LMFDB - Embedded newform 6348.2.a.s.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6348,2,Mod(1,6348)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6348, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6348.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6348 = 2^{2} \cdot 3 \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6348.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:	$50.6890352031$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 4x^{9} - 14x^{8} + 65x^{7} + 57x^{6} - 354x^{5} - 46x^{4} + 714x^{3} - 74x^{2} - 323x - 23 $ x^10 - 4x^9 - 14x^8 + 65x^7 + 57x^6 - 354x^5 - 46x^4 + 714x^3 - 74x^2 - 323*x - 23
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 276)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$2.75582$ of defining polynomial
Character	$\chi$	$=$	6348.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +3.63668 q^{5} -4.36939 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +3.63668 q^{5} -4.36939 q^{7} +1.00000 q^{9} +2.73571 q^{11} -6.14956 q^{13} +3.63668 q^{15} -1.31551 q^{17} -5.38609 q^{19} -4.36939 q^{21} +8.22547 q^{25} +1.00000 q^{27} -6.53557 q^{29} +4.09134 q^{31} +2.73571 q^{33} -15.8901 q^{35} +5.76661 q^{37} -6.14956 q^{39} +7.77719 q^{41} -9.41225 q^{43} +3.63668 q^{45} -5.07522 q^{47} +12.0916 q^{49} -1.31551 q^{51} +4.08357 q^{53} +9.94893 q^{55} -5.38609 q^{57} -3.45514 q^{59} -2.65460 q^{61} -4.36939 q^{63} -22.3640 q^{65} -4.40601 q^{67} -10.2287 q^{71} -6.49597 q^{73} +8.22547 q^{75} -11.9534 q^{77} -3.37885 q^{79} +1.00000 q^{81} -7.19420 q^{83} -4.78410 q^{85} -6.53557 q^{87} -9.80998 q^{89} +26.8698 q^{91} +4.09134 q^{93} -19.5875 q^{95} +2.44980 q^{97} +2.73571 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 10 q^{3} - 2 q^{5} - 11 q^{7} + 10 q^{9}+O(q^{10}) $ 10 * q + 10 * q^3 - 2 * q^5 - 11 * q^7 + 10 * q^9	$ 10 q + 10 q^{3} - 2 q^{5} - 11 q^{7} + 10 q^{9} - 11 q^{11} - 2 q^{15} - 13 q^{17} - 18 q^{19} - 11 q^{21} + 10 q^{25} + 10 q^{27} + 5 q^{29} + 15 q^{31} - 11 q^{33} - 13 q^{35} - 5 q^{37} + 24 q^{41} - 40 q^{43} - 2 q^{45} - 9 q^{47} + 5 q^{49} - 13 q^{51} - 6 q^{53} - 14 q^{55} - 18 q^{57} + 28 q^{59} - 39 q^{61} - 11 q^{63} - 14 q^{65} - 32 q^{67} - 33 q^{71} - 50 q^{73} + 10 q^{75} + 19 q^{77} - 33 q^{79} + 10 q^{81} - 29 q^{83} - 21 q^{85} + 5 q^{87} - 17 q^{89} - 36 q^{91} + 15 q^{93} - 42 q^{95} - 46 q^{97} - 11 q^{99}+O(q^{100}) $ 10 * q + 10 * q^3 - 2 * q^5 - 11 * q^7 + 10 * q^9 - 11 * q^11 - 2 * q^15 - 13 * q^17 - 18 * q^19 - 11 * q^21 + 10 * q^25 + 10 * q^27 + 5 * q^29 + 15 * q^31 - 11 * q^33 - 13 * q^35 - 5 * q^37 + 24 * q^41 - 40 * q^43 - 2 * q^45 - 9 * q^47 + 5 * q^49 - 13 * q^51 - 6 * q^53 - 14 * q^55 - 18 * q^57 + 28 * q^59 - 39 * q^61 - 11 * q^63 - 14 * q^65 - 32 * q^67 - 33 * q^71 - 50 * q^73 + 10 * q^75 + 19 * q^77 - 33 * q^79 + 10 * q^81 - 29 * q^83 - 21 * q^85 + 5 * q^87 - 17 * q^89 - 36 * q^91 + 15 * q^93 - 42 * q^95 - 46 * q^97 - 11 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	3.63668	1.62637	0.813187	−	0.582002i	$-0.197730\pi$
$5$	3.63668	1.62637	0.813187	+	0.582002i	$0.197730\pi$
$6$	0	0
$7$	−4.36939	−1.65147	−0.825737	−	0.564055i	$-0.809241\pi$
$7$	−4.36939	−1.65147	−0.825737	+	0.564055i	$0.809241\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.73571	0.824849	0.412424	−	0.910992i	$-0.364682\pi$
$11$	2.73571	0.824849	0.412424	+	0.910992i	$0.364682\pi$
$12$	0	0
$13$	−6.14956	−1.70558	−0.852791	−	0.522253i	$-0.825092\pi$
$13$	−6.14956	−1.70558	−0.852791	+	0.522253i	$0.825092\pi$
$14$	0	0
$15$	3.63668	0.938988
$16$	0	0
$17$	−1.31551	−0.319059	−0.159529	−	0.987193i	$-0.550998\pi$
$17$	−1.31551	−0.319059	−0.159529	+	0.987193i	$0.550998\pi$
$18$	0	0
$19$	−5.38609	−1.23565	−0.617827	−	0.786314i	$-0.711987\pi$
$19$	−5.38609	−1.23565	−0.617827	+	0.786314i	$0.711987\pi$
$20$	0	0
$21$	−4.36939	−0.953479
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	8.22547	1.64509
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−6.53557	−1.21362	−0.606812	−	0.794845i	$-0.707552\pi$
$29$	−6.53557	−1.21362	−0.606812	+	0.794845i	$0.707552\pi$
$30$	0	0
$31$	4.09134	0.734827	0.367413	−	0.930058i	$-0.380243\pi$
$31$	4.09134	0.734827	0.367413	+	0.930058i	$0.380243\pi$
$32$	0	0
$33$	2.73571	0.476227
$34$	0	0
$35$	−15.8901	−2.68592
$36$	0	0
$37$	5.76661	0.948025	0.474012	−	0.880518i	$-0.342805\pi$
$37$	5.76661	0.948025	0.474012	+	0.880518i	$0.342805\pi$
$38$	0	0
$39$	−6.14956	−0.984718
$40$	0	0
$41$	7.77719	1.21459	0.607297	−	0.794475i	$-0.292254\pi$
$41$	7.77719	1.21459	0.607297	+	0.794475i	$0.292254\pi$
$42$	0	0
$43$	−9.41225	−1.43536	−0.717678	−	0.696375i	$-0.754795\pi$
$43$	−9.41225	−1.43536	−0.717678	+	0.696375i	$0.754795\pi$
$44$	0	0
$45$	3.63668	0.542125
$46$	0	0
$47$	−5.07522	−0.740297	−0.370148	−	0.928973i	$-0.620693\pi$
$47$	−5.07522	−0.740297	−0.370148	+	0.928973i	$0.620693\pi$
$48$	0	0
$49$	12.0916	1.72737
$50$	0	0
$51$	−1.31551	−0.184209
$52$	0	0
$53$	4.08357	0.560921	0.280460	−	0.959866i	$-0.409513\pi$
$53$	4.08357	0.560921	0.280460	+	0.959866i	$0.409513\pi$
$54$	0	0
$55$	9.94893	1.34151
$56$	0	0
$57$	−5.38609	−0.713406
$58$	0	0
$59$	−3.45514	−0.449822	−0.224911	−	0.974379i	$-0.572209\pi$
$59$	−3.45514	−0.449822	−0.224911	+	0.974379i	$0.572209\pi$
$60$	0	0
$61$	−2.65460	−0.339887	−0.169944	−	0.985454i	$-0.554359\pi$
$61$	−2.65460	−0.339887	−0.169944	+	0.985454i	$0.554359\pi$
$62$	0	0
$63$	−4.36939	−0.550492
$64$	0	0
$65$	−22.3640	−2.77391
$66$	0	0
$67$	−4.40601	−0.538280	−0.269140	−	0.963101i	$-0.586739\pi$
$67$	−4.40601	−0.538280	−0.269140	+	0.963101i	$0.586739\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−10.2287	−1.21392	−0.606959	−	0.794733i	$-0.707611\pi$
$71$	−10.2287	−1.21392	−0.606959	+	0.794733i	$0.707611\pi$
$72$	0	0
$73$	−6.49597	−0.760296	−0.380148	−	0.924926i	$-0.624127\pi$
$73$	−6.49597	−0.760296	−0.380148	+	0.924926i	$0.624127\pi$
$74$	0	0
$75$	8.22547	0.949796
$76$	0	0
$77$	−11.9534	−1.36222
$78$	0	0
$79$	−3.37885	−0.380151	−0.190075	−	0.981769i	$-0.560873\pi$
$79$	−3.37885	−0.380151	−0.190075	+	0.981769i	$0.560873\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−7.19420	−0.789667	−0.394833	−	0.918753i	$-0.629198\pi$
$83$	−7.19420	−0.789667	−0.394833	+	0.918753i	$0.629198\pi$
$84$	0	0
$85$	−4.78410	−0.518909
$86$	0	0
$87$	−6.53557	−0.700686
$88$	0	0
$89$	−9.80998	−1.03986	−0.519928	−	0.854210i	$-0.674041\pi$
$89$	−9.80998	−1.03986	−0.519928	+	0.854210i	$0.674041\pi$
$90$	0	0
$91$	26.8698	2.81672
$92$	0	0
$93$	4.09134	0.424252
$94$	0	0
$95$	−19.5875	−2.00964
$96$	0	0
$97$	2.44980	0.248740	0.124370	−	0.992236i	$-0.460309\pi$
$97$	2.44980	0.248740	0.124370	+	0.992236i	$0.460309\pi$
$98$	0	0
$99$	2.73571	0.274950
$100$	0	0
$101$	16.4481	1.63665	0.818324	−	0.574757i	$-0.194903\pi$
$101$	16.4481	1.63665	0.818324	+	0.574757i	$0.194903\pi$
$102$	0	0
$103$	−5.14962	−0.507407	−0.253703	−	0.967282i	$-0.581649\pi$
$103$	−5.14962	−0.507407	−0.253703	+	0.967282i	$0.581649\pi$
$104$	0	0
$105$	−15.8901	−1.55071
$106$	0	0
$107$	−9.08411	−0.878194	−0.439097	−	0.898440i	$-0.644702\pi$
$107$	−9.08411	−0.878194	−0.439097	+	0.898440i	$0.644702\pi$
$108$	0	0
$109$	−3.15774	−0.302457	−0.151228	−	0.988499i	$-0.548323\pi$
$109$	−3.15774	−0.302457	−0.151228	+	0.988499i	$0.548323\pi$
$110$	0	0
$111$	5.76661	0.547342
$112$	0	0
$113$	−13.7325	−1.29184	−0.645920	−	0.763405i	$-0.723526\pi$
$113$	−13.7325	−1.29184	−0.645920	+	0.763405i	$0.723526\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−6.14956	−0.568527
$118$	0	0
$119$	5.74799	0.526917
$120$	0	0
$121$	−3.51587	−0.319624
$122$	0	0
$123$	7.77719	0.701246
$124$	0	0
$125$	11.7300	1.04917
$126$	0	0
$127$	9.79107	0.868817	0.434408	−	0.900716i	$-0.356957\pi$
$127$	9.79107	0.868817	0.434408	+	0.900716i	$0.356957\pi$
$128$	0	0
$129$	−9.41225	−0.828703
$130$	0	0
$131$	4.93851	0.431479	0.215740	−	0.976451i	$-0.430784\pi$
$131$	4.93851	0.431479	0.215740	+	0.976451i	$0.430784\pi$
$132$	0	0
$133$	23.5340	2.04065
$134$	0	0
$135$	3.63668	0.312996
$136$	0	0
$137$	−21.0333	−1.79699	−0.898496	−	0.438982i	$-0.855339\pi$
$137$	−21.0333	−1.79699	−0.898496	+	0.438982i	$0.855339\pi$
$138$	0	0
$139$	−8.52288	−0.722901	−0.361451	−	0.932391i	$-0.617718\pi$
$139$	−8.52288	−0.722901	−0.361451	+	0.932391i	$0.617718\pi$
$140$	0	0
$141$	−5.07522	−0.427410
$142$	0	0
$143$	−16.8234	−1.40685
$144$	0	0
$145$	−23.7678	−1.97381
$146$	0	0
$147$	12.0916	0.997297
$148$	0	0
$149$	−2.92762	−0.239840	−0.119920	−	0.992784i	$-0.538264\pi$
$149$	−2.92762	−0.239840	−0.119920	+	0.992784i	$0.538264\pi$
$150$	0	0
$151$	17.7191	1.44196	0.720980	−	0.692956i	$-0.243692\pi$
$151$	17.7191	1.44196	0.720980	+	0.692956i	$0.243692\pi$
$152$	0	0
$153$	−1.31551	−0.106353
$154$	0	0
$155$	14.8789	1.19510
$156$	0	0
$157$	−7.09513	−0.566253	−0.283127	−	0.959083i	$-0.591372\pi$
$157$	−7.09513	−0.566253	−0.283127	+	0.959083i	$0.591372\pi$
$158$	0	0
$159$	4.08357	0.323848
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0.843703	0.0660839	0.0330420	−	0.999454i	$-0.489480\pi$
$163$	0.843703	0.0660839	0.0330420	+	0.999454i	$0.489480\pi$
$164$	0	0
$165$	9.94893	0.774523
$166$	0	0
$167$	−19.0664	−1.47540	−0.737699	−	0.675129i	$-0.764088\pi$
$167$	−19.0664	−1.47540	−0.737699	+	0.675129i	$0.764088\pi$
$168$	0	0
$169$	24.8171	1.90901
$170$	0	0
$171$	−5.38609	−0.411885
$172$	0	0
$173$	−15.4323	−1.17330	−0.586648	−	0.809842i	$-0.699553\pi$
$173$	−15.4323	−1.17330	−0.586648	+	0.809842i	$0.699553\pi$
$174$	0	0
$175$	−35.9403	−2.71683
$176$	0	0
$177$	−3.45514	−0.259705
$178$	0	0
$179$	−12.1563	−0.908605	−0.454303	−	0.890847i	$-0.650112\pi$
$179$	−12.1563	−0.908605	−0.454303	+	0.890847i	$0.650112\pi$
$180$	0	0
$181$	−7.45191	−0.553896	−0.276948	−	0.960885i	$-0.589323\pi$
$181$	−7.45191	−0.553896	−0.276948	+	0.960885i	$0.589323\pi$
$182$	0	0
$183$	−2.65460	−0.196234
$184$	0	0
$185$	20.9713	1.54184
$186$	0	0
$187$	−3.59887	−0.263175
$188$	0	0
$189$	−4.36939	−0.317826
$190$	0	0
$191$	−7.71685	−0.558372	−0.279186	−	0.960237i	$-0.590065\pi$
$191$	−7.71685	−0.558372	−0.279186	+	0.960237i	$0.590065\pi$
$192$	0	0
$193$	−8.62331	−0.620719	−0.310360	−	0.950619i	$-0.600449\pi$
$193$	−8.62331	−0.620719	−0.310360	+	0.950619i	$0.600449\pi$
$194$	0	0
$195$	−22.3640	−1.60152
$196$	0	0
$197$	9.73994	0.693942	0.346971	−	0.937876i	$-0.387210\pi$
$197$	9.73994	0.693942	0.346971	+	0.937876i	$0.387210\pi$
$198$	0	0
$199$	15.9329	1.12945	0.564726	−	0.825278i	$-0.308982\pi$
$199$	15.9329	1.12945	0.564726	+	0.825278i	$0.308982\pi$
$200$	0	0
$201$	−4.40601	−0.310776
$202$	0	0
$203$	28.5564	2.00427
$204$	0	0
$205$	28.2832	1.97538
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−14.7348	−1.01923
$210$	0	0
$211$	5.17826	0.356486	0.178243	−	0.983986i	$-0.442959\pi$
$211$	5.17826	0.356486	0.178243	+	0.983986i	$0.442959\pi$
$212$	0	0
$213$	−10.2287	−0.700856
$214$	0	0
$215$	−34.2294	−2.33443
$216$	0	0
$217$	−17.8767	−1.21355
$218$	0	0
$219$	−6.49597	−0.438957
$220$	0	0
$221$	8.08983	0.544181
$222$	0	0
$223$	12.9736	0.868775	0.434388	−	0.900726i	$-0.356965\pi$
$223$	12.9736	0.868775	0.434388	+	0.900726i	$0.356965\pi$
$224$	0	0
$225$	8.22547	0.548365
$226$	0	0
$227$	−9.50561	−0.630910	−0.315455	−	0.948941i	$-0.602157\pi$
$227$	−9.50561	−0.630910	−0.315455	+	0.948941i	$0.602157\pi$
$228$	0	0
$229$	16.4978	1.09020	0.545102	−	0.838370i	$-0.316491\pi$
$229$	16.4978	1.09020	0.545102	+	0.838370i	$0.316491\pi$
$230$	0	0
$231$	−11.9534	−0.786476
$232$	0	0
$233$	1.67767	0.109908	0.0549540	−	0.998489i	$-0.482499\pi$
$233$	1.67767	0.109908	0.0549540	+	0.998489i	$0.482499\pi$
$234$	0	0
$235$	−18.4570	−1.20400
$236$	0	0
$237$	−3.37885	−0.219480
$238$	0	0
$239$	0.442505	0.0286233	0.0143116	−	0.999898i	$-0.495444\pi$
$239$	0.442505	0.0286233	0.0143116	+	0.999898i	$0.495444\pi$
$240$	0	0
$241$	−29.0794	−1.87317	−0.936584	−	0.350442i	$-0.886031\pi$
$241$	−29.0794	−1.87317	−0.936584	+	0.350442i	$0.886031\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	43.9733	2.80935
$246$	0	0
$247$	33.1221	2.10751
$248$	0	0
$249$	−7.19420	−0.455914
$250$	0	0
$251$	7.70670	0.486443	0.243221	−	0.969971i	$-0.421796\pi$
$251$	7.70670	0.486443	0.243221	+	0.969971i	$0.421796\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−4.78410	−0.299592
$256$	0	0
$257$	20.0114	1.24828	0.624138	−	0.781314i	$-0.285450\pi$
$257$	20.0114	1.24828	0.624138	+	0.781314i	$0.285450\pi$
$258$	0	0
$259$	−25.1966	−1.56564
$260$	0	0
$261$	−6.53557	−0.404541
$262$	0	0
$263$	12.8978	0.795312	0.397656	−	0.917535i	$-0.369824\pi$
$263$	12.8978	0.795312	0.397656	+	0.917535i	$0.369824\pi$
$264$	0	0
$265$	14.8506	0.912267
$266$	0	0
$267$	−9.80998	−0.600361
$268$	0	0
$269$	15.3686	0.937038	0.468519	−	0.883453i	$-0.344788\pi$
$269$	15.3686	0.937038	0.468519	+	0.883453i	$0.344788\pi$
$270$	0	0
$271$	−20.1954	−1.22678	−0.613390	−	0.789780i	$-0.710195\pi$
$271$	−20.1954	−1.22678	−0.613390	+	0.789780i	$0.710195\pi$
$272$	0	0
$273$	26.8698	1.62624
$274$	0	0
$275$	22.5025	1.35695
$276$	0	0
$277$	22.7999	1.36991	0.684957	−	0.728584i	$-0.259821\pi$
$277$	22.7999	1.36991	0.684957	+	0.728584i	$0.259821\pi$
$278$	0	0
$279$	4.09134	0.244942
$280$	0	0
$281$	14.8951	0.888565	0.444283	−	0.895887i	$-0.353459\pi$
$281$	14.8951	0.888565	0.444283	+	0.895887i	$0.353459\pi$
$282$	0	0
$283$	0.630449	0.0374763	0.0187382	−	0.999824i	$-0.494035\pi$
$283$	0.630449	0.0374763	0.0187382	+	0.999824i	$0.494035\pi$
$284$	0	0
$285$	−19.5875	−1.16026
$286$	0	0
$287$	−33.9816	−2.00587
$288$	0	0
$289$	−15.2694	−0.898202
$290$	0	0
$291$	2.44980	0.143610
$292$	0	0
$293$	13.0071	0.759882	0.379941	−	0.925011i	$-0.375944\pi$
$293$	13.0071	0.759882	0.379941	+	0.925011i	$0.375944\pi$
$294$	0	0
$295$	−12.5653	−0.731578
$296$	0	0
$297$	2.73571	0.158742
$298$	0	0
$299$	0	0
$300$	0	0
$301$	41.1258	2.37045
$302$	0	0
$303$	16.4481	0.944920
$304$	0	0
$305$	−9.65396	−0.552784
$306$	0	0
$307$	−10.4705	−0.597582	−0.298791	−	0.954319i	$-0.596583\pi$
$307$	−10.4705	−0.597582	−0.298791	+	0.954319i	$0.596583\pi$
$308$	0	0
$309$	−5.14962	−0.292952
$310$	0	0
$311$	6.56629	0.372340	0.186170	−	0.982518i	$-0.440393\pi$
$311$	6.56629	0.372340	0.186170	+	0.982518i	$0.440393\pi$
$312$	0	0
$313$	−9.24429	−0.522518	−0.261259	−	0.965269i	$-0.584138\pi$
$313$	−9.24429	−0.522518	−0.261259	+	0.965269i	$0.584138\pi$
$314$	0	0
$315$	−15.8901	−0.895305
$316$	0	0
$317$	26.2641	1.47514	0.737568	−	0.675272i	$-0.235974\pi$
$317$	26.2641	1.47514	0.737568	+	0.675272i	$0.235974\pi$
$318$	0	0
$319$	−17.8794	−1.00106
$320$	0	0
$321$	−9.08411	−0.507026
$322$	0	0
$323$	7.08548	0.394246
$324$	0	0
$325$	−50.5830	−2.80584
$326$	0	0
$327$	−3.15774	−0.174623
$328$	0	0
$329$	22.1756	1.22258
$330$	0	0
$331$	−13.2144	−0.726329	−0.363165	−	0.931725i	$-0.618304\pi$
$331$	−13.2144	−0.726329	−0.363165	+	0.931725i	$0.618304\pi$
$332$	0	0
$333$	5.76661	0.316008
$334$	0	0
$335$	−16.0233	−0.875444
$336$	0	0
$337$	−0.412554	−0.0224732	−0.0112366	−	0.999937i	$-0.503577\pi$
$337$	−0.412554	−0.0224732	−0.0112366	+	0.999937i	$0.503577\pi$
$338$	0	0
$339$	−13.7325	−0.745844
$340$	0	0
$341$	11.1927	0.606121
$342$	0	0
$343$	−22.2471	−1.20123
$344$	0	0
$345$	0	0
$346$	0	0
$347$	16.4231	0.881640	0.440820	−	0.897595i	$-0.354688\pi$
$347$	16.4231	0.881640	0.440820	+	0.897595i	$0.354688\pi$
$348$	0	0
$349$	20.2814	1.08564	0.542820	−	0.839849i	$-0.317357\pi$
$349$	20.2814	1.08564	0.542820	+	0.839849i	$0.317357\pi$
$350$	0	0
$351$	−6.14956	−0.328239
$352$	0	0
$353$	−3.21297	−0.171009	−0.0855044	−	0.996338i	$-0.527250\pi$
$353$	−3.21297	−0.171009	−0.0855044	+	0.996338i	$0.527250\pi$
$354$	0	0
$355$	−37.1984	−1.97429
$356$	0	0
$357$	5.74799	0.304216
$358$	0	0
$359$	−23.0871	−1.21849	−0.609245	−	0.792982i	$-0.708527\pi$
$359$	−23.0871	−1.21849	−0.609245	+	0.792982i	$0.708527\pi$
$360$	0	0
$361$	10.0100	0.526843
$362$	0	0
$363$	−3.51587	−0.184535
$364$	0	0
$365$	−23.6238	−1.23653
$366$	0	0
$367$	−28.8899	−1.50804	−0.754019	−	0.656852i	$-0.771888\pi$
$367$	−28.8899	−1.50804	−0.754019	+	0.656852i	$0.771888\pi$
$368$	0	0
$369$	7.77719	0.404864
$370$	0	0
$371$	−17.8427	−0.926346
$372$	0	0
$373$	−12.6405	−0.654499	−0.327250	−	0.944938i	$-0.606122\pi$
$373$	−12.6405	−0.654499	−0.327250	+	0.944938i	$0.606122\pi$
$374$	0	0
$375$	11.7300	0.605736
$376$	0	0
$377$	40.1909	2.06993
$378$	0	0
$379$	−4.34490	−0.223182	−0.111591	−	0.993754i	$-0.535595\pi$
$379$	−4.34490	−0.223182	−0.111591	+	0.993754i	$0.535595\pi$
$380$	0	0
$381$	9.79107	0.501612
$382$	0	0
$383$	8.87205	0.453341	0.226670	−	0.973972i	$-0.427216\pi$
$383$	8.87205	0.453341	0.226670	+	0.973972i	$0.427216\pi$
$384$	0	0
$385$	−43.4708	−2.21547
$386$	0	0
$387$	−9.41225	−0.478452
$388$	0	0
$389$	−6.78648	−0.344088	−0.172044	−	0.985089i	$-0.555037\pi$
$389$	−6.78648	−0.344088	−0.172044	+	0.985089i	$0.555037\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	4.93851	0.249115
$394$	0	0
$395$	−12.2878	−0.618268
$396$	0	0
$397$	26.4789	1.32894	0.664468	−	0.747317i	$-0.268658\pi$
$397$	26.4789	1.32894	0.664468	+	0.747317i	$0.268658\pi$
$398$	0	0
$399$	23.5340	1.17817
$400$	0	0
$401$	33.1875	1.65730	0.828651	−	0.559765i	$-0.189109\pi$
$401$	33.1875	1.65730	0.828651	+	0.559765i	$0.189109\pi$
$402$	0	0
$403$	−25.1600	−1.25331
$404$	0	0
$405$	3.63668	0.180708
$406$	0	0
$407$	15.7758	0.781977
$408$	0	0
$409$	28.9795	1.43294	0.716472	−	0.697616i	$-0.245756\pi$
$409$	28.9795	1.43294	0.716472	+	0.697616i	$0.245756\pi$
$410$	0	0
$411$	−21.0333	−1.03749
$412$	0	0
$413$	15.0969	0.742869
$414$	0	0
$415$	−26.1631	−1.28429
$416$	0	0
$417$	−8.52288	−0.417367
$418$	0	0
$419$	2.54469	0.124316	0.0621582	−	0.998066i	$-0.480202\pi$
$419$	2.54469	0.124316	0.0621582	+	0.998066i	$0.480202\pi$
$420$	0	0
$421$	7.81142	0.380705	0.190353	−	0.981716i	$-0.439037\pi$
$421$	7.81142	0.380705	0.190353	+	0.981716i	$0.439037\pi$
$422$	0	0
$423$	−5.07522	−0.246766
$424$	0	0
$425$	−10.8207	−0.524882
$426$	0	0
$427$	11.5990	0.561315
$428$	0	0
$429$	−16.8234	−0.812243
$430$	0	0
$431$	−2.89957	−0.139667	−0.0698337	−	0.997559i	$-0.522247\pi$
$431$	−2.89957	−0.139667	−0.0698337	+	0.997559i	$0.522247\pi$
$432$	0	0
$433$	−6.66266	−0.320187	−0.160093	−	0.987102i	$-0.551180\pi$
$433$	−6.66266	−0.320187	−0.160093	+	0.987102i	$0.551180\pi$
$434$	0	0
$435$	−23.7678	−1.13958
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−4.24503	−0.202604	−0.101302	−	0.994856i	$-0.532301\pi$
$439$	−4.24503	−0.202604	−0.101302	+	0.994856i	$0.532301\pi$
$440$	0	0
$441$	12.0916	0.575789
$442$	0	0
$443$	−5.50077	−0.261350	−0.130675	−	0.991425i	$-0.541714\pi$
$443$	−5.50077	−0.261350	−0.130675	+	0.991425i	$0.541714\pi$
$444$	0	0
$445$	−35.6758	−1.69120
$446$	0	0
$447$	−2.92762	−0.138472
$448$	0	0
$449$	16.4118	0.774520	0.387260	−	0.921970i	$-0.373422\pi$
$449$	16.4118	0.774520	0.387260	+	0.921970i	$0.373422\pi$
$450$	0	0
$451$	21.2762	1.00186
$452$	0	0
$453$	17.7191	0.832516
$454$	0	0
$455$	97.7171	4.58105
$456$	0	0
$457$	27.8646	1.30345	0.651725	−	0.758456i	$-0.274046\pi$
$457$	27.8646	1.30345	0.651725	+	0.758456i	$0.274046\pi$
$458$	0	0
$459$	−1.31551	−0.0614029
$460$	0	0
$461$	27.4201	1.27708	0.638541	−	0.769588i	$-0.279538\pi$
$461$	27.4201	1.27708	0.638541	+	0.769588i	$0.279538\pi$
$462$	0	0
$463$	33.2469	1.54511	0.772556	−	0.634946i	$-0.218978\pi$
$463$	33.2469	1.54511	0.772556	+	0.634946i	$0.218978\pi$
$464$	0	0
$465$	14.8789	0.689993
$466$	0	0
$467$	−34.1343	−1.57955	−0.789773	−	0.613399i	$-0.789802\pi$
$467$	−34.1343	−1.57955	−0.789773	+	0.613399i	$0.789802\pi$
$468$	0	0
$469$	19.2516	0.888955
$470$	0	0
$471$	−7.09513	−0.326927
$472$	0	0
$473$	−25.7492	−1.18395
$474$	0	0
$475$	−44.3032	−2.03277
$476$	0	0
$477$	4.08357	0.186974
$478$	0	0
$479$	28.6235	1.30784	0.653921	−	0.756563i	$-0.273123\pi$
$479$	28.6235	1.30784	0.653921	+	0.756563i	$0.273123\pi$
$480$	0	0
$481$	−35.4621	−1.61693
$482$	0	0
$483$	0	0
$484$	0	0
$485$	8.90915	0.404544
$486$	0	0
$487$	35.5454	1.61072	0.805358	−	0.592789i	$-0.201973\pi$
$487$	35.5454	1.61072	0.805358	+	0.592789i	$0.201973\pi$
$488$	0	0
$489$	0.843703	0.0381536
$490$	0	0
$491$	21.9500	0.990588	0.495294	−	0.868725i	$-0.335060\pi$
$491$	21.9500	0.990588	0.495294	+	0.868725i	$0.335060\pi$
$492$	0	0
$493$	8.59762	0.387217
$494$	0	0
$495$	9.94893	0.447171
$496$	0	0
$497$	44.6930	2.00476
$498$	0	0
$499$	−12.8268	−0.574205	−0.287103	−	0.957900i	$-0.592692\pi$
$499$	−12.8268	−0.574205	−0.287103	+	0.957900i	$0.592692\pi$
$500$	0	0
$501$	−19.0664	−0.851822
$502$	0	0
$503$	1.24478	0.0555019	0.0277510	−	0.999615i	$-0.491165\pi$
$503$	1.24478	0.0555019	0.0277510	+	0.999615i	$0.491165\pi$
$504$	0	0
$505$	59.8166	2.66180
$506$	0	0
$507$	24.8171	1.10217
$508$	0	0
$509$	0.571822	0.0253456	0.0126728	−	0.999920i	$-0.495966\pi$
$509$	0.571822	0.0253456	0.0126728	+	0.999920i	$0.495966\pi$
$510$	0	0
$511$	28.3834	1.25561
$512$	0	0
$513$	−5.38609	−0.237802
$514$	0	0
$515$	−18.7275	−0.825234
$516$	0	0
$517$	−13.8843	−0.610633
$518$	0	0
$519$	−15.4323	−0.677402
$520$	0	0
$521$	−20.5508	−0.900348	−0.450174	−	0.892941i	$-0.648638\pi$
$521$	−20.5508	−0.900348	−0.450174	+	0.892941i	$0.648638\pi$
$522$	0	0
$523$	15.4755	0.676698	0.338349	−	0.941021i	$-0.390132\pi$
$523$	15.4755	0.676698	0.338349	+	0.941021i	$0.390132\pi$
$524$	0	0
$525$	−35.9403	−1.56856
$526$	0	0
$527$	−5.38221	−0.234453
$528$	0	0
$529$	0	0
$530$	0	0
$531$	−3.45514	−0.149941
$532$	0	0
$533$	−47.8263	−2.07159
$534$	0	0
$535$	−33.0360	−1.42827
$536$	0	0
$537$	−12.1563	−0.524584
$538$	0	0
$539$	33.0791	1.42482
$540$	0	0
$541$	−9.33801	−0.401473	−0.200736	−	0.979645i	$-0.564333\pi$
$541$	−9.33801	−0.401473	−0.200736	+	0.979645i	$0.564333\pi$
$542$	0	0
$543$	−7.45191	−0.319792
$544$	0	0
$545$	−11.4837	−0.491908
$546$	0	0
$547$	−5.23239	−0.223721	−0.111861	−	0.993724i	$-0.535681\pi$
$547$	−5.23239	−0.223721	−0.111861	+	0.993724i	$0.535681\pi$
$548$	0	0
$549$	−2.65460	−0.113296
$550$	0	0
$551$	35.2012	1.49962
$552$	0	0
$553$	14.7635	0.627809
$554$	0	0
$555$	20.9713	0.890183
$556$	0	0
$557$	−45.9137	−1.94542	−0.972712	−	0.232015i	$-0.925468\pi$
$557$	−45.9137	−1.94542	−0.972712	+	0.232015i	$0.925468\pi$
$558$	0	0
$559$	57.8812	2.44812
$560$	0	0
$561$	−3.59887	−0.151944
$562$	0	0
$563$	23.1502	0.975665	0.487832	−	0.872937i	$-0.337788\pi$
$563$	23.1502	0.975665	0.487832	+	0.872937i	$0.337788\pi$
$564$	0	0
$565$	−49.9406	−2.10102
$566$	0	0
$567$	−4.36939	−0.183497
$568$	0	0
$569$	4.88345	0.204725	0.102363	−	0.994747i	$-0.467360\pi$
$569$	4.88345	0.204725	0.102363	+	0.994747i	$0.467360\pi$
$570$	0	0
$571$	−18.8223	−0.787688	−0.393844	−	0.919177i	$-0.628855\pi$
$571$	−18.8223	−0.787688	−0.393844	+	0.919177i	$0.628855\pi$
$572$	0	0
$573$	−7.71685	−0.322376
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−24.5307	−1.02123	−0.510614	−	0.859810i	$-0.670582\pi$
$577$	−24.5307	−1.02123	−0.510614	+	0.859810i	$0.670582\pi$
$578$	0	0
$579$	−8.62331	−0.358373
$580$	0	0
$581$	31.4343	1.30411
$582$	0	0
$583$	11.1715	0.462675
$584$	0	0
$585$	−22.3640	−0.924638
$586$	0	0
$587$	−47.1932	−1.94787	−0.973935	−	0.226826i	$-0.927165\pi$
$587$	−47.1932	−1.94787	−0.973935	+	0.226826i	$0.927165\pi$
$588$	0	0
$589$	−22.0364	−0.907992
$590$	0	0
$591$	9.73994	0.400647
$592$	0	0
$593$	−9.80233	−0.402533	−0.201267	−	0.979536i	$-0.564506\pi$
$593$	−9.80233	−0.402533	−0.201267	+	0.979536i	$0.564506\pi$
$594$	0	0
$595$	20.9036	0.856965
$596$	0	0
$597$	15.9329	0.652090
$598$	0	0
$599$	−20.6101	−0.842108	−0.421054	−	0.907036i	$-0.638340\pi$
$599$	−20.6101	−0.842108	−0.421054	+	0.907036i	$0.638340\pi$
$600$	0	0
$601$	19.0973	0.778997	0.389499	−	0.921027i	$-0.372648\pi$
$601$	19.0973	0.778997	0.389499	+	0.921027i	$0.372648\pi$
$602$	0	0
$603$	−4.40601	−0.179427
$604$	0	0
$605$	−12.7861	−0.519829
$606$	0	0
$607$	14.1132	0.572838	0.286419	−	0.958104i	$-0.407535\pi$
$607$	14.1132	0.572838	0.286419	+	0.958104i	$0.407535\pi$
$608$	0	0
$609$	28.5564	1.15717
$610$	0	0
$611$	31.2104	1.26264
$612$	0	0
$613$	46.9877	1.89782	0.948908	−	0.315554i	$-0.102190\pi$
$613$	46.9877	1.89782	0.948908	+	0.315554i	$0.102190\pi$
$614$	0	0
$615$	28.2832	1.14049
$616$	0	0
$617$	−21.7564	−0.875879	−0.437940	−	0.899004i	$-0.644292\pi$
$617$	−21.7564	−0.875879	−0.437940	+	0.899004i	$0.644292\pi$
$618$	0	0
$619$	−31.8938	−1.28192	−0.640961	−	0.767574i	$-0.721464\pi$
$619$	−31.8938	−1.28192	−0.640961	+	0.767574i	$0.721464\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	42.8636	1.71730
$624$	0	0
$625$	1.53103	0.0612413
$626$	0	0
$627$	−14.7348	−0.588452
$628$	0	0
$629$	−7.58605	−0.302475
$630$	0	0
$631$	6.29833	0.250732	0.125366	−	0.992111i	$-0.459989\pi$
$631$	6.29833	0.250732	0.125366	+	0.992111i	$0.459989\pi$
$632$	0	0
$633$	5.17826	0.205817
$634$	0	0
$635$	35.6070	1.41302
$636$	0	0
$637$	−74.3579	−2.94617
$638$	0	0
$639$	−10.2287	−0.404640
$640$	0	0
$641$	−27.6789	−1.09325	−0.546626	−	0.837377i	$-0.684088\pi$
$641$	−27.6789	−1.09325	−0.546626	+	0.837377i	$0.684088\pi$
$642$	0	0
$643$	−27.5964	−1.08830	−0.544148	−	0.838990i	$-0.683147\pi$
$643$	−27.5964	−1.08830	−0.544148	+	0.838990i	$0.683147\pi$
$644$	0	0
$645$	−34.2294	−1.34778
$646$	0	0
$647$	−0.307568	−0.0120918	−0.00604588	−	0.999982i	$-0.501924\pi$
$647$	−0.307568	−0.0120918	−0.00604588	+	0.999982i	$0.501924\pi$
$648$	0	0
$649$	−9.45229	−0.371035
$650$	0	0
$651$	−17.8767	−0.700642
$652$	0	0
$653$	17.2958	0.676838	0.338419	−	0.940996i	$-0.390108\pi$
$653$	17.2958	0.676838	0.338419	+	0.940996i	$0.390108\pi$
$654$	0	0
$655$	17.9598	0.701747
$656$	0	0
$657$	−6.49597	−0.253432
$658$	0	0
$659$	−16.2628	−0.633508	−0.316754	−	0.948508i	$-0.602593\pi$
$659$	−16.2628	−0.633508	−0.316754	+	0.948508i	$0.602593\pi$
$660$	0	0
$661$	10.2591	0.399031	0.199516	−	0.979895i	$-0.436063\pi$
$661$	10.2591	0.399031	0.199516	+	0.979895i	$0.436063\pi$
$662$	0	0
$663$	8.08983	0.314183
$664$	0	0
$665$	85.5856	3.31887
$666$	0	0
$667$	0	0
$668$	0	0
$669$	12.9736	0.501588
$670$	0	0
$671$	−7.26224	−0.280356
$672$	0	0
$673$	−26.3626	−1.01620	−0.508102	−	0.861297i	$-0.669653\pi$
$673$	−26.3626	−1.01620	−0.508102	+	0.861297i	$0.669653\pi$
$674$	0	0
$675$	8.22547	0.316599
$676$	0	0
$677$	6.72162	0.258333	0.129166	−	0.991623i	$-0.458770\pi$
$677$	6.72162	0.258333	0.129166	+	0.991623i	$0.458770\pi$
$678$	0	0
$679$	−10.7041	−0.410787
$680$	0	0
$681$	−9.50561	−0.364256
$682$	0	0
$683$	−27.8812	−1.06684	−0.533421	−	0.845850i	$-0.679094\pi$
$683$	−27.8812	−1.06684	−0.533421	+	0.845850i	$0.679094\pi$
$684$	0	0
$685$	−76.4913	−2.92258
$686$	0	0
$687$	16.4978	0.629430
$688$	0	0
$689$	−25.1121	−0.956696
$690$	0	0
$691$	28.7827	1.09495	0.547474	−	0.836823i	$-0.315590\pi$
$691$	28.7827	1.09495	0.547474	+	0.836823i	$0.315590\pi$
$692$	0	0
$693$	−11.9534	−0.454072
$694$	0	0
$695$	−30.9950	−1.17571
$696$	0	0
$697$	−10.2310	−0.387527
$698$	0	0
$699$	1.67767	0.0634554
$700$	0	0
$701$	46.7293	1.76494	0.882471	−	0.470367i	$-0.155878\pi$
$701$	46.7293	1.76494	0.882471	+	0.470367i	$0.155878\pi$
$702$	0	0
$703$	−31.0595	−1.17143
$704$	0	0
$705$	−18.4570	−0.695129
$706$	0	0
$707$	−71.8683	−2.70288
$708$	0	0
$709$	−17.7987	−0.668445	−0.334222	−	0.942494i	$-0.608474\pi$
$709$	−17.7987	−0.668445	−0.334222	+	0.942494i	$0.608474\pi$
$710$	0	0
$711$	−3.37885	−0.126717
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−61.1815	−2.28806
$716$	0	0
$717$	0.442505	0.0165257
$718$	0	0
$719$	−26.6017	−0.992075	−0.496038	−	0.868301i	$-0.665212\pi$
$719$	−26.6017	−0.992075	−0.496038	+	0.868301i	$0.665212\pi$
$720$	0	0
$721$	22.5007	0.837970
$722$	0	0
$723$	−29.0794	−1.08147
$724$	0	0
$725$	−53.7581	−1.99653
$726$	0	0
$727$	26.1533	0.969973	0.484986	−	0.874522i	$-0.338825\pi$
$727$	26.1533	0.969973	0.484986	+	0.874522i	$0.338825\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	12.3819	0.457963
$732$	0	0
$733$	15.5297	0.573603	0.286801	−	0.957990i	$-0.407408\pi$
$733$	15.5297	0.573603	0.286801	+	0.957990i	$0.407408\pi$
$734$	0	0
$735$	43.9733	1.62198
$736$	0	0
$737$	−12.0536	−0.443999
$738$	0	0
$739$	−35.1961	−1.29471	−0.647354	−	0.762189i	$-0.724124\pi$
$739$	−35.1961	−1.29471	−0.647354	+	0.762189i	$0.724124\pi$
$740$	0	0
$741$	33.1221	1.21677
$742$	0	0
$743$	−53.4678	−1.96154	−0.980772	−	0.195159i	$-0.937478\pi$
$743$	−53.4678	−1.96154	−0.980772	+	0.195159i	$0.937478\pi$
$744$	0	0
$745$	−10.6468	−0.390070
$746$	0	0
$747$	−7.19420	−0.263222
$748$	0	0
$749$	39.6920	1.45032
$750$	0	0
$751$	−32.9867	−1.20370	−0.601851	−	0.798609i	$-0.705570\pi$
$751$	−32.9867	−1.20370	−0.601851	+	0.798609i	$0.705570\pi$
$752$	0	0
$753$	7.70670	0.280848
$754$	0	0
$755$	64.4387	2.34517
$756$	0	0
$757$	29.2383	1.06268	0.531342	−	0.847157i	$-0.321688\pi$
$757$	29.2383	1.06268	0.531342	+	0.847157i	$0.321688\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	41.6189	1.50868	0.754341	−	0.656482i	$-0.227956\pi$
$761$	41.6189	1.50868	0.754341	+	0.656482i	$0.227956\pi$
$762$	0	0
$763$	13.7974	0.499499
$764$	0	0
$765$	−4.78410	−0.172970
$766$	0	0
$767$	21.2476	0.767207
$768$	0	0
$769$	38.6720	1.39455	0.697273	−	0.716805i	$-0.254396\pi$
$769$	38.6720	1.39455	0.697273	+	0.716805i	$0.254396\pi$
$770$	0	0
$771$	20.0114	0.720693
$772$	0	0
$773$	−50.0694	−1.80087	−0.900435	−	0.434990i	$-0.856752\pi$
$773$	−50.0694	−1.80087	−0.900435	+	0.434990i	$0.856752\pi$
$774$	0	0
$775$	33.6532	1.20886
$776$	0	0
$777$	−25.1966	−0.903922
$778$	0	0
$779$	−41.8887	−1.50082
$780$	0	0
$781$	−27.9827	−1.00130
$782$	0	0
$783$	−6.53557	−0.233562
$784$	0	0
$785$	−25.8028	−0.920940
$786$	0	0
$787$	33.8647	1.20715	0.603573	−	0.797308i	$-0.293743\pi$
$787$	33.8647	1.20715	0.603573	+	0.797308i	$0.293743\pi$
$788$	0	0
$789$	12.8978	0.459174
$790$	0	0
$791$	60.0025	2.13344
$792$	0	0
$793$	16.3247	0.579705
$794$	0	0
$795$	14.8506	0.526698
$796$	0	0
$797$	51.9610	1.84055	0.920277	−	0.391269i	$-0.127964\pi$
$797$	51.9610	1.84055	0.920277	+	0.391269i	$0.127964\pi$
$798$	0	0
$799$	6.67651	0.236198
$800$	0	0
$801$	−9.80998	−0.346619
$802$	0	0
$803$	−17.7711	−0.627129
$804$	0	0
$805$	0	0
$806$	0	0
$807$	15.3686	0.540999
$808$	0	0
$809$	−30.5953	−1.07567	−0.537836	−	0.843050i	$-0.680758\pi$
$809$	−30.5953	−1.07567	−0.537836	+	0.843050i	$0.680758\pi$
$810$	0	0
$811$	12.3093	0.432236	0.216118	−	0.976367i	$-0.430660\pi$
$811$	12.3093	0.432236	0.216118	+	0.976367i	$0.430660\pi$
$812$	0	0
$813$	−20.1954	−0.708282
$814$	0	0
$815$	3.06828	0.107477
$816$	0	0
$817$	50.6953	1.77360
$818$	0	0
$819$	26.8698	0.938908
$820$	0	0
$821$	−47.0363	−1.64158	−0.820790	−	0.571230i	$-0.806466\pi$
$821$	−47.0363	−1.64158	−0.820790	+	0.571230i	$0.806466\pi$
$822$	0	0
$823$	−43.5145	−1.51682	−0.758410	−	0.651778i	$-0.774023\pi$
$823$	−43.5145	−1.51682	−0.758410	+	0.651778i	$0.774023\pi$
$824$	0	0
$825$	22.5025	0.783438
$826$	0	0
$827$	−11.4035	−0.396539	−0.198270	−	0.980148i	$-0.563532\pi$
$827$	−11.4035	−0.396539	−0.198270	+	0.980148i	$0.563532\pi$
$828$	0	0
$829$	−30.0821	−1.04479	−0.522397	−	0.852702i	$-0.674962\pi$
$829$	−30.0821	−1.04479	−0.522397	+	0.852702i	$0.674962\pi$
$830$	0	0
$831$	22.7999	0.790920
$832$	0	0
$833$	−15.9066	−0.551132
$834$	0	0
$835$	−69.3383	−2.39955
$836$	0	0
$837$	4.09134	0.141417
$838$	0	0
$839$	3.41724	0.117976	0.0589882	−	0.998259i	$-0.481213\pi$
$839$	3.41724	0.117976	0.0589882	+	0.998259i	$0.481213\pi$
$840$	0	0
$841$	13.7136	0.472883
$842$	0	0
$843$	14.8951	0.513013
$844$	0	0
$845$	90.2520	3.10476
$846$	0	0
$847$	15.3622	0.527852
$848$	0	0
$849$	0.630449	0.0216370
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−53.0388	−1.81601	−0.908006	−	0.418956i	$-0.862396\pi$
$853$	−53.0388	−1.81601	−0.908006	+	0.418956i	$0.862396\pi$
$854$	0	0
$855$	−19.5875	−0.669879
$856$	0	0
$857$	−18.3647	−0.627325	−0.313663	−	0.949535i	$-0.601556\pi$
$857$	−18.3647	−0.627325	−0.313663	+	0.949535i	$0.601556\pi$
$858$	0	0
$859$	−13.5668	−0.462895	−0.231447	−	0.972847i	$-0.574346\pi$
$859$	−13.5668	−0.462895	−0.231447	+	0.972847i	$0.574346\pi$
$860$	0	0
$861$	−33.9816	−1.15809
$862$	0	0
$863$	−3.86818	−0.131674	−0.0658372	−	0.997830i	$-0.520972\pi$
$863$	−3.86818	−0.131674	−0.0658372	+	0.997830i	$0.520972\pi$
$864$	0	0
$865$	−56.1224	−1.90822
$866$	0	0
$867$	−15.2694	−0.518577
$868$	0	0
$869$	−9.24358	−0.313567
$870$	0	0
$871$	27.0950	0.918080
$872$	0	0
$873$	2.44980	0.0829132
$874$	0	0
$875$	−51.2531	−1.73267
$876$	0	0
$877$	−31.8008	−1.07384	−0.536918	−	0.843635i	$-0.680411\pi$
$877$	−31.8008	−1.07384	−0.536918	+	0.843635i	$0.680411\pi$
$878$	0	0
$879$	13.0071	0.438718
$880$	0	0
$881$	32.9702	1.11080	0.555398	−	0.831585i	$-0.312566\pi$
$881$	32.9702	1.11080	0.555398	+	0.831585i	$0.312566\pi$
$882$	0	0
$883$	−46.7465	−1.57314	−0.786572	−	0.617498i	$-0.788146\pi$
$883$	−46.7465	−1.57314	−0.786572	+	0.617498i	$0.788146\pi$
$884$	0	0
$885$	−12.5653	−0.422377
$886$	0	0
$887$	41.1437	1.38147	0.690734	−	0.723109i	$-0.257287\pi$
$887$	41.1437	1.38147	0.690734	+	0.723109i	$0.257287\pi$
$888$	0	0
$889$	−42.7810	−1.43483
$890$	0	0
$891$	2.73571	0.0916499
$892$	0	0
$893$	27.3356	0.914751
$894$	0	0
$895$	−44.2087	−1.47773
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−26.7392	−0.891803
$900$	0	0
$901$	−5.37198	−0.178967
$902$	0	0
$903$	41.1258	1.36858
$904$	0	0
$905$	−27.1003	−0.900843
$906$	0	0
$907$	53.3719	1.77219	0.886093	−	0.463508i	$-0.153410\pi$
$907$	53.3719	1.77219	0.886093	+	0.463508i	$0.153410\pi$
$908$	0	0
$909$	16.4481	0.545550
$910$	0	0
$911$	33.1417	1.09803	0.549017	−	0.835812i	$-0.315002\pi$
$911$	33.1417	1.09803	0.549017	+	0.835812i	$0.315002\pi$
$912$	0	0
$913$	−19.6813	−0.651355
$914$	0	0
$915$	−9.65396	−0.319150
$916$	0	0
$917$	−21.5783	−0.712577
$918$	0	0
$919$	−21.2533	−0.701080	−0.350540	−	0.936548i	$-0.614002\pi$
$919$	−21.2533	−0.701080	−0.350540	+	0.936548i	$0.614002\pi$
$920$	0	0
$921$	−10.4705	−0.345014
$922$	0	0
$923$	62.9018	2.07044
$924$	0	0
$925$	47.4331	1.55959
$926$	0	0
$927$	−5.14962	−0.169136
$928$	0	0
$929$	−10.7366	−0.352257	−0.176129	−	0.984367i	$-0.556357\pi$
$929$	−10.7366	−0.352257	−0.176129	+	0.984367i	$0.556357\pi$
$930$	0	0
$931$	−65.1264	−2.13443
$932$	0	0
$933$	6.56629	0.214971
$934$	0	0
$935$	−13.0879	−0.428021
$936$	0	0
$937$	−38.0546	−1.24319	−0.621594	−	0.783339i	$-0.713515\pi$
$937$	−38.0546	−1.24319	−0.621594	+	0.783339i	$0.713515\pi$
$938$	0	0
$939$	−9.24429	−0.301676
$940$	0	0
$941$	28.7234	0.936356	0.468178	−	0.883634i	$-0.344911\pi$
$941$	28.7234	0.936356	0.468178	+	0.883634i	$0.344911\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−15.8901	−0.516905
$946$	0	0
$947$	−21.3089	−0.692447	−0.346223	−	0.938152i	$-0.612536\pi$
$947$	−21.3089	−0.692447	−0.346223	+	0.938152i	$0.612536\pi$
$948$	0	0
$949$	39.9474	1.29675
$950$	0	0
$951$	26.2641	0.851671
$952$	0	0
$953$	46.7170	1.51331	0.756657	−	0.653812i	$-0.226831\pi$
$953$	46.7170	1.51331	0.756657	+	0.653812i	$0.226831\pi$
$954$	0	0
$955$	−28.0637	−0.908121
$956$	0	0
$957$	−17.8794	−0.577960
$958$	0	0
$959$	91.9025	2.96769
$960$	0	0
$961$	−14.2609	−0.460030
$962$	0	0
$963$	−9.08411	−0.292731
$964$	0	0
$965$	−31.3603	−1.00952
$966$	0	0
$967$	−16.9438	−0.544875	−0.272438	−	0.962173i	$-0.587830\pi$
$967$	−16.9438	−0.544875	−0.272438	+	0.962173i	$0.587830\pi$
$968$	0	0
$969$	7.08548	0.227618
$970$	0	0
$971$	29.1969	0.936974	0.468487	−	0.883471i	$-0.344799\pi$
$971$	29.1969	0.936974	0.468487	+	0.883471i	$0.344799\pi$
$972$	0	0
$973$	37.2398	1.19385
$974$	0	0
$975$	−50.5830	−1.61995
$976$	0	0
$977$	13.8849	0.444218	0.222109	−	0.975022i	$-0.428706\pi$
$977$	13.8849	0.444218	0.222109	+	0.975022i	$0.428706\pi$
$978$	0	0
$979$	−26.8373	−0.857724
$980$	0	0
$981$	−3.15774	−0.100819
$982$	0	0
$983$	2.62704	0.0837895	0.0418947	−	0.999122i	$-0.486661\pi$
$983$	2.62704	0.0837895	0.0418947	+	0.999122i	$0.486661\pi$
$984$	0	0
$985$	35.4211	1.12861
$986$	0	0
$987$	22.1756	0.705857
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−27.0625	−0.859670	−0.429835	−	0.902907i	$-0.641428\pi$
$991$	−27.0625	−0.859670	−0.429835	+	0.902907i	$0.641428\pi$
$992$	0	0
$993$	−13.2144	−0.419346
$994$	0	0
$995$	57.9429	1.83691
$996$	0	0
$997$	43.7740	1.38634	0.693168	−	0.720776i	$-0.256214\pi$
$997$	43.7740	1.38634	0.693168	+	0.720776i	$0.256214\pi$
$998$	0	0
$999$	5.76661	0.182447

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6348.2.a.s.1.10		10
23.13	even	11		276.2.i.a.169.1	yes	20
23.16	even	11		276.2.i.a.49.1	✓	20
23.22	odd	2		6348.2.a.t.1.1		10
69.59	odd	22		828.2.q.c.721.2		20
69.62	odd	22		828.2.q.c.325.2		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
276.2.i.a.49.1	✓	20	23.16	even	11
276.2.i.a.169.1	yes	20	23.13	even	11
828.2.q.c.325.2		20	69.62	odd	22
828.2.q.c.721.2		20	69.59	odd	22
6348.2.a.s.1.10		10	1.1	even	1	trivial
6348.2.a.t.1.1		10	23.22	odd	2

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:	\( N \)	\(=\)	\( 6348 = 2^{2} \cdot 3 \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6348.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.63668 q^{5} -4.36939 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +3.63668 q^{5} -4.36939 q^{7} +1.00000 q^{9} +2.73571 q^{11} -6.14956 q^{13} +3.63668 q^{15} -1.31551 q^{17} -5.38609 q^{19} -4.36939 q^{21} +8.22547 q^{25} +1.00000 q^{27} -6.53557 q^{29} +4.09134 q^{31} +2.73571 q^{33} -15.8901 q^{35} +5.76661 q^{37} -6.14956 q^{39} +7.77719 q^{41} -9.41225 q^{43} +3.63668 q^{45} -5.07522 q^{47} +12.0916 q^{49} -1.31551 q^{51} +4.08357 q^{53} +9.94893 q^{55} -5.38609 q^{57} -3.45514 q^{59} -2.65460 q^{61} -4.36939 q^{63} -22.3640 q^{65} -4.40601 q^{67} -10.2287 q^{71} -6.49597 q^{73} +8.22547 q^{75} -11.9534 q^{77} -3.37885 q^{79} +1.00000 q^{81} -7.19420 q^{83} -4.78410 q^{85} -6.53557 q^{87} -9.80998 q^{89} +26.8698 q^{91} +4.09134 q^{93} -19.5875 q^{95} +2.44980 q^{97} +2.73571 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 10 q^{3} - 2 q^{5} - 11 q^{7} + 10 q^{9}+O(q^{10}) \) 10 * q + 10 * q^3 - 2 * q^5 - 11 * q^7 + 10 * q^9	\( 10 q + 10 q^{3} - 2 q^{5} - 11 q^{7} + 10 q^{9} - 11 q^{11} - 2 q^{15} - 13 q^{17} - 18 q^{19} - 11 q^{21} + 10 q^{25} + 10 q^{27} + 5 q^{29} + 15 q^{31} - 11 q^{33} - 13 q^{35} - 5 q^{37} + 24 q^{41} - 40 q^{43} - 2 q^{45} - 9 q^{47} + 5 q^{49} - 13 q^{51} - 6 q^{53} - 14 q^{55} - 18 q^{57} + 28 q^{59} - 39 q^{61} - 11 q^{63} - 14 q^{65} - 32 q^{67} - 33 q^{71} - 50 q^{73} + 10 q^{75} + 19 q^{77} - 33 q^{79} + 10 q^{81} - 29 q^{83} - 21 q^{85} + 5 q^{87} - 17 q^{89} - 36 q^{91} + 15 q^{93} - 42 q^{95} - 46 q^{97} - 11 q^{99}+O(q^{100}) \) 10 * q + 10 * q^3 - 2 * q^5 - 11 * q^7 + 10 * q^9 - 11 * q^11 - 2 * q^15 - 13 * q^17 - 18 * q^19 - 11 * q^21 + 10 * q^25 + 10 * q^27 + 5 * q^29 + 15 * q^31 - 11 * q^33 - 13 * q^35 - 5 * q^37 + 24 * q^41 - 40 * q^43 - 2 * q^45 - 9 * q^47 + 5 * q^49 - 13 * q^51 - 6 * q^53 - 14 * q^55 - 18 * q^57 + 28 * q^59 - 39 * q^61 - 11 * q^63 - 14 * q^65 - 32 * q^67 - 33 * q^71 - 50 * q^73 + 10 * q^75 + 19 * q^77 - 33 * q^79 + 10 * q^81 - 29 * q^83 - 21 * q^85 + 5 * q^87 - 17 * q^89 - 36 * q^91 + 15 * q^93 - 42 * q^95 - 46 * q^97 - 11 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more