LMFDB - Embedded newform 2178.4.a.bv.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2178,4,Mod(1,2178)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2178, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2178.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2178 = 2 \cdot 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2178.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:	$128.506159993$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{67})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 35x^{2} + 256 $ x^4 - 35*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.22665$ of defining polynomial
Character	$\chi$	$=$	2178.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} +4.00000 q^{4} -7.94790 q^{5} -16.6082 q^{7} -8.00000 q^{8} +O(q^{10})$	$q-2.00000 q^{2} +4.00000 q^{4} -7.94790 q^{5} -16.6082 q^{7} -8.00000 q^{8} +15.8958 q^{10} +5.90850 q^{13} +33.2163 q^{14} +16.0000 q^{16} -5.29851 q^{17} -19.2622 q^{19} -31.7916 q^{20} -63.5832 q^{23} -61.8309 q^{25} -11.8170 q^{26} -66.4326 q^{28} +204.363 q^{29} -209.363 q^{31} -32.0000 q^{32} +10.5970 q^{34} +132.000 q^{35} +35.5970 q^{37} +38.5245 q^{38} +63.5832 q^{40} -116.831 q^{41} -70.3161 q^{43} +127.166 q^{46} +199.298 q^{47} -67.1691 q^{49} +123.662 q^{50} +23.6340 q^{52} -430.988 q^{53} +132.865 q^{56} -408.726 q^{58} -198.068 q^{59} -686.243 q^{61} +418.726 q^{62} +64.0000 q^{64} -46.9602 q^{65} +528.348 q^{67} -21.1940 q^{68} -264.000 q^{70} -690.810 q^{71} -253.006 q^{73} -71.1940 q^{74} -77.0490 q^{76} -768.933 q^{79} -127.166 q^{80} +233.662 q^{82} +93.8706 q^{83} +42.1120 q^{85} +140.632 q^{86} +572.763 q^{89} -98.1294 q^{91} -254.333 q^{92} -398.596 q^{94} +153.094 q^{95} -1495.05 q^{97} +134.338 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{2} + 16 q^{4} - 32 q^{8}+O(q^{10}) $ 4 * q - 8 * q^2 + 16 * q^4 - 32 * q^8	$ 4 q - 8 q^{2} + 16 q^{4} - 32 q^{8} + 64 q^{16} + 234 q^{17} + 178 q^{25} + 222 q^{29} - 242 q^{31} - 128 q^{32} - 468 q^{34} + 528 q^{35} - 368 q^{37} - 42 q^{41} - 694 q^{49} - 356 q^{50} - 444 q^{58} + 484 q^{62} + 256 q^{64} + 918 q^{65} - 1034 q^{67} + 936 q^{68} - 1056 q^{70} + 736 q^{74} + 84 q^{82} + 1056 q^{83} + 288 q^{91} + 5376 q^{95} - 3088 q^{97} + 1388 q^{98}+O(q^{100}) $ 4 * q - 8 * q^2 + 16 * q^4 - 32 * q^8 + 64 * q^16 + 234 * q^17 + 178 * q^25 + 222 * q^29 - 242 * q^31 - 128 * q^32 - 468 * q^34 + 528 * q^35 - 368 * q^37 - 42 * q^41 - 694 * q^49 - 356 * q^50 - 444 * q^58 + 484 * q^62 + 256 * q^64 + 918 * q^65 - 1034 * q^67 + 936 * q^68 - 1056 * q^70 + 736 * q^74 + 84 * q^82 + 1056 * q^83 + 288 * q^91 + 5376 * q^95 - 3088 * q^97 + 1388 * q^98

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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	−7.94790	−0.710882	−0.355441	−	0.934699i	$-0.615669\pi$
$5$	−7.94790	−0.710882	−0.355441	+	0.934699i	$0.615669\pi$
$6$	0	0
$7$	−16.6082	−0.896756	−0.448378	−	0.893844i	$-0.647998\pi$
$7$	−16.6082	−0.896756	−0.448378	+	0.893844i	$0.647998\pi$
$8$	−8.00000	−0.353553
$9$	0	0
$10$	15.8958	0.502669
$11$	0	0
$11$	0	0
$12$	0	0
$13$	5.90850	0.126056	0.0630279	−	0.998012i	$-0.479924\pi$
$13$	5.90850	0.126056	0.0630279	+	0.998012i	$0.479924\pi$
$14$	33.2163	0.634102
$15$	0	0
$16$	16.0000	0.250000
$17$	−5.29851	−0.0755928	−0.0377964	−	0.999285i	$-0.512034\pi$
$17$	−5.29851	−0.0755928	−0.0377964	+	0.999285i	$0.512034\pi$
$18$	0	0
$19$	−19.2622	−0.232582	−0.116291	−	0.993215i	$-0.537101\pi$
$19$	−19.2622	−0.232582	−0.116291	+	0.993215i	$0.537101\pi$
$20$	−31.7916	−0.355441
$21$	0	0
$22$	0	0
$23$	−63.5832	−0.576436	−0.288218	−	0.957565i	$-0.593063\pi$
$23$	−63.5832	−0.576436	−0.288218	+	0.957565i	$0.593063\pi$
$24$	0	0
$25$	−61.8309	−0.494647
$26$	−11.8170	−0.0891348
$27$	0	0
$28$	−66.4326	−0.448378
$29$	204.363	1.30860	0.654298	−	0.756237i	$-0.272964\pi$
$29$	204.363	1.30860	0.654298	+	0.756237i	$0.272964\pi$
$30$	0	0
$31$	−209.363	−1.21299	−0.606496	−	0.795087i	$-0.707425\pi$
$31$	−209.363	−1.21299	−0.606496	+	0.795087i	$0.707425\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	10.5970	0.0534522
$35$	132.000	0.637488
$36$	0	0
$37$	35.5970	0.158165	0.0790826	−	0.996868i	$-0.474801\pi$
$37$	35.5970	0.158165	0.0790826	+	0.996868i	$0.474801\pi$
$38$	38.5245	0.164460
$39$	0	0
$40$	63.5832	0.251335
$41$	−116.831	−0.445022	−0.222511	−	0.974930i	$-0.571425\pi$
$41$	−116.831	−0.445022	−0.222511	+	0.974930i	$0.571425\pi$
$42$	0	0
$43$	−70.3161	−0.249375	−0.124687	−	0.992196i	$-0.539793\pi$
$43$	−70.3161	−0.249375	−0.124687	+	0.992196i	$0.539793\pi$
$44$	0	0
$45$	0	0
$46$	127.166	0.407601
$47$	199.298	0.618523	0.309262	−	0.950977i	$-0.399918\pi$
$47$	199.298	0.618523	0.309262	+	0.950977i	$0.399918\pi$
$48$	0	0
$49$	−67.1691	−0.195828
$50$	123.662	0.349768
$51$	0	0
$52$	23.6340	0.0630279
$53$	−430.988	−1.11699	−0.558497	−	0.829506i	$-0.688622\pi$
$53$	−430.988	−1.11699	−0.558497	+	0.829506i	$0.688622\pi$
$54$	0	0
$55$	0	0
$56$	132.865	0.317051
$57$	0	0
$58$	−408.726	−0.925317
$59$	−198.068	−0.437056	−0.218528	−	0.975831i	$-0.570126\pi$
$59$	−198.068	−0.437056	−0.218528	+	0.975831i	$0.570126\pi$
$60$	0	0
$61$	−686.243	−1.44040	−0.720200	−	0.693767i	$-0.755950\pi$
$61$	−686.243	−1.44040	−0.720200	+	0.693767i	$0.755950\pi$
$62$	418.726	0.857715
$63$	0	0
$64$	64.0000	0.125000
$65$	−46.9602	−0.0896107
$66$	0	0
$67$	528.348	0.963403	0.481702	−	0.876335i	$-0.340019\pi$
$67$	528.348	0.963403	0.481702	+	0.876335i	$0.340019\pi$
$68$	−21.1940	−0.0377964
$69$	0	0
$70$	−264.000	−0.450772
$71$	−690.810	−1.15470	−0.577352	−	0.816495i	$-0.695914\pi$
$71$	−690.810	−1.15470	−0.577352	+	0.816495i	$0.695914\pi$
$72$	0	0
$73$	−253.006	−0.405645	−0.202823	−	0.979216i	$-0.565011\pi$
$73$	−253.006	−0.405645	−0.202823	+	0.979216i	$0.565011\pi$
$74$	−71.1940	−0.111840
$75$	0	0
$76$	−77.0490	−0.116291
$77$	0	0
$78$	0	0
$79$	−768.933	−1.09508	−0.547542	−	0.836778i	$-0.684437\pi$
$79$	−768.933	−1.09508	−0.547542	+	0.836778i	$0.684437\pi$
$80$	−127.166	−0.177720
$81$	0	0
$82$	233.662	0.314678
$83$	93.8706	0.124140	0.0620701	−	0.998072i	$-0.480230\pi$
$83$	93.8706	0.124140	0.0620701	+	0.998072i	$0.480230\pi$
$84$	0	0
$85$	42.1120	0.0537376
$86$	140.632	0.176334
$87$	0	0
$88$	0	0
$89$	572.763	0.682166	0.341083	−	0.940033i	$-0.389206\pi$
$89$	572.763	0.682166	0.341083	+	0.940033i	$0.389206\pi$
$90$	0	0
$91$	−98.1294	−0.113041
$92$	−254.333	−0.288218
$93$	0	0
$94$	−398.596	−0.437362
$95$	153.094	0.165339
$96$	0	0
$97$	−1495.05	−1.56494	−0.782471	−	0.622687i	$-0.786041\pi$
$97$	−1495.05	−1.56494	−0.782471	+	0.622687i	$0.786041\pi$
$98$	134.338	0.138472
$99$	0	0
$100$	−247.323	−0.247323
$101$	1184.57	1.16702	0.583509	−	0.812107i	$-0.301679\pi$
$101$	1184.57	1.16702	0.583509	+	0.812107i	$0.301679\pi$
$102$	0	0
$103$	−911.234	−0.871714	−0.435857	−	0.900016i	$-0.643555\pi$
$103$	−911.234	−0.871714	−0.435857	+	0.900016i	$0.643555\pi$
$104$	−47.2680	−0.0445674
$105$	0	0
$106$	861.975	0.789834
$107$	1697.71	1.53387	0.766934	−	0.641725i	$-0.221781\pi$
$107$	1697.71	1.53387	0.766934	+	0.641725i	$0.221781\pi$
$108$	0	0
$109$	60.0358	0.0527559	0.0263779	−	0.999652i	$-0.491603\pi$
$109$	60.0358	0.0527559	0.0263779	+	0.999652i	$0.491603\pi$
$110$	0	0
$111$	0	0
$112$	−265.730	−0.224189
$113$	−428.586	−0.356797	−0.178398	−	0.983958i	$-0.557092\pi$
$113$	−428.586	−0.356797	−0.178398	+	0.983958i	$0.557092\pi$
$114$	0	0
$115$	505.353	0.409778
$116$	817.453	0.654298
$117$	0	0
$118$	396.137	0.309045
$119$	87.9985	0.0677883
$120$	0	0
$121$	0	0
$122$	1372.49	1.01852
$123$	0	0
$124$	−837.453	−0.606496
$125$	1484.91	1.06252
$126$	0	0
$127$	466.413	0.325885	0.162943	−	0.986636i	$-0.447901\pi$
$127$	466.413	0.325885	0.162943	+	0.986636i	$0.447901\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	93.9204	0.0633644
$131$	1135.16	0.757097	0.378549	−	0.925581i	$-0.376423\pi$
$131$	1135.16	0.757097	0.378549	+	0.925581i	$0.376423\pi$
$132$	0	0
$133$	319.910	0.208570
$134$	−1056.70	−0.681229
$135$	0	0
$136$	42.3881	0.0267261
$137$	−2291.22	−1.42885	−0.714425	−	0.699712i	$-0.753312\pi$
$137$	−2291.22	−1.42885	−0.714425	+	0.699712i	$0.753312\pi$
$138$	0	0
$139$	−24.2200	−0.0147792	−0.00738961	−	0.999973i	$-0.502352\pi$
$139$	−24.2200	−0.0147792	−0.00738961	+	0.999973i	$0.502352\pi$
$140$	528.000	0.318744
$141$	0	0
$142$	1381.62	0.816499
$143$	0	0
$144$	0	0
$145$	−1624.26	−0.930258
$146$	506.012	0.286834
$147$	0	0
$148$	142.388	0.0790826
$149$	−11.0100	−0.00605352	−0.00302676	−	0.999995i	$-0.500963\pi$
$149$	−11.0100	−0.00605352	−0.00302676	+	0.999995i	$0.500963\pi$
$150$	0	0
$151$	3281.88	1.76871	0.884356	−	0.466814i	$-0.154598\pi$
$151$	3281.88	1.76871	0.884356	+	0.466814i	$0.154598\pi$
$152$	154.098	0.0822302
$153$	0	0
$154$	0	0
$155$	1664.00	0.862294
$156$	0	0
$157$	331.801	0.168666	0.0843331	−	0.996438i	$-0.473124\pi$
$157$	331.801	0.168666	0.0843331	+	0.996438i	$0.473124\pi$
$158$	1537.87	0.774342
$159$	0	0
$160$	254.333	0.125667
$161$	1056.00	0.516922
$162$	0	0
$163$	−934.458	−0.449033	−0.224516	−	0.974470i	$-0.572080\pi$
$163$	−934.458	−0.449033	−0.224516	+	0.974470i	$0.572080\pi$
$164$	−467.323	−0.222511
$165$	0	0
$166$	−187.741	−0.0877804
$167$	−693.970	−0.321563	−0.160782	−	0.986990i	$-0.551401\pi$
$167$	−693.970	−0.321563	−0.160782	+	0.986990i	$0.551401\pi$
$168$	0	0
$169$	−2162.09	−0.984110
$170$	−84.2241	−0.0379982
$171$	0	0
$172$	−281.264	−0.124687
$173$	2440.21	1.07240	0.536201	−	0.844090i	$-0.319859\pi$
$173$	2440.21	1.07240	0.536201	+	0.844090i	$0.319859\pi$
$174$	0	0
$175$	1026.90	0.443578
$176$	0	0
$177$	0	0
$178$	−1145.53	−0.482364
$179$	−1091.75	−0.455872	−0.227936	−	0.973676i	$-0.573198\pi$
$179$	−1091.75	−0.455872	−0.227936	+	0.973676i	$0.573198\pi$
$180$	0	0
$181$	1673.10	0.687075	0.343537	−	0.939139i	$-0.388375\pi$
$181$	1673.10	0.687075	0.343537	+	0.939139i	$0.388375\pi$
$182$	196.259	0.0799322
$183$	0	0
$184$	508.666	0.203801
$185$	−282.922	−0.112437
$186$	0	0
$187$	0	0
$188$	797.191	0.309262
$189$	0	0
$190$	−306.189	−0.116912
$191$	4828.18	1.82908	0.914541	−	0.404494i	$-0.132552\pi$
$191$	4828.18	1.82908	0.914541	+	0.404494i	$0.132552\pi$
$192$	0	0
$193$	−2002.89	−0.747001	−0.373501	−	0.927630i	$-0.621843\pi$
$193$	−2002.89	−0.747001	−0.373501	+	0.927630i	$0.621843\pi$
$194$	2990.10	1.10658
$195$	0	0
$196$	−268.677	−0.0979142
$197$	−1801.77	−0.651627	−0.325814	−	0.945434i	$-0.605638\pi$
$197$	−1801.77	−0.651627	−0.325814	+	0.945434i	$0.605638\pi$
$198$	0	0
$199$	−3390.34	−1.20771	−0.603856	−	0.797093i	$-0.706370\pi$
$199$	−3390.34	−1.20771	−0.603856	+	0.797093i	$0.706370\pi$
$200$	494.647	0.174884
$201$	0	0
$202$	−2369.13	−0.825207
$203$	−3394.10	−1.17349
$204$	0	0
$205$	928.560	0.316358
$206$	1822.47	0.616395
$207$	0	0
$208$	94.5361	0.0315139
$209$	0	0
$210$	0	0
$211$	−526.756	−0.171864	−0.0859322	−	0.996301i	$-0.527387\pi$
$211$	−526.756	−0.171864	−0.0859322	+	0.996301i	$0.527387\pi$
$212$	−1723.95	−0.558497
$213$	0	0
$214$	−3395.42	−1.08461
$215$	558.866	0.177276
$216$	0	0
$217$	3477.14	1.08776
$218$	−120.072	−0.0373040
$219$	0	0
$220$	0	0
$221$	−31.3063	−0.00952890
$222$	0	0
$223$	2740.11	0.822831	0.411415	−	0.911448i	$-0.365035\pi$
$223$	2740.11	0.822831	0.411415	+	0.911448i	$0.365035\pi$
$224$	531.461	0.158526
$225$	0	0
$226$	857.173	0.252293
$227$	5012.52	1.46561	0.732803	−	0.680441i	$-0.238212\pi$
$227$	5012.52	1.46561	0.732803	+	0.680441i	$0.238212\pi$
$228$	0	0
$229$	301.368	0.0869650	0.0434825	−	0.999054i	$-0.486155\pi$
$229$	301.368	0.0869650	0.0434825	+	0.999054i	$0.486155\pi$
$230$	−1010.71	−0.289757
$231$	0	0
$232$	−1634.91	−0.462659
$233$	−1255.96	−0.353135	−0.176567	−	0.984289i	$-0.556499\pi$
$233$	−1255.96	−0.353135	−0.176567	+	0.984289i	$0.556499\pi$
$234$	0	0
$235$	−1584.00	−0.439697
$236$	−792.274	−0.218528
$237$	0	0
$238$	−175.997	−0.0479336
$239$	4828.94	1.30694	0.653468	−	0.756954i	$-0.273313\pi$
$239$	4828.94	1.30694	0.653468	+	0.756954i	$0.273313\pi$
$240$	0	0
$241$	−2397.94	−0.640933	−0.320467	−	0.947260i	$-0.603840\pi$
$241$	−2397.94	−0.640933	−0.320467	+	0.947260i	$0.603840\pi$
$242$	0	0
$243$	0	0
$244$	−2744.97	−0.720200
$245$	533.854	0.139211
$246$	0	0
$247$	−113.811	−0.0293183
$248$	1674.91	0.428857
$249$	0	0
$250$	−2969.83	−0.751313
$251$	−7364.19	−1.85189	−0.925943	−	0.377664i	$-0.876728\pi$
$251$	−7364.19	−1.85189	−0.925943	+	0.377664i	$0.876728\pi$
$252$	0	0
$253$	0	0
$254$	−932.826	−0.230436
$255$	0	0
$256$	256.000	0.0625000
$257$	6055.07	1.46967	0.734834	−	0.678247i	$-0.237260\pi$
$257$	6055.07	1.46967	0.734834	+	0.678247i	$0.237260\pi$
$258$	0	0
$259$	−591.201	−0.141836
$260$	−187.841	−0.0448054
$261$	0	0
$262$	−2270.33	−0.535349
$263$	−6682.17	−1.56669	−0.783346	−	0.621585i	$-0.786489\pi$
$263$	−6682.17	−1.56669	−0.783346	+	0.621585i	$0.786489\pi$
$264$	0	0
$265$	3425.45	0.794051
$266$	−639.821	−0.147481
$267$	0	0
$268$	2113.39	0.481702
$269$	−5521.96	−1.25160	−0.625799	−	0.779984i	$-0.715227\pi$
$269$	−5521.96	−1.25160	−0.625799	+	0.779984i	$0.715227\pi$
$270$	0	0
$271$	−892.647	−0.200090	−0.100045	−	0.994983i	$-0.531899\pi$
$271$	−892.647	−0.200090	−0.100045	+	0.994983i	$0.531899\pi$
$272$	−84.7762	−0.0188982
$273$	0	0
$274$	4582.45	1.01035
$275$	0	0
$276$	0	0
$277$	4278.00	0.927943	0.463972	−	0.885850i	$-0.346424\pi$
$277$	4278.00	0.927943	0.463972	+	0.885850i	$0.346424\pi$
$278$	48.4399	0.0104505
$279$	0	0
$280$	−1056.00	−0.225386
$281$	8619.25	1.82983	0.914914	−	0.403649i	$-0.132258\pi$
$281$	8619.25	1.82983	0.914914	+	0.403649i	$0.132258\pi$
$282$	0	0
$283$	6261.18	1.31515	0.657577	−	0.753387i	$-0.271581\pi$
$283$	6261.18	1.31515	0.657577	+	0.753387i	$0.271581\pi$
$284$	−2763.24	−0.577352
$285$	0	0
$286$	0	0
$287$	1940.35	0.399076
$288$	0	0
$289$	−4884.93	−0.994286
$290$	3248.52	0.657791
$291$	0	0
$292$	−1012.02	−0.202823
$293$	8453.35	1.68549	0.842747	−	0.538309i	$-0.180937\pi$
$293$	8453.35	1.68549	0.842747	+	0.538309i	$0.180937\pi$
$294$	0	0
$295$	1574.23	0.310695
$296$	−284.776	−0.0559199
$297$	0	0
$298$	22.0200	0.00428049
$299$	−375.682	−0.0726630
$300$	0	0
$301$	1167.82	0.223628
$302$	−6563.75	−1.25067
$303$	0	0
$304$	−308.196	−0.0581456
$305$	5454.19	1.02395
$306$	0	0
$307$	−256.602	−0.0477037	−0.0238519	−	0.999716i	$-0.507593\pi$
$307$	−256.602	−0.0477037	−0.0238519	+	0.999716i	$0.507593\pi$
$308$	0	0
$309$	0	0
$310$	−3328.00	−0.609734
$311$	−6537.78	−1.19204	−0.596018	−	0.802971i	$-0.703251\pi$
$311$	−6537.78	−1.19204	−0.596018	+	0.802971i	$0.703251\pi$
$312$	0	0
$313$	6245.58	1.12786	0.563931	−	0.825822i	$-0.309288\pi$
$313$	6245.58	1.12786	0.563931	+	0.825822i	$0.309288\pi$
$314$	−663.602	−0.119265
$315$	0	0
$316$	−3075.73	−0.547542
$317$	3555.80	0.630011	0.315006	−	0.949090i	$-0.397994\pi$
$317$	3555.80	0.630011	0.315006	+	0.949090i	$0.397994\pi$
$318$	0	0
$319$	0	0
$320$	−508.666	−0.0888602
$321$	0	0
$322$	−2112.00	−0.365519
$323$	102.061	0.0175815
$324$	0	0
$325$	−365.328	−0.0623531
$326$	1868.92	0.317514
$327$	0	0
$328$	934.647	0.157339
$329$	−3309.97	−0.554664
$330$	0	0
$331$	584.607	0.0970783	0.0485391	−	0.998821i	$-0.484543\pi$
$331$	584.607	0.0970783	0.0485391	+	0.998821i	$0.484543\pi$
$332$	375.483	0.0620701
$333$	0	0
$334$	1387.94	0.227379
$335$	−4199.26	−0.684866
$336$	0	0
$337$	1023.79	0.165487	0.0827436	−	0.996571i	$-0.473632\pi$
$337$	1023.79	0.165487	0.0827436	+	0.996571i	$0.473632\pi$
$338$	4324.18	0.695871
$339$	0	0
$340$	168.448	0.0268688
$341$	0	0
$342$	0	0
$343$	6812.15	1.07237
$344$	562.529	0.0881672
$345$	0	0
$346$	−4880.42	−0.758303
$347$	2702.33	0.418065	0.209033	−	0.977909i	$-0.432969\pi$
$347$	2702.33	0.418065	0.209033	+	0.977909i	$0.432969\pi$
$348$	0	0
$349$	−8458.87	−1.29740	−0.648701	−	0.761044i	$-0.724687\pi$
$349$	−8458.87	−1.29740	−0.648701	+	0.761044i	$0.724687\pi$
$350$	−2053.79	−0.313657
$351$	0	0
$352$	0	0
$353$	9389.61	1.41575	0.707874	−	0.706339i	$-0.249655\pi$
$353$	9389.61	1.41575	0.707874	+	0.706339i	$0.249655\pi$
$354$	0	0
$355$	5490.49	0.820859
$356$	2291.05	0.341083
$357$	0	0
$358$	2183.50	0.322350
$359$	7582.95	1.11480	0.557399	−	0.830245i	$-0.311799\pi$
$359$	7582.95	1.11480	0.557399	+	0.830245i	$0.311799\pi$
$360$	0	0
$361$	−6487.97	−0.945905
$362$	−3346.20	−0.485835
$363$	0	0
$364$	−392.517	−0.0565206
$365$	2010.87	0.288366
$366$	0	0
$367$	3415.12	0.485744	0.242872	−	0.970058i	$-0.421911\pi$
$367$	3415.12	0.485744	0.242872	+	0.970058i	$0.421911\pi$
$368$	−1017.33	−0.144109
$369$	0	0
$370$	565.843	0.0795048
$371$	7157.91	1.00167
$372$	0	0
$373$	−12272.6	−1.70363	−0.851814	−	0.523844i	$-0.824498\pi$
$373$	−12272.6	−1.70363	−0.851814	+	0.523844i	$0.824498\pi$
$374$	0	0
$375$	0	0
$376$	−1594.38	−0.218681
$377$	1207.48	0.164956
$378$	0	0
$379$	8292.86	1.12395	0.561973	−	0.827156i	$-0.310043\pi$
$379$	8292.86	1.12395	0.561973	+	0.827156i	$0.310043\pi$
$380$	612.378	0.0826693
$381$	0	0
$382$	−9656.35	−1.29336
$383$	−2967.57	−0.395916	−0.197958	−	0.980211i	$-0.563431\pi$
$383$	−2967.57	−0.395916	−0.197958	+	0.980211i	$0.563431\pi$
$384$	0	0
$385$	0	0
$386$	4005.78	0.528210
$387$	0	0
$388$	−5980.20	−0.782471
$389$	14137.4	1.84266	0.921332	−	0.388778i	$-0.127103\pi$
$389$	14137.4	1.84266	0.921332	+	0.388778i	$0.127103\pi$
$390$	0	0
$391$	336.896	0.0435744
$392$	537.353	0.0692358
$393$	0	0
$394$	3603.53	0.460770
$395$	6111.40	0.778476
$396$	0	0
$397$	9793.78	1.23812	0.619062	−	0.785342i	$-0.287513\pi$
$397$	9793.78	1.23812	0.619062	+	0.785342i	$0.287513\pi$
$398$	6780.68	0.853981
$399$	0	0
$400$	−989.294	−0.123662
$401$	−7113.91	−0.885915	−0.442957	−	0.896543i	$-0.646071\pi$
$401$	−7113.91	−0.885915	−0.442957	+	0.896543i	$0.646071\pi$
$402$	0	0
$403$	−1237.02	−0.152905
$404$	4738.27	0.583509
$405$	0	0
$406$	6788.19	0.829784
$407$	0	0
$408$	0	0
$409$	5042.89	0.609670	0.304835	−	0.952405i	$-0.401399\pi$
$409$	5042.89	0.609670	0.304835	+	0.952405i	$0.401399\pi$
$410$	−1857.12	−0.223699
$411$	0	0
$412$	−3644.94	−0.435857
$413$	3289.55	0.391933
$414$	0	0
$415$	−746.075	−0.0882491
$416$	−189.072	−0.0222837
$417$	0	0
$418$	0	0
$419$	12390.0	1.44461	0.722303	−	0.691577i	$-0.243084\pi$
$419$	12390.0	1.44461	0.722303	+	0.691577i	$0.243084\pi$
$420$	0	0
$421$	7171.21	0.830174	0.415087	−	0.909782i	$-0.363751\pi$
$421$	7171.21	0.830174	0.415087	+	0.909782i	$0.363751\pi$
$422$	1053.51	0.121526
$423$	0	0
$424$	3447.90	0.394917
$425$	327.611	0.0373917
$426$	0	0
$427$	11397.2	1.29169
$428$	6790.85	0.766934
$429$	0	0
$430$	−1117.73	−0.125353
$431$	−10940.3	−1.22269	−0.611343	−	0.791366i	$-0.709370\pi$
$431$	−10940.3	−1.22269	−0.611343	+	0.791366i	$0.709370\pi$
$432$	0	0
$433$	9349.98	1.03772	0.518858	−	0.854860i	$-0.326357\pi$
$433$	9349.98	1.03772	0.518858	+	0.854860i	$0.326357\pi$
$434$	−6954.27	−0.769161
$435$	0	0
$436$	240.143	0.0263779
$437$	1224.76	0.134069
$438$	0	0
$439$	−6563.23	−0.713544	−0.356772	−	0.934191i	$-0.616123\pi$
$439$	−6563.23	−0.713544	−0.356772	+	0.934191i	$0.616123\pi$
$440$	0	0
$441$	0	0
$442$	62.6125	0.00673795
$443$	4818.40	0.516770	0.258385	−	0.966042i	$-0.416810\pi$
$443$	4818.40	0.516770	0.258385	+	0.966042i	$0.416810\pi$
$444$	0	0
$445$	−4552.26	−0.484939
$446$	−5480.22	−0.581829
$447$	0	0
$448$	−1062.92	−0.112095
$449$	−2562.94	−0.269382	−0.134691	−	0.990888i	$-0.543004\pi$
$449$	−2562.94	−0.269382	−0.134691	+	0.990888i	$0.543004\pi$
$450$	0	0
$451$	0	0
$452$	−1714.35	−0.178398
$453$	0	0
$454$	−10025.0	−1.03634
$455$	779.923	0.0803590
$456$	0	0
$457$	−7246.17	−0.741710	−0.370855	−	0.928691i	$-0.620935\pi$
$457$	−7246.17	−0.741710	−0.370855	+	0.928691i	$0.620935\pi$
$458$	−602.737	−0.0614935
$459$	0	0
$460$	2021.41	0.204889
$461$	6905.45	0.697655	0.348827	−	0.937187i	$-0.386580\pi$
$461$	6905.45	0.697655	0.348827	+	0.937187i	$0.386580\pi$
$462$	0	0
$463$	7396.32	0.742410	0.371205	−	0.928551i	$-0.378945\pi$
$463$	7396.32	0.742410	0.371205	+	0.928551i	$0.378945\pi$
$464$	3269.81	0.327149
$465$	0	0
$466$	2511.91	0.249704
$467$	−7175.32	−0.710994	−0.355497	−	0.934677i	$-0.615688\pi$
$467$	−7175.32	−0.710994	−0.355497	+	0.934677i	$0.615688\pi$
$468$	0	0
$469$	−8774.89	−0.863938
$470$	3168.00	0.310913
$471$	0	0
$472$	1584.55	0.154523
$473$	0	0
$474$	0	0
$475$	1191.00	0.115046
$476$	351.994	0.0338942
$477$	0	0
$478$	−9657.87	−0.924144
$479$	1377.95	0.131441	0.0657204	−	0.997838i	$-0.479065\pi$
$479$	1377.95	0.131441	0.0657204	+	0.997838i	$0.479065\pi$
$480$	0	0
$481$	210.325	0.0199376
$482$	4795.88	0.453208
$483$	0	0
$484$	0	0
$485$	11882.5	1.11249
$486$	0	0
$487$	12784.6	1.18958	0.594789	−	0.803882i	$-0.297236\pi$
$487$	12784.6	1.18958	0.594789	+	0.803882i	$0.297236\pi$
$488$	5489.94	0.509258
$489$	0	0
$490$	−1067.71	−0.0984370
$491$	11491.7	1.05624	0.528118	−	0.849171i	$-0.322898\pi$
$491$	11491.7	1.05624	0.528118	+	0.849171i	$0.322898\pi$
$492$	0	0
$493$	−1082.82	−0.0989205
$494$	227.622	0.0207312
$495$	0	0
$496$	−3349.81	−0.303248
$497$	11473.1	1.03549
$498$	0	0
$499$	3856.98	0.346017	0.173008	−	0.984920i	$-0.444651\pi$
$499$	3856.98	0.346017	0.173008	+	0.984920i	$0.444651\pi$
$500$	5939.65	0.531259
$501$	0	0
$502$	14728.4	1.30948
$503$	16184.6	1.43467	0.717333	−	0.696730i	$-0.245362\pi$
$503$	16184.6	1.43467	0.717333	+	0.696730i	$0.245362\pi$
$504$	0	0
$505$	−9414.82	−0.829612
$506$	0	0
$507$	0	0
$508$	1865.65	0.162943
$509$	−3207.57	−0.279319	−0.139659	−	0.990200i	$-0.544601\pi$
$509$	−3207.57	−0.279319	−0.139659	+	0.990200i	$0.544601\pi$
$510$	0	0
$511$	4201.96	0.363765
$512$	−512.000	−0.0441942
$513$	0	0
$514$	−12110.1	−1.03921
$515$	7242.40	0.619686
$516$	0	0
$517$	0	0
$518$	1182.40	0.100293
$519$	0	0
$520$	375.682	0.0316822
$521$	−485.365	−0.0408142	−0.0204071	−	0.999792i	$-0.506496\pi$
$521$	−485.365	−0.0408142	−0.0204071	+	0.999792i	$0.506496\pi$
$522$	0	0
$523$	19627.9	1.64105	0.820524	−	0.571612i	$-0.193682\pi$
$523$	19627.9	1.64105	0.820524	+	0.571612i	$0.193682\pi$
$524$	4540.66	0.378549
$525$	0	0
$526$	13364.3	1.10782
$527$	1109.31	0.0916934
$528$	0	0
$529$	−8124.17	−0.667722
$530$	−6850.90	−0.561479
$531$	0	0
$532$	1279.64	0.104285
$533$	−690.296	−0.0560976
$534$	0	0
$535$	−13493.2	−1.09040
$536$	−4226.79	−0.340614
$537$	0	0
$538$	11043.9	0.885013
$539$	0	0
$540$	0	0
$541$	21877.5	1.73861	0.869304	−	0.494278i	$-0.164568\pi$
$541$	21877.5	1.73861	0.869304	+	0.494278i	$0.164568\pi$
$542$	1785.29	0.141485
$543$	0	0
$544$	169.552	0.0133630
$545$	−477.159	−0.0375032
$546$	0	0
$547$	4580.08	0.358008	0.179004	−	0.983848i	$-0.442713\pi$
$547$	4580.08	0.358008	0.179004	+	0.983848i	$0.442713\pi$
$548$	−9164.90	−0.714425
$549$	0	0
$550$	0	0
$551$	−3936.49	−0.304356
$552$	0	0
$553$	12770.6	0.982024
$554$	−8556.00	−0.656155
$555$	0	0
$556$	−96.8799	−0.00738961
$557$	2785.70	0.211910	0.105955	−	0.994371i	$-0.466210\pi$
$557$	2785.70	0.211910	0.105955	+	0.994371i	$0.466210\pi$
$558$	0	0
$559$	−415.463	−0.0314351
$560$	2112.00	0.159372
$561$	0	0
$562$	−17238.5	−1.29388
$563$	4221.47	0.316010	0.158005	−	0.987438i	$-0.449494\pi$
$563$	4221.47	0.316010	0.158005	+	0.987438i	$0.449494\pi$
$564$	0	0
$565$	3406.36	0.253640
$566$	−12522.4	−0.929955
$567$	0	0
$568$	5526.48	0.408250
$569$	−11087.4	−0.816885	−0.408443	−	0.912784i	$-0.633928\pi$
$569$	−11087.4	−0.816885	−0.408443	+	0.912784i	$0.633928\pi$
$570$	0	0
$571$	−9019.42	−0.661035	−0.330518	−	0.943800i	$-0.607223\pi$
$571$	−9019.42	−0.661035	−0.330518	+	0.943800i	$0.607223\pi$
$572$	0	0
$573$	0	0
$574$	−3880.69	−0.282190
$575$	3931.40	0.285132
$576$	0	0
$577$	9853.90	0.710959	0.355479	−	0.934684i	$-0.384318\pi$
$577$	9853.90	0.710959	0.355479	+	0.934684i	$0.384318\pi$
$578$	9769.85	0.703066
$579$	0	0
$580$	−6497.03	−0.465129
$581$	−1559.02	−0.111324
$582$	0	0
$583$	0	0
$584$	2024.05	0.143417
$585$	0	0
$586$	−16906.7	−1.19182
$587$	−8320.74	−0.585066	−0.292533	−	0.956255i	$-0.594498\pi$
$587$	−8320.74	−0.585066	−0.292533	+	0.956255i	$0.594498\pi$
$588$	0	0
$589$	4032.81	0.282120
$590$	−3148.46	−0.219695
$591$	0	0
$592$	569.552	0.0395413
$593$	−15743.6	−1.09024	−0.545119	−	0.838359i	$-0.683516\pi$
$593$	−15743.6	−1.09024	−0.545119	+	0.838359i	$0.683516\pi$
$594$	0	0
$595$	−699.403	−0.0481895
$596$	−44.0400	−0.00302676
$597$	0	0
$598$	751.363	0.0513805
$599$	8359.27	0.570201	0.285101	−	0.958498i	$-0.407973\pi$
$599$	8359.27	0.570201	0.285101	+	0.958498i	$0.407973\pi$
$600$	0	0
$601$	−1458.09	−0.0989626	−0.0494813	−	0.998775i	$-0.515757\pi$
$601$	−1458.09	−0.0989626	−0.0494813	+	0.998775i	$0.515757\pi$
$602$	−2335.64	−0.158129
$603$	0	0
$604$	13127.5	0.884356
$605$	0	0
$606$	0	0
$607$	1590.27	0.106338	0.0531689	−	0.998586i	$-0.483068\pi$
$607$	1590.27	0.106338	0.0531689	+	0.998586i	$0.483068\pi$
$608$	616.392	0.0411151
$609$	0	0
$610$	−10908.4	−0.724045
$611$	1177.55	0.0779684
$612$	0	0
$613$	7543.93	0.497058	0.248529	−	0.968624i	$-0.420053\pi$
$613$	7543.93	0.497058	0.248529	+	0.968624i	$0.420053\pi$
$614$	513.204	0.0337316
$615$	0	0
$616$	0	0
$617$	−10888.9	−0.710484	−0.355242	−	0.934774i	$-0.615602\pi$
$617$	−10888.9	−0.710484	−0.355242	+	0.934774i	$0.615602\pi$
$618$	0	0
$619$	22803.5	1.48069	0.740347	−	0.672225i	$-0.234661\pi$
$619$	22803.5	1.48069	0.740347	+	0.672225i	$0.234661\pi$
$620$	6655.99	0.431147
$621$	0	0
$622$	13075.6	0.842897
$623$	−9512.54	−0.611736
$624$	0	0
$625$	−4073.09	−0.260678
$626$	−12491.2	−0.797519
$627$	0	0
$628$	1327.20	0.0843331
$629$	−188.611	−0.0119562
$630$	0	0
$631$	1023.32	0.0645608	0.0322804	−	0.999479i	$-0.489723\pi$
$631$	1023.32	0.0645608	0.0322804	+	0.999479i	$0.489723\pi$
$632$	6151.46	0.387171
$633$	0	0
$634$	−7111.60	−0.445485
$635$	−3707.00	−0.231666
$636$	0	0
$637$	−396.869	−0.0246853
$638$	0	0
$639$	0	0
$640$	1017.33	0.0628337
$641$	−87.7431	−0.00540662	−0.00270331	−	0.999996i	$-0.500860\pi$
$641$	−87.7431	−0.00540662	−0.00270331	+	0.999996i	$0.500860\pi$
$642$	0	0
$643$	−4632.67	−0.284128	−0.142064	−	0.989857i	$-0.545374\pi$
$643$	−4632.67	−0.284128	−0.142064	+	0.989857i	$0.545374\pi$
$644$	4224.00	0.258461
$645$	0	0
$646$	−204.122	−0.0124320
$647$	−22710.9	−1.37999	−0.689997	−	0.723812i	$-0.742388\pi$
$647$	−22710.9	−1.37999	−0.689997	+	0.723812i	$0.742388\pi$
$648$	0	0
$649$	0	0
$650$	730.656	0.0440903
$651$	0	0
$652$	−3737.83	−0.224516
$653$	−3238.60	−0.194083	−0.0970414	−	0.995280i	$-0.530938\pi$
$653$	−3238.60	−0.194083	−0.0970414	+	0.995280i	$0.530938\pi$
$654$	0	0
$655$	−9022.17	−0.538207
$656$	−1869.29	−0.111256
$657$	0	0
$658$	6619.94	0.392207
$659$	3727.48	0.220337	0.110169	−	0.993913i	$-0.464861\pi$
$659$	3727.48	0.220337	0.110169	+	0.993913i	$0.464861\pi$
$660$	0	0
$661$	3685.68	0.216878	0.108439	−	0.994103i	$-0.465415\pi$
$661$	3685.68	0.216878	0.108439	+	0.994103i	$0.465415\pi$
$662$	−1169.21	−0.0686447
$663$	0	0
$664$	−750.965	−0.0438902
$665$	−2542.62	−0.148268
$666$	0	0
$667$	−12994.1	−0.754321
$668$	−2775.88	−0.160782
$669$	0	0
$670$	8398.52	0.484273
$671$	0	0
$672$	0	0
$673$	−5578.51	−0.319518	−0.159759	−	0.987156i	$-0.551072\pi$
$673$	−5578.51	−0.319518	−0.159759	+	0.987156i	$0.551072\pi$
$674$	−2047.57	−0.117017
$675$	0	0
$676$	−8648.36	−0.492055
$677$	4016.30	0.228005	0.114002	−	0.993480i	$-0.463633\pi$
$677$	4016.30	0.228005	0.114002	+	0.993480i	$0.463633\pi$
$678$	0	0
$679$	24830.0	1.40337
$680$	−336.896	−0.0189991
$681$	0	0
$682$	0	0
$683$	−23381.7	−1.30992	−0.654961	−	0.755663i	$-0.727315\pi$
$683$	−23381.7	−1.30992	−0.654961	+	0.755663i	$0.727315\pi$
$684$	0	0
$685$	18210.4	1.01574
$686$	−13624.3	−0.758278
$687$	0	0
$688$	−1125.06	−0.0623436
$689$	−2546.49	−0.140804
$690$	0	0
$691$	7943.27	0.437303	0.218652	−	0.975803i	$-0.429834\pi$
$691$	7943.27	0.437303	0.218652	+	0.975803i	$0.429834\pi$
$692$	9760.84	0.536201
$693$	0	0
$694$	−5404.66	−0.295617
$695$	192.498	0.0105063
$696$	0	0
$697$	619.030	0.0336405
$698$	16917.7	0.917401
$699$	0	0
$700$	4107.59	0.221789
$701$	−24801.1	−1.33627	−0.668135	−	0.744040i	$-0.732907\pi$
$701$	−24801.1	−1.33627	−0.668135	+	0.744040i	$0.732907\pi$
$702$	0	0
$703$	−685.679	−0.0367864
$704$	0	0
$705$	0	0
$706$	−18779.2	−1.00108
$707$	−19673.5	−1.04653
$708$	0	0
$709$	18956.5	1.00413	0.502065	−	0.864830i	$-0.332574\pi$
$709$	18956.5	1.00413	0.502065	+	0.864830i	$0.332574\pi$
$710$	−10981.0	−0.580435
$711$	0	0
$712$	−4582.10	−0.241182
$713$	13312.0	0.699211
$714$	0	0
$715$	0	0
$716$	−4367.00	−0.227936
$717$	0	0
$718$	−15165.9	−0.788281
$719$	−8649.03	−0.448615	−0.224308	−	0.974518i	$-0.572012\pi$
$719$	−8649.03	−0.448615	−0.224308	+	0.974518i	$0.572012\pi$
$720$	0	0
$721$	15133.9	0.781715
$722$	12975.9	0.668856
$723$	0	0
$724$	6692.40	0.343537
$725$	−12636.0	−0.647293
$726$	0	0
$727$	8176.58	0.417129	0.208564	−	0.978009i	$-0.433121\pi$
$727$	8176.58	0.417129	0.208564	+	0.978009i	$0.433121\pi$
$728$	785.035	0.0399661
$729$	0	0
$730$	−4021.73	−0.203905
$731$	372.571	0.0188509
$732$	0	0
$733$	27043.8	1.36274	0.681368	−	0.731941i	$-0.261385\pi$
$733$	27043.8	1.36274	0.681368	+	0.731941i	$0.261385\pi$
$734$	−6830.25	−0.343473
$735$	0	0
$736$	2034.66	0.101900
$737$	0	0
$738$	0	0
$739$	14466.9	0.720128	0.360064	−	0.932928i	$-0.382755\pi$
$739$	14466.9	0.720128	0.360064	+	0.932928i	$0.382755\pi$
$740$	−1131.69	−0.0562184
$741$	0	0
$742$	−14315.8	−0.708289
$743$	−25511.2	−1.25964	−0.629821	−	0.776740i	$-0.716872\pi$
$743$	−25511.2	−1.25964	−0.629821	+	0.776740i	$0.716872\pi$
$744$	0	0
$745$	87.5064	0.00430334
$746$	24545.3	1.20465
$747$	0	0
$748$	0	0
$749$	−28195.9	−1.37551
$750$	0	0
$751$	−8941.15	−0.434443	−0.217222	−	0.976122i	$-0.569699\pi$
$751$	−8941.15	−0.434443	−0.217222	+	0.976122i	$0.569699\pi$
$752$	3188.77	0.154631
$753$	0	0
$754$	−2414.96	−0.116642
$755$	−26084.0	−1.25734
$756$	0	0
$757$	−11845.3	−0.568726	−0.284363	−	0.958717i	$-0.591782\pi$
$757$	−11845.3	−0.568726	−0.284363	+	0.958717i	$0.591782\pi$
$758$	−16585.7	−0.794750
$759$	0	0
$760$	−1224.76	−0.0584560
$761$	2065.41	0.0983850	0.0491925	−	0.998789i	$-0.484335\pi$
$761$	2065.41	0.0983850	0.0491925	+	0.998789i	$0.484335\pi$
$762$	0	0
$763$	−997.084	−0.0473091
$764$	19312.7	0.914541
$765$	0	0
$766$	5935.14	0.279955
$767$	−1170.29	−0.0550934
$768$	0	0
$769$	−38285.1	−1.79531	−0.897656	−	0.440696i	$-0.854732\pi$
$769$	−38285.1	−1.79531	−0.897656	+	0.440696i	$0.854732\pi$
$770$	0	0
$771$	0	0
$772$	−8011.56	−0.373501
$773$	26727.2	1.24361	0.621805	−	0.783172i	$-0.286400\pi$
$773$	26727.2	1.24361	0.621805	+	0.783172i	$0.286400\pi$
$774$	0	0
$775$	12945.1	0.600002
$776$	11960.4	0.553290
$777$	0	0
$778$	−28274.9	−1.30296
$779$	2250.42	0.103504
$780$	0	0
$781$	0	0
$782$	−673.793	−0.0308117
$783$	0	0
$784$	−1074.71	−0.0489571
$785$	−2637.12	−0.119902
$786$	0	0
$787$	−27594.9	−1.24987	−0.624937	−	0.780675i	$-0.714876\pi$
$787$	−27594.9	−1.24987	−0.624937	+	0.780675i	$0.714876\pi$
$788$	−7207.06	−0.325814
$789$	0	0
$790$	−12222.8	−0.550466
$791$	7118.03	0.319960
$792$	0	0
$793$	−4054.67	−0.181571
$794$	−19587.6	−0.875486
$795$	0	0
$796$	−13561.4	−0.603856
$797$	23828.3	1.05903	0.529513	−	0.848302i	$-0.322375\pi$
$797$	23828.3	1.05903	0.529513	+	0.848302i	$0.322375\pi$
$798$	0	0
$799$	−1055.98	−0.0467559
$800$	1978.59	0.0874420
$801$	0	0
$802$	14227.8	0.626436
$803$	0	0
$804$	0	0
$805$	−8392.98	−0.367471
$806$	2474.05	0.108120
$807$	0	0
$808$	−9476.54	−0.412603
$809$	−10637.6	−0.462295	−0.231148	−	0.972919i	$-0.574248\pi$
$809$	−10637.6	−0.462295	−0.231148	+	0.972919i	$0.574248\pi$
$810$	0	0
$811$	20867.8	0.903536	0.451768	−	0.892136i	$-0.350794\pi$
$811$	20867.8	0.903536	0.451768	+	0.892136i	$0.350794\pi$
$812$	−13576.4	−0.586746
$813$	0	0
$814$	0	0
$815$	7426.98	0.319209
$816$	0	0
$817$	1354.45	0.0580001
$818$	−10085.8	−0.431102
$819$	0	0
$820$	3714.24	0.158179
$821$	−28971.3	−1.23155	−0.615777	−	0.787920i	$-0.711158\pi$
$821$	−28971.3	−1.23155	−0.615777	+	0.787920i	$0.711158\pi$
$822$	0	0
$823$	−1814.90	−0.0768691	−0.0384345	−	0.999261i	$-0.512237\pi$
$823$	−1814.90	−0.0768691	−0.0384345	+	0.999261i	$0.512237\pi$
$824$	7289.87	0.308197
$825$	0	0
$826$	−6579.10	−0.277138
$827$	−21724.2	−0.913452	−0.456726	−	0.889608i	$-0.650978\pi$
$827$	−21724.2	−0.913452	−0.456726	+	0.889608i	$0.650978\pi$
$828$	0	0
$829$	−44016.6	−1.84410	−0.922050	−	0.387070i	$-0.873487\pi$
$829$	−44016.6	−1.84410	−0.922050	+	0.387070i	$0.873487\pi$
$830$	1492.15	0.0624015
$831$	0	0
$832$	378.144	0.0157570
$833$	355.896	0.0148032
$834$	0	0
$835$	5515.61	0.228593
$836$	0	0
$837$	0	0
$838$	−24779.9	−1.02149
$839$	3960.60	0.162974	0.0814870	−	0.996674i	$-0.474033\pi$
$839$	3960.60	0.162974	0.0814870	+	0.996674i	$0.474033\pi$
$840$	0	0
$841$	17375.3	0.712424
$842$	−14342.4	−0.587022
$843$	0	0
$844$	−2107.02	−0.0859322
$845$	17184.1	0.699586
$846$	0	0
$847$	0	0
$848$	−6895.80	−0.279249
$849$	0	0
$850$	−655.223	−0.0264400
$851$	−2263.37	−0.0911721
$852$	0	0
$853$	−18555.8	−0.744828	−0.372414	−	0.928067i	$-0.621470\pi$
$853$	−18555.8	−0.744828	−0.372414	+	0.928067i	$0.621470\pi$
$854$	−22794.4	−0.913361
$855$	0	0
$856$	−13581.7	−0.542305
$857$	38893.3	1.55026	0.775128	−	0.631805i	$-0.217686\pi$
$857$	38893.3	1.55026	0.775128	+	0.631805i	$0.217686\pi$
$858$	0	0
$859$	35072.8	1.39310	0.696548	−	0.717511i	$-0.254719\pi$
$859$	35072.8	1.39310	0.696548	+	0.717511i	$0.254719\pi$
$860$	2235.46	0.0886379
$861$	0	0
$862$	21880.7	0.864569
$863$	38714.6	1.52707	0.763534	−	0.645768i	$-0.223463\pi$
$863$	38714.6	1.52707	0.763534	+	0.645768i	$0.223463\pi$
$864$	0	0
$865$	−19394.5	−0.762352
$866$	−18700.0	−0.733776
$867$	0	0
$868$	13908.5	0.543879
$869$	0	0
$870$	0	0
$871$	3121.75	0.121442
$872$	−480.287	−0.0186520
$873$	0	0
$874$	−2449.51	−0.0948009
$875$	−24661.7	−0.952819
$876$	0	0
$877$	22951.6	0.883718	0.441859	−	0.897085i	$-0.354319\pi$
$877$	22951.6	0.883718	0.441859	+	0.897085i	$0.354319\pi$
$878$	13126.5	0.504552
$879$	0	0
$880$	0	0
$881$	−25301.1	−0.967555	−0.483777	−	0.875191i	$-0.660736\pi$
$881$	−25301.1	−0.967555	−0.483777	+	0.875191i	$0.660736\pi$
$882$	0	0
$883$	−48590.6	−1.85187	−0.925936	−	0.377681i	$-0.876722\pi$
$883$	−48590.6	−1.85187	−0.925936	+	0.377681i	$0.876722\pi$
$884$	−125.225	−0.00476445
$885$	0	0
$886$	−9636.80	−0.365411
$887$	−26121.4	−0.988806	−0.494403	−	0.869233i	$-0.664613\pi$
$887$	−26121.4	−0.988806	−0.494403	+	0.869233i	$0.664613\pi$
$888$	0	0
$889$	−7746.26	−0.292240
$890$	9104.53	0.342904
$891$	0	0
$892$	10960.4	0.411415
$893$	−3838.93	−0.143857
$894$	0	0
$895$	8677.12	0.324072
$896$	2125.84	0.0792628
$897$	0	0
$898$	5125.88	0.190482
$899$	−42786.1	−1.58732
$900$	0	0
$901$	2283.59	0.0844367
$902$	0	0
$903$	0	0
$904$	3428.69	0.126147
$905$	−13297.6	−0.488429
$906$	0	0
$907$	34243.7	1.25363	0.626815	−	0.779168i	$-0.284358\pi$
$907$	34243.7	1.25363	0.626815	+	0.779168i	$0.284358\pi$
$908$	20050.1	0.732803
$909$	0	0
$910$	−1559.85	−0.0568224
$911$	32331.1	1.17583	0.587913	−	0.808924i	$-0.299950\pi$
$911$	32331.1	1.17583	0.587913	+	0.808924i	$0.299950\pi$
$912$	0	0
$913$	0	0
$914$	14492.3	0.524468
$915$	0	0
$916$	1205.47	0.0434825
$917$	−18853.0	−0.678932
$918$	0	0
$919$	−22567.7	−0.810054	−0.405027	−	0.914305i	$-0.632738\pi$
$919$	−22567.7	−0.810054	−0.405027	+	0.914305i	$0.632738\pi$
$920$	−4042.83	−0.144878
$921$	0	0
$922$	−13810.9	−0.493316
$923$	−4081.65	−0.145557
$924$	0	0
$925$	−2200.99	−0.0782359
$926$	−14792.6	−0.524963
$927$	0	0
$928$	−6539.62	−0.231329
$929$	26143.7	0.923302	0.461651	−	0.887062i	$-0.347257\pi$
$929$	26143.7	0.923302	0.461651	+	0.887062i	$0.347257\pi$
$930$	0	0
$931$	1293.83	0.0455462
$932$	−5023.82	−0.176567
$933$	0	0
$934$	14350.6	0.502749
$935$	0	0
$936$	0	0
$937$	941.653	0.0328308	0.0164154	−	0.999865i	$-0.494775\pi$
$937$	941.653	0.0328308	0.0164154	+	0.999865i	$0.494775\pi$
$938$	17549.8	0.610896
$939$	0	0
$940$	−6336.00	−0.219848
$941$	−25791.3	−0.893490	−0.446745	−	0.894661i	$-0.647417\pi$
$941$	−25791.3	−0.893490	−0.446745	+	0.894661i	$0.647417\pi$
$942$	0	0
$943$	7428.48	0.256527
$944$	−3169.10	−0.109264
$945$	0	0
$946$	0	0
$947$	−21152.9	−0.725848	−0.362924	−	0.931819i	$-0.618222\pi$
$947$	−21152.9	−0.725848	−0.362924	+	0.931819i	$0.618222\pi$
$948$	0	0
$949$	−1494.89	−0.0511339
$950$	−2382.00	−0.0813499
$951$	0	0
$952$	−703.988	−0.0239668
$953$	−34854.4	−1.18473	−0.592363	−	0.805671i	$-0.701805\pi$
$953$	−34854.4	−1.18473	−0.592363	+	0.805671i	$0.701805\pi$
$954$	0	0
$955$	−38373.9	−1.30026
$956$	19315.7	0.653468
$957$	0	0
$958$	−2755.90	−0.0929427
$959$	38053.0	1.28133
$960$	0	0
$961$	14041.9	0.471349
$962$	−420.650	−0.0140980
$963$	0	0
$964$	−9591.76	−0.320467
$965$	15918.8	0.531030
$966$	0	0
$967$	400.381	0.0133148	0.00665739	−	0.999978i	$-0.497881\pi$
$967$	400.381	0.0133148	0.00665739	+	0.999978i	$0.497881\pi$
$968$	0	0
$969$	0	0
$970$	−23765.0	−0.786648
$971$	−733.069	−0.0242279	−0.0121140	−	0.999927i	$-0.503856\pi$
$971$	−733.069	−0.0242279	−0.0121140	+	0.999927i	$0.503856\pi$
$972$	0	0
$973$	402.249	0.0132533
$974$	−25569.2	−0.841159
$975$	0	0
$976$	−10979.9	−0.360100
$977$	12730.4	0.416870	0.208435	−	0.978036i	$-0.433163\pi$
$977$	12730.4	0.416870	0.208435	+	0.978036i	$0.433163\pi$
$978$	0	0
$979$	0	0
$980$	2135.42	0.0696054
$981$	0	0
$982$	−22983.3	−0.746871
$983$	−46266.1	−1.50118	−0.750590	−	0.660768i	$-0.770231\pi$
$983$	−46266.1	−1.50118	−0.750590	+	0.660768i	$0.770231\pi$
$984$	0	0
$985$	14320.3	0.463230
$986$	2165.64	0.0699473
$987$	0	0
$988$	−455.244	−0.0146592
$989$	4470.92	0.143748
$990$	0	0
$991$	17591.6	0.563890	0.281945	−	0.959431i	$-0.409020\pi$
$991$	17591.6	0.563890	0.281945	+	0.959431i	$0.409020\pi$
$992$	6699.62	0.214429
$993$	0	0
$994$	−22946.1	−0.732201
$995$	26946.1	0.858541
$996$	0	0
$997$	37402.4	1.18811	0.594055	−	0.804424i	$-0.297526\pi$
$997$	37402.4	1.18811	0.594055	+	0.804424i	$0.297526\pi$
$998$	−7713.97	−0.244671
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2178.4.a.bv.1.2	✓	4
3.2	odd	2		2178.4.a.ca.1.3	yes	4
11.10	odd	2		2178.4.a.ca.1.2	yes	4
33.32	even	2	inner	2178.4.a.bv.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2178.4.a.bv.1.2	✓	4	1.1	even	1	trivial
2178.4.a.bv.1.3	yes	4	33.32	even	2	inner
2178.4.a.ca.1.2	yes	4	11.10	odd	2
2178.4.a.ca.1.3	yes	4	3.2	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:	\( N \)	\(=\)	\( 2178 = 2 \cdot 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2178.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +4.00000 q^{4} -7.94790 q^{5} -16.6082 q^{7} -8.00000 q^{8} +O(q^{10})\)	\(q-2.00000 q^{2} +4.00000 q^{4} -7.94790 q^{5} -16.6082 q^{7} -8.00000 q^{8} +15.8958 q^{10} +5.90850 q^{13} +33.2163 q^{14} +16.0000 q^{16} -5.29851 q^{17} -19.2622 q^{19} -31.7916 q^{20} -63.5832 q^{23} -61.8309 q^{25} -11.8170 q^{26} -66.4326 q^{28} +204.363 q^{29} -209.363 q^{31} -32.0000 q^{32} +10.5970 q^{34} +132.000 q^{35} +35.5970 q^{37} +38.5245 q^{38} +63.5832 q^{40} -116.831 q^{41} -70.3161 q^{43} +127.166 q^{46} +199.298 q^{47} -67.1691 q^{49} +123.662 q^{50} +23.6340 q^{52} -430.988 q^{53} +132.865 q^{56} -408.726 q^{58} -198.068 q^{59} -686.243 q^{61} +418.726 q^{62} +64.0000 q^{64} -46.9602 q^{65} +528.348 q^{67} -21.1940 q^{68} -264.000 q^{70} -690.810 q^{71} -253.006 q^{73} -71.1940 q^{74} -77.0490 q^{76} -768.933 q^{79} -127.166 q^{80} +233.662 q^{82} +93.8706 q^{83} +42.1120 q^{85} +140.632 q^{86} +572.763 q^{89} -98.1294 q^{91} -254.333 q^{92} -398.596 q^{94} +153.094 q^{95} -1495.05 q^{97} +134.338 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{2} + 16 q^{4} - 32 q^{8}+O(q^{10}) \) 4 * q - 8 * q^2 + 16 * q^4 - 32 * q^8	\( 4 q - 8 q^{2} + 16 q^{4} - 32 q^{8} + 64 q^{16} + 234 q^{17} + 178 q^{25} + 222 q^{29} - 242 q^{31} - 128 q^{32} - 468 q^{34} + 528 q^{35} - 368 q^{37} - 42 q^{41} - 694 q^{49} - 356 q^{50} - 444 q^{58} + 484 q^{62} + 256 q^{64} + 918 q^{65} - 1034 q^{67} + 936 q^{68} - 1056 q^{70} + 736 q^{74} + 84 q^{82} + 1056 q^{83} + 288 q^{91} + 5376 q^{95} - 3088 q^{97} + 1388 q^{98}+O(q^{100}) \) 4 * q - 8 * q^2 + 16 * q^4 - 32 * q^8 + 64 * q^16 + 234 * q^17 + 178 * q^25 + 222 * q^29 - 242 * q^31 - 128 * q^32 - 468 * q^34 + 528 * q^35 - 368 * q^37 - 42 * q^41 - 694 * q^49 - 356 * q^50 - 444 * q^58 + 484 * q^62 + 256 * q^64 + 918 * q^65 - 1034 * q^67 + 936 * q^68 - 1056 * q^70 + 736 * q^74 + 84 * q^82 + 1056 * q^83 + 288 * q^91 + 5376 * q^95 - 3088 * q^97 + 1388 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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