LMFDB - Embedded newform 7448.2.a.br.1.5

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:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 15x^{6} - x^{5} + 66x^{4} + 4x^{3} - 76x^{2} + 8x + 16 $ x^8 - 15x^6 - x^5 + 66x^4 + 4x^3 - 76x^2 + 8*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1064)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.436576$ 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)

$f(q)$	$=$	$q+0.436576 q^{3} +0.299640 q^{5} -2.80940 q^{9} +O(q^{10})$	$q+0.436576 q^{3} +0.299640 q^{5} -2.80940 q^{9} -5.18938 q^{11} +1.26174 q^{13} +0.130816 q^{15} -1.95892 q^{17} -1.00000 q^{19} -1.49106 q^{23} -4.91022 q^{25} -2.53625 q^{27} -0.530372 q^{29} -1.94838 q^{31} -2.26556 q^{33} +4.19347 q^{37} +0.550846 q^{39} +2.00000 q^{41} +5.51755 q^{43} -0.841808 q^{45} +8.54096 q^{47} -0.855216 q^{51} -11.7557 q^{53} -1.55494 q^{55} -0.436576 q^{57} +12.8152 q^{59} +5.93580 q^{61} +0.378068 q^{65} +5.50772 q^{67} -0.650963 q^{69} +2.28016 q^{71} +4.69240 q^{73} -2.14368 q^{75} +5.69468 q^{79} +7.32094 q^{81} -1.46325 q^{83} -0.586969 q^{85} -0.231548 q^{87} +3.85050 q^{89} -0.850616 q^{93} -0.299640 q^{95} +9.69800 q^{97} +14.5791 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + q^{5} + 6 q^{9}+O(q^{10}) $ 8 * q + q^5 + 6 * q^9	$ 8 q + q^{5} + 6 q^{9} + 9 q^{11} + 8 q^{15} - 4 q^{17} - 8 q^{19} + 25 q^{23} + 15 q^{25} - 3 q^{27} + 6 q^{29} - 8 q^{33} + 13 q^{37} - 11 q^{39} + 16 q^{41} + 17 q^{43} - 17 q^{45} + 24 q^{47} + 5 q^{51} + 2 q^{53} - 5 q^{55} - 2 q^{59} + 13 q^{61} - 26 q^{65} + 2 q^{67} + 11 q^{69} + 10 q^{71} - 5 q^{73} + 20 q^{75} + 16 q^{79} - 12 q^{81} + 43 q^{83} + 24 q^{85} - 20 q^{87} - 8 q^{89} + 2 q^{93} - q^{95} + 12 q^{97} + 37 q^{99}+O(q^{100}) $ 8 * q + q^5 + 6 * q^9 + 9 * q^11 + 8 * q^15 - 4 * q^17 - 8 * q^19 + 25 * q^23 + 15 * q^25 - 3 * q^27 + 6 * q^29 - 8 * q^33 + 13 * q^37 - 11 * q^39 + 16 * q^41 + 17 * q^43 - 17 * q^45 + 24 * q^47 + 5 * q^51 + 2 * q^53 - 5 * q^55 - 2 * q^59 + 13 * q^61 - 26 * q^65 + 2 * q^67 + 11 * q^69 + 10 * q^71 - 5 * q^73 + 20 * q^75 + 16 * q^79 - 12 * q^81 + 43 * q^83 + 24 * q^85 - 20 * q^87 - 8 * q^89 + 2 * q^93 - q^95 + 12 * q^97 + 37 * 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
$3$	0.436576	0.252057	0.126029	−	0.992027i	$-0.459777\pi$
$3$	0.436576	0.252057	0.126029	+	0.992027i	$0.459777\pi$
$4$	0	0
$5$	0.299640	0.134003	0.0670015	−	0.997753i	$-0.478657\pi$
$5$	0.299640	0.134003	0.0670015	+	0.997753i	$0.478657\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.80940	−0.936467
$10$	0	0
$11$	−5.18938	−1.56466	−0.782329	−	0.622866i	$-0.785968\pi$
$11$	−5.18938	−1.56466	−0.782329	+	0.622866i	$0.785968\pi$
$12$	0	0
$13$	1.26174	0.349944	0.174972	−	0.984573i	$-0.444016\pi$
$13$	1.26174	0.349944	0.174972	+	0.984573i	$0.444016\pi$
$14$	0	0
$15$	0.130816	0.0337764
$16$	0	0
$17$	−1.95892	−0.475107	−0.237553	−	0.971374i	$-0.576345\pi$
$17$	−1.95892	−0.475107	−0.237553	+	0.971374i	$0.576345\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.49106	−0.310908	−0.155454	−	0.987843i	$-0.549684\pi$
$23$	−1.49106	−0.310908	−0.155454	+	0.987843i	$0.549684\pi$
$24$	0	0
$25$	−4.91022	−0.982043
$26$	0	0
$27$	−2.53625	−0.488101
$28$	0	0
$29$	−0.530372	−0.0984876	−0.0492438	−	0.998787i	$-0.515681\pi$
$29$	−0.530372	−0.0984876	−0.0492438	+	0.998787i	$0.515681\pi$
$30$	0	0
$31$	−1.94838	−0.349939	−0.174970	−	0.984574i	$-0.555983\pi$
$31$	−1.94838	−0.349939	−0.174970	+	0.984574i	$0.555983\pi$
$32$	0	0
$33$	−2.26556	−0.394383
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4.19347	0.689402	0.344701	−	0.938713i	$-0.387980\pi$
$37$	4.19347	0.689402	0.344701	+	0.938713i	$0.387980\pi$
$38$	0	0
$39$	0.550846	0.0882060
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	5.51755	0.841418	0.420709	−	0.907196i	$-0.361781\pi$
$43$	5.51755	0.841418	0.420709	+	0.907196i	$0.361781\pi$
$44$	0	0
$45$	−0.841808	−0.125489
$46$	0	0
$47$	8.54096	1.24583	0.622913	−	0.782291i	$-0.285949\pi$
$47$	8.54096	1.24583	0.622913	+	0.782291i	$0.285949\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−0.855216	−0.119754
$52$	0	0
$53$	−11.7557	−1.61477	−0.807383	−	0.590028i	$-0.799117\pi$
$53$	−11.7557	−1.61477	−0.807383	+	0.590028i	$0.799117\pi$
$54$	0	0
$55$	−1.55494	−0.209669
$56$	0	0
$57$	−0.436576	−0.0578259
$58$	0	0
$59$	12.8152	1.66840	0.834198	−	0.551465i	$-0.185931\pi$
$59$	12.8152	1.66840	0.834198	+	0.551465i	$0.185931\pi$
$60$	0	0
$61$	5.93580	0.760002	0.380001	−	0.924986i	$-0.375924\pi$
$61$	5.93580	0.760002	0.380001	+	0.924986i	$0.375924\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.378068	0.0468936
$66$	0	0
$67$	5.50772	0.672875	0.336438	−	0.941706i	$-0.390778\pi$
$67$	5.50772	0.672875	0.336438	+	0.941706i	$0.390778\pi$
$68$	0	0
$69$	−0.650963	−0.0783667
$70$	0	0
$71$	2.28016	0.270605	0.135303	−	0.990804i	$-0.456799\pi$
$71$	2.28016	0.270605	0.135303	+	0.990804i	$0.456799\pi$
$72$	0	0
$73$	4.69240	0.549203	0.274602	−	0.961558i	$-0.411454\pi$
$73$	4.69240	0.549203	0.274602	+	0.961558i	$0.411454\pi$
$74$	0	0
$75$	−2.14368	−0.247531
$76$	0	0
$77$	0	0
$78$	0	0
$79$	5.69468	0.640702	0.320351	−	0.947299i	$-0.396199\pi$
$79$	5.69468	0.640702	0.320351	+	0.947299i	$0.396199\pi$
$80$	0	0
$81$	7.32094	0.813438
$82$	0	0
$83$	−1.46325	−0.160613	−0.0803065	−	0.996770i	$-0.525590\pi$
$83$	−1.46325	−0.160613	−0.0803065	+	0.996770i	$0.525590\pi$
$84$	0	0
$85$	−0.586969	−0.0636657
$86$	0	0
$87$	−0.231548	−0.0248245
$88$	0	0
$89$	3.85050	0.408152	0.204076	−	0.978955i	$-0.434581\pi$
$89$	3.85050	0.408152	0.204076	+	0.978955i	$0.434581\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−0.850616	−0.0882048
$94$	0	0
$95$	−0.299640	−0.0307424
$96$	0	0
$97$	9.69800	0.984683	0.492341	−	0.870402i	$-0.336141\pi$
$97$	9.69800	0.984683	0.492341	+	0.870402i	$0.336141\pi$
$98$	0	0
$99$	14.5791	1.46525
$100$	0	0
$101$	−16.3807	−1.62994	−0.814969	−	0.579505i	$-0.803246\pi$
$101$	−16.3807	−1.62994	−0.814969	+	0.579505i	$0.803246\pi$
$102$	0	0
$103$	−4.87782	−0.480625	−0.240313	−	0.970696i	$-0.577250\pi$
$103$	−4.87782	−0.480625	−0.240313	+	0.970696i	$0.577250\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	7.35554	0.711087	0.355543	−	0.934660i	$-0.384296\pi$
$107$	7.35554	0.711087	0.355543	+	0.934660i	$0.384296\pi$
$108$	0	0
$109$	4.38871	0.420362	0.210181	−	0.977663i	$-0.432595\pi$
$109$	4.38871	0.420362	0.210181	+	0.977663i	$0.432595\pi$
$110$	0	0
$111$	1.83077	0.173769
$112$	0	0
$113$	6.60321	0.621178	0.310589	−	0.950544i	$-0.399474\pi$
$113$	6.60321	0.621178	0.310589	+	0.950544i	$0.399474\pi$
$114$	0	0
$115$	−0.446782	−0.0416626
$116$	0	0
$117$	−3.54474	−0.327711
$118$	0	0
$119$	0	0
$120$	0	0
$121$	15.9297	1.44815
$122$	0	0
$123$	0.873152	0.0787295
$124$	0	0
$125$	−2.96949	−0.265600
$126$	0	0
$127$	−2.94961	−0.261736	−0.130868	−	0.991400i	$-0.541776\pi$
$127$	−2.94961	−0.261736	−0.130868	+	0.991400i	$0.541776\pi$
$128$	0	0
$129$	2.40883	0.212086
$130$	0	0
$131$	12.9890	1.13486	0.567428	−	0.823423i	$-0.307938\pi$
$131$	12.9890	1.13486	0.567428	+	0.823423i	$0.307938\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−0.759960	−0.0654069
$136$	0	0
$137$	13.6324	1.16469	0.582347	−	0.812940i	$-0.302134\pi$
$137$	13.6324	1.16469	0.582347	+	0.812940i	$0.302134\pi$
$138$	0	0
$139$	−20.6698	−1.75319	−0.876596	−	0.481226i	$-0.840191\pi$
$139$	−20.6698	−1.75319	−0.876596	+	0.481226i	$0.840191\pi$
$140$	0	0
$141$	3.72878	0.314020
$142$	0	0
$143$	−6.54766	−0.547543
$144$	0	0
$145$	−0.158920	−0.0131976
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−6.67426	−0.546777	−0.273388	−	0.961904i	$-0.588144\pi$
$149$	−6.67426	−0.546777	−0.273388	+	0.961904i	$0.588144\pi$
$150$	0	0
$151$	15.6647	1.27478	0.637388	−	0.770543i	$-0.280015\pi$
$151$	15.6647	1.27478	0.637388	+	0.770543i	$0.280015\pi$
$152$	0	0
$153$	5.50338	0.444922
$154$	0	0
$155$	−0.583812	−0.0468929
$156$	0	0
$157$	−5.95923	−0.475598	−0.237799	−	0.971314i	$-0.576426\pi$
$157$	−5.95923	−0.475598	−0.237799	+	0.971314i	$0.576426\pi$
$158$	0	0
$159$	−5.13225	−0.407014
$160$	0	0
$161$	0	0
$162$	0	0
$163$	17.4627	1.36779	0.683893	−	0.729582i	$-0.260286\pi$
$163$	17.4627	1.36779	0.683893	+	0.729582i	$0.260286\pi$
$164$	0	0
$165$	−0.678852	−0.0528485
$166$	0	0
$167$	−1.72621	−0.133578	−0.0667892	−	0.997767i	$-0.521276\pi$
$167$	−1.72621	−0.133578	−0.0667892	+	0.997767i	$0.521276\pi$
$168$	0	0
$169$	−11.4080	−0.877539
$170$	0	0
$171$	2.80940	0.214840
$172$	0	0
$173$	0.879143	0.0668400	0.0334200	−	0.999441i	$-0.489360\pi$
$173$	0.879143	0.0668400	0.0334200	+	0.999441i	$0.489360\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	5.59481	0.420531
$178$	0	0
$179$	7.34176	0.548749	0.274374	−	0.961623i	$-0.411529\pi$
$179$	7.34176	0.548749	0.274374	+	0.961623i	$0.411529\pi$
$180$	0	0
$181$	18.2504	1.35654	0.678270	−	0.734813i	$-0.262730\pi$
$181$	18.2504	1.35654	0.678270	+	0.734813i	$0.262730\pi$
$182$	0	0
$183$	2.59143	0.191564
$184$	0	0
$185$	1.25653	0.0923818
$186$	0	0
$187$	10.1656	0.743379
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−7.37207	−0.533425	−0.266712	−	0.963776i	$-0.585937\pi$
$191$	−7.37207	−0.533425	−0.266712	+	0.963776i	$0.585937\pi$
$192$	0	0
$193$	1.46286	0.105299	0.0526493	−	0.998613i	$-0.483233\pi$
$193$	1.46286	0.105299	0.0526493	+	0.998613i	$0.483233\pi$
$194$	0	0
$195$	0.165055	0.0118199
$196$	0	0
$197$	2.41230	0.171870	0.0859348	−	0.996301i	$-0.472612\pi$
$197$	2.41230	0.171870	0.0859348	+	0.996301i	$0.472612\pi$
$198$	0	0
$199$	−15.9571	−1.13117	−0.565586	−	0.824689i	$-0.691350\pi$
$199$	−15.9571	−1.13117	−0.565586	+	0.824689i	$0.691350\pi$
$200$	0	0
$201$	2.40454	0.169603
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0.599279	0.0418555
$206$	0	0
$207$	4.18899	0.291155
$208$	0	0
$209$	5.18938	0.358957
$210$	0	0
$211$	13.2835	0.914474	0.457237	−	0.889345i	$-0.348839\pi$
$211$	13.2835	0.914474	0.457237	+	0.889345i	$0.348839\pi$
$212$	0	0
$213$	0.995464	0.0682081
$214$	0	0
$215$	1.65328	0.112753
$216$	0	0
$217$	0	0
$218$	0	0
$219$	2.04859	0.138431
$220$	0	0
$221$	−2.47165	−0.166261
$222$	0	0
$223$	28.1602	1.88574	0.942872	−	0.333154i	$-0.108113\pi$
$223$	28.1602	1.88574	0.942872	+	0.333154i	$0.108113\pi$
$224$	0	0
$225$	13.7948	0.919651
$226$	0	0
$227$	−18.0776	−1.19985	−0.599926	−	0.800055i	$-0.704803\pi$
$227$	−18.0776	−1.19985	−0.599926	+	0.800055i	$0.704803\pi$
$228$	0	0
$229$	−7.21993	−0.477106	−0.238553	−	0.971129i	$-0.576673\pi$
$229$	−7.21993	−0.477106	−0.238553	+	0.971129i	$0.576673\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−4.77207	−0.312629	−0.156314	−	0.987707i	$-0.549961\pi$
$233$	−4.77207	−0.312629	−0.156314	+	0.987707i	$0.549961\pi$
$234$	0	0
$235$	2.55921	0.166944
$236$	0	0
$237$	2.48616	0.161494
$238$	0	0
$239$	3.62602	0.234548	0.117274	−	0.993100i	$-0.462584\pi$
$239$	3.62602	0.234548	0.117274	+	0.993100i	$0.462584\pi$
$240$	0	0
$241$	−17.3348	−1.11663	−0.558317	−	0.829628i	$-0.688553\pi$
$241$	−17.3348	−1.11663	−0.558317	+	0.829628i	$0.688553\pi$
$242$	0	0
$243$	10.8049	0.693134
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1.26174	−0.0802827
$248$	0	0
$249$	−0.638822	−0.0404837
$250$	0	0
$251$	−9.71280	−0.613067	−0.306533	−	0.951860i	$-0.599169\pi$
$251$	−9.71280	−0.613067	−0.306533	+	0.951860i	$0.599169\pi$
$252$	0	0
$253$	7.73770	0.486465
$254$	0	0
$255$	−0.256257	−0.0160474
$256$	0	0
$257$	7.18296	0.448061	0.224030	−	0.974582i	$-0.428079\pi$
$257$	7.18296	0.448061	0.224030	+	0.974582i	$0.428079\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	1.49003	0.0922304
$262$	0	0
$263$	2.74049	0.168986	0.0844928	−	0.996424i	$-0.473073\pi$
$263$	2.74049	0.168986	0.0844928	+	0.996424i	$0.473073\pi$
$264$	0	0
$265$	−3.52247	−0.216383
$266$	0	0
$267$	1.68104	0.102878
$268$	0	0
$269$	−20.4080	−1.24430	−0.622148	−	0.782900i	$-0.713740\pi$
$269$	−20.4080	−1.24430	−0.622148	+	0.782900i	$0.713740\pi$
$270$	0	0
$271$	14.7090	0.893511	0.446755	−	0.894656i	$-0.352579\pi$
$271$	14.7090	0.893511	0.446755	+	0.894656i	$0.352579\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	25.4810	1.53656
$276$	0	0
$277$	20.1018	1.20780	0.603901	−	0.797060i	$-0.293612\pi$
$277$	20.1018	1.20780	0.603901	+	0.797060i	$0.293612\pi$
$278$	0	0
$279$	5.47378	0.327707
$280$	0	0
$281$	15.1058	0.901136	0.450568	−	0.892742i	$-0.351222\pi$
$281$	15.1058	0.901136	0.450568	+	0.892742i	$0.351222\pi$
$282$	0	0
$283$	5.05412	0.300436	0.150218	−	0.988653i	$-0.452002\pi$
$283$	5.05412	0.300436	0.150218	+	0.988653i	$0.452002\pi$
$284$	0	0
$285$	−0.130816	−0.00774884
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−13.1627	−0.774274
$290$	0	0
$291$	4.23392	0.248196
$292$	0	0
$293$	13.5367	0.790823	0.395411	−	0.918504i	$-0.370602\pi$
$293$	13.5367	0.790823	0.395411	+	0.918504i	$0.370602\pi$
$294$	0	0
$295$	3.83994	0.223570
$296$	0	0
$297$	13.1615	0.763711
$298$	0	0
$299$	−1.88134	−0.108801
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−7.15141	−0.410838
$304$	0	0
$305$	1.77860	0.101842
$306$	0	0
$307$	−0.853123	−0.0486903	−0.0243452	−	0.999704i	$-0.507750\pi$
$307$	−0.853123	−0.0486903	−0.0243452	+	0.999704i	$0.507750\pi$
$308$	0	0
$309$	−2.12954	−0.121145
$310$	0	0
$311$	−17.3930	−0.986264	−0.493132	−	0.869954i	$-0.664148\pi$
$311$	−17.3930	−0.986264	−0.493132	+	0.869954i	$0.664148\pi$
$312$	0	0
$313$	−24.0556	−1.35971	−0.679853	−	0.733349i	$-0.737956\pi$
$313$	−24.0556	−1.35971	−0.679853	+	0.733349i	$0.737956\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	7.45556	0.418746	0.209373	−	0.977836i	$-0.432858\pi$
$317$	7.45556	0.418746	0.209373	+	0.977836i	$0.432858\pi$
$318$	0	0
$319$	2.75230	0.154099
$320$	0	0
$321$	3.21125	0.179235
$322$	0	0
$323$	1.95892	0.108997
$324$	0	0
$325$	−6.19542	−0.343660
$326$	0	0
$327$	1.91600	0.105955
$328$	0	0
$329$	0	0
$330$	0	0
$331$	33.9268	1.86479	0.932393	−	0.361445i	$-0.117717\pi$
$331$	33.9268	1.86479	0.932393	+	0.361445i	$0.117717\pi$
$332$	0	0
$333$	−11.7811	−0.645602
$334$	0	0
$335$	1.65033	0.0901672
$336$	0	0
$337$	−16.1792	−0.881336	−0.440668	−	0.897670i	$-0.645258\pi$
$337$	−16.1792	−0.881336	−0.440668	+	0.897670i	$0.645258\pi$
$338$	0	0
$339$	2.88280	0.156572
$340$	0	0
$341$	10.1109	0.547535
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−0.195054	−0.0105014
$346$	0	0
$347$	−28.5014	−1.53003	−0.765016	−	0.644011i	$-0.777269\pi$
$347$	−28.5014	−1.53003	−0.765016	+	0.644011i	$0.777269\pi$
$348$	0	0
$349$	24.1291	1.29160	0.645801	−	0.763506i	$-0.276524\pi$
$349$	24.1291	1.29160	0.645801	+	0.763506i	$0.276524\pi$
$350$	0	0
$351$	−3.20009	−0.170808
$352$	0	0
$353$	−24.4536	−1.30154	−0.650768	−	0.759277i	$-0.725553\pi$
$353$	−24.4536	−1.30154	−0.650768	+	0.759277i	$0.725553\pi$
$354$	0	0
$355$	0.683227	0.0362619
$356$	0	0
$357$	0	0
$358$	0	0
$359$	29.7393	1.56958	0.784789	−	0.619763i	$-0.212771\pi$
$359$	29.7393	1.56958	0.784789	+	0.619763i	$0.212771\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	6.95452	0.365018
$364$	0	0
$365$	1.40603	0.0735949
$366$	0	0
$367$	−24.3245	−1.26973	−0.634865	−	0.772623i	$-0.718944\pi$
$367$	−24.3245	−1.26973	−0.634865	+	0.772623i	$0.718944\pi$
$368$	0	0
$369$	−5.61880	−0.292503
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−25.1701	−1.30326	−0.651628	−	0.758539i	$-0.725914\pi$
$373$	−25.1701	−1.30326	−0.651628	+	0.758539i	$0.725914\pi$
$374$	0	0
$375$	−1.29641	−0.0669463
$376$	0	0
$377$	−0.669192	−0.0344652
$378$	0	0
$379$	5.07880	0.260880	0.130440	−	0.991456i	$-0.458361\pi$
$379$	5.07880	0.260880	0.130440	+	0.991456i	$0.458361\pi$
$380$	0	0
$381$	−1.28773	−0.0659724
$382$	0	0
$383$	−4.15115	−0.212114	−0.106057	−	0.994360i	$-0.533823\pi$
$383$	−4.15115	−0.212114	−0.106057	+	0.994360i	$0.533823\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−15.5010	−0.787960
$388$	0	0
$389$	−7.83901	−0.397454	−0.198727	−	0.980055i	$-0.563681\pi$
$389$	−7.83901	−0.397454	−0.198727	+	0.980055i	$0.563681\pi$
$390$	0	0
$391$	2.92087	0.147715
$392$	0	0
$393$	5.67070	0.286049
$394$	0	0
$395$	1.70635	0.0858560
$396$	0	0
$397$	−3.07757	−0.154459	−0.0772295	−	0.997013i	$-0.524607\pi$
$397$	−3.07757	−0.154459	−0.0772295	+	0.997013i	$0.524607\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−37.4966	−1.87249	−0.936246	−	0.351344i	$-0.885725\pi$
$401$	−37.4966	−1.87249	−0.936246	+	0.351344i	$0.885725\pi$
$402$	0	0
$403$	−2.45835	−0.122459
$404$	0	0
$405$	2.19364	0.109003
$406$	0	0
$407$	−21.7615	−1.07868
$408$	0	0
$409$	34.7551	1.71853	0.859265	−	0.511530i	$-0.170921\pi$
$409$	34.7551	1.71853	0.859265	+	0.511530i	$0.170921\pi$
$410$	0	0
$411$	5.95158	0.293570
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−0.438449	−0.0215226
$416$	0	0
$417$	−9.02396	−0.441905
$418$	0	0
$419$	−21.5451	−1.05255	−0.526274	−	0.850315i	$-0.676411\pi$
$419$	−21.5451	−1.05255	−0.526274	+	0.850315i	$0.676411\pi$
$420$	0	0
$421$	2.66307	0.129790	0.0648951	−	0.997892i	$-0.479329\pi$
$421$	2.66307	0.129790	0.0648951	+	0.997892i	$0.479329\pi$
$422$	0	0
$423$	−23.9950	−1.16668
$424$	0	0
$425$	9.61870	0.466575
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−2.85855	−0.138012
$430$	0	0
$431$	−24.7176	−1.19060	−0.595302	−	0.803502i	$-0.702967\pi$
$431$	−24.7176	−1.19060	−0.595302	+	0.803502i	$0.702967\pi$
$432$	0	0
$433$	7.26963	0.349356	0.174678	−	0.984626i	$-0.444112\pi$
$433$	7.26963	0.349356	0.174678	+	0.984626i	$0.444112\pi$
$434$	0	0
$435$	−0.0693809	−0.00332656
$436$	0	0
$437$	1.49106	0.0713272
$438$	0	0
$439$	35.0201	1.67142	0.835710	−	0.549171i	$-0.185056\pi$
$439$	35.0201	1.67142	0.835710	+	0.549171i	$0.185056\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−6.11969	−0.290755	−0.145378	−	0.989376i	$-0.546440\pi$
$443$	−6.11969	−0.290755	−0.145378	+	0.989376i	$0.546440\pi$
$444$	0	0
$445$	1.15376	0.0546936
$446$	0	0
$447$	−2.91382	−0.137819
$448$	0	0
$449$	34.7695	1.64087	0.820436	−	0.571738i	$-0.193731\pi$
$449$	34.7695	1.64087	0.820436	+	0.571738i	$0.193731\pi$
$450$	0	0
$451$	−10.3788	−0.488717
$452$	0	0
$453$	6.83884	0.321317
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−22.5727	−1.05591	−0.527953	−	0.849274i	$-0.677040\pi$
$457$	−22.5727	−1.05591	−0.527953	+	0.849274i	$0.677040\pi$
$458$	0	0
$459$	4.96829	0.231900
$460$	0	0
$461$	−25.1756	−1.17254	−0.586272	−	0.810115i	$-0.699405\pi$
$461$	−25.1756	−1.17254	−0.586272	+	0.810115i	$0.699405\pi$
$462$	0	0
$463$	−7.74829	−0.360093	−0.180047	−	0.983658i	$-0.557625\pi$
$463$	−7.74829	−0.360093	−0.180047	+	0.983658i	$0.557625\pi$
$464$	0	0
$465$	−0.254878	−0.0118197
$466$	0	0
$467$	−31.0781	−1.43812	−0.719060	−	0.694948i	$-0.755428\pi$
$467$	−31.0781	−1.43812	−0.719060	+	0.694948i	$0.755428\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−2.60166	−0.119878
$472$	0	0
$473$	−28.6327	−1.31653
$474$	0	0
$475$	4.91022	0.225296
$476$	0	0
$477$	33.0264	1.51217
$478$	0	0
$479$	27.3206	1.24831	0.624154	−	0.781301i	$-0.285444\pi$
$479$	27.3206	1.24831	0.624154	+	0.781301i	$0.285444\pi$
$480$	0	0
$481$	5.29107	0.241252
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2.90591	0.131950
$486$	0	0
$487$	25.0903	1.13695	0.568475	−	0.822701i	$-0.307534\pi$
$487$	25.0903	1.13695	0.568475	+	0.822701i	$0.307534\pi$
$488$	0	0
$489$	7.62381	0.344761
$490$	0	0
$491$	29.4362	1.32844	0.664219	−	0.747538i	$-0.268764\pi$
$491$	29.4362	1.32844	0.664219	+	0.747538i	$0.268764\pi$
$492$	0	0
$493$	1.03895	0.0467921
$494$	0	0
$495$	4.36846	0.196348
$496$	0	0
$497$	0	0
$498$	0	0
$499$	42.2479	1.89127	0.945637	−	0.325225i	$-0.105440\pi$
$499$	42.2479	1.89127	0.945637	+	0.325225i	$0.105440\pi$
$500$	0	0
$501$	−0.753624	−0.0336694
$502$	0	0
$503$	9.25390	0.412611	0.206305	−	0.978488i	$-0.433856\pi$
$503$	9.25390	0.412611	0.206305	+	0.978488i	$0.433856\pi$
$504$	0	0
$505$	−4.90830	−0.218416
$506$	0	0
$507$	−4.98046	−0.221190
$508$	0	0
$509$	23.8049	1.05514	0.527568	−	0.849513i	$-0.323104\pi$
$509$	23.8049	1.05514	0.527568	+	0.849513i	$0.323104\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	2.53625	0.111978
$514$	0	0
$515$	−1.46159	−0.0644052
$516$	0	0
$517$	−44.3223	−1.94929
$518$	0	0
$519$	0.383813	0.0168475
$520$	0	0
$521$	−32.0845	−1.40565	−0.702823	−	0.711364i	$-0.748078\pi$
$521$	−32.0845	−1.40565	−0.702823	+	0.711364i	$0.748078\pi$
$522$	0	0
$523$	−17.5524	−0.767513	−0.383756	−	0.923434i	$-0.625370\pi$
$523$	−17.5524	−0.767513	−0.383756	+	0.923434i	$0.625370\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.81671	0.166259
$528$	0	0
$529$	−20.7767	−0.903336
$530$	0	0
$531$	−36.0030	−1.56240
$532$	0	0
$533$	2.52348	0.109304
$534$	0	0
$535$	2.20401	0.0952877
$536$	0	0
$537$	3.20524	0.138316
$538$	0	0
$539$	0	0
$540$	0	0
$541$	33.3770	1.43499	0.717495	−	0.696564i	$-0.245289\pi$
$541$	33.3770	1.43499	0.717495	+	0.696564i	$0.245289\pi$
$542$	0	0
$543$	7.96768	0.341926
$544$	0	0
$545$	1.31503	0.0563297
$546$	0	0
$547$	39.4470	1.68663	0.843316	−	0.537418i	$-0.180600\pi$
$547$	39.4470	1.68663	0.843316	+	0.537418i	$0.180600\pi$
$548$	0	0
$549$	−16.6761	−0.711717
$550$	0	0
$551$	0.530372	0.0225946
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0.548571	0.0232855
$556$	0	0
$557$	−35.4844	−1.50352	−0.751760	−	0.659437i	$-0.770795\pi$
$557$	−35.4844	−1.50352	−0.751760	+	0.659437i	$0.770795\pi$
$558$	0	0
$559$	6.96172	0.294449
$560$	0	0
$561$	4.43804	0.187374
$562$	0	0
$563$	44.7109	1.88434	0.942170	−	0.335135i	$-0.108782\pi$
$563$	44.7109	1.88434	0.942170	+	0.335135i	$0.108782\pi$
$564$	0	0
$565$	1.97858	0.0832396
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−9.03775	−0.378882	−0.189441	−	0.981892i	$-0.560668\pi$
$569$	−9.03775	−0.378882	−0.189441	+	0.981892i	$0.560668\pi$
$570$	0	0
$571$	16.9643	0.709933	0.354966	−	0.934879i	$-0.384492\pi$
$571$	16.9643	0.709933	0.354966	+	0.934879i	$0.384492\pi$
$572$	0	0
$573$	−3.21847	−0.134454
$574$	0	0
$575$	7.32144	0.305325
$576$	0	0
$577$	17.6676	0.735514	0.367757	−	0.929922i	$-0.380126\pi$
$577$	17.6676	0.735514	0.367757	+	0.929922i	$0.380126\pi$
$578$	0	0
$579$	0.638648	0.0265413
$580$	0	0
$581$	0	0
$582$	0	0
$583$	61.0047	2.52656
$584$	0	0
$585$	−1.06214	−0.0439143
$586$	0	0
$587$	15.6759	0.647014	0.323507	−	0.946226i	$-0.395138\pi$
$587$	15.6759	0.647014	0.323507	+	0.946226i	$0.395138\pi$
$588$	0	0
$589$	1.94838	0.0802816
$590$	0	0
$591$	1.05315	0.0433210
$592$	0	0
$593$	16.5892	0.681237	0.340619	−	0.940202i	$-0.389364\pi$
$593$	16.5892	0.681237	0.340619	+	0.940202i	$0.389364\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−6.96651	−0.285120
$598$	0	0
$599$	48.6293	1.98694	0.993469	−	0.114105i	$-0.0364001\pi$
$599$	48.6293	1.98694	0.993469	+	0.114105i	$0.0364001\pi$
$600$	0	0
$601$	23.2984	0.950362	0.475181	−	0.879888i	$-0.342383\pi$
$601$	23.2984	0.950362	0.475181	+	0.879888i	$0.342383\pi$
$602$	0	0
$603$	−15.4734	−0.630125
$604$	0	0
$605$	4.77316	0.194057
$606$	0	0
$607$	−29.0822	−1.18041	−0.590206	−	0.807253i	$-0.700953\pi$
$607$	−29.0822	−1.18041	−0.590206	+	0.807253i	$0.700953\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	10.7765	0.435970
$612$	0	0
$613$	−11.0534	−0.446443	−0.223221	−	0.974768i	$-0.571657\pi$
$613$	−11.0534	−0.446443	−0.223221	+	0.974768i	$0.571657\pi$
$614$	0	0
$615$	0.261631	0.0105500
$616$	0	0
$617$	−37.7605	−1.52018	−0.760090	−	0.649817i	$-0.774845\pi$
$617$	−37.7605	−1.52018	−0.760090	+	0.649817i	$0.774845\pi$
$618$	0	0
$619$	−0.628953	−0.0252798	−0.0126399	−	0.999920i	$-0.504024\pi$
$619$	−0.628953	−0.0252798	−0.0126399	+	0.999920i	$0.504024\pi$
$620$	0	0
$621$	3.78170	0.151755
$622$	0	0
$623$	0	0
$624$	0	0
$625$	23.6613	0.946452
$626$	0	0
$627$	2.26556	0.0904778
$628$	0	0
$629$	−8.21464	−0.327539
$630$	0	0
$631$	17.1786	0.683870	0.341935	−	0.939724i	$-0.388918\pi$
$631$	17.1786	0.683870	0.341935	+	0.939724i	$0.388918\pi$
$632$	0	0
$633$	5.79926	0.230500
$634$	0	0
$635$	−0.883821	−0.0350734
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−6.40589	−0.253413
$640$	0	0
$641$	8.13789	0.321427	0.160714	−	0.987001i	$-0.448620\pi$
$641$	8.13789	0.321427	0.160714	+	0.987001i	$0.448620\pi$
$642$	0	0
$643$	−9.21363	−0.363350	−0.181675	−	0.983359i	$-0.558152\pi$
$643$	−9.21363	−0.363350	−0.181675	+	0.983359i	$0.558152\pi$
$644$	0	0
$645$	0.721781	0.0284201
$646$	0	0
$647$	19.3022	0.758850	0.379425	−	0.925223i	$-0.376122\pi$
$647$	19.3022	0.758850	0.379425	+	0.925223i	$0.376122\pi$
$648$	0	0
$649$	−66.5029	−2.61047
$650$	0	0
$651$	0	0
$652$	0	0
$653$	4.74407	0.185650	0.0928248	−	0.995682i	$-0.470410\pi$
$653$	4.74407	0.185650	0.0928248	+	0.995682i	$0.470410\pi$
$654$	0	0
$655$	3.89203	0.152074
$656$	0	0
$657$	−13.1828	−0.514311
$658$	0	0
$659$	15.2932	0.595738	0.297869	−	0.954607i	$-0.403724\pi$
$659$	15.2932	0.595738	0.297869	+	0.954607i	$0.403724\pi$
$660$	0	0
$661$	−3.90935	−0.152056	−0.0760280	−	0.997106i	$-0.524224\pi$
$661$	−3.90935	−0.152056	−0.0760280	+	0.997106i	$0.524224\pi$
$662$	0	0
$663$	−1.07906	−0.0419073
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0.790818	0.0306206
$668$	0	0
$669$	12.2941	0.475316
$670$	0	0
$671$	−30.8031	−1.18914
$672$	0	0
$673$	−1.12569	−0.0433923	−0.0216962	−	0.999765i	$-0.506907\pi$
$673$	−1.12569	−0.0433923	−0.0216962	+	0.999765i	$0.506907\pi$
$674$	0	0
$675$	12.4535	0.479336
$676$	0	0
$677$	41.0287	1.57686	0.788430	−	0.615125i	$-0.210894\pi$
$677$	41.0287	1.57686	0.788430	+	0.615125i	$0.210894\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−7.89225	−0.302432
$682$	0	0
$683$	−5.73893	−0.219594	−0.109797	−	0.993954i	$-0.535020\pi$
$683$	−5.73893	−0.219594	−0.109797	+	0.993954i	$0.535020\pi$
$684$	0	0
$685$	4.08481	0.156073
$686$	0	0
$687$	−3.15205	−0.120258
$688$	0	0
$689$	−14.8326	−0.565078
$690$	0	0
$691$	−4.36871	−0.166194	−0.0830968	−	0.996541i	$-0.526481\pi$
$691$	−4.36871	−0.166194	−0.0830968	+	0.996541i	$0.526481\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−6.19350	−0.234933
$696$	0	0
$697$	−3.91783	−0.148398
$698$	0	0
$699$	−2.08337	−0.0788004
$700$	0	0
$701$	36.2974	1.37093	0.685466	−	0.728105i	$-0.259598\pi$
$701$	36.2974	1.37093	0.685466	+	0.728105i	$0.259598\pi$
$702$	0	0
$703$	−4.19347	−0.158160
$704$	0	0
$705$	1.11729	0.0420796
$706$	0	0
$707$	0	0
$708$	0	0
$709$	9.88287	0.371159	0.185580	−	0.982629i	$-0.440584\pi$
$709$	9.88287	0.371159	0.185580	+	0.982629i	$0.440584\pi$
$710$	0	0
$711$	−15.9987	−0.599996
$712$	0	0
$713$	2.90516	0.108799
$714$	0	0
$715$	−1.96194	−0.0733723
$716$	0	0
$717$	1.58304	0.0591196
$718$	0	0
$719$	27.9542	1.04251	0.521257	−	0.853399i	$-0.325463\pi$
$719$	27.9542	1.04251	0.521257	+	0.853399i	$0.325463\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−7.56797	−0.281456
$724$	0	0
$725$	2.60424	0.0967191
$726$	0	0
$727$	−32.8007	−1.21651	−0.608255	−	0.793742i	$-0.708130\pi$
$727$	−32.8007	−1.21651	−0.608255	+	0.793742i	$0.708130\pi$
$728$	0	0
$729$	−17.2457	−0.638728
$730$	0	0
$731$	−10.8084	−0.399763
$732$	0	0
$733$	−24.1861	−0.893335	−0.446668	−	0.894700i	$-0.647389\pi$
$733$	−24.1861	−0.893335	−0.446668	+	0.894700i	$0.647389\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−28.5817	−1.05282
$738$	0	0
$739$	26.4600	0.973346	0.486673	−	0.873584i	$-0.338210\pi$
$739$	26.4600	0.973346	0.486673	+	0.873584i	$0.338210\pi$
$740$	0	0
$741$	−0.550846	−0.0202358
$742$	0	0
$743$	33.3257	1.22260	0.611301	−	0.791398i	$-0.290646\pi$
$743$	33.3257	1.22260	0.611301	+	0.791398i	$0.290646\pi$
$744$	0	0
$745$	−1.99987	−0.0732697
$746$	0	0
$747$	4.11087	0.150409
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−4.64306	−0.169428	−0.0847138	−	0.996405i	$-0.526998\pi$
$751$	−4.64306	−0.169428	−0.0847138	+	0.996405i	$0.526998\pi$
$752$	0	0
$753$	−4.24038	−0.154528
$754$	0	0
$755$	4.69377	0.170824
$756$	0	0
$757$	45.0481	1.63730	0.818651	−	0.574291i	$-0.194722\pi$
$757$	45.0481	1.63730	0.818651	+	0.574291i	$0.194722\pi$
$758$	0	0
$759$	3.37809	0.122617
$760$	0	0
$761$	−15.0127	−0.544211	−0.272106	−	0.962267i	$-0.587720\pi$
$761$	−15.0127	−0.544211	−0.272106	+	0.962267i	$0.587720\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	1.64903	0.0596208
$766$	0	0
$767$	16.1695	0.583845
$768$	0	0
$769$	−4.05311	−0.146159	−0.0730794	−	0.997326i	$-0.523283\pi$
$769$	−4.05311	−0.146159	−0.0730794	+	0.997326i	$0.523283\pi$
$770$	0	0
$771$	3.13591	0.112937
$772$	0	0
$773$	−12.3940	−0.445782	−0.222891	−	0.974843i	$-0.571549\pi$
$773$	−12.3940	−0.445782	−0.222891	+	0.974843i	$0.571549\pi$
$774$	0	0
$775$	9.56697	0.343656
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.00000	−0.0716574
$780$	0	0
$781$	−11.8326	−0.423405
$782$	0	0
$783$	1.34515	0.0480719
$784$	0	0
$785$	−1.78562	−0.0637316
$786$	0	0
$787$	21.4231	0.763653	0.381826	−	0.924234i	$-0.375295\pi$
$787$	21.4231	0.763653	0.381826	+	0.924234i	$0.375295\pi$
$788$	0	0
$789$	1.19643	0.0425941
$790$	0	0
$791$	0	0
$792$	0	0
$793$	7.48945	0.265958
$794$	0	0
$795$	−1.53782	−0.0545410
$796$	0	0
$797$	37.1622	1.31635	0.658175	−	0.752865i	$-0.271328\pi$
$797$	37.1622	1.31635	0.658175	+	0.752865i	$0.271328\pi$
$798$	0	0
$799$	−16.7310	−0.591901
$800$	0	0
$801$	−10.8176	−0.382221
$802$	0	0
$803$	−24.3506	−0.859315
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−8.90963	−0.313634
$808$	0	0
$809$	2.08416	0.0732752	0.0366376	−	0.999329i	$-0.488335\pi$
$809$	2.08416	0.0732752	0.0366376	+	0.999329i	$0.488335\pi$
$810$	0	0
$811$	11.2776	0.396009	0.198005	−	0.980201i	$-0.436554\pi$
$811$	11.2776	0.396009	0.198005	+	0.980201i	$0.436554\pi$
$812$	0	0
$813$	6.42162	0.225216
$814$	0	0
$815$	5.23252	0.183287
$816$	0	0
$817$	−5.51755	−0.193035
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−42.5746	−1.48586	−0.742931	−	0.669368i	$-0.766565\pi$
$821$	−42.5746	−1.48586	−0.742931	+	0.669368i	$0.766565\pi$
$822$	0	0
$823$	21.6684	0.755313	0.377657	−	0.925946i	$-0.376730\pi$
$823$	21.6684	0.755313	0.377657	+	0.925946i	$0.376730\pi$
$824$	0	0
$825$	11.1244	0.387302
$826$	0	0
$827$	−31.7162	−1.10288	−0.551440	−	0.834215i	$-0.685921\pi$
$827$	−31.7162	−1.10288	−0.551440	+	0.834215i	$0.685921\pi$
$828$	0	0
$829$	−18.2900	−0.635237	−0.317618	−	0.948219i	$-0.602883\pi$
$829$	−18.2900	−0.635237	−0.317618	+	0.948219i	$0.602883\pi$
$830$	0	0
$831$	8.77598	0.304435
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−0.517242	−0.0178999
$836$	0	0
$837$	4.94157	0.170806
$838$	0	0
$839$	29.8422	1.03027	0.515134	−	0.857109i	$-0.327742\pi$
$839$	29.8422	1.03027	0.515134	+	0.857109i	$0.327742\pi$
$840$	0	0
$841$	−28.7187	−0.990300
$842$	0	0
$843$	6.59483	0.227138
$844$	0	0
$845$	−3.41829	−0.117593
$846$	0	0
$847$	0	0
$848$	0	0
$849$	2.20651	0.0757272
$850$	0	0
$851$	−6.25272	−0.214341
$852$	0	0
$853$	17.2641	0.591111	0.295555	−	0.955326i	$-0.404495\pi$
$853$	17.2641	0.591111	0.295555	+	0.955326i	$0.404495\pi$
$854$	0	0
$855$	0.841808	0.0287892
$856$	0	0
$857$	10.7193	0.366165	0.183082	−	0.983098i	$-0.441393\pi$
$857$	10.7193	0.366165	0.183082	+	0.983098i	$0.441393\pi$
$858$	0	0
$859$	30.1030	1.02710	0.513551	−	0.858059i	$-0.328330\pi$
$859$	30.1030	1.02710	0.513551	+	0.858059i	$0.328330\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	37.5256	1.27739	0.638693	−	0.769462i	$-0.279476\pi$
$863$	37.5256	1.27739	0.638693	+	0.769462i	$0.279476\pi$
$864$	0	0
$865$	0.263426	0.00895676
$866$	0	0
$867$	−5.74650	−0.195161
$868$	0	0
$869$	−29.5519	−1.00248
$870$	0	0
$871$	6.94932	0.235469
$872$	0	0
$873$	−27.2456	−0.922123
$874$	0	0
$875$	0	0
$876$	0	0
$877$	34.1851	1.15435	0.577175	−	0.816621i	$-0.304155\pi$
$877$	34.1851	1.15435	0.577175	+	0.816621i	$0.304155\pi$
$878$	0	0
$879$	5.90980	0.199333
$880$	0	0
$881$	49.6527	1.67284	0.836420	−	0.548089i	$-0.184644\pi$
$881$	49.6527	1.67284	0.836420	+	0.548089i	$0.184644\pi$
$882$	0	0
$883$	−19.4134	−0.653314	−0.326657	−	0.945143i	$-0.605922\pi$
$883$	−19.4134	−0.653314	−0.326657	+	0.945143i	$0.605922\pi$
$884$	0	0
$885$	1.67643	0.0563524
$886$	0	0
$887$	−40.2570	−1.35170	−0.675849	−	0.737040i	$-0.736223\pi$
$887$	−40.2570	−1.35170	−0.675849	+	0.737040i	$0.736223\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−37.9911	−1.27275
$892$	0	0
$893$	−8.54096	−0.285812
$894$	0	0
$895$	2.19988	0.0735340
$896$	0	0
$897$	−0.821347	−0.0274240
$898$	0	0
$899$	1.03337	0.0344647
$900$	0	0
$901$	23.0284	0.767186
$902$	0	0
$903$	0	0
$904$	0	0
$905$	5.46854	0.181780
$906$	0	0
$907$	32.6248	1.08329	0.541645	−	0.840608i	$-0.317802\pi$
$907$	32.6248	1.08329	0.541645	+	0.840608i	$0.317802\pi$
$908$	0	0
$909$	46.0199	1.52638
$910$	0	0
$911$	21.6499	0.717292	0.358646	−	0.933474i	$-0.383238\pi$
$911$	21.6499	0.717292	0.358646	+	0.933474i	$0.383238\pi$
$912$	0	0
$913$	7.59338	0.251304
$914$	0	0
$915$	0.776495	0.0256701
$916$	0	0
$917$	0	0
$918$	0	0
$919$	15.6504	0.516260	0.258130	−	0.966110i	$-0.416894\pi$
$919$	15.6504	0.516260	0.258130	+	0.966110i	$0.416894\pi$
$920$	0	0
$921$	−0.372453	−0.0122728
$922$	0	0
$923$	2.87697	0.0946967
$924$	0	0
$925$	−20.5908	−0.677022
$926$	0	0
$927$	13.7037	0.450090
$928$	0	0
$929$	37.5335	1.23143	0.615717	−	0.787968i	$-0.288867\pi$
$929$	37.5335	1.23143	0.615717	+	0.787968i	$0.288867\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−7.59335	−0.248595
$934$	0	0
$935$	3.04600	0.0996150
$936$	0	0
$937$	−47.1876	−1.54155	−0.770776	−	0.637106i	$-0.780131\pi$
$937$	−47.1876	−1.54155	−0.770776	+	0.637106i	$0.780131\pi$
$938$	0	0
$939$	−10.5021	−0.342724
$940$	0	0
$941$	−55.2500	−1.80110	−0.900550	−	0.434753i	$-0.856836\pi$
$941$	−55.2500	−1.80110	−0.900550	+	0.434753i	$0.856836\pi$
$942$	0	0
$943$	−2.98213	−0.0971114
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−44.1150	−1.43354	−0.716772	−	0.697308i	$-0.754381\pi$
$947$	−44.1150	−1.43354	−0.716772	+	0.697308i	$0.754381\pi$
$948$	0	0
$949$	5.92059	0.192191
$950$	0	0
$951$	3.25492	0.105548
$952$	0	0
$953$	−27.9855	−0.906540	−0.453270	−	0.891373i	$-0.649743\pi$
$953$	−27.9855	−0.906540	−0.453270	+	0.891373i	$0.649743\pi$
$954$	0	0
$955$	−2.20897	−0.0714805
$956$	0	0
$957$	1.20159	0.0388419
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−27.2038	−0.877542
$962$	0	0
$963$	−20.6647	−0.665909
$964$	0	0
$965$	0.438330	0.0141103
$966$	0	0
$967$	−50.3705	−1.61981	−0.809904	−	0.586563i	$-0.800481\pi$
$967$	−50.3705	−1.61981	−0.809904	+	0.586563i	$0.800481\pi$
$968$	0	0
$969$	0.855216	0.0274735
$970$	0	0
$971$	10.7342	0.344478	0.172239	−	0.985055i	$-0.444900\pi$
$971$	10.7342	0.344478	0.172239	+	0.985055i	$0.444900\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−2.70478	−0.0866221
$976$	0	0
$977$	45.7183	1.46266	0.731329	−	0.682025i	$-0.238900\pi$
$977$	45.7183	1.46266	0.731329	+	0.682025i	$0.238900\pi$
$978$	0	0
$979$	−19.9817	−0.638618
$980$	0	0
$981$	−12.3296	−0.393655
$982$	0	0
$983$	32.4020	1.03346	0.516731	−	0.856148i	$-0.327149\pi$
$983$	32.4020	1.03346	0.516731	+	0.856148i	$0.327149\pi$
$984$	0	0
$985$	0.722822	0.0230310
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−8.22701	−0.261604
$990$	0	0
$991$	−16.4348	−0.522068	−0.261034	−	0.965330i	$-0.584063\pi$
$991$	−16.4348	−0.522068	−0.261034	+	0.965330i	$0.584063\pi$
$992$	0	0
$993$	14.8116	0.470033
$994$	0	0
$995$	−4.78139	−0.151580
$996$	0	0
$997$	−10.8413	−0.343347	−0.171673	−	0.985154i	$-0.554917\pi$
$997$	−10.8413	−0.343347	−0.171673	+	0.985154i	$0.554917\pi$
$998$	0	0
$999$	−10.6357	−0.336497

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7448.2.a.br.1.5		8
7.3	odd	6		1064.2.q.n.457.5	yes	16
7.5	odd	6		1064.2.q.n.305.5	✓	16
7.6	odd	2		7448.2.a.bq.1.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1064.2.q.n.305.5	✓	16	7.5	odd	6
1064.2.q.n.457.5	yes	16	7.3	odd	6
7448.2.a.bq.1.4		8	7.6	odd	2
7448.2.a.br.1.5		8	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)

\(f(q)\)	\(=\)	\(q+0.436576 q^{3} +0.299640 q^{5} -2.80940 q^{9} +O(q^{10})\)	\(q+0.436576 q^{3} +0.299640 q^{5} -2.80940 q^{9} -5.18938 q^{11} +1.26174 q^{13} +0.130816 q^{15} -1.95892 q^{17} -1.00000 q^{19} -1.49106 q^{23} -4.91022 q^{25} -2.53625 q^{27} -0.530372 q^{29} -1.94838 q^{31} -2.26556 q^{33} +4.19347 q^{37} +0.550846 q^{39} +2.00000 q^{41} +5.51755 q^{43} -0.841808 q^{45} +8.54096 q^{47} -0.855216 q^{51} -11.7557 q^{53} -1.55494 q^{55} -0.436576 q^{57} +12.8152 q^{59} +5.93580 q^{61} +0.378068 q^{65} +5.50772 q^{67} -0.650963 q^{69} +2.28016 q^{71} +4.69240 q^{73} -2.14368 q^{75} +5.69468 q^{79} +7.32094 q^{81} -1.46325 q^{83} -0.586969 q^{85} -0.231548 q^{87} +3.85050 q^{89} -0.850616 q^{93} -0.299640 q^{95} +9.69800 q^{97} +14.5791 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + q^{5} + 6 q^{9}+O(q^{10}) \) 8 * q + q^5 + 6 * q^9	\( 8 q + q^{5} + 6 q^{9} + 9 q^{11} + 8 q^{15} - 4 q^{17} - 8 q^{19} + 25 q^{23} + 15 q^{25} - 3 q^{27} + 6 q^{29} - 8 q^{33} + 13 q^{37} - 11 q^{39} + 16 q^{41} + 17 q^{43} - 17 q^{45} + 24 q^{47} + 5 q^{51} + 2 q^{53} - 5 q^{55} - 2 q^{59} + 13 q^{61} - 26 q^{65} + 2 q^{67} + 11 q^{69} + 10 q^{71} - 5 q^{73} + 20 q^{75} + 16 q^{79} - 12 q^{81} + 43 q^{83} + 24 q^{85} - 20 q^{87} - 8 q^{89} + 2 q^{93} - q^{95} + 12 q^{97} + 37 q^{99}+O(q^{100}) \) 8 * q + q^5 + 6 * q^9 + 9 * q^11 + 8 * q^15 - 4 * q^17 - 8 * q^19 + 25 * q^23 + 15 * q^25 - 3 * q^27 + 6 * q^29 - 8 * q^33 + 13 * q^37 - 11 * q^39 + 16 * q^41 + 17 * q^43 - 17 * q^45 + 24 * q^47 + 5 * q^51 + 2 * q^53 - 5 * q^55 - 2 * q^59 + 13 * q^61 - 26 * q^65 + 2 * q^67 + 11 * q^69 + 10 * q^71 - 5 * q^73 + 20 * q^75 + 16 * q^79 - 12 * q^81 + 43 * q^83 + 24 * q^85 - 20 * q^87 - 8 * q^89 + 2 * q^93 - q^95 + 12 * q^97 + 37 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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