LMFDB - Embedded newform 2178.4.a.bx.1.4

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:	$1$
Dimension:	$4$
Coefficient field:	4.4.5157648.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 123x^{2} - 132x + 2148 $ x^4 - 123x^2 - 132x + 2148
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 726)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$10.8086$ 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)

$f(q)$	$=$	$q-2.00000 q^{2} +4.00000 q^{4} +22.1852 q^{5} +2.71575 q^{7} -8.00000 q^{8} +O(q^{10})$	$q-2.00000 q^{2} +4.00000 q^{4} +22.1852 q^{5} +2.71575 q^{7} -8.00000 q^{8} -44.3703 q^{10} -84.6526 q^{13} -5.43151 q^{14} +16.0000 q^{16} -53.8335 q^{17} +61.9418 q^{19} +88.7407 q^{20} -192.185 q^{23} +367.182 q^{25} +169.305 q^{26} +10.8630 q^{28} -121.536 q^{29} +174.074 q^{31} -32.0000 q^{32} +107.667 q^{34} +60.2495 q^{35} -44.4988 q^{37} -123.884 q^{38} -177.481 q^{40} -155.002 q^{41} +292.394 q^{43} +384.369 q^{46} +268.964 q^{47} -335.625 q^{49} -734.364 q^{50} -338.610 q^{52} +241.897 q^{53} -21.7260 q^{56} +243.071 q^{58} -287.759 q^{59} +57.0430 q^{61} -348.148 q^{62} +64.0000 q^{64} -1878.03 q^{65} -497.824 q^{67} -215.334 q^{68} -120.499 q^{70} +32.5150 q^{71} +32.7188 q^{73} +88.9977 q^{74} +247.767 q^{76} -463.884 q^{79} +354.963 q^{80} +310.004 q^{82} -1311.94 q^{83} -1194.31 q^{85} -584.789 q^{86} -1141.01 q^{89} -229.896 q^{91} -768.739 q^{92} -537.928 q^{94} +1374.19 q^{95} +487.402 q^{97} +671.249 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{2} + 16 q^{4} + 6 q^{5} + 12 q^{7} - 32 q^{8}+O(q^{10}) $ 4 * q - 8 * q^2 + 16 * q^4 + 6 * q^5 + 12 * q^7 - 32 * q^8	$ 4 q - 8 q^{2} + 16 q^{4} + 6 q^{5} + 12 q^{7} - 32 q^{8} - 12 q^{10} + 12 q^{13} - 24 q^{14} + 64 q^{16} - 108 q^{17} + 192 q^{19} + 24 q^{20} - 156 q^{23} + 286 q^{25} - 24 q^{26} + 48 q^{28} - 408 q^{29} - 26 q^{31} - 128 q^{32} + 216 q^{34} - 588 q^{35} - 224 q^{37} - 384 q^{38} - 48 q^{40} - 348 q^{41} + 540 q^{43} + 312 q^{46} - 132 q^{47} + 674 q^{49} - 572 q^{50} + 48 q^{52} + 1470 q^{53} - 96 q^{56} + 816 q^{58} + 684 q^{59} + 1320 q^{61} + 52 q^{62} + 256 q^{64} - 2556 q^{65} - 530 q^{67} - 432 q^{68} + 1176 q^{70} - 936 q^{71} - 2352 q^{73} + 448 q^{74} + 768 q^{76} - 192 q^{79} + 96 q^{80} + 696 q^{82} - 3192 q^{83} - 2340 q^{85} - 1080 q^{86} - 726 q^{89} + 360 q^{91} - 624 q^{92} + 264 q^{94} + 2064 q^{95} - 100 q^{97} - 1348 q^{98}+O(q^{100}) $ 4 * q - 8 * q^2 + 16 * q^4 + 6 * q^5 + 12 * q^7 - 32 * q^8 - 12 * q^10 + 12 * q^13 - 24 * q^14 + 64 * q^16 - 108 * q^17 + 192 * q^19 + 24 * q^20 - 156 * q^23 + 286 * q^25 - 24 * q^26 + 48 * q^28 - 408 * q^29 - 26 * q^31 - 128 * q^32 + 216 * q^34 - 588 * q^35 - 224 * q^37 - 384 * q^38 - 48 * q^40 - 348 * q^41 + 540 * q^43 + 312 * q^46 - 132 * q^47 + 674 * q^49 - 572 * q^50 + 48 * q^52 + 1470 * q^53 - 96 * q^56 + 816 * q^58 + 684 * q^59 + 1320 * q^61 + 52 * q^62 + 256 * q^64 - 2556 * q^65 - 530 * q^67 - 432 * q^68 + 1176 * q^70 - 936 * q^71 - 2352 * q^73 + 448 * q^74 + 768 * q^76 - 192 * q^79 + 96 * q^80 + 696 * q^82 - 3192 * q^83 - 2340 * q^85 - 1080 * q^86 - 726 * q^89 + 360 * q^91 - 624 * q^92 + 264 * q^94 + 2064 * q^95 - 100 * q^97 - 1348 * 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
$5$	22.1852	1.98430	0.992151	−	0.125044i	$-0.0399073\pi$
$5$	22.1852	1.98430	0.992151	+	0.125044i	$0.0399073\pi$
$6$	0	0
$7$	2.71575	0.146637	0.0733185	−	0.997309i	$-0.476641\pi$
$7$	2.71575	0.146637	0.0733185	+	0.997309i	$0.476641\pi$
$8$	−8.00000	−0.353553
$9$	0	0
$10$	−44.3703	−1.40311
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−84.6526	−1.80603	−0.903016	−	0.429608i	$-0.858652\pi$
$13$	−84.6526	−1.80603	−0.903016	+	0.429608i	$0.858652\pi$
$14$	−5.43151	−0.103688
$15$	0	0
$16$	16.0000	0.250000
$17$	−53.8335	−0.768032	−0.384016	−	0.923326i	$-0.625459\pi$
$17$	−53.8335	−0.768032	−0.384016	+	0.923326i	$0.625459\pi$
$18$	0	0
$19$	61.9418	0.747917	0.373958	−	0.927445i	$-0.378000\pi$
$19$	61.9418	0.747917	0.373958	+	0.927445i	$0.378000\pi$
$20$	88.7407	0.992151
$21$	0	0
$22$	0	0
$23$	−192.185	−1.74232	−0.871158	−	0.491002i	$-0.836631\pi$
$23$	−192.185	−1.74232	−0.871158	+	0.491002i	$0.836631\pi$
$24$	0	0
$25$	367.182	2.93746
$26$	169.305	1.27706
$27$	0	0
$28$	10.8630	0.0733185
$29$	−121.536	−0.778228	−0.389114	−	0.921190i	$-0.627219\pi$
$29$	−121.536	−0.778228	−0.389114	+	0.921190i	$0.627219\pi$
$30$	0	0
$31$	174.074	1.00854	0.504269	−	0.863547i	$-0.331762\pi$
$31$	174.074	1.00854	0.504269	+	0.863547i	$0.331762\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	107.667	0.543081
$35$	60.2495	0.290972
$36$	0	0
$37$	−44.4988	−0.197718	−0.0988590	−	0.995101i	$-0.531519\pi$
$37$	−44.4988	−0.197718	−0.0988590	+	0.995101i	$0.531519\pi$
$38$	−123.884	−0.528857
$39$	0	0
$40$	−177.481	−0.701557
$41$	−155.002	−0.590420	−0.295210	−	0.955432i	$-0.595390\pi$
$41$	−155.002	−0.590420	−0.295210	+	0.955432i	$0.595390\pi$
$42$	0	0
$43$	292.394	1.03697	0.518485	−	0.855087i	$-0.326496\pi$
$43$	292.394	1.03697	0.518485	+	0.855087i	$0.326496\pi$
$44$	0	0
$45$	0	0
$46$	384.369	1.23200
$47$	268.964	0.834733	0.417367	−	0.908738i	$-0.362953\pi$
$47$	268.964	0.834733	0.417367	+	0.908738i	$0.362953\pi$
$48$	0	0
$49$	−335.625	−0.978498
$50$	−734.364	−2.07709
$51$	0	0
$52$	−338.610	−0.903016
$53$	241.897	0.626928	0.313464	−	0.949600i	$-0.398511\pi$
$53$	241.897	0.626928	0.313464	+	0.949600i	$0.398511\pi$
$54$	0	0
$55$	0	0
$56$	−21.7260	−0.0518440
$57$	0	0
$58$	243.071	0.550290
$59$	−287.759	−0.634966	−0.317483	−	0.948264i	$-0.602838\pi$
$59$	−287.759	−0.634966	−0.317483	+	0.948264i	$0.602838\pi$
$60$	0	0
$61$	57.0430	0.119731	0.0598656	−	0.998206i	$-0.480933\pi$
$61$	57.0430	0.119731	0.0598656	+	0.998206i	$0.480933\pi$
$62$	−348.148	−0.713143
$63$	0	0
$64$	64.0000	0.125000
$65$	−1878.03	−3.58371
$66$	0	0
$67$	−497.824	−0.907745	−0.453873	−	0.891067i	$-0.649958\pi$
$67$	−497.824	−0.907745	−0.453873	+	0.891067i	$0.649958\pi$
$68$	−215.334	−0.384016
$69$	0	0
$70$	−120.499	−0.205748
$71$	32.5150	0.0543496	0.0271748	−	0.999631i	$-0.491349\pi$
$71$	32.5150	0.0543496	0.0271748	+	0.999631i	$0.491349\pi$
$72$	0	0
$73$	32.7188	0.0524582	0.0262291	−	0.999656i	$-0.491650\pi$
$73$	32.7188	0.0524582	0.0262291	+	0.999656i	$0.491650\pi$
$74$	88.9977	0.139808
$75$	0	0
$76$	247.767	0.373958
$77$	0	0
$78$	0	0
$79$	−463.884	−0.660646	−0.330323	−	0.943868i	$-0.607158\pi$
$79$	−463.884	−0.660646	−0.330323	+	0.943868i	$0.607158\pi$
$80$	354.963	0.496076
$81$	0	0
$82$	310.004	0.417490
$83$	−1311.94	−1.73499	−0.867494	−	0.497448i	$-0.834271\pi$
$83$	−1311.94	−1.73499	−0.867494	+	0.497448i	$0.834271\pi$
$84$	0	0
$85$	−1194.31	−1.52401
$86$	−584.789	−0.733248
$87$	0	0
$88$	0	0
$89$	−1141.01	−1.35895	−0.679476	−	0.733698i	$-0.737793\pi$
$89$	−1141.01	−1.35895	−0.679476	+	0.733698i	$0.737793\pi$
$90$	0	0
$91$	−229.896	−0.264831
$92$	−768.739	−0.871158
$93$	0	0
$94$	−537.928	−0.590246
$95$	1374.19	1.48409
$96$	0	0
$97$	487.402	0.510187	0.255094	−	0.966916i	$-0.417894\pi$
$97$	487.402	0.510187	0.255094	+	0.966916i	$0.417894\pi$
$98$	671.249	0.691902
$99$	0	0
$100$	1468.73	1.46873
$101$	−1445.34	−1.42392	−0.711962	−	0.702218i	$-0.752193\pi$
$101$	−1445.34	−1.42392	−0.711962	+	0.702218i	$0.752193\pi$
$102$	0	0
$103$	−1432.07	−1.36996	−0.684982	−	0.728560i	$-0.740190\pi$
$103$	−1432.07	−1.36996	−0.684982	+	0.728560i	$0.740190\pi$
$104$	677.221	0.638529
$105$	0	0
$106$	−483.795	−0.443305
$107$	−554.019	−0.500552	−0.250276	−	0.968175i	$-0.580521\pi$
$107$	−554.019	−0.500552	−0.250276	+	0.968175i	$0.580521\pi$
$108$	0	0
$109$	921.427	0.809694	0.404847	−	0.914384i	$-0.367325\pi$
$109$	921.427	0.809694	0.404847	+	0.914384i	$0.367325\pi$
$110$	0	0
$111$	0	0
$112$	43.4521	0.0366592
$113$	−70.2921	−0.0585179	−0.0292590	−	0.999572i	$-0.509315\pi$
$113$	−70.2921	−0.0585179	−0.0292590	+	0.999572i	$0.509315\pi$
$114$	0	0
$115$	−4263.65	−3.45728
$116$	−486.143	−0.389114
$117$	0	0
$118$	575.518	0.448989
$119$	−146.199	−0.112622
$120$	0	0
$121$	0	0
$122$	−114.086	−0.0846628
$123$	0	0
$124$	696.297	0.504269
$125$	5372.85	3.84450
$126$	0	0
$127$	1198.39	0.837324	0.418662	−	0.908142i	$-0.362499\pi$
$127$	1198.39	0.837324	0.418662	+	0.908142i	$0.362499\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	3756.07	2.53407
$131$	2129.21	1.42008	0.710038	−	0.704164i	$-0.248678\pi$
$131$	2129.21	1.42008	0.710038	+	0.704164i	$0.248678\pi$
$132$	0	0
$133$	168.219	0.109672
$134$	995.649	0.641873
$135$	0	0
$136$	430.668	0.271540
$137$	−1419.13	−0.884997	−0.442498	−	0.896769i	$-0.645908\pi$
$137$	−1419.13	−0.884997	−0.442498	+	0.896769i	$0.645908\pi$
$138$	0	0
$139$	−2294.37	−1.40004	−0.700022	−	0.714121i	$-0.746826\pi$
$139$	−2294.37	−1.40004	−0.700022	+	0.714121i	$0.746826\pi$
$140$	240.998	0.145486
$141$	0	0
$142$	−65.0300	−0.0384310
$143$	0	0
$144$	0	0
$145$	−2696.29	−1.54424
$146$	−65.4376	−0.0370935
$147$	0	0
$148$	−177.995	−0.0988590
$149$	−1208.23	−0.664306	−0.332153	−	0.943225i	$-0.607775\pi$
$149$	−1208.23	−0.664306	−0.332153	+	0.943225i	$0.607775\pi$
$150$	0	0
$151$	1632.07	0.879573	0.439787	−	0.898102i	$-0.355054\pi$
$151$	1632.07	0.879573	0.439787	+	0.898102i	$0.355054\pi$
$152$	−495.534	−0.264429
$153$	0	0
$154$	0	0
$155$	3861.87	2.00124
$156$	0	0
$157$	360.362	0.183185	0.0915924	−	0.995797i	$-0.470804\pi$
$157$	360.362	0.183185	0.0915924	+	0.995797i	$0.470804\pi$
$158$	927.768	0.467147
$159$	0	0
$160$	−709.926	−0.350778
$161$	−521.927	−0.255488
$162$	0	0
$163$	−2844.77	−1.36699	−0.683495	−	0.729955i	$-0.739541\pi$
$163$	−2844.77	−1.36699	−0.683495	+	0.729955i	$0.739541\pi$
$164$	−620.008	−0.295210
$165$	0	0
$166$	2623.88	1.22682
$167$	−3574.20	−1.65617	−0.828083	−	0.560605i	$-0.810568\pi$
$167$	−3574.20	−1.65617	−0.828083	+	0.560605i	$0.810568\pi$
$168$	0	0
$169$	4969.06	2.26175
$170$	2388.61	1.07764
$171$	0	0
$172$	1169.58	0.518485
$173$	2515.83	1.10564	0.552819	−	0.833302i	$-0.313552\pi$
$173$	2515.83	1.10564	0.552819	+	0.833302i	$0.313552\pi$
$174$	0	0
$175$	997.176	0.430740
$176$	0	0
$177$	0	0
$178$	2282.02	0.960924
$179$	−1268.93	−0.529855	−0.264927	−	0.964268i	$-0.585348\pi$
$179$	−1268.93	−0.529855	−0.264927	+	0.964268i	$0.585348\pi$
$180$	0	0
$181$	−621.543	−0.255243	−0.127621	−	0.991823i	$-0.540734\pi$
$181$	−621.543	−0.255243	−0.127621	+	0.991823i	$0.540734\pi$
$182$	459.791	0.187264
$183$	0	0
$184$	1537.48	0.616002
$185$	−987.214	−0.392332
$186$	0	0
$187$	0	0
$188$	1075.86	0.417367
$189$	0	0
$190$	−2748.38	−1.04941
$191$	1051.39	0.398305	0.199152	−	0.979969i	$-0.436181\pi$
$191$	1051.39	0.398305	0.199152	+	0.979969i	$0.436181\pi$
$192$	0	0
$193$	258.645	0.0964646	0.0482323	−	0.998836i	$-0.484641\pi$
$193$	258.645	0.0964646	0.0482323	+	0.998836i	$0.484641\pi$
$194$	−974.804	−0.360757
$195$	0	0
$196$	−1342.50	−0.489249
$197$	−3285.51	−1.18824	−0.594120	−	0.804377i	$-0.702499\pi$
$197$	−3285.51	−1.18824	−0.594120	+	0.804377i	$0.702499\pi$
$198$	0	0
$199$	−2394.01	−0.852799	−0.426399	−	0.904535i	$-0.640218\pi$
$199$	−2394.01	−0.852799	−0.426399	+	0.904535i	$0.640218\pi$
$200$	−2937.46	−1.03855
$201$	0	0
$202$	2890.67	1.00687
$203$	−330.061	−0.114117
$204$	0	0
$205$	−3438.74	−1.17157
$206$	2864.14	0.968711
$207$	0	0
$208$	−1354.44	−0.451508
$209$	0	0
$210$	0	0
$211$	5372.66	1.75293	0.876467	−	0.481462i	$-0.159894\pi$
$211$	5372.66	1.75293	0.876467	+	0.481462i	$0.159894\pi$
$212$	967.590	0.313464
$213$	0	0
$214$	1108.04	0.353944
$215$	6486.82	2.05766
$216$	0	0
$217$	472.743	0.147889
$218$	−1842.85	−0.572540
$219$	0	0
$220$	0	0
$221$	4557.15	1.38709
$222$	0	0
$223$	−2176.24	−0.653505	−0.326752	−	0.945110i	$-0.605954\pi$
$223$	−2176.24	−0.653505	−0.326752	+	0.945110i	$0.605954\pi$
$224$	−86.9041	−0.0259220
$225$	0	0
$226$	140.584	0.0413784
$227$	−1088.27	−0.318198	−0.159099	−	0.987263i	$-0.550859\pi$
$227$	−1088.27	−0.318198	−0.159099	+	0.987263i	$0.550859\pi$
$228$	0	0
$229$	−633.624	−0.182843	−0.0914215	−	0.995812i	$-0.529141\pi$
$229$	−633.624	−0.182843	−0.0914215	+	0.995812i	$0.529141\pi$
$230$	8527.30	2.44467
$231$	0	0
$232$	972.285	0.275145
$233$	846.924	0.238128	0.119064	−	0.992887i	$-0.462011\pi$
$233$	846.924	0.238128	0.119064	+	0.992887i	$0.462011\pi$
$234$	0	0
$235$	5967.02	1.65636
$236$	−1151.04	−0.317483
$237$	0	0
$238$	292.397	0.0796357
$239$	−3020.93	−0.817606	−0.408803	−	0.912623i	$-0.634054\pi$
$239$	−3020.93	−0.817606	−0.408803	+	0.912623i	$0.634054\pi$
$240$	0	0
$241$	5174.07	1.38295	0.691475	−	0.722401i	$-0.256961\pi$
$241$	5174.07	1.38295	0.691475	+	0.722401i	$0.256961\pi$
$242$	0	0
$243$	0	0
$244$	228.172	0.0598656
$245$	−7445.89	−1.94164
$246$	0	0
$247$	−5243.53	−1.35076
$248$	−1392.59	−0.356572
$249$	0	0
$250$	−10745.7	−2.71847
$251$	4850.49	1.21976	0.609881	−	0.792493i	$-0.291217\pi$
$251$	4850.49	1.21976	0.609881	+	0.792493i	$0.291217\pi$
$252$	0	0
$253$	0	0
$254$	−2396.78	−0.592077
$255$	0	0
$256$	256.000	0.0625000
$257$	−484.975	−0.117712	−0.0588558	−	0.998266i	$-0.518745\pi$
$257$	−484.975	−0.117712	−0.0588558	+	0.998266i	$0.518745\pi$
$258$	0	0
$259$	−120.848	−0.0289928
$260$	−7512.13	−1.79186
$261$	0	0
$262$	−4258.42	−1.00415
$263$	2491.36	0.584121	0.292061	−	0.956400i	$-0.405659\pi$
$263$	2491.36	0.584121	0.292061	+	0.956400i	$0.405659\pi$
$264$	0	0
$265$	5366.54	1.24401
$266$	−336.437	−0.0775500
$267$	0	0
$268$	−1991.30	−0.453873
$269$	−4646.10	−1.05308	−0.526538	−	0.850152i	$-0.676510\pi$
$269$	−4646.10	−1.05308	−0.526538	+	0.850152i	$0.676510\pi$
$270$	0	0
$271$	−4273.01	−0.957812	−0.478906	−	0.877866i	$-0.658967\pi$
$271$	−4273.01	−0.957812	−0.478906	+	0.877866i	$0.658967\pi$
$272$	−861.336	−0.192008
$273$	0	0
$274$	2838.26	0.625787
$275$	0	0
$276$	0	0
$277$	−6411.75	−1.39078	−0.695388	−	0.718635i	$-0.744767\pi$
$277$	−6411.75	−1.39078	−0.695388	+	0.718635i	$0.744767\pi$
$278$	4588.74	0.989980
$279$	0	0
$280$	−481.996	−0.102874
$281$	−2337.68	−0.496280	−0.248140	−	0.968724i	$-0.579819\pi$
$281$	−2337.68	−0.496280	−0.248140	+	0.968724i	$0.579819\pi$
$282$	0	0
$283$	−5854.19	−1.22967	−0.614833	−	0.788657i	$-0.710777\pi$
$283$	−5854.19	−1.22967	−0.614833	+	0.788657i	$0.710777\pi$
$284$	130.060	0.0271748
$285$	0	0
$286$	0	0
$287$	−420.947	−0.0865774
$288$	0	0
$289$	−2014.95	−0.410127
$290$	5392.58	1.09194
$291$	0	0
$292$	130.875	0.0262291
$293$	4896.29	0.976261	0.488130	−	0.872771i	$-0.337679\pi$
$293$	4896.29	0.976261	0.488130	+	0.872771i	$0.337679\pi$
$294$	0	0
$295$	−6383.98	−1.25996
$296$	355.991	0.0699038
$297$	0	0
$298$	2416.45	0.469736
$299$	16268.9	3.14668
$300$	0	0
$301$	794.071	0.152058
$302$	−3264.13	−0.621952
$303$	0	0
$304$	991.069	0.186979
$305$	1265.51	0.237583
$306$	0	0
$307$	−8542.73	−1.58814	−0.794071	−	0.607825i	$-0.792042\pi$
$307$	−8542.73	−1.58814	−0.794071	+	0.607825i	$0.792042\pi$
$308$	0	0
$309$	0	0
$310$	−7723.73	−1.41509
$311$	1833.99	0.334393	0.167197	−	0.985924i	$-0.446529\pi$
$311$	1833.99	0.334393	0.167197	+	0.985924i	$0.446529\pi$
$312$	0	0
$313$	2257.65	0.407700	0.203850	−	0.979002i	$-0.434655\pi$
$313$	2257.65	0.407700	0.203850	+	0.979002i	$0.434655\pi$
$314$	−720.724	−0.129531
$315$	0	0
$316$	−1855.54	−0.330323
$317$	6563.38	1.16289	0.581445	−	0.813586i	$-0.302487\pi$
$317$	6563.38	1.16289	0.581445	+	0.813586i	$0.302487\pi$
$318$	0	0
$319$	0	0
$320$	1419.85	0.248038
$321$	0	0
$322$	1043.85	0.180657
$323$	−3334.54	−0.574424
$324$	0	0
$325$	−31082.9	−5.30514
$326$	5689.54	0.966608
$327$	0	0
$328$	1240.02	0.208745
$329$	730.441	0.122403
$330$	0	0
$331$	1579.05	0.262213	0.131106	−	0.991368i	$-0.458147\pi$
$331$	1579.05	0.262213	0.131106	+	0.991368i	$0.458147\pi$
$332$	−5247.75	−0.867494
$333$	0	0
$334$	7148.40	1.17109
$335$	−11044.3	−1.80124
$336$	0	0
$337$	−1623.16	−0.262371	−0.131186	−	0.991358i	$-0.541878\pi$
$337$	−1623.16	−0.262371	−0.131186	+	0.991358i	$0.541878\pi$
$338$	−9938.13	−1.59930
$339$	0	0
$340$	−4777.22	−0.762004
$341$	0	0
$342$	0	0
$343$	−1842.98	−0.290121
$344$	−2339.15	−0.366624
$345$	0	0
$346$	−5031.67	−0.781804
$347$	−8407.83	−1.30074	−0.650369	−	0.759619i	$-0.725386\pi$
$347$	−8407.83	−1.30074	−0.650369	+	0.759619i	$0.725386\pi$
$348$	0	0
$349$	5926.88	0.909051	0.454526	−	0.890734i	$-0.349809\pi$
$349$	5926.88	0.909051	0.454526	+	0.890734i	$0.349809\pi$
$350$	−1994.35	−0.304579
$351$	0	0
$352$	0	0
$353$	−10481.9	−1.58044	−0.790220	−	0.612824i	$-0.790034\pi$
$353$	−10481.9	−1.58044	−0.790220	+	0.612824i	$0.790034\pi$
$354$	0	0
$355$	721.351	0.107846
$356$	−4564.04	−0.679476
$357$	0	0
$358$	2537.85	0.374664
$359$	−8756.01	−1.28725	−0.643627	−	0.765339i	$-0.722571\pi$
$359$	−8756.01	−1.28725	−0.643627	+	0.765339i	$0.722571\pi$
$360$	0	0
$361$	−3022.22	−0.440620
$362$	1243.09	0.180484
$363$	0	0
$364$	−919.583	−0.132415
$365$	725.872	0.104093
$366$	0	0
$367$	907.593	0.129090	0.0645449	−	0.997915i	$-0.479440\pi$
$367$	907.593	0.129090	0.0645449	+	0.997915i	$0.479440\pi$
$368$	−3074.96	−0.435579
$369$	0	0
$370$	1974.43	0.277421
$371$	656.934	0.0919308
$372$	0	0
$373$	8077.50	1.12128	0.560640	−	0.828060i	$-0.310555\pi$
$373$	8077.50	1.12128	0.560640	+	0.828060i	$0.310555\pi$
$374$	0	0
$375$	0	0
$376$	−2151.71	−0.295123
$377$	10288.3	1.40550
$378$	0	0
$379$	13620.9	1.84607	0.923035	−	0.384716i	$-0.125700\pi$
$379$	13620.9	1.84607	0.923035	+	0.384716i	$0.125700\pi$
$380$	5496.76	0.742047
$381$	0	0
$382$	−2102.79	−0.281644
$383$	2192.55	0.292518	0.146259	−	0.989246i	$-0.453277\pi$
$383$	2192.55	0.292518	0.146259	+	0.989246i	$0.453277\pi$
$384$	0	0
$385$	0	0
$386$	−517.290	−0.0682108
$387$	0	0
$388$	1949.61	0.255094
$389$	1725.12	0.224851	0.112425	−	0.993660i	$-0.464138\pi$
$389$	1725.12	0.224851	0.112425	+	0.993660i	$0.464138\pi$
$390$	0	0
$391$	10346.0	1.33815
$392$	2685.00	0.345951
$393$	0	0
$394$	6571.03	0.840212
$395$	−10291.3	−1.31092
$396$	0	0
$397$	−5181.09	−0.654991	−0.327496	−	0.944853i	$-0.606205\pi$
$397$	−5181.09	−0.654991	−0.327496	+	0.944853i	$0.606205\pi$
$398$	4788.02	0.603020
$399$	0	0
$400$	5874.91	0.734364
$401$	−8298.54	−1.03344	−0.516720	−	0.856154i	$-0.672847\pi$
$401$	−8298.54	−1.03344	−0.516720	+	0.856154i	$0.672847\pi$
$402$	0	0
$403$	−14735.8	−1.82145
$404$	−5781.35	−0.711962
$405$	0	0
$406$	660.122	0.0806929
$407$	0	0
$408$	0	0
$409$	11768.9	1.42282	0.711411	−	0.702777i	$-0.248057\pi$
$409$	11768.9	1.42282	0.711411	+	0.702777i	$0.248057\pi$
$410$	6877.49	0.828427
$411$	0	0
$412$	−5728.29	−0.684982
$413$	−781.482	−0.0931095
$414$	0	0
$415$	−29105.6	−3.44274
$416$	2708.88	0.319264
$417$	0	0
$418$	0	0
$419$	4770.89	0.556260	0.278130	−	0.960543i	$-0.410285\pi$
$419$	4770.89	0.556260	0.278130	+	0.960543i	$0.410285\pi$
$420$	0	0
$421$	−3615.71	−0.418573	−0.209286	−	0.977854i	$-0.567114\pi$
$421$	−3615.71	−0.418573	−0.209286	+	0.977854i	$0.567114\pi$
$422$	−10745.3	−1.23951
$423$	0	0
$424$	−1935.18	−0.221652
$425$	−19766.7	−2.25606
$426$	0	0
$427$	154.915	0.0175570
$428$	−2216.08	−0.250276
$429$	0	0
$430$	−12973.6	−1.45499
$431$	−1574.44	−0.175959	−0.0879794	−	0.996122i	$-0.528041\pi$
$431$	−1574.44	−0.175959	−0.0879794	+	0.996122i	$0.528041\pi$
$432$	0	0
$433$	7608.27	0.844412	0.422206	−	0.906500i	$-0.361256\pi$
$433$	7608.27	0.844412	0.422206	+	0.906500i	$0.361256\pi$
$434$	−945.486	−0.104573
$435$	0	0
$436$	3685.71	0.404847
$437$	−11904.3	−1.30311
$438$	0	0
$439$	−7172.23	−0.779754	−0.389877	−	0.920867i	$-0.627482\pi$
$439$	−7172.23	−0.779754	−0.389877	+	0.920867i	$0.627482\pi$
$440$	0	0
$441$	0	0
$442$	−9114.29	−0.980820
$443$	−6996.78	−0.750400	−0.375200	−	0.926944i	$-0.622426\pi$
$443$	−6996.78	−0.750400	−0.375200	+	0.926944i	$0.622426\pi$
$444$	0	0
$445$	−25313.5	−2.69657
$446$	4352.47	0.462098
$447$	0	0
$448$	173.808	0.0183296
$449$	7054.98	0.741526	0.370763	−	0.928728i	$-0.379096\pi$
$449$	7054.98	0.741526	0.370763	+	0.928728i	$0.379096\pi$
$450$	0	0
$451$	0	0
$452$	−281.168	−0.0292590
$453$	0	0
$454$	2176.53	0.225000
$455$	−5100.28	−0.525505
$456$	0	0
$457$	3129.05	0.320287	0.160143	−	0.987094i	$-0.448804\pi$
$457$	3129.05	0.320287	0.160143	+	0.987094i	$0.448804\pi$
$458$	1267.25	0.129289
$459$	0	0
$460$	−17054.6	−1.72864
$461$	−2311.17	−0.233496	−0.116748	−	0.993162i	$-0.537247\pi$
$461$	−2311.17	−0.233496	−0.116748	+	0.993162i	$0.537247\pi$
$462$	0	0
$463$	13556.2	1.36071	0.680355	−	0.732883i	$-0.261826\pi$
$463$	13556.2	1.36071	0.680355	+	0.732883i	$0.261826\pi$
$464$	−1944.57	−0.194557
$465$	0	0
$466$	−1693.85	−0.168382
$467$	−1021.12	−0.101182	−0.0505910	−	0.998719i	$-0.516110\pi$
$467$	−1021.12	−0.101182	−0.0505910	+	0.998719i	$0.516110\pi$
$468$	0	0
$469$	−1351.97	−0.133109
$470$	−11934.0	−1.17123
$471$	0	0
$472$	2302.07	0.224494
$473$	0	0
$474$	0	0
$475$	22743.9	2.19697
$476$	−584.794	−0.0563109
$477$	0	0
$478$	6041.87	0.578135
$479$	−4464.29	−0.425843	−0.212921	−	0.977069i	$-0.568298\pi$
$479$	−4464.29	−0.425843	−0.212921	+	0.977069i	$0.568298\pi$
$480$	0	0
$481$	3766.94	0.357085
$482$	−10348.1	−0.977893
$483$	0	0
$484$	0	0
$485$	10813.1	1.01237
$486$	0	0
$487$	−7985.30	−0.743015	−0.371508	−	0.928430i	$-0.621159\pi$
$487$	−7985.30	−0.743015	−0.371508	+	0.928430i	$0.621159\pi$
$488$	−456.344	−0.0423314
$489$	0	0
$490$	14891.8	1.37294
$491$	−4620.80	−0.424712	−0.212356	−	0.977192i	$-0.568114\pi$
$491$	−4620.80	−0.424712	−0.212356	+	0.977192i	$0.568114\pi$
$492$	0	0
$493$	6542.69	0.597704
$494$	10487.1	0.955132
$495$	0	0
$496$	2785.19	0.252134
$497$	88.3028	0.00796966
$498$	0	0
$499$	−12341.7	−1.10719	−0.553597	−	0.832785i	$-0.686745\pi$
$499$	−12341.7	−1.10719	−0.553597	+	0.832785i	$0.686745\pi$
$500$	21491.4	1.92225
$501$	0	0
$502$	−9700.98	−0.862502
$503$	−11592.6	−1.02761	−0.513804	−	0.857908i	$-0.671764\pi$
$503$	−11592.6	−1.02761	−0.513804	+	0.857908i	$0.671764\pi$
$504$	0	0
$505$	−32065.0	−2.82550
$506$	0	0
$507$	0	0
$508$	4793.57	0.418662
$509$	13437.2	1.17012	0.585062	−	0.810989i	$-0.301070\pi$
$509$	13437.2	1.17012	0.585062	+	0.810989i	$0.301070\pi$
$510$	0	0
$511$	88.8562	0.00769231
$512$	−512.000	−0.0441942
$513$	0	0
$514$	969.950	0.0832347
$515$	−31770.8	−2.71842
$516$	0	0
$517$	0	0
$518$	241.696	0.0205010
$519$	0	0
$520$	15024.3	1.26703
$521$	10737.2	0.902892	0.451446	−	0.892298i	$-0.350908\pi$
$521$	10737.2	0.902892	0.451446	+	0.892298i	$0.350908\pi$
$522$	0	0
$523$	−10768.8	−0.900358	−0.450179	−	0.892938i	$-0.648640\pi$
$523$	−10768.8	−0.900358	−0.450179	+	0.892938i	$0.648640\pi$
$524$	8516.84	0.710038
$525$	0	0
$526$	−4982.72	−0.413036
$527$	−9371.02	−0.774589
$528$	0	0
$529$	24768.0	2.03567
$530$	−10733.1	−0.879651
$531$	0	0
$532$	672.875	0.0548361
$533$	13121.3	1.06632
$534$	0	0
$535$	−12291.0	−0.993246
$536$	3982.59	0.320936
$537$	0	0
$538$	9292.19	0.744637
$539$	0	0
$540$	0	0
$541$	−8704.54	−0.691751	−0.345876	−	0.938280i	$-0.612418\pi$
$541$	−8704.54	−0.691751	−0.345876	+	0.938280i	$0.612418\pi$
$542$	8546.03	0.677275
$543$	0	0
$544$	1722.67	0.135770
$545$	20442.0	1.60668
$546$	0	0
$547$	9905.34	0.774263	0.387131	−	0.922025i	$-0.373466\pi$
$547$	9905.34	0.774263	0.387131	+	0.922025i	$0.373466\pi$
$548$	−5676.52	−0.442498
$549$	0	0
$550$	0	0
$551$	−7528.14	−0.582050
$552$	0	0
$553$	−1259.79	−0.0968751
$554$	12823.5	0.983427
$555$	0	0
$556$	−9177.49	−0.700022
$557$	8431.80	0.641413	0.320706	−	0.947179i	$-0.396080\pi$
$557$	8431.80	0.641413	0.320706	+	0.947179i	$0.396080\pi$
$558$	0	0
$559$	−24751.9	−1.87280
$560$	963.992	0.0727430
$561$	0	0
$562$	4675.37	0.350923
$563$	22724.1	1.70108	0.850540	−	0.525911i	$-0.176276\pi$
$563$	22724.1	1.70108	0.850540	+	0.525911i	$0.176276\pi$
$564$	0	0
$565$	−1559.44	−0.116117
$566$	11708.4	0.869506
$567$	0	0
$568$	−260.120	−0.0192155
$569$	22220.6	1.63714	0.818572	−	0.574404i	$-0.194766\pi$
$569$	22220.6	1.63714	0.818572	+	0.574404i	$0.194766\pi$
$570$	0	0
$571$	14233.3	1.04316	0.521582	−	0.853201i	$-0.325342\pi$
$571$	14233.3	1.04316	0.521582	+	0.853201i	$0.325342\pi$
$572$	0	0
$573$	0	0
$574$	841.894	0.0612195
$575$	−70566.8	−5.11798
$576$	0	0
$577$	−22389.3	−1.61539	−0.807693	−	0.589603i	$-0.799284\pi$
$577$	−22389.3	−1.61539	−0.807693	+	0.589603i	$0.799284\pi$
$578$	4029.91	0.290004
$579$	0	0
$580$	−10785.2	−0.772120
$581$	−3562.90	−0.254413
$582$	0	0
$583$	0	0
$584$	−261.750	−0.0185468
$585$	0	0
$586$	−9792.58	−0.690320
$587$	−24178.2	−1.70007	−0.850036	−	0.526725i	$-0.823420\pi$
$587$	−24178.2	−1.70007	−0.850036	+	0.526725i	$0.823420\pi$
$588$	0	0
$589$	10782.5	0.754302
$590$	12768.0	0.890930
$591$	0	0
$592$	−711.981	−0.0494295
$593$	8857.21	0.613359	0.306680	−	0.951813i	$-0.400782\pi$
$593$	8857.21	0.613359	0.306680	+	0.951813i	$0.400782\pi$
$594$	0	0
$595$	−3243.44	−0.223476
$596$	−4832.90	−0.332153
$597$	0	0
$598$	−32537.9	−2.22504
$599$	321.654	0.0219406	0.0109703	−	0.999940i	$-0.496508\pi$
$599$	321.654	0.0219406	0.0109703	+	0.999940i	$0.496508\pi$
$600$	0	0
$601$	13491.3	0.915678	0.457839	−	0.889035i	$-0.348624\pi$
$601$	13491.3	0.915678	0.457839	+	0.889035i	$0.348624\pi$
$602$	−1588.14	−0.107521
$603$	0	0
$604$	6528.26	0.439787
$605$	0	0
$606$	0	0
$607$	−7134.85	−0.477092	−0.238546	−	0.971131i	$-0.576671\pi$
$607$	−7134.85	−0.477092	−0.238546	+	0.971131i	$0.576671\pi$
$608$	−1982.14	−0.132214
$609$	0	0
$610$	−2531.02	−0.167997
$611$	−22768.5	−1.50755
$612$	0	0
$613$	28848.2	1.90076	0.950382	−	0.311086i	$-0.100693\pi$
$613$	28848.2	1.90076	0.950382	+	0.311086i	$0.100693\pi$
$614$	17085.5	1.12299
$615$	0	0
$616$	0	0
$617$	−7124.72	−0.464879	−0.232439	−	0.972611i	$-0.574671\pi$
$617$	−7124.72	−0.464879	−0.232439	+	0.972611i	$0.574671\pi$
$618$	0	0
$619$	3243.35	0.210600	0.105300	−	0.994441i	$-0.466420\pi$
$619$	3243.35	0.210600	0.105300	+	0.994441i	$0.466420\pi$
$620$	15447.5	1.00062
$621$	0	0
$622$	−3667.99	−0.236452
$623$	−3098.70	−0.199273
$624$	0	0
$625$	73299.9	4.69119
$626$	−4515.31	−0.288287
$627$	0	0
$628$	1441.45	0.0915924
$629$	2395.53	0.151854
$630$	0	0
$631$	2515.55	0.158704	0.0793522	−	0.996847i	$-0.474715\pi$
$631$	2515.55	0.158704	0.0793522	+	0.996847i	$0.474715\pi$
$632$	3711.07	0.233574
$633$	0	0
$634$	−13126.8	−0.822288
$635$	26586.5	1.66150
$636$	0	0
$637$	28411.5	1.76720
$638$	0	0
$639$	0	0
$640$	−2839.70	−0.175389
$641$	8736.08	0.538306	0.269153	−	0.963097i	$-0.413256\pi$
$641$	8736.08	0.538306	0.269153	+	0.963097i	$0.413256\pi$
$642$	0	0
$643$	−17984.6	−1.10302	−0.551511	−	0.834168i	$-0.685949\pi$
$643$	−17984.6	−1.10302	−0.551511	+	0.834168i	$0.685949\pi$
$644$	−2087.71	−0.127744
$645$	0	0
$646$	6669.09	0.406179
$647$	−23633.0	−1.43603	−0.718013	−	0.696029i	$-0.754948\pi$
$647$	−23633.0	−1.43603	−0.718013	+	0.696029i	$0.754948\pi$
$648$	0	0
$649$	0	0
$650$	62165.8	3.75130
$651$	0	0
$652$	−11379.1	−0.683495
$653$	22379.1	1.34114	0.670568	−	0.741848i	$-0.266051\pi$
$653$	22379.1	1.34114	0.670568	+	0.741848i	$0.266051\pi$
$654$	0	0
$655$	47236.9	2.81786
$656$	−2480.03	−0.147605
$657$	0	0
$658$	−1460.88	−0.0865518
$659$	5188.26	0.306686	0.153343	−	0.988173i	$-0.450996\pi$
$659$	5188.26	0.306686	0.153343	+	0.988173i	$0.450996\pi$
$660$	0	0
$661$	−18160.8	−1.06864	−0.534322	−	0.845281i	$-0.679433\pi$
$661$	−18160.8	−1.06864	−0.534322	+	0.845281i	$0.679433\pi$
$662$	−3158.10	−0.185412
$663$	0	0
$664$	10495.5	0.613411
$665$	3731.96	0.217623
$666$	0	0
$667$	23357.3	1.35592
$668$	−14296.8	−0.828083
$669$	0	0
$670$	22088.6	1.27367
$671$	0	0
$672$	0	0
$673$	−6859.31	−0.392878	−0.196439	−	0.980516i	$-0.562938\pi$
$673$	−6859.31	−0.392878	−0.196439	+	0.980516i	$0.562938\pi$
$674$	3246.32	0.185525
$675$	0	0
$676$	19876.3	1.13087
$677$	16449.2	0.933818	0.466909	−	0.884305i	$-0.345368\pi$
$677$	16449.2	0.933818	0.466909	+	0.884305i	$0.345368\pi$
$678$	0	0
$679$	1323.66	0.0748123
$680$	9554.44	0.538818
$681$	0	0
$682$	0	0
$683$	−11590.9	−0.649363	−0.324682	−	0.945823i	$-0.605257\pi$
$683$	−11590.9	−0.649363	−0.324682	+	0.945823i	$0.605257\pi$
$684$	0	0
$685$	−31483.7	−1.75610
$686$	3685.96	0.205146
$687$	0	0
$688$	4678.31	0.259242
$689$	−20477.2	−1.13225
$690$	0	0
$691$	−3851.07	−0.212014	−0.106007	−	0.994365i	$-0.533807\pi$
$691$	−3851.07	−0.212014	−0.106007	+	0.994365i	$0.533807\pi$
$692$	10063.3	0.552819
$693$	0	0
$694$	16815.7	0.919760
$695$	−50901.0	−2.77811
$696$	0	0
$697$	8344.29	0.453462
$698$	−11853.8	−0.642796
$699$	0	0
$700$	3988.70	0.215370
$701$	15419.8	0.830813	0.415406	−	0.909636i	$-0.363639\pi$
$701$	15419.8	0.830813	0.415406	+	0.909636i	$0.363639\pi$
$702$	0	0
$703$	−2756.34	−0.147877
$704$	0	0
$705$	0	0
$706$	20963.8	1.11754
$707$	−3925.18	−0.208800
$708$	0	0
$709$	9817.67	0.520043	0.260021	−	0.965603i	$-0.416270\pi$
$709$	9817.67	0.520043	0.260021	+	0.965603i	$0.416270\pi$
$710$	−1442.70	−0.0762587
$711$	0	0
$712$	9128.07	0.480462
$713$	−33454.4	−1.75719
$714$	0	0
$715$	0	0
$716$	−5075.70	−0.264927
$717$	0	0
$718$	17512.0	0.910226
$719$	−21161.7	−1.09763	−0.548816	−	0.835943i	$-0.684921\pi$
$719$	−21161.7	−1.09763	−0.548816	+	0.835943i	$0.684921\pi$
$720$	0	0
$721$	−3889.16	−0.200887
$722$	6044.43	0.311566
$723$	0	0
$724$	−2486.17	−0.127621
$725$	−44625.7	−2.28601
$726$	0	0
$727$	−23781.5	−1.21322	−0.606608	−	0.795001i	$-0.707470\pi$
$727$	−23781.5	−1.21322	−0.606608	+	0.795001i	$0.707470\pi$
$728$	1839.17	0.0936319
$729$	0	0
$730$	−1451.74	−0.0736048
$731$	−15740.6	−0.796426
$732$	0	0
$733$	27158.3	1.36851	0.684253	−	0.729245i	$-0.260128\pi$
$733$	27158.3	1.36851	0.684253	+	0.729245i	$0.260128\pi$
$734$	−1815.19	−0.0912803
$735$	0	0
$736$	6149.91	0.308001
$737$	0	0
$738$	0	0
$739$	5086.89	0.253213	0.126606	−	0.991953i	$-0.459592\pi$
$739$	5086.89	0.253213	0.126606	+	0.991953i	$0.459592\pi$
$740$	−3948.86	−0.196166
$741$	0	0
$742$	−1313.87	−0.0650049
$743$	−21412.5	−1.05727	−0.528633	−	0.848850i	$-0.677295\pi$
$743$	−21412.5	−1.05727	−0.528633	+	0.848850i	$0.677295\pi$
$744$	0	0
$745$	−26804.7	−1.31818
$746$	−16155.0	−0.792864
$747$	0	0
$748$	0	0
$749$	−1504.58	−0.0733994
$750$	0	0
$751$	19921.3	0.967958	0.483979	−	0.875079i	$-0.339191\pi$
$751$	19921.3	0.967958	0.483979	+	0.875079i	$0.339191\pi$
$752$	4303.43	0.208683
$753$	0	0
$754$	−20576.6	−0.993841
$755$	36207.6	1.74534
$756$	0	0
$757$	−28445.7	−1.36576	−0.682878	−	0.730532i	$-0.739272\pi$
$757$	−28445.7	−1.36576	−0.682878	+	0.730532i	$0.739272\pi$
$758$	−27241.9	−1.30537
$759$	0	0
$760$	−10993.5	−0.524706
$761$	18747.8	0.893046	0.446523	−	0.894772i	$-0.352662\pi$
$761$	18747.8	0.893046	0.446523	+	0.894772i	$0.352662\pi$
$762$	0	0
$763$	2502.37	0.118731
$764$	4205.58	0.199152
$765$	0	0
$766$	−4385.11	−0.206841
$767$	24359.5	1.14677
$768$	0	0
$769$	6553.31	0.307306	0.153653	−	0.988125i	$-0.450896\pi$
$769$	6553.31	0.307306	0.153653	+	0.988125i	$0.450896\pi$
$770$	0	0
$771$	0	0
$772$	1034.58	0.0482323
$773$	−4327.49	−0.201357	−0.100679	−	0.994919i	$-0.532101\pi$
$773$	−4327.49	−0.201357	−0.100679	+	0.994919i	$0.532101\pi$
$774$	0	0
$775$	63916.9	2.96253
$776$	−3899.21	−0.180378
$777$	0	0
$778$	−3450.24	−0.158994
$779$	−9601.09	−0.441585
$780$	0	0
$781$	0	0
$782$	−20692.0	−0.946218
$783$	0	0
$784$	−5369.99	−0.244624
$785$	7994.70	0.363494
$786$	0	0
$787$	25602.8	1.15965	0.579824	−	0.814742i	$-0.303121\pi$
$787$	25602.8	1.15965	0.579824	+	0.814742i	$0.303121\pi$
$788$	−13142.1	−0.594120
$789$	0	0
$790$	20582.7	0.926961
$791$	−190.896	−0.00858089
$792$	0	0
$793$	−4828.84	−0.216238
$794$	10362.2	0.463149
$795$	0	0
$796$	−9576.04	−0.426399
$797$	−13395.4	−0.595345	−0.297672	−	0.954668i	$-0.596210\pi$
$797$	−13395.4	−0.595345	−0.297672	+	0.954668i	$0.596210\pi$
$798$	0	0
$799$	−14479.3	−0.641102
$800$	−11749.8	−0.519274
$801$	0	0
$802$	16597.1	0.730753
$803$	0	0
$804$	0	0
$805$	−11579.0	−0.506966
$806$	29471.7	1.28796
$807$	0	0
$808$	11562.7	0.503433
$809$	3696.86	0.160661	0.0803305	−	0.996768i	$-0.474402\pi$
$809$	3696.86	0.160661	0.0803305	+	0.996768i	$0.474402\pi$
$810$	0	0
$811$	−300.497	−0.0130110	−0.00650548	−	0.999979i	$-0.502071\pi$
$811$	−300.497	−0.0130110	−0.00650548	+	0.999979i	$0.502071\pi$
$812$	−1320.24	−0.0570585
$813$	0	0
$814$	0	0
$815$	−63111.7	−2.71252
$816$	0	0
$817$	18111.4	0.775567
$818$	−23537.8	−1.00609
$819$	0	0
$820$	−13755.0	−0.585786
$821$	4441.49	0.188805	0.0944026	−	0.995534i	$-0.469906\pi$
$821$	4441.49	0.188805	0.0944026	+	0.995534i	$0.469906\pi$
$822$	0	0
$823$	7879.65	0.333739	0.166870	−	0.985979i	$-0.446634\pi$
$823$	7879.65	0.333739	0.166870	+	0.985979i	$0.446634\pi$
$824$	11456.6	0.484355
$825$	0	0
$826$	1562.96	0.0658384
$827$	13483.1	0.566935	0.283467	−	0.958982i	$-0.408515\pi$
$827$	13483.1	0.566935	0.283467	+	0.958982i	$0.408515\pi$
$828$	0	0
$829$	−23614.5	−0.989341	−0.494671	−	0.869081i	$-0.664711\pi$
$829$	−23614.5	−0.989341	−0.494671	+	0.869081i	$0.664711\pi$
$830$	58211.2	2.43438
$831$	0	0
$832$	−5417.77	−0.225754
$833$	18067.9	0.751517
$834$	0	0
$835$	−79294.2	−3.28633
$836$	0	0
$837$	0	0
$838$	−9541.78	−0.393336
$839$	28227.7	1.16154	0.580768	−	0.814069i	$-0.302753\pi$
$839$	28227.7	1.16154	0.580768	+	0.814069i	$0.302753\pi$
$840$	0	0
$841$	−9618.08	−0.394361
$842$	7231.43	0.295976
$843$	0	0
$844$	21490.6	0.876467
$845$	110240.	4.48799
$846$	0	0
$847$	0	0
$848$	3870.36	0.156732
$849$	0	0
$850$	39533.4	1.59528
$851$	8552.00	0.344487
$852$	0	0
$853$	24374.0	0.978372	0.489186	−	0.872180i	$-0.337294\pi$
$853$	24374.0	0.978372	0.489186	+	0.872180i	$0.337294\pi$
$854$	−309.830	−0.0124147
$855$	0	0
$856$	4432.15	0.176972
$857$	11208.4	0.446759	0.223380	−	0.974732i	$-0.428291\pi$
$857$	11208.4	0.446759	0.223380	+	0.974732i	$0.428291\pi$
$858$	0	0
$859$	−21799.5	−0.865879	−0.432939	−	0.901423i	$-0.642524\pi$
$859$	−21799.5	−0.865879	−0.432939	+	0.901423i	$0.642524\pi$
$860$	25947.3	1.02883
$861$	0	0
$862$	3148.88	0.124422
$863$	8731.52	0.344408	0.172204	−	0.985061i	$-0.444911\pi$
$863$	8731.52	0.344408	0.172204	+	0.985061i	$0.444911\pi$
$864$	0	0
$865$	55814.2	2.19392
$866$	−15216.5	−0.597089
$867$	0	0
$868$	1890.97	0.0739444
$869$	0	0
$870$	0	0
$871$	42142.1	1.63942
$872$	−7371.41	−0.286270
$873$	0	0
$874$	23808.5	0.921437
$875$	14591.3	0.563745
$876$	0	0
$877$	2711.43	0.104400	0.0521998	−	0.998637i	$-0.483377\pi$
$877$	2711.43	0.104400	0.0521998	+	0.998637i	$0.483377\pi$
$878$	14344.5	0.551369
$879$	0	0
$880$	0	0
$881$	−7616.78	−0.291278	−0.145639	−	0.989338i	$-0.546524\pi$
$881$	−7616.78	−0.291278	−0.145639	+	0.989338i	$0.546524\pi$
$882$	0	0
$883$	12501.3	0.476447	0.238223	−	0.971210i	$-0.423435\pi$
$883$	12501.3	0.476447	0.238223	+	0.971210i	$0.423435\pi$
$884$	18228.6	0.693545
$885$	0	0
$886$	13993.6	0.530613
$887$	18367.8	0.695299	0.347650	−	0.937625i	$-0.386980\pi$
$887$	18367.8	0.695299	0.347650	+	0.937625i	$0.386980\pi$
$888$	0	0
$889$	3254.54	0.122783
$890$	50627.0	1.90676
$891$	0	0
$892$	−8704.94	−0.326752
$893$	16660.1	0.624311
$894$	0	0
$895$	−28151.3	−1.05139
$896$	−347.617	−0.0129610
$897$	0	0
$898$	−14110.0	−0.524338
$899$	−21156.2	−0.784872
$900$	0	0
$901$	−13022.2	−0.481500
$902$	0	0
$903$	0	0
$904$	562.337	0.0206892
$905$	−13789.0	−0.506479
$906$	0	0
$907$	−46089.2	−1.68728	−0.843641	−	0.536907i	$-0.819592\pi$
$907$	−46089.2	−1.68728	−0.843641	+	0.536907i	$0.819592\pi$
$908$	−4353.07	−0.159099
$909$	0	0
$910$	10200.6	0.371588
$911$	36109.8	1.31325	0.656626	−	0.754216i	$-0.271983\pi$
$911$	36109.8	1.31325	0.656626	+	0.754216i	$0.271983\pi$
$912$	0	0
$913$	0	0
$914$	−6258.11	−0.226477
$915$	0	0
$916$	−2534.49	−0.0914215
$917$	5782.41	0.208236
$918$	0	0
$919$	−17516.4	−0.628741	−0.314370	−	0.949300i	$-0.601793\pi$
$919$	−17516.4	−0.628741	−0.314370	+	0.949300i	$0.601793\pi$
$920$	34109.2	1.22233
$921$	0	0
$922$	4622.33	0.165107
$923$	−2752.48	−0.0981571
$924$	0	0
$925$	−16339.2	−0.580788
$926$	−27112.3	−0.962167
$927$	0	0
$928$	3889.14	0.137573
$929$	−23730.5	−0.838074	−0.419037	−	0.907969i	$-0.637632\pi$
$929$	−23730.5	−0.838074	−0.419037	+	0.907969i	$0.637632\pi$
$930$	0	0
$931$	−20789.2	−0.731835
$932$	3387.70	0.119064
$933$	0	0
$934$	2042.25	0.0715464
$935$	0	0
$936$	0	0
$937$	−39993.8	−1.39439	−0.697194	−	0.716883i	$-0.745568\pi$
$937$	−39993.8	−1.39439	−0.697194	+	0.716883i	$0.745568\pi$
$938$	2703.94	0.0941223
$939$	0	0
$940$	23868.1	0.828182
$941$	−50377.4	−1.74522	−0.872612	−	0.488415i	$-0.837575\pi$
$941$	−50377.4	−1.74522	−0.872612	+	0.488415i	$0.837575\pi$
$942$	0	0
$943$	29789.0	1.02870
$944$	−4604.14	−0.158742
$945$	0	0
$946$	0	0
$947$	−29786.7	−1.02211	−0.511054	−	0.859548i	$-0.670745\pi$
$947$	−29786.7	−1.02211	−0.511054	+	0.859548i	$0.670745\pi$
$948$	0	0
$949$	−2769.73	−0.0947411
$950$	−45487.8	−1.55349
$951$	0	0
$952$	1169.59	0.0398178
$953$	−22063.1	−0.749942	−0.374971	−	0.927037i	$-0.622347\pi$
$953$	−22063.1	−0.749942	−0.374971	+	0.927037i	$0.622347\pi$
$954$	0	0
$955$	23325.4	0.790357
$956$	−12083.7	−0.408803
$957$	0	0
$958$	8928.59	0.301116
$959$	−3854.01	−0.129773
$960$	0	0
$961$	510.834	0.0171473
$962$	−7533.88	−0.252497
$963$	0	0
$964$	20696.3	0.691475
$965$	5738.09	0.191415
$966$	0	0
$967$	5411.35	0.179956	0.0899780	−	0.995944i	$-0.471320\pi$
$967$	5411.35	0.179956	0.0899780	+	0.995944i	$0.471320\pi$
$968$	0	0
$969$	0	0
$970$	−21626.2	−0.715851
$971$	55447.8	1.83255	0.916274	−	0.400552i	$-0.131182\pi$
$971$	55447.8	1.83255	0.916274	+	0.400552i	$0.131182\pi$
$972$	0	0
$973$	−6230.95	−0.205298
$974$	15970.6	0.525391
$975$	0	0
$976$	912.688	0.0299328
$977$	28592.5	0.936290	0.468145	−	0.883652i	$-0.344923\pi$
$977$	28592.5	0.936290	0.468145	+	0.883652i	$0.344923\pi$
$978$	0	0
$979$	0	0
$980$	−29783.6	−0.970818
$981$	0	0
$982$	9241.60	0.300317
$983$	2118.66	0.0687432	0.0343716	−	0.999409i	$-0.489057\pi$
$983$	2118.66	0.0687432	0.0343716	+	0.999409i	$0.489057\pi$
$984$	0	0
$985$	−72889.7	−2.35783
$986$	−13085.4	−0.422640
$987$	0	0
$988$	−20974.1	−0.675381
$989$	−56193.7	−1.80673
$990$	0	0
$991$	−25789.9	−0.826684	−0.413342	−	0.910576i	$-0.635639\pi$
$991$	−25789.9	−0.826684	−0.413342	+	0.910576i	$0.635639\pi$
$992$	−5570.38	−0.178286
$993$	0	0
$994$	−176.606	−0.00563540
$995$	−53111.6	−1.69221
$996$	0	0
$997$	16456.1	0.522737	0.261368	−	0.965239i	$-0.415826\pi$
$997$	16456.1	0.522737	0.261368	+	0.965239i	$0.415826\pi$
$998$	24683.4	0.782904
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2178.4.a.bx.1.4		4
3.2	odd	2		726.4.a.y.1.1	yes	4
11.10	odd	2		2178.4.a.cb.1.4		4
33.32	even	2		726.4.a.v.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
726.4.a.v.1.1	✓	4	33.32	even	2
726.4.a.y.1.1	yes	4	3.2	odd	2
2178.4.a.bx.1.4		4	1.1	even	1	trivial
2178.4.a.cb.1.4		4	11.10	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)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +4.00000 q^{4} +22.1852 q^{5} +2.71575 q^{7} -8.00000 q^{8} +O(q^{10})\)	\(q-2.00000 q^{2} +4.00000 q^{4} +22.1852 q^{5} +2.71575 q^{7} -8.00000 q^{8} -44.3703 q^{10} -84.6526 q^{13} -5.43151 q^{14} +16.0000 q^{16} -53.8335 q^{17} +61.9418 q^{19} +88.7407 q^{20} -192.185 q^{23} +367.182 q^{25} +169.305 q^{26} +10.8630 q^{28} -121.536 q^{29} +174.074 q^{31} -32.0000 q^{32} +107.667 q^{34} +60.2495 q^{35} -44.4988 q^{37} -123.884 q^{38} -177.481 q^{40} -155.002 q^{41} +292.394 q^{43} +384.369 q^{46} +268.964 q^{47} -335.625 q^{49} -734.364 q^{50} -338.610 q^{52} +241.897 q^{53} -21.7260 q^{56} +243.071 q^{58} -287.759 q^{59} +57.0430 q^{61} -348.148 q^{62} +64.0000 q^{64} -1878.03 q^{65} -497.824 q^{67} -215.334 q^{68} -120.499 q^{70} +32.5150 q^{71} +32.7188 q^{73} +88.9977 q^{74} +247.767 q^{76} -463.884 q^{79} +354.963 q^{80} +310.004 q^{82} -1311.94 q^{83} -1194.31 q^{85} -584.789 q^{86} -1141.01 q^{89} -229.896 q^{91} -768.739 q^{92} -537.928 q^{94} +1374.19 q^{95} +487.402 q^{97} +671.249 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{2} + 16 q^{4} + 6 q^{5} + 12 q^{7} - 32 q^{8}+O(q^{10}) \) 4 * q - 8 * q^2 + 16 * q^4 + 6 * q^5 + 12 * q^7 - 32 * q^8	\( 4 q - 8 q^{2} + 16 q^{4} + 6 q^{5} + 12 q^{7} - 32 q^{8} - 12 q^{10} + 12 q^{13} - 24 q^{14} + 64 q^{16} - 108 q^{17} + 192 q^{19} + 24 q^{20} - 156 q^{23} + 286 q^{25} - 24 q^{26} + 48 q^{28} - 408 q^{29} - 26 q^{31} - 128 q^{32} + 216 q^{34} - 588 q^{35} - 224 q^{37} - 384 q^{38} - 48 q^{40} - 348 q^{41} + 540 q^{43} + 312 q^{46} - 132 q^{47} + 674 q^{49} - 572 q^{50} + 48 q^{52} + 1470 q^{53} - 96 q^{56} + 816 q^{58} + 684 q^{59} + 1320 q^{61} + 52 q^{62} + 256 q^{64} - 2556 q^{65} - 530 q^{67} - 432 q^{68} + 1176 q^{70} - 936 q^{71} - 2352 q^{73} + 448 q^{74} + 768 q^{76} - 192 q^{79} + 96 q^{80} + 696 q^{82} - 3192 q^{83} - 2340 q^{85} - 1080 q^{86} - 726 q^{89} + 360 q^{91} - 624 q^{92} + 264 q^{94} + 2064 q^{95} - 100 q^{97} - 1348 q^{98}+O(q^{100}) \) 4 * q - 8 * q^2 + 16 * q^4 + 6 * q^5 + 12 * q^7 - 32 * q^8 - 12 * q^10 + 12 * q^13 - 24 * q^14 + 64 * q^16 - 108 * q^17 + 192 * q^19 + 24 * q^20 - 156 * q^23 + 286 * q^25 - 24 * q^26 + 48 * q^28 - 408 * q^29 - 26 * q^31 - 128 * q^32 + 216 * q^34 - 588 * q^35 - 224 * q^37 - 384 * q^38 - 48 * q^40 - 348 * q^41 + 540 * q^43 + 312 * q^46 - 132 * q^47 + 674 * q^49 - 572 * q^50 + 48 * q^52 + 1470 * q^53 - 96 * q^56 + 816 * q^58 + 684 * q^59 + 1320 * q^61 + 52 * q^62 + 256 * q^64 - 2556 * q^65 - 530 * q^67 - 432 * q^68 + 1176 * q^70 - 936 * q^71 - 2352 * q^73 + 448 * q^74 + 768 * q^76 - 192 * q^79 + 96 * q^80 + 696 * q^82 - 3192 * q^83 - 2340 * q^85 - 1080 * q^86 - 726 * q^89 + 360 * q^91 - 624 * q^92 + 264 * q^94 + 2064 * q^95 - 100 * q^97 - 1348 * q^98

Introduction

L-functions

Modular forms

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Database

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