LMFDB - Embedded newform 608.4.b.b.303.30

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [608,4,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, 4, names="a")

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

chi := DirichletCharacter("608.303");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 608 = 2^{5} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
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:	$35.8731612835$
Analytic rank:	$0$
Dimension:	$56$
Twist minimal:	no (minimal twist has level 152)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			303.30
Character	$\chi$	$=$	608.303
Dual form			608.4.b.b.303.27

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.255308i q^{3} +11.8859i q^{5} +31.1204i q^{7} +26.9348 q^{9} +O(q^{10})$	$q+0.255308i q^{3} +11.8859i q^{5} +31.1204i q^{7} +26.9348 q^{9} +21.9564 q^{11} +38.7293 q^{13} -3.03456 q^{15} +101.750 q^{17} +(20.2224 - 80.3122i) q^{19} -7.94528 q^{21} +57.3504i q^{23} -16.2739 q^{25} +13.7700i q^{27} -206.088 q^{29} +312.889 q^{31} +5.60563i q^{33} -369.893 q^{35} -156.076 q^{37} +9.88790i q^{39} +399.236i q^{41} -96.2576 q^{43} +320.144i q^{45} -209.396i q^{47} -625.479 q^{49} +25.9775i q^{51} +455.858 q^{53} +260.970i q^{55} +(20.5043 + 5.16293i) q^{57} -451.215i q^{59} +385.501i q^{61} +838.222i q^{63} +460.332i q^{65} -261.804i q^{67} -14.6420 q^{69} -546.989 q^{71} -443.977 q^{73} -4.15485i q^{75} +683.291i q^{77} +950.449 q^{79} +723.724 q^{81} +537.548 q^{83} +1209.38i q^{85} -52.6158i q^{87} -1404.85i q^{89} +1205.27i q^{91} +79.8830i q^{93} +(954.581 + 240.361i) q^{95} +398.059i q^{97} +591.391 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q - 528 q^{9}+O(q^{10}) $ 56 * q - 528 * q^9	$ 56 q - 528 q^{9} + 40 q^{11} - 184 q^{17} + 84 q^{19} - 1504 q^{25} - 40 q^{35} - 576 q^{43} - 3664 q^{49} - 648 q^{57} - 432 q^{73} + 2296 q^{81} - 5376 q^{83} + 6152 q^{99}+O(q^{100}) $ 56 * q - 528 * q^9 + 40 * q^11 - 184 * q^17 + 84 * q^19 - 1504 * q^25 - 40 * q^35 - 576 * q^43 - 3664 * q^49 - 648 * q^57 - 432 * q^73 + 2296 * q^81 - 5376 * q^83 + 6152 * 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
$3$			0.255308i			0.0491340i	0.999698	+	0.0245670i	$0.00782071\pi$
$3$			0.255308i			0.0491340i	−0.999698	+	0.0245670i	$0.992179\pi$
$4$		0			0
$5$			11.8859i			1.06310i	0.847026	+	0.531552i	$0.178391\pi$
$5$			11.8859i			1.06310i	−0.847026	+	0.531552i	$0.821609\pi$
$6$		0			0
$7$			31.1204i			1.68034i	0.542320	+	0.840172i	$0.317546\pi$
$7$			31.1204i			1.68034i	−0.542320	+	0.840172i	$0.682454\pi$
$8$		0			0
$9$	26.9348			0.997586
$10$		0			0
$11$	21.9564			0.601827			0.300913	−	0.953652i	$-0.402709\pi$
$11$	21.9564			0.601827			0.300913	+	0.953652i	$0.402709\pi$
$12$		0			0
$13$	38.7293			0.826276			0.413138	−	0.910669i	$-0.364433\pi$
$13$	38.7293			0.826276			0.413138	+	0.910669i	$0.364433\pi$
$14$		0			0
$15$	−3.03456			−0.0522346
$16$		0			0
$17$	101.750			1.45164			0.725821	−	0.687883i	$-0.241460\pi$
$17$	101.750			1.45164			0.725821	+	0.687883i	$0.241460\pi$
$18$		0			0
$19$	20.2224	−	80.3122i	0.244175	−	0.969731i
$19$	20.2224	−	80.3122i	0.244175	−	0.969731i
$20$		0			0
$21$	−7.94528			−0.0825620
$22$		0			0
$23$			57.3504i			0.519929i	0.965618	+	0.259965i	$0.0837109\pi$
$23$			57.3504i			0.519929i	−0.965618	+	0.259965i	$0.916289\pi$
$24$		0			0
$25$	−16.2739			−0.130191
$26$		0			0
$27$			13.7700i			0.0981494i
$28$		0			0
$29$	−206.088			−1.31964			−0.659819	−	0.751424i	$-0.729367\pi$
$29$	−206.088			−1.31964			−0.659819	+	0.751424i	$0.729367\pi$
$30$		0			0
$31$	312.889			1.81279			0.906395	−	0.422431i	$-0.138823\pi$
$31$	312.889			1.81279			0.906395	+	0.422431i	$0.138823\pi$
$32$		0			0
$33$			5.60563i			0.0295702i
$34$		0			0
$35$	−369.893			−1.78638
$36$		0			0
$37$	−156.076			−0.693481			−0.346741	−	0.937961i	$-0.612712\pi$
$37$	−156.076			−0.693481			−0.346741	+	0.937961i	$0.612712\pi$
$38$		0			0
$39$			9.88790i			0.0405982i
$40$		0			0
$41$			399.236i			1.52074i	0.649493	+	0.760368i	$0.274981\pi$
$41$			399.236i			1.52074i	−0.649493	+	0.760368i	$0.725019\pi$
$42$		0			0
$43$	−96.2576			−0.341375			−0.170688	−	0.985325i	$-0.554599\pi$
$43$	−96.2576			−0.341375			−0.170688	+	0.985325i	$0.554599\pi$
$44$		0			0
$45$			320.144i			1.06054i
$46$		0			0
$47$		−	209.396i		−	0.649863i	−0.945737	−	0.324932i	$-0.894659\pi$
$47$		−	209.396i		−	0.649863i	0.945737	−	0.324932i	$-0.105341\pi$
$48$		0			0
$49$	−625.479			−1.82356
$50$		0			0
$51$			25.9775i			0.0713250i
$52$		0			0
$53$	455.858			1.18145			0.590725	−	0.806873i	$-0.298842\pi$
$53$	455.858			1.18145			0.590725	+	0.806873i	$0.298842\pi$
$54$		0			0
$55$			260.970i			0.639804i
$56$		0			0
$57$	20.5043	+	5.16293i	0.0476468	+	0.0119973i
$58$		0			0
$59$		−	451.215i		−	0.995647i	−0.867279	−	0.497823i	$-0.834133\pi$
$59$		−	451.215i		−	0.995647i	0.867279	−	0.497823i	$-0.165867\pi$
$60$		0			0
$61$			385.501i			0.809154i	0.914504	+	0.404577i	$0.132581\pi$
$61$			385.501i			0.809154i	−0.914504	+	0.404577i	$0.867419\pi$
$62$		0			0
$63$			838.222i			1.67629i
$64$		0			0
$65$			460.332i			0.878417i
$66$		0			0
$67$		−	261.804i		−	0.477380i	−0.971096	−	0.238690i	$-0.923282\pi$
$67$		−	261.804i		−	0.477380i	0.971096	−	0.238690i	$-0.0767179\pi$
$68$		0			0
$69$	−14.6420			−0.0255462
$70$		0			0
$71$	−546.989			−0.914306			−0.457153	−	0.889388i	$-0.651131\pi$
$71$	−546.989			−0.914306			−0.457153	+	0.889388i	$0.651131\pi$
$72$		0			0
$73$	−443.977			−0.711830			−0.355915	−	0.934518i	$-0.615831\pi$
$73$	−443.977			−0.711830			−0.355915	+	0.934518i	$0.615831\pi$
$74$		0			0
$75$		−	4.15485i		−	0.00639680i
$76$		0			0
$77$			683.291i			1.01128i
$78$		0			0
$79$	950.449			1.35359			0.676797	−	0.736170i	$-0.263368\pi$
$79$	950.449			1.35359			0.676797	+	0.736170i	$0.263368\pi$
$80$		0			0
$81$	723.724			0.992763
$82$		0			0
$83$	537.548			0.710886			0.355443	−	0.934698i	$-0.384330\pi$
$83$	537.548			0.710886			0.355443	+	0.934698i	$0.384330\pi$
$84$		0			0
$85$			1209.38i			1.54325i
$86$		0			0
$87$		−	52.6158i		−	0.0648391i
$88$		0			0
$89$		−	1404.85i		−	1.67319i	−0.547822	−	0.836595i	$-0.684543\pi$
$89$		−	1404.85i		−	1.67319i	0.547822	−	0.836595i	$-0.315457\pi$
$90$		0			0
$91$			1205.27i			1.38843i
$92$		0			0
$93$			79.8830i			0.0890697i
$94$		0			0
$95$	954.581	+	240.361i	1.03093	+	0.259584i
$96$		0			0
$97$			398.059i			0.416668i	0.978058	+	0.208334i	$0.0668041\pi$
$97$			398.059i			0.416668i	−0.978058	+	0.208334i	$0.933196\pi$
$98$		0			0
$99$	591.391			0.600374
$100$		0			0
$101$		−	1111.11i		−	1.09464i	−0.836922	−	0.547322i	$-0.815647\pi$
$101$		−	1111.11i		−	1.09464i	0.836922	−	0.547322i	$-0.184353\pi$
$102$		0			0
$103$	−998.893			−0.955571			−0.477786	−	0.878477i	$-0.658560\pi$
$103$	−998.893			−0.955571			−0.477786	+	0.878477i	$0.658560\pi$
$104$		0			0
$105$		−	94.4366i		−	0.0877721i
$106$		0			0
$107$		−	82.6255i		−	0.0746515i	−0.999303	−	0.0373257i	$-0.988116\pi$
$107$		−	82.6255i		−	0.0746515i	0.999303	−	0.0373257i	$-0.0118839\pi$
$108$		0			0
$109$	−836.605			−0.735158			−0.367579	−	0.929992i	$-0.619813\pi$
$109$	−836.605			−0.735158			−0.367579	+	0.929992i	$0.619813\pi$
$110$		0			0
$111$		−	39.8475i		−	0.0340735i
$112$		0			0
$113$			580.927i			0.483620i	0.970324	+	0.241810i	$0.0777411\pi$
$113$			580.927i			0.483620i	−0.970324	+	0.241810i	$0.922259\pi$
$114$		0			0
$115$	−681.659			−0.552739
$116$		0			0
$117$	1043.17			0.824281
$118$		0			0
$119$			3166.49i			2.43926i
$120$		0			0
$121$	−848.918			−0.637805
$122$		0			0
$123$	−101.928			−0.0747199
$124$		0			0
$125$			1292.30i			0.924698i
$126$		0			0
$127$	−1015.44			−0.709496			−0.354748	−	0.934962i	$-0.615433\pi$
$127$	−1015.44			−0.709496			−0.354748	+	0.934962i	$0.615433\pi$
$128$		0			0
$129$		−	24.5753i		−	0.0167732i
$130$		0			0
$131$	−1970.99			−1.31455			−0.657276	−	0.753650i	$-0.728292\pi$
$131$	−1970.99			−1.31455			−0.657276	+	0.753650i	$0.728292\pi$
$132$		0			0
$133$	2499.35	+	629.329i	1.62948	+	0.410299i
$134$		0			0
$135$	−163.668			−0.104343
$136$		0			0
$137$	−755.460			−0.471119			−0.235559	−	0.971860i	$-0.575692\pi$
$137$	−755.460			−0.471119			−0.235559	+	0.971860i	$0.575692\pi$
$138$		0			0
$139$	−2128.52			−1.29884			−0.649419	−	0.760431i	$-0.724988\pi$
$139$	−2128.52			−1.29884			−0.649419	+	0.760431i	$0.724988\pi$
$140$		0			0
$141$	53.4605			0.0319304
$142$		0			0
$143$	850.355			0.497275
$144$		0			0
$145$		−	2449.53i		−	1.40291i
$146$		0			0
$147$		−	159.690i		−	0.0895986i
$148$		0			0
$149$			1029.90i			0.566258i	0.959082	+	0.283129i	$0.0913724\pi$
$149$			1029.90i			0.566258i	−0.959082	+	0.283129i	$0.908628\pi$
$150$		0			0
$151$	−1189.40			−0.641005			−0.320502	−	0.947248i	$-0.603852\pi$
$151$	−1189.40			−0.641005			−0.320502	+	0.947248i	$0.603852\pi$
$152$		0			0
$153$	2740.61			1.44814
$154$		0			0
$155$			3718.96i			1.92719i
$156$		0			0
$157$		−	3448.86i		−	1.75318i	−0.481241	−	0.876588i	$-0.659814\pi$
$157$		−	3448.86i		−	1.75318i	0.481241	−	0.876588i	$-0.340186\pi$
$158$		0			0
$159$			116.384i			0.0580494i
$160$		0			0
$161$	−1784.77			−0.873660
$162$		0			0
$163$	2774.87			1.33340			0.666701	−	0.745325i	$-0.267706\pi$
$163$	2774.87			1.33340			0.666701	+	0.745325i	$0.267706\pi$
$164$		0			0
$165$	−66.6278			−0.0314362
$166$		0			0
$167$	−150.985			−0.0699615			−0.0349807	−	0.999388i	$-0.511137\pi$
$167$	−150.985			−0.0699615			−0.0349807	+	0.999388i	$0.511137\pi$
$168$		0			0
$169$	−697.039			−0.317269
$170$		0			0
$171$	544.686	−	2163.20i	0.243586	−	0.967390i
$172$		0			0
$173$	−1161.80			−0.510579			−0.255289	−	0.966865i	$-0.582171\pi$
$173$	−1161.80			−0.510579			−0.255289	+	0.966865i	$0.582171\pi$
$174$		0			0
$175$		−	506.449i		−	0.218766i
$176$		0			0
$177$	115.199			0.0489201
$178$		0			0
$179$			4506.82i			1.88188i	0.338580	+	0.940938i	$0.390053\pi$
$179$			4506.82i			1.88188i	−0.338580	+	0.940938i	$0.609947\pi$
$180$		0			0
$181$	1300.89			0.534222			0.267111	−	0.963666i	$-0.413931\pi$
$181$	1300.89			0.534222			0.267111	+	0.963666i	$0.413931\pi$
$182$		0			0
$183$	−98.4215			−0.0397570
$184$		0			0
$185$		−	1855.10i		−	0.737243i
$186$		0			0
$187$	2234.05			0.873637
$188$		0			0
$189$	−428.527			−0.164925
$190$		0			0
$191$		−	1713.84i		−	0.649263i	−0.945841	−	0.324631i	$-0.894760\pi$
$191$		−	1713.84i		−	0.649263i	0.945841	−	0.324631i	$-0.105240\pi$
$192$		0			0
$193$			836.962i			0.312154i	0.987745	+	0.156077i	$0.0498849\pi$
$193$			836.962i			0.312154i	−0.987745	+	0.156077i	$0.950115\pi$
$194$		0			0
$195$	−117.526			−0.0431602
$196$		0			0
$197$		−	567.220i		−	0.205141i	−0.994726	−	0.102570i	$-0.967293\pi$
$197$		−	567.220i		−	0.205141i	0.994726	−	0.102570i	$-0.0327067\pi$
$198$		0			0
$199$		−	3533.62i		−	1.25875i	−0.777101	−	0.629376i	$-0.783310\pi$
$199$		−	3533.62i		−	1.25875i	0.777101	−	0.629376i	$-0.216690\pi$
$200$		0			0
$201$	66.8406			0.0234556
$202$		0			0
$203$		−	6413.53i		−	2.21745i
$204$		0			0
$205$	−4745.27			−1.61670
$206$		0			0
$207$			1544.72i			0.518674i
$208$		0			0
$209$	444.010	−	1763.36i	0.146951	−	0.583610i
$210$		0			0
$211$		−	3990.60i		−	1.30201i	−0.759073	−	0.651006i	$-0.774347\pi$
$211$		−	3990.60i		−	1.30201i	0.759073	−	0.651006i	$-0.225653\pi$
$212$		0			0
$213$		−	139.651i		−	0.0449235i
$214$		0			0
$215$		−	1144.11i		−	0.362918i
$216$		0			0
$217$			9737.23i			3.04611i
$218$		0			0
$219$		−	113.351i		−	0.0349751i
$220$		0			0
$221$	3940.70			1.19946
$222$		0			0
$223$	1073.95			0.322498			0.161249	−	0.986914i	$-0.448448\pi$
$223$	1073.95			0.322498			0.161249	+	0.986914i	$0.448448\pi$
$224$		0			0
$225$	−438.334			−0.129877
$226$		0			0
$227$			3165.76i			0.925633i	0.886454	+	0.462816i	$0.153161\pi$
$227$			3165.76i			0.925633i	−0.886454	+	0.462816i	$0.846839\pi$
$228$		0			0
$229$			2500.47i			0.721553i	0.932652	+	0.360776i	$0.117488\pi$
$229$			2500.47i			0.721553i	−0.932652	+	0.360776i	$0.882512\pi$
$230$		0			0
$231$	−174.450			−0.0496880
$232$		0			0
$233$	1034.30			0.290812			0.145406	−	0.989372i	$-0.453551\pi$
$233$	1034.30			0.290812			0.145406	+	0.989372i	$0.453551\pi$
$234$		0			0
$235$	2488.86			0.690873
$236$		0			0
$237$			242.657i			0.0665075i
$238$		0			0
$239$		−	1446.06i		−	0.391370i	−0.980667	−	0.195685i	$-0.937307\pi$
$239$		−	1446.06i		−	0.391370i	0.980667	−	0.195685i	$-0.0626931\pi$
$240$		0			0
$241$		−	3098.89i		−	0.828288i	−0.910211	−	0.414144i	$-0.864081\pi$
$241$		−	3098.89i		−	0.828288i	0.910211	−	0.414144i	$-0.135919\pi$
$242$		0			0
$243$			556.562i			0.146928i
$244$		0			0
$245$		−	7434.37i		−	1.93863i
$246$		0			0
$247$	783.199	−	3110.44i	0.201756	−	0.801265i
$248$		0			0
$249$			137.240i			0.0349287i
$250$		0			0
$251$	−3586.47			−0.901897			−0.450948	−	0.892550i	$-0.648914\pi$
$251$	−3586.47			−0.901897			−0.450948	+	0.892550i	$0.648914\pi$
$252$		0			0
$253$			1259.21i			0.312907i
$254$		0			0
$255$	−308.765			−0.0758260
$256$		0			0
$257$		−	6739.57i		−	1.63581i	−0.575355	−	0.817904i	$-0.695136\pi$
$257$		−	6739.57i		−	1.63581i	0.575355	−	0.817904i	$-0.304864\pi$
$258$		0			0
$259$		−	4857.16i		−	1.16529i
$260$		0			0
$261$	−5550.93			−1.31645
$262$		0			0
$263$		−	2998.75i		−	0.703083i	−0.936172	−	0.351541i	$-0.885658\pi$
$263$		−	2998.75i		−	0.703083i	0.936172	−	0.351541i	$-0.114342\pi$
$264$		0			0
$265$			5418.26i			1.25600i
$266$		0			0
$267$	358.669			0.0822105
$268$		0			0
$269$	−3587.92			−0.813231			−0.406615	−	0.913599i	$-0.633291\pi$
$269$	−3587.92			−0.813231			−0.406615	+	0.913599i	$0.633291\pi$
$270$		0			0
$271$		−	2610.23i		−	0.585092i	−0.956251	−	0.292546i	$-0.905498\pi$
$271$		−	2610.23i		−	0.585092i	0.956251	−	0.292546i	$-0.0945025\pi$
$272$		0			0
$273$	−307.715			−0.0682190
$274$		0			0
$275$	−357.315			−0.0783524
$276$		0			0
$277$		−	5163.37i		−	1.11999i	−0.828497	−	0.559994i	$-0.810803\pi$
$277$		−	5163.37i		−	1.11999i	0.828497	−	0.559994i	$-0.189197\pi$
$278$		0			0
$279$	8427.60			1.80841
$280$		0			0
$281$			5390.26i			1.14433i	0.820140	+	0.572163i	$0.193896\pi$
$281$			5390.26i			1.14433i	−0.820140	+	0.572163i	$0.806104\pi$
$282$		0			0
$283$	7541.59			1.58410			0.792051	−	0.610455i	$-0.209013\pi$
$283$	7541.59			1.58410			0.792051	+	0.610455i	$0.209013\pi$
$284$		0			0
$285$	−61.3659	+	243.712i	−0.0127544	+	0.0506535i
$286$		0			0
$287$	−12424.4			−2.55536
$288$		0			0
$289$	5440.00			1.10727
$290$		0			0
$291$	−101.628			−0.0204726
$292$		0			0
$293$	−2730.96			−0.544520			−0.272260	−	0.962224i	$-0.587771\pi$
$293$	−2730.96			−0.544520			−0.272260	+	0.962224i	$0.587771\pi$
$294$		0			0
$295$	5363.08			1.05848
$296$		0			0
$297$			302.339i			0.0590689i
$298$		0			0
$299$			2221.14i			0.429605i
$300$		0			0
$301$		−	2995.58i		−	0.573628i
$302$		0			0
$303$	283.674			0.0537843
$304$		0			0
$305$	−4582.02			−0.860215
$306$		0			0
$307$			1024.35i			0.190433i	0.995457	+	0.0952163i	$0.0303543\pi$
$307$			1024.35i			0.190433i	−0.995457	+	0.0952163i	$0.969646\pi$
$308$		0			0
$309$		−	255.025i		−	0.0469510i
$310$		0			0
$311$			2526.27i			0.460617i	0.973118	+	0.230308i	$0.0739735\pi$
$311$			2526.27i			0.460617i	−0.973118	+	0.230308i	$0.926027\pi$
$312$		0			0
$313$	10532.3			1.90199			0.950994	−	0.309209i	$-0.100064\pi$
$313$	10532.3			1.90199			0.950994	+	0.309209i	$0.100064\pi$
$314$		0			0
$315$	−9963.00			−1.78207
$316$		0			0
$317$	6189.48			1.09664			0.548322	−	0.836267i	$-0.315267\pi$
$317$	6189.48			1.09664			0.548322	+	0.836267i	$0.315267\pi$
$318$		0			0
$319$	−4524.94			−0.794194
$320$		0			0
$321$	21.0949			0.00366793
$322$		0			0
$323$	2057.62	−	8171.75i	0.354455	−	1.40770i
$324$		0			0
$325$	−630.276			−0.107574
$326$		0			0
$327$		−	213.592i		−	0.0361213i
$328$		0			0
$329$	6516.50			1.09199
$330$		0			0
$331$			4035.04i			0.670048i	0.942210	+	0.335024i	$0.108745\pi$
$331$			4035.04i			0.670048i	−0.942210	+	0.335024i	$0.891255\pi$
$332$		0			0
$333$	−4203.89			−0.691807
$334$		0			0
$335$	3111.77			0.507504
$336$		0			0
$337$		−	9651.06i		−	1.56002i	−0.625767	−	0.780010i	$-0.715214\pi$
$337$		−	9651.06i		−	1.56002i	0.625767	−	0.780010i	$-0.284786\pi$
$338$		0			0
$339$	−148.315			−0.0237622
$340$		0			0
$341$	6869.90			1.09099
$342$		0			0
$343$		−	8790.88i		−	1.38386i
$344$		0			0
$345$		−	174.033i		−	0.0271583i
$346$		0			0
$347$	1311.88			0.202955			0.101478	−	0.994838i	$-0.467643\pi$
$347$	1311.88			0.202955			0.101478	+	0.994838i	$0.467643\pi$
$348$		0			0
$349$		−	7143.37i		−	1.09563i	−0.836599	−	0.547816i	$-0.815459\pi$
$349$		−	7143.37i		−	1.09563i	0.836599	−	0.547816i	$-0.184541\pi$
$350$		0			0
$351$			533.302i			0.0810985i
$352$		0			0
$353$	3355.30			0.505905			0.252952	−	0.967479i	$-0.418598\pi$
$353$	3355.30			0.505905			0.252952	+	0.967479i	$0.418598\pi$
$354$		0			0
$355$		−	6501.44i		−	0.972002i
$356$		0			0
$357$	−808.430			−0.119851
$358$		0			0
$359$			5647.41i			0.830247i	0.909765	+	0.415124i	$0.136262\pi$
$359$			5647.41i			0.830247i	−0.909765	+	0.415124i	$0.863738\pi$
$360$		0			0
$361$	−6041.11	−	3248.21i	−0.880757	−	0.473569i
$362$		0			0
$363$		−	216.735i		−	0.0313379i
$364$		0			0
$365$		−	5277.05i		−	0.756750i
$366$		0			0
$367$		−	6607.22i		−	0.939766i	−0.882729	−	0.469883i	$-0.844296\pi$
$367$		−	6607.22i		−	0.939766i	0.882729	−	0.469883i	$-0.155704\pi$
$368$		0			0
$369$			10753.3i			1.51706i
$370$		0			0
$371$			14186.5i			1.98524i
$372$		0			0
$373$	−9677.75			−1.34342			−0.671709	−	0.740815i	$-0.734439\pi$
$373$	−9677.75			−1.34342			−0.671709	+	0.740815i	$0.734439\pi$
$374$		0			0
$375$	−329.935			−0.0454341
$376$		0			0
$377$	−7981.64			−1.09039
$378$		0			0
$379$		−	7594.94i		−	1.02936i	−0.857384	−	0.514678i	$-0.827911\pi$
$379$		−	7594.94i		−	1.02936i	0.857384	−	0.514678i	$-0.172089\pi$
$380$		0			0
$381$		−	259.251i		−	0.0348604i
$382$		0			0
$383$	−689.095			−0.0919350			−0.0459675	−	0.998943i	$-0.514637\pi$
$383$	−689.095			−0.0919350			−0.0459675	+	0.998943i	$0.514637\pi$
$384$		0			0
$385$	−8121.51			−1.07509
$386$		0			0
$387$	−2592.68			−0.340551
$388$		0			0
$389$			918.749i			0.119749i	0.998206	+	0.0598746i	$0.0190701\pi$
$389$			918.749i			0.119749i	−0.998206	+	0.0598746i	$0.980930\pi$
$390$		0			0
$391$			5835.38i			0.754752i
$392$		0			0
$393$		−	503.210i		−	0.0645893i
$394$		0			0
$395$			11296.9i			1.43901i
$396$		0			0
$397$			6718.99i			0.849411i	0.905332	+	0.424706i	$0.139622\pi$
$397$			6718.99i			0.849411i	−0.905332	+	0.424706i	$0.860378\pi$
$398$		0			0
$399$	−160.673	+	638.103i	−0.0201596	+	0.0800630i
$400$		0			0
$401$		−	2657.88i		−	0.330993i	−0.986210	−	0.165496i	$-0.947077\pi$
$401$		−	2657.88i		−	0.330993i	0.986210	−	0.165496i	$-0.0529226\pi$
$402$		0			0
$403$	12118.0			1.49786
$404$		0			0
$405$			8602.09i			1.05541i
$406$		0			0
$407$	−3426.87			−0.417355
$408$		0			0
$409$		−	865.521i		−	0.104639i	−0.998630	−	0.0523194i	$-0.983339\pi$
$409$		−	865.521i		−	0.104639i	0.998630	−	0.0523194i	$-0.0166614\pi$
$410$		0			0
$411$		−	192.875i		−	0.0231480i
$412$		0			0
$413$	14042.0			1.67303
$414$		0			0
$415$			6389.23i			0.755747i
$416$		0			0
$417$		−	543.427i		−	0.0638171i
$418$		0			0
$419$	−10425.9			−1.21561			−0.607804	−	0.794087i	$-0.707949\pi$
$419$	−10425.9			−1.21561			−0.607804	+	0.794087i	$0.707949\pi$
$420$		0			0
$421$	−8091.59			−0.936722			−0.468361	−	0.883537i	$-0.655155\pi$
$421$	−8091.59			−0.936722			−0.468361	+	0.883537i	$0.655155\pi$
$422$		0			0
$423$		−	5640.05i		−	0.648295i
$424$		0			0
$425$	−1655.86			−0.188991
$426$		0			0
$427$	−11997.0			−1.35966
$428$		0			0
$429$			217.102i			0.0244331i
$430$		0			0
$431$	−4972.36			−0.555708			−0.277854	−	0.960623i	$-0.589623\pi$
$431$	−4972.36			−0.555708			−0.277854	+	0.960623i	$0.589623\pi$
$432$		0			0
$433$		−	4916.49i		−	0.545661i	−0.962062	−	0.272831i	$-0.912040\pi$
$433$		−	4916.49i		−	0.545661i	0.962062	−	0.272831i	$-0.0879599\pi$
$434$		0			0
$435$	625.384			0.0689308
$436$		0			0
$437$	4605.94	+	1159.76i	0.504192	+	0.126954i
$438$		0			0
$439$	9969.73			1.08389			0.541947	−	0.840413i	$-0.317687\pi$
$439$	9969.73			1.08389			0.541947	+	0.840413i	$0.317687\pi$
$440$		0			0
$441$	−16847.2			−1.81915
$442$		0			0
$443$	12954.5			1.38936			0.694678	−	0.719321i	$-0.255547\pi$
$443$	12954.5			1.38936			0.694678	+	0.719321i	$0.255547\pi$
$444$		0			0
$445$	16697.9			1.77878
$446$		0			0
$447$	−262.941			−0.0278225
$448$		0			0
$449$		−	1992.49i		−	0.209424i	−0.994503	−	0.104712i	$-0.966608\pi$
$449$		−	1992.49i		−	0.209424i	0.994503	−	0.104712i	$-0.0333922\pi$
$450$		0			0
$451$			8765.77i			0.915219i
$452$		0			0
$453$		−	303.662i		−	0.0314951i
$454$		0			0
$455$	−14325.7			−1.47604
$456$		0			0
$457$	12912.0			1.32166			0.660831	−	0.750535i	$-0.270204\pi$
$457$	12912.0			1.32166			0.660831	+	0.750535i	$0.270204\pi$
$458$		0			0
$459$			1401.09i			0.142478i
$460$		0			0
$461$			14539.6i			1.46893i	0.678646	+	0.734466i	$0.262567\pi$
$461$			14539.6i			1.46893i	−0.678646	+	0.734466i	$0.737433\pi$
$462$		0			0
$463$			5767.23i			0.578889i	0.957195	+	0.289445i	$0.0934706\pi$
$463$			5767.23i			0.578889i	−0.957195	+	0.289445i	$0.906529\pi$
$464$		0			0
$465$	−949.479			−0.0946904
$466$		0			0
$467$	3777.72			0.374330			0.187165	−	0.982329i	$-0.440070\pi$
$467$	3777.72			0.374330			0.187165	+	0.982329i	$0.440070\pi$
$468$		0			0
$469$	8147.44			0.802162
$470$		0			0
$471$	880.520			0.0861406
$472$		0			0
$473$	−2113.47			−0.205449
$474$		0			0
$475$	−329.096	+	1306.99i	−0.0317894	+	0.126250i
$476$		0			0
$477$	12278.4			1.17860
$478$		0			0
$479$			16308.7i			1.55566i	0.628474	+	0.777830i	$0.283680\pi$
$479$			16308.7i			1.55566i	−0.628474	+	0.777830i	$0.716320\pi$
$480$		0			0
$481$	−6044.73			−0.573006
$482$		0			0
$483$		−	455.665i		−	0.0429264i
$484$		0			0
$485$	−4731.28			−0.442962
$486$		0			0
$487$	−7770.87			−0.723063			−0.361532	−	0.932360i	$-0.617746\pi$
$487$	−7770.87			−0.723063			−0.361532	+	0.932360i	$0.617746\pi$
$488$		0			0
$489$			708.446i			0.0655154i
$490$		0			0
$491$	16888.6			1.55228			0.776141	−	0.630559i	$-0.217174\pi$
$491$	16888.6			1.55228			0.776141	+	0.630559i	$0.217174\pi$
$492$		0			0
$493$	−20969.4			−1.91564
$494$		0			0
$495$			7029.19i			0.638260i
$496$		0			0
$497$		−	17022.5i		−	1.53635i
$498$		0			0
$499$	−961.553			−0.0862626			−0.0431313	−	0.999069i	$-0.513733\pi$
$499$	−961.553			−0.0862626			−0.0431313	+	0.999069i	$0.513733\pi$
$500$		0			0
$501$		−	38.5477i		−	0.00343749i
$502$		0			0
$503$			5526.02i			0.489847i	0.969542	+	0.244924i	$0.0787629\pi$
$503$			5526.02i			0.489847i	−0.969542	+	0.244924i	$0.921237\pi$
$504$		0			0
$505$	13206.5			1.16372
$506$		0			0
$507$		−	177.960i		−	0.0155887i
$508$		0			0
$509$	2723.71			0.237184			0.118592	−	0.992943i	$-0.462162\pi$
$509$	2723.71			0.237184			0.118592	+	0.992943i	$0.462162\pi$
$510$		0			0
$511$		−	13816.7i		−	1.19612i
$512$		0			0
$513$	1105.90	+	278.462i	0.0951785	+	0.0239657i
$514$		0			0
$515$		−	11872.7i		−	1.01587i
$516$		0			0
$517$		−	4597.58i		−	0.391105i
$518$		0			0
$519$		−	296.617i		−	0.0250868i
$520$		0			0
$521$			5935.63i			0.499126i	0.968359	+	0.249563i	$0.0802869\pi$
$521$			5935.63i			0.499126i	−0.968359	+	0.249563i	$0.919713\pi$
$522$		0			0
$523$			1446.58i			0.120945i	0.998170	+	0.0604726i	$0.0192608\pi$
$523$			1446.58i			0.120945i	−0.998170	+	0.0604726i	$0.980739\pi$
$524$		0			0
$525$	129.300			0.0107488
$526$		0			0
$527$	31836.3			2.63152
$528$		0			0
$529$	8877.94			0.729673
$530$		0			0
$531$		−	12153.4i		−	0.993243i
$532$		0			0
$533$			15462.1i			1.25655i
$534$		0			0
$535$	982.075			0.0793623
$536$		0			0
$537$	−1150.63			−0.0924641
$538$		0			0
$539$	−13733.3			−1.09746
$540$		0			0
$541$			2569.96i			0.204235i	0.994772	+	0.102118i	$0.0325618\pi$
$541$			2569.96i			0.204235i	−0.994772	+	0.102118i	$0.967438\pi$
$542$		0			0
$543$			332.127i			0.0262485i
$544$		0			0
$545$		−	9943.78i		−	0.781550i
$546$		0			0
$547$			7507.83i			0.586858i	0.955981	+	0.293429i	$0.0947965\pi$
$547$			7507.83i			0.586858i	−0.955981	+	0.293429i	$0.905204\pi$
$548$		0			0
$549$			10383.4i			0.807201i
$550$		0			0
$551$	−4167.58	+	16551.4i	−0.322223	+	1.27969i
$552$		0			0
$553$			29578.4i			2.27450i
$554$		0			0
$555$	473.622			0.0362237
$556$		0			0
$557$			12498.0i			0.950728i	0.879789	+	0.475364i	$0.157684\pi$
$557$			12498.0i			0.950728i	−0.879789	+	0.475364i	$0.842316\pi$
$558$		0			0
$559$	−3727.99			−0.282070
$560$		0			0
$561$			570.371i			0.0429253i
$562$		0			0
$563$			14437.3i			1.08075i	0.841425	+	0.540374i	$0.181717\pi$
$563$			14437.3i			1.08075i	−0.841425	+	0.540374i	$0.818283\pi$
$564$		0			0
$565$	−6904.83			−0.514138
$566$		0			0
$567$			22522.6i			1.66818i
$568$		0			0
$569$		−	3732.90i		−	0.275029i	−0.990500	−	0.137514i	$-0.956089\pi$
$569$		−	3732.90i		−	0.275029i	0.990500	−	0.137514i	$-0.0439113\pi$
$570$		0			0
$571$	−7948.75			−0.582566			−0.291283	−	0.956637i	$-0.594082\pi$
$571$	−7948.75			−0.582566			−0.291283	+	0.956637i	$0.594082\pi$
$572$		0			0
$573$	437.557			0.0319009
$574$		0			0
$575$		−	933.312i		−	0.0676901i
$576$		0			0
$577$	−1091.83			−0.0787758			−0.0393879	−	0.999224i	$-0.512541\pi$
$577$	−1091.83			−0.0787758			−0.0393879	+	0.999224i	$0.512541\pi$
$578$		0			0
$579$	−213.683			−0.0153374
$580$		0			0
$581$			16728.7i			1.19453i
$582$		0			0
$583$	10009.0			0.711028
$584$		0			0
$585$			12398.9i			0.876296i
$586$		0			0
$587$	−3678.42			−0.258645			−0.129322	−	0.991603i	$-0.541280\pi$
$587$	−3678.42			−0.258645			−0.129322	+	0.991603i	$0.541280\pi$
$588$		0			0
$589$	6327.36	−	25128.8i	0.442639	−	1.75792i
$590$		0			0
$591$	144.816			0.0100794
$592$		0			0
$593$	−17520.6			−1.21330			−0.606648	−	0.794970i	$-0.707486\pi$
$593$	−17520.6			−1.21330			−0.606648	+	0.794970i	$0.707486\pi$
$594$		0			0
$595$	−37636.5			−2.59319
$596$		0			0
$597$	902.161			0.0618476
$598$		0			0
$599$	−14805.7			−1.00992			−0.504960	−	0.863143i	$-0.668493\pi$
$599$	−14805.7			−1.00992			−0.504960	+	0.863143i	$0.668493\pi$
$600$		0			0
$601$			9476.86i			0.643210i	0.946874	+	0.321605i	$0.104222\pi$
$601$			9476.86i			0.643210i	−0.946874	+	0.321605i	$0.895778\pi$
$602$		0			0
$603$		−	7051.64i		−	0.476227i
$604$		0			0
$605$		−	10090.1i		−	0.678053i
$606$		0			0
$607$	−1962.54			−0.131231			−0.0656155	−	0.997845i	$-0.520901\pi$
$607$	−1962.54			−0.131231			−0.0656155	+	0.997845i	$0.520901\pi$
$608$		0			0
$609$	1637.42			0.108952
$610$		0			0
$611$		−	8109.77i		−	0.536966i
$612$		0			0
$613$		−	9072.88i		−	0.597798i	−0.954285	−	0.298899i	$-0.903381\pi$
$613$		−	9072.88i		−	0.597798i	0.954285	−	0.298899i	$-0.0966194\pi$
$614$		0			0
$615$		−	1211.50i		−	0.0794350i
$616$		0			0
$617$	9669.18			0.630902			0.315451	−	0.948942i	$-0.397844\pi$
$617$	9669.18			0.630902			0.315451	+	0.948942i	$0.397844\pi$
$618$		0			0
$619$	−4803.50			−0.311905			−0.155952	−	0.987765i	$-0.549845\pi$
$619$	−4803.50			−0.311905			−0.155952	+	0.987765i	$0.549845\pi$
$620$		0			0
$621$	−789.714			−0.0510308
$622$		0			0
$623$	43719.5			2.81153
$624$		0			0
$625$	−17394.4			−1.11324
$626$		0			0
$627$	450.201	+	113.359i	0.0286751	+	0.00722031i
$628$		0			0
$629$	−15880.7			−1.00669
$630$		0			0
$631$			1994.73i			0.125846i	0.998018	+	0.0629232i	$0.0200423\pi$
$631$			1994.73i			0.125846i	−0.998018	+	0.0629232i	$0.979958\pi$
$632$		0			0
$633$	1018.83			0.0639731
$634$		0			0
$635$		−	12069.4i		−	0.754268i
$636$		0			0
$637$	−24224.4			−1.50676
$638$		0			0
$639$	−14733.1			−0.912098
$640$		0			0
$641$		−	12599.6i		−	0.776374i	−0.921581	−	0.388187i	$-0.873101\pi$
$641$		−	12599.6i		−	0.776374i	0.921581	−	0.388187i	$-0.126899\pi$
$642$		0			0
$643$	−13231.7			−0.811517			−0.405759	−	0.913980i	$-0.632993\pi$
$643$	−13231.7			−0.811517			−0.405759	+	0.913980i	$0.632993\pi$
$644$		0			0
$645$	292.099			0.0178316
$646$		0			0
$647$			345.667i			0.0210040i	0.999945	+	0.0105020i	$0.00334295\pi$
$647$			345.667i			0.0210040i	−0.999945	+	0.0105020i	$0.996657\pi$
$648$		0			0
$649$		−	9907.03i		−	0.599207i
$650$		0			0
$651$	−2485.99			−0.149668
$652$		0			0
$653$			28907.1i			1.73234i	0.499745	+	0.866172i	$0.333427\pi$
$653$			28907.1i			1.73234i	−0.499745	+	0.866172i	$0.666573\pi$
$654$		0			0
$655$		−	23427.0i		−	1.39751i
$656$		0			0
$657$	−11958.4			−0.710112
$658$		0			0
$659$			17748.5i			1.04914i	0.851368	+	0.524569i	$0.175773\pi$
$659$			17748.5i			1.04914i	−0.851368	+	0.524569i	$0.824227\pi$
$660$		0			0
$661$	29973.2			1.76373			0.881864	−	0.471504i	$-0.156289\pi$
$661$	29973.2			1.76373			0.881864	+	0.471504i	$0.156289\pi$
$662$		0			0
$663$			1006.09i			0.0589341i
$664$		0			0
$665$	−7480.12	+	29706.9i	−0.436190	+	1.73231i
$666$		0			0
$667$		−	11819.2i		−	0.686119i
$668$		0			0
$669$			274.188i			0.0158456i
$670$		0			0
$671$			8464.21i			0.486970i
$672$		0			0
$673$		−	26244.4i		−	1.50319i	−0.659624	−	0.751595i	$-0.729285\pi$
$673$		−	26244.4i		−	1.50319i	0.659624	−	0.751595i	$-0.270715\pi$
$674$		0			0
$675$		−	224.091i		−	0.0127782i
$676$		0			0
$677$	−12259.6			−0.695972			−0.347986	−	0.937500i	$-0.613134\pi$
$677$	−12259.6			−0.695972			−0.347986	+	0.937500i	$0.613134\pi$
$678$		0			0
$679$	−12387.8			−0.700146
$680$		0			0
$681$	−808.243			−0.0454801
$682$		0			0
$683$		−	9968.98i		−	0.558495i	−0.960219	−	0.279248i	$-0.909915\pi$
$683$		−	9968.98i		−	0.558495i	0.960219	−	0.279248i	$-0.0900850\pi$
$684$		0			0
$685$		−	8979.29i		−	0.500848i
$686$		0			0
$687$	−638.389			−0.0354528
$688$		0			0
$689$	17655.1			0.976203
$690$		0			0
$691$	24599.5			1.35428			0.677142	−	0.735852i	$-0.263218\pi$
$691$	24599.5			1.35428			0.677142	+	0.735852i	$0.263218\pi$
$692$		0			0
$693$			18404.3i			1.00883i
$694$		0			0
$695$		−	25299.3i		−	1.38080i
$696$		0			0
$697$			40622.1i			2.20757i
$698$		0			0
$699$			264.064i			0.0142887i
$700$		0			0
$701$		−	14471.8i		−	0.779734i	−0.920871	−	0.389867i	$-0.872521\pi$
$701$		−	14471.8i		−	0.779734i	0.920871	−	0.389867i	$-0.127479\pi$
$702$		0			0
$703$	−3156.24	+	12534.8i	−0.169331	+	0.672490i
$704$		0			0
$705$			635.424i			0.0339453i
$706$		0			0
$707$	34578.1			1.83938
$708$		0			0
$709$		−	18759.1i		−	0.993671i	−0.867845	−	0.496836i	$-0.834495\pi$
$709$		−	18759.1i		−	0.993671i	0.867845	−	0.496836i	$-0.165505\pi$
$710$		0			0
$711$	25600.2			1.35033
$712$		0			0
$713$			17944.3i			0.942523i
$714$		0			0
$715$			10107.2i			0.528655i
$716$		0			0
$717$	369.189			0.0192296
$718$		0			0
$719$			4210.87i			0.218413i	0.994019	+	0.109206i	$0.0348310\pi$
$719$			4210.87i			0.218413i	−0.994019	+	0.109206i	$0.965169\pi$
$720$		0			0
$721$		−	31085.9i		−	1.60569i
$722$		0			0
$723$	791.172			0.0406971
$724$		0			0
$725$	3353.84			0.171805
$726$		0			0
$727$		−	8248.16i		−	0.420780i	−0.977618	−	0.210390i	$-0.932527\pi$
$727$		−	8248.16i		−	0.420780i	0.977618	−	0.210390i	$-0.0674734\pi$
$728$		0			0
$729$	19398.5			0.985544
$730$		0			0
$731$	−9794.18			−0.495555
$732$		0			0
$733$			6704.17i			0.337823i	0.985631	+	0.168911i	$0.0540252\pi$
$733$			6704.17i			0.337823i	−0.985631	+	0.168911i	$0.945975\pi$
$734$		0			0
$735$	1898.05			0.0952527
$736$		0			0
$737$		−	5748.26i		−	0.287300i
$738$		0			0
$739$	21399.5			1.06521			0.532606	−	0.846363i	$-0.321213\pi$
$739$	21399.5			1.06521			0.532606	+	0.846363i	$0.321213\pi$
$740$		0			0
$741$	794.119	+	199.957i	0.0393694	+	0.00991309i
$742$		0			0
$743$	−27163.2			−1.34121			−0.670606	−	0.741814i	$-0.733966\pi$
$743$	−27163.2			−1.34121			−0.670606	+	0.741814i	$0.733966\pi$
$744$		0			0
$745$	−12241.2			−0.601991
$746$		0			0
$747$	14478.8			0.709170
$748$		0			0
$749$	2571.34			0.125440
$750$		0			0
$751$	2808.18			0.136448			0.0682238	−	0.997670i	$-0.478267\pi$
$751$	2808.18			0.136448			0.0682238	+	0.997670i	$0.478267\pi$
$752$		0			0
$753$		−	915.654i		−	0.0443138i
$754$		0			0
$755$		−	14137.0i		−	0.681455i
$756$		0			0
$757$		−	26808.1i		−	1.28713i	−0.765392	−	0.643565i	$-0.777455\pi$
$757$		−	26808.1i		−	1.28713i	0.765392	−	0.643565i	$-0.222545\pi$
$758$		0			0
$759$	−321.485			−0.0153744
$760$		0			0
$761$	28180.4			1.34236			0.671182	−	0.741293i	$-0.265787\pi$
$761$	28180.4			1.34236			0.671182	+	0.741293i	$0.265787\pi$
$762$		0			0
$763$		−	26035.5i		−	1.23532i
$764$		0			0
$765$			32574.5i			1.53952i
$766$		0			0
$767$		−	17475.2i		−	0.822678i
$768$		0			0
$769$	9132.92			0.428272			0.214136	−	0.976804i	$-0.431306\pi$
$769$	9132.92			0.428272			0.214136	+	0.976804i	$0.431306\pi$
$770$		0			0
$771$	1720.66			0.0803738
$772$		0			0
$773$	12663.5			0.589231			0.294616	−	0.955616i	$-0.404808\pi$
$773$	12663.5			0.589231			0.294616	+	0.955616i	$0.404808\pi$
$774$		0			0
$775$	−5091.91			−0.236009
$776$		0			0
$777$	1240.07			0.0572552
$778$		0			0
$779$	32063.5	+	8073.50i	1.47470	+	0.371326i
$780$		0			0
$781$	−12009.9			−0.550253
$782$		0			0
$783$		−	2837.82i		−	0.129522i
$784$		0			0
$785$	40992.7			1.86381
$786$		0			0
$787$		−	37712.8i		−	1.70815i	−0.520148	−	0.854076i	$-0.674123\pi$
$787$		−	37712.8i		−	1.70815i	0.520148	−	0.854076i	$-0.325877\pi$
$788$		0			0
$789$	765.604			0.0345453
$790$		0			0
$791$	−18078.7			−0.812648
$792$		0			0
$793$			14930.2i			0.668584i
$794$		0			0
$795$	−1383.33			−0.0617126
$796$		0			0
$797$	4970.27			0.220898			0.110449	−	0.993882i	$-0.464771\pi$
$797$	4970.27			0.220898			0.110449	+	0.993882i	$0.464771\pi$
$798$		0			0
$799$		−	21306.0i		−	0.943369i
$800$		0			0
$801$		−	37839.4i		−	1.66915i
$802$		0			0
$803$	−9748.12			−0.428398
$804$		0			0
$805$		−	21213.5i		−	0.928792i
$806$		0			0
$807$		−	916.023i		−	0.0399573i
$808$		0			0
$809$	−20762.7			−0.902321			−0.451160	−	0.892443i	$-0.648990\pi$
$809$	−20762.7			−0.902321			−0.451160	+	0.892443i	$0.648990\pi$
$810$		0			0
$811$			36963.0i			1.60043i	0.599715	+	0.800214i	$0.295281\pi$
$811$			36963.0i			1.60043i	−0.599715	+	0.800214i	$0.704719\pi$
$812$		0			0
$813$	666.411			0.0287479
$814$		0			0
$815$			32981.8i			1.41755i
$816$		0			0
$817$	−1946.56	+	7730.66i	−0.0833555	+	0.331042i
$818$		0			0
$819$			32463.8i			1.38508i
$820$		0			0
$821$		−	10288.3i		−	0.437351i	−0.975798	−	0.218675i	$-0.929826\pi$
$821$		−	10288.3i		−	0.437351i	0.975798	−	0.218675i	$-0.0701736\pi$
$822$		0			0
$823$		−	28596.2i		−	1.21118i	−0.795777	−	0.605590i	$-0.792937\pi$
$823$		−	28596.2i		−	1.21118i	0.795777	−	0.605590i	$-0.207063\pi$
$824$		0			0
$825$		−	91.2253i		−	0.00384977i
$826$		0			0
$827$			24872.9i			1.04585i	0.852379	+	0.522924i	$0.175159\pi$
$827$			24872.9i			1.04585i	−0.852379	+	0.522924i	$0.824841\pi$
$828$		0			0
$829$	−25851.3			−1.08306			−0.541528	−	0.840683i	$-0.682154\pi$
$829$	−25851.3			−1.08306			−0.541528	+	0.840683i	$0.682154\pi$
$830$		0			0
$831$	1318.25			0.0550295
$832$		0			0
$833$	−63642.3			−2.64715
$834$		0			0
$835$		−	1794.59i		−	0.0743764i
$836$		0			0
$837$			4308.47i			0.177924i
$838$		0			0
$839$	−182.166			−0.00749590			−0.00374795	−	0.999993i	$-0.501193\pi$
$839$	−182.166			−0.00749590			−0.00374795	+	0.999993i	$0.501193\pi$
$840$		0			0
$841$	18083.1			0.741446
$842$		0			0
$843$	−1376.17			−0.0562254
$844$		0			0
$845$		−	8284.92i		−	0.337290i
$846$		0			0
$847$		−	26418.7i		−	1.07173i
$848$		0			0
$849$			1925.43i			0.0778333i
$850$		0			0
$851$		−	8951.04i		−	0.360561i
$852$		0			0
$853$		−	27467.2i		−	1.10253i	−0.834330	−	0.551266i	$-0.814145\pi$
$853$		−	27467.2i		−	1.10253i	0.834330	−	0.551266i	$-0.185855\pi$
$854$		0			0
$855$	25711.5	+	6474.07i	1.02844	+	0.258957i
$856$		0			0
$857$		−	37989.6i		−	1.51424i	−0.653278	−	0.757118i	$-0.726607\pi$
$857$		−	37989.6i		−	1.51424i	0.653278	−	0.757118i	$-0.273393\pi$
$858$		0			0
$859$	15744.7			0.625382			0.312691	−	0.949855i	$-0.398770\pi$
$859$	15744.7			0.625382			0.312691	+	0.949855i	$0.398770\pi$
$860$		0			0
$861$		−	3172.04i		−	0.125555i
$862$		0			0
$863$	18939.0			0.747033			0.373517	−	0.927624i	$-0.378152\pi$
$863$	18939.0			0.747033			0.373517	+	0.927624i	$0.378152\pi$
$864$		0			0
$865$		−	13809.0i		−	0.542799i
$866$		0			0
$867$			1388.88i			0.0544045i
$868$		0			0
$869$	20868.4			0.814629
$870$		0			0
$871$		−	10139.5i		−	0.394447i
$872$		0			0
$873$			10721.7i			0.415662i
$874$		0			0
$875$	−40217.0			−1.55381
$876$		0			0
$877$	−21287.1			−0.819630			−0.409815	−	0.912169i	$-0.634407\pi$
$877$	−21287.1			−0.819630			−0.409815	+	0.912169i	$0.634407\pi$
$878$		0			0
$879$		−	697.236i		−	0.0267545i
$880$		0			0
$881$	−32856.5			−1.25649			−0.628243	−	0.778017i	$-0.716226\pi$
$881$	−32856.5			−1.25649			−0.628243	+	0.778017i	$0.716226\pi$
$882$		0			0
$883$	−7030.80			−0.267956			−0.133978	−	0.990984i	$-0.542775\pi$
$883$	−7030.80			−0.267956			−0.133978	+	0.990984i	$0.542775\pi$
$884$		0			0
$885$			1369.24i			0.0520072i
$886$		0			0
$887$	13987.6			0.529492			0.264746	−	0.964318i	$-0.414712\pi$
$887$	13987.6			0.529492			0.264746	+	0.964318i	$0.414712\pi$
$888$		0			0
$889$		−	31601.0i		−	1.19220i
$890$		0			0
$891$	15890.4			0.597471
$892$		0			0
$893$	−16817.1	−	4234.49i	−0.630193	−	0.158681i
$894$		0			0
$895$	−53567.5			−2.00063
$896$		0			0
$897$	−567.075			−0.0211082
$898$		0			0
$899$	−64482.5			−2.39223
$900$		0			0
$901$	46383.4			1.71504
$902$		0			0
$903$	764.794			0.0281847
$904$		0			0
$905$			15462.2i			0.567934i
$906$		0			0
$907$			32555.9i			1.19184i	0.803043	+	0.595922i	$0.203213\pi$
$907$			32555.9i			1.19184i	−0.803043	+	0.595922i	$0.796787\pi$
$908$		0			0
$909$		−	29927.4i		−	1.09200i
$910$		0			0
$911$	−42685.1			−1.55238			−0.776191	−	0.630498i	$-0.782851\pi$
$911$	−42685.1			−1.55238			−0.776191	+	0.630498i	$0.782851\pi$
$912$		0			0
$913$	11802.6			0.427830
$914$		0			0
$915$		−	1169.83i		−	0.0422658i
$916$		0			0
$917$		−	61338.1i		−	2.20890i
$918$		0			0
$919$		−	26396.2i		−	0.947477i	−0.880666	−	0.473739i	$-0.842904\pi$
$919$		−	26396.2i		−	0.947477i	0.880666	−	0.473739i	$-0.157096\pi$
$920$		0			0
$921$	−261.525			−0.00935672
$922$		0			0
$923$	−21184.5			−0.755468
$924$		0			0
$925$	2539.97			0.0902849
$926$		0			0
$927$	−26905.0			−0.953264
$928$		0			0
$929$	−15313.5			−0.540818			−0.270409	−	0.962746i	$-0.587159\pi$
$929$	−15313.5			−0.540818			−0.270409	+	0.962746i	$0.587159\pi$
$930$		0			0
$931$	−12648.7	+	50233.7i	−0.445267	+	1.76836i
$932$		0			0
$933$	−644.977			−0.0226320
$934$		0			0
$935$			26553.7i			0.928768i
$936$		0			0
$937$	13341.6			0.465156			0.232578	−	0.972578i	$-0.425284\pi$
$937$	13341.6			0.465156			0.232578	+	0.972578i	$0.425284\pi$
$938$		0			0
$939$			2688.99i			0.0934523i
$940$		0			0
$941$	50527.3			1.75042			0.875209	−	0.483745i	$-0.160724\pi$
$941$	50527.3			1.75042			0.875209	+	0.483745i	$0.160724\pi$
$942$		0			0
$943$	−22896.3			−0.790675
$944$		0			0
$945$		−	5093.42i		−	0.175332i
$946$		0			0
$947$	24534.8			0.841895			0.420947	−	0.907085i	$-0.361698\pi$
$947$	24534.8			0.841895			0.420947	+	0.907085i	$0.361698\pi$
$948$		0			0
$949$	−17194.9			−0.588168
$950$		0			0
$951$			1580.22i			0.0538825i
$952$		0			0
$953$			42716.0i			1.45195i	0.687723	+	0.725974i	$0.258611\pi$
$953$			42716.0i			1.45195i	−0.687723	+	0.725974i	$0.741389\pi$
$954$		0			0
$955$	20370.5			0.690234
$956$		0			0
$957$		−	1155.25i		−	0.0390219i
$958$		0			0
$959$		−	23510.2i		−	0.791641i
$960$		0			0
$961$	68108.4			2.28621
$962$		0			0
$963$		−	2225.50i		−	0.0744712i
$964$		0			0
$965$	−9948.01			−0.331853
$966$		0			0
$967$		−	15724.2i		−	0.522912i	−0.965215	−	0.261456i	$-0.915797\pi$
$967$		−	15724.2i		−	0.522912i	0.965215	−	0.261456i	$-0.0842027\pi$
$968$		0			0
$969$	2086.31	+	525.327i	0.0691661	+	0.0174158i
$970$		0			0
$971$		−	11018.1i		−	0.364148i	−0.983285	−	0.182074i	$-0.941719\pi$
$971$		−	11018.1i		−	0.364148i	0.983285	−	0.182074i	$-0.0582810\pi$
$972$		0			0
$973$		−	66240.3i		−	2.18249i
$974$		0			0
$975$		−	160.914i		−	0.00528552i
$976$		0			0
$977$		−	20874.0i		−	0.683540i	−0.939784	−	0.341770i	$-0.888974\pi$
$977$		−	20874.0i		−	0.683540i	0.939784	−	0.341770i	$-0.111026\pi$
$978$		0			0
$979$		−	30845.4i		−	1.00697i
$980$		0			0
$981$	−22533.8			−0.733383
$982$		0			0
$983$	−19012.5			−0.616891			−0.308445	−	0.951242i	$-0.599809\pi$
$983$	−19012.5			−0.616891			−0.308445	+	0.951242i	$0.599809\pi$
$984$		0			0
$985$	6741.90			0.218086
$986$		0			0
$987$			1663.71i			0.0536541i
$988$		0			0
$989$		−	5520.41i		−	0.177491i
$990$		0			0
$991$	−35028.4			−1.12282			−0.561410	−	0.827538i	$-0.689741\pi$
$991$	−35028.4			−1.12282			−0.561410	+	0.827538i	$0.689741\pi$
$992$		0			0
$993$	−1030.18			−0.0329222
$994$		0			0
$995$	42000.2			1.33819
$996$		0			0
$997$			17746.0i			0.563711i	0.959457	+	0.281856i	$0.0909499\pi$
$997$			17746.0i			0.563711i	−0.959457	+	0.281856i	$0.909050\pi$
$998$		0			0
$999$		−	2149.17i		−	0.0680648i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	608.4.b.b.303.30		56
4.3	odd	2		152.4.b.b.75.51	yes	56
8.3	odd	2	inner	608.4.b.b.303.29		56
8.5	even	2		152.4.b.b.75.5	✓	56
19.18	odd	2	inner	608.4.b.b.303.28		56
76.75	even	2		152.4.b.b.75.6	yes	56
152.37	odd	2		152.4.b.b.75.52	yes	56
152.75	even	2	inner	608.4.b.b.303.27		56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.4.b.b.75.5	✓	56	8.5	even	2
152.4.b.b.75.6	yes	56	76.75	even	2
152.4.b.b.75.51	yes	56	4.3	odd	2
152.4.b.b.75.52	yes	56	152.37	odd	2
608.4.b.b.303.27		56	152.75	even	2	inner
608.4.b.b.303.28		56	19.18	odd	2	inner
608.4.b.b.303.29		56	8.3	odd	2	inner
608.4.b.b.303.30		56	1.1	even	1	trivial

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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	608.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.255308i q^{3} +11.8859i q^{5} +31.1204i q^{7} +26.9348 q^{9} +O(q^{10})\)	\(q+0.255308i q^{3} +11.8859i q^{5} +31.1204i q^{7} +26.9348 q^{9} +21.9564 q^{11} +38.7293 q^{13} -3.03456 q^{15} +101.750 q^{17} +(20.2224 - 80.3122i) q^{19} -7.94528 q^{21} +57.3504i q^{23} -16.2739 q^{25} +13.7700i q^{27} -206.088 q^{29} +312.889 q^{31} +5.60563i q^{33} -369.893 q^{35} -156.076 q^{37} +9.88790i q^{39} +399.236i q^{41} -96.2576 q^{43} +320.144i q^{45} -209.396i q^{47} -625.479 q^{49} +25.9775i q^{51} +455.858 q^{53} +260.970i q^{55} +(20.5043 + 5.16293i) q^{57} -451.215i q^{59} +385.501i q^{61} +838.222i q^{63} +460.332i q^{65} -261.804i q^{67} -14.6420 q^{69} -546.989 q^{71} -443.977 q^{73} -4.15485i q^{75} +683.291i q^{77} +950.449 q^{79} +723.724 q^{81} +537.548 q^{83} +1209.38i q^{85} -52.6158i q^{87} -1404.85i q^{89} +1205.27i q^{91} +79.8830i q^{93} +(954.581 + 240.361i) q^{95} +398.059i q^{97} +591.391 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 528 q^{9}+O(q^{10}) \) 56 * q - 528 * q^9	\( 56 q - 528 q^{9} + 40 q^{11} - 184 q^{17} + 84 q^{19} - 1504 q^{25} - 40 q^{35} - 576 q^{43} - 3664 q^{49} - 648 q^{57} - 432 q^{73} + 2296 q^{81} - 5376 q^{83} + 6152 q^{99}+O(q^{100}) \) 56 * q - 528 * q^9 + 40 * q^11 - 184 * q^17 + 84 * q^19 - 1504 * q^25 - 40 * q^35 - 576 * q^43 - 3664 * q^49 - 648 * q^57 - 432 * q^73 + 2296 * q^81 - 5376 * q^83 + 6152 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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