LMFDB - Embedded newform 608.6.b.b.303.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [608,6,Mod(303,608)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(608, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("608.303");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 608 = 2^{5} \cdot 19 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	608.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$97.5133624463$
Analytic rank:	$0$
Dimension:	$96$
Twist minimal:	no (minimal twist has level 152)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			303.5
Character	$\chi$	$=$	608.303
Dual form			608.6.b.b.303.92

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-26.7990i q^{3} -21.0026i q^{5} -145.610i q^{7} -475.186 q^{9} +O(q^{10})$	$q-26.7990i q^{3} -21.0026i q^{5} -145.610i q^{7} -475.186 q^{9} -493.253 q^{11} -1139.17 q^{13} -562.849 q^{15} -224.335 q^{17} +(-1276.47 + 920.174i) q^{19} -3902.19 q^{21} -1407.82i q^{23} +2683.89 q^{25} +6222.35i q^{27} +4135.14 q^{29} +4725.25 q^{31} +13218.7i q^{33} -3058.18 q^{35} -9780.32 q^{37} +30528.6i q^{39} -19921.2i q^{41} +9031.93 q^{43} +9980.15i q^{45} +1705.85i q^{47} -4395.16 q^{49} +6011.95i q^{51} +14052.8 q^{53} +10359.6i q^{55} +(24659.7 + 34208.1i) q^{57} -50708.3i q^{59} +14964.0i q^{61} +69191.7i q^{63} +23925.5i q^{65} -31416.7i q^{67} -37728.2 q^{69} -41786.5 q^{71} +24873.4 q^{73} -71925.6i q^{75} +71822.4i q^{77} -94780.2 q^{79} +51282.6 q^{81} -80919.0 q^{83} +4711.62i q^{85} -110818. i q^{87} +74956.5i q^{89} +165874. i q^{91} -126632. i q^{93} +(19326.1 + 26809.2i) q^{95} -73252.3i q^{97} +234387. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 96 q - 6168 q^{9}+O(q^{10}) $ 96 * q - 6168 * q^9	$ 96 q - 6168 q^{9} - 944 q^{11} - 3832 q^{17} - 5240 q^{19} - 62504 q^{25} - 7720 q^{35} - 45096 q^{43} - 210840 q^{49} - 36336 q^{57} - 4336 q^{73} - 20624 q^{81} - 52152 q^{83} + 752768 q^{99}+O(q^{100}) $ 96 * q - 6168 * q^9 - 944 * q^11 - 3832 * q^17 - 5240 * q^19 - 62504 * q^25 - 7720 * q^35 - 45096 * q^43 - 210840 * q^49 - 36336 * q^57 - 4336 * q^73 - 20624 * q^81 - 52152 * q^83 + 752768 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/608\mathbb{Z}\right)^\times$.

$n$	$97$	$191$	$229$
$\chi(n)$	$-1$	$-1$	$-1$

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
$3$		−	26.7990i		−	1.71916i	−0.511004	−	0.859578i	$-0.670726\pi$
$3$		−	26.7990i		−	1.71916i	0.511004	−	0.859578i	$-0.329274\pi$
$4$		0			0
$5$		−	21.0026i		−	0.375706i	−0.982197	−	0.187853i	$-0.939847\pi$
$5$		−	21.0026i		−	0.375706i	0.982197	−	0.187853i	$-0.0601529\pi$
$6$		0			0
$7$		−	145.610i		−	1.12317i	−0.827420	−	0.561584i	$-0.810192\pi$
$7$		−	145.610i		−	1.12317i	0.827420	−	0.561584i	$-0.189808\pi$
$8$		0			0
$9$	−475.186			−1.95550
$10$		0			0
$11$	−493.253			−1.22910			−0.614552	−	0.788877i	$-0.710663\pi$
$11$	−493.253			−1.22910			−0.614552	+	0.788877i	$0.710663\pi$
$12$		0			0
$13$	−1139.17			−1.86952			−0.934758	−	0.355284i	$-0.884384\pi$
$13$	−1139.17			−1.86952			−0.934758	+	0.355284i	$0.884384\pi$
$14$		0			0
$15$	−562.849			−0.645898
$16$		0			0
$17$	−224.335			−0.188267			−0.0941336	−	0.995560i	$-0.530008\pi$
$17$	−224.335			−0.188267			−0.0941336	+	0.995560i	$0.530008\pi$
$18$		0			0
$19$	−1276.47	+	920.174i	−0.811198	+	0.584771i
$19$	−1276.47	+	920.174i	−0.811198	+	0.584771i
$20$		0			0
$21$	−3902.19			−1.93090
$22$		0			0
$23$		−	1407.82i		−	0.554918i	−0.960738	−	0.277459i	$-0.910508\pi$
$23$		−	1407.82i		−	0.554918i	0.960738	−	0.277459i	$-0.0894922\pi$
$24$		0			0
$25$	2683.89			0.858845
$26$		0			0
$27$			6222.35i			1.64265i
$28$		0			0
$29$	4135.14			0.913051			0.456526	−	0.889710i	$-0.349094\pi$
$29$	4135.14			0.913051			0.456526	+	0.889710i	$0.349094\pi$
$30$		0			0
$31$	4725.25			0.883122			0.441561	−	0.897231i	$-0.354425\pi$
$31$	4725.25			0.883122			0.441561	+	0.897231i	$0.354425\pi$
$32$		0			0
$33$			13218.7i			2.11302i
$34$		0			0
$35$	−3058.18			−0.421981
$36$		0			0
$37$	−9780.32			−1.17449			−0.587244	−	0.809410i	$-0.699787\pi$
$37$	−9780.32			−1.17449			−0.587244	+	0.809410i	$0.699787\pi$
$38$		0			0
$39$			30528.6i			3.21399i
$40$		0			0
$41$		−	19921.2i		−	1.85079i	−0.379005	−	0.925395i	$-0.623734\pi$
$41$		−	19921.2i		−	1.85079i	0.379005	−	0.925395i	$-0.376266\pi$
$42$		0			0
$43$	9031.93			0.744920			0.372460	−	0.928048i	$-0.378514\pi$
$43$	9031.93			0.744920			0.372460	+	0.928048i	$0.378514\pi$
$44$		0			0
$45$			9980.15i			0.734693i
$46$		0			0
$47$			1705.85i			0.112641i	0.998413	+	0.0563205i	$0.0179369\pi$
$47$			1705.85i			0.112641i	−0.998413	+	0.0563205i	$0.982063\pi$
$48$		0			0
$49$	−4395.16			−0.261508
$50$		0			0
$51$			6011.95i			0.323661i
$52$		0			0
$53$	14052.8			0.687186			0.343593	−	0.939119i	$-0.388356\pi$
$53$	14052.8			0.687186			0.343593	+	0.939119i	$0.388356\pi$
$54$		0			0
$55$			10359.6i			0.461782i
$56$		0			0
$57$	24659.7	+	34208.1i	1.00531	+	1.39458i
$58$		0			0
$59$		−	50708.3i		−	1.89648i	−0.317550	−	0.948241i	$-0.602860\pi$
$59$		−	50708.3i		−	1.89648i	0.317550	−	0.948241i	$-0.397140\pi$
$60$		0			0
$61$			14964.0i			0.514899i	0.966292	+	0.257450i	$0.0828822\pi$
$61$			14964.0i			0.514899i	−0.966292	+	0.257450i	$0.917118\pi$
$62$		0			0
$63$			69191.7i			2.19635i
$64$		0			0
$65$			23925.5i			0.702389i
$66$		0			0
$67$		−	31416.7i		−	0.855014i	−0.904012	−	0.427507i	$-0.859392\pi$
$67$		−	31416.7i		−	0.855014i	0.904012	−	0.427507i	$-0.140608\pi$
$68$		0			0
$69$	−37728.2			−0.953990
$70$		0			0
$71$	−41786.5			−0.983763			−0.491882	−	0.870662i	$-0.663691\pi$
$71$	−41786.5			−0.983763			−0.491882	+	0.870662i	$0.663691\pi$
$72$		0			0
$73$	24873.4			0.546297			0.273149	−	0.961972i	$-0.411935\pi$
$73$	24873.4			0.546297			0.273149	+	0.961972i	$0.411935\pi$
$74$		0			0
$75$		−	71925.6i		−	1.47649i
$76$		0			0
$77$			71822.4i			1.38049i
$78$		0			0
$79$	−94780.2			−1.70864			−0.854319	−	0.519749i	$-0.826026\pi$
$79$	−94780.2			−1.70864			−0.854319	+	0.519749i	$0.826026\pi$
$80$		0			0
$81$	51282.6			0.868476
$82$		0			0
$83$	−80919.0			−1.28930			−0.644652	−	0.764476i	$-0.722998\pi$
$83$	−80919.0			−1.28930			−0.644652	+	0.764476i	$0.722998\pi$
$84$		0			0
$85$			4711.62i			0.0707331i
$86$		0			0
$87$		−	110818.i		−	1.56968i
$88$		0			0
$89$			74956.5i			1.00308i	0.865135	+	0.501539i	$0.167233\pi$
$89$			74956.5i			1.00308i	−0.865135	+	0.501539i	$0.832767\pi$
$90$		0			0
$91$			165874.i			2.09978i
$92$		0			0
$93$		−	126632.i		−	1.51822i
$94$		0			0
$95$	19326.1	+	26809.2i	0.219702	+	0.304772i
$96$		0			0
$97$		−	73252.3i		−	0.790481i	−0.918578	−	0.395241i	$-0.870661\pi$
$97$		−	73252.3i		−	0.790481i	0.918578	−	0.395241i	$-0.129339\pi$
$98$		0			0
$99$	234387.			2.40351
$100$		0			0
$101$			142888.i			1.39377i	0.717183	+	0.696885i	$0.245431\pi$
$101$			142888.i			1.39377i	−0.717183	+	0.696885i	$0.754569\pi$
$102$		0			0
$103$	115182.			1.06977			0.534887	−	0.844923i	$-0.320354\pi$
$103$	115182.			1.06977			0.534887	+	0.844923i	$0.320354\pi$
$104$		0			0
$105$			81956.2i			0.725452i
$106$		0			0
$107$		−	98713.8i		−	0.833525i	−0.909015	−	0.416762i	$-0.863165\pi$
$107$		−	98713.8i		−	0.833525i	0.909015	−	0.416762i	$-0.136835\pi$
$108$		0			0
$109$	−5533.19			−0.0446076			−0.0223038	−	0.999751i	$-0.507100\pi$
$109$	−5533.19			−0.0446076			−0.0223038	+	0.999751i	$0.507100\pi$
$110$		0			0
$111$			262103.i			2.01913i
$112$		0			0
$113$		−	81496.1i		−	0.600400i	−0.953876	−	0.300200i	$-0.902947\pi$
$113$		−	81496.1i		−	0.600400i	0.953876	−	0.300200i	$-0.0970534\pi$
$114$		0			0
$115$	−29568.0			−0.208486
$116$		0			0
$117$	541317.			3.65584
$118$		0			0
$119$			32665.3i			0.211456i
$120$		0			0
$121$	82247.8			0.510694
$122$		0			0
$123$	−533869.			−3.18180
$124$		0			0
$125$		−	122002.i		−	0.698379i
$126$		0			0
$127$	79964.3			0.439933			0.219967	−	0.975507i	$-0.429405\pi$
$127$	79964.3			0.439933			0.219967	+	0.975507i	$0.429405\pi$
$128$		0			0
$129$		−	242047.i		−	1.28063i
$130$		0			0
$131$	−255293.			−1.29975			−0.649875	−	0.760041i	$-0.725179\pi$
$131$	−255293.			−1.29975			−0.649875	+	0.760041i	$0.725179\pi$
$132$		0			0
$133$	133986.	+	185866.i	0.656797	+	0.911112i
$134$		0			0
$135$	130686.			0.617154
$136$		0			0
$137$	205724.			0.936449			0.468225	−	0.883609i	$-0.344894\pi$
$137$	205724.			0.936449			0.468225	+	0.883609i	$0.344894\pi$
$138$		0			0
$139$	39802.2			0.174731			0.0873655	−	0.996176i	$-0.472155\pi$
$139$	39802.2			0.174731			0.0873655	+	0.996176i	$0.472155\pi$
$140$		0			0
$141$	45715.1			0.193648
$142$		0			0
$143$	561898.			2.29783
$144$		0			0
$145$		−	86848.7i		−	0.343039i
$146$		0			0
$147$			117786.i			0.449573i
$148$		0			0
$149$		−	78750.6i		−	0.290595i	−0.989388	−	0.145298i	$-0.953586\pi$
$149$		−	78750.6i		−	0.290595i	0.989388	−	0.145298i	$-0.0464140\pi$
$150$		0			0
$151$	−123465.			−0.440659			−0.220330	−	0.975425i	$-0.570713\pi$
$151$	−123465.			−0.440659			−0.220330	+	0.975425i	$0.570713\pi$
$152$		0			0
$153$	106601.			0.368156
$154$		0			0
$155$		−	99242.6i		−	0.331794i
$156$		0			0
$157$		−	346612.i		−	1.12226i	−0.827726	−	0.561132i	$-0.810366\pi$
$157$		−	346612.i		−	1.12226i	0.827726	−	0.561132i	$-0.189634\pi$
$158$		0			0
$159$		−	376602.i		−	1.18138i
$160$		0			0
$161$	−204993.			−0.623266
$162$		0			0
$163$	235957.			0.695608			0.347804	−	0.937567i	$-0.386927\pi$
$163$	235957.			0.695608			0.347804	+	0.937567i	$0.386927\pi$
$164$		0			0
$165$	277627.			0.793875
$166$		0			0
$167$	343571.			0.953292			0.476646	−	0.879095i	$-0.341852\pi$
$167$	343571.			0.953292			0.476646	+	0.879095i	$0.341852\pi$
$168$		0			0
$169$	926411.			2.49509
$170$		0			0
$171$	606561.	−	437254.i	1.58630	−	1.14352i
$172$		0			0
$173$	−164315.			−0.417410			−0.208705	−	0.977979i	$-0.566925\pi$
$173$	−164315.			−0.417410			−0.208705	+	0.977979i	$0.566925\pi$
$174$		0			0
$175$		−	390800.i		−	0.964628i
$176$		0			0
$177$	−1.35893e6			−3.26035
$178$		0			0
$179$			180154.i			0.420254i	0.977674	+	0.210127i	$0.0673877\pi$
$179$			180154.i			0.420254i	−0.977674	+	0.210127i	$0.932612\pi$
$180$		0			0
$181$	−43451.8			−0.0985852			−0.0492926	−	0.998784i	$-0.515697\pi$
$181$	−43451.8			−0.0985852			−0.0492926	+	0.998784i	$0.515697\pi$
$182$		0			0
$183$	401019.			0.885192
$184$		0			0
$185$			205412.i			0.441262i
$186$		0			0
$187$	110654.			0.231400
$188$		0			0
$189$	906035.			1.84497
$190$		0			0
$191$			615794.i			1.22138i	0.791868	+	0.610692i	$0.209109\pi$
$191$			615794.i			1.22138i	−0.791868	+	0.610692i	$0.790891\pi$
$192$		0			0
$193$			432576.i			0.835929i	0.908463	+	0.417964i	$0.137256\pi$
$193$			432576.i			0.835929i	−0.908463	+	0.417964i	$0.862744\pi$
$194$		0			0
$195$	641179.			1.20752
$196$		0			0
$197$			164024.i			0.301121i	0.988601	+	0.150561i	$0.0481079\pi$
$197$			164024.i			0.301121i	−0.988601	+	0.150561i	$0.951892\pi$
$198$		0			0
$199$			113211.i			0.202655i	0.994853	+	0.101327i	$0.0323090\pi$
$199$			113211.i			0.202655i	−0.994853	+	0.101327i	$0.967691\pi$
$200$		0			0
$201$	−841936.			−1.46990
$202$		0			0
$203$		−	602116.i		−	1.02551i
$204$		0			0
$205$	−418398.			−0.695353
$206$		0			0
$207$			668978.i			1.08514i
$208$		0			0
$209$	629624.	−	453879.i	0.997046	−	0.718744i
$210$		0			0
$211$		−	372916.i		−	0.576641i	−0.957534	−	0.288320i	$-0.906903\pi$
$211$		−	372916.i		−	0.576641i	0.957534	−	0.288320i	$-0.0930968\pi$
$212$		0			0
$213$			1.11984e6i			1.69124i
$214$		0			0
$215$		−	189694.i		−	0.279871i
$216$		0			0
$217$		−	688042.i		−	0.991895i
$218$		0			0
$219$		−	666583.i		−	0.939170i
$220$		0			0
$221$	255555.			0.351969
$222$		0			0
$223$	−756854.			−1.01918			−0.509589	−	0.860418i	$-0.670202\pi$
$223$	−756854.			−1.01918			−0.509589	+	0.860418i	$0.670202\pi$
$224$		0			0
$225$	−1.27535e6			−1.67947
$226$		0			0
$227$			1.47405e6i			1.89866i	0.314284	+	0.949329i	$0.398236\pi$
$227$			1.47405e6i			1.89866i	−0.314284	+	0.949329i	$0.601764\pi$
$228$		0			0
$229$		−	666013.i		−	0.839255i	−0.907696	−	0.419628i	$-0.862161\pi$
$229$		−	666013.i		−	0.839255i	0.907696	−	0.419628i	$-0.137839\pi$
$230$		0			0
$231$	1.92477e6			2.37328
$232$		0			0
$233$	−50108.9			−0.0604680			−0.0302340	−	0.999543i	$-0.509625\pi$
$233$	−50108.9			−0.0604680			−0.0302340	+	0.999543i	$0.509625\pi$
$234$		0			0
$235$	35827.3			0.0423199
$236$		0			0
$237$			2.54002e6i			2.93742i
$238$		0			0
$239$			795963.i			0.901360i	0.892686	+	0.450680i	$0.148818\pi$
$239$			795963.i			0.901360i	−0.892686	+	0.450680i	$0.851182\pi$
$240$		0			0
$241$		−	1.26884e6i		−	1.40723i	−0.710582	−	0.703614i	$-0.751568\pi$
$241$		−	1.26884e6i		−	1.40723i	0.710582	−	0.703614i	$-0.248432\pi$
$242$		0			0
$243$			137710.i			0.149606i
$244$		0			0
$245$			92309.9i			0.0982501i
$246$		0			0
$247$	1.45411e6	−	1.04823e6i	1.51655	−	1.09324i
$248$		0			0
$249$			2.16855e6i			2.21652i
$250$		0			0
$251$	1.17545e6			1.17766			0.588830	−	0.808257i	$-0.299589\pi$
$251$	1.17545e6			1.17766			0.588830	+	0.808257i	$0.299589\pi$
$252$		0			0
$253$			694413.i			0.682051i
$254$		0			0
$255$	126267.			0.121601
$256$		0			0
$257$		−	238571.i		−	0.225312i	−0.993634	−	0.112656i	$-0.964064\pi$
$257$		−	238571.i		−	0.225312i	0.993634	−	0.112656i	$-0.0359359\pi$
$258$		0			0
$259$			1.42411e6i			1.31915i
$260$		0			0
$261$	−1.96496e6			−1.78547
$262$		0			0
$263$			838872.i			0.747836i	0.927462	+	0.373918i	$0.121986\pi$
$263$			838872.i			0.747836i	−0.927462	+	0.373918i	$0.878014\pi$
$264$		0			0
$265$		−	295146.i		−	0.258180i
$266$		0			0
$267$	2.00876e6			1.72445
$268$		0			0
$269$	1.14494e6			0.964720			0.482360	−	0.875973i	$-0.339780\pi$
$269$	1.14494e6			0.964720			0.482360	+	0.875973i	$0.339780\pi$
$270$		0			0
$271$			2.34706e6i			1.94134i	0.240420	+	0.970669i	$0.422715\pi$
$271$			2.34706e6i			1.94134i	−0.240420	+	0.970669i	$0.577285\pi$
$272$		0			0
$273$	4.44525e6			3.60985
$274$		0			0
$275$	−1.32384e6			−1.05561
$276$		0			0
$277$			2.31818e6i			1.81530i	0.419733	+	0.907648i	$0.362124\pi$
$277$			2.31818e6i			1.81530i	−0.419733	+	0.907648i	$0.637876\pi$
$278$		0			0
$279$	−2.24537e6			−1.72694
$280$		0			0
$281$			94475.4i			0.0713761i	0.999363	+	0.0356881i	$0.0113623\pi$
$281$			94475.4i			0.0713761i	−0.999363	+	0.0356881i	$0.988638\pi$
$282$		0			0
$283$	−1.26821e6			−0.941297			−0.470648	−	0.882321i	$-0.655980\pi$
$283$	−1.26821e6			−0.941297			−0.470648	+	0.882321i	$0.655980\pi$
$284$		0			0
$285$	718460.	−	517919.i	0.523951	−	0.377702i
$286$		0			0
$287$	−2.90073e6			−2.07875
$288$		0			0
$289$	−1.36953e6			−0.964555
$290$		0			0
$291$	−1.96309e6			−1.35896
$292$		0			0
$293$	134861.			0.0917733			0.0458867	−	0.998947i	$-0.485389\pi$
$293$	134861.			0.0917733			0.0458867	+	0.998947i	$0.485389\pi$
$294$		0			0
$295$	−1.06501e6			−0.712520
$296$		0			0
$297$		−	3.06920e6i		−	2.01899i
$298$		0			0
$299$			1.60375e6i			1.03743i
$300$		0			0
$301$		−	1.31514e6i		−	0.836670i
$302$		0			0
$303$	3.82924e6			2.39611
$304$		0			0
$305$	314282.			0.193451
$306$		0			0
$307$		−	511744.i		−	0.309889i	−0.987923	−	0.154945i	$-0.950480\pi$
$307$		−	511744.i		−	0.309889i	0.987923	−	0.154945i	$-0.0495199\pi$
$308$		0			0
$309$		−	3.08677e6i		−	1.83911i
$310$		0			0
$311$			1.11518e6i			0.653796i	0.945060	+	0.326898i	$0.106003\pi$
$311$			1.11518e6i			0.653796i	−0.945060	+	0.326898i	$0.893997\pi$
$312$		0			0
$313$	529817.			0.305679			0.152839	−	0.988251i	$-0.451158\pi$
$313$	529817.			0.305679			0.152839	+	0.988251i	$0.451158\pi$
$314$		0			0
$315$	1.45321e6			0.825184
$316$		0			0
$317$	−1.19430e6			−0.667519			−0.333759	−	0.942658i	$-0.608317\pi$
$317$	−1.19430e6			−0.667519			−0.333759	+	0.942658i	$0.608317\pi$
$318$		0			0
$319$	−2.03967e6			−1.12223
$320$		0			0
$321$	−2.64543e6			−1.43296
$322$		0			0
$323$	286357.	−	206427.i	0.152722	−	0.110093i
$324$		0			0
$325$	−3.05740e6			−1.60562
$326$		0			0
$327$			148284.i			0.0766875i
$328$		0			0
$329$	248388.			0.126515
$330$		0			0
$331$		−	104325.i		−	0.0523382i	−0.999658	−	0.0261691i	$-0.991669\pi$
$331$		−	104325.i		−	0.0523382i	0.999658	−	0.0261691i	$-0.00833083\pi$
$332$		0			0
$333$	4.64747e6			2.29671
$334$		0			0
$335$	−659832.			−0.321234
$336$		0			0
$337$		−	1.32414e6i		−	0.635126i	−0.948237	−	0.317563i	$-0.897136\pi$
$337$		−	1.32414e6i		−	0.635126i	0.948237	−	0.317563i	$-0.102864\pi$
$338$		0			0
$339$	−2.18401e6			−1.03218
$340$		0			0
$341$	−2.33075e6			−1.08545
$342$		0			0
$343$		−	1.80728e6i		−	0.829451i
$344$		0			0
$345$			792392.i			0.358420i
$346$		0			0
$347$	1.87691e6			0.836795			0.418398	−	0.908264i	$-0.362592\pi$
$347$	1.87691e6			0.836795			0.418398	+	0.908264i	$0.362592\pi$
$348$		0			0
$349$			3.88979e6i			1.70947i	0.519061	+	0.854737i	$0.326282\pi$
$349$			3.88979e6i			1.70947i	−0.519061	+	0.854737i	$0.673718\pi$
$350$		0			0
$351$		−	7.08831e6i		−	3.07096i
$352$		0			0
$353$	2.11291e6			0.902494			0.451247	−	0.892399i	$-0.350979\pi$
$353$	2.11291e6			0.902494			0.451247	+	0.892399i	$0.350979\pi$
$354$		0			0
$355$			877627.i			0.369606i
$356$		0			0
$357$	875398.			0.363525
$358$		0			0
$359$		−	1.05579e6i		−	0.432355i	−0.976354	−	0.216177i	$-0.930641\pi$
$359$		−	1.05579e6i		−	0.432355i	0.976354	−	0.216177i	$-0.0693589\pi$
$360$		0			0
$361$	782657.	−	2.34915e6i	0.316085	−	0.948731i
$362$		0			0
$363$		−	2.20416e6i		−	0.877963i
$364$		0			0
$365$		−	522407.i		−	0.205247i
$366$		0			0
$367$			2.65352e6i			1.02839i	0.857674	+	0.514194i	$0.171909\pi$
$367$			2.65352e6i			1.02839i	−0.857674	+	0.514194i	$0.828091\pi$
$368$		0			0
$369$			9.46630e6i			3.61922i
$370$		0			0
$371$		−	2.04623e6i		−	0.771826i
$372$		0			0
$373$	2.37251e6			0.882950			0.441475	−	0.897274i	$-0.354455\pi$
$373$	2.37251e6			0.882950			0.441475	+	0.897274i	$0.354455\pi$
$374$		0			0
$375$	−3.26953e6			−1.20062
$376$		0			0
$377$	−4.71062e6			−1.70696
$378$		0			0
$379$		−	3.06924e6i		−	1.09757i	−0.835963	−	0.548786i	$-0.815090\pi$
$379$		−	3.06924e6i		−	1.09757i	0.835963	−	0.548786i	$-0.184910\pi$
$380$		0			0
$381$		−	2.14296e6i		−	0.756314i
$382$		0			0
$383$	816237.			0.284328			0.142164	−	0.989843i	$-0.454594\pi$
$383$	816237.			0.284328			0.142164	+	0.989843i	$0.454594\pi$
$384$		0			0
$385$	1.50846e6			0.518659
$386$		0			0
$387$	−4.29185e6			−1.45669
$388$		0			0
$389$		−	4.42722e6i		−	1.48340i	−0.670733	−	0.741699i	$-0.734021\pi$
$389$		−	4.42722e6i		−	1.48340i	0.670733	−	0.741699i	$-0.265979\pi$
$390$		0			0
$391$			315824.i			0.104473i
$392$		0			0
$393$			6.84158e6i			2.23447i
$394$		0			0
$395$			1.99063e6i			0.641946i
$396$		0			0
$397$		−	1.59231e6i		−	0.507049i	−0.967329	−	0.253525i	$-0.918410\pi$
$397$		−	1.59231e6i		−	0.507049i	0.967329	−	0.253525i	$-0.0815899\pi$
$398$		0			0
$399$	4.98103e6	−	3.59070e6i	1.56634	−	1.12914i
$400$		0			0
$401$		−	392603.i		−	0.121925i	−0.998140	−	0.0609624i	$-0.980583\pi$
$401$		−	392603.i		−	0.121925i	0.998140	−	0.0609624i	$-0.0194170\pi$
$402$		0			0
$403$	−5.38285e6			−1.65101
$404$		0			0
$405$		−	1.07707e6i		−	0.326292i
$406$		0			0
$407$	4.82418e6			1.44357
$408$		0			0
$409$		−	2.62661e6i		−	0.776404i	−0.921574	−	0.388202i	$-0.873096\pi$
$409$		−	2.62661e6i		−	0.776404i	0.921574	−	0.388202i	$-0.126904\pi$
$410$		0			0
$411$		−	5.51321e6i		−	1.60990i
$412$		0			0
$413$	−7.38362e6			−2.13007
$414$		0			0
$415$			1.69951e6i			0.484400i
$416$		0			0
$417$		−	1.06666e6i		−	0.300390i
$418$		0			0
$419$	5.96529e6			1.65996			0.829978	−	0.557796i	$-0.188353\pi$
$419$	5.96529e6			1.65996			0.829978	+	0.557796i	$0.188353\pi$
$420$		0			0
$421$	6.34217e6			1.74394			0.871972	−	0.489555i	$-0.162841\pi$
$421$	6.34217e6			1.74394			0.871972	+	0.489555i	$0.162841\pi$
$422$		0			0
$423$		−	810597.i		−	0.220269i
$424$		0			0
$425$	−602090.			−0.161692
$426$		0			0
$427$	2.17890e6			0.578319
$428$		0			0
$429$		−	1.50583e7i		−	3.95033i
$430$		0			0
$431$	−3.62170e6			−0.939117			−0.469558	−	0.882901i	$-0.655587\pi$
$431$	−3.62170e6			−0.939117			−0.469558	+	0.882901i	$0.655587\pi$
$432$		0			0
$433$			4.27544e6i			1.09588i	0.836519	+	0.547938i	$0.184587\pi$
$433$			4.27544e6i			1.09588i	−0.836519	+	0.547938i	$0.815413\pi$
$434$		0			0
$435$	−2.32746e6			−0.589738
$436$		0			0
$437$	1.29544e6	+	1.79705e6i	0.324500	+	0.450148i
$438$		0			0
$439$	−7.02642e6			−1.74009			−0.870047	−	0.492969i	$-0.835912\pi$
$439$	−7.02642e6			−1.74009			−0.870047	+	0.492969i	$0.835912\pi$
$440$		0			0
$441$	2.08852e6			0.511378
$442$		0			0
$443$	933828.			0.226078			0.113039	−	0.993591i	$-0.463942\pi$
$443$	933828.			0.226078			0.113039	+	0.993591i	$0.463942\pi$
$444$		0			0
$445$	1.57428e6			0.376862
$446$		0			0
$447$	−2.11044e6			−0.499578
$448$		0			0
$449$		−	4.86595e6i		−	1.13907i	−0.821966	−	0.569537i	$-0.807123\pi$
$449$		−	4.86595e6i		−	1.13907i	0.821966	−	0.569537i	$-0.192877\pi$
$450$		0			0
$451$			9.82622e6i			2.27481i
$452$		0			0
$453$			3.30875e6i			0.757562i
$454$		0			0
$455$	3.48378e6			0.788901
$456$		0			0
$457$	−1.42961e6			−0.320205			−0.160102	−	0.987100i	$-0.551182\pi$
$457$	−1.42961e6			−0.320205			−0.160102	+	0.987100i	$0.551182\pi$
$458$		0			0
$459$		−	1.39589e6i		−	0.309257i
$460$		0			0
$461$		−	2.46248e6i		−	0.539660i	−0.962908	−	0.269830i	$-0.913033\pi$
$461$		−	2.46248e6i		−	0.539660i	0.962908	−	0.269830i	$-0.0869675\pi$
$462$		0			0
$463$			504497.i			0.109372i	0.998504	+	0.0546860i	$0.0174158\pi$
$463$			504497.i			0.109372i	−0.998504	+	0.0546860i	$0.982584\pi$
$464$		0			0
$465$	−2.65960e6			−0.570406
$466$		0			0
$467$	−5.85370e6			−1.24205			−0.621024	−	0.783792i	$-0.713283\pi$
$467$	−5.85370e6			−1.24205			−0.621024	+	0.783792i	$0.713283\pi$
$468$		0			0
$469$	−4.57457e6			−0.960325
$470$		0			0
$471$	−9.28887e6			−1.92935
$472$		0			0
$473$	−4.45503e6			−0.915583
$474$		0			0
$475$	−3.42591e6	+	2.46965e6i	−0.696693	+	0.502228i
$476$		0			0
$477$	−6.67772e6			−1.34379
$478$		0			0
$479$			522241.i			0.104000i	0.998647	+	0.0519999i	$0.0165596\pi$
$479$			522241.i			0.104000i	−0.998647	+	0.0519999i	$0.983440\pi$
$480$		0			0
$481$	1.11414e7			2.19573
$482$		0			0
$483$			5.49360e6i			1.07149i
$484$		0			0
$485$	−1.53849e6			−0.296989
$486$		0			0
$487$	3.26179e6			0.623210			0.311605	−	0.950212i	$-0.399134\pi$
$487$	3.26179e6			0.623210			0.311605	+	0.950212i	$0.399134\pi$
$488$		0			0
$489$		−	6.32342e6i		−	1.19586i
$490$		0			0
$491$	3.10184e6			0.580651			0.290326	−	0.956928i	$-0.406236\pi$
$491$	3.10184e6			0.580651			0.290326	+	0.956928i	$0.406236\pi$
$492$		0			0
$493$	−927656.			−0.171898
$494$		0			0
$495$		−	4.92274e6i		−	0.903013i
$496$		0			0
$497$			6.08452e6i			1.10493i
$498$		0			0
$499$	−4.36981e6			−0.785618			−0.392809	−	0.919620i	$-0.628497\pi$
$499$	−4.36981e6			−0.785618			−0.392809	+	0.919620i	$0.628497\pi$
$500$		0			0
$501$		−	9.20737e6i		−	1.63886i
$502$		0			0
$503$		−	1.70975e6i		−	0.301310i	−0.988586	−	0.150655i	$-0.951862\pi$
$503$		−	1.70975e6i		−	0.301310i	0.988586	−	0.150655i	$-0.0481382\pi$
$504$		0			0
$505$	3.00101e6			0.523648
$506$		0			0
$507$		−	2.48269e7i		−	4.28945i
$508$		0			0
$509$	−9.97226e6			−1.70608			−0.853040	−	0.521846i	$-0.825244\pi$
$509$	−9.97226e6			−1.70608			−0.853040	+	0.521846i	$0.825244\pi$
$510$		0			0
$511$		−	3.62181e6i		−	0.613584i
$512$		0			0
$513$	−5.72565e6	−	7.94266e6i	−0.960575	−	1.33252i
$514$		0			0
$515$		−	2.41913e6i		−	0.401921i
$516$		0			0
$517$		−	841417.i		−	0.138447i
$518$		0			0
$519$			4.40348e6i			0.717592i
$520$		0			0
$521$		−	2.56863e6i		−	0.414579i	−0.978280	−	0.207289i	$-0.933536\pi$
$521$		−	2.56863e6i		−	0.414579i	0.978280	−	0.207289i	$-0.0664642\pi$
$522$		0			0
$523$		−	4.26748e6i		−	0.682208i	−0.940025	−	0.341104i	$-0.889199\pi$
$523$		−	4.26748e6i		−	0.682208i	0.940025	−	0.341104i	$-0.110801\pi$
$524$		0			0
$525$	−1.04731e7			−1.65835
$526$		0			0
$527$	−1.06004e6			−0.166263
$528$		0			0
$529$	4.45438e6			0.692066
$530$		0			0
$531$			2.40959e7i			3.70857i
$532$		0			0
$533$			2.26936e7i			3.46008i
$534$		0			0
$535$	−2.07325e6			−0.313160
$536$		0			0
$537$	4.82795e6			0.722482
$538$		0			0
$539$	2.16793e6			0.321420
$540$		0			0
$541$			4.75656e6i			0.698714i	0.936990	+	0.349357i	$0.113600\pi$
$541$			4.75656e6i			0.698714i	−0.936990	+	0.349357i	$0.886400\pi$
$542$		0			0
$543$			1.16447e6i			0.169483i
$544$		0			0
$545$			116211.i			0.0167594i
$546$		0			0
$547$			9.55912e6i			1.36600i	0.730420	+	0.682998i	$0.239324\pi$
$547$			9.55912e6i			1.36600i	−0.730420	+	0.682998i	$0.760676\pi$
$548$		0			0
$549$		−	7.11067e6i		−	1.00688i
$550$		0			0
$551$	−5.27839e6	+	3.80505e6i	−0.740666	+	0.533926i
$552$		0			0
$553$			1.38009e7i			1.91909i
$554$		0			0
$555$	5.50484e6			0.758599
$556$		0			0
$557$			7.72306e6i			1.05475i	0.849631	+	0.527377i	$0.176824\pi$
$557$			7.72306e6i			1.05475i	−0.849631	+	0.527377i	$0.823176\pi$
$558$		0			0
$559$	−1.02889e7			−1.39264
$560$		0			0
$561$		−	2.96541e6i		−	0.397812i
$562$		0			0
$563$			5.73714e6i			0.762825i	0.924405	+	0.381412i	$0.124562\pi$
$563$			5.73714e6i			0.762825i	−0.924405	+	0.381412i	$0.875438\pi$
$564$		0			0
$565$	−1.71163e6			−0.225574
$566$		0			0
$567$		−	7.46724e6i		−	0.975445i
$568$		0			0
$569$			2.68314e6i			0.347426i	0.984796	+	0.173713i	$0.0555765\pi$
$569$			2.68314e6i			0.347426i	−0.984796	+	0.173713i	$0.944423\pi$
$570$		0			0
$571$	3.41086e6			0.437797			0.218899	−	0.975748i	$-0.429754\pi$
$571$	3.41086e6			0.437797			0.218899	+	0.975748i	$0.429754\pi$
$572$		0			0
$573$	1.65027e7			2.09975
$574$		0			0
$575$		−	3.77844e6i		−	0.476588i
$576$		0			0
$577$	−1.36156e7			−1.70253			−0.851267	−	0.524733i	$-0.824165\pi$
$577$	−1.36156e7			−1.70253			−0.851267	+	0.524733i	$0.824165\pi$
$578$		0			0
$579$	1.15926e7			1.43709
$580$		0			0
$581$			1.17826e7i			1.44811i
$582$		0			0
$583$	−6.93161e6			−0.844623
$584$		0			0
$585$		−	1.13691e7i		−	1.37352i
$586$		0			0
$587$	1.32830e7			1.59111			0.795554	−	0.605882i	$-0.207180\pi$
$587$	1.32830e7			1.59111			0.795554	+	0.605882i	$0.207180\pi$
$588$		0			0
$589$	−6.03165e6	+	4.34805e6i	−0.716387	+	0.516424i
$590$		0			0
$591$	4.39568e6			0.517675
$592$		0			0
$593$	563393.			0.0657923			0.0328961	−	0.999459i	$-0.489527\pi$
$593$	563393.			0.0657923			0.0328961	+	0.999459i	$0.489527\pi$
$594$		0			0
$595$	686057.			0.0794452
$596$		0			0
$597$	3.03395e6			0.348395
$598$		0			0
$599$	−2.01388e6			−0.229333			−0.114667	−	0.993404i	$-0.536580\pi$
$599$	−2.01388e6			−0.229333			−0.114667	+	0.993404i	$0.536580\pi$
$600$		0			0
$601$			3.24098e6i			0.366008i	0.983112	+	0.183004i	$0.0585821\pi$
$601$			3.24098e6i			0.366008i	−0.983112	+	0.183004i	$0.941418\pi$
$602$		0			0
$603$			1.49288e7i			1.67198i
$604$		0			0
$605$		−	1.72742e6i		−	0.191871i
$606$		0			0
$607$	−6.08533e6			−0.670367			−0.335183	−	0.942153i	$-0.608798\pi$
$607$	−6.08533e6			−0.670367			−0.335183	+	0.942153i	$0.608798\pi$
$608$		0			0
$609$	−1.61361e7			−1.76301
$610$		0			0
$611$		−	1.94325e6i		−	0.210584i
$612$		0			0
$613$		−	5.83100e6i		−	0.626747i	−0.949630	−	0.313373i	$-0.898541\pi$
$613$		−	5.83100e6i		−	0.626747i	0.949630	−	0.313373i	$-0.101459\pi$
$614$		0			0
$615$			1.12127e7i			1.19542i
$616$		0			0
$617$	−2.21336e6			−0.234066			−0.117033	−	0.993128i	$-0.537338\pi$
$617$	−2.21336e6			−0.234066			−0.117033	+	0.993128i	$0.537338\pi$
$618$		0			0
$619$	−2.99462e6			−0.314134			−0.157067	−	0.987588i	$-0.550204\pi$
$619$	−2.99462e6			−0.314134			−0.157067	+	0.987588i	$0.550204\pi$
$620$		0			0
$621$	8.75997e6			0.911536
$622$		0			0
$623$	1.09144e7			1.12663
$624$		0			0
$625$	5.82480e6			0.596460
$626$		0			0
$627$	−1.21635e7	−	1.68733e7i	−1.23563	−	1.71408i
$628$		0			0
$629$	2.19407e6			0.221118
$630$		0			0
$631$			1.17471e7i			1.17451i	0.809401	+	0.587257i	$0.199792\pi$
$631$			1.17471e7i			1.17451i	−0.809401	+	0.587257i	$0.800208\pi$
$632$		0			0
$633$	−9.99379e6			−0.991336
$634$		0			0
$635$		−	1.67946e6i		−	0.165286i
$636$		0			0
$637$	5.00683e6			0.488893
$638$		0			0
$639$	1.98564e7			1.92375
$640$		0			0
$641$		−	1.79920e7i		−	1.72956i	−0.502153	−	0.864779i	$-0.667459\pi$
$641$		−	1.79920e7i		−	1.72956i	0.502153	−	0.864779i	$-0.332541\pi$
$642$		0			0
$643$	5.76427e6			0.549816			0.274908	−	0.961471i	$-0.411353\pi$
$643$	5.76427e6			0.549816			0.274908	+	0.961471i	$0.411353\pi$
$644$		0			0
$645$	−5.08361e6			−0.481142
$646$		0			0
$647$			6.59568e6i			0.619440i	0.950828	+	0.309720i	$0.100235\pi$
$647$			6.59568e6i			0.619440i	−0.950828	+	0.309720i	$0.899765\pi$
$648$		0			0
$649$			2.50120e7i			2.33097i
$650$		0			0
$651$	−1.84388e7			−1.70522
$652$		0			0
$653$			1.68301e7i			1.54456i	0.635285	+	0.772278i	$0.280883\pi$
$653$			1.68301e7i			1.54456i	−0.635285	+	0.772278i	$0.719117\pi$
$654$		0			0
$655$			5.36181e6i			0.488324i
$656$		0			0
$657$	−1.18195e7			−1.06828
$658$		0			0
$659$		−	2.69661e6i		−	0.241882i	−0.992660	−	0.120941i	$-0.961409\pi$
$659$		−	2.69661e6i		−	0.241882i	0.992660	−	0.120941i	$-0.0385913\pi$
$660$		0			0
$661$	−1.64214e7			−1.46186			−0.730932	−	0.682451i	$-0.760914\pi$
$661$	−1.64214e7			−1.46186			−0.730932	+	0.682451i	$0.760914\pi$
$662$		0			0
$663$		−	6.84862e6i		−	0.605089i
$664$		0			0
$665$	3.90368e6	−	2.81406e6i	0.342310	−	0.246763i
$666$		0			0
$667$		−	5.82155e6i		−	0.506668i
$668$		0			0
$669$			2.02829e7i			1.75213i
$670$		0			0
$671$		−	7.38103e6i		−	0.632864i
$672$		0			0
$673$			4.26643e6i			0.363100i	0.983382	+	0.181550i	$0.0581115\pi$
$673$			4.26643e6i			0.363100i	−0.983382	+	0.181550i	$0.941889\pi$
$674$		0			0
$675$			1.67001e7i			1.41078i
$676$		0			0
$677$	5.64948e6			0.473737			0.236868	−	0.971542i	$-0.423879\pi$
$677$	5.64948e6			0.473737			0.236868	+	0.971542i	$0.423879\pi$
$678$		0			0
$679$	−1.06662e7			−0.887844
$680$		0			0
$681$	3.95030e7			3.26409
$682$		0			0
$683$		−	1.25549e7i		−	1.02982i	−0.857244	−	0.514910i	$-0.827825\pi$
$683$		−	1.25549e7i		−	1.02982i	0.857244	−	0.514910i	$-0.172175\pi$
$684$		0			0
$685$		−	4.32075e6i		−	0.351830i
$686$		0			0
$687$	−1.78485e7			−1.44281
$688$		0			0
$689$	−1.60085e7			−1.28471
$690$		0			0
$691$	1.53817e7			1.22549			0.612743	−	0.790282i	$-0.290066\pi$
$691$	1.53817e7			1.22549			0.612743	+	0.790282i	$0.290066\pi$
$692$		0			0
$693$		−	3.41290e7i		−	2.69955i
$694$		0			0
$695$		−	835950.i		−	0.0656475i
$696$		0			0
$697$			4.46903e6i			0.348443i
$698$		0			0
$699$			1.34287e6i			0.103954i
$700$		0			0
$701$		−	1.49505e7i		−	1.14910i	−0.818468	−	0.574552i	$-0.805176\pi$
$701$		−	1.49505e7i		−	1.14910i	0.818468	−	0.574552i	$-0.194824\pi$
$702$		0			0
$703$	1.24843e7	−	8.99960e6i	0.952743	−	0.686807i
$704$		0			0
$705$		−	960137.i		−	0.0727546i
$706$		0			0
$707$	2.08058e7			1.56544
$708$		0			0
$709$			2.38754e6i			0.178375i	0.996015	+	0.0891875i	$0.0284271\pi$
$709$			2.38754e6i			0.178375i	−0.996015	+	0.0891875i	$0.971573\pi$
$710$		0			0
$711$	4.50383e7			3.34124
$712$		0			0
$713$		−	6.65232e6i		−	0.490060i
$714$		0			0
$715$		−	1.18013e7i		−	0.863308i
$716$		0			0
$717$	2.13310e7			1.54958
$718$		0			0
$719$		−	2.04965e7i		−	1.47863i	−0.673362	−	0.739313i	$-0.735150\pi$
$719$		−	2.04965e7i		−	1.47863i	0.673362	−	0.739313i	$-0.264850\pi$
$720$		0			0
$721$		−	1.67716e7i		−	1.20154i
$722$		0			0
$723$	−3.40037e7			−2.41925
$724$		0			0
$725$	1.10983e7			0.784170
$726$		0			0
$727$		−	1.82750e7i		−	1.28239i	−0.767376	−	0.641197i	$-0.778438\pi$
$727$		−	1.82750e7i		−	1.28239i	0.767376	−	0.641197i	$-0.221562\pi$
$728$		0			0
$729$	1.61522e7			1.12567
$730$		0			0
$731$	−2.02618e6			−0.140244
$732$		0			0
$733$		−	5.13204e6i		−	0.352801i	−0.984318	−	0.176400i	$-0.943555\pi$
$733$		−	5.13204e6i		−	0.352801i	0.984318	−	0.176400i	$-0.0564454\pi$
$734$		0			0
$735$	2.47381e6			0.168907
$736$		0			0
$737$			1.54964e7i			1.05090i
$738$		0			0
$739$	−1.85372e7			−1.24863			−0.624315	−	0.781172i	$-0.714622\pi$
$739$	−1.85372e7			−1.24863			−0.624315	+	0.781172i	$0.714622\pi$
$740$		0			0
$741$	−2.80916e7	−	3.89688e7i	−1.87945	−	2.60718i
$742$		0			0
$743$	2.53021e7			1.68145			0.840727	−	0.541459i	$-0.182128\pi$
$743$	2.53021e7			1.68145			0.840727	+	0.541459i	$0.182128\pi$
$744$		0			0
$745$	−1.65397e6			−0.109178
$746$		0			0
$747$	3.84516e7			2.52123
$748$		0			0
$749$	−1.43737e7			−0.936189
$750$		0			0
$751$	−1.71342e7			−1.10857			−0.554287	−	0.832325i	$-0.687009\pi$
$751$	−1.71342e7			−1.10857			−0.554287	+	0.832325i	$0.687009\pi$
$752$		0			0
$753$		−	3.15009e7i		−	2.02458i
$754$		0			0
$755$			2.59310e6i			0.165558i
$756$		0			0
$757$			2.65953e7i			1.68680i	0.537283	+	0.843402i	$0.319451\pi$
$757$			2.65953e7i			1.68680i	−0.537283	+	0.843402i	$0.680549\pi$
$758$		0			0
$759$	1.86096e7			1.17255
$760$		0			0
$761$	−7.82943e6			−0.490082			−0.245041	−	0.969513i	$-0.578801\pi$
$761$	−7.82943e6			−0.490082			−0.245041	+	0.969513i	$0.578801\pi$
$762$		0			0
$763$			805685.i			0.0501019i
$764$		0			0
$765$		−	2.23890e6i		−	0.138318i
$766$		0			0
$767$			5.77653e7i			3.54551i
$768$		0			0
$769$	−1.96822e7			−1.20021			−0.600105	−	0.799921i	$-0.704874\pi$
$769$	−1.96822e7			−1.20021			−0.600105	+	0.799921i	$0.704874\pi$
$770$		0			0
$771$	−6.39347e6			−0.387347
$772$		0			0
$773$	2.19583e7			1.32175			0.660875	−	0.750496i	$-0.270185\pi$
$773$	2.19583e7			1.32175			0.660875	+	0.750496i	$0.270185\pi$
$774$		0			0
$775$	1.26821e7			0.758465
$776$		0			0
$777$	3.81647e7			2.26782
$778$		0			0
$779$	1.83310e7	+	2.54289e7i	1.08229	+	1.50136i
$780$		0			0
$781$	2.06114e7			1.20915
$782$		0			0
$783$			2.57303e7i			1.49982i
$784$		0			0
$785$	−7.27977e6			−0.421641
$786$		0			0
$787$			5.93795e6i			0.341743i	0.985293	+	0.170871i	$0.0546583\pi$
$787$			5.93795e6i			0.341743i	−0.985293	+	0.170871i	$0.945342\pi$
$788$		0			0
$789$	2.24809e7			1.28565
$790$		0			0
$791$	−1.18666e7			−0.674351
$792$		0			0
$793$		−	1.70465e7i		−	0.962613i
$794$		0			0
$795$	−7.90963e6			−0.443852
$796$		0			0
$797$	−2.23517e7			−1.24642			−0.623210	−	0.782055i	$-0.714172\pi$
$797$	−2.23517e7			−1.24642			−0.623210	+	0.782055i	$0.714172\pi$
$798$		0			0
$799$		−	382682.i		−	0.0212066i
$800$		0			0
$801$		−	3.56183e7i		−	1.96152i
$802$		0			0
$803$	−1.22689e7			−0.671455
$804$		0			0
$805$			4.30538e6i			0.234165i
$806$		0			0
$807$		−	3.06832e7i		−	1.65850i
$808$		0			0
$809$	7.33990e6			0.394293			0.197146	−	0.980374i	$-0.436833\pi$
$809$	7.33990e6			0.394293			0.197146	+	0.980374i	$0.436833\pi$
$810$		0			0
$811$		−	3.20637e7i		−	1.71184i	−0.517112	−	0.855918i	$-0.672993\pi$
$811$		−	3.20637e7i		−	1.71184i	0.517112	−	0.855918i	$-0.327007\pi$
$812$		0			0
$813$	6.28989e7			3.33746
$814$		0			0
$815$		−	4.95572e6i		−	0.261344i
$816$		0			0
$817$	−1.15290e7	+	8.31095e6i	−0.604277	+	0.435608i
$818$		0			0
$819$		−	7.88209e7i		−	4.10612i
$820$		0			0
$821$			5.75457e6i			0.297958i	0.988840	+	0.148979i	$0.0475987\pi$
$821$			5.75457e6i			0.297958i	−0.988840	+	0.148979i	$0.952401\pi$
$822$		0			0
$823$			2.07497e7i			1.06785i	0.845531	+	0.533926i	$0.179284\pi$
$823$			2.07497e7i			1.06785i	−0.845531	+	0.533926i	$0.820716\pi$
$824$		0			0
$825$			3.54775e7i			1.81476i
$826$		0			0
$827$			3.05052e6i			0.155100i	0.996988	+	0.0775498i	$0.0247097\pi$
$827$			3.05052e6i			0.155100i	−0.996988	+	0.0775498i	$0.975290\pi$
$828$		0			0
$829$	1.74416e7			0.881453			0.440726	−	0.897642i	$-0.354721\pi$
$829$	1.74416e7			0.881453			0.440726	+	0.897642i	$0.354721\pi$
$830$		0			0
$831$	6.21248e7			3.12078
$832$		0			0
$833$	985988.			0.0492333
$834$		0			0
$835$		−	7.21589e6i		−	0.358157i
$836$		0			0
$837$			2.94022e7i			1.45066i
$838$		0			0
$839$	−1.26572e7			−0.620775			−0.310388	−	0.950610i	$-0.600459\pi$
$839$	−1.26572e7			−0.620775			−0.310388	+	0.950610i	$0.600459\pi$
$840$		0			0
$841$	−3.41177e6			−0.166337
$842$		0			0
$843$	2.53185e6			0.122707
$844$		0			0
$845$		−	1.94570e7i		−	0.937422i
$846$		0			0
$847$		−	1.19761e7i		−	0.573596i
$848$		0			0
$849$			3.39869e7i			1.61824i
$850$		0			0
$851$			1.37690e7i			0.651744i
$852$		0			0
$853$		−	3.45130e7i		−	1.62409i	−0.583594	−	0.812045i	$-0.698354\pi$
$853$		−	3.45130e7i		−	1.62409i	0.583594	−	0.812045i	$-0.301646\pi$
$854$		0			0
$855$	−9.18348e6	−	1.27394e7i	−0.429627	−	0.595981i
$856$		0			0
$857$			1.98163e7i			0.921662i	0.887488	+	0.460831i	$0.152449\pi$
$857$			1.98163e7i			0.921662i	−0.887488	+	0.460831i	$0.847551\pi$
$858$		0			0
$859$	−3.49359e7			−1.61543			−0.807716	−	0.589572i	$-0.799296\pi$
$859$	−3.49359e7			−1.61543			−0.807716	+	0.589572i	$0.799296\pi$
$860$		0			0
$861$			7.77365e7i			3.57369i
$862$		0			0
$863$	−1.55830e7			−0.712238			−0.356119	−	0.934441i	$-0.615900\pi$
$863$	−1.55830e7			−0.712238			−0.356119	+	0.934441i	$0.615900\pi$
$864$		0			0
$865$			3.45105e6i			0.156823i
$866$		0			0
$867$			3.67021e7i			1.65822i
$868$		0			0
$869$	4.67507e7			2.10009
$870$		0			0
$871$			3.57889e7i			1.59846i
$872$		0			0
$873$			3.48085e7i			1.54579i
$874$		0			0
$875$	−1.77646e7			−0.784398
$876$		0			0
$877$	−3.87361e7			−1.70066			−0.850329	−	0.526251i	$-0.823597\pi$
$877$	−3.87361e7			−1.70066			−0.850329	+	0.526251i	$0.823597\pi$
$878$		0			0
$879$		−	3.61413e6i		−	0.157773i
$880$		0			0
$881$	1.37878e6			0.0598487			0.0299244	−	0.999552i	$-0.490473\pi$
$881$	1.37878e6			0.0598487			0.0299244	+	0.999552i	$0.490473\pi$
$882$		0			0
$883$	−1.26161e7			−0.544533			−0.272267	−	0.962222i	$-0.587773\pi$
$883$	−1.26161e7			−0.544533			−0.272267	+	0.962222i	$0.587773\pi$
$884$		0			0
$885$			2.85411e7i			1.22493i
$886$		0			0
$887$	−6.28959e6			−0.268419			−0.134210	−	0.990953i	$-0.542850\pi$
$887$	−6.28959e6			−0.268419			−0.134210	+	0.990953i	$0.542850\pi$
$888$		0			0
$889$		−	1.16436e7i		−	0.494119i
$890$		0			0
$891$	−2.52953e7			−1.06745
$892$		0			0
$893$	−1.56968e6	−	2.17747e6i	−0.0658693	−	0.0913742i
$894$		0			0
$895$	3.78371e6			0.157892
$896$		0			0
$897$	4.29788e7			1.78350
$898$		0			0
$899$	1.95396e7			0.806336
$900$		0			0
$901$	−3.15254e6			−0.129375
$902$		0			0
$903$	−3.52443e7			−1.43837
$904$		0			0
$905$			912602.i			0.0370391i
$906$		0			0
$907$			1.03202e7i			0.416555i	0.978070	+	0.208277i	$0.0667856\pi$
$907$			1.03202e7i			0.416555i	−0.978070	+	0.208277i	$0.933214\pi$
$908$		0			0
$909$		−	6.78982e7i		−	2.72552i
$910$		0			0
$911$	2.75921e7			1.10151			0.550756	−	0.834667i	$-0.314340\pi$
$911$	2.75921e7			1.10151			0.550756	+	0.834667i	$0.314340\pi$
$912$		0			0
$913$	3.99136e7			1.58469
$914$		0			0
$915$		−	8.42245e6i		−	0.332572i
$916$		0			0
$917$			3.71731e7i			1.45984i
$918$		0			0
$919$			2.58814e7i			1.01088i	0.862863	+	0.505438i	$0.168669\pi$
$919$			2.58814e7i			1.01088i	−0.862863	+	0.505438i	$0.831331\pi$
$920$		0			0
$921$	−1.37142e7			−0.532748
$922$		0			0
$923$	4.76019e7			1.83916
$924$		0			0
$925$	−2.62493e7			−1.00870
$926$		0			0
$927$	−5.47330e7			−2.09194
$928$		0			0
$929$	−3.76483e7			−1.43122			−0.715610	−	0.698500i	$-0.753851\pi$
$929$	−3.76483e7			−1.43122			−0.715610	+	0.698500i	$0.753851\pi$
$930$		0			0
$931$	5.61030e6	−	4.04432e6i	0.212135	−	0.152922i
$932$		0			0
$933$	2.98856e7			1.12398
$934$		0			0
$935$		−	2.32402e6i		−	0.0869383i
$936$		0			0
$937$	1.77305e7			0.659740			0.329870	−	0.944026i	$-0.392995\pi$
$937$	1.77305e7			0.659740			0.329870	+	0.944026i	$0.392995\pi$
$938$		0			0
$939$		−	1.41986e7i		−	0.525509i
$940$		0			0
$941$	−1.48101e7			−0.545235			−0.272618	−	0.962122i	$-0.587889\pi$
$941$	−1.48101e7			−0.545235			−0.272618	+	0.962122i	$0.587889\pi$
$942$		0			0
$943$	−2.80456e7			−1.02704
$944$		0			0
$945$		−	1.90291e7i		−	0.693168i
$946$		0			0
$947$	−2.82693e7			−1.02433			−0.512165	−	0.858887i	$-0.671156\pi$
$947$	−2.82693e7			−1.02433			−0.512165	+	0.858887i	$0.671156\pi$
$948$		0			0
$949$	−2.83350e7			−1.02131
$950$		0			0
$951$			3.20059e7i			1.14757i
$952$		0			0
$953$			1.07502e7i			0.383427i	0.981451	+	0.191714i	$0.0614045\pi$
$953$			1.07502e7i			0.383427i	−0.981451	+	0.191714i	$0.938595\pi$
$954$		0			0
$955$	1.29333e7			0.458881
$956$		0			0
$957$			5.46611e7i			1.92930i
$958$		0			0
$959$		−	2.99554e7i		−	1.05179i
$960$		0			0
$961$	−6.30116e6			−0.220096
$962$		0			0
$963$			4.69074e7i			1.62996i
$964$		0			0
$965$	9.08523e6			0.314064
$966$		0			0
$967$		−	2.54249e7i		−	0.874364i	−0.899373	−	0.437182i	$-0.855976\pi$
$967$		−	2.54249e7i		−	0.874364i	0.899373	−	0.437182i	$-0.144024\pi$
$968$		0			0
$969$	−5.53204e6	−	7.67408e6i	−0.189267	−	0.262553i
$970$		0			0
$971$			4.98852e7i			1.69794i	0.528437	+	0.848972i	$0.322778\pi$
$971$			4.98852e7i			1.69794i	−0.528437	+	0.848972i	$0.677222\pi$
$972$		0			0
$973$		−	5.79558e6i		−	0.196252i
$974$		0			0
$975$			8.19353e7i			2.76032i
$976$		0			0
$977$			3.09229e7i			1.03644i	0.855247	+	0.518220i	$0.173405\pi$
$977$			3.09229e7i			1.03644i	−0.855247	+	0.518220i	$0.826595\pi$
$978$		0			0
$979$		−	3.69726e7i		−	1.23289i
$980$		0			0
$981$	2.62929e6			0.0872301
$982$		0			0
$983$	−2.14888e7			−0.709296			−0.354648	−	0.935000i	$-0.615399\pi$
$983$	−2.14888e7			−0.709296			−0.354648	+	0.935000i	$0.615399\pi$
$984$		0			0
$985$	3.44493e6			0.113133
$986$		0			0
$987$		−	6.65656e6i		−	0.217499i
$988$		0			0
$989$		−	1.27154e7i		−	0.413369i
$990$		0			0
$991$	−3.12541e7			−1.01094			−0.505468	−	0.862846i	$-0.668680\pi$
$991$	−3.12541e7			−1.01094			−0.505468	+	0.862846i	$0.668680\pi$
$992$		0			0
$993$	−2.79581e6			−0.0899775
$994$		0			0
$995$	2.37773e6			0.0761386
$996$		0			0
$997$		−	2.19290e7i		−	0.698684i	−0.936995	−	0.349342i	$-0.886405\pi$
$997$		−	2.19290e7i		−	0.698684i	0.936995	−	0.349342i	$-0.113595\pi$
$998$		0			0
$999$		−	6.08566e7i		−	1.92927i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	608.6.b.b.303.5		96
4.3	odd	2		152.6.b.b.75.90	yes	96
8.3	odd	2	inner	608.6.b.b.303.6		96
8.5	even	2		152.6.b.b.75.8	yes	96
19.18	odd	2	inner	608.6.b.b.303.91		96
76.75	even	2		152.6.b.b.75.7	✓	96
152.37	odd	2		152.6.b.b.75.89	yes	96
152.75	even	2	inner	608.6.b.b.303.92		96

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.6.b.b.75.7	✓	96	76.75	even	2
152.6.b.b.75.8	yes	96	8.5	even	2
152.6.b.b.75.89	yes	96	152.37	odd	2
152.6.b.b.75.90	yes	96	4.3	odd	2
608.6.b.b.303.5		96	1.1	even	1	trivial
608.6.b.b.303.6		96	8.3	odd	2	inner
608.6.b.b.303.91		96	19.18	odd	2	inner
608.6.b.b.303.92		96	152.75	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 608 = 2^{5} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	608.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-26.7990i q^{3} -21.0026i q^{5} -145.610i q^{7} -475.186 q^{9} +O(q^{10})\)	\(q-26.7990i q^{3} -21.0026i q^{5} -145.610i q^{7} -475.186 q^{9} -493.253 q^{11} -1139.17 q^{13} -562.849 q^{15} -224.335 q^{17} +(-1276.47 + 920.174i) q^{19} -3902.19 q^{21} -1407.82i q^{23} +2683.89 q^{25} +6222.35i q^{27} +4135.14 q^{29} +4725.25 q^{31} +13218.7i q^{33} -3058.18 q^{35} -9780.32 q^{37} +30528.6i q^{39} -19921.2i q^{41} +9031.93 q^{43} +9980.15i q^{45} +1705.85i q^{47} -4395.16 q^{49} +6011.95i q^{51} +14052.8 q^{53} +10359.6i q^{55} +(24659.7 + 34208.1i) q^{57} -50708.3i q^{59} +14964.0i q^{61} +69191.7i q^{63} +23925.5i q^{65} -31416.7i q^{67} -37728.2 q^{69} -41786.5 q^{71} +24873.4 q^{73} -71925.6i q^{75} +71822.4i q^{77} -94780.2 q^{79} +51282.6 q^{81} -80919.0 q^{83} +4711.62i q^{85} -110818. i q^{87} +74956.5i q^{89} +165874. i q^{91} -126632. i q^{93} +(19326.1 + 26809.2i) q^{95} -73252.3i q^{97} +234387. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 6168 q^{9}+O(q^{10}) \) 96 * q - 6168 * q^9	\( 96 q - 6168 q^{9} - 944 q^{11} - 3832 q^{17} - 5240 q^{19} - 62504 q^{25} - 7720 q^{35} - 45096 q^{43} - 210840 q^{49} - 36336 q^{57} - 4336 q^{73} - 20624 q^{81} - 52152 q^{83} + 752768 q^{99}+O(q^{100}) \) 96 * q - 6168 * q^9 - 944 * q^11 - 3832 * q^17 - 5240 * q^19 - 62504 * q^25 - 7720 * q^35 - 45096 * q^43 - 210840 * q^49 - 36336 * q^57 - 4336 * q^73 - 20624 * q^81 - 52152 * q^83 + 752768 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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