LMFDB - Embedded newform 1104.6.a.p.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1104,6,Mod(1,1104)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1104, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1104.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1104 = 2^{4} \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1104.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$177.063737074$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 480x^{2} + 3169x + 6509 $ x^4 - x^3 - 480x^2 + 3169x + 6509
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 276)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$16.9516$ of defining polynomial
Character	$\chi$	$=$	1104.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.00000 q^{3} +19.9950 q^{5} +195.062 q^{7} +81.0000 q^{9} +O(q^{10})$	$q+9.00000 q^{3} +19.9950 q^{5} +195.062 q^{7} +81.0000 q^{9} -378.508 q^{11} +679.869 q^{13} +179.955 q^{15} -732.250 q^{17} -1120.81 q^{19} +1755.56 q^{21} -529.000 q^{23} -2725.20 q^{25} +729.000 q^{27} -6567.73 q^{29} -5420.04 q^{31} -3406.57 q^{33} +3900.26 q^{35} +2004.39 q^{37} +6118.82 q^{39} -11407.6 q^{41} -6402.81 q^{43} +1619.59 q^{45} -25666.3 q^{47} +21242.2 q^{49} -6590.25 q^{51} -23574.9 q^{53} -7568.25 q^{55} -10087.3 q^{57} +38487.4 q^{59} -52773.9 q^{61} +15800.0 q^{63} +13594.0 q^{65} +65495.5 q^{67} -4761.00 q^{69} -32841.1 q^{71} -88460.3 q^{73} -24526.8 q^{75} -73832.5 q^{77} +34192.7 q^{79} +6561.00 q^{81} -8943.35 q^{83} -14641.3 q^{85} -59109.6 q^{87} -102073. q^{89} +132617. q^{91} -48780.3 q^{93} -22410.6 q^{95} +175446. q^{97} -30659.1 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 36 q^{3} + 22 q^{5} + 154 q^{7} + 324 q^{9}+O(q^{10}) $ 4 * q + 36 * q^3 + 22 * q^5 + 154 * q^7 + 324 * q^9	$ 4 q + 36 q^{3} + 22 q^{5} + 154 q^{7} + 324 q^{9} - 392 q^{11} - 220 q^{13} + 198 q^{15} - 802 q^{17} - 2590 q^{19} + 1386 q^{21} - 2116 q^{23} + 6904 q^{25} + 2916 q^{27} - 5036 q^{29} - 20 q^{31} - 3528 q^{33} + 3752 q^{35} - 10884 q^{37} - 1980 q^{39} - 23312 q^{41} - 4406 q^{43} + 1782 q^{45} - 2472 q^{47} - 24568 q^{49} - 7218 q^{51} - 29222 q^{53} + 45928 q^{55} - 23310 q^{57} - 15136 q^{59} - 113220 q^{61} + 12474 q^{63} - 133892 q^{65} + 23214 q^{67} - 19044 q^{69} + 96096 q^{71} - 95792 q^{73} + 62136 q^{75} - 96584 q^{77} + 190094 q^{79} + 26244 q^{81} + 112280 q^{83} - 276556 q^{85} - 45324 q^{87} - 52206 q^{89} + 141548 q^{91} - 180 q^{93} - 38288 q^{95} + 9636 q^{97} - 31752 q^{99}+O(q^{100}) $ 4 * q + 36 * q^3 + 22 * q^5 + 154 * q^7 + 324 * q^9 - 392 * q^11 - 220 * q^13 + 198 * q^15 - 802 * q^17 - 2590 * q^19 + 1386 * q^21 - 2116 * q^23 + 6904 * q^25 + 2916 * q^27 - 5036 * q^29 - 20 * q^31 - 3528 * q^33 + 3752 * q^35 - 10884 * q^37 - 1980 * q^39 - 23312 * q^41 - 4406 * q^43 + 1782 * q^45 - 2472 * q^47 - 24568 * q^49 - 7218 * q^51 - 29222 * q^53 + 45928 * q^55 - 23310 * q^57 - 15136 * q^59 - 113220 * q^61 + 12474 * q^63 - 133892 * q^65 + 23214 * q^67 - 19044 * q^69 + 96096 * q^71 - 95792 * q^73 + 62136 * q^75 - 96584 * q^77 + 190094 * q^79 + 26244 * q^81 + 112280 * q^83 - 276556 * q^85 - 45324 * q^87 - 52206 * q^89 + 141548 * q^91 - 180 * q^93 - 38288 * q^95 + 9636 * q^97 - 31752 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	0	0
$5$	19.9950	0.357681	0.178840	−	0.983878i	$-0.442765\pi$
$5$	19.9950	0.357681	0.178840	+	0.983878i	$0.442765\pi$
$6$	0	0
$7$	195.062	1.50462	0.752312	−	0.658807i	$-0.228939\pi$
$7$	195.062	1.50462	0.752312	+	0.658807i	$0.228939\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−378.508	−0.943177	−0.471589	−	0.881819i	$-0.656319\pi$
$11$	−378.508	−0.943177	−0.471589	+	0.881819i	$0.656319\pi$
$12$	0	0
$13$	679.869	1.11575	0.557875	−	0.829925i	$-0.311617\pi$
$13$	679.869	1.11575	0.557875	+	0.829925i	$0.311617\pi$
$14$	0	0
$15$	179.955	0.206507
$16$	0	0
$17$	−732.250	−0.614522	−0.307261	−	0.951625i	$-0.599412\pi$
$17$	−732.250	−0.614522	−0.307261	+	0.951625i	$0.599412\pi$
$18$	0	0
$19$	−1120.81	−0.712277	−0.356139	−	0.934433i	$-0.615907\pi$
$19$	−1120.81	−0.712277	−0.356139	+	0.934433i	$0.615907\pi$
$20$	0	0
$21$	1755.56	0.868695
$22$	0	0
$23$	−529.000	−0.208514
$23$	−529.000	−0.208514
$24$	0	0
$25$	−2725.20	−0.872065
$26$	0	0
$27$	729.000	0.192450
$28$	0	0
$29$	−6567.73	−1.45018	−0.725088	−	0.688656i	$-0.758201\pi$
$29$	−6567.73	−1.45018	−0.725088	+	0.688656i	$0.758201\pi$
$30$	0	0
$31$	−5420.04	−1.01297	−0.506487	−	0.862248i	$-0.669056\pi$
$31$	−5420.04	−1.01297	−0.506487	+	0.862248i	$0.669056\pi$
$32$	0	0
$33$	−3406.57	−0.544544
$34$	0	0
$35$	3900.26	0.538175
$36$	0	0
$37$	2004.39	0.240702	0.120351	−	0.992731i	$-0.461598\pi$
$37$	2004.39	0.240702	0.120351	+	0.992731i	$0.461598\pi$
$38$	0	0
$39$	6118.82	0.644179
$40$	0	0
$41$	−11407.6	−1.05982	−0.529911	−	0.848053i	$-0.677775\pi$
$41$	−11407.6	−1.05982	−0.529911	+	0.848053i	$0.677775\pi$
$42$	0	0
$43$	−6402.81	−0.528079	−0.264040	−	0.964512i	$-0.585055\pi$
$43$	−6402.81	−0.528079	−0.264040	+	0.964512i	$0.585055\pi$
$44$	0	0
$45$	1619.59	0.119227
$46$	0	0
$47$	−25666.3	−1.69480	−0.847399	−	0.530957i	$-0.821833\pi$
$47$	−25666.3	−1.69480	−0.847399	+	0.530957i	$0.821833\pi$
$48$	0	0
$49$	21242.2	1.26389
$50$	0	0
$51$	−6590.25	−0.354794
$52$	0	0
$53$	−23574.9	−1.15282	−0.576408	−	0.817162i	$-0.695546\pi$
$53$	−23574.9	−1.15282	−0.576408	+	0.817162i	$0.695546\pi$
$54$	0	0
$55$	−7568.25	−0.337356
$56$	0	0
$57$	−10087.3	−0.411233
$58$	0	0
$59$	38487.4	1.43942	0.719711	−	0.694274i	$-0.244274\pi$
$59$	38487.4	1.43942	0.719711	+	0.694274i	$0.244274\pi$
$60$	0	0
$61$	−52773.9	−1.81591	−0.907956	−	0.419066i	$-0.862357\pi$
$61$	−52773.9	−1.81591	−0.907956	+	0.419066i	$0.862357\pi$
$62$	0	0
$63$	15800.0	0.501541
$64$	0	0
$65$	13594.0	0.399082
$66$	0	0
$67$	65495.5	1.78248	0.891239	−	0.453533i	$-0.149837\pi$
$67$	65495.5	1.78248	0.891239	+	0.453533i	$0.149837\pi$
$68$	0	0
$69$	−4761.00	−0.120386
$70$	0	0
$71$	−32841.1	−0.773164	−0.386582	−	0.922255i	$-0.626344\pi$
$71$	−32841.1	−0.773164	−0.386582	+	0.922255i	$0.626344\pi$
$72$	0	0
$73$	−88460.3	−1.94286	−0.971429	−	0.237330i	$-0.923728\pi$
$73$	−88460.3	−1.94286	−0.971429	+	0.237330i	$0.923728\pi$
$74$	0	0
$75$	−24526.8	−0.503487
$76$	0	0
$77$	−73832.5	−1.41913
$78$	0	0
$79$	34192.7	0.616404	0.308202	−	0.951321i	$-0.400273\pi$
$79$	34192.7	0.616404	0.308202	+	0.951321i	$0.400273\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−8943.35	−0.142497	−0.0712484	−	0.997459i	$-0.522698\pi$
$83$	−8943.35	−0.142497	−0.0712484	+	0.997459i	$0.522698\pi$
$84$	0	0
$85$	−14641.3	−0.219803
$86$	0	0
$87$	−59109.6	−0.837259
$88$	0	0
$89$	−102073.	−1.36595	−0.682977	−	0.730439i	$-0.739315\pi$
$89$	−102073.	−1.36595	−0.682977	+	0.730439i	$0.739315\pi$
$90$	0	0
$91$	132617.	1.67878
$92$	0	0
$93$	−48780.3	−0.584840
$94$	0	0
$95$	−22410.6	−0.254768
$96$	0	0
$97$	175446.	1.89328	0.946640	−	0.322292i	$-0.104453\pi$
$97$	175446.	1.89328	0.946640	+	0.322292i	$0.104453\pi$
$98$	0	0
$99$	−30659.1	−0.314392
$100$	0	0
$101$	−82393.2	−0.803688	−0.401844	−	0.915708i	$-0.631631\pi$
$101$	−82393.2	−0.803688	−0.401844	+	0.915708i	$0.631631\pi$
$102$	0	0
$103$	183280.	1.70225	0.851123	−	0.524966i	$-0.175922\pi$
$103$	183280.	1.70225	0.851123	+	0.524966i	$0.175922\pi$
$104$	0	0
$105$	35102.3	0.310715
$106$	0	0
$107$	−84096.4	−0.710098	−0.355049	−	0.934848i	$-0.615536\pi$
$107$	−84096.4	−0.710098	−0.355049	+	0.934848i	$0.615536\pi$
$108$	0	0
$109$	6684.08	0.0538859	0.0269430	−	0.999637i	$-0.491423\pi$
$109$	6684.08	0.0538859	0.0269430	+	0.999637i	$0.491423\pi$
$110$	0	0
$111$	18039.6	0.138969
$112$	0	0
$113$	132373.	0.975221	0.487610	−	0.873061i	$-0.337869\pi$
$113$	132373.	0.975221	0.487610	+	0.873061i	$0.337869\pi$
$114$	0	0
$115$	−10577.3	−0.0745816
$116$	0	0
$117$	55069.4	0.371917
$118$	0	0
$119$	−142834.	−0.924624
$120$	0	0
$121$	−17782.8	−0.110417
$122$	0	0
$123$	−102668.	−0.611888
$124$	0	0
$125$	−116975.	−0.669601
$126$	0	0
$127$	−69139.4	−0.380379	−0.190189	−	0.981747i	$-0.560910\pi$
$127$	−69139.4	−0.380379	−0.190189	+	0.981747i	$0.560910\pi$
$128$	0	0
$129$	−57625.3	−0.304887
$130$	0	0
$131$	280241.	1.42677	0.713384	−	0.700774i	$-0.247162\pi$
$131$	280241.	1.42677	0.713384	+	0.700774i	$0.247162\pi$
$132$	0	0
$133$	−218628.	−1.07171
$134$	0	0
$135$	14576.3	0.0688357
$136$	0	0
$137$	379242.	1.72630	0.863148	−	0.504951i	$-0.168489\pi$
$137$	379242.	1.72630	0.863148	+	0.504951i	$0.168489\pi$
$138$	0	0
$139$	−97966.2	−0.430070	−0.215035	−	0.976606i	$-0.568987\pi$
$139$	−97966.2	−0.430070	−0.215035	+	0.976606i	$0.568987\pi$
$140$	0	0
$141$	−230996.	−0.978492
$142$	0	0
$143$	−257336.	−1.05235
$144$	0	0
$145$	−131322.	−0.518700
$146$	0	0
$147$	191180.	0.729708
$148$	0	0
$149$	−219034.	−0.808250	−0.404125	−	0.914704i	$-0.632424\pi$
$149$	−219034.	−0.808250	−0.404125	+	0.914704i	$0.632424\pi$
$150$	0	0
$151$	352968.	1.25978	0.629888	−	0.776686i	$-0.283101\pi$
$151$	352968.	1.25978	0.629888	+	0.776686i	$0.283101\pi$
$152$	0	0
$153$	−59312.3	−0.204841
$154$	0	0
$155$	−108373.	−0.362321
$156$	0	0
$157$	306064.	0.990977	0.495489	−	0.868614i	$-0.334989\pi$
$157$	306064.	0.990977	0.495489	+	0.868614i	$0.334989\pi$
$158$	0	0
$159$	−212174.	−0.665578
$160$	0	0
$161$	−103188.	−0.313736
$162$	0	0
$163$	20330.2	0.0599340	0.0299670	−	0.999551i	$-0.490460\pi$
$163$	20330.2	0.0599340	0.0299670	+	0.999551i	$0.490460\pi$
$164$	0	0
$165$	−68114.2	−0.194773
$166$	0	0
$167$	438837.	1.21762	0.608810	−	0.793316i	$-0.291647\pi$
$167$	438837.	1.21762	0.608810	+	0.793316i	$0.291647\pi$
$168$	0	0
$169$	90929.4	0.244899
$170$	0	0
$171$	−90785.8	−0.237426
$172$	0	0
$173$	252731.	0.642012	0.321006	−	0.947077i	$-0.395979\pi$
$173$	252731.	0.642012	0.321006	+	0.947077i	$0.395979\pi$
$174$	0	0
$175$	−531584.	−1.31213
$176$	0	0
$177$	346386.	0.831051
$178$	0	0
$179$	−507186.	−1.18314	−0.591568	−	0.806255i	$-0.701491\pi$
$179$	−507186.	−1.18314	−0.591568	+	0.806255i	$0.701491\pi$
$180$	0	0
$181$	−214534.	−0.486743	−0.243372	−	0.969933i	$-0.578253\pi$
$181$	−214534.	−0.486743	−0.243372	+	0.969933i	$0.578253\pi$
$182$	0	0
$183$	−474965.	−1.04842
$184$	0	0
$185$	40077.8	0.0860943
$186$	0	0
$187$	277163.	0.579603
$188$	0	0
$189$	142200.	0.289565
$190$	0	0
$191$	−45770.3	−0.0907821	−0.0453910	−	0.998969i	$-0.514453\pi$
$191$	−45770.3	−0.0907821	−0.0453910	+	0.998969i	$0.514453\pi$
$192$	0	0
$193$	−293528.	−0.567226	−0.283613	−	0.958939i	$-0.591533\pi$
$193$	−293528.	−0.567226	−0.283613	+	0.958939i	$0.591533\pi$
$194$	0	0
$195$	122346.	0.230410
$196$	0	0
$197$	106574.	0.195653	0.0978264	−	0.995203i	$-0.468811\pi$
$197$	106574.	0.195653	0.0978264	+	0.995203i	$0.468811\pi$
$198$	0	0
$199$	290360.	0.519761	0.259881	−	0.965641i	$-0.416317\pi$
$199$	290360.	0.519761	0.259881	+	0.965641i	$0.416317\pi$
$200$	0	0
$201$	589459.	1.02911
$202$	0	0
$203$	−1.28112e6	−2.18197
$204$	0	0
$205$	−228094.	−0.379078
$206$	0	0
$207$	−42849.0	−0.0695048
$208$	0	0
$209$	424236.	0.671803
$210$	0	0
$211$	−779892.	−1.20595	−0.602974	−	0.797761i	$-0.706018\pi$
$211$	−779892.	−1.20595	−0.602974	+	0.797761i	$0.706018\pi$
$212$	0	0
$213$	−295570.	−0.446386
$214$	0	0
$215$	−128024.	−0.188884
$216$	0	0
$217$	−1.05724e6	−1.52414
$218$	0	0
$219$	−796142.	−1.12171
$220$	0	0
$221$	−497835.	−0.685653
$222$	0	0
$223$	401954.	0.541271	0.270635	−	0.962682i	$-0.412766\pi$
$223$	401954.	0.541271	0.270635	+	0.962682i	$0.412766\pi$
$224$	0	0
$225$	−220741.	−0.290688
$226$	0	0
$227$	−1.02179e6	−1.31613	−0.658064	−	0.752962i	$-0.728625\pi$
$227$	−1.02179e6	−1.31613	−0.658064	+	0.752962i	$0.728625\pi$
$228$	0	0
$229$	36529.0	0.0460309	0.0230155	−	0.999735i	$-0.492673\pi$
$229$	36529.0	0.0460309	0.0230155	+	0.999735i	$0.492673\pi$
$230$	0	0
$231$	−664493.	−0.819333
$232$	0	0
$233$	−819661.	−0.989110	−0.494555	−	0.869146i	$-0.664669\pi$
$233$	−819661.	−0.989110	−0.494555	+	0.869146i	$0.664669\pi$
$234$	0	0
$235$	−513196.	−0.606196
$236$	0	0
$237$	307734.	0.355881
$238$	0	0
$239$	385553.	0.436606	0.218303	−	0.975881i	$-0.429948\pi$
$239$	385553.	0.436606	0.218303	+	0.975881i	$0.429948\pi$
$240$	0	0
$241$	−1.11236e6	−1.23369	−0.616843	−	0.787086i	$-0.711589\pi$
$241$	−1.11236e6	−1.23369	−0.616843	+	0.787086i	$0.711589\pi$
$242$	0	0
$243$	59049.0	0.0641500
$244$	0	0
$245$	424737.	0.452069
$246$	0	0
$247$	−762006.	−0.794724
$248$	0	0
$249$	−80490.2	−0.0822706
$250$	0	0
$251$	973089.	0.974918	0.487459	−	0.873146i	$-0.337924\pi$
$251$	973089.	0.974918	0.487459	+	0.873146i	$0.337924\pi$
$252$	0	0
$253$	200231.	0.196666
$254$	0	0
$255$	−131772.	−0.126903
$256$	0	0
$257$	2.00809e6	1.89649	0.948243	−	0.317545i	$-0.102858\pi$
$257$	2.00809e6	1.89649	0.948243	+	0.317545i	$0.102858\pi$
$258$	0	0
$259$	390981.	0.362165
$260$	0	0
$261$	−531987.	−0.483392
$262$	0	0
$263$	−496131.	−0.442290	−0.221145	−	0.975241i	$-0.570979\pi$
$263$	−496131.	−0.442290	−0.221145	+	0.975241i	$0.570979\pi$
$264$	0	0
$265$	−471379.	−0.412340
$266$	0	0
$267$	−918658.	−0.788634
$268$	0	0
$269$	1.07989e6	0.909912	0.454956	−	0.890514i	$-0.349655\pi$
$269$	1.07989e6	0.909912	0.454956	+	0.890514i	$0.349655\pi$
$270$	0	0
$271$	503381.	0.416364	0.208182	−	0.978090i	$-0.433245\pi$
$271$	503381.	0.416364	0.208182	+	0.978090i	$0.433245\pi$
$272$	0	0
$273$	1.19355e6	0.969247
$274$	0	0
$275$	1.03151e6	0.822511
$276$	0	0
$277$	−439909.	−0.344480	−0.172240	−	0.985055i	$-0.555100\pi$
$277$	−439909.	−0.344480	−0.172240	+	0.985055i	$0.555100\pi$
$278$	0	0
$279$	−439023.	−0.337658
$280$	0	0
$281$	2.26427e6	1.71066	0.855328	−	0.518086i	$-0.173355\pi$
$281$	2.26427e6	1.71066	0.855328	+	0.518086i	$0.173355\pi$
$282$	0	0
$283$	1.89054e6	1.40320	0.701602	−	0.712569i	$-0.252469\pi$
$283$	1.89054e6	1.40320	0.701602	+	0.712569i	$0.252469\pi$
$284$	0	0
$285$	−201695.	−0.147090
$286$	0	0
$287$	−2.22518e6	−1.59463
$288$	0	0
$289$	−883666.	−0.622363
$290$	0	0
$291$	1.57902e6	1.09309
$292$	0	0
$293$	−1.36934e6	−0.931845	−0.465923	−	0.884825i	$-0.654277\pi$
$293$	−1.36934e6	−0.931845	−0.465923	+	0.884825i	$0.654277\pi$
$294$	0	0
$295$	769553.	0.514854
$296$	0	0
$297$	−275932.	−0.181515
$298$	0	0
$299$	−359651.	−0.232650
$300$	0	0
$301$	−1.24894e6	−0.794561
$302$	0	0
$303$	−741538.	−0.464010
$304$	0	0
$305$	−1.05521e6	−0.649516
$306$	0	0
$307$	−2.12052e6	−1.28409	−0.642047	−	0.766665i	$-0.721915\pi$
$307$	−2.12052e6	−1.28409	−0.642047	+	0.766665i	$0.721915\pi$
$308$	0	0
$309$	1.64952e6	0.982793
$310$	0	0
$311$	−941049.	−0.551711	−0.275855	−	0.961199i	$-0.588961\pi$
$311$	−941049.	−0.551711	−0.275855	+	0.961199i	$0.588961\pi$
$312$	0	0
$313$	2.84773e6	1.64300	0.821502	−	0.570206i	$-0.193136\pi$
$313$	2.84773e6	1.64300	0.821502	+	0.570206i	$0.193136\pi$
$314$	0	0
$315$	315921.	0.179392
$316$	0	0
$317$	−60694.3	−0.0339234	−0.0169617	−	0.999856i	$-0.505399\pi$
$317$	−60694.3	−0.0339234	−0.0169617	+	0.999856i	$0.505399\pi$
$318$	0	0
$319$	2.48594e6	1.36777
$320$	0	0
$321$	−756868.	−0.409975
$322$	0	0
$323$	820715.	0.437710
$324$	0	0
$325$	−1.85278e6	−0.973006
$326$	0	0
$327$	60156.7	0.0311111
$328$	0	0
$329$	−5.00651e6	−2.55003
$330$	0	0
$331$	2.46524e6	1.23677	0.618385	−	0.785875i	$-0.287787\pi$
$331$	2.46524e6	1.23677	0.618385	+	0.785875i	$0.287787\pi$
$332$	0	0
$333$	162356.	0.0802339
$334$	0	0
$335$	1.30958e6	0.637558
$336$	0	0
$337$	−2.29815e6	−1.10231	−0.551156	−	0.834402i	$-0.685813\pi$
$337$	−2.29815e6	−1.10231	−0.551156	+	0.834402i	$0.685813\pi$
$338$	0	0
$339$	1.19136e6	0.563044
$340$	0	0
$341$	2.05153e6	0.955413
$342$	0	0
$343$	865143.	0.397057
$344$	0	0
$345$	−95196.0	−0.0430597
$346$	0	0
$347$	1.35665e6	0.604846	0.302423	−	0.953174i	$-0.402204\pi$
$347$	1.35665e6	0.604846	0.302423	+	0.953174i	$0.402204\pi$
$348$	0	0
$349$	1.90109e6	0.835487	0.417743	−	0.908565i	$-0.362821\pi$
$349$	1.90109e6	0.835487	0.417743	+	0.908565i	$0.362821\pi$
$350$	0	0
$351$	495625.	0.214726
$352$	0	0
$353$	−2.32314e6	−0.992289	−0.496144	−	0.868240i	$-0.665251\pi$
$353$	−2.32314e6	−0.992289	−0.496144	+	0.868240i	$0.665251\pi$
$354$	0	0
$355$	−656656.	−0.276546
$356$	0	0
$357$	−1.28551e6	−0.533832
$358$	0	0
$359$	−274160.	−0.112271	−0.0561356	−	0.998423i	$-0.517878\pi$
$359$	−274160.	−0.112271	−0.0561356	+	0.998423i	$0.517878\pi$
$360$	0	0
$361$	−1.21988e6	−0.492661
$362$	0	0
$363$	−160045.	−0.0637493
$364$	0	0
$365$	−1.76876e6	−0.694923
$366$	0	0
$367$	−2.32745e6	−0.902017	−0.451008	−	0.892520i	$-0.648936\pi$
$367$	−2.32745e6	−0.902017	−0.451008	+	0.892520i	$0.648936\pi$
$368$	0	0
$369$	−924012.	−0.353274
$370$	0	0
$371$	−4.59856e6	−1.73455
$372$	0	0
$373$	−2.82705e6	−1.05211	−0.526055	−	0.850451i	$-0.676329\pi$
$373$	−2.82705e6	−1.05211	−0.526055	+	0.850451i	$0.676329\pi$
$374$	0	0
$375$	−1.05277e6	−0.386595
$376$	0	0
$377$	−4.46520e6	−1.61803
$378$	0	0
$379$	502620.	0.179739	0.0898694	−	0.995954i	$-0.471355\pi$
$379$	502620.	0.179739	0.0898694	+	0.995954i	$0.471355\pi$
$380$	0	0
$381$	−622255.	−0.219612
$382$	0	0
$383$	336894.	0.117354	0.0586768	−	0.998277i	$-0.481312\pi$
$383$	336894.	0.117354	0.0586768	+	0.998277i	$0.481312\pi$
$384$	0	0
$385$	−1.47628e6	−0.507594
$386$	0	0
$387$	−518627.	−0.176026
$388$	0	0
$389$	15415.9	0.00516531	0.00258265	−	0.999997i	$-0.499178\pi$
$389$	15415.9	0.00516531	0.00258265	+	0.999997i	$0.499178\pi$
$390$	0	0
$391$	387360.	0.128137
$392$	0	0
$393$	2.52217e6	0.823744
$394$	0	0
$395$	683681.	0.220476
$396$	0	0
$397$	−1.68282e6	−0.535872	−0.267936	−	0.963437i	$-0.586342\pi$
$397$	−1.68282e6	−0.535872	−0.267936	+	0.963437i	$0.586342\pi$
$398$	0	0
$399$	−1.96765e6	−0.618751
$400$	0	0
$401$	−3.72586e6	−1.15709	−0.578543	−	0.815652i	$-0.696378\pi$
$401$	−3.72586e6	−1.15709	−0.578543	+	0.815652i	$0.696378\pi$
$402$	0	0
$403$	−3.68492e6	−1.13023
$404$	0	0
$405$	131187.	0.0397423
$406$	0	0
$407$	−758679.	−0.227024
$408$	0	0
$409$	−1.80593e6	−0.533819	−0.266909	−	0.963722i	$-0.586002\pi$
$409$	−1.80593e6	−0.533819	−0.266909	+	0.963722i	$0.586002\pi$
$410$	0	0
$411$	3.41318e6	0.996678
$412$	0	0
$413$	7.50743e6	2.16579
$414$	0	0
$415$	−178822.	−0.0509684
$416$	0	0
$417$	−881696.	−0.248301
$418$	0	0
$419$	234194.	0.0651690	0.0325845	−	0.999469i	$-0.489626\pi$
$419$	234194.	0.0651690	0.0325845	+	0.999469i	$0.489626\pi$
$420$	0	0
$421$	−1.98385e6	−0.545510	−0.272755	−	0.962083i	$-0.587935\pi$
$421$	−1.98385e6	−0.545510	−0.272755	+	0.962083i	$0.587935\pi$
$422$	0	0
$423$	−2.07897e6	−0.564933
$424$	0	0
$425$	1.99553e6	0.535903
$426$	0	0
$427$	−1.02942e7	−2.73226
$428$	0	0
$429$	−2.31602e6	−0.607575
$430$	0	0
$431$	−1.76470e6	−0.457590	−0.228795	−	0.973475i	$-0.573479\pi$
$431$	−1.76470e6	−0.457590	−0.228795	+	0.973475i	$0.573479\pi$
$432$	0	0
$433$	−5.80779e6	−1.48864	−0.744322	−	0.667820i	$-0.767227\pi$
$433$	−5.80779e6	−1.48864	−0.744322	+	0.667820i	$0.767227\pi$
$434$	0	0
$435$	−1.18189e6	−0.299472
$436$	0	0
$437$	592910.	0.148520
$438$	0	0
$439$	4.37083e6	1.08244	0.541218	−	0.840882i	$-0.317963\pi$
$439$	4.37083e6	1.08244	0.541218	+	0.840882i	$0.317963\pi$
$440$	0	0
$441$	1.72062e6	0.421297
$442$	0	0
$443$	−4.28760e6	−1.03802	−0.519009	−	0.854769i	$-0.673699\pi$
$443$	−4.28760e6	−1.03802	−0.519009	+	0.854769i	$0.673699\pi$
$444$	0	0
$445$	−2.04095e6	−0.488576
$446$	0	0
$447$	−1.97130e6	−0.466643
$448$	0	0
$449$	−2.76030e6	−0.646161	−0.323080	−	0.946372i	$-0.604718\pi$
$449$	−2.76030e6	−0.646161	−0.323080	+	0.946372i	$0.604718\pi$
$450$	0	0
$451$	4.31785e6	0.999600
$452$	0	0
$453$	3.17671e6	0.727332
$454$	0	0
$455$	2.65167e6	0.600469
$456$	0	0
$457$	−1.92356e6	−0.430840	−0.215420	−	0.976522i	$-0.569112\pi$
$457$	−1.92356e6	−0.430840	−0.215420	+	0.976522i	$0.569112\pi$
$458$	0	0
$459$	−533810.	−0.118265
$460$	0	0
$461$	−7.05556e6	−1.54625	−0.773124	−	0.634255i	$-0.781307\pi$
$461$	−7.05556e6	−1.54625	−0.773124	+	0.634255i	$0.781307\pi$
$462$	0	0
$463$	5.16868e6	1.12054	0.560269	−	0.828310i	$-0.310698\pi$
$463$	5.16868e6	1.12054	0.560269	+	0.828310i	$0.310698\pi$
$464$	0	0
$465$	−975361.	−0.209186
$466$	0	0
$467$	1.86737e6	0.396221	0.198110	−	0.980180i	$-0.436520\pi$
$467$	1.86737e6	0.396221	0.198110	+	0.980180i	$0.436520\pi$
$468$	0	0
$469$	1.27757e7	2.68196
$470$	0	0
$471$	2.75458e6	0.572141
$472$	0	0
$473$	2.42351e6	0.498072
$474$	0	0
$475$	3.05444e6	0.621152
$476$	0	0
$477$	−1.90956e6	−0.384272
$478$	0	0
$479$	−3.79147e6	−0.755038	−0.377519	−	0.926002i	$-0.623223\pi$
$479$	−3.79147e6	−0.755038	−0.377519	+	0.926002i	$0.623223\pi$
$480$	0	0
$481$	1.36273e6	0.268563
$482$	0	0
$483$	−928691.	−0.181135
$484$	0	0
$485$	3.50804e6	0.677190
$486$	0	0
$487$	−254359.	−0.0485988	−0.0242994	−	0.999705i	$-0.507736\pi$
$487$	−254359.	−0.0485988	−0.0242994	+	0.999705i	$0.507736\pi$
$488$	0	0
$489$	182972.	0.0346029
$490$	0	0
$491$	−3.85935e6	−0.722455	−0.361228	−	0.932478i	$-0.617642\pi$
$491$	−3.85935e6	−0.722455	−0.361228	+	0.932478i	$0.617642\pi$
$492$	0	0
$493$	4.80923e6	0.891165
$494$	0	0
$495$	−613028.	−0.112452
$496$	0	0
$497$	−6.40605e6	−1.16332
$498$	0	0
$499$	6.24619e6	1.12296	0.561480	−	0.827490i	$-0.310232\pi$
$499$	6.24619e6	1.12296	0.561480	+	0.827490i	$0.310232\pi$
$500$	0	0
$501$	3.94953e6	0.702993
$502$	0	0
$503$	−7.87492e6	−1.38780	−0.693899	−	0.720073i	$-0.744108\pi$
$503$	−7.87492e6	−1.38780	−0.693899	+	0.720073i	$0.744108\pi$
$504$	0	0
$505$	−1.64745e6	−0.287464
$506$	0	0
$507$	818365.	0.141393
$508$	0	0
$509$	2.21923e6	0.379672	0.189836	−	0.981816i	$-0.439204\pi$
$509$	2.21923e6	0.379672	0.189836	+	0.981816i	$0.439204\pi$
$510$	0	0
$511$	−1.72552e7	−2.92327
$512$	0	0
$513$	−817072.	−0.137078
$514$	0	0
$515$	3.66468e6	0.608861
$516$	0	0
$517$	9.71488e6	1.59849
$518$	0	0
$519$	2.27458e6	0.370666
$520$	0	0
$521$	5.09010e6	0.821546	0.410773	−	0.911738i	$-0.365259\pi$
$521$	5.09010e6	0.821546	0.410773	+	0.911738i	$0.365259\pi$
$522$	0	0
$523$	−9.06986e6	−1.44993	−0.724964	−	0.688787i	$-0.758144\pi$
$523$	−9.06986e6	−1.44993	−0.724964	+	0.688787i	$0.758144\pi$
$524$	0	0
$525$	−4.78425e6	−0.757558
$526$	0	0
$527$	3.96882e6	0.622494
$528$	0	0
$529$	279841.	0.0434783
$530$	0	0
$531$	3.11748e6	0.479807
$532$	0	0
$533$	−7.75565e6	−1.18250
$534$	0	0
$535$	−1.68150e6	−0.253988
$536$	0	0
$537$	−4.56468e6	−0.683084
$538$	0	0
$539$	−8.04035e6	−1.19207
$540$	0	0
$541$	17613.9	0.00258739	0.00129370	−	0.999999i	$-0.499588\pi$
$541$	17613.9	0.00258739	0.00129370	+	0.999999i	$0.499588\pi$
$542$	0	0
$543$	−1.93081e6	−0.281021
$544$	0	0
$545$	133648.	0.0192740
$546$	0	0
$547$	−7.13858e6	−1.02010	−0.510051	−	0.860144i	$-0.670373\pi$
$547$	−7.13858e6	−1.02010	−0.510051	+	0.860144i	$0.670373\pi$
$548$	0	0
$549$	−4.27469e6	−0.605304
$550$	0	0
$551$	7.36120e6	1.03293
$552$	0	0
$553$	6.66969e6	0.927455
$554$	0	0
$555$	360700.	0.0497066
$556$	0	0
$557$	−7.52507e6	−1.02771	−0.513857	−	0.857876i	$-0.671784\pi$
$557$	−7.52507e6	−1.02771	−0.513857	+	0.857876i	$0.671784\pi$
$558$	0	0
$559$	−4.35307e6	−0.589205
$560$	0	0
$561$	2.49446e6	0.334634
$562$	0	0
$563$	758436.	0.100844	0.0504218	−	0.998728i	$-0.483943\pi$
$563$	758436.	0.100844	0.0504218	+	0.998728i	$0.483943\pi$
$564$	0	0
$565$	2.64679e6	0.348818
$566$	0	0
$567$	1.27980e6	0.167180
$568$	0	0
$569$	956149.	0.123807	0.0619034	−	0.998082i	$-0.480283\pi$
$569$	956149.	0.123807	0.0619034	+	0.998082i	$0.480283\pi$
$570$	0	0
$571$	−715521.	−0.0918401	−0.0459200	−	0.998945i	$-0.514622\pi$
$571$	−715521.	−0.0918401	−0.0459200	+	0.998945i	$0.514622\pi$
$572$	0	0
$573$	−411932.	−0.0524131
$574$	0	0
$575$	1.44163e6	0.181838
$576$	0	0
$577$	1.23562e7	1.54506	0.772528	−	0.634981i	$-0.218992\pi$
$577$	1.23562e7	1.54506	0.772528	+	0.634981i	$0.218992\pi$
$578$	0	0
$579$	−2.64175e6	−0.327488
$580$	0	0
$581$	−1.74451e6	−0.214404
$582$	0	0
$583$	8.92327e6	1.08731
$584$	0	0
$585$	1.10111e6	0.133027
$586$	0	0
$587$	8.35139e6	1.00038	0.500188	−	0.865917i	$-0.333264\pi$
$587$	8.35139e6	1.00038	0.500188	+	0.865917i	$0.333264\pi$
$588$	0	0
$589$	6.07484e6	0.721518
$590$	0	0
$591$	959167.	0.112960
$592$	0	0
$593$	−1.40357e7	−1.63907	−0.819535	−	0.573029i	$-0.805768\pi$
$593$	−1.40357e7	−1.63907	−0.819535	+	0.573029i	$0.805768\pi$
$594$	0	0
$595$	−2.85597e6	−0.330720
$596$	0	0
$597$	2.61324e6	0.300084
$598$	0	0
$599$	1.24765e7	1.42077	0.710386	−	0.703812i	$-0.248520\pi$
$599$	1.24765e7	1.42077	0.710386	+	0.703812i	$0.248520\pi$
$600$	0	0
$601$	9.16791e6	1.03534	0.517671	−	0.855580i	$-0.326799\pi$
$601$	9.16791e6	1.03534	0.517671	+	0.855580i	$0.326799\pi$
$602$	0	0
$603$	5.30513e6	0.594160
$604$	0	0
$605$	−355566.	−0.0394941
$606$	0	0
$607$	−3.84641e6	−0.423725	−0.211862	−	0.977300i	$-0.567953\pi$
$607$	−3.84641e6	−0.423725	−0.211862	+	0.977300i	$0.567953\pi$
$608$	0	0
$609$	−1.15300e7	−1.25976
$610$	0	0
$611$	−1.74497e7	−1.89097
$612$	0	0
$613$	−6.13142e6	−0.659038	−0.329519	−	0.944149i	$-0.606887\pi$
$613$	−6.13142e6	−0.659038	−0.329519	+	0.944149i	$0.606887\pi$
$614$	0	0
$615$	−2.05284e6	−0.218861
$616$	0	0
$617$	1.29971e7	1.37447	0.687233	−	0.726437i	$-0.258825\pi$
$617$	1.29971e7	1.37447	0.687233	+	0.726437i	$0.258825\pi$
$618$	0	0
$619$	−5.25814e6	−0.551577	−0.275788	−	0.961218i	$-0.588939\pi$
$619$	−5.25814e6	−0.551577	−0.275788	+	0.961218i	$0.588939\pi$
$620$	0	0
$621$	−385641.	−0.0401286
$622$	0	0
$623$	−1.99106e7	−2.05525
$624$	0	0
$625$	6.17735e6	0.632561
$626$	0	0
$627$	3.81813e6	0.387866
$628$	0	0
$629$	−1.46772e6	−0.147916
$630$	0	0
$631$	1.84156e7	1.84125	0.920623	−	0.390454i	$-0.127682\pi$
$631$	1.84156e7	1.84125	0.920623	+	0.390454i	$0.127682\pi$
$632$	0	0
$633$	−7.01903e6	−0.696254
$634$	0	0
$635$	−1.38244e6	−0.136054
$636$	0	0
$637$	1.44419e7	1.41019
$638$	0	0
$639$	−2.66013e6	−0.257721
$640$	0	0
$641$	1.22726e7	1.17975	0.589876	−	0.807494i	$-0.299177\pi$
$641$	1.22726e7	1.17975	0.589876	+	0.807494i	$0.299177\pi$
$642$	0	0
$643$	6.75283e6	0.644107	0.322054	−	0.946721i	$-0.395627\pi$
$643$	6.75283e6	0.644107	0.322054	+	0.946721i	$0.395627\pi$
$644$	0	0
$645$	−1.15221e6	−0.109052
$646$	0	0
$647$	−285536.	−0.0268163	−0.0134082	−	0.999910i	$-0.504268\pi$
$647$	−285536.	−0.0268163	−0.0134082	+	0.999910i	$0.504268\pi$
$648$	0	0
$649$	−1.45678e7	−1.35763
$650$	0	0
$651$	−9.51519e6	−0.879964
$652$	0	0
$653$	1.73560e7	1.59282	0.796412	−	0.604754i	$-0.206729\pi$
$653$	1.73560e7	1.59282	0.796412	+	0.604754i	$0.206729\pi$
$654$	0	0
$655$	5.60340e6	0.510327
$656$	0	0
$657$	−7.16528e6	−0.647619
$658$	0	0
$659$	−6.28171e6	−0.563462	−0.281731	−	0.959493i	$-0.590909\pi$
$659$	−6.28171e6	−0.563462	−0.281731	+	0.959493i	$0.590909\pi$
$660$	0	0
$661$	1.75599e7	1.56322	0.781608	−	0.623770i	$-0.214400\pi$
$661$	1.75599e7	1.56322	0.781608	+	0.623770i	$0.214400\pi$
$662$	0	0
$663$	−4.48051e6	−0.395862
$664$	0	0
$665$	−4.37146e6	−0.383330
$666$	0	0
$667$	3.47433e6	0.302383
$668$	0	0
$669$	3.61759e6	0.312503
$670$	0	0
$671$	1.99753e7	1.71273
$672$	0	0
$673$	1.28603e7	1.09450	0.547248	−	0.836970i	$-0.315675\pi$
$673$	1.28603e7	1.09450	0.547248	+	0.836970i	$0.315675\pi$
$674$	0	0
$675$	−1.98667e6	−0.167829
$676$	0	0
$677$	−7.77832e6	−0.652250	−0.326125	−	0.945327i	$-0.605743\pi$
$677$	−7.77832e6	−0.652250	−0.326125	+	0.945327i	$0.605743\pi$
$678$	0	0
$679$	3.42229e7	2.84867
$680$	0	0
$681$	−9.19614e6	−0.759867
$682$	0	0
$683$	2.13772e7	1.75348	0.876738	−	0.480969i	$-0.159715\pi$
$683$	2.13772e7	1.75348	0.876738	+	0.480969i	$0.159715\pi$
$684$	0	0
$685$	7.58294e6	0.617463
$686$	0	0
$687$	328761.	0.0265760
$688$	0	0
$689$	−1.60278e7	−1.28625
$690$	0	0
$691$	−1.05873e7	−0.843506	−0.421753	−	0.906711i	$-0.638585\pi$
$691$	−1.05873e7	−0.843506	−0.421753	+	0.906711i	$0.638585\pi$
$692$	0	0
$693$	−5.98044e6	−0.473042
$694$	0	0
$695$	−1.95883e6	−0.153828
$696$	0	0
$697$	8.35319e6	0.651284
$698$	0	0
$699$	−7.37695e6	−0.571063
$700$	0	0
$701$	2.03961e6	0.156766	0.0783830	−	0.996923i	$-0.475024\pi$
$701$	2.03961e6	0.156766	0.0783830	+	0.996923i	$0.475024\pi$
$702$	0	0
$703$	−2.24655e6	−0.171446
$704$	0	0
$705$	−4.61876e6	−0.349988
$706$	0	0
$707$	−1.60718e7	−1.20925
$708$	0	0
$709$	−9.39232e6	−0.701709	−0.350855	−	0.936430i	$-0.614109\pi$
$709$	−9.39232e6	−0.701709	−0.350855	+	0.936430i	$0.614109\pi$
$710$	0	0
$711$	2.76961e6	0.205468
$712$	0	0
$713$	2.86720e6	0.211219
$714$	0	0
$715$	−5.14542e6	−0.376405
$716$	0	0
$717$	3.46998e6	0.252074
$718$	0	0
$719$	−5.63055e6	−0.406190	−0.203095	−	0.979159i	$-0.565100\pi$
$719$	−5.63055e6	−0.406190	−0.203095	+	0.979159i	$0.565100\pi$
$720$	0	0
$721$	3.57510e7	2.56124
$722$	0	0
$723$	−1.00113e7	−0.712269
$724$	0	0
$725$	1.78984e7	1.26465
$726$	0	0
$727$	2.29210e7	1.60842	0.804208	−	0.594348i	$-0.202590\pi$
$727$	2.29210e7	1.60842	0.804208	+	0.594348i	$0.202590\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	4.68846e6	0.324516
$732$	0	0
$733$	−2.34174e7	−1.60982	−0.804912	−	0.593394i	$-0.797788\pi$
$733$	−2.34174e7	−1.60982	−0.804912	+	0.593394i	$0.797788\pi$
$734$	0	0
$735$	3.82264e6	0.261002
$736$	0	0
$737$	−2.47906e7	−1.68119
$738$	0	0
$739$	−7.18161e6	−0.483738	−0.241869	−	0.970309i	$-0.577760\pi$
$739$	−7.18161e6	−0.483738	−0.241869	+	0.970309i	$0.577760\pi$
$740$	0	0
$741$	−6.85806e6	−0.458834
$742$	0	0
$743$	2.06962e6	0.137537	0.0687684	−	0.997633i	$-0.478093\pi$
$743$	2.06962e6	0.137537	0.0687684	+	0.997633i	$0.478093\pi$
$744$	0	0
$745$	−4.37957e6	−0.289095
$746$	0	0
$747$	−724412.	−0.0474989
$748$	0	0
$749$	−1.64040e7	−1.06843
$750$	0	0
$751$	2.20975e6	0.142969	0.0714846	−	0.997442i	$-0.477226\pi$
$751$	2.20975e6	0.142969	0.0714846	+	0.997442i	$0.477226\pi$
$752$	0	0
$753$	8.75780e6	0.562869
$754$	0	0
$755$	7.05758e6	0.450597
$756$	0	0
$757$	−8.81675e6	−0.559202	−0.279601	−	0.960116i	$-0.590202\pi$
$757$	−8.81675e6	−0.559202	−0.279601	+	0.960116i	$0.590202\pi$
$758$	0	0
$759$	1.80208e6	0.113545
$760$	0	0
$761$	1.50476e7	0.941902	0.470951	−	0.882159i	$-0.343911\pi$
$761$	1.50476e7	0.941902	0.470951	+	0.882159i	$0.343911\pi$
$762$	0	0
$763$	1.30381e6	0.0810781
$764$	0	0
$765$	−1.18595e6	−0.0732675
$766$	0	0
$767$	2.61664e7	1.60604
$768$	0	0
$769$	2.89908e7	1.76784	0.883922	−	0.467634i	$-0.154894\pi$
$769$	2.89908e7	1.76784	0.883922	+	0.467634i	$0.154894\pi$
$770$	0	0
$771$	1.80728e7	1.09494
$772$	0	0
$773$	−2.88851e7	−1.73870	−0.869351	−	0.494195i	$-0.835463\pi$
$773$	−2.88851e7	−1.73870	−0.869351	+	0.494195i	$0.835463\pi$
$774$	0	0
$775$	1.47707e7	0.883378
$776$	0	0
$777$	3.51883e6	0.209096
$778$	0	0
$779$	1.27857e7	0.754887
$780$	0	0
$781$	1.24306e7	0.729230
$782$	0	0
$783$	−4.78788e6	−0.279086
$784$	0	0
$785$	6.11974e6	0.354453
$786$	0	0
$787$	2.08419e7	1.19950	0.599750	−	0.800187i	$-0.295267\pi$
$787$	2.08419e7	1.19950	0.599750	+	0.800187i	$0.295267\pi$
$788$	0	0
$789$	−4.46518e6	−0.255356
$790$	0	0
$791$	2.58209e7	1.46734
$792$	0	0
$793$	−3.58794e7	−2.02610
$794$	0	0
$795$	−4.24241e6	−0.238064
$796$	0	0
$797$	9.76276e6	0.544411	0.272205	−	0.962239i	$-0.412247\pi$
$797$	9.76276e6	0.544411	0.272205	+	0.962239i	$0.412247\pi$
$798$	0	0
$799$	1.87941e7	1.04149
$800$	0	0
$801$	−8.26792e6	−0.455318
$802$	0	0
$803$	3.34829e7	1.83246
$804$	0	0
$805$	−2.06324e6	−0.112217
$806$	0	0
$807$	9.71903e6	0.525338
$808$	0	0
$809$	8.91963e6	0.479155	0.239577	−	0.970877i	$-0.422991\pi$
$809$	8.91963e6	0.479155	0.239577	+	0.970877i	$0.422991\pi$
$810$	0	0
$811$	−3.39154e7	−1.81069	−0.905346	−	0.424675i	$-0.860388\pi$
$811$	−3.39154e7	−1.81069	−0.905346	+	0.424675i	$0.860388\pi$
$812$	0	0
$813$	4.53043e6	0.240388
$814$	0	0
$815$	406502.	0.0214372
$816$	0	0
$817$	7.17635e6	0.376139
$818$	0	0
$819$	1.07420e7	0.559595
$820$	0	0
$821$	−1.35849e7	−0.703393	−0.351696	−	0.936114i	$-0.614395\pi$
$821$	−1.35849e7	−0.703393	−0.351696	+	0.936114i	$0.614395\pi$
$822$	0	0
$823$	6.11526e6	0.314713	0.157357	−	0.987542i	$-0.449703\pi$
$823$	6.11526e6	0.314713	0.157357	+	0.987542i	$0.449703\pi$
$824$	0	0
$825$	9.28359e6	0.474877
$826$	0	0
$827$	−7.05329e6	−0.358615	−0.179307	−	0.983793i	$-0.557386\pi$
$827$	−7.05329e6	−0.358615	−0.179307	+	0.983793i	$0.557386\pi$
$828$	0	0
$829$	−4.04869e6	−0.204611	−0.102305	−	0.994753i	$-0.532622\pi$
$829$	−4.04869e6	−0.204611	−0.102305	+	0.994753i	$0.532622\pi$
$830$	0	0
$831$	−3.95918e6	−0.198885
$832$	0	0
$833$	−1.55546e7	−0.776689
$834$	0	0
$835$	8.77452e6	0.435519
$836$	0	0
$837$	−3.95121e6	−0.194947
$838$	0	0
$839$	−2.25052e7	−1.10377	−0.551885	−	0.833920i	$-0.686091\pi$
$839$	−2.25052e7	−1.10377	−0.551885	+	0.833920i	$0.686091\pi$
$840$	0	0
$841$	2.26240e7	1.10301
$842$	0	0
$843$	2.03784e7	0.987648
$844$	0	0
$845$	1.81813e6	0.0875958
$846$	0	0
$847$	−3.46875e6	−0.166136
$848$	0	0
$849$	1.70149e7	0.810140
$850$	0	0
$851$	−1.06032e6	−0.0501898
$852$	0	0
$853$	1.82342e7	0.858055	0.429027	−	0.903291i	$-0.358856\pi$
$853$	1.82342e7	0.858055	0.429027	+	0.903291i	$0.358856\pi$
$854$	0	0
$855$	−1.81526e6	−0.0849226
$856$	0	0
$857$	−3.56335e7	−1.65732	−0.828660	−	0.559753i	$-0.810896\pi$
$857$	−3.56335e7	−1.65732	−0.828660	+	0.559753i	$0.810896\pi$
$858$	0	0
$859$	−4.21320e6	−0.194818	−0.0974091	−	0.995244i	$-0.531056\pi$
$859$	−4.21320e6	−0.194818	−0.0974091	+	0.995244i	$0.531056\pi$
$860$	0	0
$861$	−2.00266e7	−0.920662
$862$	0	0
$863$	−3.33498e7	−1.52428	−0.762142	−	0.647410i	$-0.775852\pi$
$863$	−3.33498e7	−1.52428	−0.762142	+	0.647410i	$0.775852\pi$
$864$	0	0
$865$	5.05334e6	0.229635
$866$	0	0
$867$	−7.95300e6	−0.359321
$868$	0	0
$869$	−1.29422e7	−0.581378
$870$	0	0
$871$	4.45284e7	1.98880
$872$	0	0
$873$	1.42112e7	0.631093
$874$	0	0
$875$	−2.28173e7	−1.00750
$876$	0	0
$877$	1.57605e7	0.691946	0.345973	−	0.938245i	$-0.387549\pi$
$877$	1.57605e7	0.691946	0.345973	+	0.938245i	$0.387549\pi$
$878$	0	0
$879$	−1.23241e7	−0.538001
$880$	0	0
$881$	3.93973e7	1.71012	0.855061	−	0.518528i	$-0.173520\pi$
$881$	3.93973e7	1.71012	0.855061	+	0.518528i	$0.173520\pi$
$882$	0	0
$883$	245010.	0.0105750	0.00528751	−	0.999986i	$-0.498317\pi$
$883$	245010.	0.0105750	0.00528751	+	0.999986i	$0.498317\pi$
$884$	0	0
$885$	6.92598e6	0.297251
$886$	0	0
$887$	−2.81752e7	−1.20242	−0.601212	−	0.799089i	$-0.705315\pi$
$887$	−2.81752e7	−1.20242	−0.601212	+	0.799089i	$0.705315\pi$
$888$	0	0
$889$	−1.34865e7	−0.572327
$890$	0	0
$891$	−2.48339e6	−0.104797
$892$	0	0
$893$	2.87671e7	1.20717
$894$	0	0
$895$	−1.01412e7	−0.423185
$896$	0	0
$897$	−3.23686e6	−0.134321
$898$	0	0
$899$	3.55974e7	1.46899
$900$	0	0
$901$	1.72627e7	0.708430
$902$	0	0
$903$	−1.12405e7	−0.458740
$904$	0	0
$905$	−4.28960e6	−0.174099
$906$	0	0
$907$	−4.21901e6	−0.170291	−0.0851456	−	0.996369i	$-0.527136\pi$
$907$	−4.21901e6	−0.170291	−0.0851456	+	0.996369i	$0.527136\pi$
$908$	0	0
$909$	−6.67385e6	−0.267896
$910$	0	0
$911$	−1.56830e7	−0.626087	−0.313043	−	0.949739i	$-0.601348\pi$
$911$	−1.56830e7	−0.626087	−0.313043	+	0.949739i	$0.601348\pi$
$912$	0	0
$913$	3.38513e6	0.134400
$914$	0	0
$915$	−9.49691e6	−0.374998
$916$	0	0
$917$	5.46644e7	2.14675
$918$	0	0
$919$	1.52793e7	0.596782	0.298391	−	0.954444i	$-0.403550\pi$
$919$	1.52793e7	0.596782	0.298391	+	0.954444i	$0.403550\pi$
$920$	0	0
$921$	−1.90847e7	−0.741372
$922$	0	0
$923$	−2.23276e7	−0.862658
$924$	0	0
$925$	−5.46238e6	−0.209907
$926$	0	0
$927$	1.48457e7	0.567416
$928$	0	0
$929$	−1.46002e7	−0.555033	−0.277517	−	0.960721i	$-0.589511\pi$
$929$	−1.46002e7	−0.555033	−0.277517	+	0.960721i	$0.589511\pi$
$930$	0	0
$931$	−2.38085e7	−0.900241
$932$	0	0
$933$	−8.46944e6	−0.318530
$934$	0	0
$935$	5.54185e6	0.207313
$936$	0	0
$937$	4.43683e7	1.65091	0.825456	−	0.564467i	$-0.190918\pi$
$937$	4.43683e7	1.65091	0.825456	+	0.564467i	$0.190918\pi$
$938$	0	0
$939$	2.56296e7	0.948589
$940$	0	0
$941$	1.13266e6	0.0416988	0.0208494	−	0.999783i	$-0.493363\pi$
$941$	1.13266e6	0.0416988	0.0208494	+	0.999783i	$0.493363\pi$
$942$	0	0
$943$	6.03460e6	0.220988
$944$	0	0
$945$	2.84329e6	0.103572
$946$	0	0
$947$	1.13916e7	0.412772	0.206386	−	0.978471i	$-0.433830\pi$
$947$	1.13916e7	0.412772	0.206386	+	0.978471i	$0.433830\pi$
$948$	0	0
$949$	−6.01414e7	−2.16774
$950$	0	0
$951$	−546249.	−0.0195857
$952$	0	0
$953$	3.24469e7	1.15729	0.578644	−	0.815580i	$-0.303582\pi$
$953$	3.24469e7	1.15729	0.578644	+	0.815580i	$0.303582\pi$
$954$	0	0
$955$	−915175.	−0.0324710
$956$	0	0
$957$	2.23735e7	0.789684
$958$	0	0
$959$	7.39758e7	2.59743
$960$	0	0
$961$	747637.	0.0261145
$962$	0	0
$963$	−6.81181e6	−0.236699
$964$	0	0
$965$	−5.86908e6	−0.202886
$966$	0	0
$967$	6.52551e6	0.224413	0.112207	−	0.993685i	$-0.464208\pi$
$967$	6.52551e6	0.224413	0.112207	+	0.993685i	$0.464208\pi$
$968$	0	0
$969$	7.38644e6	0.252712
$970$	0	0
$971$	−2.24440e7	−0.763927	−0.381963	−	0.924177i	$-0.624752\pi$
$971$	−2.24440e7	−0.763927	−0.381963	+	0.924177i	$0.624752\pi$
$972$	0	0
$973$	−1.91095e7	−0.647093
$974$	0	0
$975$	−1.66750e7	−0.561766
$976$	0	0
$977$	2.57197e7	0.862043	0.431022	−	0.902342i	$-0.358153\pi$
$977$	2.57197e7	0.862043	0.431022	+	0.902342i	$0.358153\pi$
$978$	0	0
$979$	3.86355e7	1.28834
$980$	0	0
$981$	541411.	0.0179620
$982$	0	0
$983$	−4.46955e7	−1.47530	−0.737650	−	0.675184i	$-0.764064\pi$
$983$	−4.46955e7	−1.47530	−0.737650	+	0.675184i	$0.764064\pi$
$984$	0	0
$985$	2.13094e6	0.0699812
$986$	0	0
$987$	−4.50586e7	−1.47226
$988$	0	0
$989$	3.38708e6	0.110112
$990$	0	0
$991$	4.03164e7	1.30406	0.652031	−	0.758193i	$-0.273917\pi$
$991$	4.03164e7	1.30406	0.652031	+	0.758193i	$0.273917\pi$
$992$	0	0
$993$	2.21872e7	0.714050
$994$	0	0
$995$	5.80574e6	0.185909
$996$	0	0
$997$	−3.57212e7	−1.13812	−0.569060	−	0.822296i	$-0.692693\pi$
$997$	−3.57212e7	−1.13812	−0.569060	+	0.822296i	$0.692693\pi$
$998$	0	0
$999$	1.46120e6	0.0463230

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1104.6.a.p.1.3		4
4.3	odd	2		276.6.a.a.1.3	✓	4
12.11	even	2		828.6.a.a.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
276.6.a.a.1.3	✓	4	4.3	odd	2
828.6.a.a.1.2		4	12.11	even	2
1104.6.a.p.1.3		4	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:	\( N \)	\(=\)	\( 1104 = 2^{4} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1104.a (trivial)

\(f(q)\)	\(=\)	\(q+9.00000 q^{3} +19.9950 q^{5} +195.062 q^{7} +81.0000 q^{9} +O(q^{10})\)	\(q+9.00000 q^{3} +19.9950 q^{5} +195.062 q^{7} +81.0000 q^{9} -378.508 q^{11} +679.869 q^{13} +179.955 q^{15} -732.250 q^{17} -1120.81 q^{19} +1755.56 q^{21} -529.000 q^{23} -2725.20 q^{25} +729.000 q^{27} -6567.73 q^{29} -5420.04 q^{31} -3406.57 q^{33} +3900.26 q^{35} +2004.39 q^{37} +6118.82 q^{39} -11407.6 q^{41} -6402.81 q^{43} +1619.59 q^{45} -25666.3 q^{47} +21242.2 q^{49} -6590.25 q^{51} -23574.9 q^{53} -7568.25 q^{55} -10087.3 q^{57} +38487.4 q^{59} -52773.9 q^{61} +15800.0 q^{63} +13594.0 q^{65} +65495.5 q^{67} -4761.00 q^{69} -32841.1 q^{71} -88460.3 q^{73} -24526.8 q^{75} -73832.5 q^{77} +34192.7 q^{79} +6561.00 q^{81} -8943.35 q^{83} -14641.3 q^{85} -59109.6 q^{87} -102073. q^{89} +132617. q^{91} -48780.3 q^{93} -22410.6 q^{95} +175446. q^{97} -30659.1 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 36 q^{3} + 22 q^{5} + 154 q^{7} + 324 q^{9}+O(q^{10}) \) 4 * q + 36 * q^3 + 22 * q^5 + 154 * q^7 + 324 * q^9	\( 4 q + 36 q^{3} + 22 q^{5} + 154 q^{7} + 324 q^{9} - 392 q^{11} - 220 q^{13} + 198 q^{15} - 802 q^{17} - 2590 q^{19} + 1386 q^{21} - 2116 q^{23} + 6904 q^{25} + 2916 q^{27} - 5036 q^{29} - 20 q^{31} - 3528 q^{33} + 3752 q^{35} - 10884 q^{37} - 1980 q^{39} - 23312 q^{41} - 4406 q^{43} + 1782 q^{45} - 2472 q^{47} - 24568 q^{49} - 7218 q^{51} - 29222 q^{53} + 45928 q^{55} - 23310 q^{57} - 15136 q^{59} - 113220 q^{61} + 12474 q^{63} - 133892 q^{65} + 23214 q^{67} - 19044 q^{69} + 96096 q^{71} - 95792 q^{73} + 62136 q^{75} - 96584 q^{77} + 190094 q^{79} + 26244 q^{81} + 112280 q^{83} - 276556 q^{85} - 45324 q^{87} - 52206 q^{89} + 141548 q^{91} - 180 q^{93} - 38288 q^{95} + 9636 q^{97} - 31752 q^{99}+O(q^{100}) \) 4 * q + 36 * q^3 + 22 * q^5 + 154 * q^7 + 324 * q^9 - 392 * q^11 - 220 * q^13 + 198 * q^15 - 802 * q^17 - 2590 * q^19 + 1386 * q^21 - 2116 * q^23 + 6904 * q^25 + 2916 * q^27 - 5036 * q^29 - 20 * q^31 - 3528 * q^33 + 3752 * q^35 - 10884 * q^37 - 1980 * q^39 - 23312 * q^41 - 4406 * q^43 + 1782 * q^45 - 2472 * q^47 - 24568 * q^49 - 7218 * q^51 - 29222 * q^53 + 45928 * q^55 - 23310 * q^57 - 15136 * q^59 - 113220 * q^61 + 12474 * q^63 - 133892 * q^65 + 23214 * q^67 - 19044 * q^69 + 96096 * q^71 - 95792 * q^73 + 62136 * q^75 - 96584 * q^77 + 190094 * q^79 + 26244 * q^81 + 112280 * q^83 - 276556 * q^85 - 45324 * q^87 - 52206 * q^89 + 141548 * q^91 - 180 * q^93 - 38288 * q^95 + 9636 * q^97 - 31752 * q^99

Introduction

L-functions

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