LMFDB - Embedded newform 7448.2.a.bh.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7448,2,Mod(1,7448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7448, 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("7448.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7448 = 2^{3} \cdot 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7448.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:	$59.4725794254$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.1076.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 8x - 6 $ x^3 - 8*x - 6
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1064)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.14743$ of defining polynomial
Character	$\chi$	$=$	7448.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.14743 q^{3} +2.38854 q^{5} +6.90632 q^{9} +O(q^{10})$	$q-3.14743 q^{3} +2.38854 q^{5} +6.90632 q^{9} -1.00000 q^{11} +3.14743 q^{13} -7.51777 q^{15} +0.611457 q^{17} -1.00000 q^{19} -6.90632 q^{23} +0.705140 q^{25} -12.2949 q^{27} +7.81263 q^{29} +6.66520 q^{31} +3.14743 q^{33} -4.77709 q^{37} -9.90632 q^{39} -4.77709 q^{41} -2.09368 q^{43} +16.4960 q^{45} +6.90632 q^{47} -1.92452 q^{51} -12.9601 q^{53} -2.38854 q^{55} +3.14743 q^{57} -2.85257 q^{59} -10.9063 q^{61} +7.51777 q^{65} -9.51777 q^{67} +21.7372 q^{69} -4.37034 q^{71} -5.68340 q^{73} -2.21938 q^{75} -12.1830 q^{79} +17.9783 q^{81} -12.1656 q^{83} +1.46049 q^{85} -24.5897 q^{87} +2.77709 q^{89} -20.9783 q^{93} -2.38854 q^{95} +7.62966 q^{97} -6.90632 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{5} + 7 q^{9}+O(q^{10}) $ 3 * q + 2 * q^5 + 7 * q^9	$ 3 q + 2 q^{5} + 7 q^{9} - 3 q^{11} - 14 q^{15} + 7 q^{17} - 3 q^{19} - 7 q^{23} + 21 q^{25} - 18 q^{27} - 4 q^{29} + 2 q^{31} - 4 q^{37} - 16 q^{39} - 4 q^{41} - 20 q^{43} - 2 q^{45} + 7 q^{47} + 14 q^{51} - 2 q^{53} - 2 q^{55} - 18 q^{59} - 19 q^{61} + 14 q^{65} - 20 q^{67} + 18 q^{69} - 14 q^{71} + 7 q^{73} + 32 q^{75} - 10 q^{79} + 11 q^{81} - 21 q^{83} - 30 q^{85} - 36 q^{87} - 2 q^{89} - 20 q^{93} - 2 q^{95} + 22 q^{97} - 7 q^{99}+O(q^{100}) $ 3 * q + 2 * q^5 + 7 * q^9 - 3 * q^11 - 14 * q^15 + 7 * q^17 - 3 * q^19 - 7 * q^23 + 21 * q^25 - 18 * q^27 - 4 * q^29 + 2 * q^31 - 4 * q^37 - 16 * q^39 - 4 * q^41 - 20 * q^43 - 2 * q^45 + 7 * q^47 + 14 * q^51 - 2 * q^53 - 2 * q^55 - 18 * q^59 - 19 * q^61 + 14 * q^65 - 20 * q^67 + 18 * q^69 - 14 * q^71 + 7 * q^73 + 32 * q^75 - 10 * q^79 + 11 * q^81 - 21 * q^83 - 30 * q^85 - 36 * q^87 - 2 * q^89 - 20 * q^93 - 2 * q^95 + 22 * q^97 - 7 * 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
$3$	−3.14743	−1.81717	−0.908585	−	0.417700i	$-0.862836\pi$
$3$	−3.14743	−1.81717	−0.908585	+	0.417700i	$0.862836\pi$
$4$	0	0
$5$	2.38854	1.06819	0.534095	−	0.845425i	$-0.320653\pi$
$5$	2.38854	1.06819	0.534095	+	0.845425i	$0.320653\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	6.90632	2.30211
$10$	0	0
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	0	0
$13$	3.14743	0.872940	0.436470	−	0.899719i	$-0.356228\pi$
$13$	3.14743	0.872940	0.436470	+	0.899719i	$0.356228\pi$
$14$	0	0
$15$	−7.51777	−1.94108
$16$	0	0
$17$	0.611457	0.148300	0.0741500	−	0.997247i	$-0.476376\pi$
$17$	0.611457	0.148300	0.0741500	+	0.997247i	$0.476376\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−6.90632	−1.44007	−0.720033	−	0.693939i	$-0.755874\pi$
$23$	−6.90632	−1.44007	−0.720033	+	0.693939i	$0.755874\pi$
$24$	0	0
$25$	0.705140	0.141028
$26$	0	0
$27$	−12.2949	−2.36615
$28$	0	0
$29$	7.81263	1.45077	0.725385	−	0.688344i	$-0.241662\pi$
$29$	7.81263	1.45077	0.725385	+	0.688344i	$0.241662\pi$
$30$	0	0
$31$	6.66520	1.19711	0.598553	−	0.801083i	$-0.295743\pi$
$31$	6.66520	1.19711	0.598553	+	0.801083i	$0.295743\pi$
$32$	0	0
$33$	3.14743	0.547897
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−4.77709	−0.785348	−0.392674	−	0.919678i	$-0.628450\pi$
$37$	−4.77709	−0.785348	−0.392674	+	0.919678i	$0.628450\pi$
$38$	0	0
$39$	−9.90632	−1.58628
$40$	0	0
$41$	−4.77709	−0.746056	−0.373028	−	0.927820i	$-0.621680\pi$
$41$	−4.77709	−0.746056	−0.373028	+	0.927820i	$0.621680\pi$
$42$	0	0
$43$	−2.09368	−0.319284	−0.159642	−	0.987175i	$-0.551034\pi$
$43$	−2.09368	−0.319284	−0.159642	+	0.987175i	$0.551034\pi$
$44$	0	0
$45$	16.4960	2.45908
$46$	0	0
$47$	6.90632	1.00739	0.503695	−	0.863882i	$-0.331974\pi$
$47$	6.90632	1.00739	0.503695	+	0.863882i	$0.331974\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−1.92452	−0.269486
$52$	0	0
$53$	−12.9601	−1.78020	−0.890101	−	0.455764i	$-0.849366\pi$
$53$	−12.9601	−1.78020	−0.890101	+	0.455764i	$0.849366\pi$
$54$	0	0
$55$	−2.38854	−0.322071
$56$	0	0
$57$	3.14743	0.416887
$58$	0	0
$59$	−2.85257	−0.371373	−0.185686	−	0.982609i	$-0.559451\pi$
$59$	−2.85257	−0.371373	−0.185686	+	0.982609i	$0.559451\pi$
$60$	0	0
$61$	−10.9063	−1.39641	−0.698205	−	0.715897i	$-0.746018\pi$
$61$	−10.9063	−1.39641	−0.698205	+	0.715897i	$0.746018\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	7.51777	0.932465
$66$	0	0
$67$	−9.51777	−1.16278	−0.581391	−	0.813625i	$-0.697491\pi$
$67$	−9.51777	−1.16278	−0.581391	+	0.813625i	$0.697491\pi$
$68$	0	0
$69$	21.7372	2.61685
$70$	0	0
$71$	−4.37034	−0.518664	−0.259332	−	0.965788i	$-0.583502\pi$
$71$	−4.37034	−0.518664	−0.259332	+	0.965788i	$0.583502\pi$
$72$	0	0
$73$	−5.68340	−0.665192	−0.332596	−	0.943069i	$-0.607925\pi$
$73$	−5.68340	−0.665192	−0.332596	+	0.943069i	$0.607925\pi$
$74$	0	0
$75$	−2.21938	−0.256272
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−12.1830	−1.37069	−0.685346	−	0.728218i	$-0.740349\pi$
$79$	−12.1830	−1.37069	−0.685346	+	0.728218i	$0.740349\pi$
$80$	0	0
$81$	17.9783	1.99758
$82$	0	0
$83$	−12.1656	−1.33535	−0.667676	−	0.744452i	$-0.732711\pi$
$83$	−12.1656	−1.33535	−0.667676	+	0.744452i	$0.732711\pi$
$84$	0	0
$85$	1.46049	0.158412
$86$	0	0
$87$	−24.5897	−2.63629
$88$	0	0
$89$	2.77709	0.294371	0.147185	−	0.989109i	$-0.452979\pi$
$89$	2.77709	0.294371	0.147185	+	0.989109i	$0.452979\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−20.9783	−2.17534
$94$	0	0
$95$	−2.38854	−0.245059
$96$	0	0
$97$	7.62966	0.774674	0.387337	−	0.921938i	$-0.373395\pi$
$97$	7.62966	0.774674	0.387337	+	0.921938i	$0.373395\pi$
$98$	0	0
$99$	−6.90632	−0.694111
$100$	0	0
$101$	11.1075	1.10524	0.552619	−	0.833434i	$-0.313629\pi$
$101$	11.1075	1.10524	0.552619	+	0.833434i	$0.313629\pi$
$102$	0	0
$103$	13.4423	1.32451	0.662254	−	0.749279i	$-0.269600\pi$
$103$	13.4423	1.32451	0.662254	+	0.749279i	$0.269600\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−5.51777	−0.533423	−0.266712	−	0.963776i	$-0.585937\pi$
$107$	−5.51777	−0.533423	−0.266712	+	0.963776i	$0.585937\pi$
$108$	0	0
$109$	16.8846	1.61725	0.808625	−	0.588325i	$-0.200212\pi$
$109$	16.8846	1.61725	0.808625	+	0.588325i	$0.200212\pi$
$110$	0	0
$111$	15.0355	1.42711
$112$	0	0
$113$	1.33480	0.125567	0.0627835	−	0.998027i	$-0.480002\pi$
$113$	1.33480	0.125567	0.0627835	+	0.998027i	$0.480002\pi$
$114$	0	0
$115$	−16.4960	−1.53826
$116$	0	0
$117$	21.7372	2.00960
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	15.0355	1.35571
$124$	0	0
$125$	−10.2585	−0.917545
$126$	0	0
$127$	7.07195	0.627534	0.313767	−	0.949500i	$-0.398409\pi$
$127$	7.07195	0.627534	0.313767	+	0.949500i	$0.398409\pi$
$128$	0	0
$129$	6.58972	0.580193
$130$	0	0
$131$	19.5897	1.71156	0.855781	−	0.517338i	$-0.173077\pi$
$131$	19.5897	1.71156	0.855781	+	0.517338i	$0.173077\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−29.3668	−2.52749
$136$	0	0
$137$	−19.7909	−1.69085	−0.845425	−	0.534094i	$-0.820653\pi$
$137$	−19.7909	−1.69085	−0.845425	+	0.534094i	$0.820653\pi$
$138$	0	0
$139$	−2.31660	−0.196491	−0.0982456	−	0.995162i	$-0.531323\pi$
$139$	−2.31660	−0.196491	−0.0982456	+	0.995162i	$0.531323\pi$
$140$	0	0
$141$	−21.7372	−1.83060
$142$	0	0
$143$	−3.14743	−0.263201
$144$	0	0
$145$	18.6608	1.54970
$146$	0	0
$147$	0	0
$148$	0	0
$149$	13.4605	1.10273	0.551363	−	0.834265i	$-0.314108\pi$
$149$	13.4605	1.10273	0.551363	+	0.834265i	$0.314108\pi$
$150$	0	0
$151$	−7.62966	−0.620893	−0.310446	−	0.950591i	$-0.600478\pi$
$151$	−7.62966	−0.620893	−0.310446	+	0.950591i	$0.600478\pi$
$152$	0	0
$153$	4.22291	0.341402
$154$	0	0
$155$	15.9201	1.27874
$156$	0	0
$157$	4.64786	0.370939	0.185470	−	0.982650i	$-0.440619\pi$
$157$	4.64786	0.370939	0.185470	+	0.982650i	$0.440619\pi$
$158$	0	0
$159$	40.7909	3.23493
$160$	0	0
$161$	0	0
$162$	0	0
$163$	4.71895	0.369617	0.184808	−	0.982775i	$-0.440834\pi$
$163$	4.71895	0.369617	0.184808	+	0.982775i	$0.440834\pi$
$164$	0	0
$165$	7.51777	0.585258
$166$	0	0
$167$	13.4423	1.04020	0.520098	−	0.854107i	$-0.325895\pi$
$167$	13.4423	1.04020	0.520098	+	0.854107i	$0.325895\pi$
$168$	0	0
$169$	−3.09368	−0.237976
$170$	0	0
$171$	−6.90632	−0.528139
$172$	0	0
$173$	−14.9601	−1.13739	−0.568696	−	0.822548i	$-0.692552\pi$
$173$	−14.9601	−1.13739	−0.568696	+	0.822548i	$0.692552\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	8.97826	0.674848
$178$	0	0
$179$	8.18298	0.611624	0.305812	−	0.952092i	$-0.401072\pi$
$179$	8.18298	0.611624	0.305812	+	0.952092i	$0.401072\pi$
$180$	0	0
$181$	22.2949	1.65716	0.828582	−	0.559868i	$-0.189148\pi$
$181$	22.2949	1.65716	0.828582	+	0.559868i	$0.189148\pi$
$182$	0	0
$183$	34.3269	2.53752
$184$	0	0
$185$	−11.4103	−0.838900
$186$	0	0
$187$	−0.611457	−0.0447141
$188$	0	0
$189$	0	0
$190$	0	0
$191$	5.90632	0.427366	0.213683	−	0.976903i	$-0.431454\pi$
$191$	5.90632	0.427366	0.213683	+	0.976903i	$0.431454\pi$
$192$	0	0
$193$	−1.59326	−0.114685	−0.0573426	−	0.998355i	$-0.518263\pi$
$193$	−1.59326	−0.114685	−0.0573426	+	0.998355i	$0.518263\pi$
$194$	0	0
$195$	−23.6617	−1.69445
$196$	0	0
$197$	−11.3313	−0.807319	−0.403659	−	0.914909i	$-0.632262\pi$
$197$	−11.3313	−0.807319	−0.403659	+	0.914909i	$0.632262\pi$
$198$	0	0
$199$	−14.4960	−1.02760	−0.513798	−	0.857911i	$-0.671762\pi$
$199$	−14.4960	−1.02760	−0.513798	+	0.857911i	$0.671762\pi$
$200$	0	0
$201$	29.9565	2.11297
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−11.4103	−0.796928
$206$	0	0
$207$	−47.6972	−3.31519
$208$	0	0
$209$	1.00000	0.0691714
$210$	0	0
$211$	−5.73715	−0.394962	−0.197481	−	0.980307i	$-0.563276\pi$
$211$	−5.73715	−0.394962	−0.197481	+	0.980307i	$0.563276\pi$
$212$	0	0
$213$	13.7554	0.942501
$214$	0	0
$215$	−5.00085	−0.341055
$216$	0	0
$217$	0	0
$218$	0	0
$219$	17.8881	1.20877
$220$	0	0
$221$	1.92452	0.129457
$222$	0	0
$223$	−16.4024	−1.09838	−0.549191	−	0.835697i	$-0.685064\pi$
$223$	−16.4024	−1.09838	−0.549191	+	0.835697i	$0.685064\pi$
$224$	0	0
$225$	4.86992	0.324661
$226$	0	0
$227$	15.0355	0.997944	0.498972	−	0.866618i	$-0.333711\pi$
$227$	15.0355	0.997944	0.498972	+	0.866618i	$0.333711\pi$
$228$	0	0
$229$	−27.7545	−1.83407	−0.917034	−	0.398808i	$-0.869424\pi$
$229$	−27.7545	−1.83407	−0.917034	+	0.398808i	$0.869424\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−16.6253	−1.08916	−0.544579	−	0.838710i	$-0.683310\pi$
$233$	−16.6253	−1.08916	−0.544579	+	0.838710i	$0.683310\pi$
$234$	0	0
$235$	16.4960	1.07608
$236$	0	0
$237$	38.3451	2.49078
$238$	0	0
$239$	15.8846	1.02749	0.513744	−	0.857943i	$-0.328258\pi$
$239$	15.8846	1.02749	0.513744	+	0.857943i	$0.328258\pi$
$240$	0	0
$241$	−7.07195	−0.455544	−0.227772	−	0.973714i	$-0.573144\pi$
$241$	−7.07195	−0.455544	−0.227772	+	0.973714i	$0.573144\pi$
$242$	0	0
$243$	−19.7008	−1.26380
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−3.14743	−0.200266
$248$	0	0
$249$	38.2905	2.42656
$250$	0	0
$251$	−24.0138	−1.51574	−0.757869	−	0.652407i	$-0.773759\pi$
$251$	−24.0138	−1.51574	−0.757869	+	0.652407i	$0.773759\pi$
$252$	0	0
$253$	6.90632	0.434196
$254$	0	0
$255$	−4.59679	−0.287862
$256$	0	0
$257$	24.6652	1.53857	0.769287	−	0.638904i	$-0.220612\pi$
$257$	24.6652	1.53857	0.769287	+	0.638904i	$0.220612\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	53.9565	3.33983
$262$	0	0
$263$	27.9419	1.72297	0.861485	−	0.507784i	$-0.169535\pi$
$263$	27.9419	1.72297	0.861485	+	0.507784i	$0.169535\pi$
$264$	0	0
$265$	−30.9557	−1.90159
$266$	0	0
$267$	−8.74069	−0.534921
$268$	0	0
$269$	−13.8126	−0.842171	−0.421086	−	0.907021i	$-0.638351\pi$
$269$	−13.8126	−0.842171	−0.421086	+	0.907021i	$0.638351\pi$
$270$	0	0
$271$	−19.5533	−1.18778	−0.593890	−	0.804546i	$-0.702409\pi$
$271$	−19.5533	−1.18778	−0.593890	+	0.804546i	$0.702409\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−0.705140	−0.0425215
$276$	0	0
$277$	−25.8846	−1.55525	−0.777627	−	0.628726i	$-0.783577\pi$
$277$	−25.8846	−1.55525	−0.777627	+	0.628726i	$0.783577\pi$
$278$	0	0
$279$	46.0320	2.75586
$280$	0	0
$281$	21.1474	1.26155	0.630775	−	0.775966i	$-0.282737\pi$
$281$	21.1474	1.26155	0.630775	+	0.775966i	$0.282737\pi$
$282$	0	0
$283$	19.9774	1.18753	0.593767	−	0.804637i	$-0.297640\pi$
$283$	19.9774	1.18753	0.593767	+	0.804637i	$0.297640\pi$
$284$	0	0
$285$	7.51777	0.445314
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−16.6261	−0.978007
$290$	0	0
$291$	−24.0138	−1.40771
$292$	0	0
$293$	−2.48223	−0.145013	−0.0725066	−	0.997368i	$-0.523100\pi$
$293$	−2.48223	−0.145013	−0.0725066	+	0.997368i	$0.523100\pi$
$294$	0	0
$295$	−6.81349	−0.396697
$296$	0	0
$297$	12.2949	0.713420
$298$	0	0
$299$	−21.7372	−1.25709
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−34.9601	−2.00840
$304$	0	0
$305$	−26.0502	−1.49163
$306$	0	0
$307$	−1.88812	−0.107761	−0.0538803	−	0.998547i	$-0.517159\pi$
$307$	−1.88812	−0.107761	−0.0538803	+	0.998547i	$0.517159\pi$
$308$	0	0
$309$	−42.3087	−2.40686
$310$	0	0
$311$	−4.79882	−0.272116	−0.136058	−	0.990701i	$-0.543443\pi$
$311$	−4.79882	−0.272116	−0.136058	+	0.990701i	$0.543443\pi$
$312$	0	0
$313$	−8.81263	−0.498120	−0.249060	−	0.968488i	$-0.580122\pi$
$313$	−8.81263	−0.498120	−0.249060	+	0.968488i	$0.580122\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−9.70075	−0.544848	−0.272424	−	0.962177i	$-0.587825\pi$
$317$	−9.70075	−0.544848	−0.272424	+	0.962177i	$0.587825\pi$
$318$	0	0
$319$	−7.81263	−0.437424
$320$	0	0
$321$	17.3668	0.969321
$322$	0	0
$323$	−0.611457	−0.0340224
$324$	0	0
$325$	2.21938	0.123109
$326$	0	0
$327$	−53.1430	−2.93882
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−25.2185	−1.38614	−0.693068	−	0.720872i	$-0.743741\pi$
$331$	−25.2185	−1.38614	−0.693068	+	0.720872i	$0.743741\pi$
$332$	0	0
$333$	−32.9921	−1.80795
$334$	0	0
$335$	−22.7336	−1.24207
$336$	0	0
$337$	20.8091	1.13354	0.566772	−	0.823875i	$-0.308192\pi$
$337$	20.8091	1.13354	0.566772	+	0.823875i	$0.308192\pi$
$338$	0	0
$339$	−4.20118	−0.228177
$340$	0	0
$341$	−6.66520	−0.360941
$342$	0	0
$343$	0	0
$344$	0	0
$345$	51.9201	2.79529
$346$	0	0
$347$	−18.4822	−0.992178	−0.496089	−	0.868272i	$-0.665231\pi$
$347$	−18.4822	−0.992178	−0.496089	+	0.868272i	$0.665231\pi$
$348$	0	0
$349$	32.2367	1.72559	0.862796	−	0.505552i	$-0.168711\pi$
$349$	32.2367	1.72559	0.862796	+	0.505552i	$0.168711\pi$
$350$	0	0
$351$	−38.6972	−2.06550
$352$	0	0
$353$	−28.4960	−1.51669	−0.758346	−	0.651853i	$-0.773992\pi$
$353$	−28.4960	−1.51669	−0.758346	+	0.651853i	$0.773992\pi$
$354$	0	0
$355$	−10.4388	−0.554032
$356$	0	0
$357$	0	0
$358$	0	0
$359$	17.4960	0.923406	0.461703	−	0.887035i	$-0.347239\pi$
$359$	17.4960	0.923406	0.461703	+	0.887035i	$0.347239\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	31.4743	1.65197
$364$	0	0
$365$	−13.5751	−0.710551
$366$	0	0
$367$	9.94272	0.519006	0.259503	−	0.965742i	$-0.416441\pi$
$367$	9.94272	0.519006	0.259503	+	0.965742i	$0.416441\pi$
$368$	0	0
$369$	−32.9921	−1.71750
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−29.7008	−1.53785	−0.768923	−	0.639341i	$-0.779207\pi$
$373$	−29.7008	−1.53785	−0.768923	+	0.639341i	$0.779207\pi$
$374$	0	0
$375$	32.2878	1.66733
$376$	0	0
$377$	24.5897	1.26644
$378$	0	0
$379$	−4.51863	−0.232106	−0.116053	−	0.993243i	$-0.537024\pi$
$379$	−4.51863	−0.232106	−0.116053	+	0.993243i	$0.537024\pi$
$380$	0	0
$381$	−22.2585	−1.14034
$382$	0	0
$383$	11.3348	0.579181	0.289591	−	0.957151i	$-0.406481\pi$
$383$	11.3348	0.579181	0.289591	+	0.957151i	$0.406481\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−14.4596	−0.735025
$388$	0	0
$389$	−17.3885	−0.881634	−0.440817	−	0.897597i	$-0.645311\pi$
$389$	−17.3885	−0.881634	−0.440817	+	0.897597i	$0.645311\pi$
$390$	0	0
$391$	−4.22291	−0.213562
$392$	0	0
$393$	−61.6573	−3.11020
$394$	0	0
$395$	−29.0996	−1.46416
$396$	0	0
$397$	−5.29486	−0.265741	−0.132871	−	0.991133i	$-0.542420\pi$
$397$	−5.29486	−0.265741	−0.132871	+	0.991133i	$0.542420\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	4.66520	0.232969	0.116485	−	0.993192i	$-0.462837\pi$
$401$	4.66520	0.232969	0.116485	+	0.993192i	$0.462837\pi$
$402$	0	0
$403$	20.9783	1.04500
$404$	0	0
$405$	42.9419	2.13380
$406$	0	0
$407$	4.77709	0.236791
$408$	0	0
$409$	−1.84903	−0.0914288	−0.0457144	−	0.998955i	$-0.514556\pi$
$409$	−1.84903	−0.0914288	−0.0457144	+	0.998955i	$0.514556\pi$
$410$	0	0
$411$	62.2905	3.07256
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−29.0581	−1.42641
$416$	0	0
$417$	7.29132	0.357058
$418$	0	0
$419$	−6.79882	−0.332144	−0.166072	−	0.986114i	$-0.553108\pi$
$419$	−6.79882	−0.332144	−0.166072	+	0.986114i	$0.553108\pi$
$420$	0	0
$421$	−39.7328	−1.93646	−0.968228	−	0.250068i	$-0.919547\pi$
$421$	−39.7328	−1.93646	−0.968228	+	0.250068i	$0.919547\pi$
$422$	0	0
$423$	47.6972	2.31912
$424$	0	0
$425$	0.431162	0.0209144
$426$	0	0
$427$	0	0
$428$	0	0
$429$	9.90632	0.478281
$430$	0	0
$431$	21.9201	1.05586	0.527928	−	0.849289i	$-0.322969\pi$
$431$	21.9201	1.05586	0.527928	+	0.849289i	$0.322969\pi$
$432$	0	0
$433$	−1.44668	−0.0695230	−0.0347615	−	0.999396i	$-0.511067\pi$
$433$	−1.44668	−0.0695230	−0.0347615	+	0.999396i	$0.511067\pi$
$434$	0	0
$435$	−58.7336	−2.81606
$436$	0	0
$437$	6.90632	0.330374
$438$	0	0
$439$	33.4743	1.59764	0.798821	−	0.601569i	$-0.205458\pi$
$439$	33.4743	1.59764	0.798821	+	0.601569i	$0.205458\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−36.8126	−1.74902	−0.874511	−	0.485007i	$-0.838817\pi$
$443$	−36.8126	−1.74902	−0.874511	+	0.485007i	$0.838817\pi$
$444$	0	0
$445$	6.63319	0.314443
$446$	0	0
$447$	−42.3660	−2.00384
$448$	0	0
$449$	5.18383	0.244640	0.122320	−	0.992491i	$-0.460967\pi$
$449$	5.18383	0.244640	0.122320	+	0.992491i	$0.460967\pi$
$450$	0	0
$451$	4.77709	0.224944
$452$	0	0
$453$	24.0138	1.12827
$454$	0	0
$455$	0	0
$456$	0	0
$457$	19.0502	0.891131	0.445566	−	0.895249i	$-0.353003\pi$
$457$	19.0502	0.891131	0.445566	+	0.895249i	$0.353003\pi$
$458$	0	0
$459$	−7.51777	−0.350900
$460$	0	0
$461$	−17.3087	−0.806145	−0.403073	−	0.915168i	$-0.632058\pi$
$461$	−17.3087	−0.806145	−0.403073	+	0.915168i	$0.632058\pi$
$462$	0	0
$463$	−27.4388	−1.27519	−0.637594	−	0.770373i	$-0.720070\pi$
$463$	−27.4388	−1.27519	−0.637594	+	0.770373i	$0.720070\pi$
$464$	0	0
$465$	−50.1075	−2.32368
$466$	0	0
$467$	−41.4734	−1.91916	−0.959581	−	0.281432i	$-0.909191\pi$
$467$	−41.4734	−1.91916	−0.959581	+	0.281432i	$0.909191\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−14.6288	−0.674060
$472$	0	0
$473$	2.09368	0.0962676
$474$	0	0
$475$	−0.705140	−0.0323540
$476$	0	0
$477$	−89.5063	−4.09821
$478$	0	0
$479$	−29.4596	−1.34605	−0.673023	−	0.739622i	$-0.735004\pi$
$479$	−29.4596	−1.34605	−0.673023	+	0.739622i	$0.735004\pi$
$480$	0	0
$481$	−15.0355	−0.685562
$482$	0	0
$483$	0	0
$484$	0	0
$485$	18.2238	0.827499
$486$	0	0
$487$	16.5142	0.748332	0.374166	−	0.927362i	$-0.377929\pi$
$487$	16.5142	0.748332	0.374166	+	0.927362i	$0.377929\pi$
$488$	0	0
$489$	−14.8526	−0.671656
$490$	0	0
$491$	−33.2949	−1.50258	−0.751288	−	0.659974i	$-0.770567\pi$
$491$	−33.2949	−1.50258	−0.751288	+	0.659974i	$0.770567\pi$
$492$	0	0
$493$	4.77709	0.215149
$494$	0	0
$495$	−16.4960	−0.741442
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−5.38940	−0.241262	−0.120631	−	0.992697i	$-0.538492\pi$
$499$	−5.38940	−0.241262	−0.120631	+	0.992697i	$0.538492\pi$
$500$	0	0
$501$	−42.3087	−1.89021
$502$	0	0
$503$	−7.09368	−0.316292	−0.158146	−	0.987416i	$-0.550552\pi$
$503$	−7.09368	−0.316292	−0.158146	+	0.987416i	$0.550552\pi$
$504$	0	0
$505$	26.5307	1.18060
$506$	0	0
$507$	9.73715	0.432442
$508$	0	0
$509$	5.77623	0.256027	0.128014	−	0.991772i	$-0.459140\pi$
$509$	5.77623	0.256027	0.128014	+	0.991772i	$0.459140\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	12.2949	0.542831
$514$	0	0
$515$	32.1075	1.41483
$516$	0	0
$517$	−6.90632	−0.303739
$518$	0	0
$519$	47.0858	2.06684
$520$	0	0
$521$	−1.33480	−0.0584785	−0.0292392	−	0.999572i	$-0.509308\pi$
$521$	−1.33480	−0.0584785	−0.0292392	+	0.999572i	$0.509308\pi$
$522$	0	0
$523$	−7.10835	−0.310826	−0.155413	−	0.987850i	$-0.549671\pi$
$523$	−7.10835	−0.310826	−0.155413	+	0.987850i	$0.549671\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.07548	0.177531
$528$	0	0
$529$	24.6972	1.07379
$530$	0	0
$531$	−19.7008	−0.854940
$532$	0	0
$533$	−15.0355	−0.651262
$534$	0	0
$535$	−13.1794	−0.569797
$536$	0	0
$537$	−25.7554	−1.11143
$538$	0	0
$539$	0	0
$540$	0	0
$541$	23.9063	1.02781	0.513906	−	0.857846i	$-0.328198\pi$
$541$	23.9063	1.02781	0.513906	+	0.857846i	$0.328198\pi$
$542$	0	0
$543$	−70.1715	−3.01135
$544$	0	0
$545$	40.3296	1.72753
$546$	0	0
$547$	7.25931	0.310386	0.155193	−	0.987884i	$-0.450400\pi$
$547$	7.25931	0.310386	0.155193	+	0.987884i	$0.450400\pi$
$548$	0	0
$549$	−75.3225	−3.21469
$550$	0	0
$551$	−7.81263	−0.332829
$552$	0	0
$553$	0	0
$554$	0	0
$555$	35.9131	1.52442
$556$	0	0
$557$	8.68340	0.367928	0.183964	−	0.982933i	$-0.441107\pi$
$557$	8.68340	0.367928	0.183964	+	0.982933i	$0.441107\pi$
$558$	0	0
$559$	−6.58972	−0.278715
$560$	0	0
$561$	1.92452	0.0812532
$562$	0	0
$563$	−19.3624	−0.816029	−0.408014	−	0.912976i	$-0.633779\pi$
$563$	−19.3624	−0.816029	−0.408014	+	0.912976i	$0.633779\pi$
$564$	0	0
$565$	3.18822	0.134129
$566$	0	0
$567$	0	0
$568$	0	0
$569$	36.3660	1.52454	0.762270	−	0.647259i	$-0.224085\pi$
$569$	36.3660	1.52454	0.762270	+	0.647259i	$0.224085\pi$
$570$	0	0
$571$	−24.4952	−1.02509	−0.512546	−	0.858660i	$-0.671298\pi$
$571$	−24.4952	−1.02509	−0.512546	+	0.858660i	$0.671298\pi$
$572$	0	0
$573$	−18.5897	−0.776597
$574$	0	0
$575$	−4.86992	−0.203090
$576$	0	0
$577$	25.6608	1.06827	0.534137	−	0.845398i	$-0.320637\pi$
$577$	25.6608	1.06827	0.534137	+	0.845398i	$0.320637\pi$
$578$	0	0
$579$	5.01466	0.208402
$580$	0	0
$581$	0	0
$582$	0	0
$583$	12.9601	0.536751
$584$	0	0
$585$	51.9201	2.14663
$586$	0	0
$587$	−8.07195	−0.333165	−0.166582	−	0.986028i	$-0.553273\pi$
$587$	−8.07195	−0.333165	−0.166582	+	0.986028i	$0.553273\pi$
$588$	0	0
$589$	−6.66520	−0.274635
$590$	0	0
$591$	35.6644	1.46704
$592$	0	0
$593$	−30.9063	−1.26917	−0.634585	−	0.772853i	$-0.718829\pi$
$593$	−30.9063	−1.26917	−0.634585	+	0.772853i	$0.718829\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	45.6253	1.86732
$598$	0	0
$599$	−19.8490	−0.811010	−0.405505	−	0.914093i	$-0.632904\pi$
$599$	−19.8490	−0.811010	−0.405505	+	0.914093i	$0.632904\pi$
$600$	0	0
$601$	−42.9210	−1.75078	−0.875392	−	0.483414i	$-0.839396\pi$
$601$	−42.9210	−1.75078	−0.875392	+	0.483414i	$0.839396\pi$
$602$	0	0
$603$	−65.7328	−2.67685
$604$	0	0
$605$	−23.8854	−0.971081
$606$	0	0
$607$	9.33480	0.378888	0.189444	−	0.981892i	$-0.439332\pi$
$607$	9.33480	0.378888	0.189444	+	0.981892i	$0.439332\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	21.7372	0.879391
$612$	0	0
$613$	13.7771	0.556451	0.278226	−	0.960516i	$-0.410254\pi$
$613$	13.7771	0.556451	0.278226	+	0.960516i	$0.410254\pi$
$614$	0	0
$615$	35.9131	1.44815
$616$	0	0
$617$	19.1794	0.772135	0.386068	−	0.922470i	$-0.373833\pi$
$617$	19.1794	0.772135	0.386068	+	0.922470i	$0.373833\pi$
$618$	0	0
$619$	−23.4249	−0.941528	−0.470764	−	0.882259i	$-0.656022\pi$
$619$	−23.4249	−0.941528	−0.470764	+	0.882259i	$0.656022\pi$
$620$	0	0
$621$	84.9122	3.40741
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−28.0285	−1.12114
$626$	0	0
$627$	−3.14743	−0.125696
$628$	0	0
$629$	−2.92098	−0.116467
$630$	0	0
$631$	7.61146	0.303007	0.151504	−	0.988457i	$-0.451588\pi$
$631$	7.61146	0.303007	0.151504	+	0.988457i	$0.451588\pi$
$632$	0	0
$633$	18.0573	0.717713
$634$	0	0
$635$	16.8917	0.670325
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−30.1830	−1.19402
$640$	0	0
$641$	4.29047	0.169463	0.0847317	−	0.996404i	$-0.472997\pi$
$641$	4.29047	0.169463	0.0847317	+	0.996404i	$0.472997\pi$
$642$	0	0
$643$	−10.7762	−0.424973	−0.212487	−	0.977164i	$-0.568156\pi$
$643$	−10.7762	−0.424973	−0.212487	+	0.977164i	$0.568156\pi$
$644$	0	0
$645$	15.7398	0.619755
$646$	0	0
$647$	20.5533	0.808034	0.404017	−	0.914751i	$-0.367614\pi$
$647$	20.5533	0.808034	0.404017	+	0.914751i	$0.367614\pi$
$648$	0	0
$649$	2.85257	0.111973
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−16.5680	−0.648355	−0.324178	−	0.945996i	$-0.605088\pi$
$653$	−16.5680	−0.648355	−0.324178	+	0.945996i	$0.605088\pi$
$654$	0	0
$655$	46.7909	1.82827
$656$	0	0
$657$	−39.2514	−1.53134
$658$	0	0
$659$	−28.7727	−1.12083	−0.560413	−	0.828214i	$-0.689357\pi$
$659$	−28.7727	−1.12083	−0.560413	+	0.828214i	$0.689357\pi$
$660$	0	0
$661$	16.5142	0.642329	0.321165	−	0.947023i	$-0.395926\pi$
$661$	16.5142	0.642329	0.321165	+	0.947023i	$0.395926\pi$
$662$	0	0
$663$	−6.05728	−0.235245
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−53.9565	−2.08921
$668$	0	0
$669$	51.6253	1.99595
$670$	0	0
$671$	10.9063	0.421034
$672$	0	0
$673$	−22.8118	−0.879330	−0.439665	−	0.898162i	$-0.644903\pi$
$673$	−22.8118	−0.879330	−0.439665	+	0.898162i	$0.644903\pi$
$674$	0	0
$675$	−8.66959	−0.333693
$676$	0	0
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−47.3233	−1.81343
$682$	0	0
$683$	−13.7008	−0.524245	−0.262122	−	0.965035i	$-0.584422\pi$
$683$	−13.7008	−0.524245	−0.262122	+	0.965035i	$0.584422\pi$
$684$	0	0
$685$	−47.2714	−1.80615
$686$	0	0
$687$	87.3553	3.33281
$688$	0	0
$689$	−40.7909	−1.55401
$690$	0	0
$691$	−27.1213	−1.03174	−0.515872	−	0.856666i	$-0.672532\pi$
$691$	−27.1213	−1.03174	−0.515872	+	0.856666i	$0.672532\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−5.53329	−0.209890
$696$	0	0
$697$	−2.92098	−0.110640
$698$	0	0
$699$	52.3269	1.97918
$700$	0	0
$701$	17.7181	0.669203	0.334602	−	0.942360i	$-0.391398\pi$
$701$	17.7181	0.669203	0.334602	+	0.942360i	$0.391398\pi$
$702$	0	0
$703$	4.77709	0.180171
$704$	0	0
$705$	−51.9201	−1.95543
$706$	0	0
$707$	0	0
$708$	0	0
$709$	21.6261	0.812186	0.406093	−	0.913832i	$-0.366891\pi$
$709$	21.6261	0.812186	0.406093	+	0.913832i	$0.366891\pi$
$710$	0	0
$711$	−84.1395	−3.15548
$712$	0	0
$713$	−46.0320	−1.72391
$714$	0	0
$715$	−7.51777	−0.281149
$716$	0	0
$717$	−49.9956	−1.86712
$718$	0	0
$719$	42.7830	1.59554	0.797768	−	0.602965i	$-0.206014\pi$
$719$	42.7830	1.59554	0.797768	+	0.602965i	$0.206014\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	22.2585	0.827801
$724$	0	0
$725$	5.50900	0.204599
$726$	0	0
$727$	37.5316	1.39197	0.695985	−	0.718057i	$-0.254968\pi$
$727$	37.5316	1.39197	0.695985	+	0.718057i	$0.254968\pi$
$728$	0	0
$729$	8.07195	0.298961
$730$	0	0
$731$	−1.28020	−0.0473498
$732$	0	0
$733$	17.2593	0.637487	0.318744	−	0.947841i	$-0.396739\pi$
$733$	17.2593	0.637487	0.318744	+	0.947841i	$0.396739\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	9.51777	0.350592
$738$	0	0
$739$	34.7692	1.27900	0.639502	−	0.768789i	$-0.279141\pi$
$739$	34.7692	1.27900	0.639502	+	0.768789i	$0.279141\pi$
$740$	0	0
$741$	9.90632	0.363918
$742$	0	0
$743$	13.5134	0.495758	0.247879	−	0.968791i	$-0.420266\pi$
$743$	13.5134	0.495758	0.247879	+	0.968791i	$0.420266\pi$
$744$	0	0
$745$	32.1510	1.17792
$746$	0	0
$747$	−84.0197	−3.07412
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−42.2194	−1.54061	−0.770303	−	0.637677i	$-0.779895\pi$
$751$	−42.2194	−1.54061	−0.770303	+	0.637677i	$0.779895\pi$
$752$	0	0
$753$	75.5818	2.75435
$754$	0	0
$755$	−18.2238	−0.663231
$756$	0	0
$757$	39.4024	1.43210	0.716051	−	0.698047i	$-0.245948\pi$
$757$	39.4024	1.43210	0.716051	+	0.698047i	$0.245948\pi$
$758$	0	0
$759$	−21.7372	−0.789009
$760$	0	0
$761$	−40.3589	−1.46301	−0.731504	−	0.681837i	$-0.761181\pi$
$761$	−40.3589	−1.46301	−0.731504	+	0.681837i	$0.761181\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	10.0866	0.364682
$766$	0	0
$767$	−8.97826	−0.324186
$768$	0	0
$769$	22.7181	0.819236	0.409618	−	0.912257i	$-0.365662\pi$
$769$	22.7181	0.819236	0.409618	+	0.912257i	$0.365662\pi$
$770$	0	0
$771$	−77.6320	−2.79585
$772$	0	0
$773$	2.74069	0.0985757	0.0492878	−	0.998785i	$-0.484305\pi$
$773$	2.74069	0.0985757	0.0492878	+	0.998785i	$0.484305\pi$
$774$	0	0
$775$	4.69990	0.168825
$776$	0	0
$777$	0	0
$778$	0	0
$779$	4.77709	0.171157
$780$	0	0
$781$	4.37034	0.156383
$782$	0	0
$783$	−96.0552	−3.43273
$784$	0	0
$785$	11.1016	0.396233
$786$	0	0
$787$	−15.7008	−0.559671	−0.279836	−	0.960048i	$-0.590280\pi$
$787$	−15.7008	−0.559671	−0.279836	+	0.960048i	$0.590280\pi$
$788$	0	0
$789$	−87.9451	−3.13093
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−34.3269	−1.21898
$794$	0	0
$795$	97.4308	3.45552
$796$	0	0
$797$	−34.4752	−1.22117	−0.610586	−	0.791950i	$-0.709066\pi$
$797$	−34.4752	−1.22117	−0.610586	+	0.791950i	$0.709066\pi$
$798$	0	0
$799$	4.22291	0.149396
$800$	0	0
$801$	19.1794	0.677672
$802$	0	0
$803$	5.68340	0.200563
$804$	0	0
$805$	0	0
$806$	0	0
$807$	43.4743	1.53037
$808$	0	0
$809$	39.6044	1.39242	0.696208	−	0.717840i	$-0.254869\pi$
$809$	39.6044	1.39242	0.696208	+	0.717840i	$0.254869\pi$
$810$	0	0
$811$	0.589721	0.0207079	0.0103540	−	0.999946i	$-0.496704\pi$
$811$	0.589721	0.0207079	0.0103540	+	0.999946i	$0.496704\pi$
$812$	0	0
$813$	61.5427	2.15840
$814$	0	0
$815$	11.2714	0.394821
$816$	0	0
$817$	2.09368	0.0732487
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1.36596	0.0476722	0.0238361	−	0.999716i	$-0.492412\pi$
$821$	1.36596	0.0476722	0.0238361	+	0.999716i	$0.492412\pi$
$822$	0	0
$823$	32.9565	1.14879	0.574396	−	0.818577i	$-0.305237\pi$
$823$	32.9565	1.14879	0.574396	+	0.818577i	$0.305237\pi$
$824$	0	0
$825$	2.21938	0.0772688
$826$	0	0
$827$	−14.0684	−0.489207	−0.244603	−	0.969623i	$-0.578658\pi$
$827$	−14.0684	−0.489207	−0.244603	+	0.969623i	$0.578658\pi$
$828$	0	0
$829$	47.7736	1.65924	0.829622	−	0.558325i	$-0.188556\pi$
$829$	47.7736	1.65924	0.829622	+	0.558325i	$0.188556\pi$
$830$	0	0
$831$	81.4699	2.82616
$832$	0	0
$833$	0	0
$834$	0	0
$835$	32.1075	1.11113
$836$	0	0
$837$	−81.9478	−2.83253
$838$	0	0
$839$	−16.1075	−0.556092	−0.278046	−	0.960568i	$-0.589687\pi$
$839$	−16.1075	−0.556092	−0.278046	+	0.960568i	$0.589687\pi$
$840$	0	0
$841$	32.0373	1.10473
$842$	0	0
$843$	−66.5601	−2.29245
$844$	0	0
$845$	−7.38940	−0.254203
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−62.8775	−2.15795
$850$	0	0
$851$	32.9921	1.13095
$852$	0	0
$853$	22.4605	0.769033	0.384516	−	0.923118i	$-0.374368\pi$
$853$	22.4605	0.769033	0.384516	+	0.923118i	$0.374368\pi$
$854$	0	0
$855$	−16.4960	−0.564153
$856$	0	0
$857$	−48.1395	−1.64441	−0.822207	−	0.569188i	$-0.807257\pi$
$857$	−48.1395	−1.64441	−0.822207	+	0.569188i	$0.807257\pi$
$858$	0	0
$859$	−29.3166	−1.00027	−0.500135	−	0.865948i	$-0.666716\pi$
$859$	−29.3166	−1.00027	−0.500135	+	0.865948i	$0.666716\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−19.7692	−0.672950	−0.336475	−	0.941692i	$-0.609235\pi$
$863$	−19.7692	−0.672950	−0.336475	+	0.941692i	$0.609235\pi$
$864$	0	0
$865$	−35.7328	−1.21495
$866$	0	0
$867$	52.3296	1.77720
$868$	0	0
$869$	12.1830	0.413279
$870$	0	0
$871$	−29.9565	−1.01504
$872$	0	0
$873$	52.6928	1.78338
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−39.4032	−1.33055	−0.665276	−	0.746598i	$-0.731686\pi$
$877$	−39.4032	−1.33055	−0.665276	+	0.746598i	$0.731686\pi$
$878$	0	0
$879$	7.81263	0.263514
$880$	0	0
$881$	2.92891	0.0986773	0.0493387	−	0.998782i	$-0.484289\pi$
$881$	2.92891	0.0986773	0.0493387	+	0.998782i	$0.484289\pi$
$882$	0	0
$883$	−6.79882	−0.228799	−0.114399	−	0.993435i	$-0.536494\pi$
$883$	−6.79882	−0.228799	−0.114399	+	0.993435i	$0.536494\pi$
$884$	0	0
$885$	21.4450	0.720865
$886$	0	0
$887$	16.3313	0.548350	0.274175	−	0.961680i	$-0.411595\pi$
$887$	16.3313	0.548350	0.274175	+	0.961680i	$0.411595\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−17.9783	−0.602295
$892$	0	0
$893$	−6.90632	−0.231111
$894$	0	0
$895$	19.5454	0.653331
$896$	0	0
$897$	68.4162	2.28435
$898$	0	0
$899$	52.0728	1.73673
$900$	0	0
$901$	−7.92452	−0.264004
$902$	0	0
$903$	0	0
$904$	0	0
$905$	53.2522	1.77016
$906$	0	0
$907$	−55.9885	−1.85907	−0.929534	−	0.368735i	$-0.879791\pi$
$907$	−55.9885	−1.85907	−0.929534	+	0.368735i	$0.879791\pi$
$908$	0	0
$909$	76.7119	2.54437
$910$	0	0
$911$	38.5489	1.27718	0.638592	−	0.769546i	$-0.279517\pi$
$911$	38.5489	1.27718	0.638592	+	0.769546i	$0.279517\pi$
$912$	0	0
$913$	12.1656	0.402624
$914$	0	0
$915$	81.9912	2.71055
$916$	0	0
$917$	0	0
$918$	0	0
$919$	4.77623	0.157553	0.0787767	−	0.996892i	$-0.474899\pi$
$919$	4.77623	0.157553	0.0787767	+	0.996892i	$0.474899\pi$
$920$	0	0
$921$	5.94272	0.195819
$922$	0	0
$923$	−13.7554	−0.452763
$924$	0	0
$925$	−3.36851	−0.110756
$926$	0	0
$927$	92.8367	3.04916
$928$	0	0
$929$	−30.4952	−1.00051	−0.500257	−	0.865877i	$-0.666761\pi$
$929$	−30.4952	−1.00051	−0.500257	+	0.865877i	$0.666761\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	15.1040	0.494481
$934$	0	0
$935$	−1.46049	−0.0477632
$936$	0	0
$937$	−46.3798	−1.51516	−0.757580	−	0.652742i	$-0.773619\pi$
$937$	−46.3798	−1.51516	−0.757580	+	0.652742i	$0.773619\pi$
$938$	0	0
$939$	27.7372	0.905168
$940$	0	0
$941$	−30.5577	−0.996153	−0.498076	−	0.867133i	$-0.665960\pi$
$941$	−30.5577	−0.996153	−0.498076	+	0.867133i	$0.665960\pi$
$942$	0	0
$943$	32.9921	1.07437
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−40.4249	−1.31363	−0.656817	−	0.754050i	$-0.728098\pi$
$947$	−40.4249	−1.31363	−0.656817	+	0.754050i	$0.728098\pi$
$948$	0	0
$949$	−17.8881	−0.580673
$950$	0	0
$951$	30.5324	0.990082
$952$	0	0
$953$	18.7771	0.608250	0.304125	−	0.952632i	$-0.401636\pi$
$953$	18.7771	0.608250	0.304125	+	0.952632i	$0.401636\pi$
$954$	0	0
$955$	14.1075	0.456508
$956$	0	0
$957$	24.5897	0.794873
$958$	0	0
$959$	0	0
$960$	0	0
$961$	13.4249	0.433063
$962$	0	0
$963$	−38.1075	−1.22800
$964$	0	0
$965$	−3.80556	−0.122505
$966$	0	0
$967$	−27.3095	−0.878215	−0.439108	−	0.898435i	$-0.644705\pi$
$967$	−27.3095	−0.878215	−0.439108	+	0.898435i	$0.644705\pi$
$968$	0	0
$969$	1.92452	0.0618244
$970$	0	0
$971$	−7.07195	−0.226950	−0.113475	−	0.993541i	$-0.536198\pi$
$971$	−7.07195	−0.226950	−0.113475	+	0.993541i	$0.536198\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−6.98534	−0.223710
$976$	0	0
$977$	35.1838	1.12563	0.562815	−	0.826583i	$-0.309718\pi$
$977$	35.1838	1.12563	0.562815	+	0.826583i	$0.309718\pi$
$978$	0	0
$979$	−2.77709	−0.0887561
$980$	0	0
$981$	116.610	3.72308
$982$	0	0
$983$	16.7771	0.535106	0.267553	−	0.963543i	$-0.413785\pi$
$983$	16.7771	0.535106	0.267553	+	0.963543i	$0.413785\pi$
$984$	0	0
$985$	−27.0652	−0.862369
$986$	0	0
$987$	0	0
$988$	0	0
$989$	14.4596	0.459790
$990$	0	0
$991$	49.9885	1.58794	0.793969	−	0.607958i	$-0.208011\pi$
$991$	49.9885	1.58794	0.793969	+	0.607958i	$0.208011\pi$
$992$	0	0
$993$	79.3735	2.51884
$994$	0	0
$995$	−34.6244	−1.09767
$996$	0	0
$997$	−35.6261	−1.12829	−0.564145	−	0.825676i	$-0.690794\pi$
$997$	−35.6261	−1.12829	−0.564145	+	0.825676i	$0.690794\pi$
$998$	0	0
$999$	58.7336	1.85825

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7448.2.a.bh.1.1		3
7.2	even	3		1064.2.q.m.305.3	✓	6
7.4	even	3		1064.2.q.m.457.3	yes	6
7.6	odd	2		7448.2.a.bg.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1064.2.q.m.305.3	✓	6	7.2	even	3
1064.2.q.m.457.3	yes	6	7.4	even	3
7448.2.a.bg.1.3		3	7.6	odd	2
7448.2.a.bh.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:	\( N \)	\(=\)	\( 7448 = 2^{3} \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7448.a (trivial)

\(f(q)\)	\(=\)	\(q-3.14743 q^{3} +2.38854 q^{5} +6.90632 q^{9} +O(q^{10})\)	\(q-3.14743 q^{3} +2.38854 q^{5} +6.90632 q^{9} -1.00000 q^{11} +3.14743 q^{13} -7.51777 q^{15} +0.611457 q^{17} -1.00000 q^{19} -6.90632 q^{23} +0.705140 q^{25} -12.2949 q^{27} +7.81263 q^{29} +6.66520 q^{31} +3.14743 q^{33} -4.77709 q^{37} -9.90632 q^{39} -4.77709 q^{41} -2.09368 q^{43} +16.4960 q^{45} +6.90632 q^{47} -1.92452 q^{51} -12.9601 q^{53} -2.38854 q^{55} +3.14743 q^{57} -2.85257 q^{59} -10.9063 q^{61} +7.51777 q^{65} -9.51777 q^{67} +21.7372 q^{69} -4.37034 q^{71} -5.68340 q^{73} -2.21938 q^{75} -12.1830 q^{79} +17.9783 q^{81} -12.1656 q^{83} +1.46049 q^{85} -24.5897 q^{87} +2.77709 q^{89} -20.9783 q^{93} -2.38854 q^{95} +7.62966 q^{97} -6.90632 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{5} + 7 q^{9}+O(q^{10}) \) 3 * q + 2 * q^5 + 7 * q^9	\( 3 q + 2 q^{5} + 7 q^{9} - 3 q^{11} - 14 q^{15} + 7 q^{17} - 3 q^{19} - 7 q^{23} + 21 q^{25} - 18 q^{27} - 4 q^{29} + 2 q^{31} - 4 q^{37} - 16 q^{39} - 4 q^{41} - 20 q^{43} - 2 q^{45} + 7 q^{47} + 14 q^{51} - 2 q^{53} - 2 q^{55} - 18 q^{59} - 19 q^{61} + 14 q^{65} - 20 q^{67} + 18 q^{69} - 14 q^{71} + 7 q^{73} + 32 q^{75} - 10 q^{79} + 11 q^{81} - 21 q^{83} - 30 q^{85} - 36 q^{87} - 2 q^{89} - 20 q^{93} - 2 q^{95} + 22 q^{97} - 7 q^{99}+O(q^{100}) \) 3 * q + 2 * q^5 + 7 * q^9 - 3 * q^11 - 14 * q^15 + 7 * q^17 - 3 * q^19 - 7 * q^23 + 21 * q^25 - 18 * q^27 - 4 * q^29 + 2 * q^31 - 4 * q^37 - 16 * q^39 - 4 * q^41 - 20 * q^43 - 2 * q^45 + 7 * q^47 + 14 * q^51 - 2 * q^53 - 2 * q^55 - 18 * q^59 - 19 * q^61 + 14 * q^65 - 20 * q^67 + 18 * q^69 - 14 * q^71 + 7 * q^73 + 32 * q^75 - 10 * q^79 + 11 * q^81 - 21 * q^83 - 30 * q^85 - 36 * q^87 - 2 * q^89 - 20 * q^93 - 2 * q^95 + 22 * q^97 - 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more