LMFDB - Embedded newform 2475.4.a.bo.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,4,Mod(1,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2475.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2475.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:	$146.029727264$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 31x^{5} + 50x^{4} + 272x^{3} - 322x^{2} - 704x + 512 $ x^7 - 2x^6 - 31x^5 + 50x^4 + 272x^3 - 322x^2 - 704x + 512
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 5 $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-4.15324$ of defining polynomial
Character	$\chi$	$=$	2475.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.15324 q^{2} +18.5558 q^{4} -17.2148 q^{7} -54.3968 q^{8} +O(q^{10})$	$q-5.15324 q^{2} +18.5558 q^{4} -17.2148 q^{7} -54.3968 q^{8} -11.0000 q^{11} -3.57243 q^{13} +88.7118 q^{14} +131.873 q^{16} -101.723 q^{17} +87.5308 q^{19} +56.6856 q^{22} -123.329 q^{23} +18.4096 q^{26} -319.435 q^{28} +26.9059 q^{29} -174.468 q^{31} -244.397 q^{32} +524.204 q^{34} +329.042 q^{37} -451.067 q^{38} -21.1212 q^{41} +363.807 q^{43} -204.114 q^{44} +635.546 q^{46} -334.029 q^{47} -46.6513 q^{49} -66.2895 q^{52} -649.775 q^{53} +936.429 q^{56} -138.652 q^{58} -755.168 q^{59} -480.167 q^{61} +899.075 q^{62} +204.454 q^{64} -548.550 q^{67} -1887.56 q^{68} +629.256 q^{71} -1206.95 q^{73} -1695.63 q^{74} +1624.21 q^{76} +189.363 q^{77} +469.029 q^{79} +108.842 q^{82} +508.855 q^{83} -1874.78 q^{86} +598.365 q^{88} +273.521 q^{89} +61.4986 q^{91} -2288.48 q^{92} +1721.33 q^{94} -213.839 q^{97} +240.405 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 5 q^{2} + 13 q^{4} + 34 q^{7} - 75 q^{8}+O(q^{10}) $ 7 * q - 5 * q^2 + 13 * q^4 + 34 * q^7 - 75 * q^8	$ 7 q - 5 q^{2} + 13 q^{4} + 34 q^{7} - 75 q^{8} - 77 q^{11} + 80 q^{13} - 42 q^{14} - 43 q^{16} - 162 q^{17} + 58 q^{19} + 55 q^{22} - 324 q^{23} + 200 q^{26} - 168 q^{28} - 64 q^{29} - 348 q^{31} + 75 q^{32} + 206 q^{34} + 664 q^{37} - 334 q^{38} + 332 q^{41} + 774 q^{43} - 143 q^{44} - 328 q^{46} - 872 q^{47} - 417 q^{49} + 134 q^{52} - 1628 q^{53} + 1618 q^{56} + 1568 q^{58} + 332 q^{59} + 22 q^{61} + 260 q^{62} + 561 q^{64} + 1524 q^{67} - 2324 q^{68} + 516 q^{71} + 1700 q^{73} - 1628 q^{74} + 2794 q^{76} - 374 q^{77} + 1746 q^{79} + 364 q^{82} - 2344 q^{83} - 1270 q^{86} + 825 q^{88} + 2226 q^{89} + 1072 q^{91} - 4184 q^{92} + 4736 q^{94} + 1048 q^{97} - 3057 q^{98}+O(q^{100}) $ 7 * q - 5 * q^2 + 13 * q^4 + 34 * q^7 - 75 * q^8 - 77 * q^11 + 80 * q^13 - 42 * q^14 - 43 * q^16 - 162 * q^17 + 58 * q^19 + 55 * q^22 - 324 * q^23 + 200 * q^26 - 168 * q^28 - 64 * q^29 - 348 * q^31 + 75 * q^32 + 206 * q^34 + 664 * q^37 - 334 * q^38 + 332 * q^41 + 774 * q^43 - 143 * q^44 - 328 * q^46 - 872 * q^47 - 417 * q^49 + 134 * q^52 - 1628 * q^53 + 1618 * q^56 + 1568 * q^58 + 332 * q^59 + 22 * q^61 + 260 * q^62 + 561 * q^64 + 1524 * q^67 - 2324 * q^68 + 516 * q^71 + 1700 * q^73 - 1628 * q^74 + 2794 * q^76 - 374 * q^77 + 1746 * q^79 + 364 * q^82 - 2344 * q^83 - 1270 * q^86 + 825 * q^88 + 2226 * q^89 + 1072 * q^91 - 4184 * q^92 + 4736 * q^94 + 1048 * q^97 - 3057 * 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 $

$2$	−5.15324	−1.82194	−0.910972	−	0.412468i	$-0.864667\pi$
$2$	−5.15324	−1.82194	−0.910972	+	0.412468i	$0.864667\pi$
$3$	0	0
$3$	0	0
$4$	18.5558	2.31948
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−17.2148	−0.929511	−0.464755	−	0.885439i	$-0.653858\pi$
$7$	−17.2148	−0.929511	−0.464755	+	0.885439i	$0.653858\pi$
$8$	−54.3968	−2.40402
$9$	0	0
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	−3.57243	−0.0762165	−0.0381082	−	0.999274i	$-0.512133\pi$
$13$	−3.57243	−0.0762165	−0.0381082	+	0.999274i	$0.512133\pi$
$14$	88.7118	1.69352
$15$	0	0
$16$	131.873	2.06051
$17$	−101.723	−1.45127	−0.725633	−	0.688082i	$-0.758453\pi$
$17$	−101.723	−1.45127	−0.725633	+	0.688082i	$0.758453\pi$
$18$	0	0
$19$	87.5308	1.05689	0.528446	−	0.848967i	$-0.322775\pi$
$19$	87.5308	1.05689	0.528446	+	0.848967i	$0.322775\pi$
$20$	0	0
$21$	0	0
$22$	56.6856	0.549337
$23$	−123.329	−1.11809	−0.559043	−	0.829139i	$-0.688831\pi$
$23$	−123.329	−1.11809	−0.559043	+	0.829139i	$0.688831\pi$
$24$	0	0
$25$	0	0
$26$	18.4096	0.138862
$27$	0	0
$28$	−319.435	−2.15598
$29$	26.9059	0.172286	0.0861431	−	0.996283i	$-0.472546\pi$
$29$	26.9059	0.172286	0.0861431	+	0.996283i	$0.472546\pi$
$30$	0	0
$31$	−174.468	−1.01082	−0.505410	−	0.862880i	$-0.668659\pi$
$31$	−174.468	−1.01082	−0.505410	+	0.862880i	$0.668659\pi$
$32$	−244.397	−1.35012
$33$	0	0
$34$	524.204	2.64412
$35$	0	0
$36$	0	0
$37$	329.042	1.46200	0.731002	−	0.682376i	$-0.239053\pi$
$37$	329.042	1.46200	0.731002	+	0.682376i	$0.239053\pi$
$38$	−451.067	−1.92560
$39$	0	0
$40$	0	0
$41$	−21.1212	−0.0804530	−0.0402265	−	0.999191i	$-0.512808\pi$
$41$	−21.1212	−0.0804530	−0.0402265	+	0.999191i	$0.512808\pi$
$42$	0	0
$43$	363.807	1.29023	0.645117	−	0.764084i	$-0.276809\pi$
$43$	363.807	1.29023	0.645117	+	0.764084i	$0.276809\pi$
$44$	−204.114	−0.699350
$45$	0	0
$46$	635.546	2.03709
$47$	−334.029	−1.03666	−0.518331	−	0.855180i	$-0.673447\pi$
$47$	−334.029	−1.03666	−0.518331	+	0.855180i	$0.673447\pi$
$48$	0	0
$49$	−46.6513	−0.136010
$50$	0	0
$51$	0	0
$52$	−66.2895	−0.176783
$53$	−649.775	−1.68403	−0.842014	−	0.539455i	$-0.818630\pi$
$53$	−649.775	−1.68403	−0.842014	+	0.539455i	$0.818630\pi$
$54$	0	0
$55$	0	0
$56$	936.429	2.23456
$57$	0	0
$58$	−138.652	−0.313896
$59$	−755.168	−1.66635	−0.833173	−	0.553012i	$-0.813478\pi$
$59$	−755.168	−1.66635	−0.833173	+	0.553012i	$0.813478\pi$
$60$	0	0
$61$	−480.167	−1.00785	−0.503927	−	0.863746i	$-0.668112\pi$
$61$	−480.167	−1.00785	−0.503927	+	0.863746i	$0.668112\pi$
$62$	899.075	1.84166
$63$	0	0
$64$	204.454	0.399324
$65$	0	0
$66$	0	0
$67$	−548.550	−1.00024	−0.500120	−	0.865956i	$-0.666711\pi$
$67$	−548.550	−1.00024	−0.500120	+	0.865956i	$0.666711\pi$
$68$	−1887.56	−3.36618
$69$	0	0
$70$	0	0
$71$	629.256	1.05182	0.525908	−	0.850541i	$-0.323726\pi$
$71$	629.256	1.05182	0.525908	+	0.850541i	$0.323726\pi$
$72$	0	0
$73$	−1206.95	−1.93510	−0.967550	−	0.252680i	$-0.918688\pi$
$73$	−1206.95	−1.93510	−0.967550	+	0.252680i	$0.918688\pi$
$74$	−1695.63	−2.66369
$75$	0	0
$76$	1624.21	2.45144
$77$	189.363	0.280258
$78$	0	0
$79$	469.029	0.667973	0.333986	−	0.942578i	$-0.391606\pi$
$79$	469.029	0.667973	0.333986	+	0.942578i	$0.391606\pi$
$80$	0	0
$81$	0	0
$82$	108.842	0.146581
$83$	508.855	0.672941	0.336471	−	0.941694i	$-0.390767\pi$
$83$	508.855	0.672941	0.336471	+	0.941694i	$0.390767\pi$
$84$	0	0
$85$	0	0
$86$	−1874.78	−2.35073
$87$	0	0
$88$	598.365	0.724840
$89$	273.521	0.325765	0.162883	−	0.986645i	$-0.447921\pi$
$89$	273.521	0.325765	0.162883	+	0.986645i	$0.447921\pi$
$90$	0	0
$91$	61.4986	0.0708440
$92$	−2288.48	−2.59338
$93$	0	0
$94$	1721.33	1.88874
$95$	0	0
$96$	0	0
$97$	−213.839	−0.223836	−0.111918	−	0.993717i	$-0.535699\pi$
$97$	−213.839	−0.223836	−0.111918	+	0.993717i	$0.535699\pi$
$98$	240.405	0.247802
$99$	0	0
$100$	0	0
$101$	1104.08	1.08772	0.543862	−	0.839175i	$-0.316961\pi$
$101$	1104.08	1.08772	0.543862	+	0.839175i	$0.316961\pi$
$102$	0	0
$103$	1.06266	0.00101657	0.000508287	−	1.00000i	$-0.499838\pi$
$103$	1.06266	0.00101657	0.000508287		1.00000i	$0.499838\pi$
$104$	194.329	0.183226
$105$	0	0
$106$	3348.45	3.06821
$107$	−818.220	−0.739255	−0.369628	−	0.929180i	$-0.620515\pi$
$107$	−818.220	−0.739255	−0.369628	+	0.929180i	$0.620515\pi$
$108$	0	0
$109$	686.558	0.603306	0.301653	−	0.953418i	$-0.402462\pi$
$109$	686.558	0.603306	0.301653	+	0.953418i	$0.402462\pi$
$110$	0	0
$111$	0	0
$112$	−2270.16	−1.91527
$113$	1248.89	1.03969	0.519846	−	0.854260i	$-0.325989\pi$
$113$	1248.89	1.03969	0.519846	+	0.854260i	$0.325989\pi$
$114$	0	0
$115$	0	0
$116$	499.262	0.399615
$117$	0	0
$118$	3891.56	3.03599
$119$	1751.14	1.34897
$120$	0	0
$121$	121.000	0.0909091
$122$	2474.41	1.83625
$123$	0	0
$124$	−3237.40	−2.34458
$125$	0	0
$126$	0	0
$127$	36.6674	0.0256197	0.0128099	−	0.999918i	$-0.495922\pi$
$127$	36.6674	0.0256197	0.0128099	+	0.999918i	$0.495922\pi$
$128$	901.576	0.622569
$129$	0	0
$130$	0	0
$131$	−1294.70	−0.863503	−0.431751	−	0.901993i	$-0.642104\pi$
$131$	−1294.70	−0.863503	−0.431751	+	0.901993i	$0.642104\pi$
$132$	0	0
$133$	−1506.82	−0.982393
$134$	2826.81	1.82238
$135$	0	0
$136$	5533.42	3.48887
$137$	−2803.43	−1.74827	−0.874137	−	0.485679i	$-0.838572\pi$
$137$	−2803.43	−1.74827	−0.874137	+	0.485679i	$0.838572\pi$
$138$	0	0
$139$	1858.88	1.13430	0.567152	−	0.823613i	$-0.308045\pi$
$139$	1858.88	1.13430	0.567152	+	0.823613i	$0.308045\pi$
$140$	0	0
$141$	0	0
$142$	−3242.70	−1.91635
$143$	39.2967	0.0229801
$144$	0	0
$145$	0	0
$146$	6219.68	3.52564
$147$	0	0
$148$	6105.65	3.39109
$149$	1062.79	0.584343	0.292171	−	0.956366i	$-0.405622\pi$
$149$	1062.79	0.584343	0.292171	+	0.956366i	$0.405622\pi$
$150$	0	0
$151$	−1462.14	−0.787997	−0.393999	−	0.919111i	$-0.628909\pi$
$151$	−1462.14	−0.787997	−0.393999	+	0.919111i	$0.628909\pi$
$152$	−4761.40	−2.54079
$153$	0	0
$154$	−975.830	−0.510615
$155$	0	0
$156$	0	0
$157$	2880.43	1.46423	0.732114	−	0.681183i	$-0.238534\pi$
$157$	2880.43	1.46423	0.732114	+	0.681183i	$0.238534\pi$
$158$	−2417.01	−1.21701
$159$	0	0
$160$	0	0
$161$	2123.09	1.03927
$162$	0	0
$163$	−2655.52	−1.27605	−0.638025	−	0.770015i	$-0.720248\pi$
$163$	−2655.52	−1.27605	−0.638025	+	0.770015i	$0.720248\pi$
$164$	−391.921	−0.186609
$165$	0	0
$166$	−2622.25	−1.22606
$167$	−150.294	−0.0696411	−0.0348206	−	0.999394i	$-0.511086\pi$
$167$	−150.294	−0.0696411	−0.0348206	+	0.999394i	$0.511086\pi$
$168$	0	0
$169$	−2184.24	−0.994191
$170$	0	0
$171$	0	0
$172$	6750.75	2.99267
$173$	−1987.49	−0.873443	−0.436722	−	0.899597i	$-0.643861\pi$
$173$	−1987.49	−0.873443	−0.436722	+	0.899597i	$0.643861\pi$
$174$	0	0
$175$	0	0
$176$	−1450.60	−0.621267
$177$	0	0
$178$	−1409.52	−0.593526
$179$	2127.18	0.888229	0.444115	−	0.895970i	$-0.353518\pi$
$179$	2127.18	0.888229	0.444115	+	0.895970i	$0.353518\pi$
$180$	0	0
$181$	−4654.68	−1.91149	−0.955744	−	0.294198i	$-0.904947\pi$
$181$	−4654.68	−1.91149	−0.955744	+	0.294198i	$0.904947\pi$
$182$	−316.917	−0.129074
$183$	0	0
$184$	6708.73	2.68790
$185$	0	0
$186$	0	0
$187$	1118.96	0.437573
$188$	−6198.19	−2.40452
$189$	0	0
$190$	0	0
$191$	−5083.70	−1.92588	−0.962941	−	0.269712i	$-0.913071\pi$
$191$	−5083.70	−1.92588	−0.962941	+	0.269712i	$0.913071\pi$
$192$	0	0
$193$	2895.27	1.07982	0.539911	−	0.841722i	$-0.318458\pi$
$193$	2895.27	1.07982	0.539911	+	0.841722i	$0.318458\pi$
$194$	1101.96	0.407816
$195$	0	0
$196$	−865.654	−0.315472
$197$	1272.10	0.460070	0.230035	−	0.973182i	$-0.426116\pi$
$197$	1272.10	0.460070	0.230035	+	0.973182i	$0.426116\pi$
$198$	0	0
$199$	−3286.36	−1.17067	−0.585337	−	0.810790i	$-0.699038\pi$
$199$	−3286.36	−1.17067	−0.585337	+	0.810790i	$0.699038\pi$
$200$	0	0
$201$	0	0
$202$	−5689.59	−1.98177
$203$	−463.179	−0.160142
$204$	0	0
$205$	0	0
$206$	−5.47614	−0.00185214
$207$	0	0
$208$	−471.106	−0.157045
$209$	−962.839	−0.318665
$210$	0	0
$211$	30.8947	0.0100800	0.00503999	−	0.999987i	$-0.498396\pi$
$211$	30.8947	0.0100800	0.00503999	+	0.999987i	$0.498396\pi$
$212$	−12057.1	−3.90607
$213$	0	0
$214$	4216.48	1.34688
$215$	0	0
$216$	0	0
$217$	3003.43	0.939567
$218$	−3538.00	−1.09919
$219$	0	0
$220$	0	0
$221$	363.399	0.110610
$222$	0	0
$223$	−340.381	−0.102214	−0.0511068	−	0.998693i	$-0.516275\pi$
$223$	−340.381	−0.102214	−0.0511068	+	0.998693i	$0.516275\pi$
$224$	4207.24	1.25495
$225$	0	0
$226$	−6435.80	−1.89426
$227$	463.577	0.135545	0.0677724	−	0.997701i	$-0.478411\pi$
$227$	463.577	0.135545	0.0677724	+	0.997701i	$0.478411\pi$
$228$	0	0
$229$	−374.108	−0.107955	−0.0539776	−	0.998542i	$-0.517190\pi$
$229$	−374.108	−0.107955	−0.0539776	+	0.998542i	$0.517190\pi$
$230$	0	0
$231$	0	0
$232$	−1463.59	−0.414180
$233$	6254.52	1.75857	0.879286	−	0.476294i	$-0.158020\pi$
$233$	6254.52	1.75857	0.879286	+	0.476294i	$0.158020\pi$
$234$	0	0
$235$	0	0
$236$	−14012.8	−3.86506
$237$	0	0
$238$	−9024.06	−2.45774
$239$	120.575	0.0326332	0.0163166	−	0.999867i	$-0.494806\pi$
$239$	120.575	0.0326332	0.0163166	+	0.999867i	$0.494806\pi$
$240$	0	0
$241$	−5338.53	−1.42691	−0.713454	−	0.700702i	$-0.752870\pi$
$241$	−5338.53	−1.42691	−0.713454	+	0.700702i	$0.752870\pi$
$242$	−623.542	−0.165631
$243$	0	0
$244$	−8909.90	−2.33770
$245$	0	0
$246$	0	0
$247$	−312.698	−0.0805526
$248$	9490.50	2.43003
$249$	0	0
$250$	0	0
$251$	3861.52	0.971064	0.485532	−	0.874219i	$-0.338626\pi$
$251$	3861.52	0.971064	0.485532	+	0.874219i	$0.338626\pi$
$252$	0	0
$253$	1356.62	0.337116
$254$	−188.956	−0.0466777
$255$	0	0
$256$	−6281.67	−1.53361
$257$	1760.31	0.427258	0.213629	−	0.976915i	$-0.431472\pi$
$257$	1760.31	0.427258	0.213629	+	0.976915i	$0.431472\pi$
$258$	0	0
$259$	−5664.38	−1.35895
$260$	0	0
$261$	0	0
$262$	6671.92	1.57325
$263$	2664.05	0.624610	0.312305	−	0.949982i	$-0.398899\pi$
$263$	2664.05	0.624610	0.312305	+	0.949982i	$0.398899\pi$
$264$	0	0
$265$	0	0
$266$	7765.02	1.78986
$267$	0	0
$268$	−10178.8	−2.32004
$269$	3476.07	0.787881	0.393940	−	0.919136i	$-0.371112\pi$
$269$	3476.07	0.787881	0.393940	+	0.919136i	$0.371112\pi$
$270$	0	0
$271$	1991.42	0.446384	0.223192	−	0.974775i	$-0.428352\pi$
$271$	1991.42	0.446384	0.223192	+	0.974775i	$0.428352\pi$
$272$	−13414.5	−2.99035
$273$	0	0
$274$	14446.8	3.18526
$275$	0	0
$276$	0	0
$277$	−8055.12	−1.74724	−0.873619	−	0.486610i	$-0.838233\pi$
$277$	−8055.12	−1.74724	−0.873619	+	0.486610i	$0.838233\pi$
$278$	−9579.26	−2.06664
$279$	0	0
$280$	0	0
$281$	3741.65	0.794335	0.397167	−	0.917746i	$-0.369993\pi$
$281$	3741.65	0.794335	0.397167	+	0.917746i	$0.369993\pi$
$282$	0	0
$283$	8888.34	1.86699	0.933493	−	0.358595i	$-0.116744\pi$
$283$	8888.34	1.86699	0.933493	+	0.358595i	$0.116744\pi$
$284$	11676.4	2.43967
$285$	0	0
$286$	−202.505	−0.0418685
$287$	363.596	0.0747820
$288$	0	0
$289$	5434.62	1.10617
$290$	0	0
$291$	0	0
$292$	−22395.9	−4.48843
$293$	−3722.94	−0.742309	−0.371154	−	0.928571i	$-0.621038\pi$
$293$	−3722.94	−0.742309	−0.371154	+	0.928571i	$0.621038\pi$
$294$	0	0
$295$	0	0
$296$	−17898.8	−3.51469
$297$	0	0
$298$	−5476.80	−1.06464
$299$	440.586	0.0852165
$300$	0	0
$301$	−6262.86	−1.19929
$302$	7534.77	1.43569
$303$	0	0
$304$	11542.9	2.17774
$305$	0	0
$306$	0	0
$307$	−7951.42	−1.47821	−0.739106	−	0.673589i	$-0.764752\pi$
$307$	−7951.42	−1.47821	−0.739106	+	0.673589i	$0.764752\pi$
$308$	3513.78	0.650053
$309$	0	0
$310$	0	0
$311$	−2979.37	−0.543231	−0.271615	−	0.962406i	$-0.587558\pi$
$311$	−2979.37	−0.543231	−0.271615	+	0.962406i	$0.587558\pi$
$312$	0	0
$313$	−5069.70	−0.915516	−0.457758	−	0.889077i	$-0.651347\pi$
$313$	−5069.70	−0.915516	−0.457758	+	0.889077i	$0.651347\pi$
$314$	−14843.6	−2.66774
$315$	0	0
$316$	8703.22	1.54935
$317$	−5830.65	−1.03307	−0.516533	−	0.856267i	$-0.672778\pi$
$317$	−5830.65	−1.03307	−0.516533	+	0.856267i	$0.672778\pi$
$318$	0	0
$319$	−295.965	−0.0519463
$320$	0	0
$321$	0	0
$322$	−10940.8	−1.89350
$323$	−8903.92	−1.53383
$324$	0	0
$325$	0	0
$326$	13684.5	2.32489
$327$	0	0
$328$	1148.92	0.193411
$329$	5750.24	0.963589
$330$	0	0
$331$	−2586.83	−0.429563	−0.214781	−	0.976662i	$-0.568904\pi$
$331$	−2586.83	−0.429563	−0.214781	+	0.976662i	$0.568904\pi$
$332$	9442.24	1.56087
$333$	0	0
$334$	774.499	0.126882
$335$	0	0
$336$	0	0
$337$	1715.16	0.277242	0.138621	−	0.990346i	$-0.455733\pi$
$337$	1715.16	0.277242	0.138621	+	0.990346i	$0.455733\pi$
$338$	11255.9	1.81136
$339$	0	0
$340$	0	0
$341$	1919.15	0.304773
$342$	0	0
$343$	6707.76	1.05593
$344$	−19789.9	−3.10175
$345$	0	0
$346$	10242.0	1.59137
$347$	−3552.95	−0.549661	−0.274830	−	0.961493i	$-0.588622\pi$
$347$	−3552.95	−0.549661	−0.274830	+	0.961493i	$0.588622\pi$
$348$	0	0
$349$	−907.612	−0.139207	−0.0696037	−	0.997575i	$-0.522173\pi$
$349$	−907.612	−0.139207	−0.0696037	+	0.997575i	$0.522173\pi$
$350$	0	0
$351$	0	0
$352$	2688.37	0.407075
$353$	8960.39	1.35103	0.675515	−	0.737346i	$-0.263921\pi$
$353$	8960.39	1.35103	0.675515	+	0.737346i	$0.263921\pi$
$354$	0	0
$355$	0	0
$356$	5075.41	0.755607
$357$	0	0
$358$	−10961.9	−1.61830
$359$	−5230.65	−0.768978	−0.384489	−	0.923129i	$-0.625622\pi$
$359$	−5230.65	−0.768978	−0.384489	+	0.923129i	$0.625622\pi$
$360$	0	0
$361$	802.645	0.117021
$362$	23986.7	3.48263
$363$	0	0
$364$	1141.16	0.164321
$365$	0	0
$366$	0	0
$367$	−4412.00	−0.627532	−0.313766	−	0.949500i	$-0.601591\pi$
$367$	−4412.00	−0.627532	−0.313766	+	0.949500i	$0.601591\pi$
$368$	−16263.8	−2.30383
$369$	0	0
$370$	0	0
$371$	11185.7	1.56532
$372$	0	0
$373$	1588.57	0.220518	0.110259	−	0.993903i	$-0.464832\pi$
$373$	1588.57	0.220518	0.110259	+	0.993903i	$0.464832\pi$
$374$	−5766.24	−0.797234
$375$	0	0
$376$	18170.1	2.49216
$377$	−96.1195	−0.0131310
$378$	0	0
$379$	5640.53	0.764471	0.382236	−	0.924065i	$-0.375154\pi$
$379$	5640.53	0.764471	0.382236	+	0.924065i	$0.375154\pi$
$380$	0	0
$381$	0	0
$382$	26197.5	3.50885
$383$	1250.32	0.166810	0.0834052	−	0.996516i	$-0.473420\pi$
$383$	1250.32	0.166810	0.0834052	+	0.996516i	$0.473420\pi$
$384$	0	0
$385$	0	0
$386$	−14920.0	−1.96738
$387$	0	0
$388$	−3967.96	−0.519182
$389$	−8716.42	−1.13609	−0.568047	−	0.822996i	$-0.692301\pi$
$389$	−8716.42	−1.13609	−0.568047	+	0.822996i	$0.692301\pi$
$390$	0	0
$391$	12545.5	1.62264
$392$	2537.68	0.326970
$393$	0	0
$394$	−6555.46	−0.838221
$395$	0	0
$396$	0	0
$397$	9769.48	1.23505	0.617527	−	0.786550i	$-0.288135\pi$
$397$	9769.48	1.23505	0.617527	+	0.786550i	$0.288135\pi$
$398$	16935.4	2.13290
$399$	0	0
$400$	0	0
$401$	−7110.67	−0.885511	−0.442756	−	0.896642i	$-0.645999\pi$
$401$	−7110.67	−0.885511	−0.442756	+	0.896642i	$0.645999\pi$
$402$	0	0
$403$	623.275	0.0770410
$404$	20487.2	2.52296
$405$	0	0
$406$	2386.87	0.291770
$407$	−3619.46	−0.440811
$408$	0	0
$409$	−10194.8	−1.23251	−0.616257	−	0.787545i	$-0.711352\pi$
$409$	−10194.8	−1.23251	−0.616257	+	0.787545i	$0.711352\pi$
$410$	0	0
$411$	0	0
$412$	19.7186	0.00235792
$413$	13000.0	1.54889
$414$	0	0
$415$	0	0
$416$	873.091	0.102901
$417$	0	0
$418$	4961.74	0.580590
$419$	14689.4	1.71271	0.856356	−	0.516386i	$-0.172723\pi$
$419$	14689.4	1.71271	0.856356	+	0.516386i	$0.172723\pi$
$420$	0	0
$421$	7751.00	0.897293	0.448647	−	0.893709i	$-0.351906\pi$
$421$	7751.00	0.897293	0.448647	+	0.893709i	$0.351906\pi$
$422$	−159.207	−0.0183652
$423$	0	0
$424$	35345.7	4.04844
$425$	0	0
$426$	0	0
$427$	8265.97	0.936811
$428$	−15182.8	−1.71469
$429$	0	0
$430$	0	0
$431$	−14256.5	−1.59330	−0.796651	−	0.604440i	$-0.793397\pi$
$431$	−14256.5	−1.59330	−0.796651	+	0.604440i	$0.793397\pi$
$432$	0	0
$433$	6125.38	0.679831	0.339916	−	0.940456i	$-0.389601\pi$
$433$	6125.38	0.679831	0.339916	+	0.940456i	$0.389601\pi$
$434$	−15477.4	−1.71184
$435$	0	0
$436$	12739.7	1.39936
$437$	−10795.1	−1.18170
$438$	0	0
$439$	−2582.93	−0.280812	−0.140406	−	0.990094i	$-0.544841\pi$
$439$	−2582.93	−0.280812	−0.140406	+	0.990094i	$0.544841\pi$
$440$	0	0
$441$	0	0
$442$	−1872.68	−0.201526
$443$	4858.88	0.521112	0.260556	−	0.965459i	$-0.416094\pi$
$443$	4858.88	0.521112	0.260556	+	0.965459i	$0.416094\pi$
$444$	0	0
$445$	0	0
$446$	1754.07	0.186227
$447$	0	0
$448$	−3519.63	−0.371176
$449$	281.684	0.0296069	0.0148035	−	0.999890i	$-0.495288\pi$
$449$	281.684	0.0296069	0.0148035	+	0.999890i	$0.495288\pi$
$450$	0	0
$451$	232.333	0.0242575
$452$	23174.1	2.41155
$453$	0	0
$454$	−2388.92	−0.246955
$455$	0	0
$456$	0	0
$457$	9641.24	0.986866	0.493433	−	0.869784i	$-0.335742\pi$
$457$	9641.24	0.986866	0.493433	+	0.869784i	$0.335742\pi$
$458$	1927.87	0.196688
$459$	0	0
$460$	0	0
$461$	12095.4	1.22199	0.610996	−	0.791634i	$-0.290769\pi$
$461$	12095.4	1.22199	0.610996	+	0.791634i	$0.290769\pi$
$462$	0	0
$463$	19607.5	1.96812	0.984059	−	0.177843i	$-0.0569120\pi$
$463$	19607.5	1.96812	0.984059	+	0.177843i	$0.0569120\pi$
$464$	3548.15	0.354998
$465$	0	0
$466$	−32231.0	−3.20402
$467$	−9636.33	−0.954852	−0.477426	−	0.878672i	$-0.658430\pi$
$467$	−9636.33	−0.954852	−0.477426	+	0.878672i	$0.658430\pi$
$468$	0	0
$469$	9443.17	0.929734
$470$	0	0
$471$	0	0
$472$	41078.7	4.00593
$473$	−4001.88	−0.389020
$474$	0	0
$475$	0	0
$476$	32494.0	3.12890
$477$	0	0
$478$	−621.351	−0.0594560
$479$	−542.716	−0.0517689	−0.0258845	−	0.999665i	$-0.508240\pi$
$479$	−542.716	−0.0517689	−0.0258845	+	0.999665i	$0.508240\pi$
$480$	0	0
$481$	−1175.48	−0.111429
$482$	27510.7	2.59975
$483$	0	0
$484$	2245.26	0.210862
$485$	0	0
$486$	0	0
$487$	6689.76	0.622468	0.311234	−	0.950333i	$-0.399258\pi$
$487$	6689.76	0.622468	0.311234	+	0.950333i	$0.399258\pi$
$488$	26119.5	2.42290
$489$	0	0
$490$	0	0
$491$	−3148.45	−0.289384	−0.144692	−	0.989477i	$-0.546219\pi$
$491$	−3148.45	−0.289384	−0.144692	+	0.989477i	$0.546219\pi$
$492$	0	0
$493$	−2736.96	−0.250033
$494$	1611.41	0.146762
$495$	0	0
$496$	−23007.6	−2.08280
$497$	−10832.5	−0.977674
$498$	0	0
$499$	6978.29	0.626035	0.313017	−	0.949747i	$-0.398660\pi$
$499$	6978.29	0.626035	0.313017	+	0.949747i	$0.398660\pi$
$500$	0	0
$501$	0	0
$502$	−19899.3	−1.76922
$503$	20783.7	1.84234	0.921172	−	0.389156i	$-0.127233\pi$
$503$	20783.7	1.84234	0.921172	+	0.389156i	$0.127233\pi$
$504$	0	0
$505$	0	0
$506$	−6991.00	−0.614206
$507$	0	0
$508$	680.395	0.0594245
$509$	14365.4	1.25095	0.625476	−	0.780244i	$-0.284905\pi$
$509$	14365.4	1.25095	0.625476	+	0.780244i	$0.284905\pi$
$510$	0	0
$511$	20777.3	1.79870
$512$	25158.3	2.17158
$513$	0	0
$514$	−9071.31	−0.778440
$515$	0	0
$516$	0	0
$517$	3674.32	0.312566
$518$	29189.9	2.47593
$519$	0	0
$520$	0	0
$521$	14804.3	1.24489	0.622446	−	0.782663i	$-0.286139\pi$
$521$	14804.3	1.24489	0.622446	+	0.782663i	$0.286139\pi$
$522$	0	0
$523$	1055.85	0.0882776	0.0441388	−	0.999025i	$-0.485946\pi$
$523$	1055.85	0.0882776	0.0441388	+	0.999025i	$0.485946\pi$
$524$	−24024.3	−2.00288
$525$	0	0
$526$	−13728.5	−1.13800
$527$	17747.5	1.46697
$528$	0	0
$529$	3043.16	0.250116
$530$	0	0
$531$	0	0
$532$	−27960.4	−2.27864
$533$	75.4539	0.00613184
$534$	0	0
$535$	0	0
$536$	29839.4	2.40460
$537$	0	0
$538$	−17913.0	−1.43547
$539$	513.164	0.0410084
$540$	0	0
$541$	−1252.03	−0.0994993	−0.0497496	−	0.998762i	$-0.515842\pi$
$541$	−1252.03	−0.0994993	−0.0497496	+	0.998762i	$0.515842\pi$
$542$	−10262.2	−0.813286
$543$	0	0
$544$	24860.9	1.95938
$545$	0	0
$546$	0	0
$547$	−1608.10	−0.125699	−0.0628495	−	0.998023i	$-0.520019\pi$
$547$	−1608.10	−0.125699	−0.0628495	+	0.998023i	$0.520019\pi$
$548$	−52020.1	−4.05509
$549$	0	0
$550$	0	0
$551$	2355.10	0.182088
$552$	0	0
$553$	−8074.22	−0.620888
$554$	41509.9	3.18337
$555$	0	0
$556$	34493.1	2.63100
$557$	11465.4	0.872184	0.436092	−	0.899902i	$-0.356362\pi$
$557$	11465.4	0.872184	0.436092	+	0.899902i	$0.356362\pi$
$558$	0	0
$559$	−1299.68	−0.0983371
$560$	0	0
$561$	0	0
$562$	−19281.6	−1.44723
$563$	11947.0	0.894327	0.447164	−	0.894452i	$-0.352434\pi$
$563$	11947.0	0.894327	0.447164	+	0.894452i	$0.352434\pi$
$564$	0	0
$565$	0	0
$566$	−45803.7	−3.40155
$567$	0	0
$568$	−34229.5	−2.52859
$569$	−24287.5	−1.78943	−0.894713	−	0.446641i	$-0.852620\pi$
$569$	−24287.5	−1.78943	−0.894713	+	0.446641i	$0.852620\pi$
$570$	0	0
$571$	−3441.90	−0.252257	−0.126129	−	0.992014i	$-0.540255\pi$
$571$	−3441.90	−0.252257	−0.126129	+	0.992014i	$0.540255\pi$
$572$	729.184	0.0533020
$573$	0	0
$574$	−1873.70	−0.136249
$575$	0	0
$576$	0	0
$577$	11760.6	0.848525	0.424263	−	0.905539i	$-0.360533\pi$
$577$	11760.6	0.848525	0.424263	+	0.905539i	$0.360533\pi$
$578$	−28005.9	−2.01538
$579$	0	0
$580$	0	0
$581$	−8759.83	−0.625506
$582$	0	0
$583$	7147.53	0.507754
$584$	65653.9	4.65202
$585$	0	0
$586$	19185.2	1.35245
$587$	7248.40	0.509665	0.254832	−	0.966985i	$-0.417980\pi$
$587$	7248.40	0.509665	0.254832	+	0.966985i	$0.417980\pi$
$588$	0	0
$589$	−15271.3	−1.06833
$590$	0	0
$591$	0	0
$592$	43391.6	3.01247
$593$	3749.26	0.259635	0.129818	−	0.991538i	$-0.458561\pi$
$593$	3749.26	0.259635	0.129818	+	0.991538i	$0.458561\pi$
$594$	0	0
$595$	0	0
$596$	19720.9	1.35537
$597$	0	0
$598$	−2270.44	−0.155260
$599$	−8920.04	−0.608452	−0.304226	−	0.952600i	$-0.598398\pi$
$599$	−8920.04	−0.608452	−0.304226	+	0.952600i	$0.598398\pi$
$600$	0	0
$601$	2508.42	0.170251	0.0851254	−	0.996370i	$-0.472871\pi$
$601$	2508.42	0.170251	0.0851254	+	0.996370i	$0.472871\pi$
$602$	32274.0	2.18503
$603$	0	0
$604$	−27131.3	−1.82774
$605$	0	0
$606$	0	0
$607$	18233.8	1.21925	0.609627	−	0.792688i	$-0.291319\pi$
$607$	18233.8	1.21925	0.609627	+	0.792688i	$0.291319\pi$
$608$	−21392.3	−1.42693
$609$	0	0
$610$	0	0
$611$	1193.30	0.0790107
$612$	0	0
$613$	10888.0	0.717396	0.358698	−	0.933454i	$-0.383221\pi$
$613$	10888.0	0.717396	0.358698	+	0.933454i	$0.383221\pi$
$614$	40975.5	2.69322
$615$	0	0
$616$	−10300.7	−0.673746
$617$	15660.1	1.02180	0.510901	−	0.859640i	$-0.329312\pi$
$617$	15660.1	1.02180	0.510901	+	0.859640i	$0.329312\pi$
$618$	0	0
$619$	10433.7	0.677488	0.338744	−	0.940879i	$-0.389998\pi$
$619$	10433.7	0.677488	0.338744	+	0.940879i	$0.389998\pi$
$620$	0	0
$621$	0	0
$622$	15353.4	0.989736
$623$	−4708.60	−0.302802
$624$	0	0
$625$	0	0
$626$	26125.4	1.66802
$627$	0	0
$628$	53448.9	3.39625
$629$	−33471.2	−2.12176
$630$	0	0
$631$	17432.7	1.09982	0.549910	−	0.835224i	$-0.314662\pi$
$631$	17432.7	1.09982	0.549910	+	0.835224i	$0.314662\pi$
$632$	−25513.6	−1.60582
$633$	0	0
$634$	30046.7	1.88219
$635$	0	0
$636$	0	0
$637$	166.659	0.0103662
$638$	1525.18	0.0946432
$639$	0	0
$640$	0	0
$641$	−3423.99	−0.210982	−0.105491	−	0.994420i	$-0.533641\pi$
$641$	−3423.99	−0.210982	−0.105491	+	0.994420i	$0.533641\pi$
$642$	0	0
$643$	−10992.3	−0.674177	−0.337088	−	0.941473i	$-0.609442\pi$
$643$	−10992.3	−0.674177	−0.337088	+	0.941473i	$0.609442\pi$
$644$	39395.7	2.41057
$645$	0	0
$646$	45884.0	2.79455
$647$	1707.80	0.103772	0.0518861	−	0.998653i	$-0.483477\pi$
$647$	1707.80	0.103772	0.0518861	+	0.998653i	$0.483477\pi$
$648$	0	0
$649$	8306.84	0.502422
$650$	0	0
$651$	0	0
$652$	−49275.4	−2.95977
$653$	−10332.5	−0.619205	−0.309602	−	0.950866i	$-0.600196\pi$
$653$	−10332.5	−0.619205	−0.309602	+	0.950866i	$0.600196\pi$
$654$	0	0
$655$	0	0
$656$	−2785.31	−0.165774
$657$	0	0
$658$	−29632.3	−1.75561
$659$	−21363.9	−1.26285	−0.631426	−	0.775436i	$-0.717530\pi$
$659$	−21363.9	−1.26285	−0.631426	+	0.775436i	$0.717530\pi$
$660$	0	0
$661$	27607.4	1.62451	0.812257	−	0.583299i	$-0.198238\pi$
$661$	27607.4	1.62451	0.812257	+	0.583299i	$0.198238\pi$
$662$	13330.6	0.782639
$663$	0	0
$664$	−27680.1	−1.61777
$665$	0	0
$666$	0	0
$667$	−3318.29	−0.192631
$668$	−2788.83	−0.161531
$669$	0	0
$670$	0	0
$671$	5281.83	0.303879
$672$	0	0
$673$	1646.00	0.0942771	0.0471385	−	0.998888i	$-0.484990\pi$
$673$	1646.00	0.0942771	0.0471385	+	0.998888i	$0.484990\pi$
$674$	−8838.61	−0.505120
$675$	0	0
$676$	−40530.4	−2.30601
$677$	8654.56	0.491317	0.245659	−	0.969356i	$-0.420996\pi$
$677$	8654.56	0.491317	0.245659	+	0.969356i	$0.420996\pi$
$678$	0	0
$679$	3681.19	0.208058
$680$	0	0
$681$	0	0
$682$	−9889.83	−0.555280
$683$	−3915.74	−0.219373	−0.109686	−	0.993966i	$-0.534985\pi$
$683$	−3915.74	−0.219373	−0.109686	+	0.993966i	$0.534985\pi$
$684$	0	0
$685$	0	0
$686$	−34566.7	−1.92385
$687$	0	0
$688$	47976.2	2.65854
$689$	2321.28	0.128351
$690$	0	0
$691$	−23756.6	−1.30788	−0.653938	−	0.756548i	$-0.726884\pi$
$691$	−23756.6	−1.30788	−0.653938	+	0.756548i	$0.726884\pi$
$692$	−36879.5	−2.02594
$693$	0	0
$694$	18309.2	1.00145
$695$	0	0
$696$	0	0
$697$	2148.52	0.116759
$698$	4677.14	0.253628
$699$	0	0
$700$	0	0
$701$	25501.1	1.37399	0.686993	−	0.726664i	$-0.258930\pi$
$701$	25501.1	1.37399	0.686993	+	0.726664i	$0.258930\pi$
$702$	0	0
$703$	28801.3	1.54518
$704$	−2248.99	−0.120401
$705$	0	0
$706$	−46175.0	−2.46150
$707$	−19006.5	−1.01105
$708$	0	0
$709$	−7926.20	−0.419852	−0.209926	−	0.977717i	$-0.567322\pi$
$709$	−7926.20	−0.419852	−0.209926	+	0.977717i	$0.567322\pi$
$710$	0	0
$711$	0	0
$712$	−14878.6	−0.783147
$713$	21517.1	1.13018
$714$	0	0
$715$	0	0
$716$	39471.7	2.06023
$717$	0	0
$718$	26954.8	1.40104
$719$	4680.22	0.242758	0.121379	−	0.992606i	$-0.461268\pi$
$719$	4680.22	0.242758	0.121379	+	0.992606i	$0.461268\pi$
$720$	0	0
$721$	−18.2935	−0.000944916 0
$722$	−4136.22	−0.213205
$723$	0	0
$724$	−86371.5	−4.43366
$725$	0	0
$726$	0	0
$727$	−19909.9	−1.01571	−0.507853	−	0.861444i	$-0.669561\pi$
$727$	−19909.9	−1.01571	−0.507853	+	0.861444i	$0.669561\pi$
$728$	−3345.33	−0.170311
$729$	0	0
$730$	0	0
$731$	−37007.6	−1.87247
$732$	0	0
$733$	496.199	0.0250034	0.0125017	−	0.999922i	$-0.496020\pi$
$733$	496.199	0.0250034	0.0125017	+	0.999922i	$0.496020\pi$
$734$	22736.1	1.14333
$735$	0	0
$736$	30141.4	1.50954
$737$	6034.05	0.301584
$738$	0	0
$739$	−29735.0	−1.48014	−0.740069	−	0.672531i	$-0.765207\pi$
$739$	−29735.0	−1.48014	−0.740069	+	0.672531i	$0.765207\pi$
$740$	0	0
$741$	0	0
$742$	−57642.8	−2.85193
$743$	−2883.95	−0.142398	−0.0711990	−	0.997462i	$-0.522683\pi$
$743$	−2883.95	−0.142398	−0.0711990	+	0.997462i	$0.522683\pi$
$744$	0	0
$745$	0	0
$746$	−8186.28	−0.401771
$747$	0	0
$748$	20763.2	1.01494
$749$	14085.5	0.687146
$750$	0	0
$751$	22975.7	1.11637	0.558185	−	0.829716i	$-0.311498\pi$
$751$	22975.7	1.11637	0.558185	+	0.829716i	$0.311498\pi$
$752$	−44049.3	−2.13605
$753$	0	0
$754$	495.326	0.0239240
$755$	0	0
$756$	0	0
$757$	−5315.56	−0.255214	−0.127607	−	0.991825i	$-0.540730\pi$
$757$	−5315.56	−0.255214	−0.127607	+	0.991825i	$0.540730\pi$
$758$	−29067.0	−1.39282
$759$	0	0
$760$	0	0
$761$	1355.67	0.0645767	0.0322884	−	0.999479i	$-0.489721\pi$
$761$	1355.67	0.0645767	0.0322884	+	0.999479i	$0.489721\pi$
$762$	0	0
$763$	−11819.0	−0.560780
$764$	−94332.3	−4.46705
$765$	0	0
$766$	−6443.20	−0.303919
$767$	2697.78	0.127003
$768$	0	0
$769$	25270.4	1.18501	0.592505	−	0.805567i	$-0.298139\pi$
$769$	25270.4	1.18501	0.592505	+	0.805567i	$0.298139\pi$
$770$	0	0
$771$	0	0
$772$	53724.1	2.50463
$773$	2240.27	0.104239	0.0521197	−	0.998641i	$-0.483402\pi$
$773$	2240.27	0.104239	0.0521197	+	0.998641i	$0.483402\pi$
$774$	0	0
$775$	0	0
$776$	11632.1	0.538105
$777$	0	0
$778$	44917.8	2.06990
$779$	−1848.75	−0.0850302
$780$	0	0
$781$	−6921.81	−0.317134
$782$	−64649.8	−2.95636
$783$	0	0
$784$	−6152.03	−0.280249
$785$	0	0
$786$	0	0
$787$	14827.6	0.671597	0.335799	−	0.941934i	$-0.390994\pi$
$787$	14827.6	0.671597	0.335799	+	0.941934i	$0.390994\pi$
$788$	23605.0	1.06712
$789$	0	0
$790$	0	0
$791$	−21499.3	−0.966406
$792$	0	0
$793$	1715.36	0.0768150
$794$	−50344.5	−2.25020
$795$	0	0
$796$	−60981.2	−2.71536
$797$	3628.54	0.161267	0.0806334	−	0.996744i	$-0.474306\pi$
$797$	3628.54	0.161267	0.0806334	+	0.996744i	$0.474306\pi$
$798$	0	0
$799$	33978.5	1.50447
$800$	0	0
$801$	0	0
$802$	36643.0	1.61335
$803$	13276.4	0.583454
$804$	0	0
$805$	0	0
$806$	−3211.88	−0.140364
$807$	0	0
$808$	−60058.4	−2.61491
$809$	24925.9	1.08325	0.541624	−	0.840621i	$-0.317809\pi$
$809$	24925.9	1.08325	0.541624	+	0.840621i	$0.317809\pi$
$810$	0	0
$811$	−2375.63	−0.102860	−0.0514302	−	0.998677i	$-0.516378\pi$
$811$	−2375.63	−0.102860	−0.0514302	+	0.998677i	$0.516378\pi$
$812$	−8594.68	−0.371446
$813$	0	0
$814$	18651.9	0.803132
$815$	0	0
$816$	0	0
$817$	31844.3	1.36364
$818$	52536.0	2.24557
$819$	0	0
$820$	0	0
$821$	−37203.6	−1.58150	−0.790751	−	0.612137i	$-0.790310\pi$
$821$	−37203.6	−1.58150	−0.790751	+	0.612137i	$0.790310\pi$
$822$	0	0
$823$	13587.1	0.575475	0.287737	−	0.957709i	$-0.407097\pi$
$823$	13587.1	0.575475	0.287737	+	0.957709i	$0.407097\pi$
$824$	−57.8053	−0.00244387
$825$	0	0
$826$	−66992.3	−2.82199
$827$	−6366.67	−0.267704	−0.133852	−	0.991001i	$-0.542735\pi$
$827$	−6366.67	−0.267704	−0.133852	+	0.991001i	$0.542735\pi$
$828$	0	0
$829$	4317.65	0.180890	0.0904452	−	0.995901i	$-0.471171\pi$
$829$	4317.65	0.180890	0.0904452	+	0.995901i	$0.471171\pi$
$830$	0	0
$831$	0	0
$832$	−730.398	−0.0304351
$833$	4745.52	0.197386
$834$	0	0
$835$	0	0
$836$	−17866.3	−0.739137
$837$	0	0
$838$	−75698.2	−3.12047
$839$	18027.7	0.741818	0.370909	−	0.928669i	$-0.379046\pi$
$839$	18027.7	0.741818	0.370909	+	0.928669i	$0.379046\pi$
$840$	0	0
$841$	−23665.1	−0.970317
$842$	−39942.7	−1.63482
$843$	0	0
$844$	573.277	0.0233803
$845$	0	0
$846$	0	0
$847$	−2082.99	−0.0845010
$848$	−85687.6	−3.46996
$849$	0	0
$850$	0	0
$851$	−40580.5	−1.63465
$852$	0	0
$853$	20962.0	0.841413	0.420706	−	0.907197i	$-0.361782\pi$
$853$	20962.0	0.841413	0.420706	+	0.907197i	$0.361782\pi$
$854$	−42596.5	−1.70682
$855$	0	0
$856$	44508.5	1.77719
$857$	47275.4	1.88436	0.942181	−	0.335104i	$-0.108771\pi$
$857$	47275.4	1.88436	0.942181	+	0.335104i	$0.108771\pi$
$858$	0	0
$859$	2410.91	0.0957617	0.0478809	−	0.998853i	$-0.484753\pi$
$859$	2410.91	0.0957617	0.0478809	+	0.998853i	$0.484753\pi$
$860$	0	0
$861$	0	0
$862$	73467.3	2.90291
$863$	40814.5	1.60990	0.804950	−	0.593343i	$-0.202192\pi$
$863$	40814.5	1.60990	0.804950	+	0.593343i	$0.202192\pi$
$864$	0	0
$865$	0	0
$866$	−31565.5	−1.23862
$867$	0	0
$868$	55731.2	2.17931
$869$	−5159.31	−0.201401
$870$	0	0
$871$	1959.66	0.0762347
$872$	−37346.6	−1.45036
$873$	0	0
$874$	55629.9	2.15298
$875$	0	0
$876$	0	0
$877$	7035.18	0.270879	0.135440	−	0.990786i	$-0.456755\pi$
$877$	7035.18	0.270879	0.135440	+	0.990786i	$0.456755\pi$
$878$	13310.4	0.511624
$879$	0	0
$880$	0	0
$881$	32588.6	1.24624	0.623119	−	0.782127i	$-0.285865\pi$
$881$	32588.6	1.24624	0.623119	+	0.782127i	$0.285865\pi$
$882$	0	0
$883$	−40390.5	−1.53935	−0.769677	−	0.638434i	$-0.779583\pi$
$883$	−40390.5	−1.53935	−0.769677	+	0.638434i	$0.779583\pi$
$884$	6743.18	0.256559
$885$	0	0
$886$	−25039.0	−0.949437
$887$	6904.49	0.261364	0.130682	−	0.991424i	$-0.458283\pi$
$887$	6904.49	0.261364	0.130682	+	0.991424i	$0.458283\pi$
$888$	0	0
$889$	−631.221	−0.0238138
$890$	0	0
$891$	0	0
$892$	−6316.07	−0.237082
$893$	−29237.8	−1.09564
$894$	0	0
$895$	0	0
$896$	−15520.4	−0.578685
$897$	0	0
$898$	−1451.59	−0.0539421
$899$	−4694.22	−0.174150
$900$	0	0
$901$	66097.3	2.44397
$902$	−1197.27	−0.0441958
$903$	0	0
$904$	−67935.4	−2.49944
$905$	0	0
$906$	0	0
$907$	−24568.8	−0.899441	−0.449720	−	0.893169i	$-0.648476\pi$
$907$	−24568.8	−0.899441	−0.449720	+	0.893169i	$0.648476\pi$
$908$	8602.06	0.314394
$909$	0	0
$910$	0	0
$911$	−12588.0	−0.457805	−0.228902	−	0.973449i	$-0.573514\pi$
$911$	−12588.0	−0.457805	−0.228902	+	0.973449i	$0.573514\pi$
$912$	0	0
$913$	−5597.41	−0.202899
$914$	−49683.6	−1.79802
$915$	0	0
$916$	−6941.89	−0.250400
$917$	22288.1	0.802635
$918$	0	0
$919$	28899.6	1.03733	0.518666	−	0.854977i	$-0.326429\pi$
$919$	28899.6	1.03733	0.518666	+	0.854977i	$0.326429\pi$
$920$	0	0
$921$	0	0
$922$	−62330.4	−2.22640
$923$	−2247.97	−0.0801657
$924$	0	0
$925$	0	0
$926$	−101042.	−3.58580
$927$	0	0
$928$	−6575.72	−0.232606
$929$	−20170.9	−0.712363	−0.356181	−	0.934417i	$-0.615921\pi$
$929$	−20170.9	−0.712363	−0.356181	+	0.934417i	$0.615921\pi$
$930$	0	0
$931$	−4083.43	−0.143747
$932$	116058.	4.07897
$933$	0	0
$934$	49658.3	1.73969
$935$	0	0
$936$	0	0
$937$	−2665.62	−0.0929371	−0.0464685	−	0.998920i	$-0.514797\pi$
$937$	−2665.62	−0.0929371	−0.0464685	+	0.998920i	$0.514797\pi$
$938$	−48662.9	−1.69392
$939$	0	0
$940$	0	0
$941$	44964.8	1.55771	0.778857	−	0.627201i	$-0.215800\pi$
$941$	44964.8	1.55771	0.778857	+	0.627201i	$0.215800\pi$
$942$	0	0
$943$	2604.86	0.0899534
$944$	−99586.0	−3.43353
$945$	0	0
$946$	20622.6	0.708773
$947$	−21731.3	−0.745693	−0.372847	−	0.927893i	$-0.621618\pi$
$947$	−21731.3	−0.745693	−0.372847	+	0.927893i	$0.621618\pi$
$948$	0	0
$949$	4311.73	0.147486
$950$	0	0
$951$	0	0
$952$	−95256.6	−3.24294
$953$	27246.9	0.926143	0.463072	−	0.886321i	$-0.346747\pi$
$953$	27246.9	0.926143	0.463072	+	0.886321i	$0.346747\pi$
$954$	0	0
$955$	0	0
$956$	2237.37	0.0756922
$957$	0	0
$958$	2796.74	0.0943201
$959$	48260.5	1.62504
$960$	0	0
$961$	648.106	0.0217551
$962$	6057.52	0.203017
$963$	0	0
$964$	−99060.9	−3.30969
$965$	0	0
$966$	0	0
$967$	31755.3	1.05603	0.528016	−	0.849235i	$-0.322936\pi$
$967$	31755.3	1.05603	0.528016	+	0.849235i	$0.322936\pi$
$968$	−6582.01	−0.218547
$969$	0	0
$970$	0	0
$971$	13634.4	0.450617	0.225309	−	0.974287i	$-0.427661\pi$
$971$	13634.4	0.450617	0.225309	+	0.974287i	$0.427661\pi$
$972$	0	0
$973$	−32000.2	−1.05435
$974$	−34473.9	−1.13410
$975$	0	0
$976$	−63320.9	−2.07669
$977$	39260.5	1.28562	0.642811	−	0.766025i	$-0.277768\pi$
$977$	39260.5	1.28562	0.642811	+	0.766025i	$0.277768\pi$
$978$	0	0
$979$	−3008.73	−0.0982220
$980$	0	0
$981$	0	0
$982$	16224.7	0.527242
$983$	−16029.2	−0.520093	−0.260046	−	0.965596i	$-0.583738\pi$
$983$	−16029.2	−0.520093	−0.260046	+	0.965596i	$0.583738\pi$
$984$	0	0
$985$	0	0
$986$	14104.2	0.455546
$987$	0	0
$988$	−5802.37	−0.186840
$989$	−44868.1	−1.44259
$990$	0	0
$991$	−165.662	−0.00531021	−0.00265511	−	0.999996i	$-0.500845\pi$
$991$	−165.662	−0.00531021	−0.00265511	+	0.999996i	$0.500845\pi$
$992$	42639.5	1.36472
$993$	0	0
$994$	55822.4	1.78127
$995$	0	0
$996$	0	0
$997$	20199.2	0.641642	0.320821	−	0.947140i	$-0.396041\pi$
$997$	20199.2	0.641642	0.320821	+	0.947140i	$0.396041\pi$
$998$	−35960.8	−1.14060
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.4.a.bo.1.1		7
3.2	odd	2		825.4.a.bd.1.7		7
5.2	odd	4		495.4.c.d.199.1		14
5.3	odd	4		495.4.c.d.199.14		14
5.4	even	2		2475.4.a.bs.1.7		7
15.2	even	4		165.4.c.b.34.14	yes	14
15.8	even	4		165.4.c.b.34.1	✓	14
15.14	odd	2		825.4.a.ba.1.1		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.4.c.b.34.1	✓	14	15.8	even	4
165.4.c.b.34.14	yes	14	15.2	even	4
495.4.c.d.199.1		14	5.2	odd	4
495.4.c.d.199.14		14	5.3	odd	4
825.4.a.ba.1.1		7	15.14	odd	2
825.4.a.bd.1.7		7	3.2	odd	2
2475.4.a.bo.1.1		7	1.1	even	1	trivial
2475.4.a.bs.1.7		7	5.4	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:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2475.a (trivial)

\(f(q)\)	\(=\)	\(q-5.15324 q^{2} +18.5558 q^{4} -17.2148 q^{7} -54.3968 q^{8} +O(q^{10})\)	\(q-5.15324 q^{2} +18.5558 q^{4} -17.2148 q^{7} -54.3968 q^{8} -11.0000 q^{11} -3.57243 q^{13} +88.7118 q^{14} +131.873 q^{16} -101.723 q^{17} +87.5308 q^{19} +56.6856 q^{22} -123.329 q^{23} +18.4096 q^{26} -319.435 q^{28} +26.9059 q^{29} -174.468 q^{31} -244.397 q^{32} +524.204 q^{34} +329.042 q^{37} -451.067 q^{38} -21.1212 q^{41} +363.807 q^{43} -204.114 q^{44} +635.546 q^{46} -334.029 q^{47} -46.6513 q^{49} -66.2895 q^{52} -649.775 q^{53} +936.429 q^{56} -138.652 q^{58} -755.168 q^{59} -480.167 q^{61} +899.075 q^{62} +204.454 q^{64} -548.550 q^{67} -1887.56 q^{68} +629.256 q^{71} -1206.95 q^{73} -1695.63 q^{74} +1624.21 q^{76} +189.363 q^{77} +469.029 q^{79} +108.842 q^{82} +508.855 q^{83} -1874.78 q^{86} +598.365 q^{88} +273.521 q^{89} +61.4986 q^{91} -2288.48 q^{92} +1721.33 q^{94} -213.839 q^{97} +240.405 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 5 q^{2} + 13 q^{4} + 34 q^{7} - 75 q^{8}+O(q^{10}) \) 7 * q - 5 * q^2 + 13 * q^4 + 34 * q^7 - 75 * q^8	\( 7 q - 5 q^{2} + 13 q^{4} + 34 q^{7} - 75 q^{8} - 77 q^{11} + 80 q^{13} - 42 q^{14} - 43 q^{16} - 162 q^{17} + 58 q^{19} + 55 q^{22} - 324 q^{23} + 200 q^{26} - 168 q^{28} - 64 q^{29} - 348 q^{31} + 75 q^{32} + 206 q^{34} + 664 q^{37} - 334 q^{38} + 332 q^{41} + 774 q^{43} - 143 q^{44} - 328 q^{46} - 872 q^{47} - 417 q^{49} + 134 q^{52} - 1628 q^{53} + 1618 q^{56} + 1568 q^{58} + 332 q^{59} + 22 q^{61} + 260 q^{62} + 561 q^{64} + 1524 q^{67} - 2324 q^{68} + 516 q^{71} + 1700 q^{73} - 1628 q^{74} + 2794 q^{76} - 374 q^{77} + 1746 q^{79} + 364 q^{82} - 2344 q^{83} - 1270 q^{86} + 825 q^{88} + 2226 q^{89} + 1072 q^{91} - 4184 q^{92} + 4736 q^{94} + 1048 q^{97} - 3057 q^{98}+O(q^{100}) \) 7 * q - 5 * q^2 + 13 * q^4 + 34 * q^7 - 75 * q^8 - 77 * q^11 + 80 * q^13 - 42 * q^14 - 43 * q^16 - 162 * q^17 + 58 * q^19 + 55 * q^22 - 324 * q^23 + 200 * q^26 - 168 * q^28 - 64 * q^29 - 348 * q^31 + 75 * q^32 + 206 * q^34 + 664 * q^37 - 334 * q^38 + 332 * q^41 + 774 * q^43 - 143 * q^44 - 328 * q^46 - 872 * q^47 - 417 * q^49 + 134 * q^52 - 1628 * q^53 + 1618 * q^56 + 1568 * q^58 + 332 * q^59 + 22 * q^61 + 260 * q^62 + 561 * q^64 + 1524 * q^67 - 2324 * q^68 + 516 * q^71 + 1700 * q^73 - 1628 * q^74 + 2794 * q^76 - 374 * q^77 + 1746 * q^79 + 364 * q^82 - 2344 * q^83 - 1270 * q^86 + 825 * q^88 + 2226 * q^89 + 1072 * q^91 - 4184 * q^92 + 4736 * q^94 + 1048 * q^97 - 3057 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more