LMFDB - Embedded newform 4338.2.a.w.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4338,2,Mod(1,4338)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4338, 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("4338.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4338 = 2 \cdot 3^{2} \cdot 241 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4338.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:	$34.6391043968$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 20x^{5} + 26x^{4} + 95x^{3} - 121x^{2} - 126x + 158 $ x^7 - x^6 - 20x^5 + 26x^4 + 95x^3 - 121x^2 - 126*x + 158
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1446)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.85662$ of defining polynomial
Character	$\chi$	$=$	4338.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -4.06778 q^{5} +1.66241 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -4.06778 q^{5} +1.66241 q^{7} +1.00000 q^{8} -4.06778 q^{10} +3.18556 q^{11} +6.40958 q^{13} +1.66241 q^{14} +1.00000 q^{16} +0.638281 q^{17} -2.74717 q^{19} -4.06778 q^{20} +3.18556 q^{22} -8.43626 q^{23} +11.5469 q^{25} +6.40958 q^{26} +1.66241 q^{28} -3.52314 q^{29} +1.63953 q^{31} +1.00000 q^{32} +0.638281 q^{34} -6.76234 q^{35} +1.36172 q^{37} -2.74717 q^{38} -4.06778 q^{40} +8.76234 q^{41} -12.5862 q^{43} +3.18556 q^{44} -8.43626 q^{46} +0.890649 q^{47} -4.23638 q^{49} +11.5469 q^{50} +6.40958 q^{52} +11.0915 q^{53} -12.9582 q^{55} +1.66241 q^{56} -3.52314 q^{58} -0.343511 q^{59} +14.7141 q^{61} +1.63953 q^{62} +1.00000 q^{64} -26.0728 q^{65} +12.0844 q^{67} +0.638281 q^{68} -6.76234 q^{70} +3.38840 q^{71} +11.4121 q^{73} +1.36172 q^{74} -2.74717 q^{76} +5.29571 q^{77} -7.17363 q^{79} -4.06778 q^{80} +8.76234 q^{82} -3.79545 q^{83} -2.59639 q^{85} -12.5862 q^{86} +3.18556 q^{88} +13.3677 q^{89} +10.6554 q^{91} -8.43626 q^{92} +0.890649 q^{94} +11.1749 q^{95} +3.43248 q^{97} -4.23638 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 7 q^{2} + 7 q^{4} - 5 q^{5} + 7 q^{7} + 7 q^{8}+O(q^{10}) $ 7 * q + 7 * q^2 + 7 * q^4 - 5 * q^5 + 7 * q^7 + 7 * q^8	$ 7 q + 7 q^{2} + 7 q^{4} - 5 q^{5} + 7 q^{7} + 7 q^{8} - 5 q^{10} - 4 q^{11} + 14 q^{13} + 7 q^{14} + 7 q^{16} - 4 q^{17} + 7 q^{19} - 5 q^{20} - 4 q^{22} + q^{23} + 14 q^{25} + 14 q^{26} + 7 q^{28} - 3 q^{29} + 8 q^{31} + 7 q^{32} - 4 q^{34} + 17 q^{35} + 18 q^{37} + 7 q^{38} - 5 q^{40} - 3 q^{41} + 11 q^{43} - 4 q^{44} + q^{46} + 18 q^{47} + 20 q^{49} + 14 q^{50} + 14 q^{52} + 9 q^{53} - 4 q^{55} + 7 q^{56} - 3 q^{58} - 6 q^{59} + 31 q^{61} + 8 q^{62} + 7 q^{64} + 6 q^{65} + 4 q^{67} - 4 q^{68} + 17 q^{70} + 3 q^{71} + 16 q^{73} + 18 q^{74} + 7 q^{76} + 34 q^{79} - 5 q^{80} - 3 q^{82} - 10 q^{83} + 34 q^{85} + 11 q^{86} - 4 q^{88} - 24 q^{89} + 40 q^{91} + q^{92} + 18 q^{94} + q^{95} + 27 q^{97} + 20 q^{98}+O(q^{100}) $ 7 * q + 7 * q^2 + 7 * q^4 - 5 * q^5 + 7 * q^7 + 7 * q^8 - 5 * q^10 - 4 * q^11 + 14 * q^13 + 7 * q^14 + 7 * q^16 - 4 * q^17 + 7 * q^19 - 5 * q^20 - 4 * q^22 + q^23 + 14 * q^25 + 14 * q^26 + 7 * q^28 - 3 * q^29 + 8 * q^31 + 7 * q^32 - 4 * q^34 + 17 * q^35 + 18 * q^37 + 7 * q^38 - 5 * q^40 - 3 * q^41 + 11 * q^43 - 4 * q^44 + q^46 + 18 * q^47 + 20 * q^49 + 14 * q^50 + 14 * q^52 + 9 * q^53 - 4 * q^55 + 7 * q^56 - 3 * q^58 - 6 * q^59 + 31 * q^61 + 8 * q^62 + 7 * q^64 + 6 * q^65 + 4 * q^67 - 4 * q^68 + 17 * q^70 + 3 * q^71 + 16 * q^73 + 18 * q^74 + 7 * q^76 + 34 * q^79 - 5 * q^80 - 3 * q^82 - 10 * q^83 + 34 * q^85 + 11 * q^86 - 4 * q^88 - 24 * q^89 + 40 * q^91 + q^92 + 18 * q^94 + q^95 + 27 * q^97 + 20 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−4.06778	−1.81917	−0.909584	−	0.415520i	$-0.863600\pi$
$5$	−4.06778	−1.81917	−0.909584	+	0.415520i	$0.863600\pi$
$6$	0	0
$7$	1.66241	0.628333	0.314167	−	0.949368i	$-0.398275\pi$
$7$	1.66241	0.628333	0.314167	+	0.949368i	$0.398275\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−4.06778	−1.28635
$11$	3.18556	0.960482	0.480241	−	0.877137i	$-0.340549\pi$
$11$	3.18556	0.960482	0.480241	+	0.877137i	$0.340549\pi$
$12$	0	0
$13$	6.40958	1.77770	0.888849	−	0.458199i	$-0.151505\pi$
$13$	6.40958	1.77770	0.888849	+	0.458199i	$0.151505\pi$
$14$	1.66241	0.444299
$15$	0	0
$16$	1.00000	0.250000
$17$	0.638281	0.154806	0.0774030	−	0.997000i	$-0.475337\pi$
$17$	0.638281	0.154806	0.0774030	+	0.997000i	$0.475337\pi$
$18$	0	0
$19$	−2.74717	−0.630244	−0.315122	−	0.949051i	$-0.602045\pi$
$19$	−2.74717	−0.630244	−0.315122	+	0.949051i	$0.602045\pi$
$20$	−4.06778	−0.909584
$21$	0	0
$22$	3.18556	0.679163
$23$	−8.43626	−1.75908	−0.879541	−	0.475823i	$-0.842150\pi$
$23$	−8.43626	−1.75908	−0.879541	+	0.475823i	$0.842150\pi$
$24$	0	0
$25$	11.5469	2.30937
$26$	6.40958	1.25702
$27$	0	0
$28$	1.66241	0.314167
$29$	−3.52314	−0.654231	−0.327116	−	0.944984i	$-0.606077\pi$
$29$	−3.52314	−0.654231	−0.327116	+	0.944984i	$0.606077\pi$
$30$	0	0
$31$	1.63953	0.294468	0.147234	−	0.989102i	$-0.452963\pi$
$31$	1.63953	0.294468	0.147234	+	0.989102i	$0.452963\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	0.638281	0.109464
$35$	−6.76234	−1.14304
$36$	0	0
$37$	1.36172	0.223865	0.111933	−	0.993716i	$-0.464296\pi$
$37$	1.36172	0.223865	0.111933	+	0.993716i	$0.464296\pi$
$38$	−2.74717	−0.445650
$39$	0	0
$40$	−4.06778	−0.643173
$41$	8.76234	1.36845	0.684224	−	0.729272i	$-0.260141\pi$
$41$	8.76234	1.36845	0.684224	+	0.729272i	$0.260141\pi$
$42$	0	0
$43$	−12.5862	−1.91937	−0.959687	−	0.281071i	$-0.909310\pi$
$43$	−12.5862	−1.91937	−0.959687	+	0.281071i	$0.909310\pi$
$44$	3.18556	0.480241
$45$	0	0
$46$	−8.43626	−1.24386
$47$	0.890649	0.129914	0.0649572	−	0.997888i	$-0.479309\pi$
$47$	0.890649	0.129914	0.0649572	+	0.997888i	$0.479309\pi$
$48$	0	0
$49$	−4.23638	−0.605197
$50$	11.5469	1.63297
$51$	0	0
$52$	6.40958	0.888849
$53$	11.0915	1.52353	0.761767	−	0.647851i	$-0.224332\pi$
$53$	11.0915	1.52353	0.761767	+	0.647851i	$0.224332\pi$
$54$	0	0
$55$	−12.9582	−1.74728
$56$	1.66241	0.222149
$57$	0	0
$58$	−3.52314	−0.462611
$59$	−0.343511	−0.0447214	−0.0223607	−	0.999750i	$-0.507118\pi$
$59$	−0.343511	−0.0447214	−0.0223607	+	0.999750i	$0.507118\pi$
$60$	0	0
$61$	14.7141	1.88394	0.941972	−	0.335691i	$-0.108970\pi$
$61$	14.7141	1.88394	0.941972	+	0.335691i	$0.108970\pi$
$62$	1.63953	0.208221
$63$	0	0
$64$	1.00000	0.125000
$65$	−26.0728	−3.23393
$66$	0	0
$67$	12.0844	1.47634	0.738169	−	0.674615i	$-0.235691\pi$
$67$	12.0844	1.47634	0.738169	+	0.674615i	$0.235691\pi$
$68$	0.638281	0.0774030
$69$	0	0
$70$	−6.76234	−0.808254
$71$	3.38840	0.402129	0.201064	−	0.979578i	$-0.435560\pi$
$71$	3.38840	0.402129	0.201064	+	0.979578i	$0.435560\pi$
$72$	0	0
$73$	11.4121	1.33569	0.667844	−	0.744301i	$-0.267217\pi$
$73$	11.4121	1.33569	0.667844	+	0.744301i	$0.267217\pi$
$74$	1.36172	0.158297
$75$	0	0
$76$	−2.74717	−0.315122
$77$	5.29571	0.603502
$78$	0	0
$79$	−7.17363	−0.807097	−0.403548	−	0.914958i	$-0.632223\pi$
$79$	−7.17363	−0.807097	−0.403548	+	0.914958i	$0.632223\pi$
$80$	−4.06778	−0.454792
$81$	0	0
$82$	8.76234	0.967638
$83$	−3.79545	−0.416605	−0.208302	−	0.978064i	$-0.566794\pi$
$83$	−3.79545	−0.416605	−0.208302	+	0.978064i	$0.566794\pi$
$84$	0	0
$85$	−2.59639	−0.281618
$86$	−12.5862	−1.35720
$87$	0	0
$88$	3.18556	0.339582
$89$	13.3677	1.41698	0.708489	−	0.705722i	$-0.249377\pi$
$89$	13.3677	1.41698	0.708489	+	0.705722i	$0.249377\pi$
$90$	0	0
$91$	10.6554	1.11699
$92$	−8.43626	−0.879541
$93$	0	0
$94$	0.890649	0.0918634
$95$	11.1749	1.14652
$96$	0	0
$97$	3.43248	0.348515	0.174258	−	0.984700i	$-0.444247\pi$
$97$	3.43248	0.348515	0.174258	+	0.984700i	$0.444247\pi$
$98$	−4.23638	−0.427939
$99$	0	0
$100$	11.5469	1.15469
$101$	−12.8565	−1.27927	−0.639635	−	0.768679i	$-0.720915\pi$
$101$	−12.8565	−1.27927	−0.639635	+	0.768679i	$0.720915\pi$
$102$	0	0
$103$	−16.8298	−1.65829	−0.829145	−	0.559033i	$-0.811172\pi$
$103$	−16.8298	−1.65829	−0.829145	+	0.559033i	$0.811172\pi$
$104$	6.40958	0.628511
$105$	0	0
$106$	11.0915	1.07730
$107$	13.7039	1.32481	0.662404	−	0.749147i	$-0.269536\pi$
$107$	13.7039	1.32481	0.662404	+	0.749147i	$0.269536\pi$
$108$	0	0
$109$	15.8759	1.52064	0.760318	−	0.649552i	$-0.225043\pi$
$109$	15.8759	1.52064	0.760318	+	0.649552i	$0.225043\pi$
$110$	−12.9582	−1.23551
$111$	0	0
$112$	1.66241	0.157083
$113$	−1.08074	−0.101668	−0.0508339	−	0.998707i	$-0.516188\pi$
$113$	−1.08074	−0.101668	−0.0508339	+	0.998707i	$0.516188\pi$
$114$	0	0
$115$	34.3169	3.20007
$116$	−3.52314	−0.327116
$117$	0	0
$118$	−0.343511	−0.0316228
$119$	1.06109	0.0972697
$120$	0	0
$121$	−0.852226	−0.0774751
$122$	14.7141	1.33215
$123$	0	0
$124$	1.63953	0.147234
$125$	−26.6312	−2.38197
$126$	0	0
$127$	19.4710	1.72778	0.863888	−	0.503685i	$-0.168023\pi$
$127$	19.4710	1.72778	0.863888	+	0.503685i	$0.168023\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−26.0728	−2.28674
$131$	−15.8996	−1.38916	−0.694579	−	0.719417i	$-0.744409\pi$
$131$	−15.8996	−1.38916	−0.694579	+	0.719417i	$0.744409\pi$
$132$	0	0
$133$	−4.56693	−0.396003
$134$	12.0844	1.04393
$135$	0	0
$136$	0.638281	0.0547322
$137$	11.7878	1.00710	0.503548	−	0.863967i	$-0.332028\pi$
$137$	11.7878	1.00710	0.503548	+	0.863967i	$0.332028\pi$
$138$	0	0
$139$	2.83280	0.240275	0.120138	−	0.992757i	$-0.461666\pi$
$139$	2.83280	0.240275	0.120138	+	0.992757i	$0.461666\pi$
$140$	−6.76234	−0.571522
$141$	0	0
$142$	3.38840	0.284348
$143$	20.4181	1.70745
$144$	0	0
$145$	14.3314	1.19016
$146$	11.4121	0.944475
$147$	0	0
$148$	1.36172	0.111933
$149$	12.2338	1.00223	0.501116	−	0.865380i	$-0.332923\pi$
$149$	12.2338	1.00223	0.501116	+	0.865380i	$0.332923\pi$
$150$	0	0
$151$	3.87769	0.315562	0.157781	−	0.987474i	$-0.449566\pi$
$151$	3.87769	0.315562	0.157781	+	0.987474i	$0.449566\pi$
$152$	−2.74717	−0.222825
$153$	0	0
$154$	5.29571	0.426741
$155$	−6.66925	−0.535687
$156$	0	0
$157$	−10.2342	−0.816781	−0.408391	−	0.912807i	$-0.633910\pi$
$157$	−10.2342	−0.816781	−0.408391	+	0.912807i	$0.633910\pi$
$158$	−7.17363	−0.570703
$159$	0	0
$160$	−4.06778	−0.321587
$161$	−14.0246	−1.10529
$162$	0	0
$163$	−1.21142	−0.0948854	−0.0474427	−	0.998874i	$-0.515107\pi$
$163$	−1.21142	−0.0948854	−0.0474427	+	0.998874i	$0.515107\pi$
$164$	8.76234	0.684224
$165$	0	0
$166$	−3.79545	−0.294584
$167$	−6.17615	−0.477925	−0.238962	−	0.971029i	$-0.576807\pi$
$167$	−6.17615	−0.477925	−0.238962	+	0.971029i	$0.576807\pi$
$168$	0	0
$169$	28.0828	2.16021
$170$	−2.59639	−0.199134
$171$	0	0
$172$	−12.5862	−0.959687
$173$	−4.22120	−0.320932	−0.160466	−	0.987041i	$-0.551300\pi$
$173$	−4.22120	−0.320932	−0.160466	+	0.987041i	$0.551300\pi$
$174$	0	0
$175$	19.1957	1.45106
$176$	3.18556	0.240120
$177$	0	0
$178$	13.3677	1.00195
$179$	−0.162152	−0.0121198	−0.00605991	−	0.999982i	$-0.501929\pi$
$179$	−0.162152	−0.0121198	−0.00605991	+	0.999982i	$0.501929\pi$
$180$	0	0
$181$	−13.1420	−0.976836	−0.488418	−	0.872610i	$-0.662426\pi$
$181$	−13.1420	−0.976836	−0.488418	+	0.872610i	$0.662426\pi$
$182$	10.6554	0.789829
$183$	0	0
$184$	−8.43626	−0.621929
$185$	−5.53918	−0.407248
$186$	0	0
$187$	2.03328	0.148688
$188$	0.890649	0.0649572
$189$	0	0
$190$	11.1749	0.810712
$191$	1.02205	0.0739528	0.0369764	−	0.999316i	$-0.488227\pi$
$191$	1.02205	0.0739528	0.0369764	+	0.999316i	$0.488227\pi$
$192$	0	0
$193$	14.6036	1.05119	0.525595	−	0.850735i	$-0.323843\pi$
$193$	14.6036	1.05119	0.525595	+	0.850735i	$0.323843\pi$
$194$	3.43248	0.246438
$195$	0	0
$196$	−4.23638	−0.302599
$197$	24.6532	1.75647	0.878233	−	0.478233i	$-0.158723\pi$
$197$	24.6532	1.75647	0.878233	+	0.478233i	$0.158723\pi$
$198$	0	0
$199$	22.3615	1.58516	0.792582	−	0.609765i	$-0.208736\pi$
$199$	22.3615	1.58516	0.792582	+	0.609765i	$0.208736\pi$
$200$	11.5469	0.816486
$201$	0	0
$202$	−12.8565	−0.904581
$203$	−5.85692	−0.411075
$204$	0	0
$205$	−35.6433	−2.48944
$206$	−16.8298	−1.17259
$207$	0	0
$208$	6.40958	0.444425
$209$	−8.75127	−0.605338
$210$	0	0
$211$	22.8834	1.57536	0.787681	−	0.616084i	$-0.211282\pi$
$211$	22.8834	1.57536	0.787681	+	0.616084i	$0.211282\pi$
$212$	11.0915	0.761767
$213$	0	0
$214$	13.7039	0.936781
$215$	51.1978	3.49166
$216$	0	0
$217$	2.72558	0.185024
$218$	15.8759	1.07525
$219$	0	0
$220$	−12.9582	−0.873639
$221$	4.09112	0.275198
$222$	0	0
$223$	6.90753	0.462562	0.231281	−	0.972887i	$-0.425708\pi$
$223$	6.90753	0.462562	0.231281	+	0.972887i	$0.425708\pi$
$224$	1.66241	0.111075
$225$	0	0
$226$	−1.08074	−0.0718900
$227$	8.47315	0.562383	0.281191	−	0.959652i	$-0.409270\pi$
$227$	8.47315	0.562383	0.281191	+	0.959652i	$0.409270\pi$
$228$	0	0
$229$	−5.79770	−0.383123	−0.191561	−	0.981481i	$-0.561355\pi$
$229$	−5.79770	−0.383123	−0.191561	+	0.981481i	$0.561355\pi$
$230$	34.3169	2.26279
$231$	0	0
$232$	−3.52314	−0.231306
$233$	−27.1555	−1.77902	−0.889508	−	0.456919i	$-0.848953\pi$
$233$	−27.1555	−1.77902	−0.889508	+	0.456919i	$0.848953\pi$
$234$	0	0
$235$	−3.62297	−0.236336
$236$	−0.343511	−0.0223607
$237$	0	0
$238$	1.06109	0.0687801
$239$	−0.371761	−0.0240472	−0.0120236	−	0.999928i	$-0.503827\pi$
$239$	−0.371761	−0.0240472	−0.0120236	+	0.999928i	$0.503827\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	−0.852226	−0.0547832
$243$	0	0
$244$	14.7141	0.941972
$245$	17.2327	1.10096
$246$	0	0
$247$	−17.6082	−1.12038
$248$	1.63953	0.104110
$249$	0	0
$250$	−26.6312	−1.68431
$251$	30.1054	1.90024	0.950118	−	0.311890i	$-0.100962\pi$
$251$	30.1054	1.90024	0.950118	+	0.311890i	$0.100962\pi$
$252$	0	0
$253$	−26.8742	−1.68957
$254$	19.4710	1.22172
$255$	0	0
$256$	1.00000	0.0625000
$257$	16.9267	1.05586	0.527930	−	0.849288i	$-0.322968\pi$
$257$	16.9267	1.05586	0.527930	+	0.849288i	$0.322968\pi$
$258$	0	0
$259$	2.26374	0.140662
$260$	−26.0728	−1.61697
$261$	0	0
$262$	−15.8996	−0.982282
$263$	−0.677728	−0.0417905	−0.0208952	−	0.999782i	$-0.506652\pi$
$263$	−0.677728	−0.0417905	−0.0208952	+	0.999782i	$0.506652\pi$
$264$	0	0
$265$	−45.1178	−2.77157
$266$	−4.56693	−0.280017
$267$	0	0
$268$	12.0844	0.738169
$269$	1.43003	0.0871904	0.0435952	−	0.999049i	$-0.486119\pi$
$269$	1.43003	0.0871904	0.0435952	+	0.999049i	$0.486119\pi$
$270$	0	0
$271$	−13.0566	−0.793131	−0.396565	−	0.918006i	$-0.629798\pi$
$271$	−13.0566	−0.793131	−0.396565	+	0.918006i	$0.629798\pi$
$272$	0.638281	0.0387015
$273$	0	0
$274$	11.7878	0.712125
$275$	36.7832	2.21811
$276$	0	0
$277$	27.8188	1.67147	0.835736	−	0.549132i	$-0.185042\pi$
$277$	27.8188	1.67147	0.835736	+	0.549132i	$0.185042\pi$
$278$	2.83280	0.169900
$279$	0	0
$280$	−6.76234	−0.404127
$281$	−7.59944	−0.453344	−0.226672	−	0.973971i	$-0.572785\pi$
$281$	−7.59944	−0.453344	−0.226672	+	0.973971i	$0.572785\pi$
$282$	0	0
$283$	−11.6934	−0.695100	−0.347550	−	0.937662i	$-0.612986\pi$
$283$	−11.6934	−0.695100	−0.347550	+	0.937662i	$0.612986\pi$
$284$	3.38840	0.201064
$285$	0	0
$286$	20.4181	1.20735
$287$	14.5666	0.859841
$288$	0	0
$289$	−16.5926	−0.976035
$290$	14.3314	0.841568
$291$	0	0
$292$	11.4121	0.667844
$293$	14.3711	0.839569	0.419785	−	0.907624i	$-0.362106\pi$
$293$	14.3711	0.839569	0.419785	+	0.907624i	$0.362106\pi$
$294$	0	0
$295$	1.39733	0.0813557
$296$	1.36172	0.0791483
$297$	0	0
$298$	12.2338	0.708685
$299$	−54.0729	−3.12712
$300$	0	0
$301$	−20.9234	−1.20601
$302$	3.87769	0.223136
$303$	0	0
$304$	−2.74717	−0.157561
$305$	−59.8537	−3.42721
$306$	0	0
$307$	−13.9635	−0.796938	−0.398469	−	0.917182i	$-0.630458\pi$
$307$	−13.9635	−0.796938	−0.398469	+	0.917182i	$0.630458\pi$
$308$	5.29571	0.301751
$309$	0	0
$310$	−6.66925	−0.378788
$311$	−28.2534	−1.60210	−0.801052	−	0.598594i	$-0.795726\pi$
$311$	−28.2534	−1.60210	−0.801052	+	0.598594i	$0.795726\pi$
$312$	0	0
$313$	6.47877	0.366201	0.183101	−	0.983094i	$-0.441387\pi$
$313$	6.47877	0.366201	0.183101	+	0.983094i	$0.441387\pi$
$314$	−10.2342	−0.577552
$315$	0	0
$316$	−7.17363	−0.403548
$317$	3.96608	0.222757	0.111379	−	0.993778i	$-0.464473\pi$
$317$	3.96608	0.222757	0.111379	+	0.993778i	$0.464473\pi$
$318$	0	0
$319$	−11.2232	−0.628377
$320$	−4.06778	−0.227396
$321$	0	0
$322$	−14.0246	−0.781558
$323$	−1.75347	−0.0975656
$324$	0	0
$325$	74.0106	4.10537
$326$	−1.21142	−0.0670941
$327$	0	0
$328$	8.76234	0.483819
$329$	1.48063	0.0816296
$330$	0	0
$331$	−13.5451	−0.744506	−0.372253	−	0.928131i	$-0.621415\pi$
$331$	−13.5451	−0.744506	−0.372253	+	0.928131i	$0.621415\pi$
$332$	−3.79545	−0.208302
$333$	0	0
$334$	−6.17615	−0.337944
$335$	−49.1565	−2.68571
$336$	0	0
$337$	−9.06345	−0.493717	−0.246859	−	0.969051i	$-0.579398\pi$
$337$	−9.06345	−0.493717	−0.246859	+	0.969051i	$0.579398\pi$
$338$	28.0828	1.52750
$339$	0	0
$340$	−2.59639	−0.140809
$341$	5.22282	0.282831
$342$	0	0
$343$	−18.6795	−1.00860
$344$	−12.5862	−0.678601
$345$	0	0
$346$	−4.22120	−0.226933
$347$	−7.56743	−0.406241	−0.203121	−	0.979154i	$-0.565108\pi$
$347$	−7.56743	−0.406241	−0.203121	+	0.979154i	$0.565108\pi$
$348$	0	0
$349$	−25.6597	−1.37353	−0.686767	−	0.726878i	$-0.740971\pi$
$349$	−25.6597	−1.37353	−0.686767	+	0.726878i	$0.740971\pi$
$350$	19.1957	1.02605
$351$	0	0
$352$	3.18556	0.169791
$353$	21.0491	1.12033	0.560164	−	0.828382i	$-0.310738\pi$
$353$	21.0491	1.12033	0.560164	+	0.828382i	$0.310738\pi$
$354$	0	0
$355$	−13.7833	−0.731539
$356$	13.3677	0.708489
$357$	0	0
$358$	−0.162152	−0.00857000
$359$	−8.69119	−0.458703	−0.229352	−	0.973344i	$-0.573661\pi$
$359$	−8.69119	−0.458703	−0.229352	+	0.973344i	$0.573661\pi$
$360$	0	0
$361$	−11.4531	−0.602792
$362$	−13.1420	−0.690727
$363$	0	0
$364$	10.6554	0.558494
$365$	−46.4221	−2.42984
$366$	0	0
$367$	−28.7356	−1.49999	−0.749993	−	0.661445i	$-0.769943\pi$
$367$	−28.7356	−1.49999	−0.749993	+	0.661445i	$0.769943\pi$
$368$	−8.43626	−0.439771
$369$	0	0
$370$	−5.53918	−0.287968
$371$	18.4387	0.957287
$372$	0	0
$373$	−34.4749	−1.78504	−0.892521	−	0.451006i	$-0.851065\pi$
$373$	−34.4749	−1.78504	−0.892521	+	0.451006i	$0.851065\pi$
$374$	2.03328	0.105139
$375$	0	0
$376$	0.890649	0.0459317
$377$	−22.5819	−1.16303
$378$	0	0
$379$	24.1559	1.24080	0.620402	−	0.784284i	$-0.286969\pi$
$379$	24.1559	1.24080	0.620402	+	0.784284i	$0.286969\pi$
$380$	11.1749	0.573260
$381$	0	0
$382$	1.02205	0.0522925
$383$	14.5717	0.744580	0.372290	−	0.928116i	$-0.378573\pi$
$383$	14.5717	0.744580	0.372290	+	0.928116i	$0.378573\pi$
$384$	0	0
$385$	−21.5418	−1.09787
$386$	14.6036	0.743304
$387$	0	0
$388$	3.43248	0.174258
$389$	−20.4727	−1.03801	−0.519004	−	0.854772i	$-0.673697\pi$
$389$	−20.4727	−1.03801	−0.519004	+	0.854772i	$0.673697\pi$
$390$	0	0
$391$	−5.38471	−0.272316
$392$	−4.23638	−0.213970
$393$	0	0
$394$	24.6532	1.24201
$395$	29.1808	1.46824
$396$	0	0
$397$	−3.83807	−0.192627	−0.0963136	−	0.995351i	$-0.530705\pi$
$397$	−3.83807	−0.192627	−0.0963136	+	0.995351i	$0.530705\pi$
$398$	22.3615	1.12088
$399$	0	0
$400$	11.5469	0.577343
$401$	−11.5476	−0.576657	−0.288329	−	0.957531i	$-0.593100\pi$
$401$	−11.5476	−0.576657	−0.288329	+	0.957531i	$0.593100\pi$
$402$	0	0
$403$	10.5087	0.523476
$404$	−12.8565	−0.639635
$405$	0	0
$406$	−5.85692	−0.290674
$407$	4.33783	0.215018
$408$	0	0
$409$	15.2832	0.755706	0.377853	−	0.925866i	$-0.376662\pi$
$409$	15.2832	0.755706	0.377853	+	0.925866i	$0.376662\pi$
$410$	−35.6433	−1.76030
$411$	0	0
$412$	−16.8298	−0.829145
$413$	−0.571058	−0.0280999
$414$	0	0
$415$	15.4391	0.757874
$416$	6.40958	0.314256
$417$	0	0
$418$	−8.75127	−0.428039
$419$	−0.326413	−0.0159463	−0.00797317	−	0.999968i	$-0.502538\pi$
$419$	−0.326413	−0.0159463	−0.00797317	+	0.999968i	$0.502538\pi$
$420$	0	0
$421$	15.0352	0.732772	0.366386	−	0.930463i	$-0.380595\pi$
$421$	15.0352	0.732772	0.366386	+	0.930463i	$0.380595\pi$
$422$	22.8834	1.11395
$423$	0	0
$424$	11.0915	0.538651
$425$	7.37015	0.357505
$426$	0	0
$427$	24.4609	1.18374
$428$	13.7039	0.662404
$429$	0	0
$430$	51.1978	2.46898
$431$	21.9999	1.05970	0.529849	−	0.848092i	$-0.322249\pi$
$431$	21.9999	1.05970	0.529849	+	0.848092i	$0.322249\pi$
$432$	0	0
$433$	−27.7764	−1.33485	−0.667424	−	0.744678i	$-0.732603\pi$
$433$	−27.7764	−1.33485	−0.667424	+	0.744678i	$0.732603\pi$
$434$	2.72558	0.130832
$435$	0	0
$436$	15.8759	0.760318
$437$	23.1759	1.10865
$438$	0	0
$439$	−26.0316	−1.24242	−0.621211	−	0.783644i	$-0.713359\pi$
$439$	−26.0316	−1.24242	−0.621211	+	0.783644i	$0.713359\pi$
$440$	−12.9582	−0.617756
$441$	0	0
$442$	4.09112	0.194595
$443$	10.4678	0.497339	0.248669	−	0.968588i	$-0.420007\pi$
$443$	10.4678	0.497339	0.248669	+	0.968588i	$0.420007\pi$
$444$	0	0
$445$	−54.3771	−2.57772
$446$	6.90753	0.327081
$447$	0	0
$448$	1.66241	0.0785416
$449$	12.6814	0.598471	0.299235	−	0.954179i	$-0.403268\pi$
$449$	12.6814	0.598471	0.299235	+	0.954179i	$0.403268\pi$
$450$	0	0
$451$	27.9129	1.31437
$452$	−1.08074	−0.0508339
$453$	0	0
$454$	8.47315	0.397665
$455$	−43.3438	−2.03199
$456$	0	0
$457$	−12.8853	−0.602749	−0.301375	−	0.953506i	$-0.597445\pi$
$457$	−12.8853	−0.602749	−0.301375	+	0.953506i	$0.597445\pi$
$458$	−5.79770	−0.270909
$459$	0	0
$460$	34.3169	1.60003
$461$	−17.1367	−0.798138	−0.399069	−	0.916921i	$-0.630667\pi$
$461$	−17.1367	−0.798138	−0.399069	+	0.916921i	$0.630667\pi$
$462$	0	0
$463$	−17.6421	−0.819900	−0.409950	−	0.912108i	$-0.634454\pi$
$463$	−17.6421	−0.819900	−0.409950	+	0.912108i	$0.634454\pi$
$464$	−3.52314	−0.163558
$465$	0	0
$466$	−27.1555	−1.25795
$467$	−8.62304	−0.399027	−0.199513	−	0.979895i	$-0.563936\pi$
$467$	−8.62304	−0.399027	−0.199513	+	0.979895i	$0.563936\pi$
$468$	0	0
$469$	20.0892	0.927632
$470$	−3.62297	−0.167115
$471$	0	0
$472$	−0.343511	−0.0158114
$473$	−40.0940	−1.84352
$474$	0	0
$475$	−31.7212	−1.45547
$476$	1.06109	0.0486349
$477$	0	0
$478$	−0.371761	−0.0170039
$479$	1.12942	0.0516047	0.0258023	−	0.999667i	$-0.491786\pi$
$479$	1.12942	0.0516047	0.0258023	+	0.999667i	$0.491786\pi$
$480$	0	0
$481$	8.72805	0.397965
$482$	1.00000	0.0455488
$483$	0	0
$484$	−0.852226	−0.0387375
$485$	−13.9626	−0.634008
$486$	0	0
$487$	−35.6405	−1.61503	−0.807513	−	0.589850i	$-0.799187\pi$
$487$	−35.6405	−1.61503	−0.807513	+	0.589850i	$0.799187\pi$
$488$	14.7141	0.666075
$489$	0	0
$490$	17.2327	0.778493
$491$	13.9855	0.631155	0.315577	−	0.948900i	$-0.397802\pi$
$491$	13.9855	0.631155	0.315577	+	0.948900i	$0.397802\pi$
$492$	0	0
$493$	−2.24876	−0.101279
$494$	−17.6082	−0.792231
$495$	0	0
$496$	1.63953	0.0736171
$497$	5.63291	0.252671
$498$	0	0
$499$	32.1977	1.44137	0.720683	−	0.693265i	$-0.243829\pi$
$499$	32.1977	1.44137	0.720683	+	0.693265i	$0.243829\pi$
$500$	−26.6312	−1.19098
$501$	0	0
$502$	30.1054	1.34367
$503$	−10.0043	−0.446068	−0.223034	−	0.974811i	$-0.571596\pi$
$503$	−10.0043	−0.446068	−0.223034	+	0.974811i	$0.571596\pi$
$504$	0	0
$505$	52.2975	2.32721
$506$	−26.8742	−1.19470
$507$	0	0
$508$	19.4710	0.863888
$509$	−7.86938	−0.348804	−0.174402	−	0.984675i	$-0.555799\pi$
$509$	−7.86938	−0.348804	−0.174402	+	0.984675i	$0.555799\pi$
$510$	0	0
$511$	18.9717	0.839257
$512$	1.00000	0.0441942
$513$	0	0
$514$	16.9267	0.746606
$515$	68.4600	3.01671
$516$	0	0
$517$	2.83721	0.124780
$518$	2.26374	0.0994630
$519$	0	0
$520$	−26.0728	−1.14337
$521$	−9.11592	−0.399376	−0.199688	−	0.979860i	$-0.563993\pi$
$521$	−9.11592	−0.399376	−0.199688	+	0.979860i	$0.563993\pi$
$522$	0	0
$523$	−1.80118	−0.0787599	−0.0393800	−	0.999224i	$-0.512538\pi$
$523$	−1.80118	−0.0787599	−0.0393800	+	0.999224i	$0.512538\pi$
$524$	−15.8996	−0.694579
$525$	0	0
$526$	−0.677728	−0.0295503
$527$	1.04648	0.0455855
$528$	0	0
$529$	48.1705	2.09437
$530$	−45.1178	−1.95979
$531$	0	0
$532$	−4.56693	−0.198002
$533$	56.1629	2.43269
$534$	0	0
$535$	−55.7446	−2.41005
$536$	12.0844	0.521965
$537$	0	0
$538$	1.43003	0.0616529
$539$	−13.4952	−0.581281
$540$	0	0
$541$	0.960654	0.0413018	0.0206509	−	0.999787i	$-0.493426\pi$
$541$	0.960654	0.0413018	0.0206509	+	0.999787i	$0.493426\pi$
$542$	−13.0566	−0.560828
$543$	0	0
$544$	0.638281	0.0273661
$545$	−64.5797	−2.76629
$546$	0	0
$547$	−16.9435	−0.724451	−0.362225	−	0.932091i	$-0.617983\pi$
$547$	−16.9435	−0.724451	−0.362225	+	0.932091i	$0.617983\pi$
$548$	11.7878	0.503548
$549$	0	0
$550$	36.7832	1.56844
$551$	9.67868	0.412326
$552$	0	0
$553$	−11.9255	−0.507126
$554$	27.8188	1.18191
$555$	0	0
$556$	2.83280	0.120138
$557$	10.1810	0.431384	0.215692	−	0.976461i	$-0.430799\pi$
$557$	10.1810	0.431384	0.215692	+	0.976461i	$0.430799\pi$
$558$	0	0
$559$	−80.6722	−3.41207
$560$	−6.76234	−0.285761
$561$	0	0
$562$	−7.59944	−0.320563
$563$	−13.7012	−0.577435	−0.288718	−	0.957414i	$-0.593229\pi$
$563$	−13.7012	−0.577435	−0.288718	+	0.957414i	$0.593229\pi$
$564$	0	0
$565$	4.39623	0.184951
$566$	−11.6934	−0.491510
$567$	0	0
$568$	3.38840	0.142174
$569$	22.0239	0.923290	0.461645	−	0.887065i	$-0.347259\pi$
$569$	22.0239	0.923290	0.461645	+	0.887065i	$0.347259\pi$
$570$	0	0
$571$	36.4636	1.52595	0.762976	−	0.646427i	$-0.223737\pi$
$571$	36.4636	1.52595	0.762976	+	0.646427i	$0.223737\pi$
$572$	20.4181	0.853724
$573$	0	0
$574$	14.5666	0.607999
$575$	−97.4123	−4.06238
$576$	0	0
$577$	−16.7345	−0.696665	−0.348332	−	0.937371i	$-0.613252\pi$
$577$	−16.7345	−0.696665	−0.348332	+	0.937371i	$0.613252\pi$
$578$	−16.5926	−0.690161
$579$	0	0
$580$	14.3314	0.595078
$581$	−6.30961	−0.261767
$582$	0	0
$583$	35.3326	1.46333
$584$	11.4121	0.472237
$585$	0	0
$586$	14.3711	0.593665
$587$	14.9409	0.616678	0.308339	−	0.951276i	$-0.400227\pi$
$587$	14.9409	0.616678	0.308339	+	0.951276i	$0.400227\pi$
$588$	0	0
$589$	−4.50407	−0.185587
$590$	1.39733	0.0575272
$591$	0	0
$592$	1.36172	0.0559663
$593$	−26.7329	−1.09779	−0.548894	−	0.835892i	$-0.684951\pi$
$593$	−26.7329	−1.09779	−0.548894	+	0.835892i	$0.684951\pi$
$594$	0	0
$595$	−4.31627	−0.176950
$596$	12.2338	0.501116
$597$	0	0
$598$	−54.0729	−2.21121
$599$	−12.4441	−0.508450	−0.254225	−	0.967145i	$-0.581820\pi$
$599$	−12.4441	−0.508450	−0.254225	+	0.967145i	$0.581820\pi$
$600$	0	0
$601$	−19.6871	−0.803053	−0.401526	−	0.915847i	$-0.631520\pi$
$601$	−19.6871	−0.803053	−0.401526	+	0.915847i	$0.631520\pi$
$602$	−20.9234	−0.852775
$603$	0	0
$604$	3.87769	0.157781
$605$	3.46667	0.140940
$606$	0	0
$607$	−3.58955	−0.145695	−0.0728476	−	0.997343i	$-0.523209\pi$
$607$	−3.58955	−0.145695	−0.0728476	+	0.997343i	$0.523209\pi$
$608$	−2.74717	−0.111413
$609$	0	0
$610$	−59.8537	−2.42340
$611$	5.70869	0.230949
$612$	0	0
$613$	11.6811	0.471797	0.235898	−	0.971778i	$-0.424197\pi$
$613$	11.6811	0.471797	0.235898	+	0.971778i	$0.424197\pi$
$614$	−13.9635	−0.563520
$615$	0	0
$616$	5.29571	0.213370
$617$	28.8210	1.16029	0.580144	−	0.814514i	$-0.302996\pi$
$617$	28.8210	1.16029	0.580144	+	0.814514i	$0.302996\pi$
$618$	0	0
$619$	−15.7205	−0.631860	−0.315930	−	0.948783i	$-0.602316\pi$
$619$	−15.7205	−0.631860	−0.315930	+	0.948783i	$0.602316\pi$
$620$	−6.66925	−0.267844
$621$	0	0
$622$	−28.2534	−1.13286
$623$	22.2227	0.890334
$624$	0	0
$625$	50.5957	2.02383
$626$	6.47877	0.258944
$627$	0	0
$628$	−10.2342	−0.408391
$629$	0.869160	0.0346557
$630$	0	0
$631$	−35.5357	−1.41465	−0.707327	−	0.706886i	$-0.750099\pi$
$631$	−35.5357	−1.41465	−0.707327	+	0.706886i	$0.750099\pi$
$632$	−7.17363	−0.285352
$633$	0	0
$634$	3.96608	0.157513
$635$	−79.2040	−3.14311
$636$	0	0
$637$	−27.1534	−1.07586
$638$	−11.2232	−0.444330
$639$	0	0
$640$	−4.06778	−0.160793
$641$	−20.5600	−0.812073	−0.406036	−	0.913857i	$-0.633089\pi$
$641$	−20.5600	−0.812073	−0.406036	+	0.913857i	$0.633089\pi$
$642$	0	0
$643$	−16.3112	−0.643250	−0.321625	−	0.946867i	$-0.604229\pi$
$643$	−16.3112	−0.643250	−0.321625	+	0.946867i	$0.604229\pi$
$644$	−14.0246	−0.552645
$645$	0	0
$646$	−1.75347	−0.0689893
$647$	13.7928	0.542251	0.271126	−	0.962544i	$-0.412604\pi$
$647$	13.7928	0.542251	0.271126	+	0.962544i	$0.412604\pi$
$648$	0	0
$649$	−1.09428	−0.0429541
$650$	74.0106	2.90293
$651$	0	0
$652$	−1.21142	−0.0474427
$653$	−19.3038	−0.755415	−0.377708	−	0.925925i	$-0.623288\pi$
$653$	−19.3038	−0.755415	−0.377708	+	0.925925i	$0.623288\pi$
$654$	0	0
$655$	64.6763	2.52711
$656$	8.76234	0.342112
$657$	0	0
$658$	1.48063	0.0577208
$659$	−7.98276	−0.310964	−0.155482	−	0.987839i	$-0.549693\pi$
$659$	−7.98276	−0.310964	−0.155482	+	0.987839i	$0.549693\pi$
$660$	0	0
$661$	−30.1997	−1.17463	−0.587317	−	0.809357i	$-0.699816\pi$
$661$	−30.1997	−1.17463	−0.587317	+	0.809357i	$0.699816\pi$
$662$	−13.5451	−0.526445
$663$	0	0
$664$	−3.79545	−0.147292
$665$	18.5773	0.720397
$666$	0	0
$667$	29.7222	1.15085
$668$	−6.17615	−0.238962
$669$	0	0
$670$	−49.1565	−1.89908
$671$	46.8725	1.80949
$672$	0	0
$673$	19.3781	0.746970	0.373485	−	0.927636i	$-0.378163\pi$
$673$	19.3781	0.746970	0.373485	+	0.927636i	$0.378163\pi$
$674$	−9.06345	−0.349111
$675$	0	0
$676$	28.0828	1.08011
$677$	14.8618	0.571184	0.285592	−	0.958351i	$-0.407810\pi$
$677$	14.8618	0.571184	0.285592	+	0.958351i	$0.407810\pi$
$678$	0	0
$679$	5.70620	0.218984
$680$	−2.59639	−0.0995670
$681$	0	0
$682$	5.22282	0.199992
$683$	−17.3752	−0.664843	−0.332422	−	0.943131i	$-0.607866\pi$
$683$	−17.3752	−0.664843	−0.332422	+	0.943131i	$0.607866\pi$
$684$	0	0
$685$	−47.9501	−1.83208
$686$	−18.6795	−0.713187
$687$	0	0
$688$	−12.5862	−0.479843
$689$	71.0919	2.70839
$690$	0	0
$691$	6.66593	0.253584	0.126792	−	0.991929i	$-0.459532\pi$
$691$	6.66593	0.253584	0.126792	+	0.991929i	$0.459532\pi$
$692$	−4.22120	−0.160466
$693$	0	0
$694$	−7.56743	−0.287256
$695$	−11.5232	−0.437101
$696$	0	0
$697$	5.59284	0.211844
$698$	−25.6597	−0.971235
$699$	0	0
$700$	19.1957	0.725528
$701$	0.820806	0.0310014	0.0155007	−	0.999880i	$-0.495066\pi$
$701$	0.820806	0.0310014	0.0155007	+	0.999880i	$0.495066\pi$
$702$	0	0
$703$	−3.74087	−0.141090
$704$	3.18556	0.120060
$705$	0	0
$706$	21.0491	0.792192
$707$	−21.3728	−0.803808
$708$	0	0
$709$	7.49451	0.281462	0.140731	−	0.990048i	$-0.455055\pi$
$709$	7.49451	0.281462	0.140731	+	0.990048i	$0.455055\pi$
$710$	−13.7833	−0.517277
$711$	0	0
$712$	13.3677	0.500977
$713$	−13.8315	−0.517994
$714$	0	0
$715$	−83.0564	−3.10613
$716$	−0.162152	−0.00605991
$717$	0	0
$718$	−8.69119	−0.324352
$719$	4.96449	0.185144	0.0925722	−	0.995706i	$-0.470491\pi$
$719$	4.96449	0.185144	0.0925722	+	0.995706i	$0.470491\pi$
$720$	0	0
$721$	−27.9781	−1.04196
$722$	−11.4531	−0.426238
$723$	0	0
$724$	−13.1420	−0.488418
$725$	−40.6813	−1.51086
$726$	0	0
$727$	−28.4457	−1.05499	−0.527497	−	0.849557i	$-0.676869\pi$
$727$	−28.4457	−1.05499	−0.527497	+	0.849557i	$0.676869\pi$
$728$	10.6554	0.394915
$729$	0	0
$730$	−46.4221	−1.71816
$731$	−8.03352	−0.297131
$732$	0	0
$733$	0.0991409	0.00366185	0.00183093	−	0.999998i	$-0.499417\pi$
$733$	0.0991409	0.00366185	0.00183093	+	0.999998i	$0.499417\pi$
$734$	−28.7356	−1.06065
$735$	0	0
$736$	−8.43626	−0.310965
$737$	38.4954	1.41800
$738$	0	0
$739$	43.6107	1.60425	0.802123	−	0.597159i	$-0.203704\pi$
$739$	43.6107	1.60425	0.802123	+	0.597159i	$0.203704\pi$
$740$	−5.53918	−0.203624
$741$	0	0
$742$	18.4387	0.676904
$743$	14.9208	0.547391	0.273695	−	0.961816i	$-0.411754\pi$
$743$	14.9208	0.547391	0.273695	+	0.961816i	$0.411754\pi$
$744$	0	0
$745$	−49.7644	−1.82323
$746$	−34.4749	−1.26222
$747$	0	0
$748$	2.03328	0.0743441
$749$	22.7816	0.832421
$750$	0	0
$751$	−29.0874	−1.06141	−0.530707	−	0.847556i	$-0.678073\pi$
$751$	−29.0874	−1.06141	−0.530707	+	0.847556i	$0.678073\pi$
$752$	0.890649	0.0324786
$753$	0	0
$754$	−22.5819	−0.822384
$755$	−15.7736	−0.574060
$756$	0	0
$757$	−14.6605	−0.532845	−0.266423	−	0.963856i	$-0.585842\pi$
$757$	−14.6605	−0.532845	−0.266423	+	0.963856i	$0.585842\pi$
$758$	24.1559	0.877381
$759$	0	0
$760$	11.1749	0.405356
$761$	22.3122	0.808817	0.404409	−	0.914578i	$-0.367477\pi$
$761$	22.3122	0.808817	0.404409	+	0.914578i	$0.367477\pi$
$762$	0	0
$763$	26.3923	0.955465
$764$	1.02205	0.0369764
$765$	0	0
$766$	14.5717	0.526498
$767$	−2.20177	−0.0795011
$768$	0	0
$769$	−26.8558	−0.968446	−0.484223	−	0.874945i	$-0.660898\pi$
$769$	−26.8558	−0.968446	−0.484223	+	0.874945i	$0.660898\pi$
$770$	−21.5418	−0.776313
$771$	0	0
$772$	14.6036	0.525595
$773$	−30.1099	−1.08298	−0.541490	−	0.840707i	$-0.682139\pi$
$773$	−30.1099	−1.08298	−0.541490	+	0.840707i	$0.682139\pi$
$774$	0	0
$775$	18.9314	0.680037
$776$	3.43248	0.123219
$777$	0	0
$778$	−20.4727	−0.733983
$779$	−24.0716	−0.862456
$780$	0	0
$781$	10.7939	0.386237
$782$	−5.38471	−0.192557
$783$	0	0
$784$	−4.23638	−0.151299
$785$	41.6307	1.48586
$786$	0	0
$787$	−41.1616	−1.46725	−0.733627	−	0.679553i	$-0.762174\pi$
$787$	−41.1616	−1.46725	−0.733627	+	0.679553i	$0.762174\pi$
$788$	24.6532	0.878233
$789$	0	0
$790$	29.1808	1.03821
$791$	−1.79664	−0.0638812
$792$	0	0
$793$	94.3111	3.34909
$794$	−3.83807	−0.136208
$795$	0	0
$796$	22.3615	0.792582
$797$	32.3931	1.14742	0.573711	−	0.819058i	$-0.305503\pi$
$797$	32.3931	1.14742	0.573711	+	0.819058i	$0.305503\pi$
$798$	0	0
$799$	0.568485	0.0201115
$800$	11.5469	0.408243
$801$	0	0
$802$	−11.5476	−0.407758
$803$	36.3540	1.28290
$804$	0	0
$805$	57.0488	2.01071
$806$	10.5087	0.370153
$807$	0	0
$808$	−12.8565	−0.452290
$809$	44.2298	1.55504	0.777519	−	0.628859i	$-0.216478\pi$
$809$	44.2298	1.55504	0.777519	+	0.628859i	$0.216478\pi$
$810$	0	0
$811$	28.7700	1.01025	0.505125	−	0.863046i	$-0.331446\pi$
$811$	28.7700	1.01025	0.505125	+	0.863046i	$0.331446\pi$
$812$	−5.85692	−0.205538
$813$	0	0
$814$	4.33783	0.152041
$815$	4.92778	0.172612
$816$	0	0
$817$	34.5764	1.20967
$818$	15.2832	0.534365
$819$	0	0
$820$	−35.6433	−1.24472
$821$	17.5267	0.611687	0.305844	−	0.952082i	$-0.401062\pi$
$821$	17.5267	0.611687	0.305844	+	0.952082i	$0.401062\pi$
$822$	0	0
$823$	30.7209	1.07086	0.535431	−	0.844579i	$-0.320149\pi$
$823$	30.7209	1.07086	0.535431	+	0.844579i	$0.320149\pi$
$824$	−16.8298	−0.586294
$825$	0	0
$826$	−0.571058	−0.0198696
$827$	−7.00005	−0.243416	−0.121708	−	0.992566i	$-0.538837\pi$
$827$	−7.00005	−0.243416	−0.121708	+	0.992566i	$0.538837\pi$
$828$	0	0
$829$	−13.6212	−0.473084	−0.236542	−	0.971621i	$-0.576014\pi$
$829$	−13.6212	−0.473084	−0.236542	+	0.971621i	$0.576014\pi$
$830$	15.4391	0.535898
$831$	0	0
$832$	6.40958	0.222212
$833$	−2.70400	−0.0936882
$834$	0	0
$835$	25.1232	0.869425
$836$	−8.75127	−0.302669
$837$	0	0
$838$	−0.326413	−0.0112758
$839$	11.1178	0.383828	0.191914	−	0.981412i	$-0.438530\pi$
$839$	11.1178	0.383828	0.191914	+	0.981412i	$0.438530\pi$
$840$	0	0
$841$	−16.5875	−0.571981
$842$	15.0352	0.518148
$843$	0	0
$844$	22.8834	0.787681
$845$	−114.235	−3.92979
$846$	0	0
$847$	−1.41675	−0.0486802
$848$	11.0915	0.380884
$849$	0	0
$850$	7.37015	0.252794
$851$	−11.4878	−0.393797
$852$	0	0
$853$	58.0640	1.98807	0.994037	−	0.109046i	$-0.0347796\pi$
$853$	58.0640	1.98807	0.994037	+	0.109046i	$0.0347796\pi$
$854$	24.4609	0.837034
$855$	0	0
$856$	13.7039	0.468390
$857$	40.6990	1.39025	0.695126	−	0.718887i	$-0.255348\pi$
$857$	40.6990	1.39025	0.695126	+	0.718887i	$0.255348\pi$
$858$	0	0
$859$	22.7397	0.775870	0.387935	−	0.921687i	$-0.373189\pi$
$859$	22.7397	0.775870	0.387935	+	0.921687i	$0.373189\pi$
$860$	51.1978	1.74583
$861$	0	0
$862$	21.9999	0.749319
$863$	−13.8391	−0.471087	−0.235544	−	0.971864i	$-0.575687\pi$
$863$	−13.8391	−0.471087	−0.235544	+	0.971864i	$0.575687\pi$
$864$	0	0
$865$	17.1709	0.583829
$866$	−27.7764	−0.943881
$867$	0	0
$868$	2.72558	0.0925121
$869$	−22.8520	−0.775201
$870$	0	0
$871$	77.4557	2.62449
$872$	15.8759	0.537626
$873$	0	0
$874$	23.1759	0.783935
$875$	−44.2721	−1.49667
$876$	0	0
$877$	−2.61652	−0.0883537	−0.0441768	−	0.999024i	$-0.514067\pi$
$877$	−2.61652	−0.0883537	−0.0441768	+	0.999024i	$0.514067\pi$
$878$	−26.0316	−0.878524
$879$	0	0
$880$	−12.9582	−0.436819
$881$	−35.4384	−1.19395	−0.596975	−	0.802259i	$-0.703631\pi$
$881$	−35.4384	−1.19395	−0.596975	+	0.802259i	$0.703631\pi$
$882$	0	0
$883$	21.2001	0.713439	0.356720	−	0.934211i	$-0.383895\pi$
$883$	21.2001	0.713439	0.356720	+	0.934211i	$0.383895\pi$
$884$	4.09112	0.137599
$885$	0	0
$886$	10.4678	0.351672
$887$	−7.84817	−0.263516	−0.131758	−	0.991282i	$-0.542062\pi$
$887$	−7.84817	−0.263516	−0.131758	+	0.991282i	$0.542062\pi$
$888$	0	0
$889$	32.3689	1.08562
$890$	−54.3771	−1.82272
$891$	0	0
$892$	6.90753	0.231281
$893$	−2.44677	−0.0818779
$894$	0	0
$895$	0.659600	0.0220480
$896$	1.66241	0.0555373
$897$	0	0
$898$	12.6814	0.423183
$899$	−5.77630	−0.192650
$900$	0	0
$901$	7.07950	0.235852
$902$	27.9129	0.929399
$903$	0	0
$904$	−1.08074	−0.0359450
$905$	53.4588	1.77703
$906$	0	0
$907$	−23.9105	−0.793935	−0.396968	−	0.917833i	$-0.629938\pi$
$907$	−23.9105	−0.793935	−0.396968	+	0.917833i	$0.629938\pi$
$908$	8.47315	0.281191
$909$	0	0
$910$	−43.3438	−1.43683
$911$	12.9320	0.428456	0.214228	−	0.976784i	$-0.431276\pi$
$911$	12.9320	0.428456	0.214228	+	0.976784i	$0.431276\pi$
$912$	0	0
$913$	−12.0906	−0.400141
$914$	−12.8853	−0.426208
$915$	0	0
$916$	−5.79770	−0.191561
$917$	−26.4318	−0.872853
$918$	0	0
$919$	15.3088	0.504990	0.252495	−	0.967598i	$-0.418749\pi$
$919$	15.3088	0.504990	0.252495	+	0.967598i	$0.418749\pi$
$920$	34.3169	1.13139
$921$	0	0
$922$	−17.1367	−0.564369
$923$	21.7182	0.714864
$924$	0	0
$925$	15.7236	0.516988
$926$	−17.6421	−0.579757
$927$	0	0
$928$	−3.52314	−0.115653
$929$	22.7010	0.744796	0.372398	−	0.928073i	$-0.378536\pi$
$929$	22.7010	0.744796	0.372398	+	0.928073i	$0.378536\pi$
$930$	0	0
$931$	11.6381	0.381422
$932$	−27.1555	−0.889508
$933$	0	0
$934$	−8.62304	−0.282154
$935$	−8.27095	−0.270489
$936$	0	0
$937$	−41.4695	−1.35475	−0.677375	−	0.735638i	$-0.736883\pi$
$937$	−41.4695	−1.35475	−0.677375	+	0.735638i	$0.736883\pi$
$938$	20.0892	0.655935
$939$	0	0
$940$	−3.62297	−0.118168
$941$	11.5775	0.377415	0.188707	−	0.982033i	$-0.439570\pi$
$941$	11.5775	0.377415	0.188707	+	0.982033i	$0.439570\pi$
$942$	0	0
$943$	−73.9214	−2.40721
$944$	−0.343511	−0.0111803
$945$	0	0
$946$	−40.0940	−1.30357
$947$	−28.7554	−0.934424	−0.467212	−	0.884145i	$-0.654742\pi$
$947$	−28.7554	−0.934424	−0.467212	+	0.884145i	$0.654742\pi$
$948$	0	0
$949$	73.1470	2.37445
$950$	−31.7212	−1.02917
$951$	0	0
$952$	1.06109	0.0343900
$953$	50.9054	1.64899	0.824494	−	0.565870i	$-0.191460\pi$
$953$	50.9054	1.64899	0.824494	+	0.565870i	$0.191460\pi$
$954$	0	0
$955$	−4.15747	−0.134533
$956$	−0.371761	−0.0120236
$957$	0	0
$958$	1.12942	0.0364900
$959$	19.5961	0.632792
$960$	0	0
$961$	−28.3119	−0.913288
$962$	8.72805	0.281404
$963$	0	0
$964$	1.00000	0.0322078
$965$	−59.4043	−1.91229
$966$	0	0
$967$	10.8579	0.349168	0.174584	−	0.984642i	$-0.444142\pi$
$967$	10.8579	0.349168	0.174584	+	0.984642i	$0.444142\pi$
$968$	−0.852226	−0.0273916
$969$	0	0
$970$	−13.9626	−0.448311
$971$	34.1125	1.09472	0.547361	−	0.836896i	$-0.315632\pi$
$971$	34.1125	1.09472	0.547361	+	0.836896i	$0.315632\pi$
$972$	0	0
$973$	4.70929	0.150973
$974$	−35.6405	−1.14200
$975$	0	0
$976$	14.7141	0.470986
$977$	−29.9590	−0.958472	−0.479236	−	0.877686i	$-0.659086\pi$
$977$	−29.9590	−0.958472	−0.479236	+	0.877686i	$0.659086\pi$
$978$	0	0
$979$	42.5837	1.36098
$980$	17.2327	0.550478
$981$	0	0
$982$	13.9855	0.446294
$983$	−32.8246	−1.04694	−0.523471	−	0.852044i	$-0.675363\pi$
$983$	−32.8246	−1.04694	−0.523471	+	0.852044i	$0.675363\pi$
$984$	0	0
$985$	−100.284	−3.19531
$986$	−2.24876	−0.0716150
$987$	0	0
$988$	−17.6082	−0.560192
$989$	106.180	3.37634
$990$	0	0
$991$	−17.5042	−0.556039	−0.278019	−	0.960576i	$-0.589678\pi$
$991$	−17.5042	−0.556039	−0.278019	+	0.960576i	$0.589678\pi$
$992$	1.63953	0.0520551
$993$	0	0
$994$	5.63291	0.178665
$995$	−90.9617	−2.88368
$996$	0	0
$997$	−33.3584	−1.05647	−0.528236	−	0.849098i	$-0.677146\pi$
$997$	−33.3584	−1.05647	−0.528236	+	0.849098i	$0.677146\pi$
$998$	32.1977	1.01920
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4338.2.a.w.1.1		7
3.2	odd	2		1446.2.a.n.1.7	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1446.2.a.n.1.7	✓	7	3.2	odd	2
4338.2.a.w.1.1		7	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:	\( N \)	\(=\)	\( 4338 = 2 \cdot 3^{2} \cdot 241 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4338.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -4.06778 q^{5} +1.66241 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -4.06778 q^{5} +1.66241 q^{7} +1.00000 q^{8} -4.06778 q^{10} +3.18556 q^{11} +6.40958 q^{13} +1.66241 q^{14} +1.00000 q^{16} +0.638281 q^{17} -2.74717 q^{19} -4.06778 q^{20} +3.18556 q^{22} -8.43626 q^{23} +11.5469 q^{25} +6.40958 q^{26} +1.66241 q^{28} -3.52314 q^{29} +1.63953 q^{31} +1.00000 q^{32} +0.638281 q^{34} -6.76234 q^{35} +1.36172 q^{37} -2.74717 q^{38} -4.06778 q^{40} +8.76234 q^{41} -12.5862 q^{43} +3.18556 q^{44} -8.43626 q^{46} +0.890649 q^{47} -4.23638 q^{49} +11.5469 q^{50} +6.40958 q^{52} +11.0915 q^{53} -12.9582 q^{55} +1.66241 q^{56} -3.52314 q^{58} -0.343511 q^{59} +14.7141 q^{61} +1.63953 q^{62} +1.00000 q^{64} -26.0728 q^{65} +12.0844 q^{67} +0.638281 q^{68} -6.76234 q^{70} +3.38840 q^{71} +11.4121 q^{73} +1.36172 q^{74} -2.74717 q^{76} +5.29571 q^{77} -7.17363 q^{79} -4.06778 q^{80} +8.76234 q^{82} -3.79545 q^{83} -2.59639 q^{85} -12.5862 q^{86} +3.18556 q^{88} +13.3677 q^{89} +10.6554 q^{91} -8.43626 q^{92} +0.890649 q^{94} +11.1749 q^{95} +3.43248 q^{97} -4.23638 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 7 q^{2} + 7 q^{4} - 5 q^{5} + 7 q^{7} + 7 q^{8}+O(q^{10}) \) 7 * q + 7 * q^2 + 7 * q^4 - 5 * q^5 + 7 * q^7 + 7 * q^8	\( 7 q + 7 q^{2} + 7 q^{4} - 5 q^{5} + 7 q^{7} + 7 q^{8} - 5 q^{10} - 4 q^{11} + 14 q^{13} + 7 q^{14} + 7 q^{16} - 4 q^{17} + 7 q^{19} - 5 q^{20} - 4 q^{22} + q^{23} + 14 q^{25} + 14 q^{26} + 7 q^{28} - 3 q^{29} + 8 q^{31} + 7 q^{32} - 4 q^{34} + 17 q^{35} + 18 q^{37} + 7 q^{38} - 5 q^{40} - 3 q^{41} + 11 q^{43} - 4 q^{44} + q^{46} + 18 q^{47} + 20 q^{49} + 14 q^{50} + 14 q^{52} + 9 q^{53} - 4 q^{55} + 7 q^{56} - 3 q^{58} - 6 q^{59} + 31 q^{61} + 8 q^{62} + 7 q^{64} + 6 q^{65} + 4 q^{67} - 4 q^{68} + 17 q^{70} + 3 q^{71} + 16 q^{73} + 18 q^{74} + 7 q^{76} + 34 q^{79} - 5 q^{80} - 3 q^{82} - 10 q^{83} + 34 q^{85} + 11 q^{86} - 4 q^{88} - 24 q^{89} + 40 q^{91} + q^{92} + 18 q^{94} + q^{95} + 27 q^{97} + 20 q^{98}+O(q^{100}) \) 7 * q + 7 * q^2 + 7 * q^4 - 5 * q^5 + 7 * q^7 + 7 * q^8 - 5 * q^10 - 4 * q^11 + 14 * q^13 + 7 * q^14 + 7 * q^16 - 4 * q^17 + 7 * q^19 - 5 * q^20 - 4 * q^22 + q^23 + 14 * q^25 + 14 * q^26 + 7 * q^28 - 3 * q^29 + 8 * q^31 + 7 * q^32 - 4 * q^34 + 17 * q^35 + 18 * q^37 + 7 * q^38 - 5 * q^40 - 3 * q^41 + 11 * q^43 - 4 * q^44 + q^46 + 18 * q^47 + 20 * q^49 + 14 * q^50 + 14 * q^52 + 9 * q^53 - 4 * q^55 + 7 * q^56 - 3 * q^58 - 6 * q^59 + 31 * q^61 + 8 * q^62 + 7 * q^64 + 6 * q^65 + 4 * q^67 - 4 * q^68 + 17 * q^70 + 3 * q^71 + 16 * q^73 + 18 * q^74 + 7 * q^76 + 34 * q^79 - 5 * q^80 - 3 * q^82 - 10 * q^83 + 34 * q^85 + 11 * q^86 - 4 * q^88 - 24 * q^89 + 40 * q^91 + q^92 + 18 * q^94 + q^95 + 27 * q^97 + 20 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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