LMFDB - Embedded newform 7600.2.a.cj.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7600,2,Mod(1,7600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7600, 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("7600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7600 = 2^{4} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7600.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$60.6863055362$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.56310016.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 9x^{4} + 14x^{2} - 4 $ x^6 - 9x^4 + 14x^2 - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 380)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.66648$ of defining polynomial
Character	$\chi$	$=$	7600.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.750054 q^{3} -0.872810 q^{7} -2.43742 q^{9} +O(q^{10})$	$q-0.750054 q^{3} -0.872810 q^{7} -2.43742 q^{9} +1.11009 q^{11} +4.58290 q^{13} -2.96003 q^{17} -1.00000 q^{19} +0.654655 q^{21} -3.83284 q^{23} +4.07836 q^{27} +7.56553 q^{29} -9.56553 q^{31} -0.832629 q^{33} +0.750054 q^{37} -3.43742 q^{39} +8.87484 q^{41} +11.5387 q^{43} -8.53850 q^{47} -6.23820 q^{49} +2.22018 q^{51} +8.17023 q^{53} +0.750054 q^{57} -4.65465 q^{59} +0.889908 q^{61} +2.12740 q^{63} -2.59176 q^{67} +2.87484 q^{69} -7.34535 q^{71} -11.8803 q^{73} -0.968900 q^{77} -4.00000 q^{79} +4.25327 q^{81} +3.83284 q^{83} -5.67455 q^{87} +1.34535 q^{89} -4.00000 q^{91} +7.17466 q^{93} +8.17023 q^{97} -2.70576 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 10 q^{9}+O(q^{10}) $ 6 * q + 10 * q^9	$ 6 q + 10 q^{9} - 18 q^{11} - 6 q^{19} + 4 q^{21} - 4 q^{29} - 8 q^{31} + 4 q^{39} + 4 q^{41} + 12 q^{49} - 36 q^{51} - 28 q^{59} + 30 q^{61} - 32 q^{69} - 44 q^{71} - 24 q^{79} + 50 q^{81} + 8 q^{89} - 24 q^{91} - 90 q^{99}+O(q^{100}) $ 6 * q + 10 * q^9 - 18 * q^11 - 6 * q^19 + 4 * q^21 - 4 * q^29 - 8 * q^31 + 4 * q^39 + 4 * q^41 + 12 * q^49 - 36 * q^51 - 28 * q^59 + 30 * q^61 - 32 * q^69 - 44 * q^71 - 24 * q^79 + 50 * q^81 + 8 * q^89 - 24 * q^91 - 90 * 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
$3$	−0.750054	−0.433044	−0.216522	−	0.976278i	$-0.569471\pi$
$3$	−0.750054	−0.433044	−0.216522	+	0.976278i	$0.569471\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.872810	−0.329891	−0.164946	−	0.986303i	$-0.552745\pi$
$7$	−0.872810	−0.329891	−0.164946	+	0.986303i	$0.552745\pi$
$8$	0	0
$9$	−2.43742	−0.812473
$10$	0	0
$11$	1.11009	0.334705	0.167353	−	0.985897i	$-0.446478\pi$
$11$	1.11009	0.334705	0.167353	+	0.985897i	$0.446478\pi$
$12$	0	0
$13$	4.58290	1.27107	0.635534	−	0.772073i	$-0.280780\pi$
$13$	4.58290	1.27107	0.635534	+	0.772073i	$0.280780\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.96003	−0.717914	−0.358957	−	0.933354i	$-0.616867\pi$
$17$	−2.96003	−0.717914	−0.358957	+	0.933354i	$0.616867\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0.654655	0.142857
$22$	0	0
$23$	−3.83284	−0.799203	−0.399602	−	0.916689i	$-0.630852\pi$
$23$	−3.83284	−0.799203	−0.399602	+	0.916689i	$0.630852\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	4.07836	0.784880
$28$	0	0
$29$	7.56553	1.40488	0.702442	−	0.711741i	$-0.252093\pi$
$29$	7.56553	1.40488	0.702442	+	0.711741i	$0.252093\pi$
$30$	0	0
$31$	−9.56553	−1.71802	−0.859010	−	0.511959i	$-0.828920\pi$
$31$	−9.56553	−1.71802	−0.859010	+	0.511959i	$0.828920\pi$
$32$	0	0
$33$	−0.832629	−0.144942
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0.750054	0.123308	0.0616540	−	0.998098i	$-0.480362\pi$
$37$	0.750054	0.123308	0.0616540	+	0.998098i	$0.480362\pi$
$38$	0	0
$39$	−3.43742	−0.550428
$40$	0	0
$41$	8.87484	1.38602	0.693008	−	0.720929i	$-0.256285\pi$
$41$	8.87484	1.38602	0.693008	+	0.720929i	$0.256285\pi$
$42$	0	0
$43$	11.5387	1.75964	0.879819	−	0.475310i	$-0.157664\pi$
$43$	11.5387	1.75964	0.879819	+	0.475310i	$0.157664\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−8.53850	−1.24547	−0.622734	−	0.782434i	$-0.713978\pi$
$47$	−8.53850	−1.24547	−0.622734	+	0.782434i	$0.713978\pi$
$48$	0	0
$49$	−6.23820	−0.891172
$50$	0	0
$51$	2.22018	0.310888
$52$	0	0
$53$	8.17023	1.12227	0.561134	−	0.827725i	$-0.310365\pi$
$53$	8.17023	1.12227	0.561134	+	0.827725i	$0.310365\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0.750054	0.0993470
$58$	0	0
$59$	−4.65465	−0.605984	−0.302992	−	0.952993i	$-0.597986\pi$
$59$	−4.65465	−0.605984	−0.302992	+	0.952993i	$0.597986\pi$
$60$	0	0
$61$	0.889908	0.113941	0.0569705	−	0.998376i	$-0.481856\pi$
$61$	0.889908	0.113941	0.0569705	+	0.998376i	$0.481856\pi$
$62$	0	0
$63$	2.12740	0.268028
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−2.59176	−0.316634	−0.158317	−	0.987388i	$-0.550607\pi$
$67$	−2.59176	−0.316634	−0.158317	+	0.987388i	$0.550607\pi$
$68$	0	0
$69$	2.87484	0.346090
$70$	0	0
$71$	−7.34535	−0.871732	−0.435866	−	0.900012i	$-0.643558\pi$
$71$	−7.34535	−0.871732	−0.435866	+	0.900012i	$0.643558\pi$
$72$	0	0
$73$	−11.8803	−1.39049	−0.695243	−	0.718775i	$-0.744703\pi$
$73$	−11.8803	−1.39049	−0.695243	+	0.718775i	$0.744703\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.968900	−0.110416
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	4.25327	0.472586
$82$	0	0
$83$	3.83284	0.420709	0.210355	−	0.977625i	$-0.432538\pi$
$83$	3.83284	0.420709	0.210355	+	0.977625i	$0.432538\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−5.67455	−0.608376
$88$	0	0
$89$	1.34535	0.142606	0.0713032	−	0.997455i	$-0.477284\pi$
$89$	1.34535	0.142606	0.0713032	+	0.997455i	$0.477284\pi$
$90$	0	0
$91$	−4.00000	−0.419314
$92$	0	0
$93$	7.17466	0.743978
$94$	0	0
$95$	0	0
$96$	0	0
$97$	8.17023	0.829561	0.414780	−	0.909922i	$-0.363858\pi$
$97$	8.17023	0.829561	0.414780	+	0.909922i	$0.363858\pi$
$98$	0	0
$99$	−2.70576	−0.271939
$100$	0	0
$101$	5.43742	0.541043	0.270522	−	0.962714i	$-0.412804\pi$
$101$	5.43742	0.541043	0.270522	+	0.962714i	$0.412804\pi$
$102$	0	0
$103$	13.5032	1.33051	0.665254	−	0.746617i	$-0.268323\pi$
$103$	13.5032	1.33051	0.665254	+	0.746617i	$0.268323\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−10.5030	−1.01536	−0.507680	−	0.861546i	$-0.669497\pi$
$107$	−10.5030	−1.01536	−0.507680	+	0.861546i	$0.669497\pi$
$108$	0	0
$109$	−8.22018	−0.787351	−0.393675	−	0.919250i	$-0.628797\pi$
$109$	−8.22018	−0.787351	−0.393675	+	0.919250i	$0.628797\pi$
$110$	0	0
$111$	−0.562581	−0.0533978
$112$	0	0
$113$	4.82841	0.454219	0.227109	−	0.973869i	$-0.427073\pi$
$113$	4.82841	0.454219	0.227109	+	0.973869i	$0.427073\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−11.1704	−1.03271
$118$	0	0
$119$	2.58355	0.236833
$120$	0	0
$121$	−9.76770	−0.887972
$122$	0	0
$123$	−6.65661	−0.600206
$124$	0	0
$125$	0	0
$126$	0	0
$127$	12.9161	1.14612	0.573058	−	0.819515i	$-0.305757\pi$
$127$	12.9161	1.14612	0.573058	+	0.819515i	$0.305757\pi$
$128$	0	0
$129$	−8.65465	−0.762000
$130$	0	0
$131$	−13.2022	−1.15348	−0.576739	−	0.816928i	$-0.695675\pi$
$131$	−13.2022	−1.15348	−0.576739	+	0.816928i	$0.695675\pi$
$132$	0	0
$133$	0.872810	0.0756822
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−15.3716	−1.31328	−0.656640	−	0.754204i	$-0.728023\pi$
$137$	−15.3716	−1.31328	−0.656640	+	0.754204i	$0.728023\pi$
$138$	0	0
$139$	−15.9849	−1.35582	−0.677912	−	0.735143i	$-0.737115\pi$
$139$	−15.9849	−1.35582	−0.677912	+	0.735143i	$0.737115\pi$
$140$	0	0
$141$	6.40433	0.539342
$142$	0	0
$143$	5.08744	0.425433
$144$	0	0
$145$	0	0
$146$	0	0
$147$	4.67899	0.385916
$148$	0	0
$149$	−3.11009	−0.254789	−0.127394	−	0.991852i	$-0.540661\pi$
$149$	−3.11009	−0.254789	−0.127394	+	0.991852i	$0.540661\pi$
$150$	0	0
$151$	3.34535	0.272240	0.136120	−	0.990692i	$-0.456537\pi$
$151$	3.34535	0.272240	0.136120	+	0.990692i	$0.456537\pi$
$152$	0	0
$153$	7.21484	0.583285
$154$	0	0
$155$	0	0
$156$	0	0
$157$	13.3402	1.06467	0.532334	−	0.846535i	$-0.321315\pi$
$157$	13.3402	1.06467	0.532334	+	0.846535i	$0.321315\pi$
$158$	0	0
$159$	−6.12811	−0.485991
$160$	0	0
$161$	3.34535	0.263650
$162$	0	0
$163$	5.33295	0.417709	0.208854	−	0.977947i	$-0.433026\pi$
$163$	5.33295	0.417709	0.208854	+	0.977947i	$0.433026\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	18.5906	1.43859	0.719293	−	0.694707i	$-0.244466\pi$
$167$	18.5906	1.43859	0.719293	+	0.694707i	$0.244466\pi$
$168$	0	0
$169$	8.00295	0.615611
$170$	0	0
$171$	2.43742	0.186394
$172$	0	0
$173$	10.8288	0.823301	0.411651	−	0.911342i	$-0.364952\pi$
$173$	10.8288	0.823301	0.411651	+	0.911342i	$0.364952\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	3.49124	0.262418
$178$	0	0
$179$	−3.34535	−0.250043	−0.125021	−	0.992154i	$-0.539900\pi$
$179$	−3.34535	−0.250043	−0.125021	+	0.992154i	$0.539900\pi$
$180$	0	0
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	−0.667479	−0.0493415
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−3.28591	−0.240290
$188$	0	0
$189$	−3.55963	−0.258925
$190$	0	0
$191$	−8.54751	−0.618476	−0.309238	−	0.950985i	$-0.600074\pi$
$191$	−8.54751	−0.618476	−0.309238	+	0.950985i	$0.600074\pi$
$192$	0	0
$193$	2.16980	0.156185	0.0780927	−	0.996946i	$-0.475117\pi$
$193$	2.16980	0.156185	0.0780927	+	0.996946i	$0.475117\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−18.3316	−1.30607	−0.653036	−	0.757327i	$-0.726505\pi$
$197$	−18.3316	−1.30607	−0.653036	+	0.757327i	$0.726505\pi$
$198$	0	0
$199$	−5.20217	−0.368772	−0.184386	−	0.982854i	$-0.559030\pi$
$199$	−5.20217	−0.368772	−0.184386	+	0.982854i	$0.559030\pi$
$200$	0	0
$201$	1.94396	0.137117
$202$	0	0
$203$	−6.60327	−0.463459
$204$	0	0
$205$	0	0
$206$	0	0
$207$	9.34225	0.649331
$208$	0	0
$209$	−1.11009	−0.0767867
$210$	0	0
$211$	−18.6606	−1.28465	−0.642323	−	0.766434i	$-0.722029\pi$
$211$	−18.6606	−1.28465	−0.642323	+	0.766434i	$0.722029\pi$
$212$	0	0
$213$	5.50940	0.377498
$214$	0	0
$215$	0	0
$216$	0	0
$217$	8.34889	0.566760
$218$	0	0
$219$	8.91087	0.602141
$220$	0	0
$221$	−13.5655	−0.912516
$222$	0	0
$223$	−21.7560	−1.45689	−0.728444	−	0.685105i	$-0.759756\pi$
$223$	−21.7560	−1.45689	−0.728444	+	0.685105i	$0.759756\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−4.58290	−0.304178	−0.152089	−	0.988367i	$-0.548600\pi$
$227$	−4.58290	−0.304178	−0.152089	+	0.988367i	$0.548600\pi$
$228$	0	0
$229$	−28.6756	−1.89494	−0.947469	−	0.319847i	$-0.896368\pi$
$229$	−28.6756	−1.89494	−0.947469	+	0.319847i	$0.896368\pi$
$230$	0	0
$231$	0.726727	0.0478151
$232$	0	0
$233$	−25.3009	−1.65752	−0.828759	−	0.559605i	$-0.810953\pi$
$233$	−25.3009	−1.65752	−0.828759	+	0.559605i	$0.810953\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	3.00021	0.194885
$238$	0	0
$239$	−22.5835	−1.46081	−0.730404	−	0.683015i	$-0.760668\pi$
$239$	−22.5835	−1.46081	−0.730404	+	0.683015i	$0.760668\pi$
$240$	0	0
$241$	0.874839	0.0563533	0.0281767	−	0.999603i	$-0.491030\pi$
$241$	0.874839	0.0563533	0.0281767	+	0.999603i	$0.491030\pi$
$242$	0	0
$243$	−15.4253	−0.989530
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−4.58290	−0.291603
$248$	0	0
$249$	−2.87484	−0.182186
$250$	0	0
$251$	15.6786	0.989623	0.494811	−	0.869000i	$-0.335237\pi$
$251$	15.6786	0.989623	0.494811	+	0.869000i	$0.335237\pi$
$252$	0	0
$253$	−4.25481	−0.267498
$254$	0	0
$255$	0	0
$256$	0	0
$257$	10.1614	0.633848	0.316924	−	0.948451i	$-0.397350\pi$
$257$	10.1614	0.633848	0.316924	+	0.948451i	$0.397350\pi$
$258$	0	0
$259$	−0.654655	−0.0406783
$260$	0	0
$261$	−18.4404	−1.14143
$262$	0	0
$263$	5.29277	0.326366	0.163183	−	0.986596i	$-0.447824\pi$
$263$	5.29277	0.326366	0.163183	+	0.986596i	$0.447824\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−1.00908	−0.0617548
$268$	0	0
$269$	10.9109	0.665248	0.332624	−	0.943060i	$-0.392066\pi$
$269$	10.9109	0.665248	0.332624	+	0.943060i	$0.392066\pi$
$270$	0	0
$271$	18.3123	1.11239	0.556195	−	0.831052i	$-0.312261\pi$
$271$	18.3123	1.11239	0.556195	+	0.831052i	$0.312261\pi$
$272$	0	0
$273$	3.00021	0.181581
$274$	0	0
$275$	0	0
$276$	0	0
$277$	14.8805	0.894084	0.447042	−	0.894513i	$-0.352477\pi$
$277$	14.8805	0.894084	0.447042	+	0.894513i	$0.352477\pi$
$278$	0	0
$279$	23.3152	1.39584
$280$	0	0
$281$	−23.8217	−1.42109	−0.710543	−	0.703654i	$-0.751551\pi$
$281$	−23.8217	−1.42109	−0.710543	+	0.703654i	$0.751551\pi$
$282$	0	0
$283$	−2.86394	−0.170244	−0.0851219	−	0.996371i	$-0.527128\pi$
$283$	−2.86394	−0.170244	−0.0851219	+	0.996371i	$0.527128\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−7.74605	−0.457235
$288$	0	0
$289$	−8.23820	−0.484600
$290$	0	0
$291$	−6.12811	−0.359236
$292$	0	0
$293$	2.16980	0.126761	0.0633805	−	0.997989i	$-0.479812\pi$
$293$	2.16980	0.126761	0.0633805	+	0.997989i	$0.479812\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	4.52735	0.262704
$298$	0	0
$299$	−17.5655	−1.01584
$300$	0	0
$301$	−10.0711	−0.580489
$302$	0	0
$303$	−4.07836	−0.234295
$304$	0	0
$305$	0	0
$306$	0	0
$307$	25.0978	1.43241	0.716204	−	0.697891i	$-0.245878\pi$
$307$	25.0978	1.43241	0.716204	+	0.697891i	$0.245878\pi$
$308$	0	0
$309$	−10.1281	−0.576168
$310$	0	0
$311$	−17.3663	−0.984753	−0.492377	−	0.870382i	$-0.663872\pi$
$311$	−17.3663	−0.984753	−0.492377	+	0.870382i	$0.663872\pi$
$312$	0	0
$313$	−31.9977	−1.80862	−0.904309	−	0.426879i	$-0.859613\pi$
$313$	−31.9977	−1.80862	−0.904309	+	0.426879i	$0.859613\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−26.1602	−1.46930	−0.734652	−	0.678444i	$-0.762655\pi$
$317$	−26.1602	−1.46930	−0.734652	+	0.678444i	$0.762655\pi$
$318$	0	0
$319$	8.39843	0.470222
$320$	0	0
$321$	7.87779	0.439695
$322$	0	0
$323$	2.96003	0.164701
$324$	0	0
$325$	0	0
$326$	0	0
$327$	6.16558	0.340957
$328$	0	0
$329$	7.45249	0.410869
$330$	0	0
$331$	12.1841	0.669701	0.334851	−	0.942271i	$-0.391314\pi$
$331$	12.1841	0.669701	0.334851	+	0.942271i	$0.391314\pi$
$332$	0	0
$333$	−1.82820	−0.100184
$334$	0	0
$335$	0	0
$336$	0	0
$337$	20.2401	1.10255	0.551276	−	0.834323i	$-0.314141\pi$
$337$	20.2401	1.10255	0.551276	+	0.834323i	$0.314141\pi$
$338$	0	0
$339$	−3.62157	−0.196697
$340$	0	0
$341$	−10.6186	−0.575030
$342$	0	0
$343$	11.5544	0.623881
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−15.4676	−0.830347	−0.415173	−	0.909742i	$-0.636279\pi$
$347$	−15.4676	−0.830347	−0.415173	+	0.909742i	$0.636279\pi$
$348$	0	0
$349$	−1.63664	−0.0876071	−0.0438036	−	0.999040i	$-0.513948\pi$
$349$	−1.63664	−0.0876071	−0.0438036	+	0.999040i	$0.513948\pi$
$350$	0	0
$351$	18.6907	0.997635
$352$	0	0
$353$	−27.0064	−1.43740	−0.718702	−	0.695319i	$-0.755263\pi$
$353$	−27.0064	−1.43740	−0.718702	+	0.695319i	$0.755263\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−1.93780	−0.102559
$358$	0	0
$359$	−15.5446	−0.820411	−0.410205	−	0.911993i	$-0.634543\pi$
$359$	−15.5446	−0.820411	−0.410205	+	0.911993i	$0.634543\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	7.32630	0.384531
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−14.1729	−0.739818	−0.369909	−	0.929068i	$-0.620611\pi$
$367$	−14.1729	−0.739818	−0.369909	+	0.929068i	$0.620611\pi$
$368$	0	0
$369$	−21.6317	−1.12610
$370$	0	0
$371$	−7.13106	−0.370226
$372$	0	0
$373$	−28.0019	−1.44988	−0.724942	−	0.688810i	$-0.758134\pi$
$373$	−28.0019	−1.44988	−0.724942	+	0.688810i	$0.758134\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	34.6720	1.78570
$378$	0	0
$379$	29.0950	1.49451	0.747255	−	0.664537i	$-0.231371\pi$
$379$	29.0950	1.49451	0.747255	+	0.664537i	$0.231371\pi$
$380$	0	0
$381$	−9.68774	−0.496318
$382$	0	0
$383$	−29.5981	−1.51239	−0.756197	−	0.654344i	$-0.772945\pi$
$383$	−29.5981	−1.51239	−0.756197	+	0.654344i	$0.772945\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−28.1247	−1.42966
$388$	0	0
$389$	−29.6786	−1.50476	−0.752382	−	0.658727i	$-0.771095\pi$
$389$	−29.6786	−1.50476	−0.752382	+	0.658727i	$0.771095\pi$
$390$	0	0
$391$	11.3453	0.573759
$392$	0	0
$393$	9.90233	0.499507
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−5.19668	−0.260814	−0.130407	−	0.991461i	$-0.541628\pi$
$397$	−5.19668	−0.260814	−0.130407	+	0.991461i	$0.541628\pi$
$398$	0	0
$399$	−0.654655	−0.0327737
$400$	0	0
$401$	−8.22018	−0.410496	−0.205248	−	0.978710i	$-0.565800\pi$
$401$	−8.22018	−0.410496	−0.205248	+	0.978710i	$0.565800\pi$
$402$	0	0
$403$	−43.8378	−2.18372
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0.832629	0.0412719
$408$	0	0
$409$	−4.03604	−0.199569	−0.0997846	−	0.995009i	$-0.531815\pi$
$409$	−4.03604	−0.199569	−0.0997846	+	0.995009i	$0.531815\pi$
$410$	0	0
$411$	11.5295	0.568708
$412$	0	0
$413$	4.06263	0.199909
$414$	0	0
$415$	0	0
$416$	0	0
$417$	11.9896	0.587131
$418$	0	0
$419$	1.30931	0.0639639	0.0319820	−	0.999488i	$-0.489818\pi$
$419$	1.30931	0.0639639	0.0319820	+	0.999488i	$0.489818\pi$
$420$	0	0
$421$	−13.5295	−0.659387	−0.329694	−	0.944088i	$-0.606945\pi$
$421$	−13.5295	−0.659387	−0.329694	+	0.944088i	$0.606945\pi$
$422$	0	0
$423$	20.8119	1.01191
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−0.776721	−0.0375882
$428$	0	0
$429$	−3.81585	−0.184231
$430$	0	0
$431$	11.3152	0.545034	0.272517	−	0.962151i	$-0.412144\pi$
$431$	11.3152	0.545034	0.272517	+	0.962151i	$0.412144\pi$
$432$	0	0
$433$	29.1604	1.40136	0.700680	−	0.713475i	$-0.252880\pi$
$433$	29.1604	1.40136	0.700680	+	0.713475i	$0.252880\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.83284	0.183350
$438$	0	0
$439$	−19.9699	−0.953109	−0.476555	−	0.879145i	$-0.658115\pi$
$439$	−19.9699	−0.953109	−0.476555	+	0.879145i	$0.658115\pi$
$440$	0	0
$441$	15.2051	0.724053
$442$	0	0
$443$	35.6252	1.69261	0.846303	−	0.532702i	$-0.178823\pi$
$443$	35.6252	1.69261	0.846303	+	0.532702i	$0.178823\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	2.33274	0.110335
$448$	0	0
$449$	32.4043	1.52925	0.764627	−	0.644472i	$-0.222923\pi$
$449$	32.4043	1.52925	0.764627	+	0.644472i	$0.222923\pi$
$450$	0	0
$451$	9.85189	0.463907
$452$	0	0
$453$	−2.50919	−0.117892
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−6.20576	−0.290293	−0.145147	−	0.989410i	$-0.546365\pi$
$457$	−6.20576	−0.290293	−0.145147	+	0.989410i	$0.546365\pi$
$458$	0	0
$459$	−12.0721	−0.563476
$460$	0	0
$461$	−4.79783	−0.223457	−0.111729	−	0.993739i	$-0.535639\pi$
$461$	−4.79783	−0.223457	−0.111729	+	0.993739i	$0.535639\pi$
$462$	0	0
$463$	−41.5453	−1.93077	−0.965387	−	0.260823i	$-0.916006\pi$
$463$	−41.5453	−1.93077	−0.965387	+	0.260823i	$0.916006\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	18.1953	0.841979	0.420989	−	0.907066i	$-0.361683\pi$
$467$	18.1953	0.841979	0.420989	+	0.907066i	$0.361683\pi$
$468$	0	0
$469$	2.26212	0.104455
$470$	0	0
$471$	−10.0059	−0.461047
$472$	0	0
$473$	12.8090	0.588960
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−19.9143	−0.911812
$478$	0	0
$479$	−31.8778	−1.45653	−0.728267	−	0.685294i	$-0.759674\pi$
$479$	−31.8778	−1.45653	−0.728267	+	0.685294i	$0.759674\pi$
$480$	0	0
$481$	3.43742	0.156733
$482$	0	0
$483$	−2.50919	−0.114172
$484$	0	0
$485$	0	0
$486$	0	0
$487$	5.83749	0.264522	0.132261	−	0.991215i	$-0.457776\pi$
$487$	5.83749	0.264522	0.132261	+	0.991215i	$0.457776\pi$
$488$	0	0
$489$	−4.00000	−0.180886
$490$	0	0
$491$	7.13106	0.321820	0.160910	−	0.986969i	$-0.448557\pi$
$491$	7.13106	0.321820	0.160910	+	0.986969i	$0.448557\pi$
$492$	0	0
$493$	−22.3942	−1.00858
$494$	0	0
$495$	0	0
$496$	0	0
$497$	6.41109	0.287577
$498$	0	0
$499$	−3.14613	−0.140840	−0.0704200	−	0.997517i	$-0.522434\pi$
$499$	−3.14613	−0.140840	−0.0704200	+	0.997517i	$0.522434\pi$
$500$	0	0
$501$	−13.9440	−0.622970
$502$	0	0
$503$	25.1646	1.12204	0.561018	−	0.827804i	$-0.310410\pi$
$503$	25.1646	1.12204	0.561018	+	0.827804i	$0.310410\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−6.00264	−0.266587
$508$	0	0
$509$	−37.9699	−1.68298	−0.841492	−	0.540269i	$-0.818322\pi$
$509$	−37.9699	−1.68298	−0.841492	+	0.540269i	$0.818322\pi$
$510$	0	0
$511$	10.3693	0.458709
$512$	0	0
$513$	−4.07836	−0.180064
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−9.47852	−0.416865
$518$	0	0
$519$	−8.12221	−0.356526
$520$	0	0
$521$	−3.96396	−0.173664	−0.0868322	−	0.996223i	$-0.527674\pi$
$521$	−3.96396	−0.173664	−0.0868322	+	0.996223i	$0.527674\pi$
$522$	0	0
$523$	17.2557	0.754537	0.377269	−	0.926104i	$-0.376863\pi$
$523$	17.2557	0.754537	0.377269	+	0.926104i	$0.376863\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	28.3143	1.23339
$528$	0	0
$529$	−8.30931	−0.361274
$530$	0	0
$531$	11.3453	0.492346
$532$	0	0
$533$	40.6725	1.76172
$534$	0	0
$535$	0	0
$536$	0	0
$537$	2.50919	0.108279
$538$	0	0
$539$	−6.92498	−0.298280
$540$	0	0
$541$	−11.2942	−0.485577	−0.242789	−	0.970079i	$-0.578062\pi$
$541$	−11.2942	−0.485577	−0.242789	+	0.970079i	$0.578062\pi$
$542$	0	0
$543$	1.50011	0.0643758
$544$	0	0
$545$	0	0
$546$	0	0
$547$	31.1673	1.33262	0.666309	−	0.745676i	$-0.267873\pi$
$547$	31.1673	1.33262	0.666309	+	0.745676i	$0.267873\pi$
$548$	0	0
$549$	−2.16908	−0.0925740
$550$	0	0
$551$	−7.56553	−0.322302
$552$	0	0
$553$	3.49124	0.148463
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−22.6266	−0.958719	−0.479359	−	0.877619i	$-0.659131\pi$
$557$	−22.6266	−0.958719	−0.479359	+	0.877619i	$0.659131\pi$
$558$	0	0
$559$	52.8807	2.23662
$560$	0	0
$561$	2.46461	0.104056
$562$	0	0
$563$	−13.4071	−0.565041	−0.282521	−	0.959261i	$-0.591171\pi$
$563$	−13.4071	−0.565041	−0.282521	+	0.959261i	$0.591171\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−3.71230	−0.155902
$568$	0	0
$569$	−40.6606	−1.70458	−0.852290	−	0.523070i	$-0.824786\pi$
$569$	−40.6606	−1.70458	−0.852290	+	0.523070i	$0.824786\pi$
$570$	0	0
$571$	41.4433	1.73435	0.867174	−	0.498005i	$-0.165934\pi$
$571$	41.4433	1.73435	0.867174	+	0.498005i	$0.165934\pi$
$572$	0	0
$573$	6.41109	0.267827
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−24.7295	−1.02950	−0.514752	−	0.857339i	$-0.672116\pi$
$577$	−24.7295	−1.02950	−0.514752	+	0.857339i	$0.672116\pi$
$578$	0	0
$579$	−1.62747	−0.0676351
$580$	0	0
$581$	−3.34535	−0.138788
$582$	0	0
$583$	9.06971	0.375629
$584$	0	0
$585$	0	0
$586$	0	0
$587$	7.03839	0.290505	0.145253	−	0.989395i	$-0.453600\pi$
$587$	7.03839	0.290505	0.145253	+	0.989395i	$0.453600\pi$
$588$	0	0
$589$	9.56553	0.394141
$590$	0	0
$591$	13.7497	0.565586
$592$	0	0
$593$	−17.6484	−0.724732	−0.362366	−	0.932036i	$-0.618031\pi$
$593$	−17.6484	−0.724732	−0.362366	+	0.932036i	$0.618031\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	3.90190	0.159694
$598$	0	0
$599$	−33.5655	−1.37145	−0.685725	−	0.727861i	$-0.740515\pi$
$599$	−33.5655	−1.37145	−0.685725	+	0.727861i	$0.740515\pi$
$600$	0	0
$601$	28.0059	1.14238	0.571192	−	0.820816i	$-0.306481\pi$
$601$	28.0059	1.14238	0.571192	+	0.820816i	$0.306481\pi$
$602$	0	0
$603$	6.31722	0.257257
$604$	0	0
$605$	0	0
$606$	0	0
$607$	30.5802	1.24121	0.620605	−	0.784123i	$-0.286887\pi$
$607$	30.5802	1.24121	0.620605	+	0.784123i	$0.286887\pi$
$608$	0	0
$609$	4.95281	0.200698
$610$	0	0
$611$	−39.1311	−1.58307
$612$	0	0
$613$	11.8000	0.476596	0.238298	−	0.971192i	$-0.423411\pi$
$613$	11.8000	0.476596	0.238298	+	0.971192i	$0.423411\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−16.7065	−0.672579	−0.336289	−	0.941759i	$-0.609172\pi$
$617$	−16.7065	−0.672579	−0.336289	+	0.941759i	$0.609172\pi$
$618$	0	0
$619$	25.4433	1.02265	0.511327	−	0.859386i	$-0.329154\pi$
$619$	25.4433	1.02265	0.511327	+	0.859386i	$0.329154\pi$
$620$	0	0
$621$	−15.6317	−0.627279
$622$	0	0
$623$	−1.17423	−0.0470446
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0.832629	0.0332520
$628$	0	0
$629$	−2.22018	−0.0885245
$630$	0	0
$631$	−35.7706	−1.42401	−0.712003	−	0.702176i	$-0.752212\pi$
$631$	−35.7706	−1.42401	−0.712003	+	0.702176i	$0.752212\pi$
$632$	0	0
$633$	13.9964	0.556308
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−28.5890	−1.13274
$638$	0	0
$639$	17.9037	0.708259
$640$	0	0
$641$	31.3512	1.23830	0.619150	−	0.785273i	$-0.287477\pi$
$641$	31.3512	1.23830	0.619150	+	0.785273i	$0.287477\pi$
$642$	0	0
$643$	−31.5626	−1.24471	−0.622353	−	0.782736i	$-0.713823\pi$
$643$	−31.5626	−1.24471	−0.622353	+	0.782736i	$0.713823\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−41.4649	−1.63015	−0.815077	−	0.579352i	$-0.803306\pi$
$647$	−41.4649	−1.63015	−0.815077	+	0.579352i	$0.803306\pi$
$648$	0	0
$649$	−5.16710	−0.202826
$650$	0	0
$651$	−6.26212	−0.245432
$652$	0	0
$653$	−5.19668	−0.203362	−0.101681	−	0.994817i	$-0.532422\pi$
$653$	−5.19668	−0.203362	−0.101681	+	0.994817i	$0.532422\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	28.9573	1.12973
$658$	0	0
$659$	−18.2202	−0.709758	−0.354879	−	0.934912i	$-0.615478\pi$
$659$	−18.2202	−0.709758	−0.354879	+	0.934912i	$0.615478\pi$
$660$	0	0
$661$	−18.9109	−0.735548	−0.367774	−	0.929915i	$-0.619880\pi$
$661$	−18.9109	−0.735548	−0.367774	+	0.929915i	$0.619880\pi$
$662$	0	0
$663$	10.1749	0.395159
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−28.9975	−1.12279
$668$	0	0
$669$	16.3182	0.630896
$670$	0	0
$671$	0.987880	0.0381367
$672$	0	0
$673$	−4.82841	−0.186122	−0.0930608	−	0.995660i	$-0.529665\pi$
$673$	−4.82841	−0.186122	−0.0930608	+	0.995660i	$0.529665\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	36.4042	1.39913	0.699563	−	0.714571i	$-0.253378\pi$
$677$	36.4042	1.39913	0.699563	+	0.714571i	$0.253378\pi$
$678$	0	0
$679$	−7.13106	−0.273665
$680$	0	0
$681$	3.43742	0.131722
$682$	0	0
$683$	29.1762	1.11639	0.558197	−	0.829708i	$-0.311493\pi$
$683$	29.1762	1.11639	0.558197	+	0.829708i	$0.311493\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	21.5083	0.820591
$688$	0	0
$689$	37.4433	1.42648
$690$	0	0
$691$	−0.413504	−0.0157304	−0.00786521	−	0.999969i	$-0.502504\pi$
$691$	−0.413504	−0.0157304	−0.00786521	+	0.999969i	$0.502504\pi$
$692$	0	0
$693$	2.36162	0.0897103
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−26.2698	−0.995040
$698$	0	0
$699$	18.9770	0.717778
$700$	0	0
$701$	−23.2592	−0.878487	−0.439243	−	0.898368i	$-0.644753\pi$
$701$	−23.2592	−0.878487	−0.439243	+	0.898368i	$0.644753\pi$
$702$	0	0
$703$	−0.750054	−0.0282888
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−4.74584	−0.178486
$708$	0	0
$709$	28.8748	1.08442	0.542209	−	0.840244i	$-0.317588\pi$
$709$	28.8748	1.08442	0.542209	+	0.840244i	$0.317588\pi$
$710$	0	0
$711$	9.74968	0.365641
$712$	0	0
$713$	36.6632	1.37305
$714$	0	0
$715$	0	0
$716$	0	0
$717$	16.9389	0.632594
$718$	0	0
$719$	26.4613	0.986841	0.493421	−	0.869791i	$-0.335746\pi$
$719$	26.4613	0.986841	0.493421	+	0.869791i	$0.335746\pi$
$720$	0	0
$721$	−11.7857	−0.438923
$722$	0	0
$723$	−0.656176	−0.0244035
$724$	0	0
$725$	0	0
$726$	0	0
$727$	8.61886	0.319656	0.159828	−	0.987145i	$-0.448906\pi$
$727$	8.61886	0.319656	0.159828	+	0.987145i	$0.448906\pi$
$728$	0	0
$729$	−1.19005	−0.0440758
$730$	0	0
$731$	−34.1550	−1.26327
$732$	0	0
$733$	4.82620	0.178260	0.0891299	−	0.996020i	$-0.471591\pi$
$733$	4.82620	0.178260	0.0891299	+	0.996020i	$0.471591\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2.87710	−0.105979
$738$	0	0
$739$	−27.4525	−1.00986	−0.504928	−	0.863161i	$-0.668481\pi$
$739$	−27.4525	−1.00986	−0.504928	+	0.863161i	$0.668481\pi$
$740$	0	0
$741$	3.43742	0.126277
$742$	0	0
$743$	−0.830417	−0.0304650	−0.0152325	−	0.999884i	$-0.504849\pi$
$743$	−0.830417	−0.0304650	−0.0152325	+	0.999884i	$0.504849\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−9.34225	−0.341815
$748$	0	0
$749$	9.16710	0.334958
$750$	0	0
$751$	9.12516	0.332982	0.166491	−	0.986043i	$-0.446756\pi$
$751$	9.12516	0.332982	0.166491	+	0.986043i	$0.446756\pi$
$752$	0	0
$753$	−11.7598	−0.428550
$754$	0	0
$755$	0	0
$756$	0	0
$757$	1.04926	0.0381361	0.0190681	−	0.999818i	$-0.493930\pi$
$757$	1.04926	0.0381361	0.0190681	+	0.999818i	$0.493930\pi$
$758$	0	0
$759$	3.19134	0.115838
$760$	0	0
$761$	12.2353	0.443528	0.221764	−	0.975100i	$-0.428819\pi$
$761$	12.2353	0.443528	0.221764	+	0.975100i	$0.428819\pi$
$762$	0	0
$763$	7.17466	0.259740
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−21.3318	−0.770247
$768$	0	0
$769$	−25.0741	−0.904194	−0.452097	−	0.891969i	$-0.649324\pi$
$769$	−25.0741	−0.904194	−0.452097	+	0.891969i	$0.649324\pi$
$770$	0	0
$771$	−7.62157	−0.274484
$772$	0	0
$773$	−31.6696	−1.13908	−0.569539	−	0.821965i	$-0.692878\pi$
$773$	−31.6696	−1.13908	−0.569539	+	0.821965i	$0.692878\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0.491026	0.0176155
$778$	0	0
$779$	−8.87484	−0.317974
$780$	0	0
$781$	−8.15401	−0.291773
$782$	0	0
$783$	30.8549	1.10267
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−45.1593	−1.60975	−0.804877	−	0.593441i	$-0.797769\pi$
$787$	−45.1593	−1.60975	−0.804877	+	0.593441i	$0.797769\pi$
$788$	0	0
$789$	−3.96986	−0.141331
$790$	0	0
$791$	−4.21429	−0.149843
$792$	0	0
$793$	4.07836	0.144827
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−1.32144	−0.0468079	−0.0234040	−	0.999726i	$-0.507450\pi$
$797$	−1.32144	−0.0468079	−0.0234040	+	0.999726i	$0.507450\pi$
$798$	0	0
$799$	25.2742	0.894138
$800$	0	0
$801$	−3.27917	−0.115864
$802$	0	0
$803$	−13.1882	−0.465403
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−8.18374	−0.288081
$808$	0	0
$809$	43.8267	1.54086	0.770432	−	0.637522i	$-0.220041\pi$
$809$	43.8267	1.54086	0.770432	+	0.637522i	$0.220041\pi$
$810$	0	0
$811$	−52.1959	−1.83285	−0.916424	−	0.400209i	$-0.868937\pi$
$811$	−52.1959	−1.83285	−0.916424	+	0.400209i	$0.868937\pi$
$812$	0	0
$813$	−13.7352	−0.481714
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−11.5387	−0.403688
$818$	0	0
$819$	9.74968	0.340681
$820$	0	0
$821$	9.02392	0.314937	0.157468	−	0.987524i	$-0.449667\pi$
$821$	9.02392	0.314937	0.157468	+	0.987524i	$0.449667\pi$
$822$	0	0
$823$	21.0304	0.733073	0.366537	−	0.930404i	$-0.380543\pi$
$823$	21.0304	0.733073	0.366537	+	0.930404i	$0.380543\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−53.3430	−1.85492	−0.927459	−	0.373924i	$-0.878012\pi$
$827$	−53.3430	−1.85492	−0.927459	+	0.373924i	$0.878012\pi$
$828$	0	0
$829$	−10.0419	−0.348771	−0.174385	−	0.984677i	$-0.555794\pi$
$829$	−10.0419	−0.348771	−0.174385	+	0.984677i	$0.555794\pi$
$830$	0	0
$831$	−11.1612	−0.387178
$832$	0	0
$833$	18.4653	0.639784
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−39.0116	−1.34844
$838$	0	0
$839$	−46.4043	−1.60206	−0.801028	−	0.598627i	$-0.795713\pi$
$839$	−46.4043	−1.60206	−0.801028	+	0.598627i	$0.795713\pi$
$840$	0	0
$841$	28.2372	0.973698
$842$	0	0
$843$	17.8676	0.615393
$844$	0	0
$845$	0	0
$846$	0	0
$847$	8.52535	0.292934
$848$	0	0
$849$	2.14811	0.0737230
$850$	0	0
$851$	−2.87484	−0.0985482
$852$	0	0
$853$	−25.5063	−0.873317	−0.436659	−	0.899627i	$-0.643838\pi$
$853$	−25.5063	−0.873317	−0.436659	+	0.899627i	$0.643838\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−13.5993	−0.464542	−0.232271	−	0.972651i	$-0.574616\pi$
$857$	−13.5993	−0.464542	−0.232271	+	0.972651i	$0.574616\pi$
$858$	0	0
$859$	−4.83387	−0.164930	−0.0824648	−	0.996594i	$-0.526279\pi$
$859$	−4.83387	−0.164930	−0.0824648	+	0.996594i	$0.526279\pi$
$860$	0	0
$861$	5.80995	0.198003
$862$	0	0
$863$	−7.73254	−0.263219	−0.131609	−	0.991302i	$-0.542014\pi$
$863$	−7.73254	−0.263219	−0.131609	+	0.991302i	$0.542014\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	6.17909	0.209853
$868$	0	0
$869$	−4.44037	−0.150629
$870$	0	0
$871$	−11.8778	−0.402463
$872$	0	0
$873$	−19.9143	−0.673996
$874$	0	0
$875$	0	0
$876$	0	0
$877$	49.2804	1.66408	0.832040	−	0.554715i	$-0.187173\pi$
$877$	49.2804	1.66408	0.832040	+	0.554715i	$0.187173\pi$
$878$	0	0
$879$	−1.62747	−0.0548930
$880$	0	0
$881$	47.9908	1.61685	0.808426	−	0.588598i	$-0.200320\pi$
$881$	47.9908	1.61685	0.808426	+	0.588598i	$0.200320\pi$
$882$	0	0
$883$	−21.1422	−0.711492	−0.355746	−	0.934583i	$-0.615773\pi$
$883$	−21.1422	−0.711492	−0.355746	+	0.934583i	$0.615773\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−41.8175	−1.40409	−0.702047	−	0.712131i	$-0.747730\pi$
$887$	−41.8175	−1.40409	−0.702047	+	0.712131i	$0.747730\pi$
$888$	0	0
$889$	−11.2733	−0.378093
$890$	0	0
$891$	4.72152	0.158177
$892$	0	0
$893$	8.53850	0.285730
$894$	0	0
$895$	0	0
$896$	0	0
$897$	13.1751	0.439903
$898$	0	0
$899$	−72.3683	−2.41362
$900$	0	0
$901$	−24.1841	−0.805691
$902$	0	0
$903$	7.55387	0.251377
$904$	0	0
$905$	0	0
$906$	0	0
$907$	41.0809	1.36407	0.682035	−	0.731320i	$-0.261095\pi$
$907$	41.0809	1.36407	0.682035	+	0.731320i	$0.261095\pi$
$908$	0	0
$909$	−13.2533	−0.439583
$910$	0	0
$911$	−12.8388	−0.425369	−0.212684	−	0.977121i	$-0.568221\pi$
$911$	−12.8388	−0.425369	−0.212684	+	0.977121i	$0.568221\pi$
$912$	0	0
$913$	4.25481	0.140814
$914$	0	0
$915$	0	0
$916$	0	0
$917$	11.5230	0.380523
$918$	0	0
$919$	−28.8807	−0.952688	−0.476344	−	0.879259i	$-0.658038\pi$
$919$	−28.8807	−0.952688	−0.476344	+	0.879259i	$0.658038\pi$
$920$	0	0
$921$	−18.8247	−0.620295
$922$	0	0
$923$	−33.6630	−1.10803
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−32.9129	−1.08100
$928$	0	0
$929$	46.6966	1.53207	0.766033	−	0.642802i	$-0.222228\pi$
$929$	46.6966	1.53207	0.766033	+	0.642802i	$0.222228\pi$
$930$	0	0
$931$	6.23820	0.204449
$932$	0	0
$933$	13.0257	0.426441
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−20.0370	−0.654581	−0.327291	−	0.944924i	$-0.606136\pi$
$937$	−20.0370	−0.654581	−0.327291	+	0.944924i	$0.606136\pi$
$938$	0	0
$939$	24.0000	0.783210
$940$	0	0
$941$	−5.13106	−0.167268	−0.0836339	−	0.996497i	$-0.526653\pi$
$941$	−5.13106	−0.167268	−0.0836339	+	0.996497i	$0.526653\pi$
$942$	0	0
$943$	−34.0159	−1.10771
$944$	0	0
$945$	0	0
$946$	0	0
$947$	31.0044	1.00751	0.503753	−	0.863848i	$-0.331952\pi$
$947$	31.0044	1.00751	0.503753	+	0.863848i	$0.331952\pi$
$948$	0	0
$949$	−54.4463	−1.76740
$950$	0	0
$951$	19.6216	0.636273
$952$	0	0
$953$	−36.4845	−1.18185	−0.590925	−	0.806727i	$-0.701237\pi$
$953$	−36.4845	−1.18185	−0.590925	+	0.806727i	$0.701237\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−6.29928	−0.203627
$958$	0	0
$959$	13.4165	0.433240
$960$	0	0
$961$	60.4994	1.95159
$962$	0	0
$963$	25.6001	0.824953
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−3.34182	−0.107466	−0.0537328	−	0.998555i	$-0.517112\pi$
$967$	−3.34182	−0.107466	−0.0537328	+	0.998555i	$0.517112\pi$
$968$	0	0
$969$	−2.22018	−0.0713226
$970$	0	0
$971$	11.1311	0.357213	0.178606	−	0.983921i	$-0.442841\pi$
$971$	11.1311	0.357213	0.178606	+	0.983921i	$0.442841\pi$
$972$	0	0
$973$	13.9518	0.447275
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−55.2695	−1.76823	−0.884114	−	0.467271i	$-0.845237\pi$
$977$	−55.2695	−1.76823	−0.884114	+	0.467271i	$0.845237\pi$
$978$	0	0
$979$	1.49346	0.0477311
$980$	0	0
$981$	20.0360	0.639701
$982$	0	0
$983$	31.3971	1.00141	0.500706	−	0.865618i	$-0.333074\pi$
$983$	31.3971	1.00141	0.500706	+	0.865618i	$0.333074\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−5.58977	−0.177924
$988$	0	0
$989$	−44.2261	−1.40631
$990$	0	0
$991$	−16.2143	−0.515064	−0.257532	−	0.966270i	$-0.582909\pi$
$991$	−16.2143	−0.515064	−0.257532	+	0.966270i	$0.582909\pi$
$992$	0	0
$993$	−9.13877	−0.290010
$994$	0	0
$995$	0	0
$996$	0	0
$997$	5.30850	0.168122	0.0840609	−	0.996461i	$-0.473211\pi$
$997$	5.30850	0.168122	0.0840609	+	0.996461i	$0.473211\pi$
$998$	0	0
$999$	3.05899	0.0967821

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7600.2.a.cj.1.3		6
4.3	odd	2		1900.2.a.k.1.4		6
5.2	odd	4		1520.2.d.i.609.4		6
5.3	odd	4		1520.2.d.i.609.3		6
5.4	even	2	inner	7600.2.a.cj.1.4		6
20.3	even	4		380.2.c.b.229.4	yes	6
20.7	even	4		380.2.c.b.229.3	✓	6
20.19	odd	2		1900.2.a.k.1.3		6
60.23	odd	4		3420.2.f.c.1369.3		6
60.47	odd	4		3420.2.f.c.1369.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.2.c.b.229.3	✓	6	20.7	even	4
380.2.c.b.229.4	yes	6	20.3	even	4
1520.2.d.i.609.3		6	5.3	odd	4
1520.2.d.i.609.4		6	5.2	odd	4
1900.2.a.k.1.3		6	20.19	odd	2
1900.2.a.k.1.4		6	4.3	odd	2
3420.2.f.c.1369.3		6	60.23	odd	4
3420.2.f.c.1369.4		6	60.47	odd	4
7600.2.a.cj.1.3		6	1.1	even	1	trivial
7600.2.a.cj.1.4		6	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 7600 = 2^{4} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7600.a (trivial)

\(f(q)\)	\(=\)	\(q-0.750054 q^{3} -0.872810 q^{7} -2.43742 q^{9} +O(q^{10})\)	\(q-0.750054 q^{3} -0.872810 q^{7} -2.43742 q^{9} +1.11009 q^{11} +4.58290 q^{13} -2.96003 q^{17} -1.00000 q^{19} +0.654655 q^{21} -3.83284 q^{23} +4.07836 q^{27} +7.56553 q^{29} -9.56553 q^{31} -0.832629 q^{33} +0.750054 q^{37} -3.43742 q^{39} +8.87484 q^{41} +11.5387 q^{43} -8.53850 q^{47} -6.23820 q^{49} +2.22018 q^{51} +8.17023 q^{53} +0.750054 q^{57} -4.65465 q^{59} +0.889908 q^{61} +2.12740 q^{63} -2.59176 q^{67} +2.87484 q^{69} -7.34535 q^{71} -11.8803 q^{73} -0.968900 q^{77} -4.00000 q^{79} +4.25327 q^{81} +3.83284 q^{83} -5.67455 q^{87} +1.34535 q^{89} -4.00000 q^{91} +7.17466 q^{93} +8.17023 q^{97} -2.70576 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 10 q^{9}+O(q^{10}) \) 6 * q + 10 * q^9	\( 6 q + 10 q^{9} - 18 q^{11} - 6 q^{19} + 4 q^{21} - 4 q^{29} - 8 q^{31} + 4 q^{39} + 4 q^{41} + 12 q^{49} - 36 q^{51} - 28 q^{59} + 30 q^{61} - 32 q^{69} - 44 q^{71} - 24 q^{79} + 50 q^{81} + 8 q^{89} - 24 q^{91} - 90 q^{99}+O(q^{100}) \) 6 * q + 10 * q^9 - 18 * q^11 - 6 * q^19 + 4 * q^21 - 4 * q^29 - 8 * q^31 + 4 * q^39 + 4 * q^41 + 12 * q^49 - 36 * q^51 - 28 * q^59 + 30 * q^61 - 32 * q^69 - 44 * q^71 - 24 * q^79 + 50 * q^81 + 8 * q^89 - 24 * q^91 - 90 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more