LMFDB - Embedded newform 4008.2.a.m.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4008,2,Mod(1,4008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4008 = 2^{3} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4008.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:	$32.0040411301$
Analytic rank:	$0$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 2 x^{12} - 49 x^{11} + 99 x^{10} + 901 x^{9} - 1879 x^{8} - 7582 x^{7} + 16968 x^{6} + \cdots + 12144 $ x^13 - 2x^12 - 49x^11 + 99x^10 + 901x^9 - 1879x^8 - 7582x^7 + 16968x^6 + 26911x^5 - 72240x^4 - 14532x^3 + 112850x^2 - 72184x + 12144
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Root			$4.20017$ of defining polynomial
Character	$\chi$	$=$	4008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +4.20017 q^{5} -0.497670 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +4.20017 q^{5} -0.497670 q^{7} +1.00000 q^{9} -0.310660 q^{11} -0.223370 q^{13} +4.20017 q^{15} +3.33328 q^{17} -0.261095 q^{19} -0.497670 q^{21} -9.01998 q^{23} +12.6414 q^{25} +1.00000 q^{27} +3.87692 q^{29} +0.329778 q^{31} -0.310660 q^{33} -2.09030 q^{35} -0.490531 q^{37} -0.223370 q^{39} +9.61488 q^{41} +2.34275 q^{43} +4.20017 q^{45} +9.94428 q^{47} -6.75232 q^{49} +3.33328 q^{51} +2.93441 q^{53} -1.30482 q^{55} -0.261095 q^{57} +2.69784 q^{59} -1.23520 q^{61} -0.497670 q^{63} -0.938191 q^{65} +11.6982 q^{67} -9.01998 q^{69} -3.61858 q^{71} -4.33699 q^{73} +12.6414 q^{75} +0.154606 q^{77} -2.59750 q^{79} +1.00000 q^{81} +3.06581 q^{83} +14.0003 q^{85} +3.87692 q^{87} +14.4181 q^{89} +0.111164 q^{91} +0.329778 q^{93} -1.09664 q^{95} -0.603712 q^{97} -0.310660 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q + 13 q^{3} + 2 q^{5} + q^{7} + 13 q^{9}+O(q^{10}) $ 13 * q + 13 * q^3 + 2 * q^5 + q^7 + 13 * q^9	$ 13 q + 13 q^{3} + 2 q^{5} + q^{7} + 13 q^{9} + 11 q^{11} + 12 q^{13} + 2 q^{15} + 15 q^{17} + 14 q^{19} + q^{21} + 9 q^{23} + 37 q^{25} + 13 q^{27} - 3 q^{29} - 17 q^{31} + 11 q^{33} + 15 q^{35} + 16 q^{37} + 12 q^{39} + 12 q^{41} + 20 q^{43} + 2 q^{45} - 6 q^{47} + 26 q^{49} + 15 q^{51} - 12 q^{53} + 7 q^{55} + 14 q^{57} + 14 q^{59} + 24 q^{61} + q^{63} + 8 q^{65} + 3 q^{67} + 9 q^{69} + 17 q^{71} + 34 q^{73} + 37 q^{75} + 30 q^{77} + 10 q^{79} + 13 q^{81} + 44 q^{83} + 25 q^{85} - 3 q^{87} + 25 q^{89} + 29 q^{91} - 17 q^{93} - 15 q^{95} + 38 q^{97} + 11 q^{99}+O(q^{100}) $ 13 * q + 13 * q^3 + 2 * q^5 + q^7 + 13 * q^9 + 11 * q^11 + 12 * q^13 + 2 * q^15 + 15 * q^17 + 14 * q^19 + q^21 + 9 * q^23 + 37 * q^25 + 13 * q^27 - 3 * q^29 - 17 * q^31 + 11 * q^33 + 15 * q^35 + 16 * q^37 + 12 * q^39 + 12 * q^41 + 20 * q^43 + 2 * q^45 - 6 * q^47 + 26 * q^49 + 15 * q^51 - 12 * q^53 + 7 * q^55 + 14 * q^57 + 14 * q^59 + 24 * q^61 + q^63 + 8 * q^65 + 3 * q^67 + 9 * q^69 + 17 * q^71 + 34 * q^73 + 37 * q^75 + 30 * q^77 + 10 * q^79 + 13 * q^81 + 44 * q^83 + 25 * q^85 - 3 * q^87 + 25 * q^89 + 29 * q^91 - 17 * q^93 - 15 * q^95 + 38 * 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
$5$	4.20017	1.87837	0.939186	−	0.343409i	$-0.111582\pi$
$5$	4.20017	1.87837	0.939186	+	0.343409i	$0.111582\pi$
$6$	0	0
$7$	−0.497670	−0.188101	−0.0940507	−	0.995567i	$-0.529982\pi$
$7$	−0.497670	−0.188101	−0.0940507	+	0.995567i	$0.529982\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.310660	−0.0936674	−0.0468337	−	0.998903i	$-0.514913\pi$
$11$	−0.310660	−0.0936674	−0.0468337	+	0.998903i	$0.514913\pi$
$12$	0	0
$13$	−0.223370	−0.0619517	−0.0309758	−	0.999520i	$-0.509861\pi$
$13$	−0.223370	−0.0619517	−0.0309758	+	0.999520i	$0.509861\pi$
$14$	0	0
$15$	4.20017	1.08448
$16$	0	0
$17$	3.33328	0.808438	0.404219	−	0.914662i	$-0.367543\pi$
$17$	3.33328	0.808438	0.404219	+	0.914662i	$0.367543\pi$
$18$	0	0
$19$	−0.261095	−0.0598993	−0.0299496	−	0.999551i	$-0.509535\pi$
$19$	−0.261095	−0.0598993	−0.0299496	+	0.999551i	$0.509535\pi$
$20$	0	0
$21$	−0.497670	−0.108600
$22$	0	0
$23$	−9.01998	−1.88080	−0.940398	−	0.340075i	$-0.889548\pi$
$23$	−9.01998	−1.88080	−0.940398	+	0.340075i	$0.889548\pi$
$24$	0	0
$25$	12.6414	2.52828
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	3.87692	0.719927	0.359963	−	0.932966i	$-0.382789\pi$
$29$	3.87692	0.719927	0.359963	+	0.932966i	$0.382789\pi$
$30$	0	0
$31$	0.329778	0.0592299	0.0296150	−	0.999561i	$-0.490572\pi$
$31$	0.329778	0.0592299	0.0296150	+	0.999561i	$0.490572\pi$
$32$	0	0
$33$	−0.310660	−0.0540789
$34$	0	0
$35$	−2.09030	−0.353325
$36$	0	0
$37$	−0.490531	−0.0806428	−0.0403214	−	0.999187i	$-0.512838\pi$
$37$	−0.490531	−0.0806428	−0.0403214	+	0.999187i	$0.512838\pi$
$38$	0	0
$39$	−0.223370	−0.0357678
$40$	0	0
$41$	9.61488	1.50159	0.750796	−	0.660534i	$-0.229670\pi$
$41$	9.61488	1.50159	0.750796	+	0.660534i	$0.229670\pi$
$42$	0	0
$43$	2.34275	0.357267	0.178633	−	0.983916i	$-0.442832\pi$
$43$	2.34275	0.357267	0.178633	+	0.983916i	$0.442832\pi$
$44$	0	0
$45$	4.20017	0.626124
$46$	0	0
$47$	9.94428	1.45052	0.725261	−	0.688474i	$-0.241719\pi$
$47$	9.94428	1.45052	0.725261	+	0.688474i	$0.241719\pi$
$48$	0	0
$49$	−6.75232	−0.964618
$50$	0	0
$51$	3.33328	0.466752
$52$	0	0
$53$	2.93441	0.403073	0.201536	−	0.979481i	$-0.435407\pi$
$53$	2.93441	0.403073	0.201536	+	0.979481i	$0.435407\pi$
$54$	0	0
$55$	−1.30482	−0.175942
$56$	0	0
$57$	−0.261095	−0.0345829
$58$	0	0
$59$	2.69784	0.351228	0.175614	−	0.984459i	$-0.443809\pi$
$59$	2.69784	0.351228	0.175614	+	0.984459i	$0.443809\pi$
$60$	0	0
$61$	−1.23520	−0.158152	−0.0790759	−	0.996869i	$-0.525197\pi$
$61$	−1.23520	−0.158152	−0.0790759	+	0.996869i	$0.525197\pi$
$62$	0	0
$63$	−0.497670	−0.0627005
$64$	0	0
$65$	−0.938191	−0.116368
$66$	0	0
$67$	11.6982	1.42916	0.714582	−	0.699552i	$-0.246617\pi$
$67$	11.6982	1.42916	0.714582	+	0.699552i	$0.246617\pi$
$68$	0	0
$69$	−9.01998	−1.08588
$70$	0	0
$71$	−3.61858	−0.429446	−0.214723	−	0.976675i	$-0.568885\pi$
$71$	−3.61858	−0.429446	−0.214723	+	0.976675i	$0.568885\pi$
$72$	0	0
$73$	−4.33699	−0.507607	−0.253803	−	0.967256i	$-0.581682\pi$
$73$	−4.33699	−0.507607	−0.253803	+	0.967256i	$0.581682\pi$
$74$	0	0
$75$	12.6414	1.45970
$76$	0	0
$77$	0.154606	0.0176190
$78$	0	0
$79$	−2.59750	−0.292241	−0.146121	−	0.989267i	$-0.546679\pi$
$79$	−2.59750	−0.292241	−0.146121	+	0.989267i	$0.546679\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	3.06581	0.336517	0.168258	−	0.985743i	$-0.446186\pi$
$83$	3.06581	0.336517	0.168258	+	0.985743i	$0.446186\pi$
$84$	0	0
$85$	14.0003	1.51855
$86$	0	0
$87$	3.87692	0.415650
$88$	0	0
$89$	14.4181	1.52832	0.764158	−	0.645029i	$-0.223155\pi$
$89$	14.4181	1.52832	0.764158	+	0.645029i	$0.223155\pi$
$90$	0	0
$91$	0.111164	0.0116532
$92$	0	0
$93$	0.329778	0.0341964
$94$	0	0
$95$	−1.09664	−0.112513
$96$	0	0
$97$	−0.603712	−0.0612977	−0.0306489	−	0.999530i	$-0.509757\pi$
$97$	−0.603712	−0.0612977	−0.0306489	+	0.999530i	$0.509757\pi$
$98$	0	0
$99$	−0.310660	−0.0312225
$100$	0	0
$101$	−1.25046	−0.124425	−0.0622126	−	0.998063i	$-0.519816\pi$
$101$	−1.25046	−0.124425	−0.0622126	+	0.998063i	$0.519816\pi$
$102$	0	0
$103$	−15.1837	−1.49610	−0.748049	−	0.663643i	$-0.769009\pi$
$103$	−15.1837	−1.49610	−0.748049	+	0.663643i	$0.769009\pi$
$104$	0	0
$105$	−2.09030	−0.203992
$106$	0	0
$107$	−6.86387	−0.663556	−0.331778	−	0.943358i	$-0.607648\pi$
$107$	−6.86387	−0.663556	−0.331778	+	0.943358i	$0.607648\pi$
$108$	0	0
$109$	6.48328	0.620985	0.310493	−	0.950576i	$-0.399506\pi$
$109$	6.48328	0.620985	0.310493	+	0.950576i	$0.399506\pi$
$110$	0	0
$111$	−0.490531	−0.0465592
$112$	0	0
$113$	−3.94499	−0.371113	−0.185556	−	0.982634i	$-0.559409\pi$
$113$	−3.94499	−0.371113	−0.185556	+	0.982634i	$0.559409\pi$
$114$	0	0
$115$	−37.8854	−3.53284
$116$	0	0
$117$	−0.223370	−0.0206506
$118$	0	0
$119$	−1.65887	−0.152068
$120$	0	0
$121$	−10.9035	−0.991226
$122$	0	0
$123$	9.61488	0.866945
$124$	0	0
$125$	32.0952	2.87068
$126$	0	0
$127$	15.9104	1.41182	0.705911	−	0.708300i	$-0.250538\pi$
$127$	15.9104	1.41182	0.705911	+	0.708300i	$0.250538\pi$
$128$	0	0
$129$	2.34275	0.206268
$130$	0	0
$131$	5.83534	0.509836	0.254918	−	0.966963i	$-0.417952\pi$
$131$	5.83534	0.509836	0.254918	+	0.966963i	$0.417952\pi$
$132$	0	0
$133$	0.129939	0.0112671
$134$	0	0
$135$	4.20017	0.361493
$136$	0	0
$137$	18.0709	1.54390	0.771949	−	0.635684i	$-0.219282\pi$
$137$	18.0709	1.54390	0.771949	+	0.635684i	$0.219282\pi$
$138$	0	0
$139$	1.08854	0.0923285	0.0461642	−	0.998934i	$-0.485300\pi$
$139$	1.08854	0.0923285	0.0461642	+	0.998934i	$0.485300\pi$
$140$	0	0
$141$	9.94428	0.837459
$142$	0	0
$143$	0.0693920	0.00580285
$144$	0	0
$145$	16.2837	1.35229
$146$	0	0
$147$	−6.75232	−0.556922
$148$	0	0
$149$	−7.57620	−0.620667	−0.310333	−	0.950628i	$-0.600441\pi$
$149$	−7.57620	−0.620667	−0.310333	+	0.950628i	$0.600441\pi$
$150$	0	0
$151$	−6.06300	−0.493400	−0.246700	−	0.969092i	$-0.579346\pi$
$151$	−6.06300	−0.493400	−0.246700	+	0.969092i	$0.579346\pi$
$152$	0	0
$153$	3.33328	0.269479
$154$	0	0
$155$	1.38512	0.111256
$156$	0	0
$157$	−6.55015	−0.522759	−0.261379	−	0.965236i	$-0.584177\pi$
$157$	−6.55015	−0.522759	−0.261379	+	0.965236i	$0.584177\pi$
$158$	0	0
$159$	2.93441	0.232714
$160$	0	0
$161$	4.48897	0.353781
$162$	0	0
$163$	−4.61424	−0.361415	−0.180708	−	0.983537i	$-0.557839\pi$
$163$	−4.61424	−0.361415	−0.180708	+	0.983537i	$0.557839\pi$
$164$	0	0
$165$	−1.30482	−0.101580
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	−12.9501	−0.996162
$170$	0	0
$171$	−0.261095	−0.0199664
$172$	0	0
$173$	−18.3229	−1.39306	−0.696531	−	0.717526i	$-0.745274\pi$
$173$	−18.3229	−1.39306	−0.696531	+	0.717526i	$0.745274\pi$
$174$	0	0
$175$	−6.29125	−0.475574
$176$	0	0
$177$	2.69784	0.202782
$178$	0	0
$179$	−15.8576	−1.18525	−0.592625	−	0.805478i	$-0.701908\pi$
$179$	−15.8576	−1.18525	−0.592625	+	0.805478i	$0.701908\pi$
$180$	0	0
$181$	−6.26382	−0.465586	−0.232793	−	0.972526i	$-0.574786\pi$
$181$	−6.26382	−0.465586	−0.232793	+	0.972526i	$0.574786\pi$
$182$	0	0
$183$	−1.23520	−0.0913089
$184$	0	0
$185$	−2.06031	−0.151477
$186$	0	0
$187$	−1.03551	−0.0757243
$188$	0	0
$189$	−0.497670	−0.0362001
$190$	0	0
$191$	−11.8669	−0.858661	−0.429331	−	0.903147i	$-0.641250\pi$
$191$	−11.8669	−0.858661	−0.429331	+	0.903147i	$0.641250\pi$
$192$	0	0
$193$	−12.0978	−0.870818	−0.435409	−	0.900233i	$-0.643396\pi$
$193$	−12.0978	−0.870818	−0.435409	+	0.900233i	$0.643396\pi$
$194$	0	0
$195$	−0.938191	−0.0671853
$196$	0	0
$197$	−16.3800	−1.16703	−0.583513	−	0.812104i	$-0.698322\pi$
$197$	−16.3800	−1.16703	−0.583513	+	0.812104i	$0.698322\pi$
$198$	0	0
$199$	13.9684	0.990190	0.495095	−	0.868839i	$-0.335133\pi$
$199$	13.9684	0.990190	0.495095	+	0.868839i	$0.335133\pi$
$200$	0	0
$201$	11.6982	0.825128
$202$	0	0
$203$	−1.92943	−0.135419
$204$	0	0
$205$	40.3841	2.82055
$206$	0	0
$207$	−9.01998	−0.626932
$208$	0	0
$209$	0.0811117	0.00561061
$210$	0	0
$211$	−8.16435	−0.562057	−0.281028	−	0.959699i	$-0.590676\pi$
$211$	−8.16435	−0.562057	−0.281028	+	0.959699i	$0.590676\pi$
$212$	0	0
$213$	−3.61858	−0.247941
$214$	0	0
$215$	9.83996	0.671080
$216$	0	0
$217$	−0.164121	−0.0111412
$218$	0	0
$219$	−4.33699	−0.293067
$220$	0	0
$221$	−0.744554	−0.0500841
$222$	0	0
$223$	23.9773	1.60564	0.802819	−	0.596222i	$-0.203332\pi$
$223$	23.9773	1.60564	0.802819	+	0.596222i	$0.203332\pi$
$224$	0	0
$225$	12.6414	0.842761
$226$	0	0
$227$	−19.9052	−1.32115	−0.660576	−	0.750759i	$-0.729688\pi$
$227$	−19.9052	−1.32115	−0.660576	+	0.750759i	$0.729688\pi$
$228$	0	0
$229$	−4.53054	−0.299387	−0.149693	−	0.988732i	$-0.547829\pi$
$229$	−4.53054	−0.299387	−0.149693	+	0.988732i	$0.547829\pi$
$230$	0	0
$231$	0.154606	0.0101723
$232$	0	0
$233$	5.30087	0.347272	0.173636	−	0.984810i	$-0.444448\pi$
$233$	5.30087	0.347272	0.173636	+	0.984810i	$0.444448\pi$
$234$	0	0
$235$	41.7676	2.72462
$236$	0	0
$237$	−2.59750	−0.168725
$238$	0	0
$239$	7.02390	0.454338	0.227169	−	0.973855i	$-0.427053\pi$
$239$	7.02390	0.454338	0.227169	+	0.973855i	$0.427053\pi$
$240$	0	0
$241$	−6.34486	−0.408708	−0.204354	−	0.978897i	$-0.565509\pi$
$241$	−6.34486	−0.408708	−0.204354	+	0.978897i	$0.565509\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−28.3609	−1.81191
$246$	0	0
$247$	0.0583208	0.00371086
$248$	0	0
$249$	3.06581	0.194288
$250$	0	0
$251$	−25.3005	−1.59695	−0.798476	−	0.602026i	$-0.794360\pi$
$251$	−25.3005	−1.59695	−0.798476	+	0.602026i	$0.794360\pi$
$252$	0	0
$253$	2.80215	0.176169
$254$	0	0
$255$	14.0003	0.876734
$256$	0	0
$257$	−3.70489	−0.231105	−0.115552	−	0.993301i	$-0.536864\pi$
$257$	−3.70489	−0.231105	−0.115552	+	0.993301i	$0.536864\pi$
$258$	0	0
$259$	0.244122	0.0151690
$260$	0	0
$261$	3.87692	0.239976
$262$	0	0
$263$	−11.5558	−0.712563	−0.356282	−	0.934379i	$-0.615956\pi$
$263$	−11.5558	−0.712563	−0.356282	+	0.934379i	$0.615956\pi$
$264$	0	0
$265$	12.3250	0.757120
$266$	0	0
$267$	14.4181	0.882374
$268$	0	0
$269$	−27.8733	−1.69946	−0.849731	−	0.527216i	$-0.823236\pi$
$269$	−27.8733	−1.69946	−0.849731	+	0.527216i	$0.823236\pi$
$270$	0	0
$271$	−3.26815	−0.198526	−0.0992631	−	0.995061i	$-0.531649\pi$
$271$	−3.26815	−0.198526	−0.0992631	+	0.995061i	$0.531649\pi$
$272$	0	0
$273$	0.111164	0.00672798
$274$	0	0
$275$	−3.92718	−0.236818
$276$	0	0
$277$	−3.68912	−0.221658	−0.110829	−	0.993840i	$-0.535351\pi$
$277$	−3.68912	−0.221658	−0.110829	+	0.993840i	$0.535351\pi$
$278$	0	0
$279$	0.329778	0.0197433
$280$	0	0
$281$	−11.1453	−0.664874	−0.332437	−	0.943126i	$-0.607871\pi$
$281$	−11.1453	−0.664874	−0.332437	+	0.943126i	$0.607871\pi$
$282$	0	0
$283$	−25.0956	−1.49178	−0.745890	−	0.666070i	$-0.767975\pi$
$283$	−25.0956	−1.49178	−0.745890	+	0.666070i	$0.767975\pi$
$284$	0	0
$285$	−1.09664	−0.0649595
$286$	0	0
$287$	−4.78504	−0.282452
$288$	0	0
$289$	−5.88927	−0.346428
$290$	0	0
$291$	−0.603712	−0.0353902
$292$	0	0
$293$	27.4221	1.60201	0.801007	−	0.598655i	$-0.204298\pi$
$293$	27.4221	1.60201	0.801007	+	0.598655i	$0.204298\pi$
$294$	0	0
$295$	11.3314	0.659738
$296$	0	0
$297$	−0.310660	−0.0180263
$298$	0	0
$299$	2.01479	0.116518
$300$	0	0
$301$	−1.16592	−0.0672024
$302$	0	0
$303$	−1.25046	−0.0718370
$304$	0	0
$305$	−5.18807	−0.297068
$306$	0	0
$307$	30.7027	1.75229	0.876147	−	0.482044i	$-0.160106\pi$
$307$	30.7027	1.75229	0.876147	+	0.482044i	$0.160106\pi$
$308$	0	0
$309$	−15.1837	−0.863773
$310$	0	0
$311$	−1.33888	−0.0759212	−0.0379606	−	0.999279i	$-0.512086\pi$
$311$	−1.33888	−0.0759212	−0.0379606	+	0.999279i	$0.512086\pi$
$312$	0	0
$313$	10.5418	0.595856	0.297928	−	0.954588i	$-0.403704\pi$
$313$	10.5418	0.595856	0.297928	+	0.954588i	$0.403704\pi$
$314$	0	0
$315$	−2.09030	−0.117775
$316$	0	0
$317$	−4.08789	−0.229599	−0.114800	−	0.993389i	$-0.536623\pi$
$317$	−4.08789	−0.229599	−0.114800	+	0.993389i	$0.536623\pi$
$318$	0	0
$319$	−1.20440	−0.0674337
$320$	0	0
$321$	−6.86387	−0.383104
$322$	0	0
$323$	−0.870302	−0.0484249
$324$	0	0
$325$	−2.82371	−0.156631
$326$	0	0
$327$	6.48328	0.358526
$328$	0	0
$329$	−4.94897	−0.272845
$330$	0	0
$331$	18.5164	1.01776	0.508878	−	0.860839i	$-0.330061\pi$
$331$	18.5164	1.01776	0.508878	+	0.860839i	$0.330061\pi$
$332$	0	0
$333$	−0.490531	−0.0268809
$334$	0	0
$335$	49.1344	2.68450
$336$	0	0
$337$	18.2998	0.996851	0.498425	−	0.866933i	$-0.333912\pi$
$337$	18.2998	0.996851	0.498425	+	0.866933i	$0.333912\pi$
$338$	0	0
$339$	−3.94499	−0.214262
$340$	0	0
$341$	−0.102449	−0.00554791
$342$	0	0
$343$	6.84412	0.369548
$344$	0	0
$345$	−37.8854	−2.03968
$346$	0	0
$347$	−0.00397972	−0.000213642 0	−0.000106821	−	1.00000i	$-0.500034\pi$
$347$	−0.00397972	−0.000213642 0	−0.000106821		1.00000i	$0.500034\pi$
$348$	0	0
$349$	7.38445	0.395280	0.197640	−	0.980275i	$-0.436672\pi$
$349$	7.38445	0.395280	0.197640	+	0.980275i	$0.436672\pi$
$350$	0	0
$351$	−0.223370	−0.0119226
$352$	0	0
$353$	13.3841	0.712362	0.356181	−	0.934417i	$-0.384079\pi$
$353$	13.3841	0.712362	0.356181	+	0.934417i	$0.384079\pi$
$354$	0	0
$355$	−15.1986	−0.806659
$356$	0	0
$357$	−1.65887	−0.0877967
$358$	0	0
$359$	5.16101	0.272388	0.136194	−	0.990682i	$-0.456513\pi$
$359$	5.16101	0.272388	0.136194	+	0.990682i	$0.456513\pi$
$360$	0	0
$361$	−18.9318	−0.996412
$362$	0	0
$363$	−10.9035	−0.572285
$364$	0	0
$365$	−18.2161	−0.953474
$366$	0	0
$367$	2.97797	0.155449	0.0777244	−	0.996975i	$-0.475235\pi$
$367$	2.97797	0.155449	0.0777244	+	0.996975i	$0.475235\pi$
$368$	0	0
$369$	9.61488	0.500531
$370$	0	0
$371$	−1.46037	−0.0758185
$372$	0	0
$373$	34.4777	1.78519	0.892595	−	0.450860i	$-0.148883\pi$
$373$	34.4777	1.78519	0.892595	+	0.450860i	$0.148883\pi$
$374$	0	0
$375$	32.0952	1.65739
$376$	0	0
$377$	−0.865988	−0.0446007
$378$	0	0
$379$	−12.8223	−0.658639	−0.329320	−	0.944218i	$-0.606819\pi$
$379$	−12.8223	−0.658639	−0.329320	+	0.944218i	$0.606819\pi$
$380$	0	0
$381$	15.9104	0.815116
$382$	0	0
$383$	−17.6965	−0.904246	−0.452123	−	0.891955i	$-0.649333\pi$
$383$	−17.6965	−0.904246	−0.452123	+	0.891955i	$0.649333\pi$
$384$	0	0
$385$	0.649371	0.0330950
$386$	0	0
$387$	2.34275	0.119089
$388$	0	0
$389$	−1.00428	−0.0509189	−0.0254594	−	0.999676i	$-0.508105\pi$
$389$	−1.00428	−0.0509189	−0.0254594	+	0.999676i	$0.508105\pi$
$390$	0	0
$391$	−30.0661	−1.52051
$392$	0	0
$393$	5.83534	0.294354
$394$	0	0
$395$	−10.9099	−0.548937
$396$	0	0
$397$	3.25402	0.163315	0.0816573	−	0.996660i	$-0.473979\pi$
$397$	3.25402	0.163315	0.0816573	+	0.996660i	$0.473979\pi$
$398$	0	0
$399$	0.129939	0.00650509
$400$	0	0
$401$	7.93586	0.396298	0.198149	−	0.980172i	$-0.436507\pi$
$401$	7.93586	0.396298	0.198149	+	0.980172i	$0.436507\pi$
$402$	0	0
$403$	−0.0736626	−0.00366939
$404$	0	0
$405$	4.20017	0.208708
$406$	0	0
$407$	0.152388	0.00755360
$408$	0	0
$409$	16.2683	0.804417	0.402208	−	0.915548i	$-0.368243\pi$
$409$	16.2683	0.804417	0.402208	+	0.915548i	$0.368243\pi$
$410$	0	0
$411$	18.0709	0.891370
$412$	0	0
$413$	−1.34263	−0.0660666
$414$	0	0
$415$	12.8769	0.632104
$416$	0	0
$417$	1.08854	0.0533059
$418$	0	0
$419$	−10.0257	−0.489788	−0.244894	−	0.969550i	$-0.578753\pi$
$419$	−10.0257	−0.489788	−0.244894	+	0.969550i	$0.578753\pi$
$420$	0	0
$421$	33.3083	1.62335	0.811673	−	0.584112i	$-0.198557\pi$
$421$	33.3083	1.62335	0.811673	+	0.584112i	$0.198557\pi$
$422$	0	0
$423$	9.94428	0.483507
$424$	0	0
$425$	42.1373	2.04396
$426$	0	0
$427$	0.614724	0.0297486
$428$	0	0
$429$	0.0693920	0.00335028
$430$	0	0
$431$	29.2930	1.41099	0.705496	−	0.708714i	$-0.250724\pi$
$431$	29.2930	1.41099	0.705496	+	0.708714i	$0.250724\pi$
$432$	0	0
$433$	−29.8885	−1.43635	−0.718175	−	0.695862i	$-0.755022\pi$
$433$	−29.8885	−1.43635	−0.718175	+	0.695862i	$0.755022\pi$
$434$	0	0
$435$	16.2837	0.780745
$436$	0	0
$437$	2.35507	0.112658
$438$	0	0
$439$	−12.7837	−0.610131	−0.305065	−	0.952331i	$-0.598678\pi$
$439$	−12.7837	−0.610131	−0.305065	+	0.952331i	$0.598678\pi$
$440$	0	0
$441$	−6.75232	−0.321539
$442$	0	0
$443$	−1.03002	−0.0489375	−0.0244688	−	0.999701i	$-0.507789\pi$
$443$	−1.03002	−0.0489375	−0.0244688	+	0.999701i	$0.507789\pi$
$444$	0	0
$445$	60.5585	2.87075
$446$	0	0
$447$	−7.57620	−0.358342
$448$	0	0
$449$	13.9295	0.657374	0.328687	−	0.944439i	$-0.393394\pi$
$449$	13.9295	0.657374	0.328687	+	0.944439i	$0.393394\pi$
$450$	0	0
$451$	−2.98696	−0.140650
$452$	0	0
$453$	−6.06300	−0.284865
$454$	0	0
$455$	0.466909	0.0218890
$456$	0	0
$457$	9.59714	0.448935	0.224468	−	0.974482i	$-0.427936\pi$
$457$	9.59714	0.448935	0.224468	+	0.974482i	$0.427936\pi$
$458$	0	0
$459$	3.33328	0.155584
$460$	0	0
$461$	5.35477	0.249396	0.124698	−	0.992195i	$-0.460204\pi$
$461$	5.35477	0.249396	0.124698	+	0.992195i	$0.460204\pi$
$462$	0	0
$463$	17.9012	0.831941	0.415970	−	0.909378i	$-0.363442\pi$
$463$	17.9012	0.831941	0.415970	+	0.909378i	$0.363442\pi$
$464$	0	0
$465$	1.38512	0.0642336
$466$	0	0
$467$	−24.5496	−1.13602	−0.568010	−	0.823022i	$-0.692286\pi$
$467$	−24.5496	−1.13602	−0.568010	+	0.823022i	$0.692286\pi$
$468$	0	0
$469$	−5.82184	−0.268828
$470$	0	0
$471$	−6.55015	−0.301815
$472$	0	0
$473$	−0.727799	−0.0334642
$474$	0	0
$475$	−3.30061	−0.151442
$476$	0	0
$477$	2.93441	0.134358
$478$	0	0
$479$	−4.06290	−0.185639	−0.0928194	−	0.995683i	$-0.529588\pi$
$479$	−4.06290	−0.185639	−0.0928194	+	0.995683i	$0.529588\pi$
$480$	0	0
$481$	0.109570	0.00499596
$482$	0	0
$483$	4.48897	0.204255
$484$	0	0
$485$	−2.53569	−0.115140
$486$	0	0
$487$	−15.8338	−0.717496	−0.358748	−	0.933434i	$-0.616796\pi$
$487$	−15.8338	−0.717496	−0.358748	+	0.933434i	$0.616796\pi$
$488$	0	0
$489$	−4.61424	−0.208663
$490$	0	0
$491$	12.9595	0.584856	0.292428	−	0.956288i	$-0.405537\pi$
$491$	12.9595	0.584856	0.292428	+	0.956288i	$0.405537\pi$
$492$	0	0
$493$	12.9229	0.582016
$494$	0	0
$495$	−1.30482	−0.0586474
$496$	0	0
$497$	1.80086	0.0807794
$498$	0	0
$499$	12.9159	0.578194	0.289097	−	0.957300i	$-0.406645\pi$
$499$	12.9159	0.578194	0.289097	+	0.957300i	$0.406645\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	−0.878522	−0.0391713	−0.0195857	−	0.999808i	$-0.506235\pi$
$503$	−0.878522	−0.0391713	−0.0195857	+	0.999808i	$0.506235\pi$
$504$	0	0
$505$	−5.25214	−0.233717
$506$	0	0
$507$	−12.9501	−0.575134
$508$	0	0
$509$	−15.7414	−0.697727	−0.348864	−	0.937173i	$-0.613432\pi$
$509$	−15.7414	−0.697727	−0.348864	+	0.937173i	$0.613432\pi$
$510$	0	0
$511$	2.15839	0.0954815
$512$	0	0
$513$	−0.261095	−0.0115276
$514$	0	0
$515$	−63.7743	−2.81023
$516$	0	0
$517$	−3.08928	−0.135867
$518$	0	0
$519$	−18.3229	−0.804285
$520$	0	0
$521$	7.29423	0.319566	0.159783	−	0.987152i	$-0.448921\pi$
$521$	7.29423	0.319566	0.159783	+	0.987152i	$0.448921\pi$
$522$	0	0
$523$	−30.5868	−1.33747	−0.668734	−	0.743501i	$-0.733164\pi$
$523$	−30.5868	−1.33747	−0.668734	+	0.743501i	$0.733164\pi$
$524$	0	0
$525$	−6.29125	−0.274573
$526$	0	0
$527$	1.09924	0.0478837
$528$	0	0
$529$	58.3601	2.53740
$530$	0	0
$531$	2.69784	0.117076
$532$	0	0
$533$	−2.14768	−0.0930262
$534$	0	0
$535$	−28.8294	−1.24640
$536$	0	0
$537$	−15.8576	−0.684305
$538$	0	0
$539$	2.09767	0.0903532
$540$	0	0
$541$	−14.8053	−0.636531	−0.318265	−	0.948002i	$-0.603100\pi$
$541$	−14.8053	−0.636531	−0.318265	+	0.948002i	$0.603100\pi$
$542$	0	0
$543$	−6.26382	−0.268806
$544$	0	0
$545$	27.2308	1.16644
$546$	0	0
$547$	18.2584	0.780673	0.390337	−	0.920672i	$-0.372359\pi$
$547$	18.2584	0.780673	0.390337	+	0.920672i	$0.372359\pi$
$548$	0	0
$549$	−1.23520	−0.0527172
$550$	0	0
$551$	−1.01225	−0.0431231
$552$	0	0
$553$	1.29269	0.0549710
$554$	0	0
$555$	−2.06031	−0.0874554
$556$	0	0
$557$	−46.3035	−1.96194	−0.980972	−	0.194150i	$-0.937805\pi$
$557$	−46.3035	−1.96194	−0.980972	+	0.194150i	$0.937805\pi$
$558$	0	0
$559$	−0.523301	−0.0221333
$560$	0	0
$561$	−1.03551	−0.0437194
$562$	0	0
$563$	23.0152	0.969975	0.484987	−	0.874521i	$-0.338824\pi$
$563$	23.0152	0.969975	0.484987	+	0.874521i	$0.338824\pi$
$564$	0	0
$565$	−16.5696	−0.697088
$566$	0	0
$567$	−0.497670	−0.0209002
$568$	0	0
$569$	−16.9171	−0.709203	−0.354602	−	0.935017i	$-0.615384\pi$
$569$	−16.9171	−0.709203	−0.354602	+	0.935017i	$0.615384\pi$
$570$	0	0
$571$	19.9472	0.834762	0.417381	−	0.908732i	$-0.362948\pi$
$571$	19.9472	0.834762	0.417381	+	0.908732i	$0.362948\pi$
$572$	0	0
$573$	−11.8669	−0.495748
$574$	0	0
$575$	−114.025	−4.75518
$576$	0	0
$577$	37.0310	1.54162	0.770811	−	0.637064i	$-0.219851\pi$
$577$	37.0310	1.54162	0.770811	+	0.637064i	$0.219851\pi$
$578$	0	0
$579$	−12.0978	−0.502767
$580$	0	0
$581$	−1.52576	−0.0632993
$582$	0	0
$583$	−0.911603	−0.0377548
$584$	0	0
$585$	−0.938191	−0.0387894
$586$	0	0
$587$	10.8741	0.448821	0.224411	−	0.974495i	$-0.427954\pi$
$587$	10.8741	0.448821	0.224411	+	0.974495i	$0.427954\pi$
$588$	0	0
$589$	−0.0861035	−0.00354783
$590$	0	0
$591$	−16.3800	−0.673783
$592$	0	0
$593$	29.4695	1.21017	0.605084	−	0.796162i	$-0.293140\pi$
$593$	29.4695	1.21017	0.605084	+	0.796162i	$0.293140\pi$
$594$	0	0
$595$	−6.96754	−0.285641
$596$	0	0
$597$	13.9684	0.571687
$598$	0	0
$599$	−46.1804	−1.88688	−0.943441	−	0.331542i	$-0.892431\pi$
$599$	−46.1804	−1.88688	−0.943441	+	0.331542i	$0.892431\pi$
$600$	0	0
$601$	−13.9636	−0.569587	−0.284793	−	0.958589i	$-0.591925\pi$
$601$	−13.9636	−0.569587	−0.284793	+	0.958589i	$0.591925\pi$
$602$	0	0
$603$	11.6982	0.476388
$604$	0	0
$605$	−45.7965	−1.86189
$606$	0	0
$607$	−24.4257	−0.991410	−0.495705	−	0.868491i	$-0.665090\pi$
$607$	−24.4257	−0.991410	−0.495705	+	0.868491i	$0.665090\pi$
$608$	0	0
$609$	−1.92943	−0.0781844
$610$	0	0
$611$	−2.22125	−0.0898622
$612$	0	0
$613$	−27.1745	−1.09757	−0.548785	−	0.835964i	$-0.684909\pi$
$613$	−27.1745	−1.09757	−0.548785	+	0.835964i	$0.684909\pi$
$614$	0	0
$615$	40.3841	1.62845
$616$	0	0
$617$	−26.1446	−1.05254	−0.526271	−	0.850317i	$-0.676410\pi$
$617$	−26.1446	−1.05254	−0.526271	+	0.850317i	$0.676410\pi$
$618$	0	0
$619$	−24.1844	−0.972052	−0.486026	−	0.873944i	$-0.661554\pi$
$619$	−24.1844	−0.972052	−0.486026	+	0.873944i	$0.661554\pi$
$620$	0	0
$621$	−9.01998	−0.361960
$622$	0	0
$623$	−7.17545	−0.287479
$624$	0	0
$625$	71.5982	2.86393
$626$	0	0
$627$	0.0811117	0.00323929
$628$	0	0
$629$	−1.63508	−0.0651947
$630$	0	0
$631$	−13.0814	−0.520764	−0.260382	−	0.965506i	$-0.583848\pi$
$631$	−13.0814	−0.520764	−0.260382	+	0.965506i	$0.583848\pi$
$632$	0	0
$633$	−8.16435	−0.324504
$634$	0	0
$635$	66.8265	2.65193
$636$	0	0
$637$	1.50827	0.0597597
$638$	0	0
$639$	−3.61858	−0.143149
$640$	0	0
$641$	−35.8151	−1.41461	−0.707306	−	0.706908i	$-0.750090\pi$
$641$	−35.8151	−1.41461	−0.707306	+	0.706908i	$0.750090\pi$
$642$	0	0
$643$	16.5618	0.653133	0.326567	−	0.945174i	$-0.394108\pi$
$643$	16.5618	0.653133	0.326567	+	0.945174i	$0.394108\pi$
$644$	0	0
$645$	9.83996	0.387448
$646$	0	0
$647$	−37.4892	−1.47385	−0.736927	−	0.675972i	$-0.763724\pi$
$647$	−37.4892	−1.47385	−0.736927	+	0.675972i	$0.763724\pi$
$648$	0	0
$649$	−0.838109	−0.0328987
$650$	0	0
$651$	−0.164121	−0.00643240
$652$	0	0
$653$	5.82610	0.227993	0.113996	−	0.993481i	$-0.463635\pi$
$653$	5.82610	0.227993	0.113996	+	0.993481i	$0.463635\pi$
$654$	0	0
$655$	24.5094	0.957661
$656$	0	0
$657$	−4.33699	−0.169202
$658$	0	0
$659$	−8.71072	−0.339321	−0.169661	−	0.985503i	$-0.554267\pi$
$659$	−8.71072	−0.339321	−0.169661	+	0.985503i	$0.554267\pi$
$660$	0	0
$661$	3.84867	0.149696	0.0748480	−	0.997195i	$-0.476153\pi$
$661$	3.84867	0.149696	0.0748480	+	0.997195i	$0.476153\pi$
$662$	0	0
$663$	−0.744554	−0.0289161
$664$	0	0
$665$	0.545766	0.0211639
$666$	0	0
$667$	−34.9698	−1.35404
$668$	0	0
$669$	23.9773	0.927016
$670$	0	0
$671$	0.383728	0.0148137
$672$	0	0
$673$	37.8497	1.45900	0.729499	−	0.683982i	$-0.239753\pi$
$673$	37.8497	1.45900	0.729499	+	0.683982i	$0.239753\pi$
$674$	0	0
$675$	12.6414	0.486568
$676$	0	0
$677$	−31.2208	−1.19991	−0.599955	−	0.800033i	$-0.704815\pi$
$677$	−31.2208	−1.19991	−0.599955	+	0.800033i	$0.704815\pi$
$678$	0	0
$679$	0.300449	0.0115302
$680$	0	0
$681$	−19.9052	−0.762768
$682$	0	0
$683$	0.186306	0.00712880	0.00356440	−	0.999994i	$-0.498865\pi$
$683$	0.186306	0.00712880	0.00356440	+	0.999994i	$0.498865\pi$
$684$	0	0
$685$	75.9007	2.90002
$686$	0	0
$687$	−4.53054	−0.172851
$688$	0	0
$689$	−0.655459	−0.0249710
$690$	0	0
$691$	8.85083	0.336702	0.168351	−	0.985727i	$-0.446156\pi$
$691$	8.85083	0.336702	0.168351	+	0.985727i	$0.446156\pi$
$692$	0	0
$693$	0.154606	0.00587299
$694$	0	0
$695$	4.57204	0.173427
$696$	0	0
$697$	32.0491	1.21394
$698$	0	0
$699$	5.30087	0.200497
$700$	0	0
$701$	−27.7316	−1.04741	−0.523704	−	0.851900i	$-0.675450\pi$
$701$	−27.7316	−1.04741	−0.523704	+	0.851900i	$0.675450\pi$
$702$	0	0
$703$	0.128075	0.00483045
$704$	0	0
$705$	41.7676	1.57306
$706$	0	0
$707$	0.622315	0.0234046
$708$	0	0
$709$	−43.2925	−1.62588	−0.812941	−	0.582345i	$-0.802135\pi$
$709$	−43.2925	−1.62588	−0.812941	+	0.582345i	$0.802135\pi$
$710$	0	0
$711$	−2.59750	−0.0974137
$712$	0	0
$713$	−2.97460	−0.111399
$714$	0	0
$715$	0.291458	0.0108999
$716$	0	0
$717$	7.02390	0.262312
$718$	0	0
$719$	3.05404	0.113896	0.0569482	−	0.998377i	$-0.481863\pi$
$719$	3.05404	0.113896	0.0569482	+	0.998377i	$0.481863\pi$
$720$	0	0
$721$	7.55649	0.281418
$722$	0	0
$723$	−6.34486	−0.235968
$724$	0	0
$725$	49.0098	1.82018
$726$	0	0
$727$	−26.7202	−0.990998	−0.495499	−	0.868608i	$-0.665015\pi$
$727$	−26.7202	−0.990998	−0.495499	+	0.868608i	$0.665015\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	7.80905	0.288828
$732$	0	0
$733$	−41.1754	−1.52085	−0.760424	−	0.649427i	$-0.775009\pi$
$733$	−41.1754	−1.52085	−0.760424	+	0.649427i	$0.775009\pi$
$734$	0	0
$735$	−28.3609	−1.04611
$736$	0	0
$737$	−3.63416	−0.133866
$738$	0	0
$739$	20.4446	0.752067	0.376033	−	0.926606i	$-0.377288\pi$
$739$	20.4446	0.752067	0.376033	+	0.926606i	$0.377288\pi$
$740$	0	0
$741$	0.0583208	0.00214247
$742$	0	0
$743$	33.6474	1.23440	0.617202	−	0.786805i	$-0.288266\pi$
$743$	33.6474	1.23440	0.617202	+	0.786805i	$0.288266\pi$
$744$	0	0
$745$	−31.8213	−1.16584
$746$	0	0
$747$	3.06581	0.112172
$748$	0	0
$749$	3.41594	0.124816
$750$	0	0
$751$	−17.6170	−0.642853	−0.321426	−	0.946935i	$-0.604162\pi$
$751$	−17.6170	−0.642853	−0.321426	+	0.946935i	$0.604162\pi$
$752$	0	0
$753$	−25.3005	−0.922001
$754$	0	0
$755$	−25.4656	−0.926789
$756$	0	0
$757$	−33.9209	−1.23288	−0.616438	−	0.787404i	$-0.711425\pi$
$757$	−33.9209	−1.23288	−0.616438	+	0.787404i	$0.711425\pi$
$758$	0	0
$759$	2.80215	0.101711
$760$	0	0
$761$	−17.8074	−0.645519	−0.322760	−	0.946481i	$-0.604611\pi$
$761$	−17.8074	−0.645519	−0.322760	+	0.946481i	$0.604611\pi$
$762$	0	0
$763$	−3.22653	−0.116808
$764$	0	0
$765$	14.0003	0.506183
$766$	0	0
$767$	−0.602616	−0.0217592
$768$	0	0
$769$	38.5232	1.38918	0.694592	−	0.719404i	$-0.255585\pi$
$769$	38.5232	1.38918	0.694592	+	0.719404i	$0.255585\pi$
$770$	0	0
$771$	−3.70489	−0.133428
$772$	0	0
$773$	−40.8426	−1.46901	−0.734504	−	0.678605i	$-0.762585\pi$
$773$	−40.8426	−1.46901	−0.734504	+	0.678605i	$0.762585\pi$
$774$	0	0
$775$	4.16886	0.149750
$776$	0	0
$777$	0.244122	0.00875785
$778$	0	0
$779$	−2.51040	−0.0899443
$780$	0	0
$781$	1.12415	0.0402251
$782$	0	0
$783$	3.87692	0.138550
$784$	0	0
$785$	−27.5117	−0.981935
$786$	0	0
$787$	−8.76921	−0.312589	−0.156294	−	0.987711i	$-0.549955\pi$
$787$	−8.76921	−0.312589	−0.156294	+	0.987711i	$0.549955\pi$
$788$	0	0
$789$	−11.5558	−0.411399
$790$	0	0
$791$	1.96330	0.0698069
$792$	0	0
$793$	0.275907	0.00979776
$794$	0	0
$795$	12.3250	0.437124
$796$	0	0
$797$	7.40598	0.262333	0.131167	−	0.991360i	$-0.458128\pi$
$797$	7.40598	0.262333	0.131167	+	0.991360i	$0.458128\pi$
$798$	0	0
$799$	33.1470	1.17266
$800$	0	0
$801$	14.4181	0.509439
$802$	0	0
$803$	1.34733	0.0475462
$804$	0	0
$805$	18.8544	0.664532
$806$	0	0
$807$	−27.8733	−0.981185
$808$	0	0
$809$	31.6141	1.11149	0.555746	−	0.831352i	$-0.312433\pi$
$809$	31.6141	1.11149	0.555746	+	0.831352i	$0.312433\pi$
$810$	0	0
$811$	−46.7943	−1.64317	−0.821585	−	0.570086i	$-0.806910\pi$
$811$	−46.7943	−1.64317	−0.821585	+	0.570086i	$0.806910\pi$
$812$	0	0
$813$	−3.26815	−0.114619
$814$	0	0
$815$	−19.3806	−0.678872
$816$	0	0
$817$	−0.611681	−0.0214000
$818$	0	0
$819$	0.111164	0.00388440
$820$	0	0
$821$	−44.5791	−1.55582	−0.777911	−	0.628374i	$-0.783721\pi$
$821$	−44.5791	−1.55582	−0.777911	+	0.628374i	$0.783721\pi$
$822$	0	0
$823$	27.2960	0.951479	0.475739	−	0.879586i	$-0.342181\pi$
$823$	27.2960	0.951479	0.475739	+	0.879586i	$0.342181\pi$
$824$	0	0
$825$	−3.92718	−0.136727
$826$	0	0
$827$	37.4769	1.30320	0.651600	−	0.758562i	$-0.274098\pi$
$827$	37.4769	1.30320	0.651600	+	0.758562i	$0.274098\pi$
$828$	0	0
$829$	18.1688	0.631027	0.315514	−	0.948921i	$-0.397823\pi$
$829$	18.1688	0.631027	0.315514	+	0.948921i	$0.397823\pi$
$830$	0	0
$831$	−3.68912	−0.127974
$832$	0	0
$833$	−22.5074	−0.779834
$834$	0	0
$835$	−4.20017	−0.145353
$836$	0	0
$837$	0.329778	0.0113988
$838$	0	0
$839$	−46.6415	−1.61024	−0.805122	−	0.593109i	$-0.797900\pi$
$839$	−46.6415	−1.61024	−0.805122	+	0.593109i	$0.797900\pi$
$840$	0	0
$841$	−13.9695	−0.481705
$842$	0	0
$843$	−11.1453	−0.383865
$844$	0	0
$845$	−54.3926	−1.87116
$846$	0	0
$847$	5.42634	0.186451
$848$	0	0
$849$	−25.0956	−0.861279
$850$	0	0
$851$	4.42458	0.151673
$852$	0	0
$853$	24.9860	0.855504	0.427752	−	0.903896i	$-0.359306\pi$
$853$	24.9860	0.855504	0.427752	+	0.903896i	$0.359306\pi$
$854$	0	0
$855$	−1.09664	−0.0375044
$856$	0	0
$857$	−15.9695	−0.545508	−0.272754	−	0.962084i	$-0.587935\pi$
$857$	−15.9695	−0.545508	−0.272754	+	0.962084i	$0.587935\pi$
$858$	0	0
$859$	32.7854	1.11862	0.559312	−	0.828957i	$-0.311065\pi$
$859$	32.7854	1.11862	0.559312	+	0.828957i	$0.311065\pi$
$860$	0	0
$861$	−4.78504	−0.163074
$862$	0	0
$863$	−13.0578	−0.444492	−0.222246	−	0.974991i	$-0.571339\pi$
$863$	−13.0578	−0.444492	−0.222246	+	0.974991i	$0.571339\pi$
$864$	0	0
$865$	−76.9592	−2.61669
$866$	0	0
$867$	−5.88927	−0.200010
$868$	0	0
$869$	0.806937	0.0273735
$870$	0	0
$871$	−2.61303	−0.0885390
$872$	0	0
$873$	−0.603712	−0.0204326
$874$	0	0
$875$	−15.9728	−0.539980
$876$	0	0
$877$	10.8855	0.367578	0.183789	−	0.982966i	$-0.441164\pi$
$877$	10.8855	0.367578	0.183789	+	0.982966i	$0.441164\pi$
$878$	0	0
$879$	27.4221	0.924923
$880$	0	0
$881$	8.00765	0.269785	0.134892	−	0.990860i	$-0.456931\pi$
$881$	8.00765	0.269785	0.134892	+	0.990860i	$0.456931\pi$
$882$	0	0
$883$	8.54999	0.287730	0.143865	−	0.989597i	$-0.454047\pi$
$883$	8.54999	0.287730	0.143865	+	0.989597i	$0.454047\pi$
$884$	0	0
$885$	11.3314	0.380900
$886$	0	0
$887$	−5.88800	−0.197700	−0.0988499	−	0.995102i	$-0.531516\pi$
$887$	−5.88800	−0.197700	−0.0988499	+	0.995102i	$0.531516\pi$
$888$	0	0
$889$	−7.91814	−0.265566
$890$	0	0
$891$	−0.310660	−0.0104075
$892$	0	0
$893$	−2.59640	−0.0868852
$894$	0	0
$895$	−66.6045	−2.22634
$896$	0	0
$897$	2.01479	0.0672720
$898$	0	0
$899$	1.27853	0.0426412
$900$	0	0
$901$	9.78121	0.325859
$902$	0	0
$903$	−1.16592	−0.0387993
$904$	0	0
$905$	−26.3091	−0.874544
$906$	0	0
$907$	−28.9433	−0.961048	−0.480524	−	0.876982i	$-0.659553\pi$
$907$	−28.9433	−0.961048	−0.480524	+	0.876982i	$0.659553\pi$
$908$	0	0
$909$	−1.25046	−0.0414751
$910$	0	0
$911$	−31.7063	−1.05048	−0.525239	−	0.850955i	$-0.676024\pi$
$911$	−31.7063	−1.05048	−0.525239	+	0.850955i	$0.676024\pi$
$912$	0	0
$913$	−0.952424	−0.0315206
$914$	0	0
$915$	−5.18807	−0.171512
$916$	0	0
$917$	−2.90407	−0.0959009
$918$	0	0
$919$	47.1347	1.55483	0.777415	−	0.628989i	$-0.216531\pi$
$919$	47.1347	1.55483	0.777415	+	0.628989i	$0.216531\pi$
$920$	0	0
$921$	30.7027	1.01169
$922$	0	0
$923$	0.808281	0.0266049
$924$	0	0
$925$	−6.20100	−0.203888
$926$	0	0
$927$	−15.1837	−0.498700
$928$	0	0
$929$	26.7815	0.878674	0.439337	−	0.898322i	$-0.355213\pi$
$929$	26.7815	0.878674	0.439337	+	0.898322i	$0.355213\pi$
$930$	0	0
$931$	1.76300	0.0577799
$932$	0	0
$933$	−1.33888	−0.0438331
$934$	0	0
$935$	−4.34933	−0.142238
$936$	0	0
$937$	−9.24100	−0.301890	−0.150945	−	0.988542i	$-0.548232\pi$
$937$	−9.24100	−0.301890	−0.150945	+	0.988542i	$0.548232\pi$
$938$	0	0
$939$	10.5418	0.344018
$940$	0	0
$941$	38.3552	1.25034	0.625172	−	0.780487i	$-0.285029\pi$
$941$	38.3552	1.25034	0.625172	+	0.780487i	$0.285029\pi$
$942$	0	0
$943$	−86.7261	−2.82419
$944$	0	0
$945$	−2.09030	−0.0679973
$946$	0	0
$947$	−42.5887	−1.38395	−0.691973	−	0.721923i	$-0.743258\pi$
$947$	−42.5887	−1.38395	−0.691973	+	0.721923i	$0.743258\pi$
$948$	0	0
$949$	0.968753	0.0314471
$950$	0	0
$951$	−4.08789	−0.132559
$952$	0	0
$953$	−44.5963	−1.44462	−0.722308	−	0.691572i	$-0.756919\pi$
$953$	−44.5963	−1.44462	−0.722308	+	0.691572i	$0.756919\pi$
$954$	0	0
$955$	−49.8431	−1.61289
$956$	0	0
$957$	−1.20440	−0.0389328
$958$	0	0
$959$	−8.99332	−0.290410
$960$	0	0
$961$	−30.8912	−0.996492
$962$	0	0
$963$	−6.86387	−0.221185
$964$	0	0
$965$	−50.8128	−1.63572
$966$	0	0
$967$	−23.0774	−0.742118	−0.371059	−	0.928609i	$-0.621005\pi$
$967$	−23.0774	−0.742118	−0.371059	+	0.928609i	$0.621005\pi$
$968$	0	0
$969$	−0.870302	−0.0279581
$970$	0	0
$971$	45.8648	1.47187	0.735936	−	0.677051i	$-0.236743\pi$
$971$	45.8648	1.47187	0.735936	+	0.677051i	$0.236743\pi$
$972$	0	0
$973$	−0.541732	−0.0173671
$974$	0	0
$975$	−2.82371	−0.0904311
$976$	0	0
$977$	47.0962	1.50674	0.753370	−	0.657597i	$-0.228427\pi$
$977$	47.0962	1.50674	0.753370	+	0.657597i	$0.228427\pi$
$978$	0	0
$979$	−4.47912	−0.143153
$980$	0	0
$981$	6.48328	0.206995
$982$	0	0
$983$	−46.6561	−1.48810	−0.744049	−	0.668125i	$-0.767097\pi$
$983$	−46.6561	−1.48810	−0.744049	+	0.668125i	$0.767097\pi$
$984$	0	0
$985$	−68.7987	−2.19211
$986$	0	0
$987$	−4.94897	−0.157527
$988$	0	0
$989$	−21.1316	−0.671946
$990$	0	0
$991$	−59.1922	−1.88030	−0.940150	−	0.340760i	$-0.889316\pi$
$991$	−59.1922	−1.88030	−0.940150	+	0.340760i	$0.889316\pi$
$992$	0	0
$993$	18.5164	0.587601
$994$	0	0
$995$	58.6694	1.85995
$996$	0	0
$997$	58.9614	1.86733	0.933664	−	0.358150i	$-0.116593\pi$
$997$	58.9614	1.86733	0.933664	+	0.358150i	$0.116593\pi$
$998$	0	0
$999$	−0.490531	−0.0155197

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4008.2.a.m.1.13	✓	13
4.3	odd	2		8016.2.a.bg.1.13		13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.m.1.13	✓	13	1.1	even	1	trivial
8016.2.a.bg.1.13		13	4.3	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 4008 = 2^{3} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4008.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +4.20017 q^{5} -0.497670 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +4.20017 q^{5} -0.497670 q^{7} +1.00000 q^{9} -0.310660 q^{11} -0.223370 q^{13} +4.20017 q^{15} +3.33328 q^{17} -0.261095 q^{19} -0.497670 q^{21} -9.01998 q^{23} +12.6414 q^{25} +1.00000 q^{27} +3.87692 q^{29} +0.329778 q^{31} -0.310660 q^{33} -2.09030 q^{35} -0.490531 q^{37} -0.223370 q^{39} +9.61488 q^{41} +2.34275 q^{43} +4.20017 q^{45} +9.94428 q^{47} -6.75232 q^{49} +3.33328 q^{51} +2.93441 q^{53} -1.30482 q^{55} -0.261095 q^{57} +2.69784 q^{59} -1.23520 q^{61} -0.497670 q^{63} -0.938191 q^{65} +11.6982 q^{67} -9.01998 q^{69} -3.61858 q^{71} -4.33699 q^{73} +12.6414 q^{75} +0.154606 q^{77} -2.59750 q^{79} +1.00000 q^{81} +3.06581 q^{83} +14.0003 q^{85} +3.87692 q^{87} +14.4181 q^{89} +0.111164 q^{91} +0.329778 q^{93} -1.09664 q^{95} -0.603712 q^{97} -0.310660 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q + 13 q^{3} + 2 q^{5} + q^{7} + 13 q^{9}+O(q^{10}) \) 13 * q + 13 * q^3 + 2 * q^5 + q^7 + 13 * q^9	\( 13 q + 13 q^{3} + 2 q^{5} + q^{7} + 13 q^{9} + 11 q^{11} + 12 q^{13} + 2 q^{15} + 15 q^{17} + 14 q^{19} + q^{21} + 9 q^{23} + 37 q^{25} + 13 q^{27} - 3 q^{29} - 17 q^{31} + 11 q^{33} + 15 q^{35} + 16 q^{37} + 12 q^{39} + 12 q^{41} + 20 q^{43} + 2 q^{45} - 6 q^{47} + 26 q^{49} + 15 q^{51} - 12 q^{53} + 7 q^{55} + 14 q^{57} + 14 q^{59} + 24 q^{61} + q^{63} + 8 q^{65} + 3 q^{67} + 9 q^{69} + 17 q^{71} + 34 q^{73} + 37 q^{75} + 30 q^{77} + 10 q^{79} + 13 q^{81} + 44 q^{83} + 25 q^{85} - 3 q^{87} + 25 q^{89} + 29 q^{91} - 17 q^{93} - 15 q^{95} + 38 q^{97} + 11 q^{99}+O(q^{100}) \) 13 * q + 13 * q^3 + 2 * q^5 + q^7 + 13 * q^9 + 11 * q^11 + 12 * q^13 + 2 * q^15 + 15 * q^17 + 14 * q^19 + q^21 + 9 * q^23 + 37 * q^25 + 13 * q^27 - 3 * q^29 - 17 * q^31 + 11 * q^33 + 15 * q^35 + 16 * q^37 + 12 * q^39 + 12 * q^41 + 20 * q^43 + 2 * q^45 - 6 * q^47 + 26 * q^49 + 15 * q^51 - 12 * q^53 + 7 * q^55 + 14 * q^57 + 14 * q^59 + 24 * q^61 + q^63 + 8 * q^65 + 3 * q^67 + 9 * q^69 + 17 * q^71 + 34 * q^73 + 37 * q^75 + 30 * q^77 + 10 * q^79 + 13 * q^81 + 44 * q^83 + 25 * q^85 - 3 * q^87 + 25 * q^89 + 29 * q^91 - 17 * q^93 - 15 * q^95 + 38 * q^97 + 11 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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