LMFDB - Embedded newform 6036.2.a.i.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6036,2,Mod(1,6036)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6036, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("6036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6036 = 2^{2} \cdot 3 \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6036.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:	$48.1977026600$
Analytic rank:	$0$
Dimension:	$26$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Character	$\chi$	$=$	6036.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +2.09204 q^{5} -1.11995 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +2.09204 q^{5} -1.11995 q^{7} +1.00000 q^{9} -3.05871 q^{11} -6.80855 q^{13} -2.09204 q^{15} +6.88548 q^{17} -6.06064 q^{19} +1.11995 q^{21} -0.508351 q^{23} -0.623348 q^{25} -1.00000 q^{27} -4.52700 q^{29} +1.62475 q^{31} +3.05871 q^{33} -2.34300 q^{35} +4.52706 q^{37} +6.80855 q^{39} +3.40458 q^{41} -7.55017 q^{43} +2.09204 q^{45} +12.9657 q^{47} -5.74570 q^{49} -6.88548 q^{51} +5.82613 q^{53} -6.39895 q^{55} +6.06064 q^{57} +7.98329 q^{59} -11.2818 q^{61} -1.11995 q^{63} -14.2438 q^{65} +7.90749 q^{67} +0.508351 q^{69} +3.42324 q^{71} +14.4660 q^{73} +0.623348 q^{75} +3.42561 q^{77} -4.06975 q^{79} +1.00000 q^{81} -4.08357 q^{83} +14.4047 q^{85} +4.52700 q^{87} +17.2613 q^{89} +7.62527 q^{91} -1.62475 q^{93} -12.6791 q^{95} +14.8285 q^{97} -3.05871 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9}+O(q^{10}) $ 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9	$ 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9} - 11 q^{11} + 13 q^{13} - 6 q^{15} + 12 q^{17} - q^{19} - 5 q^{21} - 22 q^{23} + 48 q^{25} - 26 q^{27} + 6 q^{29} + 19 q^{31} + 11 q^{33} - 21 q^{35} + 20 q^{37} - 13 q^{39} + 25 q^{41} + 4 q^{43} + 6 q^{45} + 8 q^{47} + 67 q^{49} - 12 q^{51} - 5 q^{53} + 20 q^{55} + q^{57} - 18 q^{59} + 43 q^{61} + 5 q^{63} + 41 q^{65} + 5 q^{67} + 22 q^{69} - q^{71} + 22 q^{73} - 48 q^{75} + 23 q^{77} + 16 q^{79} + 26 q^{81} - 19 q^{83} + 29 q^{85} - 6 q^{87} + 49 q^{89} - 13 q^{91} - 19 q^{93} - 26 q^{95} + 25 q^{97} - 11 q^{99}+O(q^{100}) $ 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9 - 11 * q^11 + 13 * q^13 - 6 * q^15 + 12 * q^17 - q^19 - 5 * q^21 - 22 * q^23 + 48 * q^25 - 26 * q^27 + 6 * q^29 + 19 * q^31 + 11 * q^33 - 21 * q^35 + 20 * q^37 - 13 * q^39 + 25 * q^41 + 4 * q^43 + 6 * q^45 + 8 * q^47 + 67 * q^49 - 12 * q^51 - 5 * q^53 + 20 * q^55 + q^57 - 18 * q^59 + 43 * q^61 + 5 * q^63 + 41 * q^65 + 5 * q^67 + 22 * q^69 - q^71 + 22 * q^73 - 48 * q^75 + 23 * q^77 + 16 * q^79 + 26 * q^81 - 19 * q^83 + 29 * q^85 - 6 * q^87 + 49 * q^89 - 13 * q^91 - 19 * q^93 - 26 * q^95 + 25 * q^97 - 11 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	2.09204	0.935591	0.467795	−	0.883837i	$-0.345048\pi$
$5$	2.09204	0.935591	0.467795	+	0.883837i	$0.345048\pi$
$6$	0	0
$7$	−1.11995	−0.423303	−0.211652	−	0.977345i	$-0.567884\pi$
$7$	−1.11995	−0.423303	−0.211652	+	0.977345i	$0.567884\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.05871	−0.922234	−0.461117	−	0.887339i	$-0.652551\pi$
$11$	−3.05871	−0.922234	−0.461117	+	0.887339i	$0.652551\pi$
$12$	0	0
$13$	−6.80855	−1.88835	−0.944177	−	0.329440i	$-0.893140\pi$
$13$	−6.80855	−1.88835	−0.944177	+	0.329440i	$0.893140\pi$
$14$	0	0
$15$	−2.09204	−0.540164
$16$	0	0
$17$	6.88548	1.66997	0.834987	−	0.550270i	$-0.185475\pi$
$17$	6.88548	1.66997	0.834987	+	0.550270i	$0.185475\pi$
$18$	0	0
$19$	−6.06064	−1.39041	−0.695203	−	0.718813i	$-0.744686\pi$
$19$	−6.06064	−1.39041	−0.695203	+	0.718813i	$0.744686\pi$
$20$	0	0
$21$	1.11995	0.244394
$22$	0	0
$23$	−0.508351	−0.105998	−0.0529992	−	0.998595i	$-0.516878\pi$
$23$	−0.508351	−0.105998	−0.0529992	+	0.998595i	$0.516878\pi$
$24$	0	0
$25$	−0.623348	−0.124670
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−4.52700	−0.840643	−0.420322	−	0.907375i	$-0.638083\pi$
$29$	−4.52700	−0.840643	−0.420322	+	0.907375i	$0.638083\pi$
$30$	0	0
$31$	1.62475	0.291814	0.145907	−	0.989298i	$-0.453390\pi$
$31$	1.62475	0.291814	0.145907	+	0.989298i	$0.453390\pi$
$32$	0	0
$33$	3.05871	0.532452
$34$	0	0
$35$	−2.34300	−0.396039
$36$	0	0
$37$	4.52706	0.744244	0.372122	−	0.928184i	$-0.378630\pi$
$37$	4.52706	0.744244	0.372122	+	0.928184i	$0.378630\pi$
$38$	0	0
$39$	6.80855	1.09024
$40$	0	0
$41$	3.40458	0.531707	0.265853	−	0.964014i	$-0.414346\pi$
$41$	3.40458	0.531707	0.265853	+	0.964014i	$0.414346\pi$
$42$	0	0
$43$	−7.55017	−1.15139	−0.575695	−	0.817664i	$-0.695269\pi$
$43$	−7.55017	−1.15139	−0.575695	+	0.817664i	$0.695269\pi$
$44$	0	0
$45$	2.09204	0.311864
$46$	0	0
$47$	12.9657	1.89125	0.945624	−	0.325262i	$-0.105452\pi$
$47$	12.9657	1.89125	0.945624	+	0.325262i	$0.105452\pi$
$48$	0	0
$49$	−5.74570	−0.820815
$50$	0	0
$51$	−6.88548	−0.964159
$52$	0	0
$53$	5.82613	0.800281	0.400140	−	0.916454i	$-0.368961\pi$
$53$	5.82613	0.800281	0.400140	+	0.916454i	$0.368961\pi$
$54$	0	0
$55$	−6.39895	−0.862834
$56$	0	0
$57$	6.06064	0.802752
$58$	0	0
$59$	7.98329	1.03934	0.519668	−	0.854368i	$-0.326056\pi$
$59$	7.98329	1.03934	0.519668	+	0.854368i	$0.326056\pi$
$60$	0	0
$61$	−11.2818	−1.44449	−0.722244	−	0.691638i	$-0.756889\pi$
$61$	−11.2818	−1.44449	−0.722244	+	0.691638i	$0.756889\pi$
$62$	0	0
$63$	−1.11995	−0.141101
$64$	0	0
$65$	−14.2438	−1.76673
$66$	0	0
$67$	7.90749	0.966054	0.483027	−	0.875605i	$-0.339537\pi$
$67$	7.90749	0.966054	0.483027	+	0.875605i	$0.339537\pi$
$68$	0	0
$69$	0.508351	0.0611982
$70$	0	0
$71$	3.42324	0.406263	0.203132	−	0.979151i	$-0.434888\pi$
$71$	3.42324	0.406263	0.203132	+	0.979151i	$0.434888\pi$
$72$	0	0
$73$	14.4660	1.69311	0.846556	−	0.532299i	$-0.178672\pi$
$73$	14.4660	1.69311	0.846556	+	0.532299i	$0.178672\pi$
$74$	0	0
$75$	0.623348	0.0719780
$76$	0	0
$77$	3.42561	0.390385
$78$	0	0
$79$	−4.06975	−0.457883	−0.228941	−	0.973440i	$-0.573526\pi$
$79$	−4.06975	−0.457883	−0.228941	+	0.973440i	$0.573526\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−4.08357	−0.448230	−0.224115	−	0.974563i	$-0.571949\pi$
$83$	−4.08357	−0.448230	−0.224115	+	0.974563i	$0.571949\pi$
$84$	0	0
$85$	14.4047	1.56241
$86$	0	0
$87$	4.52700	0.485346
$88$	0	0
$89$	17.2613	1.82969	0.914845	−	0.403806i	$-0.132313\pi$
$89$	17.2613	1.82969	0.914845	+	0.403806i	$0.132313\pi$
$90$	0	0
$91$	7.62527	0.799346
$92$	0	0
$93$	−1.62475	−0.168479
$94$	0	0
$95$	−12.6791	−1.30085
$96$	0	0
$97$	14.8285	1.50561	0.752804	−	0.658244i	$-0.228701\pi$
$97$	14.8285	1.50561	0.752804	+	0.658244i	$0.228701\pi$
$98$	0	0
$99$	−3.05871	−0.307411
$100$	0	0
$101$	15.0157	1.49412	0.747060	−	0.664757i	$-0.231465\pi$
$101$	15.0157	1.49412	0.747060	+	0.664757i	$0.231465\pi$
$102$	0	0
$103$	−3.48792	−0.343675	−0.171838	−	0.985125i	$-0.554970\pi$
$103$	−3.48792	−0.343675	−0.171838	+	0.985125i	$0.554970\pi$
$104$	0	0
$105$	2.34300	0.228653
$106$	0	0
$107$	−2.50472	−0.242140	−0.121070	−	0.992644i	$-0.538633\pi$
$107$	−2.50472	−0.242140	−0.121070	+	0.992644i	$0.538633\pi$
$108$	0	0
$109$	15.5936	1.49359	0.746796	−	0.665054i	$-0.231591\pi$
$109$	15.5936	1.49359	0.746796	+	0.665054i	$0.231591\pi$
$110$	0	0
$111$	−4.52706	−0.429689
$112$	0	0
$113$	16.8534	1.58543	0.792716	−	0.609591i	$-0.208666\pi$
$113$	16.8534	1.58543	0.792716	+	0.609591i	$0.208666\pi$
$114$	0	0
$115$	−1.06349	−0.0991712
$116$	0	0
$117$	−6.80855	−0.629451
$118$	0	0
$119$	−7.71142	−0.706905
$120$	0	0
$121$	−1.64432	−0.149484
$122$	0	0
$123$	−3.40458	−0.306981
$124$	0	0
$125$	−11.7643	−1.05223
$126$	0	0
$127$	13.5676	1.20393	0.601964	−	0.798523i	$-0.294385\pi$
$127$	13.5676	1.20393	0.601964	+	0.798523i	$0.294385\pi$
$128$	0	0
$129$	7.55017	0.664756
$130$	0	0
$131$	−15.4825	−1.35272	−0.676358	−	0.736573i	$-0.736443\pi$
$131$	−15.4825	−1.35272	−0.676358	+	0.736573i	$0.736443\pi$
$132$	0	0
$133$	6.78764	0.588563
$134$	0	0
$135$	−2.09204	−0.180055
$136$	0	0
$137$	−1.68605	−0.144049	−0.0720245	−	0.997403i	$-0.522946\pi$
$137$	−1.68605	−0.144049	−0.0720245	+	0.997403i	$0.522946\pi$
$138$	0	0
$139$	−21.0299	−1.78374	−0.891868	−	0.452297i	$-0.850605\pi$
$139$	−21.0299	−1.78374	−0.891868	+	0.452297i	$0.850605\pi$
$140$	0	0
$141$	−12.9657	−1.09191
$142$	0	0
$143$	20.8254	1.74150
$144$	0	0
$145$	−9.47069	−0.786498
$146$	0	0
$147$	5.74570	0.473897
$148$	0	0
$149$	15.5986	1.27788	0.638942	−	0.769255i	$-0.279372\pi$
$149$	15.5986	1.27788	0.638942	+	0.769255i	$0.279372\pi$
$150$	0	0
$151$	−12.9393	−1.05299	−0.526494	−	0.850179i	$-0.676494\pi$
$151$	−12.9393	−1.05299	−0.526494	+	0.850179i	$0.676494\pi$
$152$	0	0
$153$	6.88548	0.556658
$154$	0	0
$155$	3.39905	0.273018
$156$	0	0
$157$	−21.1717	−1.68969	−0.844844	−	0.535012i	$-0.820307\pi$
$157$	−21.1717	−1.68969	−0.844844	+	0.535012i	$0.820307\pi$
$158$	0	0
$159$	−5.82613	−0.462042
$160$	0	0
$161$	0.569330	0.0448695
$162$	0	0
$163$	3.09855	0.242697	0.121349	−	0.992610i	$-0.461278\pi$
$163$	3.09855	0.242697	0.121349	+	0.992610i	$0.461278\pi$
$164$	0	0
$165$	6.39895	0.498158
$166$	0	0
$167$	−0.274953	−0.0212765	−0.0106383	−	0.999943i	$-0.503386\pi$
$167$	−0.274953	−0.0212765	−0.0106383	+	0.999943i	$0.503386\pi$
$168$	0	0
$169$	33.3564	2.56588
$170$	0	0
$171$	−6.06064	−0.463469
$172$	0	0
$173$	−16.5216	−1.25612	−0.628058	−	0.778166i	$-0.716150\pi$
$173$	−16.5216	−1.25612	−0.628058	+	0.778166i	$0.716150\pi$
$174$	0	0
$175$	0.698122	0.0527730
$176$	0	0
$177$	−7.98329	−0.600061
$178$	0	0
$179$	13.5841	1.01532	0.507662	−	0.861556i	$-0.330510\pi$
$179$	13.5841	1.01532	0.507662	+	0.861556i	$0.330510\pi$
$180$	0	0
$181$	−3.00685	−0.223498	−0.111749	−	0.993736i	$-0.535645\pi$
$181$	−3.00685	−0.223498	−0.111749	+	0.993736i	$0.535645\pi$
$182$	0	0
$183$	11.2818	0.833976
$184$	0	0
$185$	9.47081	0.696308
$186$	0	0
$187$	−21.0606	−1.54011
$188$	0	0
$189$	1.11995	0.0814647
$190$	0	0
$191$	6.40033	0.463111	0.231556	−	0.972822i	$-0.425618\pi$
$191$	6.40033	0.463111	0.231556	+	0.972822i	$0.425618\pi$
$192$	0	0
$193$	−10.8903	−0.783899	−0.391949	−	0.919987i	$-0.628199\pi$
$193$	−10.8903	−0.783899	−0.391949	+	0.919987i	$0.628199\pi$
$194$	0	0
$195$	14.2438	1.02002
$196$	0	0
$197$	12.6911	0.904207	0.452103	−	0.891966i	$-0.350674\pi$
$197$	12.6911	0.904207	0.452103	+	0.891966i	$0.350674\pi$
$198$	0	0
$199$	18.7361	1.32817	0.664085	−	0.747657i	$-0.268821\pi$
$199$	18.7361	1.32817	0.664085	+	0.747657i	$0.268821\pi$
$200$	0	0
$201$	−7.90749	−0.557752
$202$	0	0
$203$	5.07004	0.355847
$204$	0	0
$205$	7.12254	0.497460
$206$	0	0
$207$	−0.508351	−0.0353328
$208$	0	0
$209$	18.5377	1.28228
$210$	0	0
$211$	−9.31292	−0.641128	−0.320564	−	0.947227i	$-0.603872\pi$
$211$	−9.31292	−0.641128	−0.320564	+	0.947227i	$0.603872\pi$
$212$	0	0
$213$	−3.42324	−0.234556
$214$	0	0
$215$	−15.7953	−1.07723
$216$	0	0
$217$	−1.81965	−0.123526
$218$	0	0
$219$	−14.4660	−0.977519
$220$	0	0
$221$	−46.8801	−3.15350
$222$	0	0
$223$	3.37856	0.226245	0.113123	−	0.993581i	$-0.463915\pi$
$223$	3.37856	0.226245	0.113123	+	0.993581i	$0.463915\pi$
$224$	0	0
$225$	−0.623348	−0.0415565
$226$	0	0
$227$	14.9597	0.992912	0.496456	−	0.868062i	$-0.334634\pi$
$227$	14.9597	0.992912	0.496456	+	0.868062i	$0.334634\pi$
$228$	0	0
$229$	18.6901	1.23508	0.617538	−	0.786541i	$-0.288130\pi$
$229$	18.6901	1.23508	0.617538	+	0.786541i	$0.288130\pi$
$230$	0	0
$231$	−3.42561	−0.225389
$232$	0	0
$233$	2.10866	0.138143	0.0690713	−	0.997612i	$-0.477996\pi$
$233$	2.10866	0.138143	0.0690713	+	0.997612i	$0.477996\pi$
$234$	0	0
$235$	27.1249	1.76943
$236$	0	0
$237$	4.06975	0.264359
$238$	0	0
$239$	−7.09270	−0.458789	−0.229394	−	0.973334i	$-0.573675\pi$
$239$	−7.09270	−0.458789	−0.229394	+	0.973334i	$0.573675\pi$
$240$	0	0
$241$	6.11396	0.393835	0.196917	−	0.980420i	$-0.436907\pi$
$241$	6.11396	0.393835	0.196917	+	0.980420i	$0.436907\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−12.0203	−0.767947
$246$	0	0
$247$	41.2642	2.62558
$248$	0	0
$249$	4.08357	0.258786
$250$	0	0
$251$	2.80434	0.177008	0.0885040	−	0.996076i	$-0.471791\pi$
$251$	2.80434	0.177008	0.0885040	+	0.996076i	$0.471791\pi$
$252$	0	0
$253$	1.55489	0.0977554
$254$	0	0
$255$	−14.4047	−0.902059
$256$	0	0
$257$	−25.2183	−1.57308	−0.786538	−	0.617542i	$-0.788129\pi$
$257$	−25.2183	−1.57308	−0.786538	+	0.617542i	$0.788129\pi$
$258$	0	0
$259$	−5.07010	−0.315041
$260$	0	0
$261$	−4.52700	−0.280214
$262$	0	0
$263$	13.6148	0.839526	0.419763	−	0.907634i	$-0.362113\pi$
$263$	13.6148	0.839526	0.419763	+	0.907634i	$0.362113\pi$
$264$	0	0
$265$	12.1885	0.748735
$266$	0	0
$267$	−17.2613	−1.05637
$268$	0	0
$269$	−7.70251	−0.469631	−0.234815	−	0.972040i	$-0.575449\pi$
$269$	−7.70251	−0.469631	−0.234815	+	0.972040i	$0.575449\pi$
$270$	0	0
$271$	−25.8369	−1.56948	−0.784741	−	0.619824i	$-0.787204\pi$
$271$	−25.8369	−1.56948	−0.784741	+	0.619824i	$0.787204\pi$
$272$	0	0
$273$	−7.62527	−0.461502
$274$	0	0
$275$	1.90664	0.114975
$276$	0	0
$277$	5.94164	0.356999	0.178499	−	0.983940i	$-0.442876\pi$
$277$	5.94164	0.356999	0.178499	+	0.983940i	$0.442876\pi$
$278$	0	0
$279$	1.62475	0.0972712
$280$	0	0
$281$	−5.87340	−0.350378	−0.175189	−	0.984535i	$-0.556054\pi$
$281$	−5.87340	−0.350378	−0.175189	+	0.984535i	$0.556054\pi$
$282$	0	0
$283$	−22.8593	−1.35884	−0.679422	−	0.733747i	$-0.737770\pi$
$283$	−22.8593	−1.35884	−0.679422	+	0.733747i	$0.737770\pi$
$284$	0	0
$285$	12.6791	0.751047
$286$	0	0
$287$	−3.81298	−0.225073
$288$	0	0
$289$	30.4098	1.78881
$290$	0	0
$291$	−14.8285	−0.869264
$292$	0	0
$293$	10.1271	0.591630	0.295815	−	0.955245i	$-0.404409\pi$
$293$	10.1271	0.591630	0.295815	+	0.955245i	$0.404409\pi$
$294$	0	0
$295$	16.7014	0.972394
$296$	0	0
$297$	3.05871	0.177484
$298$	0	0
$299$	3.46113	0.200162
$300$	0	0
$301$	8.45585	0.487387
$302$	0	0
$303$	−15.0157	−0.862630
$304$	0	0
$305$	−23.6021	−1.35145
$306$	0	0
$307$	−9.08360	−0.518429	−0.259214	−	0.965820i	$-0.583464\pi$
$307$	−9.08360	−0.518429	−0.259214	+	0.965820i	$0.583464\pi$
$308$	0	0
$309$	3.48792	0.198421
$310$	0	0
$311$	28.1435	1.59587	0.797937	−	0.602741i	$-0.205925\pi$
$311$	28.1435	1.59587	0.797937	+	0.602741i	$0.205925\pi$
$312$	0	0
$313$	2.15160	0.121616	0.0608078	−	0.998149i	$-0.480632\pi$
$313$	2.15160	0.121616	0.0608078	+	0.998149i	$0.480632\pi$
$314$	0	0
$315$	−2.34300	−0.132013
$316$	0	0
$317$	−17.9505	−1.00820	−0.504100	−	0.863645i	$-0.668176\pi$
$317$	−17.9505	−1.00820	−0.504100	+	0.863645i	$0.668176\pi$
$318$	0	0
$319$	13.8468	0.775270
$320$	0	0
$321$	2.50472	0.139800
$322$	0	0
$323$	−41.7304	−2.32194
$324$	0	0
$325$	4.24410	0.235420
$326$	0	0
$327$	−15.5936	−0.862325
$328$	0	0
$329$	−14.5210	−0.800571
$330$	0	0
$331$	−0.828334	−0.0455293	−0.0227647	−	0.999741i	$-0.507247\pi$
$331$	−0.828334	−0.0455293	−0.0227647	+	0.999741i	$0.507247\pi$
$332$	0	0
$333$	4.52706	0.248081
$334$	0	0
$335$	16.5428	0.903831
$336$	0	0
$337$	15.9706	0.869974	0.434987	−	0.900437i	$-0.356753\pi$
$337$	15.9706	0.869974	0.434987	+	0.900437i	$0.356753\pi$
$338$	0	0
$339$	−16.8534	−0.915350
$340$	0	0
$341$	−4.96963	−0.269121
$342$	0	0
$343$	14.2746	0.770756
$344$	0	0
$345$	1.06349	0.0572565
$346$	0	0
$347$	−2.09974	−0.112720	−0.0563599	−	0.998411i	$-0.517949\pi$
$347$	−2.09974	−0.112720	−0.0563599	+	0.998411i	$0.517949\pi$
$348$	0	0
$349$	12.0394	0.644452	0.322226	−	0.946663i	$-0.395569\pi$
$349$	12.0394	0.644452	0.322226	+	0.946663i	$0.395569\pi$
$350$	0	0
$351$	6.80855	0.363414
$352$	0	0
$353$	−4.61169	−0.245456	−0.122728	−	0.992440i	$-0.539164\pi$
$353$	−4.61169	−0.245456	−0.122728	+	0.992440i	$0.539164\pi$
$354$	0	0
$355$	7.16157	0.380096
$356$	0	0
$357$	7.71142	0.408132
$358$	0	0
$359$	−5.45226	−0.287759	−0.143880	−	0.989595i	$-0.545958\pi$
$359$	−5.45226	−0.287759	−0.143880	+	0.989595i	$0.545958\pi$
$360$	0	0
$361$	17.7314	0.933231
$362$	0	0
$363$	1.64432	0.0863044
$364$	0	0
$365$	30.2634	1.58406
$366$	0	0
$367$	16.4642	0.859422	0.429711	−	0.902967i	$-0.358615\pi$
$367$	16.4642	0.859422	0.429711	+	0.902967i	$0.358615\pi$
$368$	0	0
$369$	3.40458	0.177236
$370$	0	0
$371$	−6.52500	−0.338761
$372$	0	0
$373$	−14.1213	−0.731171	−0.365586	−	0.930778i	$-0.619131\pi$
$373$	−14.1213	−0.731171	−0.365586	+	0.930778i	$0.619131\pi$
$374$	0	0
$375$	11.7643	0.607506
$376$	0	0
$377$	30.8223	1.58743
$378$	0	0
$379$	−1.80636	−0.0927863	−0.0463932	−	0.998923i	$-0.514773\pi$
$379$	−1.80636	−0.0927863	−0.0463932	+	0.998923i	$0.514773\pi$
$380$	0	0
$381$	−13.5676	−0.695088
$382$	0	0
$383$	−36.3146	−1.85559	−0.927794	−	0.373092i	$-0.878298\pi$
$383$	−36.3146	−1.85559	−0.927794	+	0.373092i	$0.878298\pi$
$384$	0	0
$385$	7.16653	0.365240
$386$	0	0
$387$	−7.55017	−0.383797
$388$	0	0
$389$	−16.2430	−0.823553	−0.411776	−	0.911285i	$-0.635091\pi$
$389$	−16.2430	−0.823553	−0.411776	+	0.911285i	$0.635091\pi$
$390$	0	0
$391$	−3.50024	−0.177015
$392$	0	0
$393$	15.4825	0.780991
$394$	0	0
$395$	−8.51410	−0.428391
$396$	0	0
$397$	33.4869	1.68066	0.840329	−	0.542077i	$-0.182362\pi$
$397$	33.4869	1.68066	0.840329	+	0.542077i	$0.182362\pi$
$398$	0	0
$399$	−6.78764	−0.339807
$400$	0	0
$401$	17.4558	0.871703	0.435852	−	0.900019i	$-0.356447\pi$
$401$	17.4558	0.871703	0.435852	+	0.900019i	$0.356447\pi$
$402$	0	0
$403$	−11.0622	−0.551047
$404$	0	0
$405$	2.09204	0.103955
$406$	0	0
$407$	−13.8469	−0.686367
$408$	0	0
$409$	1.44472	0.0714369	0.0357185	−	0.999362i	$-0.488628\pi$
$409$	1.44472	0.0714369	0.0357185	+	0.999362i	$0.488628\pi$
$410$	0	0
$411$	1.68605	0.0831667
$412$	0	0
$413$	−8.94093	−0.439954
$414$	0	0
$415$	−8.54302	−0.419360
$416$	0	0
$417$	21.0299	1.02984
$418$	0	0
$419$	25.1475	1.22854	0.614268	−	0.789097i	$-0.289451\pi$
$419$	25.1475	1.22854	0.614268	+	0.789097i	$0.289451\pi$
$420$	0	0
$421$	38.5780	1.88017	0.940087	−	0.340934i	$-0.110743\pi$
$421$	38.5780	1.88017	0.940087	+	0.340934i	$0.110743\pi$
$422$	0	0
$423$	12.9657	0.630416
$424$	0	0
$425$	−4.29205	−0.208195
$426$	0	0
$427$	12.6351	0.611456
$428$	0	0
$429$	−20.8254	−1.00546
$430$	0	0
$431$	34.3908	1.65655	0.828273	−	0.560324i	$-0.189324\pi$
$431$	34.3908	1.65655	0.828273	+	0.560324i	$0.189324\pi$
$432$	0	0
$433$	1.64164	0.0788923	0.0394461	−	0.999222i	$-0.487441\pi$
$433$	1.64164	0.0788923	0.0394461	+	0.999222i	$0.487441\pi$
$434$	0	0
$435$	9.47069	0.454085
$436$	0	0
$437$	3.08093	0.147381
$438$	0	0
$439$	−9.97093	−0.475887	−0.237943	−	0.971279i	$-0.576473\pi$
$439$	−9.97093	−0.475887	−0.237943	+	0.971279i	$0.576473\pi$
$440$	0	0
$441$	−5.74570	−0.273605
$442$	0	0
$443$	16.2845	0.773700	0.386850	−	0.922143i	$-0.373563\pi$
$443$	16.2845	0.773700	0.386850	+	0.922143i	$0.373563\pi$
$444$	0	0
$445$	36.1113	1.71184
$446$	0	0
$447$	−15.5986	−0.737787
$448$	0	0
$449$	0.982830	0.0463826	0.0231913	−	0.999731i	$-0.492617\pi$
$449$	0.982830	0.0463826	0.0231913	+	0.999731i	$0.492617\pi$
$450$	0	0
$451$	−10.4136	−0.490358
$452$	0	0
$453$	12.9393	0.607943
$454$	0	0
$455$	15.9524	0.747861
$456$	0	0
$457$	−4.74202	−0.221823	−0.110911	−	0.993830i	$-0.535377\pi$
$457$	−4.74202	−0.221823	−0.110911	+	0.993830i	$0.535377\pi$
$458$	0	0
$459$	−6.88548	−0.321386
$460$	0	0
$461$	11.4949	0.535369	0.267685	−	0.963507i	$-0.413742\pi$
$461$	11.4949	0.535369	0.267685	+	0.963507i	$0.413742\pi$
$462$	0	0
$463$	20.8843	0.970575	0.485287	−	0.874355i	$-0.338715\pi$
$463$	20.8843	0.970575	0.485287	+	0.874355i	$0.338715\pi$
$464$	0	0
$465$	−3.39905	−0.157627
$466$	0	0
$467$	0.744771	0.0344639	0.0172320	−	0.999852i	$-0.494515\pi$
$467$	0.744771	0.0344639	0.0172320	+	0.999852i	$0.494515\pi$
$468$	0	0
$469$	−8.85603	−0.408934
$470$	0	0
$471$	21.1717	0.975542
$472$	0	0
$473$	23.0938	1.06185
$474$	0	0
$475$	3.77789	0.173341
$476$	0	0
$477$	5.82613	0.266760
$478$	0	0
$479$	8.84492	0.404135	0.202067	−	0.979372i	$-0.435234\pi$
$479$	8.84492	0.404135	0.202067	+	0.979372i	$0.435234\pi$
$480$	0	0
$481$	−30.8227	−1.40540
$482$	0	0
$483$	−0.569330	−0.0259054
$484$	0	0
$485$	31.0219	1.40863
$486$	0	0
$487$	−25.2217	−1.14290	−0.571452	−	0.820635i	$-0.693620\pi$
$487$	−25.2217	−1.14290	−0.571452	+	0.820635i	$0.693620\pi$
$488$	0	0
$489$	−3.09855	−0.140121
$490$	0	0
$491$	19.7997	0.893546	0.446773	−	0.894647i	$-0.352573\pi$
$491$	19.7997	0.893546	0.446773	+	0.894647i	$0.352573\pi$
$492$	0	0
$493$	−31.1706	−1.40385
$494$	0	0
$495$	−6.39895	−0.287611
$496$	0	0
$497$	−3.83387	−0.171973
$498$	0	0
$499$	19.6848	0.881212	0.440606	−	0.897701i	$-0.354764\pi$
$499$	19.6848	0.881212	0.440606	+	0.897701i	$0.354764\pi$
$500$	0	0
$501$	0.274953	0.0122840
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	31.4136	1.39788
$506$	0	0
$507$	−33.3564	−1.48141
$508$	0	0
$509$	16.0171	0.709947	0.354974	−	0.934876i	$-0.384490\pi$
$509$	16.0171	0.709947	0.354974	+	0.934876i	$0.384490\pi$
$510$	0	0
$511$	−16.2012	−0.716700
$512$	0	0
$513$	6.06064	0.267584
$514$	0	0
$515$	−7.29689	−0.321540
$516$	0	0
$517$	−39.6584	−1.74417
$518$	0	0
$519$	16.5216	0.725219
$520$	0	0
$521$	−22.9713	−1.00639	−0.503195	−	0.864173i	$-0.667842\pi$
$521$	−22.9713	−1.00639	−0.503195	+	0.864173i	$0.667842\pi$
$522$	0	0
$523$	40.4831	1.77020	0.885101	−	0.465399i	$-0.154089\pi$
$523$	40.4831	1.77020	0.885101	+	0.465399i	$0.154089\pi$
$524$	0	0
$525$	−0.698122	−0.0304685
$526$	0	0
$527$	11.1872	0.487321
$528$	0	0
$529$	−22.7416	−0.988764
$530$	0	0
$531$	7.98329	0.346445
$532$	0	0
$533$	−23.1803	−1.00405
$534$	0	0
$535$	−5.23998	−0.226544
$536$	0	0
$537$	−13.5841	−0.586198
$538$	0	0
$539$	17.5744	0.756983
$540$	0	0
$541$	32.0601	1.37837	0.689186	−	0.724584i	$-0.257968\pi$
$541$	32.0601	1.37837	0.689186	+	0.724584i	$0.257968\pi$
$542$	0	0
$543$	3.00685	0.129036
$544$	0	0
$545$	32.6224	1.39739
$546$	0	0
$547$	−6.90685	−0.295316	−0.147658	−	0.989039i	$-0.547173\pi$
$547$	−6.90685	−0.295316	−0.147658	+	0.989039i	$0.547173\pi$
$548$	0	0
$549$	−11.2818	−0.481496
$550$	0	0
$551$	27.4365	1.16884
$552$	0	0
$553$	4.55794	0.193823
$554$	0	0
$555$	−9.47081	−0.402013
$556$	0	0
$557$	34.8776	1.47781	0.738906	−	0.673808i	$-0.235343\pi$
$557$	34.8776	1.47781	0.738906	+	0.673808i	$0.235343\pi$
$558$	0	0
$559$	51.4058	2.17423
$560$	0	0
$561$	21.0606	0.889181
$562$	0	0
$563$	41.8681	1.76453	0.882264	−	0.470755i	$-0.156018\pi$
$563$	41.8681	1.76453	0.882264	+	0.470755i	$0.156018\pi$
$564$	0	0
$565$	35.2580	1.48332
$566$	0	0
$567$	−1.11995	−0.0470337
$568$	0	0
$569$	5.52575	0.231651	0.115826	−	0.993270i	$-0.463049\pi$
$569$	5.52575	0.231651	0.115826	+	0.993270i	$0.463049\pi$
$570$	0	0
$571$	−22.8981	−0.958256	−0.479128	−	0.877745i	$-0.659047\pi$
$571$	−22.8981	−0.958256	−0.479128	+	0.877745i	$0.659047\pi$
$572$	0	0
$573$	−6.40033	−0.267378
$574$	0	0
$575$	0.316879	0.0132148
$576$	0	0
$577$	−10.1018	−0.420543	−0.210272	−	0.977643i	$-0.567435\pi$
$577$	−10.1018	−0.420543	−0.210272	+	0.977643i	$0.567435\pi$
$578$	0	0
$579$	10.8903	0.452584
$580$	0	0
$581$	4.57342	0.189737
$582$	0	0
$583$	−17.8204	−0.738046
$584$	0	0
$585$	−14.2438	−0.588909
$586$	0	0
$587$	−4.81788	−0.198855	−0.0994275	−	0.995045i	$-0.531701\pi$
$587$	−4.81788	−0.198855	−0.0994275	+	0.995045i	$0.531701\pi$
$588$	0	0
$589$	−9.84702	−0.405740
$590$	0	0
$591$	−12.6911	−0.522044
$592$	0	0
$593$	−23.7096	−0.973636	−0.486818	−	0.873504i	$-0.661842\pi$
$593$	−23.7096	−0.973636	−0.486818	+	0.873504i	$0.661842\pi$
$594$	0	0
$595$	−16.1326	−0.661374
$596$	0	0
$597$	−18.7361	−0.766819
$598$	0	0
$599$	10.3607	0.423325	0.211662	−	0.977343i	$-0.432112\pi$
$599$	10.3607	0.423325	0.211662	+	0.977343i	$0.432112\pi$
$600$	0	0
$601$	−34.9778	−1.42677	−0.713387	−	0.700770i	$-0.752840\pi$
$601$	−34.9778	−1.42677	−0.713387	+	0.700770i	$0.752840\pi$
$602$	0	0
$603$	7.90749	0.322018
$604$	0	0
$605$	−3.43999	−0.139856
$606$	0	0
$607$	−12.7927	−0.519241	−0.259620	−	0.965711i	$-0.583597\pi$
$607$	−12.7927	−0.519241	−0.259620	+	0.965711i	$0.583597\pi$
$608$	0	0
$609$	−5.07004	−0.205448
$610$	0	0
$611$	−88.2780	−3.57134
$612$	0	0
$613$	18.4144	0.743749	0.371874	−	0.928283i	$-0.378715\pi$
$613$	18.4144	0.743749	0.371874	+	0.928283i	$0.378715\pi$
$614$	0	0
$615$	−7.12254	−0.287209
$616$	0	0
$617$	−2.35116	−0.0946540	−0.0473270	−	0.998879i	$-0.515070\pi$
$617$	−2.35116	−0.0946540	−0.0473270	+	0.998879i	$0.515070\pi$
$618$	0	0
$619$	28.4460	1.14334	0.571671	−	0.820483i	$-0.306296\pi$
$619$	28.4460	1.14334	0.571671	+	0.820483i	$0.306296\pi$
$620$	0	0
$621$	0.508351	0.0203994
$622$	0	0
$623$	−19.3318	−0.774513
$624$	0	0
$625$	−21.4947	−0.859788
$626$	0	0
$627$	−18.5377	−0.740325
$628$	0	0
$629$	31.1709	1.24287
$630$	0	0
$631$	12.6592	0.503955	0.251977	−	0.967733i	$-0.418919\pi$
$631$	12.6592	0.503955	0.251977	+	0.967733i	$0.418919\pi$
$632$	0	0
$633$	9.31292	0.370155
$634$	0	0
$635$	28.3840	1.12638
$636$	0	0
$637$	39.1199	1.54999
$638$	0	0
$639$	3.42324	0.135421
$640$	0	0
$641$	14.8200	0.585353	0.292677	−	0.956211i	$-0.405454\pi$
$641$	14.8200	0.585353	0.292677	+	0.956211i	$0.405454\pi$
$642$	0	0
$643$	−43.9859	−1.73463	−0.867316	−	0.497757i	$-0.834157\pi$
$643$	−43.9859	−1.73463	−0.867316	+	0.497757i	$0.834157\pi$
$644$	0	0
$645$	15.7953	0.621939
$646$	0	0
$647$	−39.1606	−1.53956	−0.769781	−	0.638308i	$-0.779635\pi$
$647$	−39.1606	−1.53956	−0.769781	+	0.638308i	$0.779635\pi$
$648$	0	0
$649$	−24.4185	−0.958512
$650$	0	0
$651$	1.81965	0.0713175
$652$	0	0
$653$	2.24871	0.0879989	0.0439994	−	0.999032i	$-0.485990\pi$
$653$	2.24871	0.0879989	0.0439994	+	0.999032i	$0.485990\pi$
$654$	0	0
$655$	−32.3902	−1.26559
$656$	0	0
$657$	14.4660	0.564371
$658$	0	0
$659$	−3.93412	−0.153251	−0.0766257	−	0.997060i	$-0.524415\pi$
$659$	−3.93412	−0.153251	−0.0766257	+	0.997060i	$0.524415\pi$
$660$	0	0
$661$	20.4901	0.796971	0.398486	−	0.917175i	$-0.369536\pi$
$661$	20.4901	0.796971	0.398486	+	0.917175i	$0.369536\pi$
$662$	0	0
$663$	46.8801	1.82067
$664$	0	0
$665$	14.2001	0.550655
$666$	0	0
$667$	2.30130	0.0891069
$668$	0	0
$669$	−3.37856	−0.130623
$670$	0	0
$671$	34.5078	1.33216
$672$	0	0
$673$	−31.4063	−1.21062	−0.605312	−	0.795988i	$-0.706952\pi$
$673$	−31.4063	−1.21062	−0.605312	+	0.795988i	$0.706952\pi$
$674$	0	0
$675$	0.623348	0.0239927
$676$	0	0
$677$	30.0000	1.15299	0.576496	−	0.817100i	$-0.304420\pi$
$677$	30.0000	1.15299	0.576496	+	0.817100i	$0.304420\pi$
$678$	0	0
$679$	−16.6073	−0.637329
$680$	0	0
$681$	−14.9597	−0.573258
$682$	0	0
$683$	−15.7273	−0.601790	−0.300895	−	0.953657i	$-0.597285\pi$
$683$	−15.7273	−0.601790	−0.300895	+	0.953657i	$0.597285\pi$
$684$	0	0
$685$	−3.52729	−0.134771
$686$	0	0
$687$	−18.6901	−0.713071
$688$	0	0
$689$	−39.6675	−1.51121
$690$	0	0
$691$	28.4104	1.08078	0.540392	−	0.841414i	$-0.318276\pi$
$691$	28.4104	1.08078	0.540392	+	0.841414i	$0.318276\pi$
$692$	0	0
$693$	3.42561	0.130128
$694$	0	0
$695$	−43.9956	−1.66885
$696$	0	0
$697$	23.4422	0.887936
$698$	0	0
$699$	−2.10866	−0.0797567
$700$	0	0
$701$	8.57137	0.323736	0.161868	−	0.986812i	$-0.448248\pi$
$701$	8.57137	0.323736	0.161868	+	0.986812i	$0.448248\pi$
$702$	0	0
$703$	−27.4369	−1.03480
$704$	0	0
$705$	−27.1249	−1.02158
$706$	0	0
$707$	−16.8169	−0.632466
$708$	0	0
$709$	25.5151	0.958240	0.479120	−	0.877749i	$-0.340956\pi$
$709$	25.5151	0.958240	0.479120	+	0.877749i	$0.340956\pi$
$710$	0	0
$711$	−4.06975	−0.152628
$712$	0	0
$713$	−0.825942	−0.0309318
$714$	0	0
$715$	43.5676	1.62934
$716$	0	0
$717$	7.09270	0.264882
$718$	0	0
$719$	−41.0285	−1.53011	−0.765053	−	0.643968i	$-0.777287\pi$
$719$	−41.0285	−1.53011	−0.765053	+	0.643968i	$0.777287\pi$
$720$	0	0
$721$	3.90632	0.145479
$722$	0	0
$723$	−6.11396	−0.227380
$724$	0	0
$725$	2.82190	0.104803
$726$	0	0
$727$	−3.91694	−0.145271	−0.0726356	−	0.997359i	$-0.523141\pi$
$727$	−3.91694	−0.145271	−0.0726356	+	0.997359i	$0.523141\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−51.9865	−1.92279
$732$	0	0
$733$	−14.5673	−0.538057	−0.269028	−	0.963132i	$-0.586703\pi$
$733$	−14.5673	−0.538057	−0.269028	+	0.963132i	$0.586703\pi$
$734$	0	0
$735$	12.0203	0.443374
$736$	0	0
$737$	−24.1867	−0.890928
$738$	0	0
$739$	43.9158	1.61547	0.807734	−	0.589547i	$-0.200694\pi$
$739$	43.9158	1.61547	0.807734	+	0.589547i	$0.200694\pi$
$740$	0	0
$741$	−41.2642	−1.51588
$742$	0	0
$743$	−9.07407	−0.332895	−0.166448	−	0.986050i	$-0.553230\pi$
$743$	−9.07407	−0.332895	−0.166448	+	0.986050i	$0.553230\pi$
$744$	0	0
$745$	32.6329	1.19558
$746$	0	0
$747$	−4.08357	−0.149410
$748$	0	0
$749$	2.80517	0.102499
$750$	0	0
$751$	6.65730	0.242929	0.121464	−	0.992596i	$-0.461241\pi$
$751$	6.65730	0.242929	0.121464	+	0.992596i	$0.461241\pi$
$752$	0	0
$753$	−2.80434	−0.102196
$754$	0	0
$755$	−27.0697	−0.985166
$756$	0	0
$757$	−19.6664	−0.714788	−0.357394	−	0.933954i	$-0.616335\pi$
$757$	−19.6664	−0.714788	−0.357394	+	0.933954i	$0.616335\pi$
$758$	0	0
$759$	−1.55489	−0.0564391
$760$	0	0
$761$	5.81481	0.210787	0.105393	−	0.994431i	$-0.466390\pi$
$761$	5.81481	0.210787	0.105393	+	0.994431i	$0.466390\pi$
$762$	0	0
$763$	−17.4641	−0.632242
$764$	0	0
$765$	14.4047	0.520804
$766$	0	0
$767$	−54.3547	−1.96263
$768$	0	0
$769$	−6.58954	−0.237625	−0.118813	−	0.992917i	$-0.537909\pi$
$769$	−6.58954	−0.237625	−0.118813	+	0.992917i	$0.537909\pi$
$770$	0	0
$771$	25.2183	0.908216
$772$	0	0
$773$	19.8195	0.712857	0.356429	−	0.934323i	$-0.383994\pi$
$773$	19.8195	0.712857	0.356429	+	0.934323i	$0.383994\pi$
$774$	0	0
$775$	−1.01278	−0.0363803
$776$	0	0
$777$	5.07010	0.181889
$778$	0	0
$779$	−20.6340	−0.739288
$780$	0	0
$781$	−10.4707	−0.374670
$782$	0	0
$783$	4.52700	0.161782
$784$	0	0
$785$	−44.2922	−1.58086
$786$	0	0
$787$	17.6843	0.630376	0.315188	−	0.949029i	$-0.397932\pi$
$787$	17.6843	0.630376	0.315188	+	0.949029i	$0.397932\pi$
$788$	0	0
$789$	−13.6148	−0.484701
$790$	0	0
$791$	−18.8750	−0.671118
$792$	0	0
$793$	76.8129	2.72770
$794$	0	0
$795$	−12.1885	−0.432283
$796$	0	0
$797$	−40.7996	−1.44520	−0.722598	−	0.691269i	$-0.757052\pi$
$797$	−40.7996	−1.44520	−0.722598	+	0.691269i	$0.757052\pi$
$798$	0	0
$799$	89.2753	3.15833
$800$	0	0
$801$	17.2613	0.609897
$802$	0	0
$803$	−44.2471	−1.56145
$804$	0	0
$805$	1.19106	0.0419795
$806$	0	0
$807$	7.70251	0.271141
$808$	0	0
$809$	−2.95407	−0.103859	−0.0519297	−	0.998651i	$-0.516537\pi$
$809$	−2.95407	−0.103859	−0.0519297	+	0.998651i	$0.516537\pi$
$810$	0	0
$811$	6.87439	0.241392	0.120696	−	0.992689i	$-0.461487\pi$
$811$	6.87439	0.241392	0.120696	+	0.992689i	$0.461487\pi$
$812$	0	0
$813$	25.8369	0.906141
$814$	0	0
$815$	6.48231	0.227065
$816$	0	0
$817$	45.7589	1.60090
$818$	0	0
$819$	7.62527	0.266449
$820$	0	0
$821$	−27.1421	−0.947267	−0.473634	−	0.880722i	$-0.657058\pi$
$821$	−27.1421	−0.947267	−0.473634	+	0.880722i	$0.657058\pi$
$822$	0	0
$823$	−9.66658	−0.336956	−0.168478	−	0.985705i	$-0.553885\pi$
$823$	−9.66658	−0.336956	−0.168478	+	0.985705i	$0.553885\pi$
$824$	0	0
$825$	−1.90664	−0.0663806
$826$	0	0
$827$	−30.2680	−1.05252	−0.526260	−	0.850324i	$-0.676406\pi$
$827$	−30.2680	−1.05252	−0.526260	+	0.850324i	$0.676406\pi$
$828$	0	0
$829$	−18.0675	−0.627512	−0.313756	−	0.949504i	$-0.601587\pi$
$829$	−18.0675	−0.627512	−0.313756	+	0.949504i	$0.601587\pi$
$830$	0	0
$831$	−5.94164	−0.206113
$832$	0	0
$833$	−39.5619	−1.37074
$834$	0	0
$835$	−0.575215	−0.0199061
$836$	0	0
$837$	−1.62475	−0.0561595
$838$	0	0
$839$	13.9450	0.481434	0.240717	−	0.970595i	$-0.422617\pi$
$839$	13.9450	0.481434	0.240717	+	0.970595i	$0.422617\pi$
$840$	0	0
$841$	−8.50626	−0.293319
$842$	0	0
$843$	5.87340	0.202291
$844$	0	0
$845$	69.7831	2.40061
$846$	0	0
$847$	1.84156	0.0632769
$848$	0	0
$849$	22.8593	0.784529
$850$	0	0
$851$	−2.30133	−0.0788887
$852$	0	0
$853$	23.9459	0.819891	0.409946	−	0.912110i	$-0.365548\pi$
$853$	23.9459	0.819891	0.409946	+	0.912110i	$0.365548\pi$
$854$	0	0
$855$	−12.6791	−0.433617
$856$	0	0
$857$	12.0017	0.409971	0.204985	−	0.978765i	$-0.434285\pi$
$857$	12.0017	0.409971	0.204985	+	0.978765i	$0.434285\pi$
$858$	0	0
$859$	8.27441	0.282319	0.141160	−	0.989987i	$-0.454917\pi$
$859$	8.27441	0.282319	0.141160	+	0.989987i	$0.454917\pi$
$860$	0	0
$861$	3.81298	0.129946
$862$	0	0
$863$	−20.6687	−0.703569	−0.351785	−	0.936081i	$-0.614425\pi$
$863$	−20.6687	−0.703569	−0.351785	+	0.936081i	$0.614425\pi$
$864$	0	0
$865$	−34.5640	−1.17521
$866$	0	0
$867$	−30.4098	−1.03277
$868$	0	0
$869$	12.4482	0.422275
$870$	0	0
$871$	−53.8386	−1.82425
$872$	0	0
$873$	14.8285	0.501870
$874$	0	0
$875$	13.1755	0.445412
$876$	0	0
$877$	−52.8954	−1.78615	−0.893075	−	0.449908i	$-0.851457\pi$
$877$	−52.8954	−1.78615	−0.893075	+	0.449908i	$0.851457\pi$
$878$	0	0
$879$	−10.1271	−0.341578
$880$	0	0
$881$	48.1444	1.62203	0.811013	−	0.585029i	$-0.198917\pi$
$881$	48.1444	1.62203	0.811013	+	0.585029i	$0.198917\pi$
$882$	0	0
$883$	−24.7916	−0.834304	−0.417152	−	0.908837i	$-0.636972\pi$
$883$	−24.7916	−0.834304	−0.417152	+	0.908837i	$0.636972\pi$
$884$	0	0
$885$	−16.7014	−0.561412
$886$	0	0
$887$	−32.0672	−1.07671	−0.538355	−	0.842718i	$-0.680954\pi$
$887$	−32.0672	−1.07671	−0.538355	+	0.842718i	$0.680954\pi$
$888$	0	0
$889$	−15.1951	−0.509626
$890$	0	0
$891$	−3.05871	−0.102470
$892$	0	0
$893$	−78.5807	−2.62960
$894$	0	0
$895$	28.4186	0.949928
$896$	0	0
$897$	−3.46113	−0.115564
$898$	0	0
$899$	−7.35524	−0.245311
$900$	0	0
$901$	40.1157	1.33645
$902$	0	0
$903$	−8.45585	−0.281393
$904$	0	0
$905$	−6.29047	−0.209102
$906$	0	0
$907$	10.3373	0.343245	0.171623	−	0.985163i	$-0.445099\pi$
$907$	10.3373	0.343245	0.171623	+	0.985163i	$0.445099\pi$
$908$	0	0
$909$	15.0157	0.498040
$910$	0	0
$911$	28.4505	0.942608	0.471304	−	0.881971i	$-0.343783\pi$
$911$	28.4505	0.942608	0.471304	+	0.881971i	$0.343783\pi$
$912$	0	0
$913$	12.4904	0.413374
$914$	0	0
$915$	23.6021	0.780260
$916$	0	0
$917$	17.3397	0.572609
$918$	0	0
$919$	−36.6977	−1.21055	−0.605273	−	0.796018i	$-0.706936\pi$
$919$	−36.6977	−1.21055	−0.605273	+	0.796018i	$0.706936\pi$
$920$	0	0
$921$	9.08360	0.299315
$922$	0	0
$923$	−23.3073	−0.767169
$924$	0	0
$925$	−2.82193	−0.0927846
$926$	0	0
$927$	−3.48792	−0.114558
$928$	0	0
$929$	24.0358	0.788589	0.394295	−	0.918984i	$-0.370989\pi$
$929$	24.0358	0.788589	0.394295	+	0.918984i	$0.370989\pi$
$930$	0	0
$931$	34.8226	1.14127
$932$	0	0
$933$	−28.1435	−0.921378
$934$	0	0
$935$	−44.0598	−1.44091
$936$	0	0
$937$	40.6241	1.32713	0.663566	−	0.748118i	$-0.269042\pi$
$937$	40.6241	1.32713	0.663566	+	0.748118i	$0.269042\pi$
$938$	0	0
$939$	−2.15160	−0.0702148
$940$	0	0
$941$	−17.9401	−0.584832	−0.292416	−	0.956291i	$-0.594459\pi$
$941$	−17.9401	−0.584832	−0.292416	+	0.956291i	$0.594459\pi$
$942$	0	0
$943$	−1.73072	−0.0563601
$944$	0	0
$945$	2.34300	0.0762176
$946$	0	0
$947$	35.4771	1.15285	0.576426	−	0.817149i	$-0.304447\pi$
$947$	35.4771	1.15285	0.576426	+	0.817149i	$0.304447\pi$
$948$	0	0
$949$	−98.4923	−3.19719
$950$	0	0
$951$	17.9505	0.582084
$952$	0	0
$953$	−56.9687	−1.84540	−0.922699	−	0.385521i	$-0.874022\pi$
$953$	−56.9687	−1.84540	−0.922699	+	0.385521i	$0.874022\pi$
$954$	0	0
$955$	13.3898	0.433283
$956$	0	0
$957$	−13.8468	−0.447602
$958$	0	0
$959$	1.88830	0.0609764
$960$	0	0
$961$	−28.3602	−0.914845
$962$	0	0
$963$	−2.50472	−0.0807134
$964$	0	0
$965$	−22.7829	−0.733408
$966$	0	0
$967$	31.2070	1.00355	0.501775	−	0.864998i	$-0.332681\pi$
$967$	31.2070	1.00355	0.501775	+	0.864998i	$0.332681\pi$
$968$	0	0
$969$	41.7304	1.34057
$970$	0	0
$971$	54.2592	1.74126	0.870630	−	0.491938i	$-0.163711\pi$
$971$	54.2592	1.74126	0.870630	+	0.491938i	$0.163711\pi$
$972$	0	0
$973$	23.5526	0.755061
$974$	0	0
$975$	−4.24410	−0.135920
$976$	0	0
$977$	−3.73365	−0.119450	−0.0597251	−	0.998215i	$-0.519022\pi$
$977$	−3.73365	−0.119450	−0.0597251	+	0.998215i	$0.519022\pi$
$978$	0	0
$979$	−52.7971	−1.68740
$980$	0	0
$981$	15.5936	0.497864
$982$	0	0
$983$	48.4077	1.54397	0.771983	−	0.635643i	$-0.219265\pi$
$983$	48.4077	1.54397	0.771983	+	0.635643i	$0.219265\pi$
$984$	0	0
$985$	26.5505	0.845968
$986$	0	0
$987$	14.5210	0.462210
$988$	0	0
$989$	3.83814	0.122046
$990$	0	0
$991$	−32.3082	−1.02631	−0.513153	−	0.858297i	$-0.671523\pi$
$991$	−32.3082	−1.02631	−0.513153	+	0.858297i	$0.671523\pi$
$992$	0	0
$993$	0.828334	0.0262864
$994$	0	0
$995$	39.1968	1.24262
$996$	0	0
$997$	34.4280	1.09035	0.545173	−	0.838323i	$-0.316464\pi$
$997$	34.4280	1.09035	0.545173	+	0.838323i	$0.316464\pi$
$998$	0	0
$999$	−4.52706	−0.143230

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6036.2.a.i.1.19	✓	26

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6036.2.a.i.1.19	✓	26	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:	\( N \)	\(=\)	\( 6036 = 2^{2} \cdot 3 \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6036.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +2.09204 q^{5} -1.11995 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +2.09204 q^{5} -1.11995 q^{7} +1.00000 q^{9} -3.05871 q^{11} -6.80855 q^{13} -2.09204 q^{15} +6.88548 q^{17} -6.06064 q^{19} +1.11995 q^{21} -0.508351 q^{23} -0.623348 q^{25} -1.00000 q^{27} -4.52700 q^{29} +1.62475 q^{31} +3.05871 q^{33} -2.34300 q^{35} +4.52706 q^{37} +6.80855 q^{39} +3.40458 q^{41} -7.55017 q^{43} +2.09204 q^{45} +12.9657 q^{47} -5.74570 q^{49} -6.88548 q^{51} +5.82613 q^{53} -6.39895 q^{55} +6.06064 q^{57} +7.98329 q^{59} -11.2818 q^{61} -1.11995 q^{63} -14.2438 q^{65} +7.90749 q^{67} +0.508351 q^{69} +3.42324 q^{71} +14.4660 q^{73} +0.623348 q^{75} +3.42561 q^{77} -4.06975 q^{79} +1.00000 q^{81} -4.08357 q^{83} +14.4047 q^{85} +4.52700 q^{87} +17.2613 q^{89} +7.62527 q^{91} -1.62475 q^{93} -12.6791 q^{95} +14.8285 q^{97} -3.05871 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9}+O(q^{10}) \) 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9	\( 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9} - 11 q^{11} + 13 q^{13} - 6 q^{15} + 12 q^{17} - q^{19} - 5 q^{21} - 22 q^{23} + 48 q^{25} - 26 q^{27} + 6 q^{29} + 19 q^{31} + 11 q^{33} - 21 q^{35} + 20 q^{37} - 13 q^{39} + 25 q^{41} + 4 q^{43} + 6 q^{45} + 8 q^{47} + 67 q^{49} - 12 q^{51} - 5 q^{53} + 20 q^{55} + q^{57} - 18 q^{59} + 43 q^{61} + 5 q^{63} + 41 q^{65} + 5 q^{67} + 22 q^{69} - q^{71} + 22 q^{73} - 48 q^{75} + 23 q^{77} + 16 q^{79} + 26 q^{81} - 19 q^{83} + 29 q^{85} - 6 q^{87} + 49 q^{89} - 13 q^{91} - 19 q^{93} - 26 q^{95} + 25 q^{97} - 11 q^{99}+O(q^{100}) \) 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9 - 11 * q^11 + 13 * q^13 - 6 * q^15 + 12 * q^17 - q^19 - 5 * q^21 - 22 * q^23 + 48 * q^25 - 26 * q^27 + 6 * q^29 + 19 * q^31 + 11 * q^33 - 21 * q^35 + 20 * q^37 - 13 * q^39 + 25 * q^41 + 4 * q^43 + 6 * q^45 + 8 * q^47 + 67 * q^49 - 12 * q^51 - 5 * q^53 + 20 * q^55 + q^57 - 18 * q^59 + 43 * q^61 + 5 * q^63 + 41 * q^65 + 5 * q^67 + 22 * q^69 - q^71 + 22 * q^73 - 48 * q^75 + 23 * q^77 + 16 * q^79 + 26 * q^81 - 19 * q^83 + 29 * q^85 - 6 * q^87 + 49 * q^89 - 13 * q^91 - 19 * q^93 - 26 * q^95 + 25 * q^97 - 11 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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