LMFDB - Embedded newform 6003.2.a.o.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6003,2,Mod(1,6003)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6003, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6003.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6003 = 3^{2} \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6003.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:	$47.9341963334$
Analytic rank:	$1$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 4 x^{12} - 11 x^{11} + 58 x^{10} + 24 x^{9} - 298 x^{8} + 97 x^{7} + 641 x^{6} - 402 x^{5} + \cdots - 40 $ x^13 - 4x^12 - 11x^11 + 58x^10 + 24x^9 - 298x^8 + 97x^7 + 641x^6 - 402x^5 - 547x^4 + 352x^3 + 219x^2 - 88x - 40
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 667)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.24788$ of defining polynomial
Character	$\chi$	$=$	6003.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.24788 q^{2} +3.05297 q^{4} +0.269352 q^{5} -0.523800 q^{7} -2.36696 q^{8} +O(q^{10})$	\(q-2.24788 q^{2} +3.05297 q^{4} +0.269352 q^{5} -0.523800 q^{7} -2.36696 q^{8} -0.605470 q^{10} -5.54870 q^{11} +3.85335 q^{13} +1.17744 q^{14} -0.785308 q^{16} +1.23265 q^{17} +1.15608 q^{19} +0.822323 q^{20} +12.4728 q^{22} +1.00000 q^{23} -4.92745 q^{25} -8.66188 q^{26} -1.59915 q^{28} -1.00000 q^{29} +1.02441 q^{31} +6.49919 q^{32} -2.77086 q^{34} -0.141086 q^{35} -3.22089 q^{37} -2.59873 q^{38} -0.637543 q^{40} +3.67031 q^{41} +2.77677 q^{43} -16.9400 q^{44} -2.24788 q^{46} +6.71558 q^{47} -6.72563 q^{49} +11.0763 q^{50} +11.7642 q^{52} +2.58548 q^{53} -1.49455 q^{55} +1.23981 q^{56} +2.24788 q^{58} +9.36230 q^{59} -12.0763 q^{61} -2.30275 q^{62} -13.0388 q^{64} +1.03791 q^{65} -2.14280 q^{67} +3.76326 q^{68} +0.317146 q^{70} -4.70321 q^{71} +11.8546 q^{73} +7.24017 q^{74} +3.52948 q^{76} +2.90641 q^{77} +15.5887 q^{79} -0.211524 q^{80} -8.25042 q^{82} -8.53106 q^{83} +0.332017 q^{85} -6.24185 q^{86} +13.1335 q^{88} -12.2129 q^{89} -2.01839 q^{91} +3.05297 q^{92} -15.0958 q^{94} +0.311392 q^{95} +3.00386 q^{97} +15.1184 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 4 q^{2} + 12 q^{4} - 16 q^{5} + q^{7} - 6 q^{8}+O(q^{10}) $ 13 * q - 4 * q^2 + 12 * q^4 - 16 * q^5 + q^7 - 6 * q^8	$ 13 q - 4 q^{2} + 12 q^{4} - 16 q^{5} + q^{7} - 6 q^{8} + 10 q^{10} - 10 q^{11} + 7 q^{13} + 12 q^{14} + 2 q^{16} - 26 q^{17} - 25 q^{20} - 15 q^{22} + 13 q^{23} + 19 q^{25} + 15 q^{26} + 5 q^{28} - 13 q^{29} - 6 q^{31} - 16 q^{32} + 11 q^{34} - q^{35} + 15 q^{37} - 8 q^{38} + 14 q^{40} - 9 q^{41} + q^{43} - 29 q^{44} - 4 q^{46} - 15 q^{47} + 4 q^{49} - 31 q^{50} - 8 q^{52} - 43 q^{53} - 3 q^{55} + 5 q^{56} + 4 q^{58} + 9 q^{59} + 20 q^{61} - 11 q^{62} - 16 q^{64} + 25 q^{65} + q^{67} - 21 q^{68} - 2 q^{70} - 17 q^{71} + 26 q^{73} - 11 q^{74} + 8 q^{76} - 17 q^{77} + 5 q^{79} - 10 q^{80} - 25 q^{82} - 4 q^{83} + 20 q^{85} + 13 q^{86} - 32 q^{88} - 48 q^{89} - 9 q^{91} + 12 q^{92} - 65 q^{94} - 8 q^{95} + 30 q^{97} - 2 q^{98}+O(q^{100}) $ 13 * q - 4 * q^2 + 12 * q^4 - 16 * q^5 + q^7 - 6 * q^8 + 10 * q^10 - 10 * q^11 + 7 * q^13 + 12 * q^14 + 2 * q^16 - 26 * q^17 - 25 * q^20 - 15 * q^22 + 13 * q^23 + 19 * q^25 + 15 * q^26 + 5 * q^28 - 13 * q^29 - 6 * q^31 - 16 * q^32 + 11 * q^34 - q^35 + 15 * q^37 - 8 * q^38 + 14 * q^40 - 9 * q^41 + q^43 - 29 * q^44 - 4 * q^46 - 15 * q^47 + 4 * q^49 - 31 * q^50 - 8 * q^52 - 43 * q^53 - 3 * q^55 + 5 * q^56 + 4 * q^58 + 9 * q^59 + 20 * q^61 - 11 * q^62 - 16 * q^64 + 25 * q^65 + q^67 - 21 * q^68 - 2 * q^70 - 17 * q^71 + 26 * q^73 - 11 * q^74 + 8 * q^76 - 17 * q^77 + 5 * q^79 - 10 * q^80 - 25 * q^82 - 4 * q^83 + 20 * q^85 + 13 * q^86 - 32 * q^88 - 48 * q^89 - 9 * q^91 + 12 * q^92 - 65 * q^94 - 8 * q^95 + 30 * q^97 - 2 * q^98

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 $

$2$	−2.24788	−1.58949	−0.794746	−	0.606942i	$-0.792396\pi$
$2$	−2.24788	−1.58949	−0.794746	+	0.606942i	$0.792396\pi$
$3$	0	0
$3$	0	0
$4$	3.05297	1.52649
$5$	0.269352	0.120458	0.0602288	−	0.998185i	$-0.480817\pi$
$5$	0.269352	0.120458	0.0602288	+	0.998185i	$0.480817\pi$
$6$	0	0
$7$	−0.523800	−0.197978	−0.0989889	−	0.995089i	$-0.531561\pi$
$7$	−0.523800	−0.197978	−0.0989889	+	0.995089i	$0.531561\pi$
$8$	−2.36696	−0.836845
$9$	0	0
$10$	−0.605470	−0.191467
$11$	−5.54870	−1.67300	−0.836498	−	0.547969i	$-0.815401\pi$
$11$	−5.54870	−1.67300	−0.836498	+	0.547969i	$0.815401\pi$
$12$	0	0
$13$	3.85335	1.06873	0.534364	−	0.845255i	$-0.320551\pi$
$13$	3.85335	1.06873	0.534364	+	0.845255i	$0.320551\pi$
$14$	1.17744	0.314684
$15$	0	0
$16$	−0.785308	−0.196327
$17$	1.23265	0.298962	0.149481	−	0.988765i	$-0.452240\pi$
$17$	1.23265	0.298962	0.149481	+	0.988765i	$0.452240\pi$
$18$	0	0
$19$	1.15608	0.265223	0.132611	−	0.991168i	$-0.457664\pi$
$19$	1.15608	0.265223	0.132611	+	0.991168i	$0.457664\pi$
$20$	0.822323	0.183877
$21$	0	0
$22$	12.4728	2.65922
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−4.92745	−0.985490
$26$	−8.66188	−1.69873
$27$	0	0
$28$	−1.59915	−0.302210
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	1.02441	0.183989	0.0919945	−	0.995760i	$-0.470676\pi$
$31$	1.02441	0.183989	0.0919945	+	0.995760i	$0.470676\pi$
$32$	6.49919	1.14891
$33$	0	0
$34$	−2.77086	−0.475199
$35$	−0.141086	−0.0238480
$36$	0	0
$37$	−3.22089	−0.529511	−0.264755	−	0.964316i	$-0.585291\pi$
$37$	−3.22089	−0.529511	−0.264755	+	0.964316i	$0.585291\pi$
$38$	−2.59873	−0.421569
$39$	0	0
$40$	−0.637543	−0.100804
$41$	3.67031	0.573206	0.286603	−	0.958049i	$-0.407474\pi$
$41$	3.67031	0.573206	0.286603	+	0.958049i	$0.407474\pi$
$42$	0	0
$43$	2.77677	0.423454	0.211727	−	0.977329i	$-0.432091\pi$
$43$	2.77677	0.423454	0.211727	+	0.977329i	$0.432091\pi$
$44$	−16.9400	−2.55381
$45$	0	0
$46$	−2.24788	−0.331432
$47$	6.71558	0.979567	0.489784	−	0.871844i	$-0.337076\pi$
$47$	6.71558	0.979567	0.489784	+	0.871844i	$0.337076\pi$
$48$	0	0
$49$	−6.72563	−0.960805
$50$	11.0763	1.56643
$51$	0	0
$52$	11.7642	1.63140
$53$	2.58548	0.355143	0.177572	−	0.984108i	$-0.443176\pi$
$53$	2.58548	0.355143	0.177572	+	0.984108i	$0.443176\pi$
$54$	0	0
$55$	−1.49455	−0.201525
$56$	1.23981	0.165677
$57$	0	0
$58$	2.24788	0.295161
$59$	9.36230	1.21887	0.609434	−	0.792837i	$-0.291397\pi$
$59$	9.36230	1.21887	0.609434	+	0.792837i	$0.291397\pi$
$60$	0	0
$61$	−12.0763	−1.54621	−0.773104	−	0.634279i	$-0.781297\pi$
$61$	−12.0763	−1.54621	−0.773104	+	0.634279i	$0.781297\pi$
$62$	−2.30275	−0.292449
$63$	0	0
$64$	−13.0388	−1.62985
$65$	1.03791	0.128736
$66$	0	0
$67$	−2.14280	−0.261785	−0.130892	−	0.991397i	$-0.541784\pi$
$67$	−2.14280	−0.261785	−0.130892	+	0.991397i	$0.541784\pi$
$68$	3.76326	0.456362
$69$	0	0
$70$	0.317146	0.0379061
$71$	−4.70321	−0.558168	−0.279084	−	0.960267i	$-0.590031\pi$
$71$	−4.70321	−0.558168	−0.279084	+	0.960267i	$0.590031\pi$
$72$	0	0
$73$	11.8546	1.38748	0.693738	−	0.720227i	$-0.255962\pi$
$73$	11.8546	1.38748	0.693738	+	0.720227i	$0.255962\pi$
$74$	7.24017	0.841653
$75$	0	0
$76$	3.52948	0.404859
$77$	2.90641	0.331216
$78$	0	0
$79$	15.5887	1.75386	0.876931	−	0.480617i	$-0.159587\pi$
$79$	15.5887	1.75386	0.876931	+	0.480617i	$0.159587\pi$
$80$	−0.211524	−0.0236491
$81$	0	0
$82$	−8.25042	−0.911107
$83$	−8.53106	−0.936406	−0.468203	−	0.883621i	$-0.655098\pi$
$83$	−8.53106	−0.936406	−0.468203	+	0.883621i	$0.655098\pi$
$84$	0	0
$85$	0.332017	0.0360123
$86$	−6.24185	−0.673076
$87$	0	0
$88$	13.1335	1.40004
$89$	−12.2129	−1.29456	−0.647280	−	0.762252i	$-0.724094\pi$
$89$	−12.2129	−1.29456	−0.647280	+	0.762252i	$0.724094\pi$
$90$	0	0
$91$	−2.01839	−0.211584
$92$	3.05297	0.318294
$93$	0	0
$94$	−15.0958	−1.55701
$95$	0.311392	0.0319481
$96$	0	0
$97$	3.00386	0.304996	0.152498	−	0.988304i	$-0.451268\pi$
$97$	3.00386	0.304996	0.152498	+	0.988304i	$0.451268\pi$
$98$	15.1184	1.52719
$99$	0	0
$100$	−15.0434	−1.50434
$101$	−6.57477	−0.654214	−0.327107	−	0.944987i	$-0.606074\pi$
$101$	−6.57477	−0.654214	−0.327107	+	0.944987i	$0.606074\pi$
$102$	0	0
$103$	17.1469	1.68953	0.844766	−	0.535136i	$-0.179740\pi$
$103$	17.1469	1.68953	0.844766	+	0.535136i	$0.179740\pi$
$104$	−9.12071	−0.894359
$105$	0	0
$106$	−5.81186	−0.564497
$107$	−12.1672	−1.17625	−0.588123	−	0.808772i	$-0.700133\pi$
$107$	−12.1672	−1.17625	−0.588123	+	0.808772i	$0.700133\pi$
$108$	0	0
$109$	3.17086	0.303713	0.151857	−	0.988403i	$-0.451475\pi$
$109$	3.17086	0.303713	0.151857	+	0.988403i	$0.451475\pi$
$110$	3.35958	0.320323
$111$	0	0
$112$	0.411344	0.0388684
$113$	12.6858	1.19338	0.596688	−	0.802473i	$-0.296483\pi$
$113$	12.6858	1.19338	0.596688	+	0.802473i	$0.296483\pi$
$114$	0	0
$115$	0.269352	0.0251172
$116$	−3.05297	−0.283461
$117$	0	0
$118$	−21.0453	−1.93738
$119$	−0.645664	−0.0591879
$120$	0	0
$121$	19.7881	1.79892
$122$	27.1460	2.45769
$123$	0	0
$124$	3.12749	0.280857
$125$	−2.67397	−0.239168
$126$	0	0
$127$	−11.6051	−1.02978	−0.514892	−	0.857255i	$-0.672168\pi$
$127$	−11.6051	−1.02978	−0.514892	+	0.857255i	$0.672168\pi$
$128$	16.3113	1.44173
$129$	0	0
$130$	−2.33309	−0.204626
$131$	−9.59459	−0.838283	−0.419142	−	0.907921i	$-0.637669\pi$
$131$	−9.59459	−0.838283	−0.419142	+	0.907921i	$0.637669\pi$
$132$	0	0
$133$	−0.605554	−0.0525082
$134$	4.81677	0.416105
$135$	0	0
$136$	−2.91764	−0.250185
$137$	0.429683	0.0367103	0.0183552	−	0.999832i	$-0.494157\pi$
$137$	0.429683	0.0367103	0.0183552	+	0.999832i	$0.494157\pi$
$138$	0	0
$139$	−10.4292	−0.884597	−0.442299	−	0.896868i	$-0.645837\pi$
$139$	−10.4292	−0.884597	−0.442299	+	0.896868i	$0.645837\pi$
$140$	−0.430733	−0.0364036
$141$	0	0
$142$	10.5723	0.887204
$143$	−21.3811	−1.78798
$144$	0	0
$145$	−0.269352	−0.0223684
$146$	−26.6478	−2.20538
$147$	0	0
$148$	−9.83328	−0.808290
$149$	−12.9871	−1.06394	−0.531970	−	0.846763i	$-0.678548\pi$
$149$	−12.9871	−1.06394	−0.531970	+	0.846763i	$0.678548\pi$
$150$	0	0
$151$	−12.8395	−1.04486	−0.522431	−	0.852681i	$-0.674975\pi$
$151$	−12.8395	−1.04486	−0.522431	+	0.852681i	$0.674975\pi$
$152$	−2.73639	−0.221950
$153$	0	0
$154$	−6.53327	−0.526466
$155$	0.275926	0.0221629
$156$	0	0
$157$	−4.11732	−0.328598	−0.164299	−	0.986411i	$-0.552536\pi$
$157$	−4.11732	−0.328598	−0.164299	+	0.986411i	$0.552536\pi$
$158$	−35.0415	−2.78775
$159$	0	0
$160$	1.75057	0.138394
$161$	−0.523800	−0.0412812
$162$	0	0
$163$	−10.6963	−0.837801	−0.418900	−	0.908032i	$-0.637584\pi$
$163$	−10.6963	−0.837801	−0.418900	+	0.908032i	$0.637584\pi$
$164$	11.2054	0.874991
$165$	0	0
$166$	19.1768	1.48841
$167$	−24.3060	−1.88085	−0.940427	−	0.339997i	$-0.889574\pi$
$167$	−24.3060	−1.88085	−0.940427	+	0.339997i	$0.889574\pi$
$168$	0	0
$169$	1.84832	0.142179
$170$	−0.746335	−0.0572413
$171$	0	0
$172$	8.47740	0.646396
$173$	−3.04992	−0.231881	−0.115941	−	0.993256i	$-0.536988\pi$
$173$	−3.04992	−0.231881	−0.115941	+	0.993256i	$0.536988\pi$
$174$	0	0
$175$	2.58100	0.195105
$176$	4.35744	0.328454
$177$	0	0
$178$	27.4531	2.05769
$179$	17.7448	1.32631	0.663155	−	0.748482i	$-0.269217\pi$
$179$	17.7448	1.32631	0.663155	+	0.748482i	$0.269217\pi$
$180$	0	0
$181$	−19.4919	−1.44882	−0.724410	−	0.689369i	$-0.757888\pi$
$181$	−19.4919	−1.44882	−0.724410	+	0.689369i	$0.757888\pi$
$182$	4.53709	0.336312
$183$	0	0
$184$	−2.36696	−0.174494
$185$	−0.867551	−0.0637836
$186$	0	0
$187$	−6.83963	−0.500163
$188$	20.5025	1.49530
$189$	0	0
$190$	−0.699971	−0.0507813
$191$	14.0435	1.01615	0.508075	−	0.861313i	$-0.330357\pi$
$191$	14.0435	1.01615	0.508075	+	0.861313i	$0.330357\pi$
$192$	0	0
$193$	20.6882	1.48917	0.744585	−	0.667528i	$-0.232647\pi$
$193$	20.6882	1.48917	0.744585	+	0.667528i	$0.232647\pi$
$194$	−6.75233	−0.484789
$195$	0	0
$196$	−20.5332	−1.46665
$197$	−1.32494	−0.0943983	−0.0471991	−	0.998885i	$-0.515030\pi$
$197$	−1.32494	−0.0943983	−0.0471991	+	0.998885i	$0.515030\pi$
$198$	0	0
$199$	14.6071	1.03547	0.517736	−	0.855541i	$-0.326775\pi$
$199$	14.6071	1.03547	0.517736	+	0.855541i	$0.326775\pi$
$200$	11.6631	0.824702
$201$	0	0
$202$	14.7793	1.03987
$203$	0.523800	0.0367636
$204$	0	0
$205$	0.988604	0.0690471
$206$	−38.5441	−2.68550
$207$	0	0
$208$	−3.02607	−0.209820
$209$	−6.41474	−0.443717
$210$	0	0
$211$	−16.8423	−1.15947	−0.579734	−	0.814806i	$-0.696844\pi$
$211$	−16.8423	−1.15947	−0.579734	+	0.814806i	$0.696844\pi$
$212$	7.89340	0.542121
$213$	0	0
$214$	27.3504	1.86963
$215$	0.747928	0.0510083
$216$	0	0
$217$	−0.536585	−0.0364257
$218$	−7.12772	−0.482750
$219$	0	0
$220$	−4.56282	−0.307626
$221$	4.74985	0.319509
$222$	0	0
$223$	5.08575	0.340567	0.170284	−	0.985395i	$-0.445532\pi$
$223$	5.08575	0.340567	0.170284	+	0.985395i	$0.445532\pi$
$224$	−3.40428	−0.227458
$225$	0	0
$226$	−28.5161	−1.89686
$227$	2.37457	0.157606	0.0788030	−	0.996890i	$-0.474890\pi$
$227$	2.37457	0.157606	0.0788030	+	0.996890i	$0.474890\pi$
$228$	0	0
$229$	11.0268	0.728670	0.364335	−	0.931268i	$-0.381296\pi$
$229$	11.0268	0.728670	0.364335	+	0.931268i	$0.381296\pi$
$230$	−0.605470	−0.0399235
$231$	0	0
$232$	2.36696	0.155398
$233$	12.7890	0.837837	0.418919	−	0.908024i	$-0.362409\pi$
$233$	12.7890	0.837837	0.418919	+	0.908024i	$0.362409\pi$
$234$	0	0
$235$	1.80885	0.117996
$236$	28.5828	1.86058
$237$	0	0
$238$	1.45138	0.0940788
$239$	−7.10384	−0.459509	−0.229754	−	0.973249i	$-0.573792\pi$
$239$	−7.10384	−0.459509	−0.229754	+	0.973249i	$0.573792\pi$
$240$	0	0
$241$	2.37432	0.152943	0.0764717	−	0.997072i	$-0.475635\pi$
$241$	2.37432	0.152943	0.0764717	+	0.997072i	$0.475635\pi$
$242$	−44.4813	−2.85937
$243$	0	0
$244$	−36.8685	−2.36027
$245$	−1.81156	−0.115736
$246$	0	0
$247$	4.45478	0.283451
$248$	−2.42473	−0.153970
$249$	0	0
$250$	6.01078	0.380155
$251$	18.8107	1.18732	0.593660	−	0.804716i	$-0.297682\pi$
$251$	18.8107	1.18732	0.593660	+	0.804716i	$0.297682\pi$
$252$	0	0
$253$	−5.54870	−0.348844
$254$	26.0868	1.63683
$255$	0	0
$256$	−10.5882	−0.661765
$257$	−20.0371	−1.24988	−0.624939	−	0.780674i	$-0.714876\pi$
$257$	−20.0371	−1.24988	−0.624939	+	0.780674i	$0.714876\pi$
$258$	0	0
$259$	1.68710	0.104831
$260$	3.16870	0.196514
$261$	0	0
$262$	21.5675	1.33244
$263$	19.8158	1.22189	0.610947	−	0.791671i	$-0.290789\pi$
$263$	19.8158	1.22189	0.610947	+	0.791671i	$0.290789\pi$
$264$	0	0
$265$	0.696404	0.0427797
$266$	1.36121	0.0834614
$267$	0	0
$268$	−6.54192	−0.399611
$269$	−21.0353	−1.28254	−0.641271	−	0.767314i	$-0.721593\pi$
$269$	−21.0353	−1.28254	−0.641271	+	0.767314i	$0.721593\pi$
$270$	0	0
$271$	19.0625	1.15796	0.578982	−	0.815340i	$-0.303450\pi$
$271$	19.0625	1.15796	0.578982	+	0.815340i	$0.303450\pi$
$272$	−0.968013	−0.0586944
$273$	0	0
$274$	−0.965877	−0.0583508
$275$	27.3410	1.64872
$276$	0	0
$277$	4.40716	0.264801	0.132400	−	0.991196i	$-0.457732\pi$
$277$	4.40716	0.264801	0.132400	+	0.991196i	$0.457732\pi$
$278$	23.4437	1.40606
$279$	0	0
$280$	0.333945	0.0199570
$281$	−19.7030	−1.17538	−0.587691	−	0.809086i	$-0.699963\pi$
$281$	−19.7030	−1.17538	−0.587691	+	0.809086i	$0.699963\pi$
$282$	0	0
$283$	21.5092	1.27859	0.639294	−	0.768962i	$-0.279227\pi$
$283$	21.5092	1.27859	0.639294	+	0.768962i	$0.279227\pi$
$284$	−14.3588	−0.852036
$285$	0	0
$286$	48.0622	2.84198
$287$	−1.92251	−0.113482
$288$	0	0
$289$	−15.4806	−0.910621
$290$	0.605470	0.0355544
$291$	0	0
$292$	36.1918	2.11796
$293$	−7.30614	−0.426829	−0.213415	−	0.976962i	$-0.568458\pi$
$293$	−7.30614	−0.426829	−0.213415	+	0.976962i	$0.568458\pi$
$294$	0	0
$295$	2.52175	0.146822
$296$	7.62370	0.443118
$297$	0	0
$298$	29.1934	1.69113
$299$	3.85335	0.222845
$300$	0	0
$301$	−1.45447	−0.0838345
$302$	28.8616	1.66080
$303$	0	0
$304$	−0.907878	−0.0520704
$305$	−3.25276	−0.186253
$306$	0	0
$307$	−30.0031	−1.71237	−0.856184	−	0.516671i	$-0.827171\pi$
$307$	−30.0031	−1.71237	−0.856184	+	0.516671i	$0.827171\pi$
$308$	8.87319	0.505597
$309$	0	0
$310$	−0.620248	−0.0352277
$311$	15.3682	0.871453	0.435727	−	0.900079i	$-0.356491\pi$
$311$	15.3682	0.871453	0.435727	+	0.900079i	$0.356491\pi$
$312$	0	0
$313$	−25.4849	−1.44049	−0.720247	−	0.693718i	$-0.755971\pi$
$313$	−25.4849	−1.44049	−0.720247	+	0.693718i	$0.755971\pi$
$314$	9.25525	0.522304
$315$	0	0
$316$	47.5917	2.67724
$317$	−5.32609	−0.299143	−0.149571	−	0.988751i	$-0.547789\pi$
$317$	−5.32609	−0.299143	−0.149571	+	0.988751i	$0.547789\pi$
$318$	0	0
$319$	5.54870	0.310668
$320$	−3.51202	−0.196328
$321$	0	0
$322$	1.17744	0.0656162
$323$	1.42504	0.0792916
$324$	0	0
$325$	−18.9872	−1.05322
$326$	24.0441	1.33168
$327$	0	0
$328$	−8.68746	−0.479685
$329$	−3.51762	−0.193933
$330$	0	0
$331$	−17.6899	−0.972324	−0.486162	−	0.873869i	$-0.661603\pi$
$331$	−17.6899	−0.972324	−0.486162	+	0.873869i	$0.661603\pi$
$332$	−26.0451	−1.42941
$333$	0	0
$334$	54.6370	2.98960
$335$	−0.577167	−0.0315340
$336$	0	0
$337$	−25.5766	−1.39325	−0.696623	−	0.717437i	$-0.745315\pi$
$337$	−25.5766	−1.39325	−0.696623	+	0.717437i	$0.745315\pi$
$338$	−4.15481	−0.225992
$339$	0	0
$340$	1.01364	0.0549723
$341$	−5.68413	−0.307813
$342$	0	0
$343$	7.18949	0.388196
$344$	−6.57249	−0.354365
$345$	0	0
$346$	6.85587	0.368574
$347$	−15.8634	−0.851594	−0.425797	−	0.904819i	$-0.640006\pi$
$347$	−15.8634	−0.851594	−0.425797	+	0.904819i	$0.640006\pi$
$348$	0	0
$349$	−5.23461	−0.280202	−0.140101	−	0.990137i	$-0.544743\pi$
$349$	−5.23461	−0.280202	−0.140101	+	0.990137i	$0.544743\pi$
$350$	−5.80178	−0.310118
$351$	0	0
$352$	−36.0621	−1.92211
$353$	−10.2100	−0.543424	−0.271712	−	0.962379i	$-0.587590\pi$
$353$	−10.2100	−0.543424	−0.271712	+	0.962379i	$0.587590\pi$
$354$	0	0
$355$	−1.26682	−0.0672357
$356$	−37.2855	−1.97613
$357$	0	0
$358$	−39.8883	−2.10816
$359$	29.7321	1.56920	0.784599	−	0.620003i	$-0.212869\pi$
$359$	29.7321	1.56920	0.784599	+	0.620003i	$0.212869\pi$
$360$	0	0
$361$	−17.6635	−0.929657
$362$	43.8155	2.30289
$363$	0	0
$364$	−6.16208	−0.322981
$365$	3.19306	0.167132
$366$	0	0
$367$	16.1172	0.841310	0.420655	−	0.907221i	$-0.361800\pi$
$367$	16.1172	0.841310	0.420655	+	0.907221i	$0.361800\pi$
$368$	−0.785308	−0.0409370
$369$	0	0
$370$	1.95015	0.101384
$371$	−1.35428	−0.0703105
$372$	0	0
$373$	26.1982	1.35649	0.678247	−	0.734834i	$-0.262740\pi$
$373$	26.1982	1.35649	0.678247	+	0.734834i	$0.262740\pi$
$374$	15.3747	0.795006
$375$	0	0
$376$	−15.8955	−0.819746
$377$	−3.85335	−0.198458
$378$	0	0
$379$	25.7346	1.32190	0.660948	−	0.750432i	$-0.270155\pi$
$379$	25.7346	1.32190	0.660948	+	0.750432i	$0.270155\pi$
$380$	0.950670	0.0487683
$381$	0	0
$382$	−31.5680	−1.61516
$383$	−16.3203	−0.833926	−0.416963	−	0.908923i	$-0.636906\pi$
$383$	−16.3203	−0.833926	−0.416963	+	0.908923i	$0.636906\pi$
$384$	0	0
$385$	0.782847	0.0398976
$386$	−46.5046	−2.36702
$387$	0	0
$388$	9.17071	0.465572
$389$	−23.8266	−1.20806	−0.604029	−	0.796962i	$-0.706439\pi$
$389$	−23.8266	−1.20806	−0.604029	+	0.796962i	$0.706439\pi$
$390$	0	0
$391$	1.23265	0.0623380
$392$	15.9193	0.804045
$393$	0	0
$394$	2.97831	0.150045
$395$	4.19883	0.211266
$396$	0	0
$397$	9.78710	0.491201	0.245600	−	0.969371i	$-0.421015\pi$
$397$	9.78710	0.491201	0.245600	+	0.969371i	$0.421015\pi$
$398$	−32.8351	−1.64587
$399$	0	0
$400$	3.86956	0.193478
$401$	−8.22037	−0.410506	−0.205253	−	0.978709i	$-0.565802\pi$
$401$	−8.22037	−0.410506	−0.205253	+	0.978709i	$0.565802\pi$
$402$	0	0
$403$	3.94740	0.196634
$404$	−20.0726	−0.998649
$405$	0	0
$406$	−1.17744	−0.0584354
$407$	17.8717	0.885870
$408$	0	0
$409$	12.8235	0.634083	0.317042	−	0.948412i	$-0.397311\pi$
$409$	12.8235	0.634083	0.317042	+	0.948412i	$0.397311\pi$
$410$	−2.22226	−0.109750
$411$	0	0
$412$	52.3489	2.57905
$413$	−4.90398	−0.241309
$414$	0	0
$415$	−2.29785	−0.112797
$416$	25.0437	1.22787
$417$	0	0
$418$	14.4196	0.705284
$419$	31.8570	1.55631	0.778157	−	0.628069i	$-0.216155\pi$
$419$	31.8570	1.55631	0.778157	+	0.628069i	$0.216155\pi$
$420$	0	0
$421$	−13.4777	−0.656865	−0.328433	−	0.944527i	$-0.606520\pi$
$421$	−13.4777	−0.656865	−0.328433	+	0.944527i	$0.606520\pi$
$422$	37.8594	1.84297
$423$	0	0
$424$	−6.11972	−0.297200
$425$	−6.07384	−0.294624
$426$	0	0
$427$	6.32556	0.306115
$428$	−37.1461	−1.79552
$429$	0	0
$430$	−1.68125	−0.0810772
$431$	−9.35831	−0.450774	−0.225387	−	0.974269i	$-0.572365\pi$
$431$	−9.35831	−0.450774	−0.225387	+	0.974269i	$0.572365\pi$
$432$	0	0
$433$	−0.463834	−0.0222904	−0.0111452	−	0.999938i	$-0.503548\pi$
$433$	−0.463834	−0.0222904	−0.0111452	+	0.999938i	$0.503548\pi$
$434$	1.20618	0.0578984
$435$	0	0
$436$	9.68055	0.463614
$437$	1.15608	0.0553027
$438$	0	0
$439$	27.8294	1.32823	0.664113	−	0.747632i	$-0.268809\pi$
$439$	27.8294	1.32823	0.664113	+	0.747632i	$0.268809\pi$
$440$	3.53754	0.168645
$441$	0	0
$442$	−10.6771	−0.507858
$443$	−23.7520	−1.12849	−0.564247	−	0.825606i	$-0.690833\pi$
$443$	−23.7520	−1.12849	−0.564247	+	0.825606i	$0.690833\pi$
$444$	0	0
$445$	−3.28955	−0.155940
$446$	−11.4322	−0.541329
$447$	0	0
$448$	6.82972	0.322674
$449$	−2.07087	−0.0977304	−0.0488652	−	0.998805i	$-0.515560\pi$
$449$	−2.07087	−0.0977304	−0.0488652	+	0.998805i	$0.515560\pi$
$450$	0	0
$451$	−20.3655	−0.958972
$452$	38.7293	1.82167
$453$	0	0
$454$	−5.33776	−0.250514
$455$	−0.543656	−0.0254870
$456$	0	0
$457$	−18.4284	−0.862044	−0.431022	−	0.902341i	$-0.641847\pi$
$457$	−18.4284	−0.862044	−0.431022	+	0.902341i	$0.641847\pi$
$458$	−24.7869	−1.15822
$459$	0	0
$460$	0.822323	0.0383410
$461$	−21.6633	−1.00896	−0.504480	−	0.863423i	$-0.668316\pi$
$461$	−21.6633	−1.00896	−0.504480	+	0.863423i	$0.668316\pi$
$462$	0	0
$463$	−29.4821	−1.37015	−0.685076	−	0.728472i	$-0.740231\pi$
$463$	−29.4821	−1.37015	−0.685076	+	0.728472i	$0.740231\pi$
$464$	0.785308	0.0364570
$465$	0	0
$466$	−28.7482	−1.33174
$467$	26.6296	1.23227	0.616135	−	0.787641i	$-0.288698\pi$
$467$	26.6296	1.23227	0.616135	+	0.787641i	$0.288698\pi$
$468$	0	0
$469$	1.12240	0.0518276
$470$	−4.06608	−0.187554
$471$	0	0
$472$	−22.1602	−1.02000
$473$	−15.4075	−0.708437
$474$	0	0
$475$	−5.69652	−0.261374
$476$	−1.97119	−0.0903496
$477$	0	0
$478$	15.9686	0.730386
$479$	−5.87857	−0.268599	−0.134299	−	0.990941i	$-0.542878\pi$
$479$	−5.87857	−0.268599	−0.134299	+	0.990941i	$0.542878\pi$
$480$	0	0
$481$	−12.4112	−0.565903
$482$	−5.33719	−0.243102
$483$	0	0
$484$	60.4125	2.74602
$485$	0.809095	0.0367391
$486$	0	0
$487$	−21.3836	−0.968982	−0.484491	−	0.874796i	$-0.660995\pi$
$487$	−21.3836	−0.968982	−0.484491	+	0.874796i	$0.660995\pi$
$488$	28.5840	1.29394
$489$	0	0
$490$	4.07217	0.183962
$491$	−11.1612	−0.503698	−0.251849	−	0.967767i	$-0.581039\pi$
$491$	−11.1612	−0.503698	−0.251849	+	0.967767i	$0.581039\pi$
$492$	0	0
$493$	−1.23265	−0.0555159
$494$	−10.0138	−0.450543
$495$	0	0
$496$	−0.804475	−0.0361220
$497$	2.46354	0.110505
$498$	0	0
$499$	−14.8871	−0.666437	−0.333218	−	0.942850i	$-0.608135\pi$
$499$	−14.8871	−0.666437	−0.333218	+	0.942850i	$0.608135\pi$
$500$	−8.16357	−0.365086
$501$	0	0
$502$	−42.2842	−1.88724
$503$	−20.3724	−0.908362	−0.454181	−	0.890909i	$-0.650068\pi$
$503$	−20.3724	−0.908362	−0.454181	+	0.890909i	$0.650068\pi$
$504$	0	0
$505$	−1.77093	−0.0788051
$506$	12.4728	0.554485
$507$	0	0
$508$	−35.4299	−1.57195
$509$	5.34563	0.236941	0.118470	−	0.992958i	$-0.462201\pi$
$509$	5.34563	0.236941	0.118470	+	0.992958i	$0.462201\pi$
$510$	0	0
$511$	−6.20945	−0.274690
$512$	−8.82144	−0.389856
$513$	0	0
$514$	45.0410	1.98667
$515$	4.61854	0.203517
$516$	0	0
$517$	−37.2627	−1.63881
$518$	−3.79240	−0.166629
$519$	0	0
$520$	−2.45668	−0.107732
$521$	−9.65635	−0.423052	−0.211526	−	0.977372i	$-0.567843\pi$
$521$	−9.65635	−0.423052	−0.211526	+	0.977372i	$0.567843\pi$
$522$	0	0
$523$	1.53071	0.0669333	0.0334667	−	0.999440i	$-0.489345\pi$
$523$	1.53071	0.0669333	0.0334667	+	0.999440i	$0.489345\pi$
$524$	−29.2920	−1.27963
$525$	0	0
$526$	−44.5436	−1.94219
$527$	1.26274	0.0550058
$528$	0	0
$529$	1.00000	0.0434783
$530$	−1.56543	−0.0679981
$531$	0	0
$532$	−1.84874	−0.0801530
$533$	14.1430	0.612601
$534$	0	0
$535$	−3.27725	−0.141688
$536$	5.07192	0.219073
$537$	0	0
$538$	47.2848	2.03859
$539$	37.3185	1.60742
$540$	0	0
$541$	30.5325	1.31269	0.656347	−	0.754459i	$-0.272101\pi$
$541$	30.5325	1.31269	0.656347	+	0.754459i	$0.272101\pi$
$542$	−42.8503	−1.84058
$543$	0	0
$544$	8.01125	0.343480
$545$	0.854076	0.0365846
$546$	0	0
$547$	−38.6838	−1.65400	−0.826999	−	0.562203i	$-0.809954\pi$
$547$	−38.6838	−1.65400	−0.826999	+	0.562203i	$0.809954\pi$
$548$	1.31181	0.0560378
$549$	0	0
$550$	−61.4592	−2.62063
$551$	−1.15608	−0.0492506
$552$	0	0
$553$	−8.16534	−0.347226
$554$	−9.90678	−0.420899
$555$	0	0
$556$	−31.8402	−1.35033
$557$	−33.2707	−1.40973	−0.704863	−	0.709344i	$-0.748992\pi$
$557$	−33.2707	−1.40973	−0.704863	+	0.709344i	$0.748992\pi$
$558$	0	0
$559$	10.6999	0.452557
$560$	0.110796	0.00468200
$561$	0	0
$562$	44.2900	1.86826
$563$	28.0392	1.18171	0.590855	−	0.806778i	$-0.298790\pi$
$563$	28.0392	1.18171	0.590855	+	0.806778i	$0.298790\pi$
$564$	0	0
$565$	3.41693	0.143751
$566$	−48.3501	−2.03231
$567$	0	0
$568$	11.1323	0.467100
$569$	−12.5671	−0.526841	−0.263421	−	0.964681i	$-0.584851\pi$
$569$	−12.5671	−0.526841	−0.263421	+	0.964681i	$0.584851\pi$
$570$	0	0
$571$	−29.1558	−1.22013	−0.610065	−	0.792351i	$-0.708857\pi$
$571$	−29.1558	−1.22013	−0.610065	+	0.792351i	$0.708857\pi$
$572$	−65.2759	−2.72932
$573$	0	0
$574$	4.32157	0.180379
$575$	−4.92745	−0.205489
$576$	0	0
$577$	6.47178	0.269424	0.134712	−	0.990885i	$-0.456989\pi$
$577$	6.47178	0.269424	0.134712	+	0.990885i	$0.456989\pi$
$578$	34.7985	1.44743
$579$	0	0
$580$	−0.822323	−0.0341451
$581$	4.46857	0.185388
$582$	0	0
$583$	−14.3461	−0.594153
$584$	−28.0593	−1.16110
$585$	0	0
$586$	16.4233	0.678441
$587$	−3.91626	−0.161641	−0.0808207	−	0.996729i	$-0.525754\pi$
$587$	−3.91626	−0.161641	−0.0808207	+	0.996729i	$0.525754\pi$
$588$	0	0
$589$	1.18430	0.0487980
$590$	−5.66860	−0.233372
$591$	0	0
$592$	2.52939	0.103957
$593$	−35.0507	−1.43936	−0.719680	−	0.694306i	$-0.755711\pi$
$593$	−35.0507	−1.43936	−0.719680	+	0.694306i	$0.755711\pi$
$594$	0	0
$595$	−0.173911	−0.00712964
$596$	−39.6491	−1.62409
$597$	0	0
$598$	−8.66188	−0.354211
$599$	11.1914	0.457267	0.228633	−	0.973513i	$-0.426574\pi$
$599$	11.1914	0.457267	0.228633	+	0.973513i	$0.426574\pi$
$600$	0	0
$601$	−39.8237	−1.62444	−0.812221	−	0.583350i	$-0.801742\pi$
$601$	−39.8237	−1.62444	−0.812221	+	0.583350i	$0.801742\pi$
$602$	3.26948	0.133254
$603$	0	0
$604$	−39.1986	−1.59497
$605$	5.32996	0.216694
$606$	0	0
$607$	−1.18071	−0.0479235	−0.0239618	−	0.999713i	$-0.507628\pi$
$607$	−1.18071	−0.0479235	−0.0239618	+	0.999713i	$0.507628\pi$
$608$	7.51357	0.304716
$609$	0	0
$610$	7.31183	0.296047
$611$	25.8775	1.04689
$612$	0	0
$613$	−21.1441	−0.854001	−0.427000	−	0.904251i	$-0.640430\pi$
$613$	−21.1441	−0.854001	−0.427000	+	0.904251i	$0.640430\pi$
$614$	67.4435	2.72180
$615$	0	0
$616$	−6.87935	−0.277177
$617$	−5.73209	−0.230765	−0.115383	−	0.993321i	$-0.536809\pi$
$617$	−5.73209	−0.230765	−0.115383	+	0.993321i	$0.536809\pi$
$618$	0	0
$619$	−22.8935	−0.920169	−0.460084	−	0.887875i	$-0.652181\pi$
$619$	−22.8935	−0.920169	−0.460084	+	0.887875i	$0.652181\pi$
$620$	0.842393	0.0338313
$621$	0	0
$622$	−34.5460	−1.38517
$623$	6.39710	0.256294
$624$	0	0
$625$	23.9170	0.956680
$626$	57.2871	2.28965
$627$	0	0
$628$	−12.5701	−0.501600
$629$	−3.97024	−0.158304
$630$	0	0
$631$	−10.5614	−0.420444	−0.210222	−	0.977654i	$-0.567419\pi$
$631$	−10.5614	−0.420444	−0.210222	+	0.977654i	$0.567419\pi$
$632$	−36.8977	−1.46771
$633$	0	0
$634$	11.9724	0.475485
$635$	−3.12584	−0.124045
$636$	0	0
$637$	−25.9162	−1.02684
$638$	−12.4728	−0.493804
$639$	0	0
$640$	4.39347	0.173667
$641$	−6.79362	−0.268332	−0.134166	−	0.990959i	$-0.542836\pi$
$641$	−6.79362	−0.268332	−0.134166	+	0.990959i	$0.542836\pi$
$642$	0	0
$643$	−18.1869	−0.717221	−0.358610	−	0.933487i	$-0.616749\pi$
$643$	−18.1869	−0.717221	−0.358610	+	0.933487i	$0.616749\pi$
$644$	−1.59915	−0.0630152
$645$	0	0
$646$	−3.20333	−0.126033
$647$	−5.56073	−0.218615	−0.109307	−	0.994008i	$-0.534863\pi$
$647$	−5.56073	−0.218615	−0.109307	+	0.994008i	$0.534863\pi$
$648$	0	0
$649$	−51.9486	−2.03916
$650$	42.6810	1.67409
$651$	0	0
$652$	−32.6556	−1.27889
$653$	8.03210	0.314320	0.157160	−	0.987573i	$-0.449766\pi$
$653$	8.03210	0.314320	0.157160	+	0.987573i	$0.449766\pi$
$654$	0	0
$655$	−2.58432	−0.100978
$656$	−2.88232	−0.112536
$657$	0	0
$658$	7.90719	0.308254
$659$	−22.4860	−0.875929	−0.437965	−	0.898992i	$-0.644300\pi$
$659$	−22.4860	−0.875929	−0.437965	+	0.898992i	$0.644300\pi$
$660$	0	0
$661$	36.6866	1.42694	0.713471	−	0.700685i	$-0.247122\pi$
$661$	36.6866	1.42694	0.713471	+	0.700685i	$0.247122\pi$
$662$	39.7648	1.54550
$663$	0	0
$664$	20.1926	0.783626
$665$	−0.163107	−0.00632502
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	−74.2055	−2.87110
$669$	0	0
$670$	1.29740	0.0501231
$671$	67.0077	2.58680
$672$	0	0
$673$	−38.4756	−1.48313	−0.741563	−	0.670884i	$-0.765915\pi$
$673$	−38.4756	−1.48313	−0.741563	+	0.670884i	$0.765915\pi$
$674$	57.4932	2.21456
$675$	0	0
$676$	5.64287	0.217034
$677$	33.2780	1.27898	0.639488	−	0.768801i	$-0.279146\pi$
$677$	33.2780	1.27898	0.639488	+	0.768801i	$0.279146\pi$
$678$	0	0
$679$	−1.57342	−0.0603825
$680$	−0.785870	−0.0301367
$681$	0	0
$682$	12.7773	0.489266
$683$	−31.0886	−1.18957	−0.594786	−	0.803884i	$-0.702763\pi$
$683$	−31.0886	−1.18957	−0.594786	+	0.803884i	$0.702763\pi$
$684$	0	0
$685$	0.115736	0.00442204
$686$	−16.1611	−0.617034
$687$	0	0
$688$	−2.18062	−0.0831354
$689$	9.96277	0.379551
$690$	0	0
$691$	24.8234	0.944327	0.472163	−	0.881511i	$-0.343473\pi$
$691$	24.8234	0.944327	0.472163	+	0.881511i	$0.343473\pi$
$692$	−9.31133	−0.353964
$693$	0	0
$694$	35.6591	1.35360
$695$	−2.80913	−0.106557
$696$	0	0
$697$	4.52422	0.171367
$698$	11.7668	0.445380
$699$	0	0
$700$	7.87972	0.297825
$701$	−3.01635	−0.113926	−0.0569630	−	0.998376i	$-0.518142\pi$
$701$	−3.01635	−0.113926	−0.0569630	+	0.998376i	$0.518142\pi$
$702$	0	0
$703$	−3.72360	−0.140438
$704$	72.3484	2.72673
$705$	0	0
$706$	22.9509	0.863768
$707$	3.44387	0.129520
$708$	0	0
$709$	−12.6925	−0.476678	−0.238339	−	0.971182i	$-0.576603\pi$
$709$	−12.6925	−0.476678	−0.238339	+	0.971182i	$0.576603\pi$
$710$	2.84765	0.106871
$711$	0	0
$712$	28.9073	1.08335
$713$	1.02441	0.0383644
$714$	0	0
$715$	−5.75903	−0.215376
$716$	54.1744	2.02459
$717$	0	0
$718$	−66.8342	−2.49423
$719$	2.39678	0.0893850	0.0446925	−	0.999001i	$-0.485769\pi$
$719$	2.39678	0.0893850	0.0446925	+	0.999001i	$0.485769\pi$
$720$	0	0
$721$	−8.98154	−0.334490
$722$	39.7054	1.47768
$723$	0	0
$724$	−59.5082	−2.21160
$725$	4.92745	0.183001
$726$	0	0
$727$	−8.07624	−0.299531	−0.149766	−	0.988722i	$-0.547852\pi$
$727$	−8.07624	−0.299531	−0.149766	+	0.988722i	$0.547852\pi$
$728$	4.77743	0.177063
$729$	0	0
$730$	−7.17762	−0.265655
$731$	3.42280	0.126597
$732$	0	0
$733$	−0.860098	−0.0317684	−0.0158842	−	0.999874i	$-0.505056\pi$
$733$	−0.860098	−0.0317684	−0.0158842	+	0.999874i	$0.505056\pi$
$734$	−36.2295	−1.33726
$735$	0	0
$736$	6.49919	0.239563
$737$	11.8898	0.437965
$738$	0	0
$739$	−42.8187	−1.57511	−0.787556	−	0.616243i	$-0.788654\pi$
$739$	−42.8187	−1.57511	−0.787556	+	0.616243i	$0.788654\pi$
$740$	−2.64861	−0.0973648
$741$	0	0
$742$	3.04425	0.111758
$743$	−5.30074	−0.194465	−0.0972327	−	0.995262i	$-0.530999\pi$
$743$	−5.30074	−0.194465	−0.0972327	+	0.995262i	$0.530999\pi$
$744$	0	0
$745$	−3.49808	−0.128160
$746$	−58.8905	−2.15614
$747$	0	0
$748$	−20.8812	−0.763492
$749$	6.37317	0.232871
$750$	0	0
$751$	−44.6836	−1.63053	−0.815264	−	0.579090i	$-0.803408\pi$
$751$	−44.6836	−1.63053	−0.815264	+	0.579090i	$0.803408\pi$
$752$	−5.27379	−0.192315
$753$	0	0
$754$	8.66188	0.315447
$755$	−3.45833	−0.125862
$756$	0	0
$757$	0.442932	0.0160986	0.00804931	−	0.999968i	$-0.497438\pi$
$757$	0.442932	0.0160986	0.00804931	+	0.999968i	$0.497438\pi$
$758$	−57.8482	−2.10114
$759$	0	0
$760$	−0.737050	−0.0267356
$761$	3.31916	0.120320	0.0601598	−	0.998189i	$-0.480839\pi$
$761$	3.31916	0.120320	0.0601598	+	0.998189i	$0.480839\pi$
$762$	0	0
$763$	−1.66090	−0.0601285
$764$	42.8743	1.55114
$765$	0	0
$766$	36.6860	1.32552
$767$	36.0762	1.30264
$768$	0	0
$769$	−29.7530	−1.07292	−0.536460	−	0.843926i	$-0.680239\pi$
$769$	−29.7530	−1.07292	−0.536460	+	0.843926i	$0.680239\pi$
$770$	−1.75975	−0.0634169
$771$	0	0
$772$	63.1605	2.27320
$773$	−2.24387	−0.0807065	−0.0403532	−	0.999185i	$-0.512848\pi$
$773$	−2.24387	−0.0807065	−0.0403532	+	0.999185i	$0.512848\pi$
$774$	0	0
$775$	−5.04771	−0.181319
$776$	−7.11001	−0.255234
$777$	0	0
$778$	53.5595	1.92020
$779$	4.24317	0.152027
$780$	0	0
$781$	26.0967	0.933814
$782$	−2.77086	−0.0990857
$783$	0	0
$784$	5.28169	0.188632
$785$	−1.10901	−0.0395822
$786$	0	0
$787$	−7.21639	−0.257236	−0.128618	−	0.991694i	$-0.541054\pi$
$787$	−7.21639	−0.257236	−0.128618	+	0.991694i	$0.541054\pi$
$788$	−4.04501	−0.144098
$789$	0	0
$790$	−9.43847	−0.335806
$791$	−6.64481	−0.236262
$792$	0	0
$793$	−46.5341	−1.65248
$794$	−22.0002	−0.780759
$795$	0	0
$796$	44.5951	1.58063
$797$	−35.4558	−1.25591	−0.627954	−	0.778251i	$-0.716107\pi$
$797$	−35.4558	−1.25591	−0.627954	+	0.778251i	$0.716107\pi$
$798$	0	0
$799$	8.27798	0.292854
$800$	−32.0244	−1.13223
$801$	0	0
$802$	18.4784	0.652495
$803$	−65.7777	−2.32124
$804$	0	0
$805$	−0.141086	−0.00497264
$806$	−8.87329	−0.312548
$807$	0	0
$808$	15.5622	0.547476
$809$	−24.9091	−0.875757	−0.437879	−	0.899034i	$-0.644270\pi$
$809$	−24.9091	−0.875757	−0.437879	+	0.899034i	$0.644270\pi$
$810$	0	0
$811$	42.7906	1.50258	0.751290	−	0.659972i	$-0.229432\pi$
$811$	42.7906	1.50258	0.751290	+	0.659972i	$0.229432\pi$
$812$	1.59915	0.0561191
$813$	0	0
$814$	−40.1736	−1.40808
$815$	−2.88107	−0.100920
$816$	0	0
$817$	3.21017	0.112310
$818$	−28.8258	−1.00787
$819$	0	0
$820$	3.01818	0.105399
$821$	−24.8159	−0.866081	−0.433041	−	0.901374i	$-0.642559\pi$
$821$	−24.8159	−0.866081	−0.433041	+	0.901374i	$0.642559\pi$
$822$	0	0
$823$	−44.6078	−1.55493	−0.777466	−	0.628925i	$-0.783495\pi$
$823$	−44.6078	−1.55493	−0.777466	+	0.628925i	$0.783495\pi$
$824$	−40.5859	−1.41388
$825$	0	0
$826$	11.0236	0.383559
$827$	7.93280	0.275850	0.137925	−	0.990443i	$-0.455957\pi$
$827$	7.93280	0.275850	0.137925	+	0.990443i	$0.455957\pi$
$828$	0	0
$829$	−29.9622	−1.04063	−0.520315	−	0.853974i	$-0.674185\pi$
$829$	−29.9622	−1.04063	−0.520315	+	0.853974i	$0.674185\pi$
$830$	5.16530	0.179290
$831$	0	0
$832$	−50.2431	−1.74186
$833$	−8.29038	−0.287245
$834$	0	0
$835$	−6.54685	−0.226563
$836$	−19.5840	−0.677327
$837$	0	0
$838$	−71.6107	−2.47375
$839$	40.7880	1.40816	0.704079	−	0.710122i	$-0.251360\pi$
$839$	40.7880	1.40816	0.704079	+	0.710122i	$0.251360\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	30.2964	1.04408
$843$	0	0
$844$	−51.4189	−1.76991
$845$	0.497848	0.0171265
$846$	0	0
$847$	−10.3650	−0.356146
$848$	−2.03040	−0.0697242
$849$	0	0
$850$	13.6533	0.468303
$851$	−3.22089	−0.110411
$852$	0	0
$853$	38.3184	1.31200	0.655999	−	0.754762i	$-0.272248\pi$
$853$	38.3184	1.31200	0.655999	+	0.754762i	$0.272248\pi$
$854$	−14.2191	−0.486568
$855$	0	0
$856$	28.7992	0.984335
$857$	−3.97430	−0.135760	−0.0678798	−	0.997694i	$-0.521623\pi$
$857$	−3.97430	−0.135760	−0.0678798	+	0.997694i	$0.521623\pi$
$858$	0	0
$859$	−3.76808	−0.128565	−0.0642826	−	0.997932i	$-0.520476\pi$
$859$	−3.76808	−0.128565	−0.0642826	+	0.997932i	$0.520476\pi$
$860$	2.28340	0.0778634
$861$	0	0
$862$	21.0364	0.716502
$863$	−17.9279	−0.610274	−0.305137	−	0.952309i	$-0.598702\pi$
$863$	−17.9279	−0.610274	−0.305137	+	0.952309i	$0.598702\pi$
$864$	0	0
$865$	−0.821502	−0.0279319
$866$	1.04264	0.0354304
$867$	0	0
$868$	−1.63818	−0.0556034
$869$	−86.4968	−2.93420
$870$	0	0
$871$	−8.25697	−0.279777
$872$	−7.50529	−0.254161
$873$	0	0
$874$	−2.59873	−0.0879033
$875$	1.40063	0.0473499
$876$	0	0
$877$	−20.5426	−0.693675	−0.346838	−	0.937925i	$-0.612745\pi$
$877$	−20.5426	−0.693675	−0.346838	+	0.937925i	$0.612745\pi$
$878$	−62.5572	−2.11120
$879$	0	0
$880$	1.17368	0.0395649
$881$	−20.4710	−0.689685	−0.344843	−	0.938661i	$-0.612068\pi$
$881$	−20.4710	−0.689685	−0.344843	+	0.938661i	$0.612068\pi$
$882$	0	0
$883$	−39.4560	−1.32780	−0.663901	−	0.747821i	$-0.731100\pi$
$883$	−39.4560	−1.32780	−0.663901	+	0.747821i	$0.731100\pi$
$884$	14.5012	0.487727
$885$	0	0
$886$	53.3918	1.79373
$887$	1.87545	0.0629713	0.0314857	−	0.999504i	$-0.489976\pi$
$887$	1.87545	0.0629713	0.0314857	+	0.999504i	$0.489976\pi$
$888$	0	0
$889$	6.07874	0.203874
$890$	7.39453	0.247865
$891$	0	0
$892$	15.5267	0.519871
$893$	7.76373	0.259803
$894$	0	0
$895$	4.77960	0.159764
$896$	−8.54385	−0.285430
$897$	0	0
$898$	4.65507	0.155342
$899$	−1.02441	−0.0341659
$900$	0	0
$901$	3.18700	0.106174
$902$	45.7791	1.52428
$903$	0	0
$904$	−30.0266	−0.998671
$905$	−5.25017	−0.174522
$906$	0	0
$907$	−29.4070	−0.976443	−0.488222	−	0.872720i	$-0.662354\pi$
$907$	−29.4070	−0.976443	−0.488222	+	0.872720i	$0.662354\pi$
$908$	7.24951	0.240583
$909$	0	0
$910$	1.22207	0.0405113
$911$	−35.4638	−1.17497	−0.587483	−	0.809236i	$-0.699881\pi$
$911$	−35.4638	−1.17497	−0.587483	+	0.809236i	$0.699881\pi$
$912$	0	0
$913$	47.3363	1.56660
$914$	41.4248	1.37021
$915$	0	0
$916$	33.6645	1.11231
$917$	5.02565	0.165962
$918$	0	0
$919$	28.5662	0.942312	0.471156	−	0.882050i	$-0.343837\pi$
$919$	28.5662	0.942312	0.471156	+	0.882050i	$0.343837\pi$
$920$	−0.637543	−0.0210192
$921$	0	0
$922$	48.6965	1.60374
$923$	−18.1231	−0.596530
$924$	0	0
$925$	15.8708	0.521827
$926$	66.2724	2.17785
$927$	0	0
$928$	−6.49919	−0.213346
$929$	5.87915	0.192889	0.0964444	−	0.995338i	$-0.469253\pi$
$929$	5.87915	0.192889	0.0964444	+	0.995338i	$0.469253\pi$
$930$	0	0
$931$	−7.77536	−0.254827
$932$	39.0445	1.27895
$933$	0	0
$934$	−59.8601	−1.95868
$935$	−1.84226	−0.0602485
$936$	0	0
$937$	−21.0807	−0.688676	−0.344338	−	0.938846i	$-0.611897\pi$
$937$	−21.0807	−0.688676	−0.344338	+	0.938846i	$0.611897\pi$
$938$	−2.52302	−0.0823796
$939$	0	0
$940$	5.52237	0.180120
$941$	−31.1274	−1.01473	−0.507363	−	0.861733i	$-0.669380\pi$
$941$	−31.1274	−1.01473	−0.507363	+	0.861733i	$0.669380\pi$
$942$	0	0
$943$	3.67031	0.119522
$944$	−7.35229	−0.239297
$945$	0	0
$946$	34.6342	1.12605
$947$	24.5911	0.799103	0.399551	−	0.916711i	$-0.369166\pi$
$947$	24.5911	0.799103	0.399551	+	0.916711i	$0.369166\pi$
$948$	0	0
$949$	45.6800	1.48283
$950$	12.8051	0.415452
$951$	0	0
$952$	1.52826	0.0495311
$953$	10.9892	0.355975	0.177987	−	0.984033i	$-0.443041\pi$
$953$	10.9892	0.355975	0.177987	+	0.984033i	$0.443041\pi$
$954$	0	0
$955$	3.78263	0.122403
$956$	−21.6878	−0.701434
$957$	0	0
$958$	13.2143	0.426936
$959$	−0.225068	−0.00726783
$960$	0	0
$961$	−29.9506	−0.966148
$962$	27.8989	0.899498
$963$	0	0
$964$	7.24874	0.233466
$965$	5.57240	0.179382
$966$	0	0
$967$	−35.4663	−1.14052	−0.570260	−	0.821464i	$-0.693158\pi$
$967$	−35.4663	−1.14052	−0.570260	+	0.821464i	$0.693158\pi$
$968$	−46.8376	−1.50542
$969$	0	0
$970$	−1.81875	−0.0583965
$971$	−41.1371	−1.32015	−0.660076	−	0.751199i	$-0.729476\pi$
$971$	−41.1371	−1.32015	−0.660076	+	0.751199i	$0.729476\pi$
$972$	0	0
$973$	5.46284	0.175131
$974$	48.0677	1.54019
$975$	0	0
$976$	9.48360	0.303562
$977$	−33.2895	−1.06503	−0.532513	−	0.846422i	$-0.678752\pi$
$977$	−33.2895	−1.06503	−0.532513	+	0.846422i	$0.678752\pi$
$978$	0	0
$979$	67.7655	2.16580
$980$	−5.53064	−0.176670
$981$	0	0
$982$	25.0890	0.800624
$983$	47.5664	1.51713	0.758567	−	0.651595i	$-0.225900\pi$
$983$	47.5664	1.51713	0.758567	+	0.651595i	$0.225900\pi$
$984$	0	0
$985$	−0.356875	−0.0113710
$986$	2.77086	0.0882421
$987$	0	0
$988$	13.6003	0.432683
$989$	2.77677	0.0882962
$990$	0	0
$991$	−17.3992	−0.552705	−0.276352	−	0.961056i	$-0.589126\pi$
$991$	−17.3992	−0.552705	−0.276352	+	0.961056i	$0.589126\pi$
$992$	6.65782	0.211386
$993$	0	0
$994$	−5.53775	−0.175647
$995$	3.93445	0.124731
$996$	0	0
$997$	32.5771	1.03173	0.515863	−	0.856671i	$-0.327471\pi$
$997$	32.5771	1.03173	0.515863	+	0.856671i	$0.327471\pi$
$998$	33.4644	1.05930
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.o.1.3		13
3.2	odd	2		667.2.a.c.1.11	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
667.2.a.c.1.11	✓	13	3.2	odd	2
6003.2.a.o.1.3		13	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:	\( N \)	\(=\)	\( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6003.a (trivial)

\(f(q)\)	\(=\)	\(q-2.24788 q^{2} +3.05297 q^{4} +0.269352 q^{5} -0.523800 q^{7} -2.36696 q^{8} +O(q^{10})\)	\(q-2.24788 q^{2} +3.05297 q^{4} +0.269352 q^{5} -0.523800 q^{7} -2.36696 q^{8} -0.605470 q^{10} -5.54870 q^{11} +3.85335 q^{13} +1.17744 q^{14} -0.785308 q^{16} +1.23265 q^{17} +1.15608 q^{19} +0.822323 q^{20} +12.4728 q^{22} +1.00000 q^{23} -4.92745 q^{25} -8.66188 q^{26} -1.59915 q^{28} -1.00000 q^{29} +1.02441 q^{31} +6.49919 q^{32} -2.77086 q^{34} -0.141086 q^{35} -3.22089 q^{37} -2.59873 q^{38} -0.637543 q^{40} +3.67031 q^{41} +2.77677 q^{43} -16.9400 q^{44} -2.24788 q^{46} +6.71558 q^{47} -6.72563 q^{49} +11.0763 q^{50} +11.7642 q^{52} +2.58548 q^{53} -1.49455 q^{55} +1.23981 q^{56} +2.24788 q^{58} +9.36230 q^{59} -12.0763 q^{61} -2.30275 q^{62} -13.0388 q^{64} +1.03791 q^{65} -2.14280 q^{67} +3.76326 q^{68} +0.317146 q^{70} -4.70321 q^{71} +11.8546 q^{73} +7.24017 q^{74} +3.52948 q^{76} +2.90641 q^{77} +15.5887 q^{79} -0.211524 q^{80} -8.25042 q^{82} -8.53106 q^{83} +0.332017 q^{85} -6.24185 q^{86} +13.1335 q^{88} -12.2129 q^{89} -2.01839 q^{91} +3.05297 q^{92} -15.0958 q^{94} +0.311392 q^{95} +3.00386 q^{97} +15.1184 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 4 q^{2} + 12 q^{4} - 16 q^{5} + q^{7} - 6 q^{8}+O(q^{10}) \) 13 * q - 4 * q^2 + 12 * q^4 - 16 * q^5 + q^7 - 6 * q^8	\( 13 q - 4 q^{2} + 12 q^{4} - 16 q^{5} + q^{7} - 6 q^{8} + 10 q^{10} - 10 q^{11} + 7 q^{13} + 12 q^{14} + 2 q^{16} - 26 q^{17} - 25 q^{20} - 15 q^{22} + 13 q^{23} + 19 q^{25} + 15 q^{26} + 5 q^{28} - 13 q^{29} - 6 q^{31} - 16 q^{32} + 11 q^{34} - q^{35} + 15 q^{37} - 8 q^{38} + 14 q^{40} - 9 q^{41} + q^{43} - 29 q^{44} - 4 q^{46} - 15 q^{47} + 4 q^{49} - 31 q^{50} - 8 q^{52} - 43 q^{53} - 3 q^{55} + 5 q^{56} + 4 q^{58} + 9 q^{59} + 20 q^{61} - 11 q^{62} - 16 q^{64} + 25 q^{65} + q^{67} - 21 q^{68} - 2 q^{70} - 17 q^{71} + 26 q^{73} - 11 q^{74} + 8 q^{76} - 17 q^{77} + 5 q^{79} - 10 q^{80} - 25 q^{82} - 4 q^{83} + 20 q^{85} + 13 q^{86} - 32 q^{88} - 48 q^{89} - 9 q^{91} + 12 q^{92} - 65 q^{94} - 8 q^{95} + 30 q^{97} - 2 q^{98}+O(q^{100}) \) 13 * q - 4 * q^2 + 12 * q^4 - 16 * q^5 + q^7 - 6 * q^8 + 10 * q^10 - 10 * q^11 + 7 * q^13 + 12 * q^14 + 2 * q^16 - 26 * q^17 - 25 * q^20 - 15 * q^22 + 13 * q^23 + 19 * q^25 + 15 * q^26 + 5 * q^28 - 13 * q^29 - 6 * q^31 - 16 * q^32 + 11 * q^34 - q^35 + 15 * q^37 - 8 * q^38 + 14 * q^40 - 9 * q^41 + q^43 - 29 * q^44 - 4 * q^46 - 15 * q^47 + 4 * q^49 - 31 * q^50 - 8 * q^52 - 43 * q^53 - 3 * q^55 + 5 * q^56 + 4 * q^58 + 9 * q^59 + 20 * q^61 - 11 * q^62 - 16 * q^64 + 25 * q^65 + q^67 - 21 * q^68 - 2 * q^70 - 17 * q^71 + 26 * q^73 - 11 * q^74 + 8 * q^76 - 17 * q^77 + 5 * q^79 - 10 * q^80 - 25 * q^82 - 4 * q^83 + 20 * q^85 + 13 * q^86 - 32 * q^88 - 48 * q^89 - 9 * q^91 + 12 * q^92 - 65 * q^94 - 8 * q^95 + 30 * q^97 - 2 * q^98

Introduction

L-functions

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