LMFDB - Embedded newform 800.4.c.o.449.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,4,Mod(449,800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(800, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("800.449");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	800.c (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:	$47.2015280046$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.1135425807366400.12
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 52x^{6} + 664x^{4} + 337x^{2} + 36 $ x^8 + 52x^6 + 664x^4 + 337*x^2 + 36
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.2
Root			$5.54854i$ of defining polynomial
Character	$\chi$	$=$	800.449
Dual form			800.4.c.o.449.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-10.0971i q^{3} +10.5923i q^{7} -74.9510 q^{9} +O(q^{10})$	$q-10.0971i q^{3} +10.5923i q^{7} -74.9510 q^{9} +38.9027 q^{11} +68.9510i q^{13} +65.9510i q^{17} +49.4950 q^{19} +106.951 q^{21} -164.524i q^{23} +484.164i q^{27} +170.853 q^{29} +166.505 q^{31} -392.804i q^{33} -384.853i q^{37} +696.203 q^{39} -22.8038 q^{41} +136.709i q^{43} +307.562i q^{47} +230.804 q^{49} +665.912 q^{51} -222.000i q^{53} -499.755i q^{57} -522.765 q^{59} +393.049 q^{61} -793.901i q^{63} +476.845i q^{67} -1661.22 q^{69} -4.26352 q^{71} +601.166i q^{73} +412.068i q^{77} +1075.93 q^{79} +2864.97 q^{81} +1132.74i q^{83} -1725.11i q^{87} -479.755 q^{89} -730.347 q^{91} -1681.22i q^{93} +635.510i q^{97} -2915.80 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 192 q^{9}+O(q^{10}) $ 8 * q - 192 * q^9	$ 8 q - 192 q^{9} + 448 q^{21} + 144 q^{29} + 1448 q^{41} + 216 q^{49} + 3552 q^{61} - 6768 q^{69} + 14360 q^{81} - 1800 q^{89}+O(q^{100}) $ 8 * q - 192 * q^9 + 448 * q^21 + 144 * q^29 + 1448 * q^41 + 216 * q^49 + 3552 * q^61 - 6768 * q^69 + 14360 * q^81 - 1800 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/800\mathbb{Z}\right)^\times$.

$n$	$101$	$351$	$577$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	− 10.0971i	− 1.94318i	−0.236664	−	0.971592i	$-0.576054\pi$
$3$	− 10.0971i	− 1.94318i	0.236664	−	0.971592i	$-0.423946\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	10.5923i	0.571929i	0.958240	+	0.285964i	$0.0923139\pi$
$7$	10.5923i	0.571929i	−0.958240	+	0.285964i	$0.907686\pi$
$8$	0	0
$9$	−74.9510	−2.77596
$10$	0	0
$11$	38.9027	1.06633	0.533164	−	0.846012i	$-0.321003\pi$
$11$	38.9027	1.06633	0.533164	+	0.846012i	$0.321003\pi$
$12$	0	0
$13$	68.9510i	1.47104i	0.677502	+	0.735521i	$0.263063\pi$
$13$	68.9510i	1.47104i	−0.677502	+	0.735521i	$0.736937\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	65.9510i	0.940909i	0.882424	+	0.470455i	$0.155910\pi$
$17$	65.9510i	0.940909i	−0.882424	+	0.470455i	$0.844090\pi$
$18$	0	0
$19$	49.4950	0.597628	0.298814	−	0.954311i	$-0.403409\pi$
$19$	49.4950	0.597628	0.298814	+	0.954311i	$0.403409\pi$
$20$	0	0
$21$	106.951	1.11136
$22$	0	0
$23$	− 164.524i	− 1.49155i	−0.666197	−	0.745776i	$-0.732079\pi$
$23$	− 164.524i	− 1.49155i	0.666197	−	0.745776i	$-0.267921\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	484.164i	3.45102i
$28$	0	0
$29$	170.853	1.09402	0.547010	−	0.837126i	$-0.315766\pi$
$29$	170.853	1.09402	0.547010	+	0.837126i	$0.315766\pi$
$30$	0	0
$31$	166.505	0.964684	0.482342	−	0.875983i	$-0.339786\pi$
$31$	166.505	0.964684	0.482342	+	0.875983i	$0.339786\pi$
$32$	0	0
$33$	− 392.804i	− 2.07207i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 384.853i	− 1.70998i	−0.518641	−	0.854992i	$-0.673562\pi$
$37$	− 384.853i	− 1.70998i	0.518641	−	0.854992i	$-0.326438\pi$
$38$	0	0
$39$	696.203	2.85851
$40$	0	0
$41$	−22.8038	−0.0868624	−0.0434312	−	0.999056i	$-0.513829\pi$
$41$	−22.8038	−0.0868624	−0.0434312	+	0.999056i	$0.513829\pi$
$42$	0	0
$43$	136.709i	0.484836i	0.970172	+	0.242418i	$0.0779405\pi$
$43$	136.709i	0.484836i	−0.970172	+	0.242418i	$0.922059\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	307.562i	0.954523i	0.878761	+	0.477261i	$0.158370\pi$
$47$	307.562i	0.954523i	−0.878761	+	0.477261i	$0.841630\pi$
$48$	0	0
$49$	230.804	0.672897
$50$	0	0
$51$	665.912	1.82836
$52$	0	0
$53$	− 222.000i	− 0.575359i	−0.957727	−	0.287680i	$-0.907116\pi$
$53$	− 222.000i	− 0.575359i	0.957727	−	0.287680i	$-0.0928838\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 499.755i	− 1.16130i
$58$	0	0
$59$	−522.765	−1.15353	−0.576765	−	0.816910i	$-0.695685\pi$
$59$	−522.765	−1.15353	−0.576765	+	0.816910i	$0.695685\pi$
$60$	0	0
$61$	393.049	0.824996	0.412498	−	0.910958i	$-0.364656\pi$
$61$	393.049	0.824996	0.412498	+	0.910958i	$0.364656\pi$
$62$	0	0
$63$	− 793.901i	− 1.58765i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	476.845i	0.869492i	0.900553	+	0.434746i	$0.143162\pi$
$67$	476.845i	0.869492i	−0.900553	+	0.434746i	$0.856838\pi$
$68$	0	0
$69$	−1661.22	−2.89836
$70$	0	0
$71$	−4.26352	−0.00712657	−0.00356328	−	0.999994i	$-0.501134\pi$
$71$	−4.26352	−0.00712657	−0.00356328	+	0.999994i	$0.501134\pi$
$72$	0	0
$73$	601.166i	0.963852i	0.876212	+	0.481926i	$0.160063\pi$
$73$	601.166i	0.963852i	−0.876212	+	0.481926i	$0.839937\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	412.068i	0.609864i
$78$	0	0
$79$	1075.93	1.53230	0.766150	−	0.642662i	$-0.222170\pi$
$79$	1075.93	1.53230	0.766150	+	0.642662i	$0.222170\pi$
$80$	0	0
$81$	2864.97	3.93000
$82$	0	0
$83$	1132.74i	1.49801i	0.662564	+	0.749005i	$0.269468\pi$
$83$	1132.74i	1.49801i	−0.662564	+	0.749005i	$0.730532\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 1725.11i	− 2.12588i
$88$	0	0
$89$	−479.755	−0.571392	−0.285696	−	0.958320i	$-0.592225\pi$
$89$	−479.755	−0.571392	−0.285696	+	0.958320i	$0.592225\pi$
$90$	0	0
$91$	−730.347	−0.841332
$92$	0	0
$93$	− 1681.22i	− 1.87456i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	635.510i	0.665219i	0.943065	+	0.332609i	$0.107929\pi$
$97$	635.510i	0.665219i	−0.943065	+	0.332609i	$0.892071\pi$
$98$	0	0
$99$	−2915.80	−2.96009
$100$	0	0
$101$	−256.264	−0.252468	−0.126234	−	0.992001i	$-0.540289\pi$
$101$	−256.264	−0.252468	−0.126234	+	0.992001i	$0.540289\pi$
$102$	0	0
$103$	− 81.6825i	− 0.0781400i	−0.999236	−	0.0390700i	$-0.987560\pi$
$103$	− 81.6825i	− 0.0781400i	0.999236	−	0.0390700i	$-0.0124395\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 742.944i	− 0.671244i	−0.941997	−	0.335622i	$-0.891053\pi$
$107$	− 742.944i	− 0.671244i	0.941997	−	0.335622i	$-0.108947\pi$
$108$	0	0
$109$	495.872	0.435742	0.217871	−	0.975978i	$-0.430089\pi$
$109$	495.872	0.435742	0.217871	+	0.975978i	$0.430089\pi$
$110$	0	0
$111$	−3885.89	−3.32281
$112$	0	0
$113$	− 1878.61i	− 1.56393i	−0.623320	−	0.781967i	$-0.714217\pi$
$113$	− 1878.61i	− 1.56393i	0.623320	−	0.781967i	$-0.285783\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 5167.94i	− 4.08356i
$118$	0	0
$119$	−698.570	−0.538133
$120$	0	0
$121$	182.422	0.137057
$122$	0	0
$123$	230.252i	0.168790i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 2113.24i	− 1.47653i	−0.674511	−	0.738265i	$-0.735645\pi$
$127$	− 2113.24i	− 1.47653i	0.674511	−	0.738265i	$-0.264355\pi$
$128$	0	0
$129$	1380.36	0.942125
$130$	0	0
$131$	1105.42	0.737263	0.368631	−	0.929576i	$-0.379827\pi$
$131$	1105.42	0.737263	0.368631	+	0.929576i	$0.379827\pi$
$132$	0	0
$133$	524.264i	0.341801i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 1151.92i	− 0.718360i	−0.933268	−	0.359180i	$-0.883056\pi$
$137$	− 1151.92i	− 0.718360i	0.933268	−	0.359180i	$-0.116944\pi$
$138$	0	0
$139$	354.304	0.216199	0.108099	−	0.994140i	$-0.465524\pi$
$139$	354.304	0.216199	0.108099	+	0.994140i	$0.465524\pi$
$140$	0	0
$141$	3105.48	1.85481
$142$	0	0
$143$	2682.38i	1.56861i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 2330.44i	− 1.30756i
$148$	0	0
$149$	−715.638	−0.393472	−0.196736	−	0.980457i	$-0.563034\pi$
$149$	−715.638	−0.393472	−0.196736	+	0.980457i	$0.563034\pi$
$150$	0	0
$151$	2799.37	1.50867	0.754335	−	0.656490i	$-0.227959\pi$
$151$	2799.37	1.50867	0.754335	+	0.656490i	$0.227959\pi$
$152$	0	0
$153$	− 4943.09i	− 2.61193i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 845.275i	− 0.429683i	−0.976649	−	0.214842i	$-0.931076\pi$
$157$	− 845.275i	− 0.429683i	0.976649	−	0.214842i	$-0.0689236\pi$
$158$	0	0
$159$	−2241.55	−1.11803
$160$	0	0
$161$	1742.69	0.853062
$162$	0	0
$163$	− 539.964i	− 0.259468i	−0.991549	−	0.129734i	$-0.958588\pi$
$163$	− 539.964i	− 0.259468i	0.991549	−	0.129734i	$-0.0414123\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 1209.20i	− 0.560302i	−0.959956	−	0.280151i	$-0.909615\pi$
$167$	− 1209.20i	− 0.560302i	0.959956	−	0.280151i	$-0.0903846\pi$
$168$	0	0
$169$	−2557.23	−1.16397
$170$	0	0
$171$	−3709.70	−1.65899
$172$	0	0
$173$	4061.65i	1.78498i	0.451069	+	0.892489i	$0.351043\pi$
$173$	4061.65i	1.78498i	−0.451069	+	0.892489i	$0.648957\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	5278.40i	2.24152i
$178$	0	0
$179$	3113.67	1.30015	0.650074	−	0.759871i	$-0.274738\pi$
$179$	3113.67	1.30015	0.650074	+	0.759871i	$0.274738\pi$
$180$	0	0
$181$	473.215	0.194330	0.0971652	−	0.995268i	$-0.469022\pi$
$181$	473.215	0.194330	0.0971652	+	0.995268i	$0.469022\pi$
$182$	0	0
$183$	− 3968.65i	− 1.60312i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	2565.67i	1.00332i
$188$	0	0
$189$	−5128.40	−1.97374
$190$	0	0
$191$	2250.68	0.852637	0.426318	−	0.904573i	$-0.359810\pi$
$191$	2250.68	0.852637	0.426318	+	0.904573i	$0.359810\pi$
$192$	0	0
$193$	553.461i	0.206419i	0.994660	+	0.103210i	$0.0329113\pi$
$193$	553.461i	0.206419i	−0.994660	+	0.103210i	$0.967089\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	385.744i	0.139508i	0.997564	+	0.0697541i	$0.0222215\pi$
$197$	385.744i	0.139508i	−0.997564	+	0.0697541i	$0.977779\pi$
$198$	0	0
$199$	559.494	0.199304	0.0996520	−	0.995022i	$-0.468227\pi$
$199$	559.494	0.199304	0.0996520	+	0.995022i	$0.468227\pi$
$200$	0	0
$201$	4814.74	1.68958
$202$	0	0
$203$	1809.72i	0.625702i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	12331.3i	4.14049i
$208$	0	0
$209$	1925.49	0.637268
$210$	0	0
$211$	3721.00	1.21405	0.607025	−	0.794683i	$-0.292363\pi$
$211$	3721.00	1.21405	0.607025	+	0.794683i	$0.292363\pi$
$212$	0	0
$213$	43.0490i	0.0138482i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1763.67i	0.551731i
$218$	0	0
$219$	6070.02	1.87294
$220$	0	0
$221$	−4547.38	−1.38412
$222$	0	0
$223$	5506.41i	1.65353i	0.562550	+	0.826763i	$0.309820\pi$
$223$	5506.41i	1.65353i	−0.562550	+	0.826763i	$0.690180\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 1139.27i	− 0.333109i	−0.986032	−	0.166555i	$-0.946736\pi$
$227$	− 1139.27i	− 0.333109i	0.986032	−	0.166555i	$-0.0532642\pi$
$228$	0	0
$229$	−1727.08	−0.498378	−0.249189	−	0.968455i	$-0.580164\pi$
$229$	−1727.08	−0.498378	−0.249189	+	0.968455i	$0.580164\pi$
$230$	0	0
$231$	4160.68	1.18508
$232$	0	0
$233$	1267.84i	0.356477i	0.983987	+	0.178238i	$0.0570398\pi$
$233$	1267.84i	0.356477i	−0.983987	+	0.178238i	$0.942960\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 10863.8i	− 2.97754i
$238$	0	0
$239$	493.995	0.133698	0.0668492	−	0.997763i	$-0.478705\pi$
$239$	493.995	0.133698	0.0668492	+	0.997763i	$0.478705\pi$
$240$	0	0
$241$	−332.676	−0.0889192	−0.0444596	−	0.999011i	$-0.514157\pi$
$241$	−332.676	−0.0889192	−0.0444596	+	0.999011i	$0.514157\pi$
$242$	0	0
$243$	− 15855.4i	− 4.18569i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	3412.73i	0.879136i
$248$	0	0
$249$	11437.4	2.91091
$250$	0	0
$251$	−4365.75	−1.09786	−0.548932	−	0.835867i	$-0.684965\pi$
$251$	−4365.75	−1.09786	−0.548932	+	0.835867i	$0.684965\pi$
$252$	0	0
$253$	− 6400.45i	− 1.59048i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 3407.19i	− 0.826984i	−0.910508	−	0.413492i	$-0.864309\pi$
$257$	− 3407.19i	− 0.826984i	0.910508	−	0.413492i	$-0.135691\pi$
$258$	0	0
$259$	4076.47	0.977989
$260$	0	0
$261$	−12805.6	−3.03696
$262$	0	0
$263$	− 2388.20i	− 0.559934i	−0.960010	−	0.279967i	$-0.909676\pi$
$263$	− 2388.20i	− 0.559934i	0.960010	−	0.279967i	$-0.0903236\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	4844.12i	1.11032i
$268$	0	0
$269$	3800.33	0.861377	0.430688	−	0.902501i	$-0.358271\pi$
$269$	3800.33	0.861377	0.430688	+	0.902501i	$0.358271\pi$
$270$	0	0
$271$	2270.39	0.508917	0.254459	−	0.967084i	$-0.418103\pi$
$271$	2270.39	0.508917	0.254459	+	0.967084i	$0.418103\pi$
$272$	0	0
$273$	7374.37i	1.63486i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	7070.93i	1.53376i	0.641792	+	0.766879i	$0.278191\pi$
$277$	7070.93i	1.53376i	−0.641792	+	0.766879i	$0.721809\pi$
$278$	0	0
$279$	−12479.7	−2.67793
$280$	0	0
$281$	1536.57	0.326206	0.163103	−	0.986609i	$-0.447850\pi$
$281$	1536.57	0.326206	0.163103	+	0.986609i	$0.447850\pi$
$282$	0	0
$283$	6013.32i	1.26309i	0.775339	+	0.631545i	$0.217579\pi$
$283$	6013.32i	1.26309i	−0.775339	+	0.631545i	$0.782421\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 241.544i	− 0.0496791i
$288$	0	0
$289$	563.471	0.114690
$290$	0	0
$291$	6416.79	1.29264
$292$	0	0
$293$	− 4651.03i	− 0.927358i	−0.886003	−	0.463679i	$-0.846529\pi$
$293$	− 4651.03i	− 0.927358i	0.886003	−	0.463679i	$-0.153471\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	18835.3i	3.67992i
$298$	0	0
$299$	11344.1	2.19414
$300$	0	0
$301$	−1448.06	−0.277292
$302$	0	0
$303$	2587.52i	0.490591i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 7261.88i	− 1.35002i	−0.737807	−	0.675012i	$-0.764139\pi$
$307$	− 7261.88i	− 1.35002i	0.737807	−	0.675012i	$-0.235861\pi$
$308$	0	0
$309$	−824.755	−0.151840
$310$	0	0
$311$	6759.67	1.23249	0.616247	−	0.787553i	$-0.288652\pi$
$311$	6759.67	1.23249	0.616247	+	0.787553i	$0.288652\pi$
$312$	0	0
$313$	− 1865.55i	− 0.336891i	−0.985711	−	0.168446i	$-0.946125\pi$
$313$	− 1865.55i	− 0.336891i	0.985711	−	0.168446i	$-0.0538748\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 2759.41i	− 0.488908i	−0.969661	−	0.244454i	$-0.921391\pi$
$317$	− 2759.41i	− 0.488908i	0.969661	−	0.244454i	$-0.0786088\pi$
$318$	0	0
$319$	6646.64	1.16658
$320$	0	0
$321$	−7501.57	−1.30435
$322$	0	0
$323$	3264.24i	0.562314i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 5006.86i	− 0.846727i
$328$	0	0
$329$	−3257.78	−0.545919
$330$	0	0
$331$	−5123.99	−0.850876	−0.425438	−	0.904988i	$-0.639880\pi$
$331$	−5123.99	−0.850876	−0.425438	+	0.904988i	$0.639880\pi$
$332$	0	0
$333$	28845.1i	4.74685i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 1475.31i	− 0.238473i	−0.992866	−	0.119236i	$-0.961955\pi$
$337$	− 1475.31i	− 0.238473i	0.992866	−	0.119236i	$-0.0380446\pi$
$338$	0	0
$339$	−18968.4	−3.03901
$340$	0	0
$341$	6477.50	1.02867
$342$	0	0
$343$	6077.88i	0.956778i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 3478.67i	− 0.538170i	−0.963116	−	0.269085i	$-0.913279\pi$
$347$	− 3478.67i	− 0.538170i	0.963116	−	0.269085i	$-0.0867212\pi$
$348$	0	0
$349$	−11392.3	−1.74732	−0.873660	−	0.486537i	$-0.838260\pi$
$349$	−11392.3	−1.74732	−0.873660	+	0.486537i	$0.838260\pi$
$350$	0	0
$351$	−33383.6	−5.07660
$352$	0	0
$353$	− 2735.39i	− 0.412437i	−0.978506	−	0.206218i	$-0.933884\pi$
$353$	− 2735.39i	− 0.412437i	0.978506	−	0.206218i	$-0.0661157\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	7053.52i	1.04569i
$358$	0	0
$359$	−2390.41	−0.351424	−0.175712	−	0.984442i	$-0.556223\pi$
$359$	−2390.41	−0.351424	−0.175712	+	0.984442i	$0.556223\pi$
$360$	0	0
$361$	−4409.25	−0.642841
$362$	0	0
$363$	− 1841.93i	− 0.266326i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 9209.84i	− 1.30994i	−0.755653	−	0.654972i	$-0.772680\pi$
$367$	− 9209.84i	− 1.30994i	0.755653	−	0.654972i	$-0.227320\pi$
$368$	0	0
$369$	1709.17	0.241127
$370$	0	0
$371$	2351.48	0.329065
$372$	0	0
$373$	13768.4i	1.91127i	0.294556	+	0.955634i	$0.404828\pi$
$373$	13768.4i	1.91127i	−0.294556	+	0.955634i	$0.595172\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	11780.5i	1.60935i
$378$	0	0
$379$	−8666.50	−1.17459	−0.587293	−	0.809374i	$-0.699807\pi$
$379$	−8666.50	−1.17459	−0.587293	+	0.809374i	$0.699807\pi$
$380$	0	0
$381$	−21337.5	−2.86917
$382$	0	0
$383$	7434.12i	0.991816i	0.868375	+	0.495908i	$0.165165\pi$
$383$	7434.12i	0.991816i	−0.868375	+	0.495908i	$0.834835\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 10246.5i	− 1.34589i
$388$	0	0
$389$	7323.06	0.954483	0.477241	−	0.878772i	$-0.341637\pi$
$389$	7323.06	0.954483	0.477241	+	0.878772i	$0.341637\pi$
$390$	0	0
$391$	10850.5	1.40342
$392$	0	0
$393$	− 11161.6i	− 1.43264i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 5185.92i	− 0.655602i	−0.944747	−	0.327801i	$-0.893693\pi$
$397$	− 5185.92i	− 0.655602i	0.944747	−	0.327801i	$-0.106307\pi$
$398$	0	0
$399$	5293.54	0.664181
$400$	0	0
$401$	−9044.81	−1.12637	−0.563187	−	0.826329i	$-0.690425\pi$
$401$	−9044.81	−1.12637	−0.563187	+	0.826329i	$0.690425\pi$
$402$	0	0
$403$	11480.7i	1.41909i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 14971.8i	− 1.82341i
$408$	0	0
$409$	2311.41	0.279442	0.139721	−	0.990191i	$-0.455379\pi$
$409$	2311.41	0.279442	0.139721	+	0.990191i	$0.455379\pi$
$410$	0	0
$411$	−11631.0	−1.39590
$412$	0	0
$413$	− 5537.27i	− 0.659737i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 3577.43i	− 0.420114i
$418$	0	0
$419$	−330.860	−0.0385765	−0.0192882	−	0.999814i	$-0.506140\pi$
$419$	−330.860	−0.0385765	−0.0192882	+	0.999814i	$0.506140\pi$
$420$	0	0
$421$	1933.39	0.223819	0.111909	−	0.993718i	$-0.464303\pi$
$421$	1933.39	0.223819	0.111909	+	0.993718i	$0.464303\pi$
$422$	0	0
$423$	− 23052.1i	− 2.64972i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	4163.28i	0.471839i
$428$	0	0
$429$	27084.2	3.04811
$430$	0	0
$431$	8820.32	0.985753	0.492877	−	0.870099i	$-0.335945\pi$
$431$	8820.32	0.985753	0.492877	+	0.870099i	$0.335945\pi$
$432$	0	0
$433$	11269.7i	1.25078i	0.780312	+	0.625390i	$0.215060\pi$
$433$	11269.7i	1.25078i	−0.780312	+	0.625390i	$0.784940\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 8143.13i	− 0.891393i
$438$	0	0
$439$	−2361.79	−0.256770	−0.128385	−	0.991724i	$-0.540979\pi$
$439$	−2361.79	−0.256770	−0.128385	+	0.991724i	$0.540979\pi$
$440$	0	0
$441$	−17299.0	−1.86794
$442$	0	0
$443$	15029.6i	1.61192i	0.591971	+	0.805959i	$0.298350\pi$
$443$	15029.6i	1.61192i	−0.591971	+	0.805959i	$0.701650\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	7225.85i	0.764588i
$448$	0	0
$449$	12661.3	1.33079	0.665396	−	0.746491i	$-0.268263\pi$
$449$	12661.3	1.33079	0.665396	+	0.746491i	$0.268263\pi$
$450$	0	0
$451$	−887.131	−0.0926239
$452$	0	0
$453$	− 28265.4i	− 2.93162i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1620.47i	0.165870i	0.996555	+	0.0829348i	$0.0264293\pi$
$457$	1620.47i	0.165870i	−0.996555	+	0.0829348i	$0.973571\pi$
$458$	0	0
$459$	−31931.1	−3.24710
$460$	0	0
$461$	6893.84	0.696482	0.348241	−	0.937405i	$-0.386779\pi$
$461$	6893.84	0.696482	0.348241	+	0.937405i	$0.386779\pi$
$462$	0	0
$463$	− 12022.8i	− 1.20680i	−0.797440	−	0.603398i	$-0.793813\pi$
$463$	− 12022.8i	− 1.20680i	0.797440	−	0.603398i	$-0.206187\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	7349.60i	0.728263i	0.931348	+	0.364131i	$0.118634\pi$
$467$	7349.60i	0.728263i	−0.931348	+	0.364131i	$0.881366\pi$
$468$	0	0
$469$	−5050.87	−0.497287
$470$	0	0
$471$	−8534.81	−0.834954
$472$	0	0
$473$	5318.36i	0.516994i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	16639.1i	1.59718i
$478$	0	0
$479$	12809.4	1.22187	0.610933	−	0.791682i	$-0.290794\pi$
$479$	12809.4	1.22187	0.610933	+	0.791682i	$0.290794\pi$
$480$	0	0
$481$	26536.0	2.51546
$482$	0	0
$483$	− 17596.0i	− 1.65766i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 15782.4i	− 1.46852i	−0.678868	−	0.734260i	$-0.737529\pi$
$487$	− 15782.4i	− 1.46852i	0.678868	−	0.734260i	$-0.262471\pi$
$488$	0	0
$489$	−5452.06	−0.504193
$490$	0	0
$491$	5972.10	0.548914	0.274457	−	0.961599i	$-0.411502\pi$
$491$	5972.10	0.548914	0.274457	+	0.961599i	$0.411502\pi$
$492$	0	0
$493$	11267.9i	1.02937i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 45.1603i	− 0.00407589i
$498$	0	0
$499$	14171.8	1.27138	0.635690	−	0.771944i	$-0.280716\pi$
$499$	14171.8	1.27138	0.635690	+	0.771944i	$0.280716\pi$
$500$	0	0
$501$	−12209.4	−1.08877
$502$	0	0
$503$	5543.95i	0.491436i	0.969341	+	0.245718i	$0.0790237\pi$
$503$	5543.95i	0.491436i	−0.969341	+	0.245718i	$0.920976\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	25820.6i	2.26180i
$508$	0	0
$509$	−13228.9	−1.15199	−0.575994	−	0.817454i	$-0.695385\pi$
$509$	−13228.9	−1.15199	−0.575994	+	0.817454i	$0.695385\pi$
$510$	0	0
$511$	−6367.72	−0.551255
$512$	0	0
$513$	23963.7i	2.06243i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	11965.0i	1.01783i
$518$	0	0
$519$	41010.8	3.46854
$520$	0	0
$521$	−11092.8	−0.932795	−0.466398	−	0.884575i	$-0.654448\pi$
$521$	−11092.8	−0.932795	−0.466398	+	0.884575i	$0.654448\pi$
$522$	0	0
$523$	18550.2i	1.55094i	0.631384	+	0.775470i	$0.282487\pi$
$523$	18550.2i	1.55094i	−0.631384	+	0.775470i	$0.717513\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	10981.2i	0.907680i
$528$	0	0
$529$	−14901.3	−1.22473
$530$	0	0
$531$	39181.8	3.20215
$532$	0	0
$533$	− 1572.35i	− 0.127778i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 31438.9i	− 2.52643i
$538$	0	0
$539$	8978.90	0.717530
$540$	0	0
$541$	8146.08	0.647370	0.323685	−	0.946165i	$-0.395078\pi$
$541$	8146.08	0.647370	0.323685	+	0.946165i	$0.395078\pi$
$542$	0	0
$543$	− 4778.09i	− 0.377620i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	18001.4i	1.40710i	0.710645	+	0.703550i	$0.248403\pi$
$547$	18001.4i	1.40710i	−0.710645	+	0.703550i	$0.751597\pi$
$548$	0	0
$549$	−29459.4	−2.29016
$550$	0	0
$551$	8456.36	0.653817
$552$	0	0
$553$	11396.5i	0.876366i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	5968.52i	0.454029i	0.973891	+	0.227014i	$0.0728965\pi$
$557$	5968.52i	0.454029i	−0.973891	+	0.227014i	$0.927104\pi$
$558$	0	0
$559$	−9426.22	−0.713214
$560$	0	0
$561$	25905.8	1.94963
$562$	0	0
$563$	6021.42i	0.450751i	0.974272	+	0.225375i	$0.0723608\pi$
$563$	6021.42i	0.450751i	−0.974272	+	0.225375i	$0.927639\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	30346.5i	2.24768i
$568$	0	0
$569$	−13499.2	−0.994582	−0.497291	−	0.867584i	$-0.665672\pi$
$569$	−13499.2	−0.994582	−0.497291	+	0.867584i	$0.665672\pi$
$570$	0	0
$571$	−2413.96	−0.176920	−0.0884599	−	0.996080i	$-0.528195\pi$
$571$	−2413.96	−0.176920	−0.0884599	+	0.996080i	$0.528195\pi$
$572$	0	0
$573$	− 22725.3i	− 1.65683i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	13549.7i	0.977608i	0.872394	+	0.488804i	$0.162567\pi$
$577$	13549.7i	0.977608i	−0.872394	+	0.488804i	$0.837433\pi$
$578$	0	0
$579$	5588.33	0.401111
$580$	0	0
$581$	−11998.3	−0.856755
$582$	0	0
$583$	− 8636.41i	− 0.613522i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	19364.2i	1.36158i	0.732479	+	0.680789i	$0.238363\pi$
$587$	19364.2i	1.36158i	−0.732479	+	0.680789i	$0.761637\pi$
$588$	0	0
$589$	8241.17	0.576522
$590$	0	0
$591$	3894.89	0.271090
$592$	0	0
$593$	− 21130.1i	− 1.46325i	−0.681705	−	0.731627i	$-0.738761\pi$
$593$	− 21130.1i	− 1.46325i	0.681705	−	0.731627i	$-0.261239\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 5649.25i	− 0.387284i
$598$	0	0
$599$	−12938.1	−0.882532	−0.441266	−	0.897376i	$-0.645471\pi$
$599$	−12938.1	−0.882532	−0.441266	+	0.897376i	$0.645471\pi$
$600$	0	0
$601$	−2173.64	−0.147528	−0.0737641	−	0.997276i	$-0.523501\pi$
$601$	−2173.64	−0.147528	−0.0737641	+	0.997276i	$0.523501\pi$
$602$	0	0
$603$	− 35740.0i	− 2.41367i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 13712.9i	− 0.916951i	−0.888707	−	0.458475i	$-0.848396\pi$
$607$	− 13712.9i	− 0.916951i	0.888707	−	0.458475i	$-0.151604\pi$
$608$	0	0
$609$	18272.9	1.21585
$610$	0	0
$611$	−21206.7	−1.40414
$612$	0	0
$613$	− 3666.83i	− 0.241602i	−0.992677	−	0.120801i	$-0.961454\pi$
$613$	− 3666.83i	− 0.241602i	0.992677	−	0.120801i	$-0.0385462\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 26402.5i	− 1.72273i	−0.507984	−	0.861367i	$-0.669609\pi$
$617$	− 26402.5i	− 1.72273i	0.507984	−	0.861367i	$-0.330391\pi$
$618$	0	0
$619$	28554.8	1.85414	0.927072	−	0.374882i	$-0.122317\pi$
$619$	28554.8	1.85414	0.927072	+	0.374882i	$0.122317\pi$
$620$	0	0
$621$	79656.9	5.14737
$622$	0	0
$623$	− 5081.69i	− 0.326796i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 19441.8i	− 1.23833i
$628$	0	0
$629$	25381.4	1.60894
$630$	0	0
$631$	29567.8	1.86541	0.932706	−	0.360637i	$-0.117441\pi$
$631$	29567.8	1.86541	0.932706	+	0.360637i	$0.117441\pi$
$632$	0	0
$633$	− 37571.2i	− 2.35912i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	15914.1i	0.989861i
$638$	0	0
$639$	319.555	0.0197831
$640$	0	0
$641$	−21701.9	−1.33724	−0.668621	−	0.743604i	$-0.733115\pi$
$641$	−21701.9	−1.33724	−0.668621	+	0.743604i	$0.733115\pi$
$642$	0	0
$643$	6193.48i	0.379855i	0.981798	+	0.189928i	$0.0608254\pi$
$643$	6193.48i	0.379855i	−0.981798	+	0.189928i	$0.939175\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	2208.53i	0.134198i	0.997746	+	0.0670991i	$0.0213744\pi$
$647$	2208.53i	0.134198i	−0.997746	+	0.0670991i	$0.978626\pi$
$648$	0	0
$649$	−20337.0	−1.23004
$650$	0	0
$651$	17807.9	1.07211
$652$	0	0
$653$	20505.9i	1.22888i	0.788965	+	0.614438i	$0.210617\pi$
$653$	20505.9i	1.22888i	−0.788965	+	0.614438i	$0.789383\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 45058.0i	− 2.67562i
$658$	0	0
$659$	−26030.4	−1.53870	−0.769348	−	0.638829i	$-0.779419\pi$
$659$	−26030.4	−1.53870	−0.769348	+	0.638829i	$0.779419\pi$
$660$	0	0
$661$	−6729.32	−0.395976	−0.197988	−	0.980204i	$-0.563441\pi$
$661$	−6729.32	−0.395976	−0.197988	+	0.980204i	$0.563441\pi$
$662$	0	0
$663$	45915.3i	2.68959i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 28109.5i	− 1.63179i
$668$	0	0
$669$	55598.6	3.21311
$670$	0	0
$671$	15290.7	0.879717
$672$	0	0
$673$	− 8716.54i	− 0.499254i	−0.968342	−	0.249627i	$-0.919692\pi$
$673$	− 8716.54i	− 0.499254i	0.968342	−	0.249627i	$-0.0803080\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 24843.1i	− 1.41034i	−0.709039	−	0.705170i	$-0.750871\pi$
$677$	− 24843.1i	− 1.41034i	0.709039	−	0.705170i	$-0.249129\pi$
$678$	0	0
$679$	−6731.49	−0.380458
$680$	0	0
$681$	−11503.3	−0.647292
$682$	0	0
$683$	− 20079.1i	− 1.12490i	−0.826832	−	0.562449i	$-0.809859\pi$
$683$	− 20079.1i	− 1.12490i	0.826832	−	0.562449i	$-0.190141\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	17438.4i	0.968440i
$688$	0	0
$689$	15307.1	0.846378
$690$	0	0
$691$	−16575.8	−0.912550	−0.456275	−	0.889839i	$-0.650817\pi$
$691$	−16575.8	−0.912550	−0.456275	+	0.889839i	$0.650817\pi$
$692$	0	0
$693$	− 30884.9i	− 1.69296i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 1503.93i	− 0.0817297i
$698$	0	0
$699$	12801.5	0.692700
$700$	0	0
$701$	29052.6	1.56534	0.782668	−	0.622440i	$-0.213858\pi$
$701$	29052.6	1.56534	0.782668	+	0.622440i	$0.213858\pi$
$702$	0	0
$703$	− 19048.3i	− 1.02193i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 2714.42i	− 0.144394i
$708$	0	0
$709$	−32100.2	−1.70035	−0.850175	−	0.526499i	$-0.823504\pi$
$709$	−32100.2	−1.70035	−0.850175	+	0.526499i	$0.823504\pi$
$710$	0	0
$711$	−80642.0	−4.25360
$712$	0	0
$713$	− 27394.2i	− 1.43888i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 4987.91i	− 0.259800i
$718$	0	0
$719$	−17819.4	−0.924274	−0.462137	−	0.886808i	$-0.652917\pi$
$719$	−17819.4	−0.924274	−0.462137	+	0.886808i	$0.652917\pi$
$720$	0	0
$721$	865.203	0.0446905
$722$	0	0
$723$	3359.05i	0.172786i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	27816.4i	1.41906i	0.704677	+	0.709529i	$0.251092\pi$
$727$	27816.4i	1.41906i	−0.704677	+	0.709529i	$0.748908\pi$
$728$	0	0
$729$	−82738.8	−4.20357
$730$	0	0
$731$	−9016.10	−0.456187
$732$	0	0
$733$	2522.56i	0.127112i	0.997978	+	0.0635559i	$0.0202441\pi$
$733$	2522.56i	0.127112i	−0.997978	+	0.0635559i	$0.979756\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	18550.6i	0.927164i
$738$	0	0
$739$	−3526.91	−0.175561	−0.0877805	−	0.996140i	$-0.527977\pi$
$739$	−3526.91	−0.175561	−0.0877805	+	0.996140i	$0.527977\pi$
$740$	0	0
$741$	34458.6	1.70832
$742$	0	0
$743$	− 8557.90i	− 0.422556i	−0.977426	−	0.211278i	$-0.932237\pi$
$743$	− 8557.90i	− 0.422556i	0.977426	−	0.211278i	$-0.0677625\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 84900.3i	− 4.15842i
$748$	0	0
$749$	7869.47	0.383904
$750$	0	0
$751$	806.886	0.0392060	0.0196030	−	0.999808i	$-0.493760\pi$
$751$	806.886	0.0392060	0.0196030	+	0.999808i	$0.493760\pi$
$752$	0	0
$753$	44081.4i	2.13335i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	12898.6i	0.619298i	0.950851	+	0.309649i	$0.100212\pi$
$757$	12898.6i	0.619298i	−0.950851	+	0.309649i	$0.899788\pi$
$758$	0	0
$759$	−64625.8	−3.09060
$760$	0	0
$761$	23315.9	1.11064	0.555322	−	0.831635i	$-0.312595\pi$
$761$	23315.9	1.11064	0.555322	+	0.831635i	$0.312595\pi$
$762$	0	0
$763$	5252.41i	0.249214i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 36045.2i	− 1.69689i
$768$	0	0
$769$	−41805.3	−1.96039	−0.980193	−	0.198045i	$-0.936541\pi$
$769$	−41805.3	−1.96039	−0.980193	+	0.198045i	$0.936541\pi$
$770$	0	0
$771$	−34402.7	−1.60698
$772$	0	0
$773$	− 10587.9i	− 0.492654i	−0.969187	−	0.246327i	$-0.920776\pi$
$773$	− 10587.9i	− 0.492654i	0.969187	−	0.246327i	$-0.0792237\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 41160.4i	− 1.90041i
$778$	0	0
$779$	−1128.68	−0.0519114
$780$	0	0
$781$	−165.862	−0.00759926
$782$	0	0
$783$	82720.9i	3.77548i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	16085.8i	0.728584i	0.931285	+	0.364292i	$0.118689\pi$
$787$	16085.8i	0.728584i	−0.931285	+	0.364292i	$0.881311\pi$
$788$	0	0
$789$	−24113.8	−1.08805
$790$	0	0
$791$	19898.7	0.894459
$792$	0	0
$793$	27101.1i	1.21360i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 13813.3i	− 0.613919i	−0.951723	−	0.306959i	$-0.900688\pi$
$797$	− 13813.3i	− 0.613919i	0.951723	−	0.306959i	$-0.0993116\pi$
$798$	0	0
$799$	−20284.0	−0.898119
$800$	0	0
$801$	35958.1	1.58616
$802$	0	0
$803$	23387.0i	1.02778i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 38372.2i	− 1.67381i
$808$	0	0
$809$	14893.1	0.647234	0.323617	−	0.946188i	$-0.395101\pi$
$809$	14893.1	0.647234	0.323617	+	0.946188i	$0.395101\pi$
$810$	0	0
$811$	−24638.7	−1.06681	−0.533405	−	0.845860i	$-0.679088\pi$
$811$	−24638.7	−1.06681	−0.533405	+	0.845860i	$0.679088\pi$
$812$	0	0
$813$	− 22924.3i	− 0.988919i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	6766.42i	0.289752i
$818$	0	0
$819$	54740.2	2.33550
$820$	0	0
$821$	−11953.6	−0.508142	−0.254071	−	0.967186i	$-0.581770\pi$
$821$	−11953.6	−0.508142	−0.254071	+	0.967186i	$0.581770\pi$
$822$	0	0
$823$	4320.27i	0.182983i	0.995806	+	0.0914916i	$0.0291635\pi$
$823$	4320.27i	0.182983i	−0.995806	+	0.0914916i	$0.970837\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 17980.9i	− 0.756055i	−0.925794	−	0.378028i	$-0.876602\pi$
$827$	− 17980.9i	− 0.756055i	0.925794	−	0.378028i	$-0.123398\pi$
$828$	0	0
$829$	−29347.5	−1.22953	−0.614766	−	0.788710i	$-0.710750\pi$
$829$	−29347.5	−1.22953	−0.614766	+	0.788710i	$0.710750\pi$
$830$	0	0
$831$	71395.7	2.98037
$832$	0	0
$833$	15221.7i	0.633135i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	80615.9i	3.32914i
$838$	0	0
$839$	20819.5	0.856696	0.428348	−	0.903614i	$-0.359096\pi$
$839$	20819.5	0.856696	0.428348	+	0.903614i	$0.359096\pi$
$840$	0	0
$841$	4801.70	0.196880
$842$	0	0
$843$	− 15514.8i	− 0.633878i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1932.27i	0.0783866i
$848$	0	0
$849$	60716.9	2.45442
$850$	0	0
$851$	−63317.7	−2.55053
$852$	0	0
$853$	− 16835.3i	− 0.675766i	−0.941188	−	0.337883i	$-0.890289\pi$
$853$	− 16835.3i	− 0.675766i	0.941188	−	0.337883i	$-0.109711\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	2703.39i	0.107755i	0.998548	+	0.0538775i	$0.0171580\pi$
$857$	2703.39i	0.107755i	−0.998548	+	0.0538775i	$0.982842\pi$
$858$	0	0
$859$	−19064.6	−0.757247	−0.378624	−	0.925551i	$-0.623603\pi$
$859$	−19064.6	−0.757247	−0.378624	+	0.925551i	$0.623603\pi$
$860$	0	0
$861$	−2438.89	−0.0965356
$862$	0	0
$863$	− 41930.0i	− 1.65390i	−0.562278	−	0.826949i	$-0.690075\pi$
$863$	− 41930.0i	− 1.65390i	0.562278	−	0.826949i	$-0.309925\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 5689.41i	− 0.222863i
$868$	0	0
$869$	41856.6	1.63393
$870$	0	0
$871$	−32878.9	−1.27906
$872$	0	0
$873$	− 47632.1i	− 1.84662i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	7701.66i	0.296541i	0.988947	+	0.148270i	$0.0473706\pi$
$877$	7701.66i	0.296541i	−0.988947	+	0.148270i	$0.952629\pi$
$878$	0	0
$879$	−46961.8	−1.80203
$880$	0	0
$881$	7799.94	0.298282	0.149141	−	0.988816i	$-0.452349\pi$
$881$	7799.94	0.298282	0.149141	+	0.988816i	$0.452349\pi$
$882$	0	0
$883$	− 40922.8i	− 1.55964i	−0.626003	−	0.779820i	$-0.715310\pi$
$883$	− 40922.8i	− 1.55964i	0.626003	−	0.779820i	$-0.284690\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	9632.80i	0.364642i	0.983239	+	0.182321i	$0.0583611\pi$
$887$	9632.80i	0.364642i	−0.983239	+	0.182321i	$0.941639\pi$
$888$	0	0
$889$	22384.0	0.844470
$890$	0	0
$891$	111455.	4.19067
$892$	0	0
$893$	15222.8i	0.570449i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 114542.i	− 4.26361i
$898$	0	0
$899$	28447.9	1.05538
$900$	0	0
$901$	14641.1	0.541361
$902$	0	0
$903$	14621.2i	0.538828i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	27797.0i	1.01762i	0.860878	+	0.508812i	$0.169915\pi$
$907$	27797.0i	1.01762i	−0.860878	+	0.508812i	$0.830085\pi$
$908$	0	0
$909$	19207.3	0.700841
$910$	0	0
$911$	−20735.4	−0.754110	−0.377055	−	0.926191i	$-0.623063\pi$
$911$	−20735.4	−0.754110	−0.377055	+	0.926191i	$0.623063\pi$
$912$	0	0
$913$	44066.9i	1.59737i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	11709.0i	0.421662i
$918$	0	0
$919$	6063.46	0.217644	0.108822	−	0.994061i	$-0.465292\pi$
$919$	6063.46	0.217644	0.108822	+	0.994061i	$0.465292\pi$
$920$	0	0
$921$	−73323.7	−2.62334
$922$	0	0
$923$	− 293.973i	− 0.0104835i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	6122.18i	0.216914i
$928$	0	0
$929$	694.043	0.0245111	0.0122556	−	0.999925i	$-0.496099\pi$
$929$	694.043	0.0245111	0.0122556	+	0.999925i	$0.496099\pi$
$930$	0	0
$931$	11423.6	0.402142
$932$	0	0
$933$	− 68252.9i	− 2.39496i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	44470.5i	1.55047i	0.631674	+	0.775234i	$0.282368\pi$
$937$	44470.5i	1.55047i	−0.631674	+	0.775234i	$0.717632\pi$
$938$	0	0
$939$	−18836.6	−0.654642
$940$	0	0
$941$	−34075.3	−1.18047	−0.590235	−	0.807232i	$-0.700965\pi$
$941$	−34075.3	−1.18047	−0.590235	+	0.807232i	$0.700965\pi$
$942$	0	0
$943$	3751.79i	0.129560i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	34495.9i	1.18370i	0.806048	+	0.591851i	$0.201603\pi$
$947$	34495.9i	1.18370i	−0.806048	+	0.591851i	$0.798397\pi$
$948$	0	0
$949$	−41451.0	−1.41787
$950$	0	0
$951$	−27862.0	−0.950039
$952$	0	0
$953$	− 23975.9i	− 0.814960i	−0.913214	−	0.407480i	$-0.866408\pi$
$953$	− 23975.9i	− 0.814960i	0.913214	−	0.407480i	$-0.133592\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 67111.7i	− 2.26689i
$958$	0	0
$959$	12201.5	0.410851
$960$	0	0
$961$	−2067.03	−0.0693845
$962$	0	0
$963$	55684.4i	1.86335i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	27578.9i	0.917144i	0.888657	+	0.458572i	$0.151639\pi$
$967$	27578.9i	0.917144i	−0.888657	+	0.458572i	$0.848361\pi$
$968$	0	0
$969$	32959.3	1.09268
$970$	0	0
$971$	20834.3	0.688574	0.344287	−	0.938864i	$-0.388121\pi$
$971$	20834.3	0.688574	0.344287	+	0.938864i	$0.388121\pi$
$972$	0	0
$973$	3752.88i	0.123650i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 41636.7i	− 1.36344i	−0.731615	−	0.681718i	$-0.761233\pi$
$977$	− 41636.7i	− 1.36344i	0.731615	−	0.681718i	$-0.238767\pi$
$978$	0	0
$979$	−18663.8	−0.609292
$980$	0	0
$981$	−37166.1	−1.20960
$982$	0	0
$983$	− 18099.2i	− 0.587259i	−0.955919	−	0.293629i	$-0.905137\pi$
$983$	− 18099.2i	− 0.587259i	0.955919	−	0.293629i	$-0.0948631\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	32894.1i	1.06082i
$988$	0	0
$989$	22492.0	0.723158
$990$	0	0
$991$	55904.9	1.79201	0.896003	−	0.444049i	$-0.146458\pi$
$991$	55904.9	1.79201	0.896003	+	0.444049i	$0.146458\pi$
$992$	0	0
$993$	51737.3i	1.65341i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	15808.7i	0.502174i	0.967965	+	0.251087i	$0.0807881\pi$
$997$	15808.7i	0.502174i	−0.967965	+	0.251087i	$0.919212\pi$
$998$	0	0
$999$	186332.	5.90119

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.4.c.o.449.2		8
4.3	odd	2	inner	800.4.c.o.449.7		8
5.2	odd	4		800.4.a.bb.1.1	yes	4
5.3	odd	4		800.4.a.ba.1.4	yes	4
5.4	even	2	inner	800.4.c.o.449.8		8
20.3	even	4		800.4.a.ba.1.1	✓	4
20.7	even	4		800.4.a.bb.1.4	yes	4
20.19	odd	2	inner	800.4.c.o.449.1		8
40.3	even	4		1600.4.a.cx.1.4		4
40.13	odd	4		1600.4.a.cx.1.1		4
40.27	even	4		1600.4.a.cw.1.1		4
40.37	odd	4		1600.4.a.cw.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
800.4.a.ba.1.1	✓	4	20.3	even	4
800.4.a.ba.1.4	yes	4	5.3	odd	4
800.4.a.bb.1.1	yes	4	5.2	odd	4
800.4.a.bb.1.4	yes	4	20.7	even	4
800.4.c.o.449.1		8	20.19	odd	2	inner
800.4.c.o.449.2		8	1.1	even	1	trivial
800.4.c.o.449.7		8	4.3	odd	2	inner
800.4.c.o.449.8		8	5.4	even	2	inner
1600.4.a.cw.1.1		4	40.27	even	4
1600.4.a.cw.1.4		4	40.37	odd	4
1600.4.a.cx.1.1		4	40.13	odd	4
1600.4.a.cx.1.4		4	40.3	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:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	800.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-10.0971i q^{3} +10.5923i q^{7} -74.9510 q^{9} +O(q^{10})\)	\(q-10.0971i q^{3} +10.5923i q^{7} -74.9510 q^{9} +38.9027 q^{11} +68.9510i q^{13} +65.9510i q^{17} +49.4950 q^{19} +106.951 q^{21} -164.524i q^{23} +484.164i q^{27} +170.853 q^{29} +166.505 q^{31} -392.804i q^{33} -384.853i q^{37} +696.203 q^{39} -22.8038 q^{41} +136.709i q^{43} +307.562i q^{47} +230.804 q^{49} +665.912 q^{51} -222.000i q^{53} -499.755i q^{57} -522.765 q^{59} +393.049 q^{61} -793.901i q^{63} +476.845i q^{67} -1661.22 q^{69} -4.26352 q^{71} +601.166i q^{73} +412.068i q^{77} +1075.93 q^{79} +2864.97 q^{81} +1132.74i q^{83} -1725.11i q^{87} -479.755 q^{89} -730.347 q^{91} -1681.22i q^{93} +635.510i q^{97} -2915.80 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 192 q^{9}+O(q^{10}) \) 8 * q - 192 * q^9	\( 8 q - 192 q^{9} + 448 q^{21} + 144 q^{29} + 1448 q^{41} + 216 q^{49} + 3552 q^{61} - 6768 q^{69} + 14360 q^{81} - 1800 q^{89}+O(q^{100}) \) 8 * q - 192 * q^9 + 448 * q^21 + 144 * q^29 + 1448 * q^41 + 216 * q^49 + 3552 * q^61 - 6768 * q^69 + 14360 * q^81 - 1800 * q^89

Introduction

L-functions

Modular forms

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Database

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