LMFDB - Embedded newform 7728.2.a.bv.1.1

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:	$3$
Coefficient field:	3.3.1509.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 7x + 4 $ x^3 - x^2 - 7*x + 4
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			1.1
Root			$2.92542$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -2.92542 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -2.92542 q^{5} -1.00000 q^{7} +1.00000 q^{9} +4.85085 q^{11} -2.55810 q^{13} -2.92542 q^{15} +1.63268 q^{17} -1.00000 q^{19} -1.00000 q^{21} -1.00000 q^{23} +3.55810 q^{25} +1.00000 q^{27} -9.48352 q^{29} -0.367324 q^{31} +4.85085 q^{33} +2.92542 q^{35} +2.21817 q^{37} -2.55810 q^{39} +11.9670 q^{41} -2.70725 q^{43} -2.92542 q^{45} +2.48352 q^{47} +1.00000 q^{49} +1.63268 q^{51} -7.55810 q^{53} -14.1908 q^{55} -1.00000 q^{57} -7.55810 q^{59} +9.92542 q^{61} -1.00000 q^{63} +7.48352 q^{65} +2.92542 q^{67} -1.00000 q^{69} +7.82345 q^{71} +7.48352 q^{73} +3.55810 q^{75} -4.85085 q^{77} -12.0690 q^{79} +1.00000 q^{81} -9.48352 q^{83} -4.77627 q^{85} -9.48352 q^{87} +1.44190 q^{89} +2.55810 q^{91} -0.367324 q^{93} +2.92542 q^{95} -4.74887 q^{97} +4.85085 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} - q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 - q^5 - 3 * q^7 + 3 * q^9	$ 3 q + 3 q^{3} - q^{5} - 3 q^{7} + 3 q^{9} - q^{11} + 3 q^{13} - q^{15} + 2 q^{17} - 3 q^{19} - 3 q^{21} - 3 q^{23} + 3 q^{27} - 10 q^{29} - 4 q^{31} - q^{33} + q^{35} - 6 q^{37} + 3 q^{39} - q^{41} - 13 q^{43} - q^{45} - 11 q^{47} + 3 q^{49} + 2 q^{51} - 12 q^{53} - 29 q^{55} - 3 q^{57} - 12 q^{59} + 22 q^{61} - 3 q^{63} + 4 q^{65} + q^{67} - 3 q^{69} + 7 q^{71} + 4 q^{73} + q^{77} - 8 q^{79} + 3 q^{81} - 10 q^{83} + 9 q^{85} - 10 q^{87} + 15 q^{89} - 3 q^{91} - 4 q^{93} + q^{95} + 10 q^{97} - q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 - q^5 - 3 * q^7 + 3 * q^9 - q^11 + 3 * q^13 - q^15 + 2 * q^17 - 3 * q^19 - 3 * q^21 - 3 * q^23 + 3 * q^27 - 10 * q^29 - 4 * q^31 - q^33 + q^35 - 6 * q^37 + 3 * q^39 - q^41 - 13 * q^43 - q^45 - 11 * q^47 + 3 * q^49 + 2 * q^51 - 12 * q^53 - 29 * q^55 - 3 * q^57 - 12 * q^59 + 22 * q^61 - 3 * q^63 + 4 * q^65 + q^67 - 3 * q^69 + 7 * q^71 + 4 * q^73 + q^77 - 8 * q^79 + 3 * q^81 - 10 * q^83 + 9 * q^85 - 10 * q^87 + 15 * q^89 - 3 * q^91 - 4 * q^93 + q^95 + 10 * q^97 - 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
$5$	−2.92542	−1.30829	−0.654144	−	0.756370i	$-0.726971\pi$
$5$	−2.92542	−1.30829	−0.654144	+	0.756370i	$0.726971\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.85085	1.46259	0.731293	−	0.682064i	$-0.238917\pi$
$11$	4.85085	1.46259	0.731293	+	0.682064i	$0.238917\pi$
$12$	0	0
$13$	−2.55810	−0.709489	−0.354745	−	0.934963i	$-0.615432\pi$
$13$	−2.55810	−0.709489	−0.354745	+	0.934963i	$0.615432\pi$
$14$	0	0
$15$	−2.92542	−0.755341
$16$	0	0
$17$	1.63268	0.395982	0.197991	−	0.980204i	$-0.436558\pi$
$17$	1.63268	0.395982	0.197991	+	0.980204i	$0.436558\pi$
$18$	0	0
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	3.55810	0.711620
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−9.48352	−1.76105	−0.880523	−	0.474004i	$-0.842808\pi$
$29$	−9.48352	−1.76105	−0.880523	+	0.474004i	$0.842808\pi$
$30$	0	0
$31$	−0.367324	−0.0659733	−0.0329866	−	0.999456i	$-0.510502\pi$
$31$	−0.367324	−0.0659733	−0.0329866	+	0.999456i	$0.510502\pi$
$32$	0	0
$33$	4.85085	0.844424
$34$	0	0
$35$	2.92542	0.494487
$36$	0	0
$37$	2.21817	0.364665	0.182332	−	0.983237i	$-0.441635\pi$
$37$	2.21817	0.364665	0.182332	+	0.983237i	$0.441635\pi$
$38$	0	0
$39$	−2.55810	−0.409624
$40$	0	0
$41$	11.9670	1.86894	0.934469	−	0.356044i	$-0.115875\pi$
$41$	11.9670	1.86894	0.934469	+	0.356044i	$0.115875\pi$
$42$	0	0
$43$	−2.70725	−0.412852	−0.206426	−	0.978462i	$-0.566183\pi$
$43$	−2.70725	−0.412852	−0.206426	+	0.978462i	$0.566183\pi$
$44$	0	0
$45$	−2.92542	−0.436096
$46$	0	0
$47$	2.48352	0.362259	0.181129	−	0.983459i	$-0.442025\pi$
$47$	2.48352	0.362259	0.181129	+	0.983459i	$0.442025\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	1.63268	0.228620
$52$	0	0
$53$	−7.55810	−1.03818	−0.519092	−	0.854718i	$-0.673730\pi$
$53$	−7.55810	−1.03818	−0.519092	+	0.854718i	$0.673730\pi$
$54$	0	0
$55$	−14.1908	−1.91348
$56$	0	0
$57$	−1.00000	−0.132453
$58$	0	0
$59$	−7.55810	−0.983981	−0.491990	−	0.870601i	$-0.663730\pi$
$59$	−7.55810	−0.983981	−0.491990	+	0.870601i	$0.663730\pi$
$60$	0	0
$61$	9.92542	1.27082	0.635410	−	0.772175i	$-0.280831\pi$
$61$	9.92542	1.27082	0.635410	+	0.772175i	$0.280831\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	7.48352	0.928217
$66$	0	0
$67$	2.92542	0.357397	0.178699	−	0.983904i	$-0.442811\pi$
$67$	2.92542	0.357397	0.178699	+	0.983904i	$0.442811\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	7.82345	0.928473	0.464236	−	0.885711i	$-0.346329\pi$
$71$	7.82345	0.928473	0.464236	+	0.885711i	$0.346329\pi$
$72$	0	0
$73$	7.48352	0.875880	0.437940	−	0.899004i	$-0.355708\pi$
$73$	7.48352	0.875880	0.437940	+	0.899004i	$0.355708\pi$
$74$	0	0
$75$	3.55810	0.410854
$76$	0	0
$77$	−4.85085	−0.552805
$78$	0	0
$79$	−12.0690	−1.35787	−0.678935	−	0.734198i	$-0.737558\pi$
$79$	−12.0690	−1.35787	−0.678935	+	0.734198i	$0.737558\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−9.48352	−1.04095	−0.520476	−	0.853876i	$-0.674245\pi$
$83$	−9.48352	−1.04095	−0.520476	+	0.853876i	$0.674245\pi$
$84$	0	0
$85$	−4.77627	−0.518059
$86$	0	0
$87$	−9.48352	−1.01674
$88$	0	0
$89$	1.44190	0.152841	0.0764206	−	0.997076i	$-0.475651\pi$
$89$	1.44190	0.152841	0.0764206	+	0.997076i	$0.475651\pi$
$90$	0	0
$91$	2.55810	0.268162
$92$	0	0
$93$	−0.367324	−0.0380897
$94$	0	0
$95$	2.92542	0.300142
$96$	0	0
$97$	−4.74887	−0.482175	−0.241088	−	0.970503i	$-0.577504\pi$
$97$	−4.74887	−0.482175	−0.241088	+	0.970503i	$0.577504\pi$
$98$	0	0
$99$	4.85085	0.487528
$100$	0	0
$101$	−4.97261	−0.494793	−0.247396	−	0.968914i	$-0.579575\pi$
$101$	−4.97261	−0.494793	−0.247396	+	0.968914i	$0.579575\pi$
$102$	0	0
$103$	−9.36732	−0.922990	−0.461495	−	0.887143i	$-0.652687\pi$
$103$	−9.36732	−0.922990	−0.461495	+	0.887143i	$0.652687\pi$
$104$	0	0
$105$	2.92542	0.285492
$106$	0	0
$107$	−9.29275	−0.898364	−0.449182	−	0.893440i	$-0.648284\pi$
$107$	−9.29275	−0.898364	−0.449182	+	0.893440i	$0.648284\pi$
$108$	0	0
$109$	6.19078	0.592969	0.296484	−	0.955038i	$-0.404186\pi$
$109$	6.19078	0.592969	0.296484	+	0.955038i	$0.404186\pi$
$110$	0	0
$111$	2.21817	0.210539
$112$	0	0
$113$	−3.51092	−0.330279	−0.165140	−	0.986270i	$-0.552807\pi$
$113$	−3.51092	−0.330279	−0.165140	+	0.986270i	$0.552807\pi$
$114$	0	0
$115$	2.92542	0.272797
$116$	0	0
$117$	−2.55810	−0.236496
$118$	0	0
$119$	−1.63268	−0.149667
$120$	0	0
$121$	12.5307	1.13916
$122$	0	0
$123$	11.9670	1.07903
$124$	0	0
$125$	4.21817	0.377285
$126$	0	0
$127$	−16.8925	−1.49896	−0.749482	−	0.662025i	$-0.769697\pi$
$127$	−16.8925	−1.49896	−0.749482	+	0.662025i	$0.769697\pi$
$128$	0	0
$129$	−2.70725	−0.238360
$130$	0	0
$131$	−8.26535	−0.722147	−0.361074	−	0.932537i	$-0.617590\pi$
$131$	−8.26535	−0.722147	−0.361074	+	0.932537i	$0.617590\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	−2.92542	−0.251780
$136$	0	0
$137$	−12.1162	−1.03516	−0.517578	−	0.855636i	$-0.673166\pi$
$137$	−12.1162	−1.03516	−0.517578	+	0.855636i	$0.673166\pi$
$138$	0	0
$139$	0.776269	0.0658423	0.0329211	−	0.999458i	$-0.489519\pi$
$139$	0.776269	0.0658423	0.0329211	+	0.999458i	$0.489519\pi$
$140$	0	0
$141$	2.48352	0.209150
$142$	0	0
$143$	−12.4089	−1.03769
$144$	0	0
$145$	27.7433	2.30396
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	14.1852	1.16210	0.581049	−	0.813869i	$-0.302642\pi$
$149$	14.1852	1.16210	0.581049	+	0.813869i	$0.302642\pi$
$150$	0	0
$151$	−2.78183	−0.226382	−0.113191	−	0.993573i	$-0.536107\pi$
$151$	−2.78183	−0.226382	−0.113191	+	0.993573i	$0.536107\pi$
$152$	0	0
$153$	1.63268	0.131994
$154$	0	0
$155$	1.07458	0.0863121
$156$	0	0
$157$	−8.48352	−0.677059	−0.338529	−	0.940956i	$-0.609929\pi$
$157$	−8.48352	−0.677059	−0.338529	+	0.940956i	$0.609929\pi$
$158$	0	0
$159$	−7.55810	−0.599396
$160$	0	0
$161$	1.00000	0.0788110
$162$	0	0
$163$	−8.97261	−0.702789	−0.351394	−	0.936228i	$-0.614292\pi$
$163$	−8.97261	−0.702789	−0.351394	+	0.936228i	$0.614292\pi$
$164$	0	0
$165$	−14.1908	−1.10475
$166$	0	0
$167$	−9.00000	−0.696441	−0.348220	−	0.937413i	$-0.613214\pi$
$167$	−9.00000	−0.696441	−0.348220	+	0.937413i	$0.613214\pi$
$168$	0	0
$169$	−6.45613	−0.496625
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	21.8651	1.66237	0.831185	−	0.555995i	$-0.187663\pi$
$173$	21.8651	1.66237	0.831185	+	0.555995i	$0.187663\pi$
$174$	0	0
$175$	−3.55810	−0.268967
$176$	0	0
$177$	−7.55810	−0.568102
$178$	0	0
$179$	−10.6271	−0.794308	−0.397154	−	0.917752i	$-0.630002\pi$
$179$	−10.6271	−0.794308	−0.397154	+	0.917752i	$0.630002\pi$
$180$	0	0
$181$	−12.4506	−0.925443	−0.462722	−	0.886504i	$-0.653127\pi$
$181$	−12.4506	−0.925443	−0.462722	+	0.886504i	$0.653127\pi$
$182$	0	0
$183$	9.92542	0.733708
$184$	0	0
$185$	−6.48908	−0.477087
$186$	0	0
$187$	7.91986	0.579158
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	15.4506	1.11796	0.558982	−	0.829180i	$-0.311192\pi$
$191$	15.4506	1.11796	0.558982	+	0.829180i	$0.311192\pi$
$192$	0	0
$193$	−5.74887	−0.413813	−0.206907	−	0.978361i	$-0.566340\pi$
$193$	−5.74887	−0.413813	−0.206907	+	0.978361i	$0.566340\pi$
$194$	0	0
$195$	7.48352	0.535906
$196$	0	0
$197$	−7.29275	−0.519587	−0.259793	−	0.965664i	$-0.583654\pi$
$197$	−7.29275	−0.519587	−0.259793	+	0.965664i	$0.583654\pi$
$198$	0	0
$199$	−10.7291	−0.760565	−0.380282	−	0.924870i	$-0.624173\pi$
$199$	−10.7291	−0.760565	−0.380282	+	0.924870i	$0.624173\pi$
$200$	0	0
$201$	2.92542	0.206343
$202$	0	0
$203$	9.48352	0.665613
$204$	0	0
$205$	−35.0087	−2.44511
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−4.85085	−0.335540
$210$	0	0
$211$	7.14915	0.492168	0.246084	−	0.969248i	$-0.420856\pi$
$211$	7.14915	0.492168	0.246084	+	0.969248i	$0.420856\pi$
$212$	0	0
$213$	7.82345	0.536054
$214$	0	0
$215$	7.91986	0.540130
$216$	0	0
$217$	0.367324	0.0249356
$218$	0	0
$219$	7.48352	0.505690
$220$	0	0
$221$	−4.17655	−0.280945
$222$	0	0
$223$	25.4450	1.70392	0.851962	−	0.523604i	$-0.175413\pi$
$223$	25.4450	1.70392	0.851962	+	0.523604i	$0.175413\pi$
$224$	0	0
$225$	3.55810	0.237207
$226$	0	0
$227$	−0.925423	−0.0614225	−0.0307112	−	0.999528i	$-0.509777\pi$
$227$	−0.925423	−0.0614225	−0.0307112	+	0.999528i	$0.509777\pi$
$228$	0	0
$229$	24.3760	1.61081	0.805405	−	0.592724i	$-0.201948\pi$
$229$	24.3760	1.61081	0.805405	+	0.592724i	$0.201948\pi$
$230$	0	0
$231$	−4.85085	−0.319162
$232$	0	0
$233$	0.339930	0.0222695	0.0111348	−	0.999938i	$-0.496456\pi$
$233$	0.339930	0.0222695	0.0111348	+	0.999938i	$0.496456\pi$
$234$	0	0
$235$	−7.26535	−0.473939
$236$	0	0
$237$	−12.0690	−0.783967
$238$	0	0
$239$	−22.2740	−1.44079	−0.720393	−	0.693566i	$-0.756039\pi$
$239$	−22.2740	−1.44079	−0.720393	+	0.693566i	$0.756039\pi$
$240$	0	0
$241$	6.11620	0.393979	0.196989	−	0.980406i	$-0.436884\pi$
$241$	6.11620	0.393979	0.196989	+	0.980406i	$0.436884\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−2.92542	−0.186898
$246$	0	0
$247$	2.55810	0.162768
$248$	0	0
$249$	−9.48352	−0.600994
$250$	0	0
$251$	−0.585493	−0.0369560	−0.0184780	−	0.999829i	$-0.505882\pi$
$251$	−0.585493	−0.0369560	−0.0184780	+	0.999829i	$0.505882\pi$
$252$	0	0
$253$	−4.85085	−0.304970
$254$	0	0
$255$	−4.77627	−0.299102
$256$	0	0
$257$	−21.0142	−1.31083	−0.655416	−	0.755268i	$-0.727507\pi$
$257$	−21.0142	−1.31083	−0.655416	+	0.755268i	$0.727507\pi$
$258$	0	0
$259$	−2.21817	−0.137830
$260$	0	0
$261$	−9.48352	−0.587015
$262$	0	0
$263$	−9.43634	−0.581870	−0.290935	−	0.956743i	$-0.593966\pi$
$263$	−9.43634	−0.581870	−0.290935	+	0.956743i	$0.593966\pi$
$264$	0	0
$265$	22.1106	1.35825
$266$	0	0
$267$	1.44190	0.0882429
$268$	0	0
$269$	−27.4308	−1.67248	−0.836242	−	0.548361i	$-0.815252\pi$
$269$	−27.4308	−1.67248	−0.836242	+	0.548361i	$0.815252\pi$
$270$	0	0
$271$	17.7159	1.07617	0.538083	−	0.842892i	$-0.319149\pi$
$271$	17.7159	1.07617	0.538083	+	0.842892i	$0.319149\pi$
$272$	0	0
$273$	2.55810	0.154823
$274$	0	0
$275$	17.2598	1.04080
$276$	0	0
$277$	−21.0416	−1.26427	−0.632134	−	0.774859i	$-0.717821\pi$
$277$	−21.0416	−1.26427	−0.632134	+	0.774859i	$0.717821\pi$
$278$	0	0
$279$	−0.367324	−0.0219911
$280$	0	0
$281$	−31.1162	−1.85624	−0.928118	−	0.372285i	$-0.878574\pi$
$281$	−31.1162	−1.85624	−0.928118	+	0.372285i	$0.878574\pi$
$282$	0	0
$283$	0.339930	0.0202067	0.0101034	−	0.999949i	$-0.496784\pi$
$283$	0.339930	0.0202067	0.0101034	+	0.999949i	$0.496784\pi$
$284$	0	0
$285$	2.92542	0.173287
$286$	0	0
$287$	−11.9670	−0.706392
$288$	0	0
$289$	−14.3344	−0.843198
$290$	0	0
$291$	−4.74887	−0.278384
$292$	0	0
$293$	−10.5307	−0.615210	−0.307605	−	0.951514i	$-0.599528\pi$
$293$	−10.5307	−0.615210	−0.307605	+	0.951514i	$0.599528\pi$
$294$	0	0
$295$	22.1106	1.28733
$296$	0	0
$297$	4.85085	0.281475
$298$	0	0
$299$	2.55810	0.147939
$300$	0	0
$301$	2.70725	0.156043
$302$	0	0
$303$	−4.97261	−0.285669
$304$	0	0
$305$	−29.0361	−1.66260
$306$	0	0
$307$	−24.5997	−1.40398	−0.701990	−	0.712187i	$-0.747705\pi$
$307$	−24.5997	−1.40398	−0.701990	+	0.712187i	$0.747705\pi$
$308$	0	0
$309$	−9.36732	−0.532888
$310$	0	0
$311$	7.19078	0.407751	0.203876	−	0.978997i	$-0.434646\pi$
$311$	7.19078	0.407751	0.203876	+	0.978997i	$0.434646\pi$
$312$	0	0
$313$	19.5997	1.10784	0.553921	−	0.832569i	$-0.313131\pi$
$313$	19.5997	1.10784	0.553921	+	0.832569i	$0.313131\pi$
$314$	0	0
$315$	2.92542	0.164829
$316$	0	0
$317$	0.627115	0.0352223	0.0176111	−	0.999845i	$-0.494394\pi$
$317$	0.627115	0.0352223	0.0176111	+	0.999845i	$0.494394\pi$
$318$	0	0
$319$	−46.0031	−2.57568
$320$	0	0
$321$	−9.29275	−0.518671
$322$	0	0
$323$	−1.63268	−0.0908445
$324$	0	0
$325$	−9.10197	−0.504887
$326$	0	0
$327$	6.19078	0.342351
$328$	0	0
$329$	−2.48352	−0.136921
$330$	0	0
$331$	−4.63268	−0.254635	−0.127317	−	0.991862i	$-0.540637\pi$
$331$	−4.63268	−0.254635	−0.127317	+	0.991862i	$0.540637\pi$
$332$	0	0
$333$	2.21817	0.121555
$334$	0	0
$335$	−8.55810	−0.467579
$336$	0	0
$337$	8.70725	0.474314	0.237157	−	0.971471i	$-0.423784\pi$
$337$	8.70725	0.474314	0.237157	+	0.971471i	$0.423784\pi$
$338$	0	0
$339$	−3.51092	−0.190687
$340$	0	0
$341$	−1.78183	−0.0964915
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	2.92542	0.157499
$346$	0	0
$347$	−26.7378	−1.43536	−0.717679	−	0.696374i	$-0.754795\pi$
$347$	−26.7378	−1.43536	−0.717679	+	0.696374i	$0.754795\pi$
$348$	0	0
$349$	0.870635	0.0466040	0.0233020	−	0.999728i	$-0.492582\pi$
$349$	0.870635	0.0466040	0.0233020	+	0.999728i	$0.492582\pi$
$350$	0	0
$351$	−2.55810	−0.136541
$352$	0	0
$353$	−9.10197	−0.484449	−0.242225	−	0.970220i	$-0.577877\pi$
$353$	−9.10197	−0.484449	−0.242225	+	0.970220i	$0.577877\pi$
$354$	0	0
$355$	−22.8869	−1.21471
$356$	0	0
$357$	−1.63268	−0.0864104
$358$	0	0
$359$	−19.7433	−1.04201	−0.521006	−	0.853553i	$-0.674443\pi$
$359$	−19.7433	−1.04201	−0.521006	+	0.853553i	$0.674443\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	12.5307	0.657691
$364$	0	0
$365$	−21.8925	−1.14590
$366$	0	0
$367$	−4.00867	−0.209251	−0.104625	−	0.994512i	$-0.533364\pi$
$367$	−4.00867	−0.209251	−0.104625	+	0.994512i	$0.533364\pi$
$368$	0	0
$369$	11.9670	0.622979
$370$	0	0
$371$	7.55810	0.392397
$372$	0	0
$373$	8.74887	0.453000	0.226500	−	0.974011i	$-0.427272\pi$
$373$	8.74887	0.453000	0.226500	+	0.974011i	$0.427272\pi$
$374$	0	0
$375$	4.21817	0.217825
$376$	0	0
$377$	24.2598	1.24944
$378$	0	0
$379$	1.56366	0.0803199	0.0401599	−	0.999193i	$-0.487213\pi$
$379$	1.56366	0.0803199	0.0401599	+	0.999193i	$0.487213\pi$
$380$	0	0
$381$	−16.8925	−0.865427
$382$	0	0
$383$	−30.2182	−1.54408	−0.772038	−	0.635576i	$-0.780763\pi$
$383$	−30.2182	−1.54408	−0.772038	+	0.635576i	$0.780763\pi$
$384$	0	0
$385$	14.1908	0.723229
$386$	0	0
$387$	−2.70725	−0.137617
$388$	0	0
$389$	−24.3673	−1.23547	−0.617736	−	0.786385i	$-0.711950\pi$
$389$	−24.3673	−1.23547	−0.617736	+	0.786385i	$0.711950\pi$
$390$	0	0
$391$	−1.63268	−0.0825680
$392$	0	0
$393$	−8.26535	−0.416932
$394$	0	0
$395$	35.3070	1.77649
$396$	0	0
$397$	−2.67986	−0.134498	−0.0672491	−	0.997736i	$-0.521422\pi$
$397$	−2.67986	−0.134498	−0.0672491	+	0.997736i	$0.521422\pi$
$398$	0	0
$399$	1.00000	0.0500626
$400$	0	0
$401$	7.00000	0.349563	0.174782	−	0.984607i	$-0.444078\pi$
$401$	7.00000	0.349563	0.174782	+	0.984607i	$0.444078\pi$
$402$	0	0
$403$	0.939650	0.0468073
$404$	0	0
$405$	−2.92542	−0.145365
$406$	0	0
$407$	10.7600	0.533353
$408$	0	0
$409$	6.59972	0.326335	0.163168	−	0.986598i	$-0.447829\pi$
$409$	6.59972	0.326335	0.163168	+	0.986598i	$0.447829\pi$
$410$	0	0
$411$	−12.1162	−0.597648
$412$	0	0
$413$	7.55810	0.371910
$414$	0	0
$415$	27.7433	1.36187
$416$	0	0
$417$	0.776269	0.0380140
$418$	0	0
$419$	−18.4780	−0.902707	−0.451354	−	0.892345i	$-0.649059\pi$
$419$	−18.4780	−0.902707	−0.451354	+	0.892345i	$0.649059\pi$
$420$	0	0
$421$	15.5251	0.756649	0.378325	−	0.925673i	$-0.376500\pi$
$421$	15.5251	0.756649	0.378325	+	0.925673i	$0.376500\pi$
$422$	0	0
$423$	2.48352	0.120753
$424$	0	0
$425$	5.80922	0.281789
$426$	0	0
$427$	−9.92542	−0.480325
$428$	0	0
$429$	−12.4089	−0.599110
$430$	0	0
$431$	−13.0416	−0.628193	−0.314096	−	0.949391i	$-0.601702\pi$
$431$	−13.0416	−0.628193	−0.314096	+	0.949391i	$0.601702\pi$
$432$	0	0
$433$	29.6545	1.42510	0.712552	−	0.701619i	$-0.247539\pi$
$433$	29.6545	1.42510	0.712552	+	0.701619i	$0.247539\pi$
$434$	0	0
$435$	27.7433	1.33019
$436$	0	0
$437$	1.00000	0.0478365
$438$	0	0
$439$	16.3014	0.778024	0.389012	−	0.921233i	$-0.372816\pi$
$439$	16.3014	0.778024	0.389012	+	0.921233i	$0.372816\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−35.4034	−1.68207	−0.841033	−	0.540984i	$-0.818052\pi$
$443$	−35.4034	−1.68207	−0.841033	+	0.540984i	$0.818052\pi$
$444$	0	0
$445$	−4.21817	−0.199960
$446$	0	0
$447$	14.1852	0.670938
$448$	0	0
$449$	−29.3070	−1.38308	−0.691541	−	0.722337i	$-0.743068\pi$
$449$	−29.3070	−1.38308	−0.691541	+	0.722337i	$0.743068\pi$
$450$	0	0
$451$	58.0503	2.73348
$452$	0	0
$453$	−2.78183	−0.130702
$454$	0	0
$455$	−7.48352	−0.350833
$456$	0	0
$457$	−0.627115	−0.0293352	−0.0146676	−	0.999892i	$-0.504669\pi$
$457$	−0.627115	−0.0293352	−0.0146676	+	0.999892i	$0.504669\pi$
$458$	0	0
$459$	1.63268	0.0762068
$460$	0	0
$461$	−19.8092	−0.922608	−0.461304	−	0.887242i	$-0.652618\pi$
$461$	−19.8092	−0.922608	−0.461304	+	0.887242i	$0.652618\pi$
$462$	0	0
$463$	37.9011	1.76142	0.880708	−	0.473661i	$-0.157068\pi$
$463$	37.9011	1.76142	0.880708	+	0.473661i	$0.157068\pi$
$464$	0	0
$465$	1.07458	0.0498323
$466$	0	0
$467$	−3.57789	−0.165565	−0.0827825	−	0.996568i	$-0.526381\pi$
$467$	−3.57789	−0.165565	−0.0827825	+	0.996568i	$0.526381\pi$
$468$	0	0
$469$	−2.92542	−0.135083
$470$	0	0
$471$	−8.48352	−0.390900
$472$	0	0
$473$	−13.1325	−0.603832
$474$	0	0
$475$	−3.55810	−0.163257
$476$	0	0
$477$	−7.55810	−0.346062
$478$	0	0
$479$	38.2070	1.74572	0.872862	−	0.487967i	$-0.162261\pi$
$479$	38.2070	1.74572	0.872862	+	0.487967i	$0.162261\pi$
$480$	0	0
$481$	−5.67430	−0.258726
$482$	0	0
$483$	1.00000	0.0455016
$484$	0	0
$485$	13.8925	0.630824
$486$	0	0
$487$	2.36732	0.107274	0.0536368	−	0.998561i	$-0.482919\pi$
$487$	2.36732	0.107274	0.0536368	+	0.998561i	$0.482919\pi$
$488$	0	0
$489$	−8.97261	−0.405755
$490$	0	0
$491$	−10.6413	−0.480237	−0.240119	−	0.970744i	$-0.577186\pi$
$491$	−10.6413	−0.480237	−0.240119	+	0.970744i	$0.577186\pi$
$492$	0	0
$493$	−15.4835	−0.697343
$494$	0	0
$495$	−14.1908	−0.637828
$496$	0	0
$497$	−7.82345	−0.350930
$498$	0	0
$499$	−7.74331	−0.346638	−0.173319	−	0.984866i	$-0.555449\pi$
$499$	−7.74331	−0.346638	−0.173319	+	0.984866i	$0.555449\pi$
$500$	0	0
$501$	−9.00000	−0.402090
$502$	0	0
$503$	−22.1908	−0.989438	−0.494719	−	0.869053i	$-0.664729\pi$
$503$	−22.1908	−0.989438	−0.494719	+	0.869053i	$0.664729\pi$
$504$	0	0
$505$	14.5470	0.647332
$506$	0	0
$507$	−6.45613	−0.286727
$508$	0	0
$509$	−42.9483	−1.90365	−0.951825	−	0.306641i	$-0.900795\pi$
$509$	−42.9483	−1.90365	−0.951825	+	0.306641i	$0.900795\pi$
$510$	0	0
$511$	−7.48352	−0.331052
$512$	0	0
$513$	−1.00000	−0.0441511
$514$	0	0
$515$	27.4034	1.20754
$516$	0	0
$517$	12.0472	0.529835
$518$	0	0
$519$	21.8651	0.959770
$520$	0	0
$521$	−0.232397	−0.0101815	−0.00509075	−	0.999987i	$-0.501620\pi$
$521$	−0.232397	−0.0101815	−0.00509075	+	0.999987i	$0.501620\pi$
$522$	0	0
$523$	−17.5997	−0.769582	−0.384791	−	0.923004i	$-0.625726\pi$
$523$	−17.5997	−0.769582	−0.384791	+	0.923004i	$0.625726\pi$
$524$	0	0
$525$	−3.55810	−0.155288
$526$	0	0
$527$	−0.599721	−0.0261242
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−7.55810	−0.327994
$532$	0	0
$533$	−30.6129	−1.32599
$534$	0	0
$535$	27.1852	1.17532
$536$	0	0
$537$	−10.6271	−0.458594
$538$	0	0
$539$	4.85085	0.208941
$540$	0	0
$541$	−22.8539	−0.982568	−0.491284	−	0.870999i	$-0.663472\pi$
$541$	−22.8539	−0.982568	−0.491284	+	0.870999i	$0.663472\pi$
$542$	0	0
$543$	−12.4506	−0.534305
$544$	0	0
$545$	−18.1106	−0.775774
$546$	0	0
$547$	23.3618	0.998877	0.499438	−	0.866349i	$-0.333540\pi$
$547$	23.3618	0.998877	0.499438	+	0.866349i	$0.333540\pi$
$548$	0	0
$549$	9.92542	0.423607
$550$	0	0
$551$	9.48352	0.404012
$552$	0	0
$553$	12.0690	0.513227
$554$	0	0
$555$	−6.48908	−0.275446
$556$	0	0
$557$	−40.2872	−1.70702	−0.853511	−	0.521074i	$-0.825531\pi$
$557$	−40.2872	−1.70702	−0.853511	+	0.521074i	$0.825531\pi$
$558$	0	0
$559$	6.92542	0.292914
$560$	0	0
$561$	7.91986	0.334377
$562$	0	0
$563$	−11.8235	−0.498299	−0.249150	−	0.968465i	$-0.580151\pi$
$563$	−11.8235	−0.498299	−0.249150	+	0.968465i	$0.580151\pi$
$564$	0	0
$565$	10.2709	0.432101
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−24.8651	−1.04240	−0.521199	−	0.853435i	$-0.674515\pi$
$569$	−24.8651	−1.04240	−0.521199	+	0.853435i	$0.674515\pi$
$570$	0	0
$571$	−29.7707	−1.24587	−0.622933	−	0.782275i	$-0.714059\pi$
$571$	−29.7707	−1.24587	−0.622933	+	0.782275i	$0.714059\pi$
$572$	0	0
$573$	15.4506	0.645457
$574$	0	0
$575$	−3.55810	−0.148383
$576$	0	0
$577$	−7.17099	−0.298532	−0.149266	−	0.988797i	$-0.547691\pi$
$577$	−7.17099	−0.298532	−0.149266	+	0.988797i	$0.547691\pi$
$578$	0	0
$579$	−5.74887	−0.238915
$580$	0	0
$581$	9.48352	0.393443
$582$	0	0
$583$	−36.6632	−1.51843
$584$	0	0
$585$	7.48352	0.309406
$586$	0	0
$587$	−28.7433	−1.18636	−0.593182	−	0.805069i	$-0.702128\pi$
$587$	−28.7433	−1.18636	−0.593182	+	0.805069i	$0.702128\pi$
$588$	0	0
$589$	0.367324	0.0151353
$590$	0	0
$591$	−7.29275	−0.299984
$592$	0	0
$593$	15.1162	0.620748	0.310374	−	0.950615i	$-0.399546\pi$
$593$	15.1162	0.620748	0.310374	+	0.950615i	$0.399546\pi$
$594$	0	0
$595$	4.77627	0.195808
$596$	0	0
$597$	−10.7291	−0.439112
$598$	0	0
$599$	31.4592	1.28539	0.642695	−	0.766122i	$-0.277816\pi$
$599$	31.4592	1.28539	0.642695	+	0.766122i	$0.277816\pi$
$600$	0	0
$601$	41.5942	1.69666	0.848331	−	0.529467i	$-0.177608\pi$
$601$	41.5942	1.69666	0.848331	+	0.529467i	$0.177608\pi$
$602$	0	0
$603$	2.92542	0.119132
$604$	0	0
$605$	−36.6576	−1.49034
$606$	0	0
$607$	20.2598	0.822320	0.411160	−	0.911563i	$-0.365124\pi$
$607$	20.2598	0.822320	0.411160	+	0.911563i	$0.365124\pi$
$608$	0	0
$609$	9.48352	0.384292
$610$	0	0
$611$	−6.35310	−0.257019
$612$	0	0
$613$	30.3014	1.22386	0.611931	−	0.790911i	$-0.290393\pi$
$613$	30.3014	1.22386	0.611931	+	0.790911i	$0.290393\pi$
$614$	0	0
$615$	−35.0087	−1.41169
$616$	0	0
$617$	47.0229	1.89307	0.946535	−	0.322601i	$-0.104557\pi$
$617$	47.0229	1.89307	0.946535	+	0.322601i	$0.104557\pi$
$618$	0	0
$619$	33.7291	1.35569	0.677843	−	0.735206i	$-0.262915\pi$
$619$	33.7291	1.35569	0.677843	+	0.735206i	$0.262915\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	−1.44190	−0.0577685
$624$	0	0
$625$	−30.1304	−1.20522
$626$	0	0
$627$	−4.85085	−0.193724
$628$	0	0
$629$	3.62155	0.144401
$630$	0	0
$631$	−26.6830	−1.06223	−0.531116	−	0.847299i	$-0.678227\pi$
$631$	−26.6830	−1.06223	−0.531116	+	0.847299i	$0.678227\pi$
$632$	0	0
$633$	7.14915	0.284153
$634$	0	0
$635$	49.4176	1.96108
$636$	0	0
$637$	−2.55810	−0.101356
$638$	0	0
$639$	7.82345	0.309491
$640$	0	0
$641$	35.5612	1.40458	0.702292	−	0.711889i	$-0.252160\pi$
$641$	35.5612	1.40458	0.702292	+	0.711889i	$0.252160\pi$
$642$	0	0
$643$	−24.2126	−0.954852	−0.477426	−	0.878672i	$-0.658430\pi$
$643$	−24.2126	−0.954852	−0.477426	+	0.878672i	$0.658430\pi$
$644$	0	0
$645$	7.91986	0.311844
$646$	0	0
$647$	−17.4561	−0.686271	−0.343136	−	0.939286i	$-0.611489\pi$
$647$	−17.4561	−0.686271	−0.343136	+	0.939286i	$0.611489\pi$
$648$	0	0
$649$	−36.6632	−1.43916
$650$	0	0
$651$	0.367324	0.0143965
$652$	0	0
$653$	18.9802	0.742753	0.371377	−	0.928482i	$-0.378886\pi$
$653$	18.9802	0.742753	0.371377	+	0.928482i	$0.378886\pi$
$654$	0	0
$655$	24.1797	0.944777
$656$	0	0
$657$	7.48352	0.291960
$658$	0	0
$659$	3.78183	0.147319	0.0736596	−	0.997283i	$-0.476532\pi$
$659$	3.78183	0.147319	0.0736596	+	0.997283i	$0.476532\pi$
$660$	0	0
$661$	−36.6469	−1.42540	−0.712700	−	0.701469i	$-0.752528\pi$
$661$	−36.6469	−1.42540	−0.712700	+	0.701469i	$0.752528\pi$
$662$	0	0
$663$	−4.17655	−0.162204
$664$	0	0
$665$	−2.92542	−0.113443
$666$	0	0
$667$	9.48352	0.367203
$668$	0	0
$669$	25.4450	0.983761
$670$	0	0
$671$	48.1467	1.85868
$672$	0	0
$673$	28.7017	1.10637	0.553184	−	0.833059i	$-0.313412\pi$
$673$	28.7017	1.10637	0.553184	+	0.833059i	$0.313412\pi$
$674$	0	0
$675$	3.55810	0.136951
$676$	0	0
$677$	36.7246	1.41144	0.705720	−	0.708491i	$-0.250624\pi$
$677$	36.7246	1.41144	0.705720	+	0.708491i	$0.250624\pi$
$678$	0	0
$679$	4.74887	0.182245
$680$	0	0
$681$	−0.925423	−0.0354623
$682$	0	0
$683$	−34.2070	−1.30890	−0.654448	−	0.756107i	$-0.727099\pi$
$683$	−34.2070	−1.30890	−0.654448	+	0.756107i	$0.727099\pi$
$684$	0	0
$685$	35.4450	1.35428
$686$	0	0
$687$	24.3760	0.930002
$688$	0	0
$689$	19.3344	0.736581
$690$	0	0
$691$	28.3146	1.07714	0.538569	−	0.842582i	$-0.318965\pi$
$691$	28.3146	1.07714	0.538569	+	0.842582i	$0.318965\pi$
$692$	0	0
$693$	−4.85085	−0.184268
$694$	0	0
$695$	−2.27091	−0.0861407
$696$	0	0
$697$	19.5383	0.740066
$698$	0	0
$699$	0.339930	0.0128573
$700$	0	0
$701$	31.1654	1.17710	0.588551	−	0.808460i	$-0.299699\pi$
$701$	31.1654	1.17710	0.588551	+	0.808460i	$0.299699\pi$
$702$	0	0
$703$	−2.21817	−0.0836598
$704$	0	0
$705$	−7.26535	−0.273629
$706$	0	0
$707$	4.97261	0.187014
$708$	0	0
$709$	43.5140	1.63420	0.817102	−	0.576494i	$-0.195579\pi$
$709$	43.5140	1.63420	0.817102	+	0.576494i	$0.195579\pi$
$710$	0	0
$711$	−12.0690	−0.452623
$712$	0	0
$713$	0.367324	0.0137564
$714$	0	0
$715$	36.3014	1.35760
$716$	0	0
$717$	−22.2740	−0.831838
$718$	0	0
$719$	−1.54943	−0.0577841	−0.0288921	−	0.999583i	$-0.509198\pi$
$719$	−1.54943	−0.0577841	−0.0288921	+	0.999583i	$0.509198\pi$
$720$	0	0
$721$	9.36732	0.348857
$722$	0	0
$723$	6.11620	0.227464
$724$	0	0
$725$	−33.7433	−1.25320
$726$	0	0
$727$	43.2324	1.60340	0.801700	−	0.597726i	$-0.203929\pi$
$727$	43.2324	1.60340	0.801700	+	0.597726i	$0.203929\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−4.42007	−0.163482
$732$	0	0
$733$	2.35310	0.0869136	0.0434568	−	0.999055i	$-0.486163\pi$
$733$	2.35310	0.0869136	0.0434568	+	0.999055i	$0.486163\pi$
$734$	0	0
$735$	−2.92542	−0.107906
$736$	0	0
$737$	14.1908	0.522724
$738$	0	0
$739$	−5.54943	−0.204139	−0.102070	−	0.994777i	$-0.532546\pi$
$739$	−5.54943	−0.204139	−0.102070	+	0.994777i	$0.532546\pi$
$740$	0	0
$741$	2.55810	0.0939741
$742$	0	0
$743$	35.2598	1.29356	0.646778	−	0.762678i	$-0.276116\pi$
$743$	35.2598	1.29356	0.646778	+	0.762678i	$0.276116\pi$
$744$	0	0
$745$	−41.4977	−1.52036
$746$	0	0
$747$	−9.48352	−0.346984
$748$	0	0
$749$	9.29275	0.339550
$750$	0	0
$751$	−28.4232	−1.03718	−0.518588	−	0.855024i	$-0.673542\pi$
$751$	−28.4232	−1.03718	−0.518588	+	0.855024i	$0.673542\pi$
$752$	0	0
$753$	−0.585493	−0.0213366
$754$	0	0
$755$	8.13803	0.296173
$756$	0	0
$757$	−15.4835	−0.562758	−0.281379	−	0.959597i	$-0.590792\pi$
$757$	−15.4835	−0.562758	−0.281379	+	0.959597i	$0.590792\pi$
$758$	0	0
$759$	−4.85085	−0.176075
$760$	0	0
$761$	35.5997	1.29049	0.645244	−	0.763976i	$-0.276756\pi$
$761$	35.5997	1.29049	0.645244	+	0.763976i	$0.276756\pi$
$762$	0	0
$763$	−6.19078	−0.224121
$764$	0	0
$765$	−4.77627	−0.172686
$766$	0	0
$767$	19.3344	0.698124
$768$	0	0
$769$	25.9452	0.935608	0.467804	−	0.883832i	$-0.345045\pi$
$769$	25.9452	0.935608	0.467804	+	0.883832i	$0.345045\pi$
$770$	0	0
$771$	−21.0142	−0.756809
$772$	0	0
$773$	17.6875	0.636174	0.318087	−	0.948062i	$-0.396960\pi$
$773$	17.6875	0.636174	0.318087	+	0.948062i	$0.396960\pi$
$774$	0	0
$775$	−1.30697	−0.0469479
$776$	0	0
$777$	−2.21817	−0.0795764
$778$	0	0
$779$	−11.9670	−0.428764
$780$	0	0
$781$	37.9504	1.35797
$782$	0	0
$783$	−9.48352	−0.338913
$784$	0	0
$785$	24.8179	0.885788
$786$	0	0
$787$	−17.9285	−0.639083	−0.319541	−	0.947572i	$-0.603529\pi$
$787$	−17.9285	−0.639083	−0.319541	+	0.947572i	$0.603529\pi$
$788$	0	0
$789$	−9.43634	−0.335943
$790$	0	0
$791$	3.51092	0.124834
$792$	0	0
$793$	−25.3902	−0.901633
$794$	0	0
$795$	22.1106	0.784183
$796$	0	0
$797$	12.4475	0.440912	0.220456	−	0.975397i	$-0.429246\pi$
$797$	12.4475	0.440912	0.220456	+	0.975397i	$0.429246\pi$
$798$	0	0
$799$	4.05479	0.143448
$800$	0	0
$801$	1.44190	0.0509471
$802$	0	0
$803$	36.3014	1.28105
$804$	0	0
$805$	−2.92542	−0.103108
$806$	0	0
$807$	−27.4308	−0.965609
$808$	0	0
$809$	−14.9112	−0.524250	−0.262125	−	0.965034i	$-0.584423\pi$
$809$	−14.9112	−0.524250	−0.262125	+	0.965034i	$0.584423\pi$
$810$	0	0
$811$	−55.5556	−1.95082	−0.975411	−	0.220393i	$-0.929266\pi$
$811$	−55.5556	−1.95082	−0.975411	+	0.220393i	$0.929266\pi$
$812$	0	0
$813$	17.7159	0.621324
$814$	0	0
$815$	26.2487	0.919451
$816$	0	0
$817$	2.70725	0.0947148
$818$	0	0
$819$	2.55810	0.0893872
$820$	0	0
$821$	51.0503	1.78167	0.890834	−	0.454330i	$-0.150121\pi$
$821$	51.0503	1.78167	0.890834	+	0.454330i	$0.150121\pi$
$822$	0	0
$823$	−25.9944	−0.906109	−0.453055	−	0.891483i	$-0.649666\pi$
$823$	−25.9944	−0.906109	−0.453055	+	0.891483i	$0.649666\pi$
$824$	0	0
$825$	17.2598	0.600909
$826$	0	0
$827$	13.0558	0.453996	0.226998	−	0.973895i	$-0.427109\pi$
$827$	13.0558	0.453996	0.226998	+	0.973895i	$0.427109\pi$
$828$	0	0
$829$	32.0975	1.11479	0.557396	−	0.830247i	$-0.311801\pi$
$829$	32.0975	1.11479	0.557396	+	0.830247i	$0.311801\pi$
$830$	0	0
$831$	−21.0416	−0.729926
$832$	0	0
$833$	1.63268	0.0565689
$834$	0	0
$835$	26.3288	0.911146
$836$	0	0
$837$	−0.367324	−0.0126966
$838$	0	0
$839$	49.7464	1.71744	0.858719	−	0.512448i	$-0.171261\pi$
$839$	49.7464	1.71744	0.858719	+	0.512448i	$0.171261\pi$
$840$	0	0
$841$	60.9372	2.10128
$842$	0	0
$843$	−31.1162	−1.07170
$844$	0	0
$845$	18.8869	0.649729
$846$	0	0
$847$	−12.5307	−0.430560
$848$	0	0
$849$	0.339930	0.0116664
$850$	0	0
$851$	−2.21817	−0.0760379
$852$	0	0
$853$	−30.0173	−1.02777	−0.513887	−	0.857858i	$-0.671795\pi$
$853$	−30.0173	−1.02777	−0.513887	+	0.857858i	$0.671795\pi$
$854$	0	0
$855$	2.92542	0.100047
$856$	0	0
$857$	39.2903	1.34213	0.671065	−	0.741398i	$-0.265837\pi$
$857$	39.2903	1.34213	0.671065	+	0.741398i	$0.265837\pi$
$858$	0	0
$859$	−26.0721	−0.889569	−0.444785	−	0.895638i	$-0.646720\pi$
$859$	−26.0721	−0.889569	−0.444785	+	0.895638i	$0.646720\pi$
$860$	0	0
$861$	−11.9670	−0.407836
$862$	0	0
$863$	−24.5855	−0.836900	−0.418450	−	0.908240i	$-0.637426\pi$
$863$	−24.5855	−0.836900	−0.418450	+	0.908240i	$0.637426\pi$
$864$	0	0
$865$	−63.9646	−2.17486
$866$	0	0
$867$	−14.3344	−0.486821
$868$	0	0
$869$	−58.5449	−1.98600
$870$	0	0
$871$	−7.48352	−0.253569
$872$	0	0
$873$	−4.74887	−0.160725
$874$	0	0
$875$	−4.21817	−0.142600
$876$	0	0
$877$	2.72704	0.0920857	0.0460428	−	0.998939i	$-0.485339\pi$
$877$	2.72704	0.0920857	0.0460428	+	0.998939i	$0.485339\pi$
$878$	0	0
$879$	−10.5307	−0.355192
$880$	0	0
$881$	−10.5196	−0.354414	−0.177207	−	0.984174i	$-0.556706\pi$
$881$	−10.5196	−0.354414	−0.177207	+	0.984174i	$0.556706\pi$
$882$	0	0
$883$	43.6632	1.46938	0.734691	−	0.678401i	$-0.237327\pi$
$883$	43.6632	1.46938	0.734691	+	0.678401i	$0.237327\pi$
$884$	0	0
$885$	22.1106	0.743241
$886$	0	0
$887$	−49.6743	−1.66790	−0.833950	−	0.551840i	$-0.813926\pi$
$887$	−49.6743	−1.66790	−0.833950	+	0.551840i	$0.813926\pi$
$888$	0	0
$889$	16.8925	0.566555
$890$	0	0
$891$	4.85085	0.162509
$892$	0	0
$893$	−2.48352	−0.0831079
$894$	0	0
$895$	31.0888	1.03918
$896$	0	0
$897$	2.55810	0.0854124
$898$	0	0
$899$	3.48352	0.116182
$900$	0	0
$901$	−12.3399	−0.411103
$902$	0	0
$903$	2.70725	0.0900918
$904$	0	0
$905$	36.4232	1.21075
$906$	0	0
$907$	−4.83106	−0.160413	−0.0802063	−	0.996778i	$-0.525558\pi$
$907$	−4.83106	−0.160413	−0.0802063	+	0.996778i	$0.525558\pi$
$908$	0	0
$909$	−4.97261	−0.164931
$910$	0	0
$911$	−23.0503	−0.763690	−0.381845	−	0.924226i	$-0.624711\pi$
$911$	−23.0503	−0.763690	−0.381845	+	0.924226i	$0.624711\pi$
$912$	0	0
$913$	−46.0031	−1.52248
$914$	0	0
$915$	−29.0361	−0.959903
$916$	0	0
$917$	8.26535	0.272946
$918$	0	0
$919$	57.4835	1.89621	0.948103	−	0.317963i	$-0.102999\pi$
$919$	57.4835	1.89621	0.948103	+	0.317963i	$0.102999\pi$
$920$	0	0
$921$	−24.5997	−0.810588
$922$	0	0
$923$	−20.0132	−0.658741
$924$	0	0
$925$	7.89247	0.259503
$926$	0	0
$927$	−9.36732	−0.307663
$928$	0	0
$929$	0.135988	0.00446161	0.00223081	−	0.999998i	$-0.499290\pi$
$929$	0.135988	0.00446161	0.00223081	+	0.999998i	$0.499290\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	7.19078	0.235415
$934$	0	0
$935$	−23.1689	−0.757705
$936$	0	0
$937$	−8.63578	−0.282119	−0.141059	−	0.990001i	$-0.545051\pi$
$937$	−8.63578	−0.282119	−0.141059	+	0.990001i	$0.545051\pi$
$938$	0	0
$939$	19.5997	0.639613
$940$	0	0
$941$	37.6358	1.22689	0.613446	−	0.789737i	$-0.289783\pi$
$941$	37.6358	1.22689	0.613446	+	0.789737i	$0.289783\pi$
$942$	0	0
$943$	−11.9670	−0.389701
$944$	0	0
$945$	2.92542	0.0951640
$946$	0	0
$947$	19.2654	0.626040	0.313020	−	0.949747i	$-0.398659\pi$
$947$	19.2654	0.626040	0.313020	+	0.949747i	$0.398659\pi$
$948$	0	0
$949$	−19.1436	−0.621427
$950$	0	0
$951$	0.627115	0.0203356
$952$	0	0
$953$	−51.6444	−1.67293	−0.836464	−	0.548022i	$-0.815381\pi$
$953$	−51.6444	−1.67293	−0.836464	+	0.548022i	$0.815381\pi$
$954$	0	0
$955$	−45.1994	−1.46262
$956$	0	0
$957$	−46.0031	−1.48707
$958$	0	0
$959$	12.1162	0.391252
$960$	0	0
$961$	−30.8651	−0.995648
$962$	0	0
$963$	−9.29275	−0.299455
$964$	0	0
$965$	16.8179	0.541387
$966$	0	0
$967$	−11.8397	−0.380740	−0.190370	−	0.981712i	$-0.560969\pi$
$967$	−11.8397	−0.380740	−0.190370	+	0.981712i	$0.560969\pi$
$968$	0	0
$969$	−1.63268	−0.0524491
$970$	0	0
$971$	−4.27091	−0.137060	−0.0685301	−	0.997649i	$-0.521831\pi$
$971$	−4.27091	−0.137060	−0.0685301	+	0.997649i	$0.521831\pi$
$972$	0	0
$973$	−0.776269	−0.0248860
$974$	0	0
$975$	−9.10197	−0.291496
$976$	0	0
$977$	42.5942	1.36271	0.681354	−	0.731954i	$-0.261391\pi$
$977$	42.5942	1.36271	0.681354	+	0.731954i	$0.261391\pi$
$978$	0	0
$979$	6.99444	0.223543
$980$	0	0
$981$	6.19078	0.197656
$982$	0	0
$983$	12.3125	0.392709	0.196354	−	0.980533i	$-0.437090\pi$
$983$	12.3125	0.392709	0.196354	+	0.980533i	$0.437090\pi$
$984$	0	0
$985$	21.3344	0.679769
$986$	0	0
$987$	−2.48352	−0.0790514
$988$	0	0
$989$	2.70725	0.0860857
$990$	0	0
$991$	45.2629	1.43782	0.718912	−	0.695101i	$-0.244641\pi$
$991$	45.2629	1.43782	0.718912	+	0.695101i	$0.244641\pi$
$992$	0	0
$993$	−4.63268	−0.147014
$994$	0	0
$995$	31.3871	0.995038
$996$	0	0
$997$	−27.8651	−0.882496	−0.441248	−	0.897385i	$-0.645464\pi$
$997$	−27.8651	−0.882496	−0.441248	+	0.897385i	$0.645464\pi$
$998$	0	0
$999$	2.21817	0.0701798

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7728.2.a.bv.1.1		3
4.3	odd	2		1932.2.a.i.1.1	✓	3
12.11	even	2		5796.2.a.p.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1932.2.a.i.1.1	✓	3	4.3	odd	2
5796.2.a.p.1.3		3	12.11	even	2
7728.2.a.bv.1.1		3	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.92542 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -2.92542 q^{5} -1.00000 q^{7} +1.00000 q^{9} +4.85085 q^{11} -2.55810 q^{13} -2.92542 q^{15} +1.63268 q^{17} -1.00000 q^{19} -1.00000 q^{21} -1.00000 q^{23} +3.55810 q^{25} +1.00000 q^{27} -9.48352 q^{29} -0.367324 q^{31} +4.85085 q^{33} +2.92542 q^{35} +2.21817 q^{37} -2.55810 q^{39} +11.9670 q^{41} -2.70725 q^{43} -2.92542 q^{45} +2.48352 q^{47} +1.00000 q^{49} +1.63268 q^{51} -7.55810 q^{53} -14.1908 q^{55} -1.00000 q^{57} -7.55810 q^{59} +9.92542 q^{61} -1.00000 q^{63} +7.48352 q^{65} +2.92542 q^{67} -1.00000 q^{69} +7.82345 q^{71} +7.48352 q^{73} +3.55810 q^{75} -4.85085 q^{77} -12.0690 q^{79} +1.00000 q^{81} -9.48352 q^{83} -4.77627 q^{85} -9.48352 q^{87} +1.44190 q^{89} +2.55810 q^{91} -0.367324 q^{93} +2.92542 q^{95} -4.74887 q^{97} +4.85085 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 - q^5 - 3 * q^7 + 3 * q^9	\( 3 q + 3 q^{3} - q^{5} - 3 q^{7} + 3 q^{9} - q^{11} + 3 q^{13} - q^{15} + 2 q^{17} - 3 q^{19} - 3 q^{21} - 3 q^{23} + 3 q^{27} - 10 q^{29} - 4 q^{31} - q^{33} + q^{35} - 6 q^{37} + 3 q^{39} - q^{41} - 13 q^{43} - q^{45} - 11 q^{47} + 3 q^{49} + 2 q^{51} - 12 q^{53} - 29 q^{55} - 3 q^{57} - 12 q^{59} + 22 q^{61} - 3 q^{63} + 4 q^{65} + q^{67} - 3 q^{69} + 7 q^{71} + 4 q^{73} + q^{77} - 8 q^{79} + 3 q^{81} - 10 q^{83} + 9 q^{85} - 10 q^{87} + 15 q^{89} - 3 q^{91} - 4 q^{93} + q^{95} + 10 q^{97} - q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 - q^5 - 3 * q^7 + 3 * q^9 - q^11 + 3 * q^13 - q^15 + 2 * q^17 - 3 * q^19 - 3 * q^21 - 3 * q^23 + 3 * q^27 - 10 * q^29 - 4 * q^31 - q^33 + q^35 - 6 * q^37 + 3 * q^39 - q^41 - 13 * q^43 - q^45 - 11 * q^47 + 3 * q^49 + 2 * q^51 - 12 * q^53 - 29 * q^55 - 3 * q^57 - 12 * q^59 + 22 * q^61 - 3 * q^63 + 4 * q^65 + q^67 - 3 * q^69 + 7 * q^71 + 4 * q^73 + q^77 - 8 * q^79 + 3 * q^81 - 10 * q^83 + 9 * q^85 - 10 * q^87 + 15 * q^89 - 3 * q^91 - 4 * q^93 + q^95 + 10 * q^97 - q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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