LMFDB - Embedded newform 4004.2.a.k.1.9

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} - x^{9} - 23x^{8} + 23x^{7} + 170x^{6} - 165x^{5} - 411x^{4} + 360x^{3} + 111x^{2} - 48x - 12 $ x^10 - x^9 - 23x^8 + 23x^7 + 170x^6 - 165x^5 - 411x^4 + 360x^3 + 111x^2 - 48x - 12
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			1.9
Root			$2.85872$ 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)

$f(q)$	$=$	$q+2.85872 q^{3} +3.60422 q^{5} -1.00000 q^{7} +5.17231 q^{9} +O(q^{10})$	$q+2.85872 q^{3} +3.60422 q^{5} -1.00000 q^{7} +5.17231 q^{9} +1.00000 q^{11} +1.00000 q^{13} +10.3035 q^{15} +2.10684 q^{17} -3.22488 q^{19} -2.85872 q^{21} -1.93561 q^{23} +7.99044 q^{25} +6.21002 q^{27} +0.252925 q^{29} +1.79385 q^{31} +2.85872 q^{33} -3.60422 q^{35} -1.21508 q^{37} +2.85872 q^{39} +10.4196 q^{41} -7.61480 q^{43} +18.6422 q^{45} -1.93561 q^{47} +1.00000 q^{49} +6.02289 q^{51} +6.07722 q^{53} +3.60422 q^{55} -9.21903 q^{57} -0.0467740 q^{59} -6.26595 q^{61} -5.17231 q^{63} +3.60422 q^{65} +11.8486 q^{67} -5.53336 q^{69} -2.73428 q^{71} -15.6820 q^{73} +22.8425 q^{75} -1.00000 q^{77} +5.14817 q^{79} +2.23583 q^{81} -9.57647 q^{83} +7.59354 q^{85} +0.723043 q^{87} +4.29738 q^{89} -1.00000 q^{91} +5.12812 q^{93} -11.6232 q^{95} +1.61258 q^{97} +5.17231 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + q^{3} + 4 q^{5} - 10 q^{7} + 17 q^{9}+O(q^{10}) $ 10 * q + q^3 + 4 * q^5 - 10 * q^7 + 17 * q^9	$ 10 q + q^{3} + 4 q^{5} - 10 q^{7} + 17 q^{9} + 10 q^{11} + 10 q^{13} + 3 q^{15} + 3 q^{17} + 6 q^{19} - q^{21} + 4 q^{23} + 22 q^{25} - 5 q^{27} + 10 q^{29} - q^{31} + q^{33} - 4 q^{35} + 20 q^{37} + q^{39} + 8 q^{45} + 4 q^{47} + 10 q^{49} + 11 q^{51} - 5 q^{53} + 4 q^{55} + 16 q^{57} + 11 q^{59} + 12 q^{61} - 17 q^{63} + 4 q^{65} - 2 q^{67} + 10 q^{69} + 28 q^{71} + 11 q^{73} - 6 q^{75} - 10 q^{77} - 10 q^{79} + 46 q^{81} + 7 q^{83} + 33 q^{85} - 47 q^{87} + 30 q^{89} - 10 q^{91} + 41 q^{93} - 2 q^{95} + 55 q^{97} + 17 q^{99}+O(q^{100}) $ 10 * q + q^3 + 4 * q^5 - 10 * q^7 + 17 * q^9 + 10 * q^11 + 10 * q^13 + 3 * q^15 + 3 * q^17 + 6 * q^19 - q^21 + 4 * q^23 + 22 * q^25 - 5 * q^27 + 10 * q^29 - q^31 + q^33 - 4 * q^35 + 20 * q^37 + q^39 + 8 * q^45 + 4 * q^47 + 10 * q^49 + 11 * q^51 - 5 * q^53 + 4 * q^55 + 16 * q^57 + 11 * q^59 + 12 * q^61 - 17 * q^63 + 4 * q^65 - 2 * q^67 + 10 * q^69 + 28 * q^71 + 11 * q^73 - 6 * q^75 - 10 * q^77 - 10 * q^79 + 46 * q^81 + 7 * q^83 + 33 * q^85 - 47 * q^87 + 30 * q^89 - 10 * q^91 + 41 * q^93 - 2 * q^95 + 55 * q^97 + 17 * 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
$3$	2.85872	1.65049	0.825243	−	0.564778i	$-0.191038\pi$
$3$	2.85872	1.65049	0.825243	+	0.564778i	$0.191038\pi$
$4$	0	0
$5$	3.60422	1.61186	0.805929	−	0.592012i	$-0.201666\pi$
$5$	3.60422	1.61186	0.805929	+	0.592012i	$0.201666\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	5.17231	1.72410
$10$	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
$15$	10.3035	2.66035
$16$	0	0
$17$	2.10684	0.510985	0.255492	−	0.966811i	$-0.417762\pi$
$17$	2.10684	0.510985	0.255492	+	0.966811i	$0.417762\pi$
$18$	0	0
$19$	−3.22488	−0.739837	−0.369919	−	0.929064i	$-0.620614\pi$
$19$	−3.22488	−0.739837	−0.369919	+	0.929064i	$0.620614\pi$
$20$	0	0
$21$	−2.85872	−0.623825
$22$	0	0
$23$	−1.93561	−0.403602	−0.201801	−	0.979427i	$-0.564679\pi$
$23$	−1.93561	−0.403602	−0.201801	+	0.979427i	$0.564679\pi$
$24$	0	0
$25$	7.99044	1.59809
$26$	0	0
$27$	6.21002	1.19512
$28$	0	0
$29$	0.252925	0.0469670	0.0234835	−	0.999724i	$-0.492524\pi$
$29$	0.252925	0.0469670	0.0234835	+	0.999724i	$0.492524\pi$
$30$	0	0
$31$	1.79385	0.322185	0.161092	−	0.986939i	$-0.448498\pi$
$31$	1.79385	0.322185	0.161092	+	0.986939i	$0.448498\pi$
$32$	0	0
$33$	2.85872	0.497640
$34$	0	0
$35$	−3.60422	−0.609225
$36$	0	0
$37$	−1.21508	−0.199758	−0.0998790	−	0.995000i	$-0.531846\pi$
$37$	−1.21508	−0.199758	−0.0998790	+	0.995000i	$0.531846\pi$
$38$	0	0
$39$	2.85872	0.457762
$40$	0	0
$41$	10.4196	1.62726	0.813632	−	0.581381i	$-0.197487\pi$
$41$	10.4196	1.62726	0.813632	+	0.581381i	$0.197487\pi$
$42$	0	0
$43$	−7.61480	−1.16125	−0.580623	−	0.814172i	$-0.697191\pi$
$43$	−7.61480	−1.16125	−0.580623	+	0.814172i	$0.697191\pi$
$44$	0	0
$45$	18.6422	2.77901
$46$	0	0
$47$	−1.93561	−0.282337	−0.141169	−	0.989986i	$-0.545086\pi$
$47$	−1.93561	−0.282337	−0.141169	+	0.989986i	$0.545086\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	6.02289	0.843373
$52$	0	0
$53$	6.07722	0.834770	0.417385	−	0.908730i	$-0.362947\pi$
$53$	6.07722	0.834770	0.417385	+	0.908730i	$0.362947\pi$
$54$	0	0
$55$	3.60422	0.485994
$56$	0	0
$57$	−9.21903	−1.22109
$58$	0	0
$59$	−0.0467740	−0.00608946	−0.00304473	−	0.999995i	$-0.500969\pi$
$59$	−0.0467740	−0.00608946	−0.00304473	+	0.999995i	$0.500969\pi$
$60$	0	0
$61$	−6.26595	−0.802273	−0.401136	−	0.916018i	$-0.631385\pi$
$61$	−6.26595	−0.802273	−0.401136	+	0.916018i	$0.631385\pi$
$62$	0	0
$63$	−5.17231	−0.651649
$64$	0	0
$65$	3.60422	0.447049
$66$	0	0
$67$	11.8486	1.44754	0.723770	−	0.690041i	$-0.242408\pi$
$67$	11.8486	1.44754	0.723770	+	0.690041i	$0.242408\pi$
$68$	0	0
$69$	−5.53336	−0.666138
$70$	0	0
$71$	−2.73428	−0.324499	−0.162250	−	0.986750i	$-0.551875\pi$
$71$	−2.73428	−0.324499	−0.162250	+	0.986750i	$0.551875\pi$
$72$	0	0
$73$	−15.6820	−1.83544	−0.917718	−	0.397232i	$-0.869971\pi$
$73$	−15.6820	−1.83544	−0.917718	+	0.397232i	$0.869971\pi$
$74$	0	0
$75$	22.8425	2.63762
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	5.14817	0.579215	0.289607	−	0.957146i	$-0.406475\pi$
$79$	5.14817	0.579215	0.289607	+	0.957146i	$0.406475\pi$
$80$	0	0
$81$	2.23583	0.248426
$82$	0	0
$83$	−9.57647	−1.05115	−0.525577	−	0.850746i	$-0.676151\pi$
$83$	−9.57647	−1.05115	−0.525577	+	0.850746i	$0.676151\pi$
$84$	0	0
$85$	7.59354	0.823635
$86$	0	0
$87$	0.723043	0.0775183
$88$	0	0
$89$	4.29738	0.455521	0.227760	−	0.973717i	$-0.426860\pi$
$89$	4.29738	0.455521	0.227760	+	0.973717i	$0.426860\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	5.12812	0.531761
$94$	0	0
$95$	−11.6232	−1.19251
$96$	0	0
$97$	1.61258	0.163733	0.0818665	−	0.996643i	$-0.473912\pi$
$97$	1.61258	0.163733	0.0818665	+	0.996643i	$0.473912\pi$
$98$	0	0
$99$	5.17231	0.519836
$100$	0	0
$101$	3.21930	0.320332	0.160166	−	0.987090i	$-0.448797\pi$
$101$	3.21930	0.320332	0.160166	+	0.987090i	$0.448797\pi$
$102$	0	0
$103$	−7.53034	−0.741986	−0.370993	−	0.928636i	$-0.620983\pi$
$103$	−7.53034	−0.741986	−0.370993	+	0.928636i	$0.620983\pi$
$104$	0	0
$105$	−10.3035	−1.00552
$106$	0	0
$107$	−14.1563	−1.36854	−0.684271	−	0.729228i	$-0.739879\pi$
$107$	−14.1563	−1.36854	−0.684271	+	0.729228i	$0.739879\pi$
$108$	0	0
$109$	4.23864	0.405988	0.202994	−	0.979180i	$-0.434933\pi$
$109$	4.23864	0.405988	0.202994	+	0.979180i	$0.434933\pi$
$110$	0	0
$111$	−3.47358	−0.329698
$112$	0	0
$113$	16.6834	1.56944	0.784719	−	0.619852i	$-0.212807\pi$
$113$	16.6834	1.56944	0.784719	+	0.619852i	$0.212807\pi$
$114$	0	0
$115$	−6.97636	−0.650549
$116$	0	0
$117$	5.17231	0.478180
$118$	0	0
$119$	−2.10684	−0.193134
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	29.7867	2.68577
$124$	0	0
$125$	10.7782	0.964032
$126$	0	0
$127$	−13.1904	−1.17046	−0.585230	−	0.810867i	$-0.698996\pi$
$127$	−13.1904	−1.17046	−0.585230	+	0.810867i	$0.698996\pi$
$128$	0	0
$129$	−21.7686	−1.91662
$130$	0	0
$131$	10.8973	0.952104	0.476052	−	0.879417i	$-0.342067\pi$
$131$	10.8973	0.952104	0.476052	+	0.879417i	$0.342067\pi$
$132$	0	0
$133$	3.22488	0.279632
$134$	0	0
$135$	22.3823	1.92636
$136$	0	0
$137$	−1.01012	−0.0863003	−0.0431501	−	0.999069i	$-0.513739\pi$
$137$	−1.01012	−0.0863003	−0.0431501	+	0.999069i	$0.513739\pi$
$138$	0	0
$139$	−1.77007	−0.150135	−0.0750677	−	0.997178i	$-0.523917\pi$
$139$	−1.77007	−0.150135	−0.0750677	+	0.997178i	$0.523917\pi$
$140$	0	0
$141$	−5.53336	−0.465993
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	0.911598	0.0757041
$146$	0	0
$147$	2.85872	0.235784
$148$	0	0
$149$	6.68202	0.547412	0.273706	−	0.961813i	$-0.411750\pi$
$149$	6.68202	0.547412	0.273706	+	0.961813i	$0.411750\pi$
$150$	0	0
$151$	3.12294	0.254142	0.127071	−	0.991894i	$-0.459442\pi$
$151$	3.12294	0.254142	0.127071	+	0.991894i	$0.459442\pi$
$152$	0	0
$153$	10.8972	0.880990
$154$	0	0
$155$	6.46544	0.519316
$156$	0	0
$157$	−20.8650	−1.66521	−0.832605	−	0.553868i	$-0.813151\pi$
$157$	−20.8650	−1.66521	−0.832605	+	0.553868i	$0.813151\pi$
$158$	0	0
$159$	17.3731	1.37778
$160$	0	0
$161$	1.93561	0.152547
$162$	0	0
$163$	−4.14726	−0.324839	−0.162419	−	0.986722i	$-0.551930\pi$
$163$	−4.14726	−0.324839	−0.162419	+	0.986722i	$0.551930\pi$
$164$	0	0
$165$	10.3035	0.802125
$166$	0	0
$167$	3.38586	0.262006	0.131003	−	0.991382i	$-0.458180\pi$
$167$	3.38586	0.262006	0.131003	+	0.991382i	$0.458180\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−16.6800	−1.27555
$172$	0	0
$173$	−4.40574	−0.334962	−0.167481	−	0.985875i	$-0.553563\pi$
$173$	−4.40574	−0.334962	−0.167481	+	0.985875i	$0.553563\pi$
$174$	0	0
$175$	−7.99044	−0.604020
$176$	0	0
$177$	−0.133714	−0.0100506
$178$	0	0
$179$	12.3711	0.924660	0.462330	−	0.886708i	$-0.347013\pi$
$179$	12.3711	0.924660	0.462330	+	0.886708i	$0.347013\pi$
$180$	0	0
$181$	8.99537	0.668620	0.334310	−	0.942463i	$-0.391497\pi$
$181$	8.99537	0.668620	0.334310	+	0.942463i	$0.391497\pi$
$182$	0	0
$183$	−17.9126	−1.32414
$184$	0	0
$185$	−4.37942	−0.321982
$186$	0	0
$187$	2.10684	0.154068
$188$	0	0
$189$	−6.21002	−0.451713
$190$	0	0
$191$	−26.6081	−1.92529	−0.962647	−	0.270761i	$-0.912725\pi$
$191$	−26.6081	−1.92529	−0.962647	+	0.270761i	$0.912725\pi$
$192$	0	0
$193$	−14.9617	−1.07697	−0.538484	−	0.842636i	$-0.681003\pi$
$193$	−14.9617	−1.07697	−0.538484	+	0.842636i	$0.681003\pi$
$194$	0	0
$195$	10.3035	0.737848
$196$	0	0
$197$	0.682555	0.0486300	0.0243150	−	0.999704i	$-0.492260\pi$
$197$	0.682555	0.0486300	0.0243150	+	0.999704i	$0.492260\pi$
$198$	0	0
$199$	−15.0477	−1.06670	−0.533352	−	0.845893i	$-0.679068\pi$
$199$	−15.0477	−1.06670	−0.533352	+	0.845893i	$0.679068\pi$
$200$	0	0
$201$	33.8720	2.38914
$202$	0	0
$203$	−0.252925	−0.0177519
$204$	0	0
$205$	37.5545	2.62292
$206$	0	0
$207$	−10.0115	−0.695850
$208$	0	0
$209$	−3.22488	−0.223069
$210$	0	0
$211$	5.62379	0.387157	0.193579	−	0.981085i	$-0.437991\pi$
$211$	5.62379	0.387157	0.193579	+	0.981085i	$0.437991\pi$
$212$	0	0
$213$	−7.81656	−0.535582
$214$	0	0
$215$	−27.4455	−1.87177
$216$	0	0
$217$	−1.79385	−0.121774
$218$	0	0
$219$	−44.8305	−3.02936
$220$	0	0
$221$	2.10684	0.141722
$222$	0	0
$223$	8.96435	0.600297	0.300149	−	0.953892i	$-0.402964\pi$
$223$	8.96435	0.600297	0.300149	+	0.953892i	$0.402964\pi$
$224$	0	0
$225$	41.3290	2.75527
$226$	0	0
$227$	11.3357	0.752375	0.376187	−	0.926544i	$-0.377235\pi$
$227$	11.3357	0.752375	0.376187	+	0.926544i	$0.377235\pi$
$228$	0	0
$229$	13.6420	0.901490	0.450745	−	0.892653i	$-0.351158\pi$
$229$	13.6420	0.901490	0.450745	+	0.892653i	$0.351158\pi$
$230$	0	0
$231$	−2.85872	−0.188090
$232$	0	0
$233$	−1.21770	−0.0797742	−0.0398871	−	0.999204i	$-0.512700\pi$
$233$	−1.21770	−0.0797742	−0.0398871	+	0.999204i	$0.512700\pi$
$234$	0	0
$235$	−6.97636	−0.455087
$236$	0	0
$237$	14.7172	0.955985
$238$	0	0
$239$	−13.1556	−0.850962	−0.425481	−	0.904967i	$-0.639895\pi$
$239$	−13.1556	−0.850962	−0.425481	+	0.904967i	$0.639895\pi$
$240$	0	0
$241$	−6.90875	−0.445032	−0.222516	−	0.974929i	$-0.571427\pi$
$241$	−6.90875	−0.445032	−0.222516	+	0.974929i	$0.571427\pi$
$242$	0	0
$243$	−12.2384	−0.785097
$244$	0	0
$245$	3.60422	0.230265
$246$	0	0
$247$	−3.22488	−0.205194
$248$	0	0
$249$	−27.3765	−1.73492
$250$	0	0
$251$	9.78817	0.617824	0.308912	−	0.951091i	$-0.400035\pi$
$251$	9.78817	0.617824	0.308912	+	0.951091i	$0.400035\pi$
$252$	0	0
$253$	−1.93561	−0.121690
$254$	0	0
$255$	21.7078	1.35940
$256$	0	0
$257$	19.7121	1.22961	0.614803	−	0.788681i	$-0.289235\pi$
$257$	19.7121	1.22961	0.614803	+	0.788681i	$0.289235\pi$
$258$	0	0
$259$	1.21508	0.0755014
$260$	0	0
$261$	1.30821	0.0809759
$262$	0	0
$263$	−1.53892	−0.0948941	−0.0474471	−	0.998874i	$-0.515109\pi$
$263$	−1.53892	−0.0948941	−0.0474471	+	0.998874i	$0.515109\pi$
$264$	0	0
$265$	21.9037	1.34553
$266$	0	0
$267$	12.2850	0.751831
$268$	0	0
$269$	−8.82901	−0.538314	−0.269157	−	0.963096i	$-0.586745\pi$
$269$	−8.82901	−0.538314	−0.269157	+	0.963096i	$0.586745\pi$
$270$	0	0
$271$	−25.1103	−1.52534	−0.762670	−	0.646788i	$-0.776112\pi$
$271$	−25.1103	−1.52534	−0.762670	+	0.646788i	$0.776112\pi$
$272$	0	0
$273$	−2.85872	−0.173018
$274$	0	0
$275$	7.99044	0.481841
$276$	0	0
$277$	−21.9312	−1.31772	−0.658858	−	0.752267i	$-0.728960\pi$
$277$	−21.9312	−1.31772	−0.658858	+	0.752267i	$0.728960\pi$
$278$	0	0
$279$	9.27834	0.555479
$280$	0	0
$281$	−16.9270	−1.00978	−0.504889	−	0.863184i	$-0.668467\pi$
$281$	−16.9270	−1.00978	−0.504889	+	0.863184i	$0.668467\pi$
$282$	0	0
$283$	−15.2196	−0.904709	−0.452355	−	0.891838i	$-0.649416\pi$
$283$	−15.2196	−0.904709	−0.452355	+	0.891838i	$0.649416\pi$
$284$	0	0
$285$	−33.2275	−1.96822
$286$	0	0
$287$	−10.4196	−0.615048
$288$	0	0
$289$	−12.5612	−0.738895
$290$	0	0
$291$	4.60993	0.270239
$292$	0	0
$293$	5.62868	0.328831	0.164416	−	0.986391i	$-0.447426\pi$
$293$	5.62868	0.328831	0.164416	+	0.986391i	$0.447426\pi$
$294$	0	0
$295$	−0.168584	−0.00981535
$296$	0	0
$297$	6.21002	0.360342
$298$	0	0
$299$	−1.93561	−0.111939
$300$	0	0
$301$	7.61480	0.438910
$302$	0	0
$303$	9.20309	0.528704
$304$	0	0
$305$	−22.5839	−1.29315
$306$	0	0
$307$	20.2243	1.15426	0.577132	−	0.816651i	$-0.304172\pi$
$307$	20.2243	1.15426	0.577132	+	0.816651i	$0.304172\pi$
$308$	0	0
$309$	−21.5272	−1.22464
$310$	0	0
$311$	0.336601	0.0190869	0.00954345	−	0.999954i	$-0.496962\pi$
$311$	0.336601	0.0190869	0.00954345	+	0.999954i	$0.496962\pi$
$312$	0	0
$313$	−31.8249	−1.79885	−0.899425	−	0.437075i	$-0.856014\pi$
$313$	−31.8249	−1.79885	−0.899425	+	0.437075i	$0.856014\pi$
$314$	0	0
$315$	−18.6422	−1.05037
$316$	0	0
$317$	−0.624535	−0.0350774	−0.0175387	−	0.999846i	$-0.505583\pi$
$317$	−0.624535	−0.0350774	−0.0175387	+	0.999846i	$0.505583\pi$
$318$	0	0
$319$	0.252925	0.0141611
$320$	0	0
$321$	−40.4690	−2.25876
$322$	0	0
$323$	−6.79431	−0.378045
$324$	0	0
$325$	7.99044	0.443230
$326$	0	0
$327$	12.1171	0.670077
$328$	0	0
$329$	1.93561	0.106713
$330$	0	0
$331$	−34.5582	−1.89949	−0.949744	−	0.313028i	$-0.898657\pi$
$331$	−34.5582	−1.89949	−0.949744	+	0.313028i	$0.898657\pi$
$332$	0	0
$333$	−6.28477	−0.344403
$334$	0	0
$335$	42.7051	2.33323
$336$	0	0
$337$	−9.18670	−0.500431	−0.250216	−	0.968190i	$-0.580502\pi$
$337$	−9.18670	−0.500431	−0.250216	+	0.968190i	$0.580502\pi$
$338$	0	0
$339$	47.6931	2.59033
$340$	0	0
$341$	1.79385	0.0971424
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−19.9435	−1.07372
$346$	0	0
$347$	6.75969	0.362879	0.181440	−	0.983402i	$-0.441924\pi$
$347$	6.75969	0.362879	0.181440	+	0.983402i	$0.441924\pi$
$348$	0	0
$349$	13.4370	0.719268	0.359634	−	0.933094i	$-0.382902\pi$
$349$	13.4370	0.719268	0.359634	+	0.933094i	$0.382902\pi$
$350$	0	0
$351$	6.21002	0.331467
$352$	0	0
$353$	17.6362	0.938681	0.469341	−	0.883017i	$-0.344492\pi$
$353$	17.6362	0.938681	0.469341	+	0.883017i	$0.344492\pi$
$354$	0	0
$355$	−9.85496	−0.523047
$356$	0	0
$357$	−6.02289	−0.318765
$358$	0	0
$359$	12.4025	0.654580	0.327290	−	0.944924i	$-0.393865\pi$
$359$	12.4025	0.654580	0.327290	+	0.944924i	$0.393865\pi$
$360$	0	0
$361$	−8.60018	−0.452641
$362$	0	0
$363$	2.85872	0.150044
$364$	0	0
$365$	−56.5214	−2.95846
$366$	0	0
$367$	−10.8367	−0.565670	−0.282835	−	0.959169i	$-0.591275\pi$
$367$	−10.8367	−0.565670	−0.282835	+	0.959169i	$0.591275\pi$
$368$	0	0
$369$	53.8932	2.80557
$370$	0	0
$371$	−6.07722	−0.315513
$372$	0	0
$373$	−17.4461	−0.903326	−0.451663	−	0.892189i	$-0.649169\pi$
$373$	−17.4461	−0.903326	−0.451663	+	0.892189i	$0.649169\pi$
$374$	0	0
$375$	30.8119	1.59112
$376$	0	0
$377$	0.252925	0.0130263
$378$	0	0
$379$	22.3108	1.14603	0.573014	−	0.819546i	$-0.305774\pi$
$379$	22.3108	1.14603	0.573014	+	0.819546i	$0.305774\pi$
$380$	0	0
$381$	−37.7078	−1.93183
$382$	0	0
$383$	31.4488	1.60696	0.803479	−	0.595333i	$-0.202980\pi$
$383$	31.4488	1.60696	0.803479	+	0.595333i	$0.202980\pi$
$384$	0	0
$385$	−3.60422	−0.183688
$386$	0	0
$387$	−39.3861	−2.00211
$388$	0	0
$389$	20.8505	1.05716	0.528582	−	0.848882i	$-0.322724\pi$
$389$	20.8505	1.05716	0.528582	+	0.848882i	$0.322724\pi$
$390$	0	0
$391$	−4.07802	−0.206234
$392$	0	0
$393$	31.1525	1.57143
$394$	0	0
$395$	18.5552	0.933612
$396$	0	0
$397$	−20.1456	−1.01108	−0.505539	−	0.862804i	$-0.668706\pi$
$397$	−20.1456	−1.01108	−0.505539	+	0.862804i	$0.668706\pi$
$398$	0	0
$399$	9.21903	0.461529
$400$	0	0
$401$	24.6001	1.22847	0.614235	−	0.789123i	$-0.289465\pi$
$401$	24.6001	1.22847	0.614235	+	0.789123i	$0.289465\pi$
$402$	0	0
$403$	1.79385	0.0893580
$404$	0	0
$405$	8.05844	0.400427
$406$	0	0
$407$	−1.21508	−0.0602293
$408$	0	0
$409$	31.1649	1.54100	0.770502	−	0.637438i	$-0.220006\pi$
$409$	31.1649	1.54100	0.770502	+	0.637438i	$0.220006\pi$
$410$	0	0
$411$	−2.88765	−0.142437
$412$	0	0
$413$	0.0467740	0.00230160
$414$	0	0
$415$	−34.5158	−1.69431
$416$	0	0
$417$	−5.06014	−0.247796
$418$	0	0
$419$	25.8091	1.26086	0.630428	−	0.776247i	$-0.282879\pi$
$419$	25.8091	1.26086	0.630428	+	0.776247i	$0.282879\pi$
$420$	0	0
$421$	−39.2822	−1.91450	−0.957248	−	0.289267i	$-0.906588\pi$
$421$	−39.2822	−1.91450	−0.957248	+	0.289267i	$0.906588\pi$
$422$	0	0
$423$	−10.0115	−0.486778
$424$	0	0
$425$	16.8346	0.816598
$426$	0	0
$427$	6.26595	0.303231
$428$	0	0
$429$	2.85872	0.138021
$430$	0	0
$431$	22.9725	1.10655	0.553273	−	0.833000i	$-0.313378\pi$
$431$	22.9725	1.10655	0.553273	+	0.833000i	$0.313378\pi$
$432$	0	0
$433$	40.5210	1.94732	0.973659	−	0.228011i	$-0.0732222\pi$
$433$	40.5210	1.94732	0.973659	+	0.228011i	$0.0732222\pi$
$434$	0	0
$435$	2.60601	0.124949
$436$	0	0
$437$	6.24208	0.298599
$438$	0	0
$439$	31.3676	1.49709	0.748546	−	0.663083i	$-0.230752\pi$
$439$	31.3676	1.49709	0.748546	+	0.663083i	$0.230752\pi$
$440$	0	0
$441$	5.17231	0.246300
$442$	0	0
$443$	9.54748	0.453615	0.226807	−	0.973940i	$-0.427171\pi$
$443$	9.54748	0.453615	0.226807	+	0.973940i	$0.427171\pi$
$444$	0	0
$445$	15.4887	0.734235
$446$	0	0
$447$	19.1020	0.903496
$448$	0	0
$449$	−30.6888	−1.44829	−0.724146	−	0.689646i	$-0.757766\pi$
$449$	−30.6888	−1.44829	−0.724146	+	0.689646i	$0.757766\pi$
$450$	0	0
$451$	10.4196	0.490638
$452$	0	0
$453$	8.92764	0.419457
$454$	0	0
$455$	−3.60422	−0.168969
$456$	0	0
$457$	27.7341	1.29735	0.648673	−	0.761067i	$-0.275324\pi$
$457$	27.7341	1.29735	0.648673	+	0.761067i	$0.275324\pi$
$458$	0	0
$459$	13.0836	0.610688
$460$	0	0
$461$	−4.04012	−0.188167	−0.0940835	−	0.995564i	$-0.529992\pi$
$461$	−4.04012	−0.188167	−0.0940835	+	0.995564i	$0.529992\pi$
$462$	0	0
$463$	13.4301	0.624149	0.312074	−	0.950058i	$-0.398976\pi$
$463$	13.4301	0.624149	0.312074	+	0.950058i	$0.398976\pi$
$464$	0	0
$465$	18.4829	0.857124
$466$	0	0
$467$	−5.15432	−0.238514	−0.119257	−	0.992863i	$-0.538051\pi$
$467$	−5.15432	−0.238514	−0.119257	+	0.992863i	$0.538051\pi$
$468$	0	0
$469$	−11.8486	−0.547119
$470$	0	0
$471$	−59.6473	−2.74840
$472$	0	0
$473$	−7.61480	−0.350129
$474$	0	0
$475$	−25.7682	−1.18232
$476$	0	0
$477$	31.4332	1.43923
$478$	0	0
$479$	10.2958	0.470427	0.235213	−	0.971944i	$-0.424421\pi$
$479$	10.2958	0.470427	0.235213	+	0.971944i	$0.424421\pi$
$480$	0	0
$481$	−1.21508	−0.0554029
$482$	0	0
$483$	5.53336	0.251777
$484$	0	0
$485$	5.81211	0.263914
$486$	0	0
$487$	−6.89793	−0.312575	−0.156288	−	0.987712i	$-0.549953\pi$
$487$	−6.89793	−0.312575	−0.156288	+	0.987712i	$0.549953\pi$
$488$	0	0
$489$	−11.8559	−0.536141
$490$	0	0
$491$	−13.5602	−0.611965	−0.305982	−	0.952037i	$-0.598985\pi$
$491$	−13.5602	−0.611965	−0.305982	+	0.952037i	$0.598985\pi$
$492$	0	0
$493$	0.532873	0.0239994
$494$	0	0
$495$	18.6422	0.837903
$496$	0	0
$497$	2.73428	0.122649
$498$	0	0
$499$	9.66171	0.432517	0.216259	−	0.976336i	$-0.430615\pi$
$499$	9.66171	0.432517	0.216259	+	0.976336i	$0.430615\pi$
$500$	0	0
$501$	9.67925	0.432437
$502$	0	0
$503$	−23.0549	−1.02797	−0.513984	−	0.857800i	$-0.671831\pi$
$503$	−23.0549	−1.02797	−0.513984	+	0.857800i	$0.671831\pi$
$504$	0	0
$505$	11.6031	0.516330
$506$	0	0
$507$	2.85872	0.126960
$508$	0	0
$509$	−10.3989	−0.460921	−0.230461	−	0.973082i	$-0.574023\pi$
$509$	−10.3989	−0.460921	−0.230461	+	0.973082i	$0.574023\pi$
$510$	0	0
$511$	15.6820	0.693730
$512$	0	0
$513$	−20.0266	−0.884194
$514$	0	0
$515$	−27.1410	−1.19598
$516$	0	0
$517$	−1.93561	−0.0851278
$518$	0	0
$519$	−12.5948	−0.552850
$520$	0	0
$521$	−11.8436	−0.518877	−0.259439	−	0.965760i	$-0.583538\pi$
$521$	−11.8436	−0.518877	−0.259439	+	0.965760i	$0.583538\pi$
$522$	0	0
$523$	−8.39232	−0.366971	−0.183485	−	0.983022i	$-0.558738\pi$
$523$	−8.39232	−0.366971	−0.183485	+	0.983022i	$0.558738\pi$
$524$	0	0
$525$	−22.8425	−0.996927
$526$	0	0
$527$	3.77936	0.164632
$528$	0	0
$529$	−19.2534	−0.837106
$530$	0	0
$531$	−0.241930	−0.0104989
$532$	0	0
$533$	10.4196	0.451322
$534$	0	0
$535$	−51.0225	−2.20590
$536$	0	0
$537$	35.3656	1.52614
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	17.5760	0.755652	0.377826	−	0.925877i	$-0.376672\pi$
$541$	17.5760	0.755652	0.377826	+	0.925877i	$0.376672\pi$
$542$	0	0
$543$	25.7153	1.10355
$544$	0	0
$545$	15.2770	0.654395
$546$	0	0
$547$	5.13618	0.219607	0.109804	−	0.993953i	$-0.464978\pi$
$547$	5.13618	0.219607	0.109804	+	0.993953i	$0.464978\pi$
$548$	0	0
$549$	−32.4094	−1.38320
$550$	0	0
$551$	−0.815651	−0.0347479
$552$	0	0
$553$	−5.14817	−0.218923
$554$	0	0
$555$	−12.5196	−0.531426
$556$	0	0
$557$	−27.2507	−1.15465	−0.577324	−	0.816515i	$-0.695903\pi$
$557$	−27.2507	−1.15465	−0.577324	+	0.816515i	$0.695903\pi$
$558$	0	0
$559$	−7.61480	−0.322072
$560$	0	0
$561$	6.02289	0.254286
$562$	0	0
$563$	−14.6025	−0.615423	−0.307711	−	0.951480i	$-0.599563\pi$
$563$	−14.6025	−0.615423	−0.307711	+	0.951480i	$0.599563\pi$
$564$	0	0
$565$	60.1306	2.52971
$566$	0	0
$567$	−2.23583	−0.0938961
$568$	0	0
$569$	−22.3011	−0.934911	−0.467455	−	0.884017i	$-0.654829\pi$
$569$	−22.3011	−0.934911	−0.467455	+	0.884017i	$0.654829\pi$
$570$	0	0
$571$	22.7587	0.952423	0.476211	−	0.879331i	$-0.342010\pi$
$571$	22.7587	0.952423	0.476211	+	0.879331i	$0.342010\pi$
$572$	0	0
$573$	−76.0652	−3.17767
$574$	0	0
$575$	−15.4663	−0.644991
$576$	0	0
$577$	36.2528	1.50923	0.754613	−	0.656170i	$-0.227825\pi$
$577$	36.2528	1.50923	0.754613	+	0.656170i	$0.227825\pi$
$578$	0	0
$579$	−42.7714	−1.77752
$580$	0	0
$581$	9.57647	0.397299
$582$	0	0
$583$	6.07722	0.251693
$584$	0	0
$585$	18.6422	0.770758
$586$	0	0
$587$	−35.4671	−1.46388	−0.731942	−	0.681367i	$-0.761386\pi$
$587$	−35.4671	−1.46388	−0.731942	+	0.681367i	$0.761386\pi$
$588$	0	0
$589$	−5.78494	−0.238364
$590$	0	0
$591$	1.95124	0.0802632
$592$	0	0
$593$	−35.0827	−1.44068	−0.720338	−	0.693623i	$-0.756013\pi$
$593$	−35.0827	−1.44068	−0.720338	+	0.693623i	$0.756013\pi$
$594$	0	0
$595$	−7.59354	−0.311305
$596$	0	0
$597$	−43.0173	−1.76058
$598$	0	0
$599$	9.02421	0.368719	0.184360	−	0.982859i	$-0.440979\pi$
$599$	9.02421	0.368719	0.184360	+	0.982859i	$0.440979\pi$
$600$	0	0
$601$	−5.12595	−0.209092	−0.104546	−	0.994520i	$-0.533339\pi$
$601$	−5.12595	−0.209092	−0.104546	+	0.994520i	$0.533339\pi$
$602$	0	0
$603$	61.2847	2.49571
$604$	0	0
$605$	3.60422	0.146533
$606$	0	0
$607$	43.4859	1.76504	0.882519	−	0.470276i	$-0.155846\pi$
$607$	43.4859	1.76504	0.882519	+	0.470276i	$0.155846\pi$
$608$	0	0
$609$	−0.723043	−0.0292992
$610$	0	0
$611$	−1.93561	−0.0783062
$612$	0	0
$613$	22.9163	0.925582	0.462791	−	0.886467i	$-0.346848\pi$
$613$	22.9163	0.925582	0.462791	+	0.886467i	$0.346848\pi$
$614$	0	0
$615$	107.358	4.32909
$616$	0	0
$617$	30.8079	1.24028	0.620139	−	0.784492i	$-0.287076\pi$
$617$	30.8079	1.24028	0.620139	+	0.784492i	$0.287076\pi$
$618$	0	0
$619$	−25.4501	−1.02293	−0.511463	−	0.859305i	$-0.670896\pi$
$619$	−25.4501	−1.02293	−0.511463	+	0.859305i	$0.670896\pi$
$620$	0	0
$621$	−12.0202	−0.482352
$622$	0	0
$623$	−4.29738	−0.172171
$624$	0	0
$625$	−1.10510	−0.0442041
$626$	0	0
$627$	−9.21903	−0.368173
$628$	0	0
$629$	−2.55999	−0.102073
$630$	0	0
$631$	−46.0325	−1.83252	−0.916262	−	0.400579i	$-0.868809\pi$
$631$	−46.0325	−1.83252	−0.916262	+	0.400579i	$0.868809\pi$
$632$	0	0
$633$	16.0769	0.638998
$634$	0	0
$635$	−47.5412	−1.88662
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−14.1425	−0.559470
$640$	0	0
$641$	−5.88434	−0.232417	−0.116209	−	0.993225i	$-0.537074\pi$
$641$	−5.88434	−0.232417	−0.116209	+	0.993225i	$0.537074\pi$
$642$	0	0
$643$	−11.2566	−0.443917	−0.221959	−	0.975056i	$-0.571245\pi$
$643$	−11.2566	−0.443917	−0.221959	+	0.975056i	$0.571245\pi$
$644$	0	0
$645$	−78.4590	−3.08932
$646$	0	0
$647$	−5.59386	−0.219917	−0.109959	−	0.993936i	$-0.535072\pi$
$647$	−5.59386	−0.219917	−0.109959	+	0.993936i	$0.535072\pi$
$648$	0	0
$649$	−0.0467740	−0.00183604
$650$	0	0
$651$	−5.12812	−0.200987
$652$	0	0
$653$	−11.1985	−0.438230	−0.219115	−	0.975699i	$-0.570317\pi$
$653$	−11.1985	−0.438230	−0.219115	+	0.975699i	$0.570317\pi$
$654$	0	0
$655$	39.2764	1.53466
$656$	0	0
$657$	−81.1120	−3.16448
$658$	0	0
$659$	32.1683	1.25310	0.626550	−	0.779381i	$-0.284466\pi$
$659$	32.1683	1.25310	0.626550	+	0.779381i	$0.284466\pi$
$660$	0	0
$661$	16.8717	0.656232	0.328116	−	0.944637i	$-0.393586\pi$
$661$	16.8717	0.656232	0.328116	+	0.944637i	$0.393586\pi$
$662$	0	0
$663$	6.02289	0.233910
$664$	0	0
$665$	11.6232	0.450727
$666$	0	0
$667$	−0.489563	−0.0189559
$668$	0	0
$669$	25.6266	0.990782
$670$	0	0
$671$	−6.26595	−0.241894
$672$	0	0
$673$	7.25625	0.279708	0.139854	−	0.990172i	$-0.455337\pi$
$673$	7.25625	0.279708	0.139854	+	0.990172i	$0.455337\pi$
$674$	0	0
$675$	49.6208	1.90991
$676$	0	0
$677$	17.4509	0.670692	0.335346	−	0.942095i	$-0.391147\pi$
$677$	17.4509	0.670692	0.335346	+	0.942095i	$0.391147\pi$
$678$	0	0
$679$	−1.61258	−0.0618852
$680$	0	0
$681$	32.4056	1.24178
$682$	0	0
$683$	−2.80947	−0.107501	−0.0537507	−	0.998554i	$-0.517118\pi$
$683$	−2.80947	−0.107501	−0.0537507	+	0.998554i	$0.517118\pi$
$684$	0	0
$685$	−3.64070	−0.139104
$686$	0	0
$687$	38.9988	1.48790
$688$	0	0
$689$	6.07722	0.231524
$690$	0	0
$691$	−25.9866	−0.988578	−0.494289	−	0.869298i	$-0.664572\pi$
$691$	−25.9866	−0.988578	−0.494289	+	0.869298i	$0.664572\pi$
$692$	0	0
$693$	−5.17231	−0.196480
$694$	0	0
$695$	−6.37973	−0.241997
$696$	0	0
$697$	21.9524	0.831506
$698$	0	0
$699$	−3.48107	−0.131666
$700$	0	0
$701$	9.57164	0.361516	0.180758	−	0.983528i	$-0.442145\pi$
$701$	9.57164	0.361516	0.180758	+	0.983528i	$0.442145\pi$
$702$	0	0
$703$	3.91848	0.147788
$704$	0	0
$705$	−19.9435	−0.751115
$706$	0	0
$707$	−3.21930	−0.121074
$708$	0	0
$709$	48.3443	1.81561	0.907804	−	0.419395i	$-0.137758\pi$
$709$	48.3443	1.81561	0.907804	+	0.419395i	$0.137758\pi$
$710$	0	0
$711$	26.6279	0.998625
$712$	0	0
$713$	−3.47218	−0.130034
$714$	0	0
$715$	3.60422	0.134790
$716$	0	0
$717$	−37.6081	−1.40450
$718$	0	0
$719$	27.7756	1.03586	0.517928	−	0.855424i	$-0.326703\pi$
$719$	27.7756	1.03586	0.517928	+	0.855424i	$0.326703\pi$
$720$	0	0
$721$	7.53034	0.280444
$722$	0	0
$723$	−19.7502	−0.734518
$724$	0	0
$725$	2.02098	0.0750573
$726$	0	0
$727$	−48.2023	−1.78772	−0.893861	−	0.448343i	$-0.852014\pi$
$727$	−48.2023	−1.78772	−0.893861	+	0.448343i	$0.852014\pi$
$728$	0	0
$729$	−41.6938	−1.54422
$730$	0	0
$731$	−16.0432	−0.593379
$732$	0	0
$733$	−23.3136	−0.861109	−0.430554	−	0.902565i	$-0.641682\pi$
$733$	−23.3136	−0.861109	−0.430554	+	0.902565i	$0.641682\pi$
$734$	0	0
$735$	10.3035	0.380050
$736$	0	0
$737$	11.8486	0.436450
$738$	0	0
$739$	47.0708	1.73153	0.865764	−	0.500452i	$-0.166833\pi$
$739$	47.0708	1.73153	0.865764	+	0.500452i	$0.166833\pi$
$740$	0	0
$741$	−9.21903	−0.338670
$742$	0	0
$743$	−16.3683	−0.600494	−0.300247	−	0.953861i	$-0.597069\pi$
$743$	−16.3683	−0.600494	−0.300247	+	0.953861i	$0.597069\pi$
$744$	0	0
$745$	24.0835	0.882351
$746$	0	0
$747$	−49.5325	−1.81230
$748$	0	0
$749$	14.1563	0.517260
$750$	0	0
$751$	−11.8393	−0.432021	−0.216010	−	0.976391i	$-0.569305\pi$
$751$	−11.8393	−0.432021	−0.216010	+	0.976391i	$0.569305\pi$
$752$	0	0
$753$	27.9817	1.01971
$754$	0	0
$755$	11.2558	0.409640
$756$	0	0
$757$	−10.0137	−0.363954	−0.181977	−	0.983303i	$-0.558250\pi$
$757$	−10.0137	−0.363954	−0.181977	+	0.983303i	$0.558250\pi$
$758$	0	0
$759$	−5.53336	−0.200848
$760$	0	0
$761$	39.3234	1.42547	0.712736	−	0.701433i	$-0.247456\pi$
$761$	39.3234	1.42547	0.712736	+	0.701433i	$0.247456\pi$
$762$	0	0
$763$	−4.23864	−0.153449
$764$	0	0
$765$	39.2761	1.42003
$766$	0	0
$767$	−0.0467740	−0.00168891
$768$	0	0
$769$	52.3884	1.88917	0.944587	−	0.328262i	$-0.106463\pi$
$769$	52.3884	1.88917	0.944587	+	0.328262i	$0.106463\pi$
$770$	0	0
$771$	56.3514	2.02945
$772$	0	0
$773$	26.8014	0.963981	0.481990	−	0.876176i	$-0.339914\pi$
$773$	26.8014	0.963981	0.481990	+	0.876176i	$0.339914\pi$
$774$	0	0
$775$	14.3336	0.514879
$776$	0	0
$777$	3.47358	0.124614
$778$	0	0
$779$	−33.6018	−1.20391
$780$	0	0
$781$	−2.73428	−0.0978403
$782$	0	0
$783$	1.57067	0.0561312
$784$	0	0
$785$	−75.2022	−2.68408
$786$	0	0
$787$	46.5440	1.65911	0.829557	−	0.558422i	$-0.188593\pi$
$787$	46.5440	1.65911	0.829557	+	0.558422i	$0.188593\pi$
$788$	0	0
$789$	−4.39936	−0.156621
$790$	0	0
$791$	−16.6834	−0.593192
$792$	0	0
$793$	−6.26595	−0.222510
$794$	0	0
$795$	62.6165	2.22078
$796$	0	0
$797$	−8.98083	−0.318117	−0.159059	−	0.987269i	$-0.550846\pi$
$797$	−8.98083	−0.318117	−0.159059	+	0.987269i	$0.550846\pi$
$798$	0	0
$799$	−4.07802	−0.144270
$800$	0	0
$801$	22.2273	0.785365
$802$	0	0
$803$	−15.6820	−0.553405
$804$	0	0
$805$	6.97636	0.245884
$806$	0	0
$807$	−25.2397	−0.888479
$808$	0	0
$809$	47.7697	1.67949	0.839747	−	0.542978i	$-0.182703\pi$
$809$	47.7697	1.67949	0.839747	+	0.542978i	$0.182703\pi$
$810$	0	0
$811$	−31.4729	−1.10516	−0.552582	−	0.833458i	$-0.686357\pi$
$811$	−31.4729	−1.10516	−0.552582	+	0.833458i	$0.686357\pi$
$812$	0	0
$813$	−71.7833	−2.51755
$814$	0	0
$815$	−14.9477	−0.523594
$816$	0	0
$817$	24.5568	0.859134
$818$	0	0
$819$	−5.17231	−0.180735
$820$	0	0
$821$	−3.05820	−0.106732	−0.0533659	−	0.998575i	$-0.516995\pi$
$821$	−3.05820	−0.106732	−0.0533659	+	0.998575i	$0.516995\pi$
$822$	0	0
$823$	33.8564	1.18016	0.590079	−	0.807345i	$-0.299097\pi$
$823$	33.8564	1.18016	0.590079	+	0.807345i	$0.299097\pi$
$824$	0	0
$825$	22.8425	0.795272
$826$	0	0
$827$	−37.0364	−1.28788	−0.643941	−	0.765075i	$-0.722702\pi$
$827$	−37.0364	−1.28788	−0.643941	+	0.765075i	$0.722702\pi$
$828$	0	0
$829$	20.2930	0.704806	0.352403	−	0.935848i	$-0.385365\pi$
$829$	20.2930	0.704806	0.352403	+	0.935848i	$0.385365\pi$
$830$	0	0
$831$	−62.6951	−2.17487
$832$	0	0
$833$	2.10684	0.0729978
$834$	0	0
$835$	12.2034	0.422316
$836$	0	0
$837$	11.1398	0.385049
$838$	0	0
$839$	−12.4044	−0.428248	−0.214124	−	0.976807i	$-0.568690\pi$
$839$	−12.4044	−0.428248	−0.214124	+	0.976807i	$0.568690\pi$
$840$	0	0
$841$	−28.9360	−0.997794
$842$	0	0
$843$	−48.3895	−1.66662
$844$	0	0
$845$	3.60422	0.123989
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	−43.5086	−1.49321
$850$	0	0
$851$	2.35192	0.0806227
$852$	0	0
$853$	−19.0944	−0.653781	−0.326890	−	0.945062i	$-0.606001\pi$
$853$	−19.0944	−0.653781	−0.326890	+	0.945062i	$0.606001\pi$
$854$	0	0
$855$	−60.1186	−2.05601
$856$	0	0
$857$	−27.9482	−0.954692	−0.477346	−	0.878715i	$-0.658401\pi$
$857$	−27.9482	−0.954692	−0.477346	+	0.878715i	$0.658401\pi$
$858$	0	0
$859$	33.1681	1.13168	0.565841	−	0.824514i	$-0.308552\pi$
$859$	33.1681	1.13168	0.565841	+	0.824514i	$0.308552\pi$
$860$	0	0
$861$	−29.7867	−1.01513
$862$	0	0
$863$	24.9122	0.848021	0.424010	−	0.905657i	$-0.360622\pi$
$863$	24.9122	0.848021	0.424010	+	0.905657i	$0.360622\pi$
$864$	0	0
$865$	−15.8793	−0.539912
$866$	0	0
$867$	−35.9090	−1.21953
$868$	0	0
$869$	5.14817	0.174640
$870$	0	0
$871$	11.8486	0.401475
$872$	0	0
$873$	8.34077	0.282292
$874$	0	0
$875$	−10.7782	−0.364370
$876$	0	0
$877$	−23.7960	−0.803533	−0.401767	−	0.915742i	$-0.631604\pi$
$877$	−23.7960	−0.803533	−0.401767	+	0.915742i	$0.631604\pi$
$878$	0	0
$879$	16.0909	0.542731
$880$	0	0
$881$	11.9754	0.403462	0.201731	−	0.979441i	$-0.435343\pi$
$881$	11.9754	0.403462	0.201731	+	0.979441i	$0.435343\pi$
$882$	0	0
$883$	−15.4083	−0.518531	−0.259266	−	0.965806i	$-0.583481\pi$
$883$	−15.4083	−0.518531	−0.259266	+	0.965806i	$0.583481\pi$
$884$	0	0
$885$	−0.481936	−0.0162001
$886$	0	0
$887$	−53.5960	−1.79958	−0.899788	−	0.436327i	$-0.856279\pi$
$887$	−53.5960	−1.79958	−0.899788	+	0.436327i	$0.856279\pi$
$888$	0	0
$889$	13.1904	0.442392
$890$	0	0
$891$	2.23583	0.0749032
$892$	0	0
$893$	6.24208	0.208883
$894$	0	0
$895$	44.5883	1.49042
$896$	0	0
$897$	−5.53336	−0.184754
$898$	0	0
$899$	0.453709	0.0151320
$900$	0	0
$901$	12.8037	0.426555
$902$	0	0
$903$	21.7686	0.724415
$904$	0	0
$905$	32.4213	1.07772
$906$	0	0
$907$	55.8162	1.85335	0.926673	−	0.375868i	$-0.122655\pi$
$907$	55.8162	1.85335	0.926673	+	0.375868i	$0.122655\pi$
$908$	0	0
$909$	16.6512	0.552286
$910$	0	0
$911$	−57.1896	−1.89477	−0.947387	−	0.320090i	$-0.896287\pi$
$911$	−57.1896	−1.89477	−0.947387	+	0.320090i	$0.896287\pi$
$912$	0	0
$913$	−9.57647	−0.316935
$914$	0	0
$915$	−64.5611	−2.13432
$916$	0	0
$917$	−10.8973	−0.359862
$918$	0	0
$919$	−16.8048	−0.554340	−0.277170	−	0.960821i	$-0.589396\pi$
$919$	−16.8048	−0.554340	−0.277170	+	0.960821i	$0.589396\pi$
$920$	0	0
$921$	57.8158	1.90510
$922$	0	0
$923$	−2.73428	−0.0900000
$924$	0	0
$925$	−9.70903	−0.319231
$926$	0	0
$927$	−38.9492	−1.27926
$928$	0	0
$929$	31.2688	1.02590	0.512948	−	0.858420i	$-0.328554\pi$
$929$	31.2688	1.02590	0.512948	+	0.858420i	$0.328554\pi$
$930$	0	0
$931$	−3.22488	−0.105691
$932$	0	0
$933$	0.962250	0.0315026
$934$	0	0
$935$	7.59354	0.248335
$936$	0	0
$937$	18.4762	0.603591	0.301796	−	0.953373i	$-0.402414\pi$
$937$	18.4762	0.603591	0.301796	+	0.953373i	$0.402414\pi$
$938$	0	0
$939$	−90.9786	−2.96898
$940$	0	0
$941$	−22.0019	−0.717242	−0.358621	−	0.933483i	$-0.616753\pi$
$941$	−22.0019	−0.717242	−0.358621	+	0.933483i	$0.616753\pi$
$942$	0	0
$943$	−20.1682	−0.656766
$944$	0	0
$945$	−22.3823	−0.728097
$946$	0	0
$947$	−1.01309	−0.0329209	−0.0164604	−	0.999865i	$-0.505240\pi$
$947$	−1.01309	−0.0329209	−0.0164604	+	0.999865i	$0.505240\pi$
$948$	0	0
$949$	−15.6820	−0.509059
$950$	0	0
$951$	−1.78537	−0.0578947
$952$	0	0
$953$	−15.7093	−0.508875	−0.254437	−	0.967089i	$-0.581890\pi$
$953$	−15.7093	−0.508875	−0.254437	+	0.967089i	$0.581890\pi$
$954$	0	0
$955$	−95.9015	−3.10330
$956$	0	0
$957$	0.723043	0.0233727
$958$	0	0
$959$	1.01012	0.0326184
$960$	0	0
$961$	−27.7821	−0.896197
$962$	0	0
$963$	−73.2208	−2.35951
$964$	0	0
$965$	−53.9254	−1.73592
$966$	0	0
$967$	1.33128	0.0428111	0.0214055	−	0.999771i	$-0.493186\pi$
$967$	1.33128	0.0428111	0.0214055	+	0.999771i	$0.493186\pi$
$968$	0	0
$969$	−19.4231	−0.623958
$970$	0	0
$971$	33.3527	1.07034	0.535170	−	0.844745i	$-0.320248\pi$
$971$	33.3527	1.07034	0.535170	+	0.844745i	$0.320248\pi$
$972$	0	0
$973$	1.77007	0.0567458
$974$	0	0
$975$	22.8425	0.731544
$976$	0	0
$977$	−11.4444	−0.366139	−0.183070	−	0.983100i	$-0.558603\pi$
$977$	−11.4444	−0.366139	−0.183070	+	0.983100i	$0.558603\pi$
$978$	0	0
$979$	4.29738	0.137345
$980$	0	0
$981$	21.9235	0.699964
$982$	0	0
$983$	29.3350	0.935640	0.467820	−	0.883824i	$-0.345040\pi$
$983$	29.3350	0.935640	0.467820	+	0.883824i	$0.345040\pi$
$984$	0	0
$985$	2.46008	0.0783848
$986$	0	0
$987$	5.53336	0.176129
$988$	0	0
$989$	14.7393	0.468681
$990$	0	0
$991$	−46.3311	−1.47176	−0.735878	−	0.677115i	$-0.763230\pi$
$991$	−46.3311	−1.47176	−0.735878	+	0.677115i	$0.763230\pi$
$992$	0	0
$993$	−98.7923	−3.13508
$994$	0	0
$995$	−54.2354	−1.71938
$996$	0	0
$997$	−58.1905	−1.84291	−0.921456	−	0.388482i	$-0.873000\pi$
$997$	−58.1905	−1.84291	−0.921456	+	0.388482i	$0.873000\pi$
$998$	0	0
$999$	−7.54568	−0.238735

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.a.k.1.9	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.a.k.1.9	✓	10	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q+2.85872 q^{3} +3.60422 q^{5} -1.00000 q^{7} +5.17231 q^{9} +O(q^{10})\)	\(q+2.85872 q^{3} +3.60422 q^{5} -1.00000 q^{7} +5.17231 q^{9} +1.00000 q^{11} +1.00000 q^{13} +10.3035 q^{15} +2.10684 q^{17} -3.22488 q^{19} -2.85872 q^{21} -1.93561 q^{23} +7.99044 q^{25} +6.21002 q^{27} +0.252925 q^{29} +1.79385 q^{31} +2.85872 q^{33} -3.60422 q^{35} -1.21508 q^{37} +2.85872 q^{39} +10.4196 q^{41} -7.61480 q^{43} +18.6422 q^{45} -1.93561 q^{47} +1.00000 q^{49} +6.02289 q^{51} +6.07722 q^{53} +3.60422 q^{55} -9.21903 q^{57} -0.0467740 q^{59} -6.26595 q^{61} -5.17231 q^{63} +3.60422 q^{65} +11.8486 q^{67} -5.53336 q^{69} -2.73428 q^{71} -15.6820 q^{73} +22.8425 q^{75} -1.00000 q^{77} +5.14817 q^{79} +2.23583 q^{81} -9.57647 q^{83} +7.59354 q^{85} +0.723043 q^{87} +4.29738 q^{89} -1.00000 q^{91} +5.12812 q^{93} -11.6232 q^{95} +1.61258 q^{97} +5.17231 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + q^{3} + 4 q^{5} - 10 q^{7} + 17 q^{9}+O(q^{10}) \) 10 * q + q^3 + 4 * q^5 - 10 * q^7 + 17 * q^9	\( 10 q + q^{3} + 4 q^{5} - 10 q^{7} + 17 q^{9} + 10 q^{11} + 10 q^{13} + 3 q^{15} + 3 q^{17} + 6 q^{19} - q^{21} + 4 q^{23} + 22 q^{25} - 5 q^{27} + 10 q^{29} - q^{31} + q^{33} - 4 q^{35} + 20 q^{37} + q^{39} + 8 q^{45} + 4 q^{47} + 10 q^{49} + 11 q^{51} - 5 q^{53} + 4 q^{55} + 16 q^{57} + 11 q^{59} + 12 q^{61} - 17 q^{63} + 4 q^{65} - 2 q^{67} + 10 q^{69} + 28 q^{71} + 11 q^{73} - 6 q^{75} - 10 q^{77} - 10 q^{79} + 46 q^{81} + 7 q^{83} + 33 q^{85} - 47 q^{87} + 30 q^{89} - 10 q^{91} + 41 q^{93} - 2 q^{95} + 55 q^{97} + 17 q^{99}+O(q^{100}) \) 10 * q + q^3 + 4 * q^5 - 10 * q^7 + 17 * q^9 + 10 * q^11 + 10 * q^13 + 3 * q^15 + 3 * q^17 + 6 * q^19 - q^21 + 4 * q^23 + 22 * q^25 - 5 * q^27 + 10 * q^29 - q^31 + q^33 - 4 * q^35 + 20 * q^37 + q^39 + 8 * q^45 + 4 * q^47 + 10 * q^49 + 11 * q^51 - 5 * q^53 + 4 * q^55 + 16 * q^57 + 11 * q^59 + 12 * q^61 - 17 * q^63 + 4 * q^65 - 2 * q^67 + 10 * q^69 + 28 * q^71 + 11 * q^73 - 6 * q^75 - 10 * q^77 - 10 * q^79 + 46 * q^81 + 7 * q^83 + 33 * q^85 - 47 * q^87 + 30 * q^89 - 10 * q^91 + 41 * q^93 - 2 * q^95 + 55 * q^97 + 17 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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