LMFDB - Embedded newform 2312.4.a.h.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2312,4,Mod(1,2312)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2312, 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("2312.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2312 = 2^{3} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2312.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:	$136.412415933$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 81x^{4} + 222x^{2} - 64 $ x^6 - 81x^4 + 222x^2 - 64
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 136)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-8.84141$ of defining polynomial
Character	$\chi$	$=$	2312.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.65834 q^{3} +5.50702 q^{5} -18.8836 q^{7} -13.6165 q^{9} +O(q^{10})$	$q-3.65834 q^{3} +5.50702 q^{5} -18.8836 q^{7} -13.6165 q^{9} -24.3906 q^{11} +32.7631 q^{13} -20.1466 q^{15} -13.0865 q^{19} +69.0827 q^{21} +6.57375 q^{23} -94.6727 q^{25} +148.589 q^{27} +232.490 q^{29} +96.6635 q^{31} +89.2292 q^{33} -103.992 q^{35} +184.591 q^{37} -119.859 q^{39} +132.884 q^{41} +191.252 q^{43} -74.9865 q^{45} -215.199 q^{47} +13.5901 q^{49} +35.0788 q^{53} -134.320 q^{55} +47.8749 q^{57} +349.650 q^{59} +555.384 q^{61} +257.129 q^{63} +180.427 q^{65} -262.259 q^{67} -24.0490 q^{69} -999.505 q^{71} -412.054 q^{73} +346.345 q^{75} +460.582 q^{77} -1010.90 q^{79} -175.944 q^{81} +302.943 q^{83} -850.529 q^{87} -418.921 q^{89} -618.685 q^{91} -353.628 q^{93} -72.0676 q^{95} +1034.25 q^{97} +332.116 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 42 q^{9}+O(q^{10}) $ 6 * q + 42 * q^9	$ 6 q + 42 q^{9} - 72 q^{13} + 24 q^{15} + 24 q^{19} - 204 q^{21} + 114 q^{25} - 228 q^{33} + 408 q^{35} - 192 q^{43} - 72 q^{47} + 18 q^{49} - 924 q^{53} - 456 q^{55} - 1680 q^{59} - 312 q^{67} + 492 q^{69} + 1668 q^{77} - 1614 q^{81} - 1296 q^{83} - 456 q^{87} + 24 q^{89} - 828 q^{93}+O(q^{100}) $ 6 * q + 42 * q^9 - 72 * q^13 + 24 * q^15 + 24 * q^19 - 204 * q^21 + 114 * q^25 - 228 * q^33 + 408 * q^35 - 192 * q^43 - 72 * q^47 + 18 * q^49 - 924 * q^53 - 456 * q^55 - 1680 * q^59 - 312 * q^67 + 492 * q^69 + 1668 * q^77 - 1614 * q^81 - 1296 * q^83 - 456 * q^87 + 24 * q^89 - 828 * q^93

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−3.65834	−0.704048	−0.352024	−	0.935991i	$-0.614507\pi$
$3$	−3.65834	−0.704048	−0.352024	+	0.935991i	$0.614507\pi$
$4$	0	0
$5$	5.50702	0.492563	0.246281	−	0.969198i	$-0.420791\pi$
$5$	5.50702	0.492563	0.246281	+	0.969198i	$0.420791\pi$
$6$	0	0
$7$	−18.8836	−1.01962	−0.509809	−	0.860288i	$-0.670284\pi$
$7$	−18.8836	−1.01962	−0.509809	+	0.860288i	$0.670284\pi$
$8$	0	0
$9$	−13.6165	−0.504316
$10$	0	0
$11$	−24.3906	−0.668550	−0.334275	−	0.942476i	$-0.608491\pi$
$11$	−24.3906	−0.668550	−0.334275	+	0.942476i	$0.608491\pi$
$12$	0	0
$13$	32.7631	0.698988	0.349494	−	0.936939i	$-0.386353\pi$
$13$	32.7631	0.698988	0.349494	+	0.936939i	$0.386353\pi$
$14$	0	0
$15$	−20.1466	−0.346788
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	−13.0865	−0.158013	−0.0790065	−	0.996874i	$-0.525175\pi$
$19$	−13.0865	−0.158013	−0.0790065	+	0.996874i	$0.525175\pi$
$20$	0	0
$21$	69.0827	0.717860
$22$	0	0
$23$	6.57375	0.0595966	0.0297983	−	0.999556i	$-0.490514\pi$
$23$	6.57375	0.0595966	0.0297983	+	0.999556i	$0.490514\pi$
$24$	0	0
$25$	−94.6727	−0.757382
$26$	0	0
$27$	148.589	1.05911
$28$	0	0
$29$	232.490	1.48870	0.744351	−	0.667789i	$-0.232759\pi$
$29$	232.490	1.48870	0.744351	+	0.667789i	$0.232759\pi$
$30$	0	0
$31$	96.6635	0.560041	0.280021	−	0.959994i	$-0.409659\pi$
$31$	96.6635	0.560041	0.280021	+	0.959994i	$0.409659\pi$
$32$	0	0
$33$	89.2292	0.470691
$34$	0	0
$35$	−103.992	−0.502226
$36$	0	0
$37$	184.591	0.820176	0.410088	−	0.912046i	$-0.365498\pi$
$37$	184.591	0.820176	0.410088	+	0.912046i	$0.365498\pi$
$38$	0	0
$39$	−119.859	−0.492121
$40$	0	0
$41$	132.884	0.506171	0.253085	−	0.967444i	$-0.418555\pi$
$41$	132.884	0.506171	0.253085	+	0.967444i	$0.418555\pi$
$42$	0	0
$43$	191.252	0.678270	0.339135	−	0.940738i	$-0.389866\pi$
$43$	191.252	0.678270	0.339135	+	0.940738i	$0.389866\pi$
$44$	0	0
$45$	−74.9865	−0.248407
$46$	0	0
$47$	−215.199	−0.667872	−0.333936	−	0.942596i	$-0.608377\pi$
$47$	−215.199	−0.667872	−0.333936	+	0.942596i	$0.608377\pi$
$48$	0	0
$49$	13.5901	0.0396213
$50$	0	0
$51$	0	0
$52$	0	0
$53$	35.0788	0.0909140	0.0454570	−	0.998966i	$-0.485526\pi$
$53$	35.0788	0.0909140	0.0454570	+	0.998966i	$0.485526\pi$
$54$	0	0
$55$	−134.320	−0.329303
$56$	0	0
$57$	47.8749	0.111249
$58$	0	0
$59$	349.650	0.771534	0.385767	−	0.922596i	$-0.373937\pi$
$59$	349.650	0.771534	0.385767	+	0.922596i	$0.373937\pi$
$60$	0	0
$61$	555.384	1.16573	0.582866	−	0.812568i	$-0.301931\pi$
$61$	555.384	1.16573	0.582866	+	0.812568i	$0.301931\pi$
$62$	0	0
$63$	257.129	0.514210
$64$	0	0
$65$	180.427	0.344296
$66$	0	0
$67$	−262.259	−0.478209	−0.239105	−	0.970994i	$-0.576854\pi$
$67$	−262.259	−0.478209	−0.239105	+	0.970994i	$0.576854\pi$
$68$	0	0
$69$	−24.0490	−0.0419589
$70$	0	0
$71$	−999.505	−1.67070	−0.835348	−	0.549721i	$-0.814734\pi$
$71$	−999.505	−1.67070	−0.835348	+	0.549721i	$0.814734\pi$
$72$	0	0
$73$	−412.054	−0.660647	−0.330324	−	0.943868i	$-0.607158\pi$
$73$	−412.054	−0.660647	−0.330324	+	0.943868i	$0.607158\pi$
$74$	0	0
$75$	346.345	0.533234
$76$	0	0
$77$	460.582	0.681665
$78$	0	0
$79$	−1010.90	−1.43968	−0.719839	−	0.694141i	$-0.755785\pi$
$79$	−1010.90	−1.43968	−0.719839	+	0.694141i	$0.755785\pi$
$80$	0	0
$81$	−175.944	−0.241350
$82$	0	0
$83$	302.943	0.400631	0.200315	−	0.979731i	$-0.435803\pi$
$83$	302.943	0.400631	0.200315	+	0.979731i	$0.435803\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−850.529	−1.04812
$88$	0	0
$89$	−418.921	−0.498938	−0.249469	−	0.968383i	$-0.580256\pi$
$89$	−418.921	−0.498938	−0.249469	+	0.968383i	$0.580256\pi$
$90$	0	0
$91$	−618.685	−0.712701
$92$	0	0
$93$	−353.628	−0.394296
$94$	0	0
$95$	−72.0676	−0.0778314
$96$	0	0
$97$	1034.25	1.08260	0.541300	−	0.840829i	$-0.317932\pi$
$97$	1034.25	1.08260	0.541300	+	0.840829i	$0.317932\pi$
$98$	0	0
$99$	332.116	0.337160
$100$	0	0
$101$	−322.087	−0.317315	−0.158657	−	0.987334i	$-0.550717\pi$
$101$	−322.087	−0.317315	−0.158657	+	0.987334i	$0.550717\pi$
$102$	0	0
$103$	1472.63	1.40876	0.704381	−	0.709822i	$-0.251224\pi$
$103$	1472.63	1.40876	0.704381	+	0.709822i	$0.251224\pi$
$104$	0	0
$105$	380.439	0.353591
$106$	0	0
$107$	156.592	0.141479	0.0707397	−	0.997495i	$-0.477464\pi$
$107$	156.592	0.141479	0.0707397	+	0.997495i	$0.477464\pi$
$108$	0	0
$109$	1279.55	1.12439	0.562194	−	0.827006i	$-0.309958\pi$
$109$	1279.55	1.12439	0.562194	+	0.827006i	$0.309958\pi$
$110$	0	0
$111$	−675.296	−0.577444
$112$	0	0
$113$	1992.52	1.65876	0.829382	−	0.558682i	$-0.188693\pi$
$113$	1992.52	1.65876	0.829382	+	0.558682i	$0.188693\pi$
$114$	0	0
$115$	36.2018	0.0293551
$116$	0	0
$117$	−446.120	−0.352511
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−736.098	−0.553041
$122$	0	0
$123$	−486.136	−0.356369
$124$	0	0
$125$	−1209.74	−0.865621
$126$	0	0
$127$	−2510.51	−1.75411	−0.877053	−	0.480393i	$-0.840494\pi$
$127$	−2510.51	−1.75411	−0.877053	+	0.480393i	$0.840494\pi$
$128$	0	0
$129$	−699.665	−0.477535
$130$	0	0
$131$	−342.338	−0.228322	−0.114161	−	0.993462i	$-0.536418\pi$
$131$	−342.338	−0.228322	−0.114161	+	0.993462i	$0.536418\pi$
$132$	0	0
$133$	247.120	0.161113
$134$	0	0
$135$	818.283	0.521679
$136$	0	0
$137$	−312.728	−0.195023	−0.0975115	−	0.995234i	$-0.531088\pi$
$137$	−312.728	−0.195023	−0.0975115	+	0.995234i	$0.531088\pi$
$138$	0	0
$139$	−1854.89	−1.13187	−0.565933	−	0.824451i	$-0.691484\pi$
$139$	−1854.89	−1.13187	−0.565933	+	0.824451i	$0.691484\pi$
$140$	0	0
$141$	787.271	0.470214
$142$	0	0
$143$	−799.112	−0.467308
$144$	0	0
$145$	1280.33	0.733279
$146$	0	0
$147$	−49.7172	−0.0278953
$148$	0	0
$149$	−2059.42	−1.13231	−0.566155	−	0.824299i	$-0.691570\pi$
$149$	−2059.42	−1.13231	−0.566155	+	0.824299i	$0.691570\pi$
$150$	0	0
$151$	−1435.50	−0.773636	−0.386818	−	0.922156i	$-0.626426\pi$
$151$	−1435.50	−0.773636	−0.386818	+	0.922156i	$0.626426\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	532.328	0.275855
$156$	0	0
$157$	−389.560	−0.198027	−0.0990137	−	0.995086i	$-0.531569\pi$
$157$	−389.560	−0.198027	−0.0990137	+	0.995086i	$0.531569\pi$
$158$	0	0
$159$	−128.330	−0.0640079
$160$	0	0
$161$	−124.136	−0.0607658
$162$	0	0
$163$	794.607	0.381831	0.190915	−	0.981607i	$-0.438854\pi$
$163$	794.607	0.381831	0.190915	+	0.981607i	$0.438854\pi$
$164$	0	0
$165$	491.387	0.231845
$166$	0	0
$167$	−918.227	−0.425476	−0.212738	−	0.977109i	$-0.568238\pi$
$167$	−918.227	−0.425476	−0.212738	+	0.977109i	$0.568238\pi$
$168$	0	0
$169$	−1123.58	−0.511415
$170$	0	0
$171$	178.193	0.0796885
$172$	0	0
$173$	−2000.72	−0.879259	−0.439630	−	0.898179i	$-0.644890\pi$
$173$	−2000.72	−0.879259	−0.439630	+	0.898179i	$0.644890\pi$
$174$	0	0
$175$	1787.76	0.772240
$176$	0	0
$177$	−1279.14	−0.543197
$178$	0	0
$179$	4325.30	1.80608	0.903040	−	0.429557i	$-0.141330\pi$
$179$	4325.30	1.80608	0.903040	+	0.429557i	$0.141330\pi$
$180$	0	0
$181$	2453.80	1.00768	0.503838	−	0.863798i	$-0.331921\pi$
$181$	2453.80	1.00768	0.503838	+	0.863798i	$0.331921\pi$
$182$	0	0
$183$	−2031.79	−0.820732
$184$	0	0
$185$	1016.54	0.403988
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−2805.90	−1.07989
$190$	0	0
$191$	−3243.31	−1.22868	−0.614340	−	0.789042i	$-0.710578\pi$
$191$	−3243.31	−1.22868	−0.614340	+	0.789042i	$0.710578\pi$
$192$	0	0
$193$	−352.840	−0.131596	−0.0657978	−	0.997833i	$-0.520959\pi$
$193$	−352.840	−0.131596	−0.0657978	+	0.997833i	$0.520959\pi$
$194$	0	0
$195$	−660.064	−0.242401
$196$	0	0
$197$	2010.15	0.726991	0.363496	−	0.931596i	$-0.381583\pi$
$197$	2010.15	0.726991	0.363496	+	0.931596i	$0.381583\pi$
$198$	0	0
$199$	456.332	0.162556	0.0812778	−	0.996691i	$-0.474100\pi$
$199$	456.332	0.162556	0.0812778	+	0.996691i	$0.474100\pi$
$200$	0	0
$201$	959.433	0.336683
$202$	0	0
$203$	−4390.25	−1.51791
$204$	0	0
$205$	731.795	0.249321
$206$	0	0
$207$	−89.5117	−0.0300555
$208$	0	0
$209$	319.188	0.105640
$210$	0	0
$211$	529.191	0.172659	0.0863295	−	0.996267i	$-0.472486\pi$
$211$	529.191	0.172659	0.0863295	+	0.996267i	$0.472486\pi$
$212$	0	0
$213$	3656.53	1.17625
$214$	0	0
$215$	1053.23	0.334091
$216$	0	0
$217$	−1825.35	−0.571028
$218$	0	0
$219$	1507.43	0.465128
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−3206.15	−0.962780	−0.481390	−	0.876507i	$-0.659868\pi$
$223$	−3206.15	−0.962780	−0.481390	+	0.876507i	$0.659868\pi$
$224$	0	0
$225$	1289.11	0.381960
$226$	0	0
$227$	5875.82	1.71802	0.859012	−	0.511955i	$-0.171079\pi$
$227$	5875.82	1.71802	0.859012	+	0.511955i	$0.171079\pi$
$228$	0	0
$229$	−1037.53	−0.299397	−0.149698	−	0.988732i	$-0.547830\pi$
$229$	−1037.53	−0.299397	−0.149698	+	0.988732i	$0.547830\pi$
$230$	0	0
$231$	−1684.97	−0.479925
$232$	0	0
$233$	−3942.43	−1.10849	−0.554243	−	0.832355i	$-0.686992\pi$
$233$	−3942.43	−1.10849	−0.554243	+	0.832355i	$0.686992\pi$
$234$	0	0
$235$	−1185.10	−0.328969
$236$	0	0
$237$	3698.20	1.01360
$238$	0	0
$239$	439.384	0.118918	0.0594589	−	0.998231i	$-0.481062\pi$
$239$	439.384	0.118918	0.0594589	+	0.998231i	$0.481062\pi$
$240$	0	0
$241$	6987.83	1.86774	0.933871	−	0.357609i	$-0.116408\pi$
$241$	6987.83	1.86774	0.933871	+	0.357609i	$0.116408\pi$
$242$	0	0
$243$	−3368.25	−0.889189
$244$	0	0
$245$	74.8409	0.0195160
$246$	0	0
$247$	−428.754	−0.110449
$248$	0	0
$249$	−1108.27	−0.282063
$250$	0	0
$251$	−4692.15	−1.17994	−0.589971	−	0.807424i	$-0.700861\pi$
$251$	−4692.15	−1.17994	−0.589971	+	0.807424i	$0.700861\pi$
$252$	0	0
$253$	−160.338	−0.0398433
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−7770.28	−1.88598	−0.942990	−	0.332821i	$-0.891999\pi$
$257$	−7770.28	−1.88598	−0.942990	+	0.332821i	$0.891999\pi$
$258$	0	0
$259$	−3485.74	−0.836267
$260$	0	0
$261$	−3165.71	−0.750776
$262$	0	0
$263$	−3414.87	−0.800647	−0.400323	−	0.916374i	$-0.631102\pi$
$263$	−3414.87	−0.800647	−0.400323	+	0.916374i	$0.631102\pi$
$264$	0	0
$265$	193.180	0.0447809
$266$	0	0
$267$	1532.56	0.351277
$268$	0	0
$269$	6803.82	1.54214	0.771072	−	0.636748i	$-0.219721\pi$
$269$	6803.82	1.54214	0.771072	+	0.636748i	$0.219721\pi$
$270$	0	0
$271$	−178.804	−0.0400796	−0.0200398	−	0.999799i	$-0.506379\pi$
$271$	−178.804	−0.0400796	−0.0200398	+	0.999799i	$0.506379\pi$
$272$	0	0
$273$	2263.36	0.501776
$274$	0	0
$275$	2309.13	0.506347
$276$	0	0
$277$	3500.03	0.759193	0.379597	−	0.925152i	$-0.376063\pi$
$277$	3500.03	0.759193	0.379597	+	0.925152i	$0.376063\pi$
$278$	0	0
$279$	−1316.22	−0.282438
$280$	0	0
$281$	6663.64	1.41466	0.707330	−	0.706883i	$-0.249899\pi$
$281$	6663.64	1.41466	0.707330	+	0.706883i	$0.249899\pi$
$282$	0	0
$283$	−6363.62	−1.33667	−0.668335	−	0.743860i	$-0.732993\pi$
$283$	−6363.62	−1.33667	−0.668335	+	0.743860i	$0.732993\pi$
$284$	0	0
$285$	263.648	0.0547970
$286$	0	0
$287$	−2509.33	−0.516101
$288$	0	0
$289$	0	0
$290$	0	0
$291$	−3783.64	−0.762203
$292$	0	0
$293$	−965.112	−0.192432	−0.0962158	−	0.995361i	$-0.530674\pi$
$293$	−965.112	−0.192432	−0.0962158	+	0.995361i	$0.530674\pi$
$294$	0	0
$295$	1925.53	0.380029
$296$	0	0
$297$	−3624.18	−0.708068
$298$	0	0
$299$	215.376	0.0416573
$300$	0	0
$301$	−3611.52	−0.691577
$302$	0	0
$303$	1178.30	0.223405
$304$	0	0
$305$	3058.51	0.574196
$306$	0	0
$307$	9802.79	1.82239	0.911197	−	0.411971i	$-0.135159\pi$
$307$	9802.79	1.82239	0.911197	+	0.411971i	$0.135159\pi$
$308$	0	0
$309$	−5387.39	−0.991837
$310$	0	0
$311$	−5260.44	−0.959140	−0.479570	−	0.877504i	$-0.659207\pi$
$311$	−5260.44	−0.959140	−0.479570	+	0.877504i	$0.659207\pi$
$312$	0	0
$313$	1929.81	0.348496	0.174248	−	0.984702i	$-0.444251\pi$
$313$	1929.81	0.348496	0.174248	+	0.984702i	$0.444251\pi$
$314$	0	0
$315$	1416.01	0.253281
$316$	0	0
$317$	−9780.44	−1.73288	−0.866442	−	0.499278i	$-0.833599\pi$
$317$	−9780.44	−1.73288	−0.866442	+	0.499278i	$0.833599\pi$
$318$	0	0
$319$	−5670.58	−0.995271
$320$	0	0
$321$	−572.867	−0.0996084
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−3101.77	−0.529401
$326$	0	0
$327$	−4681.01	−0.791623
$328$	0	0
$329$	4063.73	0.680975
$330$	0	0
$331$	−9892.44	−1.64271	−0.821356	−	0.570416i	$-0.806782\pi$
$331$	−9892.44	−1.64271	−0.821356	+	0.570416i	$0.806782\pi$
$332$	0	0
$333$	−2513.48	−0.413628
$334$	0	0
$335$	−1444.27	−0.235548
$336$	0	0
$337$	−11396.1	−1.84209	−0.921043	−	0.389461i	$-0.872661\pi$
$337$	−11396.1	−1.84209	−0.921043	+	0.389461i	$0.872661\pi$
$338$	0	0
$339$	−7289.31	−1.16785
$340$	0	0
$341$	−2357.68	−0.374415
$342$	0	0
$343$	6220.44	0.979220
$344$	0	0
$345$	−132.438	−0.0206674
$346$	0	0
$347$	−8980.55	−1.38934	−0.694670	−	0.719329i	$-0.744450\pi$
$347$	−8980.55	−1.38934	−0.694670	+	0.719329i	$0.744450\pi$
$348$	0	0
$349$	10955.6	1.68034	0.840170	−	0.542323i	$-0.182455\pi$
$349$	10955.6	1.68034	0.840170	+	0.542323i	$0.182455\pi$
$350$	0	0
$351$	4868.24	0.740306
$352$	0	0
$353$	−5745.21	−0.866251	−0.433125	−	0.901334i	$-0.642589\pi$
$353$	−5745.21	−0.866251	−0.433125	+	0.901334i	$0.642589\pi$
$354$	0	0
$355$	−5504.29	−0.822923
$356$	0	0
$357$	0	0
$358$	0	0
$359$	374.516	0.0550591	0.0275295	−	0.999621i	$-0.491236\pi$
$359$	374.516	0.0550591	0.0275295	+	0.999621i	$0.491236\pi$
$360$	0	0
$361$	−6687.74	−0.975032
$362$	0	0
$363$	2692.90	0.389368
$364$	0	0
$365$	−2269.19	−0.325410
$366$	0	0
$367$	−2534.00	−0.360419	−0.180210	−	0.983628i	$-0.557678\pi$
$367$	−2534.00	−0.360419	−0.180210	+	0.983628i	$0.557678\pi$
$368$	0	0
$369$	−1809.42	−0.255270
$370$	0	0
$371$	−662.414	−0.0926976
$372$	0	0
$373$	−7221.11	−1.00240	−0.501200	−	0.865332i	$-0.667108\pi$
$373$	−7221.11	−1.00240	−0.501200	+	0.865332i	$0.667108\pi$
$374$	0	0
$375$	4425.65	0.609439
$376$	0	0
$377$	7617.10	1.04059
$378$	0	0
$379$	−966.670	−0.131015	−0.0655073	−	0.997852i	$-0.520867\pi$
$379$	−966.670	−0.131015	−0.0655073	+	0.997852i	$0.520867\pi$
$380$	0	0
$381$	9184.30	1.23498
$382$	0	0
$383$	2519.84	0.336182	0.168091	−	0.985771i	$-0.446240\pi$
$383$	2519.84	0.336182	0.168091	+	0.985771i	$0.446240\pi$
$384$	0	0
$385$	2536.44	0.335763
$386$	0	0
$387$	−2604.19	−0.342063
$388$	0	0
$389$	4613.08	0.601265	0.300633	−	0.953740i	$-0.402802\pi$
$389$	4613.08	0.601265	0.300633	+	0.953740i	$0.402802\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	1252.39	0.160750
$394$	0	0
$395$	−5567.02	−0.709132
$396$	0	0
$397$	−4890.63	−0.618271	−0.309135	−	0.951018i	$-0.600040\pi$
$397$	−4890.63	−0.618271	−0.309135	+	0.951018i	$0.600040\pi$
$398$	0	0
$399$	−904.050	−0.113431
$400$	0	0
$401$	−4149.35	−0.516730	−0.258365	−	0.966047i	$-0.583184\pi$
$401$	−4149.35	−0.516730	−0.258365	+	0.966047i	$0.583184\pi$
$402$	0	0
$403$	3166.99	0.391462
$404$	0	0
$405$	−968.926	−0.118880
$406$	0	0
$407$	−4502.28	−0.548329
$408$	0	0
$409$	−11597.0	−1.40205	−0.701023	−	0.713139i	$-0.747273\pi$
$409$	−11597.0	−1.40205	−0.701023	+	0.713139i	$0.747273\pi$
$410$	0	0
$411$	1144.07	0.137306
$412$	0	0
$413$	−6602.64	−0.786670
$414$	0	0
$415$	1668.31	0.197336
$416$	0	0
$417$	6785.81	0.796889
$418$	0	0
$419$	5559.38	0.648194	0.324097	−	0.946024i	$-0.394940\pi$
$419$	5559.38	0.648194	0.324097	+	0.946024i	$0.394940\pi$
$420$	0	0
$421$	−12464.5	−1.44295	−0.721477	−	0.692438i	$-0.756536\pi$
$421$	−12464.5	−1.44295	−0.721477	+	0.692438i	$0.756536\pi$
$422$	0	0
$423$	2930.26	0.336819
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−10487.6	−1.18860
$428$	0	0
$429$	2923.42	0.329008
$430$	0	0
$431$	−6453.05	−0.721189	−0.360594	−	0.932723i	$-0.617426\pi$
$431$	−6453.05	−0.721189	−0.360594	+	0.932723i	$0.617426\pi$
$432$	0	0
$433$	1521.27	0.168840	0.0844201	−	0.996430i	$-0.473096\pi$
$433$	1521.27	0.168840	0.0844201	+	0.996430i	$0.473096\pi$
$434$	0	0
$435$	−4683.88	−0.516264
$436$	0	0
$437$	−86.0274	−0.00941704
$438$	0	0
$439$	−13031.4	−1.41675	−0.708376	−	0.705835i	$-0.750572\pi$
$439$	−13031.4	−1.41675	−0.708376	+	0.705835i	$0.750572\pi$
$440$	0	0
$441$	−185.050	−0.0199816
$442$	0	0
$443$	13360.1	1.43286	0.716430	−	0.697659i	$-0.245775\pi$
$443$	13360.1	1.43286	0.716430	+	0.697659i	$0.245775\pi$
$444$	0	0
$445$	−2307.00	−0.245758
$446$	0	0
$447$	7534.06	0.797201
$448$	0	0
$449$	−13784.8	−1.44887	−0.724436	−	0.689342i	$-0.757900\pi$
$449$	−13784.8	−1.44887	−0.724436	+	0.689342i	$0.757900\pi$
$450$	0	0
$451$	−3241.12	−0.338400
$452$	0	0
$453$	5251.53	0.544677
$454$	0	0
$455$	−3407.11	−0.351050
$456$	0	0
$457$	−2.71275	−0.000277675 0	−0.000138837	−	1.00000i	$-0.500044\pi$
$457$	−2.71275	−0.000277675 0	−0.000138837		1.00000i	$0.500044\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−15169.3	−1.53255	−0.766276	−	0.642512i	$-0.777892\pi$
$461$	−15169.3	−1.53255	−0.766276	+	0.642512i	$0.777892\pi$
$462$	0	0
$463$	−16840.8	−1.69040	−0.845202	−	0.534447i	$-0.820520\pi$
$463$	−16840.8	−1.69040	−0.845202	+	0.534447i	$0.820520\pi$
$464$	0	0
$465$	−1947.44	−0.194215
$466$	0	0
$467$	−16695.7	−1.65436	−0.827180	−	0.561937i	$-0.810056\pi$
$467$	−16695.7	−1.65436	−0.827180	+	0.561937i	$0.810056\pi$
$468$	0	0
$469$	4952.39	0.487591
$470$	0	0
$471$	1425.15	0.139421
$472$	0	0
$473$	−4664.75	−0.453457
$474$	0	0
$475$	1238.93	0.119676
$476$	0	0
$477$	−477.652	−0.0458494
$478$	0	0
$479$	−17529.9	−1.67215	−0.836074	−	0.548616i	$-0.815155\pi$
$479$	−17529.9	−1.67215	−0.836074	+	0.548616i	$0.815155\pi$
$480$	0	0
$481$	6047.76	0.573294
$482$	0	0
$483$	454.132	0.0427820
$484$	0	0
$485$	5695.64	0.533249
$486$	0	0
$487$	−1128.94	−0.105046	−0.0525228	−	0.998620i	$-0.516726\pi$
$487$	−1128.94	−0.105046	−0.0525228	+	0.998620i	$0.516726\pi$
$488$	0	0
$489$	−2906.94	−0.268827
$490$	0	0
$491$	−3592.49	−0.330197	−0.165099	−	0.986277i	$-0.552794\pi$
$491$	−3592.49	−0.330197	−0.165099	+	0.986277i	$0.552794\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	1828.97	0.166073
$496$	0	0
$497$	18874.3	1.70347
$498$	0	0
$499$	−8778.88	−0.787568	−0.393784	−	0.919203i	$-0.628834\pi$
$499$	−8778.88	−0.787568	−0.393784	+	0.919203i	$0.628834\pi$
$500$	0	0
$501$	3359.19	0.299556
$502$	0	0
$503$	−4403.91	−0.390379	−0.195190	−	0.980766i	$-0.562532\pi$
$503$	−4403.91	−0.390379	−0.195190	+	0.980766i	$0.562532\pi$
$504$	0	0
$505$	−1773.74	−0.156298
$506$	0	0
$507$	4110.44	0.360061
$508$	0	0
$509$	15245.5	1.32760	0.663799	−	0.747911i	$-0.268943\pi$
$509$	15245.5	1.32760	0.663799	+	0.747911i	$0.268943\pi$
$510$	0	0
$511$	7781.06	0.673608
$512$	0	0
$513$	−1944.51	−0.167353
$514$	0	0
$515$	8109.80	0.693904
$516$	0	0
$517$	5248.83	0.446506
$518$	0	0
$519$	7319.32	0.619041
$520$	0	0
$521$	710.780	0.0597694	0.0298847	−	0.999553i	$-0.490486\pi$
$521$	710.780	0.0597694	0.0298847	+	0.999553i	$0.490486\pi$
$522$	0	0
$523$	3121.87	0.261013	0.130507	−	0.991447i	$-0.458340\pi$
$523$	3121.87	0.261013	0.130507	+	0.991447i	$0.458340\pi$
$524$	0	0
$525$	−6540.24	−0.543695
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−12123.8	−0.996448
$530$	0	0
$531$	−4761.02	−0.389097
$532$	0	0
$533$	4353.69	0.353808
$534$	0	0
$535$	862.354	0.0696875
$536$	0	0
$537$	−15823.4	−1.27157
$538$	0	0
$539$	−331.471	−0.0264888
$540$	0	0
$541$	4269.58	0.339304	0.169652	−	0.985504i	$-0.445736\pi$
$541$	4269.58	0.339304	0.169652	+	0.985504i	$0.445736\pi$
$542$	0	0
$543$	−8976.84	−0.709453
$544$	0	0
$545$	7046.48	0.553831
$546$	0	0
$547$	14597.0	1.14099	0.570495	−	0.821301i	$-0.306751\pi$
$547$	14597.0	1.14099	0.570495	+	0.821301i	$0.306751\pi$
$548$	0	0
$549$	−7562.40	−0.587897
$550$	0	0
$551$	−3042.48	−0.235234
$552$	0	0
$553$	19089.3	1.46792
$554$	0	0
$555$	−3718.87	−0.284427
$556$	0	0
$557$	11032.7	0.839265	0.419633	−	0.907694i	$-0.362159\pi$
$557$	11032.7	0.839265	0.419633	+	0.907694i	$0.362159\pi$
$558$	0	0
$559$	6266.00	0.474103
$560$	0	0
$561$	0	0
$562$	0	0
$563$	259.812	0.0194490	0.00972448	−	0.999953i	$-0.496905\pi$
$563$	259.812	0.0194490	0.00972448	+	0.999953i	$0.496905\pi$
$564$	0	0
$565$	10972.8	0.817045
$566$	0	0
$567$	3322.45	0.246084
$568$	0	0
$569$	3905.98	0.287781	0.143890	−	0.989594i	$-0.454039\pi$
$569$	3905.98	0.287781	0.143890	+	0.989594i	$0.454039\pi$
$570$	0	0
$571$	21303.1	1.56131	0.780653	−	0.624965i	$-0.214887\pi$
$571$	21303.1	1.56131	0.780653	+	0.624965i	$0.214887\pi$
$572$	0	0
$573$	11865.1	0.865050
$574$	0	0
$575$	−622.355	−0.0451374
$576$	0	0
$577$	13107.3	0.945691	0.472846	−	0.881145i	$-0.343227\pi$
$577$	13107.3	0.945691	0.472846	+	0.881145i	$0.343227\pi$
$578$	0	0
$579$	1290.81	0.0926497
$580$	0	0
$581$	−5720.66	−0.408490
$582$	0	0
$583$	−855.593	−0.0607806
$584$	0	0
$585$	−2456.79	−0.173634
$586$	0	0
$587$	−16260.9	−1.14337	−0.571687	−	0.820472i	$-0.693711\pi$
$587$	−16260.9	−1.14337	−0.571687	+	0.820472i	$0.693711\pi$
$588$	0	0
$589$	−1264.99	−0.0884938
$590$	0	0
$591$	−7353.82	−0.511837
$592$	0	0
$593$	−2147.79	−0.148734	−0.0743669	−	0.997231i	$-0.523694\pi$
$593$	−2147.79	−0.148734	−0.0743669	+	0.997231i	$0.523694\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−1669.42	−0.114447
$598$	0	0
$599$	−19056.4	−1.29987	−0.649937	−	0.759988i	$-0.725205\pi$
$599$	−19056.4	−1.29987	−0.649937	+	0.759988i	$0.725205\pi$
$600$	0	0
$601$	10521.3	0.714098	0.357049	−	0.934086i	$-0.383783\pi$
$601$	10521.3	0.714098	0.357049	+	0.934086i	$0.383783\pi$
$602$	0	0
$603$	3571.06	0.241169
$604$	0	0
$605$	−4053.71	−0.272408
$606$	0	0
$607$	−14117.5	−0.944006	−0.472003	−	0.881597i	$-0.656469\pi$
$607$	−14117.5	−0.944006	−0.472003	+	0.881597i	$0.656469\pi$
$608$	0	0
$609$	16061.0	1.06868
$610$	0	0
$611$	−7050.58	−0.466835
$612$	0	0
$613$	−23602.6	−1.55514	−0.777568	−	0.628799i	$-0.783547\pi$
$613$	−23602.6	−1.55514	−0.777568	+	0.628799i	$0.783547\pi$
$614$	0	0
$615$	−2677.16	−0.175534
$616$	0	0
$617$	−12867.1	−0.839561	−0.419781	−	0.907626i	$-0.637893\pi$
$617$	−12867.1	−0.839561	−0.419781	+	0.907626i	$0.637893\pi$
$618$	0	0
$619$	28514.6	1.85153	0.925765	−	0.378099i	$-0.123422\pi$
$619$	28514.6	1.85153	0.925765	+	0.378099i	$0.123422\pi$
$620$	0	0
$621$	976.788	0.0631194
$622$	0	0
$623$	7910.73	0.508727
$624$	0	0
$625$	5172.02	0.331009
$626$	0	0
$627$	−1167.70	−0.0743754
$628$	0	0
$629$	0	0
$630$	0	0
$631$	18082.0	1.14078	0.570390	−	0.821374i	$-0.306792\pi$
$631$	18082.0	1.14078	0.570390	+	0.821374i	$0.306792\pi$
$632$	0	0
$633$	−1935.96	−0.121560
$634$	0	0
$635$	−13825.4	−0.864008
$636$	0	0
$637$	445.253	0.0276948
$638$	0	0
$639$	13609.8	0.842559
$640$	0	0
$641$	17176.0	1.05836	0.529181	−	0.848509i	$-0.322499\pi$
$641$	17176.0	1.05836	0.529181	+	0.848509i	$0.322499\pi$
$642$	0	0
$643$	25908.8	1.58903	0.794514	−	0.607246i	$-0.207726\pi$
$643$	25908.8	1.58903	0.794514	+	0.607246i	$0.207726\pi$
$644$	0	0
$645$	−3853.07	−0.235216
$646$	0	0
$647$	11615.0	0.705772	0.352886	−	0.935666i	$-0.385200\pi$
$647$	11615.0	0.705772	0.352886	+	0.935666i	$0.385200\pi$
$648$	0	0
$649$	−8528.17	−0.515809
$650$	0	0
$651$	6677.77	0.402031
$652$	0	0
$653$	7300.01	0.437476	0.218738	−	0.975784i	$-0.429806\pi$
$653$	7300.01	0.437476	0.218738	+	0.975784i	$0.429806\pi$
$654$	0	0
$655$	−1885.26	−0.112463
$656$	0	0
$657$	5610.74	0.333175
$658$	0	0
$659$	−17869.2	−1.05628	−0.528138	−	0.849159i	$-0.677110\pi$
$659$	−17869.2	−1.05628	−0.528138	+	0.849159i	$0.677110\pi$
$660$	0	0
$661$	15244.6	0.897046	0.448523	−	0.893771i	$-0.351950\pi$
$661$	15244.6	0.897046	0.448523	+	0.893771i	$0.351950\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1360.90	0.0793583
$666$	0	0
$667$	1528.33	0.0887216
$668$	0	0
$669$	11729.2	0.677843
$670$	0	0
$671$	−13546.2	−0.779350
$672$	0	0
$673$	28444.1	1.62918	0.814592	−	0.580035i	$-0.196961\pi$
$673$	28444.1	1.62918	0.814592	+	0.580035i	$0.196961\pi$
$674$	0	0
$675$	−14067.3	−0.802152
$676$	0	0
$677$	7335.77	0.416450	0.208225	−	0.978081i	$-0.433231\pi$
$677$	7335.77	0.416450	0.208225	+	0.978081i	$0.433231\pi$
$678$	0	0
$679$	−19530.4	−1.10384
$680$	0	0
$681$	−21495.7	−1.20957
$682$	0	0
$683$	−12489.7	−0.699714	−0.349857	−	0.936803i	$-0.613770\pi$
$683$	−12489.7	−0.699714	−0.349857	+	0.936803i	$0.613770\pi$
$684$	0	0
$685$	−1722.20	−0.0960611
$686$	0	0
$687$	3795.64	0.210790
$688$	0	0
$689$	1149.29	0.0635478
$690$	0	0
$691$	−12743.7	−0.701580	−0.350790	−	0.936454i	$-0.614087\pi$
$691$	−12743.7	−0.701580	−0.350790	+	0.936454i	$0.614087\pi$
$692$	0	0
$693$	−6271.53	−0.343775
$694$	0	0
$695$	−10214.9	−0.557515
$696$	0	0
$697$	0	0
$698$	0	0
$699$	14422.8	0.780428
$700$	0	0
$701$	3473.00	0.187123	0.0935616	−	0.995613i	$-0.470175\pi$
$701$	3473.00	0.187123	0.0935616	+	0.995613i	$0.470175\pi$
$702$	0	0
$703$	−2415.65	−0.129599
$704$	0	0
$705$	4335.52	0.231610
$706$	0	0
$707$	6082.15	0.323540
$708$	0	0
$709$	−2864.92	−0.151755	−0.0758776	−	0.997117i	$-0.524176\pi$
$709$	−2864.92	−0.151755	−0.0758776	+	0.997117i	$0.524176\pi$
$710$	0	0
$711$	13764.9	0.726053
$712$	0	0
$713$	635.441	0.0333765
$714$	0	0
$715$	−4400.72	−0.230179
$716$	0	0
$717$	−1607.42	−0.0837239
$718$	0	0
$719$	10483.3	0.543755	0.271877	−	0.962332i	$-0.412355\pi$
$719$	10483.3	0.543755	0.271877	+	0.962332i	$0.412355\pi$
$720$	0	0
$721$	−27808.6	−1.43640
$722$	0	0
$723$	−25563.9	−1.31498
$724$	0	0
$725$	−22010.5	−1.12752
$726$	0	0
$727$	23897.4	1.21913	0.609565	−	0.792736i	$-0.291344\pi$
$727$	23897.4	1.21913	0.609565	+	0.792736i	$0.291344\pi$
$728$	0	0
$729$	17072.7	0.867382
$730$	0	0
$731$	0	0
$732$	0	0
$733$	2017.14	0.101643	0.0508217	−	0.998708i	$-0.483816\pi$
$733$	2017.14	0.101643	0.0508217	+	0.998708i	$0.483816\pi$
$734$	0	0
$735$	−273.794	−0.0137402
$736$	0	0
$737$	6396.66	0.319707
$738$	0	0
$739$	2064.30	0.102756	0.0513778	−	0.998679i	$-0.483639\pi$
$739$	2064.30	0.102756	0.0513778	+	0.998679i	$0.483639\pi$
$740$	0	0
$741$	1568.53	0.0777616
$742$	0	0
$743$	13981.0	0.690326	0.345163	−	0.938543i	$-0.387824\pi$
$743$	13981.0	0.690326	0.345163	+	0.938543i	$0.387824\pi$
$744$	0	0
$745$	−11341.3	−0.557734
$746$	0	0
$747$	−4125.04	−0.202044
$748$	0	0
$749$	−2957.02	−0.144255
$750$	0	0
$751$	7747.00	0.376421	0.188211	−	0.982129i	$-0.439731\pi$
$751$	7747.00	0.376421	0.188211	+	0.982129i	$0.439731\pi$
$752$	0	0
$753$	17165.5	0.830737
$754$	0	0
$755$	−7905.30	−0.381064
$756$	0	0
$757$	9176.47	0.440587	0.220294	−	0.975434i	$-0.429298\pi$
$757$	9176.47	0.440587	0.220294	+	0.975434i	$0.429298\pi$
$758$	0	0
$759$	586.571	0.0280516
$760$	0	0
$761$	7779.75	0.370586	0.185293	−	0.982683i	$-0.440677\pi$
$761$	7779.75	0.370586	0.185293	+	0.982683i	$0.440677\pi$
$762$	0	0
$763$	−24162.4	−1.14645
$764$	0	0
$765$	0	0
$766$	0	0
$767$	11455.6	0.539293
$768$	0	0
$769$	20539.6	0.963169	0.481584	−	0.876400i	$-0.340061\pi$
$769$	20539.6	0.963169	0.481584	+	0.876400i	$0.340061\pi$
$770$	0	0
$771$	28426.3	1.32782
$772$	0	0
$773$	32572.1	1.51557	0.757785	−	0.652504i	$-0.226282\pi$
$773$	32572.1	1.51557	0.757785	+	0.652504i	$0.226282\pi$
$774$	0	0
$775$	−9151.40	−0.424165
$776$	0	0
$777$	12752.0	0.588772
$778$	0	0
$779$	−1738.99	−0.0799816
$780$	0	0
$781$	24378.5	1.11694
$782$	0	0
$783$	34545.5	1.57670
$784$	0	0
$785$	−2145.32	−0.0975409
$786$	0	0
$787$	−15541.1	−0.703915	−0.351957	−	0.936016i	$-0.614484\pi$
$787$	−15541.1	−0.703915	−0.351957	+	0.936016i	$0.614484\pi$
$788$	0	0
$789$	12492.8	0.563694
$790$	0	0
$791$	−37625.9	−1.69131
$792$	0	0
$793$	18196.1	0.814833
$794$	0	0
$795$	−706.717	−0.0315279
$796$	0	0
$797$	1490.59	0.0662475	0.0331237	−	0.999451i	$-0.489454\pi$
$797$	1490.59	0.0662475	0.0331237	+	0.999451i	$0.489454\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	5704.25	0.251623
$802$	0	0
$803$	10050.2	0.441675
$804$	0	0
$805$	−683.619	−0.0299310
$806$	0	0
$807$	−24890.7	−1.08574
$808$	0	0
$809$	44494.6	1.93368	0.966841	−	0.255381i	$-0.0822009\pi$
$809$	44494.6	1.93368	0.966841	+	0.255381i	$0.0822009\pi$
$810$	0	0
$811$	28968.7	1.25429	0.627145	−	0.778903i	$-0.284223\pi$
$811$	28968.7	1.25429	0.627145	+	0.778903i	$0.284223\pi$
$812$	0	0
$813$	654.127	0.0282180
$814$	0	0
$815$	4375.91	0.188076
$816$	0	0
$817$	−2502.82	−0.107176
$818$	0	0
$819$	8424.34	0.359427
$820$	0	0
$821$	−20229.4	−0.859940	−0.429970	−	0.902843i	$-0.641476\pi$
$821$	−20229.4	−0.859940	−0.429970	+	0.902843i	$0.641476\pi$
$822$	0	0
$823$	16130.9	0.683216	0.341608	−	0.939843i	$-0.389029\pi$
$823$	16130.9	0.683216	0.341608	+	0.939843i	$0.389029\pi$
$824$	0	0
$825$	−8447.57	−0.356493
$826$	0	0
$827$	11669.6	0.490678	0.245339	−	0.969437i	$-0.421101\pi$
$827$	11669.6	0.490678	0.245339	+	0.969437i	$0.421101\pi$
$828$	0	0
$829$	31170.6	1.30591	0.652956	−	0.757396i	$-0.273529\pi$
$829$	31170.6	1.30591	0.652956	+	0.757396i	$0.273529\pi$
$830$	0	0
$831$	−12804.3	−0.534509
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−5056.69	−0.209574
$836$	0	0
$837$	14363.1	0.593146
$838$	0	0
$839$	−39384.8	−1.62064	−0.810319	−	0.585990i	$-0.800706\pi$
$839$	−39384.8	−1.62064	−0.810319	+	0.585990i	$0.800706\pi$
$840$	0	0
$841$	29662.7	1.21623
$842$	0	0
$843$	−24377.9	−0.995989
$844$	0	0
$845$	−6187.58	−0.251904
$846$	0	0
$847$	13900.2	0.563891
$848$	0	0
$849$	23280.3	0.941081
$850$	0	0
$851$	1213.45	0.0488797
$852$	0	0
$853$	3363.78	0.135022	0.0675110	−	0.997719i	$-0.478494\pi$
$853$	3363.78	0.135022	0.0675110	+	0.997719i	$0.478494\pi$
$854$	0	0
$855$	981.311	0.0392516
$856$	0	0
$857$	−17571.1	−0.700370	−0.350185	−	0.936681i	$-0.613881\pi$
$857$	−17571.1	−0.700370	−0.350185	+	0.936681i	$0.613881\pi$
$858$	0	0
$859$	−18291.2	−0.726527	−0.363264	−	0.931686i	$-0.618338\pi$
$859$	−18291.2	−0.726527	−0.363264	+	0.931686i	$0.618338\pi$
$860$	0	0
$861$	9179.99	0.363360
$862$	0	0
$863$	39007.1	1.53861	0.769303	−	0.638884i	$-0.220603\pi$
$863$	39007.1	1.53861	0.769303	+	0.638884i	$0.220603\pi$
$864$	0	0
$865$	−11018.0	−0.433090
$866$	0	0
$867$	0	0
$868$	0	0
$869$	24656.4	0.962497
$870$	0	0
$871$	−8592.42	−0.334263
$872$	0	0
$873$	−14082.9	−0.545973
$874$	0	0
$875$	22844.3	0.882603
$876$	0	0
$877$	−36885.4	−1.42022	−0.710110	−	0.704091i	$-0.751355\pi$
$877$	−36885.4	−1.42022	−0.710110	+	0.704091i	$0.751355\pi$
$878$	0	0
$879$	3530.71	0.135481
$880$	0	0
$881$	−26664.0	−1.01967	−0.509837	−	0.860271i	$-0.670294\pi$
$881$	−26664.0	−1.01967	−0.509837	+	0.860271i	$0.670294\pi$
$882$	0	0
$883$	−16705.1	−0.636659	−0.318330	−	0.947980i	$-0.603122\pi$
$883$	−16705.1	−0.636659	−0.318330	+	0.947980i	$0.603122\pi$
$884$	0	0
$885$	−7044.24	−0.267559
$886$	0	0
$887$	19425.9	0.735354	0.367677	−	0.929954i	$-0.380153\pi$
$887$	19425.9	0.735354	0.367677	+	0.929954i	$0.380153\pi$
$888$	0	0
$889$	47407.4	1.78852
$890$	0	0
$891$	4291.38	0.161354
$892$	0	0
$893$	2816.20	0.105533
$894$	0	0
$895$	23819.5	0.889607
$896$	0	0
$897$	−787.920	−0.0293288
$898$	0	0
$899$	22473.3	0.833734
$900$	0	0
$901$	0	0
$902$	0	0
$903$	13212.2	0.486904
$904$	0	0
$905$	13513.1	0.496344
$906$	0	0
$907$	−19614.3	−0.718062	−0.359031	−	0.933326i	$-0.616893\pi$
$907$	−19614.3	−0.718062	−0.359031	+	0.933326i	$0.616893\pi$
$908$	0	0
$909$	4385.70	0.160027
$910$	0	0
$911$	−11263.5	−0.409635	−0.204817	−	0.978800i	$-0.565660\pi$
$911$	−11263.5	−0.409635	−0.204817	+	0.978800i	$0.565660\pi$
$912$	0	0
$913$	−7388.97	−0.267841
$914$	0	0
$915$	−11189.1	−0.404262
$916$	0	0
$917$	6464.56	0.232801
$918$	0	0
$919$	−13417.2	−0.481603	−0.240802	−	0.970574i	$-0.577410\pi$
$919$	−13417.2	−0.481603	−0.240802	+	0.970574i	$0.577410\pi$
$920$	0	0
$921$	−35862.0	−1.28305
$922$	0	0
$923$	−32746.9	−1.16780
$924$	0	0
$925$	−17475.7	−0.621187
$926$	0	0
$927$	−20052.1	−0.710462
$928$	0	0
$929$	−27343.2	−0.965663	−0.482832	−	0.875713i	$-0.660392\pi$
$929$	−27343.2	−0.965663	−0.482832	+	0.875713i	$0.660392\pi$
$930$	0	0
$931$	−177.847	−0.00626068
$932$	0	0
$933$	19244.5	0.675281
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−4395.97	−0.153266	−0.0766329	−	0.997059i	$-0.524417\pi$
$937$	−4395.97	−0.153266	−0.0766329	+	0.997059i	$0.524417\pi$
$938$	0	0
$939$	−7059.90	−0.245358
$940$	0	0
$941$	5183.75	0.179581	0.0897903	−	0.995961i	$-0.471380\pi$
$941$	5183.75	0.179581	0.0897903	+	0.995961i	$0.471380\pi$
$942$	0	0
$943$	873.547	0.0301661
$944$	0	0
$945$	−15452.1	−0.531913
$946$	0	0
$947$	−37789.4	−1.29672	−0.648358	−	0.761336i	$-0.724544\pi$
$947$	−37789.4	−1.29672	−0.648358	+	0.761336i	$0.724544\pi$
$948$	0	0
$949$	−13500.2	−0.461785
$950$	0	0
$951$	35780.2	1.22003
$952$	0	0
$953$	−49462.7	−1.68127	−0.840636	−	0.541600i	$-0.817819\pi$
$953$	−49462.7	−1.68127	−0.840636	+	0.541600i	$0.817819\pi$
$954$	0	0
$955$	−17861.0	−0.605202
$956$	0	0
$957$	20744.9	0.700719
$958$	0	0
$959$	5905.43	0.198849
$960$	0	0
$961$	−20447.2	−0.686354
$962$	0	0
$963$	−2132.24	−0.0713504
$964$	0	0
$965$	−1943.10	−0.0648191
$966$	0	0
$967$	33351.3	1.10911	0.554553	−	0.832149i	$-0.312889\pi$
$967$	33351.3	1.10911	0.554553	+	0.832149i	$0.312889\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	30513.4	1.00847	0.504233	−	0.863567i	$-0.331775\pi$
$971$	30513.4	1.00847	0.504233	+	0.863567i	$0.331775\pi$
$972$	0	0
$973$	35026.9	1.15407
$974$	0	0
$975$	11347.3	0.372724
$976$	0	0
$977$	27016.6	0.884684	0.442342	−	0.896846i	$-0.354148\pi$
$977$	27016.6	0.884684	0.442342	+	0.896846i	$0.354148\pi$
$978$	0	0
$979$	10217.7	0.333565
$980$	0	0
$981$	−17423.0	−0.567046
$982$	0	0
$983$	−30916.1	−1.00312	−0.501562	−	0.865122i	$-0.667241\pi$
$983$	−30916.1	−1.00312	−0.501562	+	0.865122i	$0.667241\pi$
$984$	0	0
$985$	11069.9	0.358089
$986$	0	0
$987$	−14866.5	−0.479439
$988$	0	0
$989$	1257.24	0.0404226
$990$	0	0
$991$	−25761.2	−0.825764	−0.412882	−	0.910785i	$-0.635478\pi$
$991$	−25761.2	−0.825764	−0.412882	+	0.910785i	$0.635478\pi$
$992$	0	0
$993$	36189.9	1.15655
$994$	0	0
$995$	2513.03	0.0800688
$996$	0	0
$997$	−14573.0	−0.462921	−0.231461	−	0.972844i	$-0.574350\pi$
$997$	−14573.0	−0.462921	−0.231461	+	0.972844i	$0.574350\pi$
$998$	0	0
$999$	27428.2	0.868658

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2312.4.a.h.1.3		6
17.4	even	4		136.4.b.a.33.4	yes	6
17.13	even	4		136.4.b.a.33.3	✓	6
17.16	even	2	inner	2312.4.a.h.1.4		6
51.38	odd	4		1224.4.c.c.577.4		6
51.47	odd	4		1224.4.c.c.577.3		6
68.47	odd	4		272.4.b.e.33.4		6
68.55	odd	4		272.4.b.e.33.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.4.b.a.33.3	✓	6	17.13	even	4
136.4.b.a.33.4	yes	6	17.4	even	4
272.4.b.e.33.3		6	68.55	odd	4
272.4.b.e.33.4		6	68.47	odd	4
1224.4.c.c.577.3		6	51.47	odd	4
1224.4.c.c.577.4		6	51.38	odd	4
2312.4.a.h.1.3		6	1.1	even	1	trivial
2312.4.a.h.1.4		6	17.16	even	2	inner

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:	\( N \)	\(=\)	\( 2312 = 2^{3} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2312.a (trivial)

\(f(q)\)	\(=\)	\(q-3.65834 q^{3} +5.50702 q^{5} -18.8836 q^{7} -13.6165 q^{9} +O(q^{10})\)	\(q-3.65834 q^{3} +5.50702 q^{5} -18.8836 q^{7} -13.6165 q^{9} -24.3906 q^{11} +32.7631 q^{13} -20.1466 q^{15} -13.0865 q^{19} +69.0827 q^{21} +6.57375 q^{23} -94.6727 q^{25} +148.589 q^{27} +232.490 q^{29} +96.6635 q^{31} +89.2292 q^{33} -103.992 q^{35} +184.591 q^{37} -119.859 q^{39} +132.884 q^{41} +191.252 q^{43} -74.9865 q^{45} -215.199 q^{47} +13.5901 q^{49} +35.0788 q^{53} -134.320 q^{55} +47.8749 q^{57} +349.650 q^{59} +555.384 q^{61} +257.129 q^{63} +180.427 q^{65} -262.259 q^{67} -24.0490 q^{69} -999.505 q^{71} -412.054 q^{73} +346.345 q^{75} +460.582 q^{77} -1010.90 q^{79} -175.944 q^{81} +302.943 q^{83} -850.529 q^{87} -418.921 q^{89} -618.685 q^{91} -353.628 q^{93} -72.0676 q^{95} +1034.25 q^{97} +332.116 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 42 q^{9}+O(q^{10}) \) 6 * q + 42 * q^9	\( 6 q + 42 q^{9} - 72 q^{13} + 24 q^{15} + 24 q^{19} - 204 q^{21} + 114 q^{25} - 228 q^{33} + 408 q^{35} - 192 q^{43} - 72 q^{47} + 18 q^{49} - 924 q^{53} - 456 q^{55} - 1680 q^{59} - 312 q^{67} + 492 q^{69} + 1668 q^{77} - 1614 q^{81} - 1296 q^{83} - 456 q^{87} + 24 q^{89} - 828 q^{93}+O(q^{100}) \) 6 * q + 42 * q^9 - 72 * q^13 + 24 * q^15 + 24 * q^19 - 204 * q^21 + 114 * q^25 - 228 * q^33 + 408 * q^35 - 192 * q^43 - 72 * q^47 + 18 * q^49 - 924 * q^53 - 456 * q^55 - 1680 * q^59 - 312 * q^67 + 492 * q^69 + 1668 * q^77 - 1614 * q^81 - 1296 * q^83 - 456 * q^87 + 24 * q^89 - 828 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

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Database

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