LMFDB - Embedded newform 400.8.a.bh.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [400,8,Mod(1,400)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 8, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("400.1"); S:= CuspForms(chi, 8); N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	400.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,-97,0,0,0,1630] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$124.954010194$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.40101.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 34x + 4 $ x^3 - x^2 - 34*x + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{8}\cdot 5 $
Twist minimal:	no (minimal twist has level 200)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.117290$ of defining polynomial
Character	$\chi$	$=$	400.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+14.5033 q^{3} +1504.61 q^{7} -1976.65 q^{9} +8170.56 q^{11} -5760.03 q^{13} -18837.0 q^{17} -52325.6 q^{19} +21821.8 q^{21} +29526.1 q^{23} -60386.8 q^{27} -237925. q^{29} -178366. q^{31} +118500. q^{33} +150224. q^{37} -83539.5 q^{39} -284583. q^{41} +586286. q^{43} +20542.9 q^{47} +1.44030e6 q^{49} -273199. q^{51} -1.36322e6 q^{53} -758895. q^{57} -543514. q^{59} +1.13350e6 q^{61} -2.97408e6 q^{63} -936593. q^{67} +428227. q^{69} +1.12222e6 q^{71} -3.33421e6 q^{73} +1.22935e7 q^{77} -5.90757e6 q^{79} +3.44713e6 q^{81} +5.14032e6 q^{83} -3.45071e6 q^{87} +9.43725e6 q^{89} -8.66657e6 q^{91} -2.58690e6 q^{93} -1.59539e7 q^{97} -1.61504e7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 97 q^{3} + 1630 q^{7} + 2146 q^{9} + 2659 q^{11} + 3308 q^{13} - 24481 q^{17} - 55749 q^{19} - 61562 q^{21} - 56910 q^{23} - 300619 q^{27} + 65528 q^{29} + 144598 q^{31} + 657207 q^{33} + 204678 q^{37}+ \cdots - 53316510 q^{99}+O(q^{100}) $ 3 * q - 97 * q^3 + 1630 * q^7 + 2146 * q^9 + 2659 * q^11 + 3308 * q^13 - 24481 * q^17 - 55749 * q^19 - 61562 * q^21 - 56910 * q^23 - 300619 * q^27 + 65528 * q^29 + 144598 * q^31 + 657207 * q^33 + 204678 * q^37 - 800004 * q^39 + 299743 * q^41 + 108620 * q^43 - 670804 * q^47 + 2360487 * q^49 + 1862315 * q^51 - 1288886 * q^53 + 6703 * q^57 - 983368 * q^59 + 5710118 * q^61 + 5802452 * q^63 - 5436267 * q^67 + 5390698 * q^69 - 1933956 * q^71 - 9921215 * q^73 + 4194286 * q^77 - 5590678 * q^79 + 21712267 * q^81 + 11873921 * q^83 - 18848136 * q^87 + 22333709 * q^89 - 1031944 * q^91 - 19549890 * q^93 - 2002634 * q^97 - 53316510 * q^99

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
$3$	14.5033	0.310130	0.155065	−	0.987904i	$-0.450441\pi$
$3$	14.5033	0.310130	0.155065	+	0.987904i	$0.450441\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1504.61	1.65798	0.828991	−	0.559262i	$-0.188916\pi$
$7$	1504.61	1.65798	0.828991	+	0.559262i	$0.188916\pi$
$8$	0	0
$9$	−1976.65	−0.903820
$10$	0	0
$11$	8170.56	1.85088	0.925438	−	0.378900i	$-0.123698\pi$
$11$	8170.56	1.85088	0.925438	+	0.378900i	$0.123698\pi$
$12$	0	0
$13$	−5760.03	−0.727148	−0.363574	−	0.931565i	$-0.618444\pi$
$13$	−5760.03	−0.727148	−0.363574	+	0.931565i	$0.618444\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−18837.0	−0.929908	−0.464954	−	0.885335i	$-0.653929\pi$
$17$	−18837.0	−0.929908	−0.464954	+	0.885335i	$0.653929\pi$
$18$	0	0
$19$	−52325.6	−1.75015	−0.875077	−	0.483983i	$-0.839190\pi$
$19$	−52325.6	−1.75015	−0.875077	+	0.483983i	$0.839190\pi$
$20$	0	0
$21$	21821.8	0.514189
$22$	0	0
$23$	29526.1	0.506010	0.253005	−	0.967465i	$-0.418581\pi$
$23$	29526.1	0.506010	0.253005	+	0.967465i	$0.418581\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−60386.8	−0.590431
$28$	0	0
$29$	−237925.	−1.81154	−0.905769	−	0.423771i	$-0.860706\pi$
$29$	−237925.	−1.81154	−0.905769	+	0.423771i	$0.860706\pi$
$30$	0	0
$31$	−178366.	−1.07534	−0.537670	−	0.843156i	$-0.680695\pi$
$31$	−178366.	−1.07534	−0.537670	+	0.843156i	$0.680695\pi$
$32$	0	0
$33$	118500.	0.574011
$34$	0	0
$35$	0	0
$36$	0	0
$37$	150224.	0.487565	0.243782	−	0.969830i	$-0.421612\pi$
$37$	150224.	0.487565	0.243782	+	0.969830i	$0.421612\pi$
$38$	0	0
$39$	−83539.5	−0.225510
$40$	0	0
$41$	−284583.	−0.644861	−0.322430	−	0.946593i	$-0.604500\pi$
$41$	−284583.	−0.644861	−0.322430	+	0.946593i	$0.604500\pi$
$42$	0	0
$43$	586286.	1.12453	0.562263	−	0.826958i	$-0.309931\pi$
$43$	586286.	1.12453	0.562263	+	0.826958i	$0.309931\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	20542.9	0.0288615	0.0144307	−	0.999896i	$-0.495406\pi$
$47$	20542.9	0.0288615	0.0144307	+	0.999896i	$0.495406\pi$
$48$	0	0
$49$	1.44030e6	1.74890
$50$	0	0
$51$	−273199.	−0.288392
$52$	0	0
$53$	−1.36322e6	−1.25776	−0.628882	−	0.777501i	$-0.716487\pi$
$53$	−1.36322e6	−1.25776	−0.628882	+	0.777501i	$0.716487\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−758895.	−0.542775
$58$	0	0
$59$	−543514.	−0.344531	−0.172266	−	0.985051i	$-0.555109\pi$
$59$	−543514.	−0.344531	−0.172266	+	0.985051i	$0.555109\pi$
$60$	0	0
$61$	1.13350e6	0.639392	0.319696	−	0.947520i	$-0.396419\pi$
$61$	1.13350e6	0.639392	0.319696	+	0.947520i	$0.396419\pi$
$62$	0	0
$63$	−2.97408e6	−1.49852
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−936593.	−0.380442	−0.190221	−	0.981741i	$-0.560921\pi$
$67$	−936593.	−0.380442	−0.190221	+	0.981741i	$0.560921\pi$
$68$	0	0
$69$	428227.	0.156929
$70$	0	0
$71$	1.12222e6	0.372111	0.186055	−	0.982539i	$-0.440430\pi$
$71$	1.12222e6	0.372111	0.186055	+	0.982539i	$0.440430\pi$
$72$	0	0
$73$	−3.33421e6	−1.00314	−0.501572	−	0.865116i	$-0.667245\pi$
$73$	−3.33421e6	−1.00314	−0.501572	+	0.865116i	$0.667245\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.22935e7	3.06872
$78$	0	0
$79$	−5.90757e6	−1.34807	−0.674037	−	0.738697i	$-0.735441\pi$
$79$	−5.90757e6	−1.34807	−0.674037	+	0.738697i	$0.735441\pi$
$80$	0	0
$81$	3.44713e6	0.720710
$82$	0	0
$83$	5.14032e6	0.986771	0.493386	−	0.869811i	$-0.335759\pi$
$83$	5.14032e6	0.986771	0.493386	+	0.869811i	$0.335759\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−3.45071e6	−0.561812
$88$	0	0
$89$	9.43725e6	1.41899	0.709497	−	0.704709i	$-0.248922\pi$
$89$	9.43725e6	1.41899	0.709497	+	0.704709i	$0.248922\pi$
$90$	0	0
$91$	−8.66657e6	−1.20560
$92$	0	0
$93$	−2.58690e6	−0.333495
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−1.59539e7	−1.77487	−0.887434	−	0.460935i	$-0.847514\pi$
$97$	−1.59539e7	−1.77487	−0.887434	+	0.460935i	$0.847514\pi$
$98$	0	0
$99$	−1.61504e7	−1.67286
$100$	0	0
$101$	−1.01924e7	−0.984358	−0.492179	−	0.870494i	$-0.663800\pi$
$101$	−1.01924e7	−0.984358	−0.492179	+	0.870494i	$0.663800\pi$
$102$	0	0
$103$	1.06382e7	0.959263	0.479631	−	0.877470i	$-0.340770\pi$
$103$	1.06382e7	0.959263	0.479631	+	0.877470i	$0.340770\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.02381e7	−0.807931	−0.403966	−	0.914774i	$-0.632369\pi$
$107$	−1.02381e7	−0.807931	−0.403966	+	0.914774i	$0.632369\pi$
$108$	0	0
$109$	−2.39817e6	−0.177373	−0.0886865	−	0.996060i	$-0.528267\pi$
$109$	−2.39817e6	−0.177373	−0.0886865	+	0.996060i	$0.528267\pi$
$110$	0	0
$111$	2.17874e6	0.151208
$112$	0	0
$113$	−8.06232e6	−0.525637	−0.262818	−	0.964845i	$-0.584652\pi$
$113$	−8.06232e6	−0.525637	−0.262818	+	0.964845i	$0.584652\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1.13856e7	0.657210
$118$	0	0
$119$	−2.83422e7	−1.54177
$120$	0	0
$121$	4.72708e7	2.42574
$122$	0	0
$123$	−4.12740e6	−0.199990
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−4.13601e7	−1.79171	−0.895857	−	0.444343i	$-0.853437\pi$
$127$	−4.13601e7	−1.79171	−0.895857	+	0.444343i	$0.853437\pi$
$128$	0	0
$129$	8.50310e6	0.348749
$130$	0	0
$131$	6.39953e6	0.248713	0.124357	−	0.992238i	$-0.460313\pi$
$131$	6.39953e6	0.248713	0.124357	+	0.992238i	$0.460313\pi$
$132$	0	0
$133$	−7.87294e7	−2.90172
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.67069e7	0.887364	0.443682	−	0.896184i	$-0.353672\pi$
$137$	2.67069e7	0.887364	0.443682	+	0.896184i	$0.353672\pi$
$138$	0	0
$139$	1.41131e7	0.445727	0.222864	−	0.974850i	$-0.428460\pi$
$139$	1.41131e7	0.445727	0.222864	+	0.974850i	$0.428460\pi$
$140$	0	0
$141$	297940.	0.00895079
$142$	0	0
$143$	−4.70626e7	−1.34586
$144$	0	0
$145$	0	0
$146$	0	0
$147$	2.08891e7	0.542386
$148$	0	0
$149$	2.24351e7	0.555619	0.277809	−	0.960636i	$-0.410392\pi$
$149$	2.24351e7	0.555619	0.277809	+	0.960636i	$0.410392\pi$
$150$	0	0
$151$	1.88005e7	0.444375	0.222187	−	0.975004i	$-0.428680\pi$
$151$	1.88005e7	0.444375	0.222187	+	0.975004i	$0.428680\pi$
$152$	0	0
$153$	3.72342e7	0.840469
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−4.56503e7	−0.941444	−0.470722	−	0.882281i	$-0.656007\pi$
$157$	−4.56503e7	−0.941444	−0.470722	+	0.882281i	$0.656007\pi$
$158$	0	0
$159$	−1.97712e7	−0.390070
$160$	0	0
$161$	4.44252e7	0.838955
$162$	0	0
$163$	−1.76081e7	−0.318461	−0.159231	−	0.987241i	$-0.550901\pi$
$163$	−1.76081e7	−0.318461	−0.159231	+	0.987241i	$0.550901\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	6.91771e7	1.14936	0.574678	−	0.818380i	$-0.305127\pi$
$167$	6.91771e7	1.14936	0.574678	+	0.818380i	$0.305127\pi$
$168$	0	0
$169$	−2.95706e7	−0.471256
$170$	0	0
$171$	1.03430e8	1.58182
$172$	0	0
$173$	−3.58362e7	−0.526212	−0.263106	−	0.964767i	$-0.584747\pi$
$173$	−3.58362e7	−0.526212	−0.263106	+	0.964767i	$0.584747\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−7.88276e6	−0.106849
$178$	0	0
$179$	−6.21415e7	−0.809834	−0.404917	−	0.914353i	$-0.632700\pi$
$179$	−6.21415e7	−0.809834	−0.404917	+	0.914353i	$0.632700\pi$
$180$	0	0
$181$	−5.35093e6	−0.0670741	−0.0335370	−	0.999437i	$-0.510677\pi$
$181$	−5.35093e6	−0.0670741	−0.0335370	+	0.999437i	$0.510677\pi$
$182$	0	0
$183$	1.64395e7	0.198294
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−1.53909e8	−1.72114
$188$	0	0
$189$	−9.08584e7	−0.978923
$190$	0	0
$191$	−1.36197e6	−0.0141433	−0.00707163	−	0.999975i	$-0.502251\pi$
$191$	−1.36197e6	−0.0141433	−0.00707163	+	0.999975i	$0.502251\pi$
$192$	0	0
$193$	−4.04742e7	−0.405255	−0.202627	−	0.979256i	$-0.564948\pi$
$193$	−4.04742e7	−0.405255	−0.202627	+	0.979256i	$0.564948\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−8.57389e7	−0.798999	−0.399499	−	0.916733i	$-0.630816\pi$
$197$	−8.57389e7	−0.798999	−0.399499	+	0.916733i	$0.630816\pi$
$198$	0	0
$199$	8.83997e7	0.795179	0.397589	−	0.917563i	$-0.369847\pi$
$199$	8.83997e7	0.795179	0.397589	+	0.917563i	$0.369847\pi$
$200$	0	0
$201$	−1.35837e7	−0.117986
$202$	0	0
$203$	−3.57984e8	−3.00350
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−5.83629e7	−0.457342
$208$	0	0
$209$	−4.27529e8	−3.23932
$210$	0	0
$211$	1.36506e7	0.100037	0.0500187	−	0.998748i	$-0.484072\pi$
$211$	1.36506e7	0.100037	0.0500187	+	0.998748i	$0.484072\pi$
$212$	0	0
$213$	1.62759e7	0.115403
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−2.68370e8	−1.78289
$218$	0	0
$219$	−4.83572e7	−0.311105
$220$	0	0
$221$	1.08501e8	0.676180
$222$	0	0
$223$	−2.28723e8	−1.38116	−0.690579	−	0.723257i	$-0.742644\pi$
$223$	−2.28723e8	−1.38116	−0.690579	+	0.723257i	$0.742644\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.52906e8	0.867629	0.433814	−	0.901002i	$-0.357167\pi$
$227$	1.52906e8	0.867629	0.433814	+	0.901002i	$0.357167\pi$
$228$	0	0
$229$	−8.25002e7	−0.453974	−0.226987	−	0.973898i	$-0.572887\pi$
$229$	−8.25002e7	−0.453974	−0.226987	+	0.973898i	$0.572887\pi$
$230$	0	0
$231$	1.78296e8	0.951700
$232$	0	0
$233$	−3.46346e8	−1.79376	−0.896881	−	0.442272i	$-0.854172\pi$
$233$	−3.46346e8	−1.79376	−0.896881	+	0.442272i	$0.854172\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−8.56794e7	−0.418078
$238$	0	0
$239$	−2.10609e7	−0.0997894	−0.0498947	−	0.998754i	$-0.515889\pi$
$239$	−2.10609e7	−0.0997894	−0.0498947	+	0.998754i	$0.515889\pi$
$240$	0	0
$241$	−2.19190e8	−1.00870	−0.504349	−	0.863500i	$-0.668268\pi$
$241$	−2.19190e8	−1.00870	−0.504349	+	0.863500i	$0.668268\pi$
$242$	0	0
$243$	1.82061e8	0.813944
$244$	0	0
$245$	0	0
$246$	0	0
$247$	3.01397e8	1.27262
$248$	0	0
$249$	7.45517e7	0.306027
$250$	0	0
$251$	8.35072e7	0.333323	0.166662	−	0.986014i	$-0.446701\pi$
$251$	8.35072e7	0.333323	0.166662	+	0.986014i	$0.446701\pi$
$252$	0	0
$253$	2.41245e8	0.936561
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2.26230e8	0.831351	0.415676	−	0.909513i	$-0.363545\pi$
$257$	2.26230e8	0.831351	0.415676	+	0.909513i	$0.363545\pi$
$258$	0	0
$259$	2.26027e8	0.808373
$260$	0	0
$261$	4.70296e8	1.63730
$262$	0	0
$263$	−1.93601e8	−0.656241	−0.328121	−	0.944636i	$-0.606415\pi$
$263$	−1.93601e8	−0.656241	−0.328121	+	0.944636i	$0.606415\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	1.36872e8	0.440072
$268$	0	0
$269$	−1.56593e8	−0.490501	−0.245250	−	0.969460i	$-0.578870\pi$
$269$	−1.56593e8	−0.490501	−0.245250	+	0.969460i	$0.578870\pi$
$270$	0	0
$271$	3.46729e8	1.05827	0.529136	−	0.848537i	$-0.322516\pi$
$271$	3.46729e8	1.05827	0.529136	+	0.848537i	$0.322516\pi$
$272$	0	0
$273$	−1.25694e8	−0.373891
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1.96707e8	0.556084	0.278042	−	0.960569i	$-0.410314\pi$
$277$	1.96707e8	0.556084	0.278042	+	0.960569i	$0.410314\pi$
$278$	0	0
$279$	3.52567e8	0.971913
$280$	0	0
$281$	−5.11263e8	−1.37459	−0.687294	−	0.726379i	$-0.741202\pi$
$281$	−5.11263e8	−1.37459	−0.687294	+	0.726379i	$0.741202\pi$
$282$	0	0
$283$	−6.72694e8	−1.76427	−0.882134	−	0.470998i	$-0.843894\pi$
$283$	−6.72694e8	−1.76427	−0.882134	+	0.470998i	$0.843894\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−4.28186e8	−1.06917
$288$	0	0
$289$	−5.55071e7	−0.135271
$290$	0	0
$291$	−2.31385e8	−0.550439
$292$	0	0
$293$	−4.62188e8	−1.07345	−0.536725	−	0.843757i	$-0.680339\pi$
$293$	−4.62188e8	−1.07345	−0.536725	+	0.843757i	$0.680339\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−4.93394e8	−1.09281
$298$	0	0
$299$	−1.70071e8	−0.367944
$300$	0	0
$301$	8.82129e8	1.86444
$302$	0	0
$303$	−1.47824e8	−0.305279
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−6.24848e8	−1.23251	−0.616254	−	0.787547i	$-0.711351\pi$
$307$	−6.24848e8	−1.23251	−0.616254	+	0.787547i	$0.711351\pi$
$308$	0	0
$309$	1.54289e8	0.297496
$310$	0	0
$311$	−5.28722e8	−0.996703	−0.498352	−	0.866975i	$-0.666061\pi$
$311$	−5.28722e8	−0.996703	−0.498352	+	0.866975i	$0.666061\pi$
$312$	0	0
$313$	2.08317e8	0.383989	0.191995	−	0.981396i	$-0.438504\pi$
$313$	2.08317e8	0.383989	0.191995	+	0.981396i	$0.438504\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	9.88124e6	0.0174222	0.00871112	−	0.999962i	$-0.497227\pi$
$317$	9.88124e6	0.0174222	0.00871112	+	0.999962i	$0.497227\pi$
$318$	0	0
$319$	−1.94398e9	−3.35293
$320$	0	0
$321$	−1.48486e8	−0.250563
$322$	0	0
$323$	9.85656e8	1.62748
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−3.47815e7	−0.0550086
$328$	0	0
$329$	3.09089e7	0.0478518
$330$	0	0
$331$	5.47351e8	0.829599	0.414799	−	0.909913i	$-0.363852\pi$
$331$	5.47351e8	0.829599	0.414799	+	0.909913i	$0.363852\pi$
$332$	0	0
$333$	−2.96940e8	−0.440671
$334$	0	0
$335$	0	0
$336$	0	0
$337$	3.95141e8	0.562403	0.281201	−	0.959649i	$-0.409267\pi$
$337$	3.95141e8	0.562403	0.281201	+	0.959649i	$0.409267\pi$
$338$	0	0
$339$	−1.16930e8	−0.163015
$340$	0	0
$341$	−1.45735e9	−1.99032
$342$	0	0
$343$	9.27970e8	1.24167
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−4.53902e7	−0.0583189	−0.0291595	−	0.999575i	$-0.509283\pi$
$347$	−4.53902e7	−0.0583189	−0.0291595	+	0.999575i	$0.509283\pi$
$348$	0	0
$349$	1.07430e9	1.35281	0.676404	−	0.736531i	$-0.263537\pi$
$349$	1.07430e9	1.35281	0.676404	+	0.736531i	$0.263537\pi$
$350$	0	0
$351$	3.47830e8	0.429330
$352$	0	0
$353$	5.76841e8	0.697983	0.348991	−	0.937126i	$-0.386524\pi$
$353$	5.76841e8	0.697983	0.348991	+	0.937126i	$0.386524\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−4.11057e8	−0.478148
$358$	0	0
$359$	7.63547e8	0.870974	0.435487	−	0.900195i	$-0.356576\pi$
$359$	7.63547e8	0.870974	0.435487	+	0.900195i	$0.356576\pi$
$360$	0	0
$361$	1.84409e9	2.06304
$362$	0	0
$363$	6.85584e8	0.752294
$364$	0	0
$365$	0	0
$366$	0	0
$367$	1.55202e8	0.163895	0.0819475	−	0.996637i	$-0.473886\pi$
$367$	1.55202e8	0.163895	0.0819475	+	0.996637i	$0.473886\pi$
$368$	0	0
$369$	5.62522e8	0.582838
$370$	0	0
$371$	−2.05110e9	−2.08535
$372$	0	0
$373$	3.02866e8	0.302183	0.151091	−	0.988520i	$-0.451721\pi$
$373$	3.02866e8	0.302183	0.151091	+	0.988520i	$0.451721\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.37046e9	1.31726
$378$	0	0
$379$	1.12044e9	1.05718	0.528591	−	0.848877i	$-0.322721\pi$
$379$	1.12044e9	1.05718	0.528591	+	0.848877i	$0.322721\pi$
$380$	0	0
$381$	−5.99859e8	−0.555663
$382$	0	0
$383$	7.36644e8	0.669980	0.334990	−	0.942222i	$-0.391267\pi$
$383$	7.36644e8	0.669980	0.334990	+	0.942222i	$0.391267\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1.15888e9	−1.01637
$388$	0	0
$389$	1.81700e9	1.56506	0.782530	−	0.622612i	$-0.213929\pi$
$389$	1.81700e9	1.56506	0.782530	+	0.622612i	$0.213929\pi$
$390$	0	0
$391$	−5.56183e8	−0.470543
$392$	0	0
$393$	9.28145e7	0.0771333
$394$	0	0
$395$	0	0
$396$	0	0
$397$	2.38033e9	1.90928	0.954642	−	0.297757i	$-0.0962385\pi$
$397$	2.38033e9	1.90928	0.954642	+	0.297757i	$0.0962385\pi$
$398$	0	0
$399$	−1.14184e9	−0.899910
$400$	0	0
$401$	2.00554e9	1.55320	0.776599	−	0.629995i	$-0.216943\pi$
$401$	2.00554e9	1.55320	0.776599	+	0.629995i	$0.216943\pi$
$402$	0	0
$403$	1.02739e9	0.781931
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.22741e9	0.902422
$408$	0	0
$409$	3.92629e8	0.283760	0.141880	−	0.989884i	$-0.454685\pi$
$409$	3.92629e8	0.283760	0.141880	+	0.989884i	$0.454685\pi$
$410$	0	0
$411$	3.87339e8	0.275198
$412$	0	0
$413$	−8.17775e8	−0.571227
$414$	0	0
$415$	0	0
$416$	0	0
$417$	2.04686e8	0.138233
$418$	0	0
$419$	−1.08234e9	−0.718808	−0.359404	−	0.933182i	$-0.617020\pi$
$419$	−1.08234e9	−0.718808	−0.359404	+	0.933182i	$0.617020\pi$
$420$	0	0
$421$	−1.90157e9	−1.24201	−0.621004	−	0.783808i	$-0.713275\pi$
$421$	−1.90157e9	−1.24201	−0.621004	+	0.783808i	$0.713275\pi$
$422$	0	0
$423$	−4.06061e7	−0.0260856
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1.70547e9	1.06010
$428$	0	0
$429$	−6.82564e8	−0.417391
$430$	0	0
$431$	1.07717e9	0.648060	0.324030	−	0.946047i	$-0.394962\pi$
$431$	1.07717e9	0.648060	0.324030	+	0.946047i	$0.394962\pi$
$432$	0	0
$433$	4.78186e8	0.283067	0.141533	−	0.989933i	$-0.454797\pi$
$433$	4.78186e8	0.283067	0.141533	+	0.989933i	$0.454797\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.54497e9	−0.885595
$438$	0	0
$439$	3.30634e9	1.86519	0.932593	−	0.360929i	$-0.117540\pi$
$439$	3.30634e9	1.86519	0.932593	+	0.360929i	$0.117540\pi$
$440$	0	0
$441$	−2.84697e9	−1.58069
$442$	0	0
$443$	−2.15148e9	−1.17578	−0.587888	−	0.808942i	$-0.700040\pi$
$443$	−2.15148e9	−1.17578	−0.587888	+	0.808942i	$0.700040\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	3.25384e8	0.172314
$448$	0	0
$449$	2.08607e9	1.08759	0.543796	−	0.839217i	$-0.316987\pi$
$449$	2.08607e9	1.08759	0.543796	+	0.839217i	$0.316987\pi$
$450$	0	0
$451$	−2.32520e9	−1.19356
$452$	0	0
$453$	2.72669e8	0.137814
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−3.50830e9	−1.71945	−0.859727	−	0.510755i	$-0.829366\pi$
$457$	−3.50830e9	−1.71945	−0.859727	+	0.510755i	$0.829366\pi$
$458$	0	0
$459$	1.13751e9	0.549046
$460$	0	0
$461$	−1.29629e9	−0.616237	−0.308118	−	0.951348i	$-0.599699\pi$
$461$	−1.29629e9	−0.616237	−0.308118	+	0.951348i	$0.599699\pi$
$462$	0	0
$463$	3.89525e9	1.82390	0.911952	−	0.410298i	$-0.134575\pi$
$463$	3.89525e9	1.82390	0.911952	+	0.410298i	$0.134575\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−3.07448e9	−1.39689	−0.698446	−	0.715663i	$-0.746125\pi$
$467$	−3.07448e9	−1.39689	−0.698446	+	0.715663i	$0.746125\pi$
$468$	0	0
$469$	−1.40920e9	−0.630766
$470$	0	0
$471$	−6.62081e8	−0.291970
$472$	0	0
$473$	4.79028e9	2.08136
$474$	0	0
$475$	0	0
$476$	0	0
$477$	2.69460e9	1.13679
$478$	0	0
$479$	3.39731e8	0.141241	0.0706205	−	0.997503i	$-0.477502\pi$
$479$	3.39731e8	0.141241	0.0706205	+	0.997503i	$0.477502\pi$
$480$	0	0
$481$	−8.65292e8	−0.354532
$482$	0	0
$483$	6.44313e8	0.260185
$484$	0	0
$485$	0	0
$486$	0	0
$487$	4.11700e9	1.61521	0.807606	−	0.589723i	$-0.200763\pi$
$487$	4.11700e9	1.61521	0.807606	+	0.589723i	$0.200763\pi$
$488$	0	0
$489$	−2.55376e8	−0.0987642
$490$	0	0
$491$	3.89331e8	0.148434	0.0742170	−	0.997242i	$-0.476354\pi$
$491$	3.89331e8	0.148434	0.0742170	+	0.997242i	$0.476354\pi$
$492$	0	0
$493$	4.48179e9	1.68456
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.68849e9	0.616953
$498$	0	0
$499$	8.82005e7	0.0317775	0.0158887	−	0.999874i	$-0.494942\pi$
$499$	8.82005e7	0.0317775	0.0158887	+	0.999874i	$0.494942\pi$
$500$	0	0
$501$	1.00330e9	0.356449
$502$	0	0
$503$	1.01491e9	0.355583	0.177792	−	0.984068i	$-0.443105\pi$
$503$	1.01491e9	0.355583	0.177792	+	0.984068i	$0.443105\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−4.28872e8	−0.146151
$508$	0	0
$509$	−4.60125e9	−1.54655	−0.773275	−	0.634071i	$-0.781383\pi$
$509$	−4.60125e9	−1.54655	−0.773275	+	0.634071i	$0.781383\pi$
$510$	0	0
$511$	−5.01667e9	−1.66319
$512$	0	0
$513$	3.15978e9	1.03335
$514$	0	0
$515$	0	0
$516$	0	0
$517$	1.67847e8	0.0534190
$518$	0	0
$519$	−5.19744e8	−0.163194
$520$	0	0
$521$	−4.06999e9	−1.26084	−0.630422	−	0.776252i	$-0.717118\pi$
$521$	−4.06999e9	−1.26084	−0.630422	+	0.776252i	$0.717118\pi$
$522$	0	0
$523$	−2.22584e9	−0.680360	−0.340180	−	0.940360i	$-0.610488\pi$
$523$	−2.22584e9	−0.680360	−0.340180	+	0.940360i	$0.610488\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.35987e9	0.999967
$528$	0	0
$529$	−2.53303e9	−0.743954
$530$	0	0
$531$	1.07434e9	0.311394
$532$	0	0
$533$	1.63921e9	0.468909
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−9.01258e8	−0.251153
$538$	0	0
$539$	1.17680e10	3.23700
$540$	0	0
$541$	−4.94448e7	−0.0134255	−0.00671275	−	0.999977i	$-0.502137\pi$
$541$	−4.94448e7	−0.0134255	−0.00671275	+	0.999977i	$0.502137\pi$
$542$	0	0
$543$	−7.76063e7	−0.0208017
$544$	0	0
$545$	0	0
$546$	0	0
$547$	1.84946e9	0.483158	0.241579	−	0.970381i	$-0.422335\pi$
$547$	1.84946e9	0.483158	0.241579	+	0.970381i	$0.422335\pi$
$548$	0	0
$549$	−2.24054e9	−0.577895
$550$	0	0
$551$	1.24496e10	3.17047
$552$	0	0
$553$	−8.88856e9	−2.23508
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−5.03461e9	−1.23445	−0.617224	−	0.786788i	$-0.711743\pi$
$557$	−5.03461e9	−1.23445	−0.617224	+	0.786788i	$0.711743\pi$
$558$	0	0
$559$	−3.37702e9	−0.817697
$560$	0	0
$561$	−2.23219e9	−0.533778
$562$	0	0
$563$	4.54798e7	0.0107409	0.00537043	−	0.999986i	$-0.498291\pi$
$563$	4.54798e7	0.0107409	0.00537043	+	0.999986i	$0.498291\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	5.18657e9	1.19492
$568$	0	0
$569$	3.35165e9	0.762722	0.381361	−	0.924426i	$-0.375455\pi$
$569$	3.35165e9	0.762722	0.381361	+	0.924426i	$0.375455\pi$
$570$	0	0
$571$	−4.29937e9	−0.966448	−0.483224	−	0.875497i	$-0.660534\pi$
$571$	−4.29937e9	−0.966448	−0.483224	+	0.875497i	$0.660534\pi$
$572$	0	0
$573$	−1.97530e7	−0.00438625
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−3.41602e9	−0.740296	−0.370148	−	0.928973i	$-0.620693\pi$
$577$	−3.41602e9	−0.740296	−0.370148	+	0.928973i	$0.620693\pi$
$578$	0	0
$579$	−5.87011e8	−0.125681
$580$	0	0
$581$	7.73415e9	1.63605
$582$	0	0
$583$	−1.11382e10	−2.32796
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−4.45218e9	−0.908530	−0.454265	−	0.890867i	$-0.650098\pi$
$587$	−4.45218e9	−0.908530	−0.454265	+	0.890867i	$0.650098\pi$
$588$	0	0
$589$	9.33309e9	1.88201
$590$	0	0
$591$	−1.24350e9	−0.247793
$592$	0	0
$593$	−5.98158e9	−1.17794	−0.588972	−	0.808154i	$-0.700467\pi$
$593$	−5.98158e9	−1.17794	−0.588972	+	0.808154i	$0.700467\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1.28209e9	0.246609
$598$	0	0
$599$	9.84141e7	0.0187096	0.00935478	−	0.999956i	$-0.497022\pi$
$599$	9.84141e7	0.0187096	0.00935478	+	0.999956i	$0.497022\pi$
$600$	0	0
$601$	2.67831e9	0.503269	0.251635	−	0.967822i	$-0.419032\pi$
$601$	2.67831e9	0.503269	0.251635	+	0.967822i	$0.419032\pi$
$602$	0	0
$603$	1.85132e9	0.343851
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−3.14307e9	−0.570419	−0.285209	−	0.958465i	$-0.592063\pi$
$607$	−3.14307e9	−0.570419	−0.285209	+	0.958465i	$0.592063\pi$
$608$	0	0
$609$	−5.19196e9	−0.931473
$610$	0	0
$611$	−1.18327e8	−0.0209865
$612$	0	0
$613$	1.02572e10	1.79852	0.899260	−	0.437415i	$-0.144106\pi$
$613$	1.02572e10	1.79852	0.899260	+	0.437415i	$0.144106\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	9.20512e9	1.57772	0.788862	−	0.614570i	$-0.210670\pi$
$617$	9.20512e9	1.57772	0.788862	+	0.614570i	$0.210670\pi$
$618$	0	0
$619$	5.97483e9	1.01253	0.506266	−	0.862377i	$-0.331025\pi$
$619$	5.97483e9	1.01253	0.506266	+	0.862377i	$0.331025\pi$
$620$	0	0
$621$	−1.78299e9	−0.298764
$622$	0	0
$623$	1.41993e10	2.35266
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−6.20059e9	−1.00461
$628$	0	0
$629$	−2.82976e9	−0.453390
$630$	0	0
$631$	6.00181e8	0.0950997	0.0475498	−	0.998869i	$-0.484859\pi$
$631$	6.00181e8	0.0950997	0.0475498	+	0.998869i	$0.484859\pi$
$632$	0	0
$633$	1.97979e8	0.0310246
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−8.29614e9	−1.27171
$638$	0	0
$639$	−2.21823e9	−0.336321
$640$	0	0
$641$	6.33607e8	0.0950204	0.0475102	−	0.998871i	$-0.484871\pi$
$641$	6.33607e8	0.0950204	0.0475102	+	0.998871i	$0.484871\pi$
$642$	0	0
$643$	−1.03486e10	−1.53513	−0.767563	−	0.640973i	$-0.778531\pi$
$643$	−1.03486e10	−1.53513	−0.767563	+	0.640973i	$0.778531\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.73480e9	0.251817	0.125909	−	0.992042i	$-0.459815\pi$
$647$	1.73480e9	0.251817	0.125909	+	0.992042i	$0.459815\pi$
$648$	0	0
$649$	−4.44081e9	−0.637685
$650$	0	0
$651$	−3.89226e9	−0.552928
$652$	0	0
$653$	−3.78848e9	−0.532438	−0.266219	−	0.963913i	$-0.585774\pi$
$653$	−3.78848e9	−0.532438	−0.266219	+	0.963913i	$0.585774\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	6.59058e9	0.906661
$658$	0	0
$659$	−1.23763e10	−1.68457	−0.842287	−	0.539029i	$-0.818791\pi$
$659$	−1.23763e10	−1.68457	−0.842287	+	0.539029i	$0.818791\pi$
$660$	0	0
$661$	9.16365e8	0.123414	0.0617068	−	0.998094i	$-0.480346\pi$
$661$	9.16365e8	0.123414	0.0617068	+	0.998094i	$0.480346\pi$
$662$	0	0
$663$	1.57363e9	0.209704
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−7.02501e9	−0.916656
$668$	0	0
$669$	−3.31725e9	−0.428338
$670$	0	0
$671$	9.26133e9	1.18344
$672$	0	0
$673$	4.23791e8	0.0535919	0.0267959	−	0.999641i	$-0.491470\pi$
$673$	4.23791e8	0.0535919	0.0267959	+	0.999641i	$0.491470\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−7.40587e9	−0.917310	−0.458655	−	0.888614i	$-0.651669\pi$
$677$	−7.40587e9	−0.917310	−0.458655	+	0.888614i	$0.651669\pi$
$678$	0	0
$679$	−2.40043e10	−2.94270
$680$	0	0
$681$	2.21765e9	0.269077
$682$	0	0
$683$	3.56998e9	0.428740	0.214370	−	0.976753i	$-0.431230\pi$
$683$	3.56998e9	0.428740	0.214370	+	0.976753i	$0.431230\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−1.19653e9	−0.140791
$688$	0	0
$689$	7.85215e9	0.914580
$690$	0	0
$691$	−4.70529e9	−0.542517	−0.271259	−	0.962507i	$-0.587440\pi$
$691$	−4.70529e9	−0.542517	−0.271259	+	0.962507i	$0.587440\pi$
$692$	0	0
$693$	−2.42999e10	−2.77357
$694$	0	0
$695$	0	0
$696$	0	0
$697$	5.36069e9	0.599661
$698$	0	0
$699$	−5.02317e9	−0.556299
$700$	0	0
$701$	7.33356e7	0.00804084	0.00402042	−	0.999992i	$-0.498720\pi$
$701$	7.33356e7	0.00804084	0.00402042	+	0.999992i	$0.498720\pi$
$702$	0	0
$703$	−7.86054e9	−0.853314
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1.53356e10	−1.63205
$708$	0	0
$709$	1.43028e10	1.50716	0.753580	−	0.657356i	$-0.228325\pi$
$709$	1.43028e10	1.50716	0.753580	+	0.657356i	$0.228325\pi$
$710$	0	0
$711$	1.16772e10	1.21842
$712$	0	0
$713$	−5.26645e9	−0.544132
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−3.05453e8	−0.0309477
$718$	0	0
$719$	−8.95195e9	−0.898186	−0.449093	−	0.893485i	$-0.648253\pi$
$719$	−8.95195e9	−0.898186	−0.449093	+	0.893485i	$0.648253\pi$
$720$	0	0
$721$	1.60063e10	1.59044
$722$	0	0
$723$	−3.17899e9	−0.312827
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1.69358e10	1.63469	0.817345	−	0.576149i	$-0.195445\pi$
$727$	1.69358e10	1.63469	0.817345	+	0.576149i	$0.195445\pi$
$728$	0	0
$729$	−4.89839e9	−0.468281
$730$	0	0
$731$	−1.10439e10	−1.04571
$732$	0	0
$733$	−1.11259e10	−1.04345	−0.521723	−	0.853115i	$-0.674711\pi$
$733$	−1.11259e10	−1.04345	−0.521723	+	0.853115i	$0.674711\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−7.65248e9	−0.704151
$738$	0	0
$739$	−2.21789e9	−0.202155	−0.101078	−	0.994879i	$-0.532229\pi$
$739$	−2.21789e9	−0.202155	−0.101078	+	0.994879i	$0.532229\pi$
$740$	0	0
$741$	4.37125e9	0.394677
$742$	0	0
$743$	−1.05759e10	−0.945928	−0.472964	−	0.881082i	$-0.656816\pi$
$743$	−1.05759e10	−0.945928	−0.472964	+	0.881082i	$0.656816\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−1.01606e10	−0.891863
$748$	0	0
$749$	−1.54042e10	−1.33954
$750$	0	0
$751$	8.78857e9	0.757144	0.378572	−	0.925572i	$-0.376415\pi$
$751$	8.78857e9	0.757144	0.378572	+	0.925572i	$0.376415\pi$
$752$	0	0
$753$	1.21113e9	0.103373
$754$	0	0
$755$	0	0
$756$	0	0
$757$	1.48486e10	1.24408	0.622042	−	0.782984i	$-0.286303\pi$
$757$	1.48486e10	1.24408	0.622042	+	0.782984i	$0.286303\pi$
$758$	0	0
$759$	3.49885e9	0.290455
$760$	0	0
$761$	7.44711e9	0.612550	0.306275	−	0.951943i	$-0.400917\pi$
$761$	7.44711e9	0.612550	0.306275	+	0.951943i	$0.400917\pi$
$762$	0	0
$763$	−3.60830e9	−0.294081
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3.13066e9	0.250525
$768$	0	0
$769$	6.62137e9	0.525056	0.262528	−	0.964924i	$-0.415444\pi$
$769$	6.62137e9	0.525056	0.262528	+	0.964924i	$0.415444\pi$
$770$	0	0
$771$	3.28109e9	0.257827
$772$	0	0
$773$	−9.09632e9	−0.708333	−0.354166	−	0.935182i	$-0.615235\pi$
$773$	−9.09632e9	−0.708333	−0.354166	+	0.935182i	$0.615235\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	3.27815e9	0.250700
$778$	0	0
$779$	1.48910e10	1.12861
$780$	0	0
$781$	9.16913e9	0.688731
$782$	0	0
$783$	1.43676e10	1.06959
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−1.90436e10	−1.39264	−0.696319	−	0.717732i	$-0.745180\pi$
$787$	−1.90436e10	−1.39264	−0.696319	+	0.717732i	$0.745180\pi$
$788$	0	0
$789$	−2.80786e9	−0.203520
$790$	0	0
$791$	−1.21306e10	−0.871496
$792$	0	0
$793$	−6.52899e9	−0.464933
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.49063e10	1.04295	0.521477	−	0.853265i	$-0.325381\pi$
$797$	1.49063e10	1.04295	0.521477	+	0.853265i	$0.325381\pi$
$798$	0	0
$799$	−3.86965e8	−0.0268385
$800$	0	0
$801$	−1.86542e10	−1.28251
$802$	0	0
$803$	−2.72424e10	−1.85669
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−2.27112e9	−0.152119
$808$	0	0
$809$	2.51640e10	1.67094	0.835469	−	0.549538i	$-0.185196\pi$
$809$	2.51640e10	1.67094	0.835469	+	0.549538i	$0.185196\pi$
$810$	0	0
$811$	−3.37190e9	−0.221974	−0.110987	−	0.993822i	$-0.535401\pi$
$811$	−3.37190e9	−0.221974	−0.110987	+	0.993822i	$0.535401\pi$
$812$	0	0
$813$	5.02872e9	0.328201
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−3.06777e10	−1.96810
$818$	0	0
$819$	1.71308e10	1.08964
$820$	0	0
$821$	−3.02251e10	−1.90619	−0.953096	−	0.302669i	$-0.902122\pi$
$821$	−3.02251e10	−1.90619	−0.953096	+	0.302669i	$0.902122\pi$
$822$	0	0
$823$	1.81107e10	1.13249	0.566246	−	0.824236i	$-0.308395\pi$
$823$	1.81107e10	1.13249	0.566246	+	0.824236i	$0.308395\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−8.31258e9	−0.511054	−0.255527	−	0.966802i	$-0.582249\pi$
$827$	−8.31258e9	−0.511054	−0.255527	+	0.966802i	$0.582249\pi$
$828$	0	0
$829$	3.19126e9	0.194545	0.0972726	−	0.995258i	$-0.468988\pi$
$829$	3.19126e9	0.194545	0.0972726	+	0.995258i	$0.468988\pi$
$830$	0	0
$831$	2.85291e9	0.172458
$832$	0	0
$833$	−2.71308e10	−1.62632
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.07709e10	0.634914
$838$	0	0
$839$	−2.11602e10	−1.23695	−0.618477	−	0.785803i	$-0.712250\pi$
$839$	−2.11602e10	−1.23695	−0.618477	+	0.785803i	$0.712250\pi$
$840$	0	0
$841$	3.93586e10	2.28167
$842$	0	0
$843$	−7.41502e9	−0.426300
$844$	0	0
$845$	0	0
$846$	0	0
$847$	7.11239e10	4.02183
$848$	0	0
$849$	−9.75630e9	−0.547152
$850$	0	0
$851$	4.43552e9	0.246713
$852$	0	0
$853$	2.57804e10	1.42222	0.711112	−	0.703079i	$-0.248192\pi$
$853$	2.57804e10	1.42222	0.711112	+	0.703079i	$0.248192\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−2.67728e10	−1.45298	−0.726491	−	0.687176i	$-0.758850\pi$
$857$	−2.67728e10	−1.45298	−0.726491	+	0.687176i	$0.758850\pi$
$858$	0	0
$859$	−1.88350e10	−1.01389	−0.506944	−	0.861979i	$-0.669225\pi$
$859$	−1.88350e10	−1.01389	−0.506944	+	0.861979i	$0.669225\pi$
$860$	0	0
$861$	−6.21012e9	−0.331580
$862$	0	0
$863$	−4.36753e9	−0.231312	−0.115656	−	0.993289i	$-0.536897\pi$
$863$	−4.36753e9	−0.231312	−0.115656	+	0.993289i	$0.536897\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−8.05037e8	−0.0419517
$868$	0	0
$869$	−4.82681e10	−2.49512
$870$	0	0
$871$	5.39480e9	0.276638
$872$	0	0
$873$	3.15353e10	1.60416
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−5.63412e9	−0.282051	−0.141025	−	0.990006i	$-0.545040\pi$
$877$	−5.63412e9	−0.282051	−0.141025	+	0.990006i	$0.545040\pi$
$878$	0	0
$879$	−6.70326e9	−0.332909
$880$	0	0
$881$	−2.65172e10	−1.30651	−0.653254	−	0.757139i	$-0.726597\pi$
$881$	−2.65172e10	−1.30651	−0.653254	+	0.757139i	$0.726597\pi$
$882$	0	0
$883$	−1.44647e10	−0.707044	−0.353522	−	0.935426i	$-0.615016\pi$
$883$	−1.44647e10	−0.707044	−0.353522	+	0.935426i	$0.615016\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−2.22645e10	−1.07123	−0.535613	−	0.844464i	$-0.679919\pi$
$887$	−2.22645e10	−1.07123	−0.535613	+	0.844464i	$0.679919\pi$
$888$	0	0
$889$	−6.22307e10	−2.97063
$890$	0	0
$891$	2.81650e10	1.33394
$892$	0	0
$893$	−1.07492e9	−0.0505120
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−2.46660e9	−0.114110
$898$	0	0
$899$	4.24377e10	1.94802
$900$	0	0
$901$	2.56789e10	1.16960
$902$	0	0
$903$	1.27938e10	0.578220
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−1.72710e10	−0.768583	−0.384292	−	0.923212i	$-0.625554\pi$
$907$	−1.72710e10	−0.768583	−0.384292	+	0.923212i	$0.625554\pi$
$908$	0	0
$909$	2.01469e10	0.889682
$910$	0	0
$911$	4.28756e10	1.87887	0.939433	−	0.342733i	$-0.111353\pi$
$911$	4.28756e10	1.87887	0.939433	+	0.342733i	$0.111353\pi$
$912$	0	0
$913$	4.19992e10	1.82639
$914$	0	0
$915$	0	0
$916$	0	0
$917$	9.62877e9	0.412362
$918$	0	0
$919$	4.37453e9	0.185921	0.0929603	−	0.995670i	$-0.470367\pi$
$919$	4.37453e9	0.185921	0.0929603	+	0.995670i	$0.470367\pi$
$920$	0	0
$921$	−9.06237e9	−0.382237
$922$	0	0
$923$	−6.46400e9	−0.270580
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−2.10280e10	−0.867000
$928$	0	0
$929$	1.65493e10	0.677213	0.338606	−	0.940928i	$-0.390044\pi$
$929$	1.65493e10	0.677213	0.338606	+	0.940928i	$0.390044\pi$
$930$	0	0
$931$	−7.53643e10	−3.06085
$932$	0	0
$933$	−7.66823e9	−0.309107
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−6.78612e9	−0.269484	−0.134742	−	0.990881i	$-0.543021\pi$
$937$	−6.78612e9	−0.269484	−0.134742	+	0.990881i	$0.543021\pi$
$938$	0	0
$939$	3.02129e9	0.119086
$940$	0	0
$941$	−3.05358e10	−1.19466	−0.597332	−	0.801994i	$-0.703773\pi$
$941$	−3.05358e10	−1.19466	−0.597332	+	0.801994i	$0.703773\pi$
$942$	0	0
$943$	−8.40264e9	−0.326306
$944$	0	0
$945$	0	0
$946$	0	0
$947$	2.02600e9	0.0775203	0.0387601	−	0.999249i	$-0.487659\pi$
$947$	2.02600e9	0.0775203	0.0387601	+	0.999249i	$0.487659\pi$
$948$	0	0
$949$	1.92051e10	0.729434
$950$	0	0
$951$	1.43311e8	0.00540315
$952$	0	0
$953$	6.05056e9	0.226449	0.113224	−	0.993569i	$-0.463882\pi$
$953$	6.05056e9	0.226449	0.113224	+	0.993569i	$0.463882\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−2.81942e10	−1.03984
$958$	0	0
$959$	4.01834e10	1.47123
$960$	0	0
$961$	4.30174e9	0.156355
$962$	0	0
$963$	2.02371e10	0.730224
$964$	0	0
$965$	0	0
$966$	0	0
$967$	4.99403e10	1.77606	0.888032	−	0.459782i	$-0.152073\pi$
$967$	4.99403e10	1.77606	0.888032	+	0.459782i	$0.152073\pi$
$968$	0	0
$969$	1.42953e10	0.504731
$970$	0	0
$971$	−1.51420e10	−0.530780	−0.265390	−	0.964141i	$-0.585501\pi$
$971$	−1.51420e10	−0.530780	−0.265390	+	0.964141i	$0.585501\pi$
$972$	0	0
$973$	2.12346e10	0.739007
$974$	0	0
$975$	0	0
$976$	0	0
$977$	5.39541e10	1.85095	0.925473	−	0.378815i	$-0.123668\pi$
$977$	5.39541e10	1.85095	0.925473	+	0.378815i	$0.123668\pi$
$978$	0	0
$979$	7.71076e10	2.62638
$980$	0	0
$981$	4.74035e9	0.160313
$982$	0	0
$983$	2.37883e10	0.798777	0.399389	−	0.916782i	$-0.369222\pi$
$983$	2.37883e10	0.798777	0.399389	+	0.916782i	$0.369222\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	4.48282e8	0.0148402
$988$	0	0
$989$	1.73108e10	0.569022
$990$	0	0
$991$	2.15204e10	0.702412	0.351206	−	0.936298i	$-0.385772\pi$
$991$	2.15204e10	0.702412	0.351206	+	0.936298i	$0.385772\pi$
$992$	0	0
$993$	7.93841e9	0.257283
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−2.83526e10	−0.906067	−0.453034	−	0.891493i	$-0.649658\pi$
$997$	−2.83526e10	−0.906067	−0.453034	+	0.891493i	$0.649658\pi$
$998$	0	0
$999$	−9.07153e9	−0.287873

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.8.a.bh.1.3		3
4.3	odd	2		200.8.a.p.1.1	yes	3
5.2	odd	4		400.8.c.u.49.3		6
5.3	odd	4		400.8.c.u.49.4		6
5.4	even	2		400.8.a.bi.1.1		3
20.3	even	4		200.8.c.j.49.3		6
20.7	even	4		200.8.c.j.49.4		6
20.19	odd	2		200.8.a.o.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
200.8.a.o.1.3	✓	3	20.19	odd	2
200.8.a.p.1.1	yes	3	4.3	odd	2
200.8.c.j.49.3		6	20.3	even	4
200.8.c.j.49.4		6	20.7	even	4
400.8.a.bh.1.3		3	1.1	even	1	trivial
400.8.a.bi.1.1		3	5.4	even	2
400.8.c.u.49.3		6	5.2	odd	4
400.8.c.u.49.4		6	5.3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	400.a (trivial)

\(f(q)\)	\(=\)	\(q+14.5033 q^{3} +1504.61 q^{7} -1976.65 q^{9} +8170.56 q^{11} -5760.03 q^{13} -18837.0 q^{17} -52325.6 q^{19} +21821.8 q^{21} +29526.1 q^{23} -60386.8 q^{27} -237925. q^{29} -178366. q^{31} +118500. q^{33} +150224. q^{37} -83539.5 q^{39} -284583. q^{41} +586286. q^{43} +20542.9 q^{47} +1.44030e6 q^{49} -273199. q^{51} -1.36322e6 q^{53} -758895. q^{57} -543514. q^{59} +1.13350e6 q^{61} -2.97408e6 q^{63} -936593. q^{67} +428227. q^{69} +1.12222e6 q^{71} -3.33421e6 q^{73} +1.22935e7 q^{77} -5.90757e6 q^{79} +3.44713e6 q^{81} +5.14032e6 q^{83} -3.45071e6 q^{87} +9.43725e6 q^{89} -8.66657e6 q^{91} -2.58690e6 q^{93} -1.59539e7 q^{97} -1.61504e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 97 q^{3} + 1630 q^{7} + 2146 q^{9} + 2659 q^{11} + 3308 q^{13} - 24481 q^{17} - 55749 q^{19} - 61562 q^{21} - 56910 q^{23} - 300619 q^{27} + 65528 q^{29} + 144598 q^{31} + 657207 q^{33} + 204678 q^{37}+ \cdots - 53316510 q^{99}+O(q^{100}) \) 3 * q - 97 * q^3 + 1630 * q^7 + 2146 * q^9 + 2659 * q^11 + 3308 * q^13 - 24481 * q^17 - 55749 * q^19 - 61562 * q^21 - 56910 * q^23 - 300619 * q^27 + 65528 * q^29 + 144598 * q^31 + 657207 * q^33 + 204678 * q^37 - 800004 * q^39 + 299743 * q^41 + 108620 * q^43 - 670804 * q^47 + 2360487 * q^49 + 1862315 * q^51 - 1288886 * q^53 + 6703 * q^57 - 983368 * q^59 + 5710118 * q^61 + 5802452 * q^63 - 5436267 * q^67 + 5390698 * q^69 - 1933956 * q^71 - 9921215 * q^73 + 4194286 * q^77 - 5590678 * q^79 + 21712267 * q^81 + 11873921 * q^83 - 18848136 * q^87 + 22333709 * q^89 - 1031944 * q^91 - 19549890 * q^93 - 2002634 * q^97 - 53316510 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more