LMFDB - Embedded newform 7146.2.a.be.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7146,2,Mod(1,7146)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7146, 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("7146.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7146 = 2 \cdot 3^{2} \cdot 397 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7146.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:	$57.0610972844$
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} - x^{13} - 32 x^{12} + 33 x^{11} + 399 x^{10} - 423 x^{9} - 2413 x^{8} + 2601 x^{7} + 7136 x^{6} + \cdots - 1498 $ x^14 - x^13 - 32x^12 + 33x^11 + 399x^10 - 423x^9 - 2413x^8 + 2601x^7 + 7136x^6 - 7553x^5 - 9310x^4 + 8511x^3 + 5545x^2 - 3002x - 1498
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 794)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-0.447625$ of defining polynomial
Character	$\chi$	$=$	7146.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.34910 q^{5} +4.95731 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -1.34910 q^{5} +4.95731 q^{7} -1.00000 q^{8} +1.34910 q^{10} -2.19805 q^{11} +0.925316 q^{13} -4.95731 q^{14} +1.00000 q^{16} +4.56127 q^{17} -3.63175 q^{19} -1.34910 q^{20} +2.19805 q^{22} -7.58553 q^{23} -3.17994 q^{25} -0.925316 q^{26} +4.95731 q^{28} +5.05344 q^{29} -0.162725 q^{31} -1.00000 q^{32} -4.56127 q^{34} -6.68789 q^{35} +5.61399 q^{37} +3.63175 q^{38} +1.34910 q^{40} -8.67270 q^{41} -1.17630 q^{43} -2.19805 q^{44} +7.58553 q^{46} +8.70886 q^{47} +17.5749 q^{49} +3.17994 q^{50} +0.925316 q^{52} -4.17360 q^{53} +2.96538 q^{55} -4.95731 q^{56} -5.05344 q^{58} +7.77000 q^{59} -10.8481 q^{61} +0.162725 q^{62} +1.00000 q^{64} -1.24834 q^{65} +11.0660 q^{67} +4.56127 q^{68} +6.68789 q^{70} +11.6965 q^{71} +9.21305 q^{73} -5.61399 q^{74} -3.63175 q^{76} -10.8964 q^{77} -4.73532 q^{79} -1.34910 q^{80} +8.67270 q^{82} +3.16799 q^{83} -6.15359 q^{85} +1.17630 q^{86} +2.19805 q^{88} +8.78993 q^{89} +4.58707 q^{91} -7.58553 q^{92} -8.70886 q^{94} +4.89958 q^{95} -3.24337 q^{97} -17.5749 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 14 q^{2} + 14 q^{4} - 7 q^{5} + 7 q^{7} - 14 q^{8}+O(q^{10}) $ 14 * q - 14 * q^2 + 14 * q^4 - 7 * q^5 + 7 * q^7 - 14 * q^8	$ 14 q - 14 q^{2} + 14 q^{4} - 7 q^{5} + 7 q^{7} - 14 q^{8} + 7 q^{10} - 3 q^{11} + 7 q^{13} - 7 q^{14} + 14 q^{16} - 10 q^{17} + 7 q^{19} - 7 q^{20} + 3 q^{22} - 14 q^{23} + 27 q^{25} - 7 q^{26} + 7 q^{28} - 20 q^{29} + 6 q^{31} - 14 q^{32} + 10 q^{34} + 16 q^{35} + 20 q^{37} - 7 q^{38} + 7 q^{40} - 4 q^{41} + 3 q^{43} - 3 q^{44} + 14 q^{46} + 23 q^{49} - 27 q^{50} + 7 q^{52} - 21 q^{53} - 14 q^{55} - 7 q^{56} + 20 q^{58} + 30 q^{59} + 19 q^{61} - 6 q^{62} + 14 q^{64} - 12 q^{65} - 21 q^{67} - 10 q^{68} - 16 q^{70} + 11 q^{71} + 9 q^{73} - 20 q^{74} + 7 q^{76} + 26 q^{77} + 22 q^{79} - 7 q^{80} + 4 q^{82} + 51 q^{83} - 10 q^{85} - 3 q^{86} + 3 q^{88} + 6 q^{89} - 8 q^{91} - 14 q^{92} + 40 q^{95} - 15 q^{97} - 23 q^{98}+O(q^{100}) $ 14 * q - 14 * q^2 + 14 * q^4 - 7 * q^5 + 7 * q^7 - 14 * q^8 + 7 * q^10 - 3 * q^11 + 7 * q^13 - 7 * q^14 + 14 * q^16 - 10 * q^17 + 7 * q^19 - 7 * q^20 + 3 * q^22 - 14 * q^23 + 27 * q^25 - 7 * q^26 + 7 * q^28 - 20 * q^29 + 6 * q^31 - 14 * q^32 + 10 * q^34 + 16 * q^35 + 20 * q^37 - 7 * q^38 + 7 * q^40 - 4 * q^41 + 3 * q^43 - 3 * q^44 + 14 * q^46 + 23 * q^49 - 27 * q^50 + 7 * q^52 - 21 * q^53 - 14 * q^55 - 7 * q^56 + 20 * q^58 + 30 * q^59 + 19 * q^61 - 6 * q^62 + 14 * q^64 - 12 * q^65 - 21 * q^67 - 10 * q^68 - 16 * q^70 + 11 * q^71 + 9 * q^73 - 20 * q^74 + 7 * q^76 + 26 * q^77 + 22 * q^79 - 7 * q^80 + 4 * q^82 + 51 * q^83 - 10 * q^85 - 3 * q^86 + 3 * q^88 + 6 * q^89 - 8 * q^91 - 14 * q^92 + 40 * q^95 - 15 * q^97 - 23 * 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
$5$	−1.34910	−0.603335	−0.301667	−	0.953413i	$-0.597543\pi$
$5$	−1.34910	−0.603335	−0.301667	+	0.953413i	$0.597543\pi$
$6$	0	0
$7$	4.95731	1.87369	0.936843	−	0.349750i	$-0.113733\pi$
$7$	4.95731	1.87369	0.936843	+	0.349750i	$0.113733\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.34910	0.426622
$11$	−2.19805	−0.662737	−0.331368	−	0.943501i	$-0.607510\pi$
$11$	−2.19805	−0.662737	−0.331368	+	0.943501i	$0.607510\pi$
$12$	0	0
$13$	0.925316	0.256636	0.128318	−	0.991733i	$-0.459042\pi$
$13$	0.925316	0.256636	0.128318	+	0.991733i	$0.459042\pi$
$14$	−4.95731	−1.32490
$15$	0	0
$16$	1.00000	0.250000
$17$	4.56127	1.10627	0.553135	−	0.833092i	$-0.313431\pi$
$17$	4.56127	1.10627	0.553135	+	0.833092i	$0.313431\pi$
$18$	0	0
$19$	−3.63175	−0.833181	−0.416590	−	0.909094i	$-0.636775\pi$
$19$	−3.63175	−0.833181	−0.416590	+	0.909094i	$0.636775\pi$
$20$	−1.34910	−0.301667
$21$	0	0
$22$	2.19805	0.468626
$23$	−7.58553	−1.58169	−0.790847	−	0.612015i	$-0.790359\pi$
$23$	−7.58553	−1.58169	−0.790847	+	0.612015i	$0.790359\pi$
$24$	0	0
$25$	−3.17994	−0.635987
$26$	−0.925316	−0.181469
$27$	0	0
$28$	4.95731	0.936843
$29$	5.05344	0.938401	0.469200	−	0.883092i	$-0.344542\pi$
$29$	5.05344	0.938401	0.469200	+	0.883092i	$0.344542\pi$
$30$	0	0
$31$	−0.162725	−0.0292263	−0.0146132	−	0.999893i	$-0.504652\pi$
$31$	−0.162725	−0.0292263	−0.0146132	+	0.999893i	$0.504652\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−4.56127	−0.782251
$35$	−6.68789	−1.13046
$36$	0	0
$37$	5.61399	0.922934	0.461467	−	0.887157i	$-0.347323\pi$
$37$	5.61399	0.922934	0.461467	+	0.887157i	$0.347323\pi$
$38$	3.63175	0.589148
$39$	0	0
$40$	1.34910	0.213311
$41$	−8.67270	−1.35445	−0.677224	−	0.735777i	$-0.736817\pi$
$41$	−8.67270	−1.35445	−0.677224	+	0.735777i	$0.736817\pi$
$42$	0	0
$43$	−1.17630	−0.179384	−0.0896918	−	0.995970i	$-0.528588\pi$
$43$	−1.17630	−0.179384	−0.0896918	+	0.995970i	$0.528588\pi$
$44$	−2.19805	−0.331368
$45$	0	0
$46$	7.58553	1.11843
$47$	8.70886	1.27032	0.635159	−	0.772382i	$-0.280935\pi$
$47$	8.70886	1.27032	0.635159	+	0.772382i	$0.280935\pi$
$48$	0	0
$49$	17.5749	2.51070
$50$	3.17994	0.449711
$51$	0	0
$52$	0.925316	0.128318
$53$	−4.17360	−0.573288	−0.286644	−	0.958037i	$-0.592540\pi$
$53$	−4.17360	−0.573288	−0.286644	+	0.958037i	$0.592540\pi$
$54$	0	0
$55$	2.96538	0.399852
$56$	−4.95731	−0.662448
$57$	0	0
$58$	−5.05344	−0.663549
$59$	7.77000	1.01157	0.505784	−	0.862660i	$-0.331203\pi$
$59$	7.77000	1.01157	0.505784	+	0.862660i	$0.331203\pi$
$60$	0	0
$61$	−10.8481	−1.38896	−0.694480	−	0.719512i	$-0.744366\pi$
$61$	−10.8481	−1.38896	−0.694480	+	0.719512i	$0.744366\pi$
$62$	0.162725	0.0206661
$63$	0	0
$64$	1.00000	0.125000
$65$	−1.24834	−0.154838
$66$	0	0
$67$	11.0660	1.35193	0.675964	−	0.736935i	$-0.263727\pi$
$67$	11.0660	1.35193	0.675964	+	0.736935i	$0.263727\pi$
$68$	4.56127	0.553135
$69$	0	0
$70$	6.68789	0.799356
$71$	11.6965	1.38812	0.694058	−	0.719919i	$-0.255821\pi$
$71$	11.6965	1.38812	0.694058	+	0.719919i	$0.255821\pi$
$72$	0	0
$73$	9.21305	1.07831	0.539153	−	0.842208i	$-0.318744\pi$
$73$	9.21305	1.07831	0.539153	+	0.842208i	$0.318744\pi$
$74$	−5.61399	−0.652613
$75$	0	0
$76$	−3.63175	−0.416590
$77$	−10.8964	−1.24176
$78$	0	0
$79$	−4.73532	−0.532766	−0.266383	−	0.963867i	$-0.585829\pi$
$79$	−4.73532	−0.532766	−0.266383	+	0.963867i	$0.585829\pi$
$80$	−1.34910	−0.150834
$81$	0	0
$82$	8.67270	0.957740
$83$	3.16799	0.347732	0.173866	−	0.984769i	$-0.444374\pi$
$83$	3.16799	0.347732	0.173866	+	0.984769i	$0.444374\pi$
$84$	0	0
$85$	−6.15359	−0.667451
$86$	1.17630	0.126843
$87$	0	0
$88$	2.19805	0.234313
$89$	8.78993	0.931731	0.465865	−	0.884856i	$-0.345743\pi$
$89$	8.78993	0.931731	0.465865	+	0.884856i	$0.345743\pi$
$90$	0	0
$91$	4.58707	0.480856
$92$	−7.58553	−0.790847
$93$	0	0
$94$	−8.70886	−0.898250
$95$	4.89958	0.502687
$96$	0	0
$97$	−3.24337	−0.329314	−0.164657	−	0.986351i	$-0.552652\pi$
$97$	−3.24337	−0.329314	−0.164657	+	0.986351i	$0.552652\pi$
$98$	−17.5749	−1.77533
$99$	0	0
$100$	−3.17994	−0.317994
$101$	5.99714	0.596738	0.298369	−	0.954451i	$-0.403557\pi$
$101$	5.99714	0.596738	0.298369	+	0.954451i	$0.403557\pi$
$102$	0	0
$103$	8.61138	0.848504	0.424252	−	0.905544i	$-0.360537\pi$
$103$	8.61138	0.848504	0.424252	+	0.905544i	$0.360537\pi$
$104$	−0.925316	−0.0907347
$105$	0	0
$106$	4.17360	0.405376
$107$	17.2040	1.66317	0.831585	−	0.555397i	$-0.187434\pi$
$107$	17.2040	1.66317	0.831585	+	0.555397i	$0.187434\pi$
$108$	0	0
$109$	0.289805	0.0277583	0.0138791	−	0.999904i	$-0.495582\pi$
$109$	0.289805	0.0277583	0.0138791	+	0.999904i	$0.495582\pi$
$110$	−2.96538	−0.282738
$111$	0	0
$112$	4.95731	0.468422
$113$	−8.25002	−0.776096	−0.388048	−	0.921639i	$-0.626851\pi$
$113$	−8.25002	−0.776096	−0.388048	+	0.921639i	$0.626851\pi$
$114$	0	0
$115$	10.2336	0.954290
$116$	5.05344	0.469200
$117$	0	0
$118$	−7.77000	−0.715286
$119$	22.6116	2.07280
$120$	0	0
$121$	−6.16858	−0.560780
$122$	10.8481	0.982143
$123$	0	0
$124$	−0.162725	−0.0146132
$125$	11.0355	0.987048
$126$	0	0
$127$	−17.9409	−1.59200	−0.795999	−	0.605298i	$-0.793054\pi$
$127$	−17.9409	−1.59200	−0.795999	+	0.605298i	$0.793054\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	1.24834	0.109487
$131$	−14.1561	−1.23682	−0.618410	−	0.785856i	$-0.712223\pi$
$131$	−14.1561	−1.23682	−0.618410	+	0.785856i	$0.712223\pi$
$132$	0	0
$133$	−18.0037	−1.56112
$134$	−11.0660	−0.955957
$135$	0	0
$136$	−4.56127	−0.391125
$137$	1.08012	0.0922810	0.0461405	−	0.998935i	$-0.485308\pi$
$137$	1.08012	0.0922810	0.0461405	+	0.998935i	$0.485308\pi$
$138$	0	0
$139$	−15.8941	−1.34812	−0.674058	−	0.738678i	$-0.735450\pi$
$139$	−15.8941	−1.34812	−0.674058	+	0.738678i	$0.735450\pi$
$140$	−6.68789	−0.565230
$141$	0	0
$142$	−11.6965	−0.981546
$143$	−2.03389	−0.170082
$144$	0	0
$145$	−6.81758	−0.566169
$146$	−9.21305	−0.762477
$147$	0	0
$148$	5.61399	0.461467
$149$	18.9510	1.55252	0.776262	−	0.630411i	$-0.217114\pi$
$149$	18.9510	1.55252	0.776262	+	0.630411i	$0.217114\pi$
$150$	0	0
$151$	−5.65661	−0.460328	−0.230164	−	0.973152i	$-0.573926\pi$
$151$	−5.65661	−0.460328	−0.230164	+	0.973152i	$0.573926\pi$
$152$	3.63175	0.294574
$153$	0	0
$154$	10.8964	0.878057
$155$	0.219532	0.0176332
$156$	0	0
$157$	20.6012	1.64415	0.822077	−	0.569376i	$-0.192815\pi$
$157$	20.6012	1.64415	0.822077	+	0.569376i	$0.192815\pi$
$158$	4.73532	0.376722
$159$	0	0
$160$	1.34910	0.106655
$161$	−37.6038	−2.96360
$162$	0	0
$163$	22.4091	1.75522	0.877608	−	0.479378i	$-0.159138\pi$
$163$	22.4091	1.75522	0.877608	+	0.479378i	$0.159138\pi$
$164$	−8.67270	−0.677224
$165$	0	0
$166$	−3.16799	−0.245884
$167$	18.0954	1.40027	0.700133	−	0.714013i	$-0.253124\pi$
$167$	18.0954	1.40027	0.700133	+	0.714013i	$0.253124\pi$
$168$	0	0
$169$	−12.1438	−0.934138
$170$	6.15359	0.471959
$171$	0	0
$172$	−1.17630	−0.0896918
$173$	2.17952	0.165706	0.0828528	−	0.996562i	$-0.473597\pi$
$173$	2.17952	0.165706	0.0828528	+	0.996562i	$0.473597\pi$
$174$	0	0
$175$	−15.7639	−1.19164
$176$	−2.19805	−0.165684
$177$	0	0
$178$	−8.78993	−0.658833
$179$	15.7330	1.17594	0.587969	−	0.808883i	$-0.299928\pi$
$179$	15.7330	1.17594	0.587969	+	0.808883i	$0.299928\pi$
$180$	0	0
$181$	10.4414	0.776103	0.388052	−	0.921638i	$-0.373148\pi$
$181$	10.4414	0.776103	0.388052	+	0.921638i	$0.373148\pi$
$182$	−4.58707	−0.340017
$183$	0	0
$184$	7.58553	0.559213
$185$	−7.57382	−0.556838
$186$	0	0
$187$	−10.0259	−0.733166
$188$	8.70886	0.635159
$189$	0	0
$190$	−4.89958	−0.355453
$191$	−11.1156	−0.804299	−0.402150	−	0.915574i	$-0.631737\pi$
$191$	−11.1156	−0.804299	−0.402150	+	0.915574i	$0.631737\pi$
$192$	0	0
$193$	3.89523	0.280385	0.140193	−	0.990124i	$-0.455228\pi$
$193$	3.89523	0.280385	0.140193	+	0.990124i	$0.455228\pi$
$194$	3.24337	0.232860
$195$	0	0
$196$	17.5749	1.25535
$197$	−1.50902	−0.107513	−0.0537565	−	0.998554i	$-0.517119\pi$
$197$	−1.50902	−0.107513	−0.0537565	+	0.998554i	$0.517119\pi$
$198$	0	0
$199$	11.9161	0.844712	0.422356	−	0.906430i	$-0.361203\pi$
$199$	11.9161	0.844712	0.422356	+	0.906430i	$0.361203\pi$
$200$	3.17994	0.224856
$201$	0	0
$202$	−5.99714	−0.421957
$203$	25.0515	1.75827
$204$	0	0
$205$	11.7003	0.817186
$206$	−8.61138	−0.599983
$207$	0	0
$208$	0.925316	0.0641591
$209$	7.98277	0.552180
$210$	0	0
$211$	−0.999252	−0.0687913	−0.0343957	−	0.999408i	$-0.510951\pi$
$211$	−0.999252	−0.0687913	−0.0343957	+	0.999408i	$0.510951\pi$
$212$	−4.17360	−0.286644
$213$	0	0
$214$	−17.2040	−1.17604
$215$	1.58694	0.108228
$216$	0	0
$217$	−0.806679	−0.0547609
$218$	−0.289805	−0.0196281
$219$	0	0
$220$	2.96538	0.199926
$221$	4.22061	0.283909
$222$	0	0
$223$	1.01101	0.0677022	0.0338511	−	0.999427i	$-0.489223\pi$
$223$	1.01101	0.0677022	0.0338511	+	0.999427i	$0.489223\pi$
$224$	−4.95731	−0.331224
$225$	0	0
$226$	8.25002	0.548783
$227$	17.8679	1.18593	0.592967	−	0.805227i	$-0.297957\pi$
$227$	17.8679	1.18593	0.592967	+	0.805227i	$0.297957\pi$
$228$	0	0
$229$	0.979883	0.0647525	0.0323763	−	0.999476i	$-0.489693\pi$
$229$	0.979883	0.0647525	0.0323763	+	0.999476i	$0.489693\pi$
$230$	−10.2336	−0.674785
$231$	0	0
$232$	−5.05344	−0.331775
$233$	−7.29869	−0.478153	−0.239076	−	0.971001i	$-0.576845\pi$
$233$	−7.29869	−0.478153	−0.239076	+	0.971001i	$0.576845\pi$
$234$	0	0
$235$	−11.7491	−0.766426
$236$	7.77000	0.505784
$237$	0	0
$238$	−22.6116	−1.46569
$239$	−23.0334	−1.48991	−0.744953	−	0.667117i	$-0.767528\pi$
$239$	−23.0334	−1.48991	−0.744953	+	0.667117i	$0.767528\pi$
$240$	0	0
$241$	−15.8573	−1.02146	−0.510728	−	0.859742i	$-0.670624\pi$
$241$	−15.8573	−1.02146	−0.510728	+	0.859742i	$0.670624\pi$
$242$	6.16858	0.396531
$243$	0	0
$244$	−10.8481	−0.694480
$245$	−23.7103	−1.51479
$246$	0	0
$247$	−3.36052	−0.213825
$248$	0.162725	0.0103331
$249$	0	0
$250$	−11.0355	−0.697948
$251$	7.97624	0.503456	0.251728	−	0.967798i	$-0.419001\pi$
$251$	7.97624	0.503456	0.251728	+	0.967798i	$0.419001\pi$
$252$	0	0
$253$	16.6734	1.04825
$254$	17.9409	1.12571
$255$	0	0
$256$	1.00000	0.0625000
$257$	−4.87093	−0.303840	−0.151920	−	0.988393i	$-0.548546\pi$
$257$	−4.87093	−0.303840	−0.151920	+	0.988393i	$0.548546\pi$
$258$	0	0
$259$	27.8303	1.72929
$260$	−1.24834	−0.0774188
$261$	0	0
$262$	14.1561	0.874564
$263$	−31.5063	−1.94276	−0.971382	−	0.237525i	$-0.923664\pi$
$263$	−31.5063	−1.94276	−0.971382	+	0.237525i	$0.923664\pi$
$264$	0	0
$265$	5.63059	0.345884
$266$	18.0037	1.10388
$267$	0	0
$268$	11.0660	0.675964
$269$	−5.70493	−0.347836	−0.173918	−	0.984760i	$-0.555643\pi$
$269$	−5.70493	−0.347836	−0.173918	+	0.984760i	$0.555643\pi$
$270$	0	0
$271$	6.23357	0.378662	0.189331	−	0.981913i	$-0.439368\pi$
$271$	6.23357	0.378662	0.189331	+	0.981913i	$0.439368\pi$
$272$	4.56127	0.276567
$273$	0	0
$274$	−1.08012	−0.0652525
$275$	6.98966	0.421492
$276$	0	0
$277$	12.3348	0.741129	0.370564	−	0.928807i	$-0.379164\pi$
$277$	12.3348	0.741129	0.370564	+	0.928807i	$0.379164\pi$
$278$	15.8941	0.953262
$279$	0	0
$280$	6.68789	0.399678
$281$	−23.8730	−1.42415	−0.712073	−	0.702105i	$-0.752244\pi$
$281$	−23.8730	−1.42415	−0.712073	+	0.702105i	$0.752244\pi$
$282$	0	0
$283$	−28.8683	−1.71604	−0.858022	−	0.513614i	$-0.828307\pi$
$283$	−28.8683	−1.71604	−0.858022	+	0.513614i	$0.828307\pi$
$284$	11.6965	0.694058
$285$	0	0
$286$	2.03389	0.120266
$287$	−42.9933	−2.53781
$288$	0	0
$289$	3.80516	0.223833
$290$	6.81758	0.400342
$291$	0	0
$292$	9.21305	0.539153
$293$	22.9256	1.33933	0.669663	−	0.742665i	$-0.266438\pi$
$293$	22.9256	1.33933	0.669663	+	0.742665i	$0.266438\pi$
$294$	0	0
$295$	−10.4825	−0.610313
$296$	−5.61399	−0.326307
$297$	0	0
$298$	−18.9510	−1.09780
$299$	−7.01901	−0.405920
$300$	0	0
$301$	−5.83126	−0.336108
$302$	5.65661	0.325501
$303$	0	0
$304$	−3.63175	−0.208295
$305$	14.6352	0.838008
$306$	0	0
$307$	23.8143	1.35915	0.679576	−	0.733605i	$-0.262164\pi$
$307$	23.8143	1.35915	0.679576	+	0.733605i	$0.262164\pi$
$308$	−10.8964	−0.620880
$309$	0	0
$310$	−0.219532	−0.0124686
$311$	10.7820	0.611394	0.305697	−	0.952129i	$-0.401111\pi$
$311$	10.7820	0.611394	0.305697	+	0.952129i	$0.401111\pi$
$312$	0	0
$313$	19.5918	1.10739	0.553696	−	0.832719i	$-0.313217\pi$
$313$	19.5918	1.10739	0.553696	+	0.832719i	$0.313217\pi$
$314$	−20.6012	−1.16259
$315$	0	0
$316$	−4.73532	−0.266383
$317$	22.5910	1.26884	0.634418	−	0.772990i	$-0.281240\pi$
$317$	22.5910	1.26884	0.634418	+	0.772990i	$0.281240\pi$
$318$	0	0
$319$	−11.1077	−0.621912
$320$	−1.34910	−0.0754168
$321$	0	0
$322$	37.6038	2.09558
$323$	−16.5654	−0.921723
$324$	0	0
$325$	−2.94245	−0.163218
$326$	−22.4091	−1.24113
$327$	0	0
$328$	8.67270	0.478870
$329$	43.1725	2.38018
$330$	0	0
$331$	12.2636	0.674070	0.337035	−	0.941492i	$-0.390576\pi$
$331$	12.2636	0.674070	0.337035	+	0.941492i	$0.390576\pi$
$332$	3.16799	0.173866
$333$	0	0
$334$	−18.0954	−0.990137
$335$	−14.9291	−0.815665
$336$	0	0
$337$	1.44495	0.0787114	0.0393557	−	0.999225i	$-0.487469\pi$
$337$	1.44495	0.0787114	0.0393557	+	0.999225i	$0.487469\pi$
$338$	12.1438	0.660535
$339$	0	0
$340$	−6.15359	−0.333725
$341$	0.357678	0.0193693
$342$	0	0
$343$	52.4231	2.83058
$344$	1.17630	0.0634216
$345$	0	0
$346$	−2.17952	−0.117172
$347$	−3.24257	−0.174070	−0.0870352	−	0.996205i	$-0.527739\pi$
$347$	−3.24257	−0.174070	−0.0870352	+	0.996205i	$0.527739\pi$
$348$	0	0
$349$	−24.2741	−1.29936	−0.649682	−	0.760206i	$-0.725098\pi$
$349$	−24.2741	−1.29936	−0.649682	+	0.760206i	$0.725098\pi$
$350$	15.7639	0.842617
$351$	0	0
$352$	2.19805	0.117156
$353$	−22.5168	−1.19845	−0.599224	−	0.800581i	$-0.704524\pi$
$353$	−22.5168	−1.19845	−0.599224	+	0.800581i	$0.704524\pi$
$354$	0	0
$355$	−15.7797	−0.837498
$356$	8.78993	0.465865
$357$	0	0
$358$	−15.7330	−0.831514
$359$	3.03152	0.159998	0.0799988	−	0.996795i	$-0.474508\pi$
$359$	3.03152	0.159998	0.0799988	+	0.996795i	$0.474508\pi$
$360$	0	0
$361$	−5.81038	−0.305810
$362$	−10.4414	−0.548788
$363$	0	0
$364$	4.58707	0.240428
$365$	−12.4293	−0.650579
$366$	0	0
$367$	−13.9669	−0.729067	−0.364534	−	0.931190i	$-0.618772\pi$
$367$	−13.9669	−0.729067	−0.364534	+	0.931190i	$0.618772\pi$
$368$	−7.58553	−0.395423
$369$	0	0
$370$	7.57382	0.393744
$371$	−20.6898	−1.07416
$372$	0	0
$373$	22.7430	1.17759	0.588794	−	0.808283i	$-0.299603\pi$
$373$	22.7430	1.17759	0.588794	+	0.808283i	$0.299603\pi$
$374$	10.0259	0.518426
$375$	0	0
$376$	−8.70886	−0.449125
$377$	4.67603	0.240828
$378$	0	0
$379$	23.3257	1.19816	0.599081	−	0.800688i	$-0.295533\pi$
$379$	23.3257	1.19816	0.599081	+	0.800688i	$0.295533\pi$
$380$	4.89958	0.251343
$381$	0	0
$382$	11.1156	0.568726
$383$	15.7808	0.806359	0.403180	−	0.915121i	$-0.367905\pi$
$383$	15.7808	0.806359	0.403180	+	0.915121i	$0.367905\pi$
$384$	0	0
$385$	14.7003	0.749197
$386$	−3.89523	−0.198262
$387$	0	0
$388$	−3.24337	−0.164657
$389$	19.9086	1.00941	0.504703	−	0.863293i	$-0.331602\pi$
$389$	19.9086	1.00941	0.504703	+	0.863293i	$0.331602\pi$
$390$	0	0
$391$	−34.5996	−1.74978
$392$	−17.5749	−0.887667
$393$	0	0
$394$	1.50902	0.0760232
$395$	6.38841	0.321436
$396$	0	0
$397$	−1.00000	−0.0501886
$397$	−1.00000	−0.0501886
$398$	−11.9161	−0.597301
$399$	0	0
$400$	−3.17994	−0.158997
$401$	−12.0474	−0.601617	−0.300809	−	0.953685i	$-0.597257\pi$
$401$	−12.0474	−0.601617	−0.300809	+	0.953685i	$0.597257\pi$
$402$	0	0
$403$	−0.150572	−0.00750053
$404$	5.99714	0.298369
$405$	0	0
$406$	−25.0515	−1.24328
$407$	−12.3398	−0.611662
$408$	0	0
$409$	35.2892	1.74494	0.872470	−	0.488668i	$-0.162517\pi$
$409$	35.2892	1.74494	0.872470	+	0.488668i	$0.162517\pi$
$410$	−11.7003	−0.577837
$411$	0	0
$412$	8.61138	0.424252
$413$	38.5183	1.89536
$414$	0	0
$415$	−4.27393	−0.209799
$416$	−0.925316	−0.0453673
$417$	0	0
$418$	−7.98277	−0.390450
$419$	4.51542	0.220593	0.110296	−	0.993899i	$-0.464820\pi$
$419$	4.51542	0.220593	0.110296	+	0.993899i	$0.464820\pi$
$420$	0	0
$421$	18.3797	0.895771	0.447885	−	0.894091i	$-0.352177\pi$
$421$	18.3797	0.895771	0.447885	+	0.894091i	$0.352177\pi$
$422$	0.999252	0.0486428
$423$	0	0
$424$	4.17360	0.202688
$425$	−14.5045	−0.703574
$426$	0	0
$427$	−53.7775	−2.60248
$428$	17.2040	0.831585
$429$	0	0
$430$	−1.58694	−0.0765289
$431$	7.46835	0.359738	0.179869	−	0.983691i	$-0.442433\pi$
$431$	7.46835	0.359738	0.179869	+	0.983691i	$0.442433\pi$
$432$	0	0
$433$	4.02751	0.193550	0.0967750	−	0.995306i	$-0.469147\pi$
$433$	4.02751	0.193550	0.0967750	+	0.995306i	$0.469147\pi$
$434$	0.806679	0.0387218
$435$	0	0
$436$	0.289805	0.0138791
$437$	27.5488	1.31784
$438$	0	0
$439$	−8.00670	−0.382139	−0.191069	−	0.981577i	$-0.561196\pi$
$439$	−8.00670	−0.382139	−0.191069	+	0.981577i	$0.561196\pi$
$440$	−2.96538	−0.141369
$441$	0	0
$442$	−4.22061	−0.200754
$443$	17.6097	0.836664	0.418332	−	0.908294i	$-0.362615\pi$
$443$	17.6097	0.836664	0.418332	+	0.908294i	$0.362615\pi$
$444$	0	0
$445$	−11.8585	−0.562145
$446$	−1.01101	−0.0478727
$447$	0	0
$448$	4.95731	0.234211
$449$	7.33333	0.346081	0.173041	−	0.984915i	$-0.444641\pi$
$449$	7.33333	0.346081	0.173041	+	0.984915i	$0.444641\pi$
$450$	0	0
$451$	19.0630	0.897643
$452$	−8.25002	−0.388048
$453$	0	0
$454$	−17.8679	−0.838581
$455$	−6.18841	−0.290117
$456$	0	0
$457$	35.9698	1.68260	0.841298	−	0.540571i	$-0.181792\pi$
$457$	35.9698	1.68260	0.841298	+	0.540571i	$0.181792\pi$
$458$	−0.979883	−0.0457869
$459$	0	0
$460$	10.2336	0.477145
$461$	16.5204	0.769434	0.384717	−	0.923035i	$-0.374299\pi$
$461$	16.5204	0.769434	0.384717	+	0.923035i	$0.374299\pi$
$462$	0	0
$463$	23.9280	1.11203	0.556015	−	0.831172i	$-0.312330\pi$
$463$	23.9280	1.11203	0.556015	+	0.831172i	$0.312330\pi$
$464$	5.05344	0.234600
$465$	0	0
$466$	7.29869	0.338105
$467$	13.9916	0.647454	0.323727	−	0.946150i	$-0.395064\pi$
$467$	13.9916	0.647454	0.323727	+	0.946150i	$0.395064\pi$
$468$	0	0
$469$	54.8576	2.53309
$470$	11.7491	0.541945
$471$	0	0
$472$	−7.77000	−0.357643
$473$	2.58556	0.118884
$474$	0	0
$475$	11.5487	0.529893
$476$	22.6116	1.03640
$477$	0	0
$478$	23.0334	1.05352
$479$	−16.6626	−0.761333	−0.380666	−	0.924712i	$-0.624305\pi$
$479$	−16.6626	−0.761333	−0.380666	+	0.924712i	$0.624305\pi$
$480$	0	0
$481$	5.19471	0.236859
$482$	15.8573	0.722279
$483$	0	0
$484$	−6.16858	−0.280390
$485$	4.37562	0.198687
$486$	0	0
$487$	22.1080	1.00181	0.500904	−	0.865503i	$-0.333001\pi$
$487$	22.1080	1.00181	0.500904	+	0.865503i	$0.333001\pi$
$488$	10.8481	0.491072
$489$	0	0
$490$	23.7103	1.07112
$491$	−41.9288	−1.89222	−0.946109	−	0.323848i	$-0.895023\pi$
$491$	−41.9288	−1.89222	−0.946109	+	0.323848i	$0.895023\pi$
$492$	0	0
$493$	23.0501	1.03812
$494$	3.36052	0.151197
$495$	0	0
$496$	−0.162725	−0.00730658
$497$	57.9830	2.60089
$498$	0	0
$499$	27.7340	1.24154	0.620772	−	0.783991i	$-0.286819\pi$
$499$	27.7340	1.24154	0.620772	+	0.783991i	$0.286819\pi$
$500$	11.0355	0.493524
$501$	0	0
$502$	−7.97624	−0.355997
$503$	15.6652	0.698478	0.349239	−	0.937034i	$-0.386440\pi$
$503$	15.6652	0.698478	0.349239	+	0.937034i	$0.386440\pi$
$504$	0	0
$505$	−8.09072	−0.360032
$506$	−16.6734	−0.741222
$507$	0	0
$508$	−17.9409	−0.795999
$509$	−26.4941	−1.17433	−0.587165	−	0.809467i	$-0.699756\pi$
$509$	−26.4941	−1.17433	−0.587165	+	0.809467i	$0.699756\pi$
$510$	0	0
$511$	45.6719	2.02041
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	4.87093	0.214848
$515$	−11.6176	−0.511932
$516$	0	0
$517$	−19.1425	−0.841886
$518$	−27.8303	−1.22279
$519$	0	0
$520$	1.24834	0.0547434
$521$	−41.6071	−1.82284	−0.911420	−	0.411477i	$-0.865013\pi$
$521$	−41.6071	−1.82284	−0.911420	+	0.411477i	$0.865013\pi$
$522$	0	0
$523$	−23.2913	−1.01846	−0.509229	−	0.860631i	$-0.670069\pi$
$523$	−23.2913	−1.01846	−0.509229	+	0.860631i	$0.670069\pi$
$524$	−14.1561	−0.618410
$525$	0	0
$526$	31.5063	1.37374
$527$	−0.742233	−0.0323322
$528$	0	0
$529$	34.5403	1.50175
$530$	−5.63059	−0.244577
$531$	0	0
$532$	−18.0037	−0.780560
$533$	−8.02499	−0.347601
$534$	0	0
$535$	−23.2098	−1.00345
$536$	−11.0660	−0.477979
$537$	0	0
$538$	5.70493	0.245957
$539$	−38.6305	−1.66393
$540$	0	0
$541$	−32.1227	−1.38106	−0.690531	−	0.723303i	$-0.742623\pi$
$541$	−32.1227	−1.38106	−0.690531	+	0.723303i	$0.742623\pi$
$542$	−6.23357	−0.267755
$543$	0	0
$544$	−4.56127	−0.195563
$545$	−0.390975	−0.0167475
$546$	0	0
$547$	−15.8786	−0.678922	−0.339461	−	0.940620i	$-0.610245\pi$
$547$	−15.8786	−0.678922	−0.339461	+	0.940620i	$0.610245\pi$
$548$	1.08012	0.0461405
$549$	0	0
$550$	−6.98966	−0.298040
$551$	−18.3528	−0.781857
$552$	0	0
$553$	−23.4745	−0.998236
$554$	−12.3348	−0.524057
$555$	0	0
$556$	−15.8941	−0.674058
$557$	−36.1116	−1.53010	−0.765049	−	0.643972i	$-0.777285\pi$
$557$	−36.1116	−1.53010	−0.765049	+	0.643972i	$0.777285\pi$
$558$	0	0
$559$	−1.08845	−0.0460363
$560$	−6.68789	−0.282615
$561$	0	0
$562$	23.8730	1.00702
$563$	12.7816	0.538679	0.269339	−	0.963045i	$-0.413195\pi$
$563$	12.7816	0.538679	0.269339	+	0.963045i	$0.413195\pi$
$564$	0	0
$565$	11.1301	0.468246
$566$	28.8683	1.21343
$567$	0	0
$568$	−11.6965	−0.490773
$569$	−6.77043	−0.283831	−0.141916	−	0.989879i	$-0.545326\pi$
$569$	−6.77043	−0.283831	−0.141916	+	0.989879i	$0.545326\pi$
$570$	0	0
$571$	−17.6287	−0.737739	−0.368870	−	0.929481i	$-0.620255\pi$
$571$	−17.6287	−0.737739	−0.368870	+	0.929481i	$0.620255\pi$
$572$	−2.03389	−0.0850412
$573$	0	0
$574$	42.9933	1.79450
$575$	24.1215	1.00594
$576$	0	0
$577$	14.6547	0.610083	0.305042	−	0.952339i	$-0.401330\pi$
$577$	14.6547	0.610083	0.305042	+	0.952339i	$0.401330\pi$
$578$	−3.80516	−0.158274
$579$	0	0
$580$	−6.81758	−0.283085
$581$	15.7047	0.651541
$582$	0	0
$583$	9.17377	0.379939
$584$	−9.21305	−0.381239
$585$	0	0
$586$	−22.9256	−0.947047
$587$	26.5437	1.09557	0.547787	−	0.836618i	$-0.315470\pi$
$587$	26.5437	1.09557	0.547787	+	0.836618i	$0.315470\pi$
$588$	0	0
$589$	0.590977	0.0243508
$590$	10.4825	0.431557
$591$	0	0
$592$	5.61399	0.230734
$593$	−15.6534	−0.642808	−0.321404	−	0.946942i	$-0.604155\pi$
$593$	−15.6534	−0.642808	−0.321404	+	0.946942i	$0.604155\pi$
$594$	0	0
$595$	−30.5053	−1.25059
$596$	18.9510	0.776262
$597$	0	0
$598$	7.01901	0.287029
$599$	−28.6057	−1.16880	−0.584398	−	0.811467i	$-0.698669\pi$
$599$	−28.6057	−1.16880	−0.584398	+	0.811467i	$0.698669\pi$
$600$	0	0
$601$	39.4979	1.61115	0.805577	−	0.592491i	$-0.201855\pi$
$601$	39.4979	1.61115	0.805577	+	0.592491i	$0.201855\pi$
$602$	5.83126	0.237665
$603$	0	0
$604$	−5.65661	−0.230164
$605$	8.32201	0.338338
$606$	0	0
$607$	33.1804	1.34675	0.673376	−	0.739300i	$-0.264844\pi$
$607$	33.1804	1.34675	0.673376	+	0.739300i	$0.264844\pi$
$608$	3.63175	0.147287
$609$	0	0
$610$	−14.6352	−0.592561
$611$	8.05844	0.326010
$612$	0	0
$613$	3.06851	0.123936	0.0619680	−	0.998078i	$-0.480262\pi$
$613$	3.06851	0.123936	0.0619680	+	0.998078i	$0.480262\pi$
$614$	−23.8143	−0.961065
$615$	0	0
$616$	10.8964	0.439029
$617$	−3.64432	−0.146715	−0.0733573	−	0.997306i	$-0.523371\pi$
$617$	−3.64432	−0.146715	−0.0733573	+	0.997306i	$0.523371\pi$
$618$	0	0
$619$	−3.14773	−0.126518	−0.0632590	−	0.997997i	$-0.520149\pi$
$619$	−3.14773	−0.126518	−0.0632590	+	0.997997i	$0.520149\pi$
$620$	0.219532	0.00881662
$621$	0	0
$622$	−10.7820	−0.432321
$623$	43.5744	1.74577
$624$	0	0
$625$	1.01169	0.0404675
$626$	−19.5918	−0.783044
$627$	0	0
$628$	20.6012	0.822077
$629$	25.6069	1.02101
$630$	0	0
$631$	−1.09172	−0.0434607	−0.0217304	−	0.999764i	$-0.506918\pi$
$631$	−1.09172	−0.0434607	−0.0217304	+	0.999764i	$0.506918\pi$
$632$	4.73532	0.188361
$633$	0	0
$634$	−22.5910	−0.897202
$635$	24.2040	0.960507
$636$	0	0
$637$	16.2623	0.644337
$638$	11.1077	0.439759
$639$	0	0
$640$	1.34910	0.0533277
$641$	20.6028	0.813763	0.406881	−	0.913481i	$-0.366616\pi$
$641$	20.6028	0.813763	0.406881	+	0.913481i	$0.366616\pi$
$642$	0	0
$643$	−10.7036	−0.422110	−0.211055	−	0.977474i	$-0.567690\pi$
$643$	−10.7036	−0.422110	−0.211055	+	0.977474i	$0.567690\pi$
$644$	−37.6038	−1.48180
$645$	0	0
$646$	16.5654	0.651756
$647$	45.6576	1.79498	0.897492	−	0.441031i	$-0.145387\pi$
$647$	45.6576	1.79498	0.897492	+	0.441031i	$0.145387\pi$
$648$	0	0
$649$	−17.0788	−0.670403
$650$	2.94245	0.115412
$651$	0	0
$652$	22.4091	0.877608
$653$	22.1323	0.866103	0.433051	−	0.901369i	$-0.357437\pi$
$653$	22.1323	0.866103	0.433051	+	0.901369i	$0.357437\pi$
$654$	0	0
$655$	19.0979	0.746216
$656$	−8.67270	−0.338612
$657$	0	0
$658$	−43.1725	−1.68304
$659$	−0.699095	−0.0272329	−0.0136164	−	0.999907i	$-0.504334\pi$
$659$	−0.699095	−0.0272329	−0.0136164	+	0.999907i	$0.504334\pi$
$660$	0	0
$661$	25.4178	0.988637	0.494319	−	0.869281i	$-0.335418\pi$
$661$	25.4178	0.988637	0.494319	+	0.869281i	$0.335418\pi$
$662$	−12.2636	−0.476639
$663$	0	0
$664$	−3.16799	−0.122942
$665$	24.2888	0.941877
$666$	0	0
$667$	−38.3330	−1.48426
$668$	18.0954	0.700133
$669$	0	0
$670$	14.9291	0.576762
$671$	23.8447	0.920515
$672$	0	0
$673$	−34.1269	−1.31549	−0.657747	−	0.753239i	$-0.728490\pi$
$673$	−34.1269	−1.31549	−0.657747	+	0.753239i	$0.728490\pi$
$674$	−1.44495	−0.0556574
$675$	0	0
$676$	−12.1438	−0.467069
$677$	31.1955	1.19894	0.599471	−	0.800397i	$-0.295378\pi$
$677$	31.1955	1.19894	0.599471	+	0.800397i	$0.295378\pi$
$678$	0	0
$679$	−16.0784	−0.617032
$680$	6.15359	0.235979
$681$	0	0
$682$	−0.357678	−0.0136962
$683$	39.4692	1.51025	0.755124	−	0.655582i	$-0.227577\pi$
$683$	39.4692	1.51025	0.755124	+	0.655582i	$0.227577\pi$
$684$	0	0
$685$	−1.45719	−0.0556763
$686$	−52.4231	−2.00152
$687$	0	0
$688$	−1.17630	−0.0448459
$689$	−3.86190	−0.147127
$690$	0	0
$691$	−10.4221	−0.396477	−0.198239	−	0.980154i	$-0.563522\pi$
$691$	−10.4221	−0.396477	−0.198239	+	0.980154i	$0.563522\pi$
$692$	2.17952	0.0828528
$693$	0	0
$694$	3.24257	0.123086
$695$	21.4426	0.813365
$696$	0	0
$697$	−39.5585	−1.49839
$698$	24.2741	0.918789
$699$	0	0
$700$	−15.7639	−0.595821
$701$	−0.766152	−0.0289372	−0.0144686	−	0.999895i	$-0.504606\pi$
$701$	−0.766152	−0.0289372	−0.0144686	+	0.999895i	$0.504606\pi$
$702$	0	0
$703$	−20.3886	−0.768971
$704$	−2.19805	−0.0828421
$705$	0	0
$706$	22.5168	0.847431
$707$	29.7297	1.11810
$708$	0	0
$709$	−16.5164	−0.620288	−0.310144	−	0.950690i	$-0.600377\pi$
$709$	−16.5164	−0.620288	−0.310144	+	0.950690i	$0.600377\pi$
$710$	15.7797	0.592201
$711$	0	0
$712$	−8.78993	−0.329417
$713$	1.23436	0.0462270
$714$	0	0
$715$	2.74391	0.102617
$716$	15.7330	0.587969
$717$	0	0
$718$	−3.03152	−0.113135
$719$	7.50127	0.279750	0.139875	−	0.990169i	$-0.455330\pi$
$719$	7.50127	0.279750	0.139875	+	0.990169i	$0.455330\pi$
$720$	0	0
$721$	42.6892	1.58983
$722$	5.81038	0.216240
$723$	0	0
$724$	10.4414	0.388052
$725$	−16.0696	−0.596811
$726$	0	0
$727$	−12.6957	−0.470858	−0.235429	−	0.971892i	$-0.575650\pi$
$727$	−12.6957	−0.470858	−0.235429	+	0.971892i	$0.575650\pi$
$728$	−4.58707	−0.170008
$729$	0	0
$730$	12.4293	0.460029
$731$	−5.36540	−0.198447
$732$	0	0
$733$	8.89614	0.328586	0.164293	−	0.986412i	$-0.447466\pi$
$733$	8.89614	0.328586	0.164293	+	0.986412i	$0.447466\pi$
$734$	13.9669	0.515528
$735$	0	0
$736$	7.58553	0.279606
$737$	−24.3236	−0.895972
$738$	0	0
$739$	48.6678	1.79027	0.895137	−	0.445792i	$-0.147078\pi$
$739$	48.6678	1.79027	0.895137	+	0.445792i	$0.147078\pi$
$740$	−7.57382	−0.278419
$741$	0	0
$742$	20.6898	0.759547
$743$	−3.83348	−0.140637	−0.0703184	−	0.997525i	$-0.522402\pi$
$743$	−3.83348	−0.140637	−0.0703184	+	0.997525i	$0.522402\pi$
$744$	0	0
$745$	−25.5667	−0.936691
$746$	−22.7430	−0.832681
$747$	0	0
$748$	−10.0259	−0.366583
$749$	85.2854	3.11626
$750$	0	0
$751$	6.09920	0.222563	0.111281	−	0.993789i	$-0.464504\pi$
$751$	6.09920	0.222563	0.111281	+	0.993789i	$0.464504\pi$
$752$	8.70886	0.317579
$753$	0	0
$754$	−4.67603	−0.170291
$755$	7.63131	0.277732
$756$	0	0
$757$	32.1732	1.16936	0.584678	−	0.811266i	$-0.301221\pi$
$757$	32.1732	1.16936	0.584678	+	0.811266i	$0.301221\pi$
$758$	−23.3257	−0.847229
$759$	0	0
$760$	−4.89958	−0.177727
$761$	−2.33017	−0.0844686	−0.0422343	−	0.999108i	$-0.513448\pi$
$761$	−2.33017	−0.0844686	−0.0422343	+	0.999108i	$0.513448\pi$
$762$	0	0
$763$	1.43665	0.0520103
$764$	−11.1156	−0.402150
$765$	0	0
$766$	−15.7808	−0.570182
$767$	7.18970	0.259605
$768$	0	0
$769$	14.2764	0.514819	0.257409	−	0.966302i	$-0.417131\pi$
$769$	14.2764	0.514819	0.257409	+	0.966302i	$0.417131\pi$
$770$	−14.7003	−0.529762
$771$	0	0
$772$	3.89523	0.140193
$773$	−12.3992	−0.445969	−0.222984	−	0.974822i	$-0.571580\pi$
$773$	−12.3992	−0.445969	−0.222984	+	0.974822i	$0.571580\pi$
$774$	0	0
$775$	0.517456	0.0185876
$776$	3.24337	0.116430
$777$	0	0
$778$	−19.9086	−0.713758
$779$	31.4971	1.12850
$780$	0	0
$781$	−25.7094	−0.919955
$782$	34.5996	1.23728
$783$	0	0
$784$	17.5749	0.627675
$785$	−27.7930	−0.991975
$786$	0	0
$787$	−44.3249	−1.58001	−0.790006	−	0.613099i	$-0.789923\pi$
$787$	−44.3249	−1.58001	−0.790006	+	0.613099i	$0.789923\pi$
$788$	−1.50902	−0.0537565
$789$	0	0
$790$	−6.38841	−0.227290
$791$	−40.8979	−1.45416
$792$	0	0
$793$	−10.0379	−0.356458
$794$	1.00000	0.0354887
$795$	0	0
$796$	11.9161	0.422356
$797$	−2.94940	−0.104473	−0.0522365	−	0.998635i	$-0.516635\pi$
$797$	−2.94940	−0.104473	−0.0522365	+	0.998635i	$0.516635\pi$
$798$	0	0
$799$	39.7234	1.40531
$800$	3.17994	0.112428
$801$	0	0
$802$	12.0474	0.425408
$803$	−20.2507	−0.714633
$804$	0	0
$805$	50.7312	1.78804
$806$	0.150572	0.00530368
$807$	0	0
$808$	−5.99714	−0.210979
$809$	−3.40440	−0.119692	−0.0598461	−	0.998208i	$-0.519061\pi$
$809$	−3.40440	−0.119692	−0.0598461	+	0.998208i	$0.519061\pi$
$810$	0	0
$811$	−51.6006	−1.81194	−0.905970	−	0.423341i	$-0.860857\pi$
$811$	−51.6006	−1.81194	−0.905970	+	0.423341i	$0.860857\pi$
$812$	25.0515	0.879134
$813$	0	0
$814$	12.3398	0.432511
$815$	−30.2321	−1.05898
$816$	0	0
$817$	4.27202	0.149459
$818$	−35.2892	−1.23386
$819$	0	0
$820$	11.7003	0.408593
$821$	0.954654	0.0333177	0.0166588	−	0.999861i	$-0.494697\pi$
$821$	0.954654	0.0333177	0.0166588	+	0.999861i	$0.494697\pi$
$822$	0	0
$823$	51.8997	1.80911	0.904556	−	0.426355i	$-0.140203\pi$
$823$	51.8997	1.80911	0.904556	+	0.426355i	$0.140203\pi$
$824$	−8.61138	−0.299991
$825$	0	0
$826$	−38.5183	−1.34022
$827$	−26.4333	−0.919175	−0.459588	−	0.888132i	$-0.652003\pi$
$827$	−26.4333	−0.919175	−0.459588	+	0.888132i	$0.652003\pi$
$828$	0	0
$829$	21.5046	0.746887	0.373444	−	0.927653i	$-0.378177\pi$
$829$	21.5046	0.746887	0.373444	+	0.927653i	$0.378177\pi$
$830$	4.27393	0.148350
$831$	0	0
$832$	0.925316	0.0320795
$833$	80.1638	2.77751
$834$	0	0
$835$	−24.4125	−0.844828
$836$	7.98277	0.276090
$837$	0	0
$838$	−4.51542	−0.155983
$839$	18.5077	0.638958	0.319479	−	0.947593i	$-0.396492\pi$
$839$	18.5077	0.638958	0.319479	+	0.947593i	$0.396492\pi$
$840$	0	0
$841$	−3.46273	−0.119404
$842$	−18.3797	−0.633405
$843$	0	0
$844$	−0.999252	−0.0343957
$845$	16.3832	0.563598
$846$	0	0
$847$	−30.5796	−1.05073
$848$	−4.17360	−0.143322
$849$	0	0
$850$	14.5045	0.497502
$851$	−42.5851	−1.45980
$852$	0	0
$853$	−24.8092	−0.849451	−0.424725	−	0.905322i	$-0.639629\pi$
$853$	−24.8092	−0.849451	−0.424725	+	0.905322i	$0.639629\pi$
$854$	53.7775	1.84023
$855$	0	0
$856$	−17.2040	−0.588020
$857$	−46.1840	−1.57762	−0.788808	−	0.614639i	$-0.789302\pi$
$857$	−46.1840	−1.57762	−0.788808	+	0.614639i	$0.789302\pi$
$858$	0	0
$859$	−9.54974	−0.325833	−0.162916	−	0.986640i	$-0.552090\pi$
$859$	−9.54974	−0.325833	−0.162916	+	0.986640i	$0.552090\pi$
$860$	1.58694	0.0541141
$861$	0	0
$862$	−7.46835	−0.254373
$863$	24.1830	0.823200	0.411600	−	0.911365i	$-0.364970\pi$
$863$	24.1830	0.823200	0.411600	+	0.911365i	$0.364970\pi$
$864$	0	0
$865$	−2.94038	−0.0999760
$866$	−4.02751	−0.136860
$867$	0	0
$868$	−0.806679	−0.0273805
$869$	10.4085	0.353083
$870$	0	0
$871$	10.2395	0.346954
$872$	−0.289805	−0.00981403
$873$	0	0
$874$	−27.5488	−0.931851
$875$	54.7065	1.84942
$876$	0	0
$877$	15.4877	0.522981	0.261491	−	0.965206i	$-0.415786\pi$
$877$	15.4877	0.522981	0.261491	+	0.965206i	$0.415786\pi$
$878$	8.00670	0.270213
$879$	0	0
$880$	2.96538	0.0999630
$881$	−19.1469	−0.645076	−0.322538	−	0.946556i	$-0.604536\pi$
$881$	−19.1469	−0.645076	−0.322538	+	0.946556i	$0.604536\pi$
$882$	0	0
$883$	0.0844106	0.00284064	0.00142032	−	0.999999i	$-0.499548\pi$
$883$	0.0844106	0.00284064	0.00142032	+	0.999999i	$0.499548\pi$
$884$	4.22061	0.141955
$885$	0	0
$886$	−17.6097	−0.591611
$887$	−43.5197	−1.46125	−0.730624	−	0.682780i	$-0.760771\pi$
$887$	−43.5197	−1.46125	−0.730624	+	0.682780i	$0.760771\pi$
$888$	0	0
$889$	−88.9386	−2.98290
$890$	11.8585	0.397497
$891$	0	0
$892$	1.01101	0.0338511
$893$	−31.6284	−1.05840
$894$	0	0
$895$	−21.2253	−0.709485
$896$	−4.95731	−0.165612
$897$	0	0
$898$	−7.33333	−0.244716
$899$	−0.822322	−0.0274260
$900$	0	0
$901$	−19.0369	−0.634211
$902$	−19.0630	−0.634729
$903$	0	0
$904$	8.25002	0.274392
$905$	−14.0865	−0.468250
$906$	0	0
$907$	−11.2618	−0.373943	−0.186971	−	0.982365i	$-0.559867\pi$
$907$	−11.2618	−0.373943	−0.186971	+	0.982365i	$0.559867\pi$
$908$	17.8679	0.592967
$909$	0	0
$910$	6.18841	0.205144
$911$	43.5635	1.44332	0.721661	−	0.692247i	$-0.243379\pi$
$911$	43.5635	1.44332	0.721661	+	0.692247i	$0.243379\pi$
$912$	0	0
$913$	−6.96340	−0.230455
$914$	−35.9698	−1.18978
$915$	0	0
$916$	0.979883	0.0323763
$917$	−70.1759	−2.31741
$918$	0	0
$919$	−26.5306	−0.875162	−0.437581	−	0.899179i	$-0.644165\pi$
$919$	−26.5306	−0.875162	−0.437581	+	0.899179i	$0.644165\pi$
$920$	−10.2336	−0.337392
$921$	0	0
$922$	−16.5204	−0.544072
$923$	10.8229	0.356241
$924$	0	0
$925$	−17.8521	−0.586975
$926$	−23.9280	−0.786324
$927$	0	0
$928$	−5.05344	−0.165887
$929$	23.4384	0.768989	0.384494	−	0.923127i	$-0.374376\pi$
$929$	23.4384	0.768989	0.384494	+	0.923127i	$0.374376\pi$
$930$	0	0
$931$	−63.8277	−2.09187
$932$	−7.29869	−0.239076
$933$	0	0
$934$	−13.9916	−0.457819
$935$	13.5259	0.442344
$936$	0	0
$937$	−41.7500	−1.36391	−0.681957	−	0.731392i	$-0.738871\pi$
$937$	−41.7500	−1.36391	−0.681957	+	0.731392i	$0.738871\pi$
$938$	−54.8576	−1.79116
$939$	0	0
$940$	−11.7491	−0.383213
$941$	−28.8674	−0.941051	−0.470525	−	0.882386i	$-0.655936\pi$
$941$	−28.8674	−0.941051	−0.470525	+	0.882386i	$0.655936\pi$
$942$	0	0
$943$	65.7871	2.14232
$944$	7.77000	0.252892
$945$	0	0
$946$	−2.58556	−0.0840637
$947$	−18.6445	−0.605865	−0.302933	−	0.953012i	$-0.597966\pi$
$947$	−18.6445	−0.605865	−0.302933	+	0.953012i	$0.597966\pi$
$948$	0	0
$949$	8.52498	0.276733
$950$	−11.5487	−0.374691
$951$	0	0
$952$	−22.6116	−0.732846
$953$	47.1351	1.52685	0.763427	−	0.645894i	$-0.223515\pi$
$953$	47.1351	1.52685	0.763427	+	0.645894i	$0.223515\pi$
$954$	0	0
$955$	14.9961	0.485262
$956$	−23.0334	−0.744953
$957$	0	0
$958$	16.6626	0.538344
$959$	5.35449	0.172906
$960$	0	0
$961$	−30.9735	−0.999146
$962$	−5.19471	−0.167484
$963$	0	0
$964$	−15.8573	−0.510728
$965$	−5.25505	−0.169166
$966$	0	0
$967$	19.5434	0.628475	0.314237	−	0.949344i	$-0.398251\pi$
$967$	19.5434	0.628475	0.314237	+	0.949344i	$0.398251\pi$
$968$	6.16858	0.198266
$969$	0	0
$970$	−4.37562	−0.140493
$971$	2.05287	0.0658796	0.0329398	−	0.999457i	$-0.489513\pi$
$971$	2.05287	0.0658796	0.0329398	+	0.999457i	$0.489513\pi$
$972$	0	0
$973$	−78.7918	−2.52595
$974$	−22.1080	−0.708385
$975$	0	0
$976$	−10.8481	−0.347240
$977$	−49.2010	−1.57408	−0.787040	−	0.616902i	$-0.788387\pi$
$977$	−49.2010	−1.57408	−0.787040	+	0.616902i	$0.788387\pi$
$978$	0	0
$979$	−19.3207	−0.617492
$980$	−23.7103	−0.757396
$981$	0	0
$982$	41.9288	1.33800
$983$	−5.60019	−0.178618	−0.0893091	−	0.996004i	$-0.528466\pi$
$983$	−5.60019	−0.178618	−0.0893091	+	0.996004i	$0.528466\pi$
$984$	0	0
$985$	2.03581	0.0648663
$986$	−23.0501	−0.734065
$987$	0	0
$988$	−3.36052	−0.106912
$989$	8.92284	0.283730
$990$	0	0
$991$	13.6582	0.433867	0.216934	−	0.976186i	$-0.430394\pi$
$991$	13.6582	0.433867	0.216934	+	0.976186i	$0.430394\pi$
$992$	0.162725	0.00516653
$993$	0	0
$994$	−57.9830	−1.83911
$995$	−16.0760	−0.509644
$996$	0	0
$997$	55.2248	1.74899	0.874493	−	0.485037i	$-0.161194\pi$
$997$	55.2248	1.74899	0.874493	+	0.485037i	$0.161194\pi$
$998$	−27.7340	−0.877905
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7146.2.a.be.1.7		14
3.2	odd	2		794.2.a.h.1.7	✓	14
12.11	even	2		6352.2.a.t.1.8		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
794.2.a.h.1.7	✓	14	3.2	odd	2
6352.2.a.t.1.8		14	12.11	even	2
7146.2.a.be.1.7		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:	\( N \)	\(=\)	\( 7146 = 2 \cdot 3^{2} \cdot 397 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7146.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.34910 q^{5} +4.95731 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.34910 q^{5} +4.95731 q^{7} -1.00000 q^{8} +1.34910 q^{10} -2.19805 q^{11} +0.925316 q^{13} -4.95731 q^{14} +1.00000 q^{16} +4.56127 q^{17} -3.63175 q^{19} -1.34910 q^{20} +2.19805 q^{22} -7.58553 q^{23} -3.17994 q^{25} -0.925316 q^{26} +4.95731 q^{28} +5.05344 q^{29} -0.162725 q^{31} -1.00000 q^{32} -4.56127 q^{34} -6.68789 q^{35} +5.61399 q^{37} +3.63175 q^{38} +1.34910 q^{40} -8.67270 q^{41} -1.17630 q^{43} -2.19805 q^{44} +7.58553 q^{46} +8.70886 q^{47} +17.5749 q^{49} +3.17994 q^{50} +0.925316 q^{52} -4.17360 q^{53} +2.96538 q^{55} -4.95731 q^{56} -5.05344 q^{58} +7.77000 q^{59} -10.8481 q^{61} +0.162725 q^{62} +1.00000 q^{64} -1.24834 q^{65} +11.0660 q^{67} +4.56127 q^{68} +6.68789 q^{70} +11.6965 q^{71} +9.21305 q^{73} -5.61399 q^{74} -3.63175 q^{76} -10.8964 q^{77} -4.73532 q^{79} -1.34910 q^{80} +8.67270 q^{82} +3.16799 q^{83} -6.15359 q^{85} +1.17630 q^{86} +2.19805 q^{88} +8.78993 q^{89} +4.58707 q^{91} -7.58553 q^{92} -8.70886 q^{94} +4.89958 q^{95} -3.24337 q^{97} -17.5749 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 14 q^{2} + 14 q^{4} - 7 q^{5} + 7 q^{7} - 14 q^{8}+O(q^{10}) \) 14 * q - 14 * q^2 + 14 * q^4 - 7 * q^5 + 7 * q^7 - 14 * q^8	\( 14 q - 14 q^{2} + 14 q^{4} - 7 q^{5} + 7 q^{7} - 14 q^{8} + 7 q^{10} - 3 q^{11} + 7 q^{13} - 7 q^{14} + 14 q^{16} - 10 q^{17} + 7 q^{19} - 7 q^{20} + 3 q^{22} - 14 q^{23} + 27 q^{25} - 7 q^{26} + 7 q^{28} - 20 q^{29} + 6 q^{31} - 14 q^{32} + 10 q^{34} + 16 q^{35} + 20 q^{37} - 7 q^{38} + 7 q^{40} - 4 q^{41} + 3 q^{43} - 3 q^{44} + 14 q^{46} + 23 q^{49} - 27 q^{50} + 7 q^{52} - 21 q^{53} - 14 q^{55} - 7 q^{56} + 20 q^{58} + 30 q^{59} + 19 q^{61} - 6 q^{62} + 14 q^{64} - 12 q^{65} - 21 q^{67} - 10 q^{68} - 16 q^{70} + 11 q^{71} + 9 q^{73} - 20 q^{74} + 7 q^{76} + 26 q^{77} + 22 q^{79} - 7 q^{80} + 4 q^{82} + 51 q^{83} - 10 q^{85} - 3 q^{86} + 3 q^{88} + 6 q^{89} - 8 q^{91} - 14 q^{92} + 40 q^{95} - 15 q^{97} - 23 q^{98}+O(q^{100}) \) 14 * q - 14 * q^2 + 14 * q^4 - 7 * q^5 + 7 * q^7 - 14 * q^8 + 7 * q^10 - 3 * q^11 + 7 * q^13 - 7 * q^14 + 14 * q^16 - 10 * q^17 + 7 * q^19 - 7 * q^20 + 3 * q^22 - 14 * q^23 + 27 * q^25 - 7 * q^26 + 7 * q^28 - 20 * q^29 + 6 * q^31 - 14 * q^32 + 10 * q^34 + 16 * q^35 + 20 * q^37 - 7 * q^38 + 7 * q^40 - 4 * q^41 + 3 * q^43 - 3 * q^44 + 14 * q^46 + 23 * q^49 - 27 * q^50 + 7 * q^52 - 21 * q^53 - 14 * q^55 - 7 * q^56 + 20 * q^58 + 30 * q^59 + 19 * q^61 - 6 * q^62 + 14 * q^64 - 12 * q^65 - 21 * q^67 - 10 * q^68 - 16 * q^70 + 11 * q^71 + 9 * q^73 - 20 * q^74 + 7 * q^76 + 26 * q^77 + 22 * q^79 - 7 * q^80 + 4 * q^82 + 51 * q^83 - 10 * q^85 - 3 * q^86 + 3 * q^88 + 6 * q^89 - 8 * q^91 - 14 * q^92 + 40 * q^95 - 15 * q^97 - 23 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more