LMFDB - Embedded newform 3648.2.a.bs.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3648,2,Mod(1,3648)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3648, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3648.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3648 = 2^{6} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3648.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:	$29.1294266574$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{41}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 10 $ x^2 - x - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 456)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.70156$ of defining polynomial
Character	$\chi$	$=$	3648.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +3.70156 q^{5} +1.70156 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +3.70156 q^{5} +1.70156 q^{7} +1.00000 q^{9} +1.70156 q^{11} -6.00000 q^{13} +3.70156 q^{15} +3.70156 q^{17} -1.00000 q^{19} +1.70156 q^{21} +4.00000 q^{23} +8.70156 q^{25} +1.00000 q^{27} -2.00000 q^{29} -3.40312 q^{31} +1.70156 q^{33} +6.29844 q^{35} -9.40312 q^{37} -6.00000 q^{39} +9.40312 q^{41} +9.10469 q^{43} +3.70156 q^{45} +5.70156 q^{47} -4.10469 q^{49} +3.70156 q^{51} +6.00000 q^{53} +6.29844 q^{55} -1.00000 q^{57} +4.00000 q^{59} -7.70156 q^{61} +1.70156 q^{63} -22.2094 q^{65} +12.0000 q^{67} +4.00000 q^{69} +0.298438 q^{73} +8.70156 q^{75} +2.89531 q^{77} +14.8062 q^{79} +1.00000 q^{81} -14.8062 q^{83} +13.7016 q^{85} -2.00000 q^{87} +12.8062 q^{89} -10.2094 q^{91} -3.40312 q^{93} -3.70156 q^{95} -6.00000 q^{97} +1.70156 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + q^{5} - 3 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + q^5 - 3 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} + q^{5} - 3 q^{7} + 2 q^{9} - 3 q^{11} - 12 q^{13} + q^{15} + q^{17} - 2 q^{19} - 3 q^{21} + 8 q^{23} + 11 q^{25} + 2 q^{27} - 4 q^{29} + 6 q^{31} - 3 q^{33} + 19 q^{35} - 6 q^{37} - 12 q^{39} + 6 q^{41} - q^{43} + q^{45} + 5 q^{47} + 11 q^{49} + q^{51} + 12 q^{53} + 19 q^{55} - 2 q^{57} + 8 q^{59} - 9 q^{61} - 3 q^{63} - 6 q^{65} + 24 q^{67} + 8 q^{69} + 7 q^{73} + 11 q^{75} + 25 q^{77} + 4 q^{79} + 2 q^{81} - 4 q^{83} + 21 q^{85} - 4 q^{87} + 18 q^{91} + 6 q^{93} - q^{95} - 12 q^{97} - 3 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + q^5 - 3 * q^7 + 2 * q^9 - 3 * q^11 - 12 * q^13 + q^15 + q^17 - 2 * q^19 - 3 * q^21 + 8 * q^23 + 11 * q^25 + 2 * q^27 - 4 * q^29 + 6 * q^31 - 3 * q^33 + 19 * q^35 - 6 * q^37 - 12 * q^39 + 6 * q^41 - q^43 + q^45 + 5 * q^47 + 11 * q^49 + q^51 + 12 * q^53 + 19 * q^55 - 2 * q^57 + 8 * q^59 - 9 * q^61 - 3 * q^63 - 6 * q^65 + 24 * q^67 + 8 * q^69 + 7 * q^73 + 11 * q^75 + 25 * q^77 + 4 * q^79 + 2 * q^81 - 4 * q^83 + 21 * q^85 - 4 * q^87 + 18 * q^91 + 6 * q^93 - q^95 - 12 * q^97 - 3 * 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
$5$	3.70156	1.65539	0.827694	−	0.561179i	$-0.189652\pi$
$5$	3.70156	1.65539	0.827694	+	0.561179i	$0.189652\pi$
$6$	0	0
$7$	1.70156	0.643130	0.321565	−	0.946888i	$-0.395791\pi$
$7$	1.70156	0.643130	0.321565	+	0.946888i	$0.395791\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.70156	0.513040	0.256520	−	0.966539i	$-0.417424\pi$
$11$	1.70156	0.513040	0.256520	+	0.966539i	$0.417424\pi$
$12$	0	0
$13$	−6.00000	−1.66410	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000	−1.66410	−0.832050	+	0.554700i	$0.812833\pi$
$14$	0	0
$15$	3.70156	0.955739
$16$	0	0
$17$	3.70156	0.897761	0.448880	−	0.893592i	$-0.351823\pi$
$17$	3.70156	0.897761	0.448880	+	0.893592i	$0.351823\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	1.70156	0.371311
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	8.70156	1.74031
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	−3.40312	−0.611219	−0.305610	−	0.952157i	$-0.598860\pi$
$31$	−3.40312	−0.611219	−0.305610	+	0.952157i	$0.598860\pi$
$32$	0	0
$33$	1.70156	0.296204
$34$	0	0
$35$	6.29844	1.06463
$36$	0	0
$37$	−9.40312	−1.54586	−0.772932	−	0.634489i	$-0.781211\pi$
$37$	−9.40312	−1.54586	−0.772932	+	0.634489i	$0.781211\pi$
$38$	0	0
$39$	−6.00000	−0.960769
$40$	0	0
$41$	9.40312	1.46852	0.734261	−	0.678868i	$-0.237529\pi$
$41$	9.40312	1.46852	0.734261	+	0.678868i	$0.237529\pi$
$42$	0	0
$43$	9.10469	1.38845	0.694226	−	0.719757i	$-0.255747\pi$
$43$	9.10469	1.38845	0.694226	+	0.719757i	$0.255747\pi$
$44$	0	0
$45$	3.70156	0.551796
$46$	0	0
$47$	5.70156	0.831658	0.415829	−	0.909443i	$-0.363491\pi$
$47$	5.70156	0.831658	0.415829	+	0.909443i	$0.363491\pi$
$48$	0	0
$49$	−4.10469	−0.586384
$50$	0	0
$51$	3.70156	0.518322
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	6.29844	0.849281
$56$	0	0
$57$	−1.00000	−0.132453
$58$	0	0
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	−7.70156	−0.986084	−0.493042	−	0.870006i	$-0.664115\pi$
$61$	−7.70156	−0.986084	−0.493042	+	0.870006i	$0.664115\pi$
$62$	0	0
$63$	1.70156	0.214377
$64$	0	0
$65$	−22.2094	−2.75473
$66$	0	0
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	0	0
$69$	4.00000	0.481543
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	0.298438	0.0349295	0.0174648	−	0.999847i	$-0.494441\pi$
$73$	0.298438	0.0349295	0.0174648	+	0.999847i	$0.494441\pi$
$74$	0	0
$75$	8.70156	1.00477
$76$	0	0
$77$	2.89531	0.329952
$78$	0	0
$79$	14.8062	1.66583	0.832917	−	0.553399i	$-0.186669\pi$
$79$	14.8062	1.66583	0.832917	+	0.553399i	$0.186669\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−14.8062	−1.62520	−0.812598	−	0.582824i	$-0.801948\pi$
$83$	−14.8062	−1.62520	−0.812598	+	0.582824i	$0.801948\pi$
$84$	0	0
$85$	13.7016	1.48614
$86$	0	0
$87$	−2.00000	−0.214423
$88$	0	0
$89$	12.8062	1.35746	0.678730	−	0.734388i	$-0.262531\pi$
$89$	12.8062	1.35746	0.678730	+	0.734388i	$0.262531\pi$
$90$	0	0
$91$	−10.2094	−1.07023
$92$	0	0
$93$	−3.40312	−0.352888
$94$	0	0
$95$	−3.70156	−0.379772
$96$	0	0
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	0	0
$99$	1.70156	0.171013
$100$	0	0
$101$	8.80625	0.876254	0.438127	−	0.898913i	$-0.355642\pi$
$101$	8.80625	0.876254	0.438127	+	0.898913i	$0.355642\pi$
$102$	0	0
$103$	3.40312	0.335320	0.167660	−	0.985845i	$-0.446379\pi$
$103$	3.40312	0.335320	0.167660	+	0.985845i	$0.446379\pi$
$104$	0	0
$105$	6.29844	0.614665
$106$	0	0
$107$	7.40312	0.715687	0.357844	−	0.933782i	$-0.383512\pi$
$107$	7.40312	0.715687	0.357844	+	0.933782i	$0.383512\pi$
$108$	0	0
$109$	−9.40312	−0.900656	−0.450328	−	0.892863i	$-0.648693\pi$
$109$	−9.40312	−0.900656	−0.450328	+	0.892863i	$0.648693\pi$
$110$	0	0
$111$	−9.40312	−0.892505
$112$	0	0
$113$	−18.0000	−1.69330	−0.846649	−	0.532152i	$-0.821383\pi$
$113$	−18.0000	−1.69330	−0.846649	+	0.532152i	$0.821383\pi$
$114$	0	0
$115$	14.8062	1.38069
$116$	0	0
$117$	−6.00000	−0.554700
$118$	0	0
$119$	6.29844	0.577377
$120$	0	0
$121$	−8.10469	−0.736790
$122$	0	0
$123$	9.40312	0.847851
$124$	0	0
$125$	13.7016	1.22550
$126$	0	0
$127$	−3.40312	−0.301978	−0.150989	−	0.988535i	$-0.548246\pi$
$127$	−3.40312	−0.301978	−0.150989	+	0.988535i	$0.548246\pi$
$128$	0	0
$129$	9.10469	0.801623
$130$	0	0
$131$	6.29844	0.550297	0.275149	−	0.961402i	$-0.411273\pi$
$131$	6.29844	0.550297	0.275149	+	0.961402i	$0.411273\pi$
$132$	0	0
$133$	−1.70156	−0.147544
$134$	0	0
$135$	3.70156	0.318580
$136$	0	0
$137$	−4.29844	−0.367240	−0.183620	−	0.982997i	$-0.558782\pi$
$137$	−4.29844	−0.367240	−0.183620	+	0.982997i	$0.558782\pi$
$138$	0	0
$139$	−21.7016	−1.84070	−0.920351	−	0.391093i	$-0.872097\pi$
$139$	−21.7016	−1.84070	−0.920351	+	0.391093i	$0.872097\pi$
$140$	0	0
$141$	5.70156	0.480158
$142$	0	0
$143$	−10.2094	−0.853751
$144$	0	0
$145$	−7.40312	−0.614796
$146$	0	0
$147$	−4.10469	−0.338549
$148$	0	0
$149$	−0.895314	−0.0733470	−0.0366735	−	0.999327i	$-0.511676\pi$
$149$	−0.895314	−0.0733470	−0.0366735	+	0.999327i	$0.511676\pi$
$150$	0	0
$151$	−22.8062	−1.85595	−0.927973	−	0.372647i	$-0.878450\pi$
$151$	−22.8062	−1.85595	−0.927973	+	0.372647i	$0.878450\pi$
$152$	0	0
$153$	3.70156	0.299254
$154$	0	0
$155$	−12.5969	−1.01181
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	0	0
$159$	6.00000	0.475831
$160$	0	0
$161$	6.80625	0.536408
$162$	0	0
$163$	−10.8062	−0.846411	−0.423205	−	0.906034i	$-0.639095\pi$
$163$	−10.8062	−0.846411	−0.423205	+	0.906034i	$0.639095\pi$
$164$	0	0
$165$	6.29844	0.490333
$166$	0	0
$167$	−18.2094	−1.40908	−0.704542	−	0.709663i	$-0.748847\pi$
$167$	−18.2094	−1.40908	−0.704542	+	0.709663i	$0.748847\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	−16.8062	−1.27776	−0.638878	−	0.769308i	$-0.720601\pi$
$173$	−16.8062	−1.27776	−0.638878	+	0.769308i	$0.720601\pi$
$174$	0	0
$175$	14.8062	1.11925
$176$	0	0
$177$	4.00000	0.300658
$178$	0	0
$179$	−7.40312	−0.553335	−0.276668	−	0.960966i	$-0.589230\pi$
$179$	−7.40312	−0.553335	−0.276668	+	0.960966i	$0.589230\pi$
$180$	0	0
$181$	−20.8062	−1.54652	−0.773258	−	0.634091i	$-0.781374\pi$
$181$	−20.8062	−1.54652	−0.773258	+	0.634091i	$0.781374\pi$
$182$	0	0
$183$	−7.70156	−0.569316
$184$	0	0
$185$	−34.8062	−2.55901
$186$	0	0
$187$	6.29844	0.460587
$188$	0	0
$189$	1.70156	0.123770
$190$	0	0
$191$	23.9109	1.73013	0.865067	−	0.501656i	$-0.167276\pi$
$191$	23.9109	1.73013	0.865067	+	0.501656i	$0.167276\pi$
$192$	0	0
$193$	−1.40312	−0.100999	−0.0504995	−	0.998724i	$-0.516081\pi$
$193$	−1.40312	−0.100999	−0.0504995	+	0.998724i	$0.516081\pi$
$194$	0	0
$195$	−22.2094	−1.59045
$196$	0	0
$197$	−22.0000	−1.56744	−0.783718	−	0.621117i	$-0.786679\pi$
$197$	−22.0000	−1.56744	−0.783718	+	0.621117i	$0.786679\pi$
$198$	0	0
$199$	9.70156	0.687726	0.343863	−	0.939020i	$-0.388265\pi$
$199$	9.70156	0.687726	0.343863	+	0.939020i	$0.388265\pi$
$200$	0	0
$201$	12.0000	0.846415
$202$	0	0
$203$	−3.40312	−0.238852
$204$	0	0
$205$	34.8062	2.43097
$206$	0	0
$207$	4.00000	0.278019
$208$	0	0
$209$	−1.70156	−0.117700
$210$	0	0
$211$	18.8062	1.29468	0.647338	−	0.762203i	$-0.275882\pi$
$211$	18.8062	1.29468	0.647338	+	0.762203i	$0.275882\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	33.7016	2.29843
$216$	0	0
$217$	−5.79063	−0.393093
$218$	0	0
$219$	0.298438	0.0201666
$220$	0	0
$221$	−22.2094	−1.49396
$222$	0	0
$223$	1.19375	0.0799395	0.0399698	−	0.999201i	$-0.487274\pi$
$223$	1.19375	0.0799395	0.0399698	+	0.999201i	$0.487274\pi$
$224$	0	0
$225$	8.70156	0.580104
$226$	0	0
$227$	23.4031	1.55332	0.776660	−	0.629920i	$-0.216912\pi$
$227$	23.4031	1.55332	0.776660	+	0.629920i	$0.216912\pi$
$228$	0	0
$229$	26.5078	1.75169	0.875843	−	0.482597i	$-0.160306\pi$
$229$	26.5078	1.75169	0.875843	+	0.482597i	$0.160306\pi$
$230$	0	0
$231$	2.89531	0.190498
$232$	0	0
$233$	7.10469	0.465443	0.232722	−	0.972543i	$-0.425237\pi$
$233$	7.10469	0.465443	0.232722	+	0.972543i	$0.425237\pi$
$234$	0	0
$235$	21.1047	1.37672
$236$	0	0
$237$	14.8062	0.961769
$238$	0	0
$239$	17.1047	1.10641	0.553205	−	0.833045i	$-0.313405\pi$
$239$	17.1047	1.10641	0.553205	+	0.833045i	$0.313405\pi$
$240$	0	0
$241$	−2.59688	−0.167279	−0.0836397	−	0.996496i	$-0.526654\pi$
$241$	−2.59688	−0.167279	−0.0836397	+	0.996496i	$0.526654\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−15.1938	−0.970693
$246$	0	0
$247$	6.00000	0.381771
$248$	0	0
$249$	−14.8062	−0.938308
$250$	0	0
$251$	−25.7016	−1.62227	−0.811134	−	0.584860i	$-0.801149\pi$
$251$	−25.7016	−1.62227	−0.811134	+	0.584860i	$0.801149\pi$
$252$	0	0
$253$	6.80625	0.427905
$254$	0	0
$255$	13.7016	0.858025
$256$	0	0
$257$	−6.59688	−0.411502	−0.205751	−	0.978604i	$-0.565964\pi$
$257$	−6.59688	−0.411502	−0.205751	+	0.978604i	$0.565964\pi$
$258$	0	0
$259$	−16.0000	−0.994192
$260$	0	0
$261$	−2.00000	−0.123797
$262$	0	0
$263$	5.70156	0.351573	0.175787	−	0.984428i	$-0.443753\pi$
$263$	5.70156	0.351573	0.175787	+	0.984428i	$0.443753\pi$
$264$	0	0
$265$	22.2094	1.36431
$266$	0	0
$267$	12.8062	0.783730
$268$	0	0
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	0	0
$271$	−14.8062	−0.899416	−0.449708	−	0.893176i	$-0.648472\pi$
$271$	−14.8062	−0.899416	−0.449708	+	0.893176i	$0.648472\pi$
$272$	0	0
$273$	−10.2094	−0.617899
$274$	0	0
$275$	14.8062	0.892850
$276$	0	0
$277$	3.70156	0.222405	0.111203	−	0.993798i	$-0.464530\pi$
$277$	3.70156	0.222405	0.111203	+	0.993798i	$0.464530\pi$
$278$	0	0
$279$	−3.40312	−0.203740
$280$	0	0
$281$	−24.8062	−1.47982	−0.739908	−	0.672708i	$-0.765131\pi$
$281$	−24.8062	−1.47982	−0.739908	+	0.672708i	$0.765131\pi$
$282$	0	0
$283$	2.29844	0.136628	0.0683140	−	0.997664i	$-0.478238\pi$
$283$	2.29844	0.136628	0.0683140	+	0.997664i	$0.478238\pi$
$284$	0	0
$285$	−3.70156	−0.219262
$286$	0	0
$287$	16.0000	0.944450
$288$	0	0
$289$	−3.29844	−0.194026
$290$	0	0
$291$	−6.00000	−0.351726
$292$	0	0
$293$	10.5969	0.619076	0.309538	−	0.950887i	$-0.399826\pi$
$293$	10.5969	0.619076	0.309538	+	0.950887i	$0.399826\pi$
$294$	0	0
$295$	14.8062	0.862053
$296$	0	0
$297$	1.70156	0.0987346
$298$	0	0
$299$	−24.0000	−1.38796
$300$	0	0
$301$	15.4922	0.892955
$302$	0	0
$303$	8.80625	0.505906
$304$	0	0
$305$	−28.5078	−1.63235
$306$	0	0
$307$	26.8062	1.52991	0.764957	−	0.644082i	$-0.222760\pi$
$307$	26.8062	1.52991	0.764957	+	0.644082i	$0.222760\pi$
$308$	0	0
$309$	3.40312	0.193597
$310$	0	0
$311$	4.50781	0.255614	0.127807	−	0.991799i	$-0.459206\pi$
$311$	4.50781	0.255614	0.127807	+	0.991799i	$0.459206\pi$
$312$	0	0
$313$	3.19375	0.180522	0.0902608	−	0.995918i	$-0.471230\pi$
$313$	3.19375	0.180522	0.0902608	+	0.995918i	$0.471230\pi$
$314$	0	0
$315$	6.29844	0.354877
$316$	0	0
$317$	9.40312	0.528132	0.264066	−	0.964505i	$-0.414936\pi$
$317$	9.40312	0.528132	0.264066	+	0.964505i	$0.414936\pi$
$318$	0	0
$319$	−3.40312	−0.190538
$320$	0	0
$321$	7.40312	0.413202
$322$	0	0
$323$	−3.70156	−0.205960
$324$	0	0
$325$	−52.2094	−2.89605
$326$	0	0
$327$	−9.40312	−0.519994
$328$	0	0
$329$	9.70156	0.534864
$330$	0	0
$331$	5.19375	0.285474	0.142737	−	0.989761i	$-0.454410\pi$
$331$	5.19375	0.285474	0.142737	+	0.989761i	$0.454410\pi$
$332$	0	0
$333$	−9.40312	−0.515288
$334$	0	0
$335$	44.4187	2.42686
$336$	0	0
$337$	−12.8062	−0.697601	−0.348800	−	0.937197i	$-0.613411\pi$
$337$	−12.8062	−0.697601	−0.348800	+	0.937197i	$0.613411\pi$
$338$	0	0
$339$	−18.0000	−0.977626
$340$	0	0
$341$	−5.79063	−0.313580
$342$	0	0
$343$	−18.8953	−1.02025
$344$	0	0
$345$	14.8062	0.797142
$346$	0	0
$347$	14.2984	0.767580	0.383790	−	0.923420i	$-0.374619\pi$
$347$	14.2984	0.767580	0.383790	+	0.923420i	$0.374619\pi$
$348$	0	0
$349$	4.89531	0.262040	0.131020	−	0.991380i	$-0.458175\pi$
$349$	4.89531	0.262040	0.131020	+	0.991380i	$0.458175\pi$
$350$	0	0
$351$	−6.00000	−0.320256
$352$	0	0
$353$	32.8062	1.74610	0.873050	−	0.487630i	$-0.162139\pi$
$353$	32.8062	1.74610	0.873050	+	0.487630i	$0.162139\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	6.29844	0.333349
$358$	0	0
$359$	−36.5078	−1.92681	−0.963404	−	0.268053i	$-0.913620\pi$
$359$	−36.5078	−1.92681	−0.963404	+	0.268053i	$0.913620\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−8.10469	−0.425386
$364$	0	0
$365$	1.10469	0.0578219
$366$	0	0
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	0	0
$369$	9.40312	0.489507
$370$	0	0
$371$	10.2094	0.530044
$372$	0	0
$373$	−9.40312	−0.486875	−0.243438	−	0.969917i	$-0.578275\pi$
$373$	−9.40312	−0.486875	−0.243438	+	0.969917i	$0.578275\pi$
$374$	0	0
$375$	13.7016	0.707546
$376$	0	0
$377$	12.0000	0.618031
$378$	0	0
$379$	8.59688	0.441592	0.220796	−	0.975320i	$-0.429135\pi$
$379$	8.59688	0.441592	0.220796	+	0.975320i	$0.429135\pi$
$380$	0	0
$381$	−3.40312	−0.174347
$382$	0	0
$383$	−18.2094	−0.930455	−0.465228	−	0.885191i	$-0.654028\pi$
$383$	−18.2094	−0.930455	−0.465228	+	0.885191i	$0.654028\pi$
$384$	0	0
$385$	10.7172	0.546198
$386$	0	0
$387$	9.10469	0.462817
$388$	0	0
$389$	−9.91093	−0.502504	−0.251252	−	0.967922i	$-0.580842\pi$
$389$	−9.91093	−0.502504	−0.251252	+	0.967922i	$0.580842\pi$
$390$	0	0
$391$	14.8062	0.748784
$392$	0	0
$393$	6.29844	0.317714
$394$	0	0
$395$	54.8062	2.75760
$396$	0	0
$397$	2.50781	0.125863	0.0629317	−	0.998018i	$-0.479955\pi$
$397$	2.50781	0.125863	0.0629317	+	0.998018i	$0.479955\pi$
$398$	0	0
$399$	−1.70156	−0.0851847
$400$	0	0
$401$	−28.2094	−1.40871	−0.704354	−	0.709848i	$-0.748763\pi$
$401$	−28.2094	−1.40871	−0.704354	+	0.709848i	$0.748763\pi$
$402$	0	0
$403$	20.4187	1.01713
$404$	0	0
$405$	3.70156	0.183932
$406$	0	0
$407$	−16.0000	−0.793091
$408$	0	0
$409$	2.00000	0.0988936	0.0494468	−	0.998777i	$-0.484254\pi$
$409$	2.00000	0.0988936	0.0494468	+	0.998777i	$0.484254\pi$
$410$	0	0
$411$	−4.29844	−0.212026
$412$	0	0
$413$	6.80625	0.334914
$414$	0	0
$415$	−54.8062	−2.69033
$416$	0	0
$417$	−21.7016	−1.06273
$418$	0	0
$419$	6.80625	0.332507	0.166253	−	0.986083i	$-0.446833\pi$
$419$	6.80625	0.332507	0.166253	+	0.986083i	$0.446833\pi$
$420$	0	0
$421$	24.8062	1.20898	0.604491	−	0.796612i	$-0.293376\pi$
$421$	24.8062	1.20898	0.604491	+	0.796612i	$0.293376\pi$
$422$	0	0
$423$	5.70156	0.277219
$424$	0	0
$425$	32.2094	1.56238
$426$	0	0
$427$	−13.1047	−0.634180
$428$	0	0
$429$	−10.2094	−0.492913
$430$	0	0
$431$	−27.4031	−1.31996	−0.659981	−	0.751282i	$-0.729436\pi$
$431$	−27.4031	−1.31996	−0.659981	+	0.751282i	$0.729436\pi$
$432$	0	0
$433$	−27.6125	−1.32697	−0.663486	−	0.748189i	$-0.730924\pi$
$433$	−27.6125	−1.32697	−0.663486	+	0.748189i	$0.730924\pi$
$434$	0	0
$435$	−7.40312	−0.354953
$436$	0	0
$437$	−4.00000	−0.191346
$438$	0	0
$439$	19.4031	0.926061	0.463030	−	0.886342i	$-0.346762\pi$
$439$	19.4031	0.926061	0.463030	+	0.886342i	$0.346762\pi$
$440$	0	0
$441$	−4.10469	−0.195461
$442$	0	0
$443$	−24.5078	−1.16440	−0.582201	−	0.813045i	$-0.697808\pi$
$443$	−24.5078	−1.16440	−0.582201	+	0.813045i	$0.697808\pi$
$444$	0	0
$445$	47.4031	2.24712
$446$	0	0
$447$	−0.895314	−0.0423469
$448$	0	0
$449$	−13.4031	−0.632533	−0.316266	−	0.948670i	$-0.602429\pi$
$449$	−13.4031	−0.632533	−0.316266	+	0.948670i	$0.602429\pi$
$450$	0	0
$451$	16.0000	0.753411
$452$	0	0
$453$	−22.8062	−1.07153
$454$	0	0
$455$	−37.7906	−1.77165
$456$	0	0
$457$	−35.1047	−1.64213	−0.821064	−	0.570836i	$-0.806619\pi$
$457$	−35.1047	−1.64213	−0.821064	+	0.570836i	$0.806619\pi$
$458$	0	0
$459$	3.70156	0.172774
$460$	0	0
$461$	−8.89531	−0.414296	−0.207148	−	0.978310i	$-0.566418\pi$
$461$	−8.89531	−0.414296	−0.207148	+	0.978310i	$0.566418\pi$
$462$	0	0
$463$	−24.5078	−1.13897	−0.569487	−	0.822000i	$-0.692858\pi$
$463$	−24.5078	−1.13897	−0.569487	+	0.822000i	$0.692858\pi$
$464$	0	0
$465$	−12.5969	−0.584166
$466$	0	0
$467$	37.1047	1.71700	0.858500	−	0.512813i	$-0.171397\pi$
$467$	37.1047	1.71700	0.858500	+	0.512813i	$0.171397\pi$
$468$	0	0
$469$	20.4187	0.942850
$470$	0	0
$471$	2.00000	0.0921551
$472$	0	0
$473$	15.4922	0.712332
$474$	0	0
$475$	−8.70156	−0.399255
$476$	0	0
$477$	6.00000	0.274721
$478$	0	0
$479$	−2.80625	−0.128221	−0.0641104	−	0.997943i	$-0.520421\pi$
$479$	−2.80625	−0.128221	−0.0641104	+	0.997943i	$0.520421\pi$
$480$	0	0
$481$	56.4187	2.57247
$482$	0	0
$483$	6.80625	0.309695
$484$	0	0
$485$	−22.2094	−1.00848
$486$	0	0
$487$	30.8062	1.39596	0.697982	−	0.716115i	$-0.254081\pi$
$487$	30.8062	1.39596	0.697982	+	0.716115i	$0.254081\pi$
$488$	0	0
$489$	−10.8062	−0.488675
$490$	0	0
$491$	8.00000	0.361035	0.180517	−	0.983572i	$-0.442223\pi$
$491$	8.00000	0.361035	0.180517	+	0.983572i	$0.442223\pi$
$492$	0	0
$493$	−7.40312	−0.333420
$494$	0	0
$495$	6.29844	0.283094
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−23.9109	−1.07040	−0.535200	−	0.844725i	$-0.679764\pi$
$499$	−23.9109	−1.07040	−0.535200	+	0.844725i	$0.679764\pi$
$500$	0	0
$501$	−18.2094	−0.813535
$502$	0	0
$503$	−13.1938	−0.588280	−0.294140	−	0.955762i	$-0.595033\pi$
$503$	−13.1938	−0.588280	−0.294140	+	0.955762i	$0.595033\pi$
$504$	0	0
$505$	32.5969	1.45054
$506$	0	0
$507$	23.0000	1.02147
$508$	0	0
$509$	−7.61250	−0.337418	−0.168709	−	0.985666i	$-0.553960\pi$
$509$	−7.61250	−0.337418	−0.168709	+	0.985666i	$0.553960\pi$
$510$	0	0
$511$	0.507811	0.0224642
$512$	0	0
$513$	−1.00000	−0.0441511
$514$	0	0
$515$	12.5969	0.555085
$516$	0	0
$517$	9.70156	0.426674
$518$	0	0
$519$	−16.8062	−0.737712
$520$	0	0
$521$	25.4031	1.11293	0.556466	−	0.830871i	$-0.312157\pi$
$521$	25.4031	1.11293	0.556466	+	0.830871i	$0.312157\pi$
$522$	0	0
$523$	−22.2094	−0.971148	−0.485574	−	0.874196i	$-0.661389\pi$
$523$	−22.2094	−0.971148	−0.485574	+	0.874196i	$0.661389\pi$
$524$	0	0
$525$	14.8062	0.646198
$526$	0	0
$527$	−12.5969	−0.548729
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	4.00000	0.173585
$532$	0	0
$533$	−56.4187	−2.44377
$534$	0	0
$535$	27.4031	1.18474
$536$	0	0
$537$	−7.40312	−0.319468
$538$	0	0
$539$	−6.98438	−0.300838
$540$	0	0
$541$	44.7172	1.92254	0.961271	−	0.275605i	$-0.0888782\pi$
$541$	44.7172	1.92254	0.961271	+	0.275605i	$0.0888782\pi$
$542$	0	0
$543$	−20.8062	−0.892882
$544$	0	0
$545$	−34.8062	−1.49094
$546$	0	0
$547$	−45.0156	−1.92473	−0.962364	−	0.271762i	$-0.912394\pi$
$547$	−45.0156	−1.92473	−0.962364	+	0.271762i	$0.912394\pi$
$548$	0	0
$549$	−7.70156	−0.328695
$550$	0	0
$551$	2.00000	0.0852029
$552$	0	0
$553$	25.1938	1.07135
$554$	0	0
$555$	−34.8062	−1.47744
$556$	0	0
$557$	32.2984	1.36853	0.684264	−	0.729234i	$-0.260123\pi$
$557$	32.2984	1.36853	0.684264	+	0.729234i	$0.260123\pi$
$558$	0	0
$559$	−54.6281	−2.31052
$560$	0	0
$561$	6.29844	0.265920
$562$	0	0
$563$	4.00000	0.168580	0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000	0.168580	0.0842900	+	0.996441i	$0.473138\pi$
$564$	0	0
$565$	−66.6281	−2.80307
$566$	0	0
$567$	1.70156	0.0714589
$568$	0	0
$569$	−24.8062	−1.03993	−0.519966	−	0.854187i	$-0.674055\pi$
$569$	−24.8062	−1.03993	−0.519966	+	0.854187i	$0.674055\pi$
$570$	0	0
$571$	−26.8062	−1.12181	−0.560903	−	0.827881i	$-0.689546\pi$
$571$	−26.8062	−1.12181	−0.560903	+	0.827881i	$0.689546\pi$
$572$	0	0
$573$	23.9109	0.998894
$574$	0	0
$575$	34.8062	1.45152
$576$	0	0
$577$	34.5078	1.43658	0.718289	−	0.695744i	$-0.244925\pi$
$577$	34.5078	1.43658	0.718289	+	0.695744i	$0.244925\pi$
$578$	0	0
$579$	−1.40312	−0.0583119
$580$	0	0
$581$	−25.1938	−1.04521
$582$	0	0
$583$	10.2094	0.422829
$584$	0	0
$585$	−22.2094	−0.918245
$586$	0	0
$587$	−9.70156	−0.400426	−0.200213	−	0.979752i	$-0.564163\pi$
$587$	−9.70156	−0.400426	−0.200213	+	0.979752i	$0.564163\pi$
$588$	0	0
$589$	3.40312	0.140223
$590$	0	0
$591$	−22.0000	−0.904959
$592$	0	0
$593$	2.00000	0.0821302	0.0410651	−	0.999156i	$-0.486925\pi$
$593$	2.00000	0.0821302	0.0410651	+	0.999156i	$0.486925\pi$
$594$	0	0
$595$	23.3141	0.955783
$596$	0	0
$597$	9.70156	0.397059
$598$	0	0
$599$	14.8062	0.604967	0.302483	−	0.953155i	$-0.402184\pi$
$599$	14.8062	0.604967	0.302483	+	0.953155i	$0.402184\pi$
$600$	0	0
$601$	−16.2094	−0.661194	−0.330597	−	0.943772i	$-0.607250\pi$
$601$	−16.2094	−0.661194	−0.330597	+	0.943772i	$0.607250\pi$
$602$	0	0
$603$	12.0000	0.488678
$604$	0	0
$605$	−30.0000	−1.21967
$606$	0	0
$607$	−10.2094	−0.414386	−0.207193	−	0.978300i	$-0.566433\pi$
$607$	−10.2094	−0.414386	−0.207193	+	0.978300i	$0.566433\pi$
$608$	0	0
$609$	−3.40312	−0.137902
$610$	0	0
$611$	−34.2094	−1.38396
$612$	0	0
$613$	42.5078	1.71687	0.858437	−	0.512919i	$-0.171436\pi$
$613$	42.5078	1.71687	0.858437	+	0.512919i	$0.171436\pi$
$614$	0	0
$615$	34.8062	1.40352
$616$	0	0
$617$	2.50781	0.100961	0.0504803	−	0.998725i	$-0.483925\pi$
$617$	2.50781	0.100961	0.0504803	+	0.998725i	$0.483925\pi$
$618$	0	0
$619$	−12.0000	−0.482321	−0.241160	−	0.970485i	$-0.577528\pi$
$619$	−12.0000	−0.482321	−0.241160	+	0.970485i	$0.577528\pi$
$620$	0	0
$621$	4.00000	0.160514
$622$	0	0
$623$	21.7906	0.873023
$624$	0	0
$625$	7.20937	0.288375
$626$	0	0
$627$	−1.70156	−0.0679538
$628$	0	0
$629$	−34.8062	−1.38782
$630$	0	0
$631$	40.5078	1.61259	0.806295	−	0.591513i	$-0.201469\pi$
$631$	40.5078	1.61259	0.806295	+	0.591513i	$0.201469\pi$
$632$	0	0
$633$	18.8062	0.747481
$634$	0	0
$635$	−12.5969	−0.499892
$636$	0	0
$637$	24.6281	0.975802
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−2.00000	−0.0789953	−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000	−0.0789953	−0.0394976	+	0.999220i	$0.512576\pi$
$642$	0	0
$643$	−12.5078	−0.493260	−0.246630	−	0.969110i	$-0.579323\pi$
$643$	−12.5078	−0.493260	−0.246630	+	0.969110i	$0.579323\pi$
$644$	0	0
$645$	33.7016	1.32700
$646$	0	0
$647$	−37.7016	−1.48220	−0.741101	−	0.671394i	$-0.765696\pi$
$647$	−37.7016	−1.48220	−0.741101	+	0.671394i	$0.765696\pi$
$648$	0	0
$649$	6.80625	0.267169
$650$	0	0
$651$	−5.79063	−0.226953
$652$	0	0
$653$	−8.89531	−0.348101	−0.174050	−	0.984737i	$-0.555686\pi$
$653$	−8.89531	−0.348101	−0.174050	+	0.984737i	$0.555686\pi$
$654$	0	0
$655$	23.3141	0.910956
$656$	0	0
$657$	0.298438	0.0116432
$658$	0	0
$659$	−31.4031	−1.22329	−0.611646	−	0.791132i	$-0.709492\pi$
$659$	−31.4031	−1.22329	−0.611646	+	0.791132i	$0.709492\pi$
$660$	0	0
$661$	−0.209373	−0.00814365	−0.00407183	−	0.999992i	$-0.501296\pi$
$661$	−0.209373	−0.00814365	−0.00407183	+	0.999992i	$0.501296\pi$
$662$	0	0
$663$	−22.2094	−0.862541
$664$	0	0
$665$	−6.29844	−0.244243
$666$	0	0
$667$	−8.00000	−0.309761
$668$	0	0
$669$	1.19375	0.0461531
$670$	0	0
$671$	−13.1047	−0.505901
$672$	0	0
$673$	5.40312	0.208275	0.104138	−	0.994563i	$-0.466792\pi$
$673$	5.40312	0.208275	0.104138	+	0.994563i	$0.466792\pi$
$674$	0	0
$675$	8.70156	0.334923
$676$	0	0
$677$	12.8062	0.492184	0.246092	−	0.969246i	$-0.420853\pi$
$677$	12.8062	0.492184	0.246092	+	0.969246i	$0.420853\pi$
$678$	0	0
$679$	−10.2094	−0.391800
$680$	0	0
$681$	23.4031	0.896810
$682$	0	0
$683$	9.79063	0.374628	0.187314	−	0.982300i	$-0.440022\pi$
$683$	9.79063	0.374628	0.187314	+	0.982300i	$0.440022\pi$
$684$	0	0
$685$	−15.9109	−0.607926
$686$	0	0
$687$	26.5078	1.01134
$688$	0	0
$689$	−36.0000	−1.37149
$690$	0	0
$691$	20.5078	0.780154	0.390077	−	0.920782i	$-0.372448\pi$
$691$	20.5078	0.780154	0.390077	+	0.920782i	$0.372448\pi$
$692$	0	0
$693$	2.89531	0.109984
$694$	0	0
$695$	−80.3297	−3.04708
$696$	0	0
$697$	34.8062	1.31838
$698$	0	0
$699$	7.10469	0.268724
$700$	0	0
$701$	−28.8062	−1.08800	−0.543998	−	0.839086i	$-0.683090\pi$
$701$	−28.8062	−1.08800	−0.543998	+	0.839086i	$0.683090\pi$
$702$	0	0
$703$	9.40312	0.354646
$704$	0	0
$705$	21.1047	0.794848
$706$	0	0
$707$	14.9844	0.563546
$708$	0	0
$709$	23.6125	0.886786	0.443393	−	0.896327i	$-0.353775\pi$
$709$	23.6125	0.886786	0.443393	+	0.896327i	$0.353775\pi$
$710$	0	0
$711$	14.8062	0.555278
$712$	0	0
$713$	−13.6125	−0.509792
$714$	0	0
$715$	−37.7906	−1.41329
$716$	0	0
$717$	17.1047	0.638786
$718$	0	0
$719$	18.2984	0.682417	0.341208	−	0.939988i	$-0.389164\pi$
$719$	18.2984	0.682417	0.341208	+	0.939988i	$0.389164\pi$
$720$	0	0
$721$	5.79063	0.215654
$722$	0	0
$723$	−2.59688	−0.0965788
$724$	0	0
$725$	−17.4031	−0.646336
$726$	0	0
$727$	−16.5078	−0.612241	−0.306120	−	0.951993i	$-0.599031\pi$
$727$	−16.5078	−0.612241	−0.306120	+	0.951993i	$0.599031\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	33.7016	1.24650
$732$	0	0
$733$	−4.80625	−0.177523	−0.0887614	−	0.996053i	$-0.528291\pi$
$733$	−4.80625	−0.177523	−0.0887614	+	0.996053i	$0.528291\pi$
$734$	0	0
$735$	−15.1938	−0.560430
$736$	0	0
$737$	20.4187	0.752134
$738$	0	0
$739$	−29.7016	−1.09259	−0.546295	−	0.837593i	$-0.683962\pi$
$739$	−29.7016	−1.09259	−0.546295	+	0.837593i	$0.683962\pi$
$740$	0	0
$741$	6.00000	0.220416
$742$	0	0
$743$	−22.8062	−0.836680	−0.418340	−	0.908290i	$-0.637388\pi$
$743$	−22.8062	−0.836680	−0.418340	+	0.908290i	$0.637388\pi$
$744$	0	0
$745$	−3.31406	−0.121418
$746$	0	0
$747$	−14.8062	−0.541732
$748$	0	0
$749$	12.5969	0.460280
$750$	0	0
$751$	−21.6125	−0.788651	−0.394326	−	0.918971i	$-0.629022\pi$
$751$	−21.6125	−0.788651	−0.394326	+	0.918971i	$0.629022\pi$
$752$	0	0
$753$	−25.7016	−0.936617
$754$	0	0
$755$	−84.4187	−3.07231
$756$	0	0
$757$	−31.7016	−1.15221	−0.576106	−	0.817375i	$-0.695429\pi$
$757$	−31.7016	−1.15221	−0.576106	+	0.817375i	$0.695429\pi$
$758$	0	0
$759$	6.80625	0.247051
$760$	0	0
$761$	−17.9109	−0.649271	−0.324635	−	0.945839i	$-0.605242\pi$
$761$	−17.9109	−0.649271	−0.324635	+	0.945839i	$0.605242\pi$
$762$	0	0
$763$	−16.0000	−0.579239
$764$	0	0
$765$	13.7016	0.495381
$766$	0	0
$767$	−24.0000	−0.866590
$768$	0	0
$769$	48.1203	1.73526	0.867631	−	0.497208i	$-0.165641\pi$
$769$	48.1203	1.73526	0.867631	+	0.497208i	$0.165641\pi$
$770$	0	0
$771$	−6.59688	−0.237581
$772$	0	0
$773$	31.0156	1.11555	0.557777	−	0.829991i	$-0.311654\pi$
$773$	31.0156	1.11555	0.557777	+	0.829991i	$0.311654\pi$
$774$	0	0
$775$	−29.6125	−1.06371
$776$	0	0
$777$	−16.0000	−0.573997
$778$	0	0
$779$	−9.40312	−0.336902
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−2.00000	−0.0714742
$784$	0	0
$785$	7.40312	0.264229
$786$	0	0
$787$	−13.1938	−0.470306	−0.235153	−	0.971958i	$-0.575559\pi$
$787$	−13.1938	−0.470306	−0.235153	+	0.971958i	$0.575559\pi$
$788$	0	0
$789$	5.70156	0.202981
$790$	0	0
$791$	−30.6281	−1.08901
$792$	0	0
$793$	46.2094	1.64094
$794$	0	0
$795$	22.2094	0.787685
$796$	0	0
$797$	25.4031	0.899825	0.449912	−	0.893073i	$-0.351455\pi$
$797$	25.4031	0.899825	0.449912	+	0.893073i	$0.351455\pi$
$798$	0	0
$799$	21.1047	0.746630
$800$	0	0
$801$	12.8062	0.452487
$802$	0	0
$803$	0.507811	0.0179202
$804$	0	0
$805$	25.1938	0.887963
$806$	0	0
$807$	−10.0000	−0.352017
$808$	0	0
$809$	20.7172	0.728377	0.364189	−	0.931325i	$-0.381346\pi$
$809$	20.7172	0.728377	0.364189	+	0.931325i	$0.381346\pi$
$810$	0	0
$811$	15.4031	0.540877	0.270438	−	0.962737i	$-0.412831\pi$
$811$	15.4031	0.540877	0.270438	+	0.962737i	$0.412831\pi$
$812$	0	0
$813$	−14.8062	−0.519278
$814$	0	0
$815$	−40.0000	−1.40114
$816$	0	0
$817$	−9.10469	−0.318533
$818$	0	0
$819$	−10.2094	−0.356744
$820$	0	0
$821$	−46.5078	−1.62313	−0.811567	−	0.584260i	$-0.801385\pi$
$821$	−46.5078	−1.62313	−0.811567	+	0.584260i	$0.801385\pi$
$822$	0	0
$823$	47.3141	1.64926	0.824632	−	0.565669i	$-0.191382\pi$
$823$	47.3141	1.64926	0.824632	+	0.565669i	$0.191382\pi$
$824$	0	0
$825$	14.8062	0.515487
$826$	0	0
$827$	−32.4187	−1.12731	−0.563655	−	0.826010i	$-0.690605\pi$
$827$	−32.4187	−1.12731	−0.563655	+	0.826010i	$0.690605\pi$
$828$	0	0
$829$	−16.2094	−0.562975	−0.281487	−	0.959565i	$-0.590828\pi$
$829$	−16.2094	−0.562975	−0.281487	+	0.959565i	$0.590828\pi$
$830$	0	0
$831$	3.70156	0.128406
$832$	0	0
$833$	−15.1938	−0.526432
$834$	0	0
$835$	−67.4031	−2.33258
$836$	0	0
$837$	−3.40312	−0.117629
$838$	0	0
$839$	33.0156	1.13983	0.569913	−	0.821705i	$-0.306977\pi$
$839$	33.0156	1.13983	0.569913	+	0.821705i	$0.306977\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	−24.8062	−0.854373
$844$	0	0
$845$	85.1359	2.92876
$846$	0	0
$847$	−13.7906	−0.473852
$848$	0	0
$849$	2.29844	0.0788822
$850$	0	0
$851$	−37.6125	−1.28934
$852$	0	0
$853$	−28.8062	−0.986307	−0.493154	−	0.869942i	$-0.664156\pi$
$853$	−28.8062	−0.986307	−0.493154	+	0.869942i	$0.664156\pi$
$854$	0	0
$855$	−3.70156	−0.126591
$856$	0	0
$857$	−11.0156	−0.376286	−0.188143	−	0.982142i	$-0.560247\pi$
$857$	−11.0156	−0.376286	−0.188143	+	0.982142i	$0.560247\pi$
$858$	0	0
$859$	−37.7016	−1.28636	−0.643180	−	0.765715i	$-0.722385\pi$
$859$	−37.7016	−1.28636	−0.643180	+	0.765715i	$0.722385\pi$
$860$	0	0
$861$	16.0000	0.545279
$862$	0	0
$863$	−1.19375	−0.0406358	−0.0203179	−	0.999794i	$-0.506468\pi$
$863$	−1.19375	−0.0406358	−0.0203179	+	0.999794i	$0.506468\pi$
$864$	0	0
$865$	−62.2094	−2.11518
$866$	0	0
$867$	−3.29844	−0.112021
$868$	0	0
$869$	25.1938	0.854639
$870$	0	0
$871$	−72.0000	−2.43963
$872$	0	0
$873$	−6.00000	−0.203069
$874$	0	0
$875$	23.3141	0.788159
$876$	0	0
$877$	27.1938	0.918268	0.459134	−	0.888367i	$-0.348160\pi$
$877$	27.1938	0.918268	0.459134	+	0.888367i	$0.348160\pi$
$878$	0	0
$879$	10.5969	0.357424
$880$	0	0
$881$	12.8953	0.434454	0.217227	−	0.976121i	$-0.430299\pi$
$881$	12.8953	0.434454	0.217227	+	0.976121i	$0.430299\pi$
$882$	0	0
$883$	−0.0890652	−0.00299728	−0.00149864	−	0.999999i	$-0.500477\pi$
$883$	−0.0890652	−0.00299728	−0.00149864	+	0.999999i	$0.500477\pi$
$884$	0	0
$885$	14.8062	0.497707
$886$	0	0
$887$	21.6125	0.725677	0.362838	−	0.931852i	$-0.381808\pi$
$887$	21.6125	0.725677	0.362838	+	0.931852i	$0.381808\pi$
$888$	0	0
$889$	−5.79063	−0.194211
$890$	0	0
$891$	1.70156	0.0570045
$892$	0	0
$893$	−5.70156	−0.190796
$894$	0	0
$895$	−27.4031	−0.915985
$896$	0	0
$897$	−24.0000	−0.801337
$898$	0	0
$899$	6.80625	0.227001
$900$	0	0
$901$	22.2094	0.739901
$902$	0	0
$903$	15.4922	0.515548
$904$	0	0
$905$	−77.0156	−2.56009
$906$	0	0
$907$	37.0156	1.22908	0.614542	−	0.788884i	$-0.289341\pi$
$907$	37.0156	1.22908	0.614542	+	0.788884i	$0.289341\pi$
$908$	0	0
$909$	8.80625	0.292085
$910$	0	0
$911$	10.2094	0.338252	0.169126	−	0.985594i	$-0.445906\pi$
$911$	10.2094	0.338252	0.169126	+	0.985594i	$0.445906\pi$
$912$	0	0
$913$	−25.1938	−0.833791
$914$	0	0
$915$	−28.5078	−0.942439
$916$	0	0
$917$	10.7172	0.353913
$918$	0	0
$919$	21.6125	0.712930	0.356465	−	0.934309i	$-0.383982\pi$
$919$	21.6125	0.712930	0.356465	+	0.934309i	$0.383982\pi$
$920$	0	0
$921$	26.8062	0.883296
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−81.8219	−2.69029
$926$	0	0
$927$	3.40312	0.111773
$928$	0	0
$929$	11.1938	0.367255	0.183628	−	0.982996i	$-0.441216\pi$
$929$	11.1938	0.367255	0.183628	+	0.982996i	$0.441216\pi$
$930$	0	0
$931$	4.10469	0.134526
$932$	0	0
$933$	4.50781	0.147579
$934$	0	0
$935$	23.3141	0.762451
$936$	0	0
$937$	0.298438	0.00974954	0.00487477	−	0.999988i	$-0.498448\pi$
$937$	0.298438	0.00974954	0.00487477	+	0.999988i	$0.498448\pi$
$938$	0	0
$939$	3.19375	0.104224
$940$	0	0
$941$	−10.0000	−0.325991	−0.162995	−	0.986627i	$-0.552116\pi$
$941$	−10.0000	−0.325991	−0.162995	+	0.986627i	$0.552116\pi$
$942$	0	0
$943$	37.6125	1.22483
$944$	0	0
$945$	6.29844	0.204888
$946$	0	0
$947$	6.80625	0.221173	0.110587	−	0.993866i	$-0.464727\pi$
$947$	6.80625	0.221173	0.110587	+	0.993866i	$0.464727\pi$
$948$	0	0
$949$	−1.79063	−0.0581262
$950$	0	0
$951$	9.40312	0.304917
$952$	0	0
$953$	−5.40312	−0.175024	−0.0875122	−	0.996163i	$-0.527892\pi$
$953$	−5.40312	−0.175024	−0.0875122	+	0.996163i	$0.527892\pi$
$954$	0	0
$955$	88.5078	2.86405
$956$	0	0
$957$	−3.40312	−0.110007
$958$	0	0
$959$	−7.31406	−0.236183
$960$	0	0
$961$	−19.4187	−0.626411
$962$	0	0
$963$	7.40312	0.238562
$964$	0	0
$965$	−5.19375	−0.167193
$966$	0	0
$967$	40.0000	1.28631	0.643157	−	0.765735i	$-0.277624\pi$
$967$	40.0000	1.28631	0.643157	+	0.765735i	$0.277624\pi$
$968$	0	0
$969$	−3.70156	−0.118911
$970$	0	0
$971$	−20.0000	−0.641831	−0.320915	−	0.947108i	$-0.603990\pi$
$971$	−20.0000	−0.641831	−0.320915	+	0.947108i	$0.603990\pi$
$972$	0	0
$973$	−36.9266	−1.18381
$974$	0	0
$975$	−52.2094	−1.67204
$976$	0	0
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	0	0
$979$	21.7906	0.696431
$980$	0	0
$981$	−9.40312	−0.300219
$982$	0	0
$983$	−44.4187	−1.41674	−0.708369	−	0.705842i	$-0.750569\pi$
$983$	−44.4187	−1.41674	−0.708369	+	0.705842i	$0.750569\pi$
$984$	0	0
$985$	−81.4344	−2.59471
$986$	0	0
$987$	9.70156	0.308804
$988$	0	0
$989$	36.4187	1.15805
$990$	0	0
$991$	−48.0000	−1.52477	−0.762385	−	0.647124i	$-0.775972\pi$
$991$	−48.0000	−1.52477	−0.762385	+	0.647124i	$0.775972\pi$
$992$	0	0
$993$	5.19375	0.164819
$994$	0	0
$995$	35.9109	1.13845
$996$	0	0
$997$	−35.1047	−1.11178	−0.555888	−	0.831257i	$-0.687622\pi$
$997$	−35.1047	−1.11178	−0.555888	+	0.831257i	$0.687622\pi$
$998$	0	0
$999$	−9.40312	−0.297502

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3648.2.a.bs.1.2		2
4.3	odd	2		3648.2.a.bn.1.2		2
8.3	odd	2		912.2.a.o.1.1		2
8.5	even	2		456.2.a.e.1.1	✓	2
24.5	odd	2		1368.2.a.l.1.2		2
24.11	even	2		2736.2.a.bb.1.2		2
152.37	odd	2		8664.2.a.v.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
456.2.a.e.1.1	✓	2	8.5	even	2
912.2.a.o.1.1		2	8.3	odd	2
1368.2.a.l.1.2		2	24.5	odd	2
2736.2.a.bb.1.2		2	24.11	even	2
3648.2.a.bn.1.2		2	4.3	odd	2
3648.2.a.bs.1.2		2	1.1	even	1	trivial
8664.2.a.v.1.1		2	152.37	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3648 = 2^{6} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3648.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.70156 q^{5} +1.70156 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +3.70156 q^{5} +1.70156 q^{7} +1.00000 q^{9} +1.70156 q^{11} -6.00000 q^{13} +3.70156 q^{15} +3.70156 q^{17} -1.00000 q^{19} +1.70156 q^{21} +4.00000 q^{23} +8.70156 q^{25} +1.00000 q^{27} -2.00000 q^{29} -3.40312 q^{31} +1.70156 q^{33} +6.29844 q^{35} -9.40312 q^{37} -6.00000 q^{39} +9.40312 q^{41} +9.10469 q^{43} +3.70156 q^{45} +5.70156 q^{47} -4.10469 q^{49} +3.70156 q^{51} +6.00000 q^{53} +6.29844 q^{55} -1.00000 q^{57} +4.00000 q^{59} -7.70156 q^{61} +1.70156 q^{63} -22.2094 q^{65} +12.0000 q^{67} +4.00000 q^{69} +0.298438 q^{73} +8.70156 q^{75} +2.89531 q^{77} +14.8062 q^{79} +1.00000 q^{81} -14.8062 q^{83} +13.7016 q^{85} -2.00000 q^{87} +12.8062 q^{89} -10.2094 q^{91} -3.40312 q^{93} -3.70156 q^{95} -6.00000 q^{97} +1.70156 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + q^{5} - 3 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + q^5 - 3 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} + q^{5} - 3 q^{7} + 2 q^{9} - 3 q^{11} - 12 q^{13} + q^{15} + q^{17} - 2 q^{19} - 3 q^{21} + 8 q^{23} + 11 q^{25} + 2 q^{27} - 4 q^{29} + 6 q^{31} - 3 q^{33} + 19 q^{35} - 6 q^{37} - 12 q^{39} + 6 q^{41} - q^{43} + q^{45} + 5 q^{47} + 11 q^{49} + q^{51} + 12 q^{53} + 19 q^{55} - 2 q^{57} + 8 q^{59} - 9 q^{61} - 3 q^{63} - 6 q^{65} + 24 q^{67} + 8 q^{69} + 7 q^{73} + 11 q^{75} + 25 q^{77} + 4 q^{79} + 2 q^{81} - 4 q^{83} + 21 q^{85} - 4 q^{87} + 18 q^{91} + 6 q^{93} - q^{95} - 12 q^{97} - 3 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + q^5 - 3 * q^7 + 2 * q^9 - 3 * q^11 - 12 * q^13 + q^15 + q^17 - 2 * q^19 - 3 * q^21 + 8 * q^23 + 11 * q^25 + 2 * q^27 - 4 * q^29 + 6 * q^31 - 3 * q^33 + 19 * q^35 - 6 * q^37 - 12 * q^39 + 6 * q^41 - q^43 + q^45 + 5 * q^47 + 11 * q^49 + q^51 + 12 * q^53 + 19 * q^55 - 2 * q^57 + 8 * q^59 - 9 * q^61 - 3 * q^63 - 6 * q^65 + 24 * q^67 + 8 * q^69 + 7 * q^73 + 11 * q^75 + 25 * q^77 + 4 * q^79 + 2 * q^81 - 4 * q^83 + 21 * q^85 - 4 * q^87 + 18 * q^91 + 6 * q^93 - q^95 - 12 * q^97 - 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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