LMFDB - Embedded newform 3380.2.a.s.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3380,2,Mod(1,3380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3380, 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("3380.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3380 = 2^{2} \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3380.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:	$26.9894358832$
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} - 19x^{7} + 16x^{6} + 106x^{5} - 87x^{4} - 153x^{3} + 149x^{2} - 26x + 1 $ x^9 - x^8 - 19x^7 + 16x^6 + 106x^5 - 87x^4 - 153x^3 + 149x^2 - 26*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.0545075$ of defining polynomial
Character	$\chi$	$=$	3380.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.0545075 q^{3} +1.00000 q^{5} +3.71818 q^{7} -2.99703 q^{9} +O(q^{10})$	$q-0.0545075 q^{3} +1.00000 q^{5} +3.71818 q^{7} -2.99703 q^{9} +5.22019 q^{11} -0.0545075 q^{15} -2.97721 q^{17} +6.13337 q^{19} -0.202668 q^{21} +3.62333 q^{23} +1.00000 q^{25} +0.326883 q^{27} -2.03371 q^{29} +0.100239 q^{31} -0.284540 q^{33} +3.71818 q^{35} +7.32662 q^{37} -12.1830 q^{41} +8.18427 q^{43} -2.99703 q^{45} -8.41641 q^{47} +6.82483 q^{49} +0.162280 q^{51} -6.11335 q^{53} +5.22019 q^{55} -0.334315 q^{57} +1.53604 q^{59} +7.09884 q^{61} -11.1435 q^{63} +14.6928 q^{67} -0.197499 q^{69} +13.3972 q^{71} -11.5465 q^{73} -0.0545075 q^{75} +19.4096 q^{77} -10.0334 q^{79} +8.97327 q^{81} +4.21504 q^{83} -2.97721 q^{85} +0.110853 q^{87} -7.86394 q^{89} -0.00546375 q^{93} +6.13337 q^{95} -12.6051 q^{97} -15.6451 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - q^{3} + 9 q^{5} + q^{7} + 12 q^{9}+O(q^{10}) $ 9 * q - q^3 + 9 * q^5 + q^7 + 12 * q^9	$ 9 q - q^{3} + 9 q^{5} + q^{7} + 12 q^{9} - 7 q^{11} - q^{15} + 13 q^{17} - 4 q^{19} - 3 q^{21} + 12 q^{23} + 9 q^{25} - 4 q^{27} + 16 q^{29} + 13 q^{31} + 34 q^{33} + q^{35} + q^{37} - 6 q^{41} + q^{43} + 12 q^{45} - 2 q^{47} + 20 q^{49} + 11 q^{51} + 30 q^{53} - 7 q^{55} + 38 q^{57} + 15 q^{59} + 21 q^{61} - 17 q^{63} - 7 q^{67} + 15 q^{69} - 7 q^{71} - 28 q^{73} - q^{75} + 46 q^{77} + 31 q^{79} + 41 q^{81} + 45 q^{83} + 13 q^{85} + 28 q^{87} - 41 q^{89} - 11 q^{93} - 4 q^{95} + 8 q^{97} - 81 q^{99}+O(q^{100}) $ 9 * q - q^3 + 9 * q^5 + q^7 + 12 * q^9 - 7 * q^11 - q^15 + 13 * q^17 - 4 * q^19 - 3 * q^21 + 12 * q^23 + 9 * q^25 - 4 * q^27 + 16 * q^29 + 13 * q^31 + 34 * q^33 + q^35 + q^37 - 6 * q^41 + q^43 + 12 * q^45 - 2 * q^47 + 20 * q^49 + 11 * q^51 + 30 * q^53 - 7 * q^55 + 38 * q^57 + 15 * q^59 + 21 * q^61 - 17 * q^63 - 7 * q^67 + 15 * q^69 - 7 * q^71 - 28 * q^73 - q^75 + 46 * q^77 + 31 * q^79 + 41 * q^81 + 45 * q^83 + 13 * q^85 + 28 * q^87 - 41 * q^89 - 11 * q^93 - 4 * q^95 + 8 * q^97 - 81 * 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
$3$	−0.0545075	−0.0314699	−0.0157350	−	0.999876i	$-0.505009\pi$
$3$	−0.0545075	−0.0314699	−0.0157350	+	0.999876i	$0.505009\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	3.71818	1.40534	0.702669	−	0.711517i	$-0.251991\pi$
$7$	3.71818	1.40534	0.702669	+	0.711517i	$0.251991\pi$
$8$	0	0
$9$	−2.99703	−0.999010
$10$	0	0
$11$	5.22019	1.57395	0.786974	−	0.616986i	$-0.211647\pi$
$11$	5.22019	1.57395	0.786974	+	0.616986i	$0.211647\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−0.0545075	−0.0140738
$16$	0	0
$17$	−2.97721	−0.722079	−0.361039	−	0.932551i	$-0.617578\pi$
$17$	−2.97721	−0.722079	−0.361039	+	0.932551i	$0.617578\pi$
$18$	0	0
$19$	6.13337	1.40709	0.703546	−	0.710650i	$-0.251599\pi$
$19$	6.13337	1.40709	0.703546	+	0.710650i	$0.251599\pi$
$20$	0	0
$21$	−0.202668	−0.0442259
$22$	0	0
$23$	3.62333	0.755516	0.377758	−	0.925904i	$-0.376695\pi$
$23$	3.62333	0.755516	0.377758	+	0.925904i	$0.376695\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0.326883	0.0629087
$28$	0	0
$29$	−2.03371	−0.377651	−0.188825	−	0.982011i	$-0.560468\pi$
$29$	−2.03371	−0.377651	−0.188825	+	0.982011i	$0.560468\pi$
$30$	0	0
$31$	0.100239	0.0180034	0.00900169	−	0.999959i	$-0.497135\pi$
$31$	0.100239	0.0180034	0.00900169	+	0.999959i	$0.497135\pi$
$32$	0	0
$33$	−0.284540	−0.0495320
$34$	0	0
$35$	3.71818	0.628486
$36$	0	0
$37$	7.32662	1.20449	0.602244	−	0.798312i	$-0.294273\pi$
$37$	7.32662	1.20449	0.602244	+	0.798312i	$0.294273\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−12.1830	−1.90266	−0.951329	−	0.308176i	$-0.900282\pi$
$41$	−12.1830	−1.90266	−0.951329	+	0.308176i	$0.900282\pi$
$42$	0	0
$43$	8.18427	1.24809	0.624044	−	0.781389i	$-0.285488\pi$
$43$	8.18427	1.24809	0.624044	+	0.781389i	$0.285488\pi$
$44$	0	0
$45$	−2.99703	−0.446771
$46$	0	0
$47$	−8.41641	−1.22766	−0.613829	−	0.789439i	$-0.710372\pi$
$47$	−8.41641	−1.22766	−0.613829	+	0.789439i	$0.710372\pi$
$48$	0	0
$49$	6.82483	0.974976
$50$	0	0
$51$	0.162280	0.0227238
$52$	0	0
$53$	−6.11335	−0.839733	−0.419867	−	0.907586i	$-0.637923\pi$
$53$	−6.11335	−0.839733	−0.419867	+	0.907586i	$0.637923\pi$
$54$	0	0
$55$	5.22019	0.703891
$56$	0	0
$57$	−0.334315	−0.0442810
$58$	0	0
$59$	1.53604	0.199975	0.0999876	−	0.994989i	$-0.468120\pi$
$59$	1.53604	0.199975	0.0999876	+	0.994989i	$0.468120\pi$
$60$	0	0
$61$	7.09884	0.908914	0.454457	−	0.890769i	$-0.349833\pi$
$61$	7.09884	0.908914	0.454457	+	0.890769i	$0.349833\pi$
$62$	0	0
$63$	−11.1435	−1.40395
$64$	0	0
$65$	0	0
$66$	0	0
$67$	14.6928	1.79501	0.897507	−	0.441000i	$-0.145376\pi$
$67$	14.6928	1.79501	0.897507	+	0.441000i	$0.145376\pi$
$68$	0	0
$69$	−0.197499	−0.0237760
$70$	0	0
$71$	13.3972	1.58995	0.794975	−	0.606642i	$-0.207484\pi$
$71$	13.3972	1.58995	0.794975	+	0.606642i	$0.207484\pi$
$72$	0	0
$73$	−11.5465	−1.35142	−0.675708	−	0.737169i	$-0.736162\pi$
$73$	−11.5465	−1.35142	−0.675708	+	0.737169i	$0.736162\pi$
$74$	0	0
$75$	−0.0545075	−0.00629398
$76$	0	0
$77$	19.4096	2.21193
$78$	0	0
$79$	−10.0334	−1.12885	−0.564424	−	0.825485i	$-0.690902\pi$
$79$	−10.0334	−1.12885	−0.564424	+	0.825485i	$0.690902\pi$
$80$	0	0
$81$	8.97327	0.997030
$82$	0	0
$83$	4.21504	0.462661	0.231330	−	0.972875i	$-0.425692\pi$
$83$	4.21504	0.462661	0.231330	+	0.972875i	$0.425692\pi$
$84$	0	0
$85$	−2.97721	−0.322924
$86$	0	0
$87$	0.110853	0.0118846
$88$	0	0
$89$	−7.86394	−0.833576	−0.416788	−	0.909004i	$-0.636844\pi$
$89$	−7.86394	−0.833576	−0.416788	+	0.909004i	$0.636844\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−0.00546375	−0.000566565 0
$94$	0	0
$95$	6.13337	0.629270
$96$	0	0
$97$	−12.6051	−1.27986	−0.639929	−	0.768434i	$-0.721036\pi$
$97$	−12.6051	−1.27986	−0.639929	+	0.768434i	$0.721036\pi$
$98$	0	0
$99$	−15.6451	−1.57239
$100$	0	0
$101$	3.85725	0.383810	0.191905	−	0.981413i	$-0.438533\pi$
$101$	3.85725	0.383810	0.191905	+	0.981413i	$0.438533\pi$
$102$	0	0
$103$	−1.09068	−0.107467	−0.0537337	−	0.998555i	$-0.517112\pi$
$103$	−1.09068	−0.107467	−0.0537337	+	0.998555i	$0.517112\pi$
$104$	0	0
$105$	−0.202668	−0.0197784
$106$	0	0
$107$	5.55026	0.536563	0.268282	−	0.963340i	$-0.413544\pi$
$107$	5.55026	0.536563	0.268282	+	0.963340i	$0.413544\pi$
$108$	0	0
$109$	−0.0361815	−0.00346556	−0.00173278	−	0.999998i	$-0.500552\pi$
$109$	−0.0361815	−0.00346556	−0.00173278	+	0.999998i	$0.500552\pi$
$110$	0	0
$111$	−0.399355	−0.0379051
$112$	0	0
$113$	−13.3944	−1.26004	−0.630018	−	0.776581i	$-0.716952\pi$
$113$	−13.3944	−1.26004	−0.630018	+	0.776581i	$0.716952\pi$
$114$	0	0
$115$	3.62333	0.337877
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−11.0698	−1.01477
$120$	0	0
$121$	16.2504	1.47731
$122$	0	0
$123$	0.664063	0.0598765
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	11.2976	1.00250	0.501248	−	0.865303i	$-0.332874\pi$
$127$	11.2976	1.00250	0.501248	+	0.865303i	$0.332874\pi$
$128$	0	0
$129$	−0.446104	−0.0392772
$130$	0	0
$131$	1.04287	0.0911160	0.0455580	−	0.998962i	$-0.485493\pi$
$131$	1.04287	0.0911160	0.0455580	+	0.998962i	$0.485493\pi$
$132$	0	0
$133$	22.8049	1.97744
$134$	0	0
$135$	0.326883	0.0281336
$136$	0	0
$137$	17.8330	1.52358	0.761789	−	0.647825i	$-0.224321\pi$
$137$	17.8330	1.52358	0.761789	+	0.647825i	$0.224321\pi$
$138$	0	0
$139$	−17.5662	−1.48995	−0.744973	−	0.667095i	$-0.767538\pi$
$139$	−17.5662	−1.48995	−0.744973	+	0.667095i	$0.767538\pi$
$140$	0	0
$141$	0.458757	0.0386343
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−2.03371	−0.168891
$146$	0	0
$147$	−0.372004	−0.0306824
$148$	0	0
$149$	2.54599	0.208575	0.104288	−	0.994547i	$-0.466744\pi$
$149$	2.54599	0.208575	0.104288	+	0.994547i	$0.466744\pi$
$150$	0	0
$151$	0.213212	0.0173510	0.00867548	−	0.999962i	$-0.497238\pi$
$151$	0.213212	0.0173510	0.00867548	+	0.999962i	$0.497238\pi$
$152$	0	0
$153$	8.92278	0.721364
$154$	0	0
$155$	0.100239	0.00805136
$156$	0	0
$157$	−23.4111	−1.86841	−0.934205	−	0.356736i	$-0.883890\pi$
$157$	−23.4111	−1.86841	−0.934205	+	0.356736i	$0.883890\pi$
$158$	0	0
$159$	0.333223	0.0264263
$160$	0	0
$161$	13.4722	1.06176
$162$	0	0
$163$	−7.31629	−0.573056	−0.286528	−	0.958072i	$-0.592501\pi$
$163$	−7.31629	−0.573056	−0.286528	+	0.958072i	$0.592501\pi$
$164$	0	0
$165$	−0.284540	−0.0221514
$166$	0	0
$167$	13.3675	1.03441	0.517203	−	0.855863i	$-0.326973\pi$
$167$	13.3675	1.03441	0.517203	+	0.855863i	$0.326973\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−18.3819	−1.40570
$172$	0	0
$173$	−15.0657	−1.14542	−0.572711	−	0.819758i	$-0.694108\pi$
$173$	−15.0657	−1.14542	−0.572711	+	0.819758i	$0.694108\pi$
$174$	0	0
$175$	3.71818	0.281068
$176$	0	0
$177$	−0.0837256	−0.00629320
$178$	0	0
$179$	16.1015	1.20348	0.601741	−	0.798691i	$-0.294474\pi$
$179$	16.1015	1.20348	0.601741	+	0.798691i	$0.294474\pi$
$180$	0	0
$181$	19.0534	1.41623	0.708116	−	0.706096i	$-0.249546\pi$
$181$	19.0534	1.41623	0.708116	+	0.706096i	$0.249546\pi$
$182$	0	0
$183$	−0.386940	−0.0286034
$184$	0	0
$185$	7.32662	0.538664
$186$	0	0
$187$	−15.5416	−1.13651
$188$	0	0
$189$	1.21541	0.0884079
$190$	0	0
$191$	4.29896	0.311062	0.155531	−	0.987831i	$-0.450291\pi$
$191$	4.29896	0.311062	0.155531	+	0.987831i	$0.450291\pi$
$192$	0	0
$193$	23.4460	1.68768	0.843839	−	0.536596i	$-0.180290\pi$
$193$	23.4460	1.68768	0.843839	+	0.536596i	$0.180290\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0.687466	0.0489799	0.0244899	−	0.999700i	$-0.492204\pi$
$197$	0.687466	0.0489799	0.0244899	+	0.999700i	$0.492204\pi$
$198$	0	0
$199$	1.75780	0.124607	0.0623036	−	0.998057i	$-0.480155\pi$
$199$	1.75780	0.124607	0.0623036	+	0.998057i	$0.480155\pi$
$200$	0	0
$201$	−0.800869	−0.0564889
$202$	0	0
$203$	−7.56170	−0.530727
$204$	0	0
$205$	−12.1830	−0.850895
$206$	0	0
$207$	−10.8592	−0.754768
$208$	0	0
$209$	32.0174	2.21469
$210$	0	0
$211$	4.24474	0.292220	0.146110	−	0.989268i	$-0.453325\pi$
$211$	4.24474	0.292220	0.146110	+	0.989268i	$0.453325\pi$
$212$	0	0
$213$	−0.730245	−0.0500356
$214$	0	0
$215$	8.18427	0.558162
$216$	0	0
$217$	0.372705	0.0253008
$218$	0	0
$219$	0.629371	0.0425290
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−14.8505	−0.994462	−0.497231	−	0.867618i	$-0.665650\pi$
$223$	−14.8505	−0.994462	−0.497231	+	0.867618i	$0.665650\pi$
$224$	0	0
$225$	−2.99703	−0.199802
$226$	0	0
$227$	21.4849	1.42600	0.713002	−	0.701162i	$-0.247335\pi$
$227$	21.4849	1.42600	0.713002	+	0.701162i	$0.247335\pi$
$228$	0	0
$229$	−18.1462	−1.19913	−0.599566	−	0.800325i	$-0.704660\pi$
$229$	−18.1462	−1.19913	−0.599566	+	0.800325i	$0.704660\pi$
$230$	0	0
$231$	−1.05797	−0.0696092
$232$	0	0
$233$	12.3377	0.808271	0.404136	−	0.914699i	$-0.367572\pi$
$233$	12.3377	0.808271	0.404136	+	0.914699i	$0.367572\pi$
$234$	0	0
$235$	−8.41641	−0.549026
$236$	0	0
$237$	0.546896	0.0355247
$238$	0	0
$239$	5.05214	0.326795	0.163398	−	0.986560i	$-0.447755\pi$
$239$	5.05214	0.326795	0.163398	+	0.986560i	$0.447755\pi$
$240$	0	0
$241$	14.4408	0.930215	0.465108	−	0.885254i	$-0.346016\pi$
$241$	14.4408	0.930215	0.465108	+	0.885254i	$0.346016\pi$
$242$	0	0
$243$	−1.46976	−0.0942851
$244$	0	0
$245$	6.82483	0.436023
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−0.229751	−0.0145599
$250$	0	0
$251$	−4.44614	−0.280638	−0.140319	−	0.990106i	$-0.544813\pi$
$251$	−4.44614	−0.280638	−0.140319	+	0.990106i	$0.544813\pi$
$252$	0	0
$253$	18.9145	1.18914
$254$	0	0
$255$	0.162280	0.0101624
$256$	0	0
$257$	1.38523	0.0864083	0.0432042	−	0.999066i	$-0.486243\pi$
$257$	1.38523	0.0864083	0.0432042	+	0.999066i	$0.486243\pi$
$258$	0	0
$259$	27.2417	1.69271
$260$	0	0
$261$	6.09509	0.377277
$262$	0	0
$263$	−27.6676	−1.70606	−0.853028	−	0.521866i	$-0.825236\pi$
$263$	−27.6676	−1.70606	−0.853028	+	0.521866i	$0.825236\pi$
$264$	0	0
$265$	−6.11335	−0.375540
$266$	0	0
$267$	0.428644	0.0262326
$268$	0	0
$269$	19.4679	1.18698	0.593491	−	0.804841i	$-0.297749\pi$
$269$	19.4679	1.18698	0.593491	+	0.804841i	$0.297749\pi$
$270$	0	0
$271$	−5.45218	−0.331196	−0.165598	−	0.986193i	$-0.552955\pi$
$271$	−5.45218	−0.331196	−0.165598	+	0.986193i	$0.552955\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	5.22019	0.314789
$276$	0	0
$277$	26.9111	1.61693	0.808465	−	0.588545i	$-0.200299\pi$
$277$	26.9111	1.61693	0.808465	+	0.588545i	$0.200299\pi$
$278$	0	0
$279$	−0.300418	−0.0179856
$280$	0	0
$281$	−18.2499	−1.08870	−0.544348	−	0.838860i	$-0.683223\pi$
$281$	−18.2499	−1.08870	−0.544348	+	0.838860i	$0.683223\pi$
$282$	0	0
$283$	0.916102	0.0544566	0.0272283	−	0.999629i	$-0.491332\pi$
$283$	0.916102	0.0544566	0.0272283	+	0.999629i	$0.491332\pi$
$284$	0	0
$285$	−0.334315	−0.0198031
$286$	0	0
$287$	−45.2984	−2.67388
$288$	0	0
$289$	−8.13623	−0.478602
$290$	0	0
$291$	0.687074	0.0402770
$292$	0	0
$293$	8.96873	0.523959	0.261979	−	0.965073i	$-0.415625\pi$
$293$	8.96873	0.523959	0.261979	+	0.965073i	$0.415625\pi$
$294$	0	0
$295$	1.53604	0.0894316
$296$	0	0
$297$	1.70639	0.0990149
$298$	0	0
$299$	0	0
$300$	0	0
$301$	30.4305	1.75399
$302$	0	0
$303$	−0.210249	−0.0120785
$304$	0	0
$305$	7.09884	0.406479
$306$	0	0
$307$	31.1840	1.77976	0.889882	−	0.456192i	$-0.150787\pi$
$307$	31.1840	1.77976	0.889882	+	0.456192i	$0.150787\pi$
$308$	0	0
$309$	0.0594500	0.00338199
$310$	0	0
$311$	−10.1816	−0.577343	−0.288672	−	0.957428i	$-0.593214\pi$
$311$	−10.1816	−0.577343	−0.288672	+	0.957428i	$0.593214\pi$
$312$	0	0
$313$	30.3714	1.71670	0.858348	−	0.513068i	$-0.171491\pi$
$313$	30.3714	1.71670	0.858348	+	0.513068i	$0.171491\pi$
$314$	0	0
$315$	−11.1435	−0.627864
$316$	0	0
$317$	−8.20709	−0.460956	−0.230478	−	0.973078i	$-0.574029\pi$
$317$	−8.20709	−0.460956	−0.230478	+	0.973078i	$0.574029\pi$
$318$	0	0
$319$	−10.6164	−0.594403
$320$	0	0
$321$	−0.302530	−0.0168856
$322$	0	0
$323$	−18.2603	−1.01603
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0.00197216	0.000109061 0
$328$	0	0
$329$	−31.2937	−1.72528
$330$	0	0
$331$	0.911191	0.0500836	0.0250418	−	0.999686i	$-0.492028\pi$
$331$	0.911191	0.0500836	0.0250418	+	0.999686i	$0.492028\pi$
$332$	0	0
$333$	−21.9581	−1.20330
$334$	0	0
$335$	14.6928	0.802755
$336$	0	0
$337$	3.54525	0.193122	0.0965611	−	0.995327i	$-0.469216\pi$
$337$	3.54525	0.193122	0.0965611	+	0.995327i	$0.469216\pi$
$338$	0	0
$339$	0.730093	0.0396532
$340$	0	0
$341$	0.523265	0.0283364
$342$	0	0
$343$	−0.651306	−0.0351672
$344$	0	0
$345$	−0.197499	−0.0106330
$346$	0	0
$347$	1.58496	0.0850849	0.0425424	−	0.999095i	$-0.486454\pi$
$347$	1.58496	0.0850849	0.0425424	+	0.999095i	$0.486454\pi$
$348$	0	0
$349$	−5.47813	−0.293238	−0.146619	−	0.989193i	$-0.546839\pi$
$349$	−5.47813	−0.293238	−0.146619	+	0.989193i	$0.546839\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−4.38823	−0.233562	−0.116781	−	0.993158i	$-0.537258\pi$
$353$	−4.38823	−0.233562	−0.116781	+	0.993158i	$0.537258\pi$
$354$	0	0
$355$	13.3972	0.711047
$356$	0	0
$357$	0.603386	0.0319346
$358$	0	0
$359$	−14.2203	−0.750521	−0.375261	−	0.926919i	$-0.622447\pi$
$359$	−14.2203	−0.750521	−0.375261	+	0.926919i	$0.622447\pi$
$360$	0	0
$361$	18.6182	0.979906
$362$	0	0
$363$	−0.885769	−0.0464908
$364$	0	0
$365$	−11.5465	−0.604372
$366$	0	0
$367$	9.30803	0.485875	0.242938	−	0.970042i	$-0.421889\pi$
$367$	9.30803	0.485875	0.242938	+	0.970042i	$0.421889\pi$
$368$	0	0
$369$	36.5127	1.90077
$370$	0	0
$371$	−22.7305	−1.18011
$372$	0	0
$373$	30.8716	1.59847	0.799236	−	0.601018i	$-0.205238\pi$
$373$	30.8716	1.59847	0.799236	+	0.601018i	$0.205238\pi$
$374$	0	0
$375$	−0.0545075	−0.00281475
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−28.0188	−1.43923	−0.719616	−	0.694373i	$-0.755682\pi$
$379$	−28.0188	−1.43923	−0.719616	+	0.694373i	$0.755682\pi$
$380$	0	0
$381$	−0.615802	−0.0315485
$382$	0	0
$383$	−23.9075	−1.22162	−0.610809	−	0.791778i	$-0.709156\pi$
$383$	−23.9075	−1.22162	−0.610809	+	0.791778i	$0.709156\pi$
$384$	0	0
$385$	19.4096	0.989205
$386$	0	0
$387$	−24.5285	−1.24685
$388$	0	0
$389$	−34.2391	−1.73599	−0.867996	−	0.496571i	$-0.834592\pi$
$389$	−34.2391	−1.73599	−0.867996	+	0.496571i	$0.834592\pi$
$390$	0	0
$391$	−10.7874	−0.545542
$392$	0	0
$393$	−0.0568443	−0.00286741
$394$	0	0
$395$	−10.0334	−0.504836
$396$	0	0
$397$	−11.8187	−0.593163	−0.296581	−	0.955008i	$-0.595847\pi$
$397$	−11.8187	−0.593163	−0.296581	+	0.955008i	$0.595847\pi$
$398$	0	0
$399$	−1.24304	−0.0622299
$400$	0	0
$401$	−8.98456	−0.448667	−0.224334	−	0.974512i	$-0.572021\pi$
$401$	−8.98456	−0.448667	−0.224334	+	0.974512i	$0.572021\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	8.97327	0.445885
$406$	0	0
$407$	38.2464	1.89580
$408$	0	0
$409$	−27.9072	−1.37992	−0.689962	−	0.723846i	$-0.742373\pi$
$409$	−27.9072	−1.37992	−0.689962	+	0.723846i	$0.742373\pi$
$410$	0	0
$411$	−0.972033	−0.0479469
$412$	0	0
$413$	5.71126	0.281033
$414$	0	0
$415$	4.21504	0.206908
$416$	0	0
$417$	0.957489	0.0468885
$418$	0	0
$419$	25.6462	1.25290	0.626449	−	0.779462i	$-0.284508\pi$
$419$	25.6462	1.25290	0.626449	+	0.779462i	$0.284508\pi$
$420$	0	0
$421$	−17.4613	−0.851010	−0.425505	−	0.904956i	$-0.639904\pi$
$421$	−17.4613	−0.851010	−0.425505	+	0.904956i	$0.639904\pi$
$422$	0	0
$423$	25.2242	1.22644
$424$	0	0
$425$	−2.97721	−0.144416
$426$	0	0
$427$	26.3947	1.27733
$428$	0	0
$429$	0	0
$430$	0	0
$431$	12.5131	0.602735	0.301368	−	0.953508i	$-0.402557\pi$
$431$	12.5131	0.602735	0.301368	+	0.953508i	$0.402557\pi$
$432$	0	0
$433$	5.43503	0.261191	0.130595	−	0.991436i	$-0.458311\pi$
$433$	5.43503	0.261191	0.130595	+	0.991436i	$0.458311\pi$
$434$	0	0
$435$	0.110853	0.00531497
$436$	0	0
$437$	22.2232	1.06308
$438$	0	0
$439$	6.52958	0.311640	0.155820	−	0.987785i	$-0.450198\pi$
$439$	6.52958	0.311640	0.155820	+	0.987785i	$0.450198\pi$
$440$	0	0
$441$	−20.4542	−0.974010
$442$	0	0
$443$	11.9530	0.567905	0.283952	−	0.958838i	$-0.408354\pi$
$443$	11.9530	0.567905	0.283952	+	0.958838i	$0.408354\pi$
$444$	0	0
$445$	−7.86394	−0.372787
$446$	0	0
$447$	−0.138775	−0.00656385
$448$	0	0
$449$	−22.0981	−1.04288	−0.521438	−	0.853289i	$-0.674604\pi$
$449$	−22.0981	−1.04288	−0.521438	+	0.853289i	$0.674604\pi$
$450$	0	0
$451$	−63.5974	−2.99469
$452$	0	0
$453$	−0.0116217	−0.000546033 0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	24.0669	1.12580	0.562902	−	0.826524i	$-0.309685\pi$
$457$	24.0669	1.12580	0.562902	+	0.826524i	$0.309685\pi$
$458$	0	0
$459$	−0.973199	−0.0454250
$460$	0	0
$461$	−41.8691	−1.95004	−0.975020	−	0.222117i	$-0.928703\pi$
$461$	−41.8691	−1.95004	−0.975020	+	0.222117i	$0.928703\pi$
$462$	0	0
$463$	36.9718	1.71823	0.859113	−	0.511786i	$-0.171016\pi$
$463$	36.9718	1.71823	0.859113	+	0.511786i	$0.171016\pi$
$464$	0	0
$465$	−0.00546375	−0.000253376 0
$466$	0	0
$467$	−24.1048	−1.11544	−0.557718	−	0.830030i	$-0.688323\pi$
$467$	−24.1048	−1.11544	−0.557718	+	0.830030i	$0.688323\pi$
$468$	0	0
$469$	54.6305	2.52260
$470$	0	0
$471$	1.27608	0.0587987
$472$	0	0
$473$	42.7235	1.96443
$474$	0	0
$475$	6.13337	0.281418
$476$	0	0
$477$	18.3219	0.838902
$478$	0	0
$479$	−43.0487	−1.96694	−0.983472	−	0.181062i	$-0.942046\pi$
$479$	−43.0487	−1.96694	−0.983472	+	0.181062i	$0.942046\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−0.734334	−0.0334134
$484$	0	0
$485$	−12.6051	−0.572370
$486$	0	0
$487$	7.62896	0.345701	0.172851	−	0.984948i	$-0.444702\pi$
$487$	7.62896	0.345701	0.172851	+	0.984948i	$0.444702\pi$
$488$	0	0
$489$	0.398793	0.0180340
$490$	0	0
$491$	−20.6025	−0.929780	−0.464890	−	0.885369i	$-0.653906\pi$
$491$	−20.6025	−0.929780	−0.464890	+	0.885369i	$0.653906\pi$
$492$	0	0
$493$	6.05478	0.272694
$494$	0	0
$495$	−15.6451	−0.703194
$496$	0	0
$497$	49.8130	2.23442
$498$	0	0
$499$	8.06951	0.361241	0.180620	−	0.983553i	$-0.442189\pi$
$499$	8.06951	0.361241	0.180620	+	0.983553i	$0.442189\pi$
$500$	0	0
$501$	−0.728627	−0.0325527
$502$	0	0
$503$	−19.6068	−0.874225	−0.437112	−	0.899407i	$-0.643999\pi$
$503$	−19.6068	−0.874225	−0.437112	+	0.899407i	$0.643999\pi$
$504$	0	0
$505$	3.85725	0.171645
$506$	0	0
$507$	0	0
$508$	0	0
$509$	12.0479	0.534015	0.267008	−	0.963694i	$-0.413965\pi$
$509$	12.0479	0.534015	0.267008	+	0.963694i	$0.413965\pi$
$510$	0	0
$511$	−42.9320	−1.89920
$512$	0	0
$513$	2.00489	0.0885182
$514$	0	0
$515$	−1.09068	−0.0480609
$516$	0	0
$517$	−43.9353	−1.93227
$518$	0	0
$519$	0.821191	0.0360463
$520$	0	0
$521$	−13.8557	−0.607028	−0.303514	−	0.952827i	$-0.598160\pi$
$521$	−13.8557	−0.607028	−0.303514	+	0.952827i	$0.598160\pi$
$522$	0	0
$523$	−26.7263	−1.16866	−0.584329	−	0.811517i	$-0.698642\pi$
$523$	−26.7263	−1.16866	−0.584329	+	0.811517i	$0.698642\pi$
$524$	0	0
$525$	−0.202668	−0.00884517
$526$	0	0
$527$	−0.298431	−0.0129999
$528$	0	0
$529$	−9.87149	−0.429195
$530$	0	0
$531$	−4.60355	−0.199777
$532$	0	0
$533$	0	0
$534$	0	0
$535$	5.55026	0.239958
$536$	0	0
$537$	−0.877652	−0.0378735
$538$	0	0
$539$	35.6269	1.53456
$540$	0	0
$541$	−25.7586	−1.10745	−0.553725	−	0.832700i	$-0.686794\pi$
$541$	−25.7586	−1.10745	−0.553725	+	0.832700i	$0.686794\pi$
$542$	0	0
$543$	−1.03856	−0.0445687
$544$	0	0
$545$	−0.0361815	−0.00154985
$546$	0	0
$547$	−14.2898	−0.610987	−0.305493	−	0.952194i	$-0.598821\pi$
$547$	−14.2898	−0.610987	−0.305493	+	0.952194i	$0.598821\pi$
$548$	0	0
$549$	−21.2754	−0.908014
$550$	0	0
$551$	−12.4735	−0.531389
$552$	0	0
$553$	−37.3060	−1.58641
$554$	0	0
$555$	−0.399355	−0.0169517
$556$	0	0
$557$	18.9439	0.802679	0.401340	−	0.915929i	$-0.368545\pi$
$557$	18.9439	0.802679	0.401340	+	0.915929i	$0.368545\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0.847134	0.0357660
$562$	0	0
$563$	−5.58658	−0.235446	−0.117723	−	0.993046i	$-0.537560\pi$
$563$	−5.58658	−0.235446	−0.117723	+	0.993046i	$0.537560\pi$
$564$	0	0
$565$	−13.3944	−0.563505
$566$	0	0
$567$	33.3642	1.40116
$568$	0	0
$569$	−27.2514	−1.14244	−0.571219	−	0.820798i	$-0.693529\pi$
$569$	−27.2514	−1.14244	−0.571219	+	0.820798i	$0.693529\pi$
$570$	0	0
$571$	−16.8253	−0.704118	−0.352059	−	0.935978i	$-0.614518\pi$
$571$	−16.8253	−0.704118	−0.352059	+	0.935978i	$0.614518\pi$
$572$	0	0
$573$	−0.234325	−0.00978909
$574$	0	0
$575$	3.62333	0.151103
$576$	0	0
$577$	−30.8911	−1.28601	−0.643007	−	0.765861i	$-0.722313\pi$
$577$	−30.8911	−1.28601	−0.643007	+	0.765861i	$0.722313\pi$
$578$	0	0
$579$	−1.27798	−0.0531111
$580$	0	0
$581$	15.6723	0.650195
$582$	0	0
$583$	−31.9129	−1.32170
$584$	0	0
$585$	0	0
$586$	0	0
$587$	5.67152	0.234089	0.117044	−	0.993127i	$-0.462658\pi$
$587$	5.67152	0.234089	0.117044	+	0.993127i	$0.462658\pi$
$588$	0	0
$589$	0.614800	0.0253324
$590$	0	0
$591$	−0.0374720	−0.00154139
$592$	0	0
$593$	6.11872	0.251266	0.125633	−	0.992077i	$-0.459904\pi$
$593$	6.11872	0.251266	0.125633	+	0.992077i	$0.459904\pi$
$594$	0	0
$595$	−11.0698	−0.453817
$596$	0	0
$597$	−0.0958133	−0.00392138
$598$	0	0
$599$	39.6757	1.62111	0.810553	−	0.585665i	$-0.199166\pi$
$599$	39.6757	1.62111	0.810553	+	0.585665i	$0.199166\pi$
$600$	0	0
$601$	−44.2642	−1.80558	−0.902788	−	0.430086i	$-0.858483\pi$
$601$	−44.2642	−1.80558	−0.902788	+	0.430086i	$0.858483\pi$
$602$	0	0
$603$	−44.0348	−1.79324
$604$	0	0
$605$	16.2504	0.660673
$606$	0	0
$607$	21.5056	0.872884	0.436442	−	0.899732i	$-0.356238\pi$
$607$	21.5056	0.872884	0.436442	+	0.899732i	$0.356238\pi$
$608$	0	0
$609$	0.412169	0.0167019
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−14.3306	−0.578809	−0.289404	−	0.957207i	$-0.593457\pi$
$613$	−14.3306	−0.578809	−0.289404	+	0.957207i	$0.593457\pi$
$614$	0	0
$615$	0.664063	0.0267776
$616$	0	0
$617$	−31.0397	−1.24961	−0.624806	−	0.780780i	$-0.714822\pi$
$617$	−31.0397	−1.24961	−0.624806	+	0.780780i	$0.714822\pi$
$618$	0	0
$619$	0.892227	0.0358616	0.0179308	−	0.999839i	$-0.494292\pi$
$619$	0.892227	0.0358616	0.0179308	+	0.999839i	$0.494292\pi$
$620$	0	0
$621$	1.18440	0.0475285
$622$	0	0
$623$	−29.2395	−1.17146
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−1.74519	−0.0696960
$628$	0	0
$629$	−21.8129	−0.869736
$630$	0	0
$631$	−17.5918	−0.700318	−0.350159	−	0.936690i	$-0.613872\pi$
$631$	−17.5918	−0.700318	−0.350159	+	0.936690i	$0.613872\pi$
$632$	0	0
$633$	−0.231370	−0.00919613
$634$	0	0
$635$	11.2976	0.448330
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−40.1517	−1.58838
$640$	0	0
$641$	−18.3751	−0.725772	−0.362886	−	0.931834i	$-0.618209\pi$
$641$	−18.3751	−0.725772	−0.362886	+	0.931834i	$0.618209\pi$
$642$	0	0
$643$	27.4520	1.08260	0.541300	−	0.840830i	$-0.317932\pi$
$643$	27.4520	1.08260	0.541300	+	0.840830i	$0.317932\pi$
$644$	0	0
$645$	−0.446104	−0.0175653
$646$	0	0
$647$	−39.3570	−1.54728	−0.773642	−	0.633623i	$-0.781567\pi$
$647$	−39.3570	−1.54728	−0.773642	+	0.633623i	$0.781567\pi$
$648$	0	0
$649$	8.01842	0.314750
$650$	0	0
$651$	−0.0203152	−0.000796215 0
$652$	0	0
$653$	−11.8592	−0.464088	−0.232044	−	0.972705i	$-0.574541\pi$
$653$	−11.8592	−0.464088	−0.232044	+	0.972705i	$0.574541\pi$
$654$	0	0
$655$	1.04287	0.0407483
$656$	0	0
$657$	34.6052	1.35008
$658$	0	0
$659$	−23.3963	−0.911392	−0.455696	−	0.890136i	$-0.650610\pi$
$659$	−23.3963	−0.911392	−0.455696	+	0.890136i	$0.650610\pi$
$660$	0	0
$661$	9.17424	0.356837	0.178418	−	0.983955i	$-0.442902\pi$
$661$	9.17424	0.356837	0.178418	+	0.983955i	$0.442902\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	22.8049	0.884338
$666$	0	0
$667$	−7.36881	−0.285321
$668$	0	0
$669$	0.809463	0.0312956
$670$	0	0
$671$	37.0573	1.43058
$672$	0	0
$673$	9.18712	0.354138	0.177069	−	0.984198i	$-0.443338\pi$
$673$	9.18712	0.354138	0.177069	+	0.984198i	$0.443338\pi$
$674$	0	0
$675$	0.326883	0.0125817
$676$	0	0
$677$	−20.7895	−0.799007	−0.399503	−	0.916732i	$-0.630817\pi$
$677$	−20.7895	−0.799007	−0.399503	+	0.916732i	$0.630817\pi$
$678$	0	0
$679$	−46.8681	−1.79863
$680$	0	0
$681$	−1.17109	−0.0448762
$682$	0	0
$683$	−16.0263	−0.613229	−0.306614	−	0.951834i	$-0.599196\pi$
$683$	−16.0263	−0.613229	−0.306614	+	0.951834i	$0.599196\pi$
$684$	0	0
$685$	17.8330	0.681365
$686$	0	0
$687$	0.989101	0.0377366
$688$	0	0
$689$	0	0
$690$	0	0
$691$	44.9971	1.71177	0.855885	−	0.517167i	$-0.173013\pi$
$691$	44.9971	1.71177	0.855885	+	0.517167i	$0.173013\pi$
$692$	0	0
$693$	−58.1711	−2.20974
$694$	0	0
$695$	−17.5662	−0.666324
$696$	0	0
$697$	36.2712	1.37387
$698$	0	0
$699$	−0.672498	−0.0254362
$700$	0	0
$701$	32.9469	1.24439	0.622193	−	0.782864i	$-0.286242\pi$
$701$	32.9469	1.24439	0.622193	+	0.782864i	$0.286242\pi$
$702$	0	0
$703$	44.9369	1.69483
$704$	0	0
$705$	0.458757	0.0172778
$706$	0	0
$707$	14.3419	0.539383
$708$	0	0
$709$	−45.9596	−1.72605	−0.863025	−	0.505162i	$-0.831433\pi$
$709$	−45.9596	−1.72605	−0.863025	+	0.505162i	$0.831433\pi$
$710$	0	0
$711$	30.0704	1.12773
$712$	0	0
$713$	0.363197	0.0136018
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−0.275379	−0.0102842
$718$	0	0
$719$	−5.25214	−0.195872	−0.0979358	−	0.995193i	$-0.531224\pi$
$719$	−5.25214	−0.195872	−0.0979358	+	0.995193i	$0.531224\pi$
$720$	0	0
$721$	−4.05532	−0.151028
$722$	0	0
$723$	−0.787133	−0.0292738
$724$	0	0
$725$	−2.03371	−0.0755302
$726$	0	0
$727$	28.5596	1.05922	0.529608	−	0.848243i	$-0.322339\pi$
$727$	28.5596	1.05922	0.529608	+	0.848243i	$0.322339\pi$
$728$	0	0
$729$	−26.8397	−0.994063
$730$	0	0
$731$	−24.3663	−0.901219
$732$	0	0
$733$	9.25426	0.341814	0.170907	−	0.985287i	$-0.445330\pi$
$733$	9.25426	0.341814	0.170907	+	0.985287i	$0.445330\pi$
$734$	0	0
$735$	−0.372004	−0.0137216
$736$	0	0
$737$	76.6994	2.82526
$738$	0	0
$739$	−32.7910	−1.20623	−0.603117	−	0.797652i	$-0.706075\pi$
$739$	−32.7910	−1.20623	−0.603117	+	0.797652i	$0.706075\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−17.7645	−0.651716	−0.325858	−	0.945419i	$-0.605653\pi$
$743$	−17.7645	−0.651716	−0.325858	+	0.945419i	$0.605653\pi$
$744$	0	0
$745$	2.54599	0.0932777
$746$	0	0
$747$	−12.6326	−0.462202
$748$	0	0
$749$	20.6368	0.754053
$750$	0	0
$751$	38.7659	1.41459	0.707293	−	0.706920i	$-0.249916\pi$
$751$	38.7659	1.41459	0.707293	+	0.706920i	$0.249916\pi$
$752$	0	0
$753$	0.242348	0.00883165
$754$	0	0
$755$	0.213212	0.00775959
$756$	0	0
$757$	−18.0592	−0.656374	−0.328187	−	0.944613i	$-0.606438\pi$
$757$	−18.0592	−0.656374	−0.328187	+	0.944613i	$0.606438\pi$
$758$	0	0
$759$	−1.03098	−0.0374222
$760$	0	0
$761$	20.6089	0.747071	0.373535	−	0.927616i	$-0.378145\pi$
$761$	20.6089	0.747071	0.373535	+	0.927616i	$0.378145\pi$
$762$	0	0
$763$	−0.134529	−0.00487028
$764$	0	0
$765$	8.92278	0.322604
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−34.1017	−1.22974	−0.614869	−	0.788629i	$-0.710791\pi$
$769$	−34.1017	−1.22974	−0.614869	+	0.788629i	$0.710791\pi$
$770$	0	0
$771$	−0.0755054	−0.00271926
$772$	0	0
$773$	−16.6649	−0.599396	−0.299698	−	0.954034i	$-0.596886\pi$
$773$	−16.6649	−0.599396	−0.299698	+	0.954034i	$0.596886\pi$
$774$	0	0
$775$	0.100239	0.00360068
$776$	0	0
$777$	−1.48487	−0.0532696
$778$	0	0
$779$	−74.7226	−2.67722
$780$	0	0
$781$	69.9357	2.50250
$782$	0	0
$783$	−0.664786	−0.0237575
$784$	0	0
$785$	−23.4111	−0.835579
$786$	0	0
$787$	1.29371	0.0461157	0.0230579	−	0.999734i	$-0.492660\pi$
$787$	1.29371	0.0461157	0.0230579	+	0.999734i	$0.492660\pi$
$788$	0	0
$789$	1.50809	0.0536894
$790$	0	0
$791$	−49.8026	−1.77078
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0.333223	0.0118182
$796$	0	0
$797$	7.56095	0.267823	0.133911	−	0.990993i	$-0.457246\pi$
$797$	7.56095	0.267823	0.133911	+	0.990993i	$0.457246\pi$
$798$	0	0
$799$	25.0574	0.886467
$800$	0	0
$801$	23.5685	0.832751
$802$	0	0
$803$	−60.2750	−2.12706
$804$	0	0
$805$	13.4722	0.474832
$806$	0	0
$807$	−1.06115	−0.0373542
$808$	0	0
$809$	−34.4654	−1.21174	−0.605869	−	0.795564i	$-0.707174\pi$
$809$	−34.4654	−1.21174	−0.605869	+	0.795564i	$0.707174\pi$
$810$	0	0
$811$	7.78668	0.273427	0.136714	−	0.990611i	$-0.456346\pi$
$811$	7.78668	0.273427	0.136714	+	0.990611i	$0.456346\pi$
$812$	0	0
$813$	0.297185	0.0104227
$814$	0	0
$815$	−7.31629	−0.256279
$816$	0	0
$817$	50.1971	1.75618
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−36.5174	−1.27447	−0.637234	−	0.770671i	$-0.719921\pi$
$821$	−36.5174	−1.27447	−0.637234	+	0.770671i	$0.719921\pi$
$822$	0	0
$823$	46.9851	1.63780	0.818898	−	0.573939i	$-0.194585\pi$
$823$	46.9851	1.63780	0.818898	+	0.573939i	$0.194585\pi$
$824$	0	0
$825$	−0.284540	−0.00990640
$826$	0	0
$827$	36.9816	1.28598	0.642989	−	0.765876i	$-0.277694\pi$
$827$	36.9816	1.28598	0.642989	+	0.765876i	$0.277694\pi$
$828$	0	0
$829$	12.0453	0.418351	0.209175	−	0.977878i	$-0.432922\pi$
$829$	12.0453	0.418351	0.209175	+	0.977878i	$0.432922\pi$
$830$	0	0
$831$	−1.46685	−0.0508846
$832$	0	0
$833$	−20.3189	−0.704010
$834$	0	0
$835$	13.3675	0.462600
$836$	0	0
$837$	0.0327663	0.00113257
$838$	0	0
$839$	27.6041	0.953000	0.476500	−	0.879175i	$-0.341905\pi$
$839$	27.6041	0.953000	0.476500	+	0.879175i	$0.341905\pi$
$840$	0	0
$841$	−24.8640	−0.857380
$842$	0	0
$843$	0.994754	0.0342612
$844$	0	0
$845$	0	0
$846$	0	0
$847$	60.4219	2.07612
$848$	0	0
$849$	−0.0499344	−0.00171375
$850$	0	0
$851$	26.5467	0.910011
$852$	0	0
$853$	−3.78477	−0.129588	−0.0647940	−	0.997899i	$-0.520639\pi$
$853$	−3.78477	−0.129588	−0.0647940	+	0.997899i	$0.520639\pi$
$854$	0	0
$855$	−18.3819	−0.628647
$856$	0	0
$857$	20.4897	0.699915	0.349958	−	0.936765i	$-0.386196\pi$
$857$	20.4897	0.699915	0.349958	+	0.936765i	$0.386196\pi$
$858$	0	0
$859$	25.6031	0.873568	0.436784	−	0.899567i	$-0.356117\pi$
$859$	25.6031	0.873568	0.436784	+	0.899567i	$0.356117\pi$
$860$	0	0
$861$	2.46910	0.0841468
$862$	0	0
$863$	20.7175	0.705232	0.352616	−	0.935768i	$-0.385292\pi$
$863$	20.7175	0.705232	0.352616	+	0.935768i	$0.385292\pi$
$864$	0	0
$865$	−15.0657	−0.512248
$866$	0	0
$867$	0.443486	0.0150616
$868$	0	0
$869$	−52.3764	−1.77675
$870$	0	0
$871$	0	0
$872$	0	0
$873$	37.7779	1.27859
$874$	0	0
$875$	3.71818	0.125697
$876$	0	0
$877$	−43.2388	−1.46007	−0.730036	−	0.683409i	$-0.760497\pi$
$877$	−43.2388	−1.46007	−0.730036	+	0.683409i	$0.760497\pi$
$878$	0	0
$879$	−0.488863	−0.0164889
$880$	0	0
$881$	−56.7984	−1.91359	−0.956793	−	0.290768i	$-0.906089\pi$
$881$	−56.7984	−1.91359	−0.956793	+	0.290768i	$0.906089\pi$
$882$	0	0
$883$	−50.6651	−1.70502	−0.852508	−	0.522715i	$-0.824919\pi$
$883$	−50.6651	−1.70502	−0.852508	+	0.522715i	$0.824919\pi$
$884$	0	0
$885$	−0.0837256	−0.00281440
$886$	0	0
$887$	−19.8733	−0.667279	−0.333640	−	0.942701i	$-0.608277\pi$
$887$	−19.8733	−0.667279	−0.333640	+	0.942701i	$0.608277\pi$
$888$	0	0
$889$	42.0063	1.40885
$890$	0	0
$891$	46.8422	1.56927
$892$	0	0
$893$	−51.6209	−1.72743
$894$	0	0
$895$	16.1015	0.538214
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−0.203856	−0.00679899
$900$	0	0
$901$	18.2007	0.606354
$902$	0	0
$903$	−1.65869	−0.0551978
$904$	0	0
$905$	19.0534	0.633358
$906$	0	0
$907$	24.3321	0.807935	0.403967	−	0.914773i	$-0.367631\pi$
$907$	24.3321	0.807935	0.403967	+	0.914773i	$0.367631\pi$
$908$	0	0
$909$	−11.5603	−0.383430
$910$	0	0
$911$	−31.5701	−1.04596	−0.522981	−	0.852344i	$-0.675180\pi$
$911$	−31.5701	−1.04596	−0.522981	+	0.852344i	$0.675180\pi$
$912$	0	0
$913$	22.0033	0.728203
$914$	0	0
$915$	−0.386940	−0.0127918
$916$	0	0
$917$	3.87758	0.128049
$918$	0	0
$919$	−27.9720	−0.922711	−0.461356	−	0.887215i	$-0.652637\pi$
$919$	−27.9720	−0.922711	−0.461356	+	0.887215i	$0.652637\pi$
$920$	0	0
$921$	−1.69976	−0.0560090
$922$	0	0
$923$	0	0
$924$	0	0
$925$	7.32662	0.240898
$926$	0	0
$927$	3.26879	0.107361
$928$	0	0
$929$	−30.0456	−0.985765	−0.492882	−	0.870096i	$-0.664057\pi$
$929$	−30.0456	−0.985765	−0.492882	+	0.870096i	$0.664057\pi$
$930$	0	0
$931$	41.8592	1.37188
$932$	0	0
$933$	0.554971	0.0181689
$934$	0	0
$935$	−15.5416	−0.508265
$936$	0	0
$937$	35.2378	1.15117	0.575584	−	0.817743i	$-0.304775\pi$
$937$	35.2378	1.15117	0.575584	+	0.817743i	$0.304775\pi$
$938$	0	0
$939$	−1.65547	−0.0540243
$940$	0	0
$941$	19.2578	0.627785	0.313892	−	0.949459i	$-0.398367\pi$
$941$	19.2578	0.627785	0.313892	+	0.949459i	$0.398367\pi$
$942$	0	0
$943$	−44.1429	−1.43749
$944$	0	0
$945$	1.21541	0.0395372
$946$	0	0
$947$	43.7557	1.42187	0.710935	−	0.703257i	$-0.248272\pi$
$947$	43.7557	1.42187	0.710935	+	0.703257i	$0.248272\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0.447348	0.0145063
$952$	0	0
$953$	−31.5328	−1.02145	−0.510724	−	0.859745i	$-0.670623\pi$
$953$	−31.5328	−1.02145	−0.510724	+	0.859745i	$0.670623\pi$
$954$	0	0
$955$	4.29896	0.139111
$956$	0	0
$957$	0.578672	0.0187058
$958$	0	0
$959$	66.3063	2.14114
$960$	0	0
$961$	−30.9900	−0.999676
$962$	0	0
$963$	−16.6343	−0.536032
$964$	0	0
$965$	23.4460	0.754753
$966$	0	0
$967$	−24.8301	−0.798483	−0.399242	−	0.916846i	$-0.630727\pi$
$967$	−24.8301	−0.798483	−0.399242	+	0.916846i	$0.630727\pi$
$968$	0	0
$969$	0.995324	0.0319744
$970$	0	0
$971$	31.9493	1.02530	0.512651	−	0.858597i	$-0.328664\pi$
$971$	31.9493	1.02530	0.512651	+	0.858597i	$0.328664\pi$
$972$	0	0
$973$	−65.3142	−2.09388
$974$	0	0
$975$	0	0
$976$	0	0
$977$	54.9763	1.75885	0.879424	−	0.476040i	$-0.157928\pi$
$977$	54.9763	1.75885	0.879424	+	0.476040i	$0.157928\pi$
$978$	0	0
$979$	−41.0513	−1.31201
$980$	0	0
$981$	0.108437	0.00346213
$982$	0	0
$983$	−20.4505	−0.652270	−0.326135	−	0.945323i	$-0.605746\pi$
$983$	−20.4505	−0.652270	−0.326135	+	0.945323i	$0.605746\pi$
$984$	0	0
$985$	0.687466	0.0219045
$986$	0	0
$987$	1.70574	0.0542943
$988$	0	0
$989$	29.6543	0.942952
$990$	0	0
$991$	−22.7815	−0.723680	−0.361840	−	0.932240i	$-0.617851\pi$
$991$	−22.7815	−0.723680	−0.361840	+	0.932240i	$0.617851\pi$
$992$	0	0
$993$	−0.0496667	−0.00157613
$994$	0	0
$995$	1.75780	0.0557260
$996$	0	0
$997$	−41.2518	−1.30646	−0.653229	−	0.757161i	$-0.726586\pi$
$997$	−41.2518	−1.30646	−0.653229	+	0.757161i	$0.726586\pi$
$998$	0	0
$999$	2.39495	0.0757727

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3380.2.a.s.1.6	yes	9
13.5	odd	4		3380.2.f.j.3041.11		18
13.8	odd	4		3380.2.f.j.3041.12		18
13.12	even	2		3380.2.a.r.1.6	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3380.2.a.r.1.6	✓	9	13.12	even	2
3380.2.a.s.1.6	yes	9	1.1	even	1	trivial
3380.2.f.j.3041.11		18	13.5	odd	4
3380.2.f.j.3041.12		18	13.8	odd	4

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:	\( N \)	\(=\)	\( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3380.a (trivial)

\(f(q)\)	\(=\)	\(q-0.0545075 q^{3} +1.00000 q^{5} +3.71818 q^{7} -2.99703 q^{9} +O(q^{10})\)	\(q-0.0545075 q^{3} +1.00000 q^{5} +3.71818 q^{7} -2.99703 q^{9} +5.22019 q^{11} -0.0545075 q^{15} -2.97721 q^{17} +6.13337 q^{19} -0.202668 q^{21} +3.62333 q^{23} +1.00000 q^{25} +0.326883 q^{27} -2.03371 q^{29} +0.100239 q^{31} -0.284540 q^{33} +3.71818 q^{35} +7.32662 q^{37} -12.1830 q^{41} +8.18427 q^{43} -2.99703 q^{45} -8.41641 q^{47} +6.82483 q^{49} +0.162280 q^{51} -6.11335 q^{53} +5.22019 q^{55} -0.334315 q^{57} +1.53604 q^{59} +7.09884 q^{61} -11.1435 q^{63} +14.6928 q^{67} -0.197499 q^{69} +13.3972 q^{71} -11.5465 q^{73} -0.0545075 q^{75} +19.4096 q^{77} -10.0334 q^{79} +8.97327 q^{81} +4.21504 q^{83} -2.97721 q^{85} +0.110853 q^{87} -7.86394 q^{89} -0.00546375 q^{93} +6.13337 q^{95} -12.6051 q^{97} -15.6451 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - q^{3} + 9 q^{5} + q^{7} + 12 q^{9}+O(q^{10}) \) 9 * q - q^3 + 9 * q^5 + q^7 + 12 * q^9	\( 9 q - q^{3} + 9 q^{5} + q^{7} + 12 q^{9} - 7 q^{11} - q^{15} + 13 q^{17} - 4 q^{19} - 3 q^{21} + 12 q^{23} + 9 q^{25} - 4 q^{27} + 16 q^{29} + 13 q^{31} + 34 q^{33} + q^{35} + q^{37} - 6 q^{41} + q^{43} + 12 q^{45} - 2 q^{47} + 20 q^{49} + 11 q^{51} + 30 q^{53} - 7 q^{55} + 38 q^{57} + 15 q^{59} + 21 q^{61} - 17 q^{63} - 7 q^{67} + 15 q^{69} - 7 q^{71} - 28 q^{73} - q^{75} + 46 q^{77} + 31 q^{79} + 41 q^{81} + 45 q^{83} + 13 q^{85} + 28 q^{87} - 41 q^{89} - 11 q^{93} - 4 q^{95} + 8 q^{97} - 81 q^{99}+O(q^{100}) \) 9 * q - q^3 + 9 * q^5 + q^7 + 12 * q^9 - 7 * q^11 - q^15 + 13 * q^17 - 4 * q^19 - 3 * q^21 + 12 * q^23 + 9 * q^25 - 4 * q^27 + 16 * q^29 + 13 * q^31 + 34 * q^33 + q^35 + q^37 - 6 * q^41 + q^43 + 12 * q^45 - 2 * q^47 + 20 * q^49 + 11 * q^51 + 30 * q^53 - 7 * q^55 + 38 * q^57 + 15 * q^59 + 21 * q^61 - 17 * q^63 - 7 * q^67 + 15 * q^69 - 7 * q^71 - 28 * q^73 - q^75 + 46 * q^77 + 31 * q^79 + 41 * q^81 + 45 * q^83 + 13 * q^85 + 28 * q^87 - 41 * q^89 - 11 * q^93 - 4 * q^95 + 8 * q^97 - 81 * q^99

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