LMFDB - Embedded newform 8008.2.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8008,2,Mod(1,8008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8008.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:	$63.9442019386$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 4x^{8} - 11x^{7} + 46x^{6} + 37x^{5} - 169x^{4} - 18x^{3} + 195x^{2} - 72x + 7 $ x^9 - 4x^8 - 11x^7 + 46x^6 + 37x^5 - 169x^4 - 18x^3 + 195x^2 - 72x + 7
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.16724$ of defining polynomial
Character	$\chi$	$=$	8008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.16724 q^{3} +3.81158 q^{5} -1.00000 q^{7} +7.03141 q^{9} +O(q^{10})$	$q-3.16724 q^{3} +3.81158 q^{5} -1.00000 q^{7} +7.03141 q^{9} -1.00000 q^{11} +1.00000 q^{13} -12.0722 q^{15} +4.66290 q^{17} -0.608709 q^{19} +3.16724 q^{21} -1.18373 q^{23} +9.52813 q^{25} -12.7685 q^{27} -7.45640 q^{29} -0.641265 q^{31} +3.16724 q^{33} -3.81158 q^{35} -7.68734 q^{37} -3.16724 q^{39} +2.93823 q^{41} +5.57885 q^{43} +26.8008 q^{45} -9.83329 q^{47} +1.00000 q^{49} -14.7685 q^{51} -4.79350 q^{53} -3.81158 q^{55} +1.92793 q^{57} -4.49909 q^{59} -9.43556 q^{61} -7.03141 q^{63} +3.81158 q^{65} +6.57566 q^{67} +3.74914 q^{69} +3.48656 q^{71} -8.97801 q^{73} -30.1779 q^{75} +1.00000 q^{77} +8.44803 q^{79} +19.3465 q^{81} -8.01304 q^{83} +17.7730 q^{85} +23.6162 q^{87} -7.13793 q^{89} -1.00000 q^{91} +2.03104 q^{93} -2.32014 q^{95} -14.5012 q^{97} -7.03141 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 5 q^{3} - 5 q^{5} - 9 q^{7} + 12 q^{9}+O(q^{10}) $ 9 * q - 5 * q^3 - 5 * q^5 - 9 * q^7 + 12 * q^9	$ 9 q - 5 q^{3} - 5 q^{5} - 9 q^{7} + 12 q^{9} - 9 q^{11} + 9 q^{13} - q^{15} + 13 q^{17} - 5 q^{19} + 5 q^{21} - 7 q^{23} + 14 q^{25} - 23 q^{27} - 19 q^{29} + 2 q^{31} + 5 q^{33} + 5 q^{35} + 4 q^{37} - 5 q^{39} + 10 q^{41} - 17 q^{43} + 11 q^{45} - 5 q^{47} + 9 q^{49} - 9 q^{51} - 24 q^{53} + 5 q^{55} - 4 q^{57} - 19 q^{59} - 12 q^{63} - 5 q^{65} + 2 q^{67} - 2 q^{69} - 2 q^{71} + 34 q^{73} - 46 q^{75} + 9 q^{77} + 5 q^{79} + 37 q^{81} - 24 q^{83} + 33 q^{85} + 41 q^{87} - 11 q^{89} - 9 q^{91} + 53 q^{93} - 23 q^{95} - 12 q^{99}+O(q^{100}) $ 9 * q - 5 * q^3 - 5 * q^5 - 9 * q^7 + 12 * q^9 - 9 * q^11 + 9 * q^13 - q^15 + 13 * q^17 - 5 * q^19 + 5 * q^21 - 7 * q^23 + 14 * q^25 - 23 * q^27 - 19 * q^29 + 2 * q^31 + 5 * q^33 + 5 * q^35 + 4 * q^37 - 5 * q^39 + 10 * q^41 - 17 * q^43 + 11 * q^45 - 5 * q^47 + 9 * q^49 - 9 * q^51 - 24 * q^53 + 5 * q^55 - 4 * q^57 - 19 * q^59 - 12 * q^63 - 5 * q^65 + 2 * q^67 - 2 * q^69 - 2 * q^71 + 34 * q^73 - 46 * q^75 + 9 * q^77 + 5 * q^79 + 37 * q^81 - 24 * q^83 + 33 * q^85 + 41 * q^87 - 11 * q^89 - 9 * q^91 + 53 * q^93 - 23 * q^95 - 12 * 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
$3$	−3.16724	−1.82861	−0.914304	−	0.405030i	$-0.867261\pi$
$3$	−3.16724	−1.82861	−0.914304	+	0.405030i	$0.867261\pi$
$4$	0	0
$5$	3.81158	1.70459	0.852295	−	0.523062i	$-0.175210\pi$
$5$	3.81158	1.70459	0.852295	+	0.523062i	$0.175210\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	7.03141	2.34380
$10$	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
$15$	−12.0722	−3.11703
$16$	0	0
$17$	4.66290	1.13092	0.565459	−	0.824776i	$-0.308699\pi$
$17$	4.66290	1.13092	0.565459	+	0.824776i	$0.308699\pi$
$18$	0	0
$19$	−0.608709	−0.139647	−0.0698237	−	0.997559i	$-0.522244\pi$
$19$	−0.608709	−0.139647	−0.0698237	+	0.997559i	$0.522244\pi$
$20$	0	0
$21$	3.16724	0.691149
$22$	0	0
$23$	−1.18373	−0.246824	−0.123412	−	0.992356i	$-0.539384\pi$
$23$	−1.18373	−0.246824	−0.123412	+	0.992356i	$0.539384\pi$
$24$	0	0
$25$	9.52813	1.90563
$26$	0	0
$27$	−12.7685	−2.45729
$28$	0	0
$29$	−7.45640	−1.38462	−0.692309	−	0.721601i	$-0.743406\pi$
$29$	−7.45640	−1.38462	−0.692309	+	0.721601i	$0.743406\pi$
$30$	0	0
$31$	−0.641265	−0.115175	−0.0575873	−	0.998340i	$-0.518341\pi$
$31$	−0.641265	−0.115175	−0.0575873	+	0.998340i	$0.518341\pi$
$32$	0	0
$33$	3.16724	0.551346
$34$	0	0
$35$	−3.81158	−0.644274
$36$	0	0
$37$	−7.68734	−1.26379	−0.631896	−	0.775053i	$-0.717723\pi$
$37$	−7.68734	−1.26379	−0.631896	+	0.775053i	$0.717723\pi$
$38$	0	0
$39$	−3.16724	−0.507164
$40$	0	0
$41$	2.93823	0.458875	0.229437	−	0.973323i	$-0.426311\pi$
$41$	2.93823	0.458875	0.229437	+	0.973323i	$0.426311\pi$
$42$	0	0
$43$	5.57885	0.850767	0.425384	−	0.905013i	$-0.360139\pi$
$43$	5.57885	0.850767	0.425384	+	0.905013i	$0.360139\pi$
$44$	0	0
$45$	26.8008	3.99522
$46$	0	0
$47$	−9.83329	−1.43433	−0.717167	−	0.696902i	$-0.754561\pi$
$47$	−9.83329	−1.43433	−0.717167	+	0.696902i	$0.754561\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−14.7685	−2.06801
$52$	0	0
$53$	−4.79350	−0.658438	−0.329219	−	0.944254i	$-0.606785\pi$
$53$	−4.79350	−0.658438	−0.329219	+	0.944254i	$0.606785\pi$
$54$	0	0
$55$	−3.81158	−0.513953
$56$	0	0
$57$	1.92793	0.255360
$58$	0	0
$59$	−4.49909	−0.585731	−0.292866	−	0.956154i	$-0.594609\pi$
$59$	−4.49909	−0.585731	−0.292866	+	0.956154i	$0.594609\pi$
$60$	0	0
$61$	−9.43556	−1.20810	−0.604050	−	0.796946i	$-0.706447\pi$
$61$	−9.43556	−1.20810	−0.604050	+	0.796946i	$0.706447\pi$
$62$	0	0
$63$	−7.03141	−0.885875
$64$	0	0
$65$	3.81158	0.472768
$66$	0	0
$67$	6.57566	0.803345	0.401673	−	0.915783i	$-0.368429\pi$
$67$	6.57566	0.803345	0.401673	+	0.915783i	$0.368429\pi$
$68$	0	0
$69$	3.74914	0.451344
$70$	0	0
$71$	3.48656	0.413779	0.206890	−	0.978364i	$-0.433666\pi$
$71$	3.48656	0.413779	0.206890	+	0.978364i	$0.433666\pi$
$72$	0	0
$73$	−8.97801	−1.05080	−0.525398	−	0.850857i	$-0.676084\pi$
$73$	−8.97801	−1.05080	−0.525398	+	0.850857i	$0.676084\pi$
$74$	0	0
$75$	−30.1779	−3.48464
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	8.44803	0.950478	0.475239	−	0.879857i	$-0.342362\pi$
$79$	8.44803	0.950478	0.475239	+	0.879857i	$0.342362\pi$
$80$	0	0
$81$	19.3465	2.14961
$82$	0	0
$83$	−8.01304	−0.879546	−0.439773	−	0.898109i	$-0.644941\pi$
$83$	−8.01304	−0.879546	−0.439773	+	0.898109i	$0.644941\pi$
$84$	0	0
$85$	17.7730	1.92775
$86$	0	0
$87$	23.6162	2.53192
$88$	0	0
$89$	−7.13793	−0.756619	−0.378309	−	0.925679i	$-0.623494\pi$
$89$	−7.13793	−0.756619	−0.378309	+	0.925679i	$0.623494\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	2.03104	0.210609
$94$	0	0
$95$	−2.32014	−0.238041
$96$	0	0
$97$	−14.5012	−1.47237	−0.736185	−	0.676780i	$-0.763375\pi$
$97$	−14.5012	−1.47237	−0.736185	+	0.676780i	$0.763375\pi$
$98$	0	0
$99$	−7.03141	−0.706684
$100$	0	0
$101$	−4.68445	−0.466120	−0.233060	−	0.972462i	$-0.574874\pi$
$101$	−4.68445	−0.466120	−0.233060	+	0.972462i	$0.574874\pi$
$102$	0	0
$103$	−8.87239	−0.874223	−0.437111	−	0.899407i	$-0.643998\pi$
$103$	−8.87239	−0.874223	−0.437111	+	0.899407i	$0.643998\pi$
$104$	0	0
$105$	12.0722	1.17812
$106$	0	0
$107$	−18.8682	−1.82406	−0.912030	−	0.410125i	$-0.865485\pi$
$107$	−18.8682	−1.82406	−0.912030	+	0.410125i	$0.865485\pi$
$108$	0	0
$109$	−11.5523	−1.10651	−0.553253	−	0.833013i	$-0.686614\pi$
$109$	−11.5523	−1.10651	−0.553253	+	0.833013i	$0.686614\pi$
$110$	0	0
$111$	24.3477	2.31098
$112$	0	0
$113$	5.01128	0.471421	0.235711	−	0.971823i	$-0.424258\pi$
$113$	5.01128	0.471421	0.235711	+	0.971823i	$0.424258\pi$
$114$	0	0
$115$	−4.51186	−0.420733
$116$	0	0
$117$	7.03141	0.650054
$118$	0	0
$119$	−4.66290	−0.427447
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−9.30608	−0.839101
$124$	0	0
$125$	17.2593	1.54372
$126$	0	0
$127$	7.15140	0.634584	0.317292	−	0.948328i	$-0.397226\pi$
$127$	7.15140	0.634584	0.317292	+	0.948328i	$0.397226\pi$
$128$	0	0
$129$	−17.6696	−1.55572
$130$	0	0
$131$	13.6650	1.19392	0.596958	−	0.802272i	$-0.296376\pi$
$131$	13.6650	1.19392	0.596958	+	0.802272i	$0.296376\pi$
$132$	0	0
$133$	0.608709	0.0527817
$134$	0	0
$135$	−48.6680	−4.18867
$136$	0	0
$137$	9.53905	0.814976	0.407488	−	0.913211i	$-0.366405\pi$
$137$	9.53905	0.814976	0.407488	+	0.913211i	$0.366405\pi$
$138$	0	0
$139$	6.54607	0.555230	0.277615	−	0.960692i	$-0.410456\pi$
$139$	6.54607	0.555230	0.277615	+	0.960692i	$0.410456\pi$
$140$	0	0
$141$	31.1444	2.62283
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	−28.4206	−2.36021
$146$	0	0
$147$	−3.16724	−0.261230
$148$	0	0
$149$	−21.7559	−1.78231	−0.891155	−	0.453698i	$-0.850104\pi$
$149$	−21.7559	−1.78231	−0.891155	+	0.453698i	$0.850104\pi$
$150$	0	0
$151$	7.62497	0.620511	0.310256	−	0.950653i	$-0.399585\pi$
$151$	7.62497	0.620511	0.310256	+	0.950653i	$0.399585\pi$
$152$	0	0
$153$	32.7867	2.65065
$154$	0	0
$155$	−2.44423	−0.196326
$156$	0	0
$157$	8.26283	0.659445	0.329723	−	0.944078i	$-0.393045\pi$
$157$	8.26283	0.659445	0.329723	+	0.944078i	$0.393045\pi$
$158$	0	0
$159$	15.1822	1.20402
$160$	0	0
$161$	1.18373	0.0932907
$162$	0	0
$163$	8.63284	0.676176	0.338088	−	0.941115i	$-0.390220\pi$
$163$	8.63284	0.676176	0.338088	+	0.941115i	$0.390220\pi$
$164$	0	0
$165$	12.0722	0.939818
$166$	0	0
$167$	−6.25240	−0.483826	−0.241913	−	0.970298i	$-0.577775\pi$
$167$	−6.25240	−0.483826	−0.241913	+	0.970298i	$0.577775\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−4.28008	−0.327306
$172$	0	0
$173$	19.3989	1.47487	0.737436	−	0.675417i	$-0.236036\pi$
$173$	19.3989	1.47487	0.737436	+	0.675417i	$0.236036\pi$
$174$	0	0
$175$	−9.52813	−0.720259
$176$	0	0
$177$	14.2497	1.07107
$178$	0	0
$179$	−20.3911	−1.52410	−0.762050	−	0.647518i	$-0.775807\pi$
$179$	−20.3911	−1.52410	−0.762050	+	0.647518i	$0.775807\pi$
$180$	0	0
$181$	−8.98154	−0.667593	−0.333796	−	0.942645i	$-0.608330\pi$
$181$	−8.98154	−0.667593	−0.333796	+	0.942645i	$0.608330\pi$
$182$	0	0
$183$	29.8847	2.20914
$184$	0	0
$185$	−29.3009	−2.15425
$186$	0	0
$187$	−4.66290	−0.340985
$188$	0	0
$189$	12.7685	0.928768
$190$	0	0
$191$	−0.526477	−0.0380945	−0.0190473	−	0.999819i	$-0.506063\pi$
$191$	−0.526477	−0.0380945	−0.0190473	+	0.999819i	$0.506063\pi$
$192$	0	0
$193$	−1.40770	−0.101328	−0.0506642	−	0.998716i	$-0.516134\pi$
$193$	−1.40770	−0.101328	−0.0506642	+	0.998716i	$0.516134\pi$
$194$	0	0
$195$	−12.0722	−0.864507
$196$	0	0
$197$	15.4294	1.09930	0.549651	−	0.835394i	$-0.314761\pi$
$197$	15.4294	1.09930	0.549651	+	0.835394i	$0.314761\pi$
$198$	0	0
$199$	8.49072	0.601891	0.300946	−	0.953641i	$-0.402698\pi$
$199$	8.49072	0.601891	0.300946	+	0.953641i	$0.402698\pi$
$200$	0	0
$201$	−20.8267	−1.46900
$202$	0	0
$203$	7.45640	0.523336
$204$	0	0
$205$	11.1993	0.782193
$206$	0	0
$207$	−8.32326	−0.578507
$208$	0	0
$209$	0.608709	0.0421053
$210$	0	0
$211$	−0.905849	−0.0623612	−0.0311806	−	0.999514i	$-0.509927\pi$
$211$	−0.905849	−0.0623612	−0.0311806	+	0.999514i	$0.509927\pi$
$212$	0	0
$213$	−11.0428	−0.756639
$214$	0	0
$215$	21.2642	1.45021
$216$	0	0
$217$	0.641265	0.0435319
$218$	0	0
$219$	28.4355	1.92149
$220$	0	0
$221$	4.66290	0.313660
$222$	0	0
$223$	4.88622	0.327205	0.163603	−	0.986526i	$-0.447689\pi$
$223$	4.88622	0.327205	0.163603	+	0.986526i	$0.447689\pi$
$224$	0	0
$225$	66.9962	4.46642
$226$	0	0
$227$	23.4593	1.55705	0.778524	−	0.627615i	$-0.215969\pi$
$227$	23.4593	1.55705	0.778524	+	0.627615i	$0.215969\pi$
$228$	0	0
$229$	26.6921	1.76386	0.881932	−	0.471377i	$-0.156243\pi$
$229$	26.6921	1.76386	0.881932	+	0.471377i	$0.156243\pi$
$230$	0	0
$231$	−3.16724	−0.208389
$232$	0	0
$233$	−23.6243	−1.54768	−0.773839	−	0.633382i	$-0.781666\pi$
$233$	−23.6243	−1.54768	−0.773839	+	0.633382i	$0.781666\pi$
$234$	0	0
$235$	−37.4804	−2.44495
$236$	0	0
$237$	−26.7569	−1.73805
$238$	0	0
$239$	11.0239	0.713074	0.356537	−	0.934281i	$-0.383957\pi$
$239$	11.0239	0.713074	0.356537	+	0.934281i	$0.383957\pi$
$240$	0	0
$241$	−9.48783	−0.611165	−0.305582	−	0.952166i	$-0.598851\pi$
$241$	−9.48783	−0.611165	−0.305582	+	0.952166i	$0.598851\pi$
$242$	0	0
$243$	−22.9697	−1.47351
$244$	0	0
$245$	3.81158	0.243513
$246$	0	0
$247$	−0.608709	−0.0387312
$248$	0	0
$249$	25.3792	1.60834
$250$	0	0
$251$	27.1064	1.71094	0.855472	−	0.517850i	$-0.173267\pi$
$251$	27.1064	1.71094	0.855472	+	0.517850i	$0.173267\pi$
$252$	0	0
$253$	1.18373	0.0744202
$254$	0	0
$255$	−56.2913	−3.52510
$256$	0	0
$257$	−24.8837	−1.55220	−0.776102	−	0.630607i	$-0.782806\pi$
$257$	−24.8837	−1.55220	−0.776102	+	0.630607i	$0.782806\pi$
$258$	0	0
$259$	7.68734	0.477668
$260$	0	0
$261$	−52.4290	−3.24527
$262$	0	0
$263$	−16.1669	−0.996896	−0.498448	−	0.866920i	$-0.666097\pi$
$263$	−16.1669	−0.996896	−0.498448	+	0.866920i	$0.666097\pi$
$264$	0	0
$265$	−18.2708	−1.12237
$266$	0	0
$267$	22.6075	1.38356
$268$	0	0
$269$	7.76666	0.473542	0.236771	−	0.971566i	$-0.423911\pi$
$269$	7.76666	0.473542	0.236771	+	0.971566i	$0.423911\pi$
$270$	0	0
$271$	4.51511	0.274273	0.137137	−	0.990552i	$-0.456210\pi$
$271$	4.51511	0.274273	0.137137	+	0.990552i	$0.456210\pi$
$272$	0	0
$273$	3.16724	0.191690
$274$	0	0
$275$	−9.52813	−0.574568
$276$	0	0
$277$	−18.2322	−1.09547	−0.547733	−	0.836653i	$-0.684509\pi$
$277$	−18.2322	−1.09547	−0.547733	+	0.836653i	$0.684509\pi$
$278$	0	0
$279$	−4.50900	−0.269947
$280$	0	0
$281$	−13.7130	−0.818051	−0.409025	−	0.912523i	$-0.634131\pi$
$281$	−13.7130	−0.818051	−0.409025	+	0.912523i	$0.634131\pi$
$282$	0	0
$283$	−13.0693	−0.776888	−0.388444	−	0.921472i	$-0.626987\pi$
$283$	−13.0693	−0.776888	−0.388444	+	0.921472i	$0.626987\pi$
$284$	0	0
$285$	7.34845	0.435284
$286$	0	0
$287$	−2.93823	−0.173438
$288$	0	0
$289$	4.74259	0.278976
$290$	0	0
$291$	45.9287	2.69239
$292$	0	0
$293$	17.0757	0.997575	0.498787	−	0.866724i	$-0.333779\pi$
$293$	17.0757	0.997575	0.498787	+	0.866724i	$0.333779\pi$
$294$	0	0
$295$	−17.1486	−0.998431
$296$	0	0
$297$	12.7685	0.740901
$298$	0	0
$299$	−1.18373	−0.0684566
$300$	0	0
$301$	−5.57885	−0.321560
$302$	0	0
$303$	14.8368	0.852350
$304$	0	0
$305$	−35.9644	−2.05932
$306$	0	0
$307$	−31.7532	−1.81225	−0.906125	−	0.423011i	$-0.860973\pi$
$307$	−31.7532	−1.81225	−0.906125	+	0.423011i	$0.860973\pi$
$308$	0	0
$309$	28.1010	1.59861
$310$	0	0
$311$	27.8378	1.57854	0.789269	−	0.614047i	$-0.210460\pi$
$311$	27.8378	1.57854	0.789269	+	0.614047i	$0.210460\pi$
$312$	0	0
$313$	12.4014	0.700968	0.350484	−	0.936569i	$-0.386017\pi$
$313$	12.4014	0.700968	0.350484	+	0.936569i	$0.386017\pi$
$314$	0	0
$315$	−26.8008	−1.51005
$316$	0	0
$317$	−19.8734	−1.11620	−0.558099	−	0.829774i	$-0.688469\pi$
$317$	−19.8734	−1.11620	−0.558099	+	0.829774i	$0.688469\pi$
$318$	0	0
$319$	7.45640	0.417478
$320$	0	0
$321$	59.7602	3.33549
$322$	0	0
$323$	−2.83835	−0.157930
$324$	0	0
$325$	9.52813	0.528526
$326$	0	0
$327$	36.5888	2.02336
$328$	0	0
$329$	9.83329	0.542127
$330$	0	0
$331$	−32.1886	−1.76925	−0.884624	−	0.466306i	$-0.845585\pi$
$331$	−32.1886	−1.76925	−0.884624	+	0.466306i	$0.845585\pi$
$332$	0	0
$333$	−54.0529	−2.96208
$334$	0	0
$335$	25.0637	1.36937
$336$	0	0
$337$	−10.2854	−0.560282	−0.280141	−	0.959959i	$-0.590381\pi$
$337$	−10.2854	−0.560282	−0.280141	+	0.959959i	$0.590381\pi$
$338$	0	0
$339$	−15.8719	−0.862044
$340$	0	0
$341$	0.641265	0.0347265
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	14.2902	0.769356
$346$	0	0
$347$	−32.3517	−1.73673	−0.868366	−	0.495923i	$-0.834830\pi$
$347$	−32.3517	−1.73673	−0.868366	+	0.495923i	$0.834830\pi$
$348$	0	0
$349$	26.6282	1.42537	0.712687	−	0.701483i	$-0.247478\pi$
$349$	26.6282	1.42537	0.712687	+	0.701483i	$0.247478\pi$
$350$	0	0
$351$	−12.7685	−0.681530
$352$	0	0
$353$	33.8446	1.80136	0.900682	−	0.434479i	$-0.143067\pi$
$353$	33.8446	1.80136	0.900682	+	0.434479i	$0.143067\pi$
$354$	0	0
$355$	13.2893	0.705324
$356$	0	0
$357$	14.7685	0.781632
$358$	0	0
$359$	14.9114	0.786993	0.393496	−	0.919326i	$-0.371265\pi$
$359$	14.9114	0.786993	0.393496	+	0.919326i	$0.371265\pi$
$360$	0	0
$361$	−18.6295	−0.980499
$362$	0	0
$363$	−3.16724	−0.166237
$364$	0	0
$365$	−34.2204	−1.79118
$366$	0	0
$367$	14.9644	0.781134	0.390567	−	0.920575i	$-0.372279\pi$
$367$	14.9644	0.781134	0.390567	+	0.920575i	$0.372279\pi$
$368$	0	0
$369$	20.6599	1.07551
$370$	0	0
$371$	4.79350	0.248866
$372$	0	0
$373$	−22.0975	−1.14417	−0.572083	−	0.820196i	$-0.693865\pi$
$373$	−22.0975	−1.14417	−0.572083	+	0.820196i	$0.693865\pi$
$374$	0	0
$375$	−54.6645	−2.82286
$376$	0	0
$377$	−7.45640	−0.384024
$378$	0	0
$379$	18.1363	0.931597	0.465798	−	0.884891i	$-0.345767\pi$
$379$	18.1363	0.931597	0.465798	+	0.884891i	$0.345767\pi$
$380$	0	0
$381$	−22.6502	−1.16040
$382$	0	0
$383$	−18.3441	−0.937338	−0.468669	−	0.883374i	$-0.655266\pi$
$383$	−18.3441	−0.937338	−0.468669	+	0.883374i	$0.655266\pi$
$384$	0	0
$385$	3.81158	0.194256
$386$	0	0
$387$	39.2272	1.99403
$388$	0	0
$389$	12.8983	0.653969	0.326984	−	0.945030i	$-0.393968\pi$
$389$	12.8983	0.653969	0.326984	+	0.945030i	$0.393968\pi$
$390$	0	0
$391$	−5.51959	−0.279138
$392$	0	0
$393$	−43.2803	−2.18320
$394$	0	0
$395$	32.2003	1.62017
$396$	0	0
$397$	13.8487	0.695045	0.347522	−	0.937672i	$-0.387023\pi$
$397$	13.8487	0.695045	0.347522	+	0.937672i	$0.387023\pi$
$398$	0	0
$399$	−1.92793	−0.0965171
$400$	0	0
$401$	28.8320	1.43980	0.719902	−	0.694076i	$-0.244187\pi$
$401$	28.8320	1.43980	0.719902	+	0.694076i	$0.244187\pi$
$402$	0	0
$403$	−0.641265	−0.0319437
$404$	0	0
$405$	73.7408	3.66421
$406$	0	0
$407$	7.68734	0.381047
$408$	0	0
$409$	−8.16553	−0.403759	−0.201880	−	0.979410i	$-0.564705\pi$
$409$	−8.16553	−0.403759	−0.201880	+	0.979410i	$0.564705\pi$
$410$	0	0
$411$	−30.2125	−1.49027
$412$	0	0
$413$	4.49909	0.221386
$414$	0	0
$415$	−30.5423	−1.49926
$416$	0	0
$417$	−20.7330	−1.01530
$418$	0	0
$419$	0.794018	0.0387903	0.0193952	−	0.999812i	$-0.493826\pi$
$419$	0.794018	0.0387903	0.0193952	+	0.999812i	$0.493826\pi$
$420$	0	0
$421$	−11.5850	−0.564619	−0.282309	−	0.959323i	$-0.591100\pi$
$421$	−11.5850	−0.564619	−0.282309	+	0.959323i	$0.591100\pi$
$422$	0	0
$423$	−69.1419	−3.36180
$424$	0	0
$425$	44.4287	2.15511
$426$	0	0
$427$	9.43556	0.456619
$428$	0	0
$429$	3.16724	0.152916
$430$	0	0
$431$	6.13083	0.295312	0.147656	−	0.989039i	$-0.452827\pi$
$431$	6.13083	0.295312	0.147656	+	0.989039i	$0.452827\pi$
$432$	0	0
$433$	−22.7472	−1.09316	−0.546580	−	0.837407i	$-0.684070\pi$
$433$	−22.7472	−1.09316	−0.546580	+	0.837407i	$0.684070\pi$
$434$	0	0
$435$	90.0150	4.31589
$436$	0	0
$437$	0.720544	0.0344683
$438$	0	0
$439$	5.37984	0.256766	0.128383	−	0.991725i	$-0.459021\pi$
$439$	5.37984	0.256766	0.128383	+	0.991725i	$0.459021\pi$
$440$	0	0
$441$	7.03141	0.334829
$442$	0	0
$443$	−40.5464	−1.92642	−0.963209	−	0.268754i	$-0.913388\pi$
$443$	−40.5464	−1.92642	−0.963209	+	0.268754i	$0.913388\pi$
$444$	0	0
$445$	−27.2068	−1.28972
$446$	0	0
$447$	68.9061	3.25915
$448$	0	0
$449$	15.2103	0.717818	0.358909	−	0.933373i	$-0.383149\pi$
$449$	15.2103	0.717818	0.358909	+	0.933373i	$0.383149\pi$
$450$	0	0
$451$	−2.93823	−0.138356
$452$	0	0
$453$	−24.1501	−1.13467
$454$	0	0
$455$	−3.81158	−0.178690
$456$	0	0
$457$	25.5333	1.19440	0.597199	−	0.802093i	$-0.296280\pi$
$457$	25.5333	1.19440	0.597199	+	0.802093i	$0.296280\pi$
$458$	0	0
$459$	−59.5380	−2.77899
$460$	0	0
$461$	−20.9336	−0.974975	−0.487488	−	0.873130i	$-0.662087\pi$
$461$	−20.9336	−0.974975	−0.487488	+	0.873130i	$0.662087\pi$
$462$	0	0
$463$	−12.6178	−0.586401	−0.293201	−	0.956051i	$-0.594720\pi$
$463$	−12.6178	−0.586401	−0.293201	+	0.956051i	$0.594720\pi$
$464$	0	0
$465$	7.74147	0.359002
$466$	0	0
$467$	−35.0888	−1.62372	−0.811859	−	0.583854i	$-0.801544\pi$
$467$	−35.0888	−1.62372	−0.811859	+	0.583854i	$0.801544\pi$
$468$	0	0
$469$	−6.57566	−0.303636
$470$	0	0
$471$	−26.1704	−1.20587
$472$	0	0
$473$	−5.57885	−0.256516
$474$	0	0
$475$	−5.79986	−0.266116
$476$	0	0
$477$	−33.7051	−1.54325
$478$	0	0
$479$	9.76880	0.446348	0.223174	−	0.974779i	$-0.428358\pi$
$479$	9.76880	0.446348	0.223174	+	0.974779i	$0.428358\pi$
$480$	0	0
$481$	−7.68734	−0.350513
$482$	0	0
$483$	−3.74914	−0.170592
$484$	0	0
$485$	−55.2724	−2.50979
$486$	0	0
$487$	−34.9532	−1.58388	−0.791940	−	0.610599i	$-0.790929\pi$
$487$	−34.9532	−1.58388	−0.791940	+	0.610599i	$0.790929\pi$
$488$	0	0
$489$	−27.3423	−1.23646
$490$	0	0
$491$	−38.1976	−1.72383	−0.861916	−	0.507051i	$-0.830735\pi$
$491$	−38.1976	−1.72383	−0.861916	+	0.507051i	$0.830735\pi$
$492$	0	0
$493$	−34.7684	−1.56589
$494$	0	0
$495$	−26.8008	−1.20461
$496$	0	0
$497$	−3.48656	−0.156394
$498$	0	0
$499$	−5.89229	−0.263775	−0.131888	−	0.991265i	$-0.542104\pi$
$499$	−5.89229	−0.263775	−0.131888	+	0.991265i	$0.542104\pi$
$500$	0	0
$501$	19.8029	0.884727
$502$	0	0
$503$	−21.3094	−0.950139	−0.475070	−	0.879948i	$-0.657577\pi$
$503$	−21.3094	−0.950139	−0.475070	+	0.879948i	$0.657577\pi$
$504$	0	0
$505$	−17.8551	−0.794543
$506$	0	0
$507$	−3.16724	−0.140662
$508$	0	0
$509$	3.10079	0.137440	0.0687201	−	0.997636i	$-0.478108\pi$
$509$	3.10079	0.137440	0.0687201	+	0.997636i	$0.478108\pi$
$510$	0	0
$511$	8.97801	0.397164
$512$	0	0
$513$	7.77227	0.343154
$514$	0	0
$515$	−33.8178	−1.49019
$516$	0	0
$517$	9.83329	0.432468
$518$	0	0
$519$	−61.4410	−2.69696
$520$	0	0
$521$	−23.8147	−1.04334	−0.521671	−	0.853147i	$-0.674691\pi$
$521$	−23.8147	−1.04334	−0.521671	+	0.853147i	$0.674691\pi$
$522$	0	0
$523$	−33.8805	−1.48149	−0.740745	−	0.671786i	$-0.765527\pi$
$523$	−33.8805	−1.48149	−0.740745	+	0.671786i	$0.765527\pi$
$524$	0	0
$525$	30.1779	1.31707
$526$	0	0
$527$	−2.99015	−0.130253
$528$	0	0
$529$	−21.5988	−0.939078
$530$	0	0
$531$	−31.6349	−1.37284
$532$	0	0
$533$	2.93823	0.127269
$534$	0	0
$535$	−71.9177	−3.10927
$536$	0	0
$537$	64.5834	2.78698
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−17.5755	−0.755630	−0.377815	−	0.925881i	$-0.623325\pi$
$541$	−17.5755	−0.755630	−0.377815	+	0.925881i	$0.623325\pi$
$542$	0	0
$543$	28.4467	1.22076
$544$	0	0
$545$	−44.0323	−1.88614
$546$	0	0
$547$	1.62359	0.0694197	0.0347099	−	0.999397i	$-0.488949\pi$
$547$	1.62359	0.0694197	0.0347099	+	0.999397i	$0.488949\pi$
$548$	0	0
$549$	−66.3453	−2.83155
$550$	0	0
$551$	4.53877	0.193358
$552$	0	0
$553$	−8.44803	−0.359247
$554$	0	0
$555$	92.8030	3.93927
$556$	0	0
$557$	5.46489	0.231555	0.115777	−	0.993275i	$-0.463064\pi$
$557$	5.46489	0.231555	0.115777	+	0.993275i	$0.463064\pi$
$558$	0	0
$559$	5.57885	0.235960
$560$	0	0
$561$	14.7685	0.623527
$562$	0	0
$563$	15.4790	0.652361	0.326180	−	0.945308i	$-0.394238\pi$
$563$	15.4790	0.652361	0.326180	+	0.945308i	$0.394238\pi$
$564$	0	0
$565$	19.1009	0.803580
$566$	0	0
$567$	−19.3465	−0.812478
$568$	0	0
$569$	0.996420	0.0417721	0.0208860	−	0.999782i	$-0.493351\pi$
$569$	0.996420	0.0417721	0.0208860	+	0.999782i	$0.493351\pi$
$570$	0	0
$571$	−5.63293	−0.235731	−0.117865	−	0.993030i	$-0.537605\pi$
$571$	−5.63293	−0.235731	−0.117865	+	0.993030i	$0.537605\pi$
$572$	0	0
$573$	1.66748	0.0696599
$574$	0	0
$575$	−11.2787	−0.470354
$576$	0	0
$577$	−46.8130	−1.94885	−0.974426	−	0.224711i	$-0.927856\pi$
$577$	−46.8130	−1.94885	−0.974426	+	0.224711i	$0.927856\pi$
$578$	0	0
$579$	4.45852	0.185290
$580$	0	0
$581$	8.01304	0.332437
$582$	0	0
$583$	4.79350	0.198526
$584$	0	0
$585$	26.8008	1.10808
$586$	0	0
$587$	−22.3166	−0.921104	−0.460552	−	0.887633i	$-0.652348\pi$
$587$	−22.3166	−0.921104	−0.460552	+	0.887633i	$0.652348\pi$
$588$	0	0
$589$	0.390344	0.0160838
$590$	0	0
$591$	−48.8688	−2.01019
$592$	0	0
$593$	26.2784	1.07913	0.539563	−	0.841945i	$-0.318590\pi$
$593$	26.2784	1.07913	0.539563	+	0.841945i	$0.318590\pi$
$594$	0	0
$595$	−17.7730	−0.728622
$596$	0	0
$597$	−26.8922	−1.10062
$598$	0	0
$599$	−18.4550	−0.754051	−0.377025	−	0.926203i	$-0.623053\pi$
$599$	−18.4550	−0.754051	−0.377025	+	0.926203i	$0.623053\pi$
$600$	0	0
$601$	−4.52770	−0.184689	−0.0923444	−	0.995727i	$-0.529436\pi$
$601$	−4.52770	−0.184689	−0.0923444	+	0.995727i	$0.529436\pi$
$602$	0	0
$603$	46.2362	1.88288
$604$	0	0
$605$	3.81158	0.154963
$606$	0	0
$607$	44.0460	1.78777	0.893886	−	0.448295i	$-0.147969\pi$
$607$	44.0460	1.78777	0.893886	+	0.448295i	$0.147969\pi$
$608$	0	0
$609$	−23.6162	−0.956977
$610$	0	0
$611$	−9.83329	−0.397813
$612$	0	0
$613$	−26.7110	−1.07885	−0.539423	−	0.842035i	$-0.681357\pi$
$613$	−26.7110	−1.07885	−0.539423	+	0.842035i	$0.681357\pi$
$614$	0	0
$615$	−35.4709	−1.43032
$616$	0	0
$617$	−1.45716	−0.0586631	−0.0293316	−	0.999570i	$-0.509338\pi$
$617$	−1.45716	−0.0586631	−0.0293316	+	0.999570i	$0.509338\pi$
$618$	0	0
$619$	−24.0032	−0.964770	−0.482385	−	0.875959i	$-0.660229\pi$
$619$	−24.0032	−0.964770	−0.482385	+	0.875959i	$0.660229\pi$
$620$	0	0
$621$	15.1143	0.606518
$622$	0	0
$623$	7.13793	0.285975
$624$	0	0
$625$	18.1446	0.725786
$626$	0	0
$627$	−1.92793	−0.0769940
$628$	0	0
$629$	−35.8453	−1.42924
$630$	0	0
$631$	−16.9020	−0.672859	−0.336429	−	0.941709i	$-0.609219\pi$
$631$	−16.9020	−0.672859	−0.336429	+	0.941709i	$0.609219\pi$
$632$	0	0
$633$	2.86904	0.114034
$634$	0	0
$635$	27.2581	1.08171
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	24.5155	0.969817
$640$	0	0
$641$	35.8227	1.41491	0.707455	−	0.706758i	$-0.249843\pi$
$641$	35.8227	1.41491	0.707455	+	0.706758i	$0.249843\pi$
$642$	0	0
$643$	30.8849	1.21798	0.608991	−	0.793177i	$-0.291575\pi$
$643$	30.8849	1.21798	0.608991	+	0.793177i	$0.291575\pi$
$644$	0	0
$645$	−67.3490	−2.65186
$646$	0	0
$647$	−31.1247	−1.22364	−0.611820	−	0.790997i	$-0.709562\pi$
$647$	−31.1247	−1.22364	−0.611820	+	0.790997i	$0.709562\pi$
$648$	0	0
$649$	4.49909	0.176605
$650$	0	0
$651$	−2.03104	−0.0796028
$652$	0	0
$653$	4.39400	0.171950	0.0859752	−	0.996297i	$-0.472599\pi$
$653$	4.39400	0.171950	0.0859752	+	0.996297i	$0.472599\pi$
$654$	0	0
$655$	52.0852	2.03514
$656$	0	0
$657$	−63.1281	−2.46286
$658$	0	0
$659$	−4.75462	−0.185214	−0.0926068	−	0.995703i	$-0.529520\pi$
$659$	−4.75462	−0.185214	−0.0926068	+	0.995703i	$0.529520\pi$
$660$	0	0
$661$	11.9781	0.465893	0.232946	−	0.972490i	$-0.425163\pi$
$661$	11.9781	0.465893	0.232946	+	0.972490i	$0.425163\pi$
$662$	0	0
$663$	−14.7685	−0.573561
$664$	0	0
$665$	2.32014	0.0899712
$666$	0	0
$667$	8.82633	0.341757
$668$	0	0
$669$	−15.4758	−0.598330
$670$	0	0
$671$	9.43556	0.364256
$672$	0	0
$673$	−44.4105	−1.71190	−0.855949	−	0.517060i	$-0.827026\pi$
$673$	−44.4105	−1.71190	−0.855949	+	0.517060i	$0.827026\pi$
$674$	0	0
$675$	−121.660	−4.68268
$676$	0	0
$677$	42.3978	1.62948	0.814739	−	0.579827i	$-0.196880\pi$
$677$	42.3978	1.62948	0.814739	+	0.579827i	$0.196880\pi$
$678$	0	0
$679$	14.5012	0.556504
$680$	0	0
$681$	−74.3012	−2.84723
$682$	0	0
$683$	6.81279	0.260684	0.130342	−	0.991469i	$-0.458392\pi$
$683$	6.81279	0.260684	0.130342	+	0.991469i	$0.458392\pi$
$684$	0	0
$685$	36.3588	1.38920
$686$	0	0
$687$	−84.5403	−3.22541
$688$	0	0
$689$	−4.79350	−0.182618
$690$	0	0
$691$	−40.5283	−1.54177	−0.770885	−	0.636975i	$-0.780186\pi$
$691$	−40.5283	−1.54177	−0.770885	+	0.636975i	$0.780186\pi$
$692$	0	0
$693$	7.03141	0.267101
$694$	0	0
$695$	24.9509	0.946440
$696$	0	0
$697$	13.7007	0.518950
$698$	0	0
$699$	74.8238	2.83010
$700$	0	0
$701$	48.8066	1.84340	0.921700	−	0.387904i	$-0.126801\pi$
$701$	48.8066	1.84340	0.921700	+	0.387904i	$0.126801\pi$
$702$	0	0
$703$	4.67935	0.176485
$704$	0	0
$705$	118.709	4.47085
$706$	0	0
$707$	4.68445	0.176177
$708$	0	0
$709$	−2.83466	−0.106458	−0.0532290	−	0.998582i	$-0.516951\pi$
$709$	−2.83466	−0.106458	−0.0532290	+	0.998582i	$0.516951\pi$
$710$	0	0
$711$	59.4016	2.22773
$712$	0	0
$713$	0.759082	0.0284279
$714$	0	0
$715$	−3.81158	−0.142545
$716$	0	0
$717$	−34.9152	−1.30393
$718$	0	0
$719$	−21.0715	−0.785834	−0.392917	−	0.919574i	$-0.628534\pi$
$719$	−21.0715	−0.785834	−0.392917	+	0.919574i	$0.628534\pi$
$720$	0	0
$721$	8.87239	0.330425
$722$	0	0
$723$	30.0502	1.11758
$724$	0	0
$725$	−71.0455	−2.63856
$726$	0	0
$727$	12.7201	0.471762	0.235881	−	0.971782i	$-0.424202\pi$
$727$	12.7201	0.471762	0.235881	+	0.971782i	$0.424202\pi$
$728$	0	0
$729$	14.7111	0.544855
$730$	0	0
$731$	26.0136	0.962148
$732$	0	0
$733$	30.3748	1.12192	0.560960	−	0.827843i	$-0.310432\pi$
$733$	30.3748	1.12192	0.560960	+	0.827843i	$0.310432\pi$
$734$	0	0
$735$	−12.0722	−0.445289
$736$	0	0
$737$	−6.57566	−0.242218
$738$	0	0
$739$	29.0416	1.06831	0.534157	−	0.845385i	$-0.320629\pi$
$739$	29.0416	1.06831	0.534157	+	0.845385i	$0.320629\pi$
$740$	0	0
$741$	1.92793	0.0708242
$742$	0	0
$743$	28.2828	1.03759	0.518797	−	0.854897i	$-0.326380\pi$
$743$	28.2828	1.03759	0.518797	+	0.854897i	$0.326380\pi$
$744$	0	0
$745$	−82.9242	−3.03811
$746$	0	0
$747$	−56.3430	−2.06148
$748$	0	0
$749$	18.8682	0.689430
$750$	0	0
$751$	4.07060	0.148538	0.0742691	−	0.997238i	$-0.476338\pi$
$751$	4.07060	0.148538	0.0742691	+	0.997238i	$0.476338\pi$
$752$	0	0
$753$	−85.8526	−3.12864
$754$	0	0
$755$	29.0632	1.05772
$756$	0	0
$757$	16.8232	0.611450	0.305725	−	0.952120i	$-0.401101\pi$
$757$	16.8232	0.611450	0.305725	+	0.952120i	$0.401101\pi$
$758$	0	0
$759$	−3.74914	−0.136085
$760$	0	0
$761$	−41.8643	−1.51758	−0.758790	−	0.651335i	$-0.774209\pi$
$761$	−41.8643	−1.51758	−0.758790	+	0.651335i	$0.774209\pi$
$762$	0	0
$763$	11.5523	0.418220
$764$	0	0
$765$	124.969	4.51827
$766$	0	0
$767$	−4.49909	−0.162453
$768$	0	0
$769$	41.4854	1.49600	0.748001	−	0.663698i	$-0.231014\pi$
$769$	41.4854	1.49600	0.748001	+	0.663698i	$0.231014\pi$
$770$	0	0
$771$	78.8127	2.83837
$772$	0	0
$773$	−14.5621	−0.523763	−0.261882	−	0.965100i	$-0.584343\pi$
$773$	−14.5621	−0.523763	−0.261882	+	0.965100i	$0.584343\pi$
$774$	0	0
$775$	−6.11006	−0.219480
$776$	0	0
$777$	−24.3477	−0.873468
$778$	0	0
$779$	−1.78853	−0.0640806
$780$	0	0
$781$	−3.48656	−0.124759
$782$	0	0
$783$	95.2066	3.40241
$784$	0	0
$785$	31.4944	1.12408
$786$	0	0
$787$	24.3939	0.869550	0.434775	−	0.900539i	$-0.356828\pi$
$787$	24.3939	0.869550	0.434775	+	0.900539i	$0.356828\pi$
$788$	0	0
$789$	51.2046	1.82293
$790$	0	0
$791$	−5.01128	−0.178180
$792$	0	0
$793$	−9.43556	−0.335067
$794$	0	0
$795$	57.8680	2.05237
$796$	0	0
$797$	−32.6495	−1.15651	−0.578253	−	0.815858i	$-0.696265\pi$
$797$	−32.6495	−1.15651	−0.578253	+	0.815858i	$0.696265\pi$
$798$	0	0
$799$	−45.8516	−1.62211
$800$	0	0
$801$	−50.1897	−1.77337
$802$	0	0
$803$	8.97801	0.316827
$804$	0	0
$805$	4.51186	0.159022
$806$	0	0
$807$	−24.5989	−0.865922
$808$	0	0
$809$	−0.241056	−0.00847509	−0.00423755	−	0.999991i	$-0.501349\pi$
$809$	−0.241056	−0.00847509	−0.00423755	+	0.999991i	$0.501349\pi$
$810$	0	0
$811$	−3.21792	−0.112996	−0.0564982	−	0.998403i	$-0.517994\pi$
$811$	−3.21792	−0.112996	−0.0564982	+	0.998403i	$0.517994\pi$
$812$	0	0
$813$	−14.3004	−0.501538
$814$	0	0
$815$	32.9047	1.15260
$816$	0	0
$817$	−3.39590	−0.118807
$818$	0	0
$819$	−7.03141	−0.245697
$820$	0	0
$821$	−24.9428	−0.870510	−0.435255	−	0.900307i	$-0.643342\pi$
$821$	−24.9428	−0.870510	−0.435255	+	0.900307i	$0.643342\pi$
$822$	0	0
$823$	16.4903	0.574816	0.287408	−	0.957808i	$-0.407206\pi$
$823$	16.4903	0.574816	0.287408	+	0.957808i	$0.407206\pi$
$824$	0	0
$825$	30.1779	1.05066
$826$	0	0
$827$	11.4548	0.398322	0.199161	−	0.979967i	$-0.436178\pi$
$827$	11.4548	0.398322	0.199161	+	0.979967i	$0.436178\pi$
$828$	0	0
$829$	−55.3188	−1.92130	−0.960650	−	0.277762i	$-0.910407\pi$
$829$	−55.3188	−1.92130	−0.960650	+	0.277762i	$0.910407\pi$
$830$	0	0
$831$	57.7457	2.00318
$832$	0	0
$833$	4.66290	0.161560
$834$	0	0
$835$	−23.8315	−0.824724
$836$	0	0
$837$	8.18797	0.283017
$838$	0	0
$839$	44.8447	1.54821	0.774106	−	0.633056i	$-0.218200\pi$
$839$	44.8447	1.54821	0.774106	+	0.633056i	$0.218200\pi$
$840$	0	0
$841$	26.5978	0.917167
$842$	0	0
$843$	43.4325	1.49589
$844$	0	0
$845$	3.81158	0.131122
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	41.3936	1.42062
$850$	0	0
$851$	9.09971	0.311934
$852$	0	0
$853$	19.2816	0.660190	0.330095	−	0.943948i	$-0.392919\pi$
$853$	19.2816	0.660190	0.330095	+	0.943948i	$0.392919\pi$
$854$	0	0
$855$	−16.3139	−0.557923
$856$	0	0
$857$	32.6405	1.11498	0.557490	−	0.830184i	$-0.311765\pi$
$857$	32.6405	1.11498	0.557490	+	0.830184i	$0.311765\pi$
$858$	0	0
$859$	−43.1448	−1.47208	−0.736041	−	0.676937i	$-0.763307\pi$
$859$	−43.1448	−1.47208	−0.736041	+	0.676937i	$0.763307\pi$
$860$	0	0
$861$	9.30608	0.317150
$862$	0	0
$863$	−13.8993	−0.473138	−0.236569	−	0.971615i	$-0.576023\pi$
$863$	−13.8993	−0.473138	−0.236569	+	0.971615i	$0.576023\pi$
$864$	0	0
$865$	73.9405	2.51405
$866$	0	0
$867$	−15.0209	−0.510138
$868$	0	0
$869$	−8.44803	−0.286580
$870$	0	0
$871$	6.57566	0.222808
$872$	0	0
$873$	−101.964	−3.45095
$874$	0	0
$875$	−17.2593	−0.583472
$876$	0	0
$877$	−12.5494	−0.423764	−0.211882	−	0.977295i	$-0.567959\pi$
$877$	−12.5494	−0.423764	−0.211882	+	0.977295i	$0.567959\pi$
$878$	0	0
$879$	−54.0830	−1.82417
$880$	0	0
$881$	56.6702	1.90927	0.954633	−	0.297785i	$-0.0962477\pi$
$881$	56.6702	1.90927	0.954633	+	0.297785i	$0.0962477\pi$
$882$	0	0
$883$	−46.1701	−1.55375	−0.776873	−	0.629657i	$-0.783195\pi$
$883$	−46.1701	−1.55375	−0.776873	+	0.629657i	$0.783195\pi$
$884$	0	0
$885$	54.3138	1.82574
$886$	0	0
$887$	8.42199	0.282783	0.141391	−	0.989954i	$-0.454842\pi$
$887$	8.42199	0.282783	0.141391	+	0.989954i	$0.454842\pi$
$888$	0	0
$889$	−7.15140	−0.239850
$890$	0	0
$891$	−19.3465	−0.648133
$892$	0	0
$893$	5.98561	0.200301
$894$	0	0
$895$	−77.7222	−2.59797
$896$	0	0
$897$	3.74914	0.125180
$898$	0	0
$899$	4.78153	0.159473
$900$	0	0
$901$	−22.3516	−0.744639
$902$	0	0
$903$	17.6696	0.588006
$904$	0	0
$905$	−34.2339	−1.13797
$906$	0	0
$907$	20.1315	0.668457	0.334229	−	0.942492i	$-0.391524\pi$
$907$	20.1315	0.668457	0.334229	+	0.942492i	$0.391524\pi$
$908$	0	0
$909$	−32.9383	−1.09249
$910$	0	0
$911$	−6.97718	−0.231164	−0.115582	−	0.993298i	$-0.536873\pi$
$911$	−6.97718	−0.231164	−0.115582	+	0.993298i	$0.536873\pi$
$912$	0	0
$913$	8.01304	0.265193
$914$	0	0
$915$	113.908	3.76568
$916$	0	0
$917$	−13.6650	−0.451258
$918$	0	0
$919$	18.5506	0.611928	0.305964	−	0.952043i	$-0.401021\pi$
$919$	18.5506	0.611928	0.305964	+	0.952043i	$0.401021\pi$
$920$	0	0
$921$	100.570	3.31389
$922$	0	0
$923$	3.48656	0.114762
$924$	0	0
$925$	−73.2460	−2.40831
$926$	0	0
$927$	−62.3854	−2.04901
$928$	0	0
$929$	22.5942	0.741291	0.370645	−	0.928774i	$-0.379137\pi$
$929$	22.5942	0.741291	0.370645	+	0.928774i	$0.379137\pi$
$930$	0	0
$931$	−0.608709	−0.0199496
$932$	0	0
$933$	−88.1691	−2.88653
$934$	0	0
$935$	−17.7730	−0.581239
$936$	0	0
$937$	11.6285	0.379887	0.189944	−	0.981795i	$-0.439170\pi$
$937$	11.6285	0.379887	0.189944	+	0.981795i	$0.439170\pi$
$938$	0	0
$939$	−39.2782	−1.28179
$940$	0	0
$941$	−16.4028	−0.534717	−0.267358	−	0.963597i	$-0.586151\pi$
$941$	−16.4028	−0.534717	−0.267358	+	0.963597i	$0.586151\pi$
$942$	0	0
$943$	−3.47806	−0.113261
$944$	0	0
$945$	48.6680	1.58317
$946$	0	0
$947$	−45.1822	−1.46822	−0.734112	−	0.679028i	$-0.762401\pi$
$947$	−45.1822	−1.46822	−0.734112	+	0.679028i	$0.762401\pi$
$948$	0	0
$949$	−8.97801	−0.291438
$950$	0	0
$951$	62.9437	2.04109
$952$	0	0
$953$	18.7298	0.606717	0.303358	−	0.952877i	$-0.401892\pi$
$953$	18.7298	0.606717	0.303358	+	0.952877i	$0.401892\pi$
$954$	0	0
$955$	−2.00671	−0.0649355
$956$	0	0
$957$	−23.6162	−0.763403
$958$	0	0
$959$	−9.53905	−0.308032
$960$	0	0
$961$	−30.5888	−0.986735
$962$	0	0
$963$	−132.670	−4.27524
$964$	0	0
$965$	−5.36556	−0.172723
$966$	0	0
$967$	−19.1270	−0.615082	−0.307541	−	0.951535i	$-0.599506\pi$
$967$	−19.1270	−0.615082	−0.307541	+	0.951535i	$0.599506\pi$
$968$	0	0
$969$	8.98972	0.288791
$970$	0	0
$971$	−44.9142	−1.44137	−0.720683	−	0.693265i	$-0.756172\pi$
$971$	−44.9142	−1.44137	−0.720683	+	0.693265i	$0.756172\pi$
$972$	0	0
$973$	−6.54607	−0.209857
$974$	0	0
$975$	−30.1779	−0.966466
$976$	0	0
$977$	12.4792	0.399247	0.199623	−	0.979873i	$-0.436028\pi$
$977$	12.4792	0.399247	0.199623	+	0.979873i	$0.436028\pi$
$978$	0	0
$979$	7.13793	0.228129
$980$	0	0
$981$	−81.2287	−2.59343
$982$	0	0
$983$	−53.5234	−1.70713	−0.853566	−	0.520985i	$-0.825565\pi$
$983$	−53.5234	−1.70713	−0.853566	+	0.520985i	$0.825565\pi$
$984$	0	0
$985$	58.8106	1.87386
$986$	0	0
$987$	−31.1444	−0.991337
$988$	0	0
$989$	−6.60383	−0.209990
$990$	0	0
$991$	19.2619	0.611875	0.305937	−	0.952052i	$-0.401030\pi$
$991$	19.2619	0.611875	0.305937	+	0.952052i	$0.401030\pi$
$992$	0	0
$993$	101.949	3.23526
$994$	0	0
$995$	32.3631	1.02598
$996$	0	0
$997$	−36.1304	−1.14426	−0.572130	−	0.820163i	$-0.693883\pi$
$997$	−36.1304	−1.14426	−0.572130	+	0.820163i	$0.693883\pi$
$998$	0	0
$999$	98.1555	3.10550

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.m.1.1	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.m.1.1	✓	9	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8008.a (trivial)

\(f(q)\)	\(=\)	\(q-3.16724 q^{3} +3.81158 q^{5} -1.00000 q^{7} +7.03141 q^{9} +O(q^{10})\)	\(q-3.16724 q^{3} +3.81158 q^{5} -1.00000 q^{7} +7.03141 q^{9} -1.00000 q^{11} +1.00000 q^{13} -12.0722 q^{15} +4.66290 q^{17} -0.608709 q^{19} +3.16724 q^{21} -1.18373 q^{23} +9.52813 q^{25} -12.7685 q^{27} -7.45640 q^{29} -0.641265 q^{31} +3.16724 q^{33} -3.81158 q^{35} -7.68734 q^{37} -3.16724 q^{39} +2.93823 q^{41} +5.57885 q^{43} +26.8008 q^{45} -9.83329 q^{47} +1.00000 q^{49} -14.7685 q^{51} -4.79350 q^{53} -3.81158 q^{55} +1.92793 q^{57} -4.49909 q^{59} -9.43556 q^{61} -7.03141 q^{63} +3.81158 q^{65} +6.57566 q^{67} +3.74914 q^{69} +3.48656 q^{71} -8.97801 q^{73} -30.1779 q^{75} +1.00000 q^{77} +8.44803 q^{79} +19.3465 q^{81} -8.01304 q^{83} +17.7730 q^{85} +23.6162 q^{87} -7.13793 q^{89} -1.00000 q^{91} +2.03104 q^{93} -2.32014 q^{95} -14.5012 q^{97} -7.03141 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 5 q^{3} - 5 q^{5} - 9 q^{7} + 12 q^{9}+O(q^{10}) \) 9 * q - 5 * q^3 - 5 * q^5 - 9 * q^7 + 12 * q^9	\( 9 q - 5 q^{3} - 5 q^{5} - 9 q^{7} + 12 q^{9} - 9 q^{11} + 9 q^{13} - q^{15} + 13 q^{17} - 5 q^{19} + 5 q^{21} - 7 q^{23} + 14 q^{25} - 23 q^{27} - 19 q^{29} + 2 q^{31} + 5 q^{33} + 5 q^{35} + 4 q^{37} - 5 q^{39} + 10 q^{41} - 17 q^{43} + 11 q^{45} - 5 q^{47} + 9 q^{49} - 9 q^{51} - 24 q^{53} + 5 q^{55} - 4 q^{57} - 19 q^{59} - 12 q^{63} - 5 q^{65} + 2 q^{67} - 2 q^{69} - 2 q^{71} + 34 q^{73} - 46 q^{75} + 9 q^{77} + 5 q^{79} + 37 q^{81} - 24 q^{83} + 33 q^{85} + 41 q^{87} - 11 q^{89} - 9 q^{91} + 53 q^{93} - 23 q^{95} - 12 q^{99}+O(q^{100}) \) 9 * q - 5 * q^3 - 5 * q^5 - 9 * q^7 + 12 * q^9 - 9 * q^11 + 9 * q^13 - q^15 + 13 * q^17 - 5 * q^19 + 5 * q^21 - 7 * q^23 + 14 * q^25 - 23 * q^27 - 19 * q^29 + 2 * q^31 + 5 * q^33 + 5 * q^35 + 4 * q^37 - 5 * q^39 + 10 * q^41 - 17 * q^43 + 11 * q^45 - 5 * q^47 + 9 * q^49 - 9 * q^51 - 24 * q^53 + 5 * q^55 - 4 * q^57 - 19 * q^59 - 12 * q^63 - 5 * q^65 + 2 * q^67 - 2 * q^69 - 2 * q^71 + 34 * q^73 - 46 * q^75 + 9 * q^77 + 5 * q^79 + 37 * q^81 - 24 * q^83 + 33 * q^85 + 41 * q^87 - 11 * q^89 - 9 * q^91 + 53 * q^93 - 23 * q^95 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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