LMFDB - Embedded newform 2312.4.a.n.1.12

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:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 294 x^{16} - 14 x^{15} + 34371 x^{14} + 2670 x^{13} - 2054705 x^{12} - 160284 x^{11} + \cdots - 176969301147 $ x^18 - 294x^16 - 14x^15 + 34371x^14 + 2670x^13 - 2054705x^12 - 160284x^11 + 67981059x^10 + 2824200x^9 - 1279285428x^8 + 12108606x^7 + 13538185208x^6 - 853720902x^5 - 75816227907x^4 + 7747530606x^3 + 195817930515x^2 - 12873873186x - 176969301147
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{15}\cdot 17^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$2.65826$ 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)

$f(q)$	$=$	$q+1.65826 q^{3} -15.8272 q^{5} -7.01175 q^{7} -24.2502 q^{9} +O(q^{10})$	$q+1.65826 q^{3} -15.8272 q^{5} -7.01175 q^{7} -24.2502 q^{9} +35.1862 q^{11} +9.34098 q^{13} -26.2456 q^{15} +7.69074 q^{19} -11.6273 q^{21} -50.8433 q^{23} +125.500 q^{25} -84.9860 q^{27} +179.741 q^{29} +174.449 q^{31} +58.3477 q^{33} +110.976 q^{35} +227.334 q^{37} +15.4897 q^{39} -371.203 q^{41} +86.9814 q^{43} +383.812 q^{45} +376.535 q^{47} -293.835 q^{49} -33.4425 q^{53} -556.898 q^{55} +12.7532 q^{57} +323.825 q^{59} -685.643 q^{61} +170.036 q^{63} -147.841 q^{65} +981.231 q^{67} -84.3113 q^{69} -658.794 q^{71} -215.641 q^{73} +208.111 q^{75} -246.716 q^{77} +452.074 q^{79} +513.826 q^{81} +1441.41 q^{83} +298.056 q^{87} -549.793 q^{89} -65.4966 q^{91} +289.281 q^{93} -121.723 q^{95} +140.585 q^{97} -853.271 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 18 q^{3} - 51 q^{7} + 120 q^{9}+O(q^{10}) $ 18 * q - 18 * q^3 - 51 * q^7 + 120 * q^9	$ 18 q - 18 q^{3} - 51 q^{7} + 120 q^{9} - 132 q^{11} + 30 q^{13} + 102 q^{15} + 66 q^{19} + 144 q^{21} - 153 q^{23} + 306 q^{25} - 768 q^{27} - 51 q^{29} - 303 q^{31} + 525 q^{33} - 255 q^{35} - 717 q^{37} + 216 q^{39} + 393 q^{41} - 390 q^{43} - 558 q^{45} - 633 q^{47} + 1443 q^{49} + 1275 q^{53} + 1539 q^{55} - 810 q^{57} - 204 q^{59} - 534 q^{61} - 2556 q^{63} + 2127 q^{65} - 405 q^{67} + 2547 q^{69} + 426 q^{71} - 1149 q^{73} - 2226 q^{75} - 357 q^{77} - 1053 q^{79} + 2802 q^{81} + 66 q^{83} + 2487 q^{87} - 4119 q^{89} - 6090 q^{91} + 606 q^{93} - 2109 q^{95} - 2349 q^{97} - 1428 q^{99}+O(q^{100}) $ 18 * q - 18 * q^3 - 51 * q^7 + 120 * q^9 - 132 * q^11 + 30 * q^13 + 102 * q^15 + 66 * q^19 + 144 * q^21 - 153 * q^23 + 306 * q^25 - 768 * q^27 - 51 * q^29 - 303 * q^31 + 525 * q^33 - 255 * q^35 - 717 * q^37 + 216 * q^39 + 393 * q^41 - 390 * q^43 - 558 * q^45 - 633 * q^47 + 1443 * q^49 + 1275 * q^53 + 1539 * q^55 - 810 * q^57 - 204 * q^59 - 534 * q^61 - 2556 * q^63 + 2127 * q^65 - 405 * q^67 + 2547 * q^69 + 426 * q^71 - 1149 * q^73 - 2226 * q^75 - 357 * q^77 - 1053 * q^79 + 2802 * q^81 + 66 * q^83 + 2487 * q^87 - 4119 * q^89 - 6090 * q^91 + 606 * q^93 - 2109 * q^95 - 2349 * q^97 - 1428 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.65826	0.319132	0.159566	−	0.987187i	$-0.448991\pi$
$3$	1.65826	0.319132	0.159566	+	0.987187i	$0.448991\pi$
$4$	0	0
$5$	−15.8272	−1.41563	−0.707814	−	0.706399i	$-0.750318\pi$
$5$	−15.8272	−1.41563	−0.707814	+	0.706399i	$0.750318\pi$
$6$	0	0
$7$	−7.01175	−0.378599	−0.189299	−	0.981919i	$-0.560622\pi$
$7$	−7.01175	−0.378599	−0.189299	+	0.981919i	$0.560622\pi$
$8$	0	0
$9$	−24.2502	−0.898155
$10$	0	0
$11$	35.1862	0.964457	0.482229	−	0.876045i	$-0.339827\pi$
$11$	35.1862	0.964457	0.482229	+	0.876045i	$0.339827\pi$
$12$	0	0
$13$	9.34098	0.199286	0.0996431	−	0.995023i	$-0.468230\pi$
$13$	9.34098	0.199286	0.0996431	+	0.995023i	$0.468230\pi$
$14$	0	0
$15$	−26.2456	−0.451772
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	7.69074	0.0928620	0.0464310	−	0.998922i	$-0.485215\pi$
$19$	7.69074	0.0928620	0.0464310	+	0.998922i	$0.485215\pi$
$20$	0	0
$21$	−11.6273	−0.120823
$22$	0	0
$23$	−50.8433	−0.460937	−0.230469	−	0.973080i	$-0.574026\pi$
$23$	−50.8433	−0.460937	−0.230469	+	0.973080i	$0.574026\pi$
$24$	0	0
$25$	125.500	1.00400
$26$	0	0
$27$	−84.9860	−0.605762
$28$	0	0
$29$	179.741	1.15093	0.575466	−	0.817826i	$-0.304821\pi$
$29$	179.741	1.15093	0.575466	+	0.817826i	$0.304821\pi$
$30$	0	0
$31$	174.449	1.01071	0.505354	−	0.862912i	$-0.331362\pi$
$31$	174.449	1.01071	0.505354	+	0.862912i	$0.331362\pi$
$32$	0	0
$33$	58.3477	0.307789
$34$	0	0
$35$	110.976	0.535955
$36$	0	0
$37$	227.334	1.01010	0.505048	−	0.863091i	$-0.331475\pi$
$37$	227.334	1.01010	0.505048	+	0.863091i	$0.331475\pi$
$38$	0	0
$39$	15.4897	0.0635986
$40$	0	0
$41$	−371.203	−1.41396	−0.706978	−	0.707235i	$-0.749942\pi$
$41$	−371.203	−1.41396	−0.706978	+	0.707235i	$0.749942\pi$
$42$	0	0
$43$	86.9814	0.308478	0.154239	−	0.988034i	$-0.450707\pi$
$43$	86.9814	0.308478	0.154239	+	0.988034i	$0.450707\pi$
$44$	0	0
$45$	383.812	1.27145
$46$	0	0
$47$	376.535	1.16858	0.584290	−	0.811545i	$-0.301373\pi$
$47$	376.535	1.16858	0.584290	+	0.811545i	$0.301373\pi$
$48$	0	0
$49$	−293.835	−0.856663
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−33.4425	−0.0866733	−0.0433367	−	0.999061i	$-0.513799\pi$
$53$	−33.4425	−0.0866733	−0.0433367	+	0.999061i	$0.513799\pi$
$54$	0	0
$55$	−556.898	−1.36531
$56$	0	0
$57$	12.7532	0.0296352
$58$	0	0
$59$	323.825	0.714549	0.357274	−	0.933999i	$-0.383706\pi$
$59$	323.825	0.714549	0.357274	+	0.933999i	$0.383706\pi$
$60$	0	0
$61$	−685.643	−1.43914	−0.719570	−	0.694420i	$-0.755661\pi$
$61$	−685.643	−1.43914	−0.719570	+	0.694420i	$0.755661\pi$
$62$	0	0
$63$	170.036	0.340040
$64$	0	0
$65$	−147.841	−0.282115
$66$	0	0
$67$	981.231	1.78920	0.894600	−	0.446868i	$-0.147461\pi$
$67$	981.231	1.78920	0.894600	+	0.446868i	$0.147461\pi$
$68$	0	0
$69$	−84.3113	−0.147100
$70$	0	0
$71$	−658.794	−1.10119	−0.550595	−	0.834772i	$-0.685599\pi$
$71$	−658.794	−1.10119	−0.550595	+	0.834772i	$0.685599\pi$
$72$	0	0
$73$	−215.641	−0.345737	−0.172869	−	0.984945i	$-0.555304\pi$
$73$	−215.641	−0.345737	−0.172869	+	0.984945i	$0.555304\pi$
$74$	0	0
$75$	208.111	0.320408
$76$	0	0
$77$	−246.716	−0.365142
$78$	0	0
$79$	452.074	0.643827	0.321913	−	0.946769i	$-0.395674\pi$
$79$	452.074	0.643827	0.321913	+	0.946769i	$0.395674\pi$
$80$	0	0
$81$	513.826	0.704837
$82$	0	0
$83$	1441.41	1.90620	0.953101	−	0.302652i	$-0.0978719\pi$
$83$	1441.41	1.90620	0.953101	+	0.302652i	$0.0978719\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	298.056	0.367299
$88$	0	0
$89$	−549.793	−0.654808	−0.327404	−	0.944884i	$-0.606174\pi$
$89$	−549.793	−0.654808	−0.327404	+	0.944884i	$0.606174\pi$
$90$	0	0
$91$	−65.4966	−0.0754495
$92$	0	0
$93$	289.281	0.322549
$94$	0	0
$95$	−121.723	−0.131458
$96$	0	0
$97$	140.585	0.147157	0.0735784	−	0.997289i	$-0.476558\pi$
$97$	140.585	0.147157	0.0735784	+	0.997289i	$0.476558\pi$
$98$	0	0
$99$	−853.271	−0.866232
$100$	0	0
$101$	36.4051	0.0358658	0.0179329	−	0.999839i	$-0.494291\pi$
$101$	36.4051	0.0358658	0.0179329	+	0.999839i	$0.494291\pi$
$102$	0	0
$103$	121.952	0.116663	0.0583315	−	0.998297i	$-0.481422\pi$
$103$	121.952	0.116663	0.0583315	+	0.998297i	$0.481422\pi$
$104$	0	0
$105$	184.027	0.171040
$106$	0	0
$107$	1142.04	1.03182	0.515912	−	0.856642i	$-0.327453\pi$
$107$	1142.04	1.03182	0.515912	+	0.856642i	$0.327453\pi$
$108$	0	0
$109$	−2157.75	−1.89610	−0.948050	−	0.318121i	$-0.896948\pi$
$109$	−2157.75	−1.89610	−0.948050	+	0.318121i	$0.896948\pi$
$110$	0	0
$111$	376.979	0.322354
$112$	0	0
$113$	−460.474	−0.383343	−0.191671	−	0.981459i	$-0.561391\pi$
$113$	−460.474	−0.383343	−0.191671	+	0.981459i	$0.561391\pi$
$114$	0	0
$115$	804.707	0.652515
$116$	0	0
$117$	−226.520	−0.178990
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−92.9336	−0.0698224
$122$	0	0
$123$	−615.551	−0.451239
$124$	0	0
$125$	−7.91334	−0.00566233
$126$	0	0
$127$	−1699.17	−1.18722	−0.593610	−	0.804753i	$-0.702298\pi$
$127$	−1699.17	−1.18722	−0.593610	+	0.804753i	$0.702298\pi$
$128$	0	0
$129$	144.238	0.0984451
$130$	0	0
$131$	−2157.19	−1.43874	−0.719368	−	0.694630i	$-0.755568\pi$
$131$	−2157.19	−1.43874	−0.719368	+	0.694630i	$0.755568\pi$
$132$	0	0
$133$	−53.9255	−0.0351574
$134$	0	0
$135$	1345.09	0.857533
$136$	0	0
$137$	1052.83	0.656564	0.328282	−	0.944580i	$-0.393530\pi$
$137$	1052.83	0.656564	0.328282	+	0.944580i	$0.393530\pi$
$138$	0	0
$139$	−604.118	−0.368638	−0.184319	−	0.982867i	$-0.559008\pi$
$139$	−604.118	−0.368638	−0.184319	+	0.982867i	$0.559008\pi$
$140$	0	0
$141$	624.392	0.372931
$142$	0	0
$143$	328.673	0.192203
$144$	0	0
$145$	−2844.79	−1.62929
$146$	0	0
$147$	−487.255	−0.273388
$148$	0	0
$149$	−2040.19	−1.12173	−0.560867	−	0.827906i	$-0.689532\pi$
$149$	−2040.19	−1.12173	−0.560867	+	0.827906i	$0.689532\pi$
$150$	0	0
$151$	831.866	0.448320	0.224160	−	0.974552i	$-0.428036\pi$
$151$	831.866	0.448320	0.224160	+	0.974552i	$0.428036\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−2761.04	−1.43079
$156$	0	0
$157$	−1260.88	−0.640950	−0.320475	−	0.947257i	$-0.603842\pi$
$157$	−1260.88	−0.640950	−0.320475	+	0.947257i	$0.603842\pi$
$158$	0	0
$159$	−55.4563	−0.0276602
$160$	0	0
$161$	356.500	0.174510
$162$	0	0
$163$	476.574	0.229007	0.114503	−	0.993423i	$-0.463472\pi$
$163$	476.574	0.229007	0.114503	+	0.993423i	$0.463472\pi$
$164$	0	0
$165$	−923.481	−0.435714
$166$	0	0
$167$	−2559.34	−1.18591	−0.592957	−	0.805234i	$-0.702040\pi$
$167$	−2559.34	−1.18591	−0.592957	+	0.805234i	$0.702040\pi$
$168$	0	0
$169$	−2109.75	−0.960285
$170$	0	0
$171$	−186.502	−0.0834044
$172$	0	0
$173$	3378.83	1.48490	0.742451	−	0.669901i	$-0.233663\pi$
$173$	3378.83	1.48490	0.742451	+	0.669901i	$0.233663\pi$
$174$	0	0
$175$	−879.974	−0.380113
$176$	0	0
$177$	536.985	0.228035
$178$	0	0
$179$	−3973.59	−1.65922	−0.829609	−	0.558345i	$-0.811437\pi$
$179$	−3973.59	−1.65922	−0.829609	+	0.558345i	$0.811437\pi$
$180$	0	0
$181$	−1719.29	−0.706043	−0.353021	−	0.935615i	$-0.614846\pi$
$181$	−1719.29	−0.706043	−0.353021	+	0.935615i	$0.614846\pi$
$182$	0	0
$183$	−1136.97	−0.459275
$184$	0	0
$185$	−3598.06	−1.42992
$186$	0	0
$187$	0	0
$188$	0	0
$189$	595.900	0.229341
$190$	0	0
$191$	2693.68	1.02046	0.510229	−	0.860039i	$-0.329561\pi$
$191$	2693.68	1.02046	0.510229	+	0.860039i	$0.329561\pi$
$192$	0	0
$193$	−4400.03	−1.64104	−0.820521	−	0.571616i	$-0.806317\pi$
$193$	−4400.03	−1.64104	−0.820521	+	0.571616i	$0.806317\pi$
$194$	0	0
$195$	−245.159	−0.0900319
$196$	0	0
$197$	501.529	0.181383	0.0906916	−	0.995879i	$-0.471092\pi$
$197$	501.529	0.181383	0.0906916	+	0.995879i	$0.471092\pi$
$198$	0	0
$199$	−3969.89	−1.41416	−0.707080	−	0.707133i	$-0.749988\pi$
$199$	−3969.89	−1.41416	−0.707080	+	0.707133i	$0.749988\pi$
$200$	0	0
$201$	1627.13	0.570991
$202$	0	0
$203$	−1260.30	−0.435741
$204$	0	0
$205$	5875.11	2.00164
$206$	0	0
$207$	1232.96	0.413993
$208$	0	0
$209$	270.608	0.0895614
$210$	0	0
$211$	−36.5967	−0.0119404	−0.00597019	−	0.999982i	$-0.501900\pi$
$211$	−36.5967	−0.0119404	−0.00597019	+	0.999982i	$0.501900\pi$
$212$	0	0
$213$	−1092.45	−0.351425
$214$	0	0
$215$	−1376.67	−0.436690
$216$	0	0
$217$	−1223.19	−0.382653
$218$	0	0
$219$	−357.588	−0.110336
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−231.604	−0.0695486	−0.0347743	−	0.999395i	$-0.511071\pi$
$223$	−231.604	−0.0695486	−0.0347743	+	0.999395i	$0.511071\pi$
$224$	0	0
$225$	−3043.40	−0.901747
$226$	0	0
$227$	263.390	0.0770123	0.0385061	−	0.999258i	$-0.487740\pi$
$227$	263.390	0.0770123	0.0385061	+	0.999258i	$0.487740\pi$
$228$	0	0
$229$	643.307	0.185637	0.0928186	−	0.995683i	$-0.470412\pi$
$229$	643.307	0.185637	0.0928186	+	0.995683i	$0.470412\pi$
$230$	0	0
$231$	−409.119	−0.116529
$232$	0	0
$233$	−6213.70	−1.74709	−0.873547	−	0.486739i	$-0.838186\pi$
$233$	−6213.70	−1.74709	−0.873547	+	0.486739i	$0.838186\pi$
$234$	0	0
$235$	−5959.49	−1.65427
$236$	0	0
$237$	749.655	0.205466
$238$	0	0
$239$	6844.28	1.85238	0.926192	−	0.377052i	$-0.123062\pi$
$239$	6844.28	1.85238	0.926192	+	0.377052i	$0.123062\pi$
$240$	0	0
$241$	2181.57	0.583102	0.291551	−	0.956555i	$-0.405829\pi$
$241$	2181.57	0.583102	0.291551	+	0.956555i	$0.405829\pi$
$242$	0	0
$243$	3146.68	0.830698
$244$	0	0
$245$	4650.59	1.21272
$246$	0	0
$247$	71.8391	0.0185061
$248$	0	0
$249$	2390.22	0.608330
$250$	0	0
$251$	−2283.76	−0.574300	−0.287150	−	0.957886i	$-0.592708\pi$
$251$	−2283.76	−0.574300	−0.287150	+	0.957886i	$0.592708\pi$
$252$	0	0
$253$	−1788.98	−0.444554
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−4702.69	−1.14142	−0.570711	−	0.821151i	$-0.693332\pi$
$257$	−4702.69	−1.14142	−0.570711	+	0.821151i	$0.693332\pi$
$258$	0	0
$259$	−1594.01	−0.382421
$260$	0	0
$261$	−4358.74	−1.03371
$262$	0	0
$263$	−3264.01	−0.765276	−0.382638	−	0.923898i	$-0.624984\pi$
$263$	−3264.01	−0.765276	−0.382638	+	0.923898i	$0.624984\pi$
$264$	0	0
$265$	529.301	0.122697
$266$	0	0
$267$	−911.698	−0.208970
$268$	0	0
$269$	−3550.46	−0.804740	−0.402370	−	0.915477i	$-0.631814\pi$
$269$	−3550.46	−0.804740	−0.402370	+	0.915477i	$0.631814\pi$
$270$	0	0
$271$	7384.98	1.65537	0.827685	−	0.561192i	$-0.189657\pi$
$271$	7384.98	1.65537	0.827685	+	0.561192i	$0.189657\pi$
$272$	0	0
$273$	−108.610	−0.0240783
$274$	0	0
$275$	4415.86	0.968315
$276$	0	0
$277$	6593.88	1.43028	0.715140	−	0.698981i	$-0.246363\pi$
$277$	6593.88	1.43028	0.715140	+	0.698981i	$0.246363\pi$
$278$	0	0
$279$	−4230.42	−0.907772
$280$	0	0
$281$	−5040.82	−1.07014	−0.535071	−	0.844807i	$-0.679715\pi$
$281$	−5040.82	−1.07014	−0.535071	+	0.844807i	$0.679715\pi$
$282$	0	0
$283$	−4782.23	−1.00450	−0.502251	−	0.864722i	$-0.667495\pi$
$283$	−4782.23	−1.00450	−0.502251	+	0.864722i	$0.667495\pi$
$284$	0	0
$285$	−201.848	−0.0419524
$286$	0	0
$287$	2602.78	0.535322
$288$	0	0
$289$	0	0
$290$	0	0
$291$	233.126	0.0469624
$292$	0	0
$293$	3334.86	0.664929	0.332465	−	0.943116i	$-0.392120\pi$
$293$	3334.86	0.664929	0.332465	+	0.943116i	$0.392120\pi$
$294$	0	0
$295$	−5125.24	−1.01153
$296$	0	0
$297$	−2990.33	−0.584231
$298$	0	0
$299$	−474.926	−0.0918585
$300$	0	0
$301$	−609.892	−0.116789
$302$	0	0
$303$	60.3691	0.0114459
$304$	0	0
$305$	10851.8	2.03729
$306$	0	0
$307$	−3914.47	−0.727721	−0.363861	−	0.931453i	$-0.618542\pi$
$307$	−3914.47	−0.727721	−0.363861	+	0.931453i	$0.618542\pi$
$308$	0	0
$309$	202.228	0.0372309
$310$	0	0
$311$	−8366.73	−1.52551	−0.762755	−	0.646688i	$-0.776154\pi$
$311$	−8366.73	−1.52551	−0.762755	+	0.646688i	$0.776154\pi$
$312$	0	0
$313$	1274.66	0.230186	0.115093	−	0.993355i	$-0.463283\pi$
$313$	1274.66	0.230186	0.115093	+	0.993355i	$0.463283\pi$
$314$	0	0
$315$	−2691.19	−0.481370
$316$	0	0
$317$	3150.58	0.558216	0.279108	−	0.960260i	$-0.409961\pi$
$317$	3150.58	0.558216	0.279108	+	0.960260i	$0.409961\pi$
$318$	0	0
$319$	6324.38	1.11002
$320$	0	0
$321$	1893.80	0.329288
$322$	0	0
$323$	0	0
$324$	0	0
$325$	1172.29	0.200083
$326$	0	0
$327$	−3578.11	−0.605106
$328$	0	0
$329$	−2640.17	−0.442423
$330$	0	0
$331$	4942.46	0.820731	0.410366	−	0.911921i	$-0.365401\pi$
$331$	4942.46	0.820731	0.410366	+	0.911921i	$0.365401\pi$
$332$	0	0
$333$	−5512.90	−0.907222
$334$	0	0
$335$	−15530.1	−2.53284
$336$	0	0
$337$	−8382.20	−1.35492	−0.677459	−	0.735560i	$-0.736919\pi$
$337$	−8382.20	−1.35492	−0.677459	+	0.735560i	$0.736919\pi$
$338$	0	0
$339$	−763.585	−0.122337
$340$	0	0
$341$	6138.19	0.974785
$342$	0	0
$343$	4465.33	0.702930
$344$	0	0
$345$	1334.41	0.208238
$346$	0	0
$347$	−9053.95	−1.40070	−0.700348	−	0.713801i	$-0.746972\pi$
$347$	−9053.95	−1.40070	−0.700348	+	0.713801i	$0.746972\pi$
$348$	0	0
$349$	281.638	0.0431970	0.0215985	−	0.999767i	$-0.493124\pi$
$349$	281.638	0.0431970	0.0215985	+	0.999767i	$0.493124\pi$
$350$	0	0
$351$	−793.852	−0.120720
$352$	0	0
$353$	5450.62	0.821834	0.410917	−	0.911673i	$-0.365209\pi$
$353$	5450.62	0.821834	0.410917	+	0.911673i	$0.365209\pi$
$354$	0	0
$355$	10426.9	1.55887
$356$	0	0
$357$	0	0
$358$	0	0
$359$	10039.3	1.47591	0.737956	−	0.674849i	$-0.235791\pi$
$359$	10039.3	1.47591	0.737956	+	0.674849i	$0.235791\pi$
$360$	0	0
$361$	−6799.85	−0.991377
$362$	0	0
$363$	−154.108	−0.0222826
$364$	0	0
$365$	3412.99	0.489435
$366$	0	0
$367$	−2009.50	−0.285817	−0.142909	−	0.989736i	$-0.545646\pi$
$367$	−2009.50	−0.285817	−0.142909	+	0.989736i	$0.545646\pi$
$368$	0	0
$369$	9001.75	1.26995
$370$	0	0
$371$	234.491	0.0328144
$372$	0	0
$373$	10430.4	1.44790	0.723949	−	0.689854i	$-0.242325\pi$
$373$	10430.4	1.44790	0.723949	+	0.689854i	$0.242325\pi$
$374$	0	0
$375$	−13.1224	−0.00180703
$376$	0	0
$377$	1678.95	0.229365
$378$	0	0
$379$	11741.9	1.59140	0.795700	−	0.605690i	$-0.207103\pi$
$379$	11741.9	1.59140	0.795700	+	0.605690i	$0.207103\pi$
$380$	0	0
$381$	−2817.66	−0.378880
$382$	0	0
$383$	−9727.40	−1.29777	−0.648886	−	0.760886i	$-0.724765\pi$
$383$	−9727.40	−1.29777	−0.648886	+	0.760886i	$0.724765\pi$
$384$	0	0
$385$	3904.83	0.516905
$386$	0	0
$387$	−2109.32	−0.277061
$388$	0	0
$389$	6414.51	0.836063	0.418032	−	0.908432i	$-0.362720\pi$
$389$	6414.51	0.836063	0.418032	+	0.908432i	$0.362720\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−3577.17	−0.459146
$394$	0	0
$395$	−7155.06	−0.911418
$396$	0	0
$397$	−11652.5	−1.47311	−0.736553	−	0.676380i	$-0.763548\pi$
$397$	−11652.5	−1.47311	−0.736553	+	0.676380i	$0.763548\pi$
$398$	0	0
$399$	−89.4224	−0.0112199
$400$	0	0
$401$	−7891.85	−0.982794	−0.491397	−	0.870936i	$-0.663514\pi$
$401$	−7891.85	−0.982794	−0.491397	+	0.870936i	$0.663514\pi$
$402$	0	0
$403$	1629.52	0.201420
$404$	0	0
$405$	−8132.43	−0.997786
$406$	0	0
$407$	7999.02	0.974194
$408$	0	0
$409$	5230.36	0.632334	0.316167	−	0.948704i	$-0.397604\pi$
$409$	5230.36	0.632334	0.316167	+	0.948704i	$0.397604\pi$
$410$	0	0
$411$	1745.86	0.209530
$412$	0	0
$413$	−2270.58	−0.270527
$414$	0	0
$415$	−22813.4	−2.69847
$416$	0	0
$417$	−1001.78	−0.117644
$418$	0	0
$419$	−15088.7	−1.75926	−0.879629	−	0.475660i	$-0.842209\pi$
$419$	−15088.7	−1.75926	−0.879629	+	0.475660i	$0.842209\pi$
$420$	0	0
$421$	679.750	0.0786911	0.0393456	−	0.999226i	$-0.487473\pi$
$421$	679.750	0.0786911	0.0393456	+	0.999226i	$0.487473\pi$
$422$	0	0
$423$	−9131.05	−1.04957
$424$	0	0
$425$	0	0
$426$	0	0
$427$	4807.55	0.544857
$428$	0	0
$429$	545.025	0.0613381
$430$	0	0
$431$	5678.35	0.634609	0.317304	−	0.948324i	$-0.397222\pi$
$431$	5678.35	0.634609	0.317304	+	0.948324i	$0.397222\pi$
$432$	0	0
$433$	5469.74	0.607065	0.303533	−	0.952821i	$-0.401834\pi$
$433$	5469.74	0.607065	0.303533	+	0.952821i	$0.401834\pi$
$434$	0	0
$435$	−4717.39	−0.519958
$436$	0	0
$437$	−391.023	−0.0428036
$438$	0	0
$439$	1756.79	0.190995	0.0954975	−	0.995430i	$-0.469556\pi$
$439$	1756.79	0.190995	0.0954975	+	0.995430i	$0.469556\pi$
$440$	0	0
$441$	7125.56	0.769416
$442$	0	0
$443$	−8833.85	−0.947425	−0.473712	−	0.880680i	$-0.657086\pi$
$443$	−8833.85	−0.947425	−0.473712	+	0.880680i	$0.657086\pi$
$444$	0	0
$445$	8701.68	0.926964
$446$	0	0
$447$	−3383.15	−0.357981
$448$	0	0
$449$	3353.43	0.352469	0.176234	−	0.984348i	$-0.443608\pi$
$449$	3353.43	0.352469	0.176234	+	0.984348i	$0.443608\pi$
$450$	0	0
$451$	−13061.2	−1.36370
$452$	0	0
$453$	1379.45	0.143073
$454$	0	0
$455$	1036.63	0.106808
$456$	0	0
$457$	−8455.21	−0.865466	−0.432733	−	0.901522i	$-0.642451\pi$
$457$	−8455.21	−0.865466	−0.432733	+	0.901522i	$0.642451\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−16128.4	−1.62944	−0.814721	−	0.579853i	$-0.803110\pi$
$461$	−16128.4	−1.62944	−0.814721	+	0.579853i	$0.803110\pi$
$462$	0	0
$463$	−15851.4	−1.59110	−0.795550	−	0.605888i	$-0.792818\pi$
$463$	−15851.4	−1.59110	−0.795550	+	0.605888i	$0.792818\pi$
$464$	0	0
$465$	−4578.51	−0.456609
$466$	0	0
$467$	17530.5	1.73708	0.868541	−	0.495618i	$-0.165058\pi$
$467$	17530.5	1.73708	0.868541	+	0.495618i	$0.165058\pi$
$468$	0	0
$469$	−6880.14	−0.677389
$470$	0	0
$471$	−2090.86	−0.204548
$472$	0	0
$473$	3060.54	0.297514
$474$	0	0
$475$	965.188	0.0932334
$476$	0	0
$477$	810.988	0.0778461
$478$	0	0
$479$	8494.11	0.810242	0.405121	−	0.914263i	$-0.367229\pi$
$479$	8494.11	0.810242	0.405121	+	0.914263i	$0.367229\pi$
$480$	0	0
$481$	2123.52	0.201298
$482$	0	0
$483$	591.169	0.0556918
$484$	0	0
$485$	−2225.06	−0.208319
$486$	0	0
$487$	8154.23	0.758734	0.379367	−	0.925246i	$-0.376142\pi$
$487$	8154.23	0.758734	0.379367	+	0.925246i	$0.376142\pi$
$488$	0	0
$489$	790.282	0.0730834
$490$	0	0
$491$	8518.36	0.782949	0.391475	−	0.920189i	$-0.371965\pi$
$491$	8518.36	0.782949	0.391475	+	0.920189i	$0.371965\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	13504.9	1.22626
$496$	0	0
$497$	4619.30	0.416909
$498$	0	0
$499$	−5520.93	−0.495293	−0.247646	−	0.968851i	$-0.579657\pi$
$499$	−5520.93	−0.495293	−0.247646	+	0.968851i	$0.579657\pi$
$500$	0	0
$501$	−4244.05	−0.378463
$502$	0	0
$503$	8120.22	0.719807	0.359903	−	0.932990i	$-0.382810\pi$
$503$	8120.22	0.719807	0.359903	+	0.932990i	$0.382810\pi$
$504$	0	0
$505$	−576.191	−0.0507726
$506$	0	0
$507$	−3498.50	−0.306458
$508$	0	0
$509$	8839.27	0.769732	0.384866	−	0.922972i	$-0.374248\pi$
$509$	8839.27	0.769732	0.384866	+	0.922972i	$0.374248\pi$
$510$	0	0
$511$	1512.02	0.130896
$512$	0	0
$513$	−653.606	−0.0562522
$514$	0	0
$515$	−1930.16	−0.165151
$516$	0	0
$517$	13248.8	1.12705
$518$	0	0
$519$	5602.98	0.473879
$520$	0	0
$521$	6495.42	0.546199	0.273099	−	0.961986i	$-0.411951\pi$
$521$	6495.42	0.546199	0.273099	+	0.961986i	$0.411951\pi$
$522$	0	0
$523$	−7073.11	−0.591368	−0.295684	−	0.955286i	$-0.595548\pi$
$523$	−7073.11	−0.591368	−0.295684	+	0.955286i	$0.595548\pi$
$524$	0	0
$525$	−1459.22	−0.121306
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−9581.96	−0.787537
$530$	0	0
$531$	−7852.81	−0.641776
$532$	0	0
$533$	−3467.40	−0.281782
$534$	0	0
$535$	−18075.3	−1.46068
$536$	0	0
$537$	−6589.24	−0.529509
$538$	0	0
$539$	−10338.9	−0.826215
$540$	0	0
$541$	3548.78	0.282022	0.141011	−	0.990008i	$-0.454965\pi$
$541$	3548.78	0.282022	0.141011	+	0.990008i	$0.454965\pi$
$542$	0	0
$543$	−2851.02	−0.225321
$544$	0	0
$545$	34151.1	2.68417
$546$	0	0
$547$	−10309.2	−0.805829	−0.402914	−	0.915238i	$-0.632003\pi$
$547$	−10309.2	−0.805829	−0.402914	+	0.915238i	$0.632003\pi$
$548$	0	0
$549$	16627.0	1.29257
$550$	0	0
$551$	1382.34	0.106878
$552$	0	0
$553$	−3169.83	−0.243752
$554$	0	0
$555$	−5966.52	−0.456333
$556$	0	0
$557$	5454.69	0.414942	0.207471	−	0.978241i	$-0.433477\pi$
$557$	5454.69	0.414942	0.207471	+	0.978241i	$0.433477\pi$
$558$	0	0
$559$	812.492	0.0614754
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−15453.3	−1.15680	−0.578402	−	0.815752i	$-0.696324\pi$
$563$	−15453.3	−1.15680	−0.578402	+	0.815752i	$0.696324\pi$
$564$	0	0
$565$	7288.01	0.542671
$566$	0	0
$567$	−3602.82	−0.266850
$568$	0	0
$569$	16914.3	1.24620	0.623098	−	0.782144i	$-0.285874\pi$
$569$	16914.3	1.24620	0.623098	+	0.782144i	$0.285874\pi$
$570$	0	0
$571$	1148.04	0.0841398	0.0420699	−	0.999115i	$-0.486605\pi$
$571$	1148.04	0.0841398	0.0420699	+	0.999115i	$0.486605\pi$
$572$	0	0
$573$	4466.81	0.325661
$574$	0	0
$575$	−6380.83	−0.462781
$576$	0	0
$577$	−24631.6	−1.77717	−0.888585	−	0.458712i	$-0.848311\pi$
$577$	−24631.6	−1.77717	−0.888585	+	0.458712i	$0.848311\pi$
$578$	0	0
$579$	−7296.39	−0.523709
$580$	0	0
$581$	−10106.8	−0.721686
$582$	0	0
$583$	−1176.71	−0.0835927
$584$	0	0
$585$	3585.18	0.253383
$586$	0	0
$587$	−21562.8	−1.51617	−0.758085	−	0.652155i	$-0.773865\pi$
$587$	−21562.8	−1.51617	−0.758085	+	0.652155i	$0.773865\pi$
$588$	0	0
$589$	1341.64	0.0938564
$590$	0	0
$591$	831.665	0.0578851
$592$	0	0
$593$	3996.28	0.276741	0.138371	−	0.990381i	$-0.455813\pi$
$593$	3996.28	0.276741	0.138371	+	0.990381i	$0.455813\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−6583.10	−0.451304
$598$	0	0
$599$	24800.4	1.69168	0.845839	−	0.533438i	$-0.179100\pi$
$599$	24800.4	1.69168	0.845839	+	0.533438i	$0.179100\pi$
$600$	0	0
$601$	−13028.8	−0.884285	−0.442142	−	0.896945i	$-0.645781\pi$
$601$	−13028.8	−0.884285	−0.442142	+	0.896945i	$0.645781\pi$
$602$	0	0
$603$	−23795.0	−1.60698
$604$	0	0
$605$	1470.88	0.0988425
$606$	0	0
$607$	−12536.8	−0.838310	−0.419155	−	0.907915i	$-0.637674\pi$
$607$	−12536.8	−0.838310	−0.419155	+	0.907915i	$0.637674\pi$
$608$	0	0
$609$	−2089.90	−0.139059
$610$	0	0
$611$	3517.21	0.232882
$612$	0	0
$613$	14616.4	0.963051	0.481526	−	0.876432i	$-0.340083\pi$
$613$	14616.4	0.963051	0.481526	+	0.876432i	$0.340083\pi$
$614$	0	0
$615$	9742.44	0.638786
$616$	0	0
$617$	3721.68	0.242835	0.121417	−	0.992602i	$-0.461256\pi$
$617$	3721.68	0.242835	0.121417	+	0.992602i	$0.461256\pi$
$618$	0	0
$619$	−3799.73	−0.246727	−0.123364	−	0.992362i	$-0.539368\pi$
$619$	−3799.73	−0.246727	−0.123364	+	0.992362i	$0.539368\pi$
$620$	0	0
$621$	4320.97	0.279218
$622$	0	0
$623$	3855.01	0.247910
$624$	0	0
$625$	−15562.3	−0.995984
$626$	0	0
$627$	448.737	0.0285819
$628$	0	0
$629$	0	0
$630$	0	0
$631$	15719.3	0.991719	0.495859	−	0.868403i	$-0.334853\pi$
$631$	15719.3	0.991719	0.495859	+	0.868403i	$0.334853\pi$
$632$	0	0
$633$	−60.6868	−0.00381056
$634$	0	0
$635$	26893.1	1.68066
$636$	0	0
$637$	−2744.71	−0.170721
$638$	0	0
$639$	15975.9	0.989039
$640$	0	0
$641$	−2036.70	−0.125499	−0.0627496	−	0.998029i	$-0.519987\pi$
$641$	−2036.70	−0.125499	−0.0627496	+	0.998029i	$0.519987\pi$
$642$	0	0
$643$	−24041.3	−1.47449	−0.737243	−	0.675627i	$-0.763873\pi$
$643$	−24041.3	−1.47449	−0.737243	+	0.675627i	$0.763873\pi$
$644$	0	0
$645$	−2282.88	−0.139362
$646$	0	0
$647$	−13564.3	−0.824218	−0.412109	−	0.911135i	$-0.635208\pi$
$647$	−13564.3	−0.824218	−0.412109	+	0.911135i	$0.635208\pi$
$648$	0	0
$649$	11394.2	0.689152
$650$	0	0
$651$	−2028.37	−0.122117
$652$	0	0
$653$	26326.0	1.57767	0.788833	−	0.614608i	$-0.210686\pi$
$653$	26326.0	1.57767	0.788833	+	0.614608i	$0.210686\pi$
$654$	0	0
$655$	34142.2	2.03671
$656$	0	0
$657$	5229.33	0.310526
$658$	0	0
$659$	−10954.6	−0.647540	−0.323770	−	0.946136i	$-0.604950\pi$
$659$	−10954.6	−0.647540	−0.323770	+	0.946136i	$0.604950\pi$
$660$	0	0
$661$	−29602.5	−1.74191	−0.870956	−	0.491360i	$-0.836500\pi$
$661$	−29602.5	−1.74191	−0.870956	+	0.491360i	$0.836500\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	853.490	0.0497698
$666$	0	0
$667$	−9138.61	−0.530507
$668$	0	0
$669$	−384.059	−0.0221952
$670$	0	0
$671$	−24125.1	−1.38799
$672$	0	0
$673$	28427.1	1.62821	0.814103	−	0.580720i	$-0.197229\pi$
$673$	28427.1	1.62821	0.814103	+	0.580720i	$0.197229\pi$
$674$	0	0
$675$	−10665.7	−0.608185
$676$	0	0
$677$	16442.4	0.933430	0.466715	−	0.884408i	$-0.345437\pi$
$677$	16442.4	0.933430	0.466715	+	0.884408i	$0.345437\pi$
$678$	0	0
$679$	−985.744	−0.0557134
$680$	0	0
$681$	436.768	0.0245771
$682$	0	0
$683$	−26595.7	−1.48998	−0.744990	−	0.667076i	$-0.767546\pi$
$683$	−26595.7	−1.48998	−0.744990	+	0.667076i	$0.767546\pi$
$684$	0	0
$685$	−16663.3	−0.929450
$686$	0	0
$687$	1066.77	0.0592428
$688$	0	0
$689$	−312.386	−0.0172728
$690$	0	0
$691$	−33230.2	−1.82943	−0.914715	−	0.404099i	$-0.867585\pi$
$691$	−33230.2	−1.82943	−0.914715	+	0.404099i	$0.867585\pi$
$692$	0	0
$693$	5982.92	0.327954
$694$	0	0
$695$	9561.49	0.521853
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−10303.9	−0.557554
$700$	0	0
$701$	−1173.30	−0.0632165	−0.0316083	−	0.999500i	$-0.510063\pi$
$701$	−1173.30	−0.0632165	−0.0316083	+	0.999500i	$0.510063\pi$
$702$	0	0
$703$	1748.37	0.0937995
$704$	0	0
$705$	−9882.38	−0.527932
$706$	0	0
$707$	−255.263	−0.0135787
$708$	0	0
$709$	−6875.19	−0.364180	−0.182090	−	0.983282i	$-0.558286\pi$
$709$	−6875.19	−0.364180	−0.182090	+	0.983282i	$0.558286\pi$
$710$	0	0
$711$	−10962.9	−0.578256
$712$	0	0
$713$	−8869.56	−0.465873
$714$	0	0
$715$	−5201.97	−0.272088
$716$	0	0
$717$	11349.6	0.591155
$718$	0	0
$719$	−4348.74	−0.225564	−0.112782	−	0.993620i	$-0.535976\pi$
$719$	−4348.74	−0.225564	−0.112782	+	0.993620i	$0.535976\pi$
$720$	0	0
$721$	−855.097	−0.0441685
$722$	0	0
$723$	3617.61	0.186086
$724$	0	0
$725$	22557.4	1.15553
$726$	0	0
$727$	22592.1	1.15254	0.576269	−	0.817260i	$-0.304508\pi$
$727$	22592.1	1.15254	0.576269	+	0.817260i	$0.304508\pi$
$728$	0	0
$729$	−8655.30	−0.439735
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−9516.58	−0.479540	−0.239770	−	0.970830i	$-0.577072\pi$
$733$	−9516.58	−0.479540	−0.239770	+	0.970830i	$0.577072\pi$
$734$	0	0
$735$	7711.88	0.387016
$736$	0	0
$737$	34525.7	1.72561
$738$	0	0
$739$	−6682.15	−0.332621	−0.166311	−	0.986073i	$-0.553185\pi$
$739$	−6682.15	−0.332621	−0.166311	+	0.986073i	$0.553185\pi$
$740$	0	0
$741$	119.128	0.00590589
$742$	0	0
$743$	−16281.7	−0.803928	−0.401964	−	0.915655i	$-0.631672\pi$
$743$	−16281.7	−0.803928	−0.401964	+	0.915655i	$0.631672\pi$
$744$	0	0
$745$	32290.4	1.58796
$746$	0	0
$747$	−34954.3	−1.71207
$748$	0	0
$749$	−8007.69	−0.390647
$750$	0	0
$751$	−29633.0	−1.43985	−0.719923	−	0.694054i	$-0.755823\pi$
$751$	−29633.0	−1.43985	−0.719923	+	0.694054i	$0.755823\pi$
$752$	0	0
$753$	−3787.06	−0.183278
$754$	0	0
$755$	−13166.1	−0.634654
$756$	0	0
$757$	−12885.4	−0.618662	−0.309331	−	0.950954i	$-0.600105\pi$
$757$	−12885.4	−0.618662	−0.309331	+	0.950954i	$0.600105\pi$
$758$	0	0
$759$	−2966.59	−0.141871
$760$	0	0
$761$	−12690.1	−0.604491	−0.302245	−	0.953230i	$-0.597736\pi$
$761$	−12690.1	−0.604491	−0.302245	+	0.953230i	$0.597736\pi$
$762$	0	0
$763$	15129.6	0.717861
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3024.84	0.142400
$768$	0	0
$769$	12008.7	0.563126	0.281563	−	0.959543i	$-0.409147\pi$
$769$	12008.7	0.563126	0.281563	+	0.959543i	$0.409147\pi$
$770$	0	0
$771$	−7798.27	−0.364264
$772$	0	0
$773$	35695.8	1.66092	0.830458	−	0.557081i	$-0.188079\pi$
$773$	35695.8	1.66092	0.830458	+	0.557081i	$0.188079\pi$
$774$	0	0
$775$	21893.3	1.01475
$776$	0	0
$777$	−2643.28	−0.122043
$778$	0	0
$779$	−2854.83	−0.131303
$780$	0	0
$781$	−23180.4	−1.06205
$782$	0	0
$783$	−15275.4	−0.697190
$784$	0	0
$785$	19956.2	0.907346
$786$	0	0
$787$	2149.38	0.0973536	0.0486768	−	0.998815i	$-0.484500\pi$
$787$	2149.38	0.0973536	0.0486768	+	0.998815i	$0.484500\pi$
$788$	0	0
$789$	−5412.57	−0.244224
$790$	0	0
$791$	3228.73	0.145133
$792$	0	0
$793$	−6404.57	−0.286801
$794$	0	0
$795$	877.718	0.0391565
$796$	0	0
$797$	997.011	0.0443111	0.0221556	−	0.999755i	$-0.492947\pi$
$797$	997.011	0.0443111	0.0221556	+	0.999755i	$0.492947\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	13332.6	0.588119
$802$	0	0
$803$	−7587.57	−0.333449
$804$	0	0
$805$	−5642.40	−0.247041
$806$	0	0
$807$	−5887.57	−0.256818
$808$	0	0
$809$	−42317.4	−1.83906	−0.919530	−	0.393021i	$-0.871430\pi$
$809$	−42317.4	−1.83906	−0.919530	+	0.393021i	$0.871430\pi$
$810$	0	0
$811$	32849.8	1.42233	0.711167	−	0.703023i	$-0.248167\pi$
$811$	32849.8	1.42233	0.711167	+	0.703023i	$0.248167\pi$
$812$	0	0
$813$	12246.2	0.528282
$814$	0	0
$815$	−7542.82	−0.324188
$816$	0	0
$817$	668.952	0.0286459
$818$	0	0
$819$	1588.30	0.0677653
$820$	0	0
$821$	−28983.5	−1.23207	−0.616037	−	0.787717i	$-0.711263\pi$
$821$	−28983.5	−1.23207	−0.616037	+	0.787717i	$0.711263\pi$
$822$	0	0
$823$	−11039.9	−0.467589	−0.233795	−	0.972286i	$-0.575114\pi$
$823$	−11039.9	−0.467589	−0.233795	+	0.972286i	$0.575114\pi$
$824$	0	0
$825$	7322.64	0.309020
$826$	0	0
$827$	−22758.2	−0.956930	−0.478465	−	0.878107i	$-0.658807\pi$
$827$	−22758.2	−0.956930	−0.478465	+	0.878107i	$0.658807\pi$
$828$	0	0
$829$	−38289.8	−1.60417	−0.802087	−	0.597207i	$-0.796277\pi$
$829$	−38289.8	−1.60417	−0.802087	+	0.597207i	$0.796277\pi$
$830$	0	0
$831$	10934.3	0.456448
$832$	0	0
$833$	0	0
$834$	0	0
$835$	40507.2	1.67881
$836$	0	0
$837$	−14825.7	−0.612248
$838$	0	0
$839$	32078.4	1.31999	0.659994	−	0.751270i	$-0.270559\pi$
$839$	32078.4	1.31999	0.659994	+	0.751270i	$0.270559\pi$
$840$	0	0
$841$	7917.70	0.324642
$842$	0	0
$843$	−8358.98	−0.341517
$844$	0	0
$845$	33391.4	1.35941
$846$	0	0
$847$	651.627	0.0264347
$848$	0	0
$849$	−7930.17	−0.320569
$850$	0	0
$851$	−11558.4	−0.465591
$852$	0	0
$853$	−28627.9	−1.14912	−0.574560	−	0.818462i	$-0.694827\pi$
$853$	−28627.9	−1.14912	−0.574560	+	0.818462i	$0.694827\pi$
$854$	0	0
$855$	2951.80	0.118070
$856$	0	0
$857$	−9299.57	−0.370673	−0.185337	−	0.982675i	$-0.559338\pi$
$857$	−9299.57	−0.370673	−0.185337	+	0.982675i	$0.559338\pi$
$858$	0	0
$859$	42611.7	1.69254	0.846270	−	0.532755i	$-0.178843\pi$
$859$	42611.7	1.69254	0.846270	+	0.532755i	$0.178843\pi$
$860$	0	0
$861$	4316.09	0.170838
$862$	0	0
$863$	32886.2	1.29717	0.648585	−	0.761142i	$-0.275361\pi$
$863$	32886.2	1.29717	0.648585	+	0.761142i	$0.275361\pi$
$864$	0	0
$865$	−53477.5	−2.10207
$866$	0	0
$867$	0	0
$868$	0	0
$869$	15906.8	0.620943
$870$	0	0
$871$	9165.65	0.356563
$872$	0	0
$873$	−3409.20	−0.132170
$874$	0	0
$875$	55.4863	0.00214375
$876$	0	0
$877$	18890.0	0.727332	0.363666	−	0.931529i	$-0.381525\pi$
$877$	18890.0	0.727332	0.363666	+	0.931529i	$0.381525\pi$
$878$	0	0
$879$	5530.05	0.212200
$880$	0	0
$881$	34435.8	1.31688	0.658440	−	0.752633i	$-0.271217\pi$
$881$	34435.8	1.31688	0.658440	+	0.752633i	$0.271217\pi$
$882$	0	0
$883$	25665.3	0.978151	0.489076	−	0.872241i	$-0.337334\pi$
$883$	25665.3	0.978151	0.489076	+	0.872241i	$0.337334\pi$
$884$	0	0
$885$	−8498.96	−0.322813
$886$	0	0
$887$	4578.10	0.173301	0.0866503	−	0.996239i	$-0.472384\pi$
$887$	4578.10	0.173301	0.0866503	+	0.996239i	$0.472384\pi$
$888$	0	0
$889$	11914.2	0.449480
$890$	0	0
$891$	18079.6	0.679785
$892$	0	0
$893$	2895.84	0.108517
$894$	0	0
$895$	62890.8	2.34883
$896$	0	0
$897$	−787.550	−0.0293150
$898$	0	0
$899$	31355.6	1.16326
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−1011.36	−0.0372712
$904$	0	0
$905$	27211.5	0.999493
$906$	0	0
$907$	13083.0	0.478956	0.239478	−	0.970902i	$-0.423024\pi$
$907$	13083.0	0.478956	0.239478	+	0.970902i	$0.423024\pi$
$908$	0	0
$909$	−882.831	−0.0322130
$910$	0	0
$911$	11991.0	0.436093	0.218046	−	0.975938i	$-0.430032\pi$
$911$	11991.0	0.436093	0.218046	+	0.975938i	$0.430032\pi$
$912$	0	0
$913$	50717.5	1.83845
$914$	0	0
$915$	17995.1	0.650163
$916$	0	0
$917$	15125.6	0.544703
$918$	0	0
$919$	−37277.1	−1.33804	−0.669020	−	0.743245i	$-0.733286\pi$
$919$	−37277.1	−1.33804	−0.669020	+	0.743245i	$0.733286\pi$
$920$	0	0
$921$	−6491.19	−0.232239
$922$	0	0
$923$	−6153.78	−0.219452
$924$	0	0
$925$	28530.5	1.01414
$926$	0	0
$927$	−2957.36	−0.104781
$928$	0	0
$929$	27210.9	0.960989	0.480495	−	0.876998i	$-0.340457\pi$
$929$	27210.9	0.960989	0.480495	+	0.876998i	$0.340457\pi$
$930$	0	0
$931$	−2259.81	−0.0795514
$932$	0	0
$933$	−13874.2	−0.486839
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−40790.7	−1.42217	−0.711086	−	0.703105i	$-0.751796\pi$
$937$	−40790.7	−1.42217	−0.711086	+	0.703105i	$0.751796\pi$
$938$	0	0
$939$	2113.72	0.0734597
$940$	0	0
$941$	8364.65	0.289777	0.144888	−	0.989448i	$-0.453718\pi$
$941$	8364.65	0.289777	0.144888	+	0.989448i	$0.453718\pi$
$942$	0	0
$943$	18873.2	0.651745
$944$	0	0
$945$	−9431.43	−0.324661
$946$	0	0
$947$	34900.8	1.19760	0.598799	−	0.800899i	$-0.295645\pi$
$947$	34900.8	1.19760	0.598799	+	0.800899i	$0.295645\pi$
$948$	0	0
$949$	−2014.29	−0.0689007
$950$	0	0
$951$	5224.48	0.178144
$952$	0	0
$953$	−29662.6	−1.00825	−0.504127	−	0.863630i	$-0.668185\pi$
$953$	−29662.6	−1.00825	−0.504127	+	0.863630i	$0.668185\pi$
$954$	0	0
$955$	−42633.3	−1.44459
$956$	0	0
$957$	10487.5	0.354244
$958$	0	0
$959$	−7382.17	−0.248574
$960$	0	0
$961$	641.431	0.0215310
$962$	0	0
$963$	−27694.7	−0.926737
$964$	0	0
$965$	69640.1	2.32310
$966$	0	0
$967$	41635.8	1.38461	0.692304	−	0.721606i	$-0.256596\pi$
$967$	41635.8	1.38461	0.692304	+	0.721606i	$0.256596\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	3181.14	0.105137	0.0525683	−	0.998617i	$-0.483259\pi$
$971$	3181.14	0.105137	0.0525683	+	0.998617i	$0.483259\pi$
$972$	0	0
$973$	4235.92	0.139566
$974$	0	0
$975$	1943.96	0.0638530
$976$	0	0
$977$	46359.4	1.51808	0.759042	−	0.651042i	$-0.225668\pi$
$977$	46359.4	1.51808	0.759042	+	0.651042i	$0.225668\pi$
$978$	0	0
$979$	−19345.1	−0.631534
$980$	0	0
$981$	52325.8	1.70299
$982$	0	0
$983$	33708.5	1.09373	0.546864	−	0.837222i	$-0.315822\pi$
$983$	33708.5	1.09373	0.546864	+	0.837222i	$0.315822\pi$
$984$	0	0
$985$	−7937.80	−0.256771
$986$	0	0
$987$	−4378.08	−0.141191
$988$	0	0
$989$	−4422.42	−0.142189
$990$	0	0
$991$	11952.4	0.383129	0.191564	−	0.981480i	$-0.438644\pi$
$991$	11952.4	0.383129	0.191564	+	0.981480i	$0.438644\pi$
$992$	0	0
$993$	8195.87	0.261921
$994$	0	0
$995$	62832.2	2.00192
$996$	0	0
$997$	14645.4	0.465221	0.232611	−	0.972570i	$-0.425273\pi$
$997$	14645.4	0.465221	0.232611	+	0.972570i	$0.425273\pi$
$998$	0	0
$999$	−19320.2	−0.611877

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2312.4.a.n.1.12	✓	18
17.16	even	2		2312.4.a.q.1.7	yes	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2312.4.a.n.1.12	✓	18	1.1	even	1	trivial
2312.4.a.q.1.7	yes	18	17.16	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:	\( N \)	\(=\)	\( 2312 = 2^{3} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2312.a (trivial)

\(f(q)\)	\(=\)	\(q+1.65826 q^{3} -15.8272 q^{5} -7.01175 q^{7} -24.2502 q^{9} +O(q^{10})\)	\(q+1.65826 q^{3} -15.8272 q^{5} -7.01175 q^{7} -24.2502 q^{9} +35.1862 q^{11} +9.34098 q^{13} -26.2456 q^{15} +7.69074 q^{19} -11.6273 q^{21} -50.8433 q^{23} +125.500 q^{25} -84.9860 q^{27} +179.741 q^{29} +174.449 q^{31} +58.3477 q^{33} +110.976 q^{35} +227.334 q^{37} +15.4897 q^{39} -371.203 q^{41} +86.9814 q^{43} +383.812 q^{45} +376.535 q^{47} -293.835 q^{49} -33.4425 q^{53} -556.898 q^{55} +12.7532 q^{57} +323.825 q^{59} -685.643 q^{61} +170.036 q^{63} -147.841 q^{65} +981.231 q^{67} -84.3113 q^{69} -658.794 q^{71} -215.641 q^{73} +208.111 q^{75} -246.716 q^{77} +452.074 q^{79} +513.826 q^{81} +1441.41 q^{83} +298.056 q^{87} -549.793 q^{89} -65.4966 q^{91} +289.281 q^{93} -121.723 q^{95} +140.585 q^{97} -853.271 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 18 q^{3} - 51 q^{7} + 120 q^{9}+O(q^{10}) \) 18 * q - 18 * q^3 - 51 * q^7 + 120 * q^9	\( 18 q - 18 q^{3} - 51 q^{7} + 120 q^{9} - 132 q^{11} + 30 q^{13} + 102 q^{15} + 66 q^{19} + 144 q^{21} - 153 q^{23} + 306 q^{25} - 768 q^{27} - 51 q^{29} - 303 q^{31} + 525 q^{33} - 255 q^{35} - 717 q^{37} + 216 q^{39} + 393 q^{41} - 390 q^{43} - 558 q^{45} - 633 q^{47} + 1443 q^{49} + 1275 q^{53} + 1539 q^{55} - 810 q^{57} - 204 q^{59} - 534 q^{61} - 2556 q^{63} + 2127 q^{65} - 405 q^{67} + 2547 q^{69} + 426 q^{71} - 1149 q^{73} - 2226 q^{75} - 357 q^{77} - 1053 q^{79} + 2802 q^{81} + 66 q^{83} + 2487 q^{87} - 4119 q^{89} - 6090 q^{91} + 606 q^{93} - 2109 q^{95} - 2349 q^{97} - 1428 q^{99}+O(q^{100}) \) 18 * q - 18 * q^3 - 51 * q^7 + 120 * q^9 - 132 * q^11 + 30 * q^13 + 102 * q^15 + 66 * q^19 + 144 * q^21 - 153 * q^23 + 306 * q^25 - 768 * q^27 - 51 * q^29 - 303 * q^31 + 525 * q^33 - 255 * q^35 - 717 * q^37 + 216 * q^39 + 393 * q^41 - 390 * q^43 - 558 * q^45 - 633 * q^47 + 1443 * q^49 + 1275 * q^53 + 1539 * q^55 - 810 * q^57 - 204 * q^59 - 534 * q^61 - 2556 * q^63 + 2127 * q^65 - 405 * q^67 + 2547 * q^69 + 426 * q^71 - 1149 * q^73 - 2226 * q^75 - 357 * q^77 - 1053 * q^79 + 2802 * q^81 + 66 * q^83 + 2487 * q^87 - 4119 * q^89 - 6090 * q^91 + 606 * q^93 - 2109 * q^95 - 2349 * q^97 - 1428 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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