LMFDB - Embedded newform 240.8.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [240,8,Mod(1,240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(240, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("240.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 240 = 2^{4} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	240.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$74.9724061162$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 120)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	240.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-27.0000 q^{3} +125.000 q^{5} +776.000 q^{7} +729.000 q^{9} +O(q^{10})$

$q-27.0000 q^{3} +125.000 q^{5} +776.000 q^{7} +729.000 q^{9} -612.000 q^{11} -4506.00 q^{13} -3375.00 q^{15} -31502.0 q^{17} -14812.0 q^{19} -20952.0 q^{21} +71768.0 q^{23} +15625.0 q^{25} -19683.0 q^{27} +53142.0 q^{29} +13920.0 q^{31} +16524.0 q^{33} +97000.0 q^{35} -66930.0 q^{37} +121662. q^{39} -145958. q^{41} +281404. q^{43} +91125.0 q^{45} +635440. q^{47} -221367. q^{49} +850554. q^{51} -792770. q^{53} -76500.0 q^{55} +399924. q^{57} -1.85068e6 q^{59} -1.73678e6 q^{61} +565704. q^{63} -563250. q^{65} +661204. q^{67} -1.93774e6 q^{69} +3.67130e6 q^{71} -5.45274e6 q^{73} -421875. q^{75} -474912. q^{77} +3.08571e6 q^{79} +531441. q^{81} -4.80899e6 q^{83} -3.93775e6 q^{85} -1.43483e6 q^{87} -9.54377e6 q^{89} -3.49666e6 q^{91} -375840. q^{93} -1.85150e6 q^{95} -1.00541e6 q^{97} -446148. q^{99} +O(q^{100})$

Display 100 coefficients 10 coefficients

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
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	0	0
$5$	125.000	0.447214
$5$	125.000	0.447214
$6$	0	0
$7$	776.000	0.855103	0.427552	−	0.903991i	$-0.359376\pi$
$7$	776.000	0.855103	0.427552	+	0.903991i	$0.359376\pi$
$8$	0	0
$9$	729.000	0.333333
$10$	0	0
$11$	−612.000	−0.138636	−0.0693182	−	0.997595i	$-0.522082\pi$
$11$	−612.000	−0.138636	−0.0693182	+	0.997595i	$0.522082\pi$
$12$	0	0
$13$	−4506.00	−0.568839	−0.284420	−	0.958700i	$-0.591801\pi$
$13$	−4506.00	−0.568839	−0.284420	+	0.958700i	$0.591801\pi$
$14$	0	0
$15$	−3375.00	−0.258199
$16$	0	0
$17$	−31502.0	−1.55513	−0.777565	−	0.628802i	$-0.783546\pi$
$17$	−31502.0	−1.55513	−0.777565	+	0.628802i	$0.783546\pi$
$18$	0	0
$19$	−14812.0	−0.495423	−0.247711	−	0.968834i	$-0.579678\pi$
$19$	−14812.0	−0.495423	−0.247711	+	0.968834i	$0.579678\pi$
$20$	0	0
$21$	−20952.0	−0.493694
$22$	0	0
$23$	71768.0	1.22994	0.614969	−	0.788551i	$-0.289168\pi$
$23$	71768.0	1.22994	0.614969	+	0.788551i	$0.289168\pi$
$24$	0	0
$25$	15625.0	0.200000
$26$	0	0
$27$	−19683.0	−0.192450
$28$	0	0
$29$	53142.0	0.404618	0.202309	−	0.979322i	$-0.435156\pi$
$29$	53142.0	0.404618	0.202309	+	0.979322i	$0.435156\pi$
$30$	0	0
$31$	13920.0	0.0839215	0.0419608	−	0.999119i	$-0.486640\pi$
$31$	13920.0	0.0839215	0.0419608	+	0.999119i	$0.486640\pi$
$32$	0	0
$33$	16524.0	0.0800417
$34$	0	0
$35$	97000.0	0.382414
$36$	0	0
$37$	−66930.0	−0.217227	−0.108614	−	0.994084i	$-0.534641\pi$
$37$	−66930.0	−0.217227	−0.108614	+	0.994084i	$0.534641\pi$
$38$	0	0
$39$	121662.	0.328419
$40$	0	0
$41$	−145958.	−0.330738	−0.165369	−	0.986232i	$-0.552882\pi$
$41$	−145958.	−0.330738	−0.165369	+	0.986232i	$0.552882\pi$
$42$	0	0
$43$	281404.	0.539748	0.269874	−	0.962896i	$-0.413018\pi$
$43$	281404.	0.539748	0.269874	+	0.962896i	$0.413018\pi$
$44$	0	0
$45$	91125.0	0.149071
$46$	0	0
$47$	635440.	0.892754	0.446377	−	0.894845i	$-0.352714\pi$
$47$	635440.	0.892754	0.446377	+	0.894845i	$0.352714\pi$
$48$	0	0
$49$	−221367.	−0.268798
$50$	0	0
$51$	850554.	0.897855
$52$	0	0
$53$	−792770.	−0.731445	−0.365722	−	0.930724i	$-0.619178\pi$
$53$	−792770.	−0.731445	−0.365722	+	0.930724i	$0.619178\pi$
$54$	0	0
$55$	−76500.0	−0.0620000
$56$	0	0
$57$	399924.	0.286033
$58$	0	0
$59$	−1.85068e6	−1.17314	−0.586568	−	0.809900i	$-0.699521\pi$
$59$	−1.85068e6	−1.17314	−0.586568	+	0.809900i	$0.699521\pi$
$60$	0	0
$61$	−1.73678e6	−0.979693	−0.489846	−	0.871809i	$-0.662947\pi$
$61$	−1.73678e6	−0.979693	−0.489846	+	0.871809i	$0.662947\pi$
$62$	0	0
$63$	565704.	0.285034
$64$	0	0
$65$	−563250.	−0.254393
$66$	0	0
$67$	661204.	0.268580	0.134290	−	0.990942i	$-0.457125\pi$
$67$	661204.	0.268580	0.134290	+	0.990942i	$0.457125\pi$
$68$	0	0
$69$	−1.93774e6	−0.710105
$70$	0	0
$71$	3.67130e6	1.21735	0.608676	−	0.793419i	$-0.291701\pi$
$71$	3.67130e6	1.21735	0.608676	+	0.793419i	$0.291701\pi$
$72$	0	0
$73$	−5.45274e6	−1.64053	−0.820266	−	0.571982i	$-0.806175\pi$
$73$	−5.45274e6	−1.64053	−0.820266	+	0.571982i	$0.806175\pi$
$74$	0	0
$75$	−421875.	−0.115470
$76$	0	0
$77$	−474912.	−0.118548
$78$	0	0
$79$	3.08571e6	0.704143	0.352071	−	0.935973i	$-0.385477\pi$
$79$	3.08571e6	0.704143	0.352071	+	0.935973i	$0.385477\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	−4.80899e6	−0.923167	−0.461584	−	0.887097i	$-0.652719\pi$
$83$	−4.80899e6	−0.923167	−0.461584	+	0.887097i	$0.652719\pi$
$84$	0	0
$85$	−3.93775e6	−0.695476
$86$	0	0
$87$	−1.43483e6	−0.233606
$88$	0	0
$89$	−9.54377e6	−1.43501	−0.717505	−	0.696554i	$-0.754716\pi$
$89$	−9.54377e6	−1.43501	−0.717505	+	0.696554i	$0.754716\pi$
$90$	0	0
$91$	−3.49666e6	−0.486416
$92$	0	0
$93$	−375840.	−0.0484521
$94$	0	0
$95$	−1.85150e6	−0.221560
$96$	0	0
$97$	−1.00541e6	−0.111851	−0.0559256	−	0.998435i	$-0.517811\pi$
$97$	−1.00541e6	−0.111851	−0.0559256	+	0.998435i	$0.517811\pi$
$98$	0	0
$99$	−446148.	−0.0462121
$100$	0	0
$101$	2.73307e6	0.263953	0.131976	−	0.991253i	$-0.457868\pi$
$101$	2.73307e6	0.263953	0.131976	+	0.991253i	$0.457868\pi$
$102$	0	0
$103$	−1.49125e7	−1.34469	−0.672344	−	0.740239i	$-0.734712\pi$
$103$	−1.49125e7	−1.34469	−0.672344	+	0.740239i	$0.734712\pi$
$104$	0	0
$105$	−2.61900e6	−0.220787
$106$	0	0
$107$	−2.27451e7	−1.79492	−0.897459	−	0.441098i	$-0.854589\pi$
$107$	−2.27451e7	−1.79492	−0.897459	+	0.441098i	$0.854589\pi$
$108$	0	0
$109$	1.63981e7	1.21283	0.606416	−	0.795148i	$-0.292607\pi$
$109$	1.63981e7	1.21283	0.606416	+	0.795148i	$0.292607\pi$
$110$	0	0
$111$	1.80711e6	0.125416
$112$	0	0
$113$	−2.50597e6	−0.163381	−0.0816903	−	0.996658i	$-0.526032\pi$
$113$	−2.50597e6	−0.163381	−0.0816903	+	0.996658i	$0.526032\pi$
$114$	0	0
$115$	8.97100e6	0.550045
$116$	0	0
$117$	−3.28487e6	−0.189613
$118$	0	0
$119$	−2.44456e7	−1.32980
$120$	0	0
$121$	−1.91126e7	−0.980780
$122$	0	0
$123$	3.94087e6	0.190952
$124$	0	0
$125$	1.95312e6	0.0894427
$126$	0	0
$127$	4.74822e6	0.205692	0.102846	−	0.994697i	$-0.467205\pi$
$127$	4.74822e6	0.205692	0.102846	+	0.994697i	$0.467205\pi$
$128$	0	0
$129$	−7.59791e6	−0.311623
$130$	0	0
$131$	−2.61554e7	−1.01651	−0.508255	−	0.861207i	$-0.669709\pi$
$131$	−2.61554e7	−1.01651	−0.508255	+	0.861207i	$0.669709\pi$
$132$	0	0
$133$	−1.14941e7	−0.423638
$134$	0	0
$135$	−2.46038e6	−0.0860663
$136$	0	0
$137$	5.64822e7	1.87668	0.938338	−	0.345719i	$-0.112365\pi$
$137$	5.64822e7	1.87668	0.938338	+	0.345719i	$0.112365\pi$
$138$	0	0
$139$	−3.48240e7	−1.09983	−0.549917	−	0.835219i	$-0.685341\pi$
$139$	−3.48240e7	−1.09983	−0.549917	+	0.835219i	$0.685341\pi$
$140$	0	0
$141$	−1.71569e7	−0.515432
$142$	0	0
$143$	2.75767e6	0.0788618
$144$	0	0
$145$	6.64275e6	0.180951
$146$	0	0
$147$	5.97691e6	0.155191
$148$	0	0
$149$	7.85494e6	0.194532	0.0972660	−	0.995258i	$-0.468990\pi$
$149$	7.85494e6	0.194532	0.0972660	+	0.995258i	$0.468990\pi$
$150$	0	0
$151$	−3.33888e7	−0.789189	−0.394595	−	0.918855i	$-0.629115\pi$
$151$	−3.33888e7	−0.789189	−0.394595	+	0.918855i	$0.629115\pi$
$152$	0	0
$153$	−2.29650e7	−0.518377
$154$	0	0
$155$	1.74000e6	0.0375308
$156$	0	0
$157$	−4.18324e7	−0.862708	−0.431354	−	0.902183i	$-0.641964\pi$
$157$	−4.18324e7	−0.862708	−0.431354	+	0.902183i	$0.641964\pi$
$158$	0	0
$159$	2.14048e7	0.422300
$160$	0	0
$161$	5.56920e7	1.05172
$162$	0	0
$163$	−6.82785e6	−0.123489	−0.0617444	−	0.998092i	$-0.519666\pi$
$163$	−6.82785e6	−0.123489	−0.0617444	+	0.998092i	$0.519666\pi$
$164$	0	0
$165$	2.06550e6	0.0357957
$166$	0	0
$167$	−2.51865e7	−0.418465	−0.209233	−	0.977866i	$-0.567097\pi$
$167$	−2.51865e7	−0.418465	−0.209233	+	0.977866i	$0.567097\pi$
$168$	0	0
$169$	−4.24445e7	−0.676422
$170$	0	0
$171$	−1.07979e7	−0.165141
$172$	0	0
$173$	1.26325e7	0.185493	0.0927463	−	0.995690i	$-0.470435\pi$
$173$	1.26325e7	0.185493	0.0927463	+	0.995690i	$0.470435\pi$
$174$	0	0
$175$	1.21250e7	0.171021
$176$	0	0
$177$	4.99683e7	0.677310
$178$	0	0
$179$	−8.00473e7	−1.04318	−0.521592	−	0.853195i	$-0.674662\pi$
$179$	−8.00473e7	−1.04318	−0.521592	+	0.853195i	$0.674662\pi$
$180$	0	0
$181$	−2.03813e7	−0.255480	−0.127740	−	0.991808i	$-0.540772\pi$
$181$	−2.03813e7	−0.255480	−0.127740	+	0.991808i	$0.540772\pi$
$182$	0	0
$183$	4.68930e7	0.565626
$184$	0	0
$185$	−8.36625e6	−0.0971471
$186$	0	0
$187$	1.92792e7	0.215598
$188$	0	0
$189$	−1.52740e7	−0.164565
$190$	0	0
$191$	−6.61244e7	−0.686665	−0.343332	−	0.939214i	$-0.611556\pi$
$191$	−6.61244e7	−0.686665	−0.343332	+	0.939214i	$0.611556\pi$
$192$	0	0
$193$	−1.67406e8	−1.67617	−0.838087	−	0.545536i	$-0.816326\pi$
$193$	−1.67406e8	−1.67617	−0.838087	+	0.545536i	$0.816326\pi$
$194$	0	0
$195$	1.52078e7	0.146874
$196$	0	0
$197$	−1.30179e7	−0.121314	−0.0606569	−	0.998159i	$-0.519320\pi$
$197$	−1.30179e7	−0.121314	−0.0606569	+	0.998159i	$0.519320\pi$
$198$	0	0
$199$	1.77313e7	0.159498	0.0797488	−	0.996815i	$-0.474588\pi$
$199$	1.77313e7	0.159498	0.0797488	+	0.996815i	$0.474588\pi$
$200$	0	0
$201$	−1.78525e7	−0.155065
$202$	0	0
$203$	4.12382e7	0.345990
$204$	0	0
$205$	−1.82448e7	−0.147911
$206$	0	0
$207$	5.23189e7	0.409980
$208$	0	0
$209$	9.06494e6	0.0686836
$210$	0	0
$211$	−7.35024e7	−0.538658	−0.269329	−	0.963048i	$-0.586802\pi$
$211$	−7.35024e7	−0.538658	−0.269329	+	0.963048i	$0.586802\pi$
$212$	0	0
$213$	−9.91252e7	−0.702838
$214$	0	0
$215$	3.51755e7	0.241382
$216$	0	0
$217$	1.08019e7	0.0717616
$218$	0	0
$219$	1.47224e8	0.947162
$220$	0	0
$221$	1.41948e8	0.884619
$222$	0	0
$223$	2.90300e8	1.75300	0.876498	−	0.481406i	$-0.159874\pi$
$223$	2.90300e8	1.75300	0.876498	+	0.481406i	$0.159874\pi$
$224$	0	0
$225$	1.13906e7	0.0666667
$226$	0	0
$227$	1.47339e8	0.836042	0.418021	−	0.908437i	$-0.362724\pi$
$227$	1.47339e8	0.836042	0.418021	+	0.908437i	$0.362724\pi$
$228$	0	0
$229$	−2.70407e8	−1.48797	−0.743985	−	0.668196i	$-0.767067\pi$
$229$	−2.70407e8	−1.48797	−0.743985	+	0.668196i	$0.767067\pi$
$230$	0	0
$231$	1.28226e7	0.0684439
$232$	0	0
$233$	1.34158e8	0.694816	0.347408	−	0.937714i	$-0.387062\pi$
$233$	1.34158e8	0.694816	0.347408	+	0.937714i	$0.387062\pi$
$234$	0	0
$235$	7.94300e7	0.399252
$236$	0	0
$237$	−8.33142e7	−0.406537
$238$	0	0
$239$	8.57167e7	0.406137	0.203069	−	0.979165i	$-0.434909\pi$
$239$	8.57167e7	0.406137	0.203069	+	0.979165i	$0.434909\pi$
$240$	0	0
$241$	8.19327e7	0.377049	0.188524	−	0.982069i	$-0.439630\pi$
$241$	8.19327e7	0.377049	0.188524	+	0.982069i	$0.439630\pi$
$242$	0	0
$243$	−1.43489e7	−0.0641500
$244$	0	0
$245$	−2.76709e7	−0.120210
$246$	0	0
$247$	6.67429e7	0.281816
$248$	0	0
$249$	1.29843e8	0.532991
$250$	0	0
$251$	−2.66372e8	−1.06324	−0.531619	−	0.846984i	$-0.678416\pi$
$251$	−2.66372e8	−1.06324	−0.531619	+	0.846984i	$0.678416\pi$
$252$	0	0
$253$	−4.39220e7	−0.170514
$254$	0	0
$255$	1.06319e8	0.401533
$256$	0	0
$257$	3.75459e8	1.37974	0.689868	−	0.723935i	$-0.257668\pi$
$257$	3.75459e8	1.37974	0.689868	+	0.723935i	$0.257668\pi$
$258$	0	0
$259$	−5.19377e7	−0.185752
$260$	0	0
$261$	3.87405e7	0.134873
$262$	0	0
$263$	4.31735e8	1.46343	0.731716	−	0.681609i	$-0.238720\pi$
$263$	4.31735e8	1.46343	0.731716	+	0.681609i	$0.238720\pi$
$264$	0	0
$265$	−9.90963e7	−0.327112
$266$	0	0
$267$	2.57682e8	0.828503
$268$	0	0
$269$	1.73237e7	0.0542634	0.0271317	−	0.999632i	$-0.491363\pi$
$269$	1.73237e7	0.0542634	0.0271317	+	0.999632i	$0.491363\pi$
$270$	0	0
$271$	−3.38848e8	−1.03422	−0.517110	−	0.855919i	$-0.672992\pi$
$271$	−3.38848e8	−1.03422	−0.517110	+	0.855919i	$0.672992\pi$
$272$	0	0
$273$	9.44097e7	0.280833
$274$	0	0
$275$	−9.56250e6	−0.0277273
$276$	0	0
$277$	4.15276e8	1.17397	0.586986	−	0.809597i	$-0.300314\pi$
$277$	4.15276e8	1.17397	0.586986	+	0.809597i	$0.300314\pi$
$278$	0	0
$279$	1.01477e7	0.0279738
$280$	0	0
$281$	4.99229e8	1.34223	0.671116	−	0.741352i	$-0.265815\pi$
$281$	4.99229e8	1.34223	0.671116	+	0.741352i	$0.265815\pi$
$282$	0	0
$283$	2.79212e8	0.732287	0.366143	−	0.930558i	$-0.380678\pi$
$283$	2.79212e8	0.732287	0.366143	+	0.930558i	$0.380678\pi$
$284$	0	0
$285$	4.99905e7	0.127918
$286$	0	0
$287$	−1.13263e8	−0.282815
$288$	0	0
$289$	5.82037e8	1.41843
$290$	0	0
$291$	2.71460e7	0.0645773
$292$	0	0
$293$	−5.56612e8	−1.29275	−0.646377	−	0.763018i	$-0.723717\pi$
$293$	−5.56612e8	−1.29275	−0.646377	+	0.763018i	$0.723717\pi$
$294$	0	0
$295$	−2.31334e8	−0.524642
$296$	0	0
$297$	1.20460e7	0.0266806
$298$	0	0
$299$	−3.23387e8	−0.699637
$300$	0	0
$301$	2.18370e8	0.461540
$302$	0	0
$303$	−7.37929e7	−0.152393
$304$	0	0
$305$	−2.17097e8	−0.438132
$306$	0	0
$307$	−2.52711e8	−0.498471	−0.249236	−	0.968443i	$-0.580179\pi$
$307$	−2.52711e8	−0.498471	−0.249236	+	0.968443i	$0.580179\pi$
$308$	0	0
$309$	4.02638e8	0.776355
$310$	0	0
$311$	1.70383e8	0.321192	0.160596	−	0.987020i	$-0.448658\pi$
$311$	1.70383e8	0.321192	0.160596	+	0.987020i	$0.448658\pi$
$312$	0	0
$313$	−9.44450e8	−1.74090	−0.870450	−	0.492257i	$-0.836172\pi$
$313$	−9.44450e8	−1.74090	−0.870450	+	0.492257i	$0.836172\pi$
$314$	0	0
$315$	7.07130e7	0.127471
$316$	0	0
$317$	7.84654e8	1.38347	0.691736	−	0.722150i	$-0.256846\pi$
$317$	7.84654e8	1.38347	0.691736	+	0.722150i	$0.256846\pi$
$318$	0	0
$319$	−3.25229e7	−0.0560947
$320$	0	0
$321$	6.14117e8	1.03630
$322$	0	0
$323$	4.66608e8	0.770447
$324$	0	0
$325$	−7.04062e7	−0.113768
$326$	0	0
$327$	−4.42748e8	−0.700229
$328$	0	0
$329$	4.93101e8	0.763397
$330$	0	0
$331$	−5.09538e8	−0.772287	−0.386143	−	0.922439i	$-0.626193\pi$
$331$	−5.09538e8	−0.772287	−0.386143	+	0.922439i	$0.626193\pi$
$332$	0	0
$333$	−4.87920e7	−0.0724092
$334$	0	0
$335$	8.26505e7	0.120113
$336$	0	0
$337$	7.15903e8	1.01894	0.509471	−	0.860488i	$-0.329841\pi$
$337$	7.15903e8	1.01894	0.509471	+	0.860488i	$0.329841\pi$
$338$	0	0
$339$	6.76611e7	0.0943279
$340$	0	0
$341$	−8.51904e6	−0.0116346
$342$	0	0
$343$	−8.10850e8	−1.08495
$344$	0	0
$345$	−2.42217e8	−0.317569
$346$	0	0
$347$	1.07047e9	1.37537	0.687687	−	0.726007i	$-0.258626\pi$
$347$	1.07047e9	1.37537	0.687687	+	0.726007i	$0.258626\pi$
$348$	0	0
$349$	−4.06658e8	−0.512083	−0.256041	−	0.966666i	$-0.582418\pi$
$349$	−4.06658e8	−0.512083	−0.256041	+	0.966666i	$0.582418\pi$
$350$	0	0
$351$	8.86916e7	0.109473
$352$	0	0
$353$	6.10099e8	0.738226	0.369113	−	0.929385i	$-0.379662\pi$
$353$	6.10099e8	0.738226	0.369113	+	0.929385i	$0.379662\pi$
$354$	0	0
$355$	4.58913e8	0.544416
$356$	0	0
$357$	6.60030e8	0.767759
$358$	0	0
$359$	4.52380e8	0.516028	0.258014	−	0.966141i	$-0.416932\pi$
$359$	4.52380e8	0.516028	0.258014	+	0.966141i	$0.416932\pi$
$360$	0	0
$361$	−6.74476e8	−0.754556
$362$	0	0
$363$	5.16041e8	0.566254
$364$	0	0
$365$	−6.81593e8	−0.733669
$366$	0	0
$367$	−1.35838e9	−1.43447	−0.717234	−	0.696832i	$-0.754592\pi$
$367$	−1.35838e9	−1.43447	−0.717234	+	0.696832i	$0.754592\pi$
$368$	0	0
$369$	−1.06403e8	−0.110246
$370$	0	0
$371$	−6.15190e8	−0.625461
$372$	0	0
$373$	−1.78714e9	−1.78310	−0.891552	−	0.452918i	$-0.850383\pi$
$373$	−1.78714e9	−1.78310	−0.891552	+	0.452918i	$0.850383\pi$
$374$	0	0
$375$	−5.27344e7	−0.0516398
$376$	0	0
$377$	−2.39458e8	−0.230162
$378$	0	0
$379$	−5.95959e8	−0.562314	−0.281157	−	0.959662i	$-0.590718\pi$
$379$	−5.95959e8	−0.562314	−0.281157	+	0.959662i	$0.590718\pi$
$380$	0	0
$381$	−1.28202e8	−0.118757
$382$	0	0
$383$	9.45999e8	0.860389	0.430195	−	0.902736i	$-0.358445\pi$
$383$	9.45999e8	0.860389	0.430195	+	0.902736i	$0.358445\pi$
$384$	0	0
$385$	−5.93640e7	−0.0530164
$386$	0	0
$387$	2.05144e8	0.179916
$388$	0	0
$389$	−8.61009e8	−0.741625	−0.370812	−	0.928708i	$-0.620921\pi$
$389$	−8.61009e8	−0.741625	−0.370812	+	0.928708i	$0.620921\pi$
$390$	0	0
$391$	−2.26084e9	−1.91272
$392$	0	0
$393$	7.06195e8	0.586882
$394$	0	0
$395$	3.85714e8	0.314902
$396$	0	0
$397$	−2.16109e9	−1.73343	−0.866715	−	0.498803i	$-0.833773\pi$
$397$	−2.16109e9	−1.73343	−0.866715	+	0.498803i	$0.833773\pi$
$398$	0	0
$399$	3.10341e8	0.244587
$400$	0	0
$401$	−9.34605e8	−0.723807	−0.361904	−	0.932216i	$-0.617873\pi$
$401$	−9.34605e8	−0.723807	−0.361904	+	0.932216i	$0.617873\pi$
$402$	0	0
$403$	−6.27235e7	−0.0477378
$404$	0	0
$405$	6.64301e7	0.0496904
$406$	0	0
$407$	4.09612e7	0.0301156
$408$	0	0
$409$	1.43416e9	1.03649	0.518245	−	0.855232i	$-0.326585\pi$
$409$	1.43416e9	1.03649	0.518245	+	0.855232i	$0.326585\pi$
$410$	0	0
$411$	−1.52502e9	−1.08350
$412$	0	0
$413$	−1.43612e9	−1.00315
$414$	0	0
$415$	−6.01124e8	−0.412853
$416$	0	0
$417$	9.40249e8	0.634990
$418$	0	0
$419$	1.20979e9	0.803452	0.401726	−	0.915760i	$-0.368410\pi$
$419$	1.20979e9	0.803452	0.401726	+	0.915760i	$0.368410\pi$
$420$	0	0
$421$	−1.98061e9	−1.29363	−0.646816	−	0.762646i	$-0.723900\pi$
$421$	−1.98061e9	−1.29363	−0.646816	+	0.762646i	$0.723900\pi$
$422$	0	0
$423$	4.63236e8	0.297585
$424$	0	0
$425$	−4.92219e8	−0.311026
$426$	0	0
$427$	−1.34774e9	−0.837738
$428$	0	0
$429$	−7.44571e7	−0.0455309
$430$	0	0
$431$	−1.82217e8	−0.109627	−0.0548135	−	0.998497i	$-0.517456\pi$
$431$	−1.82217e8	−0.109627	−0.0548135	+	0.998497i	$0.517456\pi$
$432$	0	0
$433$	9.97172e8	0.590286	0.295143	−	0.955453i	$-0.404633\pi$
$433$	9.97172e8	0.590286	0.295143	+	0.955453i	$0.404633\pi$
$434$	0	0
$435$	−1.79354e8	−0.104472
$436$	0	0
$437$	−1.06303e9	−0.609340
$438$	0	0
$439$	−3.83798e8	−0.216509	−0.108255	−	0.994123i	$-0.534526\pi$
$439$	−3.83798e8	−0.216509	−0.108255	+	0.994123i	$0.534526\pi$
$440$	0	0
$441$	−1.61377e8	−0.0895995
$442$	0	0
$443$	3.02192e9	1.65147	0.825734	−	0.564060i	$-0.190761\pi$
$443$	3.02192e9	1.65147	0.825734	+	0.564060i	$0.190761\pi$
$444$	0	0
$445$	−1.19297e9	−0.641756
$446$	0	0
$447$	−2.12083e8	−0.112313
$448$	0	0
$449$	−1.24787e9	−0.650588	−0.325294	−	0.945613i	$-0.605463\pi$
$449$	−1.24787e9	−0.650588	−0.325294	+	0.945613i	$0.605463\pi$
$450$	0	0
$451$	8.93263e7	0.0458523
$452$	0	0
$453$	9.01497e8	0.455639
$454$	0	0
$455$	−4.37082e8	−0.217532
$456$	0	0
$457$	1.51650e8	0.0743254	0.0371627	−	0.999309i	$-0.488168\pi$
$457$	1.51650e8	0.0743254	0.0371627	+	0.999309i	$0.488168\pi$
$458$	0	0
$459$	6.20054e8	0.299285
$460$	0	0
$461$	−2.15143e9	−1.02276	−0.511380	−	0.859355i	$-0.670866\pi$
$461$	−2.15143e9	−1.02276	−0.511380	+	0.859355i	$0.670866\pi$
$462$	0	0
$463$	2.12703e9	0.995956	0.497978	−	0.867190i	$-0.334076\pi$
$463$	2.12703e9	0.995956	0.497978	+	0.867190i	$0.334076\pi$
$464$	0	0
$465$	−4.69800e7	−0.0216684
$466$	0	0
$467$	−4.27041e8	−0.194026	−0.0970130	−	0.995283i	$-0.530929\pi$
$467$	−4.27041e8	−0.194026	−0.0970130	+	0.995283i	$0.530929\pi$
$468$	0	0
$469$	5.13094e8	0.229664
$470$	0	0
$471$	1.12947e9	0.498084
$472$	0	0
$473$	−1.72219e8	−0.0748286
$474$	0	0
$475$	−2.31437e8	−0.0990846
$476$	0	0
$477$	−5.77929e8	−0.243815
$478$	0	0
$479$	3.71866e9	1.54601	0.773004	−	0.634401i	$-0.218753\pi$
$479$	3.71866e9	1.54601	0.773004	+	0.634401i	$0.218753\pi$
$480$	0	0
$481$	3.01587e8	0.123567
$482$	0	0
$483$	−1.50368e9	−0.607213
$484$	0	0
$485$	−1.25676e8	−0.0500213
$486$	0	0
$487$	4.21912e9	1.65528	0.827639	−	0.561261i	$-0.189684\pi$
$487$	4.21912e9	1.65528	0.827639	+	0.561261i	$0.189684\pi$
$488$	0	0
$489$	1.84352e8	0.0712963
$490$	0	0
$491$	−2.25020e9	−0.857897	−0.428948	−	0.903329i	$-0.641116\pi$
$491$	−2.25020e9	−0.857897	−0.428948	+	0.903329i	$0.641116\pi$
$492$	0	0
$493$	−1.67408e9	−0.629233
$494$	0	0
$495$	−5.57685e7	−0.0206667
$496$	0	0
$497$	2.84893e9	1.04096
$498$	0	0
$499$	−4.34769e9	−1.56642	−0.783208	−	0.621760i	$-0.786418\pi$
$499$	−4.34769e9	−1.56642	−0.783208	+	0.621760i	$0.786418\pi$
$500$	0	0
$501$	6.80034e8	0.241601
$502$	0	0
$503$	−3.62193e9	−1.26897	−0.634486	−	0.772935i	$-0.718788\pi$
$503$	−3.62193e9	−1.26897	−0.634486	+	0.772935i	$0.718788\pi$
$504$	0	0
$505$	3.41634e8	0.118043
$506$	0	0
$507$	1.14600e9	0.390532
$508$	0	0
$509$	1.85018e9	0.621872	0.310936	−	0.950431i	$-0.399357\pi$
$509$	1.85018e9	0.621872	0.310936	+	0.950431i	$0.399357\pi$
$510$	0	0
$511$	−4.23133e9	−1.40283
$512$	0	0
$513$	2.91545e8	0.0953442
$514$	0	0
$515$	−1.86407e9	−0.601362
$516$	0	0
$517$	−3.88889e8	−0.123768
$518$	0	0
$519$	−3.41076e8	−0.107094
$520$	0	0
$521$	1.76311e9	0.546195	0.273098	−	0.961986i	$-0.411952\pi$
$521$	1.76311e9	0.546195	0.273098	+	0.961986i	$0.411952\pi$
$522$	0	0
$523$	1.19055e8	0.0363909	0.0181954	−	0.999834i	$-0.494208\pi$
$523$	1.19055e8	0.0363909	0.0181954	+	0.999834i	$0.494208\pi$
$524$	0	0
$525$	−3.27375e8	−0.0987388
$526$	0	0
$527$	−4.38508e8	−0.130509
$528$	0	0
$529$	1.74582e9	0.512749
$530$	0	0
$531$	−1.34914e9	−0.391045
$532$	0	0
$533$	6.57687e8	0.188137
$534$	0	0
$535$	−2.84314e9	−0.802711
$536$	0	0
$537$	2.16128e9	0.602283
$538$	0	0
$539$	1.35477e8	0.0372652
$540$	0	0
$541$	−5.07351e9	−1.37758	−0.688791	−	0.724960i	$-0.741858\pi$
$541$	−5.07351e9	−1.37758	−0.688791	+	0.724960i	$0.741858\pi$
$542$	0	0
$543$	5.50295e8	0.147502
$544$	0	0
$545$	2.04976e9	0.542395
$546$	0	0
$547$	3.01425e9	0.787452	0.393726	−	0.919228i	$-0.371186\pi$
$547$	3.01425e9	0.787452	0.393726	+	0.919228i	$0.371186\pi$
$548$	0	0
$549$	−1.26611e9	−0.326564
$550$	0	0
$551$	−7.87139e8	−0.200457
$552$	0	0
$553$	2.39451e9	0.602115
$554$	0	0
$555$	2.25889e8	0.0560879
$556$	0	0
$557$	−9.13732e8	−0.224040	−0.112020	−	0.993706i	$-0.535732\pi$
$557$	−9.13732e8	−0.224040	−0.112020	+	0.993706i	$0.535732\pi$
$558$	0	0
$559$	−1.26801e9	−0.307030
$560$	0	0
$561$	−5.20539e8	−0.124475
$562$	0	0
$563$	−7.21616e9	−1.70423	−0.852113	−	0.523359i	$-0.824679\pi$
$563$	−7.21616e9	−1.70423	−0.852113	+	0.523359i	$0.824679\pi$
$564$	0	0
$565$	−3.13246e8	−0.0730660
$566$	0	0
$567$	4.12398e8	0.0950115
$568$	0	0
$569$	5.33850e9	1.21486	0.607430	−	0.794373i	$-0.292200\pi$
$569$	5.33850e9	1.21486	0.607430	+	0.794373i	$0.292200\pi$
$570$	0	0
$571$	−1.79662e9	−0.403860	−0.201930	−	0.979400i	$-0.564721\pi$
$571$	−1.79662e9	−0.403860	−0.201930	+	0.979400i	$0.564721\pi$
$572$	0	0
$573$	1.78536e9	0.396446
$574$	0	0
$575$	1.12138e9	0.245988
$576$	0	0
$577$	5.83946e9	1.26549	0.632743	−	0.774362i	$-0.281929\pi$
$577$	5.83946e9	1.26549	0.632743	+	0.774362i	$0.281929\pi$
$578$	0	0
$579$	4.51995e9	0.967740
$580$	0	0
$581$	−3.73177e9	−0.789403
$582$	0	0
$583$	4.85175e8	0.101405
$584$	0	0
$585$	−4.10609e8	−0.0847975
$586$	0	0
$587$	4.96397e9	1.01297	0.506484	−	0.862249i	$-0.330945\pi$
$587$	4.96397e9	1.01297	0.506484	+	0.862249i	$0.330945\pi$
$588$	0	0
$589$	−2.06183e8	−0.0415766
$590$	0	0
$591$	3.51484e8	0.0700406
$592$	0	0
$593$	7.25284e9	1.42829	0.714146	−	0.699997i	$-0.246815\pi$
$593$	7.25284e9	1.42829	0.714146	+	0.699997i	$0.246815\pi$
$594$	0	0
$595$	−3.05569e9	−0.594703
$596$	0	0
$597$	−4.78744e8	−0.0920860
$598$	0	0
$599$	4.27870e9	0.813426	0.406713	−	0.913556i	$-0.366675\pi$
$599$	4.27870e9	0.813426	0.406713	+	0.913556i	$0.366675\pi$
$600$	0	0
$601$	−2.36601e9	−0.444586	−0.222293	−	0.974980i	$-0.571354\pi$
$601$	−2.36601e9	−0.444586	−0.222293	+	0.974980i	$0.571354\pi$
$602$	0	0
$603$	4.82018e8	0.0895267
$604$	0	0
$605$	−2.38908e9	−0.438618
$606$	0	0
$607$	3.36397e9	0.610508	0.305254	−	0.952271i	$-0.401259\pi$
$607$	3.36397e9	0.610508	0.305254	+	0.952271i	$0.401259\pi$
$608$	0	0
$609$	−1.11343e9	−0.199757
$610$	0	0
$611$	−2.86329e9	−0.507834
$612$	0	0
$613$	5.05211e9	0.885851	0.442926	−	0.896558i	$-0.353941\pi$
$613$	5.05211e9	0.885851	0.442926	+	0.896558i	$0.353941\pi$
$614$	0	0
$615$	4.92608e8	0.0853962
$616$	0	0
$617$	2.68410e9	0.460045	0.230023	−	0.973185i	$-0.426120\pi$
$617$	2.68410e9	0.460045	0.230023	+	0.973185i	$0.426120\pi$
$618$	0	0
$619$	5.59615e9	0.948357	0.474178	−	0.880429i	$-0.342745\pi$
$619$	5.59615e9	0.948357	0.474178	+	0.880429i	$0.342745\pi$
$620$	0	0
$621$	−1.41261e9	−0.236702
$622$	0	0
$623$	−7.40596e9	−1.22708
$624$	0	0
$625$	2.44141e8	0.0400000
$626$	0	0
$627$	−2.44753e8	−0.0396545
$628$	0	0
$629$	2.10843e9	0.337817
$630$	0	0
$631$	1.02273e10	1.62054	0.810269	−	0.586058i	$-0.199321\pi$
$631$	1.02273e10	1.62054	0.810269	+	0.586058i	$0.199321\pi$
$632$	0	0
$633$	1.98456e9	0.310994
$634$	0	0
$635$	5.93528e8	0.0919884
$636$	0	0
$637$	9.97480e8	0.152903
$638$	0	0
$639$	2.67638e9	0.405784
$640$	0	0
$641$	1.98291e9	0.297371	0.148686	−	0.988885i	$-0.452496\pi$
$641$	1.98291e9	0.297371	0.148686	+	0.988885i	$0.452496\pi$
$642$	0	0
$643$	6.56084e9	0.973243	0.486621	−	0.873613i	$-0.338229\pi$
$643$	6.56084e9	0.973243	0.486621	+	0.873613i	$0.338229\pi$
$644$	0	0
$645$	−9.49738e8	−0.139362
$646$	0	0
$647$	8.64785e9	1.25529	0.627643	−	0.778501i	$-0.284020\pi$
$647$	8.64785e9	1.25529	0.627643	+	0.778501i	$0.284020\pi$
$648$	0	0
$649$	1.13261e9	0.162639
$650$	0	0
$651$	−2.91652e8	−0.0414316
$652$	0	0
$653$	3.34035e9	0.469458	0.234729	−	0.972061i	$-0.424580\pi$
$653$	3.34035e9	0.469458	0.234729	+	0.972061i	$0.424580\pi$
$654$	0	0
$655$	−3.26942e9	−0.454597
$656$	0	0
$657$	−3.97505e9	−0.546844
$658$	0	0
$659$	3.52240e9	0.479446	0.239723	−	0.970841i	$-0.422943\pi$
$659$	3.52240e9	0.479446	0.239723	+	0.970841i	$0.422943\pi$
$660$	0	0
$661$	−1.17841e10	−1.58705	−0.793526	−	0.608536i	$-0.791757\pi$
$661$	−1.17841e10	−1.58705	−0.793526	+	0.608536i	$0.791757\pi$
$662$	0	0
$663$	−3.83260e9	−0.510735
$664$	0	0
$665$	−1.43676e9	−0.189457
$666$	0	0
$667$	3.81390e9	0.497655
$668$	0	0
$669$	−7.83811e9	−1.01209
$670$	0	0
$671$	1.06291e9	0.135821
$672$	0	0
$673$	−1.10399e9	−0.139608	−0.0698040	−	0.997561i	$-0.522237\pi$
$673$	−1.10399e9	−0.139608	−0.0698040	+	0.997561i	$0.522237\pi$
$674$	0	0
$675$	−3.07547e8	−0.0384900
$676$	0	0
$677$	1.58398e10	1.96196	0.980980	−	0.194108i	$-0.0621814\pi$
$677$	1.58398e10	1.96196	0.980980	+	0.194108i	$0.0621814\pi$
$678$	0	0
$679$	−7.80195e8	−0.0956443
$680$	0	0
$681$	−3.97816e9	−0.482689
$682$	0	0
$683$	3.57076e8	0.0428833	0.0214416	−	0.999770i	$-0.493174\pi$
$683$	3.57076e8	0.0428833	0.0214416	+	0.999770i	$0.493174\pi$
$684$	0	0
$685$	7.06027e9	0.839275
$686$	0	0
$687$	7.30100e9	0.859080
$688$	0	0
$689$	3.57222e9	0.416075
$690$	0	0
$691$	1.43357e10	1.65290	0.826449	−	0.563012i	$-0.190357\pi$
$691$	1.43357e10	1.65290	0.826449	+	0.563012i	$0.190357\pi$
$692$	0	0
$693$	−3.46211e8	−0.0395161
$694$	0	0
$695$	−4.35300e9	−0.491861
$696$	0	0
$697$	4.59797e9	0.514341
$698$	0	0
$699$	−3.62226e9	−0.401152
$700$	0	0
$701$	−1.26602e10	−1.38812	−0.694060	−	0.719917i	$-0.744180\pi$
$701$	−1.26602e10	−1.38812	−0.694060	+	0.719917i	$0.744180\pi$
$702$	0	0
$703$	9.91367e8	0.107619
$704$	0	0
$705$	−2.14461e9	−0.230508
$706$	0	0
$707$	2.12086e9	0.225707
$708$	0	0
$709$	1.36125e10	1.43442	0.717211	−	0.696856i	$-0.245418\pi$
$709$	1.36125e10	1.43442	0.717211	+	0.696856i	$0.245418\pi$
$710$	0	0
$711$	2.24948e9	0.234714
$712$	0	0
$713$	9.99011e8	0.103218
$714$	0	0
$715$	3.44709e8	0.0352681
$716$	0	0
$717$	−2.31435e9	−0.234483
$718$	0	0
$719$	−4.25635e9	−0.427057	−0.213529	−	0.976937i	$-0.568496\pi$
$719$	−4.25635e9	−0.427057	−0.213529	+	0.976937i	$0.568496\pi$
$720$	0	0
$721$	−1.15721e10	−1.14985
$722$	0	0
$723$	−2.21218e9	−0.217689
$724$	0	0
$725$	8.30344e8	0.0809235
$726$	0	0
$727$	1.57915e10	1.52424	0.762121	−	0.647434i	$-0.224158\pi$
$727$	1.57915e10	1.52424	0.762121	+	0.647434i	$0.224158\pi$
$728$	0	0
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	−8.86479e9	−0.839378
$732$	0	0
$733$	−1.07903e10	−1.01198	−0.505988	−	0.862541i	$-0.668872\pi$
$733$	−1.07903e10	−1.01198	−0.505988	+	0.862541i	$0.668872\pi$
$734$	0	0
$735$	7.47114e8	0.0694034
$736$	0	0
$737$	−4.04657e8	−0.0372349
$738$	0	0
$739$	3.87504e9	0.353200	0.176600	−	0.984283i	$-0.443490\pi$
$739$	3.87504e9	0.353200	0.176600	+	0.984283i	$0.443490\pi$
$740$	0	0
$741$	−1.80206e9	−0.162707
$742$	0	0
$743$	−2.15396e10	−1.92653	−0.963265	−	0.268552i	$-0.913455\pi$
$743$	−2.15396e10	−1.92653	−0.963265	+	0.268552i	$0.913455\pi$
$744$	0	0
$745$	9.81868e8	0.0869974
$746$	0	0
$747$	−3.50575e9	−0.307722
$748$	0	0
$749$	−1.76502e10	−1.53484
$750$	0	0
$751$	3.99904e9	0.344521	0.172261	−	0.985051i	$-0.444893\pi$
$751$	3.99904e9	0.344521	0.172261	+	0.985051i	$0.444893\pi$
$752$	0	0
$753$	7.19204e9	0.613860
$754$	0	0
$755$	−4.17360e9	−0.352936
$756$	0	0
$757$	3.89152e9	0.326050	0.163025	−	0.986622i	$-0.447875\pi$
$757$	3.89152e9	0.326050	0.163025	+	0.986622i	$0.447875\pi$
$758$	0	0
$759$	1.18589e9	0.0984464
$760$	0	0
$761$	−1.13534e10	−0.933856	−0.466928	−	0.884295i	$-0.654639\pi$
$761$	−1.13534e10	−0.933856	−0.466928	+	0.884295i	$0.654639\pi$
$762$	0	0
$763$	1.27249e10	1.03710
$764$	0	0
$765$	−2.87062e9	−0.231825
$766$	0	0
$767$	8.33915e9	0.667326
$768$	0	0
$769$	1.12629e10	0.893114	0.446557	−	0.894755i	$-0.352650\pi$
$769$	1.12629e10	0.893114	0.446557	+	0.894755i	$0.352650\pi$
$770$	0	0
$771$	−1.01374e10	−0.796591
$772$	0	0
$773$	1.24048e10	0.965962	0.482981	−	0.875631i	$-0.339554\pi$
$773$	1.24048e10	0.965962	0.482981	+	0.875631i	$0.339554\pi$
$774$	0	0
$775$	2.17500e8	0.0167843
$776$	0	0
$777$	1.40232e9	0.107244
$778$	0	0
$779$	2.16193e9	0.163855
$780$	0	0
$781$	−2.24684e9	−0.168769
$782$	0	0
$783$	−1.04599e9	−0.0778687
$784$	0	0
$785$	−5.22905e9	−0.385815
$786$	0	0
$787$	3.26109e9	0.238479	0.119240	−	0.992865i	$-0.461954\pi$
$787$	3.26109e9	0.238479	0.119240	+	0.992865i	$0.461954\pi$
$788$	0	0
$789$	−1.16569e10	−0.844913
$790$	0	0
$791$	−1.94463e9	−0.139707
$792$	0	0
$793$	7.82592e9	0.557288
$794$	0	0
$795$	2.67560e9	0.188858
$796$	0	0
$797$	2.43714e9	0.170520	0.0852601	−	0.996359i	$-0.472828\pi$
$797$	2.43714e9	0.170520	0.0852601	+	0.996359i	$0.472828\pi$
$798$	0	0
$799$	−2.00176e10	−1.38835
$800$	0	0
$801$	−6.95741e9	−0.478336
$802$	0	0
$803$	3.33708e9	0.227437
$804$	0	0
$805$	6.96150e9	0.470346
$806$	0	0
$807$	−4.67740e8	−0.0313290
$808$	0	0
$809$	−1.83409e10	−1.21787	−0.608933	−	0.793221i	$-0.708402\pi$
$809$	−1.83409e10	−1.21787	−0.608933	+	0.793221i	$0.708402\pi$
$810$	0	0
$811$	−2.91983e10	−1.92214	−0.961070	−	0.276306i	$-0.910890\pi$
$811$	−2.91983e10	−1.92214	−0.961070	+	0.276306i	$0.910890\pi$
$812$	0	0
$813$	9.14890e9	0.597107
$814$	0	0
$815$	−8.53482e8	−0.0552259
$816$	0	0
$817$	−4.16816e9	−0.267403
$818$	0	0
$819$	−2.54906e9	−0.162139
$820$	0	0
$821$	−2.08184e10	−1.31294	−0.656471	−	0.754352i	$-0.727951\pi$
$821$	−2.08184e10	−1.31294	−0.656471	+	0.754352i	$0.727951\pi$
$822$	0	0
$823$	−6.73804e9	−0.421341	−0.210671	−	0.977557i	$-0.567565\pi$
$823$	−6.73804e9	−0.421341	−0.210671	+	0.977557i	$0.567565\pi$
$824$	0	0
$825$	2.58188e8	0.0160083
$826$	0	0
$827$	−1.90547e10	−1.17147	−0.585736	−	0.810502i	$-0.699194\pi$
$827$	−1.90547e10	−1.17147	−0.585736	+	0.810502i	$0.699194\pi$
$828$	0	0
$829$	2.56278e10	1.56232	0.781162	−	0.624329i	$-0.214627\pi$
$829$	2.56278e10	1.56232	0.781162	+	0.624329i	$0.214627\pi$
$830$	0	0
$831$	−1.12125e10	−0.677793
$832$	0	0
$833$	6.97350e9	0.418017
$834$	0	0
$835$	−3.14831e9	−0.187143
$836$	0	0
$837$	−2.73987e8	−0.0161507
$838$	0	0
$839$	−1.91353e9	−0.111859	−0.0559293	−	0.998435i	$-0.517812\pi$
$839$	−1.91353e9	−0.111859	−0.0559293	+	0.998435i	$0.517812\pi$
$840$	0	0
$841$	−1.44258e10	−0.836284
$842$	0	0
$843$	−1.34792e10	−0.774938
$844$	0	0
$845$	−5.30556e9	−0.302505
$846$	0	0
$847$	−1.48314e10	−0.838668
$848$	0	0
$849$	−7.53872e9	−0.422786
$850$	0	0
$851$	−4.80343e9	−0.267176
$852$	0	0
$853$	−6.28076e9	−0.346490	−0.173245	−	0.984879i	$-0.555425\pi$
$853$	−6.28076e9	−0.346490	−0.173245	+	0.984879i	$0.555425\pi$
$854$	0	0
$855$	−1.34974e9	−0.0738533
$856$	0	0
$857$	−7.09935e9	−0.385288	−0.192644	−	0.981269i	$-0.561706\pi$
$857$	−7.09935e9	−0.385288	−0.192644	+	0.981269i	$0.561706\pi$
$858$	0	0
$859$	−5.72657e9	−0.308261	−0.154130	−	0.988051i	$-0.549258\pi$
$859$	−5.72657e9	−0.308261	−0.154130	+	0.988051i	$0.549258\pi$
$860$	0	0
$861$	3.05811e9	0.163284
$862$	0	0
$863$	−8.14767e9	−0.431515	−0.215757	−	0.976447i	$-0.569222\pi$
$863$	−8.14767e9	−0.431515	−0.215757	+	0.976447i	$0.569222\pi$
$864$	0	0
$865$	1.57906e9	0.0829548
$866$	0	0
$867$	−1.57150e10	−0.818932
$868$	0	0
$869$	−1.88846e9	−0.0976197
$870$	0	0
$871$	−2.97939e9	−0.152779
$872$	0	0
$873$	−7.32941e8	−0.0372837
$874$	0	0
$875$	1.51562e9	0.0764828
$876$	0	0
$877$	1.01644e10	0.508844	0.254422	−	0.967093i	$-0.418115\pi$
$877$	1.01644e10	0.508844	0.254422	+	0.967093i	$0.418115\pi$
$878$	0	0
$879$	1.50285e10	0.746372
$880$	0	0
$881$	3.15577e10	1.55485	0.777427	−	0.628974i	$-0.216525\pi$
$881$	3.15577e10	1.55485	0.777427	+	0.628974i	$0.216525\pi$
$882$	0	0
$883$	2.83065e10	1.38364	0.691822	−	0.722068i	$-0.256808\pi$
$883$	2.83065e10	1.38364	0.691822	+	0.722068i	$0.256808\pi$
$884$	0	0
$885$	6.24603e9	0.302902
$886$	0	0
$887$	1.29779e10	0.624414	0.312207	−	0.950014i	$-0.398932\pi$
$887$	1.29779e10	0.624414	0.312207	+	0.950014i	$0.398932\pi$
$888$	0	0
$889$	3.68462e9	0.175888
$890$	0	0
$891$	−3.25242e8	−0.0154040
$892$	0	0
$893$	−9.41214e9	−0.442291
$894$	0	0
$895$	−1.00059e10	−0.466526
$896$	0	0
$897$	8.73144e9	0.403936
$898$	0	0
$899$	7.39737e8	0.0339561
$900$	0	0
$901$	2.49738e10	1.13749
$902$	0	0
$903$	−5.89598e9	−0.266470
$904$	0	0
$905$	−2.54766e9	−0.114254
$906$	0	0
$907$	3.48257e10	1.54980	0.774898	−	0.632086i	$-0.217801\pi$
$907$	3.48257e10	1.54980	0.774898	+	0.632086i	$0.217801\pi$
$908$	0	0
$909$	1.99241e9	0.0879842
$910$	0	0
$911$	−2.65554e10	−1.16369	−0.581846	−	0.813299i	$-0.697669\pi$
$911$	−2.65554e10	−1.16369	−0.581846	+	0.813299i	$0.697669\pi$
$912$	0	0
$913$	2.94310e9	0.127985
$914$	0	0
$915$	5.86163e9	0.252956
$916$	0	0
$917$	−2.02966e10	−0.869220
$918$	0	0
$919$	−2.92706e10	−1.24402	−0.622011	−	0.783009i	$-0.713684\pi$
$919$	−2.92706e10	−1.24402	−0.622011	+	0.783009i	$0.713684\pi$
$920$	0	0
$921$	6.82320e9	0.287792
$922$	0	0
$923$	−1.65429e10	−0.692477
$924$	0	0
$925$	−1.04578e9	−0.0434455
$926$	0	0
$927$	−1.08712e10	−0.448229
$928$	0	0
$929$	−2.90944e10	−1.19057	−0.595284	−	0.803515i	$-0.702960\pi$
$929$	−2.90944e10	−1.19057	−0.595284	+	0.803515i	$0.702960\pi$
$930$	0	0
$931$	3.27889e9	0.133169
$932$	0	0
$933$	−4.60034e9	−0.185440
$934$	0	0
$935$	2.40990e9	0.0964182
$936$	0	0
$937$	1.78798e10	0.710027	0.355013	−	0.934861i	$-0.384476\pi$
$937$	1.78798e10	0.710027	0.355013	+	0.934861i	$0.384476\pi$
$938$	0	0
$939$	2.55002e10	1.00511
$940$	0	0
$941$	2.83901e10	1.11072	0.555358	−	0.831611i	$-0.312581\pi$
$941$	2.83901e10	1.11072	0.555358	+	0.831611i	$0.312581\pi$
$942$	0	0
$943$	−1.04751e10	−0.406788
$944$	0	0
$945$	−1.90925e9	−0.0735956
$946$	0	0
$947$	−8.12148e9	−0.310749	−0.155375	−	0.987856i	$-0.549658\pi$
$947$	−8.12148e9	−0.310749	−0.155375	+	0.987856i	$0.549658\pi$
$948$	0	0
$949$	2.45701e10	0.933199
$950$	0	0
$951$	−2.11856e10	−0.798748
$952$	0	0
$953$	−5.32773e10	−1.99396	−0.996982	−	0.0776343i	$-0.975263\pi$
$953$	−5.32773e10	−1.99396	−0.996982	+	0.0776343i	$0.975263\pi$
$954$	0	0
$955$	−8.26554e9	−0.307086
$956$	0	0
$957$	8.78118e8	0.0323863
$958$	0	0
$959$	4.38302e10	1.60475
$960$	0	0
$961$	−2.73188e10	−0.992957
$962$	0	0
$963$	−1.65812e10	−0.598306
$964$	0	0
$965$	−2.09257e10	−0.749608
$966$	0	0
$967$	5.34246e9	0.189998	0.0949989	−	0.995477i	$-0.469715\pi$
$967$	5.34246e9	0.189998	0.0949989	+	0.995477i	$0.469715\pi$
$968$	0	0
$969$	−1.25984e10	−0.444818
$970$	0	0
$971$	3.70155e10	1.29753	0.648764	−	0.760990i	$-0.275286\pi$
$971$	3.70155e10	1.29753	0.648764	+	0.760990i	$0.275286\pi$
$972$	0	0
$973$	−2.70235e10	−0.940472
$974$	0	0
$975$	1.90097e9	0.0656839
$976$	0	0
$977$	−2.25193e10	−0.772544	−0.386272	−	0.922385i	$-0.626237\pi$
$977$	−2.25193e10	−0.772544	−0.386272	+	0.922385i	$0.626237\pi$
$978$	0	0
$979$	5.84078e9	0.198944
$980$	0	0
$981$	1.19542e10	0.404277
$982$	0	0
$983$	4.33313e10	1.45501	0.727503	−	0.686105i	$-0.240681\pi$
$983$	4.33313e10	1.45501	0.727503	+	0.686105i	$0.240681\pi$
$984$	0	0
$985$	−1.62724e9	−0.0542532
$986$	0	0
$987$	−1.33137e10	−0.440748
$988$	0	0
$989$	2.01958e10	0.663856
$990$	0	0
$991$	1.16405e10	0.379937	0.189969	−	0.981790i	$-0.439161\pi$
$991$	1.16405e10	0.379937	0.189969	+	0.981790i	$0.439161\pi$
$992$	0	0
$993$	1.37575e10	0.445880
$994$	0	0
$995$	2.21641e9	0.0713295
$996$	0	0
$997$	−3.18333e10	−1.01730	−0.508650	−	0.860974i	$-0.669855\pi$
$997$	−3.18333e10	−1.01730	−0.508650	+	0.860974i	$0.669855\pi$
$998$	0	0
$999$	1.31738e9	0.0418055

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	240.8.a.f.1.1		1
4.3	odd	2		120.8.a.b.1.1	✓	1
12.11	even	2		360.8.a.a.1.1		1
20.3	even	4		600.8.f.b.49.2		2
20.7	even	4		600.8.f.b.49.1		2
20.19	odd	2		600.8.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.8.a.b.1.1	✓	1	4.3	odd	2
240.8.a.f.1.1		1	1.1	even	1	trivial
360.8.a.a.1.1		1	12.11	even	2
600.8.a.d.1.1		1	20.19	odd	2
600.8.f.b.49.1		2	20.7	even	4
600.8.f.b.49.2		2	20.3	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	240.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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