LMFDB - Embedded newform 28.5.b.a.13.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [28,5,Mod(13,28)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(28, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 5, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("28.13");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 28 = 2^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	28.b (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$2.89435896635$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			13.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	28.13
Dual form			28.5.b.a.13.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6.92820i q^{3} -20.7846i q^{5} +(-7.00000 - 48.4974i) q^{7} +33.0000 q^{9} +O(q^{10})$	$q-6.92820i q^{3} -20.7846i q^{5} +(-7.00000 - 48.4974i) q^{7} +33.0000 q^{9} +18.0000 q^{11} +131.636i q^{13} -144.000 q^{15} +415.692i q^{17} -90.0666i q^{19} +(-336.000 + 48.4974i) q^{21} +738.000 q^{23} +193.000 q^{25} -789.815i q^{27} -846.000 q^{29} -1163.94i q^{31} -124.708i q^{33} +(-1008.00 + 145.492i) q^{35} +2386.00 q^{37} +912.000 q^{39} +1704.34i q^{41} -2510.00 q^{43} -685.892i q^{45} +3408.68i q^{47} +(-2303.00 + 678.964i) q^{49} +2880.00 q^{51} -270.000 q^{53} -374.123i q^{55} -624.000 q^{57} +3138.48i q^{59} -6491.73i q^{61} +(-231.000 - 1600.41i) q^{63} +2736.00 q^{65} +2450.00 q^{67} -5113.01i q^{69} -3150.00 q^{71} -235.559i q^{73} -1337.14i q^{75} +(-126.000 - 872.954i) q^{77} -3982.00 q^{79} -2799.00 q^{81} +5009.09i q^{83} +8640.00 q^{85} +5861.26i q^{87} +7607.17i q^{89} +(6384.00 - 921.451i) q^{91} -8064.00 q^{93} -1872.00 q^{95} +12581.6i q^{97} +594.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 14 q^{7} + 66 q^{9}+O(q^{10}) $ 2 * q - 14 * q^7 + 66 * q^9	$ 2 q - 14 q^{7} + 66 q^{9} + 36 q^{11} - 288 q^{15} - 672 q^{21} + 1476 q^{23} + 386 q^{25} - 1692 q^{29} - 2016 q^{35} + 4772 q^{37} + 1824 q^{39} - 5020 q^{43} - 4606 q^{49} + 5760 q^{51} - 540 q^{53} - 1248 q^{57} - 462 q^{63} + 5472 q^{65} + 4900 q^{67} - 6300 q^{71} - 252 q^{77} - 7964 q^{79} - 5598 q^{81} + 17280 q^{85} + 12768 q^{91} - 16128 q^{93} - 3744 q^{95} + 1188 q^{99}+O(q^{100}) $ 2 * q - 14 * q^7 + 66 * q^9 + 36 * q^11 - 288 * q^15 - 672 * q^21 + 1476 * q^23 + 386 * q^25 - 1692 * q^29 - 2016 * q^35 + 4772 * q^37 + 1824 * q^39 - 5020 * q^43 - 4606 * q^49 + 5760 * q^51 - 540 * q^53 - 1248 * q^57 - 462 * q^63 + 5472 * q^65 + 4900 * q^67 - 6300 * q^71 - 252 * q^77 - 7964 * q^79 - 5598 * q^81 + 17280 * q^85 + 12768 * q^91 - 16128 * q^93 - 3744 * q^95 + 1188 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/28\mathbb{Z}\right)^\times$.

$n$	$15$	$17$
$\chi(n)$	$1$	$-1$

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
$3$		−	6.92820i		−	0.769800i	−0.922958	−	0.384900i	$-0.874236\pi$
$3$		−	6.92820i		−	0.769800i	0.922958	−	0.384900i	$-0.125764\pi$
$4$		0			0
$5$		−	20.7846i		−	0.831384i	−0.909505	−	0.415692i	$-0.863539\pi$
$5$		−	20.7846i		−	0.831384i	0.909505	−	0.415692i	$-0.136461\pi$
$6$		0			0
$7$	−7.00000	−	48.4974i	−0.142857	−	0.989743i
$7$	−7.00000	−	48.4974i	−0.142857	−	0.989743i
$8$		0			0
$9$	33.0000			0.407407
$10$		0			0
$11$	18.0000			0.148760			0.0743802	−	0.997230i	$-0.476302\pi$
$11$	18.0000			0.148760			0.0743802	+	0.997230i	$0.476302\pi$
$12$		0			0
$13$			131.636i			0.778910i	0.921045	+	0.389455i	$0.127337\pi$
$13$			131.636i			0.778910i	−0.921045	+	0.389455i	$0.872663\pi$
$14$		0			0
$15$	−144.000			−0.640000
$16$		0			0
$17$			415.692i			1.43838i	0.694813	+	0.719191i	$0.255487\pi$
$17$			415.692i			1.43838i	−0.694813	+	0.719191i	$0.744513\pi$
$18$		0			0
$19$		−	90.0666i		−	0.249492i	−0.992189	−	0.124746i	$-0.960188\pi$
$19$		−	90.0666i		−	0.249492i	0.992189	−	0.124746i	$-0.0398116\pi$
$20$		0			0
$21$	−336.000	+	48.4974i	−0.761905	+	0.109971i
$22$		0			0
$23$	738.000			1.39509			0.697543	−	0.716543i	$-0.254277\pi$
$23$	738.000			1.39509			0.697543	+	0.716543i	$0.254277\pi$
$24$		0			0
$25$	193.000			0.308800
$26$		0			0
$27$		−	789.815i		−	1.08342i
$28$		0			0
$29$	−846.000			−1.00595			−0.502973	−	0.864302i	$-0.667760\pi$
$29$	−846.000			−1.00595			−0.502973	+	0.864302i	$0.667760\pi$
$30$		0			0
$31$		−	1163.94i		−	1.21117i	−0.795779	−	0.605587i	$-0.792938\pi$
$31$		−	1163.94i		−	1.21117i	0.795779	−	0.605587i	$-0.207062\pi$
$32$		0			0
$33$		−	124.708i		−	0.114516i
$34$		0			0
$35$	−1008.00	+	145.492i	−0.822857	+	0.118769i
$36$		0			0
$37$	2386.00			1.74288			0.871439	−	0.490504i	$-0.163187\pi$
$37$	2386.00			1.74288			0.871439	+	0.490504i	$0.163187\pi$
$38$		0			0
$39$	912.000			0.599606
$40$		0			0
$41$			1704.34i			1.01388i	0.861980	+	0.506942i	$0.169224\pi$
$41$			1704.34i			1.01388i	−0.861980	+	0.506942i	$0.830776\pi$
$42$		0			0
$43$	−2510.00			−1.35749			−0.678745	−	0.734374i	$-0.737476\pi$
$43$	−2510.00			−1.35749			−0.678745	+	0.734374i	$0.737476\pi$
$44$		0			0
$45$		−	685.892i		−	0.338712i
$46$		0			0
$47$			3408.68i			1.54309i	0.636177	+	0.771543i	$0.280515\pi$
$47$			3408.68i			1.54309i	−0.636177	+	0.771543i	$0.719485\pi$
$48$		0			0
$49$	−2303.00	+	678.964i	−0.959184	+	0.282784i
$50$		0			0
$51$	2880.00			1.10727
$52$		0			0
$53$	−270.000			−0.0961196			−0.0480598	−	0.998844i	$-0.515304\pi$
$53$	−270.000			−0.0961196			−0.0480598	+	0.998844i	$0.515304\pi$
$54$		0			0
$55$		−	374.123i		−	0.123677i
$56$		0			0
$57$	−624.000			−0.192059
$58$		0			0
$59$			3138.48i			0.901602i	0.892625	+	0.450801i	$0.148862\pi$
$59$			3138.48i			0.901602i	−0.892625	+	0.450801i	$0.851138\pi$
$60$		0			0
$61$		−	6491.73i		−	1.74462i	−0.488954	−	0.872309i	$-0.662622\pi$
$61$		−	6491.73i		−	1.74462i	0.488954	−	0.872309i	$-0.337378\pi$
$62$		0			0
$63$	−231.000	−	1600.41i	−0.0582011	−	0.403229i
$64$		0			0
$65$	2736.00			0.647574
$66$		0			0
$67$	2450.00			0.545779			0.272889	−	0.962045i	$-0.412021\pi$
$67$	2450.00			0.545779			0.272889	+	0.962045i	$0.412021\pi$
$68$		0			0
$69$		−	5113.01i		−	1.07394i
$70$		0			0
$71$	−3150.00			−0.624876			−0.312438	−	0.949938i	$-0.601146\pi$
$71$	−3150.00			−0.624876			−0.312438	+	0.949938i	$0.601146\pi$
$72$		0			0
$73$		−	235.559i		−	0.0442032i	−0.999756	−	0.0221016i	$-0.992964\pi$
$73$		−	235.559i		−	0.0442032i	0.999756	−	0.0221016i	$-0.00703573\pi$
$74$		0			0
$75$		−	1337.14i		−	0.237714i
$76$		0			0
$77$	−126.000	−	872.954i	−0.0212515	−	0.147235i
$78$		0			0
$79$	−3982.00			−0.638039			−0.319019	−	0.947748i	$-0.603354\pi$
$79$	−3982.00			−0.638039			−0.319019	+	0.947748i	$0.603354\pi$
$80$		0			0
$81$	−2799.00			−0.426612
$82$		0			0
$83$			5009.09i			0.727114i	0.931572	+	0.363557i	$0.118438\pi$
$83$			5009.09i			0.727114i	−0.931572	+	0.363557i	$0.881562\pi$
$84$		0			0
$85$	8640.00			1.19585
$86$		0			0
$87$			5861.26i			0.774377i
$88$		0			0
$89$			7607.17i			0.960380i	0.877165	+	0.480190i	$0.159432\pi$
$89$			7607.17i			0.960380i	−0.877165	+	0.480190i	$0.840568\pi$
$90$		0			0
$91$	6384.00	−	921.451i	0.770921	−	0.111273i
$92$		0			0
$93$	−8064.00			−0.932362
$94$		0			0
$95$	−1872.00			−0.207424
$96$		0			0
$97$			12581.6i			1.33719i	0.743627	+	0.668595i	$0.233104\pi$
$97$			12581.6i			1.33719i	−0.743627	+	0.668595i	$0.766896\pi$
$98$		0			0
$99$	594.000			0.0606061
$100$		0			0
$101$		−	11161.3i		−	1.09414i	−0.837086	−	0.547071i	$-0.815743\pi$
$101$		−	11161.3i		−	1.09414i	0.837086	−	0.547071i	$-0.184257\pi$
$102$		0			0
$103$			845.241i			0.0796721i	0.999206	+	0.0398360i	$0.0126836\pi$
$103$			845.241i			0.0796721i	−0.999206	+	0.0398360i	$0.987316\pi$
$104$		0			0
$105$	1008.00	+	6983.63i	0.0914286	+	0.633436i
$106$		0			0
$107$	−12942.0			−1.13040			−0.565202	−	0.824952i	$-0.691202\pi$
$107$	−12942.0			−1.13040			−0.565202	+	0.824952i	$0.691202\pi$
$108$		0			0
$109$	−7022.00			−0.591028			−0.295514	−	0.955338i	$-0.595491\pi$
$109$	−7022.00			−0.591028			−0.295514	+	0.955338i	$0.595491\pi$
$110$		0			0
$111$		−	16530.7i		−	1.34167i
$112$		0			0
$113$	18738.0			1.46746			0.733730	−	0.679441i	$-0.237778\pi$
$113$	18738.0			1.46746			0.733730	+	0.679441i	$0.237778\pi$
$114$		0			0
$115$		−	15339.0i		−	1.15985i
$116$		0			0
$117$			4343.98i			0.317334i
$118$		0			0
$119$	20160.0	−	2909.85i	1.42363	−	0.205483i
$120$		0			0
$121$	−14317.0			−0.977870
$122$		0			0
$123$	11808.0			0.780488
$124$		0			0
$125$		−	17001.8i		−	1.08812i
$126$		0			0
$127$	−2302.00			−0.142724			−0.0713621	−	0.997450i	$-0.522735\pi$
$127$	−2302.00			−0.142724			−0.0713621	+	0.997450i	$0.522735\pi$
$128$		0			0
$129$			17389.8i			1.04500i
$130$		0			0
$131$		−	19516.7i		−	1.13727i	−0.822589	−	0.568637i	$-0.807471\pi$
$131$		−	19516.7i		−	1.13727i	0.822589	−	0.568637i	$-0.192529\pi$
$132$		0			0
$133$	−4368.00	+	630.466i	−0.246933	+	0.0356417i
$134$		0			0
$135$	−16416.0			−0.900741
$136$		0			0
$137$	−26334.0			−1.40306			−0.701529	−	0.712641i	$-0.747499\pi$
$137$	−26334.0			−1.40306			−0.701529	+	0.712641i	$0.747499\pi$
$138$		0			0
$139$			15914.1i			0.823668i	0.911259	+	0.411834i	$0.135112\pi$
$139$			15914.1i			0.823668i	−0.911259	+	0.411834i	$0.864888\pi$
$140$		0			0
$141$	23616.0			1.18787
$142$		0			0
$143$			2369.45i			0.115871i
$144$		0			0
$145$			17583.8i			0.836327i
$146$		0			0
$147$	4704.00	+	15955.7i	0.217687	+	0.738380i
$148$		0			0
$149$	11826.0			0.532679			0.266339	−	0.963879i	$-0.414186\pi$
$149$	11826.0			0.532679			0.266339	+	0.963879i	$0.414186\pi$
$150$		0			0
$151$	15970.0			0.700408			0.350204	−	0.936673i	$-0.386112\pi$
$151$	15970.0			0.700408			0.350204	+	0.936673i	$0.386112\pi$
$152$		0			0
$153$			13717.8i			0.586007i
$154$		0			0
$155$	−24192.0			−1.00695
$156$		0			0
$157$			28703.5i			1.16449i	0.813013	+	0.582246i	$0.197826\pi$
$157$			28703.5i			1.16449i	−0.813013	+	0.582246i	$0.802174\pi$
$158$		0			0
$159$			1870.61i			0.0739929i
$160$		0			0
$161$	−5166.00	−	35791.1i	−0.199298	−	1.38078i
$162$		0			0
$163$	−7822.00			−0.294403			−0.147202	−	0.989107i	$-0.547027\pi$
$163$	−7822.00			−0.294403			−0.147202	+	0.989107i	$0.547027\pi$
$164$		0			0
$165$	−2592.00			−0.0952066
$166$		0			0
$167$			24234.9i			0.868975i	0.900678	+	0.434488i	$0.143071\pi$
$167$			24234.9i			0.868975i	−0.900678	+	0.434488i	$0.856929\pi$
$168$		0			0
$169$	11233.0			0.393299
$170$		0			0
$171$		−	2972.20i		−	0.101645i
$172$		0			0
$173$		−	28745.1i		−	0.960444i	−0.877147	−	0.480222i	$-0.840556\pi$
$173$		−	28745.1i		−	0.960444i	0.877147	−	0.480222i	$-0.159444\pi$
$174$		0			0
$175$	−1351.00	−	9360.00i	−0.0441143	−	0.305633i
$176$		0			0
$177$	21744.0			0.694053
$178$		0			0
$179$	8946.00			0.279205			0.139602	−	0.990208i	$-0.455418\pi$
$179$	8946.00			0.279205			0.139602	+	0.990208i	$0.455418\pi$
$180$		0			0
$181$		−	30920.6i		−	0.943823i	−0.881646	−	0.471911i	$-0.843564\pi$
$181$		−	30920.6i		−	0.943823i	0.881646	−	0.471911i	$-0.156436\pi$
$182$		0			0
$183$	−44976.0			−1.34301
$184$		0			0
$185$		−	49592.1i		−	1.44900i
$186$		0			0
$187$			7482.46i			0.213974i
$188$		0			0
$189$	−38304.0	+	5528.71i	−1.07231	+	0.154775i
$190$		0			0
$191$	60786.0			1.66624			0.833119	−	0.553094i	$-0.186553\pi$
$191$	60786.0			1.66624			0.833119	+	0.553094i	$0.186553\pi$
$192$		0			0
$193$	10546.0			0.283122			0.141561	−	0.989930i	$-0.454788\pi$
$193$	10546.0			0.283122			0.141561	+	0.989930i	$0.454788\pi$
$194$		0			0
$195$		−	18955.6i		−	0.498503i
$196$		0			0
$197$	18.0000			0.000463810			0.000231905	−	1.00000i	$-0.499926\pi$
$197$	18.0000			0.000463810			0.000231905		1.00000i	$0.499926\pi$
$198$		0			0
$199$		−	96.9948i		−	0.00244930i	−0.999999	−	0.00122465i	$-0.999610\pi$
$199$		−	96.9948i		−	0.00244930i	0.999999	−	0.00122465i	$-0.000389819\pi$
$200$		0			0
$201$		−	16974.1i		−	0.420141i
$202$		0			0
$203$	5922.00	+	41028.8i	0.143706	+	0.995628i
$204$		0			0
$205$	35424.0			0.842927
$206$		0			0
$207$	24354.0			0.568368
$208$		0			0
$209$		−	1621.20i		−	0.0371145i
$210$		0			0
$211$	−46190.0			−1.03749			−0.518744	−	0.854930i	$-0.673600\pi$
$211$	−46190.0			−1.03749			−0.518744	+	0.854930i	$0.673600\pi$
$212$		0			0
$213$			21823.8i			0.481030i
$214$		0			0
$215$			52169.4i			1.12860i
$216$		0			0
$217$	−56448.0	+	8147.57i	−1.19875	+	0.173025i
$218$		0			0
$219$	−1632.00			−0.0340276
$220$		0			0
$221$	−54720.0			−1.12037
$222$		0			0
$223$		−	7815.01i		−	0.157152i	−0.996908	−	0.0785760i	$-0.974963\pi$
$223$		−	7815.01i		−	0.157152i	0.996908	−	0.0785760i	$-0.0250373\pi$
$224$		0			0
$225$	6369.00			0.125807
$226$		0			0
$227$			30865.1i			0.598986i	0.954099	+	0.299493i	$0.0968175\pi$
$227$			30865.1i			0.598986i	−0.954099	+	0.299493i	$0.903182\pi$
$228$		0			0
$229$			57344.7i			1.09351i	0.837293	+	0.546755i	$0.184137\pi$
$229$			57344.7i			1.09351i	−0.837293	+	0.546755i	$0.815863\pi$
$230$		0			0
$231$	−6048.00	+	872.954i	−0.113341	+	0.0163594i
$232$		0			0
$233$	70434.0			1.29739			0.648695	−	0.761049i	$-0.275315\pi$
$233$	70434.0			1.29739			0.648695	+	0.761049i	$0.275315\pi$
$234$		0			0
$235$	70848.0			1.28290
$236$		0			0
$237$			27588.1i			0.491162i
$238$		0			0
$239$	−76158.0			−1.33327			−0.666637	−	0.745382i	$-0.732267\pi$
$239$	−76158.0			−1.33327			−0.666637	+	0.745382i	$0.732267\pi$
$240$		0			0
$241$		−	49411.9i		−	0.850742i	−0.905019	−	0.425371i	$-0.860144\pi$
$241$		−	49411.9i		−	0.850742i	0.905019	−	0.425371i	$-0.139856\pi$
$242$		0			0
$243$		−	44583.0i		−	0.755017i
$244$		0			0
$245$	14112.0	+	47867.0i	0.235102	+	0.797450i
$246$		0			0
$247$	11856.0			0.194332
$248$		0			0
$249$	34704.0			0.559733
$250$		0			0
$251$			16440.6i			0.260958i	0.991451	+	0.130479i	$0.0416515\pi$
$251$			16440.6i			0.260958i	−0.991451	+	0.130479i	$0.958348\pi$
$252$		0			0
$253$	13284.0			0.207533
$254$		0			0
$255$		−	59859.7i		−	0.920564i
$256$		0			0
$257$		−	33920.5i		−	0.513565i	−0.966469	−	0.256783i	$-0.917338\pi$
$257$		−	33920.5i		−	0.513565i	0.966469	−	0.256783i	$-0.0826625\pi$
$258$		0			0
$259$	−16702.0	−	115715.i	−0.248983	−	1.72500i
$260$		0			0
$261$	−27918.0			−0.409830
$262$		0			0
$263$	−94734.0			−1.36960			−0.684801	−	0.728730i	$-0.740111\pi$
$263$	−94734.0			−1.36960			−0.684801	+	0.728730i	$0.740111\pi$
$264$		0			0
$265$			5611.84i			0.0799123i
$266$		0			0
$267$	52704.0			0.739301
$268$		0			0
$269$			64868.8i			0.896460i	0.893918	+	0.448230i	$0.147946\pi$
$269$			64868.8i			0.896460i	−0.893918	+	0.448230i	$0.852054\pi$
$270$		0			0
$271$		−	83443.3i		−	1.13619i	−0.822961	−	0.568097i	$-0.807680\pi$
$271$		−	83443.3i		−	1.13619i	0.822961	−	0.568097i	$-0.192320\pi$
$272$		0			0
$273$	−6384.00	−	44229.6i	−0.0856579	−	0.593456i
$274$		0			0
$275$	3474.00			0.0459372
$276$		0			0
$277$	7442.00			0.0969907			0.0484954	−	0.998823i	$-0.484557\pi$
$277$	7442.00			0.0969907			0.0484954	+	0.998823i	$0.484557\pi$
$278$		0			0
$279$		−	38410.0i		−	0.493441i
$280$		0			0
$281$	14562.0			0.184420			0.0922101	−	0.995740i	$-0.470607\pi$
$281$	14562.0			0.184420			0.0922101	+	0.995740i	$0.470607\pi$
$282$		0			0
$283$			43543.8i			0.543692i	0.962341	+	0.271846i	$0.0876342\pi$
$283$			43543.8i			0.543692i	−0.962341	+	0.271846i	$0.912366\pi$
$284$		0			0
$285$			12969.6i			0.159675i
$286$		0			0
$287$	82656.0	−	11930.4i	1.00348	−	0.144840i
$288$		0			0
$289$	−89279.0			−1.06894
$290$		0			0
$291$	87168.0			1.02937
$292$		0			0
$293$			51026.2i			0.594372i	0.954820	+	0.297186i	$0.0960481\pi$
$293$			51026.2i			0.594372i	−0.954820	+	0.297186i	$0.903952\pi$
$294$		0			0
$295$	65232.0			0.749578
$296$		0			0
$297$		−	14216.7i		−	0.161170i
$298$		0			0
$299$			97147.3i			1.08665i
$300$		0			0
$301$	17570.0	+	121729.i	0.193927	+	1.34357i
$302$		0			0
$303$	−77328.0			−0.842270
$304$		0			0
$305$	−134928.			−1.45045
$306$		0			0
$307$		−	60670.3i		−	0.643723i	−0.946787	−	0.321862i	$-0.895691\pi$
$307$		−	60670.3i		−	0.643723i	0.946787	−	0.321862i	$-0.104309\pi$
$308$		0			0
$309$	5856.00			0.0613316
$310$		0			0
$311$			136222.i			1.40840i	0.709999	+	0.704202i	$0.248695\pi$
$311$			136222.i			1.40840i	−0.709999	+	0.704202i	$0.751305\pi$
$312$		0			0
$313$		−	66940.3i		−	0.683280i	−0.939831	−	0.341640i	$-0.889018\pi$
$313$		−	66940.3i		−	0.683280i	0.939831	−	0.341640i	$-0.110982\pi$
$314$		0			0
$315$	−33264.0	+	4801.24i	−0.335238	+	0.0483875i
$316$		0			0
$317$	−27918.0			−0.277821			−0.138911	−	0.990305i	$-0.544360\pi$
$317$	−27918.0			−0.277821			−0.138911	+	0.990305i	$0.544360\pi$
$318$		0			0
$319$	−15228.0			−0.149645
$320$		0			0
$321$			89664.8i			0.870186i
$322$		0			0
$323$	37440.0			0.358865
$324$		0			0
$325$			25405.7i			0.240528i
$326$		0			0
$327$			48649.8i			0.454973i
$328$		0			0
$329$	165312.	−	23860.7i	1.52726	−	0.220441i
$330$		0			0
$331$	−181166.			−1.65356			−0.826781	−	0.562523i	$-0.809831\pi$
$331$	−181166.			−1.65356			−0.826781	+	0.562523i	$0.809831\pi$
$332$		0			0
$333$	78738.0			0.710061
$334$		0			0
$335$		−	50922.3i		−	0.453752i
$336$		0			0
$337$	92338.0			0.813056			0.406528	−	0.913638i	$-0.366739\pi$
$337$	92338.0			0.813056			0.406528	+	0.913638i	$0.366739\pi$
$338$		0			0
$339$		−	129821.i		−	1.12965i
$340$		0			0
$341$		−	20950.9i		−	0.180175i
$342$		0			0
$343$	49049.0	+	106937.i	0.416910	+	0.908948i
$344$		0			0
$345$	−106272.			−0.892854
$346$		0			0
$347$	−147726.			−1.22687			−0.613434	−	0.789746i	$-0.710212\pi$
$347$	−147726.			−1.22687			−0.613434	+	0.789746i	$0.710212\pi$
$348$		0			0
$349$		−	149989.i		−	1.23142i	−0.787971	−	0.615712i	$-0.788868\pi$
$349$		−	149989.i		−	1.23142i	0.787971	−	0.615712i	$-0.211132\pi$
$350$		0			0
$351$	103968.			0.843889
$352$		0			0
$353$		−	180410.i		−	1.44781i	−0.689899	−	0.723906i	$-0.742345\pi$
$353$		−	180410.i		−	1.44781i	0.689899	−	0.723906i	$-0.257655\pi$
$354$		0			0
$355$			65471.5i			0.519512i
$356$		0			0
$357$	−20160.0	−	139673.i	−0.158181	−	1.09591i
$358$		0			0
$359$	113058.			0.877228			0.438614	−	0.898676i	$-0.355470\pi$
$359$	113058.			0.877228			0.438614	+	0.898676i	$0.355470\pi$
$360$		0			0
$361$	122209.			0.937754
$362$		0			0
$363$			99191.1i			0.752765i
$364$		0			0
$365$	−4896.00			−0.0367499
$366$		0			0
$367$		−	142472.i		−	1.05778i	−0.848690	−	0.528891i	$-0.822608\pi$
$367$		−	142472.i		−	1.05778i	0.848690	−	0.528891i	$-0.177392\pi$
$368$		0			0
$369$			56243.2i			0.413064i
$370$		0			0
$371$	1890.00	+	13094.3i	0.0137314	+	0.0951337i
$372$		0			0
$373$	−99502.0			−0.715178			−0.357589	−	0.933879i	$-0.616401\pi$
$373$	−99502.0			−0.715178			−0.357589	+	0.933879i	$0.616401\pi$
$374$		0			0
$375$	−117792.			−0.837632
$376$		0			0
$377$		−	111364.i		−	0.783541i
$378$		0			0
$379$	75538.0			0.525880			0.262940	−	0.964812i	$-0.415308\pi$
$379$	75538.0			0.525880			0.262940	+	0.964812i	$0.415308\pi$
$380$		0			0
$381$			15948.7i			0.109869i
$382$		0			0
$383$		−	99433.6i		−	0.677853i	−0.940813	−	0.338926i	$-0.889936\pi$
$383$		−	99433.6i		−	0.677853i	0.940813	−	0.338926i	$-0.110064\pi$
$384$		0			0
$385$	−18144.0	+	2618.86i	−0.122409	+	0.0176681i
$386$		0			0
$387$	−82830.0			−0.553052
$388$		0			0
$389$	−31086.0			−0.205431			−0.102715	−	0.994711i	$-0.532753\pi$
$389$	−31086.0			−0.205431			−0.102715	+	0.994711i	$0.532753\pi$
$390$		0			0
$391$			306781.i			2.00666i
$392$		0			0
$393$	−135216.			−0.875473
$394$		0			0
$395$			82764.3i			0.530455i
$396$		0			0
$397$		−	90433.8i		−	0.573786i	−0.957963	−	0.286893i	$-0.907378\pi$
$397$		−	90433.8i		−	0.573786i	0.957963	−	0.286893i	$-0.0926224\pi$
$398$		0			0
$399$	4368.00	+	30262.4i	0.0274370	+	0.190089i
$400$		0			0
$401$	100530.			0.625183			0.312591	−	0.949888i	$-0.398803\pi$
$401$	100530.			0.625183			0.312591	+	0.949888i	$0.398803\pi$
$402$		0			0
$403$	153216.			0.943396
$404$		0			0
$405$			58176.1i			0.354678i
$406$		0			0
$407$	42948.0			0.259271
$408$		0			0
$409$			305797.i			1.82804i	0.405665	+	0.914022i	$0.367040\pi$
$409$			305797.i			1.82804i	−0.405665	+	0.914022i	$0.632960\pi$
$410$		0			0
$411$			182447.i			1.08007i
$412$		0			0
$413$	152208.	−	21969.3i	0.892354	−	0.128800i
$414$		0			0
$415$	104112.			0.604512
$416$		0			0
$417$	110256.			0.634060
$418$		0			0
$419$		−	239834.i		−	1.36610i	−0.730372	−	0.683049i	$-0.760653\pi$
$419$		−	239834.i		−	1.36610i	0.730372	−	0.683049i	$-0.239347\pi$
$420$		0			0
$421$	−214222.			−1.20865			−0.604324	−	0.796739i	$-0.706557\pi$
$421$	−214222.			−1.20865			−0.604324	+	0.796739i	$0.706557\pi$
$422$		0			0
$423$			112486.i			0.628664i
$424$		0			0
$425$			80228.6i			0.444172i
$426$		0			0
$427$	−314832.	+	45442.1i	−1.72672	+	0.249231i
$428$		0			0
$429$	16416.0			0.0891975
$430$		0			0
$431$	−345150.			−1.85803			−0.929016	−	0.370039i	$-0.879344\pi$
$431$	−345150.			−1.85803			−0.929016	+	0.370039i	$0.879344\pi$
$432$		0			0
$433$			48469.7i			0.258520i	0.991611	+	0.129260i	$0.0412602\pi$
$433$			48469.7i			0.258520i	−0.991611	+	0.129260i	$0.958740\pi$
$434$		0			0
$435$	121824.			0.643805
$436$		0			0
$437$		−	66469.2i		−	0.348063i
$438$		0			0
$439$		−	185690.i		−	0.963516i	−0.876304	−	0.481758i	$-0.839998\pi$
$439$		−	185690.i		−	0.963516i	0.876304	−	0.481758i	$-0.160002\pi$
$440$		0			0
$441$	−75999.0	+	22405.8i	−0.390779	+	0.115208i
$442$		0			0
$443$	256338.			1.30619			0.653094	−	0.757277i	$-0.273471\pi$
$443$	256338.			1.30619			0.653094	+	0.757277i	$0.273471\pi$
$444$		0			0
$445$	158112.			0.798445
$446$		0			0
$447$		−	81932.9i		−	0.410056i
$448$		0			0
$449$	−319806.			−1.58633			−0.793166	−	0.609006i	$-0.791569\pi$
$449$	−319806.			−1.58633			−0.793166	+	0.609006i	$0.791569\pi$
$450$		0			0
$451$			30678.1i			0.150826i
$452$		0			0
$453$		−	110643.i		−	0.539174i
$454$		0			0
$455$	−19152.0	−	132689.i	−0.0925106	−	0.640932i
$456$		0			0
$457$	140338.			0.671959			0.335980	−	0.941869i	$-0.390933\pi$
$457$	140338.			0.671959			0.335980	+	0.941869i	$0.390933\pi$
$458$		0			0
$459$	328320.			1.55837
$460$		0			0
$461$		−	70813.2i		−	0.333205i	−0.986024	−	0.166603i	$-0.946720\pi$
$461$		−	70813.2i		−	0.333205i	0.986024	−	0.166603i	$-0.0532797\pi$
$462$		0			0
$463$	−8206.00			−0.0382798			−0.0191399	−	0.999817i	$-0.506093\pi$
$463$	−8206.00			−0.0382798			−0.0191399	+	0.999817i	$0.506093\pi$
$464$		0			0
$465$			167607.i			0.775151i
$466$		0			0
$467$		−	406360.i		−	1.86328i	−0.363388	−	0.931638i	$-0.618380\pi$
$467$		−	406360.i		−	1.86328i	0.363388	−	0.931638i	$-0.381620\pi$
$468$		0			0
$469$	−17150.0	−	118819.i	−0.0779684	−	0.540181i
$470$		0			0
$471$	198864.			0.896426
$472$		0			0
$473$	−45180.0			−0.201941
$474$		0			0
$475$		−	17382.9i		−	0.0770432i
$476$		0			0
$477$	−8910.00			−0.0391598
$478$		0			0
$479$		−	281507.i		−	1.22692i	−0.789724	−	0.613462i	$-0.789776\pi$
$479$		−	281507.i		−	1.22692i	0.789724	−	0.613462i	$-0.210224\pi$
$480$		0			0
$481$			314083.i			1.35755i
$482$		0			0
$483$	−247968.	+	35791.1i	−1.06292	+	0.153420i
$484$		0			0
$485$	261504.			1.11172
$486$		0			0
$487$	−229598.			−0.968078			−0.484039	−	0.875047i	$-0.660831\pi$
$487$	−229598.			−0.968078			−0.484039	+	0.875047i	$0.660831\pi$
$488$		0			0
$489$			54192.4i			0.226632i
$490$		0			0
$491$	−48654.0			−0.201816			−0.100908	−	0.994896i	$-0.532175\pi$
$491$	−48654.0			−0.201816			−0.100908	+	0.994896i	$0.532175\pi$
$492$		0			0
$493$		−	351676.i		−	1.44693i
$494$		0			0
$495$		−	12346.1i		−	0.0503869i
$496$		0			0
$497$	22050.0	+	152767.i	0.0892680	+	0.618467i
$498$		0			0
$499$	−91598.0			−0.367862			−0.183931	−	0.982939i	$-0.558882\pi$
$499$	−91598.0			−0.367862			−0.183931	+	0.982939i	$0.558882\pi$
$500$		0			0
$501$	167904.			0.668938
$502$		0			0
$503$			163824.i			0.647504i	0.946142	+	0.323752i	$0.104944\pi$
$503$			163824.i			0.647504i	−0.946142	+	0.323752i	$0.895056\pi$
$504$		0			0
$505$	−231984.			−0.909652
$506$		0			0
$507$		−	77824.5i		−	0.302761i
$508$		0			0
$509$			461190.i			1.78010i	0.455864	+	0.890049i	$0.349330\pi$
$509$			461190.i			1.78010i	−0.455864	+	0.890049i	$0.650670\pi$
$510$		0			0
$511$	−11424.0	+	1648.91i	−0.0437498	+	0.00631474i
$512$		0			0
$513$	−71136.0			−0.270305
$514$		0			0
$515$	17568.0			0.0662381
$516$		0			0
$517$			61356.2i			0.229550i
$518$		0			0
$519$	−199152.			−0.739350
$520$		0			0
$521$			127908.i			0.471220i	0.971848	+	0.235610i	$0.0757088\pi$
$521$			127908.i			0.471220i	−0.971848	+	0.235610i	$0.924291\pi$
$522$		0			0
$523$			175762.i			0.642570i	0.946983	+	0.321285i	$0.104115\pi$
$523$			175762.i			0.642570i	−0.946983	+	0.321285i	$0.895885\pi$
$524$		0			0
$525$	−64848.0	+	9360.00i	−0.235276	+	0.0339592i
$526$		0			0
$527$	483840.			1.74213
$528$		0			0
$529$	264803.			0.946262
$530$		0			0
$531$			103570.i			0.367319i
$532$		0			0
$533$	−224352.			−0.789724
$534$		0			0
$535$			268994.i			0.939801i
$536$		0			0
$537$		−	61979.7i		−	0.214932i
$538$		0			0
$539$	−41454.0	+	12221.4i	−0.142688	+	0.0420670i
$540$		0			0
$541$	−19054.0			−0.0651016			−0.0325508	−	0.999470i	$-0.510363\pi$
$541$	−19054.0			−0.0651016			−0.0325508	+	0.999470i	$0.510363\pi$
$542$		0			0
$543$	−214224.			−0.726555
$544$		0			0
$545$			145950.i			0.491371i
$546$		0			0
$547$	536306.			1.79241			0.896206	−	0.443637i	$-0.146312\pi$
$547$	536306.			1.79241			0.896206	+	0.443637i	$0.146312\pi$
$548$		0			0
$549$		−	214227.i		−	0.710771i
$550$		0			0
$551$			76196.4i			0.250975i
$552$		0			0
$553$	27874.0	+	193117.i	0.0911484	+	0.631495i
$554$		0			0
$555$	−343584.			−1.11544
$556$		0			0
$557$	−262638.			−0.846539			−0.423270	−	0.906004i	$-0.639118\pi$
$557$	−262638.			−0.846539			−0.423270	+	0.906004i	$0.639118\pi$
$558$		0			0
$559$		−	330406.i		−	1.05736i
$560$		0			0
$561$	51840.0			0.164717
$562$		0			0
$563$			109930.i			0.346816i	0.984850	+	0.173408i	$0.0554779\pi$
$563$			109930.i			0.346816i	−0.984850	+	0.173408i	$0.944522\pi$
$564$		0			0
$565$		−	389462.i		−	1.22002i
$566$		0			0
$567$	19593.0	+	135744.i	0.0609445	+	0.422236i
$568$		0			0
$569$	−253710.			−0.783634			−0.391817	−	0.920043i	$-0.628153\pi$
$569$	−253710.			−0.783634			−0.391817	+	0.920043i	$0.628153\pi$
$570$		0			0
$571$	164018.			0.503059			0.251530	−	0.967850i	$-0.419066\pi$
$571$	164018.			0.503059			0.251530	+	0.967850i	$0.419066\pi$
$572$		0			0
$573$		−	421138.i		−	1.28267i
$574$		0			0
$575$	142434.			0.430802
$576$		0			0
$577$		−	217601.i		−	0.653596i	−0.945094	−	0.326798i	$-0.894030\pi$
$577$		−	217601.i		−	0.653596i	0.945094	−	0.326798i	$-0.105970\pi$
$578$		0			0
$579$		−	73064.8i		−	0.217947i
$580$		0			0
$581$	242928.	−	35063.6i	0.719657	−	0.103873i
$582$		0			0
$583$	−4860.00			−0.0142988
$584$		0			0
$585$	90288.0			0.263826
$586$		0			0
$587$			251473.i			0.729819i	0.931043	+	0.364909i	$0.118900\pi$
$587$			251473.i			0.729819i	−0.931043	+	0.364909i	$0.881100\pi$
$588$		0			0
$589$	−104832.			−0.302178
$590$		0			0
$591$		−	124.708i		−	0.000357041i
$592$		0			0
$593$		−	380026.i		−	1.08070i	−0.841442	−	0.540348i	$-0.818293\pi$
$593$		−	380026.i		−	1.08070i	0.841442	−	0.540348i	$-0.181707\pi$
$594$		0			0
$595$	−60480.0	−	419018.i	−0.170835	−	1.18358i
$596$		0			0
$597$	−672.000			−0.00188547
$598$		0			0
$599$	−315342.			−0.878877			−0.439439	−	0.898273i	$-0.644823\pi$
$599$	−315342.			−0.878877			−0.439439	+	0.898273i	$0.644823\pi$
$600$		0			0
$601$		−	483907.i		−	1.33972i	−0.742489	−	0.669859i	$-0.766355\pi$
$601$		−	483907.i		−	1.33972i	0.742489	−	0.669859i	$-0.233645\pi$
$602$		0			0
$603$	80850.0			0.222354
$604$		0			0
$605$			297573.i			0.812986i
$606$		0			0
$607$			513408.i			1.39343i	0.717348	+	0.696715i	$0.245356\pi$
$607$			513408.i			1.39343i	−0.717348	+	0.696715i	$0.754644\pi$
$608$		0			0
$609$	284256.	−	41028.8i	0.766435	−	0.110625i
$610$		0			0
$611$	−448704.			−1.20193
$612$		0			0
$613$	227762.			0.606122			0.303061	−	0.952971i	$-0.401991\pi$
$613$	227762.			0.606122			0.303061	+	0.952971i	$0.401991\pi$
$614$		0			0
$615$		−	245425.i		−	0.648885i
$616$		0			0
$617$	502578.			1.32018			0.660090	−	0.751187i	$-0.270518\pi$
$617$	502578.			1.32018			0.660090	+	0.751187i	$0.270518\pi$
$618$		0			0
$619$			171605.i			0.447866i	0.974605	+	0.223933i	$0.0718896\pi$
$619$			171605.i			0.447866i	−0.974605	+	0.223933i	$0.928110\pi$
$620$		0			0
$621$		−	582884.i		−	1.51147i
$622$		0			0
$623$	368928.	−	53250.2i	0.950529	−	0.137197i
$624$		0			0
$625$	−232751.			−0.595843
$626$		0			0
$627$	−11232.0			−0.0285708
$628$		0			0
$629$			991842.i			2.50692i
$630$		0			0
$631$	−29710.0			−0.0746181			−0.0373090	−	0.999304i	$-0.511879\pi$
$631$	−29710.0			−0.0746181			−0.0373090	+	0.999304i	$0.511879\pi$
$632$		0			0
$633$			320014.i			0.798659i
$634$		0			0
$635$			47846.2i			0.118659i
$636$		0			0
$637$	−89376.0	−	303157.i	−0.220263	−	0.747118i
$638$		0			0
$639$	−103950.			−0.254579
$640$		0			0
$641$	−218574.			−0.531964			−0.265982	−	0.963978i	$-0.585696\pi$
$641$	−218574.			−0.531964			−0.265982	+	0.963978i	$0.585696\pi$
$642$		0			0
$643$			414466.i			1.00246i	0.865314	+	0.501230i	$0.167119\pi$
$643$			414466.i			1.00246i	−0.865314	+	0.501230i	$0.832881\pi$
$644$		0			0
$645$	361440.			0.868794
$646$		0			0
$647$			152767.i			0.364939i	0.983211	+	0.182470i	$0.0584091\pi$
$647$			152767.i			0.364939i	−0.983211	+	0.182470i	$0.941591\pi$
$648$		0			0
$649$			56492.6i			0.134123i
$650$		0			0
$651$	56448.0	+	391083.i	0.133195	+	0.922799i
$652$		0			0
$653$	−641070.			−1.50342			−0.751708	−	0.659496i	$-0.770770\pi$
$653$	−641070.			−1.50342			−0.751708	+	0.659496i	$0.770770\pi$
$654$		0			0
$655$	−405648.			−0.945511
$656$		0			0
$657$		−	7773.44i		−	0.0180087i
$658$		0			0
$659$	−162990.			−0.375310			−0.187655	−	0.982235i	$-0.560089\pi$
$659$	−162990.			−0.375310			−0.187655	+	0.982235i	$0.560089\pi$
$660$		0			0
$661$		−	423168.i		−	0.968522i	−0.874924	−	0.484261i	$-0.839088\pi$
$661$		−	423168.i		−	0.968522i	0.874924	−	0.484261i	$-0.160912\pi$
$662$		0			0
$663$			379111.i			0.862461i
$664$		0			0
$665$	13104.0	+	90787.2i	0.0296320	+	0.205296i
$666$		0			0
$667$	−624348.			−1.40338
$668$		0			0
$669$	−54144.0			−0.120976
$670$		0			0
$671$		−	116851.i		−	0.259530i
$672$		0			0
$673$	220738.			0.487357			0.243678	−	0.969856i	$-0.421646\pi$
$673$	220738.			0.487357			0.243678	+	0.969856i	$0.421646\pi$
$674$		0			0
$675$		−	152434.i		−	0.334561i
$676$		0			0
$677$			691192.i			1.50807i	0.656834	+	0.754035i	$0.271895\pi$
$677$			691192.i			1.50807i	−0.656834	+	0.754035i	$0.728105\pi$
$678$		0			0
$679$	610176.	−	88071.3i	1.32347	−	0.191027i
$680$		0			0
$681$	213840.			0.461100
$682$		0			0
$683$	803250.			1.72191			0.860953	−	0.508685i	$-0.169868\pi$
$683$	803250.			1.72191			0.860953	+	0.508685i	$0.169868\pi$
$684$		0			0
$685$			547342.i			1.16648i
$686$		0			0
$687$	397296.			0.841784
$688$		0			0
$689$		−	35541.7i		−	0.0748686i
$690$		0			0
$691$		−	360634.i		−	0.755284i	−0.925952	−	0.377642i	$-0.876735\pi$
$691$		−	360634.i		−	0.755284i	0.925952	−	0.377642i	$-0.123265\pi$
$692$		0			0
$693$	−4158.00	−	28807.5i	−0.00865801	−	0.0599844i
$694$		0			0
$695$	330768.			0.684784
$696$		0			0
$697$	−708480.			−1.45835
$698$		0			0
$699$		−	487981.i		−	0.998731i
$700$		0			0
$701$	247698.			0.504065			0.252032	−	0.967719i	$-0.418901\pi$
$701$	247698.			0.504065			0.252032	+	0.967719i	$0.418901\pi$
$702$		0			0
$703$		−	214899.i		−	0.434834i
$704$		0			0
$705$		−	490849.i		−	0.987575i
$706$		0			0
$707$	−541296.	+	78129.3i	−1.08292	+	0.156306i
$708$		0			0
$709$	−14414.0			−0.0286742			−0.0143371	−	0.999897i	$-0.504564\pi$
$709$	−14414.0			−0.0286742			−0.0143371	+	0.999897i	$0.504564\pi$
$710$		0			0
$711$	−131406.			−0.259942
$712$		0			0
$713$		−	858986.i		−	1.68969i
$714$		0			0
$715$	49248.0			0.0963333
$716$		0			0
$717$			527638.i			1.02636i
$718$		0			0
$719$		−	153058.i		−	0.296072i	−0.988982	−	0.148036i	$-0.952705\pi$
$719$		−	153058.i		−	0.296072i	0.988982	−	0.148036i	$-0.0472952\pi$
$720$		0			0
$721$	40992.0	−	5916.69i	0.0788549	−	0.0113817i
$722$		0			0
$723$	−342336.			−0.654901
$724$		0			0
$725$	−163278.			−0.310636
$726$		0			0
$727$		−	359782.i		−	0.680723i	−0.940295	−	0.340361i	$-0.889451\pi$
$727$		−	359782.i		−	0.680723i	0.940295	−	0.340361i	$-0.110549\pi$
$728$		0			0
$729$	−535599.			−1.00782
$730$		0			0
$731$		−	1.04339e6i		−	1.95259i
$732$		0			0
$733$			383206.i			0.713221i	0.934253	+	0.356611i	$0.116068\pi$
$733$			383206.i			0.713221i	−0.934253	+	0.356611i	$0.883932\pi$
$734$		0			0
$735$	331632.	−	97770.8i	0.613878	−	0.180982i
$736$		0			0
$737$	44100.0			0.0811902
$738$		0			0
$739$	859826.			1.57442			0.787212	−	0.616682i	$-0.211524\pi$
$739$	859826.			1.57442			0.787212	+	0.616682i	$0.211524\pi$
$740$		0			0
$741$		−	82140.8i		−	0.149597i
$742$		0			0
$743$	−669150.			−1.21212			−0.606060	−	0.795419i	$-0.707251\pi$
$743$	−669150.			−1.21212			−0.606060	+	0.795419i	$0.707251\pi$
$744$		0			0
$745$		−	245799.i		−	0.442861i
$746$		0			0
$747$			165300.i			0.296232i
$748$		0			0
$749$	90594.0	+	627654.i	0.161486	+	1.11881i
$750$		0			0
$751$	293506.			0.520400			0.260200	−	0.965555i	$-0.416212\pi$
$751$	293506.			0.520400			0.260200	+	0.965555i	$0.416212\pi$
$752$		0			0
$753$	113904.			0.200886
$754$		0			0
$755$		−	331930.i		−	0.582308i
$756$		0			0
$757$	555634.			0.969610			0.484805	−	0.874622i	$-0.338891\pi$
$757$	555634.			0.969610			0.484805	+	0.874622i	$0.338891\pi$
$758$		0			0
$759$		−	92034.3i		−	0.159759i
$760$		0			0
$761$		−	255942.i		−	0.441949i	−0.975280	−	0.220974i	$-0.929076\pi$
$761$		−	255942.i		−	0.441949i	0.975280	−	0.220974i	$-0.0709237\pi$
$762$		0			0
$763$	49154.0	+	340549.i	0.0844325	+	0.584966i
$764$		0			0
$765$	285120.			0.487197
$766$		0			0
$767$	−413136.			−0.702267
$768$		0			0
$769$			481427.i			0.814100i	0.913406	+	0.407050i	$0.133443\pi$
$769$			481427.i			0.814100i	−0.913406	+	0.407050i	$0.866557\pi$
$770$		0			0
$771$	−235008.			−0.395343
$772$		0			0
$773$			211483.i			0.353930i	0.984217	+	0.176965i	$0.0566279\pi$
$773$			211483.i			0.353930i	−0.984217	+	0.176965i	$0.943372\pi$
$774$		0			0
$775$		−	224640.i		−	0.374011i
$776$		0			0
$777$	−801696.	+	115715.i	−1.32791	+	0.191667i
$778$		0			0
$779$	153504.			0.252956
$780$		0			0
$781$	−56700.0			−0.0929568
$782$		0			0
$783$			668184.i			1.08986i
$784$		0			0
$785$	596592.			0.968140
$786$		0			0
$787$			997128.i			1.60991i	0.593336	+	0.804955i	$0.297810\pi$
$787$			997128.i			1.60991i	−0.593336	+	0.804955i	$0.702190\pi$
$788$		0			0
$789$			656336.i			1.05432i
$790$		0			0
$791$	−131166.	−	908745.i	−0.209637	−	1.45241i
$792$		0			0
$793$	854544.			1.35890
$794$		0			0
$795$	38880.0			0.0615166
$796$		0			0
$797$			258831.i			0.407473i	0.979026	+	0.203737i	$0.0653086\pi$
$797$			258831.i			0.407473i	−0.979026	+	0.203737i	$0.934691\pi$
$798$		0			0
$799$	−1.41696e6			−2.21955
$800$		0			0
$801$			251037.i			0.391266i
$802$		0			0
$803$		−	4240.06i		−	0.00657568i
$804$		0			0
$805$	−743904.	+	107373.i	−1.14796	+	0.165693i
$806$		0			0
$807$	449424.			0.690095
$808$		0			0
$809$	799794.			1.22203			0.611014	−	0.791620i	$-0.290762\pi$
$809$	799794.			1.22203			0.611014	+	0.791620i	$0.290762\pi$
$810$		0			0
$811$			254632.i			0.387143i	0.981086	+	0.193572i	$0.0620072\pi$
$811$			254632.i			0.387143i	−0.981086	+	0.193572i	$0.937993\pi$
$812$		0			0
$813$	−578112.			−0.874643
$814$		0			0
$815$			162577.i			0.244762i
$816$		0			0
$817$			226067.i			0.338683i
$818$		0			0
$819$	210672.	−	30407.9i	0.314079	−	0.0453334i
$820$		0			0
$821$	109458.			0.162391			0.0811954	−	0.996698i	$-0.474126\pi$
$821$	109458.			0.162391			0.0811954	+	0.996698i	$0.474126\pi$
$822$		0			0
$823$	−415438.			−0.613347			−0.306674	−	0.951815i	$-0.599216\pi$
$823$	−415438.			−0.613347			−0.306674	+	0.951815i	$0.599216\pi$
$824$		0			0
$825$		−	24068.6i		−	0.0353625i
$826$		0			0
$827$	−23598.0			−0.0345036			−0.0172518	−	0.999851i	$-0.505492\pi$
$827$	−23598.0			−0.0345036			−0.0172518	+	0.999851i	$0.505492\pi$
$828$		0			0
$829$			111454.i			0.162176i	0.996707	+	0.0810880i	$0.0258395\pi$
$829$			111454.i			0.162176i	−0.996707	+	0.0810880i	$0.974161\pi$
$830$		0			0
$831$		−	51559.7i		−	0.0746635i
$832$		0			0
$833$	−282240.	−	957339.i	−0.406751	−	1.37967i
$834$		0			0
$835$	503712.			0.722453
$836$		0			0
$837$	−919296.			−1.31221
$838$		0			0
$839$			477672.i			0.678587i	0.940680	+	0.339294i	$0.110188\pi$
$839$			477672.i			0.678587i	−0.940680	+	0.339294i	$0.889812\pi$
$840$		0			0
$841$	8435.00			0.0119260
$842$		0			0
$843$		−	100888.i		−	0.141967i
$844$		0			0
$845$		−	233474.i		−	0.326982i
$846$		0			0
$847$	100219.	+	694338.i	0.139696	+	0.967841i
$848$		0			0
$849$	301680.			0.418534
$850$		0			0
$851$	1.76087e6			2.43146
$852$		0			0
$853$		−	415519.i		−	0.571075i	−0.958368	−	0.285537i	$-0.907828\pi$
$853$		−	415519.i		−	0.571075i	0.958368	−	0.285537i	$-0.0921720\pi$
$854$		0			0
$855$	−61776.0			−0.0845060
$856$		0			0
$857$		−	958461.i		−	1.30501i	−0.757785	−	0.652504i	$-0.773719\pi$
$857$		−	958461.i		−	1.30501i	0.757785	−	0.652504i	$-0.226281\pi$
$858$		0			0
$859$		−	1.32647e6i		−	1.79767i	−0.438287	−	0.898835i	$-0.644415\pi$
$859$		−	1.32647e6i		−	1.79767i	0.438287	−	0.898835i	$-0.355585\pi$
$860$		0			0
$861$	−82656.0	−	572658.i	−0.111498	−	0.772483i
$862$		0			0
$863$	319986.			0.429645			0.214822	−	0.976653i	$-0.431083\pi$
$863$	319986.			0.429645			0.214822	+	0.976653i	$0.431083\pi$
$864$		0			0
$865$	−597456.			−0.798498
$866$		0			0
$867$			618543.i			0.822871i
$868$		0			0
$869$	−71676.0			−0.0949149
$870$		0			0
$871$			322508.i			0.425113i
$872$		0			0
$873$			415193.i			0.544781i
$874$		0			0
$875$	−824544.	+	119013.i	−1.07696	+	0.155445i
$876$		0			0
$877$	1.08735e6			1.41374			0.706868	−	0.707345i	$-0.250107\pi$
$877$	1.08735e6			1.41374			0.706868	+	0.707345i	$0.250107\pi$
$878$		0			0
$879$	353520.			0.457548
$880$		0			0
$881$		−	1.41344e6i		−	1.82106i	−0.413442	−	0.910531i	$-0.635673\pi$
$881$		−	1.41344e6i		−	1.82106i	0.413442	−	0.910531i	$-0.364327\pi$
$882$		0			0
$883$	−1.03902e6			−1.33261			−0.666305	−	0.745679i	$-0.732125\pi$
$883$	−1.03902e6			−1.33261			−0.666305	+	0.745679i	$0.732125\pi$
$884$		0			0
$885$		−	451941.i		−	0.577025i
$886$		0			0
$887$		−	847472.i		−	1.07715i	−0.842576	−	0.538577i	$-0.818962\pi$
$887$		−	847472.i		−	1.07715i	0.842576	−	0.538577i	$-0.181038\pi$
$888$		0			0
$889$	16114.0	+	111641.i	0.0203892	+	0.141260i
$890$		0			0
$891$	−50382.0			−0.0634629
$892$		0			0
$893$	307008.			0.384988
$894$		0			0
$895$		−	185939.i		−	0.232126i
$896$		0			0
$897$	673056.			0.836501
$898$		0			0
$899$			984692.i			1.21837i
$900$		0			0
$901$		−	112237.i		−	0.138257i
$902$		0			0
$903$	843360.	−	121729.i	1.03428	−	0.149285i
$904$		0			0
$905$	−642672.			−0.784679
$906$		0			0
$907$	−157550.			−0.191515			−0.0957577	−	0.995405i	$-0.530527\pi$
$907$	−157550.			−0.191515			−0.0957577	+	0.995405i	$0.530527\pi$
$908$		0			0
$909$		−	368324.i		−	0.445761i
$910$		0			0
$911$	−962046.			−1.15920			−0.579601	−	0.814900i	$-0.696792\pi$
$911$	−962046.			−1.15920			−0.579601	+	0.814900i	$0.696792\pi$
$912$		0			0
$913$			90163.6i			0.108166i
$914$		0			0
$915$			934809.i			1.11656i
$916$		0			0
$917$	−946512.	+	136617.i	−1.12561	+	0.162468i
$918$		0			0
$919$	−109838.			−0.130053			−0.0650267	−	0.997884i	$-0.520713\pi$
$919$	−109838.			−0.130053			−0.0650267	+	0.997884i	$0.520713\pi$
$920$		0			0
$921$	−420336.			−0.495538
$922$		0			0
$923$		−	414653.i		−	0.486722i
$924$		0			0
$925$	460498.			0.538201
$926$		0			0
$927$			27892.9i			0.0324590i
$928$		0			0
$929$		−	134518.i		−	0.155865i	−0.996959	−	0.0779326i	$-0.975168\pi$
$929$		−	134518.i		−	0.155865i	0.996959	−	0.0779326i	$-0.0248319\pi$
$930$		0			0
$931$	61152.0	+	207423.i	0.0705523	+	0.239309i
$932$		0			0
$933$	943776.			1.08419
$934$		0			0
$935$	155520.			0.177895
$936$		0			0
$937$			799057.i			0.910120i	0.890461	+	0.455060i	$0.150382\pi$
$937$			799057.i			0.910120i	−0.890461	+	0.455060i	$0.849618\pi$
$938$		0			0
$939$	−463776.			−0.525990
$940$		0			0
$941$			341720.i			0.385914i	0.981207	+	0.192957i	$0.0618078\pi$
$941$			341720.i			0.385914i	−0.981207	+	0.192957i	$0.938192\pi$
$942$		0			0
$943$			1.25780e6i			1.41445i
$944$		0			0
$945$	114912.	+	796134.i	0.128677	+	0.891502i
$946$		0			0
$947$	−620910.			−0.692355			−0.346177	−	0.938169i	$-0.612520\pi$
$947$	−620910.			−0.692355			−0.346177	+	0.938169i	$0.612520\pi$
$948$		0			0
$949$	31008.0			0.0344303
$950$		0			0
$951$			193422.i			0.213867i
$952$		0			0
$953$	855522.			0.941988			0.470994	−	0.882136i	$-0.343895\pi$
$953$	855522.			0.941988			0.470994	+	0.882136i	$0.343895\pi$
$954$		0			0
$955$		−	1.26341e6i		−	1.38528i
$956$		0			0
$957$			105503.i			0.115197i
$958$		0			0
$959$	184338.	+	1.27713e6i	0.200437	+	1.38867i
$960$		0			0
$961$	−431231.			−0.466942
$962$		0			0
$963$	−427086.			−0.460535
$964$		0			0
$965$		−	219194.i		−	0.235383i
$966$		0			0
$967$	1.68221e6			1.79898			0.899492	−	0.436937i	$-0.143937\pi$
$967$	1.68221e6			1.79898			0.899492	+	0.436937i	$0.143937\pi$
$968$		0			0
$969$		−	259392.i		−	0.276254i
$970$		0			0
$971$			1.66350e6i			1.76434i	0.470927	+	0.882172i	$0.343920\pi$
$971$			1.66350e6i			1.76434i	−0.470927	+	0.882172i	$0.656080\pi$
$972$		0			0
$973$	771792.	−	111399.i	0.815220	−	0.117667i
$974$		0			0
$975$	176016.			0.185158
$976$		0			0
$977$	1602.00			0.00167831			0.000839157	−	1.00000i	$-0.499733\pi$
$977$	1602.00			0.00167831			0.000839157		1.00000i	$0.499733\pi$
$978$		0			0
$979$			136929.i			0.142866i
$980$		0			0
$981$	−231726.			−0.240789
$982$		0			0
$983$			164323.i			0.170056i	0.996379	+	0.0850279i	$0.0270980\pi$
$983$			164323.i			0.170056i	−0.996379	+	0.0850279i	$0.972902\pi$
$984$		0			0
$985$		−	374.123i		−	0.000385604i
$986$		0			0
$987$	−165312.	−	1.14532e6i	−0.169695	−	1.17568i
$988$		0			0
$989$	−1.85238e6			−1.89381
$990$		0			0
$991$	−734606.			−0.748010			−0.374005	−	0.927427i	$-0.622016\pi$
$991$	−734606.			−0.748010			−0.374005	+	0.927427i	$0.622016\pi$
$992$		0			0
$993$			1.25515e6i			1.27291i
$994$		0			0
$995$	−2016.00			−0.00203631
$996$		0			0
$997$			159439.i			0.160400i	0.996779	+	0.0801998i	$0.0255559\pi$
$997$			159439.i			0.160400i	−0.996779	+	0.0801998i	$0.974444\pi$
$998$		0			0
$999$		−	1.88450e6i		−	1.88827i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	28.5.b.a.13.1	✓	2
3.2	odd	2		252.5.d.a.181.2		2
4.3	odd	2		112.5.c.b.97.2		2
5.2	odd	4		700.5.h.a.349.1		4
5.3	odd	4		700.5.h.a.349.4		4
5.4	even	2		700.5.d.a.601.2		2
7.2	even	3		196.5.h.b.129.1		2
7.3	odd	6		196.5.h.b.117.1		2
7.4	even	3		196.5.h.a.117.1		2
7.5	odd	6		196.5.h.a.129.1		2
7.6	odd	2	inner	28.5.b.a.13.2	yes	2
8.3	odd	2		448.5.c.d.321.1		2
8.5	even	2		448.5.c.c.321.2		2
12.11	even	2		1008.5.f.c.433.2		2
21.20	even	2		252.5.d.a.181.1		2
28.27	even	2		112.5.c.b.97.1		2
35.13	even	4		700.5.h.a.349.2		4
35.27	even	4		700.5.h.a.349.3		4
35.34	odd	2		700.5.d.a.601.1		2
56.13	odd	2		448.5.c.c.321.1		2
56.27	even	2		448.5.c.d.321.2		2
84.83	odd	2		1008.5.f.c.433.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.5.b.a.13.1	✓	2	1.1	even	1	trivial
28.5.b.a.13.2	yes	2	7.6	odd	2	inner
112.5.c.b.97.1		2	28.27	even	2
112.5.c.b.97.2		2	4.3	odd	2
196.5.h.a.117.1		2	7.4	even	3
196.5.h.a.129.1		2	7.5	odd	6
196.5.h.b.117.1		2	7.3	odd	6
196.5.h.b.129.1		2	7.2	even	3
252.5.d.a.181.1		2	21.20	even	2
252.5.d.a.181.2		2	3.2	odd	2
448.5.c.c.321.1		2	56.13	odd	2
448.5.c.c.321.2		2	8.5	even	2
448.5.c.d.321.1		2	8.3	odd	2
448.5.c.d.321.2		2	56.27	even	2
700.5.d.a.601.1		2	35.34	odd	2
700.5.d.a.601.2		2	5.4	even	2
700.5.h.a.349.1		4	5.2	odd	4
700.5.h.a.349.2		4	35.13	even	4
700.5.h.a.349.3		4	35.27	even	4
700.5.h.a.349.4		4	5.3	odd	4
1008.5.f.c.433.1		2	84.83	odd	2
1008.5.f.c.433.2		2	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 28 = 2^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	28.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-6.92820i q^{3} -20.7846i q^{5} +(-7.00000 - 48.4974i) q^{7} +33.0000 q^{9} +O(q^{10})\)	\(q-6.92820i q^{3} -20.7846i q^{5} +(-7.00000 - 48.4974i) q^{7} +33.0000 q^{9} +18.0000 q^{11} +131.636i q^{13} -144.000 q^{15} +415.692i q^{17} -90.0666i q^{19} +(-336.000 + 48.4974i) q^{21} +738.000 q^{23} +193.000 q^{25} -789.815i q^{27} -846.000 q^{29} -1163.94i q^{31} -124.708i q^{33} +(-1008.00 + 145.492i) q^{35} +2386.00 q^{37} +912.000 q^{39} +1704.34i q^{41} -2510.00 q^{43} -685.892i q^{45} +3408.68i q^{47} +(-2303.00 + 678.964i) q^{49} +2880.00 q^{51} -270.000 q^{53} -374.123i q^{55} -624.000 q^{57} +3138.48i q^{59} -6491.73i q^{61} +(-231.000 - 1600.41i) q^{63} +2736.00 q^{65} +2450.00 q^{67} -5113.01i q^{69} -3150.00 q^{71} -235.559i q^{73} -1337.14i q^{75} +(-126.000 - 872.954i) q^{77} -3982.00 q^{79} -2799.00 q^{81} +5009.09i q^{83} +8640.00 q^{85} +5861.26i q^{87} +7607.17i q^{89} +(6384.00 - 921.451i) q^{91} -8064.00 q^{93} -1872.00 q^{95} +12581.6i q^{97} +594.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 14 q^{7} + 66 q^{9}+O(q^{10}) \) 2 * q - 14 * q^7 + 66 * q^9	\( 2 q - 14 q^{7} + 66 q^{9} + 36 q^{11} - 288 q^{15} - 672 q^{21} + 1476 q^{23} + 386 q^{25} - 1692 q^{29} - 2016 q^{35} + 4772 q^{37} + 1824 q^{39} - 5020 q^{43} - 4606 q^{49} + 5760 q^{51} - 540 q^{53} - 1248 q^{57} - 462 q^{63} + 5472 q^{65} + 4900 q^{67} - 6300 q^{71} - 252 q^{77} - 7964 q^{79} - 5598 q^{81} + 17280 q^{85} + 12768 q^{91} - 16128 q^{93} - 3744 q^{95} + 1188 q^{99}+O(q^{100}) \) 2 * q - 14 * q^7 + 66 * q^9 + 36 * q^11 - 288 * q^15 - 672 * q^21 + 1476 * q^23 + 386 * q^25 - 1692 * q^29 - 2016 * q^35 + 4772 * q^37 + 1824 * q^39 - 5020 * q^43 - 4606 * q^49 + 5760 * q^51 - 540 * q^53 - 1248 * q^57 - 462 * q^63 + 5472 * q^65 + 4900 * q^67 - 6300 * q^71 - 252 * q^77 - 7964 * q^79 - 5598 * q^81 + 17280 * q^85 + 12768 * q^91 - 16128 * q^93 - 3744 * q^95 + 1188 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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