LMFDB - Embedded newform 1344.4.a.br.1.2

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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 672)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.56155$ 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)

$f(q)$	$=$	$q+3.00000 q^{3} +9.12311 q^{5} -7.00000 q^{7} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} +9.12311 q^{5} -7.00000 q^{7} +9.00000 q^{9} +34.3542 q^{11} +20.2462 q^{13} +27.3693 q^{15} -120.354 q^{17} -127.939 q^{19} -21.0000 q^{21} -95.0625 q^{23} -41.7689 q^{25} +27.0000 q^{27} -113.939 q^{29} -126.462 q^{31} +103.062 q^{33} -63.8617 q^{35} -201.723 q^{37} +60.7386 q^{39} -444.509 q^{41} +141.356 q^{43} +82.1080 q^{45} +591.542 q^{47} +49.0000 q^{49} -361.062 q^{51} -5.78410 q^{53} +313.417 q^{55} -383.818 q^{57} -348.337 q^{59} +532.220 q^{61} -63.0000 q^{63} +184.708 q^{65} -661.261 q^{67} -285.187 q^{69} +324.229 q^{71} +613.913 q^{73} -125.307 q^{75} -240.479 q^{77} +643.049 q^{79} +81.0000 q^{81} -908.189 q^{83} -1098.00 q^{85} -341.818 q^{87} -662.722 q^{89} -141.723 q^{91} -379.386 q^{93} -1167.20 q^{95} +221.845 q^{97} +309.187 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{3} + 10 q^{5} - 14 q^{7} + 18 q^{9}+O(q^{10}) $ 2 * q + 6 * q^3 + 10 * q^5 - 14 * q^7 + 18 * q^9	$ 2 q + 6 q^{3} + 10 q^{5} - 14 q^{7} + 18 q^{9} - 22 q^{11} + 24 q^{13} + 30 q^{15} - 150 q^{17} + 8 q^{19} - 42 q^{21} + 82 q^{23} - 166 q^{25} + 54 q^{27} + 36 q^{29} - 88 q^{31} - 66 q^{33} - 70 q^{35} - 288 q^{37} + 72 q^{39} - 386 q^{41} - 344 q^{43} + 90 q^{45} + 276 q^{47} + 98 q^{49} - 450 q^{51} - 160 q^{53} + 264 q^{55} + 24 q^{57} - 1076 q^{59} - 156 q^{61} - 126 q^{63} + 188 q^{65} - 1372 q^{67} + 246 q^{69} + 1102 q^{71} - 240 q^{73} - 498 q^{75} + 154 q^{77} + 412 q^{79} + 162 q^{81} - 464 q^{83} - 1124 q^{85} + 108 q^{87} - 1746 q^{89} - 168 q^{91} - 264 q^{93} - 1048 q^{95} + 856 q^{97} - 198 q^{99}+O(q^{100}) $ 2 * q + 6 * q^3 + 10 * q^5 - 14 * q^7 + 18 * q^9 - 22 * q^11 + 24 * q^13 + 30 * q^15 - 150 * q^17 + 8 * q^19 - 42 * q^21 + 82 * q^23 - 166 * q^25 + 54 * q^27 + 36 * q^29 - 88 * q^31 - 66 * q^33 - 70 * q^35 - 288 * q^37 + 72 * q^39 - 386 * q^41 - 344 * q^43 + 90 * q^45 + 276 * q^47 + 98 * q^49 - 450 * q^51 - 160 * q^53 + 264 * q^55 + 24 * q^57 - 1076 * q^59 - 156 * q^61 - 126 * q^63 + 188 * q^65 - 1372 * q^67 + 246 * q^69 + 1102 * q^71 - 240 * q^73 - 498 * q^75 + 154 * q^77 + 412 * q^79 + 162 * q^81 - 464 * q^83 - 1124 * q^85 + 108 * q^87 - 1746 * q^89 - 168 * q^91 - 264 * q^93 - 1048 * q^95 + 856 * q^97 - 198 * 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
$5$	9.12311	0.815995	0.407998	−	0.912983i	$-0.366227\pi$
$5$	9.12311	0.815995	0.407998	+	0.912983i	$0.366227\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	34.3542	0.941652	0.470826	−	0.882226i	$-0.343956\pi$
$11$	34.3542	0.941652	0.470826	+	0.882226i	$0.343956\pi$
$12$	0	0
$13$	20.2462	0.431945	0.215973	−	0.976399i	$-0.430708\pi$
$13$	20.2462	0.431945	0.215973	+	0.976399i	$0.430708\pi$
$14$	0	0
$15$	27.3693	0.471115
$16$	0	0
$17$	−120.354	−1.71707	−0.858534	−	0.512756i	$-0.828625\pi$
$17$	−120.354	−1.71707	−0.858534	+	0.512756i	$0.828625\pi$
$18$	0	0
$19$	−127.939	−1.54481	−0.772403	−	0.635133i	$-0.780945\pi$
$19$	−127.939	−1.54481	−0.772403	+	0.635133i	$0.780945\pi$
$20$	0	0
$21$	−21.0000	−0.218218
$22$	0	0
$23$	−95.0625	−0.861822	−0.430911	−	0.902395i	$-0.641808\pi$
$23$	−95.0625	−0.861822	−0.430911	+	0.902395i	$0.641808\pi$
$24$	0	0
$25$	−41.7689	−0.334152
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−113.939	−0.729587	−0.364793	−	0.931089i	$-0.618860\pi$
$29$	−113.939	−0.729587	−0.364793	+	0.931089i	$0.618860\pi$
$30$	0	0
$31$	−126.462	−0.732686	−0.366343	−	0.930480i	$-0.619390\pi$
$31$	−126.462	−0.732686	−0.366343	+	0.930480i	$0.619390\pi$
$32$	0	0
$33$	103.062	0.543663
$34$	0	0
$35$	−63.8617	−0.308417
$36$	0	0
$37$	−201.723	−0.896301	−0.448150	−	0.893958i	$-0.647917\pi$
$37$	−201.723	−0.896301	−0.448150	+	0.893958i	$0.647917\pi$
$38$	0	0
$39$	60.7386	0.249384
$40$	0	0
$41$	−444.509	−1.69319	−0.846594	−	0.532239i	$-0.821351\pi$
$41$	−444.509	−1.69319	−0.846594	+	0.532239i	$0.821351\pi$
$42$	0	0
$43$	141.356	0.501316	0.250658	−	0.968076i	$-0.419353\pi$
$43$	141.356	0.501316	0.250658	+	0.968076i	$0.419353\pi$
$44$	0	0
$45$	82.1080	0.271998
$46$	0	0
$47$	591.542	1.83586	0.917928	−	0.396747i	$-0.129861\pi$
$47$	591.542	1.83586	0.917928	+	0.396747i	$0.129861\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−361.062	−0.991350
$52$	0	0
$53$	−5.78410	−0.0149907	−0.00749535	−	0.999972i	$-0.502386\pi$
$53$	−5.78410	−0.0149907	−0.00749535	+	0.999972i	$0.502386\pi$
$54$	0	0
$55$	313.417	0.768383
$56$	0	0
$57$	−383.818	−0.891894
$58$	0	0
$59$	−348.337	−0.768638	−0.384319	−	0.923200i	$-0.625564\pi$
$59$	−348.337	−0.768638	−0.384319	+	0.923200i	$0.625564\pi$
$60$	0	0
$61$	532.220	1.11711	0.558555	−	0.829467i	$-0.311356\pi$
$61$	532.220	1.11711	0.558555	+	0.829467i	$0.311356\pi$
$62$	0	0
$63$	−63.0000	−0.125988
$64$	0	0
$65$	184.708	0.352465
$66$	0	0
$67$	−661.261	−1.20576	−0.602880	−	0.797832i	$-0.705980\pi$
$67$	−661.261	−1.20576	−0.602880	+	0.797832i	$0.705980\pi$
$68$	0	0
$69$	−285.187	−0.497573
$70$	0	0
$71$	324.229	0.541957	0.270978	−	0.962585i	$-0.412653\pi$
$71$	324.229	0.541957	0.270978	+	0.962585i	$0.412653\pi$
$72$	0	0
$73$	613.913	0.984288	0.492144	−	0.870514i	$-0.336213\pi$
$73$	613.913	0.984288	0.492144	+	0.870514i	$0.336213\pi$
$74$	0	0
$75$	−125.307	−0.192922
$76$	0	0
$77$	−240.479	−0.355911
$78$	0	0
$79$	643.049	0.915806	0.457903	−	0.889002i	$-0.348601\pi$
$79$	643.049	0.915806	0.457903	+	0.889002i	$0.348601\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−908.189	−1.20104	−0.600522	−	0.799608i	$-0.705041\pi$
$83$	−908.189	−1.20104	−0.600522	+	0.799608i	$0.705041\pi$
$84$	0	0
$85$	−1098.00	−1.40112
$86$	0	0
$87$	−341.818	−0.421227
$88$	0	0
$89$	−662.722	−0.789307	−0.394654	−	0.918830i	$-0.629135\pi$
$89$	−662.722	−0.789307	−0.394654	+	0.918830i	$0.629135\pi$
$90$	0	0
$91$	−141.723	−0.163260
$92$	0	0
$93$	−379.386	−0.423016
$94$	0	0
$95$	−1167.20	−1.26055
$96$	0	0
$97$	221.845	0.232216	0.116108	−	0.993237i	$-0.462958\pi$
$97$	221.845	0.232216	0.116108	+	0.993237i	$0.462958\pi$
$98$	0	0
$99$	309.187	0.313884
$100$	0	0
$101$	1230.64	1.21241	0.606205	−	0.795308i	$-0.292691\pi$
$101$	1230.64	1.21241	0.606205	+	0.795308i	$0.292691\pi$
$102$	0	0
$103$	289.349	0.276800	0.138400	−	0.990376i	$-0.455804\pi$
$103$	289.349	0.276800	0.138400	+	0.990376i	$0.455804\pi$
$104$	0	0
$105$	−191.585	−0.178065
$106$	0	0
$107$	−1613.28	−1.45759	−0.728794	−	0.684733i	$-0.759919\pi$
$107$	−1613.28	−1.45759	−0.728794	+	0.684733i	$0.759919\pi$
$108$	0	0
$109$	−1944.83	−1.70900	−0.854501	−	0.519450i	$-0.826137\pi$
$109$	−1944.83	−1.70900	−0.854501	+	0.519450i	$0.826137\pi$
$110$	0	0
$111$	−605.170	−0.517480
$112$	0	0
$113$	−1531.79	−1.27521	−0.637604	−	0.770364i	$-0.720074\pi$
$113$	−1531.79	−1.27521	−0.637604	+	0.770364i	$0.720074\pi$
$114$	0	0
$115$	−867.265	−0.703242
$116$	0	0
$117$	182.216	0.143982
$118$	0	0
$119$	842.479	0.648991
$120$	0	0
$121$	−150.792	−0.113292
$122$	0	0
$123$	−1333.53	−0.977563
$124$	0	0
$125$	−1521.45	−1.08866
$126$	0	0
$127$	1439.23	1.00560	0.502800	−	0.864403i	$-0.332303\pi$
$127$	1439.23	1.00560	0.502800	+	0.864403i	$0.332303\pi$
$128$	0	0
$129$	424.068	0.289435
$130$	0	0
$131$	1582.23	1.05527	0.527633	−	0.849472i	$-0.323080\pi$
$131$	1582.23	1.05527	0.527633	+	0.849472i	$0.323080\pi$
$132$	0	0
$133$	895.576	0.583882
$134$	0	0
$135$	246.324	0.157038
$136$	0	0
$137$	1411.70	0.880363	0.440181	−	0.897909i	$-0.354914\pi$
$137$	1411.70	0.880363	0.440181	+	0.897909i	$0.354914\pi$
$138$	0	0
$139$	2596.26	1.58426	0.792128	−	0.610355i	$-0.208973\pi$
$139$	2596.26	1.58426	0.792128	+	0.610355i	$0.208973\pi$
$140$	0	0
$141$	1774.62	1.05993
$142$	0	0
$143$	695.542	0.406742
$144$	0	0
$145$	−1039.48	−0.595339
$146$	0	0
$147$	147.000	0.0824786
$148$	0	0
$149$	−1191.27	−0.654984	−0.327492	−	0.944854i	$-0.606203\pi$
$149$	−1191.27	−0.654984	−0.327492	+	0.944854i	$0.606203\pi$
$150$	0	0
$151$	1661.30	0.895331	0.447665	−	0.894201i	$-0.352256\pi$
$151$	1661.30	0.895331	0.447665	+	0.894201i	$0.352256\pi$
$152$	0	0
$153$	−1083.19	−0.572356
$154$	0	0
$155$	−1153.73	−0.597868
$156$	0	0
$157$	−3743.06	−1.90273	−0.951365	−	0.308065i	$-0.900319\pi$
$157$	−3743.06	−1.90273	−0.951365	+	0.308065i	$0.900319\pi$
$158$	0	0
$159$	−17.3523	−0.00865488
$160$	0	0
$161$	665.437	0.325738
$162$	0	0
$163$	647.488	0.311136	0.155568	−	0.987825i	$-0.450279\pi$
$163$	647.488	0.311136	0.155568	+	0.987825i	$0.450279\pi$
$164$	0	0
$165$	940.250	0.443626
$166$	0	0
$167$	1294.49	0.599824	0.299912	−	0.953967i	$-0.403043\pi$
$167$	1294.49	0.599824	0.299912	+	0.953967i	$0.403043\pi$
$168$	0	0
$169$	−1787.09	−0.813423
$170$	0	0
$171$	−1151.45	−0.514935
$172$	0	0
$173$	−210.654	−0.0925763	−0.0462881	−	0.998928i	$-0.514739\pi$
$173$	−210.654	−0.0925763	−0.0462881	+	0.998928i	$0.514739\pi$
$174$	0	0
$175$	292.383	0.126297
$176$	0	0
$177$	−1045.01	−0.443773
$178$	0	0
$179$	2830.39	1.18186	0.590931	−	0.806722i	$-0.298761\pi$
$179$	2830.39	1.18186	0.590931	+	0.806722i	$0.298761\pi$
$180$	0	0
$181$	2175.89	0.893550	0.446775	−	0.894646i	$-0.352572\pi$
$181$	2175.89	0.893550	0.446775	+	0.894646i	$0.352572\pi$
$182$	0	0
$183$	1596.66	0.644964
$184$	0	0
$185$	−1840.34	−0.731377
$186$	0	0
$187$	−4134.67	−1.61688
$188$	0	0
$189$	−189.000	−0.0727393
$190$	0	0
$191$	−82.4903	−0.0312502	−0.0156251	−	0.999878i	$-0.504974\pi$
$191$	−82.4903	−0.0312502	−0.0156251	+	0.999878i	$0.504974\pi$
$192$	0	0
$193$	1631.94	0.608649	0.304325	−	0.952568i	$-0.401569\pi$
$193$	1631.94	0.608649	0.304325	+	0.952568i	$0.401569\pi$
$194$	0	0
$195$	554.125	0.203496
$196$	0	0
$197$	152.299	0.0550806	0.0275403	−	0.999621i	$-0.491233\pi$
$197$	152.299	0.0550806	0.0275403	+	0.999621i	$0.491233\pi$
$198$	0	0
$199$	−783.568	−0.279124	−0.139562	−	0.990213i	$-0.544569\pi$
$199$	−783.568	−0.279124	−0.139562	+	0.990213i	$0.544569\pi$
$200$	0	0
$201$	−1983.78	−0.696146
$202$	0	0
$203$	797.576	0.275758
$204$	0	0
$205$	−4055.31	−1.38163
$206$	0	0
$207$	−855.562	−0.287274
$208$	0	0
$209$	−4395.25	−1.45467
$210$	0	0
$211$	−2566.55	−0.837385	−0.418693	−	0.908128i	$-0.637511\pi$
$211$	−2566.55	−0.837385	−0.418693	+	0.908128i	$0.637511\pi$
$212$	0	0
$213$	972.688	0.312899
$214$	0	0
$215$	1289.61	0.409072
$216$	0	0
$217$	885.235	0.276929
$218$	0	0
$219$	1841.74	0.568279
$220$	0	0
$221$	−2436.72	−0.741680
$222$	0	0
$223$	−2658.14	−0.798217	−0.399109	−	0.916904i	$-0.630680\pi$
$223$	−2658.14	−0.798217	−0.399109	+	0.916904i	$0.630680\pi$
$224$	0	0
$225$	−375.920	−0.111384
$226$	0	0
$227$	1979.59	0.578812	0.289406	−	0.957206i	$-0.406542\pi$
$227$	1979.59	0.578812	0.289406	+	0.957206i	$0.406542\pi$
$228$	0	0
$229$	−5526.78	−1.59485	−0.797423	−	0.603421i	$-0.793804\pi$
$229$	−5526.78	−1.59485	−0.797423	+	0.603421i	$0.793804\pi$
$230$	0	0
$231$	−721.437	−0.205485
$232$	0	0
$233$	−4717.14	−1.32631	−0.663155	−	0.748482i	$-0.730783\pi$
$233$	−4717.14	−1.32631	−0.663155	+	0.748482i	$0.730783\pi$
$234$	0	0
$235$	5396.70	1.49805
$236$	0	0
$237$	1929.15	0.528741
$238$	0	0
$239$	−446.684	−0.120894	−0.0604468	−	0.998171i	$-0.519253\pi$
$239$	−446.684	−0.120894	−0.0604468	+	0.998171i	$0.519253\pi$
$240$	0	0
$241$	−3950.11	−1.05580	−0.527902	−	0.849305i	$-0.677021\pi$
$241$	−3950.11	−1.05580	−0.527902	+	0.849305i	$0.677021\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	447.032	0.116571
$246$	0	0
$247$	−2590.29	−0.667271
$248$	0	0
$249$	−2724.57	−0.693424
$250$	0	0
$251$	5008.76	1.25956	0.629781	−	0.776772i	$-0.283144\pi$
$251$	5008.76	1.25956	0.629781	+	0.776772i	$0.283144\pi$
$252$	0	0
$253$	−3265.79	−0.811536
$254$	0	0
$255$	−3294.01	−0.808937
$256$	0	0
$257$	−4401.45	−1.06831	−0.534153	−	0.845388i	$-0.679370\pi$
$257$	−4401.45	−1.06831	−0.534153	+	0.845388i	$0.679370\pi$
$258$	0	0
$259$	1412.06	0.338770
$260$	0	0
$261$	−1025.45	−0.243196
$262$	0	0
$263$	3598.74	0.843756	0.421878	−	0.906653i	$-0.361371\pi$
$263$	3598.74	0.843756	0.421878	+	0.906653i	$0.361371\pi$
$264$	0	0
$265$	−52.7689	−0.0122323
$266$	0	0
$267$	−1988.16	−0.455707
$268$	0	0
$269$	1988.94	0.450810	0.225405	−	0.974265i	$-0.427629\pi$
$269$	1988.94	0.450810	0.225405	+	0.974265i	$0.427629\pi$
$270$	0	0
$271$	−6837.82	−1.53272	−0.766361	−	0.642410i	$-0.777935\pi$
$271$	−6837.82	−1.53272	−0.766361	+	0.642410i	$0.777935\pi$
$272$	0	0
$273$	−425.170	−0.0942582
$274$	0	0
$275$	−1434.94	−0.314654
$276$	0	0
$277$	−4169.28	−0.904359	−0.452180	−	0.891927i	$-0.649353\pi$
$277$	−4169.28	−0.904359	−0.452180	+	0.891927i	$0.649353\pi$
$278$	0	0
$279$	−1138.16	−0.244229
$280$	0	0
$281$	4279.67	0.908554	0.454277	−	0.890860i	$-0.349898\pi$
$281$	4279.67	0.908554	0.454277	+	0.890860i	$0.349898\pi$
$282$	0	0
$283$	199.212	0.0418442	0.0209221	−	0.999781i	$-0.493340\pi$
$283$	199.212	0.0418442	0.0209221	+	0.999781i	$0.493340\pi$
$284$	0	0
$285$	−3501.61	−0.727781
$286$	0	0
$287$	3111.57	0.639965
$288$	0	0
$289$	9572.12	1.94833
$290$	0	0
$291$	665.534	0.134070
$292$	0	0
$293$	1063.11	0.211972	0.105986	−	0.994368i	$-0.466200\pi$
$293$	1063.11	0.211972	0.105986	+	0.994368i	$0.466200\pi$
$294$	0	0
$295$	−3177.92	−0.627205
$296$	0	0
$297$	927.562	0.181221
$298$	0	0
$299$	−1924.66	−0.372260
$300$	0	0
$301$	−989.492	−0.189480
$302$	0	0
$303$	3691.93	0.699985
$304$	0	0
$305$	4855.50	0.911557
$306$	0	0
$307$	4210.60	0.782774	0.391387	−	0.920226i	$-0.371995\pi$
$307$	4210.60	0.782774	0.391387	+	0.920226i	$0.371995\pi$
$308$	0	0
$309$	868.046	0.159810
$310$	0	0
$311$	−7253.90	−1.32261	−0.661304	−	0.750118i	$-0.729996\pi$
$311$	−7253.90	−1.32261	−0.661304	+	0.750118i	$0.729996\pi$
$312$	0	0
$313$	9330.03	1.68487	0.842435	−	0.538798i	$-0.181121\pi$
$313$	9330.03	1.68487	0.842435	+	0.538798i	$0.181121\pi$
$314$	0	0
$315$	−574.756	−0.102806
$316$	0	0
$317$	3939.11	0.697926	0.348963	−	0.937137i	$-0.386534\pi$
$317$	3939.11	0.697926	0.348963	+	0.937137i	$0.386534\pi$
$318$	0	0
$319$	−3914.29	−0.687017
$320$	0	0
$321$	−4839.85	−0.841539
$322$	0	0
$323$	15398.0	2.65254
$324$	0	0
$325$	−845.663	−0.144335
$326$	0	0
$327$	−5834.50	−0.986693
$328$	0	0
$329$	−4140.79	−0.693888
$330$	0	0
$331$	−3930.26	−0.652648	−0.326324	−	0.945258i	$-0.605810\pi$
$331$	−3930.26	−0.652648	−0.326324	+	0.945258i	$0.605810\pi$
$332$	0	0
$333$	−1815.51	−0.298767
$334$	0	0
$335$	−6032.76	−0.983895
$336$	0	0
$337$	−8233.31	−1.33085	−0.665426	−	0.746464i	$-0.731750\pi$
$337$	−8233.31	−1.33085	−0.665426	+	0.746464i	$0.731750\pi$
$338$	0	0
$339$	−4595.36	−0.736242
$340$	0	0
$341$	−4344.50	−0.689935
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	−2601.80	−0.406017
$346$	0	0
$347$	1075.55	0.166394	0.0831971	−	0.996533i	$-0.473487\pi$
$347$	1075.55	0.166394	0.0831971	+	0.996533i	$0.473487\pi$
$348$	0	0
$349$	−6611.19	−1.01401	−0.507004	−	0.861944i	$-0.669247\pi$
$349$	−6611.19	−1.01401	−0.507004	+	0.861944i	$0.669247\pi$
$350$	0	0
$351$	546.648	0.0831279
$352$	0	0
$353$	−6108.07	−0.920962	−0.460481	−	0.887670i	$-0.652323\pi$
$353$	−6108.07	−0.920962	−0.460481	+	0.887670i	$0.652323\pi$
$354$	0	0
$355$	2957.98	0.442234
$356$	0	0
$357$	2527.44	0.374695
$358$	0	0
$359$	4367.68	0.642110	0.321055	−	0.947061i	$-0.395963\pi$
$359$	4367.68	0.642110	0.321055	+	0.947061i	$0.395963\pi$
$360$	0	0
$361$	9509.48	1.38642
$362$	0	0
$363$	−452.375	−0.0654091
$364$	0	0
$365$	5600.79	0.803175
$366$	0	0
$367$	−5107.68	−0.726482	−0.363241	−	0.931695i	$-0.618330\pi$
$367$	−5107.68	−0.726482	−0.363241	+	0.931695i	$0.618330\pi$
$368$	0	0
$369$	−4000.58	−0.564396
$370$	0	0
$371$	40.4887	0.00566595
$372$	0	0
$373$	11662.4	1.61891	0.809457	−	0.587179i	$-0.199762\pi$
$373$	11662.4	1.61891	0.809457	+	0.587179i	$0.199762\pi$
$374$	0	0
$375$	−4564.35	−0.628539
$376$	0	0
$377$	−2306.84	−0.315141
$378$	0	0
$379$	−9357.41	−1.26823	−0.634113	−	0.773240i	$-0.718635\pi$
$379$	−9357.41	−1.26823	−0.634113	+	0.773240i	$0.718635\pi$
$380$	0	0
$381$	4317.69	0.580583
$382$	0	0
$383$	10979.4	1.46480	0.732401	−	0.680873i	$-0.238400\pi$
$383$	10979.4	1.46480	0.732401	+	0.680873i	$0.238400\pi$
$384$	0	0
$385$	−2193.92	−0.290422
$386$	0	0
$387$	1272.20	0.167105
$388$	0	0
$389$	695.788	0.0906885	0.0453443	−	0.998971i	$-0.485562\pi$
$389$	695.788	0.0906885	0.0453443	+	0.998971i	$0.485562\pi$
$390$	0	0
$391$	11441.2	1.47981
$392$	0	0
$393$	4746.68	0.609258
$394$	0	0
$395$	5866.61	0.747294
$396$	0	0
$397$	−11080.3	−1.40076	−0.700382	−	0.713768i	$-0.746987\pi$
$397$	−11080.3	−1.40076	−0.700382	+	0.713768i	$0.746987\pi$
$398$	0	0
$399$	2686.73	0.337104
$400$	0	0
$401$	3854.03	0.479952	0.239976	−	0.970779i	$-0.422860\pi$
$401$	3854.03	0.479952	0.239976	+	0.970779i	$0.422860\pi$
$402$	0	0
$403$	−2560.38	−0.316480
$404$	0	0
$405$	738.972	0.0906662
$406$	0	0
$407$	−6930.04	−0.844003
$408$	0	0
$409$	−172.125	−0.0208094	−0.0104047	−	0.999946i	$-0.503312\pi$
$409$	−172.125	−0.0208094	−0.0104047	+	0.999946i	$0.503312\pi$
$410$	0	0
$411$	4235.10	0.508278
$412$	0	0
$413$	2438.36	0.290518
$414$	0	0
$415$	−8285.51	−0.980047
$416$	0	0
$417$	7788.77	0.914671
$418$	0	0
$419$	−1629.97	−0.190046	−0.0950231	−	0.995475i	$-0.530293\pi$
$419$	−1629.97	−0.190046	−0.0950231	+	0.995475i	$0.530293\pi$
$420$	0	0
$421$	−3803.44	−0.440305	−0.220153	−	0.975465i	$-0.570656\pi$
$421$	−3803.44	−0.440305	−0.220153	+	0.975465i	$0.570656\pi$
$422$	0	0
$423$	5323.87	0.611952
$424$	0	0
$425$	5027.07	0.573761
$426$	0	0
$427$	−3725.54	−0.422228
$428$	0	0
$429$	2086.62	0.234833
$430$	0	0
$431$	5785.45	0.646578	0.323289	−	0.946300i	$-0.395211\pi$
$431$	5785.45	0.646578	0.323289	+	0.946300i	$0.395211\pi$
$432$	0	0
$433$	8945.75	0.992853	0.496426	−	0.868079i	$-0.334645\pi$
$433$	8945.75	0.992853	0.496426	+	0.868079i	$0.334645\pi$
$434$	0	0
$435$	−3118.44	−0.343719
$436$	0	0
$437$	12162.2	1.33135
$438$	0	0
$439$	6804.09	0.739730	0.369865	−	0.929085i	$-0.379404\pi$
$439$	6804.09	0.739730	0.369865	+	0.929085i	$0.379404\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	2883.84	0.309290	0.154645	−	0.987970i	$-0.450577\pi$
$443$	2883.84	0.309290	0.154645	+	0.987970i	$0.450577\pi$
$444$	0	0
$445$	−6046.08	−0.644071
$446$	0	0
$447$	−3573.81	−0.378155
$448$	0	0
$449$	−17216.9	−1.80962	−0.904808	−	0.425819i	$-0.859986\pi$
$449$	−17216.9	−1.80962	−0.904808	+	0.425819i	$0.859986\pi$
$450$	0	0
$451$	−15270.7	−1.59439
$452$	0	0
$453$	4983.91	0.516919
$454$	0	0
$455$	−1292.96	−0.133219
$456$	0	0
$457$	13605.2	1.39261	0.696307	−	0.717744i	$-0.254825\pi$
$457$	13605.2	1.39261	0.696307	+	0.717744i	$0.254825\pi$
$458$	0	0
$459$	−3249.56	−0.330450
$460$	0	0
$461$	7069.91	0.714271	0.357135	−	0.934053i	$-0.383754\pi$
$461$	7069.91	0.714271	0.357135	+	0.934053i	$0.383754\pi$
$462$	0	0
$463$	−7217.73	−0.724485	−0.362242	−	0.932084i	$-0.617989\pi$
$463$	−7217.73	−0.724485	−0.362242	+	0.932084i	$0.617989\pi$
$464$	0	0
$465$	−3461.18	−0.345179
$466$	0	0
$467$	5280.66	0.523255	0.261627	−	0.965169i	$-0.415741\pi$
$467$	5280.66	0.523255	0.261627	+	0.965169i	$0.415741\pi$
$468$	0	0
$469$	4628.83	0.455735
$470$	0	0
$471$	−11229.2	−1.09854
$472$	0	0
$473$	4856.17	0.472065
$474$	0	0
$475$	5343.89	0.516199
$476$	0	0
$477$	−52.0569	−0.00499690
$478$	0	0
$479$	−112.056	−0.0106889	−0.00534446	−	0.999986i	$-0.501701\pi$
$479$	−112.056	−0.0106889	−0.00534446	+	0.999986i	$0.501701\pi$
$480$	0	0
$481$	−4084.14	−0.387153
$482$	0	0
$483$	1996.31	0.188065
$484$	0	0
$485$	2023.91	0.189487
$486$	0	0
$487$	−8790.75	−0.817961	−0.408980	−	0.912543i	$-0.634116\pi$
$487$	−8790.75	−0.817961	−0.408980	+	0.912543i	$0.634116\pi$
$488$	0	0
$489$	1942.47	0.179635
$490$	0	0
$491$	−3141.85	−0.288778	−0.144389	−	0.989521i	$-0.546122\pi$
$491$	−3141.85	−0.288778	−0.144389	+	0.989521i	$0.546122\pi$
$492$	0	0
$493$	13713.1	1.25275
$494$	0	0
$495$	2820.75	0.256128
$496$	0	0
$497$	−2269.60	−0.204840
$498$	0	0
$499$	−3007.39	−0.269798	−0.134899	−	0.990859i	$-0.543071\pi$
$499$	−3007.39	−0.269798	−0.134899	+	0.990859i	$0.543071\pi$
$500$	0	0
$501$	3883.47	0.346308
$502$	0	0
$503$	19885.9	1.76276	0.881379	−	0.472410i	$-0.156616\pi$
$503$	19885.9	1.76276	0.881379	+	0.472410i	$0.156616\pi$
$504$	0	0
$505$	11227.3	0.989321
$506$	0	0
$507$	−5361.27	−0.469630
$508$	0	0
$509$	−12602.1	−1.09740	−0.548702	−	0.836018i	$-0.684878\pi$
$509$	−12602.1	−1.09740	−0.548702	+	0.836018i	$0.684878\pi$
$510$	0	0
$511$	−4297.39	−0.372026
$512$	0	0
$513$	−3454.36	−0.297298
$514$	0	0
$515$	2639.76	0.225867
$516$	0	0
$517$	20321.9	1.72874
$518$	0	0
$519$	−631.961	−0.0534489
$520$	0	0
$521$	10670.9	0.897316	0.448658	−	0.893703i	$-0.351902\pi$
$521$	10670.9	0.897316	0.448658	+	0.893703i	$0.351902\pi$
$522$	0	0
$523$	614.727	0.0513960	0.0256980	−	0.999670i	$-0.491819\pi$
$523$	614.727	0.0513960	0.0256980	+	0.999670i	$0.491819\pi$
$524$	0	0
$525$	877.148	0.0729178
$526$	0	0
$527$	15220.2	1.25807
$528$	0	0
$529$	−3130.12	−0.257263
$530$	0	0
$531$	−3135.03	−0.256213
$532$	0	0
$533$	−8999.63	−0.731365
$534$	0	0
$535$	−14718.1	−1.18938
$536$	0	0
$537$	8491.16	0.682348
$538$	0	0
$539$	1683.35	0.134522
$540$	0	0
$541$	297.163	0.0236156	0.0118078	−	0.999930i	$-0.496241\pi$
$541$	297.163	0.0236156	0.0118078	+	0.999930i	$0.496241\pi$
$542$	0	0
$543$	6527.67	0.515892
$544$	0	0
$545$	−17742.9	−1.39454
$546$	0	0
$547$	−20270.5	−1.58447	−0.792235	−	0.610216i	$-0.791083\pi$
$547$	−20270.5	−1.58447	−0.792235	+	0.610216i	$0.791083\pi$
$548$	0	0
$549$	4789.98	0.372370
$550$	0	0
$551$	14577.3	1.12707
$552$	0	0
$553$	−4501.34	−0.346142
$554$	0	0
$555$	−5521.03	−0.422261
$556$	0	0
$557$	−4058.60	−0.308740	−0.154370	−	0.988013i	$-0.549335\pi$
$557$	−4058.60	−0.308740	−0.154370	+	0.988013i	$0.549335\pi$
$558$	0	0
$559$	2861.92	0.216541
$560$	0	0
$561$	−12404.0	−0.933507
$562$	0	0
$563$	−6429.13	−0.481271	−0.240636	−	0.970616i	$-0.577356\pi$
$563$	−6429.13	−0.481271	−0.240636	+	0.970616i	$0.577356\pi$
$564$	0	0
$565$	−13974.7	−1.04056
$566$	0	0
$567$	−567.000	−0.0419961
$568$	0	0
$569$	23740.2	1.74910	0.874552	−	0.484932i	$-0.161156\pi$
$569$	23740.2	1.74910	0.874552	+	0.484932i	$0.161156\pi$
$570$	0	0
$571$	−1846.03	−0.135296	−0.0676478	−	0.997709i	$-0.521549\pi$
$571$	−1846.03	−0.135296	−0.0676478	+	0.997709i	$0.521549\pi$
$572$	0	0
$573$	−247.471	−0.0180423
$574$	0	0
$575$	3970.66	0.287979
$576$	0	0
$577$	−6725.07	−0.485214	−0.242607	−	0.970125i	$-0.578002\pi$
$577$	−6725.07	−0.485214	−0.242607	+	0.970125i	$0.578002\pi$
$578$	0	0
$579$	4895.81	0.351404
$580$	0	0
$581$	6357.33	0.453952
$582$	0	0
$583$	−198.708	−0.0141160
$584$	0	0
$585$	1662.37	0.117488
$586$	0	0
$587$	−2233.76	−0.157065	−0.0785325	−	0.996912i	$-0.525023\pi$
$587$	−2233.76	−0.157065	−0.0785325	+	0.996912i	$0.525023\pi$
$588$	0	0
$589$	16179.5	1.13186
$590$	0	0
$591$	456.898	0.0318008
$592$	0	0
$593$	26298.0	1.82113	0.910564	−	0.413368i	$-0.135648\pi$
$593$	26298.0	1.82113	0.910564	+	0.413368i	$0.135648\pi$
$594$	0	0
$595$	7686.03	0.529574
$596$	0	0
$597$	−2350.70	−0.161152
$598$	0	0
$599$	1508.25	0.102881	0.0514403	−	0.998676i	$-0.483619\pi$
$599$	1508.25	0.102881	0.0514403	+	0.998676i	$0.483619\pi$
$600$	0	0
$601$	−22110.1	−1.50065	−0.750325	−	0.661069i	$-0.770103\pi$
$601$	−22110.1	−1.50065	−0.750325	+	0.661069i	$0.770103\pi$
$602$	0	0
$603$	−5951.35	−0.401920
$604$	0	0
$605$	−1375.69	−0.0924457
$606$	0	0
$607$	19435.2	1.29959	0.649795	−	0.760110i	$-0.274855\pi$
$607$	19435.2	1.29959	0.649795	+	0.760110i	$0.274855\pi$
$608$	0	0
$609$	2392.73	0.159209
$610$	0	0
$611$	11976.5	0.792989
$612$	0	0
$613$	−28632.7	−1.88657	−0.943283	−	0.331990i	$-0.892280\pi$
$613$	−28632.7	−1.88657	−0.943283	+	0.331990i	$0.892280\pi$
$614$	0	0
$615$	−12165.9	−0.797687
$616$	0	0
$617$	507.633	0.0331224	0.0165612	−	0.999863i	$-0.494728\pi$
$617$	507.633	0.0331224	0.0165612	+	0.999863i	$0.494728\pi$
$618$	0	0
$619$	2710.27	0.175985	0.0879925	−	0.996121i	$-0.471955\pi$
$619$	2710.27	0.175985	0.0879925	+	0.996121i	$0.471955\pi$
$620$	0	0
$621$	−2566.69	−0.165858
$622$	0	0
$623$	4639.05	0.298330
$624$	0	0
$625$	−8659.24	−0.554191
$626$	0	0
$627$	−13185.8	−0.839854
$628$	0	0
$629$	24278.3	1.53901
$630$	0	0
$631$	26902.9	1.69729	0.848644	−	0.528965i	$-0.177420\pi$
$631$	26902.9	1.69729	0.848644	+	0.528965i	$0.177420\pi$
$632$	0	0
$633$	−7699.64	−0.483465
$634$	0	0
$635$	13130.3	0.820564
$636$	0	0
$637$	992.064	0.0617065
$638$	0	0
$639$	2918.06	0.180652
$640$	0	0
$641$	27113.5	1.67070	0.835350	−	0.549719i	$-0.185265\pi$
$641$	27113.5	1.67070	0.835350	+	0.549719i	$0.185265\pi$
$642$	0	0
$643$	−15828.1	−0.970763	−0.485382	−	0.874302i	$-0.661319\pi$
$643$	−15828.1	−0.970763	−0.485382	+	0.874302i	$0.661319\pi$
$644$	0	0
$645$	3868.82	0.236178
$646$	0	0
$647$	19093.9	1.16021	0.580106	−	0.814541i	$-0.303011\pi$
$647$	19093.9	1.16021	0.580106	+	0.814541i	$0.303011\pi$
$648$	0	0
$649$	−11966.8	−0.723789
$650$	0	0
$651$	2655.70	0.159885
$652$	0	0
$653$	−25935.2	−1.55424	−0.777122	−	0.629349i	$-0.783321\pi$
$653$	−25935.2	−1.55424	−0.777122	+	0.629349i	$0.783321\pi$
$654$	0	0
$655$	14434.8	0.861092
$656$	0	0
$657$	5525.22	0.328096
$658$	0	0
$659$	4095.04	0.242064	0.121032	−	0.992649i	$-0.461380\pi$
$659$	4095.04	0.242064	0.121032	+	0.992649i	$0.461380\pi$
$660$	0	0
$661$	19017.9	1.11908	0.559539	−	0.828804i	$-0.310978\pi$
$661$	19017.9	1.11908	0.559539	+	0.828804i	$0.310978\pi$
$662$	0	0
$663$	−7310.15	−0.428209
$664$	0	0
$665$	8170.43	0.476445
$666$	0	0
$667$	10831.4	0.628774
$668$	0	0
$669$	−7974.43	−0.460851
$670$	0	0
$671$	18284.0	1.05193
$672$	0	0
$673$	19467.4	1.11502	0.557512	−	0.830169i	$-0.311756\pi$
$673$	19467.4	1.11502	0.557512	+	0.830169i	$0.311756\pi$
$674$	0	0
$675$	−1127.76	−0.0643075
$676$	0	0
$677$	10546.2	0.598707	0.299353	−	0.954142i	$-0.403229\pi$
$677$	10546.2	0.598707	0.299353	+	0.954142i	$0.403229\pi$
$678$	0	0
$679$	−1552.91	−0.0877693
$680$	0	0
$681$	5938.78	0.334177
$682$	0	0
$683$	−16948.7	−0.949524	−0.474762	−	0.880114i	$-0.657466\pi$
$683$	−16948.7	−0.949524	−0.474762	+	0.880114i	$0.657466\pi$
$684$	0	0
$685$	12879.1	0.718372
$686$	0	0
$687$	−16580.3	−0.920785
$688$	0	0
$689$	−117.106	−0.00647516
$690$	0	0
$691$	7745.23	0.426400	0.213200	−	0.977009i	$-0.431611\pi$
$691$	7745.23	0.426400	0.213200	+	0.977009i	$0.431611\pi$
$692$	0	0
$693$	−2164.31	−0.118637
$694$	0	0
$695$	23685.9	1.29275
$696$	0	0
$697$	53498.6	2.90732
$698$	0	0
$699$	−14151.4	−0.765745
$700$	0	0
$701$	10485.0	0.564925	0.282462	−	0.959278i	$-0.408849\pi$
$701$	10485.0	0.564925	0.282462	+	0.959278i	$0.408849\pi$
$702$	0	0
$703$	25808.4	1.38461
$704$	0	0
$705$	16190.1	0.864899
$706$	0	0
$707$	−8614.49	−0.458248
$708$	0	0
$709$	9120.76	0.483127	0.241564	−	0.970385i	$-0.422340\pi$
$709$	9120.76	0.483127	0.241564	+	0.970385i	$0.422340\pi$
$710$	0	0
$711$	5787.44	0.305269
$712$	0	0
$713$	12021.8	0.631445
$714$	0	0
$715$	6345.50	0.331900
$716$	0	0
$717$	−1340.05	−0.0697980
$718$	0	0
$719$	−34496.1	−1.78928	−0.894638	−	0.446792i	$-0.852566\pi$
$719$	−34496.1	−1.78928	−0.894638	+	0.446792i	$0.852566\pi$
$720$	0	0
$721$	−2025.44	−0.104620
$722$	0	0
$723$	−11850.3	−0.609569
$724$	0	0
$725$	4759.13	0.243792
$726$	0	0
$727$	−3896.62	−0.198786	−0.0993931	−	0.995048i	$-0.531690\pi$
$727$	−3896.62	−0.198786	−0.0993931	+	0.995048i	$0.531690\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−17012.8	−0.860794
$732$	0	0
$733$	9066.77	0.456874	0.228437	−	0.973559i	$-0.426638\pi$
$733$	9066.77	0.456874	0.228437	+	0.973559i	$0.426638\pi$
$734$	0	0
$735$	1341.10	0.0673022
$736$	0	0
$737$	−22717.1	−1.13541
$738$	0	0
$739$	−25558.8	−1.27226	−0.636128	−	0.771584i	$-0.719465\pi$
$739$	−25558.8	−1.27226	−0.636128	+	0.771584i	$0.719465\pi$
$740$	0	0
$741$	−7770.86	−0.385249
$742$	0	0
$743$	−31302.6	−1.54560	−0.772801	−	0.634648i	$-0.781145\pi$
$743$	−31302.6	−1.54560	−0.772801	+	0.634648i	$0.781145\pi$
$744$	0	0
$745$	−10868.1	−0.534464
$746$	0	0
$747$	−8173.70	−0.400348
$748$	0	0
$749$	11293.0	0.550916
$750$	0	0
$751$	1792.59	0.0871006	0.0435503	−	0.999051i	$-0.486133\pi$
$751$	1792.59	0.0871006	0.0435503	+	0.999051i	$0.486133\pi$
$752$	0	0
$753$	15026.3	0.727209
$754$	0	0
$755$	15156.2	0.730586
$756$	0	0
$757$	40992.2	1.96814	0.984072	−	0.177768i	$-0.0568877\pi$
$757$	40992.2	1.96814	0.984072	+	0.177768i	$0.0568877\pi$
$758$	0	0
$759$	−9797.38	−0.468540
$760$	0	0
$761$	−7423.43	−0.353613	−0.176806	−	0.984246i	$-0.556577\pi$
$761$	−7423.43	−0.353613	−0.176806	+	0.984246i	$0.556577\pi$
$762$	0	0
$763$	13613.8	0.645942
$764$	0	0
$765$	−9882.03	−0.467040
$766$	0	0
$767$	−7052.51	−0.332009
$768$	0	0
$769$	−5371.42	−0.251883	−0.125942	−	0.992038i	$-0.540195\pi$
$769$	−5371.42	−0.251883	−0.125942	+	0.992038i	$0.540195\pi$
$770$	0	0
$771$	−13204.3	−0.616787
$772$	0	0
$773$	11435.3	0.532081	0.266041	−	0.963962i	$-0.414284\pi$
$773$	11435.3	0.532081	0.266041	+	0.963962i	$0.414284\pi$
$774$	0	0
$775$	5282.19	0.244828
$776$	0	0
$777$	4236.19	0.195589
$778$	0	0
$779$	56870.3	2.61565
$780$	0	0
$781$	11138.6	0.510334
$782$	0	0
$783$	−3076.36	−0.140409
$784$	0	0
$785$	−34148.3	−1.55262
$786$	0	0
$787$	28885.7	1.30834	0.654171	−	0.756346i	$-0.273018\pi$
$787$	28885.7	1.30834	0.654171	+	0.756346i	$0.273018\pi$
$788$	0	0
$789$	10796.2	0.487143
$790$	0	0
$791$	10722.5	0.481983
$792$	0	0
$793$	10775.4	0.482531
$794$	0	0
$795$	−158.307	−0.00706235
$796$	0	0
$797$	−28117.7	−1.24966	−0.624831	−	0.780760i	$-0.714832\pi$
$797$	−28117.7	−1.24966	−0.624831	+	0.780760i	$0.714832\pi$
$798$	0	0
$799$	−71194.5	−3.15229
$800$	0	0
$801$	−5964.49	−0.263102
$802$	0	0
$803$	21090.5	0.926857
$804$	0	0
$805$	6070.86	0.265801
$806$	0	0
$807$	5966.82	0.260275
$808$	0	0
$809$	38887.5	1.69000	0.845001	−	0.534765i	$-0.179600\pi$
$809$	38887.5	1.69000	0.845001	+	0.534765i	$0.179600\pi$
$810$	0	0
$811$	729.281	0.0315765	0.0157882	−	0.999875i	$-0.494974\pi$
$811$	729.281	0.0315765	0.0157882	+	0.999875i	$0.494974\pi$
$812$	0	0
$813$	−20513.5	−0.884918
$814$	0	0
$815$	5907.11	0.253886
$816$	0	0
$817$	−18085.0	−0.774436
$818$	0	0
$819$	−1275.51	−0.0544200
$820$	0	0
$821$	9877.25	0.419876	0.209938	−	0.977715i	$-0.432674\pi$
$821$	9877.25	0.419876	0.209938	+	0.977715i	$0.432674\pi$
$822$	0	0
$823$	−23349.1	−0.988941	−0.494470	−	0.869194i	$-0.664638\pi$
$823$	−23349.1	−0.988941	−0.494470	+	0.869194i	$0.664638\pi$
$824$	0	0
$825$	−4304.81	−0.181666
$826$	0	0
$827$	−30446.5	−1.28020	−0.640101	−	0.768290i	$-0.721108\pi$
$827$	−30446.5	−1.28020	−0.640101	+	0.768290i	$0.721108\pi$
$828$	0	0
$829$	−22525.2	−0.943704	−0.471852	−	0.881678i	$-0.656414\pi$
$829$	−22525.2	−0.943704	−0.471852	+	0.881678i	$0.656414\pi$
$830$	0	0
$831$	−12507.8	−0.522132
$832$	0	0
$833$	−5897.35	−0.245296
$834$	0	0
$835$	11809.8	0.489453
$836$	0	0
$837$	−3414.48	−0.141005
$838$	0	0
$839$	22906.8	0.942588	0.471294	−	0.881976i	$-0.343787\pi$
$839$	22906.8	0.942588	0.471294	+	0.881976i	$0.343787\pi$
$840$	0	0
$841$	−11406.8	−0.467703
$842$	0	0
$843$	12839.0	0.524554
$844$	0	0
$845$	−16303.8	−0.663750
$846$	0	0
$847$	1055.54	0.0428203
$848$	0	0
$849$	597.636	0.0241588
$850$	0	0
$851$	19176.3	0.772452
$852$	0	0
$853$	36516.4	1.46576	0.732882	−	0.680355i	$-0.238175\pi$
$853$	36516.4	1.46576	0.732882	+	0.680355i	$0.238175\pi$
$854$	0	0
$855$	−10504.8	−0.420185
$856$	0	0
$857$	9314.66	0.371275	0.185637	−	0.982618i	$-0.440565\pi$
$857$	9314.66	0.371275	0.185637	+	0.982618i	$0.440565\pi$
$858$	0	0
$859$	−38080.2	−1.51255	−0.756274	−	0.654255i	$-0.772982\pi$
$859$	−38080.2	−1.51255	−0.756274	+	0.654255i	$0.772982\pi$
$860$	0	0
$861$	9334.70	0.369484
$862$	0	0
$863$	12538.3	0.494562	0.247281	−	0.968944i	$-0.420463\pi$
$863$	12538.3	0.494562	0.247281	+	0.968944i	$0.420463\pi$
$864$	0	0
$865$	−1921.81	−0.0755418
$866$	0	0
$867$	28716.4	1.12487
$868$	0	0
$869$	22091.4	0.862371
$870$	0	0
$871$	−13388.0	−0.520822
$872$	0	0
$873$	1996.60	0.0774052
$874$	0	0
$875$	10650.2	0.411475
$876$	0	0
$877$	6364.32	0.245049	0.122524	−	0.992466i	$-0.460901\pi$
$877$	6364.32	0.245049	0.122524	+	0.992466i	$0.460901\pi$
$878$	0	0
$879$	3189.34	0.122382
$880$	0	0
$881$	−19041.5	−0.728177	−0.364088	−	0.931364i	$-0.618619\pi$
$881$	−19041.5	−0.728177	−0.364088	+	0.931364i	$0.618619\pi$
$882$	0	0
$883$	−45100.7	−1.71887	−0.859433	−	0.511249i	$-0.829183\pi$
$883$	−45100.7	−1.71887	−0.859433	+	0.511249i	$0.829183\pi$
$884$	0	0
$885$	−9533.75	−0.362117
$886$	0	0
$887$	25166.9	0.952673	0.476336	−	0.879263i	$-0.341964\pi$
$887$	25166.9	0.952673	0.476336	+	0.879263i	$0.341964\pi$
$888$	0	0
$889$	−10074.6	−0.380081
$890$	0	0
$891$	2782.69	0.104628
$892$	0	0
$893$	−75681.5	−2.83604
$894$	0	0
$895$	25821.9	0.964393
$896$	0	0
$897$	−5773.97	−0.214924
$898$	0	0
$899$	14409.0	0.534558
$900$	0	0
$901$	696.140	0.0257401
$902$	0	0
$903$	−2968.48	−0.109396
$904$	0	0
$905$	19850.9	0.729133
$906$	0	0
$907$	43820.6	1.60423	0.802117	−	0.597167i	$-0.203707\pi$
$907$	43820.6	1.60423	0.802117	+	0.597167i	$0.203707\pi$
$908$	0	0
$909$	11075.8	0.404137
$910$	0	0
$911$	−18066.6	−0.657051	−0.328525	−	0.944495i	$-0.606552\pi$
$911$	−18066.6	−0.657051	−0.328525	+	0.944495i	$0.606552\pi$
$912$	0	0
$913$	−31200.1	−1.13097
$914$	0	0
$915$	14566.5	0.526288
$916$	0	0
$917$	−11075.6	−0.398853
$918$	0	0
$919$	−41554.8	−1.49158	−0.745792	−	0.666179i	$-0.767929\pi$
$919$	−41554.8	−1.49158	−0.745792	+	0.666179i	$0.767929\pi$
$920$	0	0
$921$	12631.8	0.451935
$922$	0	0
$923$	6564.41	0.234096
$924$	0	0
$925$	8425.78	0.299500
$926$	0	0
$927$	2604.14	0.0922665
$928$	0	0
$929$	7481.51	0.264220	0.132110	−	0.991235i	$-0.457825\pi$
$929$	7481.51	0.264220	0.132110	+	0.991235i	$0.457825\pi$
$930$	0	0
$931$	−6269.03	−0.220687
$932$	0	0
$933$	−21761.7	−0.763608
$934$	0	0
$935$	−37721.0	−1.31937
$936$	0	0
$937$	−36682.6	−1.27894	−0.639471	−	0.768815i	$-0.720847\pi$
$937$	−36682.6	−1.27894	−0.639471	+	0.768815i	$0.720847\pi$
$938$	0	0
$939$	27990.1	0.972760
$940$	0	0
$941$	55856.3	1.93503	0.967516	−	0.252810i	$-0.0813549\pi$
$941$	55856.3	1.93503	0.967516	+	0.252810i	$0.0813549\pi$
$942$	0	0
$943$	42256.2	1.45923
$944$	0	0
$945$	−1724.27	−0.0593549
$946$	0	0
$947$	−29174.5	−1.00110	−0.500551	−	0.865707i	$-0.666869\pi$
$947$	−29174.5	−1.00110	−0.500551	+	0.865707i	$0.666869\pi$
$948$	0	0
$949$	12429.4	0.425159
$950$	0	0
$951$	11817.3	0.402948
$952$	0	0
$953$	−3095.44	−0.105216	−0.0526081	−	0.998615i	$-0.516753\pi$
$953$	−3095.44	−0.105216	−0.0526081	+	0.998615i	$0.516753\pi$
$954$	0	0
$955$	−752.568	−0.0255000
$956$	0	0
$957$	−11742.9	−0.396649
$958$	0	0
$959$	−9881.90	−0.332746
$960$	0	0
$961$	−13798.3	−0.463171
$962$	0	0
$963$	−14519.5	−0.485863
$964$	0	0
$965$	14888.3	0.496655
$966$	0	0
$967$	−27542.5	−0.915932	−0.457966	−	0.888970i	$-0.651422\pi$
$967$	−27542.5	−0.915932	−0.457966	+	0.888970i	$0.651422\pi$
$968$	0	0
$969$	46194.1	1.53144
$970$	0	0
$971$	23338.2	0.771325	0.385663	−	0.922640i	$-0.373973\pi$
$971$	23338.2	0.771325	0.385663	+	0.922640i	$0.373973\pi$
$972$	0	0
$973$	−18173.8	−0.598793
$974$	0	0
$975$	−2536.99	−0.0833320
$976$	0	0
$977$	−19936.0	−0.652824	−0.326412	−	0.945228i	$-0.605840\pi$
$977$	−19936.0	−0.652824	−0.326412	+	0.945228i	$0.605840\pi$
$978$	0	0
$979$	−22767.2	−0.743253
$980$	0	0
$981$	−17503.5	−0.569667
$982$	0	0
$983$	−40496.3	−1.31397	−0.656985	−	0.753904i	$-0.728168\pi$
$983$	−40496.3	−1.31397	−0.656985	+	0.753904i	$0.728168\pi$
$984$	0	0
$985$	1389.44	0.0449455
$986$	0	0
$987$	−12422.4	−0.400617
$988$	0	0
$989$	−13437.7	−0.432045
$990$	0	0
$991$	−35765.9	−1.14646	−0.573229	−	0.819395i	$-0.694310\pi$
$991$	−35765.9	−1.14646	−0.573229	+	0.819395i	$0.694310\pi$
$992$	0	0
$993$	−11790.8	−0.376806
$994$	0	0
$995$	−7148.58	−0.227764
$996$	0	0
$997$	27364.4	0.869246	0.434623	−	0.900612i	$-0.356882\pi$
$997$	27364.4	0.869246	0.434623	+	0.900612i	$0.356882\pi$
$998$	0	0
$999$	−5446.53	−0.172493

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.4.a.br.1.2		2
4.3	odd	2		1344.4.a.bj.1.2		2
8.3	odd	2		672.4.a.j.1.1	yes	2
8.5	even	2		672.4.a.e.1.1	✓	2
24.5	odd	2		2016.4.a.q.1.2		2
24.11	even	2		2016.4.a.r.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.4.a.e.1.1	✓	2	8.5	even	2
672.4.a.j.1.1	yes	2	8.3	odd	2
1344.4.a.bj.1.2		2	4.3	odd	2
1344.4.a.br.1.2		2	1.1	even	1	trivial
2016.4.a.q.1.2		2	24.5	odd	2
2016.4.a.r.1.2		2	24.11	even	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)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +9.12311 q^{5} -7.00000 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} +9.12311 q^{5} -7.00000 q^{7} +9.00000 q^{9} +34.3542 q^{11} +20.2462 q^{13} +27.3693 q^{15} -120.354 q^{17} -127.939 q^{19} -21.0000 q^{21} -95.0625 q^{23} -41.7689 q^{25} +27.0000 q^{27} -113.939 q^{29} -126.462 q^{31} +103.062 q^{33} -63.8617 q^{35} -201.723 q^{37} +60.7386 q^{39} -444.509 q^{41} +141.356 q^{43} +82.1080 q^{45} +591.542 q^{47} +49.0000 q^{49} -361.062 q^{51} -5.78410 q^{53} +313.417 q^{55} -383.818 q^{57} -348.337 q^{59} +532.220 q^{61} -63.0000 q^{63} +184.708 q^{65} -661.261 q^{67} -285.187 q^{69} +324.229 q^{71} +613.913 q^{73} -125.307 q^{75} -240.479 q^{77} +643.049 q^{79} +81.0000 q^{81} -908.189 q^{83} -1098.00 q^{85} -341.818 q^{87} -662.722 q^{89} -141.723 q^{91} -379.386 q^{93} -1167.20 q^{95} +221.845 q^{97} +309.187 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} + 10 q^{5} - 14 q^{7} + 18 q^{9}+O(q^{10}) \) 2 * q + 6 * q^3 + 10 * q^5 - 14 * q^7 + 18 * q^9	\( 2 q + 6 q^{3} + 10 q^{5} - 14 q^{7} + 18 q^{9} - 22 q^{11} + 24 q^{13} + 30 q^{15} - 150 q^{17} + 8 q^{19} - 42 q^{21} + 82 q^{23} - 166 q^{25} + 54 q^{27} + 36 q^{29} - 88 q^{31} - 66 q^{33} - 70 q^{35} - 288 q^{37} + 72 q^{39} - 386 q^{41} - 344 q^{43} + 90 q^{45} + 276 q^{47} + 98 q^{49} - 450 q^{51} - 160 q^{53} + 264 q^{55} + 24 q^{57} - 1076 q^{59} - 156 q^{61} - 126 q^{63} + 188 q^{65} - 1372 q^{67} + 246 q^{69} + 1102 q^{71} - 240 q^{73} - 498 q^{75} + 154 q^{77} + 412 q^{79} + 162 q^{81} - 464 q^{83} - 1124 q^{85} + 108 q^{87} - 1746 q^{89} - 168 q^{91} - 264 q^{93} - 1048 q^{95} + 856 q^{97} - 198 q^{99}+O(q^{100}) \) 2 * q + 6 * q^3 + 10 * q^5 - 14 * q^7 + 18 * q^9 - 22 * q^11 + 24 * q^13 + 30 * q^15 - 150 * q^17 + 8 * q^19 - 42 * q^21 + 82 * q^23 - 166 * q^25 + 54 * q^27 + 36 * q^29 - 88 * q^31 - 66 * q^33 - 70 * q^35 - 288 * q^37 + 72 * q^39 - 386 * q^41 - 344 * q^43 + 90 * q^45 + 276 * q^47 + 98 * q^49 - 450 * q^51 - 160 * q^53 + 264 * q^55 + 24 * q^57 - 1076 * q^59 - 156 * q^61 - 126 * q^63 + 188 * q^65 - 1372 * q^67 + 246 * q^69 + 1102 * q^71 - 240 * q^73 - 498 * q^75 + 154 * q^77 + 412 * q^79 + 162 * q^81 - 464 * q^83 - 1124 * q^85 + 108 * q^87 - 1746 * q^89 - 168 * q^91 - 264 * q^93 - 1048 * q^95 + 856 * q^97 - 198 * q^99

Introduction

L-functions

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