LMFDB - Embedded newform 4004.2.a.j.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4004,2,Mod(1,4004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4004, 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("4004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4004.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:	$31.9721009693$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 18x^{8} + 32x^{7} + 93x^{6} - 119x^{5} - 174x^{4} + 107x^{3} + 51x^{2} - 17x - 2 $ x^10 - 2x^9 - 18x^8 + 32x^7 + 93x^6 - 119x^5 - 174x^4 + 107x^3 + 51x^2 - 17*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.78432$ of defining polynomial
Character	$\chi$	$=$	4004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.78432 q^{3} +2.77311 q^{5} -1.00000 q^{7} +4.75245 q^{9} +O(q^{10})$	$q-2.78432 q^{3} +2.77311 q^{5} -1.00000 q^{7} +4.75245 q^{9} -1.00000 q^{11} -1.00000 q^{13} -7.72124 q^{15} -2.84666 q^{17} -6.75777 q^{19} +2.78432 q^{21} +0.282040 q^{23} +2.69016 q^{25} -4.87938 q^{27} +8.65828 q^{29} -2.61339 q^{31} +2.78432 q^{33} -2.77311 q^{35} -2.19155 q^{37} +2.78432 q^{39} +9.09314 q^{41} +6.50862 q^{43} +13.1791 q^{45} +0.120836 q^{47} +1.00000 q^{49} +7.92601 q^{51} +5.22807 q^{53} -2.77311 q^{55} +18.8158 q^{57} +4.04489 q^{59} -9.87513 q^{61} -4.75245 q^{63} -2.77311 q^{65} -2.78400 q^{67} -0.785291 q^{69} -9.02475 q^{71} -9.94348 q^{73} -7.49028 q^{75} +1.00000 q^{77} +10.1171 q^{79} -0.671584 q^{81} -8.81845 q^{83} -7.89411 q^{85} -24.1074 q^{87} +17.7949 q^{89} +1.00000 q^{91} +7.27652 q^{93} -18.7401 q^{95} -8.81902 q^{97} -4.75245 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 2 q^{3} - 2 q^{5} - 10 q^{7} + 10 q^{9}+O(q^{10}) $ 10 * q - 2 * q^3 - 2 * q^5 - 10 * q^7 + 10 * q^9	$ 10 q - 2 q^{3} - 2 q^{5} - 10 q^{7} + 10 q^{9} - 10 q^{11} - 10 q^{13} - 4 q^{15} + 9 q^{17} + 3 q^{19} + 2 q^{21} - 4 q^{23} + 22 q^{25} - 8 q^{27} + 13 q^{29} - 17 q^{31} + 2 q^{33} + 2 q^{35} + 11 q^{37} + 2 q^{39} + 8 q^{41} + 21 q^{43} - 15 q^{45} + q^{47} + 10 q^{49} + 19 q^{51} + 22 q^{53} + 2 q^{55} + 8 q^{57} - 24 q^{59} + 16 q^{61} - 10 q^{63} + 2 q^{65} + 19 q^{67} + 51 q^{69} - 25 q^{71} + 22 q^{73} + 28 q^{75} + 10 q^{77} + 41 q^{79} + 54 q^{81} + 8 q^{83} + 9 q^{85} - 5 q^{87} + 36 q^{89} + 10 q^{91} + 12 q^{93} + 13 q^{95} - 5 q^{97} - 10 q^{99}+O(q^{100}) $ 10 * q - 2 * q^3 - 2 * q^5 - 10 * q^7 + 10 * q^9 - 10 * q^11 - 10 * q^13 - 4 * q^15 + 9 * q^17 + 3 * q^19 + 2 * q^21 - 4 * q^23 + 22 * q^25 - 8 * q^27 + 13 * q^29 - 17 * q^31 + 2 * q^33 + 2 * q^35 + 11 * q^37 + 2 * q^39 + 8 * q^41 + 21 * q^43 - 15 * q^45 + q^47 + 10 * q^49 + 19 * q^51 + 22 * q^53 + 2 * q^55 + 8 * q^57 - 24 * q^59 + 16 * q^61 - 10 * q^63 + 2 * q^65 + 19 * q^67 + 51 * q^69 - 25 * q^71 + 22 * q^73 + 28 * q^75 + 10 * q^77 + 41 * q^79 + 54 * q^81 + 8 * q^83 + 9 * q^85 - 5 * q^87 + 36 * q^89 + 10 * q^91 + 12 * q^93 + 13 * q^95 - 5 * q^97 - 10 * 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
$3$	−2.78432	−1.60753	−0.803764	−	0.594948i	$-0.797173\pi$
$3$	−2.78432	−1.60753	−0.803764	+	0.594948i	$0.797173\pi$
$4$	0	0
$5$	2.77311	1.24017	0.620087	−	0.784533i	$-0.287097\pi$
$5$	2.77311	1.24017	0.620087	+	0.784533i	$0.287097\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	4.75245	1.58415
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−7.72124	−1.99362
$16$	0	0
$17$	−2.84666	−0.690416	−0.345208	−	0.938526i	$-0.612192\pi$
$17$	−2.84666	−0.690416	−0.345208	+	0.938526i	$0.612192\pi$
$18$	0	0
$19$	−6.75777	−1.55034	−0.775169	−	0.631754i	$-0.782335\pi$
$19$	−6.75777	−1.55034	−0.775169	+	0.631754i	$0.782335\pi$
$20$	0	0
$21$	2.78432	0.607589
$22$	0	0
$23$	0.282040	0.0588095	0.0294047	−	0.999568i	$-0.490639\pi$
$23$	0.282040	0.0588095	0.0294047	+	0.999568i	$0.490639\pi$
$24$	0	0
$25$	2.69016	0.538033
$26$	0	0
$27$	−4.87938	−0.939037
$28$	0	0
$29$	8.65828	1.60780	0.803901	−	0.594763i	$-0.202754\pi$
$29$	8.65828	1.60780	0.803901	+	0.594763i	$0.202754\pi$
$30$	0	0
$31$	−2.61339	−0.469379	−0.234689	−	0.972070i	$-0.575407\pi$
$31$	−2.61339	−0.469379	−0.234689	+	0.972070i	$0.575407\pi$
$32$	0	0
$33$	2.78432	0.484688
$34$	0	0
$35$	−2.77311	−0.468742
$36$	0	0
$37$	−2.19155	−0.360288	−0.180144	−	0.983640i	$-0.557656\pi$
$37$	−2.19155	−0.360288	−0.180144	+	0.983640i	$0.557656\pi$
$38$	0	0
$39$	2.78432	0.445848
$40$	0	0
$41$	9.09314	1.42011	0.710055	−	0.704146i	$-0.248670\pi$
$41$	9.09314	1.42011	0.710055	+	0.704146i	$0.248670\pi$
$42$	0	0
$43$	6.50862	0.992556	0.496278	−	0.868164i	$-0.334700\pi$
$43$	6.50862	0.992556	0.496278	+	0.868164i	$0.334700\pi$
$44$	0	0
$45$	13.1791	1.96462
$46$	0	0
$47$	0.120836	0.0176257	0.00881286	−	0.999961i	$-0.497195\pi$
$47$	0.120836	0.0176257	0.00881286	+	0.999961i	$0.497195\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	7.92601	1.10986
$52$	0	0
$53$	5.22807	0.718131	0.359066	−	0.933312i	$-0.383095\pi$
$53$	5.22807	0.718131	0.359066	+	0.933312i	$0.383095\pi$
$54$	0	0
$55$	−2.77311	−0.373927
$56$	0	0
$57$	18.8158	2.49221
$58$	0	0
$59$	4.04489	0.526600	0.263300	−	0.964714i	$-0.415189\pi$
$59$	4.04489	0.526600	0.263300	+	0.964714i	$0.415189\pi$
$60$	0	0
$61$	−9.87513	−1.26438	−0.632190	−	0.774813i	$-0.717844\pi$
$61$	−9.87513	−1.26438	−0.632190	+	0.774813i	$0.717844\pi$
$62$	0	0
$63$	−4.75245	−0.598752
$64$	0	0
$65$	−2.77311	−0.343962
$66$	0	0
$67$	−2.78400	−0.340119	−0.170060	−	0.985434i	$-0.554396\pi$
$67$	−2.78400	−0.340119	−0.170060	+	0.985434i	$0.554396\pi$
$68$	0	0
$69$	−0.785291	−0.0945380
$70$	0	0
$71$	−9.02475	−1.07104	−0.535520	−	0.844522i	$-0.679884\pi$
$71$	−9.02475	−1.07104	−0.535520	+	0.844522i	$0.679884\pi$
$72$	0	0
$73$	−9.94348	−1.16380	−0.581898	−	0.813262i	$-0.697690\pi$
$73$	−9.94348	−1.16380	−0.581898	+	0.813262i	$0.697690\pi$
$74$	0	0
$75$	−7.49028	−0.864903
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	10.1171	1.13827	0.569134	−	0.822245i	$-0.307279\pi$
$79$	10.1171	1.13827	0.569134	+	0.822245i	$0.307279\pi$
$80$	0	0
$81$	−0.671584	−0.0746204
$82$	0	0
$83$	−8.81845	−0.967951	−0.483975	−	0.875082i	$-0.660808\pi$
$83$	−8.81845	−0.967951	−0.483975	+	0.875082i	$0.660808\pi$
$84$	0	0
$85$	−7.89411	−0.856236
$86$	0	0
$87$	−24.1074	−2.58459
$88$	0	0
$89$	17.7949	1.88626	0.943128	−	0.332431i	$-0.107869\pi$
$89$	17.7949	1.88626	0.943128	+	0.332431i	$0.107869\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	7.27652	0.754540
$94$	0	0
$95$	−18.7401	−1.92269
$96$	0	0
$97$	−8.81902	−0.895436	−0.447718	−	0.894175i	$-0.647763\pi$
$97$	−8.81902	−0.895436	−0.447718	+	0.894175i	$0.647763\pi$
$98$	0	0
$99$	−4.75245	−0.477639
$100$	0	0
$101$	15.5773	1.55000	0.775000	−	0.631961i	$-0.217750\pi$
$101$	15.5773	1.55000	0.775000	+	0.631961i	$0.217750\pi$
$102$	0	0
$103$	7.45574	0.734636	0.367318	−	0.930095i	$-0.380276\pi$
$103$	7.45574	0.734636	0.367318	+	0.930095i	$0.380276\pi$
$104$	0	0
$105$	7.72124	0.753516
$106$	0	0
$107$	0.367658	0.0355428	0.0177714	−	0.999842i	$-0.494343\pi$
$107$	0.367658	0.0355428	0.0177714	+	0.999842i	$0.494343\pi$
$108$	0	0
$109$	−8.31826	−0.796745	−0.398372	−	0.917224i	$-0.630425\pi$
$109$	−8.31826	−0.796745	−0.398372	+	0.917224i	$0.630425\pi$
$110$	0	0
$111$	6.10198	0.579174
$112$	0	0
$113$	5.40717	0.508664	0.254332	−	0.967117i	$-0.418144\pi$
$113$	5.40717	0.508664	0.254332	+	0.967117i	$0.418144\pi$
$114$	0	0
$115$	0.782130	0.0729340
$116$	0	0
$117$	−4.75245	−0.439364
$118$	0	0
$119$	2.84666	0.260953
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−25.3182	−2.28287
$124$	0	0
$125$	−6.40544	−0.572920
$126$	0	0
$127$	18.7598	1.66466	0.832330	−	0.554280i	$-0.187006\pi$
$127$	18.7598	1.66466	0.832330	+	0.554280i	$0.187006\pi$
$128$	0	0
$129$	−18.1221	−1.59556
$130$	0	0
$131$	8.13395	0.710666	0.355333	−	0.934740i	$-0.384367\pi$
$131$	8.13395	0.710666	0.355333	+	0.934740i	$0.384367\pi$
$132$	0	0
$133$	6.75777	0.585973
$134$	0	0
$135$	−13.5311	−1.16457
$136$	0	0
$137$	−12.8557	−1.09834	−0.549169	−	0.835711i	$-0.685056\pi$
$137$	−12.8557	−1.09834	−0.549169	+	0.835711i	$0.685056\pi$
$138$	0	0
$139$	22.5614	1.91364	0.956819	−	0.290686i	$-0.0938834\pi$
$139$	22.5614	1.91364	0.956819	+	0.290686i	$0.0938834\pi$
$140$	0	0
$141$	−0.336446	−0.0283338
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	24.0104	1.99396
$146$	0	0
$147$	−2.78432	−0.229647
$148$	0	0
$149$	16.9874	1.39166	0.695831	−	0.718205i	$-0.255036\pi$
$149$	16.9874	1.39166	0.695831	+	0.718205i	$0.255036\pi$
$150$	0	0
$151$	−15.3459	−1.24883	−0.624415	−	0.781093i	$-0.714663\pi$
$151$	−15.3459	−1.24883	−0.624415	+	0.781093i	$0.714663\pi$
$152$	0	0
$153$	−13.5286	−1.09372
$154$	0	0
$155$	−7.24723	−0.582111
$156$	0	0
$157$	5.03408	0.401763	0.200882	−	0.979616i	$-0.435619\pi$
$157$	5.03408	0.401763	0.200882	+	0.979616i	$0.435619\pi$
$158$	0	0
$159$	−14.5566	−1.15442
$160$	0	0
$161$	−0.282040	−0.0222279
$162$	0	0
$163$	−3.26771	−0.255947	−0.127973	−	0.991778i	$-0.540847\pi$
$163$	−3.26771	−0.255947	−0.127973	+	0.991778i	$0.540847\pi$
$164$	0	0
$165$	7.72124	0.601098
$166$	0	0
$167$	16.9303	1.31011	0.655054	−	0.755582i	$-0.272646\pi$
$167$	16.9303	1.31011	0.655054	+	0.755582i	$0.272646\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−32.1159	−2.45597
$172$	0	0
$173$	−17.0625	−1.29724	−0.648618	−	0.761114i	$-0.724653\pi$
$173$	−17.0625	−1.29724	−0.648618	+	0.761114i	$0.724653\pi$
$174$	0	0
$175$	−2.69016	−0.203357
$176$	0	0
$177$	−11.2623	−0.846524
$178$	0	0
$179$	−7.73835	−0.578391	−0.289196	−	0.957270i	$-0.593388\pi$
$179$	−7.73835	−0.578391	−0.289196	+	0.957270i	$0.593388\pi$
$180$	0	0
$181$	2.18521	0.162425	0.0812125	−	0.996697i	$-0.474121\pi$
$181$	2.18521	0.162425	0.0812125	+	0.996697i	$0.474121\pi$
$182$	0	0
$183$	27.4955	2.03253
$184$	0	0
$185$	−6.07742	−0.446820
$186$	0	0
$187$	2.84666	0.208168
$188$	0	0
$189$	4.87938	0.354923
$190$	0	0
$191$	24.1931	1.75055	0.875276	−	0.483623i	$-0.160680\pi$
$191$	24.1931	1.75055	0.875276	+	0.483623i	$0.160680\pi$
$192$	0	0
$193$	6.37682	0.459013	0.229507	−	0.973307i	$-0.426289\pi$
$193$	6.37682	0.459013	0.229507	+	0.973307i	$0.426289\pi$
$194$	0	0
$195$	7.72124	0.552930
$196$	0	0
$197$	23.1068	1.64629	0.823146	−	0.567830i	$-0.192217\pi$
$197$	23.1068	1.64629	0.823146	+	0.567830i	$0.192217\pi$
$198$	0	0
$199$	−10.8710	−0.770626	−0.385313	−	0.922786i	$-0.625907\pi$
$199$	−10.8710	−0.770626	−0.385313	+	0.922786i	$0.625907\pi$
$200$	0	0
$201$	7.75154	0.546752
$202$	0	0
$203$	−8.65828	−0.607692
$204$	0	0
$205$	25.2163	1.76118
$206$	0	0
$207$	1.34038	0.0931630
$208$	0	0
$209$	6.75777	0.467445
$210$	0	0
$211$	17.5059	1.20515	0.602577	−	0.798061i	$-0.294141\pi$
$211$	17.5059	1.20515	0.602577	+	0.798061i	$0.294141\pi$
$212$	0	0
$213$	25.1278	1.72173
$214$	0	0
$215$	18.0492	1.23094
$216$	0	0
$217$	2.61339	0.177408
$218$	0	0
$219$	27.6858	1.87084
$220$	0	0
$221$	2.84666	0.191487
$222$	0	0
$223$	−21.8649	−1.46418	−0.732092	−	0.681206i	$-0.761456\pi$
$223$	−21.8649	−1.46418	−0.732092	+	0.681206i	$0.761456\pi$
$224$	0	0
$225$	12.7849	0.852324
$226$	0	0
$227$	4.85493	0.322233	0.161117	−	0.986935i	$-0.448490\pi$
$227$	4.85493	0.322233	0.161117	+	0.986935i	$0.448490\pi$
$228$	0	0
$229$	2.87378	0.189905	0.0949525	−	0.995482i	$-0.469730\pi$
$229$	2.87378	0.189905	0.0949525	+	0.995482i	$0.469730\pi$
$230$	0	0
$231$	−2.78432	−0.183195
$232$	0	0
$233$	0.953404	0.0624596	0.0312298	−	0.999512i	$-0.490058\pi$
$233$	0.953404	0.0624596	0.0312298	+	0.999512i	$0.490058\pi$
$234$	0	0
$235$	0.335091	0.0218590
$236$	0	0
$237$	−28.1694	−1.82980
$238$	0	0
$239$	2.47685	0.160214	0.0801071	−	0.996786i	$-0.474474\pi$
$239$	2.47685	0.160214	0.0801071	+	0.996786i	$0.474474\pi$
$240$	0	0
$241$	−16.2269	−1.04527	−0.522634	−	0.852557i	$-0.675050\pi$
$241$	−16.2269	−1.04527	−0.522634	+	0.852557i	$0.675050\pi$
$242$	0	0
$243$	16.5080	1.05899
$244$	0	0
$245$	2.77311	0.177168
$246$	0	0
$247$	6.75777	0.429986
$248$	0	0
$249$	24.5534	1.55601
$250$	0	0
$251$	−3.84609	−0.242763	−0.121381	−	0.992606i	$-0.538732\pi$
$251$	−3.84609	−0.242763	−0.121381	+	0.992606i	$0.538732\pi$
$252$	0	0
$253$	−0.282040	−0.0177317
$254$	0	0
$255$	21.9797	1.37642
$256$	0	0
$257$	27.2096	1.69729	0.848645	−	0.528963i	$-0.177419\pi$
$257$	27.2096	1.69729	0.848645	+	0.528963i	$0.177419\pi$
$258$	0	0
$259$	2.19155	0.136176
$260$	0	0
$261$	41.1480	2.54700
$262$	0	0
$263$	16.2641	1.00289	0.501445	−	0.865190i	$-0.332802\pi$
$263$	16.2641	1.00289	0.501445	+	0.865190i	$0.332802\pi$
$264$	0	0
$265$	14.4980	0.890608
$266$	0	0
$267$	−49.5467	−3.03221
$268$	0	0
$269$	21.0343	1.28249	0.641243	−	0.767338i	$-0.278419\pi$
$269$	21.0343	1.28249	0.641243	+	0.767338i	$0.278419\pi$
$270$	0	0
$271$	−14.6237	−0.888325	−0.444163	−	0.895946i	$-0.646499\pi$
$271$	−14.6237	−0.888325	−0.444163	+	0.895946i	$0.646499\pi$
$272$	0	0
$273$	−2.78432	−0.168515
$274$	0	0
$275$	−2.69016	−0.162223
$276$	0	0
$277$	26.4258	1.58777	0.793885	−	0.608067i	$-0.208055\pi$
$277$	26.4258	1.58777	0.793885	+	0.608067i	$0.208055\pi$
$278$	0	0
$279$	−12.4200	−0.743566
$280$	0	0
$281$	5.10347	0.304448	0.152224	−	0.988346i	$-0.451357\pi$
$281$	5.10347	0.304448	0.152224	+	0.988346i	$0.451357\pi$
$282$	0	0
$283$	12.2991	0.731104	0.365552	−	0.930791i	$-0.380880\pi$
$283$	12.2991	0.731104	0.365552	+	0.930791i	$0.380880\pi$
$284$	0	0
$285$	52.1784	3.09078
$286$	0	0
$287$	−9.09314	−0.536751
$288$	0	0
$289$	−8.89653	−0.523326
$290$	0	0
$291$	24.5550	1.43944
$292$	0	0
$293$	11.6569	0.681001	0.340500	−	0.940244i	$-0.389404\pi$
$293$	11.6569	0.681001	0.340500	+	0.940244i	$0.389404\pi$
$294$	0	0
$295$	11.2169	0.653076
$296$	0	0
$297$	4.87938	0.283130
$298$	0	0
$299$	−0.282040	−0.0163108
$300$	0	0
$301$	−6.50862	−0.375151
$302$	0	0
$303$	−43.3722	−2.49167
$304$	0	0
$305$	−27.3849	−1.56805
$306$	0	0
$307$	7.07652	0.403878	0.201939	−	0.979398i	$-0.435276\pi$
$307$	7.07652	0.403878	0.201939	+	0.979398i	$0.435276\pi$
$308$	0	0
$309$	−20.7592	−1.18095
$310$	0	0
$311$	−3.96382	−0.224768	−0.112384	−	0.993665i	$-0.535849\pi$
$311$	−3.96382	−0.224768	−0.112384	+	0.993665i	$0.535849\pi$
$312$	0	0
$313$	13.3806	0.756316	0.378158	−	0.925741i	$-0.376558\pi$
$313$	13.3806	0.756316	0.378158	+	0.925741i	$0.376558\pi$
$314$	0	0
$315$	−13.1791	−0.742557
$316$	0	0
$317$	−31.2650	−1.75602	−0.878008	−	0.478646i	$-0.841128\pi$
$317$	−31.2650	−1.75602	−0.878008	+	0.478646i	$0.841128\pi$
$318$	0	0
$319$	−8.65828	−0.484771
$320$	0	0
$321$	−1.02368	−0.0571361
$322$	0	0
$323$	19.2371	1.07038
$324$	0	0
$325$	−2.69016	−0.149223
$326$	0	0
$327$	23.1607	1.28079
$328$	0	0
$329$	−0.120836	−0.00666189
$330$	0	0
$331$	8.97026	0.493050	0.246525	−	0.969136i	$-0.420711\pi$
$331$	8.97026	0.493050	0.246525	+	0.969136i	$0.420711\pi$
$332$	0	0
$333$	−10.4152	−0.570751
$334$	0	0
$335$	−7.72034	−0.421807
$336$	0	0
$337$	21.0738	1.14796	0.573981	−	0.818869i	$-0.305398\pi$
$337$	21.0738	1.14796	0.573981	+	0.818869i	$0.305398\pi$
$338$	0	0
$339$	−15.0553	−0.817692
$340$	0	0
$341$	2.61339	0.141523
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−2.17770	−0.117244
$346$	0	0
$347$	6.15858	0.330610	0.165305	−	0.986243i	$-0.447139\pi$
$347$	6.15858	0.330610	0.165305	+	0.986243i	$0.447139\pi$
$348$	0	0
$349$	−10.1993	−0.545956	−0.272978	−	0.962020i	$-0.588009\pi$
$349$	−10.1993	−0.545956	−0.272978	+	0.962020i	$0.588009\pi$
$350$	0	0
$351$	4.87938	0.260442
$352$	0	0
$353$	−15.5664	−0.828518	−0.414259	−	0.910159i	$-0.635959\pi$
$353$	−15.5664	−0.828518	−0.414259	+	0.910159i	$0.635959\pi$
$354$	0	0
$355$	−25.0267	−1.32828
$356$	0	0
$357$	−7.92601	−0.419489
$358$	0	0
$359$	17.2948	0.912784	0.456392	−	0.889779i	$-0.349142\pi$
$359$	17.2948	0.912784	0.456392	+	0.889779i	$0.349142\pi$
$360$	0	0
$361$	26.6674	1.40355
$362$	0	0
$363$	−2.78432	−0.146139
$364$	0	0
$365$	−27.5744	−1.44331
$366$	0	0
$367$	−27.7144	−1.44668	−0.723339	−	0.690493i	$-0.757394\pi$
$367$	−27.7144	−1.44668	−0.723339	+	0.690493i	$0.757394\pi$
$368$	0	0
$369$	43.2147	2.24967
$370$	0	0
$371$	−5.22807	−0.271428
$372$	0	0
$373$	1.19168	0.0617030	0.0308515	−	0.999524i	$-0.490178\pi$
$373$	1.19168	0.0617030	0.0308515	+	0.999524i	$0.490178\pi$
$374$	0	0
$375$	17.8348	0.920986
$376$	0	0
$377$	−8.65828	−0.445924
$378$	0	0
$379$	13.2714	0.681706	0.340853	−	0.940117i	$-0.389284\pi$
$379$	13.2714	0.681706	0.340853	+	0.940117i	$0.389284\pi$
$380$	0	0
$381$	−52.2333	−2.67599
$382$	0	0
$383$	−3.88242	−0.198383	−0.0991913	−	0.995068i	$-0.531626\pi$
$383$	−3.88242	−0.198383	−0.0991913	+	0.995068i	$0.531626\pi$
$384$	0	0
$385$	2.77311	0.141331
$386$	0	0
$387$	30.9319	1.57236
$388$	0	0
$389$	18.0201	0.913657	0.456829	−	0.889555i	$-0.348985\pi$
$389$	18.0201	0.913657	0.456829	+	0.889555i	$0.348985\pi$
$390$	0	0
$391$	−0.802873	−0.0406030
$392$	0	0
$393$	−22.6475	−1.14242
$394$	0	0
$395$	28.0560	1.41165
$396$	0	0
$397$	36.6325	1.83853	0.919266	−	0.393637i	$-0.128783\pi$
$397$	36.6325	1.83853	0.919266	+	0.393637i	$0.128783\pi$
$398$	0	0
$399$	−18.8158	−0.941968
$400$	0	0
$401$	24.6288	1.22990	0.614952	−	0.788564i	$-0.289175\pi$
$401$	24.6288	1.22990	0.614952	+	0.788564i	$0.289175\pi$
$402$	0	0
$403$	2.61339	0.130182
$404$	0	0
$405$	−1.86238	−0.0925423
$406$	0	0
$407$	2.19155	0.108631
$408$	0	0
$409$	−15.2688	−0.754992	−0.377496	−	0.926011i	$-0.623215\pi$
$409$	−15.2688	−0.754992	−0.377496	+	0.926011i	$0.623215\pi$
$410$	0	0
$411$	35.7945	1.76561
$412$	0	0
$413$	−4.04489	−0.199036
$414$	0	0
$415$	−24.4546	−1.20043
$416$	0	0
$417$	−62.8183	−3.07623
$418$	0	0
$419$	−21.5160	−1.05113	−0.525564	−	0.850754i	$-0.676146\pi$
$419$	−21.5160	−1.05113	−0.525564	+	0.850754i	$0.676146\pi$
$420$	0	0
$421$	26.8404	1.30812	0.654059	−	0.756443i	$-0.273065\pi$
$421$	26.8404	1.30812	0.654059	+	0.756443i	$0.273065\pi$
$422$	0	0
$423$	0.574266	0.0279218
$424$	0	0
$425$	−7.65798	−0.371466
$426$	0	0
$427$	9.87513	0.477891
$428$	0	0
$429$	−2.78432	−0.134428
$430$	0	0
$431$	37.9193	1.82651	0.913254	−	0.407390i	$-0.133561\pi$
$431$	37.9193	1.82651	0.913254	+	0.407390i	$0.133561\pi$
$432$	0	0
$433$	−30.1766	−1.45020	−0.725098	−	0.688646i	$-0.758206\pi$
$433$	−30.1766	−1.45020	−0.725098	+	0.688646i	$0.758206\pi$
$434$	0	0
$435$	−66.8527	−3.20534
$436$	0	0
$437$	−1.90596	−0.0911746
$438$	0	0
$439$	13.0983	0.625149	0.312575	−	0.949893i	$-0.398809\pi$
$439$	13.0983	0.625149	0.312575	+	0.949893i	$0.398809\pi$
$440$	0	0
$441$	4.75245	0.226307
$442$	0	0
$443$	−14.1509	−0.672329	−0.336164	−	0.941803i	$-0.609130\pi$
$443$	−14.1509	−0.672329	−0.336164	+	0.941803i	$0.609130\pi$
$444$	0	0
$445$	49.3473	2.33929
$446$	0	0
$447$	−47.2984	−2.23714
$448$	0	0
$449$	1.37721	0.0649946	0.0324973	−	0.999472i	$-0.489654\pi$
$449$	1.37721	0.0649946	0.0324973	+	0.999472i	$0.489654\pi$
$450$	0	0
$451$	−9.09314	−0.428179
$452$	0	0
$453$	42.7279	2.00753
$454$	0	0
$455$	2.77311	0.130006
$456$	0	0
$457$	−3.44269	−0.161042	−0.0805212	−	0.996753i	$-0.525658\pi$
$457$	−3.44269	−0.161042	−0.0805212	+	0.996753i	$0.525658\pi$
$458$	0	0
$459$	13.8899	0.648326
$460$	0	0
$461$	−10.5969	−0.493547	−0.246774	−	0.969073i	$-0.579370\pi$
$461$	−10.5969	−0.493547	−0.246774	+	0.969073i	$0.579370\pi$
$462$	0	0
$463$	17.4442	0.810699	0.405349	−	0.914162i	$-0.367150\pi$
$463$	17.4442	0.810699	0.405349	+	0.914162i	$0.367150\pi$
$464$	0	0
$465$	20.1786	0.935761
$466$	0	0
$467$	−3.55369	−0.164445	−0.0822226	−	0.996614i	$-0.526202\pi$
$467$	−3.55369	−0.164445	−0.0822226	+	0.996614i	$0.526202\pi$
$468$	0	0
$469$	2.78400	0.128553
$470$	0	0
$471$	−14.0165	−0.645846
$472$	0	0
$473$	−6.50862	−0.299267
$474$	0	0
$475$	−18.1795	−0.834133
$476$	0	0
$477$	24.8462	1.13763
$478$	0	0
$479$	12.3749	0.565422	0.282711	−	0.959205i	$-0.408766\pi$
$479$	12.3749	0.565422	0.282711	+	0.959205i	$0.408766\pi$
$480$	0	0
$481$	2.19155	0.0999260
$482$	0	0
$483$	0.785291	0.0357320
$484$	0	0
$485$	−24.4562	−1.11050
$486$	0	0
$487$	−33.8091	−1.53204	−0.766018	−	0.642819i	$-0.777765\pi$
$487$	−33.8091	−1.53204	−0.766018	+	0.642819i	$0.777765\pi$
$488$	0	0
$489$	9.09836	0.411442
$490$	0	0
$491$	39.7317	1.79307	0.896533	−	0.442977i	$-0.146078\pi$
$491$	39.7317	1.79307	0.896533	+	0.442977i	$0.146078\pi$
$492$	0	0
$493$	−24.6472	−1.11005
$494$	0	0
$495$	−13.1791	−0.592356
$496$	0	0
$497$	9.02475	0.404815
$498$	0	0
$499$	−13.2331	−0.592393	−0.296196	−	0.955127i	$-0.595718\pi$
$499$	−13.2331	−0.592393	−0.296196	+	0.955127i	$0.595718\pi$
$500$	0	0
$501$	−47.1395	−2.10604
$502$	0	0
$503$	−9.47397	−0.422423	−0.211212	−	0.977440i	$-0.567741\pi$
$503$	−9.47397	−0.422423	−0.211212	+	0.977440i	$0.567741\pi$
$504$	0	0
$505$	43.1977	1.92227
$506$	0	0
$507$	−2.78432	−0.123656
$508$	0	0
$509$	5.62864	0.249485	0.124743	−	0.992189i	$-0.460190\pi$
$509$	5.62864	0.249485	0.124743	+	0.992189i	$0.460190\pi$
$510$	0	0
$511$	9.94348	0.439874
$512$	0	0
$513$	32.9737	1.45582
$514$	0	0
$515$	20.6756	0.911076
$516$	0	0
$517$	−0.120836	−0.00531435
$518$	0	0
$519$	47.5075	2.08535
$520$	0	0
$521$	34.8219	1.52557	0.762786	−	0.646650i	$-0.223831\pi$
$521$	34.8219	1.52557	0.762786	+	0.646650i	$0.223831\pi$
$522$	0	0
$523$	−9.12215	−0.398884	−0.199442	−	0.979910i	$-0.563913\pi$
$523$	−9.12215	−0.398884	−0.199442	+	0.979910i	$0.563913\pi$
$524$	0	0
$525$	7.49028	0.326903
$526$	0	0
$527$	7.43943	0.324067
$528$	0	0
$529$	−22.9205	−0.996541
$530$	0	0
$531$	19.2231	0.834213
$532$	0	0
$533$	−9.09314	−0.393868
$534$	0	0
$535$	1.01956	0.0440793
$536$	0	0
$537$	21.5460	0.929781
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	23.7767	1.02224	0.511119	−	0.859510i	$-0.329231\pi$
$541$	23.7767	1.02224	0.511119	+	0.859510i	$0.329231\pi$
$542$	0	0
$543$	−6.08432	−0.261103
$544$	0	0
$545$	−23.0675	−0.988102
$546$	0	0
$547$	−38.6086	−1.65079	−0.825393	−	0.564558i	$-0.809046\pi$
$547$	−38.6086	−1.65079	−0.825393	+	0.564558i	$0.809046\pi$
$548$	0	0
$549$	−46.9310	−2.00297
$550$	0	0
$551$	−58.5106	−2.49264
$552$	0	0
$553$	−10.1171	−0.430225
$554$	0	0
$555$	16.9215	0.718277
$556$	0	0
$557$	0.470484	0.0199351	0.00996753	−	0.999950i	$-0.496827\pi$
$557$	0.470484	0.0199351	0.00996753	+	0.999950i	$0.496827\pi$
$558$	0	0
$559$	−6.50862	−0.275285
$560$	0	0
$561$	−7.92601	−0.334637
$562$	0	0
$563$	30.9553	1.30461	0.652305	−	0.757957i	$-0.273802\pi$
$563$	30.9553	1.30461	0.652305	+	0.757957i	$0.273802\pi$
$564$	0	0
$565$	14.9947	0.630832
$566$	0	0
$567$	0.671584	0.0282039
$568$	0	0
$569$	−4.38784	−0.183948	−0.0919740	−	0.995761i	$-0.529318\pi$
$569$	−4.38784	−0.183948	−0.0919740	+	0.995761i	$0.529318\pi$
$570$	0	0
$571$	27.5135	1.15140	0.575702	−	0.817659i	$-0.304729\pi$
$571$	27.5135	1.15140	0.575702	+	0.817659i	$0.304729\pi$
$572$	0	0
$573$	−67.3614	−2.81406
$574$	0	0
$575$	0.758735	0.0316414
$576$	0	0
$577$	−21.5703	−0.897985	−0.448992	−	0.893536i	$-0.648217\pi$
$577$	−21.5703	−0.897985	−0.448992	+	0.893536i	$0.648217\pi$
$578$	0	0
$579$	−17.7551	−0.737877
$580$	0	0
$581$	8.81845	0.365851
$582$	0	0
$583$	−5.22807	−0.216525
$584$	0	0
$585$	−13.1791	−0.544888
$586$	0	0
$587$	−35.9394	−1.48338	−0.741689	−	0.670744i	$-0.765975\pi$
$587$	−35.9394	−1.48338	−0.741689	+	0.670744i	$0.765975\pi$
$588$	0	0
$589$	17.6607	0.727696
$590$	0	0
$591$	−64.3368	−2.64646
$592$	0	0
$593$	6.58399	0.270372	0.135186	−	0.990820i	$-0.456837\pi$
$593$	6.58399	0.270372	0.135186	+	0.990820i	$0.456837\pi$
$594$	0	0
$595$	7.89411	0.323627
$596$	0	0
$597$	30.2684	1.23880
$598$	0	0
$599$	29.8135	1.21814	0.609072	−	0.793115i	$-0.291542\pi$
$599$	29.8135	1.21814	0.609072	+	0.793115i	$0.291542\pi$
$600$	0	0
$601$	−8.33001	−0.339788	−0.169894	−	0.985462i	$-0.554343\pi$
$601$	−8.33001	−0.339788	−0.169894	+	0.985462i	$0.554343\pi$
$602$	0	0
$603$	−13.2308	−0.538800
$604$	0	0
$605$	2.77311	0.112743
$606$	0	0
$607$	12.4632	0.505864	0.252932	−	0.967484i	$-0.418605\pi$
$607$	12.4632	0.505864	0.252932	+	0.967484i	$0.418605\pi$
$608$	0	0
$609$	24.1074	0.976883
$610$	0	0
$611$	−0.120836	−0.00488849
$612$	0	0
$613$	35.4426	1.43151	0.715757	−	0.698350i	$-0.246082\pi$
$613$	35.4426	1.43151	0.715757	+	0.698350i	$0.246082\pi$
$614$	0	0
$615$	−70.2103	−2.83115
$616$	0	0
$617$	−22.5862	−0.909288	−0.454644	−	0.890673i	$-0.650233\pi$
$617$	−22.5862	−0.909288	−0.454644	+	0.890673i	$0.650233\pi$
$618$	0	0
$619$	25.4026	1.02101	0.510507	−	0.859873i	$-0.329458\pi$
$619$	25.4026	1.02101	0.510507	+	0.859873i	$0.329458\pi$
$620$	0	0
$621$	−1.37618	−0.0552243
$622$	0	0
$623$	−17.7949	−0.712937
$624$	0	0
$625$	−31.2138	−1.24855
$626$	0	0
$627$	−18.8158	−0.751431
$628$	0	0
$629$	6.23859	0.248749
$630$	0	0
$631$	27.1228	1.07974	0.539871	−	0.841748i	$-0.318473\pi$
$631$	27.1228	1.07974	0.539871	+	0.841748i	$0.318473\pi$
$632$	0	0
$633$	−48.7420	−1.93732
$634$	0	0
$635$	52.0230	2.06447
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−42.8896	−1.69669
$640$	0	0
$641$	21.1087	0.833742	0.416871	−	0.908966i	$-0.363127\pi$
$641$	21.1087	0.833742	0.416871	+	0.908966i	$0.363127\pi$
$642$	0	0
$643$	44.7611	1.76520	0.882602	−	0.470120i	$-0.155789\pi$
$643$	44.7611	1.76520	0.882602	+	0.470120i	$0.155789\pi$
$644$	0	0
$645$	−50.2547	−1.97878
$646$	0	0
$647$	11.8839	0.467202	0.233601	−	0.972332i	$-0.424949\pi$
$647$	11.8839	0.467202	0.233601	+	0.972332i	$0.424949\pi$
$648$	0	0
$649$	−4.04489	−0.158776
$650$	0	0
$651$	−7.27652	−0.285189
$652$	0	0
$653$	−41.5792	−1.62712	−0.813559	−	0.581482i	$-0.802473\pi$
$653$	−41.5792	−1.62712	−0.813559	+	0.581482i	$0.802473\pi$
$654$	0	0
$655$	22.5564	0.881350
$656$	0	0
$657$	−47.2559	−1.84363
$658$	0	0
$659$	−13.5696	−0.528598	−0.264299	−	0.964441i	$-0.585141\pi$
$659$	−13.5696	−0.528598	−0.264299	+	0.964441i	$0.585141\pi$
$660$	0	0
$661$	4.59269	0.178635	0.0893175	−	0.996003i	$-0.471531\pi$
$661$	4.59269	0.178635	0.0893175	+	0.996003i	$0.471531\pi$
$662$	0	0
$663$	−7.92601	−0.307821
$664$	0	0
$665$	18.7401	0.726708
$666$	0	0
$667$	2.44198	0.0945540
$668$	0	0
$669$	60.8790	2.35372
$670$	0	0
$671$	9.87513	0.381225
$672$	0	0
$673$	−40.5588	−1.56343	−0.781714	−	0.623637i	$-0.785654\pi$
$673$	−40.5588	−1.56343	−0.781714	+	0.623637i	$0.785654\pi$
$674$	0	0
$675$	−13.1263	−0.505232
$676$	0	0
$677$	−39.9393	−1.53499	−0.767496	−	0.641053i	$-0.778498\pi$
$677$	−39.9393	−1.53499	−0.767496	+	0.641053i	$0.778498\pi$
$678$	0	0
$679$	8.81902	0.338443
$680$	0	0
$681$	−13.5177	−0.517999
$682$	0	0
$683$	−16.6904	−0.638639	−0.319319	−	0.947647i	$-0.603454\pi$
$683$	−16.6904	−0.638639	−0.319319	+	0.947647i	$0.603454\pi$
$684$	0	0
$685$	−35.6504	−1.36213
$686$	0	0
$687$	−8.00154	−0.305278
$688$	0	0
$689$	−5.22807	−0.199174
$690$	0	0
$691$	−18.0484	−0.686593	−0.343297	−	0.939227i	$-0.611544\pi$
$691$	−18.0484	−0.686593	−0.343297	+	0.939227i	$0.611544\pi$
$692$	0	0
$693$	4.75245	0.180531
$694$	0	0
$695$	62.5655	2.37324
$696$	0	0
$697$	−25.8851	−0.980467
$698$	0	0
$699$	−2.65458	−0.100406
$700$	0	0
$701$	13.1741	0.497578	0.248789	−	0.968558i	$-0.419967\pi$
$701$	13.1741	0.497578	0.248789	+	0.968558i	$0.419967\pi$
$702$	0	0
$703$	14.8100	0.558569
$704$	0	0
$705$	−0.933002	−0.0351389
$706$	0	0
$707$	−15.5773	−0.585845
$708$	0	0
$709$	−8.31848	−0.312407	−0.156204	−	0.987725i	$-0.549926\pi$
$709$	−8.31848	−0.312407	−0.156204	+	0.987725i	$0.549926\pi$
$710$	0	0
$711$	48.0812	1.80318
$712$	0	0
$713$	−0.737081	−0.0276039
$714$	0	0
$715$	2.77311	0.103709
$716$	0	0
$717$	−6.89635	−0.257549
$718$	0	0
$719$	−11.6582	−0.434778	−0.217389	−	0.976085i	$-0.569754\pi$
$719$	−11.6582	−0.434778	−0.217389	+	0.976085i	$0.569754\pi$
$720$	0	0
$721$	−7.45574	−0.277666
$722$	0	0
$723$	45.1809	1.68030
$724$	0	0
$725$	23.2922	0.865050
$726$	0	0
$727$	−5.08052	−0.188426	−0.0942131	−	0.995552i	$-0.530034\pi$
$727$	−5.08052	−0.188426	−0.0942131	+	0.995552i	$0.530034\pi$
$728$	0	0
$729$	−43.9489	−1.62774
$730$	0	0
$731$	−18.5278	−0.685277
$732$	0	0
$733$	9.48835	0.350460	0.175230	−	0.984528i	$-0.443933\pi$
$733$	9.48835	0.350460	0.175230	+	0.984528i	$0.443933\pi$
$734$	0	0
$735$	−7.72124	−0.284802
$736$	0	0
$737$	2.78400	0.102550
$738$	0	0
$739$	−33.5555	−1.23436	−0.617179	−	0.786823i	$-0.711724\pi$
$739$	−33.5555	−1.23436	−0.617179	+	0.786823i	$0.711724\pi$
$740$	0	0
$741$	−18.8158	−0.691216
$742$	0	0
$743$	−10.5003	−0.385218	−0.192609	−	0.981276i	$-0.561695\pi$
$743$	−10.5003	−0.385218	−0.192609	+	0.981276i	$0.561695\pi$
$744$	0	0
$745$	47.1080	1.72590
$746$	0	0
$747$	−41.9092	−1.53338
$748$	0	0
$749$	−0.367658	−0.0134339
$750$	0	0
$751$	23.5350	0.858803	0.429401	−	0.903114i	$-0.358725\pi$
$751$	23.5350	0.858803	0.429401	+	0.903114i	$0.358725\pi$
$752$	0	0
$753$	10.7087	0.390248
$754$	0	0
$755$	−42.5559	−1.54877
$756$	0	0
$757$	−20.6595	−0.750884	−0.375442	−	0.926846i	$-0.622509\pi$
$757$	−20.6595	−0.750884	−0.375442	+	0.926846i	$0.622509\pi$
$758$	0	0
$759$	0.785291	0.0285043
$760$	0	0
$761$	−12.1303	−0.439723	−0.219862	−	0.975531i	$-0.570561\pi$
$761$	−12.1303	−0.439723	−0.219862	+	0.975531i	$0.570561\pi$
$762$	0	0
$763$	8.31826	0.301141
$764$	0	0
$765$	−37.5163	−1.35641
$766$	0	0
$767$	−4.04489	−0.146052
$768$	0	0
$769$	−11.8917	−0.428827	−0.214414	−	0.976743i	$-0.568784\pi$
$769$	−11.8917	−0.428827	−0.214414	+	0.976743i	$0.568784\pi$
$770$	0	0
$771$	−75.7604	−2.72844
$772$	0	0
$773$	−28.0728	−1.00971	−0.504854	−	0.863205i	$-0.668454\pi$
$773$	−28.0728	−1.00971	−0.504854	+	0.863205i	$0.668454\pi$
$774$	0	0
$775$	−7.03044	−0.252541
$776$	0	0
$777$	−6.10198	−0.218907
$778$	0	0
$779$	−61.4493	−2.20165
$780$	0	0
$781$	9.02475	0.322931
$782$	0	0
$783$	−42.2470	−1.50979
$784$	0	0
$785$	13.9601	0.498256
$786$	0	0
$787$	−2.73641	−0.0975426	−0.0487713	−	0.998810i	$-0.515531\pi$
$787$	−2.73641	−0.0975426	−0.0487713	+	0.998810i	$0.515531\pi$
$788$	0	0
$789$	−45.2846	−1.61217
$790$	0	0
$791$	−5.40717	−0.192257
$792$	0	0
$793$	9.87513	0.350676
$794$	0	0
$795$	−40.3672	−1.43168
$796$	0	0
$797$	26.7258	0.946676	0.473338	−	0.880881i	$-0.343049\pi$
$797$	26.7258	0.946676	0.473338	+	0.880881i	$0.343049\pi$
$798$	0	0
$799$	−0.343978	−0.0121691
$800$	0	0
$801$	84.5693	2.98811
$802$	0	0
$803$	9.94348	0.350898
$804$	0	0
$805$	−0.782130	−0.0275665
$806$	0	0
$807$	−58.5663	−2.06163
$808$	0	0
$809$	15.7871	0.555045	0.277523	−	0.960719i	$-0.410487\pi$
$809$	15.7871	0.555045	0.277523	+	0.960719i	$0.410487\pi$
$810$	0	0
$811$	13.0434	0.458014	0.229007	−	0.973425i	$-0.426452\pi$
$811$	13.0434	0.458014	0.229007	+	0.973425i	$0.426452\pi$
$812$	0	0
$813$	40.7170	1.42801
$814$	0	0
$815$	−9.06173	−0.317419
$816$	0	0
$817$	−43.9838	−1.53880
$818$	0	0
$819$	4.75245	0.166064
$820$	0	0
$821$	18.8379	0.657448	0.328724	−	0.944426i	$-0.393381\pi$
$821$	18.8379	0.657448	0.328724	+	0.944426i	$0.393381\pi$
$822$	0	0
$823$	25.7084	0.896139	0.448069	−	0.893999i	$-0.352112\pi$
$823$	25.7084	0.896139	0.448069	+	0.893999i	$0.352112\pi$
$824$	0	0
$825$	7.49028	0.260778
$826$	0	0
$827$	−13.5146	−0.469949	−0.234975	−	0.972002i	$-0.575501\pi$
$827$	−13.5146	−0.469949	−0.234975	+	0.972002i	$0.575501\pi$
$828$	0	0
$829$	−28.1334	−0.977114	−0.488557	−	0.872532i	$-0.662477\pi$
$829$	−28.1334	−0.977114	−0.488557	+	0.872532i	$0.662477\pi$
$830$	0	0
$831$	−73.5779	−2.55239
$832$	0	0
$833$	−2.84666	−0.0986309
$834$	0	0
$835$	46.9497	1.62476
$836$	0	0
$837$	12.7517	0.440764
$838$	0	0
$839$	−43.7578	−1.51069	−0.755343	−	0.655329i	$-0.772530\pi$
$839$	−43.7578	−1.51069	−0.755343	+	0.655329i	$0.772530\pi$
$840$	0	0
$841$	45.9658	1.58503
$842$	0	0
$843$	−14.2097	−0.489408
$844$	0	0
$845$	2.77311	0.0953980
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	−34.2446	−1.17527
$850$	0	0
$851$	−0.618105	−0.0211884
$852$	0	0
$853$	20.8971	0.715502	0.357751	−	0.933817i	$-0.383544\pi$
$853$	20.8971	0.715502	0.357751	+	0.933817i	$0.383544\pi$
$854$	0	0
$855$	−89.0612	−3.04583
$856$	0	0
$857$	−57.0628	−1.94923	−0.974614	−	0.223894i	$-0.928123\pi$
$857$	−57.0628	−1.94923	−0.974614	+	0.223894i	$0.928123\pi$
$858$	0	0
$859$	27.1516	0.926399	0.463200	−	0.886254i	$-0.346701\pi$
$859$	27.1516	0.926399	0.463200	+	0.886254i	$0.346701\pi$
$860$	0	0
$861$	25.3182	0.862843
$862$	0	0
$863$	1.68942	0.0575084	0.0287542	−	0.999587i	$-0.490846\pi$
$863$	1.68942	0.0575084	0.0287542	+	0.999587i	$0.490846\pi$
$864$	0	0
$865$	−47.3162	−1.60880
$866$	0	0
$867$	24.7708	0.841261
$868$	0	0
$869$	−10.1171	−0.343200
$870$	0	0
$871$	2.78400	0.0943321
$872$	0	0
$873$	−41.9120	−1.41850
$874$	0	0
$875$	6.40544	0.216543
$876$	0	0
$877$	−31.6006	−1.06708	−0.533539	−	0.845776i	$-0.679138\pi$
$877$	−31.6006	−1.06708	−0.533539	+	0.845776i	$0.679138\pi$
$878$	0	0
$879$	−32.4564	−1.09473
$880$	0	0
$881$	−42.6087	−1.43552	−0.717762	−	0.696289i	$-0.754833\pi$
$881$	−42.6087	−1.43552	−0.717762	+	0.696289i	$0.754833\pi$
$882$	0	0
$883$	47.7599	1.60725	0.803624	−	0.595137i	$-0.202902\pi$
$883$	47.7599	1.60725	0.803624	+	0.595137i	$0.202902\pi$
$884$	0	0
$885$	−31.2316	−1.04984
$886$	0	0
$887$	−35.8277	−1.20298	−0.601488	−	0.798882i	$-0.705425\pi$
$887$	−35.8277	−1.20298	−0.601488	+	0.798882i	$0.705425\pi$
$888$	0	0
$889$	−18.7598	−0.629183
$890$	0	0
$891$	0.671584	0.0224989
$892$	0	0
$893$	−0.816580	−0.0273258
$894$	0	0
$895$	−21.4593	−0.717306
$896$	0	0
$897$	0.785291	0.0262201
$898$	0	0
$899$	−22.6275	−0.754668
$900$	0	0
$901$	−14.8825	−0.495809
$902$	0	0
$903$	18.1221	0.603066
$904$	0	0
$905$	6.05982	0.201435
$906$	0	0
$907$	−10.7825	−0.358027	−0.179013	−	0.983847i	$-0.557291\pi$
$907$	−10.7825	−0.358027	−0.179013	+	0.983847i	$0.557291\pi$
$908$	0	0
$909$	74.0303	2.45543
$910$	0	0
$911$	18.7105	0.619905	0.309953	−	0.950752i	$-0.399687\pi$
$911$	18.7105	0.619905	0.309953	+	0.950752i	$0.399687\pi$
$912$	0	0
$913$	8.81845	0.291848
$914$	0	0
$915$	76.2483	2.52069
$916$	0	0
$917$	−8.13395	−0.268607
$918$	0	0
$919$	−38.8166	−1.28044	−0.640221	−	0.768191i	$-0.721157\pi$
$919$	−38.8166	−1.28044	−0.640221	+	0.768191i	$0.721157\pi$
$920$	0	0
$921$	−19.7033	−0.649246
$922$	0	0
$923$	9.02475	0.297053
$924$	0	0
$925$	−5.89562	−0.193847
$926$	0	0
$927$	35.4330	1.16377
$928$	0	0
$929$	−7.44831	−0.244371	−0.122186	−	0.992507i	$-0.538990\pi$
$929$	−7.44831	−0.244371	−0.122186	+	0.992507i	$0.538990\pi$
$930$	0	0
$931$	−6.75777	−0.221477
$932$	0	0
$933$	11.0365	0.361320
$934$	0	0
$935$	7.89411	0.258165
$936$	0	0
$937$	−5.01746	−0.163913	−0.0819567	−	0.996636i	$-0.526117\pi$
$937$	−5.01746	−0.163913	−0.0819567	+	0.996636i	$0.526117\pi$
$938$	0	0
$939$	−37.2559	−1.21580
$940$	0	0
$941$	39.4944	1.28748	0.643740	−	0.765244i	$-0.277382\pi$
$941$	39.4944	1.28748	0.643740	+	0.765244i	$0.277382\pi$
$942$	0	0
$943$	2.56463	0.0835159
$944$	0	0
$945$	13.5311	0.440166
$946$	0	0
$947$	39.0353	1.26848	0.634239	−	0.773137i	$-0.281314\pi$
$947$	39.0353	1.26848	0.634239	+	0.773137i	$0.281314\pi$
$948$	0	0
$949$	9.94348	0.322779
$950$	0	0
$951$	87.0518	2.82285
$952$	0	0
$953$	−20.2554	−0.656136	−0.328068	−	0.944654i	$-0.606398\pi$
$953$	−20.2554	−0.656136	−0.328068	+	0.944654i	$0.606398\pi$
$954$	0	0
$955$	67.0903	2.17099
$956$	0	0
$957$	24.1074	0.779283
$958$	0	0
$959$	12.8557	0.415133
$960$	0	0
$961$	−24.1702	−0.779684
$962$	0	0
$963$	1.74728	0.0563052
$964$	0	0
$965$	17.6836	0.569256
$966$	0	0
$967$	8.74658	0.281271	0.140635	−	0.990061i	$-0.455085\pi$
$967$	8.74658	0.281271	0.140635	+	0.990061i	$0.455085\pi$
$968$	0	0
$969$	−53.5622	−1.72066
$970$	0	0
$971$	−12.3035	−0.394839	−0.197420	−	0.980319i	$-0.563256\pi$
$971$	−12.3035	−0.394839	−0.197420	+	0.980319i	$0.563256\pi$
$972$	0	0
$973$	−22.5614	−0.723287
$974$	0	0
$975$	7.49028	0.239881
$976$	0	0
$977$	−10.6015	−0.339172	−0.169586	−	0.985515i	$-0.554243\pi$
$977$	−10.6015	−0.339172	−0.169586	+	0.985515i	$0.554243\pi$
$978$	0	0
$979$	−17.7949	−0.568727
$980$	0	0
$981$	−39.5321	−1.26216
$982$	0	0
$983$	57.3622	1.82957	0.914785	−	0.403940i	$-0.132360\pi$
$983$	57.3622	1.82957	0.914785	+	0.403940i	$0.132360\pi$
$984$	0	0
$985$	64.0778	2.04169
$986$	0	0
$987$	0.336446	0.0107092
$988$	0	0
$989$	1.83570	0.0583717
$990$	0	0
$991$	−33.1355	−1.05258	−0.526291	−	0.850304i	$-0.676418\pi$
$991$	−33.1355	−1.05258	−0.526291	+	0.850304i	$0.676418\pi$
$992$	0	0
$993$	−24.9761	−0.792592
$994$	0	0
$995$	−30.1466	−0.955711
$996$	0	0
$997$	4.24210	0.134349	0.0671743	−	0.997741i	$-0.478602\pi$
$997$	4.24210	0.134349	0.0671743	+	0.997741i	$0.478602\pi$
$998$	0	0
$999$	10.6934	0.338324

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.a.j.1.2	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.a.j.1.2	✓	10	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:	\( N \)	\(=\)	\( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4004.a (trivial)

\(f(q)\)	\(=\)	\(q-2.78432 q^{3} +2.77311 q^{5} -1.00000 q^{7} +4.75245 q^{9} +O(q^{10})\)	\(q-2.78432 q^{3} +2.77311 q^{5} -1.00000 q^{7} +4.75245 q^{9} -1.00000 q^{11} -1.00000 q^{13} -7.72124 q^{15} -2.84666 q^{17} -6.75777 q^{19} +2.78432 q^{21} +0.282040 q^{23} +2.69016 q^{25} -4.87938 q^{27} +8.65828 q^{29} -2.61339 q^{31} +2.78432 q^{33} -2.77311 q^{35} -2.19155 q^{37} +2.78432 q^{39} +9.09314 q^{41} +6.50862 q^{43} +13.1791 q^{45} +0.120836 q^{47} +1.00000 q^{49} +7.92601 q^{51} +5.22807 q^{53} -2.77311 q^{55} +18.8158 q^{57} +4.04489 q^{59} -9.87513 q^{61} -4.75245 q^{63} -2.77311 q^{65} -2.78400 q^{67} -0.785291 q^{69} -9.02475 q^{71} -9.94348 q^{73} -7.49028 q^{75} +1.00000 q^{77} +10.1171 q^{79} -0.671584 q^{81} -8.81845 q^{83} -7.89411 q^{85} -24.1074 q^{87} +17.7949 q^{89} +1.00000 q^{91} +7.27652 q^{93} -18.7401 q^{95} -8.81902 q^{97} -4.75245 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 2 q^{3} - 2 q^{5} - 10 q^{7} + 10 q^{9}+O(q^{10}) \) 10 * q - 2 * q^3 - 2 * q^5 - 10 * q^7 + 10 * q^9	\( 10 q - 2 q^{3} - 2 q^{5} - 10 q^{7} + 10 q^{9} - 10 q^{11} - 10 q^{13} - 4 q^{15} + 9 q^{17} + 3 q^{19} + 2 q^{21} - 4 q^{23} + 22 q^{25} - 8 q^{27} + 13 q^{29} - 17 q^{31} + 2 q^{33} + 2 q^{35} + 11 q^{37} + 2 q^{39} + 8 q^{41} + 21 q^{43} - 15 q^{45} + q^{47} + 10 q^{49} + 19 q^{51} + 22 q^{53} + 2 q^{55} + 8 q^{57} - 24 q^{59} + 16 q^{61} - 10 q^{63} + 2 q^{65} + 19 q^{67} + 51 q^{69} - 25 q^{71} + 22 q^{73} + 28 q^{75} + 10 q^{77} + 41 q^{79} + 54 q^{81} + 8 q^{83} + 9 q^{85} - 5 q^{87} + 36 q^{89} + 10 q^{91} + 12 q^{93} + 13 q^{95} - 5 q^{97} - 10 q^{99}+O(q^{100}) \) 10 * q - 2 * q^3 - 2 * q^5 - 10 * q^7 + 10 * q^9 - 10 * q^11 - 10 * q^13 - 4 * q^15 + 9 * q^17 + 3 * q^19 + 2 * q^21 - 4 * q^23 + 22 * q^25 - 8 * q^27 + 13 * q^29 - 17 * q^31 + 2 * q^33 + 2 * q^35 + 11 * q^37 + 2 * q^39 + 8 * q^41 + 21 * q^43 - 15 * q^45 + q^47 + 10 * q^49 + 19 * q^51 + 22 * q^53 + 2 * q^55 + 8 * q^57 - 24 * q^59 + 16 * q^61 - 10 * q^63 + 2 * q^65 + 19 * q^67 + 51 * q^69 - 25 * q^71 + 22 * q^73 + 28 * q^75 + 10 * q^77 + 41 * q^79 + 54 * q^81 + 8 * q^83 + 9 * q^85 - 5 * q^87 + 36 * q^89 + 10 * q^91 + 12 * q^93 + 13 * q^95 - 5 * q^97 - 10 * q^99

Introduction

L-functions

Modular forms

Varieties

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