LMFDB - Embedded newform 160.6.d.a.81.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [160,6,Mod(81,160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(160, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("160.81");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$160 = 2^{5} \cdot 5$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	160.d (of order $2$ , degree $1$ , not 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:	$25.6614111701$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{20} - 2 x^{19} - 17 x^{18} + 78 x^{17} + 253 x^{16} - 884 x^{15} + 2396 x^{14} + 19376 x^{13} + \cdots + 1099511627776$ x^20 - 2x^19 - 17x^18 + 78x^17 + 253x^16 - 884x^15 + 2396x^14 + 19376x^13 - 109104x^12 - 96128x^11 + 3580672x^10 - 1538048x^9 - 27930624x^8 + 79364096x^7 + 157024256x^6 - 926941184x^5 + 4244635648x^4 + 20937965568x^3 - 73014444032x^2 - 137438953472*x + 1099511627776
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$2^{90}\cdot 3^{4}\cdot 5^{12}$
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			81.18
Root			$0.236693 + 3.99299i$ of defining polynomial
Character	$\chi$	$=$	160.81
Dual form			160.6.d.a.81.3

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+25.0521i q^{3} +25.0000i q^{5} -103.624 q^{7} -384.607 q^{9} -740.776i q^{11} -892.067i q^{13} -626.302 q^{15} +1138.81 q^{17} +1155.46i q^{19} -2596.00i q^{21} -1602.57 q^{23} -625.000 q^{25} -3547.55i q^{27} +2158.47i q^{29} -4955.24 q^{31} +18558.0 q^{33} -2590.60i q^{35} +4403.89i q^{37} +22348.1 q^{39} +3780.62 q^{41} -13068.3i q^{43} -9615.17i q^{45} +8000.58 q^{47} -6069.06 q^{49} +28529.6i q^{51} -34313.6i q^{53} +18519.4 q^{55} -28946.6 q^{57} -22065.0i q^{59} +2822.53i q^{61} +39854.5 q^{63} +22301.7 q^{65} -54981.5i q^{67} -40147.8i q^{69} -42879.2 q^{71} -20893.2 q^{73} -15657.6i q^{75} +76762.2i q^{77} -30227.6 q^{79} -4586.03 q^{81} -93949.9i q^{83} +28470.3i q^{85} -54074.1 q^{87} +1178.06 q^{89} +92439.5i q^{91} -124139. i q^{93} -28886.4 q^{95} +74100.8 q^{97} +284908. i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$20 q + 196 q^{7} - 1620 q^{9} - 900 q^{15} + 4676 q^{23} - 12500 q^{25} - 7160 q^{31} + 5672 q^{33} + 44904 q^{39} + 11608 q^{41} - 44180 q^{47} + 18756 q^{49} + 24200 q^{55} + 5032 q^{57} - 240620 q^{63}+ \cdots + 147376 q^{97}+O(q^{100})$ 20 * q + 196 * q^7 - 1620 * q^9 - 900 * q^15 + 4676 * q^23 - 12500 * q^25 - 7160 * q^31 + 5672 * q^33 + 44904 * q^39 + 11608 * q^41 - 44180 * q^47 + 18756 * q^49 + 24200 * q^55 + 5032 * q^57 - 240620 * q^63 + 200312 * q^71 - 105136 * q^73 - 282080 * q^79 + 65172 * q^81 + 332592 * q^87 - 3160 * q^89 - 144400 * q^95 + 147376 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/160\mathbb{Z}\right)^\times$ .

$n$	$31$	$97$	$101$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	25.0521i	1.60709i	0.595243	+	0.803546i	$0.297056\pi$
$3$	25.0521i	1.60709i	−0.595243	+	0.803546i	$0.702944\pi$
$4$	0	0
$5$	25.0000i	0.447214i
$5$	25.0000i	0.447214i
$6$	0	0
$7$	−103.624	−0.799310	−0.399655	−	0.916666i	$-0.630870\pi$
$7$	−103.624	−0.799310	−0.399655	+	0.916666i	$0.630870\pi$
$8$	0	0
$9$	−384.607	−1.58274
$10$	0	0
$11$	− 740.776i	− 1.84589i	−0.384935	−	0.922944i	$-0.625776\pi$
$11$	− 740.776i	− 1.84589i	0.384935	−	0.922944i	$-0.374224\pi$
$12$	0	0
$13$	− 892.067i	− 1.46399i	−0.681309	−	0.731996i	$-0.738589\pi$
$13$	− 892.067i	− 1.46399i	0.681309	−	0.731996i	$-0.261411\pi$
$14$	0	0
$15$	−626.302	−0.718713
$16$	0	0
$17$	1138.81	0.955717	0.477859	−	0.878437i	$-0.341413\pi$
$17$	1138.81	0.955717	0.477859	+	0.878437i	$0.341413\pi$
$18$	0	0
$19$	1155.46i	0.734294i	0.930163	+	0.367147i	$0.119665\pi$
$19$	1155.46i	0.734294i	−0.930163	+	0.367147i	$0.880335\pi$
$20$	0	0
$21$	− 2596.00i	− 1.28457i
$22$	0	0
$23$	−1602.57	−0.631682	−0.315841	−	0.948812i	$-0.602287\pi$
$23$	−1602.57	−0.631682	−0.315841	+	0.948812i	$0.602287\pi$
$24$	0	0
$25$	−625.000	−0.200000
$26$	0	0
$27$	− 3547.55i	− 0.936524i
$28$	0	0
$29$	2158.47i	0.476596i	0.971192	+	0.238298i	$0.0765896\pi$
$29$	2158.47i	0.476596i	−0.971192	+	0.238298i	$0.923410\pi$
$30$	0	0
$31$	−4955.24	−0.926106	−0.463053	−	0.886331i	$-0.653246\pi$
$31$	−4955.24	−0.926106	−0.463053	+	0.886331i	$0.653246\pi$
$32$	0	0
$33$	18558.0	2.96651
$34$	0	0
$35$	− 2590.60i	− 0.357462i
$36$	0	0
$37$	4403.89i	0.528850i	0.964406	+	0.264425i	$0.0851821\pi$
$37$	4403.89i	0.528850i	−0.964406	+	0.264425i	$0.914818\pi$
$38$	0	0
$39$	22348.1	2.35277
$40$	0	0
$41$	3780.62	0.351240	0.175620	−	0.984458i	$-0.443807\pi$
$41$	3780.62	0.351240	0.175620	+	0.984458i	$0.443807\pi$
$42$	0	0
$43$	− 13068.3i	− 1.07783i	−0.842361	−	0.538913i	$-0.818835\pi$
$43$	− 13068.3i	− 1.07783i	0.842361	−	0.538913i	$-0.181165\pi$
$44$	0	0
$45$	− 9615.17i	− 0.707825i
$46$	0	0
$47$	8000.58	0.528296	0.264148	−	0.964482i	$-0.414909\pi$
$47$	8000.58	0.528296	0.264148	+	0.964482i	$0.414909\pi$
$48$	0	0
$49$	−6069.06	−0.361103
$50$	0	0
$51$	28529.6i	1.53593i
$52$	0	0
$53$	− 34313.6i	− 1.67794i	−0.544175	−	0.838971i	$-0.683157\pi$
$53$	− 34313.6i	− 1.67794i	0.544175	−	0.838971i	$-0.316843\pi$
$54$	0	0
$55$	18519.4	0.825506
$56$	0	0
$57$	−28946.6	−1.18008
$58$	0	0
$59$	− 22065.0i	− 0.825227i	−0.910906	−	0.412613i	$-0.864616\pi$
$59$	− 22065.0i	− 0.825227i	0.910906	−	0.412613i	$-0.135384\pi$
$60$	0	0
$61$	2822.53i	0.0971212i	0.998820	+	0.0485606i	$0.0154634\pi$
$61$	2822.53i	0.0971212i	−0.998820	+	0.0485606i	$0.984537\pi$
$62$	0	0
$63$	39854.5	1.26510
$64$	0	0
$65$	22301.7	0.654717
$66$	0	0
$67$	− 54981.5i	− 1.49634i	−0.663509	−	0.748168i	$-0.730934\pi$
$67$	− 54981.5i	− 1.49634i	0.663509	−	0.748168i	$-0.269066\pi$
$68$	0	0
$69$	− 40147.8i	− 1.01517i
$70$	0	0
$71$	−42879.2	−1.00949	−0.504744	−	0.863269i	$-0.668413\pi$
$71$	−42879.2	−1.00949	−0.504744	+	0.863269i	$0.668413\pi$
$72$	0	0
$73$	−20893.2	−0.458879	−0.229440	−	0.973323i	$-0.573689\pi$
$73$	−20893.2	−0.458879	−0.229440	+	0.973323i	$0.573689\pi$
$74$	0	0
$75$	− 15657.6i	− 0.321418i
$76$	0	0
$77$	76762.2i	1.47544i
$78$	0	0
$79$	−30227.6	−0.544925	−0.272462	−	0.962166i	$-0.587838\pi$
$79$	−30227.6	−0.544925	−0.272462	+	0.962166i	$0.587838\pi$
$80$	0	0
$81$	−4586.03	−0.0776648
$82$	0	0
$83$	− 93949.9i	− 1.49693i	−0.663175	−	0.748465i	$-0.730791\pi$
$83$	− 93949.9i	− 1.49693i	0.663175	−	0.748465i	$-0.269209\pi$
$84$	0	0
$85$	28470.3i	0.427410i
$86$	0	0
$87$	−54074.1	−0.765934
$88$	0	0
$89$	1178.06	0.0157650	0.00788248	−	0.999969i	$-0.497491\pi$
$89$	1178.06	0.0157650	0.00788248	+	0.999969i	$0.497491\pi$
$90$	0	0
$91$	92439.5i	1.17018i
$92$	0	0
$93$	− 124139.i	− 1.48834i
$94$	0	0
$95$	−28886.4	−0.328386
$96$	0	0
$97$	74100.8	0.799638	0.399819	−	0.916594i	$-0.369073\pi$
$97$	74100.8	0.799638	0.399819	+	0.916594i	$0.369073\pi$
$98$	0	0
$99$	284908.i	2.92157i
$100$	0	0
$101$	31104.3i	0.303401i	0.988427	+	0.151700i	$0.0484749\pi$
$101$	31104.3i	0.303401i	−0.988427	+	0.151700i	$0.951525\pi$
$102$	0	0
$103$	140298.	1.30305	0.651523	−	0.758629i	$-0.274130\pi$
$103$	140298.	1.30305	0.651523	+	0.758629i	$0.274130\pi$
$104$	0	0
$105$	64900.0	0.574475
$106$	0	0
$107$	34792.6i	0.293784i	0.989153	+	0.146892i	$0.0469269\pi$
$107$	34792.6i	0.293784i	−0.989153	+	0.146892i	$0.953073\pi$
$108$	0	0
$109$	83759.2i	0.675252i	0.941280	+	0.337626i	$0.109624\pi$
$109$	83759.2i	0.675252i	−0.941280	+	0.337626i	$0.890376\pi$
$110$	0	0
$111$	−110327.	−0.849910
$112$	0	0
$113$	57262.1	0.421863	0.210931	−	0.977501i	$-0.432350\pi$
$113$	57262.1	0.421863	0.210931	+	0.977501i	$0.432350\pi$
$114$	0	0
$115$	− 40064.3i	− 0.282497i
$116$	0	0
$117$	343095.i	2.31713i
$118$	0	0
$119$	−118008.	−0.763914
$120$	0	0
$121$	−387698.	−2.40730
$122$	0	0
$123$	94712.5i	0.564475i
$124$	0	0
$125$	− 15625.0i	− 0.0894427i
$126$	0	0
$127$	8848.12	0.0486790	0.0243395	−	0.999704i	$-0.492252\pi$
$127$	8848.12	0.0486790	0.0243395	+	0.999704i	$0.492252\pi$
$128$	0	0
$129$	327389.	1.73217
$130$	0	0
$131$	− 110458.i	− 0.562365i	−0.959654	−	0.281183i	$-0.909273\pi$
$131$	− 110458.i	− 0.562365i	0.959654	−	0.281183i	$-0.0907267\pi$
$132$	0	0
$133$	− 119733.i	− 0.586929i
$134$	0	0
$135$	88688.7	0.418826
$136$	0	0
$137$	−304365.	−1.38546	−0.692728	−	0.721199i	$-0.743591\pi$
$137$	−304365.	−1.38546	−0.692728	+	0.721199i	$0.743591\pi$
$138$	0	0
$139$	59379.5i	0.260675i	0.991470	+	0.130338i	$0.0416061\pi$
$139$	59379.5i	0.260675i	−0.991470	+	0.130338i	$0.958394\pi$
$140$	0	0
$141$	200431.i	0.849020i
$142$	0	0
$143$	−660822.	−2.70237
$144$	0	0
$145$	−53961.7	−0.213140
$146$	0	0
$147$	− 152043.i	− 0.580326i
$148$	0	0
$149$	− 233744.i	− 0.862529i	−0.902225	−	0.431265i	$-0.858068\pi$
$149$	− 233744.i	− 0.862529i	0.902225	−	0.431265i	$-0.141932\pi$
$150$	0	0
$151$	89068.4	0.317893	0.158947	−	0.987287i	$-0.449190\pi$
$151$	89068.4	0.317893	0.158947	+	0.987287i	$0.449190\pi$
$152$	0	0
$153$	−437994.	−1.51266
$154$	0	0
$155$	− 123881.i	− 0.414167i
$156$	0	0
$157$	− 608909.i	− 1.97153i	−0.168132	−	0.985764i	$-0.553773\pi$
$157$	− 608909.i	− 1.97153i	0.168132	−	0.985764i	$-0.446227\pi$
$158$	0	0
$159$	859628.	2.69661
$160$	0	0
$161$	166065.	0.504910
$162$	0	0
$163$	172801.i	0.509421i	0.967017	+	0.254711i	$0.0819802\pi$
$163$	172801.i	0.509421i	−0.967017	+	0.254711i	$0.918020\pi$
$164$	0	0
$165$	463950.i	1.32666i
$166$	0	0
$167$	−441506.	−1.22503	−0.612513	−	0.790461i	$-0.709841\pi$
$167$	−441506.	−1.22503	−0.612513	+	0.790461i	$0.709841\pi$
$168$	0	0
$169$	−424490.	−1.14327
$170$	0	0
$171$	− 444397.i	− 1.16220i
$172$	0	0
$173$	281414.i	0.714874i	0.933937	+	0.357437i	$0.116349\pi$
$173$	281414.i	0.714874i	−0.933937	+	0.357437i	$0.883651\pi$
$174$	0	0
$175$	64765.0	0.159862
$176$	0	0
$177$	552774.	1.32622
$178$	0	0
$179$	805768.i	1.87965i	0.341655	+	0.939825i	$0.389013\pi$
$179$	805768.i	1.87965i	−0.341655	+	0.939825i	$0.610987\pi$
$180$	0	0
$181$	− 610344.i	− 1.38477i	−0.721528	−	0.692385i	$-0.756560\pi$
$181$	− 610344.i	− 1.38477i	0.721528	−	0.692385i	$-0.243440\pi$
$182$	0	0
$183$	−70710.3	−0.156083
$184$	0	0
$185$	−110097.	−0.236509
$186$	0	0
$187$	− 843604.i	− 1.76415i
$188$	0	0
$189$	367611.i	0.748573i
$190$	0	0
$191$	845737.	1.67746	0.838729	−	0.544549i	$-0.183299\pi$
$191$	845737.	1.67746	0.838729	+	0.544549i	$0.183299\pi$
$192$	0	0
$193$	−184089.	−0.355742	−0.177871	−	0.984054i	$-0.556921\pi$
$193$	−184089.	−0.355742	−0.177871	+	0.984054i	$0.556921\pi$
$194$	0	0
$195$	558703.i	1.05219i
$196$	0	0
$197$	390522.i	0.716935i	0.933542	+	0.358468i	$0.116701\pi$
$197$	390522.i	0.716935i	−0.933542	+	0.358468i	$0.883299\pi$
$198$	0	0
$199$	584263.	1.04587	0.522933	−	0.852374i	$-0.324838\pi$
$199$	584263.	1.04587	0.522933	+	0.852374i	$0.324838\pi$
$200$	0	0
$201$	1.37740e6	2.40475
$202$	0	0
$203$	− 223669.i	− 0.380948i
$204$	0	0
$205$	94515.6i	0.157079i
$206$	0	0
$207$	616361.	0.999791
$208$	0	0
$209$	855935.	1.35542
$210$	0	0
$211$	− 428258.i	− 0.662215i	−0.943593	−	0.331107i	$-0.892578\pi$
$211$	− 428258.i	− 0.662215i	0.943593	−	0.331107i	$-0.107422\pi$
$212$	0	0
$213$	− 1.07421e6i	− 1.62234i
$214$	0	0
$215$	326708.	0.482018
$216$	0	0
$217$	513482.	0.740246
$218$	0	0
$219$	− 523419.i	− 0.737461i
$220$	0	0
$221$	− 1.01590e6i	− 1.39916i
$222$	0	0
$223$	−955618.	−1.28683	−0.643416	−	0.765517i	$-0.722484\pi$
$223$	−955618.	−1.28683	−0.643416	+	0.765517i	$0.722484\pi$
$224$	0	0
$225$	240379.	0.316549
$226$	0	0
$227$	866233.i	1.11576i	0.829922	+	0.557879i	$0.188385\pi$
$227$	866233.i	1.11576i	−0.829922	+	0.557879i	$0.811615\pi$
$228$	0	0
$229$	654035.i	0.824161i	0.911147	+	0.412080i	$0.135198\pi$
$229$	654035.i	0.824161i	−0.911147	+	0.412080i	$0.864802\pi$
$230$	0	0
$231$	−1.92305e6	−2.37116
$232$	0	0
$233$	−820513.	−0.990138	−0.495069	−	0.868854i	$-0.664857\pi$
$233$	−820513.	−0.990138	−0.495069	+	0.868854i	$0.664857\pi$
$234$	0	0
$235$	200015.i	0.236261i
$236$	0	0
$237$	− 757265.i	− 0.875744i
$238$	0	0
$239$	−867314.	−0.982159	−0.491079	−	0.871115i	$-0.663397\pi$
$239$	−867314.	−0.982159	−0.491079	+	0.871115i	$0.663397\pi$
$240$	0	0
$241$	−1.65377e6	−1.83414	−0.917072	−	0.398722i	$-0.869454\pi$
$241$	−1.65377e6	−1.83414	−0.917072	+	0.398722i	$0.869454\pi$
$242$	0	0
$243$	− 976943.i	− 1.06134i
$244$	0	0
$245$	− 151726.i	− 0.161490i
$246$	0	0
$247$	1.03075e6	1.07500
$248$	0	0
$249$	2.35364e6	2.40570
$250$	0	0
$251$	− 860879.i	− 0.862497i	−0.902233	−	0.431249i	$-0.858073\pi$
$251$	− 860879.i	− 0.862497i	0.902233	−	0.431249i	$-0.141927\pi$
$252$	0	0
$253$	1.18715e6i	1.16601i
$254$	0	0
$255$	−713240.	−0.686887
$256$	0	0
$257$	−765335.	−0.722801	−0.361401	−	0.932411i	$-0.617701\pi$
$257$	−765335.	−0.722801	−0.361401	+	0.932411i	$0.617701\pi$
$258$	0	0
$259$	− 456349.i	− 0.422715i
$260$	0	0
$261$	− 830162.i	− 0.754330i
$262$	0	0
$263$	−260746.	−0.232449	−0.116225	−	0.993223i	$-0.537079\pi$
$263$	−260746.	−0.232449	−0.116225	+	0.993223i	$0.537079\pi$
$264$	0	0
$265$	857841.	0.750399
$266$	0	0
$267$	29512.9i	0.0253357i
$268$	0	0
$269$	1.12412e6i	0.947175i	0.880747	+	0.473587i	$0.157041\pi$
$269$	1.12412e6i	0.947175i	−0.880747	+	0.473587i	$0.842959\pi$
$270$	0	0
$271$	1.07277e6	0.887330	0.443665	−	0.896193i	$-0.353678\pi$
$271$	1.07277e6	0.887330	0.443665	+	0.896193i	$0.353678\pi$
$272$	0	0
$273$	−2.31580e6	−1.88059
$274$	0	0
$275$	462985.i	0.369177i
$276$	0	0
$277$	− 615080.i	− 0.481651i	−0.970568	−	0.240825i	$-0.922582\pi$
$277$	− 615080.i	− 0.481651i	0.970568	−	0.240825i	$-0.0774181\pi$
$278$	0	0
$279$	1.90582e6	1.46579
$280$	0	0
$281$	−652198.	−0.492736	−0.246368	−	0.969176i	$-0.579237\pi$
$281$	−652198.	−0.492736	−0.246368	+	0.969176i	$0.579237\pi$
$282$	0	0
$283$	− 908596.i	− 0.674380i	−0.941437	−	0.337190i	$-0.890524\pi$
$283$	− 908596.i	− 0.674380i	0.941437	−	0.337190i	$-0.109476\pi$
$284$	0	0
$285$	− 723665.i	− 0.527747i
$286$	0	0
$287$	−391763.	−0.280750
$288$	0	0
$289$	−122967.	−0.0866050
$290$	0	0
$291$	1.85638e6i	1.28509i
$292$	0	0
$293$	− 514047.i	− 0.349811i	−0.984585	−	0.174906i	$-0.944038\pi$
$293$	− 514047.i	− 0.349811i	0.984585	−	0.174906i	$-0.0559621\pi$
$294$	0	0
$295$	551624.	0.369053
$296$	0	0
$297$	−2.62794e6	−1.72872
$298$	0	0
$299$	1.42960e6i	0.924778i
$300$	0	0
$301$	1.35419e6i	0.861517i
$302$	0	0
$303$	−779227.	−0.487593
$304$	0	0
$305$	−70563.3	−0.0434339
$306$	0	0
$307$	1.46693e6i	0.888309i	0.895950	+	0.444155i	$0.146496\pi$
$307$	1.46693e6i	0.888309i	−0.895950	+	0.444155i	$0.853504\pi$
$308$	0	0
$309$	3.51477e6i	2.09411i
$310$	0	0
$311$	−547919.	−0.321229	−0.160615	−	0.987017i	$-0.551348\pi$
$311$	−547919.	−0.321229	−0.160615	+	0.987017i	$0.551348\pi$
$312$	0	0
$313$	1.70857e6	0.985762	0.492881	−	0.870097i	$-0.335944\pi$
$313$	1.70857e6	0.985762	0.492881	+	0.870097i	$0.335944\pi$
$314$	0	0
$315$	996363.i	0.565772i
$316$	0	0
$317$	978608.i	0.546966i	0.961877	+	0.273483i	$0.0881758\pi$
$317$	978608.i	0.546966i	−0.961877	+	0.273483i	$0.911824\pi$
$318$	0	0
$319$	1.59894e6	0.879743
$320$	0	0
$321$	−871627.	−0.472137
$322$	0	0
$323$	1.31585e6i	0.701777i
$324$	0	0
$325$	557542.i	0.292799i
$326$	0	0
$327$	−2.09834e6	−1.08519
$328$	0	0
$329$	−829053.	−0.422272
$330$	0	0
$331$	345907.i	0.173536i	0.996229	+	0.0867679i	$0.0276539\pi$
$331$	345907.i	0.173536i	−0.996229	+	0.0867679i	$0.972346\pi$
$332$	0	0
$333$	− 1.69377e6i	− 0.837034i
$334$	0	0
$335$	1.37454e6	0.669182
$336$	0	0
$337$	376513.	0.180595	0.0902975	−	0.995915i	$-0.471218\pi$
$337$	376513.	0.180595	0.0902975	+	0.995915i	$0.471218\pi$
$338$	0	0
$339$	1.43454e6i	0.677972i
$340$	0	0
$341$	3.67073e6i	1.70949i
$342$	0	0
$343$	2.37051e6	1.08794
$344$	0	0
$345$	1.00370e6	0.453998
$346$	0	0
$347$	164656.i	0.0734099i	0.999326	+	0.0367049i	$0.0116862\pi$
$347$	164656.i	0.0734099i	−0.999326	+	0.0367049i	$0.988314\pi$
$348$	0	0
$349$	35942.6i	0.0157959i	0.999969	+	0.00789797i	$0.00251403\pi$
$349$	35942.6i	0.0157959i	−0.999969	+	0.00789797i	$0.997486\pi$
$350$	0	0
$351$	−3.16465e6	−1.37106
$352$	0	0
$353$	1.39430e6	0.595550	0.297775	−	0.954636i	$-0.403756\pi$
$353$	1.39430e6	0.595550	0.297775	+	0.954636i	$0.403756\pi$
$354$	0	0
$355$	− 1.07198e6i	− 0.451457i
$356$	0	0
$357$	− 2.95635e6i	− 1.22768i
$358$	0	0
$359$	−3.50589e6	−1.43569	−0.717847	−	0.696201i	$-0.754872\pi$
$359$	−3.50589e6	−1.43569	−0.717847	+	0.696201i	$0.754872\pi$
$360$	0	0
$361$	1.14102e6	0.460812
$362$	0	0
$363$	− 9.71265e6i	− 3.86875i
$364$	0	0
$365$	− 522331.i	− 0.205217i
$366$	0	0
$367$	−3.64440e6	−1.41241	−0.706206	−	0.708007i	$-0.749595\pi$
$367$	−3.64440e6	−1.41241	−0.706206	+	0.708007i	$0.749595\pi$
$368$	0	0
$369$	−1.45405e6	−0.555923
$370$	0	0
$371$	3.55572e6i	1.34120i
$372$	0	0
$373$	− 4.19252e6i	− 1.56028i	−0.625603	−	0.780142i	$-0.715147\pi$
$373$	− 4.19252e6i	− 1.56028i	0.625603	−	0.780142i	$-0.284853\pi$
$374$	0	0
$375$	391439.	0.143743
$376$	0	0
$377$	1.92550e6	0.697734
$378$	0	0
$379$	− 2.31089e6i	− 0.826384i	−0.910644	−	0.413192i	$-0.864414\pi$
$379$	− 2.31089e6i	− 0.826384i	0.910644	−	0.413192i	$-0.135586\pi$
$380$	0	0
$381$	221664.i	0.0782316i
$382$	0	0
$383$	−1.18240e6	−0.411876	−0.205938	−	0.978565i	$-0.566024\pi$
$383$	−1.18240e6	−0.411876	−0.205938	+	0.978565i	$0.566024\pi$
$384$	0	0
$385$	−1.91906e6	−0.659835
$386$	0	0
$387$	5.02617e6i	1.70592i
$388$	0	0
$389$	5.09219e6i	1.70620i	0.521745	+	0.853102i	$0.325281\pi$
$389$	5.09219e6i	1.70620i	−0.521745	+	0.853102i	$0.674719\pi$
$390$	0	0
$391$	−1.82503e6	−0.603709
$392$	0	0
$393$	2.76720e6	0.903773
$394$	0	0
$395$	− 755691.i	− 0.243698i
$396$	0	0
$397$	63006.9i	0.0200637i	0.999950	+	0.0100319i	$0.00319330\pi$
$397$	63006.9i	0.0200637i	−0.999950	+	0.0100319i	$0.996807\pi$
$398$	0	0
$399$	2.99957e6	0.943248
$400$	0	0
$401$	−4.48960e6	−1.39427	−0.697135	−	0.716940i	$-0.745542\pi$
$401$	−4.48960e6	−1.39427	−0.697135	+	0.716940i	$0.745542\pi$
$402$	0	0
$403$	4.42041e6i	1.35581i
$404$	0	0
$405$	− 114651.i	− 0.0347327i
$406$	0	0
$407$	3.26230e6	0.976197
$408$	0	0
$409$	2.03416e6	0.601281	0.300641	−	0.953738i	$-0.402800\pi$
$409$	2.03416e6	0.601281	0.300641	+	0.953738i	$0.402800\pi$
$410$	0	0
$411$	− 7.62497e6i	− 2.22656i
$412$	0	0
$413$	2.28646e6i	0.659612i
$414$	0	0
$415$	2.34875e6	0.669447
$416$	0	0
$417$	−1.48758e6	−0.418929
$418$	0	0
$419$	1.53143e6i	0.426150i	0.977036	+	0.213075i	$0.0683479\pi$
$419$	1.53143e6i	0.426150i	−0.977036	+	0.213075i	$0.931652\pi$
$420$	0	0
$421$	− 4.86857e6i	− 1.33874i	−0.742929	−	0.669370i	$-0.766564\pi$
$421$	− 4.86857e6i	− 1.33874i	0.742929	−	0.669370i	$-0.233436\pi$
$422$	0	0
$423$	−3.07708e6	−0.836157
$424$	0	0
$425$	−711757.	−0.191143
$426$	0	0
$427$	− 292482.i	− 0.0776300i
$428$	0	0
$429$	− 1.65550e7i	− 4.34295i
$430$	0	0
$431$	−537754.	−0.139441	−0.0697204	−	0.997567i	$-0.522211\pi$
$431$	−537754.	−0.139441	−0.0697204	+	0.997567i	$0.522211\pi$
$432$	0	0
$433$	1.48235e6	0.379954	0.189977	−	0.981789i	$-0.439159\pi$
$433$	1.48235e6	0.379954	0.189977	+	0.981789i	$0.439159\pi$
$434$	0	0
$435$	− 1.35185e6i	− 0.342536i
$436$	0	0
$437$	− 1.85171e6i	− 0.463840i
$438$	0	0
$439$	2.49966e6	0.619041	0.309520	−	0.950893i	$-0.399832\pi$
$439$	2.49966e6	0.619041	0.309520	+	0.950893i	$0.399832\pi$
$440$	0	0
$441$	2.33420e6	0.571534
$442$	0	0
$443$	5.00804e6i	1.21244i	0.795299	+	0.606218i	$0.207314\pi$
$443$	5.00804e6i	1.21244i	−0.795299	+	0.606218i	$0.792686\pi$
$444$	0	0
$445$	29451.5i	0.00705030i
$446$	0	0
$447$	5.85576e6	1.38616
$448$	0	0
$449$	3.04025e6	0.711694	0.355847	−	0.934544i	$-0.384192\pi$
$449$	3.04025e6	0.711694	0.355847	+	0.934544i	$0.384192\pi$
$450$	0	0
$451$	− 2.80059e6i	− 0.648349i
$452$	0	0
$453$	2.23135e6i	0.510884i
$454$	0	0
$455$	−2.31099e6	−0.523322
$456$	0	0
$457$	7.20014e6	1.61269	0.806345	−	0.591446i	$-0.201443\pi$
$457$	7.20014e6	1.61269	0.806345	+	0.591446i	$0.201443\pi$
$458$	0	0
$459$	− 4.03999e6i	− 0.895052i
$460$	0	0
$461$	7.87732e6i	1.72634i	0.504914	+	0.863170i	$0.331524\pi$
$461$	7.87732e6i	1.72634i	−0.504914	+	0.863170i	$0.668476\pi$
$462$	0	0
$463$	6.18812e6	1.34155	0.670774	−	0.741662i	$-0.265962\pi$
$463$	6.18812e6	1.34155	0.670774	+	0.741662i	$0.265962\pi$
$464$	0	0
$465$	3.10348e6	0.665605
$466$	0	0
$467$	− 7.21143e6i	− 1.53013i	−0.643952	−	0.765066i	$-0.722706\pi$
$467$	− 7.21143e6i	− 1.53013i	0.643952	−	0.765066i	$-0.277294\pi$
$468$	0	0
$469$	5.69740e6i	1.19604i
$470$	0	0
$471$	1.52544e7	3.16843
$472$	0	0
$473$	−9.68070e6	−1.98955
$474$	0	0
$475$	− 722161.i	− 0.146859i
$476$	0	0
$477$	1.31973e7i	2.65575i
$478$	0	0
$479$	−4.05783e6	−0.808081	−0.404040	−	0.914741i	$-0.632395\pi$
$479$	−4.05783e6	−0.808081	−0.404040	+	0.914741i	$0.632395\pi$
$480$	0	0
$481$	3.92856e6	0.774232
$482$	0	0
$483$	4.16028e6i	0.811437i
$484$	0	0
$485$	1.85252e6i	0.357609i
$486$	0	0
$487$	−121792.	−0.0232700	−0.0116350	−	0.999932i	$-0.503704\pi$
$487$	−121792.	−0.0232700	−0.0116350	+	0.999932i	$0.503704\pi$
$488$	0	0
$489$	−4.32902e6	−0.818687
$490$	0	0
$491$	2.80649e6i	0.525363i	0.964883	+	0.262682i	$0.0846069\pi$
$491$	2.80649e6i	0.525363i	−0.964883	+	0.262682i	$0.915393\pi$
$492$	0	0
$493$	2.45809e6i	0.455491i
$494$	0	0
$495$	−7.12269e6	−1.30656
$496$	0	0
$497$	4.44332e6	0.806894
$498$	0	0
$499$	2.59476e6i	0.466494i	0.972418	+	0.233247i	$0.0749351\pi$
$499$	2.59476e6i	0.466494i	−0.972418	+	0.233247i	$0.925065\pi$
$500$	0	0
$501$	− 1.10606e7i	− 1.96873i
$502$	0	0
$503$	−1.04638e6	−0.184404	−0.0922021	−	0.995740i	$-0.529391\pi$
$503$	−1.04638e6	−0.184404	−0.0922021	+	0.995740i	$0.529391\pi$
$504$	0	0
$505$	−777607.	−0.135685
$506$	0	0
$507$	− 1.06344e7i	− 1.83735i
$508$	0	0
$509$	− 8.38369e6i	− 1.43430i	−0.696918	−	0.717151i	$-0.745446\pi$
$509$	− 8.38369e6i	− 1.43430i	0.696918	−	0.717151i	$-0.254554\pi$
$510$	0	0
$511$	2.16504e6	0.366787
$512$	0	0
$513$	4.09904e6	0.687684
$514$	0	0
$515$	3.50746e6i	0.582740i
$516$	0	0
$517$	− 5.92664e6i	− 0.975174i
$518$	0	0
$519$	−7.05000e6	−1.14887
$520$	0	0
$521$	−4.38397e6	−0.707577	−0.353788	−	0.935325i	$-0.615107\pi$
$521$	−4.38397e6	−0.707577	−0.353788	+	0.935325i	$0.615107\pi$
$522$	0	0
$523$	915789.i	0.146400i	0.997317	+	0.0732000i	$0.0233211\pi$
$523$	915789.i	0.146400i	−0.997317	+	0.0732000i	$0.976679\pi$
$524$	0	0
$525$	1.62250e6i	0.256913i
$526$	0	0
$527$	−5.64309e6	−0.885096
$528$	0	0
$529$	−3.86810e6	−0.600978
$530$	0	0
$531$	8.48634e6i	1.30612i
$532$	0	0
$533$	− 3.37257e6i	− 0.514212i
$534$	0	0
$535$	−869815.	−0.131384
$536$	0	0
$537$	−2.01862e7	−3.02077
$538$	0	0
$539$	4.49581e6i	0.666556i
$540$	0	0
$541$	− 6.98594e6i	− 1.02620i	−0.858329	−	0.513100i	$-0.828497\pi$
$541$	− 6.98594e6i	− 1.02620i	0.858329	−	0.513100i	$-0.171503\pi$
$542$	0	0
$543$	1.52904e7	2.22545
$544$	0	0
$545$	−2.09398e6	−0.301982
$546$	0	0
$547$	− 4.72947e6i	− 0.675841i	−0.941175	−	0.337920i	$-0.890277\pi$
$547$	− 4.72947e6i	− 0.675841i	0.941175	−	0.337920i	$-0.109723\pi$
$548$	0	0
$549$	− 1.08556e6i	− 0.153718i
$550$	0	0
$551$	−2.49402e6	−0.349962
$552$	0	0
$553$	3.13231e6	0.435564
$554$	0	0
$555$	− 2.75817e6i	− 0.380091i
$556$	0	0
$557$	− 3.26312e6i	− 0.445651i	−0.974858	−	0.222825i	$-0.928472\pi$
$557$	− 3.26312e6i	− 0.445651i	0.974858	−	0.222825i	$-0.0715280\pi$
$558$	0	0
$559$	−1.16578e7	−1.57793
$560$	0	0
$561$	2.11340e7	2.83514
$562$	0	0
$563$	5.81898e6i	0.773706i	0.922141	+	0.386853i	$0.126438\pi$
$563$	5.81898e6i	0.773706i	−0.922141	+	0.386853i	$0.873562\pi$
$564$	0	0
$565$	1.43155e6i	0.188663i
$566$	0	0
$567$	475223.	0.0620782
$568$	0	0
$569$	−9.08770e6	−1.17672	−0.588360	−	0.808599i	$-0.700226\pi$
$569$	−9.08770e6	−1.17672	−0.588360	+	0.808599i	$0.700226\pi$
$570$	0	0
$571$	258870.i	0.0332270i	0.999862	+	0.0166135i	$0.00528849\pi$
$571$	258870.i	0.0332270i	−0.999862	+	0.0166135i	$0.994712\pi$
$572$	0	0
$573$	2.11875e7i	2.69583i
$574$	0	0
$575$	1.00161e6	0.126336
$576$	0	0
$577$	2.45070e6	0.306444	0.153222	−	0.988192i	$-0.451035\pi$
$577$	2.45070e6	0.306444	0.153222	+	0.988192i	$0.451035\pi$
$578$	0	0
$579$	− 4.61182e6i	− 0.571710i
$580$	0	0
$581$	9.73547e6i	1.19651i
$582$	0	0
$583$	−2.54187e7	−3.09729
$584$	0	0
$585$	−8.57737e6	−1.03625
$586$	0	0
$587$	− 3.61982e6i	− 0.433602i	−0.976216	−	0.216801i	$-0.930438\pi$
$587$	− 3.61982e6i	− 0.433602i	0.976216	−	0.216801i	$-0.0695623\pi$
$588$	0	0
$589$	− 5.72557e6i	− 0.680034i
$590$	0	0
$591$	−9.78339e6	−1.15218
$592$	0	0
$593$	−7.07347e6	−0.826030	−0.413015	−	0.910724i	$-0.635524\pi$
$593$	−7.07347e6	−0.826030	−0.413015	+	0.910724i	$0.635524\pi$
$594$	0	0
$595$	− 2.95020e6i	− 0.341633i
$596$	0	0
$597$	1.46370e7i	1.68080i
$598$	0	0
$599$	2.71813e6	0.309530	0.154765	−	0.987951i	$-0.450538\pi$
$599$	2.71813e6	0.309530	0.154765	+	0.987951i	$0.450538\pi$
$600$	0	0
$601$	1.67171e6	0.188788	0.0943942	−	0.995535i	$-0.469909\pi$
$601$	1.67171e6	0.188788	0.0943942	+	0.995535i	$0.469909\pi$
$602$	0	0
$603$	2.11462e7i	2.36832i
$604$	0	0
$605$	− 9.69245e6i	− 1.07658i
$606$	0	0
$607$	2.35562e6	0.259498	0.129749	−	0.991547i	$-0.458583\pi$
$607$	2.35562e6	0.259498	0.129749	+	0.991547i	$0.458583\pi$
$608$	0	0
$609$	5.60338e6	0.612219
$610$	0	0
$611$	− 7.13705e6i	− 0.773421i
$612$	0	0
$613$	− 1.88749e6i	− 0.202877i	−0.994842	−	0.101439i	$-0.967655\pi$
$613$	− 1.88749e6i	− 0.202877i	0.994842	−	0.101439i	$-0.0323445\pi$
$614$	0	0
$615$	−2.36781e6	−0.252441
$616$	0	0
$617$	1.45446e7	1.53812	0.769058	−	0.639178i	$-0.220725\pi$
$617$	1.45446e7	1.53812	0.769058	+	0.639178i	$0.220725\pi$
$618$	0	0
$619$	− 1.08497e7i	− 1.13813i	−0.822292	−	0.569066i	$-0.807305\pi$
$619$	− 1.08497e7i	− 1.13813i	0.822292	−	0.569066i	$-0.192695\pi$
$620$	0	0
$621$	5.68521e6i	0.591585i
$622$	0	0
$623$	−122075.	−0.0126011
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	2.14430e7i	2.17829i
$628$	0	0
$629$	5.01520e6i	0.505431i
$630$	0	0
$631$	−3.76561e6	−0.376497	−0.188249	−	0.982121i	$-0.560281\pi$
$631$	−3.76561e6	−0.376497	−0.188249	+	0.982121i	$0.560281\pi$
$632$	0	0
$633$	1.07287e7	1.06424
$634$	0	0
$635$	221203.i	0.0217699i
$636$	0	0
$637$	5.41400e6i	0.528652i
$638$	0	0
$639$	1.64917e7	1.59776
$640$	0	0
$641$	−223711.	−0.0215052	−0.0107526	−	0.999942i	$-0.503423\pi$
$641$	−223711.	−0.0215052	−0.0107526	+	0.999942i	$0.503423\pi$
$642$	0	0
$643$	2.53011e6i	0.241330i	0.992693	+	0.120665i	$0.0385028\pi$
$643$	2.53011e6i	0.241330i	−0.992693	+	0.120665i	$0.961497\pi$
$644$	0	0
$645$	8.18472e6i	0.774648i
$646$	0	0
$647$	8.44825e6	0.793425	0.396713	−	0.917943i	$-0.370151\pi$
$647$	8.44825e6	0.793425	0.396713	+	0.917943i	$0.370151\pi$
$648$	0	0
$649$	−1.63452e7	−1.52328
$650$	0	0
$651$	1.28638e7i	1.18964i
$652$	0	0
$653$	1.72303e7i	1.58128i	0.612280	+	0.790641i	$0.290253\pi$
$653$	1.72303e7i	1.58128i	−0.612280	+	0.790641i	$0.709747\pi$
$654$	0	0
$655$	2.76145e6	0.251497
$656$	0	0
$657$	8.03568e6	0.726289
$658$	0	0
$659$	5.43243e6i	0.487282i	0.969865	+	0.243641i	$0.0783419\pi$
$659$	5.43243e6i	0.487282i	−0.969865	+	0.243641i	$0.921658\pi$
$660$	0	0
$661$	− 1.12770e7i	− 1.00390i	−0.864897	−	0.501949i	$-0.832616\pi$
$661$	− 1.12770e7i	− 1.00390i	0.864897	−	0.501949i	$-0.167384\pi$
$662$	0	0
$663$	2.54503e7	2.24858
$664$	0	0
$665$	2.99333e6	0.262483
$666$	0	0
$667$	− 3.45911e6i	− 0.301057i
$668$	0	0
$669$	− 2.39402e7i	− 2.06806i
$670$	0	0
$671$	2.09086e6	0.179275
$672$	0	0
$673$	1.15475e7	0.982769	0.491384	−	0.870943i	$-0.336491\pi$
$673$	1.15475e7	0.982769	0.491384	+	0.870943i	$0.336491\pi$
$674$	0	0
$675$	2.21722e6i	0.187305i
$676$	0	0
$677$	8.72804e6i	0.731889i	0.930637	+	0.365944i	$0.119254\pi$
$677$	8.72804e6i	0.731889i	−0.930637	+	0.365944i	$0.880746\pi$
$678$	0	0
$679$	−7.67862e6	−0.639159
$680$	0	0
$681$	−2.17009e7	−1.79313
$682$	0	0
$683$	− 5.37492e6i	− 0.440880i	−0.975401	−	0.220440i	$-0.929251\pi$
$683$	− 5.37492e6i	− 0.440880i	0.975401	−	0.220440i	$-0.0707493\pi$
$684$	0	0
$685$	− 7.60912e6i	− 0.619595i
$686$	0	0
$687$	−1.63849e7	−1.32450
$688$	0	0
$689$	−3.06101e7	−2.45650
$690$	0	0
$691$	1.03106e7i	0.821467i	0.911756	+	0.410733i	$0.134727\pi$
$691$	1.03106e7i	0.821467i	−0.911756	+	0.410733i	$0.865273\pi$
$692$	0	0
$693$	− 2.95233e7i	− 2.33524i
$694$	0	0
$695$	−1.48449e6	−0.116577
$696$	0	0
$697$	4.30541e6	0.335686
$698$	0	0
$699$	− 2.05556e7i	− 1.59124i
$700$	0	0
$701$	7.87993e6i	0.605658i	0.953045	+	0.302829i	$0.0979311\pi$
$701$	7.87993e6i	0.605658i	−0.953045	+	0.302829i	$0.902069\pi$
$702$	0	0
$703$	−5.08851e6	−0.388331
$704$	0	0
$705$	−5.01078e6	−0.379693
$706$	0	0
$707$	− 3.22315e6i	− 0.242511i
$708$	0	0
$709$	2.06172e7i	1.54033i	0.637846	+	0.770164i	$0.279826\pi$
$709$	2.06172e7i	1.54033i	−0.637846	+	0.770164i	$0.720174\pi$
$710$	0	0
$711$	1.16258e7	0.862476
$712$	0	0
$713$	7.94115e6	0.585005
$714$	0	0
$715$	− 1.65205e7i	− 1.20853i
$716$	0	0
$717$	− 2.17280e7i	− 1.57842i
$718$	0	0
$719$	−1.37162e7	−0.989490	−0.494745	−	0.869038i	$-0.664738\pi$
$719$	−1.37162e7	−0.989490	−0.494745	+	0.869038i	$0.664738\pi$
$720$	0	0
$721$	−1.45383e7	−1.04154
$722$	0	0
$723$	− 4.14305e7i	− 2.94764i
$724$	0	0
$725$	− 1.34904e6i	− 0.0953193i
$726$	0	0
$727$	2.33681e7	1.63979	0.819895	−	0.572514i	$-0.194032\pi$
$727$	2.33681e7	1.63979	0.819895	+	0.572514i	$0.194032\pi$
$728$	0	0
$729$	2.33601e7	1.62800
$730$	0	0
$731$	− 1.48823e7i	− 1.03010i
$732$	0	0
$733$	− 2.94411e6i	− 0.202392i	−0.994866	−	0.101196i	$-0.967733\pi$
$733$	− 2.94411e6i	− 0.202392i	0.994866	−	0.101196i	$-0.0322670\pi$
$734$	0	0
$735$	3.80106e6	0.259530
$736$	0	0
$737$	−4.07289e7	−2.76207
$738$	0	0
$739$	3.66027e6i	0.246548i	0.992373	+	0.123274i	$0.0393395\pi$
$739$	3.66027e6i	0.246548i	−0.992373	+	0.123274i	$0.960661\pi$
$740$	0	0
$741$	2.58223e7i	1.72763i
$742$	0	0
$743$	2.60824e7	1.73331	0.866653	−	0.498912i	$-0.166267\pi$
$743$	2.60824e7	1.73331	0.866653	+	0.498912i	$0.166267\pi$
$744$	0	0
$745$	5.84359e6	0.385735
$746$	0	0
$747$	3.61338e7i	2.36926i
$748$	0	0
$749$	− 3.60535e6i	− 0.234824i
$750$	0	0
$751$	1.03196e7	0.667673	0.333836	−	0.942631i	$-0.391657\pi$
$751$	1.03196e7	0.667673	0.333836	+	0.942631i	$0.391657\pi$
$752$	0	0
$753$	2.15668e7	1.38611
$754$	0	0
$755$	2.22671e6i	0.142166i
$756$	0	0
$757$	− 2.70313e7i	− 1.71446i	−0.514936	−	0.857228i	$-0.672184\pi$
$757$	− 2.70313e7i	− 1.71446i	0.514936	−	0.857228i	$-0.327816\pi$
$758$	0	0
$759$	−2.97405e7	−1.87389
$760$	0	0
$761$	1.29548e7	0.810905	0.405452	−	0.914116i	$-0.367114\pi$
$761$	1.29548e7	0.810905	0.405452	+	0.914116i	$0.367114\pi$
$762$	0	0
$763$	− 8.67946e6i	− 0.539736i
$764$	0	0
$765$	− 1.09499e7i	− 0.676480i
$766$	0	0
$767$	−1.96834e7	−1.20813
$768$	0	0
$769$	8.35551e6	0.509515	0.254757	−	0.967005i	$-0.418004\pi$
$769$	8.35551e6	0.509515	0.254757	+	0.967005i	$0.418004\pi$
$770$	0	0
$771$	− 1.91732e7i	− 1.16161i
$772$	0	0
$773$	− 3.12215e7i	− 1.87934i	−0.342087	−	0.939668i	$-0.611134\pi$
$773$	− 3.12215e7i	− 1.87934i	0.342087	−	0.939668i	$-0.388866\pi$
$774$	0	0
$775$	3.09703e6	0.185221
$776$	0	0
$777$	1.14325e7	0.679342
$778$	0	0
$779$	4.36835e6i	0.257913i
$780$	0	0
$781$	3.17639e7i	1.86340i
$782$	0	0
$783$	7.65727e6	0.446344
$784$	0	0
$785$	1.52227e7	0.881695
$786$	0	0
$787$	− 1.14664e7i	− 0.659920i	−0.943995	−	0.329960i	$-0.892965\pi$
$787$	− 1.14664e7i	− 0.659920i	0.943995	−	0.329960i	$-0.107035\pi$
$788$	0	0
$789$	− 6.53222e6i	− 0.373567i
$790$	0	0
$791$	−5.93373e6	−0.337199
$792$	0	0
$793$	2.51789e6	0.142185
$794$	0	0
$795$	2.14907e7i	1.20596i
$796$	0	0
$797$	− 2.28998e7i	− 1.27699i	−0.769628	−	0.638493i	$-0.779558\pi$
$797$	− 2.28998e7i	− 1.27699i	0.769628	−	0.638493i	$-0.220442\pi$
$798$	0	0
$799$	9.11115e6	0.504901
$800$	0	0
$801$	−453090.	−0.0249519
$802$	0	0
$803$	1.54772e7i	0.847040i
$804$	0	0
$805$	4.15163e6i	0.225803i
$806$	0	0
$807$	−2.81614e7	−1.52220
$808$	0	0
$809$	2.82761e7	1.51896	0.759482	−	0.650528i	$-0.225452\pi$
$809$	2.82761e7	1.51896	0.759482	+	0.650528i	$0.225452\pi$
$810$	0	0
$811$	2.97657e7i	1.58915i	0.607167	+	0.794574i	$0.292306\pi$
$811$	2.97657e7i	1.58915i	−0.607167	+	0.794574i	$0.707694\pi$
$812$	0	0
$813$	2.68752e7i	1.42602i
$814$	0	0
$815$	−4.32002e6	−0.227820
$816$	0	0
$817$	1.50999e7	0.791441
$818$	0	0
$819$	− 3.55529e7i	− 1.85210i
$820$	0	0
$821$	− 2.33663e7i	− 1.20985i	−0.796282	−	0.604926i	$-0.793203\pi$
$821$	− 2.33663e7i	− 1.20985i	0.796282	−	0.604926i	$-0.206797\pi$
$822$	0	0
$823$	3.90864e6	0.201153	0.100576	−	0.994929i	$-0.467931\pi$
$823$	3.90864e6	0.201153	0.100576	+	0.994929i	$0.467931\pi$
$824$	0	0
$825$	−1.15987e7	−0.593302
$826$	0	0
$827$	1.34407e7i	0.683371i	0.939814	+	0.341686i	$0.110998\pi$
$827$	1.34407e7i	0.683371i	−0.939814	+	0.341686i	$0.889002\pi$
$828$	0	0
$829$	2.39273e7i	1.20923i	0.796519	+	0.604613i	$0.206672\pi$
$829$	2.39273e7i	1.20923i	−0.796519	+	0.604613i	$0.793328\pi$
$830$	0	0
$831$	1.54090e7	0.774057
$832$	0	0
$833$	−6.91151e6	−0.345112
$834$	0	0
$835$	− 1.10376e7i	− 0.547848i
$836$	0	0
$837$	1.75790e7i	0.867320i
$838$	0	0
$839$	4.44572e6	0.218041	0.109020	−	0.994040i	$-0.465229\pi$
$839$	4.44572e6	0.218041	0.109020	+	0.994040i	$0.465229\pi$
$840$	0	0
$841$	1.58522e7	0.772856
$842$	0	0
$843$	− 1.63389e7i	− 0.791871i
$844$	0	0
$845$	− 1.06122e7i	− 0.511288i
$846$	0	0
$847$	4.01748e7	1.92418
$848$	0	0
$849$	2.27622e7	1.08379
$850$	0	0
$851$	− 7.05756e6i	− 0.334065i
$852$	0	0
$853$	2.57954e7i	1.21386i	0.794754	+	0.606931i	$0.207600\pi$
$853$	2.57954e7i	1.21386i	−0.794754	+	0.606931i	$0.792400\pi$
$854$	0	0
$855$	1.11099e7	0.519752
$856$	0	0
$857$	1.79012e7	0.832590	0.416295	−	0.909230i	$-0.363328\pi$
$857$	1.79012e7	0.832590	0.416295	+	0.909230i	$0.363328\pi$
$858$	0	0
$859$	2.50747e7i	1.15945i	0.814811	+	0.579727i	$0.196841\pi$
$859$	2.50747e7i	1.15945i	−0.814811	+	0.579727i	$0.803159\pi$
$860$	0	0
$861$	− 9.81449e6i	− 0.451190i
$862$	0	0
$863$	−8.98511e6	−0.410673	−0.205337	−	0.978691i	$-0.565829\pi$
$863$	−8.98511e6	−0.410673	−0.205337	+	0.978691i	$0.565829\pi$
$864$	0	0
$865$	−7.03534e6	−0.319702
$866$	0	0
$867$	− 3.08057e6i	− 0.139182i
$868$	0	0
$869$	2.23919e7i	1.00587i
$870$	0	0
$871$	−4.90471e7	−2.19063
$872$	0	0
$873$	−2.84997e7	−1.26562
$874$	0	0
$875$	1.61913e6i	0.0714925i
$876$	0	0
$877$	− 1.08381e7i	− 0.475834i	−0.971286	−	0.237917i	$-0.923535\pi$
$877$	− 1.08381e7i	− 0.475834i	0.971286	−	0.237917i	$-0.0764646\pi$
$878$	0	0
$879$	1.28780e7	0.562179
$880$	0	0
$881$	−816876.	−0.0354582	−0.0177291	−	0.999843i	$-0.505644\pi$
$881$	−816876.	−0.0354582	−0.0177291	+	0.999843i	$0.505644\pi$
$882$	0	0
$883$	2.50842e6i	0.108268i	0.998534	+	0.0541339i	$0.0172398\pi$
$883$	2.50842e6i	0.108268i	−0.998534	+	0.0541339i	$0.982760\pi$
$884$	0	0
$885$	1.38193e7i	0.593102i
$886$	0	0
$887$	1.58294e7	0.675545	0.337773	−	0.941228i	$-0.390327\pi$
$887$	1.58294e7	0.675545	0.337773	+	0.941228i	$0.390327\pi$
$888$	0	0
$889$	−916878.	−0.0389096
$890$	0	0
$891$	3.39722e6i	0.143360i
$892$	0	0
$893$	9.24434e6i	0.387924i
$894$	0	0
$895$	−2.01442e7	−0.840605
$896$	0	0
$897$	−3.58145e7	−1.48620
$898$	0	0
$899$	− 1.06957e7i	− 0.441379i
$900$	0	0
$901$	− 3.90768e7i	− 1.60364i
$902$	0	0
$903$	−3.39253e7	−1.38454
$904$	0	0
$905$	1.52586e7	0.619288
$906$	0	0
$907$	2.36328e7i	0.953887i	0.878934	+	0.476944i	$0.158255\pi$
$907$	2.36328e7i	0.953887i	−0.878934	+	0.476944i	$0.841745\pi$
$908$	0	0
$909$	− 1.19629e7i	− 0.480206i
$910$	0	0
$911$	9.17484e6	0.366271	0.183136	−	0.983088i	$-0.441375\pi$
$911$	9.17484e6	0.366271	0.183136	+	0.983088i	$0.441375\pi$
$912$	0	0
$913$	−6.95959e7	−2.76316
$914$	0	0
$915$	− 1.76776e6i	− 0.0698023i
$916$	0	0
$917$	1.14461e7i	0.449504i
$918$	0	0
$919$	3.58059e7	1.39851	0.699255	−	0.714872i	$-0.253515\pi$
$919$	3.58059e7	1.39851	0.699255	+	0.714872i	$0.253515\pi$
$920$	0	0
$921$	−3.67497e7	−1.42759
$922$	0	0
$923$	3.82511e7i	1.47788i
$924$	0	0
$925$	− 2.75243e6i	− 0.105770i
$926$	0	0
$927$	−5.39597e7	−2.06239
$928$	0	0
$929$	−2.41031e7	−0.916291	−0.458146	−	0.888877i	$-0.651486\pi$
$929$	−2.41031e7	−0.916291	−0.458146	+	0.888877i	$0.651486\pi$
$930$	0	0
$931$	− 7.01254e6i	− 0.265156i
$932$	0	0
$933$	− 1.37265e7i	− 0.516245i
$934$	0	0
$935$	2.10901e7	0.788950
$936$	0	0
$937$	−1.84615e7	−0.686938	−0.343469	−	0.939164i	$-0.611602\pi$
$937$	−1.84615e7	−0.686938	−0.343469	+	0.939164i	$0.611602\pi$
$938$	0	0
$939$	4.28032e7i	1.58421i
$940$	0	0
$941$	− 3.10319e6i	− 0.114244i	−0.998367	−	0.0571220i	$-0.981808\pi$
$941$	− 3.10319e6i	− 0.114244i	0.998367	−	0.0571220i	$-0.0181924\pi$
$942$	0	0
$943$	−6.05873e6	−0.221872
$944$	0	0
$945$	−9.19028e6	−0.334772
$946$	0	0
$947$	− 4.89572e7i	− 1.77395i	−0.461817	−	0.886975i	$-0.652802\pi$
$947$	− 4.89572e7i	− 1.77395i	0.461817	−	0.886975i	$-0.347198\pi$
$948$	0	0
$949$	1.86382e7i	0.671796i
$950$	0	0
$951$	−2.45162e7	−0.879025
$952$	0	0
$953$	3.94753e7	1.40797	0.703984	−	0.710216i	$-0.251403\pi$
$953$	3.94753e7	1.40797	0.703984	+	0.710216i	$0.251403\pi$
$954$	0	0
$955$	2.11434e7i	0.750182i
$956$	0	0
$957$	4.00568e7i	1.41383i
$958$	0	0
$959$	3.15395e7	1.10741
$960$	0	0
$961$	−4.07470e6	−0.142327
$962$	0	0
$963$	− 1.33815e7i	− 0.464984i
$964$	0	0
$965$	− 4.60223e6i	− 0.159093i
$966$	0	0
$967$	2.23447e7	0.768438	0.384219	−	0.923242i	$-0.374471\pi$
$967$	2.23447e7	0.768438	0.384219	+	0.923242i	$0.374471\pi$
$968$	0	0
$969$	−3.29647e7	−1.12782
$970$	0	0
$971$	− 2.63033e7i	− 0.895288i	−0.894212	−	0.447644i	$-0.852263\pi$
$971$	− 2.63033e7i	− 0.895288i	0.894212	−	0.447644i	$-0.147737\pi$
$972$	0	0
$973$	− 6.15315e6i	− 0.208360i
$974$	0	0
$975$	−1.39676e7	−0.470554
$976$	0	0
$977$	4.78338e7	1.60324	0.801619	−	0.597835i	$-0.203972\pi$
$977$	4.78338e7	1.60324	0.801619	+	0.597835i	$0.203972\pi$
$978$	0	0
$979$	− 872679.i	− 0.0291003i
$980$	0	0
$981$	− 3.22144e7i	− 1.06875i
$982$	0	0
$983$	−4.88361e7	−1.61197	−0.805986	−	0.591935i	$-0.798364\pi$
$983$	−4.88361e7	−1.61197	−0.805986	+	0.591935i	$0.798364\pi$
$984$	0	0
$985$	−9.76305e6	−0.320623
$986$	0	0
$987$	− 2.07695e7i	− 0.678630i
$988$	0	0
$989$	2.09429e7i	0.680843i
$990$	0	0
$991$	−7.49231e6	−0.242344	−0.121172	−	0.992632i	$-0.538665\pi$
$991$	−7.49231e6	−0.242344	−0.121172	+	0.992632i	$0.538665\pi$
$992$	0	0
$993$	−8.66569e6	−0.278888
$994$	0	0
$995$	1.46066e7i	0.467725i
$996$	0	0
$997$	1.46209e7i	0.465841i	0.972496	+	0.232920i	$0.0748281\pi$
$997$	1.46209e7i	0.465841i	−0.972496	+	0.232920i	$0.925172\pi$
$998$	0	0
$999$	1.56230e7	0.495280

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	160.6.d.a.81.18		20
4.3	odd	2		40.6.d.a.21.15	✓	20
5.2	odd	4		800.6.f.c.49.17		20
5.3	odd	4		800.6.f.b.49.4		20
5.4	even	2		800.6.d.c.401.3		20
8.3	odd	2		40.6.d.a.21.16	yes	20
8.5	even	2	inner	160.6.d.a.81.3		20
12.11	even	2		360.6.k.b.181.6		20
20.3	even	4		200.6.f.c.149.5		20
20.7	even	4		200.6.f.b.149.16		20
20.19	odd	2		200.6.d.b.101.6		20
24.11	even	2		360.6.k.b.181.5		20
40.3	even	4		200.6.f.b.149.15		20
40.13	odd	4		800.6.f.c.49.18		20
40.19	odd	2		200.6.d.b.101.5		20
40.27	even	4		200.6.f.c.149.6		20
40.29	even	2		800.6.d.c.401.18		20
40.37	odd	4		800.6.f.b.49.3		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.6.d.a.21.15	✓	20	4.3	odd	2
40.6.d.a.21.16	yes	20	8.3	odd	2
160.6.d.a.81.3		20	8.5	even	2	inner
160.6.d.a.81.18		20	1.1	even	1	trivial
200.6.d.b.101.5		20	40.19	odd	2
200.6.d.b.101.6		20	20.19	odd	2
200.6.f.b.149.15		20	40.3	even	4
200.6.f.b.149.16		20	20.7	even	4
200.6.f.c.149.5		20	20.3	even	4
200.6.f.c.149.6		20	40.27	even	4
360.6.k.b.181.5		20	24.11	even	2
360.6.k.b.181.6		20	12.11	even	2
800.6.d.c.401.3		20	5.4	even	2
800.6.d.c.401.18		20	40.29	even	2
800.6.f.b.49.3		20	40.37	odd	4
800.6.f.b.49.4		20	5.3	odd	4
800.6.f.c.49.17		20	5.2	odd	4
800.6.f.c.49.18		20	40.13	odd	4

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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	160.6.d.a.81.18

Level	$160$
Weight	$6$
Character	160.81
Analytic conductor	$25.661$
Analytic rank	$0$
Dimension	$20$
Inner twists	$2$