LMFDB - Embedded newform 3380.2.a.r.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	$-0.748009\pi$
$7$	−3.71818	−1.40534	−0.702669	+	0.711517i	$0.748009\pi$
$8$	0	0
$9$	−2.99703	−0.999010
$10$	0	0
$11$	−5.22019	−1.57395	−0.786974	−	0.616986i	$-0.788353\pi$
$11$	−5.22019	−1.57395	−0.786974	+	0.616986i	$0.788353\pi$
$12$	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	$-0.748401\pi$
$19$	−6.13337	−1.40709	−0.703546	+	0.710650i	$0.748401\pi$
$20$	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	$-0.502865\pi$
$31$	−0.100239	−0.0180034	−0.00900169	+	0.999959i	$0.502865\pi$
$32$	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	$-0.705727\pi$
$37$	−7.32662	−1.20449	−0.602244	+	0.798312i	$0.705727\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	12.1830	1.90266	0.951329	−	0.308176i	$-0.0997185\pi$
$41$	12.1830	1.90266	0.951329	+	0.308176i	$0.0997185\pi$
$42$	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	$-0.289628\pi$
$47$	8.41641	1.22766	0.613829	+	0.789439i	$0.289628\pi$
$48$	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	$-0.531880\pi$
$59$	−1.53604	−0.199975	−0.0999876	+	0.994989i	$0.531880\pi$
$60$	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	$-0.854624\pi$
$67$	−14.6928	−1.79501	−0.897507	+	0.441000i	$0.854624\pi$
$68$	0	0
$69$	−0.197499	−0.0237760
$70$	0	0
$71$	−13.3972	−1.58995	−0.794975	−	0.606642i	$-0.792516\pi$
$71$	−13.3972	−1.58995	−0.794975	+	0.606642i	$0.792516\pi$
$72$	0	0
$73$	11.5465	1.35142	0.675708	−	0.737169i	$-0.263838\pi$
$73$	11.5465	1.35142	0.675708	+	0.737169i	$0.263838\pi$
$74$	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	$-0.574308\pi$
$83$	−4.21504	−0.462661	−0.231330	+	0.972875i	$0.574308\pi$
$84$	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	$-0.363156\pi$
$89$	7.86394	0.833576	0.416788	+	0.909004i	$0.363156\pi$
$90$	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	$-0.278964\pi$
$97$	12.6051	1.27986	0.639929	+	0.768434i	$0.278964\pi$
$98$	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	$-0.499448\pi$
$109$	0.0361815	0.00346556	0.00173278	+	0.999998i	$0.499448\pi$
$110$	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	$-0.775679\pi$
$137$	−17.8330	−1.52358	−0.761789	+	0.647825i	$0.775679\pi$
$138$	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	$-0.533256\pi$
$149$	−2.54599	−0.208575	−0.104288	+	0.994547i	$0.533256\pi$
$150$	0	0
$151$	−0.213212	−0.0173510	−0.00867548	−	0.999962i	$-0.502762\pi$
$151$	−0.213212	−0.0173510	−0.00867548	+	0.999962i	$0.502762\pi$
$152$	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	$-0.407499\pi$
$163$	7.31629	0.573056	0.286528	+	0.958072i	$0.407499\pi$
$164$	0	0
$165$	−0.284540	−0.0221514
$166$	0	0
$167$	−13.3675	−1.03441	−0.517203	−	0.855863i	$-0.673027\pi$
$167$	−13.3675	−1.03441	−0.517203	+	0.855863i	$0.673027\pi$
$168$	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	$-0.819710\pi$
$193$	−23.4460	−1.68768	−0.843839	+	0.536596i	$0.819710\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−0.687466	−0.0489799	−0.0244899	−	0.999700i	$-0.507796\pi$
$197$	−0.687466	−0.0489799	−0.0244899	+	0.999700i	$0.507796\pi$
$198$	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	$-0.334350\pi$
$223$	14.8505	0.994462	0.497231	+	0.867618i	$0.334350\pi$
$224$	0	0
$225$	−2.99703	−0.199802
$226$	0	0
$227$	−21.4849	−1.42600	−0.713002	−	0.701162i	$-0.752665\pi$
$227$	−21.4849	−1.42600	−0.713002	+	0.701162i	$0.752665\pi$
$228$	0	0
$229$	18.1462	1.19913	0.599566	−	0.800325i	$-0.295340\pi$
$229$	18.1462	1.19913	0.599566	+	0.800325i	$0.295340\pi$
$230$	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	$-0.552245\pi$
$239$	−5.05214	−0.326795	−0.163398	+	0.986560i	$0.552245\pi$
$240$	0	0
$241$	−14.4408	−0.930215	−0.465108	−	0.885254i	$-0.653984\pi$
$241$	−14.4408	−0.930215	−0.465108	+	0.885254i	$0.653984\pi$
$242$	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	$-0.447045\pi$
$271$	5.45218	0.331196	0.165598	+	0.986193i	$0.447045\pi$
$272$	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	$-0.316777\pi$
$281$	18.2499	1.08870	0.544348	+	0.838860i	$0.316777\pi$
$282$	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	$-0.584375\pi$
$293$	−8.96873	−0.523959	−0.261979	+	0.965073i	$0.584375\pi$
$294$	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	$-0.849213\pi$
$307$	−31.1840	−1.77976	−0.889882	+	0.456192i	$0.849213\pi$
$308$	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	$-0.425971\pi$
$317$	8.20709	0.460956	0.230478	+	0.973078i	$0.425971\pi$
$318$	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	$-0.507972\pi$
$331$	−0.911191	−0.0500836	−0.0250418	+	0.999686i	$0.507972\pi$
$332$	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	$-0.453161\pi$
$349$	5.47813	0.293238	0.146619	+	0.989193i	$0.453161\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	4.38823	0.233562	0.116781	−	0.993158i	$-0.462742\pi$
$353$	4.38823	0.233562	0.116781	+	0.993158i	$0.462742\pi$
$354$	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	$-0.377553\pi$
$359$	14.2203	0.750521	0.375261	+	0.926919i	$0.377553\pi$
$360$	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	$-0.244318\pi$
$379$	28.0188	1.43923	0.719616	+	0.694373i	$0.244318\pi$
$380$	0	0
$381$	−0.615802	−0.0315485
$382$	0	0
$383$	23.9075	1.22162	0.610809	−	0.791778i	$-0.290844\pi$
$383$	23.9075	1.22162	0.610809	+	0.791778i	$0.290844\pi$
$384$	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	$-0.404153\pi$
$397$	11.8187	0.593163	0.296581	+	0.955008i	$0.404153\pi$
$398$	0	0
$399$	−1.24304	−0.0622299
$400$	0	0
$401$	8.98456	0.448667	0.224334	−	0.974512i	$-0.427979\pi$
$401$	8.98456	0.448667	0.224334	+	0.974512i	$0.427979\pi$
$402$	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	$-0.257627\pi$
$409$	27.9072	1.37992	0.689962	+	0.723846i	$0.257627\pi$
$410$	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	$-0.360096\pi$
$421$	17.4613	0.851010	0.425505	+	0.904956i	$0.360096\pi$
$422$	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	$-0.597443\pi$
$431$	−12.5131	−0.602735	−0.301368	+	0.953508i	$0.597443\pi$
$432$	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	$-0.325396\pi$
$449$	22.0981	1.04288	0.521438	+	0.853289i	$0.325396\pi$
$450$	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	$-0.690315\pi$
$457$	−24.0669	−1.12580	−0.562902	+	0.826524i	$0.690315\pi$
$458$	0	0
$459$	−0.973199	−0.0454250
$460$	0	0
$461$	41.8691	1.95004	0.975020	−	0.222117i	$-0.0712969\pi$
$461$	41.8691	1.95004	0.975020	+	0.222117i	$0.0712969\pi$
$462$	0	0
$463$	−36.9718	−1.71823	−0.859113	−	0.511786i	$-0.828984\pi$
$463$	−36.9718	−1.71823	−0.859113	+	0.511786i	$0.828984\pi$
$464$	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	$-0.0579535\pi$
$479$	43.0487	1.96694	0.983472	+	0.181062i	$0.0579535\pi$
$480$	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	$-0.555298\pi$
$487$	−7.62896	−0.345701	−0.172851	+	0.984948i	$0.555298\pi$
$488$	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	$-0.557811\pi$
$499$	−8.06951	−0.361241	−0.180620	+	0.983553i	$0.557811\pi$
$500$	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	$-0.586035\pi$
$509$	−12.0479	−0.534015	−0.267008	+	0.963694i	$0.586035\pi$
$510$	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	$-0.313206\pi$
$541$	25.7586	1.10745	0.553725	+	0.832700i	$0.313206\pi$
$542$	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	$-0.631455\pi$
$557$	−18.9439	−0.802679	−0.401340	+	0.915929i	$0.631455\pi$
$558$	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	$-0.277687\pi$
$577$	30.8911	1.28601	0.643007	+	0.765861i	$0.277687\pi$
$578$	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	$-0.537342\pi$
$587$	−5.67152	−0.234089	−0.117044	+	0.993127i	$0.537342\pi$
$588$	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	$-0.540096\pi$
$593$	−6.11872	−0.251266	−0.125633	+	0.992077i	$0.540096\pi$
$594$	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	$-0.406543\pi$
$613$	14.3306	0.578809	0.289404	+	0.957207i	$0.406543\pi$
$614$	0	0
$615$	0.664063	0.0267776
$616$	0	0
$617$	31.0397	1.24961	0.624806	−	0.780780i	$-0.285178\pi$
$617$	31.0397	1.24961	0.624806	+	0.780780i	$0.285178\pi$
$618$	0	0
$619$	−0.892227	−0.0358616	−0.0179308	−	0.999839i	$-0.505708\pi$
$619$	−0.892227	−0.0358616	−0.0179308	+	0.999839i	$0.505708\pi$
$620$	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	$-0.386128\pi$
$631$	17.5918	0.700318	0.350159	+	0.936690i	$0.386128\pi$
$632$	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	$-0.682068\pi$
$643$	−27.4520	−1.08260	−0.541300	+	0.840830i	$0.682068\pi$
$644$	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	$-0.557098\pi$
$661$	−9.17424	−0.356837	−0.178418	+	0.983955i	$0.557098\pi$
$662$	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	$-0.400804\pi$
$683$	16.0263	0.613229	0.306614	+	0.951834i	$0.400804\pi$
$684$	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	$-0.826987\pi$
$691$	−44.9971	−1.71177	−0.855885	+	0.517167i	$0.826987\pi$
$692$	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	$-0.168567\pi$
$709$	45.9596	1.72605	0.863025	+	0.505162i	$0.168567\pi$
$710$	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	$-0.554670\pi$
$733$	−9.25426	−0.341814	−0.170907	+	0.985287i	$0.554670\pi$
$734$	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	$-0.293925\pi$
$739$	32.7910	1.20623	0.603117	+	0.797652i	$0.293925\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	17.7645	0.651716	0.325858	−	0.945419i	$-0.394347\pi$
$743$	17.7645	0.651716	0.325858	+	0.945419i	$0.394347\pi$
$744$	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	$-0.621855\pi$
$761$	−20.6089	−0.747071	−0.373535	+	0.927616i	$0.621855\pi$
$762$	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	$-0.289209\pi$
$769$	34.1017	1.22974	0.614869	+	0.788629i	$0.289209\pi$
$770$	0	0
$771$	−0.0755054	−0.00271926
$772$	0	0
$773$	16.6649	0.599396	0.299698	−	0.954034i	$-0.403114\pi$
$773$	16.6649	0.599396	0.299698	+	0.954034i	$0.403114\pi$
$774$	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	$-0.507340\pi$
$787$	−1.29371	−0.0461157	−0.0230579	+	0.999734i	$0.507340\pi$
$788$	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	$-0.543654\pi$
$811$	−7.78668	−0.273427	−0.136714	+	0.990611i	$0.543654\pi$
$812$	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	$-0.280079\pi$
$821$	36.5174	1.27447	0.637234	+	0.770671i	$0.280079\pi$
$822$	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	$-0.722306\pi$
$827$	−36.9816	−1.28598	−0.642989	+	0.765876i	$0.722306\pi$
$828$	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	$-0.658095\pi$
$839$	−27.6041	−0.953000	−0.476500	+	0.879175i	$0.658095\pi$
$840$	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	$-0.479361\pi$
$853$	3.78477	0.129588	0.0647940	+	0.997899i	$0.479361\pi$
$854$	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	$-0.614708\pi$
$863$	−20.7175	−0.705232	−0.352616	+	0.935768i	$0.614708\pi$
$864$	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	$-0.239503\pi$
$877$	43.2388	1.46007	0.730036	+	0.683409i	$0.239503\pi$
$878$	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	$-0.335943\pi$
$929$	30.0456	0.985765	0.492882	+	0.870096i	$0.335943\pi$
$930$	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	$-0.601633\pi$
$941$	−19.2578	−0.627785	−0.313892	+	0.949459i	$0.601633\pi$
$942$	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	$-0.751728\pi$
$947$	−43.7557	−1.42187	−0.710935	+	0.703257i	$0.751728\pi$
$948$	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	$-0.369273\pi$
$967$	24.8301	0.798483	0.399242	+	0.916846i	$0.369273\pi$
$968$	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	$-0.842072\pi$
$977$	−54.9763	−1.75885	−0.879424	+	0.476040i	$0.842072\pi$
$978$	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	$-0.394254\pi$
$983$	20.4505	0.652270	0.326135	+	0.945323i	$0.394254\pi$
$984$	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.r.1.6	✓	9
13.5	odd	4		3380.2.f.j.3041.12		18
13.8	odd	4		3380.2.f.j.3041.11		18
13.12	even	2		3380.2.a.s.1.6	yes	9

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more