LMFDB - Embedded newform 1344.4.a.bs.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1344,4,Mod(1,1344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1344, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1344.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1344.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$79.2985670477$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.37341.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 57x - 148 $ x^3 - 57*x - 148
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 672)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-5.47374$ of defining polynomial
Character	$\chi$	$=$	1344.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} -12.9475 q^{5} +7.00000 q^{7} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} -12.9475 q^{5} +7.00000 q^{7} +9.00000 q^{9} +50.3745 q^{11} +9.63710 q^{13} +38.8424 q^{15} +84.3745 q^{17} -61.5321 q^{19} -21.0000 q^{21} +38.5846 q^{23} +42.6371 q^{25} -27.0000 q^{27} -281.756 q^{29} +168.386 q^{31} -151.123 q^{33} -90.6323 q^{35} -58.3629 q^{37} -28.9113 q^{39} -83.0184 q^{41} -32.7258 q^{43} -116.527 q^{45} -260.634 q^{47} +49.0000 q^{49} -253.123 q^{51} -168.911 q^{53} -652.222 q^{55} +184.596 q^{57} +236.004 q^{59} +450.413 q^{61} +63.0000 q^{63} -124.776 q^{65} -542.858 q^{67} -115.754 q^{69} -437.324 q^{71} +1099.78 q^{73} -127.911 q^{75} +352.621 q^{77} +272.050 q^{79} +81.0000 q^{81} -136.371 q^{83} -1092.44 q^{85} +845.267 q^{87} +1419.65 q^{89} +67.4597 q^{91} -505.158 q^{93} +796.685 q^{95} +269.510 q^{97} +453.370 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 9 q^{3} - 6 q^{5} + 21 q^{7} + 27 q^{9}+O(q^{10}) $ 3 * q - 9 * q^3 - 6 * q^5 + 21 * q^7 + 27 * q^9	$ 3 q - 9 q^{3} - 6 q^{5} + 21 q^{7} + 27 q^{9} - 48 q^{11} - 6 q^{13} + 18 q^{15} + 54 q^{17} - 84 q^{19} - 63 q^{21} + 48 q^{23} + 93 q^{25} - 81 q^{27} - 18 q^{29} + 72 q^{31} + 144 q^{33} - 42 q^{35} - 210 q^{37} + 18 q^{39} + 414 q^{41} - 168 q^{43} - 54 q^{45} + 72 q^{47} + 147 q^{49} - 162 q^{51} - 402 q^{53} - 456 q^{55} + 252 q^{57} - 540 q^{59} - 798 q^{61} + 189 q^{63} + 1740 q^{65} - 48 q^{67} - 144 q^{69} - 456 q^{71} + 1230 q^{73} - 279 q^{75} - 336 q^{77} - 1368 q^{79} + 243 q^{81} - 60 q^{83} - 660 q^{85} + 54 q^{87} + 2742 q^{89} - 42 q^{91} - 216 q^{93} - 648 q^{95} + 1950 q^{97} - 432 q^{99}+O(q^{100}) $ 3 * q - 9 * q^3 - 6 * q^5 + 21 * q^7 + 27 * q^9 - 48 * q^11 - 6 * q^13 + 18 * q^15 + 54 * q^17 - 84 * q^19 - 63 * q^21 + 48 * q^23 + 93 * q^25 - 81 * q^27 - 18 * q^29 + 72 * q^31 + 144 * q^33 - 42 * q^35 - 210 * q^37 + 18 * q^39 + 414 * q^41 - 168 * q^43 - 54 * q^45 + 72 * q^47 + 147 * q^49 - 162 * q^51 - 402 * q^53 - 456 * q^55 + 252 * q^57 - 540 * q^59 - 798 * q^61 + 189 * q^63 + 1740 * q^65 - 48 * q^67 - 144 * q^69 - 456 * q^71 + 1230 * q^73 - 279 * q^75 - 336 * q^77 - 1368 * q^79 + 243 * q^81 - 60 * q^83 - 660 * q^85 + 54 * q^87 + 2742 * q^89 - 42 * q^91 - 216 * q^93 - 648 * q^95 + 1950 * q^97 - 432 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	−12.9475	−1.15806	−0.579029	−	0.815307i	$-0.696568\pi$
$5$	−12.9475	−1.15806	−0.579029	+	0.815307i	$0.696568\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	50.3745	1.38077	0.690385	−	0.723442i	$-0.257441\pi$
$11$	50.3745	1.38077	0.690385	+	0.723442i	$0.257441\pi$
$12$	0	0
$13$	9.63710	0.205604	0.102802	−	0.994702i	$-0.467219\pi$
$13$	9.63710	0.205604	0.102802	+	0.994702i	$0.467219\pi$
$14$	0	0
$15$	38.8424	0.668605
$16$	0	0
$17$	84.3745	1.20375	0.601877	−	0.798589i	$-0.294420\pi$
$17$	84.3745	1.20375	0.601877	+	0.798589i	$0.294420\pi$
$18$	0	0
$19$	−61.5321	−0.742970	−0.371485	−	0.928439i	$-0.621151\pi$
$19$	−61.5321	−0.742970	−0.371485	+	0.928439i	$0.621151\pi$
$20$	0	0
$21$	−21.0000	−0.218218
$22$	0	0
$23$	38.5846	0.349802	0.174901	−	0.984586i	$-0.444039\pi$
$23$	38.5846	0.349802	0.174901	+	0.984586i	$0.444039\pi$
$24$	0	0
$25$	42.6371	0.341097
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−281.756	−1.80416	−0.902082	−	0.431566i	$-0.857961\pi$
$29$	−281.756	−1.80416	−0.902082	+	0.431566i	$0.857961\pi$
$30$	0	0
$31$	168.386	0.975582	0.487791	−	0.872961i	$-0.337803\pi$
$31$	168.386	0.975582	0.487791	+	0.872961i	$0.337803\pi$
$32$	0	0
$33$	−151.123	−0.797188
$34$	0	0
$35$	−90.6323	−0.437705
$36$	0	0
$37$	−58.3629	−0.259319	−0.129659	−	0.991559i	$-0.541388\pi$
$37$	−58.3629	−0.259319	−0.129659	+	0.991559i	$0.541388\pi$
$38$	0	0
$39$	−28.9113	−0.118706
$40$	0	0
$41$	−83.0184	−0.316227	−0.158113	−	0.987421i	$-0.550541\pi$
$41$	−83.0184	−0.316227	−0.158113	+	0.987421i	$0.550541\pi$
$42$	0	0
$43$	−32.7258	−0.116061	−0.0580307	−	0.998315i	$-0.518482\pi$
$43$	−32.7258	−0.116061	−0.0580307	+	0.998315i	$0.518482\pi$
$44$	0	0
$45$	−116.527	−0.386019
$46$	0	0
$47$	−260.634	−0.808881	−0.404441	−	0.914564i	$-0.632534\pi$
$47$	−260.634	−0.808881	−0.404441	+	0.914564i	$0.632534\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−253.123	−0.694988
$52$	0	0
$53$	−168.911	−0.437769	−0.218884	−	0.975751i	$-0.570242\pi$
$53$	−168.911	−0.437769	−0.218884	+	0.975751i	$0.570242\pi$
$54$	0	0
$55$	−652.222	−1.59901
$56$	0	0
$57$	184.596	0.428954
$58$	0	0
$59$	236.004	0.520765	0.260382	−	0.965506i	$-0.416151\pi$
$59$	236.004	0.520765	0.260382	+	0.965506i	$0.416151\pi$
$60$	0	0
$61$	450.413	0.945402	0.472701	−	0.881223i	$-0.343279\pi$
$61$	450.413	0.945402	0.472701	+	0.881223i	$0.343279\pi$
$62$	0	0
$63$	63.0000	0.125988
$64$	0	0
$65$	−124.776	−0.238101
$66$	0	0
$67$	−542.858	−0.989861	−0.494930	−	0.868933i	$-0.664806\pi$
$67$	−542.858	−0.989861	−0.494930	+	0.868933i	$0.664806\pi$
$68$	0	0
$69$	−115.754	−0.201958
$70$	0	0
$71$	−437.324	−0.730997	−0.365499	−	0.930812i	$-0.619102\pi$
$71$	−437.324	−0.730997	−0.365499	+	0.930812i	$0.619102\pi$
$72$	0	0
$73$	1099.78	1.76328	0.881639	−	0.471925i	$-0.156441\pi$
$73$	1099.78	1.76328	0.881639	+	0.471925i	$0.156441\pi$
$74$	0	0
$75$	−127.911	−0.196932
$76$	0	0
$77$	352.621	0.521882
$78$	0	0
$79$	272.050	0.387444	0.193722	−	0.981056i	$-0.437944\pi$
$79$	272.050	0.387444	0.193722	+	0.981056i	$0.437944\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−136.371	−0.180345	−0.0901727	−	0.995926i	$-0.528742\pi$
$83$	−136.371	−0.180345	−0.0901727	+	0.995926i	$0.528742\pi$
$84$	0	0
$85$	−1092.44	−1.39402
$86$	0	0
$87$	845.267	1.04163
$88$	0	0
$89$	1419.65	1.69082	0.845409	−	0.534119i	$-0.179356\pi$
$89$	1419.65	1.69082	0.845409	+	0.534119i	$0.179356\pi$
$90$	0	0
$91$	67.4597	0.0777110
$92$	0	0
$93$	−505.158	−0.563252
$94$	0	0
$95$	796.685	0.860401
$96$	0	0
$97$	269.510	0.282110	0.141055	−	0.990002i	$-0.454951\pi$
$97$	269.510	0.282110	0.141055	+	0.990002i	$0.454951\pi$
$98$	0	0
$99$	453.370	0.460257
$100$	0	0
$101$	555.799	0.547565	0.273782	−	0.961792i	$-0.411725\pi$
$101$	555.799	0.547565	0.273782	+	0.961792i	$0.411725\pi$
$102$	0	0
$103$	−1912.39	−1.82945	−0.914727	−	0.404072i	$-0.867594\pi$
$103$	−1912.39	−1.82945	−0.914727	+	0.404072i	$0.867594\pi$
$104$	0	0
$105$	271.897	0.252709
$106$	0	0
$107$	921.016	0.832130	0.416065	−	0.909335i	$-0.363409\pi$
$107$	921.016	0.832130	0.416065	+	0.909335i	$0.363409\pi$
$108$	0	0
$109$	24.5484	0.0215717	0.0107858	−	0.999942i	$-0.496567\pi$
$109$	24.5484	0.0215717	0.0107858	+	0.999942i	$0.496567\pi$
$110$	0	0
$111$	175.089	0.149718
$112$	0	0
$113$	1928.06	1.60510	0.802551	−	0.596583i	$-0.203475\pi$
$113$	1928.06	1.60510	0.802551	+	0.596583i	$0.203475\pi$
$114$	0	0
$115$	−499.573	−0.405091
$116$	0	0
$117$	86.7339	0.0685347
$118$	0	0
$119$	590.621	0.454976
$120$	0	0
$121$	1206.59	0.906527
$122$	0	0
$123$	249.055	0.182574
$124$	0	0
$125$	1066.39	0.763048
$126$	0	0
$127$	−1493.51	−1.04353	−0.521763	−	0.853090i	$-0.674726\pi$
$127$	−1493.51	−1.04353	−0.521763	+	0.853090i	$0.674726\pi$
$128$	0	0
$129$	98.1774	0.0670080
$130$	0	0
$131$	2038.82	1.35979	0.679895	−	0.733309i	$-0.262025\pi$
$131$	2038.82	1.35979	0.679895	+	0.733309i	$0.262025\pi$
$132$	0	0
$133$	−430.724	−0.280816
$134$	0	0
$135$	349.582	0.222868
$136$	0	0
$137$	2320.45	1.44707	0.723537	−	0.690286i	$-0.242515\pi$
$137$	2320.45	1.44707	0.723537	+	0.690286i	$0.242515\pi$
$138$	0	0
$139$	−324.445	−0.197979	−0.0989893	−	0.995088i	$-0.531561\pi$
$139$	−324.445	−0.197979	−0.0989893	+	0.995088i	$0.531561\pi$
$140$	0	0
$141$	781.903	0.467008
$142$	0	0
$143$	485.464	0.283892
$144$	0	0
$145$	3648.03	2.08932
$146$	0	0
$147$	−147.000	−0.0824786
$148$	0	0
$149$	−2344.12	−1.28884	−0.644422	−	0.764670i	$-0.722902\pi$
$149$	−2344.12	−1.28884	−0.644422	+	0.764670i	$0.722902\pi$
$150$	0	0
$151$	−2610.09	−1.40666	−0.703331	−	0.710863i	$-0.748305\pi$
$151$	−2610.09	−1.40666	−0.703331	+	0.710863i	$0.748305\pi$
$152$	0	0
$153$	759.370	0.401251
$154$	0	0
$155$	−2180.17	−1.12978
$156$	0	0
$157$	1499.71	0.762357	0.381178	−	0.924501i	$-0.375518\pi$
$157$	1499.71	0.762357	0.381178	+	0.924501i	$0.375518\pi$
$158$	0	0
$159$	506.734	0.252746
$160$	0	0
$161$	270.092	0.132213
$162$	0	0
$163$	716.834	0.344459	0.172229	−	0.985057i	$-0.444903\pi$
$163$	716.834	0.344459	0.172229	+	0.985057i	$0.444903\pi$
$164$	0	0
$165$	1956.67	0.923190
$166$	0	0
$167$	3767.82	1.74588	0.872941	−	0.487826i	$-0.162210\pi$
$167$	3767.82	1.74588	0.872941	+	0.487826i	$0.162210\pi$
$168$	0	0
$169$	−2104.13	−0.957727
$170$	0	0
$171$	−553.788	−0.247657
$172$	0	0
$173$	1686.20	0.741036	0.370518	−	0.928825i	$-0.379180\pi$
$173$	1686.20	0.741036	0.370518	+	0.928825i	$0.379180\pi$
$174$	0	0
$175$	298.460	0.128922
$176$	0	0
$177$	−708.012	−0.300664
$178$	0	0
$179$	−4101.86	−1.71278	−0.856388	−	0.516332i	$-0.827297\pi$
$179$	−4101.86	−1.71278	−0.856388	+	0.516332i	$0.827297\pi$
$180$	0	0
$181$	992.993	0.407782	0.203891	−	0.978994i	$-0.434641\pi$
$181$	992.993	0.407782	0.203891	+	0.978994i	$0.434641\pi$
$182$	0	0
$183$	−1351.24	−0.545828
$184$	0	0
$185$	755.652	0.300306
$186$	0	0
$187$	4250.32	1.66211
$188$	0	0
$189$	−189.000	−0.0727393
$190$	0	0
$191$	3773.36	1.42948	0.714740	−	0.699390i	$-0.246545\pi$
$191$	3773.36	1.42948	0.714740	+	0.699390i	$0.246545\pi$
$192$	0	0
$193$	−2100.40	−0.783367	−0.391683	−	0.920100i	$-0.628107\pi$
$193$	−2100.40	−0.783367	−0.391683	+	0.920100i	$0.628107\pi$
$194$	0	0
$195$	374.328	0.137468
$196$	0	0
$197$	3050.45	1.10323	0.551613	−	0.834100i	$-0.314013\pi$
$197$	3050.45	1.10323	0.551613	+	0.834100i	$0.314013\pi$
$198$	0	0
$199$	2937.24	1.04631	0.523155	−	0.852238i	$-0.324755\pi$
$199$	2937.24	1.04631	0.523155	+	0.852238i	$0.324755\pi$
$200$	0	0
$201$	1628.57	0.571496
$202$	0	0
$203$	−1972.29	−0.681910
$204$	0	0
$205$	1074.88	0.366209
$206$	0	0
$207$	347.261	0.116601
$208$	0	0
$209$	−3099.65	−1.02587
$210$	0	0
$211$	1621.01	0.528887	0.264444	−	0.964401i	$-0.414812\pi$
$211$	1621.01	0.528887	0.264444	+	0.964401i	$0.414812\pi$
$212$	0	0
$213$	1311.97	0.422041
$214$	0	0
$215$	423.716	0.134406
$216$	0	0
$217$	1178.70	0.368735
$218$	0	0
$219$	−3299.33	−1.01803
$220$	0	0
$221$	813.126	0.247497
$222$	0	0
$223$	−5687.03	−1.70776	−0.853882	−	0.520466i	$-0.825758\pi$
$223$	−5687.03	−1.70776	−0.853882	+	0.520466i	$0.825758\pi$
$224$	0	0
$225$	383.734	0.113699
$226$	0	0
$227$	4029.94	1.17831	0.589156	−	0.808019i	$-0.299460\pi$
$227$	4029.94	1.17831	0.589156	+	0.808019i	$0.299460\pi$
$228$	0	0
$229$	637.547	0.183975	0.0919876	−	0.995760i	$-0.470678\pi$
$229$	637.547	0.183975	0.0919876	+	0.995760i	$0.470678\pi$
$230$	0	0
$231$	−1057.86	−0.301309
$232$	0	0
$233$	397.611	0.111796	0.0558978	−	0.998436i	$-0.482198\pi$
$233$	397.611	0.111796	0.0558978	+	0.998436i	$0.482198\pi$
$234$	0	0
$235$	3374.56	0.936731
$236$	0	0
$237$	−816.151	−0.223691
$238$	0	0
$239$	3851.90	1.04251	0.521253	−	0.853402i	$-0.325465\pi$
$239$	3851.90	1.04251	0.521253	+	0.853402i	$0.325465\pi$
$240$	0	0
$241$	−3740.16	−0.999687	−0.499844	−	0.866116i	$-0.666609\pi$
$241$	−3740.16	−0.999687	−0.499844	+	0.866116i	$0.666609\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	−634.426	−0.165437
$246$	0	0
$247$	−592.991	−0.152758
$248$	0	0
$249$	409.113	0.104122
$250$	0	0
$251$	1555.53	0.391173	0.195587	−	0.980686i	$-0.437339\pi$
$251$	1555.53	0.391173	0.195587	+	0.980686i	$0.437339\pi$
$252$	0	0
$253$	1943.68	0.482996
$254$	0	0
$255$	3277.31	0.804836
$256$	0	0
$257$	−5473.20	−1.32844	−0.664219	−	0.747538i	$-0.731236\pi$
$257$	−5473.20	−1.32844	−0.664219	+	0.747538i	$0.731236\pi$
$258$	0	0
$259$	−408.540	−0.0980133
$260$	0	0
$261$	−2535.80	−0.601388
$262$	0	0
$263$	−3930.08	−0.921442	−0.460721	−	0.887545i	$-0.652409\pi$
$263$	−3930.08	−0.921442	−0.460721	+	0.887545i	$0.652409\pi$
$264$	0	0
$265$	2186.97	0.506962
$266$	0	0
$267$	−4258.96	−0.976195
$268$	0	0
$269$	−1338.92	−0.303477	−0.151738	−	0.988421i	$-0.548487\pi$
$269$	−1338.92	−0.303477	−0.151738	+	0.988421i	$0.548487\pi$
$270$	0	0
$271$	1406.78	0.315335	0.157667	−	0.987492i	$-0.449603\pi$
$271$	1406.78	0.315335	0.157667	+	0.987492i	$0.449603\pi$
$272$	0	0
$273$	−202.379	−0.0448665
$274$	0	0
$275$	2147.82	0.470976
$276$	0	0
$277$	4261.68	0.924403	0.462202	−	0.886775i	$-0.347060\pi$
$277$	4261.68	0.924403	0.462202	+	0.886775i	$0.347060\pi$
$278$	0	0
$279$	1515.47	0.325194
$280$	0	0
$281$	7047.84	1.49622	0.748112	−	0.663573i	$-0.230961\pi$
$281$	7047.84	1.49622	0.748112	+	0.663573i	$0.230961\pi$
$282$	0	0
$283$	4048.15	0.850310	0.425155	−	0.905121i	$-0.360220\pi$
$283$	4048.15	0.850310	0.425155	+	0.905121i	$0.360220\pi$
$284$	0	0
$285$	−2390.05	−0.496753
$286$	0	0
$287$	−581.129	−0.119522
$288$	0	0
$289$	2206.05	0.449024
$290$	0	0
$291$	−808.531	−0.162876
$292$	0	0
$293$	6763.35	1.34853	0.674265	−	0.738490i	$-0.264461\pi$
$293$	6763.35	1.34853	0.674265	+	0.738490i	$0.264461\pi$
$294$	0	0
$295$	−3055.66	−0.603075
$296$	0	0
$297$	−1360.11	−0.265729
$298$	0	0
$299$	371.844	0.0719206
$300$	0	0
$301$	−229.081	−0.0438671
$302$	0	0
$303$	−1667.40	−0.316137
$304$	0	0
$305$	−5831.71	−1.09483
$306$	0	0
$307$	−8451.41	−1.57116	−0.785582	−	0.618758i	$-0.787636\pi$
$307$	−8451.41	−1.57116	−0.785582	+	0.618758i	$0.787636\pi$
$308$	0	0
$309$	5737.18	1.05624
$310$	0	0
$311$	5549.46	1.01184	0.505918	−	0.862582i	$-0.331154\pi$
$311$	5549.46	1.01184	0.505918	+	0.862582i	$0.331154\pi$
$312$	0	0
$313$	2138.08	0.386107	0.193054	−	0.981188i	$-0.438161\pi$
$313$	2138.08	0.386107	0.193054	+	0.981188i	$0.438161\pi$
$314$	0	0
$315$	−815.691	−0.145902
$316$	0	0
$317$	5408.13	0.958204	0.479102	−	0.877759i	$-0.340962\pi$
$317$	5408.13	0.958204	0.479102	+	0.877759i	$0.340962\pi$
$318$	0	0
$319$	−14193.3	−2.49114
$320$	0	0
$321$	−2763.05	−0.480431
$322$	0	0
$323$	−5191.73	−0.894353
$324$	0	0
$325$	410.898	0.0701309
$326$	0	0
$327$	−73.6453	−0.0124544
$328$	0	0
$329$	−1824.44	−0.305728
$330$	0	0
$331$	7622.95	1.26585	0.632923	−	0.774215i	$-0.281855\pi$
$331$	7622.95	1.26585	0.632923	+	0.774215i	$0.281855\pi$
$332$	0	0
$333$	−525.266	−0.0864396
$334$	0	0
$335$	7028.64	1.14632
$336$	0	0
$337$	4762.96	0.769895	0.384948	−	0.922938i	$-0.374219\pi$
$337$	4762.96	0.769895	0.384948	+	0.922938i	$0.374219\pi$
$338$	0	0
$339$	−5784.18	−0.926707
$340$	0	0
$341$	8482.36	1.34705
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	1498.72	0.233879
$346$	0	0
$347$	12112.0	1.87379	0.936894	−	0.349613i	$-0.113687\pi$
$347$	12112.0	1.87379	0.936894	+	0.349613i	$0.113687\pi$
$348$	0	0
$349$	11305.2	1.73396	0.866982	−	0.498339i	$-0.166057\pi$
$349$	11305.2	1.73396	0.866982	+	0.498339i	$0.166057\pi$
$350$	0	0
$351$	−260.202	−0.0395685
$352$	0	0
$353$	5428.79	0.818542	0.409271	−	0.912413i	$-0.365783\pi$
$353$	5428.79	0.818542	0.409271	+	0.912413i	$0.365783\pi$
$354$	0	0
$355$	5662.24	0.846537
$356$	0	0
$357$	−1771.86	−0.262681
$358$	0	0
$359$	−6609.72	−0.971720	−0.485860	−	0.874037i	$-0.661494\pi$
$359$	−6609.72	−0.971720	−0.485860	+	0.874037i	$0.661494\pi$
$360$	0	0
$361$	−3072.81	−0.447996
$362$	0	0
$363$	−3619.76	−0.523384
$364$	0	0
$365$	−14239.3	−2.04198
$366$	0	0
$367$	−2674.23	−0.380364	−0.190182	−	0.981749i	$-0.560908\pi$
$367$	−2674.23	−0.380364	−0.190182	+	0.981749i	$0.560908\pi$
$368$	0	0
$369$	−747.165	−0.105409
$370$	0	0
$371$	−1182.38	−0.165461
$372$	0	0
$373$	7916.54	1.09894	0.549468	−	0.835515i	$-0.314830\pi$
$373$	7916.54	1.09894	0.549468	+	0.835515i	$0.314830\pi$
$374$	0	0
$375$	−3199.17	−0.440546
$376$	0	0
$377$	−2715.31	−0.370943
$378$	0	0
$379$	−3311.19	−0.448772	−0.224386	−	0.974500i	$-0.572038\pi$
$379$	−3311.19	−0.448772	−0.224386	+	0.974500i	$0.572038\pi$
$380$	0	0
$381$	4480.54	0.602480
$382$	0	0
$383$	1045.40	0.139471	0.0697354	−	0.997566i	$-0.477785\pi$
$383$	1045.40	0.139471	0.0697354	+	0.997566i	$0.477785\pi$
$384$	0	0
$385$	−4565.56	−0.604370
$386$	0	0
$387$	−294.532	−0.0386871
$388$	0	0
$389$	1096.68	0.142940	0.0714702	−	0.997443i	$-0.477231\pi$
$389$	1096.68	0.142940	0.0714702	+	0.997443i	$0.477231\pi$
$390$	0	0
$391$	3255.55	0.421075
$392$	0	0
$393$	−6116.46	−0.785076
$394$	0	0
$395$	−3522.37	−0.448682
$396$	0	0
$397$	−2293.08	−0.289890	−0.144945	−	0.989440i	$-0.546301\pi$
$397$	−2293.08	−0.289890	−0.144945	+	0.989440i	$0.546301\pi$
$398$	0	0
$399$	1292.17	0.162129
$400$	0	0
$401$	−11386.3	−1.41797	−0.708983	−	0.705226i	$-0.750846\pi$
$401$	−11386.3	−1.41797	−0.708983	+	0.705226i	$0.750846\pi$
$402$	0	0
$403$	1622.75	0.200583
$404$	0	0
$405$	−1048.75	−0.128673
$406$	0	0
$407$	−2940.00	−0.358060
$408$	0	0
$409$	−13136.1	−1.58811	−0.794057	−	0.607844i	$-0.792035\pi$
$409$	−13136.1	−1.58811	−0.794057	+	0.607844i	$0.792035\pi$
$410$	0	0
$411$	−6961.34	−0.835468
$412$	0	0
$413$	1652.03	0.196831
$414$	0	0
$415$	1765.66	0.208850
$416$	0	0
$417$	973.334	0.114303
$418$	0	0
$419$	−4697.39	−0.547691	−0.273846	−	0.961774i	$-0.588296\pi$
$419$	−4697.39	−0.547691	−0.273846	+	0.961774i	$0.588296\pi$
$420$	0	0
$421$	10340.7	1.19708	0.598542	−	0.801091i	$-0.295747\pi$
$421$	10340.7	1.19708	0.598542	+	0.801091i	$0.295747\pi$
$422$	0	0
$423$	−2345.71	−0.269627
$424$	0	0
$425$	3597.48	0.410597
$426$	0	0
$427$	3152.89	0.357328
$428$	0	0
$429$	−1456.39	−0.163905
$430$	0	0
$431$	−934.495	−0.104439	−0.0522193	−	0.998636i	$-0.516629\pi$
$431$	−934.495	−0.104439	−0.0522193	+	0.998636i	$0.516629\pi$
$432$	0	0
$433$	5361.48	0.595050	0.297525	−	0.954714i	$-0.403839\pi$
$433$	5361.48	0.595050	0.297525	+	0.954714i	$0.403839\pi$
$434$	0	0
$435$	−10944.1	−1.20627
$436$	0	0
$437$	−2374.19	−0.259892
$438$	0	0
$439$	8500.70	0.924183	0.462091	−	0.886832i	$-0.347099\pi$
$439$	8500.70	0.924183	0.462091	+	0.886832i	$0.347099\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	8854.95	0.949687	0.474844	−	0.880070i	$-0.342505\pi$
$443$	8854.95	0.949687	0.474844	+	0.880070i	$0.342505\pi$
$444$	0	0
$445$	−18380.9	−1.95807
$446$	0	0
$447$	7032.36	0.744114
$448$	0	0
$449$	11097.2	1.16639	0.583195	−	0.812333i	$-0.301803\pi$
$449$	11097.2	1.16639	0.583195	+	0.812333i	$0.301803\pi$
$450$	0	0
$451$	−4182.01	−0.436636
$452$	0	0
$453$	7830.26	0.812136
$454$	0	0
$455$	−873.433	−0.0899938
$456$	0	0
$457$	12548.9	1.28449	0.642246	−	0.766499i	$-0.278003\pi$
$457$	12548.9	1.28449	0.642246	+	0.766499i	$0.278003\pi$
$458$	0	0
$459$	−2278.11	−0.231663
$460$	0	0
$461$	−2813.75	−0.284272	−0.142136	−	0.989847i	$-0.545397\pi$
$461$	−2813.75	−0.284272	−0.142136	+	0.989847i	$0.545397\pi$
$462$	0	0
$463$	−5804.23	−0.582604	−0.291302	−	0.956631i	$-0.594088\pi$
$463$	−5804.23	−0.582604	−0.291302	+	0.956631i	$0.594088\pi$
$464$	0	0
$465$	6540.52	0.652278
$466$	0	0
$467$	1048.38	0.103883	0.0519416	−	0.998650i	$-0.483459\pi$
$467$	1048.38	0.103883	0.0519416	+	0.998650i	$0.483459\pi$
$468$	0	0
$469$	−3800.01	−0.374132
$470$	0	0
$471$	−4499.14	−0.440147
$472$	0	0
$473$	−1648.54	−0.160254
$474$	0	0
$475$	−2623.55	−0.253425
$476$	0	0
$477$	−1520.20	−0.145923
$478$	0	0
$479$	16587.5	1.58226	0.791129	−	0.611650i	$-0.209494\pi$
$479$	16587.5	1.58226	0.791129	+	0.611650i	$0.209494\pi$
$480$	0	0
$481$	−562.449	−0.0533170
$482$	0	0
$483$	−810.276	−0.0763330
$484$	0	0
$485$	−3489.48	−0.326699
$486$	0	0
$487$	−11053.9	−1.02854	−0.514272	−	0.857627i	$-0.671938\pi$
$487$	−11053.9	−1.02854	−0.514272	+	0.857627i	$0.671938\pi$
$488$	0	0
$489$	−2150.50	−0.198873
$490$	0	0
$491$	−14277.1	−1.31226	−0.656128	−	0.754649i	$-0.727807\pi$
$491$	−14277.1	−1.31226	−0.656128	+	0.754649i	$0.727807\pi$
$492$	0	0
$493$	−23773.0	−2.17177
$494$	0	0
$495$	−5870.00	−0.533004
$496$	0	0
$497$	−3061.27	−0.276291
$498$	0	0
$499$	20512.8	1.84024	0.920120	−	0.391637i	$-0.128091\pi$
$499$	20512.8	1.84024	0.920120	+	0.391637i	$0.128091\pi$
$500$	0	0
$501$	−11303.4	−1.00799
$502$	0	0
$503$	9871.95	0.875087	0.437543	−	0.899197i	$-0.355849\pi$
$503$	9871.95	0.875087	0.437543	+	0.899197i	$0.355849\pi$
$504$	0	0
$505$	−7196.19	−0.634111
$506$	0	0
$507$	6312.38	0.552944
$508$	0	0
$509$	3791.10	0.330133	0.165067	−	0.986282i	$-0.447216\pi$
$509$	3791.10	0.330133	0.165067	+	0.986282i	$0.447216\pi$
$510$	0	0
$511$	7698.44	0.666456
$512$	0	0
$513$	1661.37	0.142985
$514$	0	0
$515$	24760.7	2.11861
$516$	0	0
$517$	−13129.3	−1.11688
$518$	0	0
$519$	−5058.59	−0.427837
$520$	0	0
$521$	3308.41	0.278204	0.139102	−	0.990278i	$-0.455578\pi$
$521$	3308.41	0.278204	0.139102	+	0.990278i	$0.455578\pi$
$522$	0	0
$523$	−2675.61	−0.223702	−0.111851	−	0.993725i	$-0.535678\pi$
$523$	−2675.61	−0.223702	−0.111851	+	0.993725i	$0.535678\pi$
$524$	0	0
$525$	−895.379	−0.0744334
$526$	0	0
$527$	14207.5	1.17436
$528$	0	0
$529$	−10678.2	−0.877639
$530$	0	0
$531$	2124.04	0.173588
$532$	0	0
$533$	−800.057	−0.0650175
$534$	0	0
$535$	−11924.8	−0.963655
$536$	0	0
$537$	12305.6	0.988872
$538$	0	0
$539$	2468.35	0.197253
$540$	0	0
$541$	−11480.4	−0.912349	−0.456174	−	0.889890i	$-0.650781\pi$
$541$	−11480.4	−0.912349	−0.456174	+	0.889890i	$0.650781\pi$
$542$	0	0
$543$	−2978.98	−0.235433
$544$	0	0
$545$	−317.840	−0.0249812
$546$	0	0
$547$	−23031.0	−1.80024	−0.900121	−	0.435639i	$-0.856522\pi$
$547$	−23031.0	−1.80024	−0.900121	+	0.435639i	$0.856522\pi$
$548$	0	0
$549$	4053.72	0.315134
$550$	0	0
$551$	17337.0	1.34044
$552$	0	0
$553$	1904.35	0.146440
$554$	0	0
$555$	−2266.96	−0.173382
$556$	0	0
$557$	−19636.4	−1.49376	−0.746878	−	0.664961i	$-0.768448\pi$
$557$	−19636.4	−1.49376	−0.746878	+	0.664961i	$0.768448\pi$
$558$	0	0
$559$	−315.382	−0.0238627
$560$	0	0
$561$	−12751.0	−0.959619
$562$	0	0
$563$	−14183.7	−1.06176	−0.530881	−	0.847446i	$-0.678139\pi$
$563$	−14183.7	−1.06176	−0.530881	+	0.847446i	$0.678139\pi$
$564$	0	0
$565$	−24963.5	−1.85880
$566$	0	0
$567$	567.000	0.0419961
$568$	0	0
$569$	1469.79	0.108290	0.0541449	−	0.998533i	$-0.482757\pi$
$569$	1469.79	0.108290	0.0541449	+	0.998533i	$0.482757\pi$
$570$	0	0
$571$	13826.2	1.01332	0.506662	−	0.862145i	$-0.330879\pi$
$571$	13826.2	1.01332	0.506662	+	0.862145i	$0.330879\pi$
$572$	0	0
$573$	−11320.1	−0.825310
$574$	0	0
$575$	1645.13	0.119316
$576$	0	0
$577$	−17815.7	−1.28540	−0.642702	−	0.766117i	$-0.722186\pi$
$577$	−17815.7	−1.28540	−0.642702	+	0.766117i	$0.722186\pi$
$578$	0	0
$579$	6301.19	0.452277
$580$	0	0
$581$	−954.597	−0.0681641
$582$	0	0
$583$	−8508.82	−0.604458
$584$	0	0
$585$	−1122.99	−0.0793671
$586$	0	0
$587$	912.364	0.0641521	0.0320761	−	0.999485i	$-0.489788\pi$
$587$	912.364	0.0641521	0.0320761	+	0.999485i	$0.489788\pi$
$588$	0	0
$589$	−10361.1	−0.724827
$590$	0	0
$591$	−9151.34	−0.636947
$592$	0	0
$593$	−9188.95	−0.636332	−0.318166	−	0.948035i	$-0.603067\pi$
$593$	−9188.95	−0.636332	−0.318166	+	0.948035i	$0.603067\pi$
$594$	0	0
$595$	−7647.06	−0.526889
$596$	0	0
$597$	−8811.72	−0.604087
$598$	0	0
$599$	−17432.9	−1.18913	−0.594566	−	0.804047i	$-0.702676\pi$
$599$	−17432.9	−1.18913	−0.594566	+	0.804047i	$0.702676\pi$
$600$	0	0
$601$	15003.4	1.01830	0.509152	−	0.860677i	$-0.329959\pi$
$601$	15003.4	1.01830	0.509152	+	0.860677i	$0.329959\pi$
$602$	0	0
$603$	−4885.72	−0.329954
$604$	0	0
$605$	−15622.3	−1.04981
$606$	0	0
$607$	−27923.8	−1.86720	−0.933602	−	0.358312i	$-0.883353\pi$
$607$	−27923.8	−1.86720	−0.933602	+	0.358312i	$0.883353\pi$
$608$	0	0
$609$	5916.87	0.393701
$610$	0	0
$611$	−2511.76	−0.166309
$612$	0	0
$613$	22925.3	1.51051	0.755255	−	0.655431i	$-0.227513\pi$
$613$	22925.3	1.51051	0.755255	+	0.655431i	$0.227513\pi$
$614$	0	0
$615$	−3224.64	−0.211431
$616$	0	0
$617$	−15891.8	−1.03692	−0.518460	−	0.855102i	$-0.673494\pi$
$617$	−15891.8	−1.03692	−0.518460	+	0.855102i	$0.673494\pi$
$618$	0	0
$619$	−26174.7	−1.69960	−0.849799	−	0.527106i	$-0.823277\pi$
$619$	−26174.7	−1.69960	−0.849799	+	0.527106i	$0.823277\pi$
$620$	0	0
$621$	−1041.78	−0.0673194
$622$	0	0
$623$	9937.57	0.639070
$624$	0	0
$625$	−19136.7	−1.22475
$626$	0	0
$627$	9298.94	0.592287
$628$	0	0
$629$	−4924.34	−0.312156
$630$	0	0
$631$	27500.4	1.73498	0.867491	−	0.497454i	$-0.165732\pi$
$631$	27500.4	1.73498	0.867491	+	0.497454i	$0.165732\pi$
$632$	0	0
$633$	−4863.04	−0.305353
$634$	0	0
$635$	19337.2	1.20846
$636$	0	0
$637$	472.218	0.0293720
$638$	0	0
$639$	−3935.92	−0.243666
$640$	0	0
$641$	20367.5	1.25502	0.627509	−	0.778609i	$-0.284074\pi$
$641$	20367.5	1.25502	0.627509	+	0.778609i	$0.284074\pi$
$642$	0	0
$643$	−15853.7	−0.972331	−0.486166	−	0.873867i	$-0.661605\pi$
$643$	−15853.7	−0.972331	−0.486166	+	0.873867i	$0.661605\pi$
$644$	0	0
$645$	−1271.15	−0.0775991
$646$	0	0
$647$	16200.8	0.984420	0.492210	−	0.870476i	$-0.336189\pi$
$647$	16200.8	0.984420	0.492210	+	0.870476i	$0.336189\pi$
$648$	0	0
$649$	11888.6	0.719056
$650$	0	0
$651$	−3536.11	−0.212889
$652$	0	0
$653$	−27662.8	−1.65778	−0.828889	−	0.559414i	$-0.811026\pi$
$653$	−27662.8	−1.65778	−0.828889	+	0.559414i	$0.811026\pi$
$654$	0	0
$655$	−26397.6	−1.57472
$656$	0	0
$657$	9898.00	0.587759
$658$	0	0
$659$	29422.9	1.73923	0.869614	−	0.493732i	$-0.164368\pi$
$659$	29422.9	1.73923	0.869614	+	0.493732i	$0.164368\pi$
$660$	0	0
$661$	15385.2	0.905315	0.452657	−	0.891684i	$-0.350476\pi$
$661$	15385.2	0.905315	0.452657	+	0.891684i	$0.350476\pi$
$662$	0	0
$663$	−2439.38	−0.142892
$664$	0	0
$665$	5576.79	0.325201
$666$	0	0
$667$	−10871.4	−0.631099
$668$	0	0
$669$	17061.1	0.985978
$670$	0	0
$671$	22689.3	1.30538
$672$	0	0
$673$	2014.77	0.115399	0.0576997	−	0.998334i	$-0.481623\pi$
$673$	2014.77	0.115399	0.0576997	+	0.998334i	$0.481623\pi$
$674$	0	0
$675$	−1151.20	−0.0656441
$676$	0	0
$677$	13103.2	0.743863	0.371931	−	0.928260i	$-0.378696\pi$
$677$	13103.2	0.743863	0.371931	+	0.928260i	$0.378696\pi$
$678$	0	0
$679$	1886.57	0.106627
$680$	0	0
$681$	−12089.8	−0.680298
$682$	0	0
$683$	21602.6	1.21025	0.605126	−	0.796130i	$-0.293123\pi$
$683$	21602.6	1.21025	0.605126	+	0.796130i	$0.293123\pi$
$684$	0	0
$685$	−30043.9	−1.67579
$686$	0	0
$687$	−1912.64	−0.106218
$688$	0	0
$689$	−1627.82	−0.0900070
$690$	0	0
$691$	30702.5	1.69027	0.845137	−	0.534550i	$-0.179519\pi$
$691$	30702.5	1.69027	0.845137	+	0.534550i	$0.179519\pi$
$692$	0	0
$693$	3173.59	0.173961
$694$	0	0
$695$	4200.74	0.229271
$696$	0	0
$697$	−7004.63	−0.380659
$698$	0	0
$699$	−1192.83	−0.0645452
$700$	0	0
$701$	−2311.72	−0.124554	−0.0622771	−	0.998059i	$-0.519836\pi$
$701$	−2311.72	−0.124554	−0.0622771	+	0.998059i	$0.519836\pi$
$702$	0	0
$703$	3591.19	0.192666
$704$	0	0
$705$	−10123.7	−0.540822
$706$	0	0
$707$	3890.59	0.206960
$708$	0	0
$709$	21972.8	1.16390	0.581950	−	0.813225i	$-0.302290\pi$
$709$	21972.8	1.16390	0.581950	+	0.813225i	$0.302290\pi$
$710$	0	0
$711$	2448.45	0.129148
$712$	0	0
$713$	6497.11	0.341260
$714$	0	0
$715$	−6285.53	−0.328763
$716$	0	0
$717$	−11555.7	−0.601891
$718$	0	0
$719$	−11040.1	−0.572635	−0.286318	−	0.958135i	$-0.592431\pi$
$719$	−11040.1	−0.572635	−0.286318	+	0.958135i	$0.592431\pi$
$720$	0	0
$721$	−13386.8	−0.691469
$722$	0	0
$723$	11220.5	0.577170
$724$	0	0
$725$	−12013.2	−0.615394
$726$	0	0
$727$	21573.7	1.10058	0.550291	−	0.834973i	$-0.314517\pi$
$727$	21573.7	1.10058	0.550291	+	0.834973i	$0.314517\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−2761.22	−0.139709
$732$	0	0
$733$	−5080.79	−0.256021	−0.128010	−	0.991773i	$-0.540859\pi$
$733$	−5080.79	−0.256021	−0.128010	+	0.991773i	$0.540859\pi$
$734$	0	0
$735$	1903.28	0.0955150
$736$	0	0
$737$	−27346.2	−1.36677
$738$	0	0
$739$	−10798.1	−0.537503	−0.268751	−	0.963210i	$-0.586611\pi$
$739$	−10798.1	−0.537503	−0.268751	+	0.963210i	$0.586611\pi$
$740$	0	0
$741$	1778.97	0.0881946
$742$	0	0
$743$	−11856.2	−0.585415	−0.292707	−	0.956202i	$-0.594556\pi$
$743$	−11856.2	−0.585415	−0.292707	+	0.956202i	$0.594556\pi$
$744$	0	0
$745$	30350.4	1.49256
$746$	0	0
$747$	−1227.34	−0.0601151
$748$	0	0
$749$	6447.11	0.314516
$750$	0	0
$751$	3951.40	0.191996	0.0959978	−	0.995382i	$-0.469396\pi$
$751$	3951.40	0.191996	0.0959978	+	0.995382i	$0.469396\pi$
$752$	0	0
$753$	−4666.60	−0.225844
$754$	0	0
$755$	33794.0	1.62899
$756$	0	0
$757$	17491.0	0.839791	0.419896	−	0.907572i	$-0.362067\pi$
$757$	17491.0	0.839791	0.419896	+	0.907572i	$0.362067\pi$
$758$	0	0
$759$	−5831.03	−0.278858
$760$	0	0
$761$	14552.8	0.693219	0.346610	−	0.938009i	$-0.387333\pi$
$761$	14552.8	0.693219	0.346610	+	0.938009i	$0.387333\pi$
$762$	0	0
$763$	171.839	0.00815332
$764$	0	0
$765$	−9831.93	−0.464672
$766$	0	0
$767$	2274.40	0.107071
$768$	0	0
$769$	−5902.98	−0.276810	−0.138405	−	0.990376i	$-0.544198\pi$
$769$	−5902.98	−0.276810	−0.138405	+	0.990376i	$0.544198\pi$
$770$	0	0
$771$	16419.6	0.766974
$772$	0	0
$773$	−5314.31	−0.247274	−0.123637	−	0.992328i	$-0.539456\pi$
$773$	−5314.31	−0.247274	−0.123637	+	0.992328i	$0.539456\pi$
$774$	0	0
$775$	7179.49	0.332768
$776$	0	0
$777$	1225.62	0.0565880
$778$	0	0
$779$	5108.29	0.234947
$780$	0	0
$781$	−22030.0	−1.00934
$782$	0	0
$783$	7607.41	0.347211
$784$	0	0
$785$	−19417.5	−0.882853
$786$	0	0
$787$	−40609.1	−1.83933	−0.919667	−	0.392698i	$-0.871542\pi$
$787$	−40609.1	−1.83933	−0.919667	+	0.392698i	$0.871542\pi$
$788$	0	0
$789$	11790.2	0.531995
$790$	0	0
$791$	13496.4	0.606672
$792$	0	0
$793$	4340.68	0.194378
$794$	0	0
$795$	−6560.92	−0.292694
$796$	0	0
$797$	−1698.11	−0.0754707	−0.0377353	−	0.999288i	$-0.512014\pi$
$797$	−1698.11	−0.0754707	−0.0377353	+	0.999288i	$0.512014\pi$
$798$	0	0
$799$	−21990.9	−0.973694
$800$	0	0
$801$	12776.9	0.563606
$802$	0	0
$803$	55400.7	2.43468
$804$	0	0
$805$	−3497.01	−0.153110
$806$	0	0
$807$	4016.75	0.175212
$808$	0	0
$809$	17829.7	0.774855	0.387427	−	0.921900i	$-0.373364\pi$
$809$	17829.7	0.774855	0.387427	+	0.921900i	$0.373364\pi$
$810$	0	0
$811$	−22388.2	−0.969367	−0.484683	−	0.874690i	$-0.661065\pi$
$811$	−22388.2	−0.969367	−0.484683	+	0.874690i	$0.661065\pi$
$812$	0	0
$813$	−4220.33	−0.182059
$814$	0	0
$815$	−9281.18	−0.398903
$816$	0	0
$817$	2013.69	0.0862300
$818$	0	0
$819$	607.138	0.0259037
$820$	0	0
$821$	−14841.1	−0.630889	−0.315444	−	0.948944i	$-0.602154\pi$
$821$	−14841.1	−0.630889	−0.315444	+	0.948944i	$0.602154\pi$
$822$	0	0
$823$	19799.6	0.838603	0.419301	−	0.907847i	$-0.362275\pi$
$823$	19799.6	0.838603	0.419301	+	0.907847i	$0.362275\pi$
$824$	0	0
$825$	−6443.47	−0.271918
$826$	0	0
$827$	−20537.7	−0.863561	−0.431781	−	0.901979i	$-0.642115\pi$
$827$	−20537.7	−0.863561	−0.431781	+	0.901979i	$0.642115\pi$
$828$	0	0
$829$	18146.2	0.760244	0.380122	−	0.924936i	$-0.375882\pi$
$829$	18146.2	0.760244	0.380122	+	0.924936i	$0.375882\pi$
$830$	0	0
$831$	−12785.1	−0.533705
$832$	0	0
$833$	4134.35	0.171965
$834$	0	0
$835$	−48783.7	−2.02183
$836$	0	0
$837$	−4546.42	−0.187751
$838$	0	0
$839$	23037.3	0.947956	0.473978	−	0.880537i	$-0.342818\pi$
$839$	23037.3	0.947956	0.473978	+	0.880537i	$0.342818\pi$
$840$	0	0
$841$	54997.3	2.25500
$842$	0	0
$843$	−21143.5	−0.863845
$844$	0	0
$845$	27243.1	1.10910
$846$	0	0
$847$	8446.12	0.342635
$848$	0	0
$849$	−12144.5	−0.490926
$850$	0	0
$851$	−2251.91	−0.0907102
$852$	0	0
$853$	9567.13	0.384024	0.192012	−	0.981393i	$-0.438499\pi$
$853$	9567.13	0.384024	0.192012	+	0.981393i	$0.438499\pi$
$854$	0	0
$855$	7170.16	0.286800
$856$	0	0
$857$	−17492.0	−0.697219	−0.348610	−	0.937268i	$-0.613346\pi$
$857$	−17492.0	−0.697219	−0.348610	+	0.937268i	$0.613346\pi$
$858$	0	0
$859$	−8357.19	−0.331948	−0.165974	−	0.986130i	$-0.553077\pi$
$859$	−8357.19	−0.331948	−0.165974	+	0.986130i	$0.553077\pi$
$860$	0	0
$861$	1743.39	0.0690063
$862$	0	0
$863$	−25905.9	−1.02184	−0.510920	−	0.859628i	$-0.670695\pi$
$863$	−25905.9	−1.02184	−0.510920	+	0.859628i	$0.670695\pi$
$864$	0	0
$865$	−21832.0	−0.858162
$866$	0	0
$867$	−6618.16	−0.259244
$868$	0	0
$869$	13704.4	0.534971
$870$	0	0
$871$	−5231.58	−0.203519
$872$	0	0
$873$	2425.59	0.0940365
$874$	0	0
$875$	7464.74	0.288405
$876$	0	0
$877$	41803.4	1.60958	0.804789	−	0.593561i	$-0.202278\pi$
$877$	41803.4	1.60958	0.804789	+	0.593561i	$0.202278\pi$
$878$	0	0
$879$	−20290.0	−0.778574
$880$	0	0
$881$	−33968.8	−1.29902	−0.649510	−	0.760353i	$-0.725026\pi$
$881$	−33968.8	−1.29902	−0.649510	+	0.760353i	$0.725026\pi$
$882$	0	0
$883$	−32367.3	−1.23357	−0.616787	−	0.787130i	$-0.711566\pi$
$883$	−32367.3	−1.23357	−0.616787	+	0.787130i	$0.711566\pi$
$884$	0	0
$885$	9166.97	0.348186
$886$	0	0
$887$	−4073.32	−0.154193	−0.0770963	−	0.997024i	$-0.524565\pi$
$887$	−4073.32	−0.154193	−0.0770963	+	0.997024i	$0.524565\pi$
$888$	0	0
$889$	−10454.6	−0.394416
$890$	0	0
$891$	4080.33	0.153419
$892$	0	0
$893$	16037.4	0.600974
$894$	0	0
$895$	53108.7	1.98349
$896$	0	0
$897$	−1115.53	−0.0415234
$898$	0	0
$899$	−47443.7	−1.76011
$900$	0	0
$901$	−14251.8	−0.526966
$902$	0	0
$903$	687.242	0.0253267
$904$	0	0
$905$	−12856.8	−0.472235
$906$	0	0
$907$	9202.54	0.336897	0.168448	−	0.985710i	$-0.446124\pi$
$907$	9202.54	0.336897	0.168448	+	0.985710i	$0.446124\pi$
$908$	0	0
$909$	5002.19	0.182522
$910$	0	0
$911$	−1591.42	−0.0578771	−0.0289386	−	0.999581i	$-0.509213\pi$
$911$	−1591.42	−0.0578771	−0.0289386	+	0.999581i	$0.509213\pi$
$912$	0	0
$913$	−6869.62	−0.249016
$914$	0	0
$915$	17495.1	0.632100
$916$	0	0
$917$	14271.7	0.513953
$918$	0	0
$919$	−10105.0	−0.362714	−0.181357	−	0.983417i	$-0.558049\pi$
$919$	−10105.0	−0.362714	−0.181357	+	0.983417i	$0.558049\pi$
$920$	0	0
$921$	25354.2	0.907112
$922$	0	0
$923$	−4214.54	−0.150296
$924$	0	0
$925$	−2488.42	−0.0884529
$926$	0	0
$927$	−17211.5	−0.609818
$928$	0	0
$929$	−29786.4	−1.05195	−0.525974	−	0.850500i	$-0.676299\pi$
$929$	−29786.4	−1.05195	−0.525974	+	0.850500i	$0.676299\pi$
$930$	0	0
$931$	−3015.07	−0.106139
$932$	0	0
$933$	−16648.4	−0.584183
$934$	0	0
$935$	−55030.9	−1.92482
$936$	0	0
$937$	5248.02	0.182973	0.0914863	−	0.995806i	$-0.470838\pi$
$937$	5248.02	0.182973	0.0914863	+	0.995806i	$0.470838\pi$
$938$	0	0
$939$	−6414.25	−0.222919
$940$	0	0
$941$	7592.29	0.263020	0.131510	−	0.991315i	$-0.458018\pi$
$941$	7592.29	0.263020	0.131510	+	0.991315i	$0.458018\pi$
$942$	0	0
$943$	−3203.23	−0.110617
$944$	0	0
$945$	2447.07	0.0842363
$946$	0	0
$947$	12468.6	0.427851	0.213925	−	0.976850i	$-0.431375\pi$
$947$	12468.6	0.427851	0.213925	+	0.976850i	$0.431375\pi$
$948$	0	0
$949$	10598.7	0.362537
$950$	0	0
$951$	−16224.4	−0.553219
$952$	0	0
$953$	10698.8	0.363659	0.181829	−	0.983330i	$-0.441798\pi$
$953$	10698.8	0.363659	0.181829	+	0.983330i	$0.441798\pi$
$954$	0	0
$955$	−48855.5	−1.65542
$956$	0	0
$957$	42579.9	1.43826
$958$	0	0
$959$	16243.1	0.546942
$960$	0	0
$961$	−1437.13	−0.0482406
$962$	0	0
$963$	8289.14	0.277377
$964$	0	0
$965$	27194.8	0.907184
$966$	0	0
$967$	−22602.7	−0.751657	−0.375829	−	0.926689i	$-0.622642\pi$
$967$	−22602.7	−0.751657	−0.375829	+	0.926689i	$0.622642\pi$
$968$	0	0
$969$	15575.2	0.516355
$970$	0	0
$971$	40221.9	1.32933	0.664667	−	0.747140i	$-0.268573\pi$
$971$	40221.9	1.32933	0.664667	+	0.747140i	$0.268573\pi$
$972$	0	0
$973$	−2271.11	−0.0748289
$974$	0	0
$975$	−1232.69	−0.0404901
$976$	0	0
$977$	3121.48	0.102216	0.0511080	−	0.998693i	$-0.483725\pi$
$977$	3121.48	0.102216	0.0511080	+	0.998693i	$0.483725\pi$
$978$	0	0
$979$	71514.3	2.33463
$980$	0	0
$981$	220.936	0.00719056
$982$	0	0
$983$	−60207.1	−1.95352	−0.976758	−	0.214343i	$-0.931239\pi$
$983$	−60207.1	−1.95352	−0.976758	+	0.214343i	$0.931239\pi$
$984$	0	0
$985$	−39495.6	−1.27760
$986$	0	0
$987$	5473.32	0.176512
$988$	0	0
$989$	−1262.71	−0.0405985
$990$	0	0
$991$	8.74556	0.000280335 0	0.000140167	−	1.00000i	$-0.499955\pi$
$991$	8.74556	0.000280335 0	0.000140167		1.00000i	$0.499955\pi$
$992$	0	0
$993$	−22868.8	−0.730837
$994$	0	0
$995$	−38029.9	−1.21169
$996$	0	0
$997$	9825.28	0.312106	0.156053	−	0.987749i	$-0.450123\pi$
$997$	9825.28	0.312106	0.156053	+	0.987749i	$0.450123\pi$
$998$	0	0
$999$	1575.80	0.0499060

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.4.a.bs.1.1		3
4.3	odd	2		1344.4.a.bu.1.1		3
8.3	odd	2		672.4.a.p.1.3	✓	3
8.5	even	2		672.4.a.r.1.3	yes	3
24.5	odd	2		2016.4.a.v.1.1		3
24.11	even	2		2016.4.a.u.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.4.a.p.1.3	✓	3	8.3	odd	2
672.4.a.r.1.3	yes	3	8.5	even	2
1344.4.a.bs.1.1		3	1.1	even	1	trivial
1344.4.a.bu.1.1		3	4.3	odd	2
2016.4.a.u.1.1		3	24.11	even	2
2016.4.a.v.1.1		3	24.5	odd	2

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:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1344.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -12.9475 q^{5} +7.00000 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} -12.9475 q^{5} +7.00000 q^{7} +9.00000 q^{9} +50.3745 q^{11} +9.63710 q^{13} +38.8424 q^{15} +84.3745 q^{17} -61.5321 q^{19} -21.0000 q^{21} +38.5846 q^{23} +42.6371 q^{25} -27.0000 q^{27} -281.756 q^{29} +168.386 q^{31} -151.123 q^{33} -90.6323 q^{35} -58.3629 q^{37} -28.9113 q^{39} -83.0184 q^{41} -32.7258 q^{43} -116.527 q^{45} -260.634 q^{47} +49.0000 q^{49} -253.123 q^{51} -168.911 q^{53} -652.222 q^{55} +184.596 q^{57} +236.004 q^{59} +450.413 q^{61} +63.0000 q^{63} -124.776 q^{65} -542.858 q^{67} -115.754 q^{69} -437.324 q^{71} +1099.78 q^{73} -127.911 q^{75} +352.621 q^{77} +272.050 q^{79} +81.0000 q^{81} -136.371 q^{83} -1092.44 q^{85} +845.267 q^{87} +1419.65 q^{89} +67.4597 q^{91} -505.158 q^{93} +796.685 q^{95} +269.510 q^{97} +453.370 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 9 q^{3} - 6 q^{5} + 21 q^{7} + 27 q^{9}+O(q^{10}) \) 3 * q - 9 * q^3 - 6 * q^5 + 21 * q^7 + 27 * q^9	\( 3 q - 9 q^{3} - 6 q^{5} + 21 q^{7} + 27 q^{9} - 48 q^{11} - 6 q^{13} + 18 q^{15} + 54 q^{17} - 84 q^{19} - 63 q^{21} + 48 q^{23} + 93 q^{25} - 81 q^{27} - 18 q^{29} + 72 q^{31} + 144 q^{33} - 42 q^{35} - 210 q^{37} + 18 q^{39} + 414 q^{41} - 168 q^{43} - 54 q^{45} + 72 q^{47} + 147 q^{49} - 162 q^{51} - 402 q^{53} - 456 q^{55} + 252 q^{57} - 540 q^{59} - 798 q^{61} + 189 q^{63} + 1740 q^{65} - 48 q^{67} - 144 q^{69} - 456 q^{71} + 1230 q^{73} - 279 q^{75} - 336 q^{77} - 1368 q^{79} + 243 q^{81} - 60 q^{83} - 660 q^{85} + 54 q^{87} + 2742 q^{89} - 42 q^{91} - 216 q^{93} - 648 q^{95} + 1950 q^{97} - 432 q^{99}+O(q^{100}) \) 3 * q - 9 * q^3 - 6 * q^5 + 21 * q^7 + 27 * q^9 - 48 * q^11 - 6 * q^13 + 18 * q^15 + 54 * q^17 - 84 * q^19 - 63 * q^21 + 48 * q^23 + 93 * q^25 - 81 * q^27 - 18 * q^29 + 72 * q^31 + 144 * q^33 - 42 * q^35 - 210 * q^37 + 18 * q^39 + 414 * q^41 - 168 * q^43 - 54 * q^45 + 72 * q^47 + 147 * q^49 - 162 * q^51 - 402 * q^53 - 456 * q^55 + 252 * q^57 - 540 * q^59 - 798 * q^61 + 189 * q^63 + 1740 * q^65 - 48 * q^67 - 144 * q^69 - 456 * q^71 + 1230 * q^73 - 279 * q^75 - 336 * q^77 - 1368 * q^79 + 243 * q^81 - 60 * q^83 - 660 * q^85 + 54 * q^87 + 2742 * q^89 - 42 * q^91 - 216 * q^93 - 648 * q^95 + 1950 * q^97 - 432 * q^99

Introduction

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