LMFDB - Embedded newform 4004.2.a.k.1.7

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			1.7
Root			$1.02188$ 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)

$f(q)$	$=$	$q+1.02188 q^{3} +3.35739 q^{5} -1.00000 q^{7} -1.95576 q^{9} +O(q^{10})$	$q+1.02188 q^{3} +3.35739 q^{5} -1.00000 q^{7} -1.95576 q^{9} +1.00000 q^{11} +1.00000 q^{13} +3.43085 q^{15} -7.02073 q^{17} +6.34953 q^{19} -1.02188 q^{21} +0.633218 q^{23} +6.27207 q^{25} -5.06419 q^{27} +9.44347 q^{29} +0.386753 q^{31} +1.02188 q^{33} -3.35739 q^{35} +10.7276 q^{37} +1.02188 q^{39} -10.7441 q^{41} +7.84585 q^{43} -6.56625 q^{45} +0.633218 q^{47} +1.00000 q^{49} -7.17435 q^{51} +2.46650 q^{53} +3.35739 q^{55} +6.48846 q^{57} -7.83022 q^{59} +13.5536 q^{61} +1.95576 q^{63} +3.35739 q^{65} +8.43037 q^{67} +0.647073 q^{69} +12.9556 q^{71} +2.10919 q^{73} +6.40931 q^{75} -1.00000 q^{77} +4.48960 q^{79} +0.692281 q^{81} -4.65917 q^{83} -23.5713 q^{85} +9.65009 q^{87} +9.54087 q^{89} -1.00000 q^{91} +0.395216 q^{93} +21.3179 q^{95} +7.14717 q^{97} -1.95576 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.02188	0.589983	0.294991	−	0.955500i	$-0.404683\pi$
$3$	1.02188	0.589983	0.294991	+	0.955500i	$0.404683\pi$
$4$	0	0
$5$	3.35739	1.50147	0.750735	−	0.660603i	$-0.229699\pi$
$5$	3.35739	1.50147	0.750735	+	0.660603i	$0.229699\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−1.95576	−0.651920
$10$	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
$15$	3.43085	0.885842
$16$	0	0
$17$	−7.02073	−1.70278	−0.851389	−	0.524536i	$-0.824239\pi$
$17$	−7.02073	−1.70278	−0.851389	+	0.524536i	$0.824239\pi$
$18$	0	0
$19$	6.34953	1.45668	0.728341	−	0.685215i	$-0.240292\pi$
$19$	6.34953	1.45668	0.728341	+	0.685215i	$0.240292\pi$
$20$	0	0
$21$	−1.02188	−0.222993
$22$	0	0
$23$	0.633218	0.132035	0.0660176	−	0.997818i	$-0.478971\pi$
$23$	0.633218	0.132035	0.0660176	+	0.997818i	$0.478971\pi$
$24$	0	0
$25$	6.27207	1.25441
$26$	0	0
$27$	−5.06419	−0.974605
$28$	0	0
$29$	9.44347	1.75361	0.876804	−	0.480848i	$-0.159671\pi$
$29$	9.44347	1.75361	0.876804	+	0.480848i	$0.159671\pi$
$30$	0	0
$31$	0.386753	0.0694630	0.0347315	−	0.999397i	$-0.488942\pi$
$31$	0.386753	0.0694630	0.0347315	+	0.999397i	$0.488942\pi$
$32$	0	0
$33$	1.02188	0.177887
$34$	0	0
$35$	−3.35739	−0.567503
$36$	0	0
$37$	10.7276	1.76360	0.881801	−	0.471621i	$-0.156331\pi$
$37$	10.7276	1.76360	0.881801	+	0.471621i	$0.156331\pi$
$38$	0	0
$39$	1.02188	0.163632
$40$	0	0
$41$	−10.7441	−1.67794	−0.838972	−	0.544174i	$-0.816843\pi$
$41$	−10.7441	−1.67794	−0.838972	+	0.544174i	$0.816843\pi$
$42$	0	0
$43$	7.84585	1.19648	0.598240	−	0.801317i	$-0.295867\pi$
$43$	7.84585	1.19648	0.598240	+	0.801317i	$0.295867\pi$
$44$	0	0
$45$	−6.56625	−0.978839
$46$	0	0
$47$	0.633218	0.0923644	0.0461822	−	0.998933i	$-0.485295\pi$
$47$	0.633218	0.0923644	0.0461822	+	0.998933i	$0.485295\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−7.17435	−1.00461
$52$	0	0
$53$	2.46650	0.338799	0.169400	−	0.985547i	$-0.445817\pi$
$53$	2.46650	0.338799	0.169400	+	0.985547i	$0.445817\pi$
$54$	0	0
$55$	3.35739	0.452711
$56$	0	0
$57$	6.48846	0.859417
$58$	0	0
$59$	−7.83022	−1.01941	−0.509704	−	0.860350i	$-0.670245\pi$
$59$	−7.83022	−1.01941	−0.509704	+	0.860350i	$0.670245\pi$
$60$	0	0
$61$	13.5536	1.73536	0.867679	−	0.497125i	$-0.165611\pi$
$61$	13.5536	1.73536	0.867679	+	0.497125i	$0.165611\pi$
$62$	0	0
$63$	1.95576	0.246403
$64$	0	0
$65$	3.35739	0.416433
$66$	0	0
$67$	8.43037	1.02993	0.514967	−	0.857210i	$-0.327804\pi$
$67$	8.43037	1.02993	0.514967	+	0.857210i	$0.327804\pi$
$68$	0	0
$69$	0.647073	0.0778985
$70$	0	0
$71$	12.9556	1.53754	0.768770	−	0.639525i	$-0.220869\pi$
$71$	12.9556	1.53754	0.768770	+	0.639525i	$0.220869\pi$
$72$	0	0
$73$	2.10919	0.246862	0.123431	−	0.992353i	$-0.460610\pi$
$73$	2.10919	0.246862	0.123431	+	0.992353i	$0.460610\pi$
$74$	0	0
$75$	6.40931	0.740083
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	4.48960	0.505120	0.252560	−	0.967581i	$-0.418728\pi$
$79$	4.48960	0.505120	0.252560	+	0.967581i	$0.418728\pi$
$80$	0	0
$81$	0.692281	0.0769201
$82$	0	0
$83$	−4.65917	−0.511410	−0.255705	−	0.966755i	$-0.582308\pi$
$83$	−4.65917	−0.511410	−0.255705	+	0.966755i	$0.582308\pi$
$84$	0	0
$85$	−23.5713	−2.55667
$86$	0	0
$87$	9.65009	1.03460
$88$	0	0
$89$	9.54087	1.01133	0.505665	−	0.862730i	$-0.331247\pi$
$89$	9.54087	1.01133	0.505665	+	0.862730i	$0.331247\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	0.395216	0.0409820
$94$	0	0
$95$	21.3179	2.18717
$96$	0	0
$97$	7.14717	0.725685	0.362843	−	0.931850i	$-0.381806\pi$
$97$	7.14717	0.725685	0.362843	+	0.931850i	$0.381806\pi$
$98$	0	0
$99$	−1.95576	−0.196561
$100$	0	0
$101$	5.32866	0.530221	0.265111	−	0.964218i	$-0.414591\pi$
$101$	5.32866	0.530221	0.265111	+	0.964218i	$0.414591\pi$
$102$	0	0
$103$	8.83600	0.870637	0.435318	−	0.900277i	$-0.356636\pi$
$103$	8.83600	0.870637	0.435318	+	0.900277i	$0.356636\pi$
$104$	0	0
$105$	−3.43085	−0.334817
$106$	0	0
$107$	−10.2488	−0.990788	−0.495394	−	0.868669i	$-0.664976\pi$
$107$	−10.2488	−0.990788	−0.495394	+	0.868669i	$0.664976\pi$
$108$	0	0
$109$	−11.0307	−1.05655	−0.528275	−	0.849074i	$-0.677161\pi$
$109$	−11.0307	−1.05655	−0.528275	+	0.849074i	$0.677161\pi$
$110$	0	0
$111$	10.9623	1.04050
$112$	0	0
$113$	−2.55909	−0.240739	−0.120369	−	0.992729i	$-0.538408\pi$
$113$	−2.55909	−0.240739	−0.120369	+	0.992729i	$0.538408\pi$
$114$	0	0
$115$	2.12596	0.198247
$116$	0	0
$117$	−1.95576	−0.180810
$118$	0	0
$119$	7.02073	0.643589
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−10.9792	−0.989958
$124$	0	0
$125$	4.27085	0.381996
$126$	0	0
$127$	−1.47805	−0.131155	−0.0655776	−	0.997847i	$-0.520889\pi$
$127$	−1.47805	−0.131155	−0.0655776	+	0.997847i	$0.520889\pi$
$128$	0	0
$129$	8.01752	0.705903
$130$	0	0
$131$	−14.5209	−1.26870	−0.634348	−	0.773048i	$-0.718731\pi$
$131$	−14.5209	−1.26870	−0.634348	+	0.773048i	$0.718731\pi$
$132$	0	0
$133$	−6.34953	−0.550574
$134$	0	0
$135$	−17.0025	−1.46334
$136$	0	0
$137$	−20.2045	−1.72618	−0.863091	−	0.505048i	$-0.831475\pi$
$137$	−20.2045	−1.72618	−0.863091	+	0.505048i	$0.831475\pi$
$138$	0	0
$139$	1.48834	0.126239	0.0631195	−	0.998006i	$-0.479895\pi$
$139$	1.48834	0.126239	0.0631195	+	0.998006i	$0.479895\pi$
$140$	0	0
$141$	0.647073	0.0544934
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	31.7054	2.63299
$146$	0	0
$147$	1.02188	0.0842833
$148$	0	0
$149$	15.0143	1.23002	0.615008	−	0.788521i	$-0.289153\pi$
$149$	15.0143	1.23002	0.615008	+	0.788521i	$0.289153\pi$
$150$	0	0
$151$	−21.9672	−1.78766	−0.893832	−	0.448403i	$-0.851993\pi$
$151$	−21.9672	−1.78766	−0.893832	+	0.448403i	$0.851993\pi$
$152$	0	0
$153$	13.7309	1.11007
$154$	0	0
$155$	1.29848	0.104297
$156$	0	0
$157$	13.7380	1.09641	0.548205	−	0.836344i	$-0.315311\pi$
$157$	13.7380	1.09641	0.548205	+	0.836344i	$0.315311\pi$
$158$	0	0
$159$	2.52046	0.199886
$160$	0	0
$161$	−0.633218	−0.0499046
$162$	0	0
$163$	−18.2723	−1.43120	−0.715599	−	0.698512i	$-0.753846\pi$
$163$	−18.2723	−1.43120	−0.715599	+	0.698512i	$0.753846\pi$
$164$	0	0
$165$	3.43085	0.267091
$166$	0	0
$167$	−15.0418	−1.16397	−0.581983	−	0.813201i	$-0.697723\pi$
$167$	−15.0418	−1.16397	−0.581983	+	0.813201i	$0.697723\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−12.4182	−0.949640
$172$	0	0
$173$	−9.33719	−0.709894	−0.354947	−	0.934886i	$-0.615501\pi$
$173$	−9.33719	−0.709894	−0.354947	+	0.934886i	$0.615501\pi$
$174$	0	0
$175$	−6.27207	−0.474124
$176$	0	0
$177$	−8.00155	−0.601433
$178$	0	0
$179$	12.8482	0.960322	0.480161	−	0.877180i	$-0.340578\pi$
$179$	12.8482	0.960322	0.480161	+	0.877180i	$0.340578\pi$
$180$	0	0
$181$	−9.60251	−0.713749	−0.356875	−	0.934152i	$-0.616158\pi$
$181$	−9.60251	−0.713749	−0.356875	+	0.934152i	$0.616158\pi$
$182$	0	0
$183$	13.8501	1.02383
$184$	0	0
$185$	36.0167	2.64800
$186$	0	0
$187$	−7.02073	−0.513407
$188$	0	0
$189$	5.06419	0.368366
$190$	0	0
$191$	18.1810	1.31553	0.657764	−	0.753224i	$-0.271502\pi$
$191$	18.1810	1.31553	0.657764	+	0.753224i	$0.271502\pi$
$192$	0	0
$193$	16.6558	1.19891	0.599457	−	0.800407i	$-0.295383\pi$
$193$	16.6558	1.19891	0.599457	+	0.800407i	$0.295383\pi$
$194$	0	0
$195$	3.43085	0.245688
$196$	0	0
$197$	22.2840	1.58767	0.793834	−	0.608135i	$-0.208082\pi$
$197$	22.2840	1.58767	0.793834	+	0.608135i	$0.208082\pi$
$198$	0	0
$199$	−5.95565	−0.422185	−0.211092	−	0.977466i	$-0.567702\pi$
$199$	−5.95565	−0.422185	−0.211092	+	0.977466i	$0.567702\pi$
$200$	0	0
$201$	8.61483	0.607643
$202$	0	0
$203$	−9.44347	−0.662801
$204$	0	0
$205$	−36.0721	−2.51938
$206$	0	0
$207$	−1.23842	−0.0860764
$208$	0	0
$209$	6.34953	0.439206
$210$	0	0
$211$	−20.3250	−1.39923	−0.699615	−	0.714520i	$-0.746645\pi$
$211$	−20.3250	−1.39923	−0.699615	+	0.714520i	$0.746645\pi$
$212$	0	0
$213$	13.2390	0.907123
$214$	0	0
$215$	26.3416	1.79648
$216$	0	0
$217$	−0.386753	−0.0262545
$218$	0	0
$219$	2.15534	0.145644
$220$	0	0
$221$	−7.02073	−0.472265
$222$	0	0
$223$	0.164264	0.0109999	0.00549997	−	0.999985i	$-0.498249\pi$
$223$	0.164264	0.0109999	0.00549997	+	0.999985i	$0.498249\pi$
$224$	0	0
$225$	−12.2667	−0.817778
$226$	0	0
$227$	−24.6324	−1.63491	−0.817455	−	0.575993i	$-0.804616\pi$
$227$	−24.6324	−1.63491	−0.817455	+	0.575993i	$0.804616\pi$
$228$	0	0
$229$	−5.00116	−0.330486	−0.165243	−	0.986253i	$-0.552841\pi$
$229$	−5.00116	−0.330486	−0.165243	+	0.986253i	$0.552841\pi$
$230$	0	0
$231$	−1.02188	−0.0672348
$232$	0	0
$233$	−12.9898	−0.850989	−0.425495	−	0.904961i	$-0.639900\pi$
$233$	−12.9898	−0.850989	−0.425495	+	0.904961i	$0.639900\pi$
$234$	0	0
$235$	2.12596	0.138682
$236$	0	0
$237$	4.58784	0.298012
$238$	0	0
$239$	−17.5964	−1.13822	−0.569109	−	0.822262i	$-0.692712\pi$
$239$	−17.5964	−1.13822	−0.569109	+	0.822262i	$0.692712\pi$
$240$	0	0
$241$	−16.4436	−1.05922	−0.529612	−	0.848240i	$-0.677663\pi$
$241$	−16.4436	−1.05922	−0.529612	+	0.848240i	$0.677663\pi$
$242$	0	0
$243$	15.9000	1.01999
$244$	0	0
$245$	3.35739	0.214496
$246$	0	0
$247$	6.34953	0.404011
$248$	0	0
$249$	−4.76111	−0.301723
$250$	0	0
$251$	0.426797	0.0269392	0.0134696	−	0.999909i	$-0.495712\pi$
$251$	0.426797	0.0269392	0.0134696	+	0.999909i	$0.495712\pi$
$252$	0	0
$253$	0.633218	0.0398101
$254$	0	0
$255$	−24.0871	−1.50839
$256$	0	0
$257$	−4.98072	−0.310689	−0.155344	−	0.987860i	$-0.549649\pi$
$257$	−4.98072	−0.310689	−0.155344	+	0.987860i	$0.549649\pi$
$258$	0	0
$259$	−10.7276	−0.666579
$260$	0	0
$261$	−18.4692	−1.14321
$262$	0	0
$263$	−6.93652	−0.427724	−0.213862	−	0.976864i	$-0.568604\pi$
$263$	−6.93652	−0.427724	−0.213862	+	0.976864i	$0.568604\pi$
$264$	0	0
$265$	8.28099	0.508697
$266$	0	0
$267$	9.74963	0.596668
$268$	0	0
$269$	21.1307	1.28836	0.644180	−	0.764874i	$-0.277199\pi$
$269$	21.1307	1.28836	0.644180	+	0.764874i	$0.277199\pi$
$270$	0	0
$271$	−19.4799	−1.18332	−0.591659	−	0.806188i	$-0.701527\pi$
$271$	−19.4799	−1.18332	−0.591659	+	0.806188i	$0.701527\pi$
$272$	0	0
$273$	−1.02188	−0.0618470
$274$	0	0
$275$	6.27207	0.378220
$276$	0	0
$277$	9.58754	0.576059	0.288030	−	0.957621i	$-0.407000\pi$
$277$	9.58754	0.576059	0.288030	+	0.957621i	$0.407000\pi$
$278$	0	0
$279$	−0.756397	−0.0452843
$280$	0	0
$281$	14.2149	0.847990	0.423995	−	0.905664i	$-0.360627\pi$
$281$	14.2149	0.847990	0.423995	+	0.905664i	$0.360627\pi$
$282$	0	0
$283$	−12.7813	−0.759767	−0.379883	−	0.925034i	$-0.624036\pi$
$283$	−12.7813	−0.759767	−0.379883	+	0.925034i	$0.624036\pi$
$284$	0	0
$285$	21.7843	1.29039
$286$	0	0
$287$	10.7441	0.634203
$288$	0	0
$289$	32.2906	1.89945
$290$	0	0
$291$	7.30355	0.428142
$292$	0	0
$293$	−6.82028	−0.398445	−0.199223	−	0.979954i	$-0.563842\pi$
$293$	−6.82028	−0.398445	−0.199223	+	0.979954i	$0.563842\pi$
$294$	0	0
$295$	−26.2891	−1.53061
$296$	0	0
$297$	−5.06419	−0.293854
$298$	0	0
$299$	0.633218	0.0366200
$300$	0	0
$301$	−7.84585	−0.452227
$302$	0	0
$303$	5.44525	0.312822
$304$	0	0
$305$	45.5047	2.60559
$306$	0	0
$307$	−3.64847	−0.208229	−0.104115	−	0.994565i	$-0.533201\pi$
$307$	−3.64847	−0.208229	−0.104115	+	0.994565i	$0.533201\pi$
$308$	0	0
$309$	9.02933	0.513661
$310$	0	0
$311$	2.09534	0.118816	0.0594079	−	0.998234i	$-0.481079\pi$
$311$	2.09534	0.118816	0.0594079	+	0.998234i	$0.481079\pi$
$312$	0	0
$313$	−1.17448	−0.0663856	−0.0331928	−	0.999449i	$-0.510568\pi$
$313$	−1.17448	−0.0663856	−0.0331928	+	0.999449i	$0.510568\pi$
$314$	0	0
$315$	6.56625	0.369966
$316$	0	0
$317$	−2.99527	−0.168231	−0.0841157	−	0.996456i	$-0.526807\pi$
$317$	−2.99527	−0.168231	−0.0841157	+	0.996456i	$0.526807\pi$
$318$	0	0
$319$	9.44347	0.528733
$320$	0	0
$321$	−10.4730	−0.584548
$322$	0	0
$323$	−44.5783	−2.48040
$324$	0	0
$325$	6.27207	0.347912
$326$	0	0
$327$	−11.2721	−0.623346
$328$	0	0
$329$	−0.633218	−0.0349105
$330$	0	0
$331$	−3.22817	−0.177436	−0.0887180	−	0.996057i	$-0.528277\pi$
$331$	−3.22817	−0.177436	−0.0887180	+	0.996057i	$0.528277\pi$
$332$	0	0
$333$	−20.9806	−1.14973
$334$	0	0
$335$	28.3041	1.54642
$336$	0	0
$337$	−34.7655	−1.89380	−0.946900	−	0.321530i	$-0.895803\pi$
$337$	−34.7655	−1.89380	−0.946900	+	0.321530i	$0.895803\pi$
$338$	0	0
$339$	−2.61508	−0.142032
$340$	0	0
$341$	0.386753	0.0209439
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	2.17248	0.116962
$346$	0	0
$347$	−15.0929	−0.810228	−0.405114	−	0.914266i	$-0.632768\pi$
$347$	−15.0929	−0.810228	−0.405114	+	0.914266i	$0.632768\pi$
$348$	0	0
$349$	14.6164	0.782400	0.391200	−	0.920306i	$-0.372060\pi$
$349$	14.6164	0.782400	0.391200	+	0.920306i	$0.372060\pi$
$350$	0	0
$351$	−5.06419	−0.270307
$352$	0	0
$353$	−14.4750	−0.770426	−0.385213	−	0.922828i	$-0.625872\pi$
$353$	−14.4750	−0.770426	−0.385213	+	0.922828i	$0.625872\pi$
$354$	0	0
$355$	43.4968	2.30857
$356$	0	0
$357$	7.17435	0.379707
$358$	0	0
$359$	−4.50518	−0.237774	−0.118887	−	0.992908i	$-0.537933\pi$
$359$	−4.50518	−0.237774	−0.118887	+	0.992908i	$0.537933\pi$
$360$	0	0
$361$	21.3165	1.12192
$362$	0	0
$363$	1.02188	0.0536348
$364$	0	0
$365$	7.08137	0.370656
$366$	0	0
$367$	−19.5387	−1.01991	−0.509957	−	0.860200i	$-0.670339\pi$
$367$	−19.5387	−1.01991	−0.509957	+	0.860200i	$0.670339\pi$
$368$	0	0
$369$	21.0129	1.09389
$370$	0	0
$371$	−2.46650	−0.128054
$372$	0	0
$373$	22.2819	1.15371	0.576857	−	0.816845i	$-0.304279\pi$
$373$	22.2819	1.15371	0.576857	+	0.816845i	$0.304279\pi$
$374$	0	0
$375$	4.36430	0.225371
$376$	0	0
$377$	9.44347	0.486363
$378$	0	0
$379$	35.3833	1.81752	0.908759	−	0.417321i	$-0.137031\pi$
$379$	35.3833	1.81752	0.908759	+	0.417321i	$0.137031\pi$
$380$	0	0
$381$	−1.51039	−0.0773794
$382$	0	0
$383$	−10.4308	−0.532991	−0.266495	−	0.963836i	$-0.585866\pi$
$383$	−10.4308	−0.532991	−0.266495	+	0.963836i	$0.585866\pi$
$384$	0	0
$385$	−3.35739	−0.171108
$386$	0	0
$387$	−15.3446	−0.780010
$388$	0	0
$389$	−11.3587	−0.575911	−0.287955	−	0.957644i	$-0.592975\pi$
$389$	−11.3587	−0.575911	−0.287955	+	0.957644i	$0.592975\pi$
$390$	0	0
$391$	−4.44565	−0.224826
$392$	0	0
$393$	−14.8386	−0.748509
$394$	0	0
$395$	15.0733	0.758422
$396$	0	0
$397$	−10.5660	−0.530292	−0.265146	−	0.964208i	$-0.585420\pi$
$397$	−10.5660	−0.530292	−0.265146	+	0.964208i	$0.585420\pi$
$398$	0	0
$399$	−6.48846	−0.324829
$400$	0	0
$401$	−11.6822	−0.583379	−0.291689	−	0.956513i	$-0.594217\pi$
$401$	−11.6822	−0.583379	−0.291689	+	0.956513i	$0.594217\pi$
$402$	0	0
$403$	0.386753	0.0192656
$404$	0	0
$405$	2.32426	0.115493
$406$	0	0
$407$	10.7276	0.531746
$408$	0	0
$409$	25.4396	1.25791	0.628955	−	0.777442i	$-0.283483\pi$
$409$	25.4396	1.25791	0.628955	+	0.777442i	$0.283483\pi$
$410$	0	0
$411$	−20.6465	−1.01842
$412$	0	0
$413$	7.83022	0.385300
$414$	0	0
$415$	−15.6426	−0.767867
$416$	0	0
$417$	1.52090	0.0744788
$418$	0	0
$419$	20.5002	1.00150	0.500750	−	0.865592i	$-0.333057\pi$
$419$	20.5002	1.00150	0.500750	+	0.865592i	$0.333057\pi$
$420$	0	0
$421$	19.3789	0.944471	0.472235	−	0.881472i	$-0.343447\pi$
$421$	19.3789	0.944471	0.472235	+	0.881472i	$0.343447\pi$
$422$	0	0
$423$	−1.23842	−0.0602142
$424$	0	0
$425$	−44.0345	−2.13599
$426$	0	0
$427$	−13.5536	−0.655903
$428$	0	0
$429$	1.02188	0.0493368
$430$	0	0
$431$	21.3881	1.03023	0.515115	−	0.857121i	$-0.327749\pi$
$431$	21.3881	1.03023	0.515115	+	0.857121i	$0.327749\pi$
$432$	0	0
$433$	17.5933	0.845478	0.422739	−	0.906251i	$-0.361069\pi$
$433$	17.5933	0.845478	0.422739	+	0.906251i	$0.361069\pi$
$434$	0	0
$435$	32.3991	1.55342
$436$	0	0
$437$	4.02064	0.192333
$438$	0	0
$439$	1.15284	0.0550218	0.0275109	−	0.999622i	$-0.491242\pi$
$439$	1.15284	0.0550218	0.0275109	+	0.999622i	$0.491242\pi$
$440$	0	0
$441$	−1.95576	−0.0931315
$442$	0	0
$443$	−12.0307	−0.571596	−0.285798	−	0.958290i	$-0.592259\pi$
$443$	−12.0307	−0.571596	−0.285798	+	0.958290i	$0.592259\pi$
$444$	0	0
$445$	32.0324	1.51848
$446$	0	0
$447$	15.3428	0.725689
$448$	0	0
$449$	−19.3804	−0.914619	−0.457309	−	0.889308i	$-0.651187\pi$
$449$	−19.3804	−0.914619	−0.457309	+	0.889308i	$0.651187\pi$
$450$	0	0
$451$	−10.7441	−0.505919
$452$	0	0
$453$	−22.4478	−1.05469
$454$	0	0
$455$	−3.35739	−0.157397
$456$	0	0
$457$	18.4488	0.862997	0.431498	−	0.902114i	$-0.357985\pi$
$457$	18.4488	0.862997	0.431498	+	0.902114i	$0.357985\pi$
$458$	0	0
$459$	35.5543	1.65953
$460$	0	0
$461$	1.93996	0.0903528	0.0451764	−	0.998979i	$-0.485615\pi$
$461$	1.93996	0.0903528	0.0451764	+	0.998979i	$0.485615\pi$
$462$	0	0
$463$	20.4001	0.948073	0.474036	−	0.880505i	$-0.342797\pi$
$463$	20.4001	0.948073	0.474036	+	0.880505i	$0.342797\pi$
$464$	0	0
$465$	1.32689	0.0615332
$466$	0	0
$467$	−3.17042	−0.146710	−0.0733548	−	0.997306i	$-0.523371\pi$
$467$	−3.17042	−0.146710	−0.0733548	+	0.997306i	$0.523371\pi$
$468$	0	0
$469$	−8.43037	−0.389278
$470$	0	0
$471$	14.0386	0.646863
$472$	0	0
$473$	7.84585	0.360753
$474$	0	0
$475$	39.8247	1.82728
$476$	0	0
$477$	−4.82388	−0.220870
$478$	0	0
$479$	15.1229	0.690981	0.345490	−	0.938422i	$-0.387713\pi$
$479$	15.1229	0.690981	0.345490	+	0.938422i	$0.387713\pi$
$480$	0	0
$481$	10.7276	0.489135
$482$	0	0
$483$	−0.647073	−0.0294429
$484$	0	0
$485$	23.9959	1.08960
$486$	0	0
$487$	27.8772	1.26323	0.631617	−	0.775280i	$-0.282391\pi$
$487$	27.8772	1.26323	0.631617	+	0.775280i	$0.282391\pi$
$488$	0	0
$489$	−18.6721	−0.844382
$490$	0	0
$491$	14.5461	0.656456	0.328228	−	0.944598i	$-0.393548\pi$
$491$	14.5461	0.656456	0.328228	+	0.944598i	$0.393548\pi$
$492$	0	0
$493$	−66.3000	−2.98600
$494$	0	0
$495$	−6.56625	−0.295131
$496$	0	0
$497$	−12.9556	−0.581136
$498$	0	0
$499$	−20.4939	−0.917435	−0.458717	−	0.888582i	$-0.651691\pi$
$499$	−20.4939	−0.917435	−0.458717	+	0.888582i	$0.651691\pi$
$500$	0	0
$501$	−15.3709	−0.686720
$502$	0	0
$503$	3.36059	0.149841	0.0749206	−	0.997190i	$-0.476130\pi$
$503$	3.36059	0.149841	0.0749206	+	0.997190i	$0.476130\pi$
$504$	0	0
$505$	17.8904	0.796112
$506$	0	0
$507$	1.02188	0.0453833
$508$	0	0
$509$	−3.22497	−0.142944	−0.0714722	−	0.997443i	$-0.522770\pi$
$509$	−3.22497	−0.142944	−0.0714722	+	0.997443i	$0.522770\pi$
$510$	0	0
$511$	−2.10919	−0.0933050
$512$	0	0
$513$	−32.1553	−1.41969
$514$	0	0
$515$	29.6659	1.30724
$516$	0	0
$517$	0.633218	0.0278489
$518$	0	0
$519$	−9.54150	−0.418825
$520$	0	0
$521$	25.6522	1.12384	0.561922	−	0.827190i	$-0.310062\pi$
$521$	25.6522	1.12384	0.561922	+	0.827190i	$0.310062\pi$
$522$	0	0
$523$	−37.1457	−1.62427	−0.812134	−	0.583472i	$-0.801694\pi$
$523$	−37.1457	−1.62427	−0.812134	+	0.583472i	$0.801694\pi$
$524$	0	0
$525$	−6.40931	−0.279725
$526$	0	0
$527$	−2.71529	−0.118280
$528$	0	0
$529$	−22.5990	−0.982567
$530$	0	0
$531$	15.3140	0.664572
$532$	0	0
$533$	−10.7441	−0.465378
$534$	0	0
$535$	−34.4092	−1.48764
$536$	0	0
$537$	13.1294	0.566573
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	−13.1871	−0.566956	−0.283478	−	0.958979i	$-0.591488\pi$
$541$	−13.1871	−0.566956	−0.283478	+	0.958979i	$0.591488\pi$
$542$	0	0
$543$	−9.81262	−0.421100
$544$	0	0
$545$	−37.0344	−1.58638
$546$	0	0
$547$	−13.1816	−0.563602	−0.281801	−	0.959473i	$-0.590932\pi$
$547$	−13.1816	−0.563602	−0.281801	+	0.959473i	$0.590932\pi$
$548$	0	0
$549$	−26.5075	−1.13131
$550$	0	0
$551$	59.9616	2.55445
$552$	0	0
$553$	−4.48960	−0.190917
$554$	0	0
$555$	36.8047	1.56227
$556$	0	0
$557$	13.8213	0.585627	0.292814	−	0.956170i	$-0.405408\pi$
$557$	13.8213	0.585627	0.292814	+	0.956170i	$0.405408\pi$
$558$	0	0
$559$	7.84585	0.331844
$560$	0	0
$561$	−7.17435	−0.302901
$562$	0	0
$563$	−21.4098	−0.902314	−0.451157	−	0.892445i	$-0.648989\pi$
$563$	−21.4098	−0.902314	−0.451157	+	0.892445i	$0.648989\pi$
$564$	0	0
$565$	−8.59187	−0.361463
$566$	0	0
$567$	−0.692281	−0.0290731
$568$	0	0
$569$	38.0399	1.59472	0.797359	−	0.603506i	$-0.206230\pi$
$569$	38.0399	1.59472	0.797359	+	0.603506i	$0.206230\pi$
$570$	0	0
$571$	−0.0775796	−0.00324661	−0.00162330	−	0.999999i	$-0.500517\pi$
$571$	−0.0775796	−0.00324661	−0.00162330	+	0.999999i	$0.500517\pi$
$572$	0	0
$573$	18.5788	0.776139
$574$	0	0
$575$	3.97159	0.165627
$576$	0	0
$577$	21.1506	0.880512	0.440256	−	0.897872i	$-0.354888\pi$
$577$	21.1506	0.880512	0.440256	+	0.897872i	$0.354888\pi$
$578$	0	0
$579$	17.0203	0.707338
$580$	0	0
$581$	4.65917	0.193295
$582$	0	0
$583$	2.46650	0.102152
$584$	0	0
$585$	−6.56625	−0.271481
$586$	0	0
$587$	40.3977	1.66739	0.833696	−	0.552224i	$-0.186221\pi$
$587$	40.3977	1.66739	0.833696	+	0.552224i	$0.186221\pi$
$588$	0	0
$589$	2.45570	0.101185
$590$	0	0
$591$	22.7716	0.936697
$592$	0	0
$593$	36.2908	1.49029	0.745143	−	0.666905i	$-0.232381\pi$
$593$	36.2908	1.49029	0.745143	+	0.666905i	$0.232381\pi$
$594$	0	0
$595$	23.5713	0.966330
$596$	0	0
$597$	−6.08596	−0.249082
$598$	0	0
$599$	−20.1665	−0.823979	−0.411989	−	0.911189i	$-0.635166\pi$
$599$	−20.1665	−0.823979	−0.411989	+	0.911189i	$0.635166\pi$
$600$	0	0
$601$	−14.3867	−0.586845	−0.293423	−	0.955983i	$-0.594794\pi$
$601$	−14.3867	−0.586845	−0.293423	+	0.955983i	$0.594794\pi$
$602$	0	0
$603$	−16.4878	−0.671435
$604$	0	0
$605$	3.35739	0.136497
$606$	0	0
$607$	31.0305	1.25949	0.629745	−	0.776802i	$-0.283160\pi$
$607$	31.0305	1.25949	0.629745	+	0.776802i	$0.283160\pi$
$608$	0	0
$609$	−9.65009	−0.391041
$610$	0	0
$611$	0.633218	0.0256173
$612$	0	0
$613$	−39.6830	−1.60278	−0.801391	−	0.598140i	$-0.795906\pi$
$613$	−39.6830	−1.60278	−0.801391	+	0.598140i	$0.795906\pi$
$614$	0	0
$615$	−36.8614	−1.48639
$616$	0	0
$617$	−11.0302	−0.444057	−0.222029	−	0.975040i	$-0.571268\pi$
$617$	−11.0302	−0.444057	−0.222029	+	0.975040i	$0.571268\pi$
$618$	0	0
$619$	−32.7705	−1.31716	−0.658578	−	0.752512i	$-0.728842\pi$
$619$	−32.7705	−1.31716	−0.658578	+	0.752512i	$0.728842\pi$
$620$	0	0
$621$	−3.20674	−0.128682
$622$	0	0
$623$	−9.54087	−0.382247
$624$	0	0
$625$	−17.0215	−0.680858
$626$	0	0
$627$	6.48846	0.259124
$628$	0	0
$629$	−75.3154	−3.00302
$630$	0	0
$631$	5.43573	0.216393	0.108196	−	0.994130i	$-0.465492\pi$
$631$	5.43573	0.216393	0.108196	+	0.994130i	$0.465492\pi$
$632$	0	0
$633$	−20.7697	−0.825522
$634$	0	0
$635$	−4.96238	−0.196926
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−25.3380	−1.00235
$640$	0	0
$641$	19.0844	0.753788	0.376894	−	0.926256i	$-0.376992\pi$
$641$	19.0844	0.753788	0.376894	+	0.926256i	$0.376992\pi$
$642$	0	0
$643$	26.4114	1.04157	0.520783	−	0.853689i	$-0.325640\pi$
$643$	26.4114	1.04157	0.520783	+	0.853689i	$0.325640\pi$
$644$	0	0
$645$	26.9180	1.05989
$646$	0	0
$647$	25.7257	1.01138	0.505690	−	0.862715i	$-0.331238\pi$
$647$	25.7257	1.01138	0.505690	+	0.862715i	$0.331238\pi$
$648$	0	0
$649$	−7.83022	−0.307363
$650$	0	0
$651$	−0.395216	−0.0154897
$652$	0	0
$653$	−5.22147	−0.204332	−0.102166	−	0.994767i	$-0.532577\pi$
$653$	−5.22147	−0.204332	−0.102166	+	0.994767i	$0.532577\pi$
$654$	0	0
$655$	−48.7523	−1.90491
$656$	0	0
$657$	−4.12507	−0.160934
$658$	0	0
$659$	−46.1796	−1.79890	−0.899450	−	0.437023i	$-0.856033\pi$
$659$	−46.1796	−1.79890	−0.899450	+	0.437023i	$0.856033\pi$
$660$	0	0
$661$	−27.3247	−1.06281	−0.531404	−	0.847119i	$-0.678335\pi$
$661$	−27.3247	−1.06281	−0.531404	+	0.847119i	$0.678335\pi$
$662$	0	0
$663$	−7.17435	−0.278628
$664$	0	0
$665$	−21.3179	−0.826671
$666$	0	0
$667$	5.97978	0.231538
$668$	0	0
$669$	0.167858	0.00648977
$670$	0	0
$671$	13.5536	0.523230
$672$	0	0
$673$	−28.6418	−1.10406	−0.552030	−	0.833824i	$-0.686147\pi$
$673$	−28.6418	−1.10406	−0.552030	+	0.833824i	$0.686147\pi$
$674$	0	0
$675$	−31.7630	−1.22256
$676$	0	0
$677$	−18.1161	−0.696260	−0.348130	−	0.937446i	$-0.613183\pi$
$677$	−18.1161	−0.696260	−0.348130	+	0.937446i	$0.613183\pi$
$678$	0	0
$679$	−7.14717	−0.274283
$680$	0	0
$681$	−25.1714	−0.964568
$682$	0	0
$683$	−13.6202	−0.521162	−0.260581	−	0.965452i	$-0.583914\pi$
$683$	−13.6202	−0.521162	−0.260581	+	0.965452i	$0.583914\pi$
$684$	0	0
$685$	−67.8343	−2.59181
$686$	0	0
$687$	−5.11058	−0.194981
$688$	0	0
$689$	2.46650	0.0939660
$690$	0	0
$691$	−33.4959	−1.27425	−0.637123	−	0.770762i	$-0.719876\pi$
$691$	−33.4959	−1.27425	−0.637123	+	0.770762i	$0.719876\pi$
$692$	0	0
$693$	1.95576	0.0742932
$694$	0	0
$695$	4.99692	0.189544
$696$	0	0
$697$	75.4313	2.85716
$698$	0	0
$699$	−13.2740	−0.502069
$700$	0	0
$701$	−16.3899	−0.619039	−0.309519	−	0.950893i	$-0.600168\pi$
$701$	−16.3899	−0.619039	−0.309519	+	0.950893i	$0.600168\pi$
$702$	0	0
$703$	68.1151	2.56901
$704$	0	0
$705$	2.17248	0.0818203
$706$	0	0
$707$	−5.32866	−0.200405
$708$	0	0
$709$	30.8694	1.15933	0.579663	−	0.814857i	$-0.303184\pi$
$709$	30.8694	1.15933	0.579663	+	0.814857i	$0.303184\pi$
$710$	0	0
$711$	−8.78059	−0.329298
$712$	0	0
$713$	0.244899	0.00917155
$714$	0	0
$715$	3.35739	0.125559
$716$	0	0
$717$	−17.9814	−0.671529
$718$	0	0
$719$	22.5775	0.841997	0.420998	−	0.907061i	$-0.361680\pi$
$719$	22.5775	0.841997	0.420998	+	0.907061i	$0.361680\pi$
$720$	0	0
$721$	−8.83600	−0.329070
$722$	0	0
$723$	−16.8034	−0.624924
$724$	0	0
$725$	59.2301	2.19975
$726$	0	0
$727$	−9.37601	−0.347737	−0.173868	−	0.984769i	$-0.555627\pi$
$727$	−9.37601	−0.347737	−0.173868	+	0.984769i	$0.555627\pi$
$728$	0	0
$729$	14.1711	0.524854
$730$	0	0
$731$	−55.0836	−2.03734
$732$	0	0
$733$	−13.6512	−0.504220	−0.252110	−	0.967699i	$-0.581124\pi$
$733$	−13.6512	−0.504220	−0.252110	+	0.967699i	$0.581124\pi$
$734$	0	0
$735$	3.43085	0.126549
$736$	0	0
$737$	8.43037	0.310537
$738$	0	0
$739$	28.5204	1.04914	0.524570	−	0.851368i	$-0.324226\pi$
$739$	28.5204	1.04914	0.524570	+	0.851368i	$0.324226\pi$
$740$	0	0
$741$	6.48846	0.238360
$742$	0	0
$743$	−52.6291	−1.93077	−0.965387	−	0.260821i	$-0.916007\pi$
$743$	−52.6291	−1.93077	−0.965387	+	0.260821i	$0.916007\pi$
$744$	0	0
$745$	50.4088	1.84683
$746$	0	0
$747$	9.11221	0.333398
$748$	0	0
$749$	10.2488	0.374483
$750$	0	0
$751$	−9.46762	−0.345479	−0.172739	−	0.984968i	$-0.555262\pi$
$751$	−9.46762	−0.345479	−0.172739	+	0.984968i	$0.555262\pi$
$752$	0	0
$753$	0.436135	0.0158937
$754$	0	0
$755$	−73.7524	−2.68412
$756$	0	0
$757$	6.03685	0.219413	0.109707	−	0.993964i	$-0.465009\pi$
$757$	6.03685	0.219413	0.109707	+	0.993964i	$0.465009\pi$
$758$	0	0
$759$	0.647073	0.0234873
$760$	0	0
$761$	11.1518	0.404252	0.202126	−	0.979360i	$-0.435215\pi$
$761$	11.1518	0.404252	0.202126	+	0.979360i	$0.435215\pi$
$762$	0	0
$763$	11.0307	0.399338
$764$	0	0
$765$	46.0999	1.66674
$766$	0	0
$767$	−7.83022	−0.282733
$768$	0	0
$769$	−17.4019	−0.627529	−0.313764	−	0.949501i	$-0.601590\pi$
$769$	−17.4019	−0.627529	−0.313764	+	0.949501i	$0.601590\pi$
$770$	0	0
$771$	−5.08970	−0.183301
$772$	0	0
$773$	−33.7684	−1.21456	−0.607282	−	0.794487i	$-0.707740\pi$
$773$	−33.7684	−1.21456	−0.607282	+	0.794487i	$0.707740\pi$
$774$	0	0
$775$	2.42575	0.0871354
$776$	0	0
$777$	−10.9623	−0.393270
$778$	0	0
$779$	−68.2199	−2.44423
$780$	0	0
$781$	12.9556	0.463586
$782$	0	0
$783$	−47.8235	−1.70907
$784$	0	0
$785$	46.1237	1.64623
$786$	0	0
$787$	−2.12986	−0.0759213	−0.0379606	−	0.999279i	$-0.512086\pi$
$787$	−2.12986	−0.0759213	−0.0379606	+	0.999279i	$0.512086\pi$
$788$	0	0
$789$	−7.08830	−0.252350
$790$	0	0
$791$	2.55909	0.0909908
$792$	0	0
$793$	13.5536	0.481302
$794$	0	0
$795$	8.46218	0.300123
$796$	0	0
$797$	−0.800241	−0.0283460	−0.0141730	−	0.999900i	$-0.504512\pi$
$797$	−0.800241	−0.0283460	−0.0141730	+	0.999900i	$0.504512\pi$
$798$	0	0
$799$	−4.44565	−0.157276
$800$	0	0
$801$	−18.6597	−0.659307
$802$	0	0
$803$	2.10919	0.0744316
$804$	0	0
$805$	−2.12596	−0.0749303
$806$	0	0
$807$	21.5930	0.760111
$808$	0	0
$809$	−37.1881	−1.30746	−0.653732	−	0.756727i	$-0.726797\pi$
$809$	−37.1881	−1.30746	−0.653732	+	0.756727i	$0.726797\pi$
$810$	0	0
$811$	−35.4615	−1.24522	−0.622611	−	0.782532i	$-0.713928\pi$
$811$	−35.4615	−1.24522	−0.622611	+	0.782532i	$0.713928\pi$
$812$	0	0
$813$	−19.9061	−0.698138
$814$	0	0
$815$	−61.3473	−2.14890
$816$	0	0
$817$	49.8175	1.74289
$818$	0	0
$819$	1.95576	0.0683398
$820$	0	0
$821$	19.1878	0.669657	0.334829	−	0.942279i	$-0.391322\pi$
$821$	19.1878	0.669657	0.334829	+	0.942279i	$0.391322\pi$
$822$	0	0
$823$	−28.3927	−0.989708	−0.494854	−	0.868976i	$-0.664778\pi$
$823$	−28.3927	−0.989708	−0.494854	+	0.868976i	$0.664778\pi$
$824$	0	0
$825$	6.40931	0.223144
$826$	0	0
$827$	−39.1231	−1.36044	−0.680222	−	0.733006i	$-0.738117\pi$
$827$	−39.1231	−1.36044	−0.680222	+	0.733006i	$0.738117\pi$
$828$	0	0
$829$	24.0122	0.833977	0.416989	−	0.908912i	$-0.363085\pi$
$829$	24.0122	0.833977	0.416989	+	0.908912i	$0.363085\pi$
$830$	0	0
$831$	9.79732	0.339865
$832$	0	0
$833$	−7.02073	−0.243254
$834$	0	0
$835$	−50.5011	−1.74766
$836$	0	0
$837$	−1.95859	−0.0676989
$838$	0	0
$839$	−40.1066	−1.38463	−0.692317	−	0.721594i	$-0.743410\pi$
$839$	−40.1066	−1.38463	−0.692317	+	0.721594i	$0.743410\pi$
$840$	0	0
$841$	60.1790	2.07514
$842$	0	0
$843$	14.5259	0.500300
$844$	0	0
$845$	3.35739	0.115498
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	−13.0609	−0.448249
$850$	0	0
$851$	6.79290	0.232858
$852$	0	0
$853$	13.0105	0.445470	0.222735	−	0.974879i	$-0.428502\pi$
$853$	13.0105	0.445470	0.222735	+	0.974879i	$0.428502\pi$
$854$	0	0
$855$	−41.6926	−1.42586
$856$	0	0
$857$	51.7690	1.76840	0.884198	−	0.467113i	$-0.154706\pi$
$857$	51.7690	1.76840	0.884198	+	0.467113i	$0.154706\pi$
$858$	0	0
$859$	−27.9887	−0.954962	−0.477481	−	0.878642i	$-0.658450\pi$
$859$	−27.9887	−0.954962	−0.477481	+	0.878642i	$0.658450\pi$
$860$	0	0
$861$	10.9792	0.374169
$862$	0	0
$863$	35.4182	1.20565	0.602824	−	0.797874i	$-0.294042\pi$
$863$	35.4182	1.20565	0.602824	+	0.797874i	$0.294042\pi$
$864$	0	0
$865$	−31.3486	−1.06588
$866$	0	0
$867$	32.9972	1.12064
$868$	0	0
$869$	4.48960	0.152299
$870$	0	0
$871$	8.43037	0.285652
$872$	0	0
$873$	−13.9782	−0.473089
$874$	0	0
$875$	−4.27085	−0.144381
$876$	0	0
$877$	45.4413	1.53444	0.767222	−	0.641382i	$-0.221639\pi$
$877$	45.4413	1.53444	0.767222	+	0.641382i	$0.221639\pi$
$878$	0	0
$879$	−6.96951	−0.235076
$880$	0	0
$881$	−38.3184	−1.29098	−0.645490	−	0.763769i	$-0.723347\pi$
$881$	−38.3184	−1.29098	−0.645490	+	0.763769i	$0.723347\pi$
$882$	0	0
$883$	−20.5705	−0.692251	−0.346126	−	0.938188i	$-0.612503\pi$
$883$	−20.5705	−0.692251	−0.346126	+	0.938188i	$0.612503\pi$
$884$	0	0
$885$	−26.8643	−0.903034
$886$	0	0
$887$	−36.1441	−1.21360	−0.606800	−	0.794855i	$-0.707547\pi$
$887$	−36.1441	−1.21360	−0.606800	+	0.794855i	$0.707547\pi$
$888$	0	0
$889$	1.47805	0.0495720
$890$	0	0
$891$	0.692281	0.0231923
$892$	0	0
$893$	4.02064	0.134546
$894$	0	0
$895$	43.1365	1.44190
$896$	0	0
$897$	0.647073	0.0216052
$898$	0	0
$899$	3.65229	0.121811
$900$	0	0
$901$	−17.3166	−0.576900
$902$	0	0
$903$	−8.01752	−0.266806
$904$	0	0
$905$	−32.2394	−1.07167
$906$	0	0
$907$	−52.1693	−1.73225	−0.866126	−	0.499826i	$-0.833397\pi$
$907$	−52.1693	−1.73225	−0.866126	+	0.499826i	$0.833397\pi$
$908$	0	0
$909$	−10.4216	−0.345662
$910$	0	0
$911$	10.1908	0.337635	0.168817	−	0.985647i	$-0.446005\pi$
$911$	10.1908	0.337635	0.168817	+	0.985647i	$0.446005\pi$
$912$	0	0
$913$	−4.65917	−0.154196
$914$	0	0
$915$	46.5003	1.53725
$916$	0	0
$917$	14.5209	0.479522
$918$	0	0
$919$	10.2288	0.337417	0.168709	−	0.985666i	$-0.446040\pi$
$919$	10.2288	0.337417	0.168709	+	0.985666i	$0.446040\pi$
$920$	0	0
$921$	−3.72830	−0.122852
$922$	0	0
$923$	12.9556	0.426437
$924$	0	0
$925$	67.2841	2.21229
$926$	0	0
$927$	−17.2811	−0.567586
$928$	0	0
$929$	−7.13800	−0.234190	−0.117095	−	0.993121i	$-0.537358\pi$
$929$	−7.13800	−0.234190	−0.117095	+	0.993121i	$0.537358\pi$
$930$	0	0
$931$	6.34953	0.208097
$932$	0	0
$933$	2.14119	0.0700993
$934$	0	0
$935$	−23.5713	−0.770865
$936$	0	0
$937$	−54.7895	−1.78990	−0.894948	−	0.446171i	$-0.852787\pi$
$937$	−54.7895	−1.78990	−0.894948	+	0.446171i	$0.852787\pi$
$938$	0	0
$939$	−1.20018	−0.0391664
$940$	0	0
$941$	44.6804	1.45654	0.728270	−	0.685290i	$-0.240324\pi$
$941$	44.6804	1.45654	0.728270	+	0.685290i	$0.240324\pi$
$942$	0	0
$943$	−6.80335	−0.221548
$944$	0	0
$945$	17.0025	0.553091
$946$	0	0
$947$	16.5694	0.538434	0.269217	−	0.963079i	$-0.413235\pi$
$947$	16.5694	0.538434	0.269217	+	0.963079i	$0.413235\pi$
$948$	0	0
$949$	2.10919	0.0684671
$950$	0	0
$951$	−3.06081	−0.0992536
$952$	0	0
$953$	−21.4581	−0.695095	−0.347548	−	0.937662i	$-0.612985\pi$
$953$	−21.4581	−0.695095	−0.347548	+	0.937662i	$0.612985\pi$
$954$	0	0
$955$	61.0406	1.97523
$956$	0	0
$957$	9.65009	0.311943
$958$	0	0
$959$	20.2045	0.652436
$960$	0	0
$961$	−30.8504	−0.995175
$962$	0	0
$963$	20.0442	0.645914
$964$	0	0
$965$	55.9202	1.80013
$966$	0	0
$967$	−39.0908	−1.25707	−0.628537	−	0.777779i	$-0.716346\pi$
$967$	−39.0908	−1.25707	−0.628537	+	0.777779i	$0.716346\pi$
$968$	0	0
$969$	−45.5537	−1.46340
$970$	0	0
$971$	−25.5026	−0.818419	−0.409209	−	0.912441i	$-0.634195\pi$
$971$	−25.5026	−0.818419	−0.409209	+	0.912441i	$0.634195\pi$
$972$	0	0
$973$	−1.48834	−0.0477138
$974$	0	0
$975$	6.40931	0.205262
$976$	0	0
$977$	34.2163	1.09468	0.547338	−	0.836911i	$-0.315641\pi$
$977$	34.2163	1.09468	0.547338	+	0.836911i	$0.315641\pi$
$978$	0	0
$979$	9.54087	0.304928
$980$	0	0
$981$	21.5734	0.688786
$982$	0	0
$983$	−30.3895	−0.969275	−0.484638	−	0.874715i	$-0.661048\pi$
$983$	−30.3895	−0.969275	−0.484638	+	0.874715i	$0.661048\pi$
$984$	0	0
$985$	74.8160	2.38384
$986$	0	0
$987$	−0.647073	−0.0205966
$988$	0	0
$989$	4.96814	0.157978
$990$	0	0
$991$	47.5726	1.51119	0.755597	−	0.655037i	$-0.227347\pi$
$991$	47.5726	1.51119	0.755597	+	0.655037i	$0.227347\pi$
$992$	0	0
$993$	−3.29880	−0.104684
$994$	0	0
$995$	−19.9954	−0.633898
$996$	0	0
$997$	−51.0641	−1.61722	−0.808609	−	0.588347i	$-0.799779\pi$
$997$	−51.0641	−1.61722	−0.808609	+	0.588347i	$0.799779\pi$
$998$	0	0
$999$	−54.3265	−1.71882

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.a.k.1.7	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.a.k.1.7	✓	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)

\(f(q)\)	\(=\)	\(q+1.02188 q^{3} +3.35739 q^{5} -1.00000 q^{7} -1.95576 q^{9} +O(q^{10})\)	\(q+1.02188 q^{3} +3.35739 q^{5} -1.00000 q^{7} -1.95576 q^{9} +1.00000 q^{11} +1.00000 q^{13} +3.43085 q^{15} -7.02073 q^{17} +6.34953 q^{19} -1.02188 q^{21} +0.633218 q^{23} +6.27207 q^{25} -5.06419 q^{27} +9.44347 q^{29} +0.386753 q^{31} +1.02188 q^{33} -3.35739 q^{35} +10.7276 q^{37} +1.02188 q^{39} -10.7441 q^{41} +7.84585 q^{43} -6.56625 q^{45} +0.633218 q^{47} +1.00000 q^{49} -7.17435 q^{51} +2.46650 q^{53} +3.35739 q^{55} +6.48846 q^{57} -7.83022 q^{59} +13.5536 q^{61} +1.95576 q^{63} +3.35739 q^{65} +8.43037 q^{67} +0.647073 q^{69} +12.9556 q^{71} +2.10919 q^{73} +6.40931 q^{75} -1.00000 q^{77} +4.48960 q^{79} +0.692281 q^{81} -4.65917 q^{83} -23.5713 q^{85} +9.65009 q^{87} +9.54087 q^{89} -1.00000 q^{91} +0.395216 q^{93} +21.3179 q^{95} +7.14717 q^{97} -1.95576 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + q^{3} + 4 q^{5} - 10 q^{7} + 17 q^{9}+O(q^{10}) \) 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

Groups

Database

Properties

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