LMFDB - Embedded newform 4004.2.a.k.1.8

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			1.8
Root			$2.24381$ 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)

$f(q)$	$=$	$q+2.24381 q^{3} -2.78728 q^{5} -1.00000 q^{7} +2.03468 q^{9} +O(q^{10})$	$q+2.24381 q^{3} -2.78728 q^{5} -1.00000 q^{7} +2.03468 q^{9} +1.00000 q^{11} +1.00000 q^{13} -6.25413 q^{15} +3.83090 q^{17} +5.00024 q^{19} -2.24381 q^{21} -6.83482 q^{23} +2.76895 q^{25} -2.16599 q^{27} +1.62251 q^{29} -3.96211 q^{31} +2.24381 q^{33} +2.78728 q^{35} +5.53313 q^{37} +2.24381 q^{39} +11.0414 q^{41} +6.43231 q^{43} -5.67124 q^{45} -6.83482 q^{47} +1.00000 q^{49} +8.59581 q^{51} +7.94104 q^{53} -2.78728 q^{55} +11.2196 q^{57} +4.33959 q^{59} -9.55004 q^{61} -2.03468 q^{63} -2.78728 q^{65} -2.80120 q^{67} -15.3360 q^{69} +9.59410 q^{71} +11.7631 q^{73} +6.21299 q^{75} -1.00000 q^{77} +5.73521 q^{79} -10.9641 q^{81} +10.9728 q^{83} -10.6778 q^{85} +3.64061 q^{87} +7.02589 q^{89} -1.00000 q^{91} -8.89021 q^{93} -13.9371 q^{95} +19.1400 q^{97} +2.03468 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.24381	1.29546	0.647732	−	0.761868i	$-0.275718\pi$
$3$	2.24381	1.29546	0.647732	+	0.761868i	$0.275718\pi$
$4$	0	0
$5$	−2.78728	−1.24651	−0.623255	−	0.782018i	$-0.714190\pi$
$5$	−2.78728	−1.24651	−0.623255	+	0.782018i	$0.714190\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	2.03468	0.678228
$10$	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
$15$	−6.25413	−1.61481
$16$	0	0
$17$	3.83090	0.929130	0.464565	−	0.885539i	$-0.346211\pi$
$17$	3.83090	0.929130	0.464565	+	0.885539i	$0.346211\pi$
$18$	0	0
$19$	5.00024	1.14713	0.573567	−	0.819159i	$-0.305559\pi$
$19$	5.00024	1.14713	0.573567	+	0.819159i	$0.305559\pi$
$20$	0	0
$21$	−2.24381	−0.489639
$22$	0	0
$23$	−6.83482	−1.42516	−0.712579	−	0.701591i	$-0.752473\pi$
$23$	−6.83482	−1.42516	−0.712579	+	0.701591i	$0.752473\pi$
$24$	0	0
$25$	2.76895	0.553789
$26$	0	0
$27$	−2.16599	−0.416844
$28$	0	0
$29$	1.62251	0.301293	0.150647	−	0.988588i	$-0.451864\pi$
$29$	1.62251	0.301293	0.150647	+	0.988588i	$0.451864\pi$
$30$	0	0
$31$	−3.96211	−0.711615	−0.355808	−	0.934559i	$-0.615794\pi$
$31$	−3.96211	−0.711615	−0.355808	+	0.934559i	$0.615794\pi$
$32$	0	0
$33$	2.24381	0.390597
$34$	0	0
$35$	2.78728	0.471137
$36$	0	0
$37$	5.53313	0.909640	0.454820	−	0.890583i	$-0.349704\pi$
$37$	5.53313	0.909640	0.454820	+	0.890583i	$0.349704\pi$
$38$	0	0
$39$	2.24381	0.359297
$40$	0	0
$41$	11.0414	1.72437	0.862185	−	0.506594i	$-0.169096\pi$
$41$	11.0414	1.72437	0.862185	+	0.506594i	$0.169096\pi$
$42$	0	0
$43$	6.43231	0.980918	0.490459	−	0.871464i	$-0.336829\pi$
$43$	6.43231	0.980918	0.490459	+	0.871464i	$0.336829\pi$
$44$	0	0
$45$	−5.67124	−0.845418
$46$	0	0
$47$	−6.83482	−0.996961	−0.498481	−	0.866901i	$-0.666108\pi$
$47$	−6.83482	−0.996961	−0.498481	+	0.866901i	$0.666108\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	8.59581	1.20365
$52$	0	0
$53$	7.94104	1.09079	0.545393	−	0.838181i	$-0.316381\pi$
$53$	7.94104	1.09079	0.545393	+	0.838181i	$0.316381\pi$
$54$	0	0
$55$	−2.78728	−0.375837
$56$	0	0
$57$	11.2196	1.48607
$58$	0	0
$59$	4.33959	0.564967	0.282483	−	0.959272i	$-0.408842\pi$
$59$	4.33959	0.564967	0.282483	+	0.959272i	$0.408842\pi$
$60$	0	0
$61$	−9.55004	−1.22276	−0.611379	−	0.791338i	$-0.709385\pi$
$61$	−9.55004	−1.22276	−0.611379	+	0.791338i	$0.709385\pi$
$62$	0	0
$63$	−2.03468	−0.256346
$64$	0	0
$65$	−2.78728	−0.345720
$66$	0	0
$67$	−2.80120	−0.342221	−0.171110	−	0.985252i	$-0.554735\pi$
$67$	−2.80120	−0.342221	−0.171110	+	0.985252i	$0.554735\pi$
$68$	0	0
$69$	−15.3360	−1.84624
$70$	0	0
$71$	9.59410	1.13861	0.569305	−	0.822126i	$-0.307212\pi$
$71$	9.59410	1.13861	0.569305	+	0.822126i	$0.307212\pi$
$72$	0	0
$73$	11.7631	1.37676	0.688381	−	0.725349i	$-0.258322\pi$
$73$	11.7631	1.37676	0.688381	+	0.725349i	$0.258322\pi$
$74$	0	0
$75$	6.21299	0.717414
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	5.73521	0.645262	0.322631	−	0.946525i	$-0.395433\pi$
$79$	5.73521	0.645262	0.322631	+	0.946525i	$0.395433\pi$
$80$	0	0
$81$	−10.9641	−1.21823
$82$	0	0
$83$	10.9728	1.20442	0.602211	−	0.798337i	$-0.294287\pi$
$83$	10.9728	1.20442	0.602211	+	0.798337i	$0.294287\pi$
$84$	0	0
$85$	−10.6778	−1.15817
$86$	0	0
$87$	3.64061	0.390315
$88$	0	0
$89$	7.02589	0.744742	0.372371	−	0.928084i	$-0.378545\pi$
$89$	7.02589	0.744742	0.372371	+	0.928084i	$0.378545\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−8.89021	−0.921872
$94$	0	0
$95$	−13.9371	−1.42991
$96$	0	0
$97$	19.1400	1.94337	0.971687	−	0.236271i	$-0.0759252\pi$
$97$	19.1400	1.94337	0.971687	+	0.236271i	$0.0759252\pi$
$98$	0	0
$99$	2.03468	0.204493
$100$	0	0
$101$	5.50443	0.547711	0.273856	−	0.961771i	$-0.411701\pi$
$101$	5.50443	0.547711	0.273856	+	0.961771i	$0.411701\pi$
$102$	0	0
$103$	−3.22893	−0.318156	−0.159078	−	0.987266i	$-0.550852\pi$
$103$	−3.22893	−0.318156	−0.159078	+	0.987266i	$0.550852\pi$
$104$	0	0
$105$	6.25413	0.610341
$106$	0	0
$107$	−17.7279	−1.71382	−0.856909	−	0.515467i	$-0.827618\pi$
$107$	−17.7279	−1.71382	−0.856909	+	0.515467i	$0.827618\pi$
$108$	0	0
$109$	4.28326	0.410262	0.205131	−	0.978735i	$-0.434238\pi$
$109$	4.28326	0.410262	0.205131	+	0.978735i	$0.434238\pi$
$110$	0	0
$111$	12.4153	1.17841
$112$	0	0
$113$	−15.6476	−1.47200	−0.736002	−	0.676979i	$-0.763289\pi$
$113$	−15.6476	−1.47200	−0.736002	+	0.676979i	$0.763289\pi$
$114$	0	0
$115$	19.0506	1.77648
$116$	0	0
$117$	2.03468	0.188107
$118$	0	0
$119$	−3.83090	−0.351178
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	24.7747	2.23386
$124$	0	0
$125$	6.21858	0.556207
$126$	0	0
$127$	15.6688	1.39038	0.695192	−	0.718824i	$-0.255319\pi$
$127$	15.6688	1.39038	0.695192	+	0.718824i	$0.255319\pi$
$128$	0	0
$129$	14.4329	1.27074
$130$	0	0
$131$	−14.7136	−1.28554	−0.642768	−	0.766061i	$-0.722214\pi$
$131$	−14.7136	−1.28554	−0.642768	+	0.766061i	$0.722214\pi$
$132$	0	0
$133$	−5.00024	−0.433576
$134$	0	0
$135$	6.03722	0.519601
$136$	0	0
$137$	4.90891	0.419397	0.209698	−	0.977766i	$-0.432752\pi$
$137$	4.90891	0.419397	0.209698	+	0.977766i	$0.432752\pi$
$138$	0	0
$139$	5.77409	0.489752	0.244876	−	0.969554i	$-0.421253\pi$
$139$	5.77409	0.489752	0.244876	+	0.969554i	$0.421253\pi$
$140$	0	0
$141$	−15.3360	−1.29153
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−4.52241	−0.375565
$146$	0	0
$147$	2.24381	0.185066
$148$	0	0
$149$	−14.4319	−1.18231	−0.591155	−	0.806558i	$-0.701328\pi$
$149$	−14.4319	−1.18231	−0.591155	+	0.806558i	$0.701328\pi$
$150$	0	0
$151$	19.6388	1.59818	0.799090	−	0.601212i	$-0.205315\pi$
$151$	19.6388	1.59818	0.799090	+	0.601212i	$0.205315\pi$
$152$	0	0
$153$	7.79467	0.630162
$154$	0	0
$155$	11.0435	0.887036
$156$	0	0
$157$	−7.21757	−0.576024	−0.288012	−	0.957627i	$-0.592994\pi$
$157$	−7.21757	−0.576024	−0.288012	+	0.957627i	$0.592994\pi$
$158$	0	0
$159$	17.8182	1.41307
$160$	0	0
$161$	6.83482	0.538659
$162$	0	0
$163$	13.5569	1.06186	0.530928	−	0.847417i	$-0.321843\pi$
$163$	13.5569	1.06186	0.530928	+	0.847417i	$0.321843\pi$
$164$	0	0
$165$	−6.25413	−0.486884
$166$	0	0
$167$	10.3458	0.800581	0.400290	−	0.916388i	$-0.368909\pi$
$167$	10.3458	0.800581	0.400290	+	0.916388i	$0.368909\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	10.1739	0.778018
$172$	0	0
$173$	−5.54256	−0.421393	−0.210696	−	0.977552i	$-0.567573\pi$
$173$	−5.54256	−0.421393	−0.210696	+	0.977552i	$0.567573\pi$
$174$	0	0
$175$	−2.76895	−0.209313
$176$	0	0
$177$	9.73722	0.731894
$178$	0	0
$179$	−3.34843	−0.250273	−0.125137	−	0.992140i	$-0.539937\pi$
$179$	−3.34843	−0.250273	−0.125137	+	0.992140i	$0.539937\pi$
$180$	0	0
$181$	24.9285	1.85292	0.926460	−	0.376394i	$-0.122836\pi$
$181$	24.9285	1.85292	0.926460	+	0.376394i	$0.122836\pi$
$182$	0	0
$183$	−21.4285	−1.58404
$184$	0	0
$185$	−15.4224	−1.13388
$186$	0	0
$187$	3.83090	0.280143
$188$	0	0
$189$	2.16599	0.157552
$190$	0	0
$191$	19.6983	1.42532	0.712661	−	0.701509i	$-0.247490\pi$
$191$	19.6983	1.42532	0.712661	+	0.701509i	$0.247490\pi$
$192$	0	0
$193$	23.8996	1.72033	0.860166	−	0.510014i	$-0.170360\pi$
$193$	23.8996	1.72033	0.860166	+	0.510014i	$0.170360\pi$
$194$	0	0
$195$	−6.25413	−0.447868
$196$	0	0
$197$	−17.9118	−1.27616	−0.638082	−	0.769969i	$-0.720272\pi$
$197$	−17.9118	−1.27616	−0.638082	+	0.769969i	$0.720272\pi$
$198$	0	0
$199$	−9.79040	−0.694023	−0.347011	−	0.937861i	$-0.612804\pi$
$199$	−9.79040	−0.694023	−0.347011	+	0.937861i	$0.612804\pi$
$200$	0	0
$201$	−6.28535	−0.443335
$202$	0	0
$203$	−1.62251	−0.113878
$204$	0	0
$205$	−30.7754	−2.14945
$206$	0	0
$207$	−13.9067	−0.966582
$208$	0	0
$209$	5.00024	0.345874
$210$	0	0
$211$	−22.1543	−1.52517	−0.762583	−	0.646890i	$-0.776069\pi$
$211$	−22.1543	−1.52517	−0.762583	+	0.646890i	$0.776069\pi$
$212$	0	0
$213$	21.5273	1.47503
$214$	0	0
$215$	−17.9287	−1.22272
$216$	0	0
$217$	3.96211	0.268965
$218$	0	0
$219$	26.3941	1.78355
$220$	0	0
$221$	3.83090	0.257694
$222$	0	0
$223$	−17.4357	−1.16758	−0.583790	−	0.811905i	$-0.698431\pi$
$223$	−17.4357	−1.16758	−0.583790	+	0.811905i	$0.698431\pi$
$224$	0	0
$225$	5.63393	0.375595
$226$	0	0
$227$	−22.9973	−1.52638	−0.763191	−	0.646173i	$-0.776369\pi$
$227$	−22.9973	−1.52638	−0.763191	+	0.646173i	$0.776369\pi$
$228$	0	0
$229$	15.3310	1.01310	0.506552	−	0.862209i	$-0.330920\pi$
$229$	15.3310	1.01310	0.506552	+	0.862209i	$0.330920\pi$
$230$	0	0
$231$	−2.24381	−0.147632
$232$	0	0
$233$	28.4392	1.86311	0.931557	−	0.363596i	$-0.118451\pi$
$233$	28.4392	1.86311	0.931557	+	0.363596i	$0.118451\pi$
$234$	0	0
$235$	19.0506	1.24272
$236$	0	0
$237$	12.8687	0.835913
$238$	0	0
$239$	28.9143	1.87031	0.935154	−	0.354242i	$-0.115261\pi$
$239$	28.9143	1.87031	0.935154	+	0.354242i	$0.115261\pi$
$240$	0	0
$241$	−25.4254	−1.63780	−0.818898	−	0.573938i	$-0.805415\pi$
$241$	−25.4254	−1.63780	−0.818898	+	0.573938i	$0.805415\pi$
$242$	0	0
$243$	−18.1034	−1.16134
$244$	0	0
$245$	−2.78728	−0.178073
$246$	0	0
$247$	5.00024	0.318158
$248$	0	0
$249$	24.6209	1.56029
$250$	0	0
$251$	24.3717	1.53833	0.769165	−	0.639050i	$-0.220672\pi$
$251$	24.3717	1.53833	0.769165	+	0.639050i	$0.220672\pi$
$252$	0	0
$253$	−6.83482	−0.429702
$254$	0	0
$255$	−23.9590	−1.50037
$256$	0	0
$257$	−25.2465	−1.57484	−0.787418	−	0.616420i	$-0.788582\pi$
$257$	−25.2465	−1.57484	−0.787418	+	0.616420i	$0.788582\pi$
$258$	0	0
$259$	−5.53313	−0.343812
$260$	0	0
$261$	3.30130	0.204346
$262$	0	0
$263$	−8.38928	−0.517305	−0.258653	−	0.965970i	$-0.583278\pi$
$263$	−8.38928	−0.517305	−0.258653	+	0.965970i	$0.583278\pi$
$264$	0	0
$265$	−22.1339	−1.35968
$266$	0	0
$267$	15.7648	0.964787
$268$	0	0
$269$	17.6442	1.07578	0.537892	−	0.843013i	$-0.319221\pi$
$269$	17.6442	1.07578	0.537892	+	0.843013i	$0.319221\pi$
$270$	0	0
$271$	−1.44967	−0.0880612	−0.0440306	−	0.999030i	$-0.514020\pi$
$271$	−1.44967	−0.0880612	−0.0440306	+	0.999030i	$0.514020\pi$
$272$	0	0
$273$	−2.24381	−0.135802
$274$	0	0
$275$	2.76895	0.166974
$276$	0	0
$277$	21.2414	1.27627	0.638137	−	0.769923i	$-0.279705\pi$
$277$	21.2414	1.27627	0.638137	+	0.769923i	$0.279705\pi$
$278$	0	0
$279$	−8.06163	−0.482637
$280$	0	0
$281$	−31.7251	−1.89256	−0.946279	−	0.323350i	$-0.895191\pi$
$281$	−31.7251	−1.89256	−0.946279	+	0.323350i	$0.895191\pi$
$282$	0	0
$283$	2.69946	0.160466	0.0802330	−	0.996776i	$-0.474434\pi$
$283$	2.69946	0.160466	0.0802330	+	0.996776i	$0.474434\pi$
$284$	0	0
$285$	−31.2722	−1.85240
$286$	0	0
$287$	−11.0414	−0.651751
$288$	0	0
$289$	−2.32419	−0.136717
$290$	0	0
$291$	42.9466	2.51757
$292$	0	0
$293$	−8.20140	−0.479131	−0.239566	−	0.970880i	$-0.577005\pi$
$293$	−8.20140	−0.479131	−0.239566	+	0.970880i	$0.577005\pi$
$294$	0	0
$295$	−12.0957	−0.704237
$296$	0	0
$297$	−2.16599	−0.125683
$298$	0	0
$299$	−6.83482	−0.395268
$300$	0	0
$301$	−6.43231	−0.370752
$302$	0	0
$303$	12.3509	0.709540
$304$	0	0
$305$	26.6187	1.52418
$306$	0	0
$307$	−26.1154	−1.49049	−0.745243	−	0.666793i	$-0.767666\pi$
$307$	−26.1154	−1.49049	−0.745243	+	0.666793i	$0.767666\pi$
$308$	0	0
$309$	−7.24510	−0.412159
$310$	0	0
$311$	19.1743	1.08728	0.543639	−	0.839319i	$-0.317046\pi$
$311$	19.1743	1.08728	0.543639	+	0.839319i	$0.317046\pi$
$312$	0	0
$313$	−9.15945	−0.517723	−0.258861	−	0.965914i	$-0.583347\pi$
$313$	−9.15945	−0.517723	−0.258861	+	0.965914i	$0.583347\pi$
$314$	0	0
$315$	5.67124	0.319538
$316$	0	0
$317$	−19.7519	−1.10938	−0.554689	−	0.832058i	$-0.687163\pi$
$317$	−19.7519	−1.10938	−0.554689	+	0.832058i	$0.687163\pi$
$318$	0	0
$319$	1.62251	0.0908434
$320$	0	0
$321$	−39.7780	−2.22019
$322$	0	0
$323$	19.1554	1.06584
$324$	0	0
$325$	2.76895	0.153593
$326$	0	0
$327$	9.61081	0.531479
$328$	0	0
$329$	6.83482	0.376816
$330$	0	0
$331$	31.6011	1.73695	0.868477	−	0.495729i	$-0.165099\pi$
$331$	31.6011	1.73695	0.868477	+	0.495729i	$0.165099\pi$
$332$	0	0
$333$	11.2582	0.616943
$334$	0	0
$335$	7.80773	0.426582
$336$	0	0
$337$	3.99638	0.217697	0.108848	−	0.994058i	$-0.465284\pi$
$337$	3.99638	0.217697	0.108848	+	0.994058i	$0.465284\pi$
$338$	0	0
$339$	−35.1103	−1.90693
$340$	0	0
$341$	−3.96211	−0.214560
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	42.7459	2.30136
$346$	0	0
$347$	−21.0356	−1.12925	−0.564625	−	0.825347i	$-0.690979\pi$
$347$	−21.0356	−1.12925	−0.564625	+	0.825347i	$0.690979\pi$
$348$	0	0
$349$	−15.9664	−0.854662	−0.427331	−	0.904095i	$-0.640546\pi$
$349$	−15.9664	−0.854662	−0.427331	+	0.904095i	$0.640546\pi$
$350$	0	0
$351$	−2.16599	−0.115612
$352$	0	0
$353$	1.49830	0.0797462	0.0398731	−	0.999205i	$-0.487305\pi$
$353$	1.49830	0.0797462	0.0398731	+	0.999205i	$0.487305\pi$
$354$	0	0
$355$	−26.7415	−1.41929
$356$	0	0
$357$	−8.59581	−0.454939
$358$	0	0
$359$	−14.6892	−0.775265	−0.387632	−	0.921814i	$-0.626707\pi$
$359$	−14.6892	−0.775265	−0.387632	+	0.921814i	$0.626707\pi$
$360$	0	0
$361$	6.00241	0.315916
$362$	0	0
$363$	2.24381	0.117769
$364$	0	0
$365$	−32.7870	−1.71615
$366$	0	0
$367$	28.5041	1.48790	0.743952	−	0.668233i	$-0.232949\pi$
$367$	28.5041	1.48790	0.743952	+	0.668233i	$0.232949\pi$
$368$	0	0
$369$	22.4657	1.16952
$370$	0	0
$371$	−7.94104	−0.412278
$372$	0	0
$373$	−5.05089	−0.261525	−0.130763	−	0.991414i	$-0.541743\pi$
$373$	−5.05089	−0.261525	−0.130763	+	0.991414i	$0.541743\pi$
$374$	0	0
$375$	13.9533	0.720546
$376$	0	0
$377$	1.62251	0.0835638
$378$	0	0
$379$	−8.10147	−0.416145	−0.208072	−	0.978113i	$-0.566719\pi$
$379$	−8.10147	−0.416145	−0.208072	+	0.978113i	$0.566719\pi$
$380$	0	0
$381$	35.1579	1.80119
$382$	0	0
$383$	−11.1648	−0.570496	−0.285248	−	0.958454i	$-0.592076\pi$
$383$	−11.1648	−0.570496	−0.285248	+	0.958454i	$0.592076\pi$
$384$	0	0
$385$	2.78728	0.142053
$386$	0	0
$387$	13.0877	0.665286
$388$	0	0
$389$	−15.3088	−0.776185	−0.388092	−	0.921620i	$-0.626866\pi$
$389$	−15.3088	−0.776185	−0.388092	+	0.921620i	$0.626866\pi$
$390$	0	0
$391$	−26.1835	−1.32416
$392$	0	0
$393$	−33.0146	−1.66537
$394$	0	0
$395$	−15.9857	−0.804325
$396$	0	0
$397$	−15.0148	−0.753571	−0.376786	−	0.926300i	$-0.622971\pi$
$397$	−15.0148	−0.753571	−0.376786	+	0.926300i	$0.622971\pi$
$398$	0	0
$399$	−11.2196	−0.561682
$400$	0	0
$401$	−19.8534	−0.991433	−0.495716	−	0.868484i	$-0.665094\pi$
$401$	−19.8534	−0.991433	−0.495716	+	0.868484i	$0.665094\pi$
$402$	0	0
$403$	−3.96211	−0.197367
$404$	0	0
$405$	30.5601	1.51854
$406$	0	0
$407$	5.53313	0.274267
$408$	0	0
$409$	−13.5073	−0.667895	−0.333947	−	0.942592i	$-0.608381\pi$
$409$	−13.5073	−0.667895	−0.333947	+	0.942592i	$0.608381\pi$
$410$	0	0
$411$	11.0147	0.543313
$412$	0	0
$413$	−4.33959	−0.213537
$414$	0	0
$415$	−30.5843	−1.50133
$416$	0	0
$417$	12.9560	0.634456
$418$	0	0
$419$	−1.46928	−0.0717791	−0.0358896	−	0.999356i	$-0.511426\pi$
$419$	−1.46928	−0.0717791	−0.0358896	+	0.999356i	$0.511426\pi$
$420$	0	0
$421$	−21.5988	−1.05266	−0.526330	−	0.850280i	$-0.676432\pi$
$421$	−21.5988	−1.05266	−0.526330	+	0.850280i	$0.676432\pi$
$422$	0	0
$423$	−13.9067	−0.676167
$424$	0	0
$425$	10.6076	0.514542
$426$	0	0
$427$	9.55004	0.462159
$428$	0	0
$429$	2.24381	0.108332
$430$	0	0
$431$	−15.3520	−0.739481	−0.369740	−	0.929135i	$-0.620553\pi$
$431$	−15.3520	−0.739481	−0.369740	+	0.929135i	$0.620553\pi$
$432$	0	0
$433$	−6.79531	−0.326562	−0.163281	−	0.986580i	$-0.552208\pi$
$433$	−6.79531	−0.326562	−0.163281	+	0.986580i	$0.552208\pi$
$434$	0	0
$435$	−10.1474	−0.486532
$436$	0	0
$437$	−34.1758	−1.63485
$438$	0	0
$439$	−3.30886	−0.157923	−0.0789617	−	0.996878i	$-0.525160\pi$
$439$	−3.30886	−0.157923	−0.0789617	+	0.996878i	$0.525160\pi$
$440$	0	0
$441$	2.03468	0.0968897
$442$	0	0
$443$	−15.9894	−0.759680	−0.379840	−	0.925052i	$-0.624021\pi$
$443$	−15.9894	−0.759680	−0.379840	+	0.925052i	$0.624021\pi$
$444$	0	0
$445$	−19.5831	−0.928329
$446$	0	0
$447$	−32.3825	−1.53164
$448$	0	0
$449$	−13.9140	−0.656643	−0.328321	−	0.944566i	$-0.606483\pi$
$449$	−13.9140	−0.656643	−0.328321	+	0.944566i	$0.606483\pi$
$450$	0	0
$451$	11.0414	0.519917
$452$	0	0
$453$	44.0656	2.07038
$454$	0	0
$455$	2.78728	0.130670
$456$	0	0
$457$	33.0363	1.54537	0.772687	−	0.634787i	$-0.218912\pi$
$457$	33.0363	1.54537	0.772687	+	0.634787i	$0.218912\pi$
$458$	0	0
$459$	−8.29768	−0.387303
$460$	0	0
$461$	−4.61168	−0.214787	−0.107394	−	0.994217i	$-0.534251\pi$
$461$	−4.61168	−0.214787	−0.107394	+	0.994217i	$0.534251\pi$
$462$	0	0
$463$	−19.8354	−0.921828	−0.460914	−	0.887445i	$-0.652478\pi$
$463$	−19.8354	−0.921828	−0.460914	+	0.887445i	$0.652478\pi$
$464$	0	0
$465$	24.7795	1.14912
$466$	0	0
$467$	1.14490	0.0529798	0.0264899	−	0.999649i	$-0.491567\pi$
$467$	1.14490	0.0529798	0.0264899	+	0.999649i	$0.491567\pi$
$468$	0	0
$469$	2.80120	0.129347
$470$	0	0
$471$	−16.1948	−0.746219
$472$	0	0
$473$	6.43231	0.295758
$474$	0	0
$475$	13.8454	0.635270
$476$	0	0
$477$	16.1575	0.739801
$478$	0	0
$479$	−27.7594	−1.26836	−0.634179	−	0.773186i	$-0.718662\pi$
$479$	−27.7594	−1.26836	−0.634179	+	0.773186i	$0.718662\pi$
$480$	0	0
$481$	5.53313	0.252289
$482$	0	0
$483$	15.3360	0.697814
$484$	0	0
$485$	−53.3486	−2.42244
$486$	0	0
$487$	−41.7341	−1.89115	−0.945577	−	0.325397i	$-0.894502\pi$
$487$	−41.7341	−1.89115	−0.945577	+	0.325397i	$0.894502\pi$
$488$	0	0
$489$	30.4191	1.37560
$490$	0	0
$491$	−5.54197	−0.250105	−0.125053	−	0.992150i	$-0.539910\pi$
$491$	−5.54197	−0.250105	−0.125053	+	0.992150i	$0.539910\pi$
$492$	0	0
$493$	6.21569	0.279941
$494$	0	0
$495$	−5.67124	−0.254903
$496$	0	0
$497$	−9.59410	−0.430354
$498$	0	0
$499$	15.9751	0.715146	0.357573	−	0.933885i	$-0.383604\pi$
$499$	15.9751	0.715146	0.357573	+	0.933885i	$0.383604\pi$
$500$	0	0
$501$	23.2140	1.03712
$502$	0	0
$503$	−13.1940	−0.588289	−0.294145	−	0.955761i	$-0.595035\pi$
$503$	−13.1940	−0.588289	−0.294145	+	0.955761i	$0.595035\pi$
$504$	0	0
$505$	−15.3424	−0.682728
$506$	0	0
$507$	2.24381	0.0996511
$508$	0	0
$509$	36.6747	1.62558	0.812790	−	0.582557i	$-0.197948\pi$
$509$	36.6747	1.62558	0.812790	+	0.582557i	$0.197948\pi$
$510$	0	0
$511$	−11.7631	−0.520367
$512$	0	0
$513$	−10.8305	−0.478176
$514$	0	0
$515$	8.99994	0.396585
$516$	0	0
$517$	−6.83482	−0.300595
$518$	0	0
$519$	−12.4364	−0.545899
$520$	0	0
$521$	−38.3444	−1.67990	−0.839949	−	0.542665i	$-0.817415\pi$
$521$	−38.3444	−1.67990	−0.839949	+	0.542665i	$0.817415\pi$
$522$	0	0
$523$	−0.276471	−0.0120892	−0.00604462	−	0.999982i	$-0.501924\pi$
$523$	−0.276471	−0.0120892	−0.00604462	+	0.999982i	$0.501924\pi$
$524$	0	0
$525$	−6.21299	−0.271157
$526$	0	0
$527$	−15.1784	−0.661183
$528$	0	0
$529$	23.7148	1.03108
$530$	0	0
$531$	8.82969	0.383176
$532$	0	0
$533$	11.0414	0.478254
$534$	0	0
$535$	49.4126	2.13629
$536$	0	0
$537$	−7.51323	−0.324220
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	34.9452	1.50241	0.751205	−	0.660069i	$-0.229473\pi$
$541$	34.9452	1.50241	0.751205	+	0.660069i	$0.229473\pi$
$542$	0	0
$543$	55.9348	2.40039
$544$	0	0
$545$	−11.9386	−0.511395
$546$	0	0
$547$	30.4428	1.30164	0.650820	−	0.759232i	$-0.274425\pi$
$547$	30.4428	1.30164	0.650820	+	0.759232i	$0.274425\pi$
$548$	0	0
$549$	−19.4313	−0.829308
$550$	0	0
$551$	8.11296	0.345624
$552$	0	0
$553$	−5.73521	−0.243886
$554$	0	0
$555$	−34.6049	−1.46890
$556$	0	0
$557$	33.2427	1.40854	0.704268	−	0.709934i	$-0.251275\pi$
$557$	33.2427	1.40854	0.704268	+	0.709934i	$0.251275\pi$
$558$	0	0
$559$	6.43231	0.272058
$560$	0	0
$561$	8.59581	0.362916
$562$	0	0
$563$	−4.98452	−0.210072	−0.105036	−	0.994468i	$-0.533496\pi$
$563$	−4.98452	−0.210072	−0.105036	+	0.994468i	$0.533496\pi$
$564$	0	0
$565$	43.6144	1.83487
$566$	0	0
$567$	10.9641	0.460450
$568$	0	0
$569$	18.2621	0.765587	0.382793	−	0.923834i	$-0.374962\pi$
$569$	18.2621	0.765587	0.382793	+	0.923834i	$0.374962\pi$
$570$	0	0
$571$	5.87875	0.246018	0.123009	−	0.992406i	$-0.460746\pi$
$571$	5.87875	0.246018	0.123009	+	0.992406i	$0.460746\pi$
$572$	0	0
$573$	44.1993	1.84645
$574$	0	0
$575$	−18.9252	−0.789237
$576$	0	0
$577$	19.9404	0.830129	0.415064	−	0.909792i	$-0.363759\pi$
$577$	19.9404	0.830129	0.415064	+	0.909792i	$0.363759\pi$
$578$	0	0
$579$	53.6262	2.22863
$580$	0	0
$581$	−10.9728	−0.455229
$582$	0	0
$583$	7.94104	0.328884
$584$	0	0
$585$	−5.67124	−0.234477
$586$	0	0
$587$	11.0828	0.457438	0.228719	−	0.973492i	$-0.426546\pi$
$587$	11.0828	0.457438	0.228719	+	0.973492i	$0.426546\pi$
$588$	0	0
$589$	−19.8115	−0.816318
$590$	0	0
$591$	−40.1907	−1.65322
$592$	0	0
$593$	−2.82607	−0.116053	−0.0580263	−	0.998315i	$-0.518481\pi$
$593$	−2.82607	−0.116053	−0.0580263	+	0.998315i	$0.518481\pi$
$594$	0	0
$595$	10.6778	0.437747
$596$	0	0
$597$	−21.9678	−0.899082
$598$	0	0
$599$	42.7918	1.74843	0.874214	−	0.485542i	$-0.161378\pi$
$599$	42.7918	1.74843	0.874214	+	0.485542i	$0.161378\pi$
$600$	0	0
$601$	4.28011	0.174589	0.0872946	−	0.996183i	$-0.472178\pi$
$601$	4.28011	0.174589	0.0872946	+	0.996183i	$0.472178\pi$
$602$	0	0
$603$	−5.69955	−0.232103
$604$	0	0
$605$	−2.78728	−0.113319
$606$	0	0
$607$	−39.7982	−1.61536	−0.807679	−	0.589623i	$-0.799276\pi$
$607$	−39.7982	−1.61536	−0.807679	+	0.589623i	$0.799276\pi$
$608$	0	0
$609$	−3.64061	−0.147525
$610$	0	0
$611$	−6.83482	−0.276507
$612$	0	0
$613$	12.6748	0.511932	0.255966	−	0.966686i	$-0.417607\pi$
$613$	12.6748	0.511932	0.255966	+	0.966686i	$0.417607\pi$
$614$	0	0
$615$	−69.0541	−2.78453
$616$	0	0
$617$	38.6226	1.55489	0.777444	−	0.628953i	$-0.216516\pi$
$617$	38.6226	1.55489	0.777444	+	0.628953i	$0.216516\pi$
$618$	0	0
$619$	−21.6038	−0.868329	−0.434165	−	0.900834i	$-0.642956\pi$
$619$	−21.6038	−0.868329	−0.434165	+	0.900834i	$0.642956\pi$
$620$	0	0
$621$	14.8041	0.594070
$622$	0	0
$623$	−7.02589	−0.281486
$624$	0	0
$625$	−31.1777	−1.24711
$626$	0	0
$627$	11.2196	0.448067
$628$	0	0
$629$	21.1969	0.845174
$630$	0	0
$631$	−0.651632	−0.0259411	−0.0129705	−	0.999916i	$-0.504129\pi$
$631$	−0.651632	−0.0259411	−0.0129705	+	0.999916i	$0.504129\pi$
$632$	0	0
$633$	−49.7101	−1.97580
$634$	0	0
$635$	−43.6735	−1.73313
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	19.5210	0.772237
$640$	0	0
$641$	37.7521	1.49112	0.745559	−	0.666440i	$-0.232183\pi$
$641$	37.7521	1.49112	0.745559	+	0.666440i	$0.232183\pi$
$642$	0	0
$643$	23.5109	0.927180	0.463590	−	0.886050i	$-0.346561\pi$
$643$	23.5109	0.927180	0.463590	+	0.886050i	$0.346561\pi$
$644$	0	0
$645$	−40.2285	−1.58400
$646$	0	0
$647$	−38.1076	−1.49816	−0.749082	−	0.662477i	$-0.769505\pi$
$647$	−38.1076	−1.49816	−0.749082	+	0.662477i	$0.769505\pi$
$648$	0	0
$649$	4.33959	0.170344
$650$	0	0
$651$	8.89021	0.348435
$652$	0	0
$653$	−36.8320	−1.44135	−0.720674	−	0.693274i	$-0.756168\pi$
$653$	−36.8320	−1.44135	−0.720674	+	0.693274i	$0.756168\pi$
$654$	0	0
$655$	41.0111	1.60243
$656$	0	0
$657$	23.9341	0.933758
$658$	0	0
$659$	32.0125	1.24703	0.623515	−	0.781811i	$-0.285704\pi$
$659$	32.0125	1.24703	0.623515	+	0.781811i	$0.285704\pi$
$660$	0	0
$661$	36.4712	1.41857	0.709283	−	0.704924i	$-0.249019\pi$
$661$	36.4712	1.41857	0.709283	+	0.704924i	$0.249019\pi$
$662$	0	0
$663$	8.59581	0.333834
$664$	0	0
$665$	13.9371	0.540457
$666$	0	0
$667$	−11.0896	−0.429391
$668$	0	0
$669$	−39.1223	−1.51256
$670$	0	0
$671$	−9.55004	−0.368675
$672$	0	0
$673$	31.2823	1.20584	0.602922	−	0.797800i	$-0.294003\pi$
$673$	31.2823	1.20584	0.602922	+	0.797800i	$0.294003\pi$
$674$	0	0
$675$	−5.99750	−0.230844
$676$	0	0
$677$	−8.21689	−0.315801	−0.157900	−	0.987455i	$-0.550472\pi$
$677$	−8.21689	−0.315801	−0.157900	+	0.987455i	$0.550472\pi$
$678$	0	0
$679$	−19.1400	−0.734527
$680$	0	0
$681$	−51.6015	−1.97737
$682$	0	0
$683$	18.9073	0.723468	0.361734	−	0.932281i	$-0.382185\pi$
$683$	18.9073	0.723468	0.361734	+	0.932281i	$0.382185\pi$
$684$	0	0
$685$	−13.6825	−0.522782
$686$	0	0
$687$	34.4000	1.31244
$688$	0	0
$689$	7.94104	0.302529
$690$	0	0
$691$	−43.3895	−1.65062	−0.825308	−	0.564683i	$-0.808998\pi$
$691$	−43.3895	−1.65062	−0.825308	+	0.564683i	$0.808998\pi$
$692$	0	0
$693$	−2.03468	−0.0772912
$694$	0	0
$695$	−16.0940	−0.610481
$696$	0	0
$697$	42.2983	1.60216
$698$	0	0
$699$	63.8121	2.41360
$700$	0	0
$701$	10.0810	0.380756	0.190378	−	0.981711i	$-0.439029\pi$
$701$	10.0810	0.380756	0.190378	+	0.981711i	$0.439029\pi$
$702$	0	0
$703$	27.6670	1.04348
$704$	0	0
$705$	42.7459	1.60990
$706$	0	0
$707$	−5.50443	−0.207015
$708$	0	0
$709$	4.13006	0.155108	0.0775539	−	0.996988i	$-0.475289\pi$
$709$	4.13006	0.155108	0.0775539	+	0.996988i	$0.475289\pi$
$710$	0	0
$711$	11.6693	0.437634
$712$	0	0
$713$	27.0803	1.01416
$714$	0	0
$715$	−2.78728	−0.104238
$716$	0	0
$717$	64.8781	2.42292
$718$	0	0
$719$	18.3590	0.684674	0.342337	−	0.939577i	$-0.388782\pi$
$719$	18.3590	0.684674	0.342337	+	0.939577i	$0.388782\pi$
$720$	0	0
$721$	3.22893	0.120252
$722$	0	0
$723$	−57.0499	−2.12171
$724$	0	0
$725$	4.49265	0.166853
$726$	0	0
$727$	−1.51547	−0.0562058	−0.0281029	−	0.999605i	$-0.508947\pi$
$727$	−1.51547	−0.0562058	−0.0281029	+	0.999605i	$0.508947\pi$
$728$	0	0
$729$	−7.72831	−0.286234
$730$	0	0
$731$	24.6415	0.911400
$732$	0	0
$733$	1.09766	0.0405430	0.0202715	−	0.999795i	$-0.493547\pi$
$733$	1.09766	0.0405430	0.0202715	+	0.999795i	$0.493547\pi$
$734$	0	0
$735$	−6.25413	−0.230687
$736$	0	0
$737$	−2.80120	−0.103183
$738$	0	0
$739$	−3.50710	−0.129011	−0.0645053	−	0.997917i	$-0.520547\pi$
$739$	−3.50710	−0.129011	−0.0645053	+	0.997917i	$0.520547\pi$
$740$	0	0
$741$	11.2196	0.412162
$742$	0	0
$743$	−29.6701	−1.08849	−0.544244	−	0.838927i	$-0.683184\pi$
$743$	−29.6701	−1.08849	−0.544244	+	0.838927i	$0.683184\pi$
$744$	0	0
$745$	40.2259	1.47376
$746$	0	0
$747$	22.3262	0.816872
$748$	0	0
$749$	17.7279	0.647763
$750$	0	0
$751$	−2.26759	−0.0827455	−0.0413727	−	0.999144i	$-0.513173\pi$
$751$	−2.26759	−0.0827455	−0.0413727	+	0.999144i	$0.513173\pi$
$752$	0	0
$753$	54.6856	1.99285
$754$	0	0
$755$	−54.7388	−1.99215
$756$	0	0
$757$	25.0470	0.910350	0.455175	−	0.890402i	$-0.349577\pi$
$757$	25.0470	0.910350	0.455175	+	0.890402i	$0.349577\pi$
$758$	0	0
$759$	−15.3360	−0.556663
$760$	0	0
$761$	−31.3072	−1.13489	−0.567443	−	0.823412i	$-0.692067\pi$
$761$	−31.3072	−1.13489	−0.567443	+	0.823412i	$0.692067\pi$
$762$	0	0
$763$	−4.28326	−0.155064
$764$	0	0
$765$	−21.7260	−0.785503
$766$	0	0
$767$	4.33959	0.156694
$768$	0	0
$769$	3.61748	0.130450	0.0652248	−	0.997871i	$-0.479224\pi$
$769$	3.61748	0.130450	0.0652248	+	0.997871i	$0.479224\pi$
$770$	0	0
$771$	−56.6484	−2.04014
$772$	0	0
$773$	7.52612	0.270696	0.135348	−	0.990798i	$-0.456785\pi$
$773$	7.52612	0.270696	0.135348	+	0.990798i	$0.456785\pi$
$774$	0	0
$775$	−10.9709	−0.394085
$776$	0	0
$777$	−12.4153	−0.445396
$778$	0	0
$779$	55.2094	1.97808
$780$	0	0
$781$	9.59410	0.343304
$782$	0	0
$783$	−3.51435	−0.125592
$784$	0	0
$785$	20.1174	0.718021
$786$	0	0
$787$	−3.74298	−0.133423	−0.0667115	−	0.997772i	$-0.521251\pi$
$787$	−3.74298	−0.133423	−0.0667115	+	0.997772i	$0.521251\pi$
$788$	0	0
$789$	−18.8240	−0.670150
$790$	0	0
$791$	15.6476	0.556365
$792$	0	0
$793$	−9.55004	−0.339132
$794$	0	0
$795$	−49.6643	−1.76141
$796$	0	0
$797$	2.64002	0.0935144	0.0467572	−	0.998906i	$-0.485111\pi$
$797$	2.64002	0.0935144	0.0467572	+	0.998906i	$0.485111\pi$
$798$	0	0
$799$	−26.1835	−0.926307
$800$	0	0
$801$	14.2955	0.505105
$802$	0	0
$803$	11.7631	0.415109
$804$	0	0
$805$	−19.0506	−0.671445
$806$	0	0
$807$	39.5902	1.39364
$808$	0	0
$809$	−42.4359	−1.49197	−0.745983	−	0.665965i	$-0.768020\pi$
$809$	−42.4359	−1.49197	−0.745983	+	0.665965i	$0.768020\pi$
$810$	0	0
$811$	3.12085	0.109588	0.0547940	−	0.998498i	$-0.482550\pi$
$811$	3.12085	0.109588	0.0547940	+	0.998498i	$0.482550\pi$
$812$	0	0
$813$	−3.25278	−0.114080
$814$	0	0
$815$	−37.7868	−1.32362
$816$	0	0
$817$	32.1631	1.12524
$818$	0	0
$819$	−2.03468	−0.0710976
$820$	0	0
$821$	36.8246	1.28519	0.642594	−	0.766207i	$-0.277858\pi$
$821$	36.8246	1.28519	0.642594	+	0.766207i	$0.277858\pi$
$822$	0	0
$823$	−45.1291	−1.57310	−0.786551	−	0.617525i	$-0.788135\pi$
$823$	−45.1291	−1.57310	−0.786551	+	0.617525i	$0.788135\pi$
$824$	0	0
$825$	6.21299	0.216308
$826$	0	0
$827$	−7.18686	−0.249912	−0.124956	−	0.992162i	$-0.539879\pi$
$827$	−7.18686	−0.249912	−0.124956	+	0.992162i	$0.539879\pi$
$828$	0	0
$829$	21.1524	0.734653	0.367327	−	0.930092i	$-0.380273\pi$
$829$	21.1524	0.734653	0.367327	+	0.930092i	$0.380273\pi$
$830$	0	0
$831$	47.6618	1.65337
$832$	0	0
$833$	3.83090	0.132733
$834$	0	0
$835$	−28.8366	−0.997932
$836$	0	0
$837$	8.58187	0.296633
$838$	0	0
$839$	−32.1220	−1.10898	−0.554488	−	0.832192i	$-0.687086\pi$
$839$	−32.1220	−1.10898	−0.554488	+	0.832192i	$0.687086\pi$
$840$	0	0
$841$	−26.3674	−0.909222
$842$	0	0
$843$	−71.1850	−2.45174
$844$	0	0
$845$	−2.78728	−0.0958854
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	6.05707	0.207878
$850$	0	0
$851$	−37.8179	−1.29638
$852$	0	0
$853$	−39.5477	−1.35409	−0.677044	−	0.735943i	$-0.736739\pi$
$853$	−39.5477	−1.35409	−0.677044	+	0.735943i	$0.736739\pi$
$854$	0	0
$855$	−28.3576	−0.969808
$856$	0	0
$857$	11.1933	0.382356	0.191178	−	0.981555i	$-0.438769\pi$
$857$	11.1933	0.382356	0.191178	+	0.981555i	$0.438769\pi$
$858$	0	0
$859$	−29.0708	−0.991884	−0.495942	−	0.868356i	$-0.665177\pi$
$859$	−29.0708	−0.991884	−0.495942	+	0.868356i	$0.665177\pi$
$860$	0	0
$861$	−24.7747	−0.844320
$862$	0	0
$863$	−55.8342	−1.90062	−0.950310	−	0.311306i	$-0.899233\pi$
$863$	−55.8342	−1.90062	−0.950310	+	0.311306i	$0.899233\pi$
$864$	0	0
$865$	15.4487	0.525270
$866$	0	0
$867$	−5.21505	−0.177112
$868$	0	0
$869$	5.73521	0.194554
$870$	0	0
$871$	−2.80120	−0.0949149
$872$	0	0
$873$	38.9439	1.31805
$874$	0	0
$875$	−6.21858	−0.210226
$876$	0	0
$877$	41.3561	1.39650	0.698248	−	0.715856i	$-0.253963\pi$
$877$	41.3561	1.39650	0.698248	+	0.715856i	$0.253963\pi$
$878$	0	0
$879$	−18.4024	−0.620697
$880$	0	0
$881$	−46.2447	−1.55802	−0.779012	−	0.627008i	$-0.784279\pi$
$881$	−46.2447	−1.55802	−0.779012	+	0.627008i	$0.784279\pi$
$882$	0	0
$883$	−24.2973	−0.817669	−0.408834	−	0.912609i	$-0.634065\pi$
$883$	−24.2973	−0.817669	−0.408834	+	0.912609i	$0.634065\pi$
$884$	0	0
$885$	−27.1404	−0.912314
$886$	0	0
$887$	0.216897	0.00728270	0.00364135	−	0.999993i	$-0.498841\pi$
$887$	0.216897	0.00728270	0.00364135	+	0.999993i	$0.498841\pi$
$888$	0	0
$889$	−15.6688	−0.525516
$890$	0	0
$891$	−10.9641	−0.367312
$892$	0	0
$893$	−34.1758	−1.14365
$894$	0	0
$895$	9.33301	0.311968
$896$	0	0
$897$	−15.3360	−0.512055
$898$	0	0
$899$	−6.42857	−0.214405
$900$	0	0
$901$	30.4213	1.01348
$902$	0	0
$903$	−14.4329	−0.480296
$904$	0	0
$905$	−69.4827	−2.30968
$906$	0	0
$907$	24.4010	0.810221	0.405111	−	0.914268i	$-0.367233\pi$
$907$	24.4010	0.810221	0.405111	+	0.914268i	$0.367233\pi$
$908$	0	0
$909$	11.1998	0.371473
$910$	0	0
$911$	0.194858	0.00645593	0.00322797	−	0.999995i	$-0.498973\pi$
$911$	0.194858	0.00645593	0.00322797	+	0.999995i	$0.498973\pi$
$912$	0	0
$913$	10.9728	0.363147
$914$	0	0
$915$	59.7272	1.97452
$916$	0	0
$917$	14.7136	0.485887
$918$	0	0
$919$	21.2817	0.702020	0.351010	−	0.936372i	$-0.385838\pi$
$919$	21.2817	0.702020	0.351010	+	0.936372i	$0.385838\pi$
$920$	0	0
$921$	−58.5980	−1.93087
$922$	0	0
$923$	9.59410	0.315794
$924$	0	0
$925$	15.3209	0.503749
$926$	0	0
$927$	−6.56985	−0.215782
$928$	0	0
$929$	1.25433	0.0411534	0.0205767	−	0.999788i	$-0.493450\pi$
$929$	1.25433	0.0411534	0.0205767	+	0.999788i	$0.493450\pi$
$930$	0	0
$931$	5.00024	0.163876
$932$	0	0
$933$	43.0236	1.40853
$934$	0	0
$935$	−10.6778	−0.349202
$936$	0	0
$937$	1.64665	0.0537937	0.0268968	−	0.999638i	$-0.491437\pi$
$937$	1.64665	0.0537937	0.0268968	+	0.999638i	$0.491437\pi$
$938$	0	0
$939$	−20.5521	−0.670692
$940$	0	0
$941$	−7.28942	−0.237628	−0.118814	−	0.992917i	$-0.537909\pi$
$941$	−7.28942	−0.237628	−0.118814	+	0.992917i	$0.537909\pi$
$942$	0	0
$943$	−75.4657	−2.45750
$944$	0	0
$945$	−6.03722	−0.196391
$946$	0	0
$947$	59.4149	1.93072	0.965362	−	0.260912i	$-0.0840234\pi$
$947$	59.4149	1.93072	0.965362	+	0.260912i	$0.0840234\pi$
$948$	0	0
$949$	11.7631	0.381845
$950$	0	0
$951$	−44.3195	−1.43716
$952$	0	0
$953$	−41.9170	−1.35782	−0.678912	−	0.734220i	$-0.737548\pi$
$953$	−41.9170	−1.35782	−0.678912	+	0.734220i	$0.737548\pi$
$954$	0	0
$955$	−54.9048	−1.77668
$956$	0	0
$957$	3.64061	0.117684
$958$	0	0
$959$	−4.90891	−0.158517
$960$	0	0
$961$	−15.3017	−0.493604
$962$	0	0
$963$	−36.0706	−1.16236
$964$	0	0
$965$	−66.6150	−2.14441
$966$	0	0
$967$	50.6494	1.62878	0.814388	−	0.580320i	$-0.197073\pi$
$967$	50.6494	1.62878	0.814388	+	0.580320i	$0.197073\pi$
$968$	0	0
$969$	42.9811	1.38075
$970$	0	0
$971$	−18.9705	−0.608794	−0.304397	−	0.952545i	$-0.598455\pi$
$971$	−18.9705	−0.608794	−0.304397	+	0.952545i	$0.598455\pi$
$972$	0	0
$973$	−5.77409	−0.185109
$974$	0	0
$975$	6.21299	0.198975
$976$	0	0
$977$	22.4521	0.718305	0.359153	−	0.933279i	$-0.383066\pi$
$977$	22.4521	0.718305	0.359153	+	0.933279i	$0.383066\pi$
$978$	0	0
$979$	7.02589	0.224548
$980$	0	0
$981$	8.71507	0.278251
$982$	0	0
$983$	−21.2327	−0.677219	−0.338609	−	0.940927i	$-0.609957\pi$
$983$	−21.2327	−0.677219	−0.338609	+	0.940927i	$0.609957\pi$
$984$	0	0
$985$	49.9253	1.59075
$986$	0	0
$987$	15.3360	0.488152
$988$	0	0
$989$	−43.9637	−1.39796
$990$	0	0
$991$	14.0839	0.447390	0.223695	−	0.974659i	$-0.428188\pi$
$991$	14.0839	0.447390	0.223695	+	0.974659i	$0.428188\pi$
$992$	0	0
$993$	70.9069	2.25016
$994$	0	0
$995$	27.2886	0.865107
$996$	0	0
$997$	42.7424	1.35367	0.676833	−	0.736137i	$-0.263352\pi$
$997$	42.7424	1.35367	0.676833	+	0.736137i	$0.263352\pi$
$998$	0	0
$999$	−11.9847	−0.379179

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.a.k.1.8	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.a.k.1.8	✓	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)

\(f(q)\)	\(=\)	\(q+2.24381 q^{3} -2.78728 q^{5} -1.00000 q^{7} +2.03468 q^{9} +O(q^{10})\)	\(q+2.24381 q^{3} -2.78728 q^{5} -1.00000 q^{7} +2.03468 q^{9} +1.00000 q^{11} +1.00000 q^{13} -6.25413 q^{15} +3.83090 q^{17} +5.00024 q^{19} -2.24381 q^{21} -6.83482 q^{23} +2.76895 q^{25} -2.16599 q^{27} +1.62251 q^{29} -3.96211 q^{31} +2.24381 q^{33} +2.78728 q^{35} +5.53313 q^{37} +2.24381 q^{39} +11.0414 q^{41} +6.43231 q^{43} -5.67124 q^{45} -6.83482 q^{47} +1.00000 q^{49} +8.59581 q^{51} +7.94104 q^{53} -2.78728 q^{55} +11.2196 q^{57} +4.33959 q^{59} -9.55004 q^{61} -2.03468 q^{63} -2.78728 q^{65} -2.80120 q^{67} -15.3360 q^{69} +9.59410 q^{71} +11.7631 q^{73} +6.21299 q^{75} -1.00000 q^{77} +5.73521 q^{79} -10.9641 q^{81} +10.9728 q^{83} -10.6778 q^{85} +3.64061 q^{87} +7.02589 q^{89} -1.00000 q^{91} -8.89021 q^{93} -13.9371 q^{95} +19.1400 q^{97} +2.03468 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + q^{3} + 4 q^{5} - 10 q^{7} + 17 q^{9}+O(q^{10}) \) 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

Properties

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