LMFDB - Embedded newform 1380.4.a.d.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1380,4,Mod(1,1380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1380, 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("1380.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1380.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:	$81.4226358079$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 45x^{2} - 97x - 18 $ x^4 - x^3 - 45x^2 - 97x - 18
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-4.42363$ of defining polynomial
Character	$\chi$	$=$	1380.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} -5.00000 q^{5} +19.5952 q^{7} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} -5.00000 q^{5} +19.5952 q^{7} +9.00000 q^{9} -62.2285 q^{11} +13.6757 q^{13} +15.0000 q^{15} +36.6553 q^{17} +22.1955 q^{19} -58.7856 q^{21} -23.0000 q^{23} +25.0000 q^{25} -27.0000 q^{27} +10.8431 q^{29} +296.088 q^{31} +186.686 q^{33} -97.9760 q^{35} -241.697 q^{37} -41.0271 q^{39} -230.380 q^{41} -68.6500 q^{43} -45.0000 q^{45} -129.792 q^{47} +40.9717 q^{49} -109.966 q^{51} -369.181 q^{53} +311.143 q^{55} -66.5865 q^{57} +388.437 q^{59} +869.875 q^{61} +176.357 q^{63} -68.3785 q^{65} -502.151 q^{67} +69.0000 q^{69} -96.4670 q^{71} +535.587 q^{73} -75.0000 q^{75} -1219.38 q^{77} +373.886 q^{79} +81.0000 q^{81} +910.148 q^{83} -183.277 q^{85} -32.5294 q^{87} -396.126 q^{89} +267.978 q^{91} -888.263 q^{93} -110.978 q^{95} -144.547 q^{97} -560.057 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{3} - 20 q^{5} + 35 q^{7} + 36 q^{9}+O(q^{10}) $ 4 * q - 12 * q^3 - 20 * q^5 + 35 * q^7 + 36 * q^9	$ 4 q - 12 q^{3} - 20 q^{5} + 35 q^{7} + 36 q^{9} - 56 q^{11} + 42 q^{13} + 60 q^{15} - 139 q^{17} + 100 q^{19} - 105 q^{21} - 92 q^{23} + 100 q^{25} - 108 q^{27} + 125 q^{29} - 3 q^{31} + 168 q^{33} - 175 q^{35} + 243 q^{37} - 126 q^{39} - 349 q^{41} + 652 q^{43} - 180 q^{45} - 178 q^{47} + 367 q^{49} + 417 q^{51} + 157 q^{53} + 280 q^{55} - 300 q^{57} + 87 q^{59} + 428 q^{61} + 315 q^{63} - 210 q^{65} + 119 q^{67} + 276 q^{69} - 659 q^{71} + 1150 q^{73} - 300 q^{75} - 424 q^{77} + 384 q^{79} + 324 q^{81} - 127 q^{83} + 695 q^{85} - 375 q^{87} - 1278 q^{89} - 1024 q^{91} + 9 q^{93} - 500 q^{95} - 260 q^{97} - 504 q^{99}+O(q^{100}) $ 4 * q - 12 * q^3 - 20 * q^5 + 35 * q^7 + 36 * q^9 - 56 * q^11 + 42 * q^13 + 60 * q^15 - 139 * q^17 + 100 * q^19 - 105 * q^21 - 92 * q^23 + 100 * q^25 - 108 * q^27 + 125 * q^29 - 3 * q^31 + 168 * q^33 - 175 * q^35 + 243 * q^37 - 126 * q^39 - 349 * q^41 + 652 * q^43 - 180 * q^45 - 178 * q^47 + 367 * q^49 + 417 * q^51 + 157 * q^53 + 280 * q^55 - 300 * q^57 + 87 * q^59 + 428 * q^61 + 315 * q^63 - 210 * q^65 + 119 * q^67 + 276 * q^69 - 659 * q^71 + 1150 * q^73 - 300 * q^75 - 424 * q^77 + 384 * q^79 + 324 * q^81 - 127 * q^83 + 695 * q^85 - 375 * q^87 - 1278 * q^89 - 1024 * q^91 + 9 * q^93 - 500 * q^95 - 260 * q^97 - 504 * 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
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	19.5952	1.05804	0.529021	−	0.848609i	$-0.322559\pi$
$7$	19.5952	1.05804	0.529021	+	0.848609i	$0.322559\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−62.2285	−1.70569	−0.852846	−	0.522163i	$-0.825125\pi$
$11$	−62.2285	−1.70569	−0.852846	+	0.522163i	$0.825125\pi$
$12$	0	0
$13$	13.6757	0.291766	0.145883	−	0.989302i	$-0.453398\pi$
$13$	13.6757	0.291766	0.145883	+	0.989302i	$0.453398\pi$
$14$	0	0
$15$	15.0000	0.258199
$16$	0	0
$17$	36.6553	0.522954	0.261477	−	0.965210i	$-0.415790\pi$
$17$	36.6553	0.522954	0.261477	+	0.965210i	$0.415790\pi$
$18$	0	0
$19$	22.1955	0.268000	0.134000	−	0.990981i	$-0.457218\pi$
$19$	22.1955	0.268000	0.134000	+	0.990981i	$0.457218\pi$
$20$	0	0
$21$	−58.7856	−0.610860
$22$	0	0
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	10.8431	0.0694317	0.0347159	−	0.999397i	$-0.488947\pi$
$29$	10.8431	0.0694317	0.0347159	+	0.999397i	$0.488947\pi$
$30$	0	0
$31$	296.088	1.71545	0.857724	−	0.514110i	$-0.171878\pi$
$31$	296.088	1.71545	0.857724	+	0.514110i	$0.171878\pi$
$32$	0	0
$33$	186.686	0.984781
$34$	0	0
$35$	−97.9760	−0.473170
$36$	0	0
$37$	−241.697	−1.07391	−0.536955	−	0.843611i	$-0.680426\pi$
$37$	−241.697	−1.07391	−0.536955	+	0.843611i	$0.680426\pi$
$38$	0	0
$39$	−41.0271	−0.168451
$40$	0	0
$41$	−230.380	−0.877544	−0.438772	−	0.898598i	$-0.644586\pi$
$41$	−230.380	−0.877544	−0.438772	+	0.898598i	$0.644586\pi$
$42$	0	0
$43$	−68.6500	−0.243466	−0.121733	−	0.992563i	$-0.538845\pi$
$43$	−68.6500	−0.243466	−0.121733	+	0.992563i	$0.538845\pi$
$44$	0	0
$45$	−45.0000	−0.149071
$46$	0	0
$47$	−129.792	−0.402812	−0.201406	−	0.979508i	$-0.564551\pi$
$47$	−129.792	−0.402812	−0.201406	+	0.979508i	$0.564551\pi$
$48$	0	0
$49$	40.9717	0.119451
$50$	0	0
$51$	−109.966	−0.301928
$52$	0	0
$53$	−369.181	−0.956809	−0.478405	−	0.878140i	$-0.658785\pi$
$53$	−369.181	−0.956809	−0.478405	+	0.878140i	$0.658785\pi$
$54$	0	0
$55$	311.143	0.762808
$56$	0	0
$57$	−66.5865	−0.154730
$58$	0	0
$59$	388.437	0.857123	0.428561	−	0.903513i	$-0.359021\pi$
$59$	388.437	0.857123	0.428561	+	0.903513i	$0.359021\pi$
$60$	0	0
$61$	869.875	1.82584	0.912918	−	0.408143i	$-0.133823\pi$
$61$	869.875	1.82584	0.912918	+	0.408143i	$0.133823\pi$
$62$	0	0
$63$	176.357	0.352680
$64$	0	0
$65$	−68.3785	−0.130482
$66$	0	0
$67$	−502.151	−0.915634	−0.457817	−	0.889046i	$-0.651369\pi$
$67$	−502.151	−0.915634	−0.457817	+	0.889046i	$0.651369\pi$
$68$	0	0
$69$	69.0000	0.120386
$70$	0	0
$71$	−96.4670	−0.161247	−0.0806234	−	0.996745i	$-0.525691\pi$
$71$	−96.4670	−0.161247	−0.0806234	+	0.996745i	$0.525691\pi$
$72$	0	0
$73$	535.587	0.858708	0.429354	−	0.903136i	$-0.358741\pi$
$73$	535.587	0.858708	0.429354	+	0.903136i	$0.358741\pi$
$74$	0	0
$75$	−75.0000	−0.115470
$76$	0	0
$77$	−1219.38	−1.80469
$78$	0	0
$79$	373.886	0.532474	0.266237	−	0.963908i	$-0.414220\pi$
$79$	373.886	0.532474	0.266237	+	0.963908i	$0.414220\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	910.148	1.20364	0.601818	−	0.798633i	$-0.294443\pi$
$83$	910.148	1.20364	0.601818	+	0.798633i	$0.294443\pi$
$84$	0	0
$85$	−183.277	−0.233872
$86$	0	0
$87$	−32.5294	−0.0400864
$88$	0	0
$89$	−396.126	−0.471789	−0.235895	−	0.971779i	$-0.575802\pi$
$89$	−396.126	−0.471789	−0.235895	+	0.971779i	$0.575802\pi$
$90$	0	0
$91$	267.978	0.308700
$92$	0	0
$93$	−888.263	−0.990415
$94$	0	0
$95$	−110.978	−0.119853
$96$	0	0
$97$	−144.547	−0.151304	−0.0756520	−	0.997134i	$-0.524104\pi$
$97$	−144.547	−0.151304	−0.0756520	+	0.997134i	$0.524104\pi$
$98$	0	0
$99$	−560.057	−0.568564
$100$	0	0
$101$	−932.614	−0.918798	−0.459399	−	0.888230i	$-0.651935\pi$
$101$	−932.614	−0.918798	−0.459399	+	0.888230i	$0.651935\pi$
$102$	0	0
$103$	31.3767	0.0300159	0.0150080	−	0.999887i	$-0.495223\pi$
$103$	31.3767	0.0300159	0.0150080	+	0.999887i	$0.495223\pi$
$104$	0	0
$105$	293.928	0.273185
$106$	0	0
$107$	−588.458	−0.531667	−0.265834	−	0.964019i	$-0.585647\pi$
$107$	−588.458	−0.531667	−0.265834	+	0.964019i	$0.585647\pi$
$108$	0	0
$109$	464.990	0.408605	0.204302	−	0.978908i	$-0.434507\pi$
$109$	464.990	0.408605	0.204302	+	0.978908i	$0.434507\pi$
$110$	0	0
$111$	725.090	0.620022
$112$	0	0
$113$	838.250	0.697840	0.348920	−	0.937152i	$-0.386548\pi$
$113$	838.250	0.697840	0.348920	+	0.937152i	$0.386548\pi$
$114$	0	0
$115$	115.000	0.0932505
$116$	0	0
$117$	123.081	0.0972553
$118$	0	0
$119$	718.268	0.553307
$120$	0	0
$121$	2541.39	1.90938
$122$	0	0
$123$	691.140	0.506650
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−1703.16	−1.19001	−0.595005	−	0.803722i	$-0.702850\pi$
$127$	−1703.16	−1.19001	−0.595005	+	0.803722i	$0.702850\pi$
$128$	0	0
$129$	205.950	0.140565
$130$	0	0
$131$	−1174.04	−0.783025	−0.391513	−	0.920173i	$-0.628048\pi$
$131$	−1174.04	−0.783025	−0.391513	+	0.920173i	$0.628048\pi$
$132$	0	0
$133$	434.925	0.283555
$134$	0	0
$135$	135.000	0.0860663
$136$	0	0
$137$	−1593.68	−0.993847	−0.496923	−	0.867794i	$-0.665537\pi$
$137$	−1593.68	−0.993847	−0.496923	+	0.867794i	$0.665537\pi$
$138$	0	0
$139$	−2449.92	−1.49496	−0.747482	−	0.664283i	$-0.768737\pi$
$139$	−2449.92	−1.49496	−0.747482	+	0.664283i	$0.768737\pi$
$140$	0	0
$141$	389.377	0.232564
$142$	0	0
$143$	−851.019	−0.497663
$144$	0	0
$145$	−54.2157	−0.0310508
$146$	0	0
$147$	−122.915	−0.0689651
$148$	0	0
$149$	−2252.92	−1.23870	−0.619350	−	0.785115i	$-0.712604\pi$
$149$	−2252.92	−1.23870	−0.619350	+	0.785115i	$0.712604\pi$
$150$	0	0
$151$	−1580.50	−0.851785	−0.425893	−	0.904774i	$-0.640040\pi$
$151$	−1580.50	−0.851785	−0.425893	+	0.904774i	$0.640040\pi$
$152$	0	0
$153$	329.898	0.174318
$154$	0	0
$155$	−1480.44	−0.767172
$156$	0	0
$157$	−2094.55	−1.06473	−0.532367	−	0.846514i	$-0.678697\pi$
$157$	−2094.55	−1.06473	−0.532367	+	0.846514i	$0.678697\pi$
$158$	0	0
$159$	1107.54	0.552414
$160$	0	0
$161$	−450.690	−0.220617
$162$	0	0
$163$	−1678.51	−0.806571	−0.403286	−	0.915074i	$-0.632132\pi$
$163$	−1678.51	−0.806571	−0.403286	+	0.915074i	$0.632132\pi$
$164$	0	0
$165$	−933.428	−0.440408
$166$	0	0
$167$	1351.50	0.626243	0.313121	−	0.949713i	$-0.398625\pi$
$167$	1351.50	0.626243	0.313121	+	0.949713i	$0.398625\pi$
$168$	0	0
$169$	−2009.98	−0.914873
$170$	0	0
$171$	199.760	0.0893333
$172$	0	0
$173$	−3739.48	−1.64340	−0.821698	−	0.569923i	$-0.806973\pi$
$173$	−3739.48	−1.64340	−0.821698	+	0.569923i	$0.806973\pi$
$174$	0	0
$175$	489.880	0.211608
$176$	0	0
$177$	−1165.31	−0.494860
$178$	0	0
$179$	−2025.86	−0.845923	−0.422962	−	0.906148i	$-0.639009\pi$
$179$	−2025.86	−0.845923	−0.422962	+	0.906148i	$0.639009\pi$
$180$	0	0
$181$	−2131.27	−0.875226	−0.437613	−	0.899163i	$-0.644176\pi$
$181$	−2131.27	−0.875226	−0.437613	+	0.899163i	$0.644176\pi$
$182$	0	0
$183$	−2609.62	−1.05415
$184$	0	0
$185$	1208.48	0.480267
$186$	0	0
$187$	−2281.01	−0.891998
$188$	0	0
$189$	−529.070	−0.203620
$190$	0	0
$191$	−3532.74	−1.33833	−0.669163	−	0.743116i	$-0.733347\pi$
$191$	−3532.74	−1.33833	−0.669163	+	0.743116i	$0.733347\pi$
$192$	0	0
$193$	−4658.31	−1.73737	−0.868685	−	0.495366i	$-0.835034\pi$
$193$	−4658.31	−1.73737	−0.868685	+	0.495366i	$0.835034\pi$
$194$	0	0
$195$	205.136	0.0753337
$196$	0	0
$197$	2029.37	0.733944	0.366972	−	0.930232i	$-0.380395\pi$
$197$	2029.37	0.733944	0.366972	+	0.930232i	$0.380395\pi$
$198$	0	0
$199$	136.194	0.0485153	0.0242577	−	0.999706i	$-0.492278\pi$
$199$	136.194	0.0485153	0.0242577	+	0.999706i	$0.492278\pi$
$200$	0	0
$201$	1506.45	0.528642
$202$	0	0
$203$	212.473	0.0734616
$204$	0	0
$205$	1151.90	0.392450
$206$	0	0
$207$	−207.000	−0.0695048
$208$	0	0
$209$	−1381.19	−0.457125
$210$	0	0
$211$	−6103.93	−1.99152	−0.995762	−	0.0919653i	$-0.970685\pi$
$211$	−6103.93	−1.99152	−0.995762	+	0.0919653i	$0.970685\pi$
$212$	0	0
$213$	289.401	0.0930959
$214$	0	0
$215$	343.250	0.108881
$216$	0	0
$217$	5801.90	1.81502
$218$	0	0
$219$	−1606.76	−0.495775
$220$	0	0
$221$	501.287	0.152580
$222$	0	0
$223$	3931.36	1.18055	0.590277	−	0.807201i	$-0.299019\pi$
$223$	3931.36	1.18055	0.590277	+	0.807201i	$0.299019\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	4661.67	1.36302	0.681510	−	0.731809i	$-0.261324\pi$
$227$	4661.67	1.36302	0.681510	+	0.731809i	$0.261324\pi$
$228$	0	0
$229$	−3949.35	−1.13965	−0.569827	−	0.821765i	$-0.692990\pi$
$229$	−3949.35	−1.13965	−0.569827	+	0.821765i	$0.692990\pi$
$230$	0	0
$231$	3658.14	1.04194
$232$	0	0
$233$	3662.71	1.02984	0.514919	−	0.857239i	$-0.327822\pi$
$233$	3662.71	1.02984	0.514919	+	0.857239i	$0.327822\pi$
$234$	0	0
$235$	648.962	0.180143
$236$	0	0
$237$	−1121.66	−0.307424
$238$	0	0
$239$	−4957.57	−1.34175	−0.670876	−	0.741570i	$-0.734082\pi$
$239$	−4957.57	−1.34175	−0.670876	+	0.741570i	$0.734082\pi$
$240$	0	0
$241$	1148.54	0.306987	0.153493	−	0.988150i	$-0.450948\pi$
$241$	1148.54	0.306987	0.153493	+	0.988150i	$0.450948\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	−204.859	−0.0534201
$246$	0	0
$247$	303.539	0.0781933
$248$	0	0
$249$	−2730.45	−0.694919
$250$	0	0
$251$	3251.47	0.817654	0.408827	−	0.912612i	$-0.365938\pi$
$251$	3251.47	0.817654	0.408827	+	0.912612i	$0.365938\pi$
$252$	0	0
$253$	1431.26	0.355661
$254$	0	0
$255$	549.830	0.135026
$256$	0	0
$257$	1231.08	0.298805	0.149403	−	0.988776i	$-0.452265\pi$
$257$	1231.08	0.298805	0.149403	+	0.988776i	$0.452265\pi$
$258$	0	0
$259$	−4736.09	−1.13624
$260$	0	0
$261$	97.5882	0.0231439
$262$	0	0
$263$	−5986.98	−1.40370	−0.701850	−	0.712325i	$-0.747642\pi$
$263$	−5986.98	−1.40370	−0.701850	+	0.712325i	$0.747642\pi$
$264$	0	0
$265$	1845.90	0.427898
$266$	0	0
$267$	1188.38	0.272388
$268$	0	0
$269$	−1099.61	−0.249235	−0.124617	−	0.992205i	$-0.539770\pi$
$269$	−1099.61	−0.249235	−0.124617	+	0.992205i	$0.539770\pi$
$270$	0	0
$271$	5962.39	1.33649	0.668246	−	0.743940i	$-0.267045\pi$
$271$	5962.39	1.33649	0.668246	+	0.743940i	$0.267045\pi$
$272$	0	0
$273$	−803.934	−0.178228
$274$	0	0
$275$	−1555.71	−0.341138
$276$	0	0
$277$	−3953.58	−0.857574	−0.428787	−	0.903406i	$-0.641059\pi$
$277$	−3953.58	−0.857574	−0.428787	+	0.903406i	$0.641059\pi$
$278$	0	0
$279$	2664.79	0.571816
$280$	0	0
$281$	−1814.50	−0.385210	−0.192605	−	0.981276i	$-0.561694\pi$
$281$	−1814.50	−0.385210	−0.192605	+	0.981276i	$0.561694\pi$
$282$	0	0
$283$	2556.91	0.537076	0.268538	−	0.963269i	$-0.413460\pi$
$283$	2556.91	0.537076	0.268538	+	0.963269i	$0.413460\pi$
$284$	0	0
$285$	332.933	0.0691973
$286$	0	0
$287$	−4514.34	−0.928478
$288$	0	0
$289$	−3569.39	−0.726519
$290$	0	0
$291$	433.640	0.0873554
$292$	0	0
$293$	2505.58	0.499582	0.249791	−	0.968300i	$-0.419638\pi$
$293$	2505.58	0.499582	0.249791	+	0.968300i	$0.419638\pi$
$294$	0	0
$295$	−1942.19	−0.383317
$296$	0	0
$297$	1680.17	0.328260
$298$	0	0
$299$	−314.541	−0.0608374
$300$	0	0
$301$	−1345.21	−0.257597
$302$	0	0
$303$	2797.84	0.530468
$304$	0	0
$305$	−4349.37	−0.816539
$306$	0	0
$307$	8994.27	1.67208	0.836042	−	0.548665i	$-0.184864\pi$
$307$	8994.27	1.67208	0.836042	+	0.548665i	$0.184864\pi$
$308$	0	0
$309$	−94.1302	−0.0173297
$310$	0	0
$311$	5259.39	0.958947	0.479474	−	0.877556i	$-0.340828\pi$
$311$	5259.39	0.958947	0.479474	+	0.877556i	$0.340828\pi$
$312$	0	0
$313$	7589.24	1.37051	0.685255	−	0.728304i	$-0.259691\pi$
$313$	7589.24	1.37051	0.685255	+	0.728304i	$0.259691\pi$
$314$	0	0
$315$	−881.784	−0.157723
$316$	0	0
$317$	−5659.31	−1.00271	−0.501354	−	0.865242i	$-0.667165\pi$
$317$	−5659.31	−1.00271	−0.501354	+	0.865242i	$0.667165\pi$
$318$	0	0
$319$	−674.752	−0.118429
$320$	0	0
$321$	1765.37	0.306958
$322$	0	0
$323$	813.583	0.140152
$324$	0	0
$325$	341.893	0.0583532
$326$	0	0
$327$	−1394.97	−0.235908
$328$	0	0
$329$	−2543.31	−0.426192
$330$	0	0
$331$	5319.24	0.883299	0.441650	−	0.897188i	$-0.354393\pi$
$331$	5319.24	0.883299	0.441650	+	0.897188i	$0.354393\pi$
$332$	0	0
$333$	−2175.27	−0.357970
$334$	0	0
$335$	2510.75	0.409484
$336$	0	0
$337$	3231.08	0.522279	0.261139	−	0.965301i	$-0.415902\pi$
$337$	3231.08	0.522279	0.261139	+	0.965301i	$0.415902\pi$
$338$	0	0
$339$	−2514.75	−0.402898
$340$	0	0
$341$	−18425.1	−2.92603
$342$	0	0
$343$	−5918.30	−0.931657
$344$	0	0
$345$	−345.000	−0.0538382
$346$	0	0
$347$	10431.7	1.61384	0.806920	−	0.590661i	$-0.201133\pi$
$347$	10431.7	1.61384	0.806920	+	0.590661i	$0.201133\pi$
$348$	0	0
$349$	−2565.61	−0.393506	−0.196753	−	0.980453i	$-0.563040\pi$
$349$	−2565.61	−0.393506	−0.196753	+	0.980453i	$0.563040\pi$
$350$	0	0
$351$	−369.244	−0.0561504
$352$	0	0
$353$	−7392.38	−1.11461	−0.557304	−	0.830308i	$-0.688164\pi$
$353$	−7392.38	−1.11461	−0.557304	+	0.830308i	$0.688164\pi$
$354$	0	0
$355$	482.335	0.0721118
$356$	0	0
$357$	−2154.80	−0.319452
$358$	0	0
$359$	10705.2	1.57382	0.786909	−	0.617069i	$-0.211680\pi$
$359$	10705.2	1.57382	0.786909	+	0.617069i	$0.211680\pi$
$360$	0	0
$361$	−6366.36	−0.928176
$362$	0	0
$363$	−7624.16	−1.10238
$364$	0	0
$365$	−2677.93	−0.384026
$366$	0	0
$367$	−3680.46	−0.523484	−0.261742	−	0.965138i	$-0.584297\pi$
$367$	−3680.46	−0.523484	−0.261742	+	0.965138i	$0.584297\pi$
$368$	0	0
$369$	−2073.42	−0.292515
$370$	0	0
$371$	−7234.17	−1.01234
$372$	0	0
$373$	7625.09	1.05848	0.529239	−	0.848473i	$-0.322477\pi$
$373$	7625.09	1.05848	0.529239	+	0.848473i	$0.322477\pi$
$374$	0	0
$375$	375.000	0.0516398
$376$	0	0
$377$	148.288	0.0202578
$378$	0	0
$379$	645.515	0.0874878	0.0437439	−	0.999043i	$-0.486071\pi$
$379$	645.515	0.0874878	0.0437439	+	0.999043i	$0.486071\pi$
$380$	0	0
$381$	5109.48	0.687052
$382$	0	0
$383$	−9355.08	−1.24810	−0.624050	−	0.781385i	$-0.714514\pi$
$383$	−9355.08	−1.24810	−0.624050	+	0.781385i	$0.714514\pi$
$384$	0	0
$385$	6096.90	0.807082
$386$	0	0
$387$	−617.850	−0.0811552
$388$	0	0
$389$	−11547.9	−1.50515	−0.752576	−	0.658506i	$-0.771189\pi$
$389$	−11547.9	−1.50515	−0.752576	+	0.658506i	$0.771189\pi$
$390$	0	0
$391$	−843.072	−0.109043
$392$	0	0
$393$	3522.12	0.452080
$394$	0	0
$395$	−1869.43	−0.238129
$396$	0	0
$397$	−12792.6	−1.61723	−0.808614	−	0.588339i	$-0.799782\pi$
$397$	−12792.6	−1.61723	−0.808614	+	0.588339i	$0.799782\pi$
$398$	0	0
$399$	−1304.78	−0.163711
$400$	0	0
$401$	−1169.27	−0.145612	−0.0728060	−	0.997346i	$-0.523195\pi$
$401$	−1169.27	−0.145612	−0.0728060	+	0.997346i	$0.523195\pi$
$402$	0	0
$403$	4049.21	0.500510
$404$	0	0
$405$	−405.000	−0.0496904
$406$	0	0
$407$	15040.4	1.83176
$408$	0	0
$409$	−984.381	−0.119009	−0.0595043	−	0.998228i	$-0.518952\pi$
$409$	−984.381	−0.119009	−0.0595043	+	0.998228i	$0.518952\pi$
$410$	0	0
$411$	4781.03	0.573798
$412$	0	0
$413$	7611.51	0.906871
$414$	0	0
$415$	−4550.74	−0.538282
$416$	0	0
$417$	7349.77	0.863117
$418$	0	0
$419$	2951.51	0.344130	0.172065	−	0.985086i	$-0.444956\pi$
$419$	2951.51	0.344130	0.172065	+	0.985086i	$0.444956\pi$
$420$	0	0
$421$	−11187.5	−1.29512	−0.647560	−	0.762015i	$-0.724210\pi$
$421$	−11187.5	−1.29512	−0.647560	+	0.762015i	$0.724210\pi$
$422$	0	0
$423$	−1168.13	−0.134271
$424$	0	0
$425$	916.383	0.104591
$426$	0	0
$427$	17045.4	1.93181
$428$	0	0
$429$	2553.06	0.287326
$430$	0	0
$431$	10473.1	1.17047	0.585236	−	0.810863i	$-0.301002\pi$
$431$	10473.1	1.17047	0.585236	+	0.810863i	$0.301002\pi$
$432$	0	0
$433$	−9309.52	−1.03323	−0.516613	−	0.856219i	$-0.672807\pi$
$433$	−9309.52	−1.03323	−0.516613	+	0.856219i	$0.672807\pi$
$434$	0	0
$435$	162.647	0.0179272
$436$	0	0
$437$	−510.497	−0.0558819
$438$	0	0
$439$	8918.58	0.969614	0.484807	−	0.874621i	$-0.338890\pi$
$439$	8918.58	0.969614	0.484807	+	0.874621i	$0.338890\pi$
$440$	0	0
$441$	368.745	0.0398170
$442$	0	0
$443$	−12357.4	−1.32532	−0.662661	−	0.748920i	$-0.730573\pi$
$443$	−12357.4	−1.32532	−0.662661	+	0.748920i	$0.730573\pi$
$444$	0	0
$445$	1980.63	0.210991
$446$	0	0
$447$	6758.76	0.715164
$448$	0	0
$449$	−12224.5	−1.28488	−0.642439	−	0.766337i	$-0.722077\pi$
$449$	−12224.5	−1.28488	−0.642439	+	0.766337i	$0.722077\pi$
$450$	0	0
$451$	14336.2	1.49682
$452$	0	0
$453$	4741.51	0.491778
$454$	0	0
$455$	−1339.89	−0.138055
$456$	0	0
$457$	12540.3	1.28361	0.641806	−	0.766867i	$-0.278185\pi$
$457$	12540.3	1.28361	0.641806	+	0.766867i	$0.278185\pi$
$458$	0	0
$459$	−989.693	−0.100643
$460$	0	0
$461$	−12202.5	−1.23281	−0.616405	−	0.787430i	$-0.711411\pi$
$461$	−12202.5	−1.23281	−0.616405	+	0.787430i	$0.711411\pi$
$462$	0	0
$463$	9502.52	0.953822	0.476911	−	0.878952i	$-0.341756\pi$
$463$	9502.52	0.953822	0.476911	+	0.878952i	$0.341756\pi$
$464$	0	0
$465$	4441.31	0.442927
$466$	0	0
$467$	10575.4	1.04790	0.523951	−	0.851748i	$-0.324457\pi$
$467$	10575.4	1.04790	0.523951	+	0.851748i	$0.324457\pi$
$468$	0	0
$469$	−9839.74	−0.968778
$470$	0	0
$471$	6283.65	0.614725
$472$	0	0
$473$	4271.99	0.415277
$474$	0	0
$475$	554.888	0.0536000
$476$	0	0
$477$	−3322.63	−0.318936
$478$	0	0
$479$	−2898.11	−0.276446	−0.138223	−	0.990401i	$-0.544139\pi$
$479$	−2898.11	−0.276446	−0.138223	+	0.990401i	$0.544139\pi$
$480$	0	0
$481$	−3305.37	−0.313331
$482$	0	0
$483$	1352.07	0.127373
$484$	0	0
$485$	722.733	0.0676652
$486$	0	0
$487$	−2504.81	−0.233067	−0.116534	−	0.993187i	$-0.537178\pi$
$487$	−2504.81	−0.233067	−0.116534	+	0.993187i	$0.537178\pi$
$488$	0	0
$489$	5035.53	0.465674
$490$	0	0
$491$	−4861.94	−0.446876	−0.223438	−	0.974718i	$-0.571728\pi$
$491$	−4861.94	−0.446876	−0.223438	+	0.974718i	$0.571728\pi$
$492$	0	0
$493$	397.459	0.0363096
$494$	0	0
$495$	2800.28	0.254269
$496$	0	0
$497$	−1890.29	−0.170606
$498$	0	0
$499$	6339.64	0.568740	0.284370	−	0.958715i	$-0.408216\pi$
$499$	6339.64	0.568740	0.284370	+	0.958715i	$0.408216\pi$
$500$	0	0
$501$	−4054.51	−0.361561
$502$	0	0
$503$	3653.28	0.323840	0.161920	−	0.986804i	$-0.448231\pi$
$503$	3653.28	0.323840	0.161920	+	0.986804i	$0.448231\pi$
$504$	0	0
$505$	4663.07	0.410899
$506$	0	0
$507$	6029.93	0.528202
$508$	0	0
$509$	−3901.52	−0.339748	−0.169874	−	0.985466i	$-0.554336\pi$
$509$	−3901.52	−0.339748	−0.169874	+	0.985466i	$0.554336\pi$
$510$	0	0
$511$	10494.9	0.908549
$512$	0	0
$513$	−599.279	−0.0515766
$514$	0	0
$515$	−156.884	−0.0134235
$516$	0	0
$517$	8076.79	0.687073
$518$	0	0
$519$	11218.4	0.948815
$520$	0	0
$521$	1359.73	0.114339	0.0571695	−	0.998364i	$-0.481792\pi$
$521$	1359.73	0.114339	0.0571695	+	0.998364i	$0.481792\pi$
$522$	0	0
$523$	8294.94	0.693523	0.346761	−	0.937953i	$-0.387281\pi$
$523$	8294.94	0.693523	0.346761	+	0.937953i	$0.387281\pi$
$524$	0	0
$525$	−1469.64	−0.122172
$526$	0	0
$527$	10853.2	0.897101
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	3495.94	0.285708
$532$	0	0
$533$	−3150.61	−0.256038
$534$	0	0
$535$	2942.29	0.237769
$536$	0	0
$537$	6077.59	0.488394
$538$	0	0
$539$	−2549.61	−0.203747
$540$	0	0
$541$	−16645.4	−1.32281	−0.661407	−	0.750027i	$-0.730040\pi$
$541$	−16645.4	−1.32281	−0.661407	+	0.750027i	$0.730040\pi$
$542$	0	0
$543$	6393.80	0.505312
$544$	0	0
$545$	−2324.95	−0.182734
$546$	0	0
$547$	−9890.34	−0.773091	−0.386545	−	0.922270i	$-0.626332\pi$
$547$	−9890.34	−0.773091	−0.386545	+	0.922270i	$0.626332\pi$
$548$	0	0
$549$	7828.87	0.608612
$550$	0	0
$551$	240.669	0.0186077
$552$	0	0
$553$	7326.36	0.563379
$554$	0	0
$555$	−3625.45	−0.277282
$556$	0	0
$557$	1588.89	0.120868	0.0604341	−	0.998172i	$-0.480751\pi$
$557$	1588.89	0.120868	0.0604341	+	0.998172i	$0.480751\pi$
$558$	0	0
$559$	−938.837	−0.0710350
$560$	0	0
$561$	6843.02	0.514995
$562$	0	0
$563$	−12935.8	−0.968348	−0.484174	−	0.874972i	$-0.660880\pi$
$563$	−12935.8	−0.968348	−0.484174	+	0.874972i	$0.660880\pi$
$564$	0	0
$565$	−4191.25	−0.312084
$566$	0	0
$567$	1587.21	0.117560
$568$	0	0
$569$	1541.18	0.113549	0.0567746	−	0.998387i	$-0.481918\pi$
$569$	1541.18	0.113549	0.0567746	+	0.998387i	$0.481918\pi$
$570$	0	0
$571$	−10304.0	−0.755180	−0.377590	−	0.925973i	$-0.623247\pi$
$571$	−10304.0	−0.755180	−0.377590	+	0.925973i	$0.623247\pi$
$572$	0	0
$573$	10598.2	0.772683
$574$	0	0
$575$	−575.000	−0.0417029
$576$	0	0
$577$	1840.62	0.132801	0.0664003	−	0.997793i	$-0.478849\pi$
$577$	1840.62	0.132801	0.0664003	+	0.997793i	$0.478849\pi$
$578$	0	0
$579$	13974.9	1.00307
$580$	0	0
$581$	17834.5	1.27350
$582$	0	0
$583$	22973.6	1.63202
$584$	0	0
$585$	−615.407	−0.0434939
$586$	0	0
$587$	−15122.6	−1.06334	−0.531668	−	0.846953i	$-0.678435\pi$
$587$	−15122.6	−1.06334	−0.531668	+	0.846953i	$0.678435\pi$
$588$	0	0
$589$	6571.82	0.459740
$590$	0	0
$591$	−6088.12	−0.423742
$592$	0	0
$593$	1312.34	0.0908791	0.0454395	−	0.998967i	$-0.485531\pi$
$593$	1312.34	0.0908791	0.0454395	+	0.998967i	$0.485531\pi$
$594$	0	0
$595$	−3591.34	−0.247446
$596$	0	0
$597$	−408.582	−0.0280103
$598$	0	0
$599$	7081.59	0.483048	0.241524	−	0.970395i	$-0.422353\pi$
$599$	7081.59	0.483048	0.241524	+	0.970395i	$0.422353\pi$
$600$	0	0
$601$	13531.2	0.918382	0.459191	−	0.888337i	$-0.348139\pi$
$601$	13531.2	0.918382	0.459191	+	0.888337i	$0.348139\pi$
$602$	0	0
$603$	−4519.36	−0.305211
$604$	0	0
$605$	−12706.9	−0.853902
$606$	0	0
$607$	−4199.50	−0.280811	−0.140406	−	0.990094i	$-0.544841\pi$
$607$	−4199.50	−0.280811	−0.140406	+	0.990094i	$0.544841\pi$
$608$	0	0
$609$	−637.420	−0.0424131
$610$	0	0
$611$	−1775.00	−0.117527
$612$	0	0
$613$	−5636.18	−0.371359	−0.185679	−	0.982610i	$-0.559449\pi$
$613$	−5636.18	−0.371359	−0.185679	+	0.982610i	$0.559449\pi$
$614$	0	0
$615$	−3455.70	−0.226581
$616$	0	0
$617$	15168.2	0.989709	0.494854	−	0.868976i	$-0.335221\pi$
$617$	15168.2	0.989709	0.494854	+	0.868976i	$0.335221\pi$
$618$	0	0
$619$	14471.0	0.939640	0.469820	−	0.882762i	$-0.344319\pi$
$619$	14471.0	0.939640	0.469820	+	0.882762i	$0.344319\pi$
$620$	0	0
$621$	621.000	0.0401286
$622$	0	0
$623$	−7762.16	−0.499173
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	4143.58	0.263921
$628$	0	0
$629$	−8859.47	−0.561606
$630$	0	0
$631$	13016.3	0.821188	0.410594	−	0.911818i	$-0.365321\pi$
$631$	13016.3	0.821188	0.410594	+	0.911818i	$0.365321\pi$
$632$	0	0
$633$	18311.8	1.14981
$634$	0	0
$635$	8515.81	0.532188
$636$	0	0
$637$	560.317	0.0348518
$638$	0	0
$639$	−868.203	−0.0537490
$640$	0	0
$641$	14050.7	0.865787	0.432894	−	0.901445i	$-0.357493\pi$
$641$	14050.7	0.865787	0.432894	+	0.901445i	$0.357493\pi$
$642$	0	0
$643$	1427.94	0.0875778	0.0437889	−	0.999041i	$-0.486057\pi$
$643$	1427.94	0.0875778	0.0437889	+	0.999041i	$0.486057\pi$
$644$	0	0
$645$	−1029.75	−0.0628626
$646$	0	0
$647$	−2333.26	−0.141777	−0.0708887	−	0.997484i	$-0.522584\pi$
$647$	−2333.26	−0.141777	−0.0708887	+	0.997484i	$0.522584\pi$
$648$	0	0
$649$	−24171.9	−1.46199
$650$	0	0
$651$	−17405.7	−1.04790
$652$	0	0
$653$	−3504.51	−0.210018	−0.105009	−	0.994471i	$-0.533487\pi$
$653$	−3504.51	−0.210018	−0.105009	+	0.994471i	$0.533487\pi$
$654$	0	0
$655$	5870.20	0.350180
$656$	0	0
$657$	4820.28	0.286236
$658$	0	0
$659$	−2351.39	−0.138994	−0.0694970	−	0.997582i	$-0.522139\pi$
$659$	−2351.39	−0.138994	−0.0694970	+	0.997582i	$0.522139\pi$
$660$	0	0
$661$	28239.1	1.66169	0.830844	−	0.556506i	$-0.187858\pi$
$661$	28239.1	1.66169	0.830844	+	0.556506i	$0.187858\pi$
$662$	0	0
$663$	−1503.86	−0.0880922
$664$	0	0
$665$	−2174.63	−0.126810
$666$	0	0
$667$	−249.392	−0.0144775
$668$	0	0
$669$	−11794.1	−0.681593
$670$	0	0
$671$	−54131.0	−3.11431
$672$	0	0
$673$	4567.91	0.261634	0.130817	−	0.991407i	$-0.458240\pi$
$673$	4567.91	0.261634	0.130817	+	0.991407i	$0.458240\pi$
$674$	0	0
$675$	−675.000	−0.0384900
$676$	0	0
$677$	−23007.6	−1.30614	−0.653069	−	0.757299i	$-0.726519\pi$
$677$	−23007.6	−1.30614	−0.653069	+	0.757299i	$0.726519\pi$
$678$	0	0
$679$	−2832.42	−0.160086
$680$	0	0
$681$	−13985.0	−0.786940
$682$	0	0
$683$	4776.51	0.267596	0.133798	−	0.991009i	$-0.457283\pi$
$683$	4776.51	0.267596	0.133798	+	0.991009i	$0.457283\pi$
$684$	0	0
$685$	7968.38	0.444462
$686$	0	0
$687$	11848.1	0.657979
$688$	0	0
$689$	−5048.81	−0.279164
$690$	0	0
$691$	11658.0	0.641811	0.320906	−	0.947111i	$-0.396013\pi$
$691$	11658.0	0.641811	0.320906	+	0.947111i	$0.396013\pi$
$692$	0	0
$693$	−10974.4	−0.601564
$694$	0	0
$695$	12249.6	0.668568
$696$	0	0
$697$	−8444.65	−0.458915
$698$	0	0
$699$	−10988.1	−0.594577
$700$	0	0
$701$	7814.00	0.421014	0.210507	−	0.977592i	$-0.432489\pi$
$701$	7814.00	0.421014	0.210507	+	0.977592i	$0.432489\pi$
$702$	0	0
$703$	−5364.58	−0.287808
$704$	0	0
$705$	−1946.89	−0.104006
$706$	0	0
$707$	−18274.8	−0.972126
$708$	0	0
$709$	−15366.6	−0.813969	−0.406984	−	0.913435i	$-0.633420\pi$
$709$	−15366.6	−0.813969	−0.406984	+	0.913435i	$0.633420\pi$
$710$	0	0
$711$	3364.97	0.177491
$712$	0	0
$713$	−6810.02	−0.357696
$714$	0	0
$715$	4255.09	0.222562
$716$	0	0
$717$	14872.7	0.774661
$718$	0	0
$719$	10330.1	0.535809	0.267905	−	0.963445i	$-0.413669\pi$
$719$	10330.1	0.535809	0.267905	+	0.963445i	$0.413669\pi$
$720$	0	0
$721$	614.833	0.0317581
$722$	0	0
$723$	−3445.61	−0.177239
$724$	0	0
$725$	271.078	0.0138863
$726$	0	0
$727$	25819.6	1.31719	0.658594	−	0.752498i	$-0.271151\pi$
$727$	25819.6	1.31719	0.658594	+	0.752498i	$0.271151\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−2516.39	−0.127321
$732$	0	0
$733$	7945.82	0.400390	0.200195	−	0.979756i	$-0.435843\pi$
$733$	7945.82	0.400390	0.200195	+	0.979756i	$0.435843\pi$
$734$	0	0
$735$	614.576	0.0308421
$736$	0	0
$737$	31248.1	1.56179
$738$	0	0
$739$	31221.0	1.55410	0.777052	−	0.629436i	$-0.216714\pi$
$739$	31221.0	1.55410	0.777052	+	0.629436i	$0.216714\pi$
$740$	0	0
$741$	−910.618	−0.0451449
$742$	0	0
$743$	−10223.4	−0.504791	−0.252395	−	0.967624i	$-0.581218\pi$
$743$	−10223.4	−0.504791	−0.252395	+	0.967624i	$0.581218\pi$
$744$	0	0
$745$	11264.6	0.553964
$746$	0	0
$747$	8191.34	0.401212
$748$	0	0
$749$	−11531.0	−0.562526
$750$	0	0
$751$	−6344.38	−0.308269	−0.154134	−	0.988050i	$-0.549259\pi$
$751$	−6344.38	−0.308269	−0.154134	+	0.988050i	$0.549259\pi$
$752$	0	0
$753$	−9754.42	−0.472073
$754$	0	0
$755$	7902.52	0.380930
$756$	0	0
$757$	7694.17	0.369418	0.184709	−	0.982793i	$-0.440866\pi$
$757$	7694.17	0.369418	0.184709	+	0.982793i	$0.440866\pi$
$758$	0	0
$759$	−4293.77	−0.205341
$760$	0	0
$761$	27693.7	1.31918	0.659590	−	0.751625i	$-0.270730\pi$
$761$	27693.7	1.31918	0.659590	+	0.751625i	$0.270730\pi$
$762$	0	0
$763$	9111.56	0.432321
$764$	0	0
$765$	−1649.49	−0.0779574
$766$	0	0
$767$	5312.16	0.250079
$768$	0	0
$769$	−16224.4	−0.760815	−0.380408	−	0.924819i	$-0.624216\pi$
$769$	−16224.4	−0.760815	−0.380408	+	0.924819i	$0.624216\pi$
$770$	0	0
$771$	−3693.25	−0.172515
$772$	0	0
$773$	17892.3	0.832525	0.416262	−	0.909245i	$-0.363340\pi$
$773$	17892.3	0.832525	0.416262	+	0.909245i	$0.363340\pi$
$774$	0	0
$775$	7402.19	0.343090
$776$	0	0
$777$	14208.3	0.656009
$778$	0	0
$779$	−5113.40	−0.235182
$780$	0	0
$781$	6003.00	0.275037
$782$	0	0
$783$	−292.765	−0.0133621
$784$	0	0
$785$	10472.8	0.476164
$786$	0	0
$787$	24697.3	1.11863	0.559316	−	0.828955i	$-0.311064\pi$
$787$	24697.3	1.11863	0.559316	+	0.828955i	$0.311064\pi$
$788$	0	0
$789$	17960.9	0.810426
$790$	0	0
$791$	16425.7	0.738344
$792$	0	0
$793$	11896.1	0.532717
$794$	0	0
$795$	−5537.71	−0.247047
$796$	0	0
$797$	−23794.9	−1.05754	−0.528770	−	0.848765i	$-0.677347\pi$
$797$	−23794.9	−1.05754	−0.528770	+	0.848765i	$0.677347\pi$
$798$	0	0
$799$	−4757.58	−0.210652
$800$	0	0
$801$	−3565.13	−0.157263
$802$	0	0
$803$	−33328.8	−1.46469
$804$	0	0
$805$	2253.45	0.0986628
$806$	0	0
$807$	3298.82	0.143896
$808$	0	0
$809$	−41207.6	−1.79083	−0.895415	−	0.445232i	$-0.853121\pi$
$809$	−41207.6	−1.79083	−0.895415	+	0.445232i	$0.853121\pi$
$810$	0	0
$811$	−7791.79	−0.337369	−0.168685	−	0.985670i	$-0.553952\pi$
$811$	−7791.79	−0.337369	−0.168685	+	0.985670i	$0.553952\pi$
$812$	0	0
$813$	−17887.2	−0.771624
$814$	0	0
$815$	8392.55	0.360710
$816$	0	0
$817$	−1523.72	−0.0652488
$818$	0	0
$819$	2411.80	0.102900
$820$	0	0
$821$	2765.03	0.117540	0.0587699	−	0.998272i	$-0.481282\pi$
$821$	2765.03	0.117540	0.0587699	+	0.998272i	$0.481282\pi$
$822$	0	0
$823$	−7820.54	−0.331236	−0.165618	−	0.986190i	$-0.552962\pi$
$823$	−7820.54	−0.331236	−0.165618	+	0.986190i	$0.552962\pi$
$824$	0	0
$825$	4667.14	0.196956
$826$	0	0
$827$	−45509.2	−1.91356	−0.956778	−	0.290820i	$-0.906072\pi$
$827$	−45509.2	−1.91356	−0.956778	+	0.290820i	$0.906072\pi$
$828$	0	0
$829$	33687.0	1.41134	0.705669	−	0.708542i	$-0.250647\pi$
$829$	33687.0	1.41134	0.705669	+	0.708542i	$0.250647\pi$
$830$	0	0
$831$	11860.8	0.495120
$832$	0	0
$833$	1501.83	0.0624674
$834$	0	0
$835$	−6757.52	−0.280064
$836$	0	0
$837$	−7994.37	−0.330138
$838$	0	0
$839$	−30399.2	−1.25089	−0.625446	−	0.780267i	$-0.715083\pi$
$839$	−30399.2	−1.25089	−0.625446	+	0.780267i	$0.715083\pi$
$840$	0	0
$841$	−24271.4	−0.995179
$842$	0	0
$843$	5443.50	0.222401
$844$	0	0
$845$	10049.9	0.409143
$846$	0	0
$847$	49799.0	2.02020
$848$	0	0
$849$	−7670.73	−0.310081
$850$	0	0
$851$	5559.02	0.223926
$852$	0	0
$853$	−562.398	−0.0225746	−0.0112873	−	0.999936i	$-0.503593\pi$
$853$	−562.398	−0.0225746	−0.0112873	+	0.999936i	$0.503593\pi$
$854$	0	0
$855$	−998.798	−0.0399511
$856$	0	0
$857$	29165.6	1.16252	0.581259	−	0.813719i	$-0.302560\pi$
$857$	29165.6	1.16252	0.581259	+	0.813719i	$0.302560\pi$
$858$	0	0
$859$	−23463.6	−0.931977	−0.465988	−	0.884791i	$-0.654301\pi$
$859$	−23463.6	−0.931977	−0.465988	+	0.884791i	$0.654301\pi$
$860$	0	0
$861$	13543.0	0.536057
$862$	0	0
$863$	−42240.5	−1.66615	−0.833073	−	0.553163i	$-0.813421\pi$
$863$	−42240.5	−1.66615	−0.833073	+	0.553163i	$0.813421\pi$
$864$	0	0
$865$	18697.4	0.734949
$866$	0	0
$867$	10708.2	0.419456
$868$	0	0
$869$	−23266.3	−0.908235
$870$	0	0
$871$	−6867.27	−0.267151
$872$	0	0
$873$	−1300.92	−0.0504346
$874$	0	0
$875$	−2449.40	−0.0946341
$876$	0	0
$877$	25282.0	0.973447	0.486724	−	0.873556i	$-0.338192\pi$
$877$	25282.0	0.973447	0.486724	+	0.873556i	$0.338192\pi$
$878$	0	0
$879$	−7516.74	−0.288434
$880$	0	0
$881$	1102.58	0.0421644	0.0210822	−	0.999778i	$-0.493289\pi$
$881$	1102.58	0.0421644	0.0210822	+	0.999778i	$0.493289\pi$
$882$	0	0
$883$	2903.17	0.110645	0.0553225	−	0.998469i	$-0.482381\pi$
$883$	2903.17	0.110645	0.0553225	+	0.998469i	$0.482381\pi$
$884$	0	0
$885$	5826.56	0.221308
$886$	0	0
$887$	33197.8	1.25668	0.628338	−	0.777940i	$-0.283736\pi$
$887$	33197.8	1.25668	0.628338	+	0.777940i	$0.283736\pi$
$888$	0	0
$889$	−33373.8	−1.25908
$890$	0	0
$891$	−5040.51	−0.189521
$892$	0	0
$893$	−2880.81	−0.107954
$894$	0	0
$895$	10129.3	0.378308
$896$	0	0
$897$	943.624	0.0351245
$898$	0	0
$899$	3210.52	0.119107
$900$	0	0
$901$	−13532.4	−0.500367
$902$	0	0
$903$	4035.63	0.148724
$904$	0	0
$905$	10656.3	0.391413
$906$	0	0
$907$	−186.826	−0.00683952	−0.00341976	−	0.999994i	$-0.501089\pi$
$907$	−186.826	−0.00683952	−0.00341976	+	0.999994i	$0.501089\pi$
$908$	0	0
$909$	−8393.53	−0.306266
$910$	0	0
$911$	5562.74	0.202307	0.101154	−	0.994871i	$-0.467747\pi$
$911$	5562.74	0.202307	0.101154	+	0.994871i	$0.467747\pi$
$912$	0	0
$913$	−56637.2	−2.05303
$914$	0	0
$915$	13048.1	0.471429
$916$	0	0
$917$	−23005.5	−0.828473
$918$	0	0
$919$	−34715.3	−1.24608	−0.623042	−	0.782189i	$-0.714103\pi$
$919$	−34715.3	−1.24608	−0.623042	+	0.782189i	$0.714103\pi$
$920$	0	0
$921$	−26982.8	−0.965379
$922$	0	0
$923$	−1319.25	−0.0470464
$924$	0	0
$925$	−6042.42	−0.214782
$926$	0	0
$927$	282.391	0.0100053
$928$	0	0
$929$	−6910.15	−0.244041	−0.122021	−	0.992528i	$-0.538937\pi$
$929$	−6910.15	−0.244041	−0.122021	+	0.992528i	$0.538937\pi$
$930$	0	0
$931$	909.388	0.0320129
$932$	0	0
$933$	−15778.2	−0.553648
$934$	0	0
$935$	11405.0	0.398914
$936$	0	0
$937$	−25242.6	−0.880085	−0.440042	−	0.897977i	$-0.645037\pi$
$937$	−25242.6	−0.880085	−0.440042	+	0.897977i	$0.645037\pi$
$938$	0	0
$939$	−22767.7	−0.791264
$940$	0	0
$941$	39131.6	1.35564	0.677819	−	0.735229i	$-0.262925\pi$
$941$	39131.6	1.35564	0.677819	+	0.735229i	$0.262925\pi$
$942$	0	0
$943$	5298.74	0.182981
$944$	0	0
$945$	2645.35	0.0910617
$946$	0	0
$947$	39311.3	1.34894	0.674469	−	0.738303i	$-0.264373\pi$
$947$	39311.3	1.34894	0.674469	+	0.738303i	$0.264373\pi$
$948$	0	0
$949$	7324.53	0.250542
$950$	0	0
$951$	16977.9	0.578914
$952$	0	0
$953$	−28938.1	−0.983628	−0.491814	−	0.870700i	$-0.663666\pi$
$953$	−28938.1	−0.983628	−0.491814	+	0.870700i	$0.663666\pi$
$954$	0	0
$955$	17663.7	0.598518
$956$	0	0
$957$	2024.26	0.0683751
$958$	0	0
$959$	−31228.4	−1.05153
$960$	0	0
$961$	57876.9	1.94276
$962$	0	0
$963$	−5296.12	−0.177222
$964$	0	0
$965$	23291.5	0.776975
$966$	0	0
$967$	7883.28	0.262160	0.131080	−	0.991372i	$-0.458155\pi$
$967$	7883.28	0.262160	0.131080	+	0.991372i	$0.458155\pi$
$968$	0	0
$969$	−2440.75	−0.0809166
$970$	0	0
$971$	19517.5	0.645051	0.322526	−	0.946561i	$-0.395468\pi$
$971$	19517.5	0.645051	0.322526	+	0.946561i	$0.395468\pi$
$972$	0	0
$973$	−48006.7	−1.58173
$974$	0	0
$975$	−1025.68	−0.0336902
$976$	0	0
$977$	38354.8	1.25597	0.627983	−	0.778227i	$-0.283881\pi$
$977$	38354.8	1.25597	0.627983	+	0.778227i	$0.283881\pi$
$978$	0	0
$979$	24650.3	0.804727
$980$	0	0
$981$	4184.91	0.136202
$982$	0	0
$983$	18413.1	0.597443	0.298722	−	0.954340i	$-0.403440\pi$
$983$	18413.1	0.597443	0.298722	+	0.954340i	$0.403440\pi$
$984$	0	0
$985$	−10146.9	−0.328230
$986$	0	0
$987$	7629.92	0.246062
$988$	0	0
$989$	1578.95	0.0507661
$990$	0	0
$991$	−33480.8	−1.07321	−0.536605	−	0.843833i	$-0.680294\pi$
$991$	−33480.8	−1.07321	−0.536605	+	0.843833i	$0.680294\pi$
$992$	0	0
$993$	−15957.7	−0.509973
$994$	0	0
$995$	−680.971	−0.0216967
$996$	0	0
$997$	33940.7	1.07815	0.539074	−	0.842258i	$-0.318774\pi$
$997$	33940.7	1.07815	0.539074	+	0.842258i	$0.318774\pi$
$998$	0	0
$999$	6525.81	0.206674

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1380.4.a.d.1.3	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.4.a.d.1.3	✓	4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1380.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -5.00000 q^{5} +19.5952 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} -5.00000 q^{5} +19.5952 q^{7} +9.00000 q^{9} -62.2285 q^{11} +13.6757 q^{13} +15.0000 q^{15} +36.6553 q^{17} +22.1955 q^{19} -58.7856 q^{21} -23.0000 q^{23} +25.0000 q^{25} -27.0000 q^{27} +10.8431 q^{29} +296.088 q^{31} +186.686 q^{33} -97.9760 q^{35} -241.697 q^{37} -41.0271 q^{39} -230.380 q^{41} -68.6500 q^{43} -45.0000 q^{45} -129.792 q^{47} +40.9717 q^{49} -109.966 q^{51} -369.181 q^{53} +311.143 q^{55} -66.5865 q^{57} +388.437 q^{59} +869.875 q^{61} +176.357 q^{63} -68.3785 q^{65} -502.151 q^{67} +69.0000 q^{69} -96.4670 q^{71} +535.587 q^{73} -75.0000 q^{75} -1219.38 q^{77} +373.886 q^{79} +81.0000 q^{81} +910.148 q^{83} -183.277 q^{85} -32.5294 q^{87} -396.126 q^{89} +267.978 q^{91} -888.263 q^{93} -110.978 q^{95} -144.547 q^{97} -560.057 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{3} - 20 q^{5} + 35 q^{7} + 36 q^{9}+O(q^{10}) \) 4 * q - 12 * q^3 - 20 * q^5 + 35 * q^7 + 36 * q^9	\( 4 q - 12 q^{3} - 20 q^{5} + 35 q^{7} + 36 q^{9} - 56 q^{11} + 42 q^{13} + 60 q^{15} - 139 q^{17} + 100 q^{19} - 105 q^{21} - 92 q^{23} + 100 q^{25} - 108 q^{27} + 125 q^{29} - 3 q^{31} + 168 q^{33} - 175 q^{35} + 243 q^{37} - 126 q^{39} - 349 q^{41} + 652 q^{43} - 180 q^{45} - 178 q^{47} + 367 q^{49} + 417 q^{51} + 157 q^{53} + 280 q^{55} - 300 q^{57} + 87 q^{59} + 428 q^{61} + 315 q^{63} - 210 q^{65} + 119 q^{67} + 276 q^{69} - 659 q^{71} + 1150 q^{73} - 300 q^{75} - 424 q^{77} + 384 q^{79} + 324 q^{81} - 127 q^{83} + 695 q^{85} - 375 q^{87} - 1278 q^{89} - 1024 q^{91} + 9 q^{93} - 500 q^{95} - 260 q^{97} - 504 q^{99}+O(q^{100}) \) 4 * q - 12 * q^3 - 20 * q^5 + 35 * q^7 + 36 * q^9 - 56 * q^11 + 42 * q^13 + 60 * q^15 - 139 * q^17 + 100 * q^19 - 105 * q^21 - 92 * q^23 + 100 * q^25 - 108 * q^27 + 125 * q^29 - 3 * q^31 + 168 * q^33 - 175 * q^35 + 243 * q^37 - 126 * q^39 - 349 * q^41 + 652 * q^43 - 180 * q^45 - 178 * q^47 + 367 * q^49 + 417 * q^51 + 157 * q^53 + 280 * q^55 - 300 * q^57 + 87 * q^59 + 428 * q^61 + 315 * q^63 - 210 * q^65 + 119 * q^67 + 276 * q^69 - 659 * q^71 + 1150 * q^73 - 300 * q^75 - 424 * q^77 + 384 * q^79 + 324 * q^81 - 127 * q^83 + 695 * q^85 - 375 * q^87 - 1278 * q^89 - 1024 * q^91 + 9 * q^93 - 500 * q^95 - 260 * q^97 - 504 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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