LMFDB - Embedded newform 400.8.c.u.49.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [400,8,Mod(49,400)] 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, 1])) N = Newforms(chi, 8, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("400.49"); S:= CuspForms(chi, 8); N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	400.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,0,0,0,0,-4292] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$124.954010194$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 69x^{4} + 1164x^{2} + 16 $ x^6 + 69x^4 + 1164x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{16}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 200)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.3
Root			$-0.117290i$ of defining polynomial
Character	$\chi$	$=$	400.49
Dual form			400.8.c.u.49.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-14.5033i q^{3} +1504.61i q^{7} +1976.65 q^{9} +8170.56 q^{11} +5760.03i q^{13} -18837.0i q^{17} +52325.6 q^{19} +21821.8 q^{21} -29526.1i q^{23} -60386.8i q^{27} +237925. q^{29} -178366. q^{31} -118500. i q^{33} +150224. i q^{37} +83539.5 q^{39} -284583. q^{41} -586286. i q^{43} +20542.9i q^{47} -1.44030e6 q^{49} -273199. q^{51} +1.36322e6i q^{53} -758895. i q^{57} +543514. q^{59} +1.13350e6 q^{61} +2.97408e6i q^{63} -936593. i q^{67} -428227. q^{69} +1.12222e6 q^{71} +3.33421e6i q^{73} +1.22935e7i q^{77} +5.90757e6 q^{79} +3.44713e6 q^{81} -5.14032e6i q^{83} -3.45071e6i q^{87} -9.43725e6 q^{89} -8.66657e6 q^{91} +2.58690e6i q^{93} -1.59539e7i q^{97} +1.61504e7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4292 q^{9} + 5318 q^{11} + 111498 q^{19} - 123124 q^{21} - 131056 q^{29} + 289196 q^{31} + 1600008 q^{39} + 599486 q^{41} - 4720974 q^{49} + 3724630 q^{51} + 1966736 q^{59} + 11420236 q^{61} - 10781396 q^{69}+ \cdots + 106633020 q^{99}+O(q^{100}) $ 6 * q - 4292 * q^9 + 5318 * q^11 + 111498 * q^19 - 123124 * q^21 - 131056 * q^29 + 289196 * q^31 + 1600008 * q^39 + 599486 * q^41 - 4720974 * q^49 + 3724630 * q^51 + 1966736 * q^59 + 11420236 * q^61 - 10781396 * q^69 - 3867912 * q^71 + 11181356 * q^79 + 43424534 * q^81 - 44667418 * q^89 - 2063888 * q^91 + 106633020 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$.

$n$	$101$	$177$	$351$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	− 14.5033i	− 0.310130i	−0.987904	−	0.155065i	$-0.950441\pi$
$3$	− 14.5033i	− 0.310130i	0.987904	−	0.155065i	$-0.0495586\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1504.61i	1.65798i	0.559262	+	0.828991i	$0.311084\pi$
$7$	1504.61i	1.65798i	−0.559262	+	0.828991i	$0.688916\pi$
$8$	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
$13$	5760.03i	0.727148i	0.931565	+	0.363574i	$0.118444\pi$
$13$	5760.03i	0.727148i	−0.931565	+	0.363574i	$0.881556\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 18837.0i	− 0.929908i	−0.885335	−	0.464954i	$-0.846071\pi$
$17$	− 18837.0i	− 0.929908i	0.885335	−	0.464954i	$-0.153929\pi$
$18$	0	0
$19$	52325.6	1.75015	0.875077	−	0.483983i	$-0.160810\pi$
$19$	52325.6	1.75015	0.875077	+	0.483983i	$0.160810\pi$
$20$	0	0
$21$	21821.8	0.514189
$22$	0	0
$23$	− 29526.1i	− 0.506010i	−0.967465	−	0.253005i	$-0.918581\pi$
$23$	− 29526.1i	− 0.506010i	0.967465	−	0.253005i	$-0.0814189\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 60386.8i	− 0.590431i
$28$	0	0
$29$	237925.	1.81154	0.905769	−	0.423771i	$-0.139294\pi$
$29$	237925.	1.81154	0.905769	+	0.423771i	$0.139294\pi$
$30$	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.i	− 0.574011i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	150224.i	0.487565i	0.969830	+	0.243782i	$0.0783883\pi$
$37$	150224.i	0.487565i	−0.969830	+	0.243782i	$0.921612\pi$
$38$	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.i	− 1.12453i	−0.826958	−	0.562263i	$-0.809931\pi$
$43$	− 586286.i	− 1.12453i	0.826958	−	0.562263i	$-0.190069\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	20542.9i	0.0288615i	0.999896	+	0.0144307i	$0.00459360\pi$
$47$	20542.9i	0.0288615i	−0.999896	+	0.0144307i	$0.995406\pi$
$48$	0	0
$49$	−1.44030e6	−1.74890
$50$	0	0
$51$	−273199.	−0.288392
$52$	0	0
$53$	1.36322e6i	1.25776i	0.777501	+	0.628882i	$0.216487\pi$
$53$	1.36322e6i	1.25776i	−0.777501	+	0.628882i	$0.783513\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 758895.i	− 0.542775i
$58$	0	0
$59$	543514.	0.344531	0.172266	−	0.985051i	$-0.444891\pi$
$59$	543514.	0.344531	0.172266	+	0.985051i	$0.444891\pi$
$60$	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
$63$	2.97408e6i	1.49852i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 936593.i	− 0.380442i	−0.981741	−	0.190221i	$-0.939079\pi$
$67$	− 936593.i	− 0.380442i	0.981741	−	0.190221i	$-0.0609205\pi$
$68$	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
$73$	3.33421e6i	1.00314i	0.865116	+	0.501572i	$0.167245\pi$
$73$	3.33421e6i	1.00314i	−0.865116	+	0.501572i	$0.832755\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.22935e7i	3.06872i
$78$	0	0
$79$	5.90757e6	1.34807	0.674037	−	0.738697i	$-0.264559\pi$
$79$	5.90757e6	1.34807	0.674037	+	0.738697i	$0.264559\pi$
$80$	0	0
$81$	3.44713e6	0.720710
$82$	0	0
$83$	− 5.14032e6i	− 0.986771i	−0.869811	−	0.493386i	$-0.835759\pi$
$83$	− 5.14032e6i	− 0.986771i	0.869811	−	0.493386i	$-0.164241\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 3.45071e6i	− 0.561812i
$88$	0	0
$89$	−9.43725e6	−1.41899	−0.709497	−	0.704709i	$-0.751078\pi$
$89$	−9.43725e6	−1.41899	−0.709497	+	0.704709i	$0.751078\pi$
$90$	0	0
$91$	−8.66657e6	−1.20560
$92$	0	0
$93$	2.58690e6i	0.333495i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 1.59539e7i	− 1.77487i	−0.460935	−	0.887434i	$-0.652486\pi$
$97$	− 1.59539e7i	− 1.77487i	0.460935	−	0.887434i	$-0.347514\pi$
$98$	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
$103$	− 1.06382e7i	− 0.959263i	−0.877470	−	0.479631i	$-0.840770\pi$
$103$	− 1.06382e7i	− 0.959263i	0.877470	−	0.479631i	$-0.159230\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1.02381e7i	− 0.807931i	−0.914774	−	0.403966i	$-0.867631\pi$
$107$	− 1.02381e7i	− 0.807931i	0.914774	−	0.403966i	$-0.132369\pi$
$108$	0	0
$109$	2.39817e6	0.177373	0.0886865	−	0.996060i	$-0.471733\pi$
$109$	2.39817e6	0.177373	0.0886865	+	0.996060i	$0.471733\pi$
$110$	0	0
$111$	2.17874e6	0.151208
$112$	0	0
$113$	8.06232e6i	0.525637i	0.964845	+	0.262818i	$0.0846520\pi$
$113$	8.06232e6i	0.525637i	−0.964845	+	0.262818i	$0.915348\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1.13856e7i	0.657210i
$118$	0	0
$119$	2.83422e7	1.54177
$120$	0	0
$121$	4.72708e7	2.42574
$122$	0	0
$123$	4.12740e6i	0.199990i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 4.13601e7i	− 1.79171i	−0.444343	−	0.895857i	$-0.646563\pi$
$127$	− 4.13601e7i	− 1.79171i	0.444343	−	0.895857i	$-0.353437\pi$
$128$	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
$133$	7.87294e7i	2.90172i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.67069e7i	0.887364i	0.896184	+	0.443682i	$0.146328\pi$
$137$	2.67069e7i	0.887364i	−0.896184	+	0.443682i	$0.853672\pi$
$138$	0	0
$139$	−1.41131e7	−0.445727	−0.222864	−	0.974850i	$-0.571540\pi$
$139$	−1.41131e7	−0.445727	−0.222864	+	0.974850i	$0.571540\pi$
$140$	0	0
$141$	297940.	0.00895079
$142$	0	0
$143$	4.70626e7i	1.34586i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	2.08891e7i	0.542386i
$148$	0	0
$149$	−2.24351e7	−0.555619	−0.277809	−	0.960636i	$-0.589608\pi$
$149$	−2.24351e7	−0.555619	−0.277809	+	0.960636i	$0.589608\pi$
$150$	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
$153$	− 3.72342e7i	− 0.840469i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 4.56503e7i	− 0.941444i	−0.882281	−	0.470722i	$-0.843993\pi$
$157$	− 4.56503e7i	− 0.941444i	0.882281	−	0.470722i	$-0.156007\pi$
$158$	0	0
$159$	1.97712e7	0.390070
$160$	0	0
$161$	4.44252e7	0.838955
$162$	0	0
$163$	1.76081e7i	0.318461i	0.987241	+	0.159231i	$0.0509013\pi$
$163$	1.76081e7i	0.318461i	−0.987241	+	0.159231i	$0.949099\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	6.91771e7i	1.14936i	0.818380	+	0.574678i	$0.194873\pi$
$167$	6.91771e7i	1.14936i	−0.818380	+	0.574678i	$0.805127\pi$
$168$	0	0
$169$	2.95706e7	0.471256
$170$	0	0
$171$	1.03430e8	1.58182
$172$	0	0
$173$	3.58362e7i	0.526212i	0.964767	+	0.263106i	$0.0847468\pi$
$173$	3.58362e7i	0.526212i	−0.964767	+	0.263106i	$0.915253\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 7.88276e6i	− 0.106849i
$178$	0	0
$179$	6.21415e7	0.809834	0.404917	−	0.914353i	$-0.367300\pi$
$179$	6.21415e7	0.809834	0.404917	+	0.914353i	$0.367300\pi$
$180$	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
$183$	− 1.64395e7i	− 0.198294i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 1.53909e8i	− 1.72114i
$188$	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
$193$	4.04742e7i	0.405255i	0.979256	+	0.202627i	$0.0649480\pi$
$193$	4.04742e7i	0.405255i	−0.979256	+	0.202627i	$0.935052\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 8.57389e7i	− 0.798999i	−0.916733	−	0.399499i	$-0.869184\pi$
$197$	− 8.57389e7i	− 0.798999i	0.916733	−	0.399499i	$-0.130816\pi$
$198$	0	0
$199$	−8.83997e7	−0.795179	−0.397589	−	0.917563i	$-0.630153\pi$
$199$	−8.83997e7	−0.795179	−0.397589	+	0.917563i	$0.630153\pi$
$200$	0	0
$201$	−1.35837e7	−0.117986
$202$	0	0
$203$	3.57984e8i	3.00350i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 5.83629e7i	− 0.457342i
$208$	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
$213$	− 1.62759e7i	− 0.115403i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 2.68370e8i	− 1.78289i
$218$	0	0
$219$	4.83572e7	0.311105
$220$	0	0
$221$	1.08501e8	0.676180
$222$	0	0
$223$	2.28723e8i	1.38116i	0.723257	+	0.690579i	$0.242644\pi$
$223$	2.28723e8i	1.38116i	−0.723257	+	0.690579i	$0.757356\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.52906e8i	0.867629i	0.901002	+	0.433814i	$0.142833\pi$
$227$	1.52906e8i	0.867629i	−0.901002	+	0.433814i	$0.857167\pi$
$228$	0	0
$229$	8.25002e7	0.453974	0.226987	−	0.973898i	$-0.427113\pi$
$229$	8.25002e7	0.453974	0.226987	+	0.973898i	$0.427113\pi$
$230$	0	0
$231$	1.78296e8	0.951700
$232$	0	0
$233$	3.46346e8i	1.79376i	0.442272	+	0.896881i	$0.354172\pi$
$233$	3.46346e8i	1.79376i	−0.442272	+	0.896881i	$0.645828\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 8.56794e7i	− 0.418078i
$238$	0	0
$239$	2.10609e7	0.0997894	0.0498947	−	0.998754i	$-0.484111\pi$
$239$	2.10609e7	0.0997894	0.0498947	+	0.998754i	$0.484111\pi$
$240$	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
$243$	− 1.82061e8i	− 0.813944i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	3.01397e8i	1.27262i
$248$	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
$253$	− 2.41245e8i	− 0.936561i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2.26230e8i	0.831351i	0.909513	+	0.415676i	$0.136455\pi$
$257$	2.26230e8i	0.831351i	−0.909513	+	0.415676i	$0.863545\pi$
$258$	0	0
$259$	−2.26027e8	−0.808373
$260$	0	0
$261$	4.70296e8	1.63730
$262$	0	0
$263$	1.93601e8i	0.656241i	0.944636	+	0.328121i	$0.106415\pi$
$263$	1.93601e8i	0.656241i	−0.944636	+	0.328121i	$0.893585\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	1.36872e8i	0.440072i
$268$	0	0
$269$	1.56593e8	0.490501	0.245250	−	0.969460i	$-0.421130\pi$
$269$	1.56593e8	0.490501	0.245250	+	0.969460i	$0.421130\pi$
$270$	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
$273$	1.25694e8i	0.373891i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1.96707e8i	0.556084i	0.960569	+	0.278042i	$0.0896855\pi$
$277$	1.96707e8i	0.556084i	−0.960569	+	0.278042i	$0.910314\pi$
$278$	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
$283$	6.72694e8i	1.76427i	0.470998	+	0.882134i	$0.343894\pi$
$283$	6.72694e8i	1.76427i	−0.470998	+	0.882134i	$0.656106\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 4.28186e8i	− 1.06917i
$288$	0	0
$289$	5.55071e7	0.135271
$290$	0	0
$291$	−2.31385e8	−0.550439
$292$	0	0
$293$	4.62188e8i	1.07345i	0.843757	+	0.536725i	$0.180339\pi$
$293$	4.62188e8i	1.07345i	−0.843757	+	0.536725i	$0.819661\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 4.93394e8i	− 1.09281i
$298$	0	0
$299$	1.70071e8	0.367944
$300$	0	0
$301$	8.82129e8	1.86444
$302$	0	0
$303$	1.47824e8i	0.305279i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 6.24848e8i	− 1.23251i	−0.787547	−	0.616254i	$-0.788649\pi$
$307$	− 6.24848e8i	− 1.23251i	0.787547	−	0.616254i	$-0.211351\pi$
$308$	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
$313$	− 2.08317e8i	− 0.383989i	−0.981396	−	0.191995i	$-0.938504\pi$
$313$	− 2.08317e8i	− 0.383989i	0.981396	−	0.191995i	$-0.0614956\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	9.88124e6i	0.0174222i	0.999962	+	0.00871112i	$0.00277287\pi$
$317$	9.88124e6i	0.0174222i	−0.999962	+	0.00871112i	$0.997227\pi$
$318$	0	0
$319$	1.94398e9	3.35293
$320$	0	0
$321$	−1.48486e8	−0.250563
$322$	0	0
$323$	− 9.85656e8i	− 1.62748i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 3.47815e7i	− 0.0550086i
$328$	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
$333$	2.96940e8i	0.440671i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	3.95141e8i	0.562403i	0.959649	+	0.281201i	$0.0907329\pi$
$337$	3.95141e8i	0.562403i	−0.959649	+	0.281201i	$0.909267\pi$
$338$	0	0
$339$	1.16930e8	0.163015
$340$	0	0
$341$	−1.45735e9	−1.99032
$342$	0	0
$343$	− 9.27970e8i	− 1.24167i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 4.53902e7i	− 0.0583189i	−0.999575	−	0.0291595i	$-0.990717\pi$
$347$	− 4.53902e7i	− 0.0583189i	0.999575	−	0.0291595i	$-0.00928306\pi$
$348$	0	0
$349$	−1.07430e9	−1.35281	−0.676404	−	0.736531i	$-0.736463\pi$
$349$	−1.07430e9	−1.35281	−0.676404	+	0.736531i	$0.736463\pi$
$350$	0	0
$351$	3.47830e8	0.429330
$352$	0	0
$353$	− 5.76841e8i	− 0.697983i	−0.937126	−	0.348991i	$-0.886524\pi$
$353$	− 5.76841e8i	− 0.697983i	0.937126	−	0.348991i	$-0.113476\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 4.11057e8i	− 0.478148i
$358$	0	0
$359$	−7.63547e8	−0.870974	−0.435487	−	0.900195i	$-0.643424\pi$
$359$	−7.63547e8	−0.870974	−0.435487	+	0.900195i	$0.643424\pi$
$360$	0	0
$361$	1.84409e9	2.06304
$362$	0	0
$363$	− 6.85584e8i	− 0.752294i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	1.55202e8i	0.163895i	0.996637	+	0.0819475i	$0.0261140\pi$
$367$	1.55202e8i	0.163895i	−0.996637	+	0.0819475i	$0.973886\pi$
$368$	0	0
$369$	−5.62522e8	−0.582838
$370$	0	0
$371$	−2.05110e9	−2.08535
$372$	0	0
$373$	− 3.02866e8i	− 0.302183i	−0.988520	−	0.151091i	$-0.951721\pi$
$373$	− 3.02866e8i	− 0.302183i	0.988520	−	0.151091i	$-0.0482788\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.37046e9i	1.31726i
$378$	0	0
$379$	−1.12044e9	−1.05718	−0.528591	−	0.848877i	$-0.677279\pi$
$379$	−1.12044e9	−1.05718	−0.528591	+	0.848877i	$0.677279\pi$
$380$	0	0
$381$	−5.99859e8	−0.555663
$382$	0	0
$383$	− 7.36644e8i	− 0.669980i	−0.942222	−	0.334990i	$-0.891267\pi$
$383$	− 7.36644e8i	− 0.669980i	0.942222	−	0.334990i	$-0.108733\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 1.15888e9i	− 1.01637i
$388$	0	0
$389$	−1.81700e9	−1.56506	−0.782530	−	0.622612i	$-0.786071\pi$
$389$	−1.81700e9	−1.56506	−0.782530	+	0.622612i	$0.786071\pi$
$390$	0	0
$391$	−5.56183e8	−0.470543
$392$	0	0
$393$	− 9.28145e7i	− 0.0771333i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	2.38033e9i	1.90928i	0.297757	+	0.954642i	$0.403762\pi$
$397$	2.38033e9i	1.90928i	−0.297757	+	0.954642i	$0.596238\pi$
$398$	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
$403$	− 1.02739e9i	− 0.781931i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.22741e9i	0.902422i
$408$	0	0
$409$	−3.92629e8	−0.283760	−0.141880	−	0.989884i	$-0.545315\pi$
$409$	−3.92629e8	−0.283760	−0.141880	+	0.989884i	$0.545315\pi$
$410$	0	0
$411$	3.87339e8	0.275198
$412$	0	0
$413$	8.17775e8i	0.571227i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	2.04686e8i	0.138233i
$418$	0	0
$419$	1.08234e9	0.718808	0.359404	−	0.933182i	$-0.382980\pi$
$419$	1.08234e9	0.718808	0.359404	+	0.933182i	$0.382980\pi$
$420$	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
$423$	4.06061e7i	0.0260856i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1.70547e9i	1.06010i
$428$	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
$433$	− 4.78186e8i	− 0.283067i	−0.989933	−	0.141533i	$-0.954797\pi$
$433$	− 4.78186e8i	− 0.283067i	0.989933	−	0.141533i	$-0.0452033\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 1.54497e9i	− 0.885595i
$438$	0	0
$439$	−3.30634e9	−1.86519	−0.932593	−	0.360929i	$-0.882460\pi$
$439$	−3.30634e9	−1.86519	−0.932593	+	0.360929i	$0.882460\pi$
$440$	0	0
$441$	−2.84697e9	−1.58069
$442$	0	0
$443$	2.15148e9i	1.17578i	0.808942	+	0.587888i	$0.200040\pi$
$443$	2.15148e9i	1.17578i	−0.808942	+	0.587888i	$0.799960\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	3.25384e8i	0.172314i
$448$	0	0
$449$	−2.08607e9	−1.08759	−0.543796	−	0.839217i	$-0.683013\pi$
$449$	−2.08607e9	−1.08759	−0.543796	+	0.839217i	$0.683013\pi$
$450$	0	0
$451$	−2.32520e9	−1.19356
$452$	0	0
$453$	− 2.72669e8i	− 0.137814i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 3.50830e9i	− 1.71945i	−0.510755	−	0.859727i	$-0.670634\pi$
$457$	− 3.50830e9i	− 1.71945i	0.510755	−	0.859727i	$-0.329366\pi$
$458$	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
$463$	− 3.89525e9i	− 1.82390i	−0.410298	−	0.911952i	$-0.634575\pi$
$463$	− 3.89525e9i	− 1.82390i	0.410298	−	0.911952i	$-0.365425\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 3.07448e9i	− 1.39689i	−0.715663	−	0.698446i	$-0.753875\pi$
$467$	− 3.07448e9i	− 1.39689i	0.715663	−	0.698446i	$-0.246125\pi$
$468$	0	0
$469$	1.40920e9	0.630766
$470$	0	0
$471$	−6.62081e8	−0.291970
$472$	0	0
$473$	− 4.79028e9i	− 2.08136i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	2.69460e9i	1.13679i
$478$	0	0
$479$	−3.39731e8	−0.141241	−0.0706205	−	0.997503i	$-0.522498\pi$
$479$	−3.39731e8	−0.141241	−0.0706205	+	0.997503i	$0.522498\pi$
$480$	0	0
$481$	−8.65292e8	−0.354532
$482$	0	0
$483$	− 6.44313e8i	− 0.260185i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	4.11700e9i	1.61521i	0.589723	+	0.807606i	$0.299237\pi$
$487$	4.11700e9i	1.61521i	−0.589723	+	0.807606i	$0.700763\pi$
$488$	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
$493$	− 4.48179e9i	− 1.68456i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.68849e9i	0.616953i
$498$	0	0
$499$	−8.82005e7	−0.0317775	−0.0158887	−	0.999874i	$-0.505058\pi$
$499$	−8.82005e7	−0.0317775	−0.0158887	+	0.999874i	$0.505058\pi$
$500$	0	0
$501$	1.00330e9	0.356449
$502$	0	0
$503$	− 1.01491e9i	− 0.355583i	−0.984068	−	0.177792i	$-0.943105\pi$
$503$	− 1.01491e9i	− 0.355583i	0.984068	−	0.177792i	$-0.0568953\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 4.28872e8i	− 0.146151i
$508$	0	0
$509$	4.60125e9	1.54655	0.773275	−	0.634071i	$-0.218617\pi$
$509$	4.60125e9	1.54655	0.773275	+	0.634071i	$0.218617\pi$
$510$	0	0
$511$	−5.01667e9	−1.66319
$512$	0	0
$513$	− 3.15978e9i	− 1.03335i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	1.67847e8i	0.0534190i
$518$	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
$523$	2.22584e9i	0.680360i	0.940360	+	0.340180i	$0.110488\pi$
$523$	2.22584e9i	0.680360i	−0.940360	+	0.340180i	$0.889512\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.35987e9i	0.999967i
$528$	0	0
$529$	2.53303e9	0.743954
$530$	0	0
$531$	1.07434e9	0.311394
$532$	0	0
$533$	− 1.63921e9i	− 0.468909i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 9.01258e8i	− 0.251153i
$538$	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
$543$	7.76063e7i	0.0208017i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	1.84946e9i	0.483158i	0.970381	+	0.241579i	$0.0776652\pi$
$547$	1.84946e9i	0.483158i	−0.970381	+	0.241579i	$0.922335\pi$
$548$	0	0
$549$	2.24054e9	0.577895
$550$	0	0
$551$	1.24496e10	3.17047
$552$	0	0
$553$	8.88856e9i	2.23508i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 5.03461e9i	− 1.23445i	−0.786788	−	0.617224i	$-0.788257\pi$
$557$	− 5.03461e9i	− 1.23445i	0.786788	−	0.617224i	$-0.211743\pi$
$558$	0	0
$559$	3.37702e9	0.817697
$560$	0	0
$561$	−2.23219e9	−0.533778
$562$	0	0
$563$	− 4.54798e7i	− 0.0107409i	−0.999986	−	0.00537043i	$-0.998291\pi$
$563$	− 4.54798e7i	− 0.0107409i	0.999986	−	0.00537043i	$-0.00170947\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	5.18657e9i	1.19492i
$568$	0	0
$569$	−3.35165e9	−0.762722	−0.381361	−	0.924426i	$-0.624545\pi$
$569$	−3.35165e9	−0.762722	−0.381361	+	0.924426i	$0.624545\pi$
$570$	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
$573$	1.97530e7i	0.00438625i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 3.41602e9i	− 0.740296i	−0.928973	−	0.370148i	$-0.879307\pi$
$577$	− 3.41602e9i	− 0.740296i	0.928973	−	0.370148i	$-0.120693\pi$
$578$	0	0
$579$	5.87011e8	0.125681
$580$	0	0
$581$	7.73415e9	1.63605
$582$	0	0
$583$	1.11382e10i	2.32796i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 4.45218e9i	− 0.908530i	−0.890867	−	0.454265i	$-0.849902\pi$
$587$	− 4.45218e9i	− 0.908530i	0.890867	−	0.454265i	$-0.150098\pi$
$588$	0	0
$589$	−9.33309e9	−1.88201
$590$	0	0
$591$	−1.24350e9	−0.247793
$592$	0	0
$593$	5.98158e9i	1.17794i	0.808154	+	0.588972i	$0.200467\pi$
$593$	5.98158e9i	1.17794i	−0.808154	+	0.588972i	$0.799533\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1.28209e9i	0.246609i
$598$	0	0
$599$	−9.84141e7	−0.0187096	−0.00935478	−	0.999956i	$-0.502978\pi$
$599$	−9.84141e7	−0.0187096	−0.00935478	+	0.999956i	$0.502978\pi$
$600$	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
$603$	− 1.85132e9i	− 0.343851i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 3.14307e9i	− 0.570419i	−0.958465	−	0.285209i	$-0.907937\pi$
$607$	− 3.14307e9i	− 0.570419i	0.958465	−	0.285209i	$-0.0920632\pi$
$608$	0	0
$609$	5.19196e9	0.931473
$610$	0	0
$611$	−1.18327e8	−0.0209865
$612$	0	0
$613$	− 1.02572e10i	− 1.79852i	−0.437415	−	0.899260i	$-0.644106\pi$
$613$	− 1.02572e10i	− 1.79852i	0.437415	−	0.899260i	$-0.355894\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	9.20512e9i	1.57772i	0.614570	+	0.788862i	$0.289330\pi$
$617$	9.20512e9i	1.57772i	−0.614570	+	0.788862i	$0.710670\pi$
$618$	0	0
$619$	−5.97483e9	−1.01253	−0.506266	−	0.862377i	$-0.668975\pi$
$619$	−5.97483e9	−1.01253	−0.506266	+	0.862377i	$0.668975\pi$
$620$	0	0
$621$	−1.78299e9	−0.298764
$622$	0	0
$623$	− 1.41993e10i	− 2.35266i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 6.20059e9i	− 1.00461i
$628$	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
$633$	− 1.97979e8i	− 0.0310246i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 8.29614e9i	− 1.27171i
$638$	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
$643$	1.03486e10i	1.53513i	0.640973	+	0.767563i	$0.278531\pi$
$643$	1.03486e10i	1.53513i	−0.640973	+	0.767563i	$0.721469\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.73480e9i	0.251817i	0.992042	+	0.125909i	$0.0401846\pi$
$647$	1.73480e9i	0.251817i	−0.992042	+	0.125909i	$0.959815\pi$
$648$	0	0
$649$	4.44081e9	0.637685
$650$	0	0
$651$	−3.89226e9	−0.552928
$652$	0	0
$653$	3.78848e9i	0.532438i	0.963913	+	0.266219i	$0.0857745\pi$
$653$	3.78848e9i	0.532438i	−0.963913	+	0.266219i	$0.914226\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	6.59058e9i	0.906661i
$658$	0	0
$659$	1.23763e10	1.68457	0.842287	−	0.539029i	$-0.181209\pi$
$659$	1.23763e10	1.68457	0.842287	+	0.539029i	$0.181209\pi$
$660$	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
$663$	− 1.57363e9i	− 0.209704i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 7.02501e9i	− 0.916656i
$668$	0	0
$669$	3.31725e9	0.428338
$670$	0	0
$671$	9.26133e9	1.18344
$672$	0	0
$673$	− 4.23791e8i	− 0.0535919i	−0.999641	−	0.0267959i	$-0.991470\pi$
$673$	− 4.23791e8i	− 0.0535919i	0.999641	−	0.0267959i	$-0.00853043\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 7.40587e9i	− 0.917310i	−0.888614	−	0.458655i	$-0.848331\pi$
$677$	− 7.40587e9i	− 0.917310i	0.888614	−	0.458655i	$-0.151669\pi$
$678$	0	0
$679$	2.40043e10	2.94270
$680$	0	0
$681$	2.21765e9	0.269077
$682$	0	0
$683$	− 3.56998e9i	− 0.428740i	−0.976753	−	0.214370i	$-0.931230\pi$
$683$	− 3.56998e9i	− 0.428740i	0.976753	−	0.214370i	$-0.0687698\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 1.19653e9i	− 0.140791i
$688$	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
$693$	2.42999e10i	2.77357i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	5.36069e9i	0.599661i
$698$	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
$703$	7.86054e9i	0.853314i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 1.53356e10i	− 1.63205i
$708$	0	0
$709$	−1.43028e10	−1.50716	−0.753580	−	0.657356i	$-0.771675\pi$
$709$	−1.43028e10	−1.50716	−0.753580	+	0.657356i	$0.771675\pi$
$710$	0	0
$711$	1.16772e10	1.21842
$712$	0	0
$713$	5.26645e9i	0.544132i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 3.05453e8i	− 0.0309477i
$718$	0	0
$719$	8.95195e9	0.898186	0.449093	−	0.893485i	$-0.351747\pi$
$719$	8.95195e9	0.898186	0.449093	+	0.893485i	$0.351747\pi$
$720$	0	0
$721$	1.60063e10	1.59044
$722$	0	0
$723$	3.17899e9i	0.312827i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1.69358e10i	1.63469i	0.576149	+	0.817345i	$0.304555\pi$
$727$	1.69358e10i	1.63469i	−0.576149	+	0.817345i	$0.695445\pi$
$728$	0	0
$729$	4.89839e9	0.468281
$730$	0	0
$731$	−1.10439e10	−1.04571
$732$	0	0
$733$	1.11259e10i	1.04345i	0.853115	+	0.521723i	$0.174711\pi$
$733$	1.11259e10i	1.04345i	−0.853115	+	0.521723i	$0.825289\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 7.65248e9i	− 0.704151i
$738$	0	0
$739$	2.21789e9	0.202155	0.101078	−	0.994879i	$-0.467771\pi$
$739$	2.21789e9	0.202155	0.101078	+	0.994879i	$0.467771\pi$
$740$	0	0
$741$	4.37125e9	0.394677
$742$	0	0
$743$	1.05759e10i	0.945928i	0.881082	+	0.472964i	$0.156816\pi$
$743$	1.05759e10i	0.945928i	−0.881082	+	0.472964i	$0.843184\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 1.01606e10i	− 0.891863i
$748$	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
$753$	− 1.21113e9i	− 0.103373i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	1.48486e10i	1.24408i	0.782984	+	0.622042i	$0.213697\pi$
$757$	1.48486e10i	1.24408i	−0.782984	+	0.622042i	$0.786303\pi$
$758$	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
$763$	3.60830e9i	0.294081i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3.13066e9i	0.250525i
$768$	0	0
$769$	−6.62137e9	−0.525056	−0.262528	−	0.964924i	$-0.584556\pi$
$769$	−6.62137e9	−0.525056	−0.262528	+	0.964924i	$0.584556\pi$
$770$	0	0
$771$	3.28109e9	0.257827
$772$	0	0
$773$	9.09632e9i	0.708333i	0.935182	+	0.354166i	$0.115235\pi$
$773$	9.09632e9i	0.708333i	−0.935182	+	0.354166i	$0.884765\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	3.27815e9i	0.250700i
$778$	0	0
$779$	−1.48910e10	−1.12861
$780$	0	0
$781$	9.16913e9	0.688731
$782$	0	0
$783$	− 1.43676e10i	− 1.06959i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 1.90436e10i	− 1.39264i	−0.717732	−	0.696319i	$-0.754820\pi$
$787$	− 1.90436e10i	− 1.39264i	0.717732	−	0.696319i	$-0.245180\pi$
$788$	0	0
$789$	2.80786e9	0.203520
$790$	0	0
$791$	−1.21306e10	−0.871496
$792$	0	0
$793$	6.52899e9i	0.464933i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.49063e10i	1.04295i	0.853265	+	0.521477i	$0.174619\pi$
$797$	1.49063e10i	1.04295i	−0.853265	+	0.521477i	$0.825381\pi$
$798$	0	0
$799$	3.86965e8	0.0268385
$800$	0	0
$801$	−1.86542e10	−1.28251
$802$	0	0
$803$	2.72424e10i	1.85669i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 2.27112e9i	− 0.152119i
$808$	0	0
$809$	−2.51640e10	−1.67094	−0.835469	−	0.549538i	$-0.814804\pi$
$809$	−2.51640e10	−1.67094	−0.835469	+	0.549538i	$0.814804\pi$
$810$	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
$813$	− 5.02872e9i	− 0.328201i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 3.06777e10i	− 1.96810i
$818$	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
$823$	− 1.81107e10i	− 1.13249i	−0.824236	−	0.566246i	$-0.808395\pi$
$823$	− 1.81107e10i	− 1.13249i	0.824236	−	0.566246i	$-0.191605\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 8.31258e9i	− 0.511054i	−0.966802	−	0.255527i	$-0.917751\pi$
$827$	− 8.31258e9i	− 0.511054i	0.966802	−	0.255527i	$-0.0822490\pi$
$828$	0	0
$829$	−3.19126e9	−0.194545	−0.0972726	−	0.995258i	$-0.531012\pi$
$829$	−3.19126e9	−0.194545	−0.0972726	+	0.995258i	$0.531012\pi$
$830$	0	0
$831$	2.85291e9	0.172458
$832$	0	0
$833$	2.71308e10i	1.62632i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.07709e10i	0.634914i
$838$	0	0
$839$	2.11602e10	1.23695	0.618477	−	0.785803i	$-0.287750\pi$
$839$	2.11602e10	1.23695	0.618477	+	0.785803i	$0.287750\pi$
$840$	0	0
$841$	3.93586e10	2.28167
$842$	0	0
$843$	7.41502e9i	0.426300i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	7.11239e10i	4.02183i
$848$	0	0
$849$	9.75630e9	0.547152
$850$	0	0
$851$	4.43552e9	0.246713
$852$	0	0
$853$	− 2.57804e10i	− 1.42222i	−0.703079	−	0.711112i	$-0.748192\pi$
$853$	− 2.57804e10i	− 1.42222i	0.703079	−	0.711112i	$-0.251808\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 2.67728e10i	− 1.45298i	−0.687176	−	0.726491i	$-0.741150\pi$
$857$	− 2.67728e10i	− 1.45298i	0.687176	−	0.726491i	$-0.258850\pi$
$858$	0	0
$859$	1.88350e10	1.01389	0.506944	−	0.861979i	$-0.330775\pi$
$859$	1.88350e10	1.01389	0.506944	+	0.861979i	$0.330775\pi$
$860$	0	0
$861$	−6.21012e9	−0.331580
$862$	0	0
$863$	4.36753e9i	0.231312i	0.993289	+	0.115656i	$0.0368970\pi$
$863$	4.36753e9i	0.231312i	−0.993289	+	0.115656i	$0.963103\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 8.05037e8i	− 0.0419517i
$868$	0	0
$869$	4.82681e10	2.49512
$870$	0	0
$871$	5.39480e9	0.276638
$872$	0	0
$873$	− 3.15353e10i	− 1.60416i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 5.63412e9i	− 0.282051i	−0.990006	−	0.141025i	$-0.954960\pi$
$877$	− 5.63412e9i	− 0.282051i	0.990006	−	0.141025i	$-0.0450399\pi$
$878$	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
$883$	1.44647e10i	0.707044i	0.935426	+	0.353522i	$0.115016\pi$
$883$	1.44647e10i	0.707044i	−0.935426	+	0.353522i	$0.884984\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 2.22645e10i	− 1.07123i	−0.844464	−	0.535613i	$-0.820081\pi$
$887$	− 2.22645e10i	− 1.07123i	0.844464	−	0.535613i	$-0.179919\pi$
$888$	0	0
$889$	6.22307e10	2.97063
$890$	0	0
$891$	2.81650e10	1.33394
$892$	0	0
$893$	1.07492e9i	0.0505120i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 2.46660e9i	− 0.114110i
$898$	0	0
$899$	−4.24377e10	−1.94802
$900$	0	0
$901$	2.56789e10	1.16960
$902$	0	0
$903$	− 1.27938e10i	− 0.578220i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 1.72710e10i	− 0.768583i	−0.923212	−	0.384292i	$-0.874446\pi$
$907$	− 1.72710e10i	− 0.768583i	0.923212	−	0.384292i	$-0.125554\pi$
$908$	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
$913$	− 4.19992e10i	− 1.82639i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	9.62877e9i	0.412362i
$918$	0	0
$919$	−4.37453e9	−0.185921	−0.0929603	−	0.995670i	$-0.529633\pi$
$919$	−4.37453e9	−0.185921	−0.0929603	+	0.995670i	$0.529633\pi$
$920$	0	0
$921$	−9.06237e9	−0.382237
$922$	0	0
$923$	6.46400e9i	0.270580i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 2.10280e10i	− 0.867000i
$928$	0	0
$929$	−1.65493e10	−0.677213	−0.338606	−	0.940928i	$-0.609956\pi$
$929$	−1.65493e10	−0.677213	−0.338606	+	0.940928i	$0.609956\pi$
$930$	0	0
$931$	−7.53643e10	−3.06085
$932$	0	0
$933$	7.66823e9i	0.309107i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 6.78612e9i	− 0.269484i	−0.990881	−	0.134742i	$-0.956979\pi$
$937$	− 6.78612e9i	− 0.269484i	0.990881	−	0.134742i	$-0.0430205\pi$
$938$	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
$943$	8.40264e9i	0.326306i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	2.02600e9i	0.0775203i	0.999249	+	0.0387601i	$0.0123408\pi$
$947$	2.02600e9i	0.0775203i	−0.999249	+	0.0387601i	$0.987659\pi$
$948$	0	0
$949$	−1.92051e10	−0.729434
$950$	0	0
$951$	1.43311e8	0.00540315
$952$	0	0
$953$	− 6.05056e9i	− 0.226449i	−0.993569	−	0.113224i	$-0.963882\pi$
$953$	− 6.05056e9i	− 0.226449i	0.993569	−	0.113224i	$-0.0361179\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 2.81942e10i	− 1.03984i
$958$	0	0
$959$	−4.01834e10	−1.47123
$960$	0	0
$961$	4.30174e9	0.156355
$962$	0	0
$963$	− 2.02371e10i	− 0.730224i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	4.99403e10i	1.77606i	0.459782	+	0.888032i	$0.347927\pi$
$967$	4.99403e10i	1.77606i	−0.459782	+	0.888032i	$0.652073\pi$
$968$	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
$973$	− 2.12346e10i	− 0.739007i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	5.39541e10i	1.85095i	0.378815	+	0.925473i	$0.376332\pi$
$977$	5.39541e10i	1.85095i	−0.378815	+	0.925473i	$0.623668\pi$
$978$	0	0
$979$	−7.71076e10	−2.62638
$980$	0	0
$981$	4.74035e9	0.160313
$982$	0	0
$983$	− 2.37883e10i	− 0.798777i	−0.916782	−	0.399389i	$-0.869222\pi$
$983$	− 2.37883e10i	− 0.798777i	0.916782	−	0.399389i	$-0.130778\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	4.48282e8i	0.0148402i
$988$	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
$993$	− 7.93841e9i	− 0.257283i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 2.83526e10i	− 0.906067i	−0.891493	−	0.453034i	$-0.850342\pi$
$997$	− 2.83526e10i	− 0.906067i	0.891493	−	0.453034i	$-0.149658\pi$
$998$	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.c.u.49.3		6
4.3	odd	2		200.8.c.j.49.4		6
5.2	odd	4		400.8.a.bi.1.1		3
5.3	odd	4		400.8.a.bh.1.3		3
5.4	even	2	inner	400.8.c.u.49.4		6
20.3	even	4		200.8.a.p.1.1	yes	3
20.7	even	4		200.8.a.o.1.3	✓	3
20.19	odd	2		200.8.c.j.49.3		6

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

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.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-14.5033i q^{3} +1504.61i q^{7} +1976.65 q^{9} +8170.56 q^{11} +5760.03i q^{13} -18837.0i q^{17} +52325.6 q^{19} +21821.8 q^{21} -29526.1i q^{23} -60386.8i q^{27} +237925. q^{29} -178366. q^{31} -118500. i q^{33} +150224. i q^{37} +83539.5 q^{39} -284583. q^{41} -586286. i q^{43} +20542.9i q^{47} -1.44030e6 q^{49} -273199. q^{51} +1.36322e6i q^{53} -758895. i q^{57} +543514. q^{59} +1.13350e6 q^{61} +2.97408e6i q^{63} -936593. i q^{67} -428227. q^{69} +1.12222e6 q^{71} +3.33421e6i q^{73} +1.22935e7i q^{77} +5.90757e6 q^{79} +3.44713e6 q^{81} -5.14032e6i q^{83} -3.45071e6i q^{87} -9.43725e6 q^{89} -8.66657e6 q^{91} +2.58690e6i q^{93} -1.59539e7i q^{97} +1.61504e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4292 q^{9} + 5318 q^{11} + 111498 q^{19} - 123124 q^{21} - 131056 q^{29} + 289196 q^{31} + 1600008 q^{39} + 599486 q^{41} - 4720974 q^{49} + 3724630 q^{51} + 1966736 q^{59} + 11420236 q^{61} - 10781396 q^{69}+ \cdots + 106633020 q^{99}+O(q^{100}) \) 6 * q - 4292 * q^9 + 5318 * q^11 + 111498 * q^19 - 123124 * q^21 - 131056 * q^29 + 289196 * q^31 + 1600008 * q^39 + 599486 * q^41 - 4720974 * q^49 + 3724630 * q^51 + 1966736 * q^59 + 11420236 * q^61 - 10781396 * q^69 - 3867912 * q^71 + 11181356 * q^79 + 43424534 * q^81 - 44667418 * q^89 - 2063888 * q^91 + 106633020 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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