0.448510	−	0.893778i	$-0.351955\pi$
$809$	25.5139	0.897020	0.448510	+	0.893778i	$0.351955\pi$
$810$	0	0
$811$	−10.6333	−0.373386	−0.186693	−	0.982418i	$-0.559777\pi$
$811$	−10.6333	−0.373386	−0.186693	+	0.982418i	$0.559777\pi$
$812$	0	0
$813$	0.577795	0.0202642
$814$	0	0
$815$	−6.35829	−0.222721
$816$	0	0
$817$	−1.69722	−0.0593784
$818$	0	0
$819$	2.30278	0.0804655
$820$	0	0
$821$	−17.4500	−0.609008	−0.304504	−	0.952511i	$-0.598491\pi$
$821$	−17.4500	−0.609008	−0.304504	+	0.952511i	$0.598491\pi$
$822$	0	0
$823$	48.5139	1.69109	0.845544	−	0.533906i	$-0.179276\pi$
$823$	48.5139	1.69109	0.845544	+	0.533906i	$0.179276\pi$
$824$	0	0
$825$	25.3028	0.880930
$826$	0	0
$827$	−22.6972	−0.789260	−0.394630	−	0.918840i	$-0.629127\pi$
$827$	−22.6972	−0.789260	−0.394630	+	0.918840i	$0.629127\pi$
$828$	0	0
$829$	1.18335	0.0410993	0.0205497	−	0.999789i	$-0.493458\pi$
$829$	1.18335	0.0410993	0.0205497	+	0.999789i	$0.493458\pi$
$830$	0	0
$831$	24.5416	0.851340
$832$	0	0
$833$	−0.394449	−0.0136668
$834$	0	0
$835$	−9.78049	−0.338468
$836$	0	0
$837$	3.60555	0.124626
$838$	0	0
$839$	28.5416	0.985367	0.492683	−	0.870209i	$-0.336016\pi$
$839$	28.5416	0.985367	0.492683	+	0.870209i	$0.336016\pi$
$840$	0	0
$841$	2.42221	0.0835243
$842$	0	0
$843$	6.60555	0.227507
$844$	0	0
$845$	5.36669	0.184620
$846$	0	0
$847$	20.4222	0.701715
$848$	0	0
$849$	−25.3028	−0.868389
$850$	0	0
$851$	5.60555	0.192156
$852$	0	0
$853$	−2.60555	−0.0892124	−0.0446062	−	0.999005i	$-0.514203\pi$
$853$	−2.60555	−0.0892124	−0.0446062	+	0.999005i	$0.514203\pi$
$854$	0	0
$855$	−0.275019	−0.00940546
$856$	0	0
$857$	9.02776	0.308382	0.154191	−	0.988041i	$-0.450723\pi$
$857$	9.02776	0.308382	0.154191	+	0.988041i	$0.450723\pi$
$858$	0	0
$859$	−28.0555	−0.957242	−0.478621	−	0.878022i	$-0.658863\pi$
$859$	−28.0555	−0.957242	−0.478621	+	0.878022i	$0.658863\pi$
$860$	0	0
$861$	3.60555	0.122877
$862$	0	0
$863$	11.8167	0.402244	0.201122	−	0.979566i	$-0.435541\pi$
$863$	11.8167	0.402244	0.201122	+	0.979566i	$0.435541\pi$
$864$	0	0
$865$	4.45837	0.151589
$866$	0	0
$867$	−16.8444	−0.572066
$868$	0	0
$869$	75.2389	2.55230
$870$	0	0
$871$	−30.6333	−1.03797
$872$	0	0
$873$	3.78890	0.128235
$874$	0	0
$875$	6.63331	0.224247
$876$	0	0
$877$	−32.0000	−1.08056	−0.540282	−	0.841484i	$-0.681682\pi$
$877$	−32.0000	−1.08056	−0.540282	+	0.841484i	$0.681682\pi$
$878$	0	0
$879$	−18.7889	−0.633734
$880$	0	0
$881$	−2.23886	−0.0754291	−0.0377145	−	0.999289i	$-0.512008\pi$
$881$	−2.23886	−0.0754291	−0.0377145	+	0.999289i	$0.512008\pi$
$882$	0	0
$883$	6.90833	0.232484	0.116242	−	0.993221i	$-0.462915\pi$
$883$	6.90833	0.232484	0.116242	+	0.993221i	$0.462915\pi$
$884$	0	0
$885$	2.72498	0.0915992
$886$	0	0
$887$	1.51388	0.0508311	0.0254155	−	0.999677i	$-0.491909\pi$
$887$	1.51388	0.0508311	0.0254155	+	0.999677i	$0.491909\pi$
$888$	0	0
$889$	−18.1194	−0.607706
$890$	0	0
$891$	−5.60555	−0.187793
$892$	0	0
$893$	1.81665	0.0607920
$894$	0	0
$895$	17.3860	0.581151
$896$	0	0
$897$	2.30278	0.0768874
$898$	0	0
$899$	−20.2111	−0.674078
$900$	0	0
$901$	−2.33053	−0.0776413
$902$	0	0
$903$	−4.30278	−0.143187
$904$	0	0
$905$	13.2473	0.440354
$906$	0	0
$907$	49.7250	1.65109	0.825545	−	0.564336i	$-0.190868\pi$
$907$	49.7250	1.65109	0.825545	+	0.564336i	$0.190868\pi$
$908$	0	0
$909$	−12.5139	−0.415059
$910$	0	0
$911$	−24.6056	−0.815218	−0.407609	−	0.913156i	$-0.633637\pi$
$911$	−24.6056	−0.815218	−0.407609	+	0.913156i	$0.633637\pi$
$912$	0	0
$913$	90.8722	3.00743
$914$	0	0
$915$	−3.42221	−0.113135
$916$	0	0
$917$	6.81665	0.225106
$918$	0	0
$919$	−8.02776	−0.264811	−0.132406	−	0.991196i	$-0.542270\pi$
$919$	−8.02776	−0.264811	−0.132406	+	0.991196i	$0.542270\pi$
$920$	0	0
$921$	25.6056	0.843732
$922$	0	0
$923$	−29.7250	−0.978410
$924$	0	0
$925$	−25.3028	−0.831950
$926$	0	0
$927$	−2.78890	−0.0915994
$928$	0	0
$929$	−34.6972	−1.13838	−0.569190	−	0.822206i	$-0.692743\pi$
$929$	−34.6972	−1.13838	−0.569190	+	0.822206i	$0.692743\pi$
$930$	0	0
$931$	0.394449	0.0129275
$932$	0	0
$933$	−10.1194	−0.331295
$934$	0	0
$935$	−1.54163	−0.0504168
$936$	0	0
$937$	−5.97224	−0.195105	−0.0975523	−	0.995230i	$-0.531101\pi$
$937$	−5.97224	−0.195105	−0.0975523	+	0.995230i	$0.531101\pi$
$938$	0	0
$939$	−30.4222	−0.992791
$940$	0	0
$941$	−5.21110	−0.169877	−0.0849385	−	0.996386i	$-0.527069\pi$
$941$	−5.21110	−0.169877	−0.0849385	+	0.996386i	$0.527069\pi$
$942$	0	0
$943$	3.60555	0.117413
$944$	0	0
$945$	−0.697224	−0.0226807
$946$	0	0
$947$	18.8444	0.612361	0.306181	−	0.951973i	$-0.400949\pi$
$947$	18.8444	0.612361	0.306181	+	0.951973i	$0.400949\pi$
$948$	0	0
$949$	25.3305	0.822264
$950$	0	0
$951$	−12.6972	−0.411736
$952$	0	0
$953$	26.7250	0.865707	0.432854	−	0.901464i	$-0.357507\pi$
$953$	26.7250	0.865707	0.432854	+	0.901464i	$0.357507\pi$
$954$	0	0
$955$	−6.00000	−0.194155
$956$	0	0
$957$	31.4222	1.01574
$958$	0	0
$959$	−5.60555	−0.181013
$960$	0	0
$961$	−18.0000	−0.580645
$962$	0	0
$963$	−0.302776	−0.00975681
$964$	0	0
$965$	−14.6611	−0.471956
$966$	0	0
$967$	16.0555	0.516310	0.258155	−	0.966103i	$-0.416885\pi$
$967$	16.0555	0.516310	0.258155	+	0.966103i	$0.416885\pi$
$968$	0	0
$969$	−0.155590	−0.00499826
$970$	0	0
$971$	−6.06392	−0.194600	−0.0973002	−	0.995255i	$-0.531021\pi$
$971$	−6.06392	−0.194600	−0.0973002	+	0.995255i	$0.531021\pi$
$972$	0	0
$973$	15.1194	0.484707
$974$	0	0
$975$	−10.3944	−0.332889
$976$	0	0
$977$	−13.6972	−0.438213	−0.219107	−	0.975701i	$-0.570314\pi$
$977$	−13.6972	−0.438213	−0.219107	+	0.975701i	$0.570314\pi$
$978$	0	0
$979$	86.9638	2.77938
$980$	0	0
$981$	−4.51388	−0.144117
$982$	0	0
$983$	−2.57779	−0.0822189	−0.0411094	−	0.999155i	$-0.513089\pi$
$983$	−2.57779	−0.0822189	−0.0411094	+	0.999155i	$0.513089\pi$
$984$	0	0
$985$	18.0833	0.576181
$986$	0	0
$987$	4.60555	0.146596
$988$	0	0
$989$	−4.30278	−0.136820
$990$	0	0
$991$	30.9638	0.983599	0.491799	−	0.870709i	$-0.336339\pi$
$991$	30.9638	0.983599	0.491799	+	0.870709i	$0.336339\pi$
$992$	0	0
$993$	2.00000	0.0634681
$994$	0	0
$995$	−1.33053	−0.0421807
$996$	0	0
$997$	−3.18335	−0.100818	−0.0504088	−	0.998729i	$-0.516052\pi$
$997$	−3.18335	−0.100818	−0.0504088	+	0.998729i	$0.516052\pi$
$998$	0	0
$999$	5.60555	0.177352

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7728.2.a.bf.1.2		2
4.3	odd	2		1932.2.a.c.1.2	✓	2
12.11	even	2		5796.2.a.m.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1932.2.a.c.1.2	✓	2	4.3	odd	2
5796.2.a.m.1.1		2	12.11	even	2
7728.2.a.bf.1.2		2	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:	\( N \)	\(=\)	\( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7728.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -0.697224 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -0.697224 q^{5} +1.00000 q^{7} +1.00000 q^{9} -5.60555 q^{11} +2.30278 q^{13} -0.697224 q^{15} -0.394449 q^{17} +0.394449 q^{19} +1.00000 q^{21} +1.00000 q^{23} -4.51388 q^{25} +1.00000 q^{27} -5.60555 q^{29} +3.60555 q^{31} -5.60555 q^{33} -0.697224 q^{35} +5.60555 q^{37} +2.30278 q^{39} +3.60555 q^{41} -4.30278 q^{43} -0.697224 q^{45} +4.60555 q^{47} +1.00000 q^{49} -0.394449 q^{51} +5.90833 q^{53} +3.90833 q^{55} +0.394449 q^{57} -3.90833 q^{59} +4.90833 q^{61} +1.00000 q^{63} -1.60555 q^{65} -13.3028 q^{67} +1.00000 q^{69} -12.9083 q^{71} +11.0000 q^{73} -4.51388 q^{75} -5.60555 q^{77} -13.4222 q^{79} +1.00000 q^{81} -16.2111 q^{83} +0.275019 q^{85} -5.60555 q^{87} -15.5139 q^{89} +2.30278 q^{91} +3.60555 q^{93} -0.275019 q^{95} +3.78890 q^{97} -5.60555 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 5 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 5 * q^5 + 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} - 5 q^{5} + 2 q^{7} + 2 q^{9} - 4 q^{11} + q^{13} - 5 q^{15} - 8 q^{17} + 8 q^{19} + 2 q^{21} + 2 q^{23} + 9 q^{25} + 2 q^{27} - 4 q^{29} - 4 q^{33} - 5 q^{35} + 4 q^{37} + q^{39} - 5 q^{43} - 5 q^{45} + 2 q^{47} + 2 q^{49} - 8 q^{51} + q^{53} - 3 q^{55} + 8 q^{57} + 3 q^{59} - q^{61} + 2 q^{63} + 4 q^{65} - 23 q^{67} + 2 q^{69} - 15 q^{71} + 22 q^{73} + 9 q^{75} - 4 q^{77} + 2 q^{79} + 2 q^{81} - 18 q^{83} + 33 q^{85} - 4 q^{87} - 13 q^{89} + q^{91} - 33 q^{95} + 22 q^{97} - 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 5 * q^5 + 2 * q^7 + 2 * q^9 - 4 * q^11 + q^13 - 5 * q^15 - 8 * q^17 + 8 * q^19 + 2 * q^21 + 2 * q^23 + 9 * q^25 + 2 * q^27 - 4 * q^29 - 4 * q^33 - 5 * q^35 + 4 * q^37 + q^39 - 5 * q^43 - 5 * q^45 + 2 * q^47 + 2 * q^49 - 8 * q^51 + q^53 - 3 * q^55 + 8 * q^57 + 3 * q^59 - q^61 + 2 * q^63 + 4 * q^65 - 23 * q^67 + 2 * q^69 - 15 * q^71 + 22 * q^73 + 9 * q^75 - 4 * q^77 + 2 * q^79 + 2 * q^81 - 18 * q^83 + 33 * q^85 - 4 * q^87 - 13 * q^89 + q^91 - 33 * q^95 + 22 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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