LMFDB - Embedded newform 304.5.e.f.113.17

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [304,5,Mod(113,304)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("304.113"); S:= CuspForms(chi, 5); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(304, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 5, names="a")

Level:	$ N $	$=$	$ 304 = 2^{4} \cdot 19 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	304.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [20,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$31.4244687775$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 996 x^{18} + 408854 x^{16} + 89661524 x^{14} + 11414409521 x^{12} + 861580608848 x^{10} + \cdots + 34\!\cdots\!64 $ x^20 + 996x^18 + 408854x^16 + 89661524x^14 + 11414409521x^12 + 861580608848x^10 + 37911701888064x^8 + 915101881952256x^6 + 10365057927217152x^4 + 34336742150504448*x^2 + 34273490123096064
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{50} $
Twist minimal:	no (minimal twist has level 152)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			113.17
Root			$11.4302i$ of defining polynomial
Character	$\chi$	$=$	304.113
Dual form			304.5.e.f.113.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+11.4302i q^{3} -9.10929 q^{5} -75.9168 q^{7} -49.6505 q^{9} -149.183 q^{11} -188.468i q^{13} -104.121i q^{15} +322.296 q^{17} +(349.839 + 89.0705i) q^{19} -867.748i q^{21} +561.396 q^{23} -542.021 q^{25} +358.332i q^{27} -260.782i q^{29} -167.262i q^{31} -1705.20i q^{33} +691.548 q^{35} +1800.56i q^{37} +2154.24 q^{39} -1423.75i q^{41} -836.566 q^{43} +452.281 q^{45} -1086.54 q^{47} +3362.36 q^{49} +3683.92i q^{51} -4337.45i q^{53} +1358.95 q^{55} +(-1018.10 + 3998.75i) q^{57} +5831.99i q^{59} +5513.48 q^{61} +3769.31 q^{63} +1716.81i q^{65} -3107.22i q^{67} +6416.89i q^{69} -5926.54i q^{71} +2350.54 q^{73} -6195.43i q^{75} +11325.5 q^{77} -8365.30i q^{79} -8117.52 q^{81} +11086.6 q^{83} -2935.89 q^{85} +2980.80 q^{87} -834.210i q^{89} +14307.9i q^{91} +1911.84 q^{93} +(-3186.79 - 811.369i) q^{95} +2254.86i q^{97} +7407.03 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 32 q^{7} - 372 q^{9} + 24 q^{11} + 216 q^{17} + 596 q^{19} - 576 q^{23} + 1412 q^{25} + 144 q^{35} + 520 q^{39} + 1256 q^{43} + 7232 q^{45} + 3768 q^{47} - 2740 q^{49} + 10128 q^{55} - 728 q^{57}+ \cdots - 760 q^{99}+O(q^{100}) $ 20 * q - 32 * q^7 - 372 * q^9 + 24 * q^11 + 216 * q^17 + 596 * q^19 - 576 * q^23 + 1412 * q^25 + 144 * q^35 + 520 * q^39 + 1256 * q^43 + 7232 * q^45 + 3768 * q^47 - 2740 * q^49 + 10128 * q^55 - 728 * q^57 + 352 * q^61 - 6104 * q^63 + 1352 * q^73 + 9288 * q^77 - 4220 * q^81 + 16104 * q^83 + 10232 * q^85 - 2936 * q^87 + 36432 * q^93 - 14232 * q^95 - 760 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/304\mathbb{Z}\right)^\times$.

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

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
$3$			11.4302i			1.27003i	0.772501	+	0.635014i	$0.219006\pi$
$3$			11.4302i			1.27003i	−0.772501	+	0.635014i	$0.780994\pi$
$4$		0			0
$5$	−9.10929			−0.364372			−0.182186	−	0.983264i	$-0.558317\pi$
$5$	−9.10929			−0.364372			−0.182186	+	0.983264i	$0.558317\pi$
$6$		0			0
$7$	−75.9168			−1.54932			−0.774661	−	0.632376i	$-0.782080\pi$
$7$	−75.9168			−1.54932			−0.774661	+	0.632376i	$0.782080\pi$
$8$		0			0
$9$	−49.6505			−0.612969
$10$		0			0
$11$	−149.183			−1.23292			−0.616460	−	0.787386i	$-0.711434\pi$
$11$	−149.183			−1.23292			−0.616460	+	0.787386i	$0.711434\pi$
$12$		0			0
$13$		−	188.468i		−	1.11520i	−0.830111	−	0.557598i	$-0.811723\pi$
$13$		−	188.468i		−	1.11520i	0.830111	−	0.557598i	$-0.188277\pi$
$14$		0			0
$15$		−	104.121i		−	0.462762i
$16$		0			0
$17$	322.296			1.11521			0.557605	−	0.830106i	$-0.311720\pi$
$17$	322.296			1.11521			0.557605	+	0.830106i	$0.311720\pi$
$18$		0			0
$19$	349.839	+	89.0705i	0.969084	+	0.246733i
$19$	349.839	+	89.0705i	0.969084	+	0.246733i
$20$		0			0
$21$		−	867.748i		−	1.96768i
$22$		0			0
$23$	561.396			1.06124			0.530620	−	0.847610i	$-0.321959\pi$
$23$	561.396			1.06124			0.530620	+	0.847610i	$0.321959\pi$
$24$		0			0
$25$	−542.021			−0.867233
$26$		0			0
$27$			358.332i			0.491539i
$28$		0			0
$29$		−	260.782i		−	0.310085i	−0.987908	−	0.155043i	$-0.950448\pi$
$29$		−	260.782i		−	0.310085i	0.987908	−	0.155043i	$-0.0495515\pi$
$30$		0			0
$31$		−	167.262i		−	0.174050i	−0.996206	−	0.0870248i	$-0.972264\pi$
$31$		−	167.262i		−	0.174050i	0.996206	−	0.0870248i	$-0.0277359\pi$
$32$		0			0
$33$		−	1705.20i		−	1.56584i
$34$		0			0
$35$	691.548			0.564529
$36$		0			0
$37$			1800.56i			1.31524i	0.753352	+	0.657618i	$0.228436\pi$
$37$			1800.56i			1.31524i	−0.753352	+	0.657618i	$0.771564\pi$
$38$		0			0
$39$	2154.24			1.41633
$40$		0			0
$41$		−	1423.75i		−	0.846964i	−0.905904	−	0.423482i	$-0.860808\pi$
$41$		−	1423.75i		−	0.846964i	0.905904	−	0.423482i	$-0.139192\pi$
$42$		0			0
$43$	−836.566			−0.452442			−0.226221	−	0.974076i	$-0.572637\pi$
$43$	−836.566			−0.452442			−0.226221	+	0.974076i	$0.572637\pi$
$44$		0			0
$45$	452.281			0.223349
$46$		0			0
$47$	−1086.54			−0.491871			−0.245935	−	0.969286i	$-0.579095\pi$
$47$	−1086.54			−0.491871			−0.245935	+	0.969286i	$0.579095\pi$
$48$		0			0
$49$	3362.36			1.40040
$50$		0			0
$51$			3683.92i			1.41635i
$52$		0			0
$53$		−	4337.45i		−	1.54413i	−0.635545	−	0.772064i	$-0.719225\pi$
$53$		−	4337.45i		−	1.54413i	0.635545	−	0.772064i	$-0.280775\pi$
$54$		0			0
$55$	1358.95			0.449241
$56$		0			0
$57$	−1018.10	+	3998.75i	−0.313357	+	1.23076i
$58$		0			0
$59$			5831.99i			1.67538i	0.546147	+	0.837690i	$0.316094\pi$
$59$			5831.99i			1.67538i	−0.546147	+	0.837690i	$0.683906\pi$
$60$		0			0
$61$	5513.48			1.48172			0.740860	−	0.671660i	$-0.234418\pi$
$61$	5513.48			1.48172			0.740860	+	0.671660i	$0.234418\pi$
$62$		0			0
$63$	3769.31			0.949687
$64$		0			0
$65$			1716.81i			0.406346i
$66$		0			0
$67$		−	3107.22i		−	0.692186i	−0.938200	−	0.346093i	$-0.887508\pi$
$67$		−	3107.22i		−	0.692186i	0.938200	−	0.346093i	$-0.112492\pi$
$68$		0			0
$69$			6416.89i			1.34780i
$70$		0			0
$71$		−	5926.54i		−	1.17567i	−0.808982	−	0.587833i	$-0.799981\pi$
$71$		−	5926.54i		−	1.17567i	0.808982	−	0.587833i	$-0.200019\pi$
$72$		0			0
$73$	2350.54			0.441085			0.220543	−	0.975377i	$-0.429217\pi$
$73$	2350.54			0.441085			0.220543	+	0.975377i	$0.429217\pi$
$74$		0			0
$75$		−	6195.43i		−	1.10141i
$76$		0			0
$77$	11325.5			1.91019
$78$		0			0
$79$		−	8365.30i		−	1.34038i	−0.742190	−	0.670189i	$-0.766213\pi$
$79$		−	8365.30i		−	1.34038i	0.742190	−	0.670189i	$-0.233787\pi$
$80$		0			0
$81$	−8117.52			−1.23724
$82$		0			0
$83$	11086.6			1.60932			0.804660	−	0.593736i	$-0.202348\pi$
$83$	11086.6			1.60932			0.804660	+	0.593736i	$0.202348\pi$
$84$		0			0
$85$	−2935.89			−0.406351
$86$		0			0
$87$	2980.80			0.393817
$88$		0			0
$89$		−	834.210i		−	0.105316i	−0.998613	−	0.0526582i	$-0.983231\pi$
$89$		−	834.210i		−	0.105316i	0.998613	−	0.0526582i	$-0.0167694\pi$
$90$		0			0
$91$			14307.9i			1.72780i
$92$		0			0
$93$	1911.84			0.221048
$94$		0			0
$95$	−3186.79	−	811.369i	−0.353106	−	0.0899024i
$96$		0			0
$97$			2254.86i			0.239649i	0.992795	+	0.119825i	$0.0382332\pi$
$97$			2254.86i			0.239649i	−0.992795	+	0.119825i	$0.961767\pi$
$98$		0			0
$99$	7407.03			0.755742
$100$		0			0
$101$	783.930			0.0768484			0.0384242	−	0.999262i	$-0.487766\pi$
$101$	783.930			0.0768484			0.0384242	+	0.999262i	$0.487766\pi$
$102$		0			0
$103$		−	17833.0i		−	1.68093i	−0.541862	−	0.840467i	$-0.682280\pi$
$103$		−	17833.0i		−	1.68093i	0.541862	−	0.840467i	$-0.317720\pi$
$104$		0			0
$105$			7904.56i			0.716967i
$106$		0			0
$107$		−	17730.5i		−	1.54865i	−0.632787	−	0.774326i	$-0.718089\pi$
$107$		−	17730.5i		−	1.54865i	0.632787	−	0.774326i	$-0.281911\pi$
$108$		0			0
$109$		−	13138.6i		−	1.10585i	−0.833232	−	0.552924i	$-0.813512\pi$
$109$		−	13138.6i		−	1.10585i	0.833232	−	0.552924i	$-0.186488\pi$
$110$		0			0
$111$	−20580.8			−1.67038
$112$		0			0
$113$		−	4495.21i		−	0.352041i	−0.984387	−	0.176021i	$-0.943677\pi$
$113$		−	4495.21i		−	0.352041i	0.984387	−	0.176021i	$-0.0563225\pi$
$114$		0			0
$115$	−5113.92			−0.386686
$116$		0			0
$117$			9357.55i			0.683582i
$118$		0			0
$119$	−24467.7			−1.72782
$120$		0			0
$121$	7614.66			0.520092
$122$		0			0
$123$	16273.8			1.07567
$124$		0			0
$125$	10630.7			0.680367
$126$		0			0
$127$			3478.63i			0.215676i	0.994168	+	0.107838i	$0.0343927\pi$
$127$			3478.63i			0.215676i	−0.994168	+	0.107838i	$0.965607\pi$
$128$		0			0
$129$		−	9562.15i		−	0.574614i
$130$		0			0
$131$	13454.4			0.784013			0.392006	−	0.919963i	$-0.371781\pi$
$131$	13454.4			0.784013			0.392006	+	0.919963i	$0.371781\pi$
$132$		0			0
$133$	−26558.7	−	6761.95i	−1.50142	−	0.382268i
$134$		0			0
$135$		−	3264.15i		−	0.179103i
$136$		0			0
$137$	5102.78			0.271873			0.135936	−	0.990718i	$-0.456596\pi$
$137$	5102.78			0.271873			0.135936	+	0.990718i	$0.456596\pi$
$138$		0			0
$139$	−17146.7			−0.887464			−0.443732	−	0.896160i	$-0.646346\pi$
$139$	−17146.7			−0.887464			−0.443732	+	0.896160i	$0.646346\pi$
$140$		0			0
$141$		−	12419.4i		−	0.624689i
$142$		0			0
$143$			28116.3i			1.37495i
$144$		0			0
$145$			2375.54i			0.112986i
$146$		0			0
$147$			38432.6i			1.77855i
$148$		0			0
$149$	5776.87			0.260208			0.130104	−	0.991500i	$-0.458469\pi$
$149$	5776.87			0.260208			0.130104	+	0.991500i	$0.458469\pi$
$150$		0			0
$151$			2852.68i			0.125112i	0.998041	+	0.0625561i	$0.0199252\pi$
$151$			2852.68i			0.125112i	−0.998041	+	0.0625561i	$0.980075\pi$
$152$		0			0
$153$	−16002.2			−0.683590
$154$		0			0
$155$			1523.63i			0.0634187i
$156$		0			0
$157$	17754.0			0.720274			0.360137	−	0.932900i	$-0.382730\pi$
$157$	17754.0			0.720274			0.360137	+	0.932900i	$0.382730\pi$
$158$		0			0
$159$	49578.2			1.96108
$160$		0			0
$161$	−42619.4			−1.64420
$162$		0			0
$163$	−32885.4			−1.23774			−0.618868	−	0.785495i	$-0.712408\pi$
$163$	−32885.4			−1.23774			−0.618868	+	0.785495i	$0.712408\pi$
$164$		0			0
$165$			15533.2i			0.570548i
$166$		0			0
$167$		−	13740.4i		−	0.492680i	−0.969183	−	0.246340i	$-0.920772\pi$
$167$		−	13740.4i		−	0.492680i	0.969183	−	0.246340i	$-0.0792280\pi$
$168$		0			0
$169$	−6959.29			−0.243664
$170$		0			0
$171$	−17369.7	−	4422.40i	−0.594019	−	0.151240i
$172$		0			0
$173$			10712.9i			0.357944i	0.983854	+	0.178972i	$0.0572772\pi$
$173$			10712.9i			0.357944i	−0.983854	+	0.178972i	$0.942723\pi$
$174$		0			0
$175$	41148.5			1.34362
$176$		0			0
$177$	−66661.1			−2.12778
$178$		0			0
$179$			39711.4i			1.23939i	0.784841	+	0.619697i	$0.212744\pi$
$179$			39711.4i			1.23939i	−0.784841	+	0.619697i	$0.787256\pi$
$180$		0			0
$181$			13456.0i			0.410733i	0.978685	+	0.205366i	$0.0658386\pi$
$181$			13456.0i			0.410733i	−0.978685	+	0.205366i	$0.934161\pi$
$182$		0			0
$183$			63020.4i			1.88182i
$184$		0			0
$185$		−	16401.8i		−	0.479234i
$186$		0			0
$187$	−48081.2			−1.37497
$188$		0			0
$189$		−	27203.4i		−	0.761553i
$190$		0			0
$191$	−59296.4			−1.62541			−0.812703	−	0.582678i	$-0.802005\pi$
$191$	−59296.4			−1.62541			−0.812703	+	0.582678i	$0.802005\pi$
$192$		0			0
$193$			33032.9i			0.886813i	0.896321	+	0.443407i	$0.146230\pi$
$193$			33032.9i			0.886813i	−0.896321	+	0.443407i	$0.853770\pi$
$194$		0			0
$195$	−19623.6			−0.516071
$196$		0			0
$197$	−29861.3			−0.769442			−0.384721	−	0.923033i	$-0.625702\pi$
$197$	−29861.3			−0.769442			−0.384721	+	0.923033i	$0.625702\pi$
$198$		0			0
$199$	30968.8			0.782021			0.391010	−	0.920386i	$-0.372126\pi$
$199$	30968.8			0.782021			0.391010	+	0.920386i	$0.372126\pi$
$200$		0			0
$201$	35516.3			0.879095
$202$		0			0
$203$			19797.7i			0.480422i
$204$		0			0
$205$			12969.3i			0.308610i
$206$		0			0
$207$	−27873.6			−0.650508
$208$		0			0
$209$	−52190.2	−	13287.8i	−1.19480	−	0.304202i
$210$		0			0
$211$		−	76035.8i		−	1.70786i	−0.520385	−	0.853932i	$-0.674212\pi$
$211$		−	76035.8i		−	1.70786i	0.520385	−	0.853932i	$-0.325788\pi$
$212$		0			0
$213$	67741.8			1.49313
$214$		0			0
$215$	7620.52			0.164857
$216$		0			0
$217$			12698.0i			0.269659i
$218$		0			0
$219$			26867.3i			0.560190i
$220$		0			0
$221$		−	60742.6i		−	1.24368i
$222$		0			0
$223$		−	66035.2i		−	1.32790i	−0.747777	−	0.663950i	$-0.768879\pi$
$223$		−	66035.2i		−	1.32790i	0.747777	−	0.663950i	$-0.231121\pi$
$224$		0			0
$225$	26911.6			0.531588
$226$		0			0
$227$		−	70838.7i		−	1.37473i	−0.726310	−	0.687367i	$-0.758766\pi$
$227$		−	70838.7i		−	1.37473i	0.726310	−	0.687367i	$-0.241234\pi$
$228$		0			0
$229$	69086.9			1.31742			0.658710	−	0.752397i	$-0.271102\pi$
$229$	69086.9			1.31742			0.658710	+	0.752397i	$0.271102\pi$
$230$		0			0
$231$			129453.i			2.42599i
$232$		0			0
$233$	−72230.6			−1.33048			−0.665242	−	0.746628i	$-0.731672\pi$
$233$	−72230.6			−1.33048			−0.665242	+	0.746628i	$0.731672\pi$
$234$		0			0
$235$	9897.63			0.179224
$236$		0			0
$237$	95617.4			1.70232
$238$		0			0
$239$	−14753.9			−0.258292			−0.129146	−	0.991626i	$-0.541224\pi$
$239$	−14753.9			−0.258292			−0.129146	+	0.991626i	$0.541224\pi$
$240$		0			0
$241$			45140.5i			0.777198i	0.921407	+	0.388599i	$0.127041\pi$
$241$			45140.5i			0.777198i	−0.921407	+	0.388599i	$0.872959\pi$
$242$		0			0
$243$		−	63760.3i		−	1.07979i
$244$		0			0
$245$	−30628.7			−0.510266
$246$		0			0
$247$	16787.0	−	65933.6i	0.275155	−	1.08072i
$248$		0			0
$249$			126723.i			2.04388i
$250$		0			0
$251$	−46493.4			−0.737980			−0.368990	−	0.929433i	$-0.620296\pi$
$251$	−46493.4			−0.737980			−0.368990	+	0.929433i	$0.620296\pi$
$252$		0			0
$253$	−83750.9			−1.30842
$254$		0			0
$255$		−	33557.9i		−	0.516077i
$256$		0			0
$257$		−	123613.i		−	1.87154i	−0.352617	−	0.935768i	$-0.614708\pi$
$257$		−	123613.i		−	1.87154i	0.352617	−	0.935768i	$-0.385292\pi$
$258$		0			0
$259$		−	136693.i		−	2.03772i
$260$		0			0
$261$			12948.0i			0.190073i
$262$		0			0
$263$	−51498.3			−0.744528			−0.372264	−	0.928127i	$-0.621418\pi$
$263$	−51498.3			−0.744528			−0.372264	+	0.928127i	$0.621418\pi$
$264$		0			0
$265$			39511.1i			0.562636i
$266$		0			0
$267$	9535.23			0.133755
$268$		0			0
$269$		−	76030.5i		−	1.05071i	−0.850883	−	0.525355i	$-0.823932\pi$
$269$		−	76030.5i		−	1.05071i	0.850883	−	0.525355i	$-0.176068\pi$
$270$		0			0
$271$	10469.5			0.142557			0.0712785	−	0.997456i	$-0.477292\pi$
$271$	10469.5			0.142557			0.0712785	+	0.997456i	$0.477292\pi$
$272$		0			0
$273$	−163543.			−2.19435
$274$		0			0
$275$	80860.5			1.06923
$276$		0			0
$277$	117299.			1.52874			0.764369	−	0.644779i	$-0.223050\pi$
$277$	117299.			1.52874			0.764369	+	0.644779i	$0.223050\pi$
$278$		0			0
$279$			8304.63i			0.106687i
$280$		0			0
$281$			76392.4i			0.967470i	0.875215	+	0.483735i	$0.160720\pi$
$281$			76392.4i			0.967470i	−0.875215	+	0.483735i	$0.839280\pi$
$282$		0			0
$283$	133039.			1.66114			0.830569	−	0.556916i	$-0.188016\pi$
$283$	133039.			1.66114			0.830569	+	0.556916i	$0.188016\pi$
$284$		0			0
$285$	9274.14	−	36425.7i	0.114178	−	0.448455i
$286$		0			0
$287$			108086.i			1.31222i
$288$		0			0
$289$	20353.6			0.243695
$290$		0			0
$291$	−25773.6			−0.304361
$292$		0			0
$293$			132667.i			1.54535i	0.634800	+	0.772677i	$0.281083\pi$
$293$			132667.i			1.54535i	−0.634800	+	0.772677i	$0.718917\pi$
$294$		0			0
$295$		−	53125.3i		−	0.610460i
$296$		0			0
$297$		−	53457.2i		−	0.606029i
$298$		0			0
$299$		−	105805.i		−	1.18349i
$300$		0			0
$301$	63509.4			0.700979
$302$		0			0
$303$			8960.52i			0.0975996i
$304$		0			0
$305$	−50223.9			−0.539896
$306$		0			0
$307$			116528.i			1.23638i	0.786028	+	0.618192i	$0.212134\pi$
$307$			116528.i			1.23638i	−0.786028	+	0.618192i	$0.787866\pi$
$308$		0			0
$309$	203836.			2.13483
$310$		0			0
$311$	−74587.7			−0.771164			−0.385582	−	0.922674i	$-0.625999\pi$
$311$	−74587.7			−0.771164			−0.385582	+	0.922674i	$0.625999\pi$
$312$		0			0
$313$	−140470.			−1.43383			−0.716913	−	0.697163i	$-0.754445\pi$
$313$	−140470.			−1.43383			−0.716913	+	0.697163i	$0.754445\pi$
$314$		0			0
$315$	−34335.7			−0.346039
$316$		0			0
$317$			78316.2i			0.779351i	0.920952	+	0.389676i	$0.127413\pi$
$317$			78316.2i			0.779351i	−0.920952	+	0.389676i	$0.872587\pi$
$318$		0			0
$319$			38904.3i			0.382310i
$320$		0			0
$321$	202664.			1.96683
$322$		0			0
$323$	112752.	+	28707.1i	1.08073	+	0.275159i
$324$		0			0
$325$			102154.i			0.967136i
$326$		0			0
$327$	150177.			1.40446
$328$		0			0
$329$	82486.8			0.762066
$330$		0			0
$331$			30840.1i			0.281488i	0.990046	+	0.140744i	$0.0449494\pi$
$331$			30840.1i			0.281488i	−0.990046	+	0.140744i	$0.955051\pi$
$332$		0			0
$333$		−	89398.6i		−	0.806199i
$334$		0			0
$335$			28304.6i			0.252213i
$336$		0			0
$337$		−	179800.i		−	1.58318i	−0.611052	−	0.791590i	$-0.709253\pi$
$337$		−	179800.i		−	1.58318i	0.611052	−	0.791590i	$-0.290747\pi$
$338$		0			0
$339$	51381.4			0.447102
$340$		0			0
$341$			24952.6i			0.214589i
$342$		0			0
$343$	−72983.4			−0.620349
$344$		0			0
$345$		−	58453.3i		−	0.491101i
$346$		0			0
$347$	4348.77			0.0361166			0.0180583	−	0.999837i	$-0.494252\pi$
$347$	4348.77			0.0361166			0.0180583	+	0.999837i	$0.494252\pi$
$348$		0			0
$349$	165030.			1.35492			0.677459	−	0.735561i	$-0.263081\pi$
$349$	165030.			1.35492			0.677459	+	0.735561i	$0.263081\pi$
$350$		0			0
$351$	67534.3			0.548163
$352$		0			0
$353$	−27708.9			−0.222367			−0.111183	−	0.993800i	$-0.535464\pi$
$353$	−27708.9			−0.222367			−0.111183	+	0.993800i	$0.535464\pi$
$354$		0			0
$355$			53986.5i			0.428379i
$356$		0			0
$357$		−	279672.i		−	2.19438i
$358$		0			0
$359$	30040.7			0.233089			0.116544	−	0.993185i	$-0.462818\pi$
$359$	30040.7			0.233089			0.116544	+	0.993185i	$0.462818\pi$
$360$		0			0
$361$	114454.	+	62320.7i	0.878246	+	0.478209i
$362$		0			0
$363$			87037.4i			0.660530i
$364$		0			0
$365$	−21411.8			−0.160719
$366$		0			0
$367$	51235.7			0.380400			0.190200	−	0.981745i	$-0.439086\pi$
$367$	51235.7			0.380400			0.190200	+	0.981745i	$0.439086\pi$
$368$		0			0
$369$			70689.8i			0.519163i
$370$		0			0
$371$			329286.i			2.39235i
$372$		0			0
$373$		−	75674.3i		−	0.543914i	−0.962309	−	0.271957i	$-0.912329\pi$
$373$		−	75674.3i		−	0.543914i	0.962309	−	0.271957i	$-0.0876709\pi$
$374$		0			0
$375$			121512.i			0.864084i
$376$		0			0
$377$	−49149.1			−0.345806
$378$		0			0
$379$		−	216796.i		−	1.50929i	−0.656132	−	0.754646i	$-0.727809\pi$
$379$		−	216796.i		−	1.50929i	0.656132	−	0.754646i	$-0.272191\pi$
$380$		0			0
$381$	−39761.6			−0.273914
$382$		0			0
$383$			56907.1i			0.387944i	0.981007	+	0.193972i	$0.0621371\pi$
$383$			56907.1i			0.387944i	−0.981007	+	0.193972i	$0.937863\pi$
$384$		0			0
$385$	−103167.			−0.696019
$386$		0			0
$387$	41535.9			0.277333
$388$		0			0
$389$	−43485.7			−0.287374			−0.143687	−	0.989623i	$-0.545896\pi$
$389$	−43485.7			−0.287374			−0.143687	+	0.989623i	$0.545896\pi$
$390$		0			0
$391$	180936.			1.18351
$392$		0			0
$393$			153788.i			0.995718i
$394$		0			0
$395$			76201.9i			0.488396i
$396$		0			0
$397$	−148863.			−0.944507			−0.472253	−	0.881463i	$-0.656559\pi$
$397$	−148863.			−0.944507			−0.472253	+	0.881463i	$0.656559\pi$
$398$		0			0
$399$	77290.7	−	303572.i	0.485491	−	1.90685i
$400$		0			0
$401$		−	54439.7i		−	0.338553i	−0.985569	−	0.169277i	$-0.945857\pi$
$401$		−	54439.7i		−	0.338553i	0.985569	−	0.169277i	$-0.0541431\pi$
$402$		0			0
$403$	−31523.5			−0.194100
$404$		0			0
$405$	73944.8			0.450814
$406$		0			0
$407$		−	268613.i		−	1.62158i
$408$		0			0
$409$			262595.i			1.56978i	0.619632	+	0.784892i	$0.287282\pi$
$409$			262595.i			1.56978i	−0.619632	+	0.784892i	$0.712718\pi$
$410$		0			0
$411$			58326.0i			0.345286i
$412$		0			0
$413$		−	442746.i		−	2.59570i
$414$		0			0
$415$	−100991.			−0.586390
$416$		0			0
$417$		−	195991.i		−	1.12710i
$418$		0			0
$419$	−21837.6			−0.124387			−0.0621937	−	0.998064i	$-0.519810\pi$
$419$	−21837.6			−0.124387			−0.0621937	+	0.998064i	$0.519810\pi$
$420$		0			0
$421$			243876.i			1.37596i	0.725732	+	0.687978i	$0.241501\pi$
$421$			243876.i			1.37596i	−0.725732	+	0.687978i	$0.758499\pi$
$422$		0			0
$423$	53947.4			0.301502
$424$		0			0
$425$	−174691.			−0.967148
$426$		0			0
$427$	−418566.			−2.29566
$428$		0			0
$429$	−321376.			−1.74622
$430$		0			0
$431$			150402.i			0.809655i	0.914393	+	0.404827i	$0.132668\pi$
$431$			150402.i			0.809655i	−0.914393	+	0.404827i	$0.867332\pi$
$432$		0			0
$433$		−	172534.i		−	0.920237i	−0.887858	−	0.460118i	$-0.847807\pi$
$433$		−	172534.i		−	0.920237i	0.887858	−	0.460118i	$-0.152193\pi$
$434$		0			0
$435$	−27153.0			−0.143496
$436$		0			0
$437$	196398.	+	50003.8i	1.02843	+	0.261843i
$438$		0			0
$439$		−	339644.i		−	1.76236i	−0.472781	−	0.881180i	$-0.656750\pi$
$439$		−	339644.i		−	1.76236i	0.472781	−	0.881180i	$-0.343250\pi$
$440$		0			0
$441$	−166943.			−0.858402
$442$		0			0
$443$	69761.2			0.355473			0.177736	−	0.984078i	$-0.443123\pi$
$443$	69761.2			0.355473			0.177736	+	0.984078i	$0.443123\pi$
$444$		0			0
$445$			7599.06i			0.0383743i
$446$		0			0
$447$			66031.1i			0.330471i
$448$		0			0
$449$		−	105494.i		−	0.523280i	−0.965166	−	0.261640i	$-0.915737\pi$
$449$		−	105494.i		−	0.523280i	0.965166	−	0.261640i	$-0.0842633\pi$
$450$		0			0
$451$			212399.i			1.04424i
$452$		0			0
$453$	−32606.9			−0.158896
$454$		0			0
$455$		−	130335.i		−	0.629561i
$456$		0			0
$457$	−134453.			−0.643781			−0.321890	−	0.946777i	$-0.604318\pi$
$457$	−134453.			−0.643781			−0.321890	+	0.946777i	$0.604318\pi$
$458$		0			0
$459$			115489.i			0.548170i
$460$		0			0
$461$	196297.			0.923657			0.461829	−	0.886969i	$-0.347194\pi$
$461$	196297.			0.923657			0.461829	+	0.886969i	$0.347194\pi$
$462$		0			0
$463$	−14619.6			−0.0681982			−0.0340991	−	0.999418i	$-0.510856\pi$
$463$	−14619.6			−0.0681982			−0.0340991	+	0.999418i	$0.510856\pi$
$464$		0			0
$465$	−17415.5			−0.0805435
$466$		0			0
$467$	−214407.			−0.983115			−0.491557	−	0.870845i	$-0.663572\pi$
$467$	−214407.			−0.983115			−0.491557	+	0.870845i	$0.663572\pi$
$468$		0			0
$469$			235890.i			1.07242i
$470$		0			0
$471$			202933.i			0.914767i
$472$		0			0
$473$	124802.			0.557825
$474$		0			0
$475$	−189620.	−	48278.1i	−0.840422	−	0.213975i
$476$		0			0
$477$			215357.i			0.946503i
$478$		0			0
$479$	−349544.			−1.52346			−0.761729	−	0.647895i	$-0.775649\pi$
$479$	−349544.			−1.52346			−0.761729	+	0.647895i	$0.775649\pi$
$480$		0			0
$481$	339348.			1.46675
$482$		0			0
$483$		−	487150.i		−	2.08818i
$484$		0			0
$485$		−	20540.2i		−	0.0873213i
$486$		0			0
$487$		−	334987.i		−	1.41244i	−0.707992	−	0.706220i	$-0.750399\pi$
$487$		−	334987.i		−	1.41244i	0.707992	−	0.706220i	$-0.249601\pi$
$488$		0			0
$489$		−	375888.i		−	1.57196i
$490$		0			0
$491$	474897.			1.96987			0.984933	−	0.172938i	$-0.0553260\pi$
$491$	474897.			1.96987			0.984933	+	0.172938i	$0.0553260\pi$
$492$		0			0
$493$		−	84048.9i		−	0.345810i
$494$		0			0
$495$	−67472.8			−0.275371
$496$		0			0
$497$			449924.i			1.82149i
$498$		0			0
$499$	422704.			1.69760			0.848801	−	0.528713i	$-0.177325\pi$
$499$	422704.			1.69760			0.848801	+	0.528713i	$0.177325\pi$
$500$		0			0
$501$	157056.			0.625717
$502$		0			0
$503$	−341698.			−1.35054			−0.675268	−	0.737572i	$-0.735972\pi$
$503$	−341698.			−1.35054			−0.675268	+	0.737572i	$0.735972\pi$
$504$		0			0
$505$	−7141.05			−0.0280014
$506$		0			0
$507$		−	79546.4i		−	0.309460i
$508$		0			0
$509$			174479.i			0.673452i	0.941603	+	0.336726i	$0.109320\pi$
$509$			174479.i			0.673452i	−0.941603	+	0.336726i	$0.890680\pi$
$510$		0			0
$511$	−178446.			−0.683383
$512$		0			0
$513$	−31916.8	+	125359.i	−0.121279	+	0.476343i
$514$		0			0
$515$			162446.i			0.612485i
$516$		0			0
$517$	162094.			0.606437
$518$		0			0
$519$	−122451.			−0.454599
$520$		0			0
$521$			255132.i			0.939917i	0.882689	+	0.469958i	$0.155731\pi$
$521$			255132.i			0.939917i	−0.882689	+	0.469958i	$0.844269\pi$
$522$		0			0
$523$			440253.i			1.60953i	0.593593	+	0.804765i	$0.297709\pi$
$523$			440253.i			1.60953i	−0.593593	+	0.804765i	$0.702291\pi$
$524$		0			0
$525$			470337.i			1.70644i
$526$		0			0
$527$		−	53907.7i		−	0.194102i
$528$		0			0
$529$	35324.4			0.126230
$530$		0			0
$531$		−	289562.i		−	1.02696i
$532$		0			0
$533$	−268331.			−0.944532
$534$		0			0
$535$			161512.i			0.564285i
$536$		0			0
$537$	−453911.			−1.57406
$538$		0			0
$539$	−501608.			−1.72658
$540$		0			0
$541$	−137088.			−0.468386			−0.234193	−	0.972190i	$-0.575245\pi$
$541$	−137088.			−0.468386			−0.234193	+	0.972190i	$0.575245\pi$
$542$		0			0
$543$	−153806.			−0.521642
$544$		0			0
$545$			119683.i			0.402940i
$546$		0			0
$547$			93456.1i			0.312344i	0.987730	+	0.156172i	$0.0499154\pi$
$547$			93456.1i			0.312344i	−0.987730	+	0.156172i	$0.950085\pi$
$548$		0			0
$549$	−273747.			−0.908249
$550$		0			0
$551$	23228.0	−	91231.7i	0.0765082	−	0.300499i
$552$		0			0
$553$			635067.i			2.07668i
$554$		0			0
$555$	187477.			0.608641
$556$		0			0
$557$	−511715.			−1.64937			−0.824684	−	0.565594i	$-0.808647\pi$
$557$	−511715.			−1.64937			−0.824684	+	0.565594i	$0.808647\pi$
$558$		0			0
$559$			157666.i			0.504562i
$560$		0			0
$561$		−	549580.i		−	1.74624i
$562$		0			0
$563$			27109.4i			0.0855271i	0.999085	+	0.0427636i	$0.0136162\pi$
$563$			27109.4i			0.0855271i	−0.999085	+	0.0427636i	$0.986384\pi$
$564$		0			0
$565$			40948.2i			0.128274i
$566$		0			0
$567$	616256.			1.91688
$568$		0			0
$569$			90925.6i			0.280842i	0.990092	+	0.140421i	$0.0448456\pi$
$569$			90925.6i			0.280842i	−0.990092	+	0.140421i	$0.955154\pi$
$570$		0			0
$571$	374122.			1.14747			0.573735	−	0.819041i	$-0.305494\pi$
$571$	374122.			1.14747			0.573735	+	0.819041i	$0.305494\pi$
$572$		0			0
$573$		−	677773.i		−	2.06431i
$574$		0			0
$575$	−304288.			−0.920343
$576$		0			0
$577$	618133.			1.85665			0.928325	−	0.371769i	$-0.121249\pi$
$577$	618133.			1.85665			0.928325	+	0.371769i	$0.121249\pi$
$578$		0			0
$579$	−377574.			−1.12628
$580$		0			0
$581$	−841659.			−2.49336
$582$		0			0
$583$			647076.i			1.90379i
$584$		0			0
$585$		−	85240.6i		−	0.249078i
$586$		0			0
$587$	−546495.			−1.58603			−0.793013	−	0.609205i	$-0.791489\pi$
$587$	−546495.			−1.58603			−0.793013	+	0.609205i	$0.791489\pi$
$588$		0			0
$589$	14898.1	−	58514.7i	0.0429437	−	0.168669i
$590$		0			0
$591$		−	341322.i		−	0.977212i
$592$		0			0
$593$	−126105.			−0.358610			−0.179305	−	0.983794i	$-0.557385\pi$
$593$	−126105.			−0.358610			−0.179305	+	0.983794i	$0.557385\pi$
$594$		0			0
$595$	222883.			0.629569
$596$		0			0
$597$			353981.i			0.993188i
$598$		0			0
$599$			592405.i			1.65107i	0.564351	+	0.825535i	$0.309126\pi$
$599$			592405.i			1.65107i	−0.564351	+	0.825535i	$0.690874\pi$
$600$		0			0
$601$		−	150697.i		−	0.417211i	−0.978000	−	0.208606i	$-0.933108\pi$
$601$		−	150697.i		−	0.417211i	0.978000	−	0.208606i	$-0.0668925\pi$
$602$		0			0
$603$			154275.i			0.424289i
$604$		0			0
$605$	−69364.1			−0.189507
$606$		0			0
$607$		−	442114.i		−	1.19993i	−0.800025	−	0.599967i	$-0.795180\pi$
$607$		−	442114.i		−	1.19993i	0.800025	−	0.599967i	$-0.204820\pi$
$608$		0			0
$609$	−226293.			−0.610149
$610$		0			0
$611$			204779.i			0.548533i
$612$		0			0
$613$	419179.			1.11552			0.557761	−	0.830001i	$-0.311660\pi$
$613$	419179.			1.11552			0.557761	+	0.830001i	$0.311660\pi$
$614$		0			0
$615$	−148243.			−0.391943
$616$		0			0
$617$	104140.			0.273558			0.136779	−	0.990602i	$-0.456325\pi$
$617$	104140.			0.273558			0.136779	+	0.990602i	$0.456325\pi$
$618$		0			0
$619$	−184995.			−0.482812			−0.241406	−	0.970424i	$-0.577608\pi$
$619$	−184995.			−0.482812			−0.241406	+	0.970424i	$0.577608\pi$
$620$		0			0
$621$			201166.i			0.521641i
$622$		0			0
$623$			63330.6i			0.163169i
$624$		0			0
$625$	241925.			0.619327
$626$		0			0
$627$	151883.	−	596546.i	0.386344	−	1.51743i
$628$		0			0
$629$			580312.i			1.46676i
$630$		0			0
$631$	677804.			1.70234			0.851169	−	0.524892i	$-0.175894\pi$
$631$	677804.			1.70234			0.851169	+	0.524892i	$0.175894\pi$
$632$		0			0
$633$	869108.			2.16903
$634$		0			0
$635$		−	31687.9i		−	0.0785861i
$636$		0			0
$637$		−	633698.i		−	1.56172i
$638$		0			0
$639$			294256.i			0.720648i
$640$		0			0
$641$		−	272570.i		−	0.663379i	−0.943389	−	0.331690i	$-0.892381\pi$
$641$		−	272570.i		−	0.663379i	0.943389	−	0.331690i	$-0.107619\pi$
$642$		0			0
$643$	436968.			1.05689			0.528443	−	0.848969i	$-0.322776\pi$
$643$	436968.			1.05689			0.528443	+	0.848969i	$0.322776\pi$
$644$		0			0
$645$			87104.4i			0.209373i
$646$		0			0
$647$	80856.5			0.193155			0.0965776	−	0.995325i	$-0.469210\pi$
$647$	80856.5			0.193155			0.0965776	+	0.995325i	$0.469210\pi$
$648$		0			0
$649$		−	870036.i		−	2.06561i
$650$		0			0
$651$	−145141.			−0.342474
$652$		0			0
$653$	252641.			0.592485			0.296243	−	0.955113i	$-0.404266\pi$
$653$	252641.			0.592485			0.296243	+	0.955113i	$0.404266\pi$
$654$		0			0
$655$	−122560.			−0.285672
$656$		0			0
$657$	−116706.			−0.270372
$658$		0			0
$659$		−	290788.i		−	0.669585i	−0.942292	−	0.334793i	$-0.891334\pi$
$659$		−	290788.i		−	0.669585i	0.942292	−	0.334793i	$-0.108666\pi$
$660$		0			0
$661$		−	676941.i		−	1.54934i	−0.632364	−	0.774672i	$-0.717915\pi$
$661$		−	676941.i		−	1.54934i	0.632364	−	0.774672i	$-0.282085\pi$
$662$		0			0
$663$	694302.			1.57951
$664$		0			0
$665$	241931.	+	61596.5i	0.547076	+	0.139288i
$666$		0			0
$667$		−	146402.i		−	0.329075i
$668$		0			0
$669$	754798.			1.68647
$670$		0			0
$671$	−822519.			−1.82684
$672$		0			0
$673$		−	643778.i		−	1.42137i	−0.703512	−	0.710683i	$-0.748386\pi$
$673$		−	643778.i		−	1.42137i	0.703512	−	0.710683i	$-0.251614\pi$
$674$		0			0
$675$		−	194224.i		−	0.426279i
$676$		0			0
$677$		−	813059.i		−	1.77396i	−0.461804	−	0.886982i	$-0.652798\pi$
$677$		−	813059.i		−	1.77396i	0.461804	−	0.886982i	$-0.347202\pi$
$678$		0			0
$679$		−	171182.i		−	0.371294i
$680$		0			0
$681$	809704.			1.74595
$682$		0			0
$683$			374698.i			0.803231i	0.915808	+	0.401615i	$0.131551\pi$
$683$			374698.i			0.803231i	−0.915808	+	0.401615i	$0.868449\pi$
$684$		0			0
$685$	−46482.7			−0.0990626
$686$		0			0
$687$			789680.i			1.67316i
$688$		0			0
$689$	−817472.			−1.72201
$690$		0			0
$691$	578886.			1.21238			0.606188	−	0.795321i	$-0.292698\pi$
$691$	578886.			1.21238			0.606188	+	0.795321i	$0.292698\pi$
$692$		0			0
$693$	−562318.			−1.17089
$694$		0			0
$695$	156194.			0.323366
$696$		0			0
$697$		−	458868.i		−	0.944543i
$698$		0			0
$699$		−	825614.i		−	1.68975i
$700$		0			0
$701$	−325316.			−0.662018			−0.331009	−	0.943628i	$-0.607389\pi$
$701$	−325316.			−0.662018			−0.331009	+	0.943628i	$0.607389\pi$
$702$		0			0
$703$	−160376.	+	629905.i	−0.324511	+	1.27457i
$704$		0			0
$705$			113132.i			0.227619i
$706$		0			0
$707$	−59513.5			−0.119063
$708$		0			0
$709$	−805646.			−1.60270			−0.801350	−	0.598196i	$-0.795884\pi$
$709$	−805646.			−1.60270			−0.801350	+	0.598196i	$0.795884\pi$
$710$		0			0
$711$			415342.i			0.821611i
$712$		0			0
$713$		−	93900.0i		−	0.184708i
$714$		0			0
$715$		−	256120.i		−	0.500992i
$716$		0			0
$717$		−	168641.i		−	0.328038i
$718$		0			0
$719$	234223.			0.453077			0.226539	−	0.974002i	$-0.427259\pi$
$719$	234223.			0.453077			0.226539	+	0.974002i	$0.427259\pi$
$720$		0			0
$721$			1.35383e6i			2.60431i
$722$		0			0
$723$	−515967.			−0.987063
$724$		0			0
$725$			141349.i			0.268916i
$726$		0			0
$727$	464219.			0.878322			0.439161	−	0.898408i	$-0.355276\pi$
$727$	464219.			0.878322			0.439161	+	0.898408i	$0.355276\pi$
$728$		0			0
$729$	71277.1			0.134121
$730$		0			0
$731$	−269622.			−0.504568
$732$		0			0
$733$	−82410.5			−0.153382			−0.0766910	−	0.997055i	$-0.524435\pi$
$733$	−82410.5			−0.153382			−0.0766910	+	0.997055i	$0.524435\pi$
$734$		0			0
$735$		−	350094.i		−	0.648052i
$736$		0			0
$737$			463546.i			0.853410i
$738$		0			0
$739$	326895.			0.598576			0.299288	−	0.954163i	$-0.403251\pi$
$739$	326895.			0.598576			0.299288	+	0.954163i	$0.403251\pi$
$740$		0			0
$741$	753637.	+	191879.i	1.37254	+	0.349455i
$742$		0			0
$743$		−	735492.i		−	1.33230i	−0.745820	−	0.666148i	$-0.767942\pi$
$743$		−	735492.i		−	1.33230i	0.745820	−	0.666148i	$-0.232058\pi$
$744$		0			0
$745$	−52623.2			−0.0948123
$746$		0			0
$747$	−550456.			−0.986464
$748$		0			0
$749$			1.34604e6i			2.39936i
$750$		0			0
$751$			30777.9i			0.0545706i	0.999628	+	0.0272853i	$0.00868626\pi$
$751$			30777.9i			0.0545706i	−0.999628	+	0.0272853i	$0.991314\pi$
$752$		0			0
$753$		−	531432.i		−	0.937254i
$754$		0			0
$755$		−	25985.9i		−	0.0455873i
$756$		0			0
$757$	−829936.			−1.44828			−0.724141	−	0.689652i	$-0.757763\pi$
$757$	−829936.			−1.44828			−0.724141	+	0.689652i	$0.757763\pi$
$758$		0			0
$759$		−	957294.i		−	1.66173i
$760$		0			0
$761$	−170283.			−0.294037			−0.147018	−	0.989134i	$-0.546968\pi$
$761$	−170283.			−0.294037			−0.147018	+	0.989134i	$0.546968\pi$
$762$		0			0
$763$			997439.i			1.71332i
$764$		0			0
$765$	145768.			0.249081
$766$		0			0
$767$	1.09915e6			1.86838
$768$		0			0
$769$	770218.			1.30245			0.651225	−	0.758885i	$-0.274255\pi$
$769$	770218.			1.30245			0.651225	+	0.758885i	$0.274255\pi$
$770$		0			0
$771$	1.41293e6			2.37690
$772$		0			0
$773$			23021.2i			0.0385273i	0.999814	+	0.0192637i	$0.00613219\pi$
$773$			23021.2i			0.0385273i	−0.999814	+	0.0192637i	$0.993868\pi$
$774$		0			0
$775$			90659.3i			0.150942i
$776$		0			0
$777$	1.56243e6			2.58796
$778$		0			0
$779$	126814.	−	498082.i	0.208974	−	0.820779i
$780$		0			0
$781$			884140.i			1.44950i
$782$		0			0
$783$	93446.5			0.152419
$784$		0			0
$785$	−161727.			−0.262447
$786$		0			0
$787$		−	67840.5i		−	0.109532i	−0.998499	−	0.0547658i	$-0.982559\pi$
$787$		−	67840.5i		−	0.109532i	0.998499	−	0.0547658i	$-0.0174412\pi$
$788$		0			0
$789$		−	588638.i		−	0.945571i
$790$		0			0
$791$			341262.i			0.545425i
$792$		0			0
$793$		−	1.03912e6i		−	1.65241i
$794$		0			0
$795$	−451622.			−0.714563
$796$		0			0
$797$		−	296624.i		−	0.466971i	−0.972360	−	0.233485i	$-0.924987\pi$
$797$		−	296624.i		−	0.466971i	0.972360	−	0.233485i	$-0.0750131\pi$
$798$		0			0
$799$	−350188.			−0.548540
$800$		0			0
$801$			41419.0i			0.0645557i
$802$		0			0
$803$	−350662.			−0.543823
$804$		0			0
$805$	388232.			0.599101
$806$		0			0
$807$	869047.			1.33443
$808$		0			0
$809$	822479.			1.25669			0.628344	−	0.777935i	$-0.283733\pi$
$809$	822479.			1.25669			0.628344	+	0.777935i	$0.283733\pi$
$810$		0			0
$811$		−	539509.i		−	0.820270i	−0.912025	−	0.410135i	$-0.865482\pi$
$811$		−	539509.i		−	0.820270i	0.912025	−	0.410135i	$-0.134518\pi$
$812$		0			0
$813$			119669.i			0.181051i
$814$		0			0
$815$	299562.			0.450995
$816$		0			0
$817$	−292663.	−	74513.3i	−0.438454	−	0.111632i
$818$		0			0
$819$		−	710395.i		−	1.05909i
$820$		0			0
$821$	−443727.			−0.658309			−0.329154	−	0.944276i	$-0.606764\pi$
$821$	−443727.			−0.658309			−0.329154	+	0.944276i	$0.606764\pi$
$822$		0			0
$823$	−171466.			−0.253150			−0.126575	−	0.991957i	$-0.540398\pi$
$823$	−171466.			−0.253150			−0.126575	+	0.991957i	$0.540398\pi$
$824$		0			0
$825$			924255.i			1.35795i
$826$		0			0
$827$			62968.3i			0.0920685i	0.998940	+	0.0460342i	$0.0146583\pi$
$827$			62968.3i			0.0920685i	−0.998940	+	0.0460342i	$0.985342\pi$
$828$		0			0
$829$			1.00097e6i			1.45650i	0.685312	+	0.728250i	$0.259666\pi$
$829$			1.00097e6i			1.45650i	−0.685312	+	0.728250i	$0.740334\pi$
$830$		0			0
$831$			1.34075e6i			1.94154i
$832$		0			0
$833$	1.08367e6			1.56174
$834$		0			0
$835$			125165.i			0.179519i
$836$		0			0
$837$	59935.2			0.0855522
$838$		0			0
$839$		−	579085.i		−	0.822656i	−0.911487	−	0.411328i	$-0.865065\pi$
$839$		−	579085.i		−	0.822656i	0.911487	−	0.411328i	$-0.134935\pi$
$840$		0			0
$841$	639274.			0.903847
$842$		0			0
$843$	−873184.			−1.22871
$844$		0			0
$845$	63394.2			0.0887843
$846$		0			0
$847$	−578081.			−0.805789
$848$		0			0
$849$			1.52067e6i			2.10969i
$850$		0			0
$851$			1.01083e6i			1.39578i
$852$		0			0
$853$	252867.			0.347532			0.173766	−	0.984787i	$-0.444406\pi$
$853$	252867.			0.347532			0.173766	+	0.984787i	$0.444406\pi$
$854$		0			0
$855$	158226.	+	40284.9i	0.216443	+	0.0551074i
$856$		0			0
$857$		−	466322.i		−	0.634928i	−0.948270	−	0.317464i	$-0.897169\pi$
$857$		−	466322.i		−	0.634928i	0.948270	−	0.317464i	$-0.102831\pi$
$858$		0			0
$859$	484650.			0.656814			0.328407	−	0.944536i	$-0.393488\pi$
$859$	484650.			0.656814			0.328407	+	0.944536i	$0.393488\pi$
$860$		0			0
$861$	−1.23545e6			−1.66656
$862$		0			0
$863$		−	1.22166e6i		−	1.64032i	−0.572137	−	0.820158i	$-0.693886\pi$
$863$		−	1.22166e6i		−	1.64032i	0.572137	−	0.820158i	$-0.306114\pi$
$864$		0			0
$865$		−	97587.0i		−	0.130425i
$866$		0			0
$867$			232647.i			0.309499i
$868$		0			0
$869$			1.24796e6i			1.65258i
$870$		0			0
$871$	−585613.			−0.771923
$872$		0			0
$873$		−	111955.i		−	0.146898i
$874$		0			0
$875$	−807051.			−1.05411
$876$		0			0
$877$			264511.i			0.343910i	0.985105	+	0.171955i	$0.0550083\pi$
$877$			264511.i			0.343910i	−0.985105	+	0.171955i	$0.944992\pi$
$878$		0			0
$879$	−1.51642e6			−1.96264
$880$		0			0
$881$	−396673.			−0.511071			−0.255535	−	0.966800i	$-0.582252\pi$
$881$	−396673.			−0.511071			−0.255535	+	0.966800i	$0.582252\pi$
$882$		0			0
$883$	−380255.			−0.487701			−0.243851	−	0.969813i	$-0.578411\pi$
$883$	−380255.			−0.487701			−0.243851	+	0.969813i	$0.578411\pi$
$884$		0			0
$885$	607235.			0.775301
$886$		0			0
$887$		−	1.55369e6i		−	1.97477i	−0.158334	−	0.987386i	$-0.550612\pi$
$887$		−	1.55369e6i		−	1.97477i	0.158334	−	0.987386i	$-0.449388\pi$
$888$		0			0
$889$		−	264087.i		−	0.334151i
$890$		0			0
$891$	1.21100e6			1.52542
$892$		0			0
$893$	−380115.	−	96778.9i	−0.476664	−	0.121361i
$894$		0			0
$895$		−	361743.i		−	0.451600i
$896$		0			0
$897$	1.20938e6			1.50307
$898$		0			0
$899$	−43618.8			−0.0539702
$900$		0			0
$901$		−	1.39794e6i		−	1.72203i
$902$		0			0
$903$			725928.i			0.890262i
$904$		0			0
$905$		−	122575.i		−	0.149659i
$906$		0			0
$907$			1.19038e6i			1.44701i	0.690321	+	0.723503i	$0.257469\pi$
$907$			1.19038e6i			1.44701i	−0.690321	+	0.723503i	$0.742531\pi$
$908$		0			0
$909$	−38922.6			−0.0471057
$910$		0			0
$911$			680660.i			0.820150i	0.912052	+	0.410075i	$0.134497\pi$
$911$			680660.i			0.820150i	−0.912052	+	0.410075i	$0.865503\pi$
$912$		0			0
$913$	−1.65394e6			−1.98416
$914$		0			0
$915$		−	574071.i		−	0.685683i
$916$		0			0
$917$	−1.02142e6			−1.21469
$918$		0			0
$919$	527917.			0.625079			0.312539	−	0.949905i	$-0.398820\pi$
$919$	527917.			0.625079			0.312539	+	0.949905i	$0.398820\pi$
$920$		0			0
$921$	−1.33194e6			−1.57024
$922$		0			0
$923$	−1.11696e6			−1.31110
$924$		0			0
$925$		−	975939.i		−	1.14062i
$926$		0			0
$927$			885420.i			1.03036i
$928$		0			0
$929$	−28451.8			−0.0329669			−0.0164834	−	0.999864i	$-0.505247\pi$
$929$	−28451.8			−0.0329669			−0.0164834	+	0.999864i	$0.505247\pi$
$930$		0			0
$931$	1.17629e6	+	299487.i	1.35710	+	0.345524i
$932$		0			0
$933$		−	852556.i		−	0.979399i
$934$		0			0
$935$	437985.			0.500998
$936$		0			0
$937$	−1.32816e6			−1.51276			−0.756380	−	0.654133i	$-0.773034\pi$
$937$	−1.32816e6			−1.51276			−0.756380	+	0.654133i	$0.773034\pi$
$938$		0			0
$939$		−	1.60561e6i		−	1.82100i
$940$		0			0
$941$			171072.i			0.193196i	0.995323	+	0.0965981i	$0.0307962\pi$
$941$			171072.i			0.193196i	−0.995323	+	0.0965981i	$0.969204\pi$
$942$		0			0
$943$		−	799286.i		−	0.898832i
$944$		0			0
$945$			247804.i			0.277488i
$946$		0			0
$947$	853096.			0.951257			0.475628	−	0.879646i	$-0.342221\pi$
$947$	853096.			0.951257			0.475628	+	0.879646i	$0.342221\pi$
$948$		0			0
$949$		−	443003.i		−	0.491897i
$950$		0			0
$951$	−895174.			−0.989797
$952$		0			0
$953$		−	796329.i		−	0.876812i	−0.898777	−	0.438406i	$-0.855543\pi$
$953$		−	796329.i		−	0.876812i	0.898777	−	0.438406i	$-0.144457\pi$
$954$		0			0
$955$	540148.			0.592252
$956$		0			0
$957$	−444686.			−0.485545
$958$		0			0
$959$	−387386.			−0.421218
$960$		0			0
$961$	895545.			0.969707
$962$		0			0
$963$			880330.i			0.949277i
$964$		0			0
$965$		−	300906.i		−	0.323130i
$966$		0			0
$967$	−331903.			−0.354943			−0.177471	−	0.984126i	$-0.556792\pi$
$967$	−331903.			−0.354943			−0.177471	+	0.984126i	$0.556792\pi$
$968$		0			0
$969$	−328129.	+	1.28878e6i	−0.349459	+	1.37256i
$970$		0			0
$971$		−	718821.i		−	0.762399i	−0.924493	−	0.381200i	$-0.875511\pi$
$971$		−	718821.i		−	0.762399i	0.924493	−	0.381200i	$-0.124489\pi$
$972$		0			0
$973$	1.30172e6			1.37497
$974$		0			0
$975$	−1.16764e6			−1.22829
$976$		0			0
$977$		−	314771.i		−	0.329765i	−0.986313	−	0.164883i	$-0.947275\pi$
$977$		−	314771.i		−	0.329765i	0.986313	−	0.164883i	$-0.0527246\pi$
$978$		0			0
$979$			124450.i			0.129847i
$980$		0			0
$981$			652338.i			0.677851i
$982$		0			0
$983$			641055.i			0.663420i	0.943381	+	0.331710i	$0.107626\pi$
$983$			641055.i			0.663420i	−0.943381	+	0.331710i	$0.892374\pi$
$984$		0			0
$985$	272015.			0.280363
$986$		0			0
$987$			942845.i			0.967845i
$988$		0			0
$989$	−469645.			−0.480150
$990$		0			0
$991$		−	1.36712e6i		−	1.39207i	−0.718009	−	0.696034i	$-0.754946\pi$
$991$		−	1.36712e6i		−	1.39207i	0.718009	−	0.696034i	$-0.245054\pi$
$992$		0			0
$993$	−352510.			−0.357497
$994$		0			0
$995$	−282104.			−0.284946
$996$		0			0
$997$	−1.66593e6			−1.67597			−0.837985	−	0.545694i	$-0.816266\pi$
$997$	−1.66593e6			−1.67597			−0.837985	+	0.545694i	$0.816266\pi$
$998$		0			0
$999$	−645198.			−0.646490

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.5.e.f.113.17		20
4.3	odd	2		152.5.e.a.113.4	✓	20
12.11	even	2		1368.5.o.a.721.13		20
19.18	odd	2	inner	304.5.e.f.113.4		20
76.75	even	2		152.5.e.a.113.17	yes	20
228.227	odd	2		1368.5.o.a.721.14		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.5.e.a.113.4	✓	20	4.3	odd	2
152.5.e.a.113.17	yes	20	76.75	even	2
304.5.e.f.113.4		20	19.18	odd	2	inner
304.5.e.f.113.17		20	1.1	even	1	trivial
1368.5.o.a.721.13		20	12.11	even	2
1368.5.o.a.721.14		20	228.227	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 304 = 2^{4} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	304.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+11.4302i q^{3} -9.10929 q^{5} -75.9168 q^{7} -49.6505 q^{9} -149.183 q^{11} -188.468i q^{13} -104.121i q^{15} +322.296 q^{17} +(349.839 + 89.0705i) q^{19} -867.748i q^{21} +561.396 q^{23} -542.021 q^{25} +358.332i q^{27} -260.782i q^{29} -167.262i q^{31} -1705.20i q^{33} +691.548 q^{35} +1800.56i q^{37} +2154.24 q^{39} -1423.75i q^{41} -836.566 q^{43} +452.281 q^{45} -1086.54 q^{47} +3362.36 q^{49} +3683.92i q^{51} -4337.45i q^{53} +1358.95 q^{55} +(-1018.10 + 3998.75i) q^{57} +5831.99i q^{59} +5513.48 q^{61} +3769.31 q^{63} +1716.81i q^{65} -3107.22i q^{67} +6416.89i q^{69} -5926.54i q^{71} +2350.54 q^{73} -6195.43i q^{75} +11325.5 q^{77} -8365.30i q^{79} -8117.52 q^{81} +11086.6 q^{83} -2935.89 q^{85} +2980.80 q^{87} -834.210i q^{89} +14307.9i q^{91} +1911.84 q^{93} +(-3186.79 - 811.369i) q^{95} +2254.86i q^{97} +7407.03 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 32 q^{7} - 372 q^{9} + 24 q^{11} + 216 q^{17} + 596 q^{19} - 576 q^{23} + 1412 q^{25} + 144 q^{35} + 520 q^{39} + 1256 q^{43} + 7232 q^{45} + 3768 q^{47} - 2740 q^{49} + 10128 q^{55} - 728 q^{57}+ \cdots - 760 q^{99}+O(q^{100}) \) 20 * q - 32 * q^7 - 372 * q^9 + 24 * q^11 + 216 * q^17 + 596 * q^19 - 576 * q^23 + 1412 * q^25 + 144 * q^35 + 520 * q^39 + 1256 * q^43 + 7232 * q^45 + 3768 * q^47 - 2740 * q^49 + 10128 * q^55 - 728 * q^57 + 352 * q^61 - 6104 * q^63 + 1352 * q^73 + 9288 * q^77 - 4220 * q^81 + 16104 * q^83 + 10232 * q^85 - 2936 * q^87 + 36432 * q^93 - 14232 * q^95 - 760 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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