LMFDB - Embedded newform 8379.2.a.cs.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8379,2,Mod(1,8379)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8379, 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("8379.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8379 = 3^{2} \cdot 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8379.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:	$66.9066518536$
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} - 2x^{9} - 13x^{8} + 24x^{7} + 57x^{6} - 98x^{5} - 93x^{4} + 152x^{3} + 39x^{2} - 58x - 7 $ x^10 - 2x^9 - 13x^8 + 24x^7 + 57x^6 - 98x^5 - 93x^4 + 152x^3 + 39x^2 - 58*x - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 931)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$1.31483$ of defining polynomial
Character	$\chi$	$=$	8379.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.31483 q^{2} -0.271216 q^{4} -4.02523 q^{5} -2.98627 q^{8} +O(q^{10})$	$q+1.31483 q^{2} -0.271216 q^{4} -4.02523 q^{5} -2.98627 q^{8} -5.29250 q^{10} -1.61490 q^{11} +5.21945 q^{13} -3.38401 q^{16} -2.70633 q^{17} +1.00000 q^{19} +1.09171 q^{20} -2.12332 q^{22} +2.88811 q^{23} +11.2025 q^{25} +6.86270 q^{26} +6.81005 q^{29} +0.440553 q^{31} +1.52313 q^{32} -3.55837 q^{34} -6.68583 q^{37} +1.31483 q^{38} +12.0204 q^{40} -4.42729 q^{41} +8.88620 q^{43} +0.437987 q^{44} +3.79738 q^{46} -0.655937 q^{47} +14.7294 q^{50} -1.41560 q^{52} +4.75749 q^{53} +6.50034 q^{55} +8.95407 q^{58} -11.1446 q^{59} +10.5050 q^{61} +0.579254 q^{62} +8.77068 q^{64} -21.0095 q^{65} -11.8711 q^{67} +0.734000 q^{68} +3.63982 q^{71} -6.49287 q^{73} -8.79074 q^{74} -0.271216 q^{76} +8.78112 q^{79} +13.6214 q^{80} -5.82115 q^{82} -0.609081 q^{83} +10.8936 q^{85} +11.6839 q^{86} +4.82252 q^{88} -11.4165 q^{89} -0.783303 q^{92} -0.862447 q^{94} -4.02523 q^{95} -6.91655 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 2 q^{2} + 10 q^{4} - 16 q^{5} + 6 q^{8}+O(q^{10}) $ 10 * q + 2 * q^2 + 10 * q^4 - 16 * q^5 + 6 * q^8	$ 10 q + 2 q^{2} + 10 q^{4} - 16 q^{5} + 6 q^{8} - 12 q^{10} + 12 q^{13} + 2 q^{16} - 16 q^{17} + 10 q^{19} - 32 q^{20} + 4 q^{22} - 12 q^{23} + 14 q^{25} - 24 q^{26} + 12 q^{29} + 8 q^{31} + 34 q^{32} + 16 q^{34} + 4 q^{37} + 2 q^{38} - 20 q^{40} - 40 q^{41} + 4 q^{43} + 20 q^{44} - 32 q^{46} - 16 q^{47} + 34 q^{50} - 40 q^{52} + 16 q^{55} - 8 q^{58} - 36 q^{59} + 16 q^{61} + 16 q^{62} + 18 q^{64} - 8 q^{65} - 28 q^{67} - 40 q^{68} + 12 q^{71} - 24 q^{74} + 10 q^{76} - 8 q^{79} - 8 q^{80} - 8 q^{82} + 40 q^{85} + 52 q^{86} - 4 q^{88} - 48 q^{89} - 28 q^{92} - 36 q^{94} - 16 q^{95} - 16 q^{97}+O(q^{100}) $ 10 * q + 2 * q^2 + 10 * q^4 - 16 * q^5 + 6 * q^8 - 12 * q^10 + 12 * q^13 + 2 * q^16 - 16 * q^17 + 10 * q^19 - 32 * q^20 + 4 * q^22 - 12 * q^23 + 14 * q^25 - 24 * q^26 + 12 * q^29 + 8 * q^31 + 34 * q^32 + 16 * q^34 + 4 * q^37 + 2 * q^38 - 20 * q^40 - 40 * q^41 + 4 * q^43 + 20 * q^44 - 32 * q^46 - 16 * q^47 + 34 * q^50 - 40 * q^52 + 16 * q^55 - 8 * q^58 - 36 * q^59 + 16 * q^61 + 16 * q^62 + 18 * q^64 - 8 * q^65 - 28 * q^67 - 40 * q^68 + 12 * q^71 - 24 * q^74 + 10 * q^76 - 8 * q^79 - 8 * q^80 - 8 * q^82 + 40 * q^85 + 52 * q^86 - 4 * q^88 - 48 * q^89 - 28 * q^92 - 36 * q^94 - 16 * q^95 - 16 * q^97

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 $

$2$	1.31483	0.929727	0.464863	−	0.885382i	$-0.346103\pi$
$2$	1.31483	0.929727	0.464863	+	0.885382i	$0.346103\pi$
$3$	0	0
$3$	0	0
$4$	−0.271216	−0.135608
$5$	−4.02523	−1.80014	−0.900069	−	0.435748i	$-0.856484\pi$
$5$	−4.02523	−1.80014	−0.900069	+	0.435748i	$0.856484\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−2.98627	−1.05581
$9$	0	0
$10$	−5.29250	−1.67364
$11$	−1.61490	−0.486911	−0.243455	−	0.969912i	$-0.578281\pi$
$11$	−1.61490	−0.486911	−0.243455	+	0.969912i	$0.578281\pi$
$12$	0	0
$13$	5.21945	1.44762	0.723808	−	0.690002i	$-0.242390\pi$
$13$	5.21945	1.44762	0.723808	+	0.690002i	$0.242390\pi$
$14$	0	0
$15$	0	0
$16$	−3.38401	−0.846002
$17$	−2.70633	−0.656381	−0.328190	−	0.944612i	$-0.606439\pi$
$17$	−2.70633	−0.656381	−0.328190	+	0.944612i	$0.606439\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	1.09171	0.244113
$21$	0	0
$22$	−2.12332	−0.452694
$23$	2.88811	0.602212	0.301106	−	0.953591i	$-0.402644\pi$
$23$	2.88811	0.602212	0.301106	+	0.953591i	$0.402644\pi$
$24$	0	0
$25$	11.2025	2.24050
$26$	6.86270	1.34589
$27$	0	0
$28$	0	0
$29$	6.81005	1.26459	0.632297	−	0.774726i	$-0.282112\pi$
$29$	6.81005	1.26459	0.632297	+	0.774726i	$0.282112\pi$
$30$	0	0
$31$	0.440553	0.0791257	0.0395629	−	0.999217i	$-0.487403\pi$
$31$	0.440553	0.0791257	0.0395629	+	0.999217i	$0.487403\pi$
$32$	1.52313	0.269254
$33$	0	0
$34$	−3.55837	−0.610255
$35$	0	0
$36$	0	0
$37$	−6.68583	−1.09914	−0.549572	−	0.835447i	$-0.685209\pi$
$37$	−6.68583	−1.09914	−0.549572	+	0.835447i	$0.685209\pi$
$38$	1.31483	0.213294
$39$	0	0
$40$	12.0204	1.90059
$41$	−4.42729	−0.691427	−0.345714	−	0.938340i	$-0.612363\pi$
$41$	−4.42729	−0.691427	−0.345714	+	0.938340i	$0.612363\pi$
$42$	0	0
$43$	8.88620	1.35513	0.677567	−	0.735461i	$-0.263035\pi$
$43$	8.88620	1.35513	0.677567	+	0.735461i	$0.263035\pi$
$44$	0.437987	0.0660291
$45$	0	0
$46$	3.79738	0.559893
$47$	−0.655937	−0.0956782	−0.0478391	−	0.998855i	$-0.515233\pi$
$47$	−0.655937	−0.0956782	−0.0478391	+	0.998855i	$0.515233\pi$
$48$	0	0
$49$	0	0
$50$	14.7294	2.08305
$51$	0	0
$52$	−1.41560	−0.196309
$53$	4.75749	0.653491	0.326746	−	0.945112i	$-0.394048\pi$
$53$	4.75749	0.653491	0.326746	+	0.945112i	$0.394048\pi$
$54$	0	0
$55$	6.50034	0.876506
$56$	0	0
$57$	0	0
$58$	8.95407	1.17573
$59$	−11.1446	−1.45090	−0.725450	−	0.688275i	$-0.758368\pi$
$59$	−11.1446	−1.45090	−0.725450	+	0.688275i	$0.758368\pi$
$60$	0	0
$61$	10.5050	1.34503	0.672514	−	0.740084i	$-0.265214\pi$
$61$	10.5050	1.34503	0.672514	+	0.740084i	$0.265214\pi$
$62$	0.579254	0.0735653
$63$	0	0
$64$	8.77068	1.09634
$65$	−21.0095	−2.60591
$66$	0	0
$67$	−11.8711	−1.45029	−0.725145	−	0.688596i	$-0.758227\pi$
$67$	−11.8711	−1.45029	−0.725145	+	0.688596i	$0.758227\pi$
$68$	0.734000	0.0890106
$69$	0	0
$70$	0	0
$71$	3.63982	0.431967	0.215983	−	0.976397i	$-0.430704\pi$
$71$	3.63982	0.431967	0.215983	+	0.976397i	$0.430704\pi$
$72$	0	0
$73$	−6.49287	−0.759932	−0.379966	−	0.925000i	$-0.624064\pi$
$73$	−6.49287	−0.759932	−0.379966	+	0.925000i	$0.624064\pi$
$74$	−8.79074	−1.02190
$75$	0	0
$76$	−0.271216	−0.0311107
$77$	0	0
$78$	0	0
$79$	8.78112	0.987953	0.493977	−	0.869475i	$-0.335543\pi$
$79$	8.78112	0.987953	0.493977	+	0.869475i	$0.335543\pi$
$80$	13.6214	1.52292
$81$	0	0
$82$	−5.82115	−0.642838
$83$	−0.609081	−0.0668553	−0.0334277	−	0.999441i	$-0.510642\pi$
$83$	−0.609081	−0.0668553	−0.0334277	+	0.999441i	$0.510642\pi$
$84$	0	0
$85$	10.8936	1.18158
$86$	11.6839	1.25990
$87$	0	0
$88$	4.82252	0.514083
$89$	−11.4165	−1.21014	−0.605072	−	0.796171i	$-0.706855\pi$
$89$	−11.4165	−1.21014	−0.605072	+	0.796171i	$0.706855\pi$
$90$	0	0
$91$	0	0
$92$	−0.783303	−0.0816649
$93$	0	0
$94$	−0.862447	−0.0889546
$95$	−4.02523	−0.412980
$96$	0	0
$97$	−6.91655	−0.702269	−0.351134	−	0.936325i	$-0.614204\pi$
$97$	−6.91655	−0.702269	−0.351134	+	0.936325i	$0.614204\pi$
$98$	0	0
$99$	0	0
$100$	−3.03829	−0.303829
$101$	−0.0670634	−0.00667306	−0.00333653	−	0.999994i	$-0.501062\pi$
$101$	−0.0670634	−0.00667306	−0.00333653	+	0.999994i	$0.501062\pi$
$102$	0	0
$103$	−11.5525	−1.13830	−0.569149	−	0.822234i	$-0.692727\pi$
$103$	−11.5525	−1.13830	−0.569149	+	0.822234i	$0.692727\pi$
$104$	−15.5867	−1.52840
$105$	0	0
$106$	6.25530	0.607568
$107$	13.3790	1.29339	0.646697	−	0.762747i	$-0.276150\pi$
$107$	13.3790	1.29339	0.646697	+	0.762747i	$0.276150\pi$
$108$	0	0
$109$	17.8212	1.70696	0.853481	−	0.521124i	$-0.174487\pi$
$109$	17.8212	1.70696	0.853481	+	0.521124i	$0.174487\pi$
$110$	8.54686	0.814911
$111$	0	0
$112$	0	0
$113$	−5.43115	−0.510919	−0.255460	−	0.966820i	$-0.582227\pi$
$113$	−5.43115	−0.510919	−0.255460	+	0.966820i	$0.582227\pi$
$114$	0	0
$115$	−11.6253	−1.08407
$116$	−1.84700	−0.171489
$117$	0	0
$118$	−14.6532	−1.34894
$119$	0	0
$120$	0	0
$121$	−8.39210	−0.762918
$122$	13.8123	1.25051
$123$	0	0
$124$	−0.119485	−0.0107301
$125$	−24.9664	−2.23306
$126$	0	0
$127$	6.09351	0.540711	0.270356	−	0.962761i	$-0.412859\pi$
$127$	6.09351	0.540711	0.270356	+	0.962761i	$0.412859\pi$
$128$	8.48571	0.750038
$129$	0	0
$130$	−27.6240	−2.42278
$131$	−8.54360	−0.746457	−0.373229	−	0.927739i	$-0.621749\pi$
$131$	−8.54360	−0.746457	−0.373229	+	0.927739i	$0.621749\pi$
$132$	0	0
$133$	0	0
$134$	−15.6086	−1.34837
$135$	0	0
$136$	8.08182	0.693010
$137$	−1.87908	−0.160540	−0.0802702	−	0.996773i	$-0.525578\pi$
$137$	−1.87908	−0.160540	−0.0802702	+	0.996773i	$0.525578\pi$
$138$	0	0
$139$	−8.66413	−0.734882	−0.367441	−	0.930047i	$-0.619766\pi$
$139$	−8.66413	−0.734882	−0.367441	+	0.930047i	$0.619766\pi$
$140$	0	0
$141$	0	0
$142$	4.78575	0.401611
$143$	−8.42889	−0.704860
$144$	0	0
$145$	−27.4120	−2.27644
$146$	−8.53703	−0.706529
$147$	0	0
$148$	1.81331	0.149053
$149$	−2.29567	−0.188069	−0.0940344	−	0.995569i	$-0.529976\pi$
$149$	−2.29567	−0.188069	−0.0940344	+	0.995569i	$0.529976\pi$
$150$	0	0
$151$	−3.19706	−0.260173	−0.130087	−	0.991503i	$-0.541526\pi$
$151$	−3.19706	−0.260173	−0.130087	+	0.991503i	$0.541526\pi$
$152$	−2.98627	−0.242218
$153$	0	0
$154$	0	0
$155$	−1.77333	−0.142437
$156$	0	0
$157$	−12.4884	−0.996687	−0.498343	−	0.866980i	$-0.666058\pi$
$157$	−12.4884	−0.996687	−0.498343	+	0.866980i	$0.666058\pi$
$158$	11.5457	0.918527
$159$	0	0
$160$	−6.13096	−0.484695
$161$	0	0
$162$	0	0
$163$	−14.1958	−1.11190	−0.555952	−	0.831214i	$-0.687646\pi$
$163$	−14.1958	−1.11190	−0.555952	+	0.831214i	$0.687646\pi$
$164$	1.20075	0.0937632
$165$	0	0
$166$	−0.800839	−0.0621572
$167$	−0.193288	−0.0149570	−0.00747852	−	0.999972i	$-0.502381\pi$
$167$	−0.193288	−0.0149570	−0.00747852	+	0.999972i	$0.502381\pi$
$168$	0	0
$169$	14.2427	1.09559
$170$	14.3232	1.09854
$171$	0	0
$172$	−2.41008	−0.183767
$173$	17.7970	1.35308	0.676540	−	0.736406i	$-0.263479\pi$
$173$	17.7970	1.35308	0.676540	+	0.736406i	$0.263479\pi$
$174$	0	0
$175$	0	0
$176$	5.46484	0.411927
$177$	0	0
$178$	−15.0107	−1.12510
$179$	−15.2728	−1.14154	−0.570772	−	0.821109i	$-0.693356\pi$
$179$	−15.2728	−1.14154	−0.570772	+	0.821109i	$0.693356\pi$
$180$	0	0
$181$	−7.40779	−0.550617	−0.275308	−	0.961356i	$-0.588780\pi$
$181$	−7.40779	−0.550617	−0.275308	+	0.961356i	$0.588780\pi$
$182$	0	0
$183$	0	0
$184$	−8.62467	−0.635819
$185$	26.9120	1.97861
$186$	0	0
$187$	4.37045	0.319599
$188$	0.177901	0.0129748
$189$	0	0
$190$	−5.29250	−0.383958
$191$	−15.7629	−1.14056	−0.570282	−	0.821449i	$-0.693166\pi$
$191$	−15.7629	−1.14056	−0.570282	+	0.821449i	$0.693166\pi$
$192$	0	0
$193$	12.3796	0.891100	0.445550	−	0.895257i	$-0.353008\pi$
$193$	12.3796	0.891100	0.445550	+	0.895257i	$0.353008\pi$
$194$	−9.09410	−0.652918
$195$	0	0
$196$	0	0
$197$	−12.5826	−0.896475	−0.448238	−	0.893914i	$-0.647948\pi$
$197$	−12.5826	−0.896475	−0.448238	+	0.893914i	$0.647948\pi$
$198$	0	0
$199$	0.806681	0.0571841	0.0285921	−	0.999591i	$-0.490898\pi$
$199$	0.806681	0.0571841	0.0285921	+	0.999591i	$0.490898\pi$
$200$	−33.4536	−2.36553
$201$	0	0
$202$	−0.0881771	−0.00620412
$203$	0	0
$204$	0	0
$205$	17.8209	1.24466
$206$	−15.1895	−1.05831
$207$	0	0
$208$	−17.6627	−1.22469
$209$	−1.61490	−0.111705
$210$	0	0
$211$	−4.01368	−0.276313	−0.138157	−	0.990410i	$-0.544118\pi$
$211$	−4.01368	−0.276313	−0.138157	+	0.990410i	$0.544118\pi$
$212$	−1.29031	−0.0886188
$213$	0	0
$214$	17.5911	1.20250
$215$	−35.7690	−2.43943
$216$	0	0
$217$	0	0
$218$	23.4319	1.58701
$219$	0	0
$220$	−1.76300	−0.118861
$221$	−14.1255	−0.950187
$222$	0	0
$223$	25.4284	1.70281	0.851407	−	0.524505i	$-0.175750\pi$
$223$	25.4284	1.70281	0.851407	+	0.524505i	$0.175750\pi$
$224$	0	0
$225$	0	0
$226$	−7.14104	−0.475015
$227$	−21.2387	−1.40966	−0.704832	−	0.709375i	$-0.748977\pi$
$227$	−21.2387	−1.40966	−0.704832	+	0.709375i	$0.748977\pi$
$228$	0	0
$229$	−9.41348	−0.622060	−0.311030	−	0.950400i	$-0.600674\pi$
$229$	−9.41348	−0.622060	−0.311030	+	0.950400i	$0.600674\pi$
$230$	−15.2853	−1.00788
$231$	0	0
$232$	−20.3366	−1.33517
$233$	−13.0170	−0.852770	−0.426385	−	0.904542i	$-0.640213\pi$
$233$	−13.0170	−0.852770	−0.426385	+	0.904542i	$0.640213\pi$
$234$	0	0
$235$	2.64030	0.172234
$236$	3.02259	0.196754
$237$	0	0
$238$	0	0
$239$	−12.1280	−0.784497	−0.392248	−	0.919859i	$-0.628303\pi$
$239$	−12.1280	−0.784497	−0.392248	+	0.919859i	$0.628303\pi$
$240$	0	0
$241$	−9.76645	−0.629113	−0.314556	−	0.949239i	$-0.601856\pi$
$241$	−9.76645	−0.629113	−0.314556	+	0.949239i	$0.601856\pi$
$242$	−11.0342	−0.709305
$243$	0	0
$244$	−2.84913	−0.182397
$245$	0	0
$246$	0	0
$247$	5.21945	0.332106
$248$	−1.31561	−0.0835413
$249$	0	0
$250$	−32.8266	−2.07614
$251$	1.99555	0.125958	0.0629789	−	0.998015i	$-0.479940\pi$
$251$	1.99555	0.125958	0.0629789	+	0.998015i	$0.479940\pi$
$252$	0	0
$253$	−4.66401	−0.293224
$254$	8.01194	0.502714
$255$	0	0
$256$	−6.38408	−0.399005
$257$	2.42885	0.151507	0.0757537	−	0.997127i	$-0.475864\pi$
$257$	2.42885	0.151507	0.0757537	+	0.997127i	$0.475864\pi$
$258$	0	0
$259$	0	0
$260$	5.69812	0.353382
$261$	0	0
$262$	−11.2334	−0.694001
$263$	−1.81948	−0.112194	−0.0560971	−	0.998425i	$-0.517866\pi$
$263$	−1.81948	−0.112194	−0.0560971	+	0.998425i	$0.517866\pi$
$264$	0	0
$265$	−19.1500	−1.17637
$266$	0	0
$267$	0	0
$268$	3.21965	0.196671
$269$	−25.6076	−1.56132	−0.780662	−	0.624954i	$-0.785118\pi$
$269$	−25.6076	−1.56132	−0.780662	+	0.624954i	$0.785118\pi$
$270$	0	0
$271$	13.2977	0.807775	0.403888	−	0.914809i	$-0.367659\pi$
$271$	13.2977	0.807775	0.403888	+	0.914809i	$0.367659\pi$
$272$	9.15824	0.555300
$273$	0	0
$274$	−2.47067	−0.149259
$275$	−18.0909	−1.09092
$276$	0	0
$277$	−7.01712	−0.421618	−0.210809	−	0.977527i	$-0.567610\pi$
$277$	−7.01712	−0.421618	−0.210809	+	0.977527i	$0.567610\pi$
$278$	−11.3919	−0.683240
$279$	0	0
$280$	0	0
$281$	23.8630	1.42354	0.711772	−	0.702410i	$-0.247893\pi$
$281$	23.8630	1.42354	0.711772	+	0.702410i	$0.247893\pi$
$282$	0	0
$283$	31.9668	1.90023	0.950115	−	0.311899i	$-0.100965\pi$
$283$	31.9668	1.90023	0.950115	+	0.311899i	$0.100965\pi$
$284$	−0.987178	−0.0585783
$285$	0	0
$286$	−11.0826	−0.655327
$287$	0	0
$288$	0	0
$289$	−9.67579	−0.569164
$290$	−36.0422	−2.11647
$291$	0	0
$292$	1.76097	0.103053
$293$	10.4989	0.613355	0.306677	−	0.951814i	$-0.400783\pi$
$293$	10.4989	0.613355	0.306677	+	0.951814i	$0.400783\pi$
$294$	0	0
$295$	44.8594	2.61182
$296$	19.9657	1.16048
$297$	0	0
$298$	−3.01842	−0.174853
$299$	15.0744	0.871772
$300$	0	0
$301$	0	0
$302$	−4.20360	−0.241890
$303$	0	0
$304$	−3.38401	−0.194086
$305$	−42.2851	−2.42124
$306$	0	0
$307$	11.5671	0.660168	0.330084	−	0.943952i	$-0.392923\pi$
$307$	11.5671	0.660168	0.330084	+	0.943952i	$0.392923\pi$
$308$	0	0
$309$	0	0
$310$	−2.33163	−0.132428
$311$	18.8327	1.06790	0.533952	−	0.845515i	$-0.320706\pi$
$311$	18.8327	1.06790	0.533952	+	0.845515i	$0.320706\pi$
$312$	0	0
$313$	−5.03403	−0.284540	−0.142270	−	0.989828i	$-0.545440\pi$
$313$	−5.03403	−0.284540	−0.142270	+	0.989828i	$0.545440\pi$
$314$	−16.4202	−0.926646
$315$	0	0
$316$	−2.38158	−0.133975
$317$	−18.5536	−1.04207	−0.521036	−	0.853535i	$-0.674454\pi$
$317$	−18.5536	−1.04207	−0.521036	+	0.853535i	$0.674454\pi$
$318$	0	0
$319$	−10.9975	−0.615744
$320$	−35.3040	−1.97355
$321$	0	0
$322$	0	0
$323$	−2.70633	−0.150584
$324$	0	0
$325$	58.4708	3.24338
$326$	−18.6652	−1.03377
$327$	0	0
$328$	13.2211	0.730012
$329$	0	0
$330$	0	0
$331$	−7.96438	−0.437762	−0.218881	−	0.975752i	$-0.570241\pi$
$331$	−7.96438	−0.437762	−0.218881	+	0.975752i	$0.570241\pi$
$332$	0.165193	0.00906613
$333$	0	0
$334$	−0.254141	−0.0139060
$335$	47.7841	2.61072
$336$	0	0
$337$	−8.38621	−0.456826	−0.228413	−	0.973564i	$-0.573354\pi$
$337$	−8.38621	−0.456826	−0.228413	+	0.973564i	$0.573354\pi$
$338$	18.7267	1.01860
$339$	0	0
$340$	−2.95452	−0.160231
$341$	−0.711449	−0.0385271
$342$	0	0
$343$	0	0
$344$	−26.5366	−1.43076
$345$	0	0
$346$	23.4001	1.25799
$347$	12.8792	0.691390	0.345695	−	0.938347i	$-0.387643\pi$
$347$	12.8792	0.691390	0.345695	+	0.938347i	$0.387643\pi$
$348$	0	0
$349$	−6.10735	−0.326919	−0.163460	−	0.986550i	$-0.552265\pi$
$349$	−6.10735	−0.326919	−0.163460	+	0.986550i	$0.552265\pi$
$350$	0	0
$351$	0	0
$352$	−2.45971	−0.131103
$353$	−11.8104	−0.628606	−0.314303	−	0.949323i	$-0.601771\pi$
$353$	−11.8104	−0.628606	−0.314303	+	0.949323i	$0.601771\pi$
$354$	0	0
$355$	−14.6511	−0.777600
$356$	3.09633	0.164105
$357$	0	0
$358$	−20.0812	−1.06132
$359$	−19.8856	−1.04952	−0.524762	−	0.851249i	$-0.675846\pi$
$359$	−19.8856	−1.04952	−0.524762	+	0.851249i	$0.675846\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−9.74000	−0.511923
$363$	0	0
$364$	0	0
$365$	26.1353	1.36798
$366$	0	0
$367$	−12.1539	−0.634430	−0.317215	−	0.948354i	$-0.602748\pi$
$367$	−12.1539	−0.634430	−0.317215	+	0.948354i	$0.602748\pi$
$368$	−9.77339	−0.509473
$369$	0	0
$370$	35.3848	1.83957
$371$	0	0
$372$	0	0
$373$	21.4558	1.11094	0.555471	−	0.831536i	$-0.312538\pi$
$373$	21.4558	1.11094	0.555471	+	0.831536i	$0.312538\pi$
$374$	5.74641	0.297140
$375$	0	0
$376$	1.95880	0.101018
$377$	35.5447	1.83065
$378$	0	0
$379$	−0.558193	−0.0286725	−0.0143362	−	0.999897i	$-0.504564\pi$
$379$	−0.558193	−0.0286725	−0.0143362	+	0.999897i	$0.504564\pi$
$380$	1.09171	0.0560035
$381$	0	0
$382$	−20.7256	−1.06041
$383$	−16.9350	−0.865336	−0.432668	−	0.901553i	$-0.642428\pi$
$383$	−16.9350	−0.865336	−0.432668	+	0.901553i	$0.642428\pi$
$384$	0	0
$385$	0	0
$386$	16.2770	0.828479
$387$	0	0
$388$	1.87588	0.0952334
$389$	6.24045	0.316404	0.158202	−	0.987407i	$-0.449430\pi$
$389$	6.24045	0.316404	0.158202	+	0.987407i	$0.449430\pi$
$390$	0	0
$391$	−7.81617	−0.395281
$392$	0	0
$393$	0	0
$394$	−16.5440	−0.833477
$395$	−35.3460	−1.77845
$396$	0	0
$397$	−33.3701	−1.67480	−0.837400	−	0.546591i	$-0.815925\pi$
$397$	−33.3701	−1.67480	−0.837400	+	0.546591i	$0.815925\pi$
$398$	1.06065	0.0531656
$399$	0	0
$400$	−37.9093	−1.89546
$401$	31.6007	1.57806	0.789031	−	0.614354i	$-0.210583\pi$
$401$	31.6007	1.57806	0.789031	+	0.614354i	$0.210583\pi$
$402$	0	0
$403$	2.29945	0.114544
$404$	0.0181887	0.000904921 0
$405$	0	0
$406$	0	0
$407$	10.7969	0.535185
$408$	0	0
$409$	−22.8822	−1.13145	−0.565727	−	0.824593i	$-0.691404\pi$
$409$	−22.8822	−1.13145	−0.565727	+	0.824593i	$0.691404\pi$
$410$	23.4315	1.15720
$411$	0	0
$412$	3.13322	0.154363
$413$	0	0
$414$	0	0
$415$	2.45169	0.120349
$416$	7.94992	0.389777
$417$	0	0
$418$	−2.12332	−0.103855
$419$	35.8409	1.75094	0.875471	−	0.483271i	$-0.160551\pi$
$419$	35.8409	1.75094	0.875471	+	0.483271i	$0.160551\pi$
$420$	0	0
$421$	20.4928	0.998758	0.499379	−	0.866384i	$-0.333562\pi$
$421$	20.4928	0.998758	0.499379	+	0.866384i	$0.333562\pi$
$422$	−5.27731	−0.256896
$423$	0	0
$424$	−14.2071	−0.689960
$425$	−30.3176	−1.47062
$426$	0	0
$427$	0	0
$428$	−3.62859	−0.175395
$429$	0	0
$430$	−47.0302	−2.26800
$431$	18.4643	0.889396	0.444698	−	0.895681i	$-0.353311\pi$
$431$	18.4643	0.889396	0.444698	+	0.895681i	$0.353311\pi$
$432$	0	0
$433$	−10.9024	−0.523934	−0.261967	−	0.965077i	$-0.584371\pi$
$433$	−10.9024	−0.523934	−0.261967	+	0.965077i	$0.584371\pi$
$434$	0	0
$435$	0	0
$436$	−4.83340	−0.231478
$437$	2.88811	0.138157
$438$	0	0
$439$	11.9913	0.572316	0.286158	−	0.958182i	$-0.407622\pi$
$439$	11.9913	0.572316	0.286158	+	0.958182i	$0.407622\pi$
$440$	−19.4118	−0.925420
$441$	0	0
$442$	−18.5727	−0.883414
$443$	−5.12656	−0.243570	−0.121785	−	0.992556i	$-0.538862\pi$
$443$	−5.12656	−0.243570	−0.121785	+	0.992556i	$0.538862\pi$
$444$	0	0
$445$	45.9539	2.17842
$446$	33.4341	1.58315
$447$	0	0
$448$	0	0
$449$	8.35649	0.394367	0.197184	−	0.980367i	$-0.436821\pi$
$449$	8.35649	0.394367	0.197184	+	0.980367i	$0.436821\pi$
$450$	0	0
$451$	7.14964	0.336663
$452$	1.47302	0.0692848
$453$	0	0
$454$	−27.9253	−1.31060
$455$	0	0
$456$	0	0
$457$	−13.5750	−0.635013	−0.317507	−	0.948256i	$-0.602846\pi$
$457$	−13.5750	−0.635013	−0.317507	+	0.948256i	$0.602846\pi$
$458$	−12.3771	−0.578346
$459$	0	0
$460$	3.15297	0.147008
$461$	−37.0681	−1.72643	−0.863216	−	0.504835i	$-0.831553\pi$
$461$	−37.0681	−1.72643	−0.863216	+	0.504835i	$0.831553\pi$
$462$	0	0
$463$	10.6201	0.493556	0.246778	−	0.969072i	$-0.420628\pi$
$463$	10.6201	0.493556	0.246778	+	0.969072i	$0.420628\pi$
$464$	−23.0453	−1.06985
$465$	0	0
$466$	−17.1151	−0.792843
$467$	9.65297	0.446686	0.223343	−	0.974740i	$-0.428303\pi$
$467$	9.65297	0.446686	0.223343	+	0.974740i	$0.428303\pi$
$468$	0	0
$469$	0	0
$470$	3.47155	0.160131
$471$	0	0
$472$	33.2807	1.53187
$473$	−14.3503	−0.659829
$474$	0	0
$475$	11.2025	0.514005
$476$	0	0
$477$	0	0
$478$	−15.9463	−0.729367
$479$	−5.14848	−0.235240	−0.117620	−	0.993059i	$-0.537526\pi$
$479$	−5.14848	−0.235240	−0.117620	+	0.993059i	$0.537526\pi$
$480$	0	0
$481$	−34.8964	−1.59114
$482$	−12.8412	−0.584903
$483$	0	0
$484$	2.27607	0.103458
$485$	27.8407	1.26418
$486$	0	0
$487$	−41.3799	−1.87510	−0.937551	−	0.347847i	$-0.886913\pi$
$487$	−41.3799	−1.87510	−0.937551	+	0.347847i	$0.886913\pi$
$488$	−31.3708	−1.42009
$489$	0	0
$490$	0	0
$491$	−0.425814	−0.0192167	−0.00960837	−	0.999954i	$-0.503058\pi$
$491$	−0.425814	−0.0192167	−0.00960837	+	0.999954i	$0.503058\pi$
$492$	0	0
$493$	−18.4302	−0.830055
$494$	6.86270	0.308768
$495$	0	0
$496$	−1.49084	−0.0669405
$497$	0	0
$498$	0	0
$499$	−36.7654	−1.64584	−0.822922	−	0.568154i	$-0.807658\pi$
$499$	−36.7654	−1.64584	−0.822922	+	0.568154i	$0.807658\pi$
$500$	6.77129	0.302821
$501$	0	0
$502$	2.62381	0.117106
$503$	7.98940	0.356230	0.178115	−	0.984010i	$-0.443000\pi$
$503$	7.98940	0.356230	0.178115	+	0.984010i	$0.443000\pi$
$504$	0	0
$505$	0.269946	0.0120124
$506$	−6.13239	−0.272618
$507$	0	0
$508$	−1.65266	−0.0733249
$509$	−39.7557	−1.76214	−0.881070	−	0.472987i	$-0.843176\pi$
$509$	−39.7557	−1.76214	−0.881070	+	0.472987i	$0.843176\pi$
$510$	0	0
$511$	0	0
$512$	−25.3654	−1.12100
$513$	0	0
$514$	3.19353	0.140861
$515$	46.5013	2.04909
$516$	0	0
$517$	1.05927	0.0465868
$518$	0	0
$519$	0	0
$520$	62.7400	2.75133
$521$	−19.6450	−0.860664	−0.430332	−	0.902671i	$-0.641604\pi$
$521$	−19.6450	−0.860664	−0.430332	+	0.902671i	$0.641604\pi$
$522$	0	0
$523$	−3.20781	−0.140268	−0.0701338	−	0.997538i	$-0.522343\pi$
$523$	−3.20781	−0.140268	−0.0701338	+	0.997538i	$0.522343\pi$
$524$	2.31716	0.101226
$525$	0	0
$526$	−2.39231	−0.104310
$527$	−1.19228	−0.0519366
$528$	0	0
$529$	−14.6588	−0.637340
$530$	−25.1790	−1.09371
$531$	0	0
$532$	0	0
$533$	−23.1081	−1.00092
$534$	0	0
$535$	−53.8534	−2.32829
$536$	35.4504	1.53122
$537$	0	0
$538$	−33.6697	−1.45160
$539$	0	0
$540$	0	0
$541$	−17.8218	−0.766220	−0.383110	−	0.923703i	$-0.625147\pi$
$541$	−17.8218	−0.766220	−0.383110	+	0.923703i	$0.625147\pi$
$542$	17.4842	0.751010
$543$	0	0
$544$	−4.12210	−0.176733
$545$	−71.7345	−3.07277
$546$	0	0
$547$	5.56646	0.238005	0.119002	−	0.992894i	$-0.462030\pi$
$547$	5.56646	0.238005	0.119002	+	0.992894i	$0.462030\pi$
$548$	0.509636	0.0217706
$549$	0	0
$550$	−23.7865	−1.01426
$551$	6.81005	0.290118
$552$	0	0
$553$	0	0
$554$	−9.22633	−0.391989
$555$	0	0
$556$	2.34985	0.0996560
$557$	12.0788	0.511795	0.255897	−	0.966704i	$-0.417629\pi$
$557$	12.0788	0.511795	0.255897	+	0.966704i	$0.417629\pi$
$558$	0	0
$559$	46.3811	1.96171
$560$	0	0
$561$	0	0
$562$	31.3758	1.32351
$563$	−17.9554	−0.756730	−0.378365	−	0.925656i	$-0.623514\pi$
$563$	−17.9554	−0.756730	−0.378365	+	0.925656i	$0.623514\pi$
$564$	0	0
$565$	21.8616	0.919725
$566$	42.0310	1.76670
$567$	0	0
$568$	−10.8695	−0.456073
$569$	−23.2302	−0.973860	−0.486930	−	0.873441i	$-0.661883\pi$
$569$	−23.2302	−0.973860	−0.486930	+	0.873441i	$0.661883\pi$
$570$	0	0
$571$	−2.60448	−0.108994	−0.0544970	−	0.998514i	$-0.517356\pi$
$571$	−2.60448	−0.108994	−0.0544970	+	0.998514i	$0.517356\pi$
$572$	2.28605	0.0955847
$573$	0	0
$574$	0	0
$575$	32.3540	1.34925
$576$	0	0
$577$	35.5104	1.47832	0.739159	−	0.673531i	$-0.235223\pi$
$577$	35.5104	1.47832	0.739159	+	0.673531i	$0.235223\pi$
$578$	−12.7220	−0.529167
$579$	0	0
$580$	7.43459	0.308704
$581$	0	0
$582$	0	0
$583$	−7.68287	−0.318192
$584$	19.3894	0.802341
$585$	0	0
$586$	13.8043	0.570252
$587$	−13.2781	−0.548047	−0.274024	−	0.961723i	$-0.588355\pi$
$587$	−13.2781	−0.548047	−0.274024	+	0.961723i	$0.588355\pi$
$588$	0	0
$589$	0.440553	0.0181527
$590$	58.9826	2.42828
$591$	0	0
$592$	22.6249	0.929878
$593$	31.7766	1.30491	0.652455	−	0.757827i	$-0.273739\pi$
$593$	31.7766	1.30491	0.652455	+	0.757827i	$0.273739\pi$
$594$	0	0
$595$	0	0
$596$	0.622624	0.0255037
$597$	0	0
$598$	19.8202	0.810510
$599$	15.6980	0.641405	0.320702	−	0.947180i	$-0.396081\pi$
$599$	15.6980	0.641405	0.320702	+	0.947180i	$0.396081\pi$
$600$	0	0
$601$	−8.25276	−0.336637	−0.168319	−	0.985733i	$-0.553834\pi$
$601$	−8.25276	−0.336637	−0.168319	+	0.985733i	$0.553834\pi$
$602$	0	0
$603$	0	0
$604$	0.867095	0.0352816
$605$	33.7801	1.37336
$606$	0	0
$607$	−10.5423	−0.427899	−0.213949	−	0.976845i	$-0.568633\pi$
$607$	−10.5423	−0.427899	−0.213949	+	0.976845i	$0.568633\pi$
$608$	1.52313	0.0617712
$609$	0	0
$610$	−55.5978	−2.25109
$611$	−3.42363	−0.138505
$612$	0	0
$613$	10.5497	0.426100	0.213050	−	0.977041i	$-0.431660\pi$
$613$	10.5497	0.426100	0.213050	+	0.977041i	$0.431660\pi$
$614$	15.2088	0.613776
$615$	0	0
$616$	0	0
$617$	31.8232	1.28116	0.640578	−	0.767893i	$-0.278695\pi$
$617$	31.8232	1.28116	0.640578	+	0.767893i	$0.278695\pi$
$618$	0	0
$619$	48.2962	1.94119	0.970595	−	0.240720i	$-0.0773834\pi$
$619$	48.2962	1.94119	0.970595	+	0.240720i	$0.0773834\pi$
$620$	0.480956	0.0193156
$621$	0	0
$622$	24.7618	0.992858
$623$	0	0
$624$	0	0
$625$	44.4831	1.77932
$626$	−6.61891	−0.264545
$627$	0	0
$628$	3.38707	0.135159
$629$	18.0940	0.721457
$630$	0	0
$631$	4.94230	0.196750	0.0983750	−	0.995149i	$-0.468636\pi$
$631$	4.94230	0.196750	0.0983750	+	0.995149i	$0.468636\pi$
$632$	−26.2228	−1.04309
$633$	0	0
$634$	−24.3948	−0.968842
$635$	−24.5278	−0.973355
$636$	0	0
$637$	0	0
$638$	−14.4599	−0.572474
$639$	0	0
$640$	−34.1569	−1.35017
$641$	−38.0533	−1.50302	−0.751508	−	0.659724i	$-0.770673\pi$
$641$	−38.0533	−1.50302	−0.751508	+	0.659724i	$0.770673\pi$
$642$	0	0
$643$	−12.8821	−0.508019	−0.254009	−	0.967202i	$-0.581749\pi$
$643$	−12.8821	−0.508019	−0.254009	+	0.967202i	$0.581749\pi$
$644$	0	0
$645$	0	0
$646$	−3.55837	−0.140002
$647$	24.9667	0.981543	0.490772	−	0.871288i	$-0.336715\pi$
$647$	24.9667	0.981543	0.490772	+	0.871288i	$0.336715\pi$
$648$	0	0
$649$	17.9974	0.706458
$650$	76.8793	3.01545
$651$	0	0
$652$	3.85015	0.150783
$653$	35.0795	1.37277	0.686383	−	0.727240i	$-0.259197\pi$
$653$	35.0795	1.37277	0.686383	+	0.727240i	$0.259197\pi$
$654$	0	0
$655$	34.3899	1.34373
$656$	14.9820	0.584949
$657$	0	0
$658$	0	0
$659$	25.9992	1.01278	0.506392	−	0.862303i	$-0.330979\pi$
$659$	25.9992	1.01278	0.506392	+	0.862303i	$0.330979\pi$
$660$	0	0
$661$	34.5021	1.34198	0.670988	−	0.741468i	$-0.265870\pi$
$661$	34.5021	1.34198	0.670988	+	0.741468i	$0.265870\pi$
$662$	−10.4718	−0.406999
$663$	0	0
$664$	1.81888	0.0705862
$665$	0	0
$666$	0	0
$667$	19.6682	0.761554
$668$	0.0524227	0.00202830
$669$	0	0
$670$	62.8280	2.42726
$671$	−16.9645	−0.654908
$672$	0	0
$673$	−14.0570	−0.541858	−0.270929	−	0.962599i	$-0.587331\pi$
$673$	−14.0570	−0.541858	−0.270929	+	0.962599i	$0.587331\pi$
$674$	−11.0265	−0.424723
$675$	0	0
$676$	−3.86285	−0.148571
$677$	−2.58630	−0.0993997	−0.0496999	−	0.998764i	$-0.515826\pi$
$677$	−2.58630	−0.0993997	−0.0496999	+	0.998764i	$0.515826\pi$
$678$	0	0
$679$	0	0
$680$	−32.5312	−1.24751
$681$	0	0
$682$	−0.935437	−0.0358197
$683$	−43.6180	−1.66900	−0.834498	−	0.551012i	$-0.814242\pi$
$683$	−43.6180	−1.66900	−0.834498	+	0.551012i	$0.814242\pi$
$684$	0	0
$685$	7.56372	0.288995
$686$	0	0
$687$	0	0
$688$	−30.0710	−1.14645
$689$	24.8315	0.946004
$690$	0	0
$691$	−8.52917	−0.324465	−0.162232	−	0.986753i	$-0.551869\pi$
$691$	−8.52917	−0.324465	−0.162232	+	0.986753i	$0.551869\pi$
$692$	−4.82683	−0.183489
$693$	0	0
$694$	16.9340	0.642804
$695$	34.8751	1.32289
$696$	0	0
$697$	11.9817	0.453840
$698$	−8.03014	−0.303945
$699$	0	0
$700$	0	0
$701$	36.6789	1.38534	0.692671	−	0.721254i	$-0.256434\pi$
$701$	36.6789	1.38534	0.692671	+	0.721254i	$0.256434\pi$
$702$	0	0
$703$	−6.68583	−0.252161
$704$	−14.1638	−0.533817
$705$	0	0
$706$	−15.5287	−0.584431
$707$	0	0
$708$	0	0
$709$	25.2002	0.946415	0.473208	−	0.880951i	$-0.343096\pi$
$709$	25.2002	0.946415	0.473208	+	0.880951i	$0.343096\pi$
$710$	−19.2637	−0.722955
$711$	0	0
$712$	34.0926	1.27768
$713$	1.27237	0.0476505
$714$	0	0
$715$	33.9282	1.26884
$716$	4.14224	0.154803
$717$	0	0
$718$	−26.1463	−0.975771
$719$	−6.45827	−0.240853	−0.120426	−	0.992722i	$-0.538426\pi$
$719$	−6.45827	−0.240853	−0.120426	+	0.992722i	$0.538426\pi$
$720$	0	0
$721$	0	0
$722$	1.31483	0.0489330
$723$	0	0
$724$	2.00911	0.0746681
$725$	76.2894	2.83332
$726$	0	0
$727$	−37.8391	−1.40338	−0.701688	−	0.712485i	$-0.747570\pi$
$727$	−37.8391	−1.40338	−0.701688	+	0.712485i	$0.747570\pi$
$728$	0	0
$729$	0	0
$730$	34.3635	1.27185
$731$	−24.0490	−0.889483
$732$	0	0
$733$	3.75562	0.138717	0.0693584	−	0.997592i	$-0.477905\pi$
$733$	3.75562	0.138717	0.0693584	+	0.997592i	$0.477905\pi$
$734$	−15.9804	−0.589846
$735$	0	0
$736$	4.39897	0.162148
$737$	19.1707	0.706162
$738$	0	0
$739$	−44.6922	−1.64403	−0.822015	−	0.569466i	$-0.807150\pi$
$739$	−44.6922	−1.64403	−0.822015	+	0.569466i	$0.807150\pi$
$740$	−7.29897	−0.268316
$741$	0	0
$742$	0	0
$743$	−28.4487	−1.04368	−0.521841	−	0.853042i	$-0.674755\pi$
$743$	−28.4487	−1.04368	−0.521841	+	0.853042i	$0.674755\pi$
$744$	0	0
$745$	9.24061	0.338550
$746$	28.2108	1.03287
$747$	0	0
$748$	−1.18534	−0.0433402
$749$	0	0
$750$	0	0
$751$	−45.3886	−1.65625	−0.828126	−	0.560541i	$-0.810593\pi$
$751$	−45.3886	−1.65625	−0.828126	+	0.560541i	$0.810593\pi$
$752$	2.21970	0.0809440
$753$	0	0
$754$	46.7354	1.70200
$755$	12.8689	0.468347
$756$	0	0
$757$	−23.7118	−0.861820	−0.430910	−	0.902395i	$-0.641807\pi$
$757$	−23.7118	−0.861820	−0.430910	+	0.902395i	$0.641807\pi$
$758$	−0.733930	−0.0266576
$759$	0	0
$760$	12.0204	0.436026
$761$	42.0135	1.52299	0.761494	−	0.648172i	$-0.224466\pi$
$761$	42.0135	1.52299	0.761494	+	0.648172i	$0.224466\pi$
$762$	0	0
$763$	0	0
$764$	4.27515	0.154670
$765$	0	0
$766$	−22.2666	−0.804526
$767$	−58.1685	−2.10034
$768$	0	0
$769$	−13.8751	−0.500348	−0.250174	−	0.968201i	$-0.580488\pi$
$769$	−13.8751	−0.500348	−0.250174	+	0.968201i	$0.580488\pi$
$770$	0	0
$771$	0	0
$772$	−3.35754	−0.120840
$773$	−22.7374	−0.817807	−0.408904	−	0.912578i	$-0.634089\pi$
$773$	−22.7374	−0.817807	−0.408904	+	0.912578i	$0.634089\pi$
$774$	0	0
$775$	4.93529	0.177281
$776$	20.6547	0.741459
$777$	0	0
$778$	8.20515	0.294169
$779$	−4.42729	−0.158624
$780$	0	0
$781$	−5.87794	−0.210329
$782$	−10.2770	−0.367503
$783$	0	0
$784$	0	0
$785$	50.2689	1.79417
$786$	0	0
$787$	−41.8083	−1.49030	−0.745152	−	0.666895i	$-0.767623\pi$
$787$	−41.8083	−1.49030	−0.745152	+	0.666895i	$0.767623\pi$
$788$	3.41262	0.121569
$789$	0	0
$790$	−46.4741	−1.65347
$791$	0	0
$792$	0	0
$793$	54.8304	1.94708
$794$	−43.8761	−1.55711
$795$	0	0
$796$	−0.218785	−0.00775463
$797$	3.43467	0.121662	0.0608310	−	0.998148i	$-0.480625\pi$
$797$	3.43467	0.121662	0.0608310	+	0.998148i	$0.480625\pi$
$798$	0	0
$799$	1.77518	0.0628014
$800$	17.0629	0.603263
$801$	0	0
$802$	41.5496	1.46717
$803$	10.4853	0.370019
$804$	0	0
$805$	0	0
$806$	3.02339	0.106494
$807$	0	0
$808$	0.200269	0.00704545
$809$	−38.2871	−1.34610	−0.673052	−	0.739595i	$-0.735017\pi$
$809$	−38.2871	−1.34610	−0.673052	+	0.739595i	$0.735017\pi$
$810$	0	0
$811$	−35.4750	−1.24570	−0.622848	−	0.782343i	$-0.714024\pi$
$811$	−35.4750	−1.24570	−0.622848	+	0.782343i	$0.714024\pi$
$812$	0	0
$813$	0	0
$814$	14.1962	0.497575
$815$	57.1415	2.00158
$816$	0	0
$817$	8.88620	0.310889
$818$	−30.0863	−1.05194
$819$	0	0
$820$	−4.83331	−0.168787
$821$	8.01294	0.279653	0.139827	−	0.990176i	$-0.455345\pi$
$821$	8.01294	0.279653	0.139827	+	0.990176i	$0.455345\pi$
$822$	0	0
$823$	−30.1673	−1.05157	−0.525783	−	0.850619i	$-0.676228\pi$
$823$	−30.1673	−1.05157	−0.525783	+	0.850619i	$0.676228\pi$
$824$	34.4988	1.20182
$825$	0	0
$826$	0	0
$827$	−30.4531	−1.05896	−0.529479	−	0.848323i	$-0.677612\pi$
$827$	−30.4531	−1.05896	−0.529479	+	0.848323i	$0.677612\pi$
$828$	0	0
$829$	6.26242	0.217503	0.108751	−	0.994069i	$-0.465315\pi$
$829$	6.26242	0.217503	0.108751	+	0.994069i	$0.465315\pi$
$830$	3.22356	0.111891
$831$	0	0
$832$	45.7782	1.58707
$833$	0	0
$834$	0	0
$835$	0.778027	0.0269247
$836$	0.437987	0.0151481
$837$	0	0
$838$	47.1247	1.62790
$839$	6.86060	0.236854	0.118427	−	0.992963i	$-0.462215\pi$
$839$	6.86060	0.236854	0.118427	+	0.992963i	$0.462215\pi$
$840$	0	0
$841$	17.3768	0.599199
$842$	26.9446	0.928572
$843$	0	0
$844$	1.08858	0.0374703
$845$	−57.3301	−1.97222
$846$	0	0
$847$	0	0
$848$	−16.0994	−0.552855
$849$	0	0
$850$	−39.8625	−1.36727
$851$	−19.3094	−0.661918
$852$	0	0
$853$	−26.9735	−0.923557	−0.461778	−	0.886995i	$-0.652788\pi$
$853$	−26.9735	−0.923557	−0.461778	+	0.886995i	$0.652788\pi$
$854$	0	0
$855$	0	0
$856$	−39.9532	−1.36557
$857$	5.06697	0.173084	0.0865422	−	0.996248i	$-0.472418\pi$
$857$	5.06697	0.173084	0.0865422	+	0.996248i	$0.472418\pi$
$858$	0	0
$859$	34.4157	1.17425	0.587125	−	0.809497i	$-0.300260\pi$
$859$	34.4157	1.17425	0.587125	+	0.809497i	$0.300260\pi$
$860$	9.70114	0.330806
$861$	0	0
$862$	24.2775	0.826895
$863$	37.7836	1.28617	0.643084	−	0.765795i	$-0.277655\pi$
$863$	37.7836	1.28617	0.643084	+	0.765795i	$0.277655\pi$
$864$	0	0
$865$	−71.6370	−2.43573
$866$	−14.3348	−0.487115
$867$	0	0
$868$	0	0
$869$	−14.1806	−0.481045
$870$	0	0
$871$	−61.9609	−2.09946
$872$	−53.2189	−1.80222
$873$	0	0
$874$	3.79738	0.128448
$875$	0	0
$876$	0	0
$877$	−6.94742	−0.234598	−0.117299	−	0.993097i	$-0.537424\pi$
$877$	−6.94742	−0.234598	−0.117299	+	0.993097i	$0.537424\pi$
$878$	15.7666	0.532097
$879$	0	0
$880$	−21.9972	−0.741526
$881$	−55.8959	−1.88318	−0.941591	−	0.336758i	$-0.890669\pi$
$881$	−55.8959	−1.88318	−0.941591	+	0.336758i	$0.890669\pi$
$882$	0	0
$883$	42.1647	1.41895	0.709477	−	0.704729i	$-0.248931\pi$
$883$	42.1647	1.41895	0.709477	+	0.704729i	$0.248931\pi$
$884$	3.83108	0.128853
$885$	0	0
$886$	−6.74057	−0.226454
$887$	44.7081	1.50115	0.750575	−	0.660786i	$-0.229777\pi$
$887$	44.7081	1.50115	0.750575	+	0.660786i	$0.229777\pi$
$888$	0	0
$889$	0	0
$890$	60.4217	2.02534
$891$	0	0
$892$	−6.89661	−0.230916
$893$	−0.655937	−0.0219501
$894$	0	0
$895$	61.4766	2.05494
$896$	0	0
$897$	0	0
$898$	10.9874	0.366654
$899$	3.00019	0.100062
$900$	0	0
$901$	−12.8753	−0.428939
$902$	9.40057	0.313005
$903$	0	0
$904$	16.2189	0.539431
$905$	29.8181	0.991186
$906$	0	0
$907$	20.3975	0.677287	0.338643	−	0.940915i	$-0.390032\pi$
$907$	20.3975	0.677287	0.338643	+	0.940915i	$0.390032\pi$
$908$	5.76029	0.191162
$909$	0	0
$910$	0	0
$911$	42.3351	1.40262	0.701312	−	0.712854i	$-0.252598\pi$
$911$	42.3351	1.40262	0.701312	+	0.712854i	$0.252598\pi$
$912$	0	0
$913$	0.983605	0.0325526
$914$	−17.8489	−0.590389
$915$	0	0
$916$	2.55309	0.0843564
$917$	0	0
$918$	0	0
$919$	48.4005	1.59659	0.798293	−	0.602270i	$-0.205737\pi$
$919$	48.4005	1.59659	0.798293	+	0.602270i	$0.205737\pi$
$920$	34.7163	1.14456
$921$	0	0
$922$	−48.7383	−1.60511
$923$	18.9979	0.625322
$924$	0	0
$925$	−74.8978	−2.46263
$926$	13.9636	0.458872
$927$	0	0
$928$	10.3726	0.340498
$929$	58.1426	1.90760	0.953798	−	0.300449i	$-0.0971365\pi$
$929$	58.1426	1.90760	0.953798	+	0.300449i	$0.0971365\pi$
$930$	0	0
$931$	0	0
$932$	3.53041	0.115643
$933$	0	0
$934$	12.6920	0.415296
$935$	−17.5921	−0.575322
$936$	0	0
$937$	0.198191	0.00647462	0.00323731	−	0.999995i	$-0.498970\pi$
$937$	0.198191	0.00647462	0.00323731	+	0.999995i	$0.498970\pi$
$938$	0	0
$939$	0	0
$940$	−0.716092	−0.0233563
$941$	−11.8456	−0.386157	−0.193079	−	0.981183i	$-0.561847\pi$
$941$	−11.8456	−0.386157	−0.193079	+	0.981183i	$0.561847\pi$
$942$	0	0
$943$	−12.7865	−0.416386
$944$	37.7133	1.22746
$945$	0	0
$946$	−18.8683	−0.613460
$947$	−20.3829	−0.662354	−0.331177	−	0.943569i	$-0.607446\pi$
$947$	−20.3829	−0.662354	−0.331177	+	0.943569i	$0.607446\pi$
$948$	0	0
$949$	−33.8892	−1.10009
$950$	14.7294	0.477884
$951$	0	0
$952$	0	0
$953$	−25.0309	−0.810831	−0.405415	−	0.914133i	$-0.632873\pi$
$953$	−25.0309	−0.810831	−0.405415	+	0.914133i	$0.632873\pi$
$954$	0	0
$955$	63.4493	2.05317
$956$	3.28932	0.106384
$957$	0	0
$958$	−6.76939	−0.218709
$959$	0	0
$960$	0	0
$961$	−30.8059	−0.993739
$962$	−45.8829	−1.47932
$963$	0	0
$964$	2.64882	0.0853128
$965$	−49.8305	−1.60410
$966$	0	0
$967$	−55.7276	−1.79208	−0.896040	−	0.443973i	$-0.853569\pi$
$967$	−55.7276	−1.79208	−0.896040	+	0.443973i	$0.853569\pi$
$968$	25.0611	0.805493
$969$	0	0
$970$	36.6058	1.17534
$971$	−38.0762	−1.22192	−0.610961	−	0.791661i	$-0.709217\pi$
$971$	−38.0762	−1.22192	−0.610961	+	0.791661i	$0.709217\pi$
$972$	0	0
$973$	0	0
$974$	−54.4076	−1.74333
$975$	0	0
$976$	−35.5490	−1.13790
$977$	38.7229	1.23886	0.619428	−	0.785053i	$-0.287365\pi$
$977$	38.7229	1.23886	0.619428	+	0.785053i	$0.287365\pi$
$978$	0	0
$979$	18.4365	0.589232
$980$	0	0
$981$	0	0
$982$	−0.559874	−0.0178663
$983$	−30.0635	−0.958876	−0.479438	−	0.877576i	$-0.659160\pi$
$983$	−30.0635	−0.958876	−0.479438	+	0.877576i	$0.659160\pi$
$984$	0	0
$985$	50.6480	1.61378
$986$	−24.2326	−0.771725
$987$	0	0
$988$	−1.41560	−0.0450363
$989$	25.6643	0.816078
$990$	0	0
$991$	−19.1158	−0.607233	−0.303617	−	0.952794i	$-0.598194\pi$
$991$	−19.1158	−0.607233	−0.303617	+	0.952794i	$0.598194\pi$
$992$	0.671021	0.0213049
$993$	0	0
$994$	0	0
$995$	−3.24708	−0.102939
$996$	0	0
$997$	43.3237	1.37208	0.686038	−	0.727565i	$-0.259348\pi$
$997$	43.3237	1.37208	0.686038	+	0.727565i	$0.259348\pi$
$998$	−48.3403	−1.53019
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8379.2.a.cs.1.7		10
3.2	odd	2		931.2.a.q.1.4	yes	10
7.6	odd	2		8379.2.a.ct.1.7		10
21.2	odd	6		931.2.f.q.704.7		20
21.5	even	6		931.2.f.r.704.7		20
21.11	odd	6		931.2.f.q.324.7		20
21.17	even	6		931.2.f.r.324.7		20
21.20	even	2		931.2.a.p.1.4	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
931.2.a.p.1.4	✓	10	21.20	even	2
931.2.a.q.1.4	yes	10	3.2	odd	2
931.2.f.q.324.7		20	21.11	odd	6
931.2.f.q.704.7		20	21.2	odd	6
931.2.f.r.324.7		20	21.17	even	6
931.2.f.r.704.7		20	21.5	even	6
8379.2.a.cs.1.7		10	1.1	even	1	trivial
8379.2.a.ct.1.7		10	7.6	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:	\( N \)	\(=\)	\( 8379 = 3^{2} \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8379.a (trivial)

\(f(q)\)	\(=\)	\(q+1.31483 q^{2} -0.271216 q^{4} -4.02523 q^{5} -2.98627 q^{8} +O(q^{10})\)	\(q+1.31483 q^{2} -0.271216 q^{4} -4.02523 q^{5} -2.98627 q^{8} -5.29250 q^{10} -1.61490 q^{11} +5.21945 q^{13} -3.38401 q^{16} -2.70633 q^{17} +1.00000 q^{19} +1.09171 q^{20} -2.12332 q^{22} +2.88811 q^{23} +11.2025 q^{25} +6.86270 q^{26} +6.81005 q^{29} +0.440553 q^{31} +1.52313 q^{32} -3.55837 q^{34} -6.68583 q^{37} +1.31483 q^{38} +12.0204 q^{40} -4.42729 q^{41} +8.88620 q^{43} +0.437987 q^{44} +3.79738 q^{46} -0.655937 q^{47} +14.7294 q^{50} -1.41560 q^{52} +4.75749 q^{53} +6.50034 q^{55} +8.95407 q^{58} -11.1446 q^{59} +10.5050 q^{61} +0.579254 q^{62} +8.77068 q^{64} -21.0095 q^{65} -11.8711 q^{67} +0.734000 q^{68} +3.63982 q^{71} -6.49287 q^{73} -8.79074 q^{74} -0.271216 q^{76} +8.78112 q^{79} +13.6214 q^{80} -5.82115 q^{82} -0.609081 q^{83} +10.8936 q^{85} +11.6839 q^{86} +4.82252 q^{88} -11.4165 q^{89} -0.783303 q^{92} -0.862447 q^{94} -4.02523 q^{95} -6.91655 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 2 q^{2} + 10 q^{4} - 16 q^{5} + 6 q^{8}+O(q^{10}) \) 10 * q + 2 * q^2 + 10 * q^4 - 16 * q^5 + 6 * q^8	\( 10 q + 2 q^{2} + 10 q^{4} - 16 q^{5} + 6 q^{8} - 12 q^{10} + 12 q^{13} + 2 q^{16} - 16 q^{17} + 10 q^{19} - 32 q^{20} + 4 q^{22} - 12 q^{23} + 14 q^{25} - 24 q^{26} + 12 q^{29} + 8 q^{31} + 34 q^{32} + 16 q^{34} + 4 q^{37} + 2 q^{38} - 20 q^{40} - 40 q^{41} + 4 q^{43} + 20 q^{44} - 32 q^{46} - 16 q^{47} + 34 q^{50} - 40 q^{52} + 16 q^{55} - 8 q^{58} - 36 q^{59} + 16 q^{61} + 16 q^{62} + 18 q^{64} - 8 q^{65} - 28 q^{67} - 40 q^{68} + 12 q^{71} - 24 q^{74} + 10 q^{76} - 8 q^{79} - 8 q^{80} - 8 q^{82} + 40 q^{85} + 52 q^{86} - 4 q^{88} - 48 q^{89} - 28 q^{92} - 36 q^{94} - 16 q^{95} - 16 q^{97}+O(q^{100}) \) 10 * q + 2 * q^2 + 10 * q^4 - 16 * q^5 + 6 * q^8 - 12 * q^10 + 12 * q^13 + 2 * q^16 - 16 * q^17 + 10 * q^19 - 32 * q^20 + 4 * q^22 - 12 * q^23 + 14 * q^25 - 24 * q^26 + 12 * q^29 + 8 * q^31 + 34 * q^32 + 16 * q^34 + 4 * q^37 + 2 * q^38 - 20 * q^40 - 40 * q^41 + 4 * q^43 + 20 * q^44 - 32 * q^46 - 16 * q^47 + 34 * q^50 - 40 * q^52 + 16 * q^55 - 8 * q^58 - 36 * q^59 + 16 * q^61 + 16 * q^62 + 18 * q^64 - 8 * q^65 - 28 * q^67 - 40 * q^68 + 12 * q^71 - 24 * q^74 + 10 * q^76 - 8 * q^79 - 8 * q^80 - 8 * q^82 + 40 * q^85 + 52 * q^86 - 4 * q^88 - 48 * q^89 - 28 * q^92 - 36 * q^94 - 16 * q^95 - 16 * q^97

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