LMFDB - Embedded newform 2007.2.a.n.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2007,2,Mod(1,2007)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2007, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2007.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2007 = 3^{2} \cdot 223 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2007.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:	$16.0259756857$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 22 x^{12} - x^{11} + 187 x^{10} + 14 x^{9} - 774 x^{8} - 70 x^{7} + 1622 x^{6} + 168 x^{5} + \cdots - 11 $ x^14 - 22x^12 - x^11 + 187x^10 + 14x^9 - 774x^8 - 70x^7 + 1622x^6 + 168x^5 - 1617x^4 - 219x^3 + 604x^2 + 126*x - 11
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 669)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$1.51262$ of defining polynomial
Character	$\chi$	$=$	2007.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.51262 q^{2} +0.288027 q^{4} -3.35334 q^{5} -3.15808 q^{7} -2.58957 q^{8} +O(q^{10})$	\(q+1.51262 q^{2} +0.288027 q^{4} -3.35334 q^{5} -3.15808 q^{7} -2.58957 q^{8} -5.07234 q^{10} -2.95339 q^{11} +2.79394 q^{13} -4.77698 q^{14} -4.49310 q^{16} +7.55008 q^{17} +7.69789 q^{19} -0.965854 q^{20} -4.46736 q^{22} +1.53661 q^{23} +6.24488 q^{25} +4.22618 q^{26} -0.909613 q^{28} -9.10880 q^{29} +5.22466 q^{31} -1.61722 q^{32} +11.4204 q^{34} +10.5901 q^{35} -11.8289 q^{37} +11.6440 q^{38} +8.68370 q^{40} +5.38611 q^{41} +5.51397 q^{43} -0.850657 q^{44} +2.32432 q^{46} +3.00222 q^{47} +2.97344 q^{49} +9.44615 q^{50} +0.804732 q^{52} -4.85408 q^{53} +9.90371 q^{55} +8.17805 q^{56} -13.7782 q^{58} +1.39802 q^{59} +7.53074 q^{61} +7.90294 q^{62} +6.53995 q^{64} -9.36903 q^{65} +12.2116 q^{67} +2.17463 q^{68} +16.0188 q^{70} -5.82004 q^{71} +14.4246 q^{73} -17.8927 q^{74} +2.21720 q^{76} +9.32703 q^{77} +4.43909 q^{79} +15.0669 q^{80} +8.14715 q^{82} -9.68881 q^{83} -25.3180 q^{85} +8.34055 q^{86} +7.64800 q^{88} +15.4310 q^{89} -8.82348 q^{91} +0.442587 q^{92} +4.54122 q^{94} -25.8136 q^{95} +0.876292 q^{97} +4.49769 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 16 q^{4} - q^{5} + 14 q^{7} + 3 q^{8}+O(q^{10}) $ 14 * q + 16 * q^4 - q^5 + 14 * q^7 + 3 * q^8	$ 14 q + 16 q^{4} - q^{5} + 14 q^{7} + 3 q^{8} + 4 q^{10} - 13 q^{11} + 10 q^{13} + 12 q^{16} - q^{17} + 32 q^{19} + 10 q^{20} - 3 q^{22} + 7 q^{23} + 25 q^{25} + 3 q^{26} + 28 q^{28} + q^{29} + 18 q^{31} + 16 q^{32} + q^{34} + 12 q^{35} + 16 q^{37} + 31 q^{38} + q^{40} - 6 q^{41} + 15 q^{43} + 12 q^{44} + 6 q^{46} + 21 q^{47} + 28 q^{49} + 32 q^{50} - 7 q^{52} + 17 q^{53} - q^{55} + 33 q^{56} - 27 q^{58} - 12 q^{59} + 28 q^{61} + 16 q^{62} + 3 q^{64} + 4 q^{65} + 17 q^{67} - 5 q^{68} - 14 q^{70} + 3 q^{71} + 17 q^{73} + 16 q^{74} + 14 q^{76} + 14 q^{77} + 23 q^{79} + 36 q^{80} + 5 q^{82} - 18 q^{83} - 11 q^{85} - 9 q^{86} - 35 q^{88} - 10 q^{89} + 16 q^{91} + 45 q^{92} - 43 q^{94} + 20 q^{95} + 29 q^{97} + 19 q^{98}+O(q^{100}) $ 14 * q + 16 * q^4 - q^5 + 14 * q^7 + 3 * q^8 + 4 * q^10 - 13 * q^11 + 10 * q^13 + 12 * q^16 - q^17 + 32 * q^19 + 10 * q^20 - 3 * q^22 + 7 * q^23 + 25 * q^25 + 3 * q^26 + 28 * q^28 + q^29 + 18 * q^31 + 16 * q^32 + q^34 + 12 * q^35 + 16 * q^37 + 31 * q^38 + q^40 - 6 * q^41 + 15 * q^43 + 12 * q^44 + 6 * q^46 + 21 * q^47 + 28 * q^49 + 32 * q^50 - 7 * q^52 + 17 * q^53 - q^55 + 33 * q^56 - 27 * q^58 - 12 * q^59 + 28 * q^61 + 16 * q^62 + 3 * q^64 + 4 * q^65 + 17 * q^67 - 5 * q^68 - 14 * q^70 + 3 * q^71 + 17 * q^73 + 16 * q^74 + 14 * q^76 + 14 * q^77 + 23 * q^79 + 36 * q^80 + 5 * q^82 - 18 * q^83 - 11 * q^85 - 9 * q^86 - 35 * q^88 - 10 * q^89 + 16 * q^91 + 45 * q^92 - 43 * q^94 + 20 * q^95 + 29 * q^97 + 19 * 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 $

$2$	1.51262	1.06959	0.534793	−	0.844983i	$-0.320390\pi$
$2$	1.51262	1.06959	0.534793	+	0.844983i	$0.320390\pi$
$3$	0	0
$3$	0	0
$4$	0.288027	0.144014
$5$	−3.35334	−1.49966	−0.749829	−	0.661631i	$-0.769864\pi$
$5$	−3.35334	−1.49966	−0.749829	+	0.661631i	$0.769864\pi$
$6$	0	0
$7$	−3.15808	−1.19364	−0.596820	−	0.802375i	$-0.703569\pi$
$7$	−3.15808	−1.19364	−0.596820	+	0.802375i	$0.703569\pi$
$8$	−2.58957	−0.915551
$9$	0	0
$10$	−5.07234	−1.60401
$11$	−2.95339	−0.890480	−0.445240	−	0.895411i	$-0.646882\pi$
$11$	−2.95339	−0.890480	−0.445240	+	0.895411i	$0.646882\pi$
$12$	0	0
$13$	2.79394	0.774900	0.387450	−	0.921891i	$-0.373356\pi$
$13$	2.79394	0.774900	0.387450	+	0.921891i	$0.373356\pi$
$14$	−4.77698	−1.27670
$15$	0	0
$16$	−4.49310	−1.12327
$17$	7.55008	1.83116	0.915581	−	0.402133i	$-0.131731\pi$
$17$	7.55008	1.83116	0.915581	+	0.402133i	$0.131731\pi$
$18$	0	0
$19$	7.69789	1.76602	0.883008	−	0.469357i	$-0.155514\pi$
$19$	7.69789	1.76602	0.883008	+	0.469357i	$0.155514\pi$
$20$	−0.965854	−0.215971
$21$	0	0
$22$	−4.46736	−0.952445
$23$	1.53661	0.320406	0.160203	−	0.987084i	$-0.448785\pi$
$23$	1.53661	0.320406	0.160203	+	0.987084i	$0.448785\pi$
$24$	0	0
$25$	6.24488	1.24898
$26$	4.22618	0.828822
$27$	0	0
$28$	−0.909613	−0.171901
$29$	−9.10880	−1.69146	−0.845731	−	0.533610i	$-0.820835\pi$
$29$	−9.10880	−1.69146	−0.845731	+	0.533610i	$0.820835\pi$
$30$	0	0
$31$	5.22466	0.938377	0.469188	−	0.883098i	$-0.344547\pi$
$31$	5.22466	0.938377	0.469188	+	0.883098i	$0.344547\pi$
$32$	−1.61722	−0.285887
$33$	0	0
$34$	11.4204	1.95859
$35$	10.5901	1.79005
$36$	0	0
$37$	−11.8289	−1.94466	−0.972332	−	0.233603i	$-0.924948\pi$
$37$	−11.8289	−1.94466	−0.972332	+	0.233603i	$0.924948\pi$
$38$	11.6440	1.88891
$39$	0	0
$40$	8.68370	1.37301
$41$	5.38611	0.841169	0.420585	−	0.907253i	$-0.361825\pi$
$41$	5.38611	0.841169	0.420585	+	0.907253i	$0.361825\pi$
$42$	0	0
$43$	5.51397	0.840872	0.420436	−	0.907322i	$-0.361877\pi$
$43$	5.51397	0.840872	0.420436	+	0.907322i	$0.361877\pi$
$44$	−0.850657	−0.128241
$45$	0	0
$46$	2.32432	0.342702
$47$	3.00222	0.437918	0.218959	−	0.975734i	$-0.429734\pi$
$47$	3.00222	0.437918	0.218959	+	0.975734i	$0.429734\pi$
$48$	0	0
$49$	2.97344	0.424777
$50$	9.44615	1.33589
$51$	0	0
$52$	0.804732	0.111596
$53$	−4.85408	−0.666759	−0.333380	−	0.942793i	$-0.608189\pi$
$53$	−4.85408	−0.666759	−0.333380	+	0.942793i	$0.608189\pi$
$54$	0	0
$55$	9.90371	1.33542
$56$	8.17805	1.09284
$57$	0	0
$58$	−13.7782	−1.80916
$59$	1.39802	0.182007	0.0910034	−	0.995851i	$-0.470993\pi$
$59$	1.39802	0.182007	0.0910034	+	0.995851i	$0.470993\pi$
$60$	0	0
$61$	7.53074	0.964212	0.482106	−	0.876113i	$-0.339872\pi$
$61$	7.53074	0.964212	0.482106	+	0.876113i	$0.339872\pi$
$62$	7.90294	1.00367
$63$	0	0
$64$	6.53995	0.817493
$65$	−9.36903	−1.16209
$66$	0	0
$67$	12.2116	1.49188	0.745941	−	0.666012i	$-0.232000\pi$
$67$	12.2116	1.49188	0.745941	+	0.666012i	$0.232000\pi$
$68$	2.17463	0.263713
$69$	0	0
$70$	16.0188	1.91462
$71$	−5.82004	−0.690712	−0.345356	−	0.938472i	$-0.612242\pi$
$71$	−5.82004	−0.690712	−0.345356	+	0.938472i	$0.612242\pi$
$72$	0	0
$73$	14.4246	1.68828	0.844138	−	0.536126i	$-0.180113\pi$
$73$	14.4246	1.68828	0.844138	+	0.536126i	$0.180113\pi$
$74$	−17.8927	−2.07998
$75$	0	0
$76$	2.21720	0.254331
$77$	9.32703	1.06291
$78$	0	0
$79$	4.43909	0.499436	0.249718	−	0.968319i	$-0.419662\pi$
$79$	4.43909	0.499436	0.249718	+	0.968319i	$0.419662\pi$
$80$	15.0669	1.68453
$81$	0	0
$82$	8.14715	0.899703
$83$	−9.68881	−1.06349	−0.531743	−	0.846906i	$-0.678463\pi$
$83$	−9.68881	−1.06349	−0.531743	+	0.846906i	$0.678463\pi$
$84$	0	0
$85$	−25.3180	−2.74612
$86$	8.34055	0.899385
$87$	0	0
$88$	7.64800	0.815280
$89$	15.4310	1.63569	0.817843	−	0.575441i	$-0.195170\pi$
$89$	15.4310	1.63569	0.817843	+	0.575441i	$0.195170\pi$
$90$	0	0
$91$	−8.82348	−0.924952
$92$	0.442587	0.0461429
$93$	0	0
$94$	4.54122	0.468391
$95$	−25.8136	−2.64842
$96$	0	0
$97$	0.876292	0.0889739	0.0444870	−	0.999010i	$-0.485835\pi$
$97$	0.876292	0.0889739	0.0444870	+	0.999010i	$0.485835\pi$
$98$	4.49769	0.454336
$99$	0	0
$100$	1.79870	0.179870
$101$	−4.78547	−0.476172	−0.238086	−	0.971244i	$-0.576520\pi$
$101$	−4.78547	−0.476172	−0.238086	+	0.971244i	$0.576520\pi$
$102$	0	0
$103$	−16.1716	−1.59344	−0.796719	−	0.604349i	$-0.793433\pi$
$103$	−16.1716	−1.59344	−0.796719	+	0.604349i	$0.793433\pi$
$104$	−7.23510	−0.709460
$105$	0	0
$106$	−7.34239	−0.713156
$107$	−3.94673	−0.381545	−0.190773	−	0.981634i	$-0.561099\pi$
$107$	−3.94673	−0.381545	−0.190773	+	0.981634i	$0.561099\pi$
$108$	0	0
$109$	−17.9662	−1.72085	−0.860424	−	0.509579i	$-0.829801\pi$
$109$	−17.9662	−1.72085	−0.860424	+	0.509579i	$0.829801\pi$
$110$	14.9806	1.42834
$111$	0	0
$112$	14.1895	1.34078
$113$	10.7144	1.00793	0.503963	−	0.863725i	$-0.331875\pi$
$113$	10.7144	1.00793	0.503963	+	0.863725i	$0.331875\pi$
$114$	0	0
$115$	−5.15279	−0.480500
$116$	−2.62358	−0.243594
$117$	0	0
$118$	2.11468	0.194672
$119$	−23.8437	−2.18575
$120$	0	0
$121$	−2.27749	−0.207045
$122$	11.3912	1.03131
$123$	0	0
$124$	1.50485	0.135139
$125$	−4.17450	−0.373379
$126$	0	0
$127$	7.39679	0.656359	0.328179	−	0.944615i	$-0.393565\pi$
$127$	7.39679	0.656359	0.328179	+	0.944615i	$0.393565\pi$
$128$	13.1269	1.16027
$129$	0	0
$130$	−14.1718	−1.24295
$131$	−0.639704	−0.0558912	−0.0279456	−	0.999609i	$-0.508897\pi$
$131$	−0.639704	−0.0558912	−0.0279456	+	0.999609i	$0.508897\pi$
$132$	0	0
$133$	−24.3105	−2.10799
$134$	18.4715	1.59570
$135$	0	0
$136$	−19.5514	−1.67652
$137$	9.53091	0.814281	0.407140	−	0.913366i	$-0.366526\pi$
$137$	9.53091	0.814281	0.407140	+	0.913366i	$0.366526\pi$
$138$	0	0
$139$	−5.17152	−0.438643	−0.219321	−	0.975653i	$-0.570384\pi$
$139$	−5.17152	−0.438643	−0.219321	+	0.975653i	$0.570384\pi$
$140$	3.05024	0.257792
$141$	0	0
$142$	−8.80352	−0.738775
$143$	−8.25160	−0.690033
$144$	0	0
$145$	30.5449	2.53662
$146$	21.8190	1.80576
$147$	0	0
$148$	−3.40706	−0.280058
$149$	15.2630	1.25039	0.625196	−	0.780468i	$-0.285019\pi$
$149$	15.2630	1.25039	0.625196	+	0.780468i	$0.285019\pi$
$150$	0	0
$151$	13.2108	1.07508	0.537539	−	0.843239i	$-0.319354\pi$
$151$	13.2108	1.07508	0.537539	+	0.843239i	$0.319354\pi$
$152$	−19.9342	−1.61688
$153$	0	0
$154$	14.1083	1.13688
$155$	−17.5201	−1.40724
$156$	0	0
$157$	0.691319	0.0551732	0.0275866	−	0.999619i	$-0.491218\pi$
$157$	0.691319	0.0551732	0.0275866	+	0.999619i	$0.491218\pi$
$158$	6.71467	0.534190
$159$	0	0
$160$	5.42309	0.428733
$161$	−4.85275	−0.382450
$162$	0	0
$163$	−21.3861	−1.67509	−0.837543	−	0.546372i	$-0.816008\pi$
$163$	−21.3861	−1.67509	−0.837543	+	0.546372i	$0.816008\pi$
$164$	1.55135	0.121140
$165$	0	0
$166$	−14.6555	−1.13749
$167$	8.72242	0.674961	0.337481	−	0.941332i	$-0.390425\pi$
$167$	8.72242	0.674961	0.337481	+	0.941332i	$0.390425\pi$
$168$	0	0
$169$	−5.19390	−0.399530
$170$	−38.2965	−2.93721
$171$	0	0
$172$	1.58817	0.121097
$173$	12.5556	0.954581	0.477291	−	0.878746i	$-0.341619\pi$
$173$	12.5556	0.954581	0.477291	+	0.878746i	$0.341619\pi$
$174$	0	0
$175$	−19.7218	−1.49083
$176$	13.2699	1.00025
$177$	0	0
$178$	23.3413	1.74951
$179$	21.1850	1.58344	0.791720	−	0.610884i	$-0.209186\pi$
$179$	21.1850	1.58344	0.791720	+	0.610884i	$0.209186\pi$
$180$	0	0
$181$	10.1122	0.751636	0.375818	−	0.926693i	$-0.377362\pi$
$181$	10.1122	0.751636	0.375818	+	0.926693i	$0.377362\pi$
$182$	−13.3466	−0.989315
$183$	0	0
$184$	−3.97917	−0.293348
$185$	39.6664	2.91633
$186$	0	0
$187$	−22.2983	−1.63061
$188$	0.864721	0.0630663
$189$	0	0
$190$	−39.0463	−2.83271
$191$	2.44789	0.177123	0.0885615	−	0.996071i	$-0.471773\pi$
$191$	2.44789	0.177123	0.0885615	+	0.996071i	$0.471773\pi$
$192$	0	0
$193$	8.31334	0.598407	0.299204	−	0.954189i	$-0.403279\pi$
$193$	8.31334	0.598407	0.299204	+	0.954189i	$0.403279\pi$
$194$	1.32550	0.0951653
$195$	0	0
$196$	0.856433	0.0611738
$197$	18.6761	1.33061	0.665307	−	0.746570i	$-0.268301\pi$
$197$	18.6761	1.33061	0.665307	+	0.746570i	$0.268301\pi$
$198$	0	0
$199$	1.01579	0.0720071	0.0360036	−	0.999352i	$-0.488537\pi$
$199$	1.01579	0.0720071	0.0360036	+	0.999352i	$0.488537\pi$
$200$	−16.1715	−1.14350
$201$	0	0
$202$	−7.23861	−0.509307
$203$	28.7663	2.01900
$204$	0	0
$205$	−18.0615	−1.26147
$206$	−24.4616	−1.70432
$207$	0	0
$208$	−12.5534	−0.870425
$209$	−22.7349	−1.57260
$210$	0	0
$211$	7.25723	0.499608	0.249804	−	0.968296i	$-0.419634\pi$
$211$	7.25723	0.499608	0.249804	+	0.968296i	$0.419634\pi$
$212$	−1.39811	−0.0960225
$213$	0	0
$214$	−5.96992	−0.408095
$215$	−18.4902	−1.26102
$216$	0	0
$217$	−16.4999	−1.12008
$218$	−27.1761	−1.84059
$219$	0	0
$220$	2.85254	0.192318
$221$	21.0945	1.41897
$222$	0	0
$223$	−1.00000	−0.0669650
$223$	−1.00000	−0.0669650
$224$	5.10731	0.341246
$225$	0	0
$226$	16.2068	1.07806
$227$	−5.03218	−0.333997	−0.166999	−	0.985957i	$-0.553408\pi$
$227$	−5.03218	−0.333997	−0.166999	+	0.985957i	$0.553408\pi$
$228$	0	0
$229$	11.3951	0.753010	0.376505	−	0.926415i	$-0.377126\pi$
$229$	11.3951	0.753010	0.376505	+	0.926415i	$0.377126\pi$
$230$	−7.79423	−0.513936
$231$	0	0
$232$	23.5879	1.54862
$233$	3.47765	0.227829	0.113914	−	0.993491i	$-0.463661\pi$
$233$	3.47765	0.227829	0.113914	+	0.993491i	$0.463661\pi$
$234$	0	0
$235$	−10.0675	−0.656728
$236$	0.402668	0.0262115
$237$	0	0
$238$	−36.0665	−2.33785
$239$	7.59950	0.491571	0.245785	−	0.969324i	$-0.420954\pi$
$239$	7.59950	0.491571	0.245785	+	0.969324i	$0.420954\pi$
$240$	0	0
$241$	9.76931	0.629296	0.314648	−	0.949208i	$-0.398113\pi$
$241$	9.76931	0.629296	0.314648	+	0.949208i	$0.398113\pi$
$242$	−3.44498	−0.221452
$243$	0	0
$244$	2.16906	0.138860
$245$	−9.97095	−0.637021
$246$	0	0
$247$	21.5074	1.36849
$248$	−13.5296	−0.859132
$249$	0	0
$250$	−6.31444	−0.399360
$251$	−4.77966	−0.301689	−0.150845	−	0.988557i	$-0.548199\pi$
$251$	−4.77966	−0.301689	−0.150845	+	0.988557i	$0.548199\pi$
$252$	0	0
$253$	−4.53822	−0.285316
$254$	11.1886	0.702032
$255$	0	0
$256$	6.77617	0.423511
$257$	−7.16834	−0.447149	−0.223574	−	0.974687i	$-0.571773\pi$
$257$	−7.16834	−0.447149	−0.223574	+	0.974687i	$0.571773\pi$
$258$	0	0
$259$	37.3566	2.32123
$260$	−2.69854	−0.167356
$261$	0	0
$262$	−0.967631	−0.0597805
$263$	−8.47871	−0.522820	−0.261410	−	0.965228i	$-0.584187\pi$
$263$	−8.47871	−0.522820	−0.261410	+	0.965228i	$0.584187\pi$
$264$	0	0
$265$	16.2774	0.999911
$266$	−36.7726	−2.25467
$267$	0	0
$268$	3.51727	0.214851
$269$	18.8346	1.14837	0.574184	−	0.818727i	$-0.305320\pi$
$269$	18.8346	1.14837	0.574184	+	0.818727i	$0.305320\pi$
$270$	0	0
$271$	−29.3405	−1.78231	−0.891153	−	0.453702i	$-0.850103\pi$
$271$	−29.3405	−1.78231	−0.891153	+	0.453702i	$0.850103\pi$
$272$	−33.9232	−2.05690
$273$	0	0
$274$	14.4167	0.870943
$275$	−18.4436	−1.11219
$276$	0	0
$277$	−30.2687	−1.81867	−0.909335	−	0.416064i	$-0.863409\pi$
$277$	−30.2687	−1.81867	−0.909335	+	0.416064i	$0.863409\pi$
$278$	−7.82256	−0.469166
$279$	0	0
$280$	−27.4238	−1.63888
$281$	2.90357	0.173212	0.0866061	−	0.996243i	$-0.472398\pi$
$281$	2.90357	0.173212	0.0866061	+	0.996243i	$0.472398\pi$
$282$	0	0
$283$	22.5190	1.33862	0.669308	−	0.742985i	$-0.266591\pi$
$283$	22.5190	1.33862	0.669308	+	0.742985i	$0.266591\pi$
$284$	−1.67633	−0.0994720
$285$	0	0
$286$	−12.4816	−0.738050
$287$	−17.0097	−1.00405
$288$	0	0
$289$	40.0036	2.35316
$290$	46.2029	2.71313
$291$	0	0
$292$	4.15469	0.243135
$293$	20.3356	1.18802	0.594008	−	0.804459i	$-0.297545\pi$
$293$	20.3356	1.18802	0.594008	+	0.804459i	$0.297545\pi$
$294$	0	0
$295$	−4.68803	−0.272948
$296$	30.6318	1.78044
$297$	0	0
$298$	23.0871	1.33740
$299$	4.29321	0.248283
$300$	0	0
$301$	−17.4135	−1.00370
$302$	19.9829	1.14989
$303$	0	0
$304$	−34.5873	−1.98372
$305$	−25.2531	−1.44599
$306$	0	0
$307$	−8.62950	−0.492512	−0.246256	−	0.969205i	$-0.579200\pi$
$307$	−8.62950	−0.492512	−0.246256	+	0.969205i	$0.579200\pi$
$308$	2.68644	0.153074
$309$	0	0
$310$	−26.5012	−1.50517
$311$	17.6849	1.00282	0.501410	−	0.865210i	$-0.332815\pi$
$311$	17.6849	1.00282	0.501410	+	0.865210i	$0.332815\pi$
$312$	0	0
$313$	−4.29240	−0.242621	−0.121310	−	0.992615i	$-0.538710\pi$
$313$	−4.29240	−0.242621	−0.121310	+	0.992615i	$0.538710\pi$
$314$	1.04570	0.0590125
$315$	0	0
$316$	1.27858	0.0719257
$317$	−3.38696	−0.190231	−0.0951154	−	0.995466i	$-0.530322\pi$
$317$	−3.38696	−0.190231	−0.0951154	+	0.995466i	$0.530322\pi$
$318$	0	0
$319$	26.9018	1.50621
$320$	−21.9307	−1.22596
$321$	0	0
$322$	−7.34037	−0.409063
$323$	58.1196	3.23386
$324$	0	0
$325$	17.4478	0.967831
$326$	−32.3490	−1.79165
$327$	0	0
$328$	−13.9477	−0.770133
$329$	−9.48123	−0.522717
$330$	0	0
$331$	−13.3833	−0.735612	−0.367806	−	0.929903i	$-0.619891\pi$
$331$	−13.3833	−0.735612	−0.367806	+	0.929903i	$0.619891\pi$
$332$	−2.79064	−0.153157
$333$	0	0
$334$	13.1937	0.721929
$335$	−40.9496	−2.23731
$336$	0	0
$337$	−5.43105	−0.295848	−0.147924	−	0.988999i	$-0.547259\pi$
$337$	−5.43105	−0.295848	−0.147924	+	0.988999i	$0.547259\pi$
$338$	−7.85640	−0.427332
$339$	0	0
$340$	−7.29227	−0.395479
$341$	−15.4305	−0.835606
$342$	0	0
$343$	12.7162	0.686609
$344$	−14.2788	−0.769861
$345$	0	0
$346$	18.9918	1.02101
$347$	−13.4193	−0.720386	−0.360193	−	0.932878i	$-0.617289\pi$
$347$	−13.4193	−0.720386	−0.360193	+	0.932878i	$0.617289\pi$
$348$	0	0
$349$	12.6553	0.677423	0.338712	−	0.940890i	$-0.390009\pi$
$349$	12.6553	0.677423	0.338712	+	0.940890i	$0.390009\pi$
$350$	−29.8316	−1.59457
$351$	0	0
$352$	4.77628	0.254577
$353$	−13.4949	−0.718260	−0.359130	−	0.933288i	$-0.616927\pi$
$353$	−13.4949	−0.718260	−0.359130	+	0.933288i	$0.616927\pi$
$354$	0	0
$355$	19.5166	1.03583
$356$	4.44456	0.235561
$357$	0	0
$358$	32.0449	1.69362
$359$	7.43330	0.392314	0.196157	−	0.980572i	$-0.437154\pi$
$359$	7.43330	0.392314	0.196157	+	0.980572i	$0.437154\pi$
$360$	0	0
$361$	40.2575	2.11881
$362$	15.2960	0.803940
$363$	0	0
$364$	−2.54140	−0.133206
$365$	−48.3707	−2.53184
$366$	0	0
$367$	−29.2320	−1.52590	−0.762949	−	0.646459i	$-0.776249\pi$
$367$	−29.2320	−1.52590	−0.762949	+	0.646459i	$0.776249\pi$
$368$	−6.90416	−0.359904
$369$	0	0
$370$	60.0003	3.11927
$371$	15.3296	0.795871
$372$	0	0
$373$	14.6388	0.757970	0.378985	−	0.925403i	$-0.376273\pi$
$373$	14.6388	0.757970	0.378985	+	0.925403i	$0.376273\pi$
$374$	−33.7289	−1.74408
$375$	0	0
$376$	−7.77445	−0.400937
$377$	−25.4494	−1.31071
$378$	0	0
$379$	−15.7476	−0.808898	−0.404449	−	0.914561i	$-0.632537\pi$
$379$	−15.7476	−0.808898	−0.404449	+	0.914561i	$0.632537\pi$
$380$	−7.43503	−0.381409
$381$	0	0
$382$	3.70273	0.189448
$383$	38.8692	1.98612	0.993062	−	0.117591i	$-0.0375173\pi$
$383$	38.8692	1.98612	0.993062	+	0.117591i	$0.0375173\pi$
$384$	0	0
$385$	−31.2767	−1.59401
$386$	12.5749	0.640048
$387$	0	0
$388$	0.252396	0.0128135
$389$	−22.4120	−1.13633	−0.568166	−	0.822914i	$-0.692347\pi$
$389$	−22.4120	−1.13633	−0.568166	+	0.822914i	$0.692347\pi$
$390$	0	0
$391$	11.6016	0.586716
$392$	−7.69993	−0.388905
$393$	0	0
$394$	28.2498	1.42321
$395$	−14.8858	−0.748984
$396$	0	0
$397$	11.2563	0.564939	0.282470	−	0.959276i	$-0.408846\pi$
$397$	11.2563	0.564939	0.282470	+	0.959276i	$0.408846\pi$
$398$	1.53650	0.0770178
$399$	0	0
$400$	−28.0588	−1.40294
$401$	−10.0667	−0.502709	−0.251355	−	0.967895i	$-0.580876\pi$
$401$	−10.0667	−0.502709	−0.251355	+	0.967895i	$0.580876\pi$
$402$	0	0
$403$	14.5974	0.727148
$404$	−1.37835	−0.0685753
$405$	0	0
$406$	43.5125	2.15949
$407$	34.9354	1.73169
$408$	0	0
$409$	−9.30468	−0.460087	−0.230043	−	0.973180i	$-0.573887\pi$
$409$	−9.30468	−0.460087	−0.230043	+	0.973180i	$0.573887\pi$
$410$	−27.3202	−1.34925
$411$	0	0
$412$	−4.65788	−0.229477
$413$	−4.41505	−0.217251
$414$	0	0
$415$	32.4899	1.59487
$416$	−4.51842	−0.221534
$417$	0	0
$418$	−34.3893	−1.68203
$419$	−18.5787	−0.907630	−0.453815	−	0.891096i	$-0.649937\pi$
$419$	−18.5787	−0.907630	−0.453815	+	0.891096i	$0.649937\pi$
$420$	0	0
$421$	3.21597	0.156737	0.0783684	−	0.996924i	$-0.475029\pi$
$421$	3.21597	0.156737	0.0783684	+	0.996924i	$0.475029\pi$
$422$	10.9774	0.534374
$423$	0	0
$424$	12.5700	0.610452
$425$	47.1493	2.28708
$426$	0	0
$427$	−23.7826	−1.15092
$428$	−1.13677	−0.0549477
$429$	0	0
$430$	−27.9687	−1.34877
$431$	16.3489	0.787499	0.393749	−	0.919218i	$-0.371178\pi$
$431$	16.3489	0.787499	0.393749	+	0.919218i	$0.371178\pi$
$432$	0	0
$433$	19.5096	0.937571	0.468786	−	0.883312i	$-0.344692\pi$
$433$	19.5096	0.937571	0.468786	+	0.883312i	$0.344692\pi$
$434$	−24.9581	−1.19803
$435$	0	0
$436$	−5.17475	−0.247826
$437$	11.8287	0.565843
$438$	0	0
$439$	−0.902131	−0.0430563	−0.0215282	−	0.999768i	$-0.506853\pi$
$439$	−0.902131	−0.0430563	−0.0215282	+	0.999768i	$0.506853\pi$
$440$	−25.6463	−1.22264
$441$	0	0
$442$	31.9080	1.51771
$443$	0.874698	0.0415582	0.0207791	−	0.999784i	$-0.493385\pi$
$443$	0.874698	0.0415582	0.0207791	+	0.999784i	$0.493385\pi$
$444$	0	0
$445$	−51.7455	−2.45297
$446$	−1.51262	−0.0716248
$447$	0	0
$448$	−20.6536	−0.975793
$449$	2.04596	0.0965547	0.0482773	−	0.998834i	$-0.484627\pi$
$449$	2.04596	0.0965547	0.0482773	+	0.998834i	$0.484627\pi$
$450$	0	0
$451$	−15.9073	−0.749045
$452$	3.08604	0.145155
$453$	0	0
$454$	−7.61178	−0.357239
$455$	29.5881	1.38711
$456$	0	0
$457$	22.7347	1.06348	0.531741	−	0.846907i	$-0.321538\pi$
$457$	22.7347	1.06348	0.531741	+	0.846907i	$0.321538\pi$
$458$	17.2365	0.805409
$459$	0	0
$460$	−1.48415	−0.0691986
$461$	6.95254	0.323812	0.161906	−	0.986806i	$-0.448236\pi$
$461$	6.95254	0.323812	0.161906	+	0.986806i	$0.448236\pi$
$462$	0	0
$463$	13.5053	0.627643	0.313821	−	0.949482i	$-0.398391\pi$
$463$	13.5053	0.627643	0.313821	+	0.949482i	$0.398391\pi$
$464$	40.9267	1.89997
$465$	0	0
$466$	5.26038	0.243682
$467$	−19.2689	−0.891659	−0.445829	−	0.895118i	$-0.647091\pi$
$467$	−19.2689	−0.891659	−0.445829	+	0.895118i	$0.647091\pi$
$468$	0	0
$469$	−38.5651	−1.78077
$470$	−15.2283	−0.702427
$471$	0	0
$472$	−3.62027	−0.166636
$473$	−16.2849	−0.748780
$474$	0	0
$475$	48.0724	2.20571
$476$	−6.86764	−0.314778
$477$	0	0
$478$	11.4952	0.525777
$479$	−25.8403	−1.18067	−0.590337	−	0.807157i	$-0.701005\pi$
$479$	−25.8403	−1.18067	−0.590337	+	0.807157i	$0.701005\pi$
$480$	0	0
$481$	−33.0493	−1.50692
$482$	14.7773	0.673087
$483$	0	0
$484$	−0.655980	−0.0298173
$485$	−2.93850	−0.133431
$486$	0	0
$487$	16.6843	0.756037	0.378018	−	0.925798i	$-0.376606\pi$
$487$	16.6843	0.756037	0.378018	+	0.925798i	$0.376606\pi$
$488$	−19.5014	−0.882785
$489$	0	0
$490$	−15.0823	−0.681348
$491$	−21.5762	−0.973722	−0.486861	−	0.873479i	$-0.661858\pi$
$491$	−21.5762	−0.973722	−0.486861	+	0.873479i	$0.661858\pi$
$492$	0	0
$493$	−68.7721	−3.09734
$494$	32.5326	1.46371
$495$	0	0
$496$	−23.4749	−1.05405
$497$	18.3801	0.824461
$498$	0	0
$499$	−2.27085	−0.101657	−0.0508286	−	0.998707i	$-0.516186\pi$
$499$	−2.27085	−0.101657	−0.0508286	+	0.998707i	$0.516186\pi$
$500$	−1.20237	−0.0537716
$501$	0	0
$502$	−7.22982	−0.322683
$503$	−11.9167	−0.531338	−0.265669	−	0.964064i	$-0.585593\pi$
$503$	−11.9167	−0.531338	−0.265669	+	0.964064i	$0.585593\pi$
$504$	0	0
$505$	16.0473	0.714095
$506$	−6.86462	−0.305170
$507$	0	0
$508$	2.13048	0.0945247
$509$	27.6093	1.22376	0.611881	−	0.790950i	$-0.290413\pi$
$509$	27.6093	1.22376	0.611881	+	0.790950i	$0.290413\pi$
$510$	0	0
$511$	−45.5541	−2.01519
$512$	−16.0040	−0.707285
$513$	0	0
$514$	−10.8430	−0.478264
$515$	54.2290	2.38961
$516$	0	0
$517$	−8.86672	−0.389958
$518$	56.5065	2.48275
$519$	0	0
$520$	24.2617	1.06395
$521$	33.2408	1.45631	0.728153	−	0.685415i	$-0.240379\pi$
$521$	33.2408	1.45631	0.728153	+	0.685415i	$0.240379\pi$
$522$	0	0
$523$	0.985614	0.0430979	0.0215489	−	0.999768i	$-0.493140\pi$
$523$	0.985614	0.0430979	0.0215489	+	0.999768i	$0.493140\pi$
$524$	−0.184252	−0.00804910
$525$	0	0
$526$	−12.8251	−0.559200
$527$	39.4466	1.71832
$528$	0	0
$529$	−20.6388	−0.897340
$530$	24.6215	1.06949
$531$	0	0
$532$	−7.00210	−0.303579
$533$	15.0485	0.651822
$534$	0	0
$535$	13.2347	0.572187
$536$	−31.6227	−1.36589
$537$	0	0
$538$	28.4897	1.22828
$539$	−8.78173	−0.378256
$540$	0	0
$541$	36.8503	1.58432	0.792159	−	0.610315i	$-0.208957\pi$
$541$	36.8503	1.58432	0.792159	+	0.610315i	$0.208957\pi$
$542$	−44.3811	−1.90633
$543$	0	0
$544$	−12.2101	−0.523505
$545$	60.2467	2.58068
$546$	0	0
$547$	−20.4345	−0.873717	−0.436859	−	0.899530i	$-0.643909\pi$
$547$	−20.4345	−0.873717	−0.436859	+	0.899530i	$0.643909\pi$
$548$	2.74516	0.117268
$549$	0	0
$550$	−27.8981	−1.18958
$551$	−70.1185	−2.98715
$552$	0	0
$553$	−14.0190	−0.596148
$554$	−45.7851	−1.94522
$555$	0	0
$556$	−1.48954	−0.0631706
$557$	13.4374	0.569362	0.284681	−	0.958622i	$-0.408112\pi$
$557$	13.4374	0.569362	0.284681	+	0.958622i	$0.408112\pi$
$558$	0	0
$559$	15.4057	0.651591
$560$	−47.5823	−2.01072
$561$	0	0
$562$	4.39200	0.185265
$563$	26.6681	1.12393	0.561963	−	0.827162i	$-0.310046\pi$
$563$	26.6681	1.12393	0.561963	+	0.827162i	$0.310046\pi$
$564$	0	0
$565$	−35.9290	−1.51154
$566$	34.0628	1.43177
$567$	0	0
$568$	15.0714	0.632382
$569$	30.0890	1.26140	0.630699	−	0.776027i	$-0.282768\pi$
$569$	30.0890	1.26140	0.630699	+	0.776027i	$0.282768\pi$
$570$	0	0
$571$	−38.7356	−1.62104	−0.810518	−	0.585714i	$-0.800814\pi$
$571$	−38.7356	−1.62104	−0.810518	+	0.585714i	$0.800814\pi$
$572$	−2.37669	−0.0993743
$573$	0	0
$574$	−25.7293	−1.07392
$575$	9.59597	0.400180
$576$	0	0
$577$	−16.0471	−0.668048	−0.334024	−	0.942564i	$-0.608407\pi$
$577$	−16.0471	−0.668048	−0.334024	+	0.942564i	$0.608407\pi$
$578$	60.5104	2.51690
$579$	0	0
$580$	8.79777	0.365307
$581$	30.5980	1.26942
$582$	0	0
$583$	14.3360	0.593736
$584$	−37.3536	−1.54570
$585$	0	0
$586$	30.7600	1.27068
$587$	7.90079	0.326101	0.163050	−	0.986618i	$-0.447867\pi$
$587$	7.90079	0.326101	0.163050	+	0.986618i	$0.447867\pi$
$588$	0	0
$589$	40.2189	1.65719
$590$	−7.09123	−0.291941
$591$	0	0
$592$	53.1485	2.18439
$593$	5.08733	0.208912	0.104456	−	0.994530i	$-0.466690\pi$
$593$	5.08733	0.208912	0.104456	+	0.994530i	$0.466690\pi$
$594$	0	0
$595$	79.9560	3.27788
$596$	4.39616	0.180074
$597$	0	0
$598$	6.49401	0.265560
$599$	−17.9410	−0.733048	−0.366524	−	0.930409i	$-0.619452\pi$
$599$	−17.9410	−0.733048	−0.366524	+	0.930409i	$0.619452\pi$
$600$	0	0
$601$	−4.15678	−0.169558	−0.0847792	−	0.996400i	$-0.527018\pi$
$601$	−4.15678	−0.169558	−0.0847792	+	0.996400i	$0.527018\pi$
$602$	−26.3401	−1.07354
$603$	0	0
$604$	3.80507	0.154826
$605$	7.63719	0.310496
$606$	0	0
$607$	0.682865	0.0277166	0.0138583	−	0.999904i	$-0.495589\pi$
$607$	0.682865	0.0277166	0.0138583	+	0.999904i	$0.495589\pi$
$608$	−12.4492	−0.504881
$609$	0	0
$610$	−38.1984	−1.54661
$611$	8.38802	0.339343
$612$	0	0
$613$	−3.83636	−0.154949	−0.0774745	−	0.996994i	$-0.524686\pi$
$613$	−3.83636	−0.154949	−0.0774745	+	0.996994i	$0.524686\pi$
$614$	−13.0532	−0.526784
$615$	0	0
$616$	−24.1530	−0.973151
$617$	−32.3577	−1.30267	−0.651337	−	0.758789i	$-0.725791\pi$
$617$	−32.3577	−1.30267	−0.651337	+	0.758789i	$0.725791\pi$
$618$	0	0
$619$	19.4862	0.783217	0.391609	−	0.920132i	$-0.371919\pi$
$619$	19.4862	0.783217	0.391609	+	0.920132i	$0.371919\pi$
$620$	−5.04626	−0.202663
$621$	0	0
$622$	26.7506	1.07260
$623$	−48.7324	−1.95242
$624$	0	0
$625$	−17.2259	−0.689035
$626$	−6.49278	−0.259504
$627$	0	0
$628$	0.199119	0.00794571
$629$	−89.3093	−3.56100
$630$	0	0
$631$	15.3890	0.612625	0.306313	−	0.951931i	$-0.400905\pi$
$631$	15.3890	0.612625	0.306313	+	0.951931i	$0.400905\pi$
$632$	−11.4953	−0.457259
$633$	0	0
$634$	−5.12320	−0.203468
$635$	−24.8039	−0.984314
$636$	0	0
$637$	8.30762	0.329160
$638$	40.6923	1.61102
$639$	0	0
$640$	−44.0190	−1.74000
$641$	−17.2286	−0.680491	−0.340245	−	0.940337i	$-0.610510\pi$
$641$	−17.2286	−0.680491	−0.340245	+	0.940337i	$0.610510\pi$
$642$	0	0
$643$	18.9314	0.746583	0.373292	−	0.927714i	$-0.378229\pi$
$643$	18.9314	0.746583	0.373292	+	0.927714i	$0.378229\pi$
$644$	−1.39772	−0.0550780
$645$	0	0
$646$	87.9131	3.45889
$647$	−13.1097	−0.515395	−0.257697	−	0.966226i	$-0.582964\pi$
$647$	−13.1097	−0.515395	−0.257697	+	0.966226i	$0.582964\pi$
$648$	0	0
$649$	−4.12890	−0.162073
$650$	26.3920	1.03518
$651$	0	0
$652$	−6.15977	−0.241235
$653$	10.5607	0.413272	0.206636	−	0.978418i	$-0.433748\pi$
$653$	10.5607	0.413272	0.206636	+	0.978418i	$0.433748\pi$
$654$	0	0
$655$	2.14514	0.0838177
$656$	−24.2003	−0.944863
$657$	0	0
$658$	−14.3415	−0.559091
$659$	−47.3977	−1.84635	−0.923175	−	0.384379i	$-0.874415\pi$
$659$	−47.3977	−1.84635	−0.923175	+	0.384379i	$0.874415\pi$
$660$	0	0
$661$	−4.57218	−0.177837	−0.0889186	−	0.996039i	$-0.528341\pi$
$661$	−4.57218	−0.177837	−0.0889186	+	0.996039i	$0.528341\pi$
$662$	−20.2439	−0.786800
$663$	0	0
$664$	25.0898	0.973675
$665$	81.5214	3.16126
$666$	0	0
$667$	−13.9967	−0.541955
$668$	2.51230	0.0972037
$669$	0	0
$670$	−61.9412	−2.39300
$671$	−22.2412	−0.858612
$672$	0	0
$673$	−23.8901	−0.920896	−0.460448	−	0.887687i	$-0.652311\pi$
$673$	−23.8901	−0.920896	−0.460448	+	0.887687i	$0.652311\pi$
$674$	−8.21513	−0.316435
$675$	0	0
$676$	−1.49598	−0.0575379
$677$	11.9321	0.458586	0.229293	−	0.973357i	$-0.426359\pi$
$677$	11.9321	0.458586	0.229293	+	0.973357i	$0.426359\pi$
$678$	0	0
$679$	−2.76740	−0.106203
$680$	65.5626	2.51421
$681$	0	0
$682$	−23.3405	−0.893753
$683$	−1.90503	−0.0728939	−0.0364470	−	0.999336i	$-0.511604\pi$
$683$	−1.90503	−0.0728939	−0.0364470	+	0.999336i	$0.511604\pi$
$684$	0	0
$685$	−31.9604	−1.22114
$686$	19.2348	0.734387
$687$	0	0
$688$	−24.7748	−0.944529
$689$	−13.5620	−0.516672
$690$	0	0
$691$	35.7644	1.36054	0.680270	−	0.732961i	$-0.261862\pi$
$691$	35.7644	1.36054	0.680270	+	0.732961i	$0.261862\pi$
$692$	3.61635	0.137473
$693$	0	0
$694$	−20.2984	−0.770515
$695$	17.3419	0.657814
$696$	0	0
$697$	40.6655	1.54032
$698$	19.1427	0.724562
$699$	0	0
$700$	−5.68042	−0.214700
$701$	−41.6427	−1.57282	−0.786411	−	0.617704i	$-0.788063\pi$
$701$	−41.6427	−1.57282	−0.786411	+	0.617704i	$0.788063\pi$
$702$	0	0
$703$	−91.0578	−3.43431
$704$	−19.3150	−0.727962
$705$	0	0
$706$	−20.4127	−0.768241
$707$	15.1129	0.568378
$708$	0	0
$709$	−12.9732	−0.487220	−0.243610	−	0.969873i	$-0.578332\pi$
$709$	−12.9732	−0.487220	−0.243610	+	0.969873i	$0.578332\pi$
$710$	29.5212	1.10791
$711$	0	0
$712$	−39.9597	−1.49755
$713$	8.02829	0.300662
$714$	0	0
$715$	27.6704	1.03481
$716$	6.10186	0.228037
$717$	0	0
$718$	11.2438	0.419614
$719$	35.0316	1.30646	0.653229	−	0.757160i	$-0.273414\pi$
$719$	35.0316	1.30646	0.653229	+	0.757160i	$0.273414\pi$
$720$	0	0
$721$	51.0712	1.90199
$722$	60.8944	2.26625
$723$	0	0
$724$	2.91260	0.108246
$725$	−56.8834	−2.11259
$726$	0	0
$727$	2.53229	0.0939175	0.0469587	−	0.998897i	$-0.485047\pi$
$727$	2.53229	0.0939175	0.0469587	+	0.998897i	$0.485047\pi$
$728$	22.8490	0.846840
$729$	0	0
$730$	−73.1666	−2.70802
$731$	41.6309	1.53977
$732$	0	0
$733$	9.16166	0.338394	0.169197	−	0.985582i	$-0.445883\pi$
$733$	9.16166	0.338394	0.169197	+	0.985582i	$0.445883\pi$
$734$	−44.2170	−1.63208
$735$	0	0
$736$	−2.48505	−0.0916000
$737$	−36.0655	−1.32849
$738$	0	0
$739$	0.454268	0.0167105	0.00835526	−	0.999965i	$-0.497340\pi$
$739$	0.454268	0.0167105	0.00835526	+	0.999965i	$0.497340\pi$
$740$	11.4250	0.419992
$741$	0	0
$742$	23.1878	0.851252
$743$	−26.1468	−0.959234	−0.479617	−	0.877478i	$-0.659224\pi$
$743$	−26.1468	−0.959234	−0.479617	+	0.877478i	$0.659224\pi$
$744$	0	0
$745$	−51.1819	−1.87516
$746$	22.1430	0.810714
$747$	0	0
$748$	−6.42253	−0.234831
$749$	12.4641	0.455428
$750$	0	0
$751$	28.8921	1.05429	0.527143	−	0.849776i	$-0.323263\pi$
$751$	28.8921	1.05429	0.527143	+	0.849776i	$0.323263\pi$
$752$	−13.4893	−0.491902
$753$	0	0
$754$	−38.4954	−1.40192
$755$	−44.3002	−1.61225
$756$	0	0
$757$	−12.1211	−0.440549	−0.220275	−	0.975438i	$-0.570695\pi$
$757$	−12.1211	−0.440549	−0.220275	+	0.975438i	$0.570695\pi$
$758$	−23.8201	−0.865185
$759$	0	0
$760$	66.8461	2.42476
$761$	−18.7609	−0.680082	−0.340041	−	0.940411i	$-0.610441\pi$
$761$	−18.7609	−0.680082	−0.340041	+	0.940411i	$0.610441\pi$
$762$	0	0
$763$	56.7386	2.05407
$764$	0.705059	0.0255082
$765$	0	0
$766$	58.7945	2.12433
$767$	3.90599	0.141037
$768$	0	0
$769$	−35.8014	−1.29103	−0.645516	−	0.763747i	$-0.723358\pi$
$769$	−35.8014	−1.29103	−0.645516	+	0.763747i	$0.723358\pi$
$770$	−47.3098	−1.70493
$771$	0	0
$772$	2.39447	0.0861789
$773$	53.5666	1.92666	0.963329	−	0.268323i	$-0.0864695\pi$
$773$	53.5666	1.92666	0.963329	+	0.268323i	$0.0864695\pi$
$774$	0	0
$775$	32.6274	1.17201
$776$	−2.26922	−0.0814602
$777$	0	0
$778$	−33.9009	−1.21541
$779$	41.4617	1.48552
$780$	0	0
$781$	17.1888	0.615065
$782$	17.5488	0.627543
$783$	0	0
$784$	−13.3600	−0.477141
$785$	−2.31823	−0.0827410
$786$	0	0
$787$	29.4356	1.04927	0.524633	−	0.851329i	$-0.324203\pi$
$787$	29.4356	1.04927	0.524633	+	0.851329i	$0.324203\pi$
$788$	5.37922	0.191627
$789$	0	0
$790$	−22.5165	−0.801103
$791$	−33.8369	−1.20310
$792$	0	0
$793$	21.0404	0.747168
$794$	17.0266	0.604251
$795$	0	0
$796$	0.292574	0.0103700
$797$	45.7598	1.62089	0.810447	−	0.585812i	$-0.199224\pi$
$797$	45.7598	1.62089	0.810447	+	0.585812i	$0.199224\pi$
$798$	0	0
$799$	22.6670	0.801900
$800$	−10.0993	−0.357066
$801$	0	0
$802$	−15.2272	−0.537690
$803$	−42.6016	−1.50338
$804$	0	0
$805$	16.2729	0.573544
$806$	22.0803	0.777747
$807$	0	0
$808$	12.3923	0.435959
$809$	11.0849	0.389724	0.194862	−	0.980831i	$-0.437574\pi$
$809$	11.0849	0.389724	0.194862	+	0.980831i	$0.437574\pi$
$810$	0	0
$811$	−28.4320	−0.998384	−0.499192	−	0.866491i	$-0.666370\pi$
$811$	−28.4320	−0.998384	−0.499192	+	0.866491i	$0.666370\pi$
$812$	8.28548	0.290763
$813$	0	0
$814$	52.8441	1.85219
$815$	71.7147	2.51206
$816$	0	0
$817$	42.4459	1.48499
$818$	−14.0745	−0.492102
$819$	0	0
$820$	−5.20220	−0.181669
$821$	20.8747	0.728533	0.364266	−	0.931295i	$-0.381320\pi$
$821$	20.8747	0.728533	0.364266	+	0.931295i	$0.381320\pi$
$822$	0	0
$823$	−45.4370	−1.58384	−0.791918	−	0.610628i	$-0.790917\pi$
$823$	−45.4370	−1.58384	−0.791918	+	0.610628i	$0.790917\pi$
$824$	41.8776	1.45887
$825$	0	0
$826$	−6.67831	−0.232368
$827$	−6.73141	−0.234074	−0.117037	−	0.993128i	$-0.537340\pi$
$827$	−6.73141	−0.234074	−0.117037	+	0.993128i	$0.537340\pi$
$828$	0	0
$829$	32.9679	1.14502	0.572511	−	0.819897i	$-0.305969\pi$
$829$	32.9679	1.14502	0.572511	+	0.819897i	$0.305969\pi$
$830$	49.1449	1.70585
$831$	0	0
$832$	18.2722	0.633475
$833$	22.4497	0.777836
$834$	0	0
$835$	−29.2492	−1.01221
$836$	−6.54827	−0.226477
$837$	0	0
$838$	−28.1026	−0.970788
$839$	4.23811	0.146316	0.0731580	−	0.997320i	$-0.476692\pi$
$839$	4.23811	0.146316	0.0731580	+	0.997320i	$0.476692\pi$
$840$	0	0
$841$	53.9702	1.86104
$842$	4.86455	0.167643
$843$	0	0
$844$	2.09028	0.0719505
$845$	17.4169	0.599159
$846$	0	0
$847$	7.19248	0.247137
$848$	21.8099	0.748953
$849$	0	0
$850$	71.3191	2.44623
$851$	−18.1765	−0.623083
$852$	0	0
$853$	48.8383	1.67219	0.836095	−	0.548584i	$-0.184833\pi$
$853$	48.8383	1.67219	0.836095	+	0.548584i	$0.184833\pi$
$854$	−35.9741	−1.23101
$855$	0	0
$856$	10.2203	0.349324
$857$	−4.42286	−0.151082	−0.0755410	−	0.997143i	$-0.524068\pi$
$857$	−4.42286	−0.151082	−0.0755410	+	0.997143i	$0.524068\pi$
$858$	0	0
$859$	−52.8833	−1.80435	−0.902177	−	0.431366i	$-0.858032\pi$
$859$	−52.8833	−1.80435	−0.902177	+	0.431366i	$0.858032\pi$
$860$	−5.32568	−0.181604
$861$	0	0
$862$	24.7297	0.842297
$863$	7.21199	0.245499	0.122749	−	0.992438i	$-0.460829\pi$
$863$	7.21199	0.245499	0.122749	+	0.992438i	$0.460829\pi$
$864$	0	0
$865$	−42.1030	−1.43155
$866$	29.5107	1.00281
$867$	0	0
$868$	−4.75242	−0.161308
$869$	−13.1104	−0.444738
$870$	0	0
$871$	34.1184	1.15606
$872$	46.5247	1.57552
$873$	0	0
$874$	17.8923	0.605218
$875$	13.1834	0.445680
$876$	0	0
$877$	26.9780	0.910981	0.455491	−	0.890241i	$-0.349464\pi$
$877$	26.9780	0.910981	0.455491	+	0.890241i	$0.349464\pi$
$878$	−1.36458	−0.0460524
$879$	0	0
$880$	−44.4983	−1.50004
$881$	5.29097	0.178257	0.0891286	−	0.996020i	$-0.471592\pi$
$881$	5.29097	0.178257	0.0891286	+	0.996020i	$0.471592\pi$
$882$	0	0
$883$	−13.8366	−0.465640	−0.232820	−	0.972520i	$-0.574795\pi$
$883$	−13.8366	−0.465640	−0.232820	+	0.972520i	$0.574795\pi$
$884$	6.07579	0.204351
$885$	0	0
$886$	1.32309	0.0444500
$887$	−4.51400	−0.151565	−0.0757827	−	0.997124i	$-0.524146\pi$
$887$	−4.51400	−0.151565	−0.0757827	+	0.997124i	$0.524146\pi$
$888$	0	0
$889$	−23.3596	−0.783457
$890$	−78.2714	−2.62366
$891$	0	0
$892$	−0.288027	−0.00964387
$893$	23.1107	0.773371
$894$	0	0
$895$	−71.0404	−2.37462
$896$	−41.4558	−1.38494
$897$	0	0
$898$	3.09476	0.103274
$899$	−47.5904	−1.58723
$900$	0	0
$901$	−36.6487	−1.22094
$902$	−24.0617	−0.801168
$903$	0	0
$904$	−27.7457	−0.922807
$905$	−33.9097	−1.12720
$906$	0	0
$907$	30.3503	1.00777	0.503883	−	0.863772i	$-0.331904\pi$
$907$	30.3503	1.00777	0.503883	+	0.863772i	$0.331904\pi$
$908$	−1.44941	−0.0481002
$909$	0	0
$910$	44.7556	1.48363
$911$	−49.5420	−1.64140	−0.820699	−	0.571361i	$-0.806416\pi$
$911$	−49.5420	−1.64140	−0.820699	+	0.571361i	$0.806416\pi$
$912$	0	0
$913$	28.6148	0.947013
$914$	34.3890	1.13749
$915$	0	0
$916$	3.28211	0.108444
$917$	2.02023	0.0667140
$918$	0	0
$919$	−21.1065	−0.696240	−0.348120	−	0.937450i	$-0.613180\pi$
$919$	−21.1065	−0.696240	−0.348120	+	0.937450i	$0.613180\pi$
$920$	13.3435	0.439922
$921$	0	0
$922$	10.5166	0.346345
$923$	−16.2608	−0.535232
$924$	0	0
$925$	−73.8702	−2.42884
$926$	20.4284	0.671318
$927$	0	0
$928$	14.7309	0.483567
$929$	−19.5353	−0.640931	−0.320466	−	0.947260i	$-0.603839\pi$
$929$	−19.5353	−0.640931	−0.320466	+	0.947260i	$0.603839\pi$
$930$	0	0
$931$	22.8892	0.750164
$932$	1.00166	0.0328105
$933$	0	0
$934$	−29.1466	−0.953706
$935$	74.7738	2.44536
$936$	0	0
$937$	11.8539	0.387251	0.193625	−	0.981076i	$-0.437975\pi$
$937$	11.8539	0.387251	0.193625	+	0.981076i	$0.437975\pi$
$938$	−58.3344	−1.90469
$939$	0	0
$940$	−2.89970	−0.0945779
$941$	−10.5964	−0.345434	−0.172717	−	0.984972i	$-0.555255\pi$
$941$	−10.5964	−0.345434	−0.172717	+	0.984972i	$0.555255\pi$
$942$	0	0
$943$	8.27638	0.269516
$944$	−6.28144	−0.204443
$945$	0	0
$946$	−24.6329	−0.800884
$947$	−34.6300	−1.12532	−0.562661	−	0.826687i	$-0.690223\pi$
$947$	−34.6300	−1.12532	−0.562661	+	0.826687i	$0.690223\pi$
$948$	0	0
$949$	40.3016	1.30824
$950$	72.7154	2.35920
$951$	0	0
$952$	61.7449	2.00116
$953$	22.7248	0.736127	0.368064	−	0.929801i	$-0.380021\pi$
$953$	22.7248	0.736127	0.368064	+	0.929801i	$0.380021\pi$
$954$	0	0
$955$	−8.20860	−0.265624
$956$	2.18886	0.0707929
$957$	0	0
$958$	−39.0866	−1.26283
$959$	−30.0993	−0.971958
$960$	0	0
$961$	−3.70292	−0.119449
$962$	−49.9912	−1.61178
$963$	0	0
$964$	2.81383	0.0906273
$965$	−27.8774	−0.897407
$966$	0	0
$967$	−19.3569	−0.622475	−0.311238	−	0.950332i	$-0.600743\pi$
$967$	−19.3569	−0.622475	−0.311238	+	0.950332i	$0.600743\pi$
$968$	5.89772	0.189560
$969$	0	0
$970$	−4.44485	−0.142715
$971$	−5.45220	−0.174970	−0.0874848	−	0.996166i	$-0.527883\pi$
$971$	−5.45220	−0.174970	−0.0874848	+	0.996166i	$0.527883\pi$
$972$	0	0
$973$	16.3321	0.523581
$974$	25.2370	0.808646
$975$	0	0
$976$	−33.8363	−1.08307
$977$	−36.6280	−1.17183	−0.585916	−	0.810372i	$-0.699265\pi$
$977$	−36.6280	−1.17183	−0.585916	+	0.810372i	$0.699265\pi$
$978$	0	0
$979$	−45.5739	−1.45655
$980$	−2.87191	−0.0917397
$981$	0	0
$982$	−32.6367	−1.04148
$983$	10.9256	0.348472	0.174236	−	0.984704i	$-0.444255\pi$
$983$	10.9256	0.348472	0.174236	+	0.984704i	$0.444255\pi$
$984$	0	0
$985$	−62.6271	−1.99547
$986$	−104.026	−3.31287
$987$	0	0
$988$	6.19474	0.197081
$989$	8.47284	0.269421
$990$	0	0
$991$	−11.7781	−0.374144	−0.187072	−	0.982346i	$-0.559900\pi$
$991$	−11.7781	−0.374144	−0.187072	+	0.982346i	$0.559900\pi$
$992$	−8.44943	−0.268270
$993$	0	0
$994$	27.8022	0.881832
$995$	−3.40627	−0.107986
$996$	0	0
$997$	−26.4900	−0.838947	−0.419473	−	0.907768i	$-0.637785\pi$
$997$	−26.4900	−0.838947	−0.419473	+	0.907768i	$0.637785\pi$
$998$	−3.43494	−0.108731
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2007.2.a.n.1.11		14
3.2	odd	2		669.2.a.i.1.4	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
669.2.a.i.1.4	✓	14	3.2	odd	2
2007.2.a.n.1.11		14	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:	\( N \)	\(=\)	\( 2007 = 3^{2} \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2007.a (trivial)

\(f(q)\)	\(=\)	\(q+1.51262 q^{2} +0.288027 q^{4} -3.35334 q^{5} -3.15808 q^{7} -2.58957 q^{8} +O(q^{10})\)	\(q+1.51262 q^{2} +0.288027 q^{4} -3.35334 q^{5} -3.15808 q^{7} -2.58957 q^{8} -5.07234 q^{10} -2.95339 q^{11} +2.79394 q^{13} -4.77698 q^{14} -4.49310 q^{16} +7.55008 q^{17} +7.69789 q^{19} -0.965854 q^{20} -4.46736 q^{22} +1.53661 q^{23} +6.24488 q^{25} +4.22618 q^{26} -0.909613 q^{28} -9.10880 q^{29} +5.22466 q^{31} -1.61722 q^{32} +11.4204 q^{34} +10.5901 q^{35} -11.8289 q^{37} +11.6440 q^{38} +8.68370 q^{40} +5.38611 q^{41} +5.51397 q^{43} -0.850657 q^{44} +2.32432 q^{46} +3.00222 q^{47} +2.97344 q^{49} +9.44615 q^{50} +0.804732 q^{52} -4.85408 q^{53} +9.90371 q^{55} +8.17805 q^{56} -13.7782 q^{58} +1.39802 q^{59} +7.53074 q^{61} +7.90294 q^{62} +6.53995 q^{64} -9.36903 q^{65} +12.2116 q^{67} +2.17463 q^{68} +16.0188 q^{70} -5.82004 q^{71} +14.4246 q^{73} -17.8927 q^{74} +2.21720 q^{76} +9.32703 q^{77} +4.43909 q^{79} +15.0669 q^{80} +8.14715 q^{82} -9.68881 q^{83} -25.3180 q^{85} +8.34055 q^{86} +7.64800 q^{88} +15.4310 q^{89} -8.82348 q^{91} +0.442587 q^{92} +4.54122 q^{94} -25.8136 q^{95} +0.876292 q^{97} +4.49769 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 16 q^{4} - q^{5} + 14 q^{7} + 3 q^{8}+O(q^{10}) \) 14 * q + 16 * q^4 - q^5 + 14 * q^7 + 3 * q^8	\( 14 q + 16 q^{4} - q^{5} + 14 q^{7} + 3 q^{8} + 4 q^{10} - 13 q^{11} + 10 q^{13} + 12 q^{16} - q^{17} + 32 q^{19} + 10 q^{20} - 3 q^{22} + 7 q^{23} + 25 q^{25} + 3 q^{26} + 28 q^{28} + q^{29} + 18 q^{31} + 16 q^{32} + q^{34} + 12 q^{35} + 16 q^{37} + 31 q^{38} + q^{40} - 6 q^{41} + 15 q^{43} + 12 q^{44} + 6 q^{46} + 21 q^{47} + 28 q^{49} + 32 q^{50} - 7 q^{52} + 17 q^{53} - q^{55} + 33 q^{56} - 27 q^{58} - 12 q^{59} + 28 q^{61} + 16 q^{62} + 3 q^{64} + 4 q^{65} + 17 q^{67} - 5 q^{68} - 14 q^{70} + 3 q^{71} + 17 q^{73} + 16 q^{74} + 14 q^{76} + 14 q^{77} + 23 q^{79} + 36 q^{80} + 5 q^{82} - 18 q^{83} - 11 q^{85} - 9 q^{86} - 35 q^{88} - 10 q^{89} + 16 q^{91} + 45 q^{92} - 43 q^{94} + 20 q^{95} + 29 q^{97} + 19 q^{98}+O(q^{100}) \) 14 * q + 16 * q^4 - q^5 + 14 * q^7 + 3 * q^8 + 4 * q^10 - 13 * q^11 + 10 * q^13 + 12 * q^16 - q^17 + 32 * q^19 + 10 * q^20 - 3 * q^22 + 7 * q^23 + 25 * q^25 + 3 * q^26 + 28 * q^28 + q^29 + 18 * q^31 + 16 * q^32 + q^34 + 12 * q^35 + 16 * q^37 + 31 * q^38 + q^40 - 6 * q^41 + 15 * q^43 + 12 * q^44 + 6 * q^46 + 21 * q^47 + 28 * q^49 + 32 * q^50 - 7 * q^52 + 17 * q^53 - q^55 + 33 * q^56 - 27 * q^58 - 12 * q^59 + 28 * q^61 + 16 * q^62 + 3 * q^64 + 4 * q^65 + 17 * q^67 - 5 * q^68 - 14 * q^70 + 3 * q^71 + 17 * q^73 + 16 * q^74 + 14 * q^76 + 14 * q^77 + 23 * q^79 + 36 * q^80 + 5 * q^82 - 18 * q^83 - 11 * q^85 - 9 * q^86 - 35 * q^88 - 10 * q^89 + 16 * q^91 + 45 * q^92 - 43 * q^94 + 20 * q^95 + 29 * q^97 + 19 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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