LMFDB - Embedded newform 315.8.a.g.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [315,8,Mod(1,315)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(315, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("315.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 315 = 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	315.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:	$98.4012830275$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 277x^{2} + 80x + 15348 $ x^4 - 2x^3 - 277x^2 + 80*x + 15348
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$8.50252$ of defining polynomial
Character	$\chi$	$=$	315.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+13.9037 q^{2} +65.3140 q^{4} -125.000 q^{5} +343.000 q^{7} -871.570 q^{8} -1737.97 q^{10} -2958.70 q^{11} +2184.94 q^{13} +4768.98 q^{14} -20478.3 q^{16} -1861.36 q^{17} +5038.34 q^{19} -8164.25 q^{20} -41137.0 q^{22} +73411.5 q^{23} +15625.0 q^{25} +30378.9 q^{26} +22402.7 q^{28} +52747.6 q^{29} -533.745 q^{31} -173164. q^{32} -25879.8 q^{34} -42875.0 q^{35} +136474. q^{37} +70051.8 q^{38} +108946. q^{40} +452012. q^{41} +730329. q^{43} -193244. q^{44} +1.02069e6 q^{46} -580714. q^{47} +117649. q^{49} +217246. q^{50} +142707. q^{52} -824779. q^{53} +369837. q^{55} -298949. q^{56} +733389. q^{58} -324469. q^{59} +1.11728e6 q^{61} -7421.05 q^{62} +213598. q^{64} -273118. q^{65} +2.52040e6 q^{67} -121573. q^{68} -596123. q^{70} +3.84285e6 q^{71} +2.16857e6 q^{73} +1.89750e6 q^{74} +329074. q^{76} -1.01483e6 q^{77} -1.05542e6 q^{79} +2.55978e6 q^{80} +6.28465e6 q^{82} -2.53493e6 q^{83} +232669. q^{85} +1.01543e7 q^{86} +2.57871e6 q^{88} +6.17151e6 q^{89} +749435. q^{91} +4.79479e6 q^{92} -8.07410e6 q^{94} -629793. q^{95} +5.36213e6 q^{97} +1.63576e6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 11 q^{2} + 141 q^{4} - 500 q^{5} + 1372 q^{7} - 2133 q^{8} + 1375 q^{10} + 2708 q^{11} - 2212 q^{13} - 3773 q^{14} - 9599 q^{16} + 17016 q^{17} + 32668 q^{19} - 17625 q^{20} - 7196 q^{22} - 87696 q^{23}+ \cdots - 1294139 q^{98}+O(q^{100}) $ 4 * q - 11 * q^2 + 141 * q^4 - 500 * q^5 + 1372 * q^7 - 2133 * q^8 + 1375 * q^10 + 2708 * q^11 - 2212 * q^13 - 3773 * q^14 - 9599 * q^16 + 17016 * q^17 + 32668 * q^19 - 17625 * q^20 - 7196 * q^22 - 87696 * q^23 + 62500 * q^25 - 67354 * q^26 + 48363 * q^28 - 200488 * q^29 + 258852 * q^31 - 39445 * q^32 + 225518 * q^34 - 171500 * q^35 + 666856 * q^37 - 846556 * q^38 + 266625 * q^40 - 833000 * q^41 + 1051896 * q^43 - 1918236 * q^44 + 1786160 * q^46 - 529824 * q^47 + 470596 * q^49 - 171875 * q^50 + 211718 * q^52 - 791764 * q^53 - 338500 * q^55 - 731619 * q^56 + 3500242 * q^58 + 3825904 * q^59 + 212432 * q^61 - 882400 * q^62 - 1515063 * q^64 + 276500 * q^65 + 5165272 * q^67 - 7423202 * q^68 + 471625 * q^70 + 623260 * q^71 - 2083228 * q^73 - 1059642 * q^74 + 6990964 * q^76 + 928844 * q^77 - 1325016 * q^79 + 1199875 * q^80 + 11798982 * q^82 - 11447616 * q^83 - 2127000 * q^85 - 7525948 * q^86 + 19484572 * q^88 + 5805808 * q^89 - 758716 * q^91 + 2716288 * q^92 - 15552560 * q^94 - 4083500 * q^95 + 5857516 * q^97 - 1294139 * q^98

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 $

$2$	13.9037	1.22893	0.614464	−	0.788945i	$-0.289372\pi$
$2$	13.9037	1.22893	0.614464	+	0.788945i	$0.289372\pi$
$3$	0	0
$3$	0	0
$4$	65.3140	0.510265
$5$	−125.000	−0.447214
$5$	−125.000	−0.447214
$6$	0	0
$7$	343.000	0.377964
$7$	343.000	0.377964
$8$	−871.570	−0.601849
$9$	0	0
$10$	−1737.97	−0.549594
$11$	−2958.70	−0.670234	−0.335117	−	0.942177i	$-0.608776\pi$
$11$	−2958.70	−0.670234	−0.335117	+	0.942177i	$0.608776\pi$
$12$	0	0
$13$	2184.94	0.275828	0.137914	−	0.990444i	$-0.455960\pi$
$13$	2184.94	0.275828	0.137914	+	0.990444i	$0.455960\pi$
$14$	4768.98	0.464491
$15$	0	0
$16$	−20478.3	−1.24989
$17$	−1861.36	−0.0918879	−0.0459439	−	0.998944i	$-0.514630\pi$
$17$	−1861.36	−0.0918879	−0.0459439	+	0.998944i	$0.514630\pi$
$18$	0	0
$19$	5038.34	0.168519	0.0842597	−	0.996444i	$-0.473147\pi$
$19$	5038.34	0.168519	0.0842597	+	0.996444i	$0.473147\pi$
$20$	−8164.25	−0.228198
$21$	0	0
$22$	−41137.0	−0.823670
$23$	73411.5	1.25810	0.629052	−	0.777363i	$-0.283443\pi$
$23$	73411.5	1.25810	0.629052	+	0.777363i	$0.283443\pi$
$24$	0	0
$25$	15625.0	0.200000
$26$	30378.9	0.338973
$27$	0	0
$28$	22402.7	0.192862
$29$	52747.6	0.401615	0.200807	−	0.979631i	$-0.435643\pi$
$29$	52747.6	0.401615	0.200807	+	0.979631i	$0.435643\pi$
$30$	0	0
$31$	−533.745	−0.00321786	−0.00160893	−	0.999999i	$-0.500512\pi$
$31$	−533.745	−0.00321786	−0.00160893	+	0.999999i	$0.500512\pi$
$32$	−173164.	−0.934182
$33$	0	0
$34$	−25879.8	−0.112924
$35$	−42875.0	−0.169031
$36$	0	0
$37$	136474.	0.442938	0.221469	−	0.975167i	$-0.428915\pi$
$37$	136474.	0.442938	0.221469	+	0.975167i	$0.428915\pi$
$38$	70051.8	0.207098
$39$	0	0
$40$	108946.	0.269155
$41$	452012.	1.02425	0.512125	−	0.858911i	$-0.328858\pi$
$41$	452012.	1.02425	0.512125	+	0.858911i	$0.328858\pi$
$42$	0	0
$43$	730329.	1.40081	0.700405	−	0.713746i	$-0.253003\pi$
$43$	730329.	1.40081	0.700405	+	0.713746i	$0.253003\pi$
$44$	−193244.	−0.341997
$45$	0	0
$46$	1.02069e6	1.54612
$47$	−580714.	−0.815868	−0.407934	−	0.913011i	$-0.633751\pi$
$47$	−580714.	−0.815868	−0.407934	+	0.913011i	$0.633751\pi$
$48$	0	0
$49$	117649.	0.142857
$50$	217246.	0.245786
$51$	0	0
$52$	142707.	0.140745
$53$	−824779.	−0.760978	−0.380489	−	0.924785i	$-0.624244\pi$
$53$	−824779.	−0.760978	−0.380489	+	0.924785i	$0.624244\pi$
$54$	0	0
$55$	369837.	0.299738
$56$	−298949.	−0.227477
$57$	0	0
$58$	733389.	0.493556
$59$	−324469.	−0.205679	−0.102840	−	0.994698i	$-0.532793\pi$
$59$	−324469.	−0.205679	−0.102840	+	0.994698i	$0.532793\pi$
$60$	0	0
$61$	1.11728e6	0.630242	0.315121	−	0.949051i	$-0.397955\pi$
$61$	1.11728e6	0.630242	0.315121	+	0.949051i	$0.397955\pi$
$62$	−7421.05	−0.00395453
$63$	0	0
$64$	213598.	0.101851
$65$	−273118.	−0.123354
$66$	0	0
$67$	2.52040e6	1.02378	0.511891	−	0.859050i	$-0.328945\pi$
$67$	2.52040e6	1.02378	0.511891	+	0.859050i	$0.328945\pi$
$68$	−121573.	−0.0468872
$69$	0	0
$70$	−596123.	−0.207727
$71$	3.84285e6	1.27423	0.637117	−	0.770767i	$-0.280127\pi$
$71$	3.84285e6	1.27423	0.637117	+	0.770767i	$0.280127\pi$
$72$	0	0
$73$	2.16857e6	0.652446	0.326223	−	0.945293i	$-0.394224\pi$
$73$	2.16857e6	0.652446	0.326223	+	0.945293i	$0.394224\pi$
$74$	1.89750e6	0.544340
$75$	0	0
$76$	329074.	0.0859896
$77$	−1.01483e6	−0.253325
$78$	0	0
$79$	−1.05542e6	−0.240840	−0.120420	−	0.992723i	$-0.538424\pi$
$79$	−1.05542e6	−0.240840	−0.120420	+	0.992723i	$0.538424\pi$
$80$	2.55978e6	0.558970
$81$	0	0
$82$	6.28465e6	1.25873
$83$	−2.53493e6	−0.486623	−0.243312	−	0.969948i	$-0.578234\pi$
$83$	−2.53493e6	−0.486623	−0.243312	+	0.969948i	$0.578234\pi$
$84$	0	0
$85$	232669.	0.0410935
$86$	1.01543e7	1.72149
$87$	0	0
$88$	2.57871e6	0.403380
$89$	6.17151e6	0.927954	0.463977	−	0.885847i	$-0.346422\pi$
$89$	6.17151e6	0.927954	0.463977	+	0.885847i	$0.346422\pi$
$90$	0	0
$91$	749435.	0.104253
$92$	4.79479e6	0.641967
$93$	0	0
$94$	−8.07410e6	−1.00264
$95$	−629793.	−0.0753642
$96$	0	0
$97$	5.36213e6	0.596535	0.298268	−	0.954482i	$-0.403591\pi$
$97$	5.36213e6	0.596535	0.298268	+	0.954482i	$0.403591\pi$
$98$	1.63576e6	0.175561
$99$	0	0
$100$	1.02053e6	0.102053
$101$	6.68767e6	0.645877	0.322939	−	0.946420i	$-0.395329\pi$
$101$	6.68767e6	0.645877	0.322939	+	0.946420i	$0.395329\pi$
$102$	0	0
$103$	493600.	0.0445087	0.0222543	−	0.999752i	$-0.492916\pi$
$103$	493600.	0.0445087	0.0222543	+	0.999752i	$0.492916\pi$
$104$	−1.90433e6	−0.166007
$105$	0	0
$106$	−1.14675e7	−0.935188
$107$	8.53994e6	0.673925	0.336963	−	0.941518i	$-0.390600\pi$
$107$	8.53994e6	0.673925	0.336963	+	0.941518i	$0.390600\pi$
$108$	0	0
$109$	9.37440e6	0.693348	0.346674	−	0.937986i	$-0.387311\pi$
$109$	9.37440e6	0.693348	0.346674	+	0.937986i	$0.387311\pi$
$110$	5.14212e6	0.368356
$111$	0	0
$112$	−7.02405e6	−0.472416
$113$	2.90770e6	0.189572	0.0947862	−	0.995498i	$-0.469783\pi$
$113$	2.90770e6	0.189572	0.0947862	+	0.995498i	$0.469783\pi$
$114$	0	0
$115$	−9.17643e6	−0.562641
$116$	3.44515e6	0.204930
$117$	0	0
$118$	−4.51133e6	−0.252765
$119$	−638445.	−0.0347303
$120$	0	0
$121$	−1.07333e7	−0.550787
$122$	1.55344e7	0.774523
$123$	0	0
$124$	−34861.0	−0.00164196
$125$	−1.95312e6	−0.0894427
$126$	0	0
$127$	3.09234e7	1.33960	0.669798	−	0.742543i	$-0.266381\pi$
$127$	3.09234e7	1.33960	0.669798	+	0.742543i	$0.266381\pi$
$128$	2.51347e7	1.05935
$129$	0	0
$130$	−3.79736e6	−0.151593
$131$	1.31403e6	0.0510690	0.0255345	−	0.999674i	$-0.491871\pi$
$131$	1.31403e6	0.0510690	0.0255345	+	0.999674i	$0.491871\pi$
$132$	0	0
$133$	1.72815e6	0.0636943
$134$	3.50430e7	1.25816
$135$	0	0
$136$	1.62230e6	0.0553026
$137$	−5.31860e7	−1.76716	−0.883579	−	0.468282i	$-0.844873\pi$
$137$	−5.31860e7	−1.76716	−0.883579	+	0.468282i	$0.844873\pi$
$138$	0	0
$139$	5.10228e7	1.61143	0.805717	−	0.592301i	$-0.201780\pi$
$139$	5.10228e7	1.61143	0.805717	+	0.592301i	$0.201780\pi$
$140$	−2.80034e6	−0.0862506
$141$	0	0
$142$	5.34300e7	1.56594
$143$	−6.46458e6	−0.184869
$144$	0	0
$145$	−6.59345e6	−0.179608
$146$	3.01513e7	0.801809
$147$	0	0
$148$	8.91364e6	0.226016
$149$	−6.25124e7	−1.54815	−0.774077	−	0.633091i	$-0.781786\pi$
$149$	−6.25124e7	−1.54815	−0.774077	+	0.633091i	$0.781786\pi$
$150$	0	0
$151$	3.77217e7	0.891603	0.445801	−	0.895132i	$-0.352919\pi$
$151$	3.77217e7	0.891603	0.445801	+	0.895132i	$0.352919\pi$
$152$	−4.39127e6	−0.101423
$153$	0	0
$154$	−1.41100e7	−0.311318
$155$	66718.1	0.00143907
$156$	0	0
$157$	9.00303e7	1.85669	0.928347	−	0.371716i	$-0.121230\pi$
$157$	9.00303e7	1.85669	0.928347	+	0.371716i	$0.121230\pi$
$158$	−1.46742e7	−0.295975
$159$	0	0
$160$	2.16454e7	0.417779
$161$	2.51801e7	0.475519
$162$	0	0
$163$	−6.11773e7	−1.10645	−0.553227	−	0.833030i	$-0.686604\pi$
$163$	−6.11773e7	−1.10645	−0.553227	+	0.833030i	$0.686604\pi$
$164$	2.95227e7	0.522639
$165$	0	0
$166$	−3.52450e7	−0.598025
$167$	−1.16060e8	−1.92830	−0.964149	−	0.265360i	$-0.914509\pi$
$167$	−1.16060e8	−1.92830	−0.964149	+	0.265360i	$0.914509\pi$
$168$	0	0
$169$	−5.79745e7	−0.923919
$170$	3.23498e6	0.0505010
$171$	0	0
$172$	4.77007e7	0.714785
$173$	4.58752e7	0.673622	0.336811	−	0.941572i	$-0.390652\pi$
$173$	4.58752e7	0.673622	0.336811	+	0.941572i	$0.390652\pi$
$174$	0	0
$175$	5.35938e6	0.0755929
$176$	6.05890e7	0.837722
$177$	0	0
$178$	8.58071e7	1.14039
$179$	−9.29665e7	−1.21155	−0.605774	−	0.795636i	$-0.707137\pi$
$179$	−9.29665e7	−1.21155	−0.605774	+	0.795636i	$0.707137\pi$
$180$	0	0
$181$	1.86276e7	0.233498	0.116749	−	0.993161i	$-0.462753\pi$
$181$	1.86276e7	0.233498	0.116749	+	0.993161i	$0.462753\pi$
$182$	1.04199e7	0.128120
$183$	0	0
$184$	−6.39833e7	−0.757188
$185$	−1.70592e7	−0.198088
$186$	0	0
$187$	5.50719e6	0.0615864
$188$	−3.79288e7	−0.416309
$189$	0	0
$190$	−8.75647e6	−0.0926172
$191$	2.44240e7	0.253630	0.126815	−	0.991926i	$-0.459525\pi$
$191$	2.44240e7	0.253630	0.126815	+	0.991926i	$0.459525\pi$
$192$	0	0
$193$	−1.13841e7	−0.113985	−0.0569923	−	0.998375i	$-0.518151\pi$
$193$	−1.13841e7	−0.113985	−0.0569923	+	0.998375i	$0.518151\pi$
$194$	7.45536e7	0.733099
$195$	0	0
$196$	7.68412e6	0.0728951
$197$	−1.26299e7	−0.117698	−0.0588488	−	0.998267i	$-0.518743\pi$
$197$	−1.26299e7	−0.117698	−0.0588488	+	0.998267i	$0.518743\pi$
$198$	0	0
$199$	1.01401e8	0.912129	0.456064	−	0.889947i	$-0.349259\pi$
$199$	1.01401e8	0.912129	0.456064	+	0.889947i	$0.349259\pi$
$200$	−1.36183e7	−0.120370
$201$	0	0
$202$	9.29836e7	0.793737
$203$	1.80924e7	0.151796
$204$	0	0
$205$	−5.65015e7	−0.458059
$206$	6.86288e6	0.0546980
$207$	0	0
$208$	−4.47438e7	−0.344756
$209$	−1.49069e7	−0.112947
$210$	0	0
$211$	−2.41298e7	−0.176834	−0.0884170	−	0.996084i	$-0.528181\pi$
$211$	−2.41298e7	−0.176834	−0.0884170	+	0.996084i	$0.528181\pi$
$212$	−5.38696e7	−0.388301
$213$	0	0
$214$	1.18737e8	0.828206
$215$	−9.12912e7	−0.626461
$216$	0	0
$217$	−183074.	−0.00121624
$218$	1.30339e8	0.852075
$219$	0	0
$220$	2.41555e7	0.152946
$221$	−4.06695e6	−0.0253452
$222$	0	0
$223$	−2.32800e7	−0.140578	−0.0702889	−	0.997527i	$-0.522392\pi$
$223$	−2.32800e7	−0.140578	−0.0702889	+	0.997527i	$0.522392\pi$
$224$	−5.93951e7	−0.353088
$225$	0	0
$226$	4.04279e7	0.232971
$227$	−1.61931e8	−0.918841	−0.459420	−	0.888219i	$-0.651943\pi$
$227$	−1.61931e8	−0.918841	−0.459420	+	0.888219i	$0.651943\pi$
$228$	0	0
$229$	−9.71497e7	−0.534586	−0.267293	−	0.963615i	$-0.586129\pi$
$229$	−9.71497e7	−0.534586	−0.267293	+	0.963615i	$0.586129\pi$
$230$	−1.27587e8	−0.691446
$231$	0	0
$232$	−4.59732e7	−0.241711
$233$	−6.18305e7	−0.320226	−0.160113	−	0.987099i	$-0.551186\pi$
$233$	−6.18305e7	−0.320226	−0.160113	+	0.987099i	$0.551186\pi$
$234$	0	0
$235$	7.25893e7	0.364867
$236$	−2.11923e7	−0.104951
$237$	0	0
$238$	−8.87677e6	−0.0426811
$239$	−1.09923e8	−0.520831	−0.260415	−	0.965497i	$-0.583859\pi$
$239$	−1.09923e8	−0.520831	−0.260415	+	0.965497i	$0.583859\pi$
$240$	0	0
$241$	2.03146e8	0.934865	0.467433	−	0.884029i	$-0.345179\pi$
$241$	2.03146e8	0.934865	0.467433	+	0.884029i	$0.345179\pi$
$242$	−1.49233e8	−0.676877
$243$	0	0
$244$	7.29740e7	0.321591
$245$	−1.47061e7	−0.0638877
$246$	0	0
$247$	1.10085e7	0.0464823
$248$	465196.	0.00193667
$249$	0	0
$250$	−2.71557e7	−0.109919
$251$	−8.87979e7	−0.354442	−0.177221	−	0.984171i	$-0.556711\pi$
$251$	−8.87979e7	−0.354442	−0.177221	+	0.984171i	$0.556711\pi$
$252$	0	0
$253$	−2.17202e8	−0.843224
$254$	4.29950e8	1.64627
$255$	0	0
$256$	3.22126e8	1.20001
$257$	4.87641e8	1.79198	0.895991	−	0.444071i	$-0.146466\pi$
$257$	4.87641e8	1.79198	0.895991	+	0.444071i	$0.146466\pi$
$258$	0	0
$259$	4.68105e7	0.167415
$260$	−1.78384e7	−0.0629432
$261$	0	0
$262$	1.82700e7	0.0627601
$263$	1.12181e8	0.380254	0.190127	−	0.981760i	$-0.439110\pi$
$263$	1.12181e8	0.380254	0.190127	+	0.981760i	$0.439110\pi$
$264$	0	0
$265$	1.03097e8	0.340320
$266$	2.40278e7	0.0782758
$267$	0	0
$268$	1.64617e8	0.522401
$269$	4.95137e8	1.55093	0.775466	−	0.631389i	$-0.217515\pi$
$269$	4.95137e8	1.55093	0.775466	+	0.631389i	$0.217515\pi$
$270$	0	0
$271$	6.97464e7	0.212877	0.106439	−	0.994319i	$-0.466055\pi$
$271$	6.97464e7	0.212877	0.106439	+	0.994319i	$0.466055\pi$
$272$	3.81173e7	0.114850
$273$	0	0
$274$	−7.39484e8	−2.17171
$275$	−4.62297e7	−0.134047
$276$	0	0
$277$	2.29654e8	0.649225	0.324612	−	0.945847i	$-0.394766\pi$
$277$	2.29654e8	0.649225	0.324612	+	0.945847i	$0.394766\pi$
$278$	7.09408e8	1.98034
$279$	0	0
$280$	3.73686e7	0.101731
$281$	8.26555e7	0.222228	0.111114	−	0.993808i	$-0.464558\pi$
$281$	8.26555e7	0.222228	0.111114	+	0.993808i	$0.464558\pi$
$282$	0	0
$283$	−1.35905e7	−0.0356437	−0.0178219	−	0.999841i	$-0.505673\pi$
$283$	−1.35905e7	−0.0356437	−0.0178219	+	0.999841i	$0.505673\pi$
$284$	2.50992e8	0.650198
$285$	0	0
$286$	−8.98819e7	−0.227191
$287$	1.55040e8	0.387130
$288$	0	0
$289$	−4.06874e8	−0.991557
$290$	−9.16736e7	−0.220725
$291$	0	0
$292$	1.41638e8	0.332920
$293$	−1.02087e8	−0.237101	−0.118551	−	0.992948i	$-0.537825\pi$
$293$	−1.02087e8	−0.237101	−0.118551	+	0.992948i	$0.537825\pi$
$294$	0	0
$295$	4.05586e7	0.0919826
$296$	−1.18947e8	−0.266582
$297$	0	0
$298$	−8.69156e8	−1.90257
$299$	1.60400e8	0.347020
$300$	0	0
$301$	2.50503e8	0.529456
$302$	5.24472e8	1.09572
$303$	0	0
$304$	−1.03177e8	−0.210631
$305$	−1.39660e8	−0.281853
$306$	0	0
$307$	−3.74499e8	−0.738696	−0.369348	−	0.929291i	$-0.620419\pi$
$307$	−3.74499e8	−0.738696	−0.369348	+	0.929291i	$0.620419\pi$
$308$	−6.62828e7	−0.129263
$309$	0	0
$310$	927631.	0.00176852
$311$	2.97703e8	0.561205	0.280603	−	0.959824i	$-0.409466\pi$
$311$	2.97703e8	0.561205	0.280603	+	0.959824i	$0.409466\pi$
$312$	0	0
$313$	−6.51149e8	−1.20026	−0.600130	−	0.799903i	$-0.704884\pi$
$313$	−6.51149e8	−1.20026	−0.600130	+	0.799903i	$0.704884\pi$
$314$	1.25176e9	2.28174
$315$	0	0
$316$	−6.89334e7	−0.122892
$317$	−5.04553e8	−0.889610	−0.444805	−	0.895627i	$-0.646727\pi$
$317$	−5.04553e8	−0.889610	−0.444805	+	0.895627i	$0.646727\pi$
$318$	0	0
$319$	−1.56064e8	−0.269176
$320$	−2.66997e7	−0.0455493
$321$	0	0
$322$	3.50098e8	0.584378
$323$	−9.37814e6	−0.0154849
$324$	0	0
$325$	3.41397e7	0.0551656
$326$	−8.50593e8	−1.35975
$327$	0	0
$328$	−3.93960e8	−0.616444
$329$	−1.99185e8	−0.308369
$330$	0	0
$331$	8.69824e8	1.31836	0.659179	−	0.751986i	$-0.270904\pi$
$331$	8.69824e8	1.31836	0.659179	+	0.751986i	$0.270904\pi$
$332$	−1.65566e8	−0.248307
$333$	0	0
$334$	−1.61367e9	−2.36974
$335$	−3.15050e8	−0.457850
$336$	0	0
$337$	1.22083e9	1.73760	0.868802	−	0.495159i	$-0.164890\pi$
$337$	1.22083e9	1.73760	0.868802	+	0.495159i	$0.164890\pi$
$338$	−8.06063e8	−1.13543
$339$	0	0
$340$	1.51966e7	0.0209686
$341$	1.57919e6	0.00215672
$342$	0	0
$343$	4.03536e7	0.0539949
$344$	−6.36533e8	−0.843076
$345$	0	0
$346$	6.37837e8	0.827834
$347$	−1.06220e9	−1.36474	−0.682372	−	0.731005i	$-0.739052\pi$
$347$	−1.06220e9	−1.36474	−0.682372	+	0.731005i	$0.739052\pi$
$348$	0	0
$349$	2.69763e8	0.339698	0.169849	−	0.985470i	$-0.445672\pi$
$349$	2.69763e8	0.339698	0.169849	+	0.985470i	$0.445672\pi$
$350$	7.45154e7	0.0928983
$351$	0	0
$352$	5.12339e8	0.626121
$353$	−4.95774e8	−0.599891	−0.299946	−	0.953956i	$-0.596969\pi$
$353$	−4.95774e8	−0.599891	−0.299946	+	0.953956i	$0.596969\pi$
$354$	0	0
$355$	−4.80356e8	−0.569855
$356$	4.03086e8	0.473503
$357$	0	0
$358$	−1.29258e9	−1.48891
$359$	−6.53192e8	−0.745093	−0.372546	−	0.928014i	$-0.621515\pi$
$359$	−6.53192e8	−0.745093	−0.372546	+	0.928014i	$0.621515\pi$
$360$	0	0
$361$	−8.68487e8	−0.971601
$362$	2.58994e8	0.286952
$363$	0	0
$364$	4.89486e7	0.0531967
$365$	−2.71072e8	−0.291783
$366$	0	0
$367$	−7.53440e7	−0.0795641	−0.0397821	−	0.999208i	$-0.512666\pi$
$367$	−7.53440e7	−0.0795641	−0.0397821	+	0.999208i	$0.512666\pi$
$368$	−1.50334e9	−1.57250
$369$	0	0
$370$	−2.37187e8	−0.243436
$371$	−2.82899e8	−0.287623
$372$	0	0
$373$	−1.34806e9	−1.34502	−0.672508	−	0.740090i	$-0.734783\pi$
$373$	−1.34806e9	−1.34502	−0.672508	+	0.740090i	$0.734783\pi$
$374$	7.65705e7	0.0756852
$375$	0	0
$376$	5.06133e8	0.491029
$377$	1.15250e8	0.110776
$378$	0	0
$379$	1.55120e9	1.46363	0.731814	−	0.681505i	$-0.238674\pi$
$379$	1.55120e9	1.46363	0.731814	+	0.681505i	$0.238674\pi$
$380$	−4.11342e7	−0.0384557
$381$	0	0
$382$	3.39585e8	0.311693
$383$	−1.65628e9	−1.50639	−0.753194	−	0.657798i	$-0.771488\pi$
$383$	−1.65628e9	−1.50639	−0.753194	+	0.657798i	$0.771488\pi$
$384$	0	0
$385$	1.26854e8	0.113290
$386$	−1.58281e8	−0.140079
$387$	0	0
$388$	3.50222e8	0.304391
$389$	−1.72150e9	−1.48280	−0.741399	−	0.671064i	$-0.765838\pi$
$389$	−1.72150e9	−1.48280	−0.741399	+	0.671064i	$0.765838\pi$
$390$	0	0
$391$	−1.36645e8	−0.115604
$392$	−1.02539e8	−0.0859784
$393$	0	0
$394$	−1.75603e8	−0.144642
$395$	1.31927e8	0.107707
$396$	0	0
$397$	−7.19442e8	−0.577071	−0.288536	−	0.957469i	$-0.593168\pi$
$397$	−7.19442e8	−0.577071	−0.288536	+	0.957469i	$0.593168\pi$
$398$	1.40985e9	1.12094
$399$	0	0
$400$	−3.19973e8	−0.249979
$401$	−1.80953e9	−1.40139	−0.700697	−	0.713459i	$-0.747128\pi$
$401$	−1.80953e9	−1.40139	−0.700697	+	0.713459i	$0.747128\pi$
$402$	0	0
$403$	−1.16620e6	−0.000887576 0
$404$	4.36798e8	0.329569
$405$	0	0
$406$	2.51552e8	0.186547
$407$	−4.03785e8	−0.296872
$408$	0	0
$409$	−2.23865e8	−0.161791	−0.0808955	−	0.996723i	$-0.525778\pi$
$409$	−2.23865e8	−0.161791	−0.0808955	+	0.996723i	$0.525778\pi$
$410$	−7.85581e8	−0.562921
$411$	0	0
$412$	3.22389e7	0.0227112
$413$	−1.11293e8	−0.0777395
$414$	0	0
$415$	3.16866e8	0.217624
$416$	−3.78352e8	−0.257673
$417$	0	0
$418$	−2.07262e8	−0.138804
$419$	2.13566e9	1.41835	0.709175	−	0.705032i	$-0.249067\pi$
$419$	2.13566e9	1.41835	0.709175	+	0.705032i	$0.249067\pi$
$420$	0	0
$421$	−1.69012e9	−1.10390	−0.551951	−	0.833877i	$-0.686116\pi$
$421$	−1.69012e9	−1.10390	−0.551951	+	0.833877i	$0.686116\pi$
$422$	−3.35495e8	−0.217316
$423$	0	0
$424$	7.18853e8	0.457994
$425$	−2.90837e7	−0.0183776
$426$	0	0
$427$	3.83227e8	0.238209
$428$	5.57777e8	0.343881
$429$	0	0
$430$	−1.26929e9	−0.769876
$431$	7.17422e8	0.431623	0.215811	−	0.976435i	$-0.430760\pi$
$431$	7.17422e8	0.431623	0.215811	+	0.976435i	$0.430760\pi$
$432$	0	0
$433$	−2.65820e9	−1.57355	−0.786773	−	0.617242i	$-0.788250\pi$
$433$	−2.65820e9	−1.57355	−0.786773	+	0.617242i	$0.788250\pi$
$434$	−2.54542e6	−0.00149467
$435$	0	0
$436$	6.12280e8	0.353791
$437$	3.69872e8	0.212015
$438$	0	0
$439$	1.39177e9	0.785129	0.392565	−	0.919724i	$-0.371588\pi$
$439$	1.39177e9	0.785129	0.392565	+	0.919724i	$0.371588\pi$
$440$	−3.22339e8	−0.180397
$441$	0	0
$442$	−5.65458e7	−0.0311475
$443$	−5.19996e8	−0.284176	−0.142088	−	0.989854i	$-0.545382\pi$
$443$	−5.19996e8	−0.284176	−0.142088	+	0.989854i	$0.545382\pi$
$444$	0	0
$445$	−7.71439e8	−0.414994
$446$	−3.23680e8	−0.172760
$447$	0	0
$448$	7.32640e7	0.0384962
$449$	9.46400e8	0.493415	0.246708	−	0.969090i	$-0.420651\pi$
$449$	9.46400e8	0.493415	0.246708	+	0.969090i	$0.420651\pi$
$450$	0	0
$451$	−1.33737e9	−0.686487
$452$	1.89914e8	0.0967323
$453$	0	0
$454$	−2.25145e9	−1.12919
$455$	−9.36794e7	−0.0466234
$456$	0	0
$457$	2.55740e9	1.25341	0.626703	−	0.779259i	$-0.284404\pi$
$457$	2.55740e9	1.25341	0.626703	+	0.779259i	$0.284404\pi$
$458$	−1.35074e9	−0.656968
$459$	0	0
$460$	−5.99349e8	−0.287096
$461$	3.65768e9	1.73881	0.869405	−	0.494101i	$-0.164503\pi$
$461$	3.65768e9	1.73881	0.869405	+	0.494101i	$0.164503\pi$
$462$	0	0
$463$	−1.54707e9	−0.724398	−0.362199	−	0.932101i	$-0.617974\pi$
$463$	−1.54707e9	−0.724398	−0.362199	+	0.932101i	$0.617974\pi$
$464$	−1.08018e9	−0.501976
$465$	0	0
$466$	−8.59675e8	−0.393535
$467$	2.07885e9	0.944524	0.472262	−	0.881458i	$-0.343438\pi$
$467$	2.07885e9	0.944524	0.472262	+	0.881458i	$0.343438\pi$
$468$	0	0
$469$	8.64497e8	0.386953
$470$	1.00926e9	0.448396
$471$	0	0
$472$	2.82797e8	0.123788
$473$	−2.16082e9	−0.938870
$474$	0	0
$475$	7.87241e7	0.0337039
$476$	−4.16994e7	−0.0177217
$477$	0	0
$478$	−1.52834e9	−0.640064
$479$	6.28097e7	0.0261128	0.0130564	−	0.999915i	$-0.495844\pi$
$479$	6.28097e7	0.0261128	0.0130564	+	0.999915i	$0.495844\pi$
$480$	0	0
$481$	2.98187e8	0.122175
$482$	2.82449e9	1.14888
$483$	0	0
$484$	−7.01033e8	−0.281047
$485$	−6.70266e8	−0.266779
$486$	0	0
$487$	−5.65337e8	−0.221797	−0.110899	−	0.993832i	$-0.535373\pi$
$487$	−5.65337e8	−0.221797	−0.110899	+	0.993832i	$0.535373\pi$
$488$	−9.73788e8	−0.379311
$489$	0	0
$490$	−2.04470e8	−0.0785134
$491$	4.05243e9	1.54501	0.772504	−	0.635010i	$-0.219004\pi$
$491$	4.05243e9	1.54501	0.772504	+	0.635010i	$0.219004\pi$
$492$	0	0
$493$	−9.81820e7	−0.0369035
$494$	1.53059e8	0.0571235
$495$	0	0
$496$	1.09302e7	0.00402199
$497$	1.31810e9	0.481615
$498$	0	0
$499$	3.62634e9	1.30652	0.653260	−	0.757133i	$-0.273401\pi$
$499$	3.62634e9	1.30652	0.653260	+	0.757133i	$0.273401\pi$
$500$	−1.27566e8	−0.0456395
$501$	0	0
$502$	−1.23462e9	−0.435584
$503$	2.17925e8	0.0763517	0.0381759	−	0.999271i	$-0.487845\pi$
$503$	2.17925e8	0.0763517	0.0381759	+	0.999271i	$0.487845\pi$
$504$	0	0
$505$	−8.35959e8	−0.288845
$506$	−3.01993e9	−1.03626
$507$	0	0
$508$	2.01973e9	0.683549
$509$	−1.12646e9	−0.378622	−0.189311	−	0.981917i	$-0.560625\pi$
$509$	−1.12646e9	−0.378622	−0.189311	+	0.981917i	$0.560625\pi$
$510$	0	0
$511$	7.43821e8	0.246601
$512$	1.26151e9	0.415382
$513$	0	0
$514$	6.78003e9	2.20222
$515$	−6.16999e7	−0.0199049
$516$	0	0
$517$	1.71816e9	0.546823
$518$	6.50841e8	0.205741
$519$	0	0
$520$	2.38041e8	0.0742404
$521$	3.45164e9	1.06928	0.534642	−	0.845079i	$-0.320446\pi$
$521$	3.45164e9	1.06928	0.534642	+	0.845079i	$0.320446\pi$
$522$	0	0
$523$	−2.96843e9	−0.907343	−0.453672	−	0.891169i	$-0.649886\pi$
$523$	−2.96843e9	−0.907343	−0.453672	+	0.891169i	$0.649886\pi$
$524$	8.58248e7	0.0260587
$525$	0	0
$526$	1.55973e9	0.467305
$527$	993489.	0.000295683 0
$528$	0	0
$529$	1.98442e9	0.582826
$530$	1.43344e9	0.418229
$531$	0	0
$532$	1.12872e8	0.0325010
$533$	9.87619e8	0.282517
$534$	0	0
$535$	−1.06749e9	−0.301388
$536$	−2.19671e9	−0.616162
$537$	0	0
$538$	6.88426e9	1.90598
$539$	−3.48088e8	−0.0957477
$540$	0	0
$541$	−3.14949e8	−0.0855164	−0.0427582	−	0.999085i	$-0.513615\pi$
$541$	−3.14949e8	−0.0855164	−0.0427582	+	0.999085i	$0.513615\pi$
$542$	9.69736e8	0.261611
$543$	0	0
$544$	3.22319e8	0.0858400
$545$	−1.17180e9	−0.310074
$546$	0	0
$547$	5.61159e9	1.46599	0.732994	−	0.680235i	$-0.238122\pi$
$547$	5.61159e9	1.46599	0.732994	+	0.680235i	$0.238122\pi$
$548$	−3.47379e9	−0.901720
$549$	0	0
$550$	−6.42765e8	−0.164734
$551$	2.65760e8	0.0676799
$552$	0	0
$553$	−3.62008e8	−0.0910290
$554$	3.19305e9	0.797851
$555$	0	0
$556$	3.33250e9	0.822259
$557$	−3.05994e7	−0.00750273	−0.00375136	−	0.999993i	$-0.501194\pi$
$557$	−3.05994e7	−0.00750273	−0.00375136	+	0.999993i	$0.501194\pi$
$558$	0	0
$559$	1.59573e9	0.386382
$560$	8.78006e8	0.211271
$561$	0	0
$562$	1.14922e9	0.273103
$563$	8.06283e9	1.90418	0.952090	−	0.305818i	$-0.0989298\pi$
$563$	8.06283e9	1.90418	0.952090	+	0.305818i	$0.0989298\pi$
$564$	0	0
$565$	−3.63463e8	−0.0847794
$566$	−1.88959e8	−0.0438036
$567$	0	0
$568$	−3.34931e9	−0.766896
$569$	6.34627e9	1.44419	0.722097	−	0.691792i	$-0.243178\pi$
$569$	6.34627e9	1.44419	0.722097	+	0.691792i	$0.243178\pi$
$570$	0	0
$571$	−8.59406e8	−0.193184	−0.0965921	−	0.995324i	$-0.530794\pi$
$571$	−8.59406e8	−0.193184	−0.0965921	+	0.995324i	$0.530794\pi$
$572$	−4.22228e8	−0.0943323
$573$	0	0
$574$	2.15564e9	0.475755
$575$	1.14705e9	0.251621
$576$	0	0
$577$	−6.06450e9	−1.31426	−0.657128	−	0.753779i	$-0.728229\pi$
$577$	−6.06450e9	−1.31426	−0.657128	+	0.753779i	$0.728229\pi$
$578$	−5.65707e9	−1.21855
$579$	0	0
$580$	−4.30644e8	−0.0916475
$581$	−8.69481e8	−0.183926
$582$	0	0
$583$	2.44027e9	0.510033
$584$	−1.89007e9	−0.392674
$585$	0	0
$586$	−1.41939e9	−0.291381
$587$	4.77389e9	0.974180	0.487090	−	0.873352i	$-0.338058\pi$
$587$	4.77389e9	0.974180	0.487090	+	0.873352i	$0.338058\pi$
$588$	0	0
$589$	−2.68919e6	−0.000542273 0
$590$	5.63916e8	0.113040
$591$	0	0
$592$	−2.79475e9	−0.553626
$593$	4.59686e9	0.905253	0.452627	−	0.891700i	$-0.350487\pi$
$593$	4.59686e9	0.905253	0.452627	+	0.891700i	$0.350487\pi$
$594$	0	0
$595$	7.98056e7	0.0155319
$596$	−4.08293e9	−0.789970
$597$	0	0
$598$	2.23016e9	0.426463
$599$	−5.98241e9	−1.13732	−0.568659	−	0.822573i	$-0.692538\pi$
$599$	−5.98241e9	−1.13732	−0.568659	+	0.822573i	$0.692538\pi$
$600$	0	0
$601$	9.86903e9	1.85444	0.927222	−	0.374513i	$-0.122190\pi$
$601$	9.86903e9	1.85444	0.927222	+	0.374513i	$0.122190\pi$
$602$	3.48293e9	0.650664
$603$	0	0
$604$	2.46375e9	0.454954
$605$	1.34166e9	0.246319
$606$	0	0
$607$	−3.18007e9	−0.577134	−0.288567	−	0.957460i	$-0.593179\pi$
$607$	−3.18007e9	−0.577134	−0.288567	+	0.957460i	$0.593179\pi$
$608$	−8.72457e8	−0.157428
$609$	0	0
$610$	−1.94180e9	−0.346377
$611$	−1.26883e9	−0.225039
$612$	0	0
$613$	6.44084e7	0.0112936	0.00564678	−	0.999984i	$-0.498203\pi$
$613$	6.44084e7	0.0112936	0.00564678	+	0.999984i	$0.498203\pi$
$614$	−5.20693e9	−0.907805
$615$	0	0
$616$	8.84499e8	0.152463
$617$	7.16777e9	1.22853	0.614266	−	0.789099i	$-0.289452\pi$
$617$	7.16777e9	1.22853	0.614266	+	0.789099i	$0.289452\pi$
$618$	0	0
$619$	8.16795e9	1.38419	0.692095	−	0.721807i	$-0.256688\pi$
$619$	8.16795e9	1.38419	0.692095	+	0.721807i	$0.256688\pi$
$620$	4.35762e6	0.000734309 0
$621$	0	0
$622$	4.13918e9	0.689681
$623$	2.11683e9	0.350734
$624$	0	0
$625$	2.44141e8	0.0400000
$626$	−9.05340e9	−1.47503
$627$	0	0
$628$	5.88024e9	0.947406
$629$	−2.54026e8	−0.0407006
$630$	0	0
$631$	4.18586e9	0.663257	0.331628	−	0.943410i	$-0.392402\pi$
$631$	4.18586e9	0.663257	0.331628	+	0.943410i	$0.392402\pi$
$632$	9.19869e8	0.144949
$633$	0	0
$634$	−7.01518e9	−1.09327
$635$	−3.86542e9	−0.599085
$636$	0	0
$637$	2.57056e8	0.0394040
$638$	−2.16988e9	−0.330798
$639$	0	0
$640$	−3.14184e9	−0.473756
$641$	−8.24535e9	−1.23653	−0.618267	−	0.785968i	$-0.712165\pi$
$641$	−8.24535e9	−1.23653	−0.618267	+	0.785968i	$0.712165\pi$
$642$	0	0
$643$	1.13998e10	1.69105	0.845526	−	0.533934i	$-0.179287\pi$
$643$	1.13998e10	1.69105	0.845526	+	0.533934i	$0.179287\pi$
$644$	1.64461e9	0.242641
$645$	0	0
$646$	−1.30391e8	−0.0190298
$647$	1.19774e10	1.73859	0.869295	−	0.494293i	$-0.164573\pi$
$647$	1.19774e10	1.73859	0.869295	+	0.494293i	$0.164573\pi$
$648$	0	0
$649$	9.60005e8	0.137853
$650$	4.74670e8	0.0677945
$651$	0	0
$652$	−3.99573e9	−0.564586
$653$	1.30013e10	1.82722	0.913610	−	0.406591i	$-0.133283\pi$
$653$	1.30013e10	1.82722	0.913610	+	0.406591i	$0.133283\pi$
$654$	0	0
$655$	−1.64254e8	−0.0228387
$656$	−9.25642e9	−1.28020
$657$	0	0
$658$	−2.76942e9	−0.378964
$659$	−4.16352e9	−0.566711	−0.283356	−	0.959015i	$-0.591448\pi$
$659$	−4.16352e9	−0.566711	−0.283356	+	0.959015i	$0.591448\pi$
$660$	0	0
$661$	−6.85816e9	−0.923640	−0.461820	−	0.886974i	$-0.652803\pi$
$661$	−6.85816e9	−0.923640	−0.461820	+	0.886974i	$0.652803\pi$
$662$	1.20938e10	1.62017
$663$	0	0
$664$	2.20937e9	0.292874
$665$	−2.16019e8	−0.0284850
$666$	0	0
$667$	3.87228e9	0.505273
$668$	−7.58033e9	−0.983944
$669$	0	0
$670$	−4.38037e9	−0.562664
$671$	−3.30570e9	−0.422410
$672$	0	0
$673$	1.17148e10	1.48144	0.740718	−	0.671816i	$-0.234486\pi$
$673$	1.17148e10	1.48144	0.740718	+	0.671816i	$0.234486\pi$
$674$	1.69741e10	2.13539
$675$	0	0
$676$	−3.78655e9	−0.471444
$677$	9.82931e9	1.21748	0.608741	−	0.793369i	$-0.291675\pi$
$677$	9.82931e9	1.21748	0.608741	+	0.793369i	$0.291675\pi$
$678$	0	0
$679$	1.83921e9	0.225469
$680$	−2.02788e8	−0.0247321
$681$	0	0
$682$	2.19566e7	0.00265046
$683$	−1.53027e10	−1.83779	−0.918894	−	0.394504i	$-0.870916\pi$
$683$	−1.53027e10	−1.83779	−0.918894	+	0.394504i	$0.870916\pi$
$684$	0	0
$685$	6.64825e9	0.790297
$686$	5.61066e8	0.0663559
$687$	0	0
$688$	−1.49559e10	−1.75086
$689$	−1.80209e9	−0.209899
$690$	0	0
$691$	7.32282e9	0.844316	0.422158	−	0.906522i	$-0.361273\pi$
$691$	7.32282e9	0.844316	0.422158	+	0.906522i	$0.361273\pi$
$692$	2.99629e9	0.343726
$693$	0	0
$694$	−1.47685e10	−1.67717
$695$	−6.37785e9	−0.720655
$696$	0	0
$697$	−8.41354e8	−0.0941162
$698$	3.75071e9	0.417465
$699$	0	0
$700$	3.50042e8	0.0385724
$701$	−8.02517e9	−0.879916	−0.439958	−	0.898018i	$-0.645007\pi$
$701$	−8.02517e9	−0.879916	−0.439958	+	0.898018i	$0.645007\pi$
$702$	0	0
$703$	6.87601e8	0.0746437
$704$	−6.31971e8	−0.0682642
$705$	0	0
$706$	−6.89312e9	−0.737224
$707$	2.29387e9	0.244119
$708$	0	0
$709$	−5.32921e9	−0.561567	−0.280783	−	0.959771i	$-0.590594\pi$
$709$	−5.32921e9	−0.561567	−0.280783	+	0.959771i	$0.590594\pi$
$710$	−6.67875e9	−0.700311
$711$	0	0
$712$	−5.37891e9	−0.558488
$713$	−3.91830e7	−0.00404841
$714$	0	0
$715$	8.08073e8	0.0826760
$716$	−6.07201e9	−0.618211
$717$	0	0
$718$	−9.08181e9	−0.915666
$719$	−9.89593e9	−0.992900	−0.496450	−	0.868065i	$-0.665363\pi$
$719$	−9.89593e9	−0.992900	−0.496450	+	0.868065i	$0.665363\pi$
$720$	0	0
$721$	1.69305e8	0.0168227
$722$	−1.20752e10	−1.19403
$723$	0	0
$724$	1.21664e9	0.119146
$725$	8.24181e8	0.0803229
$726$	0	0
$727$	1.67155e10	1.61343	0.806715	−	0.590941i	$-0.201243\pi$
$727$	1.67155e10	1.61343	0.806715	+	0.590941i	$0.201243\pi$
$728$	−6.53185e8	−0.0627446
$729$	0	0
$730$	−3.76891e9	−0.358580
$731$	−1.35940e9	−0.128717
$732$	0	0
$733$	1.23126e10	1.15475	0.577374	−	0.816480i	$-0.304077\pi$
$733$	1.23126e10	1.15475	0.577374	+	0.816480i	$0.304077\pi$
$734$	−1.04756e9	−0.0977786
$735$	0	0
$736$	−1.27122e10	−1.17530
$737$	−7.45711e9	−0.686174
$738$	0	0
$739$	−5.35923e9	−0.488480	−0.244240	−	0.969715i	$-0.578539\pi$
$739$	−5.35923e9	−0.488480	−0.244240	+	0.969715i	$0.578539\pi$
$740$	−1.11421e9	−0.101077
$741$	0	0
$742$	−3.93336e9	−0.353468
$743$	2.33377e9	0.208736	0.104368	−	0.994539i	$-0.466718\pi$
$743$	2.33377e9	0.208736	0.104368	+	0.994539i	$0.466718\pi$
$744$	0	0
$745$	7.81405e9	0.692356
$746$	−1.87430e10	−1.65293
$747$	0	0
$748$	3.59696e8	0.0314254
$749$	2.92920e9	0.254720
$750$	0	0
$751$	−8.49871e9	−0.732172	−0.366086	−	0.930581i	$-0.619302\pi$
$751$	−8.49871e9	−0.732172	−0.366086	+	0.930581i	$0.619302\pi$
$752$	1.18920e10	1.01975
$753$	0	0
$754$	1.60241e9	0.136136
$755$	−4.71521e9	−0.398737
$756$	0	0
$757$	−4.51420e9	−0.378220	−0.189110	−	0.981956i	$-0.560560\pi$
$757$	−4.51420e9	−0.378220	−0.189110	+	0.981956i	$0.560560\pi$
$758$	2.15675e10	1.79869
$759$	0	0
$760$	5.48908e8	0.0453578
$761$	−6.64246e9	−0.546364	−0.273182	−	0.961962i	$-0.588076\pi$
$761$	−6.64246e9	−0.546364	−0.273182	+	0.961962i	$0.588076\pi$
$762$	0	0
$763$	3.21542e9	0.262061
$764$	1.59523e9	0.129418
$765$	0	0
$766$	−2.30284e10	−1.85124
$767$	−7.08945e8	−0.0567321
$768$	0	0
$769$	−8.71913e8	−0.0691402	−0.0345701	−	0.999402i	$-0.511006\pi$
$769$	−8.71913e8	−0.0691402	−0.0345701	+	0.999402i	$0.511006\pi$
$770$	1.76375e9	0.139226
$771$	0	0
$772$	−7.43538e8	−0.0581624
$773$	2.09485e9	0.163127	0.0815633	−	0.996668i	$-0.474009\pi$
$773$	2.09485e9	0.163127	0.0815633	+	0.996668i	$0.474009\pi$
$774$	0	0
$775$	−8.33976e6	−0.000643573 0
$776$	−4.67347e9	−0.359024
$777$	0	0
$778$	−2.39352e10	−1.82225
$779$	2.27739e9	0.172606
$780$	0	0
$781$	−1.13698e10	−0.854035
$782$	−1.89987e9	−0.142070
$783$	0	0
$784$	−2.40925e9	−0.178556
$785$	−1.12538e10	−0.830338
$786$	0	0
$787$	−1.99449e10	−1.45855	−0.729273	−	0.684223i	$-0.760142\pi$
$787$	−1.99449e10	−1.45855	−0.729273	+	0.684223i	$0.760142\pi$
$788$	−8.24908e8	−0.0600570
$789$	0	0
$790$	1.83428e9	0.132364
$791$	9.97342e8	0.0716517
$792$	0	0
$793$	2.44119e9	0.173838
$794$	−1.00029e10	−0.709179
$795$	0	0
$796$	6.62290e9	0.465428
$797$	1.59998e10	1.11947	0.559733	−	0.828673i	$-0.310904\pi$
$797$	1.59998e10	1.11947	0.559733	+	0.828673i	$0.310904\pi$
$798$	0	0
$799$	1.08092e9	0.0749684
$800$	−2.70568e9	−0.186836
$801$	0	0
$802$	−2.51592e10	−1.72221
$803$	−6.41616e9	−0.437291
$804$	0	0
$805$	−3.14752e9	−0.212658
$806$	−1.62146e7	−0.00109077
$807$	0	0
$808$	−5.82878e9	−0.388721
$809$	−3.39534e9	−0.225457	−0.112728	−	0.993626i	$-0.535959\pi$
$809$	−3.39534e9	−0.225457	−0.112728	+	0.993626i	$0.535959\pi$
$810$	0	0
$811$	−1.53461e10	−1.01024	−0.505119	−	0.863050i	$-0.668551\pi$
$811$	−1.53461e10	−1.01024	−0.505119	+	0.863050i	$0.668551\pi$
$812$	1.18169e9	0.0774563
$813$	0	0
$814$	−5.61412e9	−0.364835
$815$	7.64716e9	0.494822
$816$	0	0
$817$	3.67965e9	0.236064
$818$	−3.11256e9	−0.198830
$819$	0	0
$820$	−3.69033e9	−0.233731
$821$	1.04411e10	0.658484	0.329242	−	0.944246i	$-0.393207\pi$
$821$	1.04411e10	0.658484	0.329242	+	0.944246i	$0.393207\pi$
$822$	0	0
$823$	6.71827e9	0.420105	0.210052	−	0.977690i	$-0.432637\pi$
$823$	6.71827e9	0.420105	0.210052	+	0.977690i	$0.432637\pi$
$824$	−4.30207e8	−0.0267875
$825$	0	0
$826$	−1.54739e9	−0.0955363
$827$	−9.51821e9	−0.585175	−0.292588	−	0.956239i	$-0.594516\pi$
$827$	−9.51821e9	−0.585175	−0.292588	+	0.956239i	$0.594516\pi$
$828$	0	0
$829$	−1.09730e10	−0.668939	−0.334469	−	0.942407i	$-0.608557\pi$
$829$	−1.09730e10	−0.668939	−0.334469	+	0.942407i	$0.608557\pi$
$830$	4.40563e9	0.267445
$831$	0	0
$832$	4.66698e8	0.0280934
$833$	−2.18987e8	−0.0131268
$834$	0	0
$835$	1.45075e10	0.862361
$836$	−9.73631e8	−0.0576332
$837$	0	0
$838$	2.96937e10	1.74305
$839$	1.13557e10	0.663815	0.331907	−	0.943312i	$-0.392308\pi$
$839$	1.13557e10	0.663815	0.331907	+	0.943312i	$0.392308\pi$
$840$	0	0
$841$	−1.44676e10	−0.838706
$842$	−2.34990e10	−1.35662
$843$	0	0
$844$	−1.57601e9	−0.0902322
$845$	7.24682e9	0.413189
$846$	0	0
$847$	−3.68151e9	−0.208178
$848$	1.68901e10	0.951142
$849$	0	0
$850$	−4.04372e8	−0.0225847
$851$	1.00187e10	0.557262
$852$	0	0
$853$	−4.87491e9	−0.268934	−0.134467	−	0.990918i	$-0.542932\pi$
$853$	−4.87491e9	−0.268934	−0.134467	+	0.990918i	$0.542932\pi$
$854$	5.32829e9	0.292742
$855$	0	0
$856$	−7.44316e9	−0.405601
$857$	1.58245e10	0.858809	0.429404	−	0.903112i	$-0.358724\pi$
$857$	1.58245e10	0.858809	0.429404	+	0.903112i	$0.358724\pi$
$858$	0	0
$859$	1.93322e10	1.04065	0.520325	−	0.853968i	$-0.325811\pi$
$859$	1.93322e10	1.04065	0.520325	+	0.853968i	$0.325811\pi$
$860$	−5.96259e9	−0.319661
$861$	0	0
$862$	9.97485e9	0.530434
$863$	7.29451e9	0.386330	0.193165	−	0.981166i	$-0.438125\pi$
$863$	7.29451e9	0.386330	0.193165	+	0.981166i	$0.438125\pi$
$864$	0	0
$865$	−5.73440e9	−0.301253
$866$	−3.69589e10	−1.93378
$867$	0	0
$868$	−1.19573e7	−0.000620604 0
$869$	3.12266e9	0.161419
$870$	0	0
$871$	5.50693e9	0.282388
$872$	−8.17045e9	−0.417290
$873$	0	0
$874$	5.14260e9	0.260551
$875$	−6.69922e8	−0.0338062
$876$	0	0
$877$	3.68227e10	1.84339	0.921694	−	0.387918i	$-0.126806\pi$
$877$	3.68227e10	1.84339	0.921694	+	0.387918i	$0.126806\pi$
$878$	1.93508e10	0.964868
$879$	0	0
$880$	−7.57363e9	−0.374641
$881$	2.06876e10	1.01928	0.509640	−	0.860388i	$-0.329778\pi$
$881$	2.06876e10	1.01928	0.509640	+	0.860388i	$0.329778\pi$
$882$	0	0
$883$	1.05121e10	0.513841	0.256920	−	0.966433i	$-0.417292\pi$
$883$	1.05121e10	0.513841	0.256920	+	0.966433i	$0.417292\pi$
$884$	−2.65629e8	−0.0129328
$885$	0	0
$886$	−7.22989e9	−0.349232
$887$	2.30445e9	0.110875	0.0554376	−	0.998462i	$-0.482345\pi$
$887$	2.30445e9	0.110875	0.0554376	+	0.998462i	$0.482345\pi$
$888$	0	0
$889$	1.06067e10	0.506320
$890$	−1.07259e10	−0.509998
$891$	0	0
$892$	−1.52051e9	−0.0717320
$893$	−2.92584e9	−0.137490
$894$	0	0
$895$	1.16208e10	0.541821
$896$	8.62122e9	0.400397
$897$	0	0
$898$	1.31585e10	0.606372
$899$	−2.81537e7	−0.00129234
$900$	0	0
$901$	1.53521e9	0.0699246
$902$	−1.85944e10	−0.843644
$903$	0	0
$904$	−2.53427e9	−0.114094
$905$	−2.32845e9	−0.104423
$906$	0	0
$907$	1.70138e10	0.757138	0.378569	−	0.925573i	$-0.376416\pi$
$907$	1.70138e10	0.757138	0.378569	+	0.925573i	$0.376416\pi$
$908$	−1.05764e10	−0.468853
$909$	0	0
$910$	−1.30249e9	−0.0572968
$911$	−4.07352e10	−1.78507	−0.892536	−	0.450976i	$-0.851076\pi$
$911$	−4.07352e10	−1.78507	−0.892536	+	0.450976i	$0.851076\pi$
$912$	0	0
$913$	7.50009e9	0.326151
$914$	3.55574e10	1.54035
$915$	0	0
$916$	−6.34523e9	−0.272781
$917$	4.50714e8	0.0193023
$918$	0	0
$919$	−4.03604e10	−1.71534	−0.857671	−	0.514198i	$-0.828090\pi$
$919$	−4.03604e10	−1.71534	−0.857671	+	0.514198i	$0.828090\pi$
$920$	7.99791e9	0.338625
$921$	0	0
$922$	5.08554e10	2.13687
$923$	8.39640e9	0.351469
$924$	0	0
$925$	2.13240e9	0.0885877
$926$	−2.15101e10	−0.890234
$927$	0	0
$928$	−9.13396e9	−0.375181
$929$	8.81475e9	0.360707	0.180354	−	0.983602i	$-0.442276\pi$
$929$	8.81475e9	0.360707	0.180354	+	0.983602i	$0.442276\pi$
$930$	0	0
$931$	5.92756e8	0.0240742
$932$	−4.03839e9	−0.163400
$933$	0	0
$934$	2.89037e10	1.16075
$935$	−6.88399e8	−0.0275423
$936$	0	0
$937$	7.28321e9	0.289224	0.144612	−	0.989488i	$-0.453807\pi$
$937$	7.28321e9	0.289224	0.144612	+	0.989488i	$0.453807\pi$
$938$	1.20197e10	0.475538
$939$	0	0
$940$	4.74110e9	0.186179
$941$	−5.48013e9	−0.214401	−0.107201	−	0.994237i	$-0.534189\pi$
$941$	−5.48013e9	−0.214401	−0.107201	+	0.994237i	$0.534189\pi$
$942$	0	0
$943$	3.31828e10	1.28861
$944$	6.64456e9	0.257078
$945$	0	0
$946$	−3.00435e10	−1.15380
$947$	−4.26478e10	−1.63182	−0.815909	−	0.578180i	$-0.803763\pi$
$947$	−4.26478e10	−1.63182	−0.815909	+	0.578180i	$0.803763\pi$
$948$	0	0
$949$	4.73821e9	0.179963
$950$	1.09456e9	0.0414197
$951$	0	0
$952$	5.56450e8	0.0209024
$953$	−2.61130e10	−0.977307	−0.488654	−	0.872478i	$-0.662512\pi$
$953$	−2.61130e10	−0.977307	−0.488654	+	0.872478i	$0.662512\pi$
$954$	0	0
$955$	−3.05300e9	−0.113427
$956$	−7.17952e9	−0.265762
$957$	0	0
$958$	8.73290e8	0.0320907
$959$	−1.82428e10	−0.667923
$960$	0	0
$961$	−2.75123e10	−0.999990
$962$	4.14592e9	0.150144
$963$	0	0
$964$	1.32683e10	0.477029
$965$	1.42301e9	0.0509755
$966$	0	0
$967$	1.73017e10	0.615313	0.307656	−	0.951498i	$-0.400455\pi$
$967$	1.73017e10	0.615313	0.307656	+	0.951498i	$0.400455\pi$
$968$	9.35480e9	0.331490
$969$	0	0
$970$	−9.31920e9	−0.327852
$971$	2.51678e8	0.00882221	0.00441111	−	0.999990i	$-0.498596\pi$
$971$	2.51678e8	0.00882221	0.00441111	+	0.999990i	$0.498596\pi$
$972$	0	0
$973$	1.75008e10	0.609065
$974$	−7.86029e9	−0.272573
$975$	0	0
$976$	−2.28800e10	−0.787737
$977$	3.94226e10	1.35243	0.676215	−	0.736705i	$-0.263619\pi$
$977$	3.94226e10	1.35243	0.676215	+	0.736705i	$0.263619\pi$
$978$	0	0
$979$	−1.82596e10	−0.621946
$980$	−9.60515e8	−0.0325997
$981$	0	0
$982$	5.63440e10	1.89870
$983$	2.90217e9	0.0974507	0.0487253	−	0.998812i	$-0.484484\pi$
$983$	2.90217e9	0.0974507	0.0487253	+	0.998812i	$0.484484\pi$
$984$	0	0
$985$	1.57874e9	0.0526359
$986$	−1.36510e9	−0.0453518
$987$	0	0
$988$	7.19007e8	0.0237183
$989$	5.36145e10	1.76236
$990$	0	0
$991$	−3.08412e10	−1.00664	−0.503319	−	0.864100i	$-0.667888\pi$
$991$	−3.08412e10	−1.00664	−0.503319	+	0.864100i	$0.667888\pi$
$992$	9.24252e7	0.00300607
$993$	0	0
$994$	1.83265e10	0.591871
$995$	−1.26751e10	−0.407916
$996$	0	0
$997$	−4.01320e10	−1.28250	−0.641251	−	0.767331i	$-0.721584\pi$
$997$	−4.01320e10	−1.28250	−0.641251	+	0.767331i	$0.721584\pi$
$998$	5.04196e10	1.60562
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.8.a.g.1.4		4
3.2	odd	2		105.8.a.h.1.1	✓	4
15.14	odd	2		525.8.a.i.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.8.a.h.1.1	✓	4	3.2	odd	2
315.8.a.g.1.4		4	1.1	even	1	trivial
525.8.a.i.1.4		4	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	315.a (trivial)

\(f(q)\)	\(=\)	\(q+13.9037 q^{2} +65.3140 q^{4} -125.000 q^{5} +343.000 q^{7} -871.570 q^{8} -1737.97 q^{10} -2958.70 q^{11} +2184.94 q^{13} +4768.98 q^{14} -20478.3 q^{16} -1861.36 q^{17} +5038.34 q^{19} -8164.25 q^{20} -41137.0 q^{22} +73411.5 q^{23} +15625.0 q^{25} +30378.9 q^{26} +22402.7 q^{28} +52747.6 q^{29} -533.745 q^{31} -173164. q^{32} -25879.8 q^{34} -42875.0 q^{35} +136474. q^{37} +70051.8 q^{38} +108946. q^{40} +452012. q^{41} +730329. q^{43} -193244. q^{44} +1.02069e6 q^{46} -580714. q^{47} +117649. q^{49} +217246. q^{50} +142707. q^{52} -824779. q^{53} +369837. q^{55} -298949. q^{56} +733389. q^{58} -324469. q^{59} +1.11728e6 q^{61} -7421.05 q^{62} +213598. q^{64} -273118. q^{65} +2.52040e6 q^{67} -121573. q^{68} -596123. q^{70} +3.84285e6 q^{71} +2.16857e6 q^{73} +1.89750e6 q^{74} +329074. q^{76} -1.01483e6 q^{77} -1.05542e6 q^{79} +2.55978e6 q^{80} +6.28465e6 q^{82} -2.53493e6 q^{83} +232669. q^{85} +1.01543e7 q^{86} +2.57871e6 q^{88} +6.17151e6 q^{89} +749435. q^{91} +4.79479e6 q^{92} -8.07410e6 q^{94} -629793. q^{95} +5.36213e6 q^{97} +1.63576e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 11 q^{2} + 141 q^{4} - 500 q^{5} + 1372 q^{7} - 2133 q^{8} + 1375 q^{10} + 2708 q^{11} - 2212 q^{13} - 3773 q^{14} - 9599 q^{16} + 17016 q^{17} + 32668 q^{19} - 17625 q^{20} - 7196 q^{22} - 87696 q^{23}+ \cdots - 1294139 q^{98}+O(q^{100}) \) 4 * q - 11 * q^2 + 141 * q^4 - 500 * q^5 + 1372 * q^7 - 2133 * q^8 + 1375 * q^10 + 2708 * q^11 - 2212 * q^13 - 3773 * q^14 - 9599 * q^16 + 17016 * q^17 + 32668 * q^19 - 17625 * q^20 - 7196 * q^22 - 87696 * q^23 + 62500 * q^25 - 67354 * q^26 + 48363 * q^28 - 200488 * q^29 + 258852 * q^31 - 39445 * q^32 + 225518 * q^34 - 171500 * q^35 + 666856 * q^37 - 846556 * q^38 + 266625 * q^40 - 833000 * q^41 + 1051896 * q^43 - 1918236 * q^44 + 1786160 * q^46 - 529824 * q^47 + 470596 * q^49 - 171875 * q^50 + 211718 * q^52 - 791764 * q^53 - 338500 * q^55 - 731619 * q^56 + 3500242 * q^58 + 3825904 * q^59 + 212432 * q^61 - 882400 * q^62 - 1515063 * q^64 + 276500 * q^65 + 5165272 * q^67 - 7423202 * q^68 + 471625 * q^70 + 623260 * q^71 - 2083228 * q^73 - 1059642 * q^74 + 6990964 * q^76 + 928844 * q^77 - 1325016 * q^79 + 1199875 * q^80 + 11798982 * q^82 - 11447616 * q^83 - 2127000 * q^85 - 7525948 * q^86 + 19484572 * q^88 + 5805808 * q^89 - 758716 * q^91 + 2716288 * q^92 - 15552560 * q^94 - 4083500 * q^95 + 5857516 * q^97 - 1294139 * q^98

Introduction

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