LMFDB - Embedded newform 1400.4.g.c.449.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1400,4,Mod(449,1400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1400, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1400.449");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1400 = 2^{3} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1400.g (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$82.6026740080$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 280)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1400.449
Dual form			1400.4.g.c.449.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.00000i q^{3} +7.00000i q^{7} +2.00000 q^{9} +O(q^{10})$	$q+5.00000i q^{3} +7.00000i q^{7} +2.00000 q^{9} -39.0000 q^{11} -19.0000i q^{13} +37.0000i q^{17} +18.0000 q^{19} -35.0000 q^{21} -90.0000i q^{23} +145.000i q^{27} -99.0000 q^{29} -32.0000 q^{31} -195.000i q^{33} -46.0000i q^{37} +95.0000 q^{39} -248.000 q^{41} +178.000i q^{43} -429.000i q^{47} -49.0000 q^{49} -185.000 q^{51} -652.000i q^{53} +90.0000i q^{57} -40.0000 q^{59} -36.0000 q^{61} +14.0000i q^{63} +348.000i q^{67} +450.000 q^{69} +72.0000 q^{71} -1190.00i q^{73} -273.000i q^{77} -699.000 q^{79} -671.000 q^{81} -116.000i q^{83} -495.000i q^{87} +704.000 q^{89} +133.000 q^{91} -160.000i q^{93} -223.000i q^{97} -78.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{9}+O(q^{10}) $ 2 * q + 4 * q^9	$ 2 q + 4 q^{9} - 78 q^{11} + 36 q^{19} - 70 q^{21} - 198 q^{29} - 64 q^{31} + 190 q^{39} - 496 q^{41} - 98 q^{49} - 370 q^{51} - 80 q^{59} - 72 q^{61} + 900 q^{69} + 144 q^{71} - 1398 q^{79} - 1342 q^{81} + 1408 q^{89} + 266 q^{91} - 156 q^{99}+O(q^{100}) $ 2 * q + 4 * q^9 - 78 * q^11 + 36 * q^19 - 70 * q^21 - 198 * q^29 - 64 * q^31 + 190 * q^39 - 496 * q^41 - 98 * q^49 - 370 * q^51 - 80 * q^59 - 72 * q^61 + 900 * q^69 + 144 * q^71 - 1398 * q^79 - 1342 * q^81 + 1408 * q^89 + 266 * q^91 - 156 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1400\mathbb{Z}\right)^\times$.

$n$	$351$	$701$	$801$	$1177$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	5.00000i	0.962250i	0.876652	+	0.481125i	$0.159772\pi$
$3$	5.00000i	0.962250i	−0.876652	+	0.481125i	$0.840228\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	7.00000i	0.377964i
$7$	7.00000i	0.377964i
$8$	0	0
$9$	2.00000	0.0740741
$10$	0	0
$11$	−39.0000	−1.06899	−0.534497	−	0.845170i	$-0.679499\pi$
$11$	−39.0000	−1.06899	−0.534497	+	0.845170i	$0.679499\pi$
$12$	0	0
$13$	− 19.0000i	− 0.405358i	−0.979245	−	0.202679i	$-0.935035\pi$
$13$	− 19.0000i	− 0.405358i	0.979245	−	0.202679i	$-0.0649648\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	37.0000i	0.527872i	0.964540	+	0.263936i	$0.0850207\pi$
$17$	37.0000i	0.527872i	−0.964540	+	0.263936i	$0.914979\pi$
$18$	0	0
$19$	18.0000	0.217341	0.108671	−	0.994078i	$-0.465341\pi$
$19$	18.0000	0.217341	0.108671	+	0.994078i	$0.465341\pi$
$20$	0	0
$21$	−35.0000	−0.363696
$22$	0	0
$23$	− 90.0000i	− 0.815926i	−0.912998	−	0.407963i	$-0.866239\pi$
$23$	− 90.0000i	− 0.815926i	0.912998	−	0.407963i	$-0.133761\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	145.000i	1.03353i
$28$	0	0
$29$	−99.0000	−0.633925	−0.316963	−	0.948438i	$-0.602663\pi$
$29$	−99.0000	−0.633925	−0.316963	+	0.948438i	$0.602663\pi$
$30$	0	0
$31$	−32.0000	−0.185399	−0.0926995	−	0.995694i	$-0.529550\pi$
$31$	−32.0000	−0.185399	−0.0926995	+	0.995694i	$0.529550\pi$
$32$	0	0
$33$	− 195.000i	− 1.02864i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 46.0000i	− 0.204388i	−0.994764	−	0.102194i	$-0.967414\pi$
$37$	− 46.0000i	− 0.204388i	0.994764	−	0.102194i	$-0.0325862\pi$
$38$	0	0
$39$	95.0000	0.390056
$40$	0	0
$41$	−248.000	−0.944661	−0.472330	−	0.881422i	$-0.656587\pi$
$41$	−248.000	−0.944661	−0.472330	+	0.881422i	$0.656587\pi$
$42$	0	0
$43$	178.000i	0.631273i	0.948880	+	0.315637i	$0.102218\pi$
$43$	178.000i	0.631273i	−0.948880	+	0.315637i	$0.897782\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 429.000i	− 1.33141i	−0.746217	−	0.665703i	$-0.768132\pi$
$47$	− 429.000i	− 1.33141i	0.746217	−	0.665703i	$-0.231868\pi$
$48$	0	0
$49$	−49.0000	−0.142857
$50$	0	0
$51$	−185.000	−0.507945
$52$	0	0
$53$	− 652.000i	− 1.68979i	−0.534929	−	0.844897i	$-0.679662\pi$
$53$	− 652.000i	− 1.68979i	0.534929	−	0.844897i	$-0.320338\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	90.0000i	0.209137i
$58$	0	0
$59$	−40.0000	−0.0882637	−0.0441318	−	0.999026i	$-0.514052\pi$
$59$	−40.0000	−0.0882637	−0.0441318	+	0.999026i	$0.514052\pi$
$60$	0	0
$61$	−36.0000	−0.0755627	−0.0377814	−	0.999286i	$-0.512029\pi$
$61$	−36.0000	−0.0755627	−0.0377814	+	0.999286i	$0.512029\pi$
$62$	0	0
$63$	14.0000i	0.0279974i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	348.000i	0.634552i	0.948333	+	0.317276i	$0.102768\pi$
$67$	348.000i	0.634552i	−0.948333	+	0.317276i	$0.897232\pi$
$68$	0	0
$69$	450.000	0.785125
$70$	0	0
$71$	72.0000	0.120350	0.0601748	−	0.998188i	$-0.480834\pi$
$71$	72.0000	0.120350	0.0601748	+	0.998188i	$0.480834\pi$
$72$	0	0
$73$	− 1190.00i	− 1.90793i	−0.299916	−	0.953966i	$-0.596959\pi$
$73$	− 1190.00i	− 1.90793i	0.299916	−	0.953966i	$-0.403041\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 273.000i	− 0.404042i
$78$	0	0
$79$	−699.000	−0.995489	−0.497745	−	0.867324i	$-0.665838\pi$
$79$	−699.000	−0.995489	−0.497745	+	0.867324i	$0.665838\pi$
$80$	0	0
$81$	−671.000	−0.920439
$82$	0	0
$83$	− 116.000i	− 0.153405i	−0.997054	−	0.0767027i	$-0.975561\pi$
$83$	− 116.000i	− 0.153405i	0.997054	−	0.0767027i	$-0.0244392\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 495.000i	− 0.609995i
$88$	0	0
$89$	704.000	0.838470	0.419235	−	0.907878i	$-0.362298\pi$
$89$	704.000	0.838470	0.419235	+	0.907878i	$0.362298\pi$
$90$	0	0
$91$	133.000	0.153211
$92$	0	0
$93$	− 160.000i	− 0.178400i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 223.000i	− 0.233425i	−0.993166	−	0.116712i	$-0.962764\pi$
$97$	− 223.000i	− 0.233425i	0.993166	−	0.116712i	$-0.0372356\pi$
$98$	0	0
$99$	−78.0000	−0.0791848
$100$	0	0
$101$	522.000	0.514267	0.257133	−	0.966376i	$-0.417222\pi$
$101$	522.000	0.514267	0.257133	+	0.966376i	$0.417222\pi$
$102$	0	0
$103$	1697.00i	1.62340i	0.584073	+	0.811701i	$0.301458\pi$
$103$	1697.00i	1.62340i	−0.584073	+	0.811701i	$0.698542\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 246.000i	− 0.222259i	−0.993806	−	0.111130i	$-0.964553\pi$
$107$	− 246.000i	− 0.222259i	0.993806	−	0.111130i	$-0.0354468\pi$
$108$	0	0
$109$	−345.000	−0.303165	−0.151583	−	0.988445i	$-0.548437\pi$
$109$	−345.000	−0.303165	−0.151583	+	0.988445i	$0.548437\pi$
$110$	0	0
$111$	230.000	0.196672
$112$	0	0
$113$	186.000i	0.154844i	0.996998	+	0.0774222i	$0.0246689\pi$
$113$	186.000i	0.154844i	−0.996998	+	0.0774222i	$0.975331\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 38.0000i	− 0.0300265i
$118$	0	0
$119$	−259.000	−0.199517
$120$	0	0
$121$	190.000	0.142750
$122$	0	0
$123$	− 1240.00i	− 0.909000i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 2344.00i	− 1.63777i	−0.573960	−	0.818883i	$-0.694594\pi$
$127$	− 2344.00i	− 1.63777i	0.573960	−	0.818883i	$-0.305406\pi$
$128$	0	0
$129$	−890.000	−0.607443
$130$	0	0
$131$	958.000	0.638938	0.319469	−	0.947597i	$-0.396496\pi$
$131$	958.000	0.638938	0.319469	+	0.947597i	$0.396496\pi$
$132$	0	0
$133$	126.000i	0.0821473i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 768.000i	− 0.478939i	−0.970904	−	0.239470i	$-0.923026\pi$
$137$	− 768.000i	− 0.478939i	0.970904	−	0.239470i	$-0.0769735\pi$
$138$	0	0
$139$	122.000	0.0744454	0.0372227	−	0.999307i	$-0.488149\pi$
$139$	122.000	0.0744454	0.0372227	+	0.999307i	$0.488149\pi$
$140$	0	0
$141$	2145.00	1.28115
$142$	0	0
$143$	741.000i	0.433325i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 245.000i	− 0.137464i
$148$	0	0
$149$	166.000	0.0912701	0.0456351	−	0.998958i	$-0.485469\pi$
$149$	166.000	0.0912701	0.0456351	+	0.998958i	$0.485469\pi$
$150$	0	0
$151$	−1725.00	−0.929659	−0.464830	−	0.885400i	$-0.653884\pi$
$151$	−1725.00	−0.929659	−0.464830	+	0.885400i	$0.653884\pi$
$152$	0	0
$153$	74.0000i	0.0391016i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 3686.00i	− 1.87373i	−0.349698	−	0.936863i	$-0.613716\pi$
$157$	− 3686.00i	− 1.87373i	0.349698	−	0.936863i	$-0.386284\pi$
$158$	0	0
$159$	3260.00	1.62601
$160$	0	0
$161$	630.000	0.308391
$162$	0	0
$163$	2594.00i	1.24649i	0.782027	+	0.623245i	$0.214186\pi$
$163$	2594.00i	1.24649i	−0.782027	+	0.623245i	$0.785814\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 1457.00i	− 0.675126i	−0.941303	−	0.337563i	$-0.890397\pi$
$167$	− 1457.00i	− 0.675126i	0.941303	−	0.337563i	$-0.109603\pi$
$168$	0	0
$169$	1836.00	0.835685
$170$	0	0
$171$	36.0000	0.0160993
$172$	0	0
$173$	− 3241.00i	− 1.42433i	−0.702013	−	0.712164i	$-0.747715\pi$
$173$	− 3241.00i	− 1.42433i	0.702013	−	0.712164i	$-0.252285\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 200.000i	− 0.0849318i
$178$	0	0
$179$	4596.00	1.91911	0.959556	−	0.281517i	$-0.0908375\pi$
$179$	4596.00	1.91911	0.959556	+	0.281517i	$0.0908375\pi$
$180$	0	0
$181$	−1140.00	−0.468152	−0.234076	−	0.972218i	$-0.575206\pi$
$181$	−1140.00	−0.468152	−0.234076	+	0.972218i	$0.575206\pi$
$182$	0	0
$183$	− 180.000i	− 0.0727103i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 1443.00i	− 0.564292i
$188$	0	0
$189$	−1015.00	−0.390637
$190$	0	0
$191$	−4131.00	−1.56497	−0.782483	−	0.622671i	$-0.786047\pi$
$191$	−4131.00	−1.56497	−0.782483	+	0.622671i	$0.786047\pi$
$192$	0	0
$193$	− 2792.00i	− 1.04131i	−0.853768	−	0.520654i	$-0.825688\pi$
$193$	− 2792.00i	− 1.04131i	0.853768	−	0.520654i	$-0.174312\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 2692.00i	− 0.973589i	−0.873516	−	0.486795i	$-0.838166\pi$
$197$	− 2692.00i	− 0.973589i	0.873516	−	0.486795i	$-0.161834\pi$
$198$	0	0
$199$	2640.00	0.940425	0.470213	−	0.882553i	$-0.344177\pi$
$199$	2640.00	0.940425	0.470213	+	0.882553i	$0.344177\pi$
$200$	0	0
$201$	−1740.00	−0.610598
$202$	0	0
$203$	− 693.000i	− 0.239601i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 180.000i	− 0.0604390i
$208$	0	0
$209$	−702.000	−0.232337
$210$	0	0
$211$	327.000	0.106690	0.0533450	−	0.998576i	$-0.483012\pi$
$211$	327.000	0.106690	0.0533450	+	0.998576i	$0.483012\pi$
$212$	0	0
$213$	360.000i	0.115807i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 224.000i	− 0.0700742i
$218$	0	0
$219$	5950.00	1.83591
$220$	0	0
$221$	703.000	0.213977
$222$	0	0
$223$	4665.00i	1.40086i	0.713722	+	0.700429i	$0.247008\pi$
$223$	4665.00i	1.40086i	−0.713722	+	0.700429i	$0.752992\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1071.00i	0.313149i	0.987666	+	0.156574i	$0.0500451\pi$
$227$	1071.00i	0.313149i	−0.987666	+	0.156574i	$0.949955\pi$
$228$	0	0
$229$	3920.00	1.13118	0.565591	−	0.824686i	$-0.308648\pi$
$229$	3920.00	1.13118	0.565591	+	0.824686i	$0.308648\pi$
$230$	0	0
$231$	1365.00	0.388790
$232$	0	0
$233$	− 2616.00i	− 0.735536i	−0.929918	−	0.367768i	$-0.880122\pi$
$233$	− 2616.00i	− 0.735536i	0.929918	−	0.367768i	$-0.119878\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 3495.00i	− 0.957910i
$238$	0	0
$239$	6913.00	1.87098	0.935491	−	0.353350i	$-0.114958\pi$
$239$	6913.00	1.87098	0.935491	+	0.353350i	$0.114958\pi$
$240$	0	0
$241$	2690.00	0.718996	0.359498	−	0.933146i	$-0.382948\pi$
$241$	2690.00	0.718996	0.359498	+	0.933146i	$0.382948\pi$
$242$	0	0
$243$	560.000i	0.147835i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 342.000i	− 0.0881010i
$248$	0	0
$249$	580.000	0.147614
$250$	0	0
$251$	3034.00	0.762966	0.381483	−	0.924376i	$-0.375414\pi$
$251$	3034.00	0.762966	0.381483	+	0.924376i	$0.375414\pi$
$252$	0	0
$253$	3510.00i	0.872221i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2266.00i	0.549997i	0.961445	+	0.274998i	$0.0886773\pi$
$257$	2266.00i	0.549997i	−0.961445	+	0.274998i	$0.911323\pi$
$258$	0	0
$259$	322.000	0.0772514
$260$	0	0
$261$	−198.000	−0.0469574
$262$	0	0
$263$	− 7058.00i	− 1.65481i	−0.561606	−	0.827405i	$-0.689816\pi$
$263$	− 7058.00i	− 1.65481i	0.561606	−	0.827405i	$-0.310184\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	3520.00i	0.806818i
$268$	0	0
$269$	−3474.00	−0.787411	−0.393705	−	0.919237i	$-0.628807\pi$
$269$	−3474.00	−0.787411	−0.393705	+	0.919237i	$0.628807\pi$
$270$	0	0
$271$	−20.0000	−0.00448308	−0.00224154	−	0.999997i	$-0.500714\pi$
$271$	−20.0000	−0.00448308	−0.00224154	+	0.999997i	$0.500714\pi$
$272$	0	0
$273$	665.000i	0.147427i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	4186.00i	0.907987i	0.891005	+	0.453993i	$0.150001\pi$
$277$	4186.00i	0.907987i	−0.891005	+	0.453993i	$0.849999\pi$
$278$	0	0
$279$	−64.0000	−0.0137333
$280$	0	0
$281$	−7221.00	−1.53298	−0.766492	−	0.642253i	$-0.778000\pi$
$281$	−7221.00	−1.53298	−0.766492	+	0.642253i	$0.778000\pi$
$282$	0	0
$283$	23.0000i	0.00483112i	0.999997	+	0.00241556i	$0.000768898\pi$
$283$	23.0000i	0.00483112i	−0.999997	+	0.00241556i	$0.999231\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 1736.00i	− 0.357048i
$288$	0	0
$289$	3544.00	0.721352
$290$	0	0
$291$	1115.00	0.224613
$292$	0	0
$293$	139.000i	0.0277149i	0.999904	+	0.0138575i	$0.00441111\pi$
$293$	139.000i	0.0277149i	−0.999904	+	0.0138575i	$0.995589\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 5655.00i	− 1.10484i
$298$	0	0
$299$	−1710.00	−0.330742
$300$	0	0
$301$	−1246.00	−0.238599
$302$	0	0
$303$	2610.00i	0.494853i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 6759.00i	− 1.25654i	−0.777997	−	0.628268i	$-0.783764\pi$
$307$	− 6759.00i	− 1.25654i	0.777997	−	0.628268i	$-0.216236\pi$
$308$	0	0
$309$	−8485.00	−1.56212
$310$	0	0
$311$	1038.00	0.189259	0.0946295	−	0.995513i	$-0.469833\pi$
$311$	1038.00	0.189259	0.0946295	+	0.995513i	$0.469833\pi$
$312$	0	0
$313$	− 9507.00i	− 1.71683i	−0.512957	−	0.858414i	$-0.671450\pi$
$313$	− 9507.00i	− 1.71683i	0.512957	−	0.858414i	$-0.328550\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1290.00i	0.228560i	0.993449	+	0.114280i	$0.0364562\pi$
$317$	1290.00i	0.228560i	−0.993449	+	0.114280i	$0.963544\pi$
$318$	0	0
$319$	3861.00	0.677663
$320$	0	0
$321$	1230.00	0.213869
$322$	0	0
$323$	666.000i	0.114728i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 1725.00i	− 0.291721i
$328$	0	0
$329$	3003.00	0.503224
$330$	0	0
$331$	−5404.00	−0.897374	−0.448687	−	0.893689i	$-0.648108\pi$
$331$	−5404.00	−0.897374	−0.448687	+	0.893689i	$0.648108\pi$
$332$	0	0
$333$	− 92.0000i	− 0.0151398i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 3094.00i	− 0.500121i	−0.968230	−	0.250061i	$-0.919549\pi$
$337$	− 3094.00i	− 0.500121i	0.968230	−	0.250061i	$-0.0804506\pi$
$338$	0	0
$339$	−930.000	−0.148999
$340$	0	0
$341$	1248.00	0.198191
$342$	0	0
$343$	− 343.000i	− 0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1486.00i	0.229892i	0.993372	+	0.114946i	$0.0366696\pi$
$347$	1486.00i	0.229892i	−0.993372	+	0.114946i	$0.963330\pi$
$348$	0	0
$349$	−6630.00	−1.01689	−0.508447	−	0.861093i	$-0.669780\pi$
$349$	−6630.00	−1.01689	−0.508447	+	0.861093i	$0.669780\pi$
$350$	0	0
$351$	2755.00	0.418949
$352$	0	0
$353$	− 5417.00i	− 0.816764i	−0.912811	−	0.408382i	$-0.866093\pi$
$353$	− 5417.00i	− 0.816764i	0.912811	−	0.408382i	$-0.133907\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 1295.00i	− 0.191985i
$358$	0	0
$359$	9200.00	1.35253	0.676264	−	0.736660i	$-0.263598\pi$
$359$	9200.00	1.35253	0.676264	+	0.736660i	$0.263598\pi$
$360$	0	0
$361$	−6535.00	−0.952763
$362$	0	0
$363$	950.000i	0.137361i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 4157.00i	− 0.591263i	−0.955302	−	0.295632i	$-0.904470\pi$
$367$	− 4157.00i	− 0.591263i	0.955302	−	0.295632i	$-0.0955301\pi$
$368$	0	0
$369$	−496.000	−0.0699749
$370$	0	0
$371$	4564.00	0.638682
$372$	0	0
$373$	8592.00i	1.19270i	0.802725	+	0.596350i	$0.203383\pi$
$373$	8592.00i	1.19270i	−0.802725	+	0.596350i	$0.796617\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1881.00i	0.256967i
$378$	0	0
$379$	−2084.00	−0.282448	−0.141224	−	0.989978i	$-0.545104\pi$
$379$	−2084.00	−0.282448	−0.141224	+	0.989978i	$0.545104\pi$
$380$	0	0
$381$	11720.0	1.57594
$382$	0	0
$383$	− 9292.00i	− 1.23968i	−0.784727	−	0.619842i	$-0.787197\pi$
$383$	− 9292.00i	− 1.23968i	0.784727	−	0.619842i	$-0.212803\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	356.000i	0.0467610i
$388$	0	0
$389$	1723.00	0.224575	0.112287	−	0.993676i	$-0.464182\pi$
$389$	1723.00	0.224575	0.112287	+	0.993676i	$0.464182\pi$
$390$	0	0
$391$	3330.00	0.430704
$392$	0	0
$393$	4790.00i	0.614818i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 3767.00i	− 0.476222i	−0.971238	−	0.238111i	$-0.923472\pi$
$397$	− 3767.00i	− 0.476222i	0.971238	−	0.238111i	$-0.0765283\pi$
$398$	0	0
$399$	−630.000	−0.0790462
$400$	0	0
$401$	−14967.0	−1.86388	−0.931941	−	0.362611i	$-0.881885\pi$
$401$	−14967.0	−1.86388	−0.931941	+	0.362611i	$0.881885\pi$
$402$	0	0
$403$	608.000i	0.0751529i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1794.00i	0.218490i
$408$	0	0
$409$	−11374.0	−1.37508	−0.687540	−	0.726146i	$-0.741310\pi$
$409$	−11374.0	−1.37508	−0.687540	+	0.726146i	$0.741310\pi$
$410$	0	0
$411$	3840.00	0.460859
$412$	0	0
$413$	− 280.000i	− 0.0333605i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	610.000i	0.0716351i
$418$	0	0
$419$	−14076.0	−1.64119	−0.820594	−	0.571512i	$-0.806357\pi$
$419$	−14076.0	−1.64119	−0.820594	+	0.571512i	$0.806357\pi$
$420$	0	0
$421$	3769.00	0.436318	0.218159	−	0.975913i	$-0.429995\pi$
$421$	3769.00	0.436318	0.218159	+	0.975913i	$0.429995\pi$
$422$	0	0
$423$	− 858.000i	− 0.0986227i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 252.000i	− 0.0285600i
$428$	0	0
$429$	−3705.00	−0.416968
$430$	0	0
$431$	−7423.00	−0.829590	−0.414795	−	0.909915i	$-0.636147\pi$
$431$	−7423.00	−0.829590	−0.414795	+	0.909915i	$0.636147\pi$
$432$	0	0
$433$	− 670.000i	− 0.0743606i	−0.999309	−	0.0371803i	$-0.988162\pi$
$433$	− 670.000i	− 0.0743606i	0.999309	−	0.0371803i	$-0.0118376\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 1620.00i	− 0.177334i
$438$	0	0
$439$	−6466.00	−0.702973	−0.351487	−	0.936193i	$-0.614324\pi$
$439$	−6466.00	−0.702973	−0.351487	+	0.936193i	$0.614324\pi$
$440$	0	0
$441$	−98.0000	−0.0105820
$442$	0	0
$443$	14618.0i	1.56777i	0.620906	+	0.783885i	$0.286765\pi$
$443$	14618.0i	1.56777i	−0.620906	+	0.783885i	$0.713235\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	830.000i	0.0878247i
$448$	0	0
$449$	−7551.00	−0.793661	−0.396830	−	0.917892i	$-0.629890\pi$
$449$	−7551.00	−0.793661	−0.396830	+	0.917892i	$0.629890\pi$
$450$	0	0
$451$	9672.00	1.00984
$452$	0	0
$453$	− 8625.00i	− 0.894565i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	14076.0i	1.44080i	0.693557	+	0.720402i	$0.256043\pi$
$457$	14076.0i	1.44080i	−0.693557	+	0.720402i	$0.743957\pi$
$458$	0	0
$459$	−5365.00	−0.545570
$460$	0	0
$461$	−18360.0	−1.85490	−0.927452	−	0.373943i	$-0.878006\pi$
$461$	−18360.0	−1.85490	−0.927452	+	0.373943i	$0.878006\pi$
$462$	0	0
$463$	2812.00i	0.282256i	0.989991	+	0.141128i	$0.0450730\pi$
$463$	2812.00i	0.282256i	−0.989991	+	0.141128i	$0.954927\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 2619.00i	− 0.259514i	−0.991546	−	0.129757i	$-0.958580\pi$
$467$	− 2619.00i	− 0.259514i	0.991546	−	0.129757i	$-0.0414197\pi$
$468$	0	0
$469$	−2436.00	−0.239838
$470$	0	0
$471$	18430.0	1.80299
$472$	0	0
$473$	− 6942.00i	− 0.674828i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 1304.00i	− 0.125170i
$478$	0	0
$479$	7546.00	0.719803	0.359901	−	0.932990i	$-0.382810\pi$
$479$	7546.00	0.719803	0.359901	+	0.932990i	$0.382810\pi$
$480$	0	0
$481$	−874.000	−0.0828502
$482$	0	0
$483$	3150.00i	0.296749i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 9398.00i	− 0.874464i	−0.899349	−	0.437232i	$-0.855959\pi$
$487$	− 9398.00i	− 0.874464i	0.899349	−	0.437232i	$-0.144041\pi$
$488$	0	0
$489$	−12970.0	−1.19943
$490$	0	0
$491$	−5397.00	−0.496055	−0.248028	−	0.968753i	$-0.579782\pi$
$491$	−5397.00	−0.496055	−0.248028	+	0.968753i	$0.579782\pi$
$492$	0	0
$493$	− 3663.00i	− 0.334631i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	504.000i	0.0454879i
$498$	0	0
$499$	−11669.0	−1.04685	−0.523423	−	0.852073i	$-0.675345\pi$
$499$	−11669.0	−1.04685	−0.523423	+	0.852073i	$0.675345\pi$
$500$	0	0
$501$	7285.00	0.649640
$502$	0	0
$503$	4055.00i	0.359450i	0.983717	+	0.179725i	$0.0575208\pi$
$503$	4055.00i	0.359450i	−0.983717	+	0.179725i	$0.942479\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	9180.00i	0.804138i
$508$	0	0
$509$	−12026.0	−1.04724	−0.523618	−	0.851953i	$-0.675418\pi$
$509$	−12026.0	−1.04724	−0.523618	+	0.851953i	$0.675418\pi$
$510$	0	0
$511$	8330.00	0.721130
$512$	0	0
$513$	2610.00i	0.224628i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	16731.0i	1.42327i
$518$	0	0
$519$	16205.0	1.37056
$520$	0	0
$521$	−17622.0	−1.48183	−0.740915	−	0.671598i	$-0.765608\pi$
$521$	−17622.0	−1.48183	−0.740915	+	0.671598i	$0.765608\pi$
$522$	0	0
$523$	14684.0i	1.22770i	0.789423	+	0.613849i	$0.210380\pi$
$523$	14684.0i	1.22770i	−0.789423	+	0.613849i	$0.789620\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 1184.00i	− 0.0978669i
$528$	0	0
$529$	4067.00	0.334265
$530$	0	0
$531$	−80.0000	−0.00653805
$532$	0	0
$533$	4712.00i	0.382926i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	22980.0i	1.84667i
$538$	0	0
$539$	1911.00	0.152714
$540$	0	0
$541$	−5741.00	−0.456238	−0.228119	−	0.973633i	$-0.573258\pi$
$541$	−5741.00	−0.456238	−0.228119	+	0.973633i	$0.573258\pi$
$542$	0	0
$543$	− 5700.00i	− 0.450480i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	9268.00i	0.724444i	0.932092	+	0.362222i	$0.117982\pi$
$547$	9268.00i	0.724444i	−0.932092	+	0.362222i	$0.882018\pi$
$548$	0	0
$549$	−72.0000	−0.00559724
$550$	0	0
$551$	−1782.00	−0.137778
$552$	0	0
$553$	− 4893.00i	− 0.376260i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	23112.0i	1.75815i	0.476688	+	0.879073i	$0.341837\pi$
$557$	23112.0i	1.75815i	−0.476688	+	0.879073i	$0.658163\pi$
$558$	0	0
$559$	3382.00	0.255892
$560$	0	0
$561$	7215.00	0.542990
$562$	0	0
$563$	− 13284.0i	− 0.994412i	−0.867633	−	0.497206i	$-0.834359\pi$
$563$	− 13284.0i	− 0.994412i	0.867633	−	0.497206i	$-0.165641\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 4697.00i	− 0.347893i
$568$	0	0
$569$	10446.0	0.769629	0.384815	−	0.922994i	$-0.374265\pi$
$569$	10446.0	0.769629	0.384815	+	0.922994i	$0.374265\pi$
$570$	0	0
$571$	−2252.00	−0.165050	−0.0825248	−	0.996589i	$-0.526298\pi$
$571$	−2252.00	−0.165050	−0.0825248	+	0.996589i	$0.526298\pi$
$572$	0	0
$573$	− 20655.0i	− 1.50589i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 4457.00i	− 0.321573i	−0.986989	−	0.160786i	$-0.948597\pi$
$577$	− 4457.00i	− 0.321573i	0.986989	−	0.160786i	$-0.0514030\pi$
$578$	0	0
$579$	13960.0	1.00200
$580$	0	0
$581$	812.000	0.0579818
$582$	0	0
$583$	25428.0i	1.80638i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	25048.0i	1.76123i	0.473833	+	0.880615i	$0.342870\pi$
$587$	25048.0i	1.76123i	−0.473833	+	0.880615i	$0.657130\pi$
$588$	0	0
$589$	−576.000	−0.0402948
$590$	0	0
$591$	13460.0	0.936837
$592$	0	0
$593$	177.000i	0.0122572i	0.999981	+	0.00612860i	$0.00195081\pi$
$593$	177.000i	0.0122572i	−0.999981	+	0.00612860i	$0.998049\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	13200.0i	0.904925i
$598$	0	0
$599$	−9009.00	−0.614520	−0.307260	−	0.951626i	$-0.599412\pi$
$599$	−9009.00	−0.614520	−0.307260	+	0.951626i	$0.599412\pi$
$600$	0	0
$601$	−25794.0	−1.75068	−0.875340	−	0.483507i	$-0.839363\pi$
$601$	−25794.0	−1.75068	−0.875340	+	0.483507i	$0.839363\pi$
$602$	0	0
$603$	696.000i	0.0470038i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 4319.00i	− 0.288802i	−0.989519	−	0.144401i	$-0.953874\pi$
$607$	− 4319.00i	− 0.288802i	0.989519	−	0.144401i	$-0.0461255\pi$
$608$	0	0
$609$	3465.00	0.230556
$610$	0	0
$611$	−8151.00	−0.539696
$612$	0	0
$613$	− 15134.0i	− 0.997156i	−0.866845	−	0.498578i	$-0.833856\pi$
$613$	− 15134.0i	− 0.997156i	0.866845	−	0.498578i	$-0.166144\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 8394.00i	− 0.547698i	−0.961773	−	0.273849i	$-0.911703\pi$
$617$	− 8394.00i	− 0.547698i	0.961773	−	0.273849i	$-0.0882969\pi$
$618$	0	0
$619$	1658.00	0.107659	0.0538293	−	0.998550i	$-0.482857\pi$
$619$	1658.00	0.107659	0.0538293	+	0.998550i	$0.482857\pi$
$620$	0	0
$621$	13050.0	0.843283
$622$	0	0
$623$	4928.00i	0.316912i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 3510.00i	− 0.223566i
$628$	0	0
$629$	1702.00	0.107891
$630$	0	0
$631$	−8071.00	−0.509194	−0.254597	−	0.967047i	$-0.581943\pi$
$631$	−8071.00	−0.509194	−0.254597	+	0.967047i	$0.581943\pi$
$632$	0	0
$633$	1635.00i	0.102663i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	931.000i	0.0579083i
$638$	0	0
$639$	144.000	0.00891479
$640$	0	0
$641$	28130.0	1.73334	0.866668	−	0.498886i	$-0.166257\pi$
$641$	28130.0	1.73334	0.866668	+	0.498886i	$0.166257\pi$
$642$	0	0
$643$	2693.00i	0.165166i	0.996584	+	0.0825829i	$0.0263169\pi$
$643$	2693.00i	0.165166i	−0.996584	+	0.0825829i	$0.973683\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 1376.00i	− 0.0836107i	−0.999126	−	0.0418054i	$-0.986689\pi$
$647$	− 1376.00i	− 0.0836107i	0.999126	−	0.0418054i	$-0.0133109\pi$
$648$	0	0
$649$	1560.00	0.0943534
$650$	0	0
$651$	1120.00	0.0674290
$652$	0	0
$653$	30342.0i	1.81834i	0.416428	+	0.909169i	$0.363282\pi$
$653$	30342.0i	1.81834i	−0.416428	+	0.909169i	$0.636718\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 2380.00i	− 0.141328i
$658$	0	0
$659$	15163.0	0.896307	0.448154	−	0.893957i	$-0.352082\pi$
$659$	15163.0	0.896307	0.448154	+	0.893957i	$0.352082\pi$
$660$	0	0
$661$	−5688.00	−0.334701	−0.167351	−	0.985897i	$-0.553521\pi$
$661$	−5688.00	−0.334701	−0.167351	+	0.985897i	$0.553521\pi$
$662$	0	0
$663$	3515.00i	0.205899i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	8910.00i	0.517236i
$668$	0	0
$669$	−23325.0	−1.34798
$670$	0	0
$671$	1404.00	0.0807762
$672$	0	0
$673$	21488.0i	1.23076i	0.788231	+	0.615380i	$0.210997\pi$
$673$	21488.0i	1.23076i	−0.788231	+	0.615380i	$0.789003\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 16453.0i	− 0.934033i	−0.884249	−	0.467016i	$-0.845329\pi$
$677$	− 16453.0i	− 0.934033i	0.884249	−	0.467016i	$-0.154671\pi$
$678$	0	0
$679$	1561.00	0.0882263
$680$	0	0
$681$	−5355.00	−0.301328
$682$	0	0
$683$	− 3372.00i	− 0.188911i	−0.995529	−	0.0944553i	$-0.969889\pi$
$683$	− 3372.00i	− 0.188911i	0.995529	−	0.0944553i	$-0.0301110\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	19600.0i	1.08848i
$688$	0	0
$689$	−12388.0	−0.684971
$690$	0	0
$691$	14172.0	0.780215	0.390107	−	0.920769i	$-0.372438\pi$
$691$	14172.0	0.780215	0.390107	+	0.920769i	$0.372438\pi$
$692$	0	0
$693$	− 546.000i	− 0.0299290i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 9176.00i	− 0.498660i
$698$	0	0
$699$	13080.0	0.707770
$700$	0	0
$701$	30663.0	1.65210	0.826052	−	0.563593i	$-0.190581\pi$
$701$	30663.0	1.65210	0.826052	+	0.563593i	$0.190581\pi$
$702$	0	0
$703$	− 828.000i	− 0.0444219i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3654.00i	0.194375i
$708$	0	0
$709$	14485.0	0.767272	0.383636	−	0.923484i	$-0.374672\pi$
$709$	14485.0	0.767272	0.383636	+	0.923484i	$0.374672\pi$
$710$	0	0
$711$	−1398.00	−0.0737399
$712$	0	0
$713$	2880.00i	0.151272i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	34565.0i	1.80035i
$718$	0	0
$719$	8818.00	0.457380	0.228690	−	0.973499i	$-0.426556\pi$
$719$	8818.00	0.457380	0.228690	+	0.973499i	$0.426556\pi$
$720$	0	0
$721$	−11879.0	−0.613588
$722$	0	0
$723$	13450.0i	0.691855i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	11408.0i	0.581980i	0.956726	+	0.290990i	$0.0939846\pi$
$727$	11408.0i	0.581980i	−0.956726	+	0.290990i	$0.906015\pi$
$728$	0	0
$729$	−20917.0	−1.06269
$730$	0	0
$731$	−6586.00	−0.333231
$732$	0	0
$733$	15233.0i	0.767590i	0.923418	+	0.383795i	$0.125383\pi$
$733$	15233.0i	0.767590i	−0.923418	+	0.383795i	$0.874617\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 13572.0i	− 0.678332i
$738$	0	0
$739$	−32113.0	−1.59851	−0.799253	−	0.600995i	$-0.794771\pi$
$739$	−32113.0	−1.59851	−0.799253	+	0.600995i	$0.794771\pi$
$740$	0	0
$741$	1710.00	0.0847752
$742$	0	0
$743$	4596.00i	0.226933i	0.993542	+	0.113466i	$0.0361954\pi$
$743$	4596.00i	0.226933i	−0.993542	+	0.113466i	$0.963805\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 232.000i	− 0.0113634i
$748$	0	0
$749$	1722.00	0.0840060
$750$	0	0
$751$	−15893.0	−0.772229	−0.386114	−	0.922451i	$-0.626183\pi$
$751$	−15893.0	−0.772229	−0.386114	+	0.922451i	$0.626183\pi$
$752$	0	0
$753$	15170.0i	0.734164i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 15336.0i	− 0.736323i	−0.929762	−	0.368161i	$-0.879987\pi$
$757$	− 15336.0i	− 0.736323i	0.929762	−	0.368161i	$-0.120013\pi$
$758$	0	0
$759$	−17550.0	−0.839295
$760$	0	0
$761$	33010.0	1.57242	0.786210	−	0.617959i	$-0.212040\pi$
$761$	33010.0	1.57242	0.786210	+	0.617959i	$0.212040\pi$
$762$	0	0
$763$	− 2415.00i	− 0.114586i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	760.000i	0.0357784i
$768$	0	0
$769$	−30982.0	−1.45285	−0.726424	−	0.687247i	$-0.758819\pi$
$769$	−30982.0	−1.45285	−0.726424	+	0.687247i	$0.758819\pi$
$770$	0	0
$771$	−11330.0	−0.529235
$772$	0	0
$773$	− 5915.00i	− 0.275223i	−0.990486	−	0.137612i	$-0.956057\pi$
$773$	− 5915.00i	− 0.275223i	0.990486	−	0.137612i	$-0.0439426\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	1610.00i	0.0743352i
$778$	0	0
$779$	−4464.00	−0.205314
$780$	0	0
$781$	−2808.00	−0.128653
$782$	0	0
$783$	− 14355.0i	− 0.655180i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	24615.0i	1.11490i	0.830209	+	0.557452i	$0.188221\pi$
$787$	24615.0i	1.11490i	−0.830209	+	0.557452i	$0.811779\pi$
$788$	0	0
$789$	35290.0	1.59234
$790$	0	0
$791$	−1302.00	−0.0585257
$792$	0	0
$793$	684.000i	0.0306300i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 6055.00i	− 0.269108i	−0.990906	−	0.134554i	$-0.957040\pi$
$797$	− 6055.00i	− 0.269108i	0.990906	−	0.134554i	$-0.0429602\pi$
$798$	0	0
$799$	15873.0	0.702811
$800$	0	0
$801$	1408.00	0.0621089
$802$	0	0
$803$	46410.0i	2.03957i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 17370.0i	− 0.757686i
$808$	0	0
$809$	−27809.0	−1.20854	−0.604272	−	0.796778i	$-0.706536\pi$
$809$	−27809.0	−1.20854	−0.604272	+	0.796778i	$0.706536\pi$
$810$	0	0
$811$	−18654.0	−0.807683	−0.403841	−	0.914829i	$-0.632325\pi$
$811$	−18654.0	−0.807683	−0.403841	+	0.914829i	$0.632325\pi$
$812$	0	0
$813$	− 100.000i	− 0.00431384i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	3204.00i	0.137202i
$818$	0	0
$819$	266.000	0.0113490
$820$	0	0
$821$	27285.0	1.15987	0.579935	−	0.814663i	$-0.303078\pi$
$821$	27285.0	1.15987	0.579935	+	0.814663i	$0.303078\pi$
$822$	0	0
$823$	18592.0i	0.787456i	0.919227	+	0.393728i	$0.128815\pi$
$823$	18592.0i	0.787456i	−0.919227	+	0.393728i	$0.871185\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	37146.0i	1.56190i	0.624592	+	0.780951i	$0.285265\pi$
$827$	37146.0i	1.56190i	−0.624592	+	0.780951i	$0.714735\pi$
$828$	0	0
$829$	−11024.0	−0.461857	−0.230928	−	0.972971i	$-0.574176\pi$
$829$	−11024.0	−0.461857	−0.230928	+	0.972971i	$0.574176\pi$
$830$	0	0
$831$	−20930.0	−0.873711
$832$	0	0
$833$	− 1813.00i	− 0.0754102i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 4640.00i	− 0.191615i
$838$	0	0
$839$	−42858.0	−1.76355	−0.881777	−	0.471666i	$-0.843653\pi$
$839$	−42858.0	−1.76355	−0.881777	+	0.471666i	$0.843653\pi$
$840$	0	0
$841$	−14588.0	−0.598139
$842$	0	0
$843$	− 36105.0i	− 1.47512i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1330.00i	0.0539544i
$848$	0	0
$849$	−115.000	−0.00464875
$850$	0	0
$851$	−4140.00	−0.166765
$852$	0	0
$853$	− 31522.0i	− 1.26529i	−0.774442	−	0.632645i	$-0.781969\pi$
$853$	− 31522.0i	− 1.26529i	0.774442	−	0.632645i	$-0.218031\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 37026.0i	− 1.47583i	−0.674895	−	0.737914i	$-0.735811\pi$
$857$	− 37026.0i	− 1.47583i	0.674895	−	0.737914i	$-0.264189\pi$
$858$	0	0
$859$	−34380.0	−1.36558	−0.682788	−	0.730616i	$-0.739233\pi$
$859$	−34380.0	−1.36558	−0.682788	+	0.730616i	$0.739233\pi$
$860$	0	0
$861$	8680.00	0.343570
$862$	0	0
$863$	− 29172.0i	− 1.15067i	−0.817919	−	0.575334i	$-0.804872\pi$
$863$	− 29172.0i	− 1.15067i	0.817919	−	0.575334i	$-0.195128\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	17720.0i	0.694121i
$868$	0	0
$869$	27261.0	1.06417
$870$	0	0
$871$	6612.00	0.257221
$872$	0	0
$873$	− 446.000i	− 0.0172907i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	18178.0i	0.699917i	0.936765	+	0.349959i	$0.113804\pi$
$877$	18178.0i	0.699917i	−0.936765	+	0.349959i	$0.886196\pi$
$878$	0	0
$879$	−695.000	−0.0266687
$880$	0	0
$881$	2608.00	0.0997341	0.0498671	−	0.998756i	$-0.484120\pi$
$881$	2608.00	0.0997341	0.0498671	+	0.998756i	$0.484120\pi$
$882$	0	0
$883$	26880.0i	1.02444i	0.858853	+	0.512222i	$0.171177\pi$
$883$	26880.0i	1.02444i	−0.858853	+	0.512222i	$0.828823\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 31208.0i	− 1.18136i	−0.806908	−	0.590678i	$-0.798861\pi$
$887$	− 31208.0i	− 1.18136i	0.806908	−	0.590678i	$-0.201139\pi$
$888$	0	0
$889$	16408.0	0.619018
$890$	0	0
$891$	26169.0	0.983944
$892$	0	0
$893$	− 7722.00i	− 0.289369i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 8550.00i	− 0.318257i
$898$	0	0
$899$	3168.00	0.117529
$900$	0	0
$901$	24124.0	0.891994
$902$	0	0
$903$	− 6230.00i	− 0.229592i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 21494.0i	− 0.786876i	−0.919351	−	0.393438i	$-0.871286\pi$
$907$	− 21494.0i	− 0.786876i	0.919351	−	0.393438i	$-0.128714\pi$
$908$	0	0
$909$	1044.00	0.0380938
$910$	0	0
$911$	3704.00	0.134708	0.0673540	−	0.997729i	$-0.478544\pi$
$911$	3704.00	0.134708	0.0673540	+	0.997729i	$0.478544\pi$
$912$	0	0
$913$	4524.00i	0.163990i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	6706.00i	0.241496i
$918$	0	0
$919$	16185.0	0.580951	0.290475	−	0.956882i	$-0.406187\pi$
$919$	16185.0	0.580951	0.290475	+	0.956882i	$0.406187\pi$
$920$	0	0
$921$	33795.0	1.20910
$922$	0	0
$923$	− 1368.00i	− 0.0487847i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	3394.00i	0.120252i
$928$	0	0
$929$	−8500.00	−0.300189	−0.150095	−	0.988672i	$-0.547958\pi$
$929$	−8500.00	−0.300189	−0.150095	+	0.988672i	$0.547958\pi$
$930$	0	0
$931$	−882.000	−0.0310487
$932$	0	0
$933$	5190.00i	0.182115i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 33415.0i	− 1.16502i	−0.812825	−	0.582508i	$-0.802071\pi$
$937$	− 33415.0i	− 1.16502i	0.812825	−	0.582508i	$-0.197929\pi$
$938$	0	0
$939$	47535.0	1.65202
$940$	0	0
$941$	−25884.0	−0.896700	−0.448350	−	0.893858i	$-0.647988\pi$
$941$	−25884.0	−0.896700	−0.448350	+	0.893858i	$0.647988\pi$
$942$	0	0
$943$	22320.0i	0.770773i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 26404.0i	− 0.906035i	−0.891502	−	0.453017i	$-0.850348\pi$
$947$	− 26404.0i	− 0.906035i	0.891502	−	0.453017i	$-0.149652\pi$
$948$	0	0
$949$	−22610.0	−0.773395
$950$	0	0
$951$	−6450.00	−0.219932
$952$	0	0
$953$	− 21068.0i	− 0.716117i	−0.933699	−	0.358058i	$-0.883439\pi$
$953$	− 21068.0i	− 0.716117i	0.933699	−	0.358058i	$-0.116561\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	19305.0i	0.652082i
$958$	0	0
$959$	5376.00	0.181022
$960$	0	0
$961$	−28767.0	−0.965627
$962$	0	0
$963$	− 492.000i	− 0.0164636i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 298.000i	− 0.00991007i	−0.999988	−	0.00495503i	$-0.998423\pi$
$967$	− 298.000i	− 0.00991007i	0.999988	−	0.00495503i	$-0.00157724\pi$
$968$	0	0
$969$	−3330.00	−0.110397
$970$	0	0
$971$	−8972.00	−0.296524	−0.148262	−	0.988948i	$-0.547368\pi$
$971$	−8972.00	−0.296524	−0.148262	+	0.988948i	$0.547368\pi$
$972$	0	0
$973$	854.000i	0.0281377i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 7874.00i	− 0.257842i	−0.991655	−	0.128921i	$-0.958849\pi$
$977$	− 7874.00i	− 0.257842i	0.991655	−	0.128921i	$-0.0411514\pi$
$978$	0	0
$979$	−27456.0	−0.896320
$980$	0	0
$981$	−690.000	−0.0224567
$982$	0	0
$983$	− 47585.0i	− 1.54397i	−0.635639	−	0.771987i	$-0.719263\pi$
$983$	− 47585.0i	− 1.54397i	0.635639	−	0.771987i	$-0.280737\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	15015.0i	0.484228i
$988$	0	0
$989$	16020.0	0.515072
$990$	0	0
$991$	−42016.0	−1.34680	−0.673402	−	0.739277i	$-0.735168\pi$
$991$	−42016.0	−1.34680	−0.673402	+	0.739277i	$0.735168\pi$
$992$	0	0
$993$	− 27020.0i	− 0.863498i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 1867.00i	− 0.0593064i	−0.999560	−	0.0296532i	$-0.990560\pi$
$997$	− 1867.00i	− 0.0593064i	0.999560	−	0.0296532i	$-0.00944029\pi$
$998$	0	0
$999$	6670.00	0.211241

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1400.4.g.c.449.2		2
5.2	odd	4		280.4.a.c.1.1	✓	1
5.3	odd	4		1400.4.a.d.1.1		1
5.4	even	2	inner	1400.4.g.c.449.1		2
20.7	even	4		560.4.a.e.1.1		1
35.27	even	4		1960.4.a.d.1.1		1
40.27	even	4		2240.4.a.be.1.1		1
40.37	odd	4		2240.4.a.j.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.4.a.c.1.1	✓	1	5.2	odd	4
560.4.a.e.1.1		1	20.7	even	4
1400.4.a.d.1.1		1	5.3	odd	4
1400.4.g.c.449.1		2	5.4	even	2	inner
1400.4.g.c.449.2		2	1.1	even	1	trivial
1960.4.a.d.1.1		1	35.27	even	4
2240.4.a.j.1.1		1	40.37	odd	4
2240.4.a.be.1.1		1	40.27	even	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:	\( N \)	\(=\)	\( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1400.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+5.00000i q^{3} +7.00000i q^{7} +2.00000 q^{9} +O(q^{10})\)	\(q+5.00000i q^{3} +7.00000i q^{7} +2.00000 q^{9} -39.0000 q^{11} -19.0000i q^{13} +37.0000i q^{17} +18.0000 q^{19} -35.0000 q^{21} -90.0000i q^{23} +145.000i q^{27} -99.0000 q^{29} -32.0000 q^{31} -195.000i q^{33} -46.0000i q^{37} +95.0000 q^{39} -248.000 q^{41} +178.000i q^{43} -429.000i q^{47} -49.0000 q^{49} -185.000 q^{51} -652.000i q^{53} +90.0000i q^{57} -40.0000 q^{59} -36.0000 q^{61} +14.0000i q^{63} +348.000i q^{67} +450.000 q^{69} +72.0000 q^{71} -1190.00i q^{73} -273.000i q^{77} -699.000 q^{79} -671.000 q^{81} -116.000i q^{83} -495.000i q^{87} +704.000 q^{89} +133.000 q^{91} -160.000i q^{93} -223.000i q^{97} -78.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{9}+O(q^{10}) \) 2 * q + 4 * q^9	\( 2 q + 4 q^{9} - 78 q^{11} + 36 q^{19} - 70 q^{21} - 198 q^{29} - 64 q^{31} + 190 q^{39} - 496 q^{41} - 98 q^{49} - 370 q^{51} - 80 q^{59} - 72 q^{61} + 900 q^{69} + 144 q^{71} - 1398 q^{79} - 1342 q^{81} + 1408 q^{89} + 266 q^{91} - 156 q^{99}+O(q^{100}) \) 2 * q + 4 * q^9 - 78 * q^11 + 36 * q^19 - 70 * q^21 - 198 * q^29 - 64 * q^31 + 190 * q^39 - 496 * q^41 - 98 * q^49 - 370 * q^51 - 80 * q^59 - 72 * q^61 + 900 * q^69 + 144 * q^71 - 1398 * q^79 - 1342 * q^81 + 1408 * q^89 + 266 * q^91 - 156 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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