LMFDB - Embedded newform 3344.2.a.bb.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3344,2,Mod(1,3344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3344, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3344.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3344 = 2^{4} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3344.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:	$26.7019744359$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 22x^{7} + 22x^{6} + 152x^{5} - 136x^{4} - 341x^{3} + 169x^{2} + 196x + 32 $ x^9 - x^8 - 22x^7 + 22x^6 + 152x^5 - 136x^4 - 341x^3 + 169x^2 + 196*x + 32
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1672)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.12163$ of defining polynomial
Character	$\chi$	$=$	3344.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.12163 q^{3} -1.30304 q^{5} -0.855540 q^{7} +6.74459 q^{9} +O(q^{10})$	$q-3.12163 q^{3} -1.30304 q^{5} -0.855540 q^{7} +6.74459 q^{9} -1.00000 q^{11} +4.34586 q^{13} +4.06761 q^{15} -4.22175 q^{17} +1.00000 q^{19} +2.67068 q^{21} -7.96501 q^{23} -3.30209 q^{25} -11.6892 q^{27} +7.89693 q^{29} +8.92448 q^{31} +3.12163 q^{33} +1.11480 q^{35} +6.88905 q^{37} -13.5662 q^{39} -8.30692 q^{41} +1.02380 q^{43} -8.78847 q^{45} -6.60608 q^{47} -6.26805 q^{49} +13.1787 q^{51} -12.7383 q^{53} +1.30304 q^{55} -3.12163 q^{57} -5.87749 q^{59} +10.8794 q^{61} -5.77027 q^{63} -5.66283 q^{65} -6.26799 q^{67} +24.8638 q^{69} -16.5117 q^{71} -8.74031 q^{73} +10.3079 q^{75} +0.855540 q^{77} +6.77925 q^{79} +16.2558 q^{81} +4.68349 q^{83} +5.50110 q^{85} -24.6513 q^{87} +16.3463 q^{89} -3.71806 q^{91} -27.8589 q^{93} -1.30304 q^{95} -14.3950 q^{97} -6.74459 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - q^{3} + 6 q^{5} - 3 q^{7} + 18 q^{9}+O(q^{10}) $ 9 * q - q^3 + 6 * q^5 - 3 * q^7 + 18 * q^9	$ 9 q - q^{3} + 6 q^{5} - 3 q^{7} + 18 q^{9} - 9 q^{11} + q^{13} - 6 q^{15} + 7 q^{17} + 9 q^{19} + 3 q^{21} - 13 q^{23} + 31 q^{25} + 5 q^{27} + 9 q^{29} + 4 q^{31} + q^{33} + 4 q^{35} + 24 q^{37} - 13 q^{39} - 6 q^{41} + 14 q^{43} + 26 q^{45} - 24 q^{47} + 20 q^{49} + 33 q^{51} + 19 q^{53} - 6 q^{55} - q^{57} + 19 q^{59} + 28 q^{61} - 16 q^{63} + 16 q^{65} - 5 q^{67} + 35 q^{69} - 16 q^{71} + 15 q^{73} - 3 q^{75} + 3 q^{77} - 2 q^{79} + 37 q^{81} - 8 q^{83} + 20 q^{85} - 23 q^{87} + 12 q^{89} + 29 q^{91} + 44 q^{93} + 6 q^{95} - 4 q^{97} - 18 q^{99}+O(q^{100}) $ 9 * q - q^3 + 6 * q^5 - 3 * q^7 + 18 * q^9 - 9 * q^11 + q^13 - 6 * q^15 + 7 * q^17 + 9 * q^19 + 3 * q^21 - 13 * q^23 + 31 * q^25 + 5 * q^27 + 9 * q^29 + 4 * q^31 + q^33 + 4 * q^35 + 24 * q^37 - 13 * q^39 - 6 * q^41 + 14 * q^43 + 26 * q^45 - 24 * q^47 + 20 * q^49 + 33 * q^51 + 19 * q^53 - 6 * q^55 - q^57 + 19 * q^59 + 28 * q^61 - 16 * q^63 + 16 * q^65 - 5 * q^67 + 35 * q^69 - 16 * q^71 + 15 * q^73 - 3 * q^75 + 3 * q^77 - 2 * q^79 + 37 * q^81 - 8 * q^83 + 20 * q^85 - 23 * q^87 + 12 * q^89 + 29 * q^91 + 44 * q^93 + 6 * q^95 - 4 * q^97 - 18 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−3.12163	−1.80228	−0.901138	−	0.433533i	$-0.857267\pi$
$3$	−3.12163	−1.80228	−0.901138	+	0.433533i	$0.857267\pi$
$4$	0	0
$5$	−1.30304	−0.582737	−0.291368	−	0.956611i	$-0.594111\pi$
$5$	−1.30304	−0.582737	−0.291368	+	0.956611i	$0.594111\pi$
$6$	0	0
$7$	−0.855540	−0.323364	−0.161682	−	0.986843i	$-0.551692\pi$
$7$	−0.855540	−0.323364	−0.161682	+	0.986843i	$0.551692\pi$
$8$	0	0
$9$	6.74459	2.24820
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	4.34586	1.20533	0.602663	−	0.797996i	$-0.294106\pi$
$13$	4.34586	1.20533	0.602663	+	0.797996i	$0.294106\pi$
$14$	0	0
$15$	4.06761	1.05025
$16$	0	0
$17$	−4.22175	−1.02392	−0.511962	−	0.859008i	$-0.671081\pi$
$17$	−4.22175	−1.02392	−0.511962	+	0.859008i	$0.671081\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	2.67068	0.582791
$22$	0	0
$23$	−7.96501	−1.66082	−0.830410	−	0.557153i	$-0.811894\pi$
$23$	−7.96501	−1.66082	−0.830410	+	0.557153i	$0.811894\pi$
$24$	0	0
$25$	−3.30209	−0.660418
$26$	0	0
$27$	−11.6892	−2.24960
$28$	0	0
$29$	7.89693	1.46642	0.733211	−	0.680001i	$-0.238021\pi$
$29$	7.89693	1.46642	0.733211	+	0.680001i	$0.238021\pi$
$30$	0	0
$31$	8.92448	1.60288	0.801442	−	0.598073i	$-0.204067\pi$
$31$	8.92448	1.60288	0.801442	+	0.598073i	$0.204067\pi$
$32$	0	0
$33$	3.12163	0.543407
$34$	0	0
$35$	1.11480	0.188436
$36$	0	0
$37$	6.88905	1.13255	0.566277	−	0.824215i	$-0.308383\pi$
$37$	6.88905	1.13255	0.566277	+	0.824215i	$0.308383\pi$
$38$	0	0
$39$	−13.5662	−2.17233
$40$	0	0
$41$	−8.30692	−1.29732	−0.648662	−	0.761077i	$-0.724671\pi$
$41$	−8.30692	−1.29732	−0.648662	+	0.761077i	$0.724671\pi$
$42$	0	0
$43$	1.02380	0.156127	0.0780637	−	0.996948i	$-0.475126\pi$
$43$	1.02380	0.156127	0.0780637	+	0.996948i	$0.475126\pi$
$44$	0	0
$45$	−8.78847	−1.31011
$46$	0	0
$47$	−6.60608	−0.963595	−0.481798	−	0.876282i	$-0.660016\pi$
$47$	−6.60608	−0.963595	−0.481798	+	0.876282i	$0.660016\pi$
$48$	0	0
$49$	−6.26805	−0.895436
$50$	0	0
$51$	13.1787	1.84539
$52$	0	0
$53$	−12.7383	−1.74974	−0.874871	−	0.484356i	$-0.839054\pi$
$53$	−12.7383	−1.74974	−0.874871	+	0.484356i	$0.839054\pi$
$54$	0	0
$55$	1.30304	0.175702
$56$	0	0
$57$	−3.12163	−0.413470
$58$	0	0
$59$	−5.87749	−0.765184	−0.382592	−	0.923917i	$-0.624969\pi$
$59$	−5.87749	−0.765184	−0.382592	+	0.923917i	$0.624969\pi$
$60$	0	0
$61$	10.8794	1.39296	0.696481	−	0.717576i	$-0.254748\pi$
$61$	10.8794	1.39296	0.696481	+	0.717576i	$0.254748\pi$
$62$	0	0
$63$	−5.77027	−0.726986
$64$	0	0
$65$	−5.66283	−0.702387
$66$	0	0
$67$	−6.26799	−0.765757	−0.382879	−	0.923799i	$-0.625067\pi$
$67$	−6.26799	−0.765757	−0.382879	+	0.923799i	$0.625067\pi$
$68$	0	0
$69$	24.8638	2.99326
$70$	0	0
$71$	−16.5117	−1.95958	−0.979788	−	0.200040i	$-0.935893\pi$
$71$	−16.5117	−1.95958	−0.979788	+	0.200040i	$0.935893\pi$
$72$	0	0
$73$	−8.74031	−1.02298	−0.511488	−	0.859291i	$-0.670905\pi$
$73$	−8.74031	−1.02298	−0.511488	+	0.859291i	$0.670905\pi$
$74$	0	0
$75$	10.3079	1.19026
$76$	0	0
$77$	0.855540	0.0974979
$78$	0	0
$79$	6.77925	0.762725	0.381362	−	0.924426i	$-0.375455\pi$
$79$	6.77925	0.762725	0.381362	+	0.924426i	$0.375455\pi$
$80$	0	0
$81$	16.2558	1.80619
$82$	0	0
$83$	4.68349	0.514080	0.257040	−	0.966401i	$-0.417253\pi$
$83$	4.68349	0.514080	0.257040	+	0.966401i	$0.417253\pi$
$84$	0	0
$85$	5.50110	0.596678
$86$	0	0
$87$	−24.6513	−2.64290
$88$	0	0
$89$	16.3463	1.73270	0.866352	−	0.499433i	$-0.166458\pi$
$89$	16.3463	1.73270	0.866352	+	0.499433i	$0.166458\pi$
$90$	0	0
$91$	−3.71806	−0.389759
$92$	0	0
$93$	−27.8589	−2.88884
$94$	0	0
$95$	−1.30304	−0.133689
$96$	0	0
$97$	−14.3950	−1.46159	−0.730797	−	0.682595i	$-0.760851\pi$
$97$	−14.3950	−1.46159	−0.730797	+	0.682595i	$0.760851\pi$
$98$	0	0
$99$	−6.74459	−0.677857
$100$	0	0
$101$	−5.66435	−0.563624	−0.281812	−	0.959470i	$-0.590935\pi$
$101$	−5.66435	−0.563624	−0.281812	+	0.959470i	$0.590935\pi$
$102$	0	0
$103$	2.77266	0.273198	0.136599	−	0.990626i	$-0.456383\pi$
$103$	2.77266	0.273198	0.136599	+	0.990626i	$0.456383\pi$
$104$	0	0
$105$	−3.48000	−0.339614
$106$	0	0
$107$	−2.04858	−0.198044	−0.0990219	−	0.995085i	$-0.531571\pi$
$107$	−2.04858	−0.198044	−0.0990219	+	0.995085i	$0.531571\pi$
$108$	0	0
$109$	−3.84531	−0.368314	−0.184157	−	0.982897i	$-0.558955\pi$
$109$	−3.84531	−0.368314	−0.184157	+	0.982897i	$0.558955\pi$
$110$	0	0
$111$	−21.5051	−2.04117
$112$	0	0
$113$	12.9610	1.21926	0.609632	−	0.792685i	$-0.291317\pi$
$113$	12.9610	1.21926	0.609632	+	0.792685i	$0.291317\pi$
$114$	0	0
$115$	10.3787	0.967821
$116$	0	0
$117$	29.3111	2.70981
$118$	0	0
$119$	3.61188	0.331100
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	25.9312	2.33813
$124$	0	0
$125$	10.8179	0.967586
$126$	0	0
$127$	18.7162	1.66079	0.830396	−	0.557174i	$-0.188114\pi$
$127$	18.7162	1.66079	0.830396	+	0.557174i	$0.188114\pi$
$128$	0	0
$129$	−3.19592	−0.281385
$130$	0	0
$131$	18.5589	1.62150	0.810751	−	0.585391i	$-0.199059\pi$
$131$	18.5589	1.62150	0.810751	+	0.585391i	$0.199059\pi$
$132$	0	0
$133$	−0.855540	−0.0741848
$134$	0	0
$135$	15.2315	1.31092
$136$	0	0
$137$	15.8437	1.35362	0.676808	−	0.736159i	$-0.263363\pi$
$137$	15.8437	1.35362	0.676808	+	0.736159i	$0.263363\pi$
$138$	0	0
$139$	14.0920	1.19526	0.597632	−	0.801770i	$-0.296108\pi$
$139$	14.0920	1.19526	0.597632	+	0.801770i	$0.296108\pi$
$140$	0	0
$141$	20.6217	1.73666
$142$	0	0
$143$	−4.34586	−0.363419
$144$	0	0
$145$	−10.2900	−0.854538
$146$	0	0
$147$	19.5666	1.61382
$148$	0	0
$149$	1.06134	0.0869487	0.0434744	−	0.999055i	$-0.486157\pi$
$149$	1.06134	0.0869487	0.0434744	+	0.999055i	$0.486157\pi$
$150$	0	0
$151$	0.955453	0.0777537	0.0388768	−	0.999244i	$-0.487622\pi$
$151$	0.955453	0.0777537	0.0388768	+	0.999244i	$0.487622\pi$
$152$	0	0
$153$	−28.4740	−2.30198
$154$	0	0
$155$	−11.6289	−0.934059
$156$	0	0
$157$	−5.40212	−0.431136	−0.215568	−	0.976489i	$-0.569160\pi$
$157$	−5.40212	−0.431136	−0.215568	+	0.976489i	$0.569160\pi$
$158$	0	0
$159$	39.7643	3.15352
$160$	0	0
$161$	6.81439	0.537049
$162$	0	0
$163$	13.5992	1.06517	0.532586	−	0.846376i	$-0.321220\pi$
$163$	13.5992	1.06517	0.532586	+	0.846376i	$0.321220\pi$
$164$	0	0
$165$	−4.06761	−0.316663
$166$	0	0
$167$	19.2336	1.48834	0.744169	−	0.667992i	$-0.232846\pi$
$167$	19.2336	1.48834	0.744169	+	0.667992i	$0.232846\pi$
$168$	0	0
$169$	5.88653	0.452810
$170$	0	0
$171$	6.74459	0.515772
$172$	0	0
$173$	−18.8647	−1.43426	−0.717130	−	0.696939i	$-0.754545\pi$
$173$	−18.8647	−1.43426	−0.717130	+	0.696939i	$0.754545\pi$
$174$	0	0
$175$	2.82507	0.213555
$176$	0	0
$177$	18.3474	1.37907
$178$	0	0
$179$	14.2291	1.06353	0.531766	−	0.846891i	$-0.321529\pi$
$179$	14.2291	1.06353	0.531766	+	0.846891i	$0.321529\pi$
$180$	0	0
$181$	−8.95342	−0.665503	−0.332751	−	0.943015i	$-0.607977\pi$
$181$	−8.95342	−0.665503	−0.332751	+	0.943015i	$0.607977\pi$
$182$	0	0
$183$	−33.9614	−2.51050
$184$	0	0
$185$	−8.97670	−0.659980
$186$	0	0
$187$	4.22175	0.308725
$188$	0	0
$189$	10.0006	0.727438
$190$	0	0
$191$	10.1721	0.736031	0.368015	−	0.929820i	$-0.380037\pi$
$191$	10.1721	0.736031	0.368015	+	0.929820i	$0.380037\pi$
$192$	0	0
$193$	15.8337	1.13973	0.569866	−	0.821738i	$-0.306995\pi$
$193$	15.8337	1.13973	0.569866	+	0.821738i	$0.306995\pi$
$194$	0	0
$195$	17.6773	1.26590
$196$	0	0
$197$	18.7225	1.33392	0.666961	−	0.745092i	$-0.267595\pi$
$197$	18.7225	1.33392	0.666961	+	0.745092i	$0.267595\pi$
$198$	0	0
$199$	−2.10904	−0.149506	−0.0747528	−	0.997202i	$-0.523817\pi$
$199$	−2.10904	−0.149506	−0.0747528	+	0.997202i	$0.523817\pi$
$200$	0	0
$201$	19.5664	1.38011
$202$	0	0
$203$	−6.75614	−0.474188
$204$	0	0
$205$	10.8242	0.755998
$206$	0	0
$207$	−53.7208	−3.73385
$208$	0	0
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	2.05955	0.141785	0.0708926	−	0.997484i	$-0.477415\pi$
$211$	2.05955	0.141785	0.0708926	+	0.997484i	$0.477415\pi$
$212$	0	0
$213$	51.5434	3.53170
$214$	0	0
$215$	−1.33405	−0.0909812
$216$	0	0
$217$	−7.63525	−0.518315
$218$	0	0
$219$	27.2840	1.84368
$220$	0	0
$221$	−18.3471	−1.23416
$222$	0	0
$223$	6.19543	0.414877	0.207438	−	0.978248i	$-0.433487\pi$
$223$	6.19543	0.414877	0.207438	+	0.978248i	$0.433487\pi$
$224$	0	0
$225$	−22.2713	−1.48475
$226$	0	0
$227$	−8.21003	−0.544919	−0.272459	−	0.962167i	$-0.587837\pi$
$227$	−8.21003	−0.544919	−0.272459	+	0.962167i	$0.587837\pi$
$228$	0	0
$229$	−5.20019	−0.343638	−0.171819	−	0.985129i	$-0.554964\pi$
$229$	−5.20019	−0.343638	−0.171819	+	0.985129i	$0.554964\pi$
$230$	0	0
$231$	−2.67068	−0.175718
$232$	0	0
$233$	−5.30160	−0.347319	−0.173660	−	0.984806i	$-0.555559\pi$
$233$	−5.30160	−0.347319	−0.173660	+	0.984806i	$0.555559\pi$
$234$	0	0
$235$	8.60798	0.561522
$236$	0	0
$237$	−21.1623	−1.37464
$238$	0	0
$239$	−1.88280	−0.121788	−0.0608940	−	0.998144i	$-0.519395\pi$
$239$	−1.88280	−0.121788	−0.0608940	+	0.998144i	$0.519395\pi$
$240$	0	0
$241$	−12.7585	−0.821848	−0.410924	−	0.911670i	$-0.634794\pi$
$241$	−12.7585	−0.821848	−0.410924	+	0.911670i	$0.634794\pi$
$242$	0	0
$243$	−15.6768	−1.00566
$244$	0	0
$245$	8.16751	0.521803
$246$	0	0
$247$	4.34586	0.276521
$248$	0	0
$249$	−14.6201	−0.926514
$250$	0	0
$251$	−13.0186	−0.821724	−0.410862	−	0.911697i	$-0.634772\pi$
$251$	−13.0186	−0.821724	−0.410862	+	0.911697i	$0.634772\pi$
$252$	0	0
$253$	7.96501	0.500756
$254$	0	0
$255$	−17.1724	−1.07538
$256$	0	0
$257$	−31.8490	−1.98669	−0.993343	−	0.115194i	$-0.963251\pi$
$257$	−31.8490	−1.98669	−0.993343	+	0.115194i	$0.963251\pi$
$258$	0	0
$259$	−5.89386	−0.366227
$260$	0	0
$261$	53.2616	3.29681
$262$	0	0
$263$	23.1522	1.42763	0.713814	−	0.700335i	$-0.246966\pi$
$263$	23.1522	1.42763	0.713814	+	0.700335i	$0.246966\pi$
$264$	0	0
$265$	16.5985	1.01964
$266$	0	0
$267$	−51.0272	−3.12281
$268$	0	0
$269$	10.8473	0.661375	0.330687	−	0.943740i	$-0.392719\pi$
$269$	10.8473	0.661375	0.330687	+	0.943740i	$0.392719\pi$
$270$	0	0
$271$	−10.2223	−0.620961	−0.310480	−	0.950580i	$-0.600490\pi$
$271$	−10.2223	−0.620961	−0.310480	+	0.950580i	$0.600490\pi$
$272$	0	0
$273$	11.6064	0.702453
$274$	0	0
$275$	3.30209	0.199124
$276$	0	0
$277$	16.7761	1.00798	0.503990	−	0.863710i	$-0.331865\pi$
$277$	16.7761	1.00798	0.503990	+	0.863710i	$0.331865\pi$
$278$	0	0
$279$	60.1920	3.60360
$280$	0	0
$281$	15.5222	0.925978	0.462989	−	0.886364i	$-0.346777\pi$
$281$	15.5222	0.925978	0.462989	+	0.886364i	$0.346777\pi$
$282$	0	0
$283$	12.1060	0.719629	0.359815	−	0.933024i	$-0.382840\pi$
$283$	12.1060	0.719629	0.359815	+	0.933024i	$0.382840\pi$
$284$	0	0
$285$	4.06761	0.240944
$286$	0	0
$287$	7.10691	0.419508
$288$	0	0
$289$	0.823158	0.0484210
$290$	0	0
$291$	44.9360	2.63420
$292$	0	0
$293$	−21.9032	−1.27960	−0.639800	−	0.768542i	$-0.720983\pi$
$293$	−21.9032	−1.27960	−0.639800	+	0.768542i	$0.720983\pi$
$294$	0	0
$295$	7.65860	0.445901
$296$	0	0
$297$	11.6892	0.678279
$298$	0	0
$299$	−34.6149	−2.00183
$300$	0	0
$301$	−0.875899	−0.0504860
$302$	0	0
$303$	17.6820	1.01581
$304$	0	0
$305$	−14.1762	−0.811730
$306$	0	0
$307$	−22.3560	−1.27593	−0.637963	−	0.770067i	$-0.720223\pi$
$307$	−22.3560	−1.27593	−0.637963	+	0.770067i	$0.720223\pi$
$308$	0	0
$309$	−8.65522	−0.492378
$310$	0	0
$311$	13.6614	0.774669	0.387335	−	0.921939i	$-0.373396\pi$
$311$	13.6614	0.774669	0.387335	+	0.921939i	$0.373396\pi$
$312$	0	0
$313$	29.0578	1.64244	0.821221	−	0.570610i	$-0.193293\pi$
$313$	29.0578	1.64244	0.821221	+	0.570610i	$0.193293\pi$
$314$	0	0
$315$	7.51889	0.423641
$316$	0	0
$317$	9.65368	0.542205	0.271102	−	0.962551i	$-0.412612\pi$
$317$	9.65368	0.542205	0.271102	+	0.962551i	$0.412612\pi$
$318$	0	0
$319$	−7.89693	−0.442143
$320$	0	0
$321$	6.39492	0.356929
$322$	0	0
$323$	−4.22175	−0.234904
$324$	0	0
$325$	−14.3504	−0.796019
$326$	0	0
$327$	12.0036	0.663803
$328$	0	0
$329$	5.65177	0.311592
$330$	0	0
$331$	−11.9654	−0.657680	−0.328840	−	0.944386i	$-0.606658\pi$
$331$	−11.9654	−0.657680	−0.328840	+	0.944386i	$0.606658\pi$
$332$	0	0
$333$	46.4639	2.54620
$334$	0	0
$335$	8.16744	0.446235
$336$	0	0
$337$	17.4792	0.952150	0.476075	−	0.879405i	$-0.342059\pi$
$337$	17.4792	0.952150	0.476075	+	0.879405i	$0.342059\pi$
$338$	0	0
$339$	−40.4593	−2.19745
$340$	0	0
$341$	−8.92448	−0.483288
$342$	0	0
$343$	11.3514	0.612916
$344$	0	0
$345$	−32.3986	−1.74428
$346$	0	0
$347$	19.3503	1.03878	0.519390	−	0.854538i	$-0.326159\pi$
$347$	19.3503	1.03878	0.519390	+	0.854538i	$0.326159\pi$
$348$	0	0
$349$	−15.9760	−0.855174	−0.427587	−	0.903974i	$-0.640636\pi$
$349$	−15.9760	−0.855174	−0.427587	+	0.903974i	$0.640636\pi$
$350$	0	0
$351$	−50.7999	−2.71150
$352$	0	0
$353$	−17.0078	−0.905235	−0.452618	−	0.891705i	$-0.649510\pi$
$353$	−17.0078	−0.905235	−0.452618	+	0.891705i	$0.649510\pi$
$354$	0	0
$355$	21.5154	1.14192
$356$	0	0
$357$	−11.2750	−0.596734
$358$	0	0
$359$	−0.300849	−0.0158782	−0.00793910	−	0.999968i	$-0.502527\pi$
$359$	−0.300849	−0.0158782	−0.00793910	+	0.999968i	$0.502527\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−3.12163	−0.163843
$364$	0	0
$365$	11.3890	0.596125
$366$	0	0
$367$	−11.0562	−0.577131	−0.288566	−	0.957460i	$-0.593178\pi$
$367$	−11.0562	−0.577131	−0.288566	+	0.957460i	$0.593178\pi$
$368$	0	0
$369$	−56.0268	−2.91664
$370$	0	0
$371$	10.8981	0.565803
$372$	0	0
$373$	9.27728	0.480359	0.240180	−	0.970728i	$-0.422794\pi$
$373$	9.27728	0.480359	0.240180	+	0.970728i	$0.422794\pi$
$374$	0	0
$375$	−33.7697	−1.74386
$376$	0	0
$377$	34.3190	1.76752
$378$	0	0
$379$	6.33683	0.325501	0.162750	−	0.986667i	$-0.447963\pi$
$379$	6.33683	0.325501	0.162750	+	0.986667i	$0.447963\pi$
$380$	0	0
$381$	−58.4250	−2.99320
$382$	0	0
$383$	−7.28273	−0.372130	−0.186065	−	0.982537i	$-0.559574\pi$
$383$	−7.28273	−0.372130	−0.186065	+	0.982537i	$0.559574\pi$
$384$	0	0
$385$	−1.11480	−0.0568156
$386$	0	0
$387$	6.90509	0.351005
$388$	0	0
$389$	−2.24945	−0.114052	−0.0570258	−	0.998373i	$-0.518162\pi$
$389$	−2.24945	−0.114052	−0.0570258	+	0.998373i	$0.518162\pi$
$390$	0	0
$391$	33.6263	1.70055
$392$	0	0
$393$	−57.9342	−2.92240
$394$	0	0
$395$	−8.83362	−0.444468
$396$	0	0
$397$	−13.6266	−0.683900	−0.341950	−	0.939718i	$-0.611087\pi$
$397$	−13.6266	−0.683900	−0.341950	+	0.939718i	$0.611087\pi$
$398$	0	0
$399$	2.67068	0.133701
$400$	0	0
$401$	12.3266	0.615559	0.307780	−	0.951458i	$-0.400414\pi$
$401$	12.3266	0.615559	0.307780	+	0.951458i	$0.400414\pi$
$402$	0	0
$403$	38.7846	1.93200
$404$	0	0
$405$	−21.1819	−1.05254
$406$	0	0
$407$	−6.88905	−0.341478
$408$	0	0
$409$	17.5885	0.869694	0.434847	−	0.900504i	$-0.356802\pi$
$409$	17.5885	0.869694	0.434847	+	0.900504i	$0.356802\pi$
$410$	0	0
$411$	−49.4581	−2.43959
$412$	0	0
$413$	5.02843	0.247433
$414$	0	0
$415$	−6.10277	−0.299573
$416$	0	0
$417$	−43.9899	−2.15420
$418$	0	0
$419$	−2.22341	−0.108621	−0.0543104	−	0.998524i	$-0.517296\pi$
$419$	−2.22341	−0.108621	−0.0543104	+	0.998524i	$0.517296\pi$
$420$	0	0
$421$	−4.19325	−0.204366	−0.102183	−	0.994766i	$-0.532583\pi$
$421$	−4.19325	−0.204366	−0.102183	+	0.994766i	$0.532583\pi$
$422$	0	0
$423$	−44.5553	−2.16635
$424$	0	0
$425$	13.9406	0.676218
$426$	0	0
$427$	−9.30775	−0.450433
$428$	0	0
$429$	13.5662	0.654982
$430$	0	0
$431$	5.16384	0.248734	0.124367	−	0.992236i	$-0.460310\pi$
$431$	5.16384	0.248734	0.124367	+	0.992236i	$0.460310\pi$
$432$	0	0
$433$	−0.202612	−0.00973689	−0.00486844	−	0.999988i	$-0.501550\pi$
$433$	−0.202612	−0.00973689	−0.00486844	+	0.999988i	$0.501550\pi$
$434$	0	0
$435$	32.1216	1.54011
$436$	0	0
$437$	−7.96501	−0.381018
$438$	0	0
$439$	11.5710	0.552254	0.276127	−	0.961121i	$-0.410949\pi$
$439$	11.5710	0.552254	0.276127	+	0.961121i	$0.410949\pi$
$440$	0	0
$441$	−42.2754	−2.01312
$442$	0	0
$443$	19.0710	0.906093	0.453046	−	0.891487i	$-0.350337\pi$
$443$	19.0710	0.906093	0.453046	+	0.891487i	$0.350337\pi$
$444$	0	0
$445$	−21.2999	−1.00971
$446$	0	0
$447$	−3.31313	−0.156706
$448$	0	0
$449$	29.5466	1.39439	0.697194	−	0.716882i	$-0.254431\pi$
$449$	29.5466	1.39439	0.697194	+	0.716882i	$0.254431\pi$
$450$	0	0
$451$	8.30692	0.391158
$452$	0	0
$453$	−2.98257	−0.140134
$454$	0	0
$455$	4.84478	0.227127
$456$	0	0
$457$	−22.9059	−1.07149	−0.535745	−	0.844380i	$-0.679969\pi$
$457$	−22.9059	−1.07149	−0.535745	+	0.844380i	$0.679969\pi$
$458$	0	0
$459$	49.3490	2.30342
$460$	0	0
$461$	30.9491	1.44144	0.720721	−	0.693225i	$-0.243811\pi$
$461$	30.9491	1.44144	0.720721	+	0.693225i	$0.243811\pi$
$462$	0	0
$463$	−6.88953	−0.320184	−0.160092	−	0.987102i	$-0.551179\pi$
$463$	−6.88953	−0.320184	−0.160092	+	0.987102i	$0.551179\pi$
$464$	0	0
$465$	36.3013	1.68343
$466$	0	0
$467$	−2.31826	−0.107276	−0.0536382	−	0.998560i	$-0.517082\pi$
$467$	−2.31826	−0.107276	−0.0536382	+	0.998560i	$0.517082\pi$
$468$	0	0
$469$	5.36252	0.247618
$470$	0	0
$471$	16.8634	0.777027
$472$	0	0
$473$	−1.02380	−0.0470742
$474$	0	0
$475$	−3.30209	−0.151510
$476$	0	0
$477$	−85.9147	−3.93377
$478$	0	0
$479$	−22.5308	−1.02946	−0.514729	−	0.857353i	$-0.672108\pi$
$479$	−22.5308	−1.02946	−0.514729	+	0.857353i	$0.672108\pi$
$480$	0	0
$481$	29.9389	1.36510
$482$	0	0
$483$	−21.2720	−0.967911
$484$	0	0
$485$	18.7573	0.851725
$486$	0	0
$487$	−9.13323	−0.413866	−0.206933	−	0.978355i	$-0.566348\pi$
$487$	−9.13323	−0.413866	−0.206933	+	0.978355i	$0.566348\pi$
$488$	0	0
$489$	−42.4517	−1.91973
$490$	0	0
$491$	6.97121	0.314606	0.157303	−	0.987550i	$-0.449720\pi$
$491$	6.97121	0.314606	0.157303	+	0.987550i	$0.449720\pi$
$492$	0	0
$493$	−33.3388	−1.50151
$494$	0	0
$495$	8.78847	0.395012
$496$	0	0
$497$	14.1264	0.633656
$498$	0	0
$499$	39.2833	1.75856	0.879281	−	0.476303i	$-0.158024\pi$
$499$	39.2833	1.75856	0.879281	+	0.476303i	$0.158024\pi$
$500$	0	0
$501$	−60.0401	−2.68239
$502$	0	0
$503$	38.8850	1.73380	0.866898	−	0.498486i	$-0.166111\pi$
$503$	38.8850	1.73380	0.866898	+	0.498486i	$0.166111\pi$
$504$	0	0
$505$	7.38086	0.328444
$506$	0	0
$507$	−18.3756	−0.816088
$508$	0	0
$509$	19.4769	0.863300	0.431650	−	0.902041i	$-0.357932\pi$
$509$	19.4769	0.863300	0.431650	+	0.902041i	$0.357932\pi$
$510$	0	0
$511$	7.47769	0.330793
$512$	0	0
$513$	−11.6892	−0.516093
$514$	0	0
$515$	−3.61288	−0.159203
$516$	0	0
$517$	6.60608	0.290535
$518$	0	0
$519$	58.8888	2.58493
$520$	0	0
$521$	13.3329	0.584124	0.292062	−	0.956399i	$-0.405659\pi$
$521$	13.3329	0.584124	0.292062	+	0.956399i	$0.405659\pi$
$522$	0	0
$523$	−15.3046	−0.669221	−0.334611	−	0.942356i	$-0.608605\pi$
$523$	−15.3046	−0.669221	−0.334611	+	0.942356i	$0.608605\pi$
$524$	0	0
$525$	−8.81884	−0.384886
$526$	0	0
$527$	−37.6769	−1.64123
$528$	0	0
$529$	40.4414	1.75832
$530$	0	0
$531$	−39.6413	−1.72029
$532$	0	0
$533$	−36.1008	−1.56370
$534$	0	0
$535$	2.66938	0.115407
$536$	0	0
$537$	−44.4180	−1.91678
$538$	0	0
$539$	6.26805	0.269984
$540$	0	0
$541$	2.19482	0.0943628	0.0471814	−	0.998886i	$-0.484976\pi$
$541$	2.19482	0.0943628	0.0471814	+	0.998886i	$0.484976\pi$
$542$	0	0
$543$	27.9493	1.19942
$544$	0	0
$545$	5.01059	0.214630
$546$	0	0
$547$	7.87590	0.336749	0.168375	−	0.985723i	$-0.446148\pi$
$547$	7.87590	0.336749	0.168375	+	0.985723i	$0.446148\pi$
$548$	0	0
$549$	73.3770	3.13165
$550$	0	0
$551$	7.89693	0.336420
$552$	0	0
$553$	−5.79992	−0.246638
$554$	0	0
$555$	28.0220	1.18947
$556$	0	0
$557$	13.0214	0.551736	0.275868	−	0.961196i	$-0.411035\pi$
$557$	13.0214	0.551736	0.275868	+	0.961196i	$0.411035\pi$
$558$	0	0
$559$	4.44928	0.188184
$560$	0	0
$561$	−13.1787	−0.556407
$562$	0	0
$563$	3.46487e−5 0	1.46027e−6 0	7.30135e−7	−	1.00000i	$-0.500000\pi$
$563$	3.46487e−5 0	1.46027e−6 0	7.30135e−7		1.00000i	$0.500000\pi$
$564$	0	0
$565$	−16.8886	−0.710510
$566$	0	0
$567$	−13.9075	−0.584058
$568$	0	0
$569$	30.2002	1.26606	0.633028	−	0.774129i	$-0.281812\pi$
$569$	30.2002	1.26606	0.633028	+	0.774129i	$0.281812\pi$
$570$	0	0
$571$	31.9475	1.33696	0.668480	−	0.743730i	$-0.266945\pi$
$571$	31.9475	1.33696	0.668480	+	0.743730i	$0.266945\pi$
$572$	0	0
$573$	−31.7537	−1.32653
$574$	0	0
$575$	26.3012	1.09684
$576$	0	0
$577$	−43.8362	−1.82492	−0.912462	−	0.409161i	$-0.865821\pi$
$577$	−43.8362	−1.82492	−0.912462	+	0.409161i	$0.865821\pi$
$578$	0	0
$579$	−49.4269	−2.05411
$580$	0	0
$581$	−4.00692	−0.166235
$582$	0	0
$583$	12.7383	0.527567
$584$	0	0
$585$	−38.1935	−1.57911
$586$	0	0
$587$	7.56696	0.312322	0.156161	−	0.987732i	$-0.450088\pi$
$587$	7.56696	0.312322	0.156161	+	0.987732i	$0.450088\pi$
$588$	0	0
$589$	8.92448	0.367727
$590$	0	0
$591$	−58.4448	−2.40410
$592$	0	0
$593$	−25.9610	−1.06609	−0.533044	−	0.846087i	$-0.678952\pi$
$593$	−25.9610	−1.06609	−0.533044	+	0.846087i	$0.678952\pi$
$594$	0	0
$595$	−4.70642	−0.192944
$596$	0	0
$597$	6.58364	0.269450
$598$	0	0
$599$	−44.6992	−1.82636	−0.913180	−	0.407557i	$-0.866381\pi$
$599$	−44.6992	−1.82636	−0.913180	+	0.407557i	$0.866381\pi$
$600$	0	0
$601$	−35.8419	−1.46202	−0.731010	−	0.682367i	$-0.760951\pi$
$601$	−35.8419	−1.46202	−0.731010	+	0.682367i	$0.760951\pi$
$602$	0	0
$603$	−42.2751	−1.72157
$604$	0	0
$605$	−1.30304	−0.0529761
$606$	0	0
$607$	25.2257	1.02388	0.511939	−	0.859022i	$-0.328927\pi$
$607$	25.2257	1.02388	0.511939	+	0.859022i	$0.328927\pi$
$608$	0	0
$609$	21.0902	0.854618
$610$	0	0
$611$	−28.7091	−1.16145
$612$	0	0
$613$	24.2334	0.978779	0.489389	−	0.872065i	$-0.337220\pi$
$613$	24.2334	0.978779	0.489389	+	0.872065i	$0.337220\pi$
$614$	0	0
$615$	−33.7893	−1.36252
$616$	0	0
$617$	1.44690	0.0582500	0.0291250	−	0.999576i	$-0.490728\pi$
$617$	1.44690	0.0582500	0.0291250	+	0.999576i	$0.490728\pi$
$618$	0	0
$619$	5.89294	0.236857	0.118429	−	0.992963i	$-0.462214\pi$
$619$	5.89294	0.236857	0.118429	+	0.992963i	$0.462214\pi$
$620$	0	0
$621$	93.1050	3.73617
$622$	0	0
$623$	−13.9849	−0.560294
$624$	0	0
$625$	2.41425	0.0965699
$626$	0	0
$627$	3.12163	0.124666
$628$	0	0
$629$	−29.0838	−1.15965
$630$	0	0
$631$	19.0898	0.759954	0.379977	−	0.924996i	$-0.375932\pi$
$631$	19.0898	0.759954	0.379977	+	0.924996i	$0.375932\pi$
$632$	0	0
$633$	−6.42915	−0.255536
$634$	0	0
$635$	−24.3879	−0.967804
$636$	0	0
$637$	−27.2401	−1.07929
$638$	0	0
$639$	−111.365	−4.40551
$640$	0	0
$641$	34.1810	1.35007	0.675033	−	0.737787i	$-0.264129\pi$
$641$	34.1810	1.35007	0.675033	+	0.737787i	$0.264129\pi$
$642$	0	0
$643$	−40.2191	−1.58609	−0.793044	−	0.609164i	$-0.791505\pi$
$643$	−40.2191	−1.58609	−0.793044	+	0.609164i	$0.791505\pi$
$644$	0	0
$645$	4.16440	0.163973
$646$	0	0
$647$	−27.0695	−1.06421	−0.532105	−	0.846678i	$-0.678599\pi$
$647$	−27.0695	−1.06421	−0.532105	+	0.846678i	$0.678599\pi$
$648$	0	0
$649$	5.87749	0.230712
$650$	0	0
$651$	23.8345	0.934146
$652$	0	0
$653$	27.7239	1.08492	0.542460	−	0.840081i	$-0.317493\pi$
$653$	27.7239	1.08492	0.542460	+	0.840081i	$0.317493\pi$
$654$	0	0
$655$	−24.1830	−0.944909
$656$	0	0
$657$	−58.9498	−2.29985
$658$	0	0
$659$	0.236087	0.00919665	0.00459832	−	0.999989i	$-0.498536\pi$
$659$	0.236087	0.00919665	0.00459832	+	0.999989i	$0.498536\pi$
$660$	0	0
$661$	−14.2520	−0.554337	−0.277169	−	0.960821i	$-0.589396\pi$
$661$	−14.2520	−0.554337	−0.277169	+	0.960821i	$0.589396\pi$
$662$	0	0
$663$	57.2730	2.22430
$664$	0	0
$665$	1.11480	0.0432302
$666$	0	0
$667$	−62.8991	−2.43546
$668$	0	0
$669$	−19.3399	−0.747722
$670$	0	0
$671$	−10.8794	−0.419994
$672$	0	0
$673$	18.9363	0.729940	0.364970	−	0.931019i	$-0.381079\pi$
$673$	18.9363	0.729940	0.364970	+	0.931019i	$0.381079\pi$
$674$	0	0
$675$	38.5989	1.48567
$676$	0	0
$677$	1.20528	0.0463228	0.0231614	−	0.999732i	$-0.492627\pi$
$677$	1.20528	0.0463228	0.0231614	+	0.999732i	$0.492627\pi$
$678$	0	0
$679$	12.3155	0.472627
$680$	0	0
$681$	25.6287	0.982093
$682$	0	0
$683$	14.7780	0.565464	0.282732	−	0.959199i	$-0.408759\pi$
$683$	14.7780	0.565464	0.282732	+	0.959199i	$0.408759\pi$
$684$	0	0
$685$	−20.6449	−0.788802
$686$	0	0
$687$	16.2331	0.619331
$688$	0	0
$689$	−55.3590	−2.10901
$690$	0	0
$691$	15.9754	0.607731	0.303866	−	0.952715i	$-0.401723\pi$
$691$	15.9754	0.607731	0.303866	+	0.952715i	$0.401723\pi$
$692$	0	0
$693$	5.77027	0.219195
$694$	0	0
$695$	−18.3624	−0.696524
$696$	0	0
$697$	35.0697	1.32836
$698$	0	0
$699$	16.5496	0.625965
$700$	0	0
$701$	12.7171	0.480318	0.240159	−	0.970734i	$-0.422800\pi$
$701$	12.7171	0.480318	0.240159	+	0.970734i	$0.422800\pi$
$702$	0	0
$703$	6.88905	0.259826
$704$	0	0
$705$	−26.8709	−1.01202
$706$	0	0
$707$	4.84608	0.182256
$708$	0	0
$709$	−25.3436	−0.951798	−0.475899	−	0.879500i	$-0.657877\pi$
$709$	−25.3436	−0.951798	−0.475899	+	0.879500i	$0.657877\pi$
$710$	0	0
$711$	45.7232	1.71476
$712$	0	0
$713$	−71.0836	−2.66210
$714$	0	0
$715$	5.66283	0.211778
$716$	0	0
$717$	5.87740	0.219496
$718$	0	0
$719$	−37.4409	−1.39631	−0.698154	−	0.715947i	$-0.745995\pi$
$719$	−37.4409	−1.39631	−0.698154	+	0.715947i	$0.745995\pi$
$720$	0	0
$721$	−2.37212	−0.0883424
$722$	0	0
$723$	39.8274	1.48120
$724$	0	0
$725$	−26.0764	−0.968452
$726$	0	0
$727$	−34.3329	−1.27334	−0.636669	−	0.771137i	$-0.719688\pi$
$727$	−34.3329	−1.27334	−0.636669	+	0.771137i	$0.719688\pi$
$728$	0	0
$729$	0.169840	0.00629039
$730$	0	0
$731$	−4.32221	−0.159863
$732$	0	0
$733$	26.4252	0.976035	0.488018	−	0.872834i	$-0.337720\pi$
$733$	26.4252	0.976035	0.488018	+	0.872834i	$0.337720\pi$
$734$	0	0
$735$	−25.4960	−0.940433
$736$	0	0
$737$	6.26799	0.230884
$738$	0	0
$739$	17.5075	0.644023	0.322011	−	0.946736i	$-0.395641\pi$
$739$	17.5075	0.644023	0.322011	+	0.946736i	$0.395641\pi$
$740$	0	0
$741$	−13.5662	−0.498366
$742$	0	0
$743$	−22.6606	−0.831338	−0.415669	−	0.909516i	$-0.636453\pi$
$743$	−22.6606	−0.831338	−0.415669	+	0.909516i	$0.636453\pi$
$744$	0	0
$745$	−1.38297	−0.0506682
$746$	0	0
$747$	31.5882	1.15575
$748$	0	0
$749$	1.75264	0.0640402
$750$	0	0
$751$	−34.5959	−1.26242	−0.631211	−	0.775611i	$-0.717442\pi$
$751$	−34.5959	−1.26242	−0.631211	+	0.775611i	$0.717442\pi$
$752$	0	0
$753$	40.6392	1.48097
$754$	0	0
$755$	−1.24499	−0.0453099
$756$	0	0
$757$	9.24594	0.336049	0.168025	−	0.985783i	$-0.446261\pi$
$757$	9.24594	0.336049	0.168025	+	0.985783i	$0.446261\pi$
$758$	0	0
$759$	−24.8638	−0.902500
$760$	0	0
$761$	50.9489	1.84690	0.923448	−	0.383725i	$-0.125359\pi$
$761$	50.9489	1.84690	0.923448	+	0.383725i	$0.125359\pi$
$762$	0	0
$763$	3.28982	0.119099
$764$	0	0
$765$	37.1027	1.34145
$766$	0	0
$767$	−25.5428	−0.922296
$768$	0	0
$769$	24.8400	0.895753	0.447877	−	0.894095i	$-0.352180\pi$
$769$	24.8400	0.895753	0.447877	+	0.894095i	$0.352180\pi$
$770$	0	0
$771$	99.4209	3.58056
$772$	0	0
$773$	37.1574	1.33646	0.668230	−	0.743955i	$-0.267052\pi$
$773$	37.1574	1.33646	0.668230	+	0.743955i	$0.267052\pi$
$774$	0	0
$775$	−29.4694	−1.05857
$776$	0	0
$777$	18.3985	0.660042
$778$	0	0
$779$	−8.30692	−0.297626
$780$	0	0
$781$	16.5117	0.590834
$782$	0	0
$783$	−92.3091	−3.29886
$784$	0	0
$785$	7.03918	0.251239
$786$	0	0
$787$	18.5192	0.660137	0.330069	−	0.943957i	$-0.392928\pi$
$787$	18.5192	0.660137	0.330069	+	0.943957i	$0.392928\pi$
$788$	0	0
$789$	−72.2728	−2.57298
$790$	0	0
$791$	−11.0886	−0.394266
$792$	0	0
$793$	47.2803	1.67897
$794$	0	0
$795$	−51.8145	−1.83767
$796$	0	0
$797$	−24.9874	−0.885101	−0.442550	−	0.896744i	$-0.645926\pi$
$797$	−24.9874	−0.885101	−0.442550	+	0.896744i	$0.645926\pi$
$798$	0	0
$799$	27.8892	0.986649
$800$	0	0
$801$	110.249	3.89546
$802$	0	0
$803$	8.74031	0.308439
$804$	0	0
$805$	−8.87941	−0.312958
$806$	0	0
$807$	−33.8614	−1.19198
$808$	0	0
$809$	−31.1570	−1.09542	−0.547712	−	0.836667i	$-0.684501\pi$
$809$	−31.1570	−1.09542	−0.547712	+	0.836667i	$0.684501\pi$
$810$	0	0
$811$	38.0147	1.33488	0.667438	−	0.744666i	$-0.267391\pi$
$811$	38.0147	1.33488	0.667438	+	0.744666i	$0.267391\pi$
$812$	0	0
$813$	31.9103	1.11914
$814$	0	0
$815$	−17.7203	−0.620715
$816$	0	0
$817$	1.02380	0.0358181
$818$	0	0
$819$	−25.0768	−0.876255
$820$	0	0
$821$	−25.3026	−0.883065	−0.441533	−	0.897245i	$-0.645565\pi$
$821$	−25.3026	−0.883065	−0.441533	+	0.897245i	$0.645565\pi$
$822$	0	0
$823$	−14.2736	−0.497547	−0.248774	−	0.968562i	$-0.580027\pi$
$823$	−14.2736	−0.497547	−0.248774	+	0.968562i	$0.580027\pi$
$824$	0	0
$825$	−10.3079	−0.358875
$826$	0	0
$827$	40.2245	1.39874	0.699372	−	0.714758i	$-0.253463\pi$
$827$	40.2245	1.39874	0.699372	+	0.714758i	$0.253463\pi$
$828$	0	0
$829$	8.42401	0.292578	0.146289	−	0.989242i	$-0.453267\pi$
$829$	8.42401	0.292578	0.146289	+	0.989242i	$0.453267\pi$
$830$	0	0
$831$	−52.3689	−1.81666
$832$	0	0
$833$	26.4621	0.916858
$834$	0	0
$835$	−25.0621	−0.867309
$836$	0	0
$837$	−104.320	−3.60584
$838$	0	0
$839$	−26.0205	−0.898326	−0.449163	−	0.893450i	$-0.648278\pi$
$839$	−26.0205	−0.898326	−0.449163	+	0.893450i	$0.648278\pi$
$840$	0	0
$841$	33.3615	1.15040
$842$	0	0
$843$	−48.4547	−1.66887
$844$	0	0
$845$	−7.67037	−0.263869
$846$	0	0
$847$	−0.855540	−0.0293967
$848$	0	0
$849$	−37.7906	−1.29697
$850$	0	0
$851$	−54.8714	−1.88097
$852$	0	0
$853$	2.74022	0.0938232	0.0469116	−	0.998899i	$-0.485062\pi$
$853$	2.74022	0.0938232	0.0469116	+	0.998899i	$0.485062\pi$
$854$	0	0
$855$	−8.78847	−0.300559
$856$	0	0
$857$	37.0504	1.26562	0.632808	−	0.774309i	$-0.281902\pi$
$857$	37.0504	1.26562	0.632808	+	0.774309i	$0.281902\pi$
$858$	0	0
$859$	−2.45499	−0.0837632	−0.0418816	−	0.999123i	$-0.513335\pi$
$859$	−2.45499	−0.0837632	−0.0418816	+	0.999123i	$0.513335\pi$
$860$	0	0
$861$	−22.1852	−0.756068
$862$	0	0
$863$	22.3602	0.761151	0.380576	−	0.924750i	$-0.375726\pi$
$863$	22.3602	0.761151	0.380576	+	0.924750i	$0.375726\pi$
$864$	0	0
$865$	24.5815	0.835796
$866$	0	0
$867$	−2.56960	−0.0872681
$868$	0	0
$869$	−6.77925	−0.229970
$870$	0	0
$871$	−27.2398	−0.922987
$872$	0	0
$873$	−97.0887	−3.28595
$874$	0	0
$875$	−9.25519	−0.312883
$876$	0	0
$877$	58.6391	1.98010	0.990051	−	0.140709i	$-0.0449383\pi$
$877$	58.6391	1.98010	0.990051	+	0.140709i	$0.0449383\pi$
$878$	0	0
$879$	68.3738	2.30619
$880$	0	0
$881$	39.1633	1.31945	0.659723	−	0.751509i	$-0.270674\pi$
$881$	39.1633	1.31945	0.659723	+	0.751509i	$0.270674\pi$
$882$	0	0
$883$	−15.8199	−0.532383	−0.266191	−	0.963920i	$-0.585765\pi$
$883$	−15.8199	−0.532383	−0.266191	+	0.963920i	$0.585765\pi$
$884$	0	0
$885$	−23.9073	−0.803636
$886$	0	0
$887$	4.07940	0.136973	0.0684864	−	0.997652i	$-0.478183\pi$
$887$	4.07940	0.136973	0.0684864	+	0.997652i	$0.478183\pi$
$888$	0	0
$889$	−16.0124	−0.537040
$890$	0	0
$891$	−16.2558	−0.544588
$892$	0	0
$893$	−6.60608	−0.221064
$894$	0	0
$895$	−18.5411	−0.619759
$896$	0	0
$897$	108.055	3.60785
$898$	0	0
$899$	70.4760	2.35051
$900$	0	0
$901$	53.7779	1.79160
$902$	0	0
$903$	2.73423	0.0909896
$904$	0	0
$905$	11.6667	0.387813
$906$	0	0
$907$	−17.9400	−0.595688	−0.297844	−	0.954615i	$-0.596268\pi$
$907$	−17.9400	−0.595688	−0.297844	+	0.954615i	$0.596268\pi$
$908$	0	0
$909$	−38.2037	−1.26714
$910$	0	0
$911$	−7.51033	−0.248828	−0.124414	−	0.992230i	$-0.539705\pi$
$911$	−7.51033	−0.248828	−0.124414	+	0.992230i	$0.539705\pi$
$912$	0	0
$913$	−4.68349	−0.155001
$914$	0	0
$915$	44.2530	1.46296
$916$	0	0
$917$	−15.8779	−0.524335
$918$	0	0
$919$	−8.54632	−0.281917	−0.140959	−	0.990016i	$-0.545018\pi$
$919$	−8.54632	−0.281917	−0.140959	+	0.990016i	$0.545018\pi$
$920$	0	0
$921$	69.7873	2.29957
$922$	0	0
$923$	−71.7575	−2.36193
$924$	0	0
$925$	−22.7483	−0.747959
$926$	0	0
$927$	18.7004	0.614203
$928$	0	0
$929$	4.20328	0.137905	0.0689526	−	0.997620i	$-0.478034\pi$
$929$	4.20328	0.137905	0.0689526	+	0.997620i	$0.478034\pi$
$930$	0	0
$931$	−6.26805	−0.205427
$932$	0	0
$933$	−42.6460	−1.39617
$934$	0	0
$935$	−5.50110	−0.179905
$936$	0	0
$937$	48.4723	1.58352	0.791760	−	0.610832i	$-0.209165\pi$
$937$	48.4723	1.58352	0.791760	+	0.610832i	$0.209165\pi$
$938$	0	0
$939$	−90.7077	−2.96013
$940$	0	0
$941$	42.8180	1.39583	0.697913	−	0.716183i	$-0.254112\pi$
$941$	42.8180	1.39583	0.697913	+	0.716183i	$0.254112\pi$
$942$	0	0
$943$	66.1648	2.15462
$944$	0	0
$945$	−13.0312	−0.423905
$946$	0	0
$947$	13.0956	0.425551	0.212776	−	0.977101i	$-0.431750\pi$
$947$	13.0956	0.425551	0.212776	+	0.977101i	$0.431750\pi$
$948$	0	0
$949$	−37.9842	−1.23302
$950$	0	0
$951$	−30.1353	−0.977203
$952$	0	0
$953$	−54.4895	−1.76509	−0.882544	−	0.470229i	$-0.844171\pi$
$953$	−54.4895	−1.76509	−0.882544	+	0.470229i	$0.844171\pi$
$954$	0	0
$955$	−13.2547	−0.428912
$956$	0	0
$957$	24.6513	0.796864
$958$	0	0
$959$	−13.5549	−0.437711
$960$	0	0
$961$	48.6463	1.56924
$962$	0	0
$963$	−13.8168	−0.445241
$964$	0	0
$965$	−20.6319	−0.664164
$966$	0	0
$967$	−43.0442	−1.38421	−0.692104	−	0.721798i	$-0.743316\pi$
$967$	−43.0442	−1.38421	−0.692104	+	0.721798i	$0.743316\pi$
$968$	0	0
$969$	13.1787	0.423362
$970$	0	0
$971$	13.0644	0.419257	0.209629	−	0.977781i	$-0.432775\pi$
$971$	13.0644	0.419257	0.209629	+	0.977781i	$0.432775\pi$
$972$	0	0
$973$	−12.0562	−0.386505
$974$	0	0
$975$	44.7968	1.43465
$976$	0	0
$977$	−15.9163	−0.509207	−0.254603	−	0.967046i	$-0.581945\pi$
$977$	−15.9163	−0.509207	−0.254603	+	0.967046i	$0.581945\pi$
$978$	0	0
$979$	−16.3463	−0.522430
$980$	0	0
$981$	−25.9351	−0.828042
$982$	0	0
$983$	−13.4431	−0.428768	−0.214384	−	0.976749i	$-0.568774\pi$
$983$	−13.4431	−0.428768	−0.214384	+	0.976749i	$0.568774\pi$
$984$	0	0
$985$	−24.3961	−0.777325
$986$	0	0
$987$	−17.6427	−0.561575
$988$	0	0
$989$	−8.15455	−0.259299
$990$	0	0
$991$	1.81920	0.0577887	0.0288944	−	0.999582i	$-0.490801\pi$
$991$	1.81920	0.0577887	0.0288944	+	0.999582i	$0.490801\pi$
$992$	0	0
$993$	37.3517	1.18532
$994$	0	0
$995$	2.74816	0.0871224
$996$	0	0
$997$	−28.4736	−0.901767	−0.450883	−	0.892583i	$-0.648891\pi$
$997$	−28.4736	−0.901767	−0.450883	+	0.892583i	$0.648891\pi$
$998$	0	0
$999$	−80.5278	−2.54779

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.bb.1.1		9
4.3	odd	2		1672.2.a.k.1.9	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1672.2.a.k.1.9	✓	9	4.3	odd	2
3344.2.a.bb.1.1		9	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:	\( N \)	\(=\)	\( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3344.a (trivial)

\(f(q)\)	\(=\)	\(q-3.12163 q^{3} -1.30304 q^{5} -0.855540 q^{7} +6.74459 q^{9} +O(q^{10})\)	\(q-3.12163 q^{3} -1.30304 q^{5} -0.855540 q^{7} +6.74459 q^{9} -1.00000 q^{11} +4.34586 q^{13} +4.06761 q^{15} -4.22175 q^{17} +1.00000 q^{19} +2.67068 q^{21} -7.96501 q^{23} -3.30209 q^{25} -11.6892 q^{27} +7.89693 q^{29} +8.92448 q^{31} +3.12163 q^{33} +1.11480 q^{35} +6.88905 q^{37} -13.5662 q^{39} -8.30692 q^{41} +1.02380 q^{43} -8.78847 q^{45} -6.60608 q^{47} -6.26805 q^{49} +13.1787 q^{51} -12.7383 q^{53} +1.30304 q^{55} -3.12163 q^{57} -5.87749 q^{59} +10.8794 q^{61} -5.77027 q^{63} -5.66283 q^{65} -6.26799 q^{67} +24.8638 q^{69} -16.5117 q^{71} -8.74031 q^{73} +10.3079 q^{75} +0.855540 q^{77} +6.77925 q^{79} +16.2558 q^{81} +4.68349 q^{83} +5.50110 q^{85} -24.6513 q^{87} +16.3463 q^{89} -3.71806 q^{91} -27.8589 q^{93} -1.30304 q^{95} -14.3950 q^{97} -6.74459 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - q^{3} + 6 q^{5} - 3 q^{7} + 18 q^{9}+O(q^{10}) \) 9 * q - q^3 + 6 * q^5 - 3 * q^7 + 18 * q^9	\( 9 q - q^{3} + 6 q^{5} - 3 q^{7} + 18 q^{9} - 9 q^{11} + q^{13} - 6 q^{15} + 7 q^{17} + 9 q^{19} + 3 q^{21} - 13 q^{23} + 31 q^{25} + 5 q^{27} + 9 q^{29} + 4 q^{31} + q^{33} + 4 q^{35} + 24 q^{37} - 13 q^{39} - 6 q^{41} + 14 q^{43} + 26 q^{45} - 24 q^{47} + 20 q^{49} + 33 q^{51} + 19 q^{53} - 6 q^{55} - q^{57} + 19 q^{59} + 28 q^{61} - 16 q^{63} + 16 q^{65} - 5 q^{67} + 35 q^{69} - 16 q^{71} + 15 q^{73} - 3 q^{75} + 3 q^{77} - 2 q^{79} + 37 q^{81} - 8 q^{83} + 20 q^{85} - 23 q^{87} + 12 q^{89} + 29 q^{91} + 44 q^{93} + 6 q^{95} - 4 q^{97} - 18 q^{99}+O(q^{100}) \) 9 * q - q^3 + 6 * q^5 - 3 * q^7 + 18 * q^9 - 9 * q^11 + q^13 - 6 * q^15 + 7 * q^17 + 9 * q^19 + 3 * q^21 - 13 * q^23 + 31 * q^25 + 5 * q^27 + 9 * q^29 + 4 * q^31 + q^33 + 4 * q^35 + 24 * q^37 - 13 * q^39 - 6 * q^41 + 14 * q^43 + 26 * q^45 - 24 * q^47 + 20 * q^49 + 33 * q^51 + 19 * q^53 - 6 * q^55 - q^57 + 19 * q^59 + 28 * q^61 - 16 * q^63 + 16 * q^65 - 5 * q^67 + 35 * q^69 - 16 * q^71 + 15 * q^73 - 3 * q^75 + 3 * q^77 - 2 * q^79 + 37 * q^81 - 8 * q^83 + 20 * q^85 - 23 * q^87 + 12 * q^89 + 29 * q^91 + 44 * q^93 + 6 * q^95 - 4 * q^97 - 18 * q^99

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