LMFDB - Embedded newform 4004.2.a.f.1.3

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:	$1$
Dimension:	$5$
Coefficient field:	5.5.463341.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 6x^{3} + 6x^{2} + 6x + 1 $ x^5 - 2x^4 - 6x^3 + 6x^2 + 6x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.231747$ 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)

$f(q)$	$=$	$q+1.23175 q^{3} -0.600512 q^{5} -1.00000 q^{7} -1.48280 q^{9} +O(q^{10})$	$q+1.23175 q^{3} -0.600512 q^{5} -1.00000 q^{7} -1.48280 q^{9} -1.00000 q^{11} +1.00000 q^{13} -0.739678 q^{15} +1.80713 q^{17} +4.27507 q^{19} -1.23175 q^{21} +3.07404 q^{23} -4.63939 q^{25} -5.52167 q^{27} -1.52279 q^{29} -6.09361 q^{31} -1.23175 q^{33} +0.600512 q^{35} +0.452078 q^{37} +1.23175 q^{39} -5.59124 q^{41} -5.23620 q^{43} +0.890439 q^{45} -7.70416 q^{47} +1.00000 q^{49} +2.22592 q^{51} +2.83671 q^{53} +0.600512 q^{55} +5.26580 q^{57} +7.62530 q^{59} +6.76817 q^{61} +1.48280 q^{63} -0.600512 q^{65} +7.57426 q^{67} +3.78644 q^{69} -4.36498 q^{71} -1.56379 q^{73} -5.71455 q^{75} +1.00000 q^{77} -11.4461 q^{79} -2.35290 q^{81} -9.00918 q^{83} -1.08520 q^{85} -1.87569 q^{87} -15.9025 q^{89} -1.00000 q^{91} -7.50578 q^{93} -2.56723 q^{95} -9.57538 q^{97} +1.48280 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 3 q^{3} - 5 q^{7} + 2 q^{9}+O(q^{10}) $ 5 * q + 3 * q^3 - 5 * q^7 + 2 * q^9	$ 5 q + 3 q^{3} - 5 q^{7} + 2 q^{9} - 5 q^{11} + 5 q^{13} - 3 q^{15} - 9 q^{17} - 4 q^{19} - 3 q^{21} - 4 q^{23} + q^{25} + 3 q^{27} - 8 q^{29} + 7 q^{31} - 3 q^{33} - 10 q^{37} + 3 q^{39} - 18 q^{41} - 22 q^{43} - 26 q^{45} + 12 q^{47} + 5 q^{49} - 7 q^{51} + 7 q^{53} - 6 q^{57} - q^{59} - 26 q^{61} - 2 q^{63} - 8 q^{67} - 12 q^{69} + 18 q^{71} - 25 q^{73} - 16 q^{75} + 5 q^{77} - 6 q^{79} - 7 q^{81} + 11 q^{83} - 7 q^{85} - 5 q^{87} - 14 q^{89} - 5 q^{91} - 41 q^{93} - 22 q^{95} - 33 q^{97} - 2 q^{99}+O(q^{100}) $ 5 * q + 3 * q^3 - 5 * q^7 + 2 * q^9 - 5 * q^11 + 5 * q^13 - 3 * q^15 - 9 * q^17 - 4 * q^19 - 3 * q^21 - 4 * q^23 + q^25 + 3 * q^27 - 8 * q^29 + 7 * q^31 - 3 * q^33 - 10 * q^37 + 3 * q^39 - 18 * q^41 - 22 * q^43 - 26 * q^45 + 12 * q^47 + 5 * q^49 - 7 * q^51 + 7 * q^53 - 6 * q^57 - q^59 - 26 * q^61 - 2 * q^63 - 8 * q^67 - 12 * q^69 + 18 * q^71 - 25 * q^73 - 16 * q^75 + 5 * q^77 - 6 * q^79 - 7 * q^81 + 11 * q^83 - 7 * q^85 - 5 * q^87 - 14 * q^89 - 5 * q^91 - 41 * q^93 - 22 * q^95 - 33 * q^97 - 2 * 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
$3$	1.23175	0.711149	0.355575	−	0.934648i	$-0.384285\pi$
$3$	1.23175	0.711149	0.355575	+	0.934648i	$0.384285\pi$
$4$	0	0
$5$	−0.600512	−0.268557	−0.134279	−	0.990944i	$-0.542872\pi$
$5$	−0.600512	−0.268557	−0.134279	+	0.990944i	$0.542872\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−1.48280	−0.494267
$10$	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
$15$	−0.739678	−0.190984
$16$	0	0
$17$	1.80713	0.438293	0.219146	−	0.975692i	$-0.429673\pi$
$17$	1.80713	0.438293	0.219146	+	0.975692i	$0.429673\pi$
$18$	0	0
$19$	4.27507	0.980768	0.490384	−	0.871506i	$-0.336857\pi$
$19$	4.27507	0.980768	0.490384	+	0.871506i	$0.336857\pi$
$20$	0	0
$21$	−1.23175	−0.268789
$22$	0	0
$23$	3.07404	0.640983	0.320491	−	0.947251i	$-0.396152\pi$
$23$	3.07404	0.640983	0.320491	+	0.947251i	$0.396152\pi$
$24$	0	0
$25$	−4.63939	−0.927877
$26$	0	0
$27$	−5.52167	−1.06265
$28$	0	0
$29$	−1.52279	−0.282775	−0.141388	−	0.989954i	$-0.545156\pi$
$29$	−1.52279	−0.282775	−0.141388	+	0.989954i	$0.545156\pi$
$30$	0	0
$31$	−6.09361	−1.09444	−0.547222	−	0.836987i	$-0.684315\pi$
$31$	−6.09361	−1.09444	−0.547222	+	0.836987i	$0.684315\pi$
$32$	0	0
$33$	−1.23175	−0.214420
$34$	0	0
$35$	0.600512	0.101505
$36$	0	0
$37$	0.452078	0.0743211	0.0371606	−	0.999309i	$-0.488169\pi$
$37$	0.452078	0.0743211	0.0371606	+	0.999309i	$0.488169\pi$
$38$	0	0
$39$	1.23175	0.197237
$40$	0	0
$41$	−5.59124	−0.873206	−0.436603	−	0.899654i	$-0.643818\pi$
$41$	−5.59124	−0.873206	−0.436603	+	0.899654i	$0.643818\pi$
$42$	0	0
$43$	−5.23620	−0.798512	−0.399256	−	0.916839i	$-0.630732\pi$
$43$	−5.23620	−0.798512	−0.399256	+	0.916839i	$0.630732\pi$
$44$	0	0
$45$	0.890439	0.132739
$46$	0	0
$47$	−7.70416	−1.12377	−0.561884	−	0.827216i	$-0.689923\pi$
$47$	−7.70416	−1.12377	−0.561884	+	0.827216i	$0.689923\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	2.22592	0.311691
$52$	0	0
$53$	2.83671	0.389652	0.194826	−	0.980838i	$-0.437586\pi$
$53$	2.83671	0.389652	0.194826	+	0.980838i	$0.437586\pi$
$54$	0	0
$55$	0.600512	0.0809730
$56$	0	0
$57$	5.26580	0.697472
$58$	0	0
$59$	7.62530	0.992729	0.496365	−	0.868114i	$-0.334668\pi$
$59$	7.62530	0.992729	0.496365	+	0.868114i	$0.334668\pi$
$60$	0	0
$61$	6.76817	0.866575	0.433288	−	0.901256i	$-0.357353\pi$
$61$	6.76817	0.866575	0.433288	+	0.901256i	$0.357353\pi$
$62$	0	0
$63$	1.48280	0.186815
$64$	0	0
$65$	−0.600512	−0.0744843
$66$	0	0
$67$	7.57426	0.925344	0.462672	−	0.886530i	$-0.346891\pi$
$67$	7.57426	0.925344	0.462672	+	0.886530i	$0.346891\pi$
$68$	0	0
$69$	3.78644	0.455834
$70$	0	0
$71$	−4.36498	−0.518027	−0.259014	−	0.965874i	$-0.583397\pi$
$71$	−4.36498	−0.518027	−0.259014	+	0.965874i	$0.583397\pi$
$72$	0	0
$73$	−1.56379	−0.183027	−0.0915137	−	0.995804i	$-0.529171\pi$
$73$	−1.56379	−0.183027	−0.0915137	+	0.995804i	$0.529171\pi$
$74$	0	0
$75$	−5.71455	−0.659859
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−11.4461	−1.28779	−0.643896	−	0.765113i	$-0.722683\pi$
$79$	−11.4461	−1.28779	−0.643896	+	0.765113i	$0.722683\pi$
$80$	0	0
$81$	−2.35290	−0.261434
$82$	0	0
$83$	−9.00918	−0.988886	−0.494443	−	0.869210i	$-0.664628\pi$
$83$	−9.00918	−0.988886	−0.494443	+	0.869210i	$0.664628\pi$
$84$	0	0
$85$	−1.08520	−0.117707
$86$	0	0
$87$	−1.87569	−0.201095
$88$	0	0
$89$	−15.9025	−1.68566	−0.842832	−	0.538177i	$-0.819113\pi$
$89$	−15.9025	−1.68566	−0.842832	+	0.538177i	$0.819113\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−7.50578	−0.778314
$94$	0	0
$95$	−2.56723	−0.263392
$96$	0	0
$97$	−9.57538	−0.972233	−0.486116	−	0.873894i	$-0.661587\pi$
$97$	−9.57538	−0.972233	−0.486116	+	0.873894i	$0.661587\pi$
$98$	0	0
$99$	1.48280	0.149027
$100$	0	0
$101$	1.17245	0.116663	0.0583315	−	0.998297i	$-0.481422\pi$
$101$	1.17245	0.116663	0.0583315	+	0.998297i	$0.481422\pi$
$102$	0	0
$103$	−14.6528	−1.44378	−0.721890	−	0.692008i	$-0.756726\pi$
$103$	−14.6528	−1.44378	−0.721890	+	0.692008i	$0.756726\pi$
$104$	0	0
$105$	0.739678	0.0721852
$106$	0	0
$107$	−6.03620	−0.583542	−0.291771	−	0.956488i	$-0.594244\pi$
$107$	−6.03620	−0.583542	−0.291771	+	0.956488i	$0.594244\pi$
$108$	0	0
$109$	−19.7912	−1.89565	−0.947824	−	0.318793i	$-0.896722\pi$
$109$	−19.7912	−1.89565	−0.947824	+	0.318793i	$0.896722\pi$
$110$	0	0
$111$	0.556845	0.0528534
$112$	0	0
$113$	−1.50896	−0.141951	−0.0709756	−	0.997478i	$-0.522611\pi$
$113$	−1.50896	−0.141951	−0.0709756	+	0.997478i	$0.522611\pi$
$114$	0	0
$115$	−1.84600	−0.172140
$116$	0	0
$117$	−1.48280	−0.137085
$118$	0	0
$119$	−1.80713	−0.165659
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−6.88700	−0.620980
$124$	0	0
$125$	5.78857	0.517745
$126$	0	0
$127$	18.4823	1.64004	0.820021	−	0.572333i	$-0.193961\pi$
$127$	18.4823	1.64004	0.820021	+	0.572333i	$0.193961\pi$
$128$	0	0
$129$	−6.44967	−0.567861
$130$	0	0
$131$	8.19164	0.715707	0.357854	−	0.933778i	$-0.383509\pi$
$131$	8.19164	0.715707	0.357854	+	0.933778i	$0.383509\pi$
$132$	0	0
$133$	−4.27507	−0.370695
$134$	0	0
$135$	3.31583	0.285381
$136$	0	0
$137$	−5.79571	−0.495161	−0.247581	−	0.968867i	$-0.579635\pi$
$137$	−5.79571	−0.495161	−0.247581	+	0.968867i	$0.579635\pi$
$138$	0	0
$139$	−17.0065	−1.44247	−0.721237	−	0.692689i	$-0.756426\pi$
$139$	−17.0065	−1.44247	−0.721237	+	0.692689i	$0.756426\pi$
$140$	0	0
$141$	−9.48958	−0.799166
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	0.914454	0.0759413
$146$	0	0
$147$	1.23175	0.101593
$148$	0	0
$149$	13.8281	1.13284	0.566422	−	0.824115i	$-0.308327\pi$
$149$	13.8281	1.13284	0.566422	+	0.824115i	$0.308327\pi$
$150$	0	0
$151$	8.31572	0.676724	0.338362	−	0.941016i	$-0.390127\pi$
$151$	8.31572	0.676724	0.338362	+	0.941016i	$0.390127\pi$
$152$	0	0
$153$	−2.67961	−0.216634
$154$	0	0
$155$	3.65929	0.293921
$156$	0	0
$157$	−17.7805	−1.41904	−0.709520	−	0.704686i	$-0.751088\pi$
$157$	−17.7805	−1.41904	−0.709520	+	0.704686i	$0.751088\pi$
$158$	0	0
$159$	3.49410	0.277100
$160$	0	0
$161$	−3.07404	−0.242269
$162$	0	0
$163$	4.68230	0.366746	0.183373	−	0.983043i	$-0.441298\pi$
$163$	4.68230	0.366746	0.183373	+	0.983043i	$0.441298\pi$
$164$	0	0
$165$	0.739678	0.0575839
$166$	0	0
$167$	−1.52594	−0.118081	−0.0590405	−	0.998256i	$-0.518804\pi$
$167$	−1.52594	−0.118081	−0.0590405	+	0.998256i	$0.518804\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−6.33907	−0.484761
$172$	0	0
$173$	−15.5296	−1.18069	−0.590346	−	0.807151i	$-0.701009\pi$
$173$	−15.5296	−1.18069	−0.590346	+	0.807151i	$0.701009\pi$
$174$	0	0
$175$	4.63939	0.350705
$176$	0	0
$177$	9.39244	0.705979
$178$	0	0
$179$	−25.0011	−1.86867	−0.934335	−	0.356396i	$-0.884005\pi$
$179$	−25.0011	−1.86867	−0.934335	+	0.356396i	$0.884005\pi$
$180$	0	0
$181$	22.2110	1.65093	0.825465	−	0.564453i	$-0.190913\pi$
$181$	22.2110	1.65093	0.825465	+	0.564453i	$0.190913\pi$
$182$	0	0
$183$	8.33667	0.616264
$184$	0	0
$185$	−0.271478	−0.0199595
$186$	0	0
$187$	−1.80713	−0.132150
$188$	0	0
$189$	5.52167	0.401643
$190$	0	0
$191$	−14.8157	−1.07202	−0.536012	−	0.844210i	$-0.680070\pi$
$191$	−14.8157	−1.07202	−0.536012	+	0.844210i	$0.680070\pi$
$192$	0	0
$193$	−12.7203	−0.915626	−0.457813	−	0.889048i	$-0.651367\pi$
$193$	−12.7203	−0.915626	−0.457813	+	0.889048i	$0.651367\pi$
$194$	0	0
$195$	−0.739678	−0.0529695
$196$	0	0
$197$	3.54336	0.252454	0.126227	−	0.992001i	$-0.459713\pi$
$197$	3.54336	0.252454	0.126227	+	0.992001i	$0.459713\pi$
$198$	0	0
$199$	−7.65617	−0.542731	−0.271366	−	0.962476i	$-0.587475\pi$
$199$	−7.65617	−0.542731	−0.271366	+	0.962476i	$0.587475\pi$
$200$	0	0
$201$	9.32957	0.658057
$202$	0	0
$203$	1.52279	0.106879
$204$	0	0
$205$	3.35761	0.234506
$206$	0	0
$207$	−4.55819	−0.316816
$208$	0	0
$209$	−4.27507	−0.295713
$210$	0	0
$211$	19.0444	1.31107	0.655535	−	0.755165i	$-0.272443\pi$
$211$	19.0444	1.31107	0.655535	+	0.755165i	$0.272443\pi$
$212$	0	0
$213$	−5.37655	−0.368395
$214$	0	0
$215$	3.14440	0.214446
$216$	0	0
$217$	6.09361	0.413661
$218$	0	0
$219$	−1.92619	−0.130160
$220$	0	0
$221$	1.80713	0.121561
$222$	0	0
$223$	8.31469	0.556793	0.278396	−	0.960466i	$-0.410197\pi$
$223$	8.31469	0.556793	0.278396	+	0.960466i	$0.410197\pi$
$224$	0	0
$225$	6.87928	0.458619
$226$	0	0
$227$	8.09427	0.537236	0.268618	−	0.963247i	$-0.413433\pi$
$227$	8.09427	0.537236	0.268618	+	0.963247i	$0.413433\pi$
$228$	0	0
$229$	−2.56342	−0.169395	−0.0846976	−	0.996407i	$-0.526992\pi$
$229$	−2.56342	−0.169395	−0.0846976	+	0.996407i	$0.526992\pi$
$230$	0	0
$231$	1.23175	0.0810430
$232$	0	0
$233$	15.5875	1.02117	0.510585	−	0.859827i	$-0.329429\pi$
$233$	15.5875	1.02117	0.510585	+	0.859827i	$0.329429\pi$
$234$	0	0
$235$	4.62644	0.301796
$236$	0	0
$237$	−14.0987	−0.915812
$238$	0	0
$239$	−12.4198	−0.803370	−0.401685	−	0.915778i	$-0.631575\pi$
$239$	−12.4198	−0.803370	−0.401685	+	0.915778i	$0.631575\pi$
$240$	0	0
$241$	−18.2173	−1.17348	−0.586741	−	0.809775i	$-0.699589\pi$
$241$	−18.2173	−1.17348	−0.586741	+	0.809775i	$0.699589\pi$
$242$	0	0
$243$	13.6668	0.876728
$244$	0	0
$245$	−0.600512	−0.0383653
$246$	0	0
$247$	4.27507	0.272016
$248$	0	0
$249$	−11.0970	−0.703246
$250$	0	0
$251$	25.8378	1.63087	0.815434	−	0.578850i	$-0.196498\pi$
$251$	25.8378	1.63087	0.815434	+	0.578850i	$0.196498\pi$
$252$	0	0
$253$	−3.07404	−0.193264
$254$	0	0
$255$	−1.33669	−0.0837070
$256$	0	0
$257$	21.6737	1.35197	0.675985	−	0.736915i	$-0.263718\pi$
$257$	21.6737	1.35197	0.675985	+	0.736915i	$0.263718\pi$
$258$	0	0
$259$	−0.452078	−0.0280907
$260$	0	0
$261$	2.25799	0.139766
$262$	0	0
$263$	19.0652	1.17561	0.587807	−	0.809002i	$-0.299992\pi$
$263$	19.0652	1.17561	0.587807	+	0.809002i	$0.299992\pi$
$264$	0	0
$265$	−1.70348	−0.104644
$266$	0	0
$267$	−19.5879	−1.19876
$268$	0	0
$269$	23.7715	1.44937	0.724686	−	0.689079i	$-0.241985\pi$
$269$	23.7715	1.44937	0.724686	+	0.689079i	$0.241985\pi$
$270$	0	0
$271$	−8.56511	−0.520293	−0.260147	−	0.965569i	$-0.583771\pi$
$271$	−8.56511	−0.520293	−0.260147	+	0.965569i	$0.583771\pi$
$272$	0	0
$273$	−1.23175	−0.0745487
$274$	0	0
$275$	4.63939	0.279765
$276$	0	0
$277$	−20.8561	−1.25312	−0.626560	−	0.779373i	$-0.715538\pi$
$277$	−20.8561	−1.25312	−0.626560	+	0.779373i	$0.715538\pi$
$278$	0	0
$279$	9.03561	0.540948
$280$	0	0
$281$	24.9860	1.49054	0.745269	−	0.666764i	$-0.232321\pi$
$281$	24.9860	1.49054	0.745269	+	0.666764i	$0.232321\pi$
$282$	0	0
$283$	−24.5835	−1.46134	−0.730668	−	0.682733i	$-0.760791\pi$
$283$	−24.5835	−1.46134	−0.730668	+	0.682733i	$0.760791\pi$
$284$	0	0
$285$	−3.16218	−0.187311
$286$	0	0
$287$	5.59124	0.330041
$288$	0	0
$289$	−13.7343	−0.807900
$290$	0	0
$291$	−11.7944	−0.691402
$292$	0	0
$293$	−11.4384	−0.668240	−0.334120	−	0.942531i	$-0.608439\pi$
$293$	−11.4384	−0.668240	−0.334120	+	0.942531i	$0.608439\pi$
$294$	0	0
$295$	−4.57908	−0.266605
$296$	0	0
$297$	5.52167	0.320400
$298$	0	0
$299$	3.07404	0.177777
$300$	0	0
$301$	5.23620	0.301809
$302$	0	0
$303$	1.44416	0.0829648
$304$	0	0
$305$	−4.06437	−0.232725
$306$	0	0
$307$	−13.5149	−0.771333	−0.385667	−	0.922638i	$-0.626029\pi$
$307$	−13.5149	−0.771333	−0.385667	+	0.922638i	$0.626029\pi$
$308$	0	0
$309$	−18.0485	−1.02674
$310$	0	0
$311$	22.7865	1.29210	0.646051	−	0.763294i	$-0.276419\pi$
$311$	22.7865	1.29210	0.646051	+	0.763294i	$0.276419\pi$
$312$	0	0
$313$	7.04952	0.398462	0.199231	−	0.979953i	$-0.436156\pi$
$313$	7.04952	0.398462	0.199231	+	0.979953i	$0.436156\pi$
$314$	0	0
$315$	−0.890439	−0.0501706
$316$	0	0
$317$	−4.02350	−0.225982	−0.112991	−	0.993596i	$-0.536043\pi$
$317$	−4.02350	−0.225982	−0.112991	+	0.993596i	$0.536043\pi$
$318$	0	0
$319$	1.52279	0.0852599
$320$	0	0
$321$	−7.43507	−0.414985
$322$	0	0
$323$	7.72559	0.429863
$324$	0	0
$325$	−4.63939	−0.257347
$326$	0	0
$327$	−24.3777	−1.34809
$328$	0	0
$329$	7.70416	0.424744
$330$	0	0
$331$	−33.1935	−1.82448	−0.912241	−	0.409654i	$-0.865649\pi$
$331$	−33.1935	−1.82448	−0.912241	+	0.409654i	$0.865649\pi$
$332$	0	0
$333$	−0.670341	−0.0367345
$334$	0	0
$335$	−4.54844	−0.248508
$336$	0	0
$337$	−8.83684	−0.481374	−0.240687	−	0.970603i	$-0.577373\pi$
$337$	−8.83684	−0.481374	−0.240687	+	0.970603i	$0.577373\pi$
$338$	0	0
$339$	−1.85866	−0.100949
$340$	0	0
$341$	6.09361	0.329988
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−2.27380	−0.122418
$346$	0	0
$347$	−26.7178	−1.43429	−0.717144	−	0.696925i	$-0.754551\pi$
$347$	−26.7178	−1.43429	−0.717144	+	0.696925i	$0.754551\pi$
$348$	0	0
$349$	10.8285	0.579638	0.289819	−	0.957081i	$-0.406405\pi$
$349$	10.8285	0.579638	0.289819	+	0.957081i	$0.406405\pi$
$350$	0	0
$351$	−5.52167	−0.294725
$352$	0	0
$353$	−27.3697	−1.45674	−0.728371	−	0.685183i	$-0.759722\pi$
$353$	−27.3697	−1.45674	−0.728371	+	0.685183i	$0.759722\pi$
$354$	0	0
$355$	2.62122	0.139120
$356$	0	0
$357$	−2.22592	−0.117808
$358$	0	0
$359$	3.97599	0.209844	0.104922	−	0.994480i	$-0.466541\pi$
$359$	3.97599	0.209844	0.104922	+	0.994480i	$0.466541\pi$
$360$	0	0
$361$	−0.723789	−0.0380941
$362$	0	0
$363$	1.23175	0.0646499
$364$	0	0
$365$	0.939072	0.0491533
$366$	0	0
$367$	29.0449	1.51613	0.758066	−	0.652178i	$-0.226145\pi$
$367$	29.0449	1.51613	0.758066	+	0.652178i	$0.226145\pi$
$368$	0	0
$369$	8.29070	0.431597
$370$	0	0
$371$	−2.83671	−0.147274
$372$	0	0
$373$	19.0534	0.986547	0.493274	−	0.869874i	$-0.335800\pi$
$373$	19.0534	0.986547	0.493274	+	0.869874i	$0.335800\pi$
$374$	0	0
$375$	7.13005	0.368194
$376$	0	0
$377$	−1.52279	−0.0784277
$378$	0	0
$379$	24.3514	1.25085	0.625425	−	0.780285i	$-0.284926\pi$
$379$	24.3514	1.25085	0.625425	+	0.780285i	$0.284926\pi$
$380$	0	0
$381$	22.7656	1.16632
$382$	0	0
$383$	−8.70995	−0.445058	−0.222529	−	0.974926i	$-0.571431\pi$
$383$	−8.70995	−0.445058	−0.222529	+	0.974926i	$0.571431\pi$
$384$	0	0
$385$	−0.600512	−0.0306049
$386$	0	0
$387$	7.76423	0.394678
$388$	0	0
$389$	−13.3170	−0.675198	−0.337599	−	0.941290i	$-0.609615\pi$
$389$	−13.3170	−0.675198	−0.337599	+	0.941290i	$0.609615\pi$
$390$	0	0
$391$	5.55519	0.280938
$392$	0	0
$393$	10.0900	0.508975
$394$	0	0
$395$	6.87354	0.345846
$396$	0	0
$397$	36.9338	1.85365	0.926827	−	0.375488i	$-0.122525\pi$
$397$	36.9338	1.85365	0.926827	+	0.375488i	$0.122525\pi$
$398$	0	0
$399$	−5.26580	−0.263620
$400$	0	0
$401$	−21.2862	−1.06298	−0.531491	−	0.847064i	$-0.678368\pi$
$401$	−21.2862	−1.06298	−0.531491	+	0.847064i	$0.678368\pi$
$402$	0	0
$403$	−6.09361	−0.303544
$404$	0	0
$405$	1.41295	0.0702098
$406$	0	0
$407$	−0.452078	−0.0224087
$408$	0	0
$409$	−12.5246	−0.619301	−0.309651	−	0.950850i	$-0.600212\pi$
$409$	−12.5246	−0.619301	−0.309651	+	0.950850i	$0.600212\pi$
$410$	0	0
$411$	−7.13885	−0.352133
$412$	0	0
$413$	−7.62530	−0.375216
$414$	0	0
$415$	5.41012	0.265572
$416$	0	0
$417$	−20.9477	−1.02581
$418$	0	0
$419$	36.4430	1.78036	0.890179	−	0.455610i	$-0.150579\pi$
$419$	36.4430	1.78036	0.890179	+	0.455610i	$0.150579\pi$
$420$	0	0
$421$	27.4483	1.33775	0.668873	−	0.743376i	$-0.266777\pi$
$421$	27.4483	1.33775	0.668873	+	0.743376i	$0.266777\pi$
$422$	0	0
$423$	11.4237	0.555441
$424$	0	0
$425$	−8.38396	−0.406682
$426$	0	0
$427$	−6.76817	−0.327535
$428$	0	0
$429$	−1.23175	−0.0594693
$430$	0	0
$431$	−6.05638	−0.291725	−0.145863	−	0.989305i	$-0.546596\pi$
$431$	−6.05638	−0.291725	−0.145863	+	0.989305i	$0.546596\pi$
$432$	0	0
$433$	−18.0372	−0.866810	−0.433405	−	0.901199i	$-0.642688\pi$
$433$	−18.0372	−0.866810	−0.433405	+	0.901199i	$0.642688\pi$
$434$	0	0
$435$	1.12638	0.0540056
$436$	0	0
$437$	13.1418	0.628655
$438$	0	0
$439$	−17.2999	−0.825677	−0.412839	−	0.910804i	$-0.635463\pi$
$439$	−17.2999	−0.825677	−0.412839	+	0.910804i	$0.635463\pi$
$440$	0	0
$441$	−1.48280	−0.0706095
$442$	0	0
$443$	25.2302	1.19872	0.599361	−	0.800479i	$-0.295421\pi$
$443$	25.2302	1.19872	0.599361	+	0.800479i	$0.295421\pi$
$444$	0	0
$445$	9.54965	0.452697
$446$	0	0
$447$	17.0327	0.805621
$448$	0	0
$449$	6.89252	0.325278	0.162639	−	0.986686i	$-0.447999\pi$
$449$	6.89252	0.325278	0.162639	+	0.986686i	$0.447999\pi$
$450$	0	0
$451$	5.59124	0.263281
$452$	0	0
$453$	10.2429	0.481251
$454$	0	0
$455$	0.600512	0.0281524
$456$	0	0
$457$	−19.1134	−0.894088	−0.447044	−	0.894512i	$-0.647523\pi$
$457$	−19.1134	−0.894088	−0.447044	+	0.894512i	$0.647523\pi$
$458$	0	0
$459$	−9.97837	−0.465750
$460$	0	0
$461$	−5.15766	−0.240216	−0.120108	−	0.992761i	$-0.538324\pi$
$461$	−5.15766	−0.240216	−0.120108	+	0.992761i	$0.538324\pi$
$462$	0	0
$463$	36.4376	1.69340	0.846699	−	0.532073i	$-0.178587\pi$
$463$	36.4376	1.69340	0.846699	+	0.532073i	$0.178587\pi$
$464$	0	0
$465$	4.50731	0.209022
$466$	0	0
$467$	25.4526	1.17781	0.588903	−	0.808203i	$-0.299560\pi$
$467$	25.4526	1.17781	0.588903	+	0.808203i	$0.299560\pi$
$468$	0	0
$469$	−7.57426	−0.349747
$470$	0	0
$471$	−21.9011	−1.00915
$472$	0	0
$473$	5.23620	0.240761
$474$	0	0
$475$	−19.8337	−0.910032
$476$	0	0
$477$	−4.20627	−0.192592
$478$	0	0
$479$	−5.44537	−0.248805	−0.124403	−	0.992232i	$-0.539701\pi$
$479$	−5.44537	−0.248805	−0.124403	+	0.992232i	$0.539701\pi$
$480$	0	0
$481$	0.452078	0.0206130
$482$	0	0
$483$	−3.78644	−0.172289
$484$	0	0
$485$	5.75013	0.261100
$486$	0	0
$487$	−32.0717	−1.45331	−0.726653	−	0.687004i	$-0.758925\pi$
$487$	−32.0717	−1.45331	−0.726653	+	0.687004i	$0.758925\pi$
$488$	0	0
$489$	5.76740	0.260811
$490$	0	0
$491$	−31.4343	−1.41861	−0.709304	−	0.704902i	$-0.750991\pi$
$491$	−31.4343	−1.41861	−0.709304	+	0.704902i	$0.750991\pi$
$492$	0	0
$493$	−2.75188	−0.123938
$494$	0	0
$495$	−0.890439	−0.0400223
$496$	0	0
$497$	4.36498	0.195796
$498$	0	0
$499$	28.4052	1.27159	0.635796	−	0.771857i	$-0.280672\pi$
$499$	28.4052	1.27159	0.635796	+	0.771857i	$0.280672\pi$
$500$	0	0
$501$	−1.87958	−0.0839732
$502$	0	0
$503$	21.9734	0.979745	0.489872	−	0.871794i	$-0.337043\pi$
$503$	21.9734	0.979745	0.489872	+	0.871794i	$0.337043\pi$
$504$	0	0
$505$	−0.704070	−0.0313307
$506$	0	0
$507$	1.23175	0.0547038
$508$	0	0
$509$	10.3369	0.458176	0.229088	−	0.973406i	$-0.426426\pi$
$509$	10.3369	0.458176	0.229088	+	0.973406i	$0.426426\pi$
$510$	0	0
$511$	1.56379	0.0691778
$512$	0	0
$513$	−23.6055	−1.04221
$514$	0	0
$515$	8.79916	0.387737
$516$	0	0
$517$	7.70416	0.338829
$518$	0	0
$519$	−19.1285	−0.839647
$520$	0	0
$521$	32.4884	1.42334	0.711671	−	0.702513i	$-0.247939\pi$
$521$	32.4884	1.42334	0.711671	+	0.702513i	$0.247939\pi$
$522$	0	0
$523$	9.21179	0.402804	0.201402	−	0.979509i	$-0.435450\pi$
$523$	9.21179	0.402804	0.201402	+	0.979509i	$0.435450\pi$
$524$	0	0
$525$	5.71455	0.249403
$526$	0	0
$527$	−11.0119	−0.479687
$528$	0	0
$529$	−13.5502	−0.589141
$530$	0	0
$531$	−11.3068	−0.490673
$532$	0	0
$533$	−5.59124	−0.242184
$534$	0	0
$535$	3.62481	0.156714
$536$	0	0
$537$	−30.7950	−1.32890
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	4.06538	0.174784	0.0873922	−	0.996174i	$-0.472147\pi$
$541$	4.06538	0.174784	0.0873922	+	0.996174i	$0.472147\pi$
$542$	0	0
$543$	27.3583	1.17406
$544$	0	0
$545$	11.8848	0.509090
$546$	0	0
$547$	−8.61562	−0.368377	−0.184189	−	0.982891i	$-0.558966\pi$
$547$	−8.61562	−0.368377	−0.184189	+	0.982891i	$0.558966\pi$
$548$	0	0
$549$	−10.0358	−0.428319
$550$	0	0
$551$	−6.51004	−0.277337
$552$	0	0
$553$	11.4461	0.486739
$554$	0	0
$555$	−0.334392	−0.0141942
$556$	0	0
$557$	−16.2610	−0.688999	−0.344499	−	0.938787i	$-0.611951\pi$
$557$	−16.2610	−0.688999	−0.344499	+	0.938787i	$0.611951\pi$
$558$	0	0
$559$	−5.23620	−0.221467
$560$	0	0
$561$	−2.22592	−0.0939785
$562$	0	0
$563$	24.6248	1.03781	0.518907	−	0.854831i	$-0.326339\pi$
$563$	24.6248	1.03781	0.518907	+	0.854831i	$0.326339\pi$
$564$	0	0
$565$	0.906150	0.0381220
$566$	0	0
$567$	2.35290	0.0988126
$568$	0	0
$569$	−0.240201	−0.0100697	−0.00503487	−	0.999987i	$-0.501603\pi$
$569$	−0.240201	−0.0100697	−0.00503487	+	0.999987i	$0.501603\pi$
$570$	0	0
$571$	9.61669	0.402446	0.201223	−	0.979545i	$-0.435508\pi$
$571$	9.61669	0.402446	0.201223	+	0.979545i	$0.435508\pi$
$572$	0	0
$573$	−18.2492	−0.762370
$574$	0	0
$575$	−14.2617	−0.594753
$576$	0	0
$577$	−31.3373	−1.30459	−0.652293	−	0.757967i	$-0.726193\pi$
$577$	−31.3373	−1.30459	−0.652293	+	0.757967i	$0.726193\pi$
$578$	0	0
$579$	−15.6682	−0.651147
$580$	0	0
$581$	9.00918	0.373764
$582$	0	0
$583$	−2.83671	−0.117484
$584$	0	0
$585$	0.890439	0.0368151
$586$	0	0
$587$	−32.6235	−1.34652	−0.673259	−	0.739407i	$-0.735106\pi$
$587$	−32.6235	−1.34652	−0.673259	+	0.739407i	$0.735106\pi$
$588$	0	0
$589$	−26.0506	−1.07340
$590$	0	0
$591$	4.36452	0.179533
$592$	0	0
$593$	−20.3134	−0.834170	−0.417085	−	0.908867i	$-0.636948\pi$
$593$	−20.3134	−0.834170	−0.417085	+	0.908867i	$0.636948\pi$
$594$	0	0
$595$	1.08520	0.0444889
$596$	0	0
$597$	−9.43046	−0.385963
$598$	0	0
$599$	41.3966	1.69142	0.845709	−	0.533644i	$-0.179178\pi$
$599$	41.3966	1.69142	0.845709	+	0.533644i	$0.179178\pi$
$600$	0	0
$601$	7.89008	0.321843	0.160922	−	0.986967i	$-0.448553\pi$
$601$	7.89008	0.321843	0.160922	+	0.986967i	$0.448553\pi$
$602$	0	0
$603$	−11.2311	−0.457367
$604$	0	0
$605$	−0.600512	−0.0244143
$606$	0	0
$607$	30.6721	1.24494	0.622470	−	0.782643i	$-0.286129\pi$
$607$	30.6721	1.24494	0.622470	+	0.782643i	$0.286129\pi$
$608$	0	0
$609$	1.87569	0.0760069
$610$	0	0
$611$	−7.70416	−0.311677
$612$	0	0
$613$	−29.4097	−1.18785	−0.593924	−	0.804521i	$-0.702422\pi$
$613$	−29.4097	−1.18785	−0.593924	+	0.804521i	$0.702422\pi$
$614$	0	0
$615$	4.13572	0.166768
$616$	0	0
$617$	−9.28443	−0.373777	−0.186889	−	0.982381i	$-0.559840\pi$
$617$	−9.28443	−0.373777	−0.186889	+	0.982381i	$0.559840\pi$
$618$	0	0
$619$	36.0378	1.44848	0.724241	−	0.689547i	$-0.242190\pi$
$619$	36.0378	1.44848	0.724241	+	0.689547i	$0.242190\pi$
$620$	0	0
$621$	−16.9739	−0.681138
$622$	0	0
$623$	15.9025	0.637121
$624$	0	0
$625$	19.7208	0.788833
$626$	0	0
$627$	−5.26580	−0.210296
$628$	0	0
$629$	0.816962	0.0325744
$630$	0	0
$631$	20.9396	0.833592	0.416796	−	0.909000i	$-0.363153\pi$
$631$	20.9396	0.833592	0.416796	+	0.909000i	$0.363153\pi$
$632$	0	0
$633$	23.4578	0.932366
$634$	0	0
$635$	−11.0989	−0.440445
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	6.47239	0.256044
$640$	0	0
$641$	−24.8280	−0.980647	−0.490323	−	0.871541i	$-0.663121\pi$
$641$	−24.8280	−0.980647	−0.490323	+	0.871541i	$0.663121\pi$
$642$	0	0
$643$	25.0243	0.986862	0.493431	−	0.869785i	$-0.335743\pi$
$643$	25.0243	0.986862	0.493431	+	0.869785i	$0.335743\pi$
$644$	0	0
$645$	3.87310	0.152503
$646$	0	0
$647$	12.9199	0.507932	0.253966	−	0.967213i	$-0.418265\pi$
$647$	12.9199	0.507932	0.253966	+	0.967213i	$0.418265\pi$
$648$	0	0
$649$	−7.62530	−0.299319
$650$	0	0
$651$	7.50578	0.294175
$652$	0	0
$653$	30.8944	1.20899	0.604496	−	0.796608i	$-0.293374\pi$
$653$	30.8944	1.20899	0.604496	+	0.796608i	$0.293374\pi$
$654$	0	0
$655$	−4.91918	−0.192208
$656$	0	0
$657$	2.31878	0.0904643
$658$	0	0
$659$	49.3215	1.92129	0.960647	−	0.277772i	$-0.0895961\pi$
$659$	49.3215	1.92129	0.960647	+	0.277772i	$0.0895961\pi$
$660$	0	0
$661$	0.677741	0.0263611	0.0131805	−	0.999913i	$-0.495804\pi$
$661$	0.677741	0.0263611	0.0131805	+	0.999913i	$0.495804\pi$
$662$	0	0
$663$	2.22592	0.0864477
$664$	0	0
$665$	2.56723	0.0995529
$666$	0	0
$667$	−4.68113	−0.181254
$668$	0	0
$669$	10.2416	0.395963
$670$	0	0
$671$	−6.76817	−0.261282
$672$	0	0
$673$	−0.323819	−0.0124823	−0.00624116	−	0.999981i	$-0.501987\pi$
$673$	−0.323819	−0.0124823	−0.00624116	+	0.999981i	$0.501987\pi$
$674$	0	0
$675$	25.6172	0.986005
$676$	0	0
$677$	−38.7424	−1.48899	−0.744495	−	0.667628i	$-0.767310\pi$
$677$	−38.7424	−1.48899	−0.744495	+	0.667628i	$0.767310\pi$
$678$	0	0
$679$	9.57538	0.367469
$680$	0	0
$681$	9.97009	0.382055
$682$	0	0
$683$	−32.2907	−1.23557	−0.617784	−	0.786348i	$-0.711969\pi$
$683$	−32.2907	−1.23557	−0.617784	+	0.786348i	$0.711969\pi$
$684$	0	0
$685$	3.48039	0.132979
$686$	0	0
$687$	−3.15748	−0.120465
$688$	0	0
$689$	2.83671	0.108070
$690$	0	0
$691$	−28.1227	−1.06984	−0.534920	−	0.844903i	$-0.679658\pi$
$691$	−28.1227	−1.06984	−0.534920	+	0.844903i	$0.679658\pi$
$692$	0	0
$693$	−1.48280	−0.0563269
$694$	0	0
$695$	10.2126	0.387386
$696$	0	0
$697$	−10.1041	−0.382720
$698$	0	0
$699$	19.1998	0.726204
$700$	0	0
$701$	4.02877	0.152165	0.0760823	−	0.997102i	$-0.475759\pi$
$701$	4.02877	0.152165	0.0760823	+	0.997102i	$0.475759\pi$
$702$	0	0
$703$	1.93266	0.0728918
$704$	0	0
$705$	5.69860	0.214622
$706$	0	0
$707$	−1.17245	−0.0440945
$708$	0	0
$709$	−0.781086	−0.0293343	−0.0146671	−	0.999892i	$-0.504669\pi$
$709$	−0.781086	−0.0293343	−0.0146671	+	0.999892i	$0.504669\pi$
$710$	0	0
$711$	16.9723	0.636513
$712$	0	0
$713$	−18.7320	−0.701520
$714$	0	0
$715$	0.600512	0.0224579
$716$	0	0
$717$	−15.2980	−0.571316
$718$	0	0
$719$	27.4332	1.02309	0.511543	−	0.859258i	$-0.329074\pi$
$719$	27.4332	1.02309	0.511543	+	0.859258i	$0.329074\pi$
$720$	0	0
$721$	14.6528	0.545697
$722$	0	0
$723$	−22.4391	−0.834520
$724$	0	0
$725$	7.06481	0.262381
$726$	0	0
$727$	−14.9877	−0.555863	−0.277931	−	0.960601i	$-0.589649\pi$
$727$	−14.9877	−0.555863	−0.277931	+	0.960601i	$0.589649\pi$
$728$	0	0
$729$	23.8928	0.884918
$730$	0	0
$731$	−9.46247	−0.349982
$732$	0	0
$733$	−29.0848	−1.07427	−0.537135	−	0.843496i	$-0.680494\pi$
$733$	−29.0848	−1.07427	−0.537135	+	0.843496i	$0.680494\pi$
$734$	0	0
$735$	−0.739678	−0.0272835
$736$	0	0
$737$	−7.57426	−0.279002
$738$	0	0
$739$	−38.7167	−1.42422	−0.712108	−	0.702070i	$-0.752259\pi$
$739$	−38.7167	−1.42422	−0.712108	+	0.702070i	$0.752259\pi$
$740$	0	0
$741$	5.26580	0.193444
$742$	0	0
$743$	47.6905	1.74960	0.874798	−	0.484488i	$-0.160994\pi$
$743$	47.6905	1.74960	0.874798	+	0.484488i	$0.160994\pi$
$744$	0	0
$745$	−8.30395	−0.304233
$746$	0	0
$747$	13.3588	0.488774
$748$	0	0
$749$	6.03620	0.220558
$750$	0	0
$751$	−22.7739	−0.831033	−0.415516	−	0.909586i	$-0.636399\pi$
$751$	−22.7739	−0.831033	−0.415516	+	0.909586i	$0.636399\pi$
$752$	0	0
$753$	31.8256	1.15979
$754$	0	0
$755$	−4.99369	−0.181739
$756$	0	0
$757$	46.0676	1.67436	0.837178	−	0.546931i	$-0.184204\pi$
$757$	46.0676	1.67436	0.837178	+	0.546931i	$0.184204\pi$
$758$	0	0
$759$	−3.78644	−0.137439
$760$	0	0
$761$	−37.5895	−1.36262	−0.681310	−	0.731995i	$-0.738589\pi$
$761$	−37.5895	−1.36262	−0.681310	+	0.731995i	$0.738589\pi$
$762$	0	0
$763$	19.7912	0.716488
$764$	0	0
$765$	1.60914	0.0581785
$766$	0	0
$767$	7.62530	0.275334
$768$	0	0
$769$	−36.4964	−1.31609	−0.658046	−	0.752978i	$-0.728617\pi$
$769$	−36.4964	−1.31609	−0.658046	+	0.752978i	$0.728617\pi$
$770$	0	0
$771$	26.6966	0.961453
$772$	0	0
$773$	−7.90564	−0.284346	−0.142173	−	0.989842i	$-0.545409\pi$
$773$	−7.90564	−0.284346	−0.142173	+	0.989842i	$0.545409\pi$
$774$	0	0
$775$	28.2706	1.01551
$776$	0	0
$777$	−0.556845	−0.0199767
$778$	0	0
$779$	−23.9030	−0.856412
$780$	0	0
$781$	4.36498	0.156191
$782$	0	0
$783$	8.40835	0.300490
$784$	0	0
$785$	10.6774	0.381093
$786$	0	0
$787$	40.8304	1.45545	0.727724	−	0.685870i	$-0.240578\pi$
$787$	40.8304	1.45545	0.727724	+	0.685870i	$0.240578\pi$
$788$	0	0
$789$	23.4835	0.836036
$790$	0	0
$791$	1.50896	0.0536525
$792$	0	0
$793$	6.76817	0.240345
$794$	0	0
$795$	−2.09825	−0.0744173
$796$	0	0
$797$	3.74317	0.132590	0.0662950	−	0.997800i	$-0.478882\pi$
$797$	3.74317	0.132590	0.0662950	+	0.997800i	$0.478882\pi$
$798$	0	0
$799$	−13.9224	−0.492539
$800$	0	0
$801$	23.5803	0.833167
$802$	0	0
$803$	1.56379	0.0551848
$804$	0	0
$805$	1.84600	0.0650630
$806$	0	0
$807$	29.2804	1.03072
$808$	0	0
$809$	−15.9541	−0.560916	−0.280458	−	0.959866i	$-0.590486\pi$
$809$	−15.9541	−0.560916	−0.280458	+	0.959866i	$0.590486\pi$
$810$	0	0
$811$	3.55737	0.124916	0.0624581	−	0.998048i	$-0.480106\pi$
$811$	3.55737	0.124916	0.0624581	+	0.998048i	$0.480106\pi$
$812$	0	0
$813$	−10.5500	−0.370006
$814$	0	0
$815$	−2.81177	−0.0984922
$816$	0	0
$817$	−22.3851	−0.783155
$818$	0	0
$819$	1.48280	0.0518132
$820$	0	0
$821$	13.9901	0.488258	0.244129	−	0.969743i	$-0.421498\pi$
$821$	13.9901	0.488258	0.244129	+	0.969743i	$0.421498\pi$
$822$	0	0
$823$	−35.5679	−1.23982	−0.619910	−	0.784673i	$-0.712831\pi$
$823$	−35.5679	−1.23982	−0.619910	+	0.784673i	$0.712831\pi$
$824$	0	0
$825$	5.71455	0.198955
$826$	0	0
$827$	−4.45727	−0.154995	−0.0774973	−	0.996993i	$-0.524693\pi$
$827$	−4.45727	−0.154995	−0.0774973	+	0.996993i	$0.524693\pi$
$828$	0	0
$829$	−43.2302	−1.50145	−0.750724	−	0.660616i	$-0.770295\pi$
$829$	−43.2302	−1.50145	−0.750724	+	0.660616i	$0.770295\pi$
$830$	0	0
$831$	−25.6894	−0.891155
$832$	0	0
$833$	1.80713	0.0626132
$834$	0	0
$835$	0.916347	0.0317115
$836$	0	0
$837$	33.6469	1.16301
$838$	0	0
$839$	−8.79291	−0.303565	−0.151783	−	0.988414i	$-0.548501\pi$
$839$	−8.79291	−0.303565	−0.151783	+	0.988414i	$0.548501\pi$
$840$	0	0
$841$	−26.6811	−0.920038
$842$	0	0
$843$	30.7764	1.05999
$844$	0	0
$845$	−0.600512	−0.0206582
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	−30.2806	−1.03923
$850$	0	0
$851$	1.38971	0.0476385
$852$	0	0
$853$	36.4266	1.24722	0.623612	−	0.781734i	$-0.285665\pi$
$853$	36.4266	1.24722	0.623612	+	0.781734i	$0.285665\pi$
$854$	0	0
$855$	3.80669	0.130186
$856$	0	0
$857$	−27.5743	−0.941920	−0.470960	−	0.882155i	$-0.656092\pi$
$857$	−27.5743	−0.941920	−0.470960	+	0.882155i	$0.656092\pi$
$858$	0	0
$859$	47.7606	1.62957	0.814786	−	0.579762i	$-0.196855\pi$
$859$	47.7606	1.62957	0.814786	+	0.579762i	$0.196855\pi$
$860$	0	0
$861$	6.88700	0.234708
$862$	0	0
$863$	27.5618	0.938216	0.469108	−	0.883141i	$-0.344575\pi$
$863$	27.5618	0.938216	0.469108	+	0.883141i	$0.344575\pi$
$864$	0	0
$865$	9.32569	0.317083
$866$	0	0
$867$	−16.9172	−0.574537
$868$	0	0
$869$	11.4461	0.388284
$870$	0	0
$871$	7.57426	0.256644
$872$	0	0
$873$	14.1984	0.480542
$874$	0	0
$875$	−5.78857	−0.195689
$876$	0	0
$877$	−38.9960	−1.31680	−0.658401	−	0.752667i	$-0.728767\pi$
$877$	−38.9960	−1.31680	−0.658401	+	0.752667i	$0.728767\pi$
$878$	0	0
$879$	−14.0892	−0.475218
$880$	0	0
$881$	8.24084	0.277641	0.138820	−	0.990318i	$-0.455669\pi$
$881$	8.24084	0.277641	0.138820	+	0.990318i	$0.455669\pi$
$882$	0	0
$883$	−43.1697	−1.45278	−0.726388	−	0.687285i	$-0.758802\pi$
$883$	−43.1697	−1.45278	−0.726388	+	0.687285i	$0.758802\pi$
$884$	0	0
$885$	−5.64027	−0.189596
$886$	0	0
$887$	44.4121	1.49121	0.745607	−	0.666386i	$-0.232160\pi$
$887$	44.4121	1.49121	0.745607	+	0.666386i	$0.232160\pi$
$888$	0	0
$889$	−18.4823	−0.619878
$890$	0	0
$891$	2.35290	0.0788252
$892$	0	0
$893$	−32.9358	−1.10216
$894$	0	0
$895$	15.0135	0.501845
$896$	0	0
$897$	3.78644	0.126426
$898$	0	0
$899$	9.27929	0.309482
$900$	0	0
$901$	5.12629	0.170781
$902$	0	0
$903$	6.44967	0.214631
$904$	0	0
$905$	−13.3380	−0.443369
$906$	0	0
$907$	−15.6037	−0.518114	−0.259057	−	0.965862i	$-0.583412\pi$
$907$	−15.6037	−0.518114	−0.259057	+	0.965862i	$0.583412\pi$
$908$	0	0
$909$	−1.73851	−0.0576627
$910$	0	0
$911$	−12.8347	−0.425233	−0.212616	−	0.977136i	$-0.568198\pi$
$911$	−12.8347	−0.425233	−0.212616	+	0.977136i	$0.568198\pi$
$912$	0	0
$913$	9.00918	0.298160
$914$	0	0
$915$	−5.00627	−0.165502
$916$	0	0
$917$	−8.19164	−0.270512
$918$	0	0
$919$	45.1800	1.49035	0.745176	−	0.666868i	$-0.232366\pi$
$919$	45.1800	1.49035	0.745176	+	0.666868i	$0.232366\pi$
$920$	0	0
$921$	−16.6469	−0.548533
$922$	0	0
$923$	−4.36498	−0.143675
$924$	0	0
$925$	−2.09736	−0.0689609
$926$	0	0
$927$	21.7271	0.713612
$928$	0	0
$929$	36.3341	1.19208	0.596042	−	0.802953i	$-0.296739\pi$
$929$	36.3341	1.19208	0.596042	+	0.802953i	$0.296739\pi$
$930$	0	0
$931$	4.27507	0.140110
$932$	0	0
$933$	28.0672	0.918877
$934$	0	0
$935$	1.08520	0.0354899
$936$	0	0
$937$	22.0889	0.721612	0.360806	−	0.932641i	$-0.382502\pi$
$937$	22.0889	0.721612	0.360806	+	0.932641i	$0.382502\pi$
$938$	0	0
$939$	8.68322	0.283366
$940$	0	0
$941$	−10.7229	−0.349557	−0.174778	−	0.984608i	$-0.555921\pi$
$941$	−10.7229	−0.349557	−0.174778	+	0.984608i	$0.555921\pi$
$942$	0	0
$943$	−17.1877	−0.559710
$944$	0	0
$945$	−3.31583	−0.107864
$946$	0	0
$947$	0.553376	0.0179823	0.00899115	−	0.999960i	$-0.497138\pi$
$947$	0.553376	0.0179823	0.00899115	+	0.999960i	$0.497138\pi$
$948$	0	0
$949$	−1.56379	−0.0507626
$950$	0	0
$951$	−4.95593	−0.160707
$952$	0	0
$953$	45.7506	1.48201	0.741004	−	0.671500i	$-0.234350\pi$
$953$	45.7506	1.48201	0.741004	+	0.671500i	$0.234350\pi$
$954$	0	0
$955$	8.89699	0.287900
$956$	0	0
$957$	1.87569	0.0606325
$958$	0	0
$959$	5.79571	0.187153
$960$	0	0
$961$	6.13209	0.197809
$962$	0	0
$963$	8.95048	0.288425
$964$	0	0
$965$	7.63868	0.245898
$966$	0	0
$967$	−23.7193	−0.762761	−0.381381	−	0.924418i	$-0.624551\pi$
$967$	−23.7193	−0.762761	−0.381381	+	0.924418i	$0.624551\pi$
$968$	0	0
$969$	9.51597	0.305697
$970$	0	0
$971$	3.73356	0.119816	0.0599078	−	0.998204i	$-0.480919\pi$
$971$	3.73356	0.119816	0.0599078	+	0.998204i	$0.480919\pi$
$972$	0	0
$973$	17.0065	0.545204
$974$	0	0
$975$	−5.71455	−0.183012
$976$	0	0
$977$	1.52523	0.0487964	0.0243982	−	0.999702i	$-0.492233\pi$
$977$	1.52523	0.0487964	0.0243982	+	0.999702i	$0.492233\pi$
$978$	0	0
$979$	15.9025	0.508247
$980$	0	0
$981$	29.3463	0.936956
$982$	0	0
$983$	52.1516	1.66338	0.831689	−	0.555241i	$-0.187374\pi$
$983$	52.1516	1.66338	0.831689	+	0.555241i	$0.187374\pi$
$984$	0	0
$985$	−2.12783	−0.0677983
$986$	0	0
$987$	9.48958	0.302057
$988$	0	0
$989$	−16.0963	−0.511832
$990$	0	0
$991$	−18.1733	−0.577295	−0.288648	−	0.957435i	$-0.593206\pi$
$991$	−18.1733	−0.577295	−0.288648	+	0.957435i	$0.593206\pi$
$992$	0	0
$993$	−40.8860	−1.29748
$994$	0	0
$995$	4.59762	0.145754
$996$	0	0
$997$	−46.7456	−1.48045	−0.740224	−	0.672361i	$-0.765280\pi$
$997$	−46.7456	−1.48045	−0.740224	+	0.672361i	$0.765280\pi$
$998$	0	0
$999$	−2.49623	−0.0789771

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.a.f.1.3	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.a.f.1.3	✓	5	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4004.a (trivial)

\(f(q)\)	\(=\)	\(q+1.23175 q^{3} -0.600512 q^{5} -1.00000 q^{7} -1.48280 q^{9} +O(q^{10})\)	\(q+1.23175 q^{3} -0.600512 q^{5} -1.00000 q^{7} -1.48280 q^{9} -1.00000 q^{11} +1.00000 q^{13} -0.739678 q^{15} +1.80713 q^{17} +4.27507 q^{19} -1.23175 q^{21} +3.07404 q^{23} -4.63939 q^{25} -5.52167 q^{27} -1.52279 q^{29} -6.09361 q^{31} -1.23175 q^{33} +0.600512 q^{35} +0.452078 q^{37} +1.23175 q^{39} -5.59124 q^{41} -5.23620 q^{43} +0.890439 q^{45} -7.70416 q^{47} +1.00000 q^{49} +2.22592 q^{51} +2.83671 q^{53} +0.600512 q^{55} +5.26580 q^{57} +7.62530 q^{59} +6.76817 q^{61} +1.48280 q^{63} -0.600512 q^{65} +7.57426 q^{67} +3.78644 q^{69} -4.36498 q^{71} -1.56379 q^{73} -5.71455 q^{75} +1.00000 q^{77} -11.4461 q^{79} -2.35290 q^{81} -9.00918 q^{83} -1.08520 q^{85} -1.87569 q^{87} -15.9025 q^{89} -1.00000 q^{91} -7.50578 q^{93} -2.56723 q^{95} -9.57538 q^{97} +1.48280 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 3 q^{3} - 5 q^{7} + 2 q^{9}+O(q^{10}) \) 5 * q + 3 * q^3 - 5 * q^7 + 2 * q^9	\( 5 q + 3 q^{3} - 5 q^{7} + 2 q^{9} - 5 q^{11} + 5 q^{13} - 3 q^{15} - 9 q^{17} - 4 q^{19} - 3 q^{21} - 4 q^{23} + q^{25} + 3 q^{27} - 8 q^{29} + 7 q^{31} - 3 q^{33} - 10 q^{37} + 3 q^{39} - 18 q^{41} - 22 q^{43} - 26 q^{45} + 12 q^{47} + 5 q^{49} - 7 q^{51} + 7 q^{53} - 6 q^{57} - q^{59} - 26 q^{61} - 2 q^{63} - 8 q^{67} - 12 q^{69} + 18 q^{71} - 25 q^{73} - 16 q^{75} + 5 q^{77} - 6 q^{79} - 7 q^{81} + 11 q^{83} - 7 q^{85} - 5 q^{87} - 14 q^{89} - 5 q^{91} - 41 q^{93} - 22 q^{95} - 33 q^{97} - 2 q^{99}+O(q^{100}) \) 5 * q + 3 * q^3 - 5 * q^7 + 2 * q^9 - 5 * q^11 + 5 * q^13 - 3 * q^15 - 9 * q^17 - 4 * q^19 - 3 * q^21 - 4 * q^23 + q^25 + 3 * q^27 - 8 * q^29 + 7 * q^31 - 3 * q^33 - 10 * q^37 + 3 * q^39 - 18 * q^41 - 22 * q^43 - 26 * q^45 + 12 * q^47 + 5 * q^49 - 7 * q^51 + 7 * q^53 - 6 * q^57 - q^59 - 26 * q^61 - 2 * q^63 - 8 * q^67 - 12 * q^69 + 18 * q^71 - 25 * q^73 - 16 * q^75 + 5 * q^77 - 6 * q^79 - 7 * q^81 + 11 * q^83 - 7 * q^85 - 5 * q^87 - 14 * q^89 - 5 * q^91 - 41 * q^93 - 22 * q^95 - 33 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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