LMFDB - Embedded newform 2800.2.g.v.449.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2800,2,Mod(449,2800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2800, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2800.449");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2800 = 2^{4} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2800.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:	$22.3581125660$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1400)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.4
Root			$2.56155i$ of defining polynomial
Character	$\chi$	$=$	2800.449
Dual form			2800.2.g.v.449.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.56155i q^{3} -1.00000i q^{7} -3.56155 q^{9} +O(q^{10})$	$q+2.56155i q^{3} -1.00000i q^{7} -3.56155 q^{9} -2.12311 q^{11} -2.00000i q^{13} -2.56155i q^{17} -0.561553 q^{19} +2.56155 q^{21} +5.56155i q^{23} -1.43845i q^{27} -7.56155 q^{29} +0.876894 q^{31} -5.43845i q^{33} -11.8078i q^{37} +5.12311 q^{39} -6.56155 q^{41} +2.43845i q^{43} -8.24621i q^{47} -1.00000 q^{49} +6.56155 q^{51} -7.12311i q^{53} -1.43845i q^{57} -13.3693 q^{59} +2.87689 q^{61} +3.56155i q^{63} -16.1231i q^{67} -14.2462 q^{69} +10.6847 q^{71} -10.5616i q^{73} +2.12311i q^{77} +6.68466 q^{79} -7.00000 q^{81} -13.6847i q^{83} -19.3693i q^{87} +10.8078 q^{89} -2.00000 q^{91} +2.24621i q^{93} +6.00000i q^{97} +7.56155 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{9}+O(q^{10}) $ 4 * q - 6 * q^9	$ 4 q - 6 q^{9} + 8 q^{11} + 6 q^{19} + 2 q^{21} - 22 q^{29} + 20 q^{31} + 4 q^{39} - 18 q^{41} - 4 q^{49} + 18 q^{51} - 4 q^{59} + 28 q^{61} - 24 q^{69} + 18 q^{71} + 2 q^{79} - 28 q^{81} + 2 q^{89} - 8 q^{91} + 22 q^{99}+O(q^{100}) $ 4 * q - 6 * q^9 + 8 * q^11 + 6 * q^19 + 2 * q^21 - 22 * q^29 + 20 * q^31 + 4 * q^39 - 18 * q^41 - 4 * q^49 + 18 * q^51 - 4 * q^59 + 28 * q^61 - 24 * q^69 + 18 * q^71 + 2 * q^79 - 28 * q^81 + 2 * q^89 - 8 * q^91 + 22 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2800\mathbb{Z}\right)^\times$.

$n$	$351$	$801$	$2101$	$2577$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	2.56155i	1.47891i	0.673204	+	0.739457i	$0.264917\pi$
$3$	2.56155i	1.47891i	−0.673204	+	0.739457i	$0.735083\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	0	0
$9$	−3.56155	−1.18718
$10$	0	0
$11$	−2.12311	−0.640140	−0.320070	−	0.947394i	$-0.603707\pi$
$11$	−2.12311	−0.640140	−0.320070	+	0.947394i	$0.603707\pi$
$12$	0	0
$13$	− 2.00000i	− 0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$	− 2.00000i	− 0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 2.56155i	− 0.621268i	−0.950530	−	0.310634i	$-0.899459\pi$
$17$	− 2.56155i	− 0.621268i	0.950530	−	0.310634i	$-0.100541\pi$
$18$	0	0
$19$	−0.561553	−0.128829	−0.0644145	−	0.997923i	$-0.520518\pi$
$19$	−0.561553	−0.128829	−0.0644145	+	0.997923i	$0.520518\pi$
$20$	0	0
$21$	2.56155	0.558977
$22$	0	0
$23$	5.56155i	1.15966i	0.814736	+	0.579832i	$0.196882\pi$
$23$	5.56155i	1.15966i	−0.814736	+	0.579832i	$0.803118\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 1.43845i	− 0.276829i
$28$	0	0
$29$	−7.56155	−1.40415	−0.702073	−	0.712105i	$-0.747742\pi$
$29$	−7.56155	−1.40415	−0.702073	+	0.712105i	$0.747742\pi$
$30$	0	0
$31$	0.876894	0.157495	0.0787474	−	0.996895i	$-0.474908\pi$
$31$	0.876894	0.157495	0.0787474	+	0.996895i	$0.474908\pi$
$32$	0	0
$33$	− 5.43845i	− 0.946712i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 11.8078i	− 1.94118i	−0.240730	−	0.970592i	$-0.577387\pi$
$37$	− 11.8078i	− 1.94118i	0.240730	−	0.970592i	$-0.422613\pi$
$38$	0	0
$39$	5.12311	0.820353
$40$	0	0
$41$	−6.56155	−1.02474	−0.512371	−	0.858764i	$-0.671233\pi$
$41$	−6.56155	−1.02474	−0.512371	+	0.858764i	$0.671233\pi$
$42$	0	0
$43$	2.43845i	0.371860i	0.982563	+	0.185930i	$0.0595297\pi$
$43$	2.43845i	0.371860i	−0.982563	+	0.185930i	$0.940470\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 8.24621i	− 1.20283i	−0.798935	−	0.601417i	$-0.794603\pi$
$47$	− 8.24621i	− 1.20283i	0.798935	−	0.601417i	$-0.205397\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	6.56155	0.918801
$52$	0	0
$53$	− 7.12311i	− 0.978434i	−0.872162	−	0.489217i	$-0.837283\pi$
$53$	− 7.12311i	− 0.978434i	0.872162	−	0.489217i	$-0.162717\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 1.43845i	− 0.190527i
$58$	0	0
$59$	−13.3693	−1.74054	−0.870268	−	0.492578i	$-0.836055\pi$
$59$	−13.3693	−1.74054	−0.870268	+	0.492578i	$0.836055\pi$
$60$	0	0
$61$	2.87689	0.368349	0.184174	−	0.982894i	$-0.441039\pi$
$61$	2.87689	0.368349	0.184174	+	0.982894i	$0.441039\pi$
$62$	0	0
$63$	3.56155i	0.448713i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 16.1231i	− 1.96975i	−0.173264	−	0.984875i	$-0.555431\pi$
$67$	− 16.1231i	− 1.96975i	0.173264	−	0.984875i	$-0.444569\pi$
$68$	0	0
$69$	−14.2462	−1.71504
$70$	0	0
$71$	10.6847	1.26804	0.634018	−	0.773318i	$-0.281405\pi$
$71$	10.6847	1.26804	0.634018	+	0.773318i	$0.281405\pi$
$72$	0	0
$73$	− 10.5616i	− 1.23614i	−0.786125	−	0.618068i	$-0.787916\pi$
$73$	− 10.5616i	− 1.23614i	0.786125	−	0.618068i	$-0.212084\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.12311i	0.241950i
$78$	0	0
$79$	6.68466	0.752083	0.376041	−	0.926603i	$-0.377285\pi$
$79$	6.68466	0.752083	0.376041	+	0.926603i	$0.377285\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	− 13.6847i	− 1.50209i	−0.660253	−	0.751043i	$-0.729551\pi$
$83$	− 13.6847i	− 1.50209i	0.660253	−	0.751043i	$-0.270449\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 19.3693i	− 2.07661i
$88$	0	0
$89$	10.8078	1.14562	0.572810	−	0.819688i	$-0.305853\pi$
$89$	10.8078	1.14562	0.572810	+	0.819688i	$0.305853\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	0	0
$93$	2.24621i	0.232921i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	6.00000i	0.609208i	0.952479	+	0.304604i	$0.0985241\pi$
$97$	6.00000i	0.609208i	−0.952479	+	0.304604i	$0.901476\pi$
$98$	0	0
$99$	7.56155	0.759965
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	7.12311i	0.701860i	0.936402	+	0.350930i	$0.114135\pi$
$103$	7.12311i	0.701860i	−0.936402	+	0.350930i	$0.885865\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	12.8078i	1.23817i	0.785323	+	0.619087i	$0.212497\pi$
$107$	12.8078i	1.23817i	−0.785323	+	0.619087i	$0.787503\pi$
$108$	0	0
$109$	8.93087	0.855422	0.427711	−	0.903915i	$-0.359320\pi$
$109$	8.93087	0.855422	0.427711	+	0.903915i	$0.359320\pi$
$110$	0	0
$111$	30.2462	2.87084
$112$	0	0
$113$	− 0.753789i	− 0.0709105i	−0.999371	−	0.0354552i	$-0.988712\pi$
$113$	− 0.753789i	− 0.0709105i	0.999371	−	0.0354552i	$-0.0112881\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	7.12311i	0.658531i
$118$	0	0
$119$	−2.56155	−0.234817
$120$	0	0
$121$	−6.49242	−0.590220
$122$	0	0
$123$	− 16.8078i	− 1.51551i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	9.80776i	0.870298i	0.900358	+	0.435149i	$0.143304\pi$
$127$	9.80776i	0.870298i	−0.900358	+	0.435149i	$0.856696\pi$
$128$	0	0
$129$	−6.24621	−0.549948
$130$	0	0
$131$	1.75379	0.153229	0.0766146	−	0.997061i	$-0.475589\pi$
$131$	1.75379	0.153229	0.0766146	+	0.997061i	$0.475589\pi$
$132$	0	0
$133$	0.561553i	0.0486928i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 2.31534i	− 0.197813i	−0.995097	−	0.0989065i	$-0.968466\pi$
$137$	− 2.31534i	− 0.197813i	0.995097	−	0.0989065i	$-0.0315345\pi$
$138$	0	0
$139$	−19.6847	−1.66963	−0.834815	−	0.550530i	$-0.814426\pi$
$139$	−19.6847	−1.66963	−0.834815	+	0.550530i	$0.814426\pi$
$140$	0	0
$141$	21.1231	1.77889
$142$	0	0
$143$	4.24621i	0.355086i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 2.56155i	− 0.211273i
$148$	0	0
$149$	−17.5616	−1.43870	−0.719349	−	0.694649i	$-0.755560\pi$
$149$	−17.5616	−1.43870	−0.719349	+	0.694649i	$0.755560\pi$
$150$	0	0
$151$	5.56155	0.452593	0.226296	−	0.974058i	$-0.427338\pi$
$151$	5.56155	0.452593	0.226296	+	0.974058i	$0.427338\pi$
$152$	0	0
$153$	9.12311i	0.737559i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	24.4924i	1.95471i	0.211611	+	0.977354i	$0.432129\pi$
$157$	24.4924i	1.95471i	−0.211611	+	0.977354i	$0.567871\pi$
$158$	0	0
$159$	18.2462	1.44702
$160$	0	0
$161$	5.56155	0.438312
$162$	0	0
$163$	− 13.9309i	− 1.09115i	−0.838062	−	0.545575i	$-0.816311\pi$
$163$	− 13.9309i	− 1.09115i	0.838062	−	0.545575i	$-0.183689\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	19.6155i	1.51790i	0.651152	+	0.758948i	$0.274286\pi$
$167$	19.6155i	1.51790i	−0.651152	+	0.758948i	$0.725714\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	2.00000	0.152944
$172$	0	0
$173$	− 14.4924i	− 1.10184i	−0.834559	−	0.550919i	$-0.814277\pi$
$173$	− 14.4924i	− 1.10184i	0.834559	−	0.550919i	$-0.185723\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 34.2462i	− 2.57410i
$178$	0	0
$179$	−8.31534	−0.621518	−0.310759	−	0.950489i	$-0.600583\pi$
$179$	−8.31534	−0.621518	−0.310759	+	0.950489i	$0.600583\pi$
$180$	0	0
$181$	−2.87689	−0.213838	−0.106919	−	0.994268i	$-0.534099\pi$
$181$	−2.87689	−0.213838	−0.106919	+	0.994268i	$0.534099\pi$
$182$	0	0
$183$	7.36932i	0.544756i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	5.43845i	0.397699i
$188$	0	0
$189$	−1.43845	−0.104632
$190$	0	0
$191$	−0.630683	−0.0456346	−0.0228173	−	0.999740i	$-0.507264\pi$
$191$	−0.630683	−0.0456346	−0.0228173	+	0.999740i	$0.507264\pi$
$192$	0	0
$193$	− 20.6155i	− 1.48394i	−0.670434	−	0.741969i	$-0.733892\pi$
$193$	− 20.6155i	− 1.48394i	0.670434	−	0.741969i	$-0.266108\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 3.56155i	− 0.253750i	−0.991919	−	0.126875i	$-0.959505\pi$
$197$	− 3.56155i	− 0.253750i	0.991919	−	0.126875i	$-0.0404947\pi$
$198$	0	0
$199$	−13.1231	−0.930272	−0.465136	−	0.885239i	$-0.653995\pi$
$199$	−13.1231	−0.930272	−0.465136	+	0.885239i	$0.653995\pi$
$200$	0	0
$201$	41.3002	2.91309
$202$	0	0
$203$	7.56155i	0.530717i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 19.8078i	− 1.37673i
$208$	0	0
$209$	1.19224	0.0824687
$210$	0	0
$211$	−10.5616	−0.727087	−0.363544	−	0.931577i	$-0.618433\pi$
$211$	−10.5616	−0.727087	−0.363544	+	0.931577i	$0.618433\pi$
$212$	0	0
$213$	27.3693i	1.87531i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 0.876894i	− 0.0595275i
$218$	0	0
$219$	27.0540	1.82814
$220$	0	0
$221$	−5.12311	−0.344617
$222$	0	0
$223$	− 0.630683i	− 0.0422337i	−0.999777	−	0.0211168i	$-0.993278\pi$
$223$	− 0.630683i	− 0.0422337i	0.999777	−	0.0211168i	$-0.00672220\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	13.3693i	0.887353i	0.896187	+	0.443676i	$0.146326\pi$
$227$	13.3693i	0.887353i	−0.896187	+	0.443676i	$0.853674\pi$
$228$	0	0
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	0	0
$231$	−5.43845	−0.357824
$232$	0	0
$233$	− 3.56155i	− 0.233325i	−0.993172	−	0.116663i	$-0.962780\pi$
$233$	− 3.56155i	− 0.233325i	0.993172	−	0.116663i	$-0.0372196\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	17.1231i	1.11227i
$238$	0	0
$239$	−4.00000	−0.258738	−0.129369	−	0.991596i	$-0.541295\pi$
$239$	−4.00000	−0.258738	−0.129369	+	0.991596i	$0.541295\pi$
$240$	0	0
$241$	25.0540	1.61387	0.806934	−	0.590641i	$-0.201125\pi$
$241$	25.0540	1.61387	0.806934	+	0.590641i	$0.201125\pi$
$242$	0	0
$243$	− 22.2462i	− 1.42710i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.12311i	0.0714615i
$248$	0	0
$249$	35.0540	2.22146
$250$	0	0
$251$	12.3153	0.777337	0.388669	−	0.921378i	$-0.372935\pi$
$251$	12.3153	0.777337	0.388669	+	0.921378i	$0.372935\pi$
$252$	0	0
$253$	− 11.8078i	− 0.742348i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 4.24621i	− 0.264871i	−0.991192	−	0.132436i	$-0.957720\pi$
$257$	− 4.24621i	− 0.264871i	0.991192	−	0.132436i	$-0.0422798\pi$
$258$	0	0
$259$	−11.8078	−0.733699
$260$	0	0
$261$	26.9309	1.66698
$262$	0	0
$263$	− 10.6847i	− 0.658844i	−0.944183	−	0.329422i	$-0.893146\pi$
$263$	− 10.6847i	− 0.658844i	0.944183	−	0.329422i	$-0.106854\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	27.6847i	1.69427i
$268$	0	0
$269$	−11.7538	−0.716641	−0.358321	−	0.933599i	$-0.616651\pi$
$269$	−11.7538	−0.716641	−0.358321	+	0.933599i	$0.616651\pi$
$270$	0	0
$271$	−16.2462	−0.986887	−0.493444	−	0.869778i	$-0.664262\pi$
$271$	−16.2462	−0.986887	−0.493444	+	0.869778i	$0.664262\pi$
$272$	0	0
$273$	− 5.12311i	− 0.310064i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 10.4924i	− 0.630429i	−0.949021	−	0.315214i	$-0.897924\pi$
$277$	− 10.4924i	− 0.630429i	0.949021	−	0.315214i	$-0.102076\pi$
$278$	0	0
$279$	−3.12311	−0.186975
$280$	0	0
$281$	11.5616	0.689704	0.344852	−	0.938657i	$-0.387929\pi$
$281$	11.5616	0.689704	0.344852	+	0.938657i	$0.387929\pi$
$282$	0	0
$283$	− 13.6847i	− 0.813469i	−0.913547	−	0.406734i	$-0.866667\pi$
$283$	− 13.6847i	− 0.813469i	0.913547	−	0.406734i	$-0.133333\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	6.56155i	0.387316i
$288$	0	0
$289$	10.4384	0.614026
$290$	0	0
$291$	−15.3693	−0.900965
$292$	0	0
$293$	30.0000i	1.75262i	0.481749	+	0.876309i	$0.340002\pi$
$293$	30.0000i	1.75262i	−0.481749	+	0.876309i	$0.659998\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	3.05398i	0.177210i
$298$	0	0
$299$	11.1231	0.643266
$300$	0	0
$301$	2.43845	0.140550
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 4.31534i	− 0.246290i	−0.992389	−	0.123145i	$-0.960702\pi$
$307$	− 4.31534i	− 0.246290i	0.992389	−	0.123145i	$-0.0392980\pi$
$308$	0	0
$309$	−18.2462	−1.03799
$310$	0	0
$311$	−2.87689	−0.163134	−0.0815669	−	0.996668i	$-0.525992\pi$
$311$	−2.87689	−0.163134	−0.0815669	+	0.996668i	$0.525992\pi$
$312$	0	0
$313$	− 22.0000i	− 1.24351i	−0.783210	−	0.621757i	$-0.786419\pi$
$313$	− 22.0000i	− 1.24351i	0.783210	−	0.621757i	$-0.213581\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 20.6847i	− 1.16177i	−0.813987	−	0.580883i	$-0.802707\pi$
$317$	− 20.6847i	− 1.16177i	0.813987	−	0.580883i	$-0.197293\pi$
$318$	0	0
$319$	16.0540	0.898850
$320$	0	0
$321$	−32.8078	−1.83115
$322$	0	0
$323$	1.43845i	0.0800373i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	22.8769i	1.26510i
$328$	0	0
$329$	−8.24621	−0.454628
$330$	0	0
$331$	−27.0000	−1.48405	−0.742027	−	0.670370i	$-0.766135\pi$
$331$	−27.0000	−1.48405	−0.742027	+	0.670370i	$0.766135\pi$
$332$	0	0
$333$	42.0540i	2.30454i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 5.68466i	− 0.309663i	−0.987941	−	0.154832i	$-0.950517\pi$
$337$	− 5.68466i	− 0.309663i	0.987941	−	0.154832i	$-0.0494835\pi$
$338$	0	0
$339$	1.93087	0.104870
$340$	0	0
$341$	−1.86174	−0.100819
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 22.6155i	− 1.21407i	−0.794677	−	0.607033i	$-0.792360\pi$
$347$	− 22.6155i	− 1.21407i	0.794677	−	0.607033i	$-0.207640\pi$
$348$	0	0
$349$	12.8769	0.689284	0.344642	−	0.938734i	$-0.388000\pi$
$349$	12.8769	0.689284	0.344642	+	0.938734i	$0.388000\pi$
$350$	0	0
$351$	−2.87689	−0.153557
$352$	0	0
$353$	11.7538i	0.625591i	0.949820	+	0.312796i	$0.101265\pi$
$353$	11.7538i	0.625591i	−0.949820	+	0.312796i	$0.898735\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 6.56155i	− 0.347274i
$358$	0	0
$359$	−14.6847	−0.775027	−0.387513	−	0.921864i	$-0.626666\pi$
$359$	−14.6847	−0.775027	−0.387513	+	0.921864i	$0.626666\pi$
$360$	0	0
$361$	−18.6847	−0.983403
$362$	0	0
$363$	− 16.6307i	− 0.872884i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 17.7538i	− 0.926740i	−0.886165	−	0.463370i	$-0.846640\pi$
$367$	− 17.7538i	− 0.926740i	0.886165	−	0.463370i	$-0.153360\pi$
$368$	0	0
$369$	23.3693	1.21656
$370$	0	0
$371$	−7.12311	−0.369813
$372$	0	0
$373$	− 14.0540i	− 0.727687i	−0.931460	−	0.363844i	$-0.881464\pi$
$373$	− 14.0540i	− 0.727687i	0.931460	−	0.363844i	$-0.118536\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	15.1231i	0.778880i
$378$	0	0
$379$	−24.8617	−1.27706	−0.638531	−	0.769596i	$-0.720458\pi$
$379$	−24.8617	−1.27706	−0.638531	+	0.769596i	$0.720458\pi$
$380$	0	0
$381$	−25.1231	−1.28710
$382$	0	0
$383$	− 20.2462i	− 1.03453i	−0.855824	−	0.517267i	$-0.826950\pi$
$383$	− 20.2462i	− 1.03453i	0.855824	−	0.517267i	$-0.173050\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 8.68466i	− 0.441466i
$388$	0	0
$389$	−2.43845	−0.123634	−0.0618171	−	0.998087i	$-0.519690\pi$
$389$	−2.43845	−0.123634	−0.0618171	+	0.998087i	$0.519690\pi$
$390$	0	0
$391$	14.2462	0.720462
$392$	0	0
$393$	4.49242i	0.226613i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	38.4924i	1.93188i	0.258767	+	0.965940i	$0.416684\pi$
$397$	38.4924i	1.93188i	−0.258767	+	0.965940i	$0.583316\pi$
$398$	0	0
$399$	−1.43845	−0.0720124
$400$	0	0
$401$	−31.4924	−1.57266	−0.786328	−	0.617809i	$-0.788021\pi$
$401$	−31.4924	−1.57266	−0.786328	+	0.617809i	$0.788021\pi$
$402$	0	0
$403$	− 1.75379i	− 0.0873624i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	25.0691i	1.24263i
$408$	0	0
$409$	5.19224	0.256740	0.128370	−	0.991726i	$-0.459026\pi$
$409$	5.19224	0.256740	0.128370	+	0.991726i	$0.459026\pi$
$410$	0	0
$411$	5.93087	0.292548
$412$	0	0
$413$	13.3693i	0.657861i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 50.4233i	− 2.46924i
$418$	0	0
$419$	−18.8078	−0.918819	−0.459410	−	0.888224i	$-0.651939\pi$
$419$	−18.8078	−0.918819	−0.459410	+	0.888224i	$0.651939\pi$
$420$	0	0
$421$	5.06913	0.247054	0.123527	−	0.992341i	$-0.460579\pi$
$421$	5.06913	0.247054	0.123527	+	0.992341i	$0.460579\pi$
$422$	0	0
$423$	29.3693i	1.42799i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 2.87689i	− 0.139223i
$428$	0	0
$429$	−10.8769	−0.525141
$430$	0	0
$431$	13.7538	0.662497	0.331248	−	0.943544i	$-0.392530\pi$
$431$	13.7538	0.662497	0.331248	+	0.943544i	$0.392530\pi$
$432$	0	0
$433$	35.5464i	1.70825i	0.520067	+	0.854125i	$0.325907\pi$
$433$	35.5464i	1.70825i	−0.520067	+	0.854125i	$0.674093\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 3.12311i	− 0.149398i
$438$	0	0
$439$	−7.12311	−0.339967	−0.169984	−	0.985447i	$-0.554371\pi$
$439$	−7.12311	−0.339967	−0.169984	+	0.985447i	$0.554371\pi$
$440$	0	0
$441$	3.56155	0.169598
$442$	0	0
$443$	22.5616i	1.07193i	0.844240	+	0.535966i	$0.180052\pi$
$443$	22.5616i	1.07193i	−0.844240	+	0.535966i	$0.819948\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 44.9848i	− 2.12771i
$448$	0	0
$449$	−10.3693	−0.489358	−0.244679	−	0.969604i	$-0.578683\pi$
$449$	−10.3693	−0.489358	−0.244679	+	0.969604i	$0.578683\pi$
$450$	0	0
$451$	13.9309	0.655979
$452$	0	0
$453$	14.2462i	0.669345i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	37.1080i	1.73584i	0.496707	+	0.867918i	$0.334542\pi$
$457$	37.1080i	1.73584i	−0.496707	+	0.867918i	$0.665458\pi$
$458$	0	0
$459$	−3.68466	−0.171985
$460$	0	0
$461$	6.49242	0.302382	0.151191	−	0.988505i	$-0.451689\pi$
$461$	6.49242	0.302382	0.151191	+	0.988505i	$0.451689\pi$
$462$	0	0
$463$	− 20.4924i	− 0.952364i	−0.879347	−	0.476182i	$-0.842020\pi$
$463$	− 20.4924i	− 0.952364i	0.879347	−	0.476182i	$-0.157980\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	24.8769i	1.15117i	0.817744	+	0.575583i	$0.195225\pi$
$467$	24.8769i	1.15117i	−0.817744	+	0.575583i	$0.804775\pi$
$468$	0	0
$469$	−16.1231	−0.744496
$470$	0	0
$471$	−62.7386	−2.89084
$472$	0	0
$473$	− 5.17708i	− 0.238042i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	25.3693i	1.16158i
$478$	0	0
$479$	−40.7386	−1.86140	−0.930698	−	0.365789i	$-0.880799\pi$
$479$	−40.7386	−1.86140	−0.930698	+	0.365789i	$0.880799\pi$
$480$	0	0
$481$	−23.6155	−1.07678
$482$	0	0
$483$	14.2462i	0.648225i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 35.5616i	− 1.61145i	−0.592291	−	0.805724i	$-0.701777\pi$
$487$	− 35.5616i	− 1.61145i	0.592291	−	0.805724i	$-0.298223\pi$
$488$	0	0
$489$	35.6847	1.61372
$490$	0	0
$491$	−20.6847	−0.933486	−0.466743	−	0.884393i	$-0.654573\pi$
$491$	−20.6847	−0.933486	−0.466743	+	0.884393i	$0.654573\pi$
$492$	0	0
$493$	19.3693i	0.872350i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 10.6847i	− 0.479272i
$498$	0	0
$499$	−16.4924	−0.738302	−0.369151	−	0.929369i	$-0.620352\pi$
$499$	−16.4924	−0.738302	−0.369151	+	0.929369i	$0.620352\pi$
$500$	0	0
$501$	−50.2462	−2.24484
$502$	0	0
$503$	19.7538i	0.880778i	0.897807	+	0.440389i	$0.145159\pi$
$503$	19.7538i	0.880778i	−0.897807	+	0.440389i	$0.854841\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	23.0540i	1.02386i
$508$	0	0
$509$	−26.0000	−1.15243	−0.576215	−	0.817298i	$-0.695471\pi$
$509$	−26.0000	−1.15243	−0.576215	+	0.817298i	$0.695471\pi$
$510$	0	0
$511$	−10.5616	−0.467216
$512$	0	0
$513$	0.807764i	0.0356637i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	17.5076i	0.769982i
$518$	0	0
$519$	37.1231	1.62952
$520$	0	0
$521$	−27.4384	−1.20210	−0.601050	−	0.799211i	$-0.705251\pi$
$521$	−27.4384	−1.20210	−0.601050	+	0.799211i	$0.705251\pi$
$522$	0	0
$523$	28.3153i	1.23814i	0.785334	+	0.619072i	$0.212491\pi$
$523$	28.3153i	1.23814i	−0.785334	+	0.619072i	$0.787509\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 2.24621i	− 0.0978465i
$528$	0	0
$529$	−7.93087	−0.344820
$530$	0	0
$531$	47.6155	2.06634
$532$	0	0
$533$	13.1231i	0.568425i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 21.3002i	− 0.919171i
$538$	0	0
$539$	2.12311	0.0914486
$540$	0	0
$541$	41.4233	1.78093	0.890463	−	0.455055i	$-0.150380\pi$
$541$	41.4233	1.78093	0.890463	+	0.455055i	$0.150380\pi$
$542$	0	0
$543$	− 7.36932i	− 0.316248i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 33.7386i	− 1.44256i	−0.692644	−	0.721280i	$-0.743554\pi$
$547$	− 33.7386i	− 1.44256i	0.692644	−	0.721280i	$-0.256446\pi$
$548$	0	0
$549$	−10.2462	−0.437298
$550$	0	0
$551$	4.24621	0.180895
$552$	0	0
$553$	− 6.68466i	− 0.284261i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	9.56155i	0.405136i	0.979268	+	0.202568i	$0.0649287\pi$
$557$	9.56155i	0.405136i	−0.979268	+	0.202568i	$0.935071\pi$
$558$	0	0
$559$	4.87689	0.206271
$560$	0	0
$561$	−13.9309	−0.588162
$562$	0	0
$563$	− 7.12311i	− 0.300203i	−0.988671	−	0.150102i	$-0.952040\pi$
$563$	− 7.12311i	− 0.300203i	0.988671	−	0.150102i	$-0.0479601\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	7.00000i	0.293972i
$568$	0	0
$569$	19.9848	0.837808	0.418904	−	0.908030i	$-0.362414\pi$
$569$	19.9848	0.837808	0.418904	+	0.908030i	$0.362414\pi$
$570$	0	0
$571$	−27.3153	−1.14311	−0.571556	−	0.820563i	$-0.693660\pi$
$571$	−27.3153	−1.14311	−0.571556	+	0.820563i	$0.693660\pi$
$572$	0	0
$573$	− 1.61553i	− 0.0674897i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 45.5464i	− 1.89612i	−0.318090	−	0.948061i	$-0.603041\pi$
$577$	− 45.5464i	− 1.89612i	0.318090	−	0.948061i	$-0.396959\pi$
$578$	0	0
$579$	52.8078	2.19462
$580$	0	0
$581$	−13.6847	−0.567735
$582$	0	0
$583$	15.1231i	0.626335i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 8.17708i	− 0.337504i	−0.985659	−	0.168752i	$-0.946026\pi$
$587$	− 8.17708i	− 0.337504i	0.985659	−	0.168752i	$-0.0539737\pi$
$588$	0	0
$589$	−0.492423	−0.0202899
$590$	0	0
$591$	9.12311	0.375274
$592$	0	0
$593$	− 12.3153i	− 0.505730i	−0.967502	−	0.252865i	$-0.918627\pi$
$593$	− 12.3153i	− 0.505730i	0.967502	−	0.252865i	$-0.0813729\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 33.6155i	− 1.37579i
$598$	0	0
$599$	24.3002	0.992879	0.496439	−	0.868071i	$-0.334641\pi$
$599$	24.3002	0.992879	0.496439	+	0.868071i	$0.334641\pi$
$600$	0	0
$601$	−4.94602	−0.201753	−0.100876	−	0.994899i	$-0.532165\pi$
$601$	−4.94602	−0.201753	−0.100876	+	0.994899i	$0.532165\pi$
$602$	0	0
$603$	57.4233i	2.33846i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	37.3693i	1.51677i	0.651805	+	0.758387i	$0.274012\pi$
$607$	37.3693i	1.51677i	−0.651805	+	0.758387i	$0.725988\pi$
$608$	0	0
$609$	−19.3693	−0.784884
$610$	0	0
$611$	−16.4924	−0.667212
$612$	0	0
$613$	6.68466i	0.269991i	0.990846	+	0.134995i	$0.0431020\pi$
$613$	6.68466i	0.269991i	−0.990846	+	0.134995i	$0.956898\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	17.8078i	0.716914i	0.933546	+	0.358457i	$0.116697\pi$
$617$	17.8078i	0.716914i	−0.933546	+	0.358457i	$0.883303\pi$
$618$	0	0
$619$	29.3693	1.18045	0.590226	−	0.807238i	$-0.299039\pi$
$619$	29.3693	1.18045	0.590226	+	0.807238i	$0.299039\pi$
$620$	0	0
$621$	8.00000	0.321029
$622$	0	0
$623$	− 10.8078i	− 0.433004i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	3.05398i	0.121964i
$628$	0	0
$629$	−30.2462	−1.20600
$630$	0	0
$631$	−34.0540	−1.35567	−0.677834	−	0.735215i	$-0.737081\pi$
$631$	−34.0540	−1.35567	−0.677834	+	0.735215i	$0.737081\pi$
$632$	0	0
$633$	− 27.0540i	− 1.07530i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	2.00000i	0.0792429i
$638$	0	0
$639$	−38.0540	−1.50539
$640$	0	0
$641$	−16.4384	−0.649280	−0.324640	−	0.945838i	$-0.605243\pi$
$641$	−16.4384	−0.649280	−0.324640	+	0.945838i	$0.605243\pi$
$642$	0	0
$643$	2.73863i	0.108001i	0.998541	+	0.0540006i	$0.0171973\pi$
$643$	2.73863i	0.108001i	−0.998541	+	0.0540006i	$0.982803\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	25.3693i	0.997371i	0.866783	+	0.498685i	$0.166184\pi$
$647$	25.3693i	0.997371i	−0.866783	+	0.498685i	$0.833816\pi$
$648$	0	0
$649$	28.3845	1.11419
$650$	0	0
$651$	2.24621	0.0880360
$652$	0	0
$653$	− 33.8617i	− 1.32511i	−0.749012	−	0.662556i	$-0.769472\pi$
$653$	− 33.8617i	− 1.32511i	0.749012	−	0.662556i	$-0.230528\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	37.6155i	1.46752i
$658$	0	0
$659$	18.5616	0.723055	0.361528	−	0.932361i	$-0.382255\pi$
$659$	18.5616	0.723055	0.361528	+	0.932361i	$0.382255\pi$
$660$	0	0
$661$	−22.4924	−0.874854	−0.437427	−	0.899254i	$-0.644110\pi$
$661$	−22.4924	−0.874854	−0.437427	+	0.899254i	$0.644110\pi$
$662$	0	0
$663$	− 13.1231i	− 0.509659i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 42.0540i	− 1.62834i
$668$	0	0
$669$	1.61553	0.0624599
$670$	0	0
$671$	−6.10795	−0.235795
$672$	0	0
$673$	− 46.4924i	− 1.79215i	−0.443901	−	0.896076i	$-0.646406\pi$
$673$	− 46.4924i	− 1.79215i	0.443901	−	0.896076i	$-0.353594\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	14.6307i	0.562303i	0.959663	+	0.281151i	$0.0907163\pi$
$677$	14.6307i	0.562303i	−0.959663	+	0.281151i	$0.909284\pi$
$678$	0	0
$679$	6.00000	0.230259
$680$	0	0
$681$	−34.2462	−1.31232
$682$	0	0
$683$	34.1231i	1.30568i	0.757494	+	0.652842i	$0.226424\pi$
$683$	34.1231i	1.30568i	−0.757494	+	0.652842i	$0.773576\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	35.8617i	1.36821i
$688$	0	0
$689$	−14.2462	−0.542737
$690$	0	0
$691$	17.4384	0.663390	0.331695	−	0.943387i	$-0.392380\pi$
$691$	17.4384	0.663390	0.331695	+	0.943387i	$0.392380\pi$
$692$	0	0
$693$	− 7.56155i	− 0.287240i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	16.8078i	0.636639i
$698$	0	0
$699$	9.12311	0.345068
$700$	0	0
$701$	44.1080	1.66593	0.832967	−	0.553322i	$-0.186640\pi$
$701$	44.1080	1.66593	0.832967	+	0.553322i	$0.186640\pi$
$702$	0	0
$703$	6.63068i	0.250081i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−37.3693	−1.40343	−0.701717	−	0.712456i	$-0.747583\pi$
$709$	−37.3693	−1.40343	−0.701717	+	0.712456i	$0.747583\pi$
$710$	0	0
$711$	−23.8078	−0.892861
$712$	0	0
$713$	4.87689i	0.182641i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 10.2462i	− 0.382652i
$718$	0	0
$719$	38.2462	1.42634	0.713171	−	0.700990i	$-0.247258\pi$
$719$	38.2462	1.42634	0.713171	+	0.700990i	$0.247258\pi$
$720$	0	0
$721$	7.12311	0.265278
$722$	0	0
$723$	64.1771i	2.38677i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1.61553i	0.0599166i	0.999551	+	0.0299583i	$0.00953745\pi$
$727$	1.61553i	0.0599166i	−0.999551	+	0.0299583i	$0.990463\pi$
$728$	0	0
$729$	35.9848	1.33277
$730$	0	0
$731$	6.24621	0.231024
$732$	0	0
$733$	− 22.2462i	− 0.821683i	−0.911707	−	0.410841i	$-0.865235\pi$
$733$	− 22.2462i	− 0.821683i	0.911707	−	0.410841i	$-0.134765\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	34.2311i	1.26092i
$738$	0	0
$739$	17.1771	0.631869	0.315935	−	0.948781i	$-0.397682\pi$
$739$	17.1771	0.631869	0.315935	+	0.948781i	$0.397682\pi$
$740$	0	0
$741$	−2.87689	−0.105685
$742$	0	0
$743$	− 18.7386i	− 0.687454i	−0.939070	−	0.343727i	$-0.888311\pi$
$743$	− 18.7386i	− 0.687454i	0.939070	−	0.343727i	$-0.111689\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	48.7386i	1.78325i
$748$	0	0
$749$	12.8078	0.467986
$750$	0	0
$751$	−27.3693	−0.998721	−0.499360	−	0.866394i	$-0.666432\pi$
$751$	−27.3693	−0.998721	−0.499360	+	0.866394i	$0.666432\pi$
$752$	0	0
$753$	31.5464i	1.14961i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	13.3153i	0.483954i	0.970282	+	0.241977i	$0.0777959\pi$
$757$	13.3153i	0.483954i	−0.970282	+	0.241977i	$0.922204\pi$
$758$	0	0
$759$	30.2462	1.09787
$760$	0	0
$761$	34.4233	1.24784	0.623922	−	0.781487i	$-0.285538\pi$
$761$	34.4233	1.24784	0.623922	+	0.781487i	$0.285538\pi$
$762$	0	0
$763$	− 8.93087i	− 0.323319i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	26.7386i	0.965476i
$768$	0	0
$769$	33.3002	1.20084	0.600418	−	0.799687i	$-0.295001\pi$
$769$	33.3002	1.20084	0.600418	+	0.799687i	$0.295001\pi$
$770$	0	0
$771$	10.8769	0.391722
$772$	0	0
$773$	11.6155i	0.417782i	0.977939	+	0.208891i	$0.0669853\pi$
$773$	11.6155i	0.417782i	−0.977939	+	0.208891i	$0.933015\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 30.2462i	− 1.08508i
$778$	0	0
$779$	3.68466	0.132017
$780$	0	0
$781$	−22.6847	−0.811721
$782$	0	0
$783$	10.8769i	0.388708i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 26.2462i	− 0.935576i	−0.883841	−	0.467788i	$-0.845051\pi$
$787$	− 26.2462i	− 0.935576i	0.883841	−	0.467788i	$-0.154949\pi$
$788$	0	0
$789$	27.3693	0.974373
$790$	0	0
$791$	−0.753789	−0.0268016
$792$	0	0
$793$	− 5.75379i	− 0.204323i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	41.6155i	1.47410i	0.675840	+	0.737049i	$0.263781\pi$
$797$	41.6155i	1.47410i	−0.675840	+	0.737049i	$0.736219\pi$
$798$	0	0
$799$	−21.1231	−0.747282
$800$	0	0
$801$	−38.4924	−1.36006
$802$	0	0
$803$	22.4233i	0.791301i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 30.1080i	− 1.05985i
$808$	0	0
$809$	−49.4233	−1.73763	−0.868815	−	0.495136i	$-0.835118\pi$
$809$	−49.4233	−1.73763	−0.868815	+	0.495136i	$0.835118\pi$
$810$	0	0
$811$	−43.6155	−1.53155	−0.765774	−	0.643110i	$-0.777644\pi$
$811$	−43.6155	−1.53155	−0.765774	+	0.643110i	$0.777644\pi$
$812$	0	0
$813$	− 41.6155i	− 1.45952i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 1.36932i	− 0.0479063i
$818$	0	0
$819$	7.12311	0.248901
$820$	0	0
$821$	45.8617	1.60059	0.800293	−	0.599609i	$-0.204677\pi$
$821$	45.8617	1.60059	0.800293	+	0.599609i	$0.204677\pi$
$822$	0	0
$823$	12.4384i	0.433577i	0.976219	+	0.216789i	$0.0695582\pi$
$823$	12.4384i	0.433577i	−0.976219	+	0.216789i	$0.930442\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	9.24621i	0.321522i	0.986993	+	0.160761i	$0.0513949\pi$
$827$	9.24621i	0.321522i	−0.986993	+	0.160761i	$0.948605\pi$
$828$	0	0
$829$	2.24621	0.0780141	0.0390071	−	0.999239i	$-0.487581\pi$
$829$	2.24621	0.0780141	0.0390071	+	0.999239i	$0.487581\pi$
$830$	0	0
$831$	26.8769	0.932349
$832$	0	0
$833$	2.56155i	0.0887525i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 1.26137i	− 0.0435992i
$838$	0	0
$839$	−1.12311	−0.0387739	−0.0193870	−	0.999812i	$-0.506171\pi$
$839$	−1.12311	−0.0387739	−0.0193870	+	0.999812i	$0.506171\pi$
$840$	0	0
$841$	28.1771	0.971623
$842$	0	0
$843$	29.6155i	1.02001i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	6.49242i	0.223082i
$848$	0	0
$849$	35.0540	1.20305
$850$	0	0
$851$	65.6695	2.25112
$852$	0	0
$853$	− 10.8769i	− 0.372418i	−0.982510	−	0.186209i	$-0.940380\pi$
$853$	− 10.8769i	− 0.372418i	0.982510	−	0.186209i	$-0.0596201\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	4.94602i	0.168953i	0.996425	+	0.0844765i	$0.0269218\pi$
$857$	4.94602i	0.168953i	−0.996425	+	0.0844765i	$0.973078\pi$
$858$	0	0
$859$	34.1771	1.16611	0.583053	−	0.812434i	$-0.301858\pi$
$859$	34.1771	1.16611	0.583053	+	0.812434i	$0.301858\pi$
$860$	0	0
$861$	−16.8078	−0.572807
$862$	0	0
$863$	− 23.8078i	− 0.810426i	−0.914222	−	0.405213i	$-0.867197\pi$
$863$	− 23.8078i	− 0.810426i	0.914222	−	0.405213i	$-0.132803\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	26.7386i	0.908092i
$868$	0	0
$869$	−14.1922	−0.481439
$870$	0	0
$871$	−32.2462	−1.09262
$872$	0	0
$873$	− 21.3693i	− 0.723242i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	23.6155i	0.797440i	0.917073	+	0.398720i	$0.130545\pi$
$877$	23.6155i	0.797440i	−0.917073	+	0.398720i	$0.869455\pi$
$878$	0	0
$879$	−76.8466	−2.59197
$880$	0	0
$881$	−40.7386	−1.37252	−0.686260	−	0.727357i	$-0.740749\pi$
$881$	−40.7386	−1.37252	−0.686260	+	0.727357i	$0.740749\pi$
$882$	0	0
$883$	− 1.49242i	− 0.0502240i	−0.999685	−	0.0251120i	$-0.992006\pi$
$883$	− 1.49242i	− 0.0502240i	0.999685	−	0.0251120i	$-0.00799424\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	49.6155i	1.66593i	0.553328	+	0.832963i	$0.313357\pi$
$887$	49.6155i	1.66593i	−0.553328	+	0.832963i	$0.686643\pi$
$888$	0	0
$889$	9.80776	0.328942
$890$	0	0
$891$	14.8617	0.497887
$892$	0	0
$893$	4.63068i	0.154960i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	28.4924i	0.951334i
$898$	0	0
$899$	−6.63068	−0.221146
$900$	0	0
$901$	−18.2462	−0.607869
$902$	0	0
$903$	6.24621i	0.207861i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 4.00000i	− 0.132818i	−0.997792	−	0.0664089i	$-0.978846\pi$
$907$	− 4.00000i	− 0.132818i	0.997792	−	0.0664089i	$-0.0211542\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	39.4233	1.30615	0.653076	−	0.757292i	$-0.273478\pi$
$911$	39.4233	1.30615	0.653076	+	0.757292i	$0.273478\pi$
$912$	0	0
$913$	29.0540i	0.961546i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 1.75379i	− 0.0579152i
$918$	0	0
$919$	18.1922	0.600106	0.300053	−	0.953922i	$-0.402996\pi$
$919$	18.1922	0.600106	0.300053	+	0.953922i	$0.402996\pi$
$920$	0	0
$921$	11.0540	0.364241
$922$	0	0
$923$	− 21.3693i	− 0.703380i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 25.3693i	− 0.833238i
$928$	0	0
$929$	−56.3542	−1.84892	−0.924460	−	0.381279i	$-0.875484\pi$
$929$	−56.3542	−1.84892	−0.924460	+	0.381279i	$0.875484\pi$
$930$	0	0
$931$	0.561553	0.0184042
$932$	0	0
$933$	− 7.36932i	− 0.241261i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	21.3002i	0.695847i	0.937523	+	0.347923i	$0.113113\pi$
$937$	21.3002i	0.695847i	−0.937523	+	0.347923i	$0.886887\pi$
$938$	0	0
$939$	56.3542	1.83905
$940$	0	0
$941$	−1.36932	−0.0446385	−0.0223192	−	0.999751i	$-0.507105\pi$
$941$	−1.36932	−0.0446385	−0.0223192	+	0.999751i	$0.507105\pi$
$942$	0	0
$943$	− 36.4924i	− 1.18836i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 16.4924i	− 0.535932i	−0.963428	−	0.267966i	$-0.913649\pi$
$947$	− 16.4924i	− 0.535932i	0.963428	−	0.267966i	$-0.0863514\pi$
$948$	0	0
$949$	−21.1231	−0.685685
$950$	0	0
$951$	52.9848	1.71815
$952$	0	0
$953$	27.8769i	0.903021i	0.892266	+	0.451511i	$0.149115\pi$
$953$	27.8769i	0.903021i	−0.892266	+	0.451511i	$0.850885\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	41.1231i	1.32932i
$958$	0	0
$959$	−2.31534	−0.0747663
$960$	0	0
$961$	−30.2311	−0.975195
$962$	0	0
$963$	− 45.6155i	− 1.46994i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	16.6307i	0.534807i	0.963585	+	0.267403i	$0.0861656\pi$
$967$	16.6307i	0.534807i	−0.963585	+	0.267403i	$0.913834\pi$
$968$	0	0
$969$	−3.68466	−0.118368
$970$	0	0
$971$	2.31534	0.0743028	0.0371514	−	0.999310i	$-0.488172\pi$
$971$	2.31534	0.0743028	0.0371514	+	0.999310i	$0.488172\pi$
$972$	0	0
$973$	19.6847i	0.631061i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 33.2462i	− 1.06364i	−0.846857	−	0.531820i	$-0.821508\pi$
$977$	− 33.2462i	− 1.06364i	0.846857	−	0.531820i	$-0.178492\pi$
$978$	0	0
$979$	−22.9460	−0.733358
$980$	0	0
$981$	−31.8078	−1.01554
$982$	0	0
$983$	− 62.3542i	− 1.98879i	−0.105734	−	0.994394i	$-0.533719\pi$
$983$	− 62.3542i	− 1.98879i	0.105734	−	0.994394i	$-0.466281\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 21.1231i	− 0.672356i
$988$	0	0
$989$	−13.5616	−0.431232
$990$	0	0
$991$	29.6695	0.942483	0.471241	−	0.882004i	$-0.343806\pi$
$991$	29.6695	0.942483	0.471241	+	0.882004i	$0.343806\pi$
$992$	0	0
$993$	− 69.1619i	− 2.19479i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	12.2462i	0.387841i	0.981017	+	0.193921i	$0.0621204\pi$
$997$	12.2462i	0.387841i	−0.981017	+	0.193921i	$0.937880\pi$
$998$	0	0
$999$	−16.9848	−0.537377

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2800.2.g.v.449.4		4
4.3	odd	2		1400.2.g.j.449.1		4
5.2	odd	4		2800.2.a.bo.1.2		2
5.3	odd	4		2800.2.a.bj.1.1		2
5.4	even	2	inner	2800.2.g.v.449.1		4
20.3	even	4		1400.2.a.q.1.2	yes	2
20.7	even	4		1400.2.a.o.1.1	✓	2
20.19	odd	2		1400.2.g.j.449.4		4
140.27	odd	4		9800.2.a.bx.1.2		2
140.83	odd	4		9800.2.a.bt.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1400.2.a.o.1.1	✓	2	20.7	even	4
1400.2.a.q.1.2	yes	2	20.3	even	4
1400.2.g.j.449.1		4	4.3	odd	2
1400.2.g.j.449.4		4	20.19	odd	2
2800.2.a.bj.1.1		2	5.3	odd	4
2800.2.a.bo.1.2		2	5.2	odd	4
2800.2.g.v.449.1		4	5.4	even	2	inner
2800.2.g.v.449.4		4	1.1	even	1	trivial
9800.2.a.bt.1.1		2	140.83	odd	4
9800.2.a.bx.1.2		2	140.27	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:	\( N \)	\(=\)	\( 2800 = 2^{4} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2800.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.56155i q^{3} -1.00000i q^{7} -3.56155 q^{9} +O(q^{10})\)	\(q+2.56155i q^{3} -1.00000i q^{7} -3.56155 q^{9} -2.12311 q^{11} -2.00000i q^{13} -2.56155i q^{17} -0.561553 q^{19} +2.56155 q^{21} +5.56155i q^{23} -1.43845i q^{27} -7.56155 q^{29} +0.876894 q^{31} -5.43845i q^{33} -11.8078i q^{37} +5.12311 q^{39} -6.56155 q^{41} +2.43845i q^{43} -8.24621i q^{47} -1.00000 q^{49} +6.56155 q^{51} -7.12311i q^{53} -1.43845i q^{57} -13.3693 q^{59} +2.87689 q^{61} +3.56155i q^{63} -16.1231i q^{67} -14.2462 q^{69} +10.6847 q^{71} -10.5616i q^{73} +2.12311i q^{77} +6.68466 q^{79} -7.00000 q^{81} -13.6847i q^{83} -19.3693i q^{87} +10.8078 q^{89} -2.00000 q^{91} +2.24621i q^{93} +6.00000i q^{97} +7.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{9}+O(q^{10}) \) 4 * q - 6 * q^9	\( 4 q - 6 q^{9} + 8 q^{11} + 6 q^{19} + 2 q^{21} - 22 q^{29} + 20 q^{31} + 4 q^{39} - 18 q^{41} - 4 q^{49} + 18 q^{51} - 4 q^{59} + 28 q^{61} - 24 q^{69} + 18 q^{71} + 2 q^{79} - 28 q^{81} + 2 q^{89} - 8 q^{91} + 22 q^{99}+O(q^{100}) \) 4 * q - 6 * q^9 + 8 * q^11 + 6 * q^19 + 2 * q^21 - 22 * q^29 + 20 * q^31 + 4 * q^39 - 18 * q^41 - 4 * q^49 + 18 * q^51 - 4 * q^59 + 28 * q^61 - 24 * q^69 + 18 * q^71 + 2 * q^79 - 28 * q^81 + 2 * q^89 - 8 * q^91 + 22 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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