LMFDB - Embedded newform 1232.4.a.u.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1232,4,Mod(1,1232)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1232, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1232.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1232 = 2^{4} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1232.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:	$72.6903531271$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 29x^{2} + 3x + 114 $ x^4 - x^3 - 29x^2 + 3x + 114
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 616)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.21517$ of defining polynomial
Character	$\chi$	$=$	1232.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.21517 q^{3} +1.66265 q^{5} -7.00000 q^{7} -16.6627 q^{9} +O(q^{10})$	$q+3.21517 q^{3} +1.66265 q^{5} -7.00000 q^{7} -16.6627 q^{9} -11.0000 q^{11} +20.0620 q^{13} +5.34571 q^{15} +32.3206 q^{17} +141.911 q^{19} -22.5062 q^{21} -69.6900 q^{23} -122.236 q^{25} -140.383 q^{27} +253.347 q^{29} -100.540 q^{31} -35.3669 q^{33} -11.6386 q^{35} -217.928 q^{37} +64.5029 q^{39} -149.139 q^{41} +397.322 q^{43} -27.7042 q^{45} +252.958 q^{47} +49.0000 q^{49} +103.916 q^{51} +223.338 q^{53} -18.2892 q^{55} +456.268 q^{57} +296.647 q^{59} +478.041 q^{61} +116.639 q^{63} +33.3562 q^{65} +491.105 q^{67} -224.066 q^{69} +402.503 q^{71} -388.314 q^{73} -393.009 q^{75} +77.0000 q^{77} +1035.36 q^{79} -1.46447 q^{81} +1039.64 q^{83} +53.7379 q^{85} +814.554 q^{87} -639.555 q^{89} -140.434 q^{91} -323.254 q^{93} +235.948 q^{95} -225.195 q^{97} +183.289 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 3 q^{3} - 13 q^{5} - 28 q^{7} - 47 q^{9}+O(q^{10}) $ 4 * q + 3 * q^3 - 13 * q^5 - 28 * q^7 - 47 * q^9	$ 4 q + 3 q^{3} - 13 q^{5} - 28 q^{7} - 47 q^{9} - 44 q^{11} - 34 q^{13} - 63 q^{15} - 58 q^{17} - 60 q^{19} - 21 q^{21} + 93 q^{23} - 19 q^{25} - 63 q^{27} - 144 q^{29} + 129 q^{31} - 33 q^{33} + 91 q^{35} - 187 q^{37} + 244 q^{39} + 110 q^{41} + 360 q^{43} - 286 q^{45} + 438 q^{47} + 196 q^{49} + 902 q^{51} + 56 q^{53} + 143 q^{55} - 94 q^{57} + 1209 q^{59} + 104 q^{61} + 329 q^{63} - 1256 q^{65} + 1075 q^{67} + 451 q^{69} + 963 q^{71} - 646 q^{73} + 804 q^{75} + 308 q^{77} + 1838 q^{79} - 656 q^{81} + 1238 q^{83} - 278 q^{85} + 914 q^{87} - 1453 q^{89} + 238 q^{91} + 521 q^{93} + 2730 q^{95} + 573 q^{97} + 517 q^{99}+O(q^{100}) $ 4 * q + 3 * q^3 - 13 * q^5 - 28 * q^7 - 47 * q^9 - 44 * q^11 - 34 * q^13 - 63 * q^15 - 58 * q^17 - 60 * q^19 - 21 * q^21 + 93 * q^23 - 19 * q^25 - 63 * q^27 - 144 * q^29 + 129 * q^31 - 33 * q^33 + 91 * q^35 - 187 * q^37 + 244 * q^39 + 110 * q^41 + 360 * q^43 - 286 * q^45 + 438 * q^47 + 196 * q^49 + 902 * q^51 + 56 * q^53 + 143 * q^55 - 94 * q^57 + 1209 * q^59 + 104 * q^61 + 329 * q^63 - 1256 * q^65 + 1075 * q^67 + 451 * q^69 + 963 * q^71 - 646 * q^73 + 804 * q^75 + 308 * q^77 + 1838 * q^79 - 656 * q^81 + 1238 * q^83 - 278 * q^85 + 914 * q^87 - 1453 * q^89 + 238 * q^91 + 521 * q^93 + 2730 * q^95 + 573 * q^97 + 517 * 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
$3$	3.21517	0.618761	0.309380	−	0.950938i	$-0.399878\pi$
$3$	3.21517	0.618761	0.309380	+	0.950938i	$0.399878\pi$
$4$	0	0
$5$	1.66265	0.148712	0.0743560	−	0.997232i	$-0.476310\pi$
$5$	1.66265	0.148712	0.0743560	+	0.997232i	$0.476310\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	−16.6627	−0.617135
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	20.0620	0.428016	0.214008	−	0.976832i	$-0.431348\pi$
$13$	20.0620	0.428016	0.214008	+	0.976832i	$0.431348\pi$
$14$	0	0
$15$	5.34571	0.0920172
$16$	0	0
$17$	32.3206	0.461112	0.230556	−	0.973059i	$-0.425946\pi$
$17$	32.3206	0.461112	0.230556	+	0.973059i	$0.425946\pi$
$18$	0	0
$19$	141.911	1.71350	0.856751	−	0.515730i	$-0.172479\pi$
$19$	141.911	1.71350	0.856751	+	0.515730i	$0.172479\pi$
$20$	0	0
$21$	−22.5062	−0.233870
$22$	0	0
$23$	−69.6900	−0.631799	−0.315899	−	0.948793i	$-0.602306\pi$
$23$	−69.6900	−0.631799	−0.315899	+	0.948793i	$0.602306\pi$
$24$	0	0
$25$	−122.236	−0.977885
$26$	0	0
$27$	−140.383	−1.00062
$28$	0	0
$29$	253.347	1.62225	0.811126	−	0.584872i	$-0.198855\pi$
$29$	253.347	1.62225	0.811126	+	0.584872i	$0.198855\pi$
$30$	0	0
$31$	−100.540	−0.582502	−0.291251	−	0.956647i	$-0.594071\pi$
$31$	−100.540	−0.582502	−0.291251	+	0.956647i	$0.594071\pi$
$32$	0	0
$33$	−35.3669	−0.186563
$34$	0	0
$35$	−11.6386	−0.0562079
$36$	0	0
$37$	−217.928	−0.968299	−0.484149	−	0.874985i	$-0.660871\pi$
$37$	−217.928	−0.968299	−0.484149	+	0.874985i	$0.660871\pi$
$38$	0	0
$39$	64.5029	0.264839
$40$	0	0
$41$	−149.139	−0.568088	−0.284044	−	0.958811i	$-0.591676\pi$
$41$	−149.139	−0.568088	−0.284044	+	0.958811i	$0.591676\pi$
$42$	0	0
$43$	397.322	1.40909	0.704547	−	0.709657i	$-0.251150\pi$
$43$	397.322	1.40909	0.704547	+	0.709657i	$0.251150\pi$
$44$	0	0
$45$	−27.7042	−0.0917754
$46$	0	0
$47$	252.958	0.785059	0.392529	−	0.919739i	$-0.371600\pi$
$47$	252.958	0.785059	0.392529	+	0.919739i	$0.371600\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	103.916	0.285318
$52$	0	0
$53$	223.338	0.578827	0.289414	−	0.957204i	$-0.406540\pi$
$53$	223.338	0.578827	0.289414	+	0.957204i	$0.406540\pi$
$54$	0	0
$55$	−18.2892	−0.0448384
$56$	0	0
$57$	456.268	1.06025
$58$	0	0
$59$	296.647	0.654578	0.327289	−	0.944924i	$-0.393865\pi$
$59$	296.647	0.654578	0.327289	+	0.944924i	$0.393865\pi$
$60$	0	0
$61$	478.041	1.00339	0.501696	−	0.865044i	$-0.332710\pi$
$61$	478.041	1.00339	0.501696	+	0.865044i	$0.332710\pi$
$62$	0	0
$63$	116.639	0.233255
$64$	0	0
$65$	33.3562	0.0636511
$66$	0	0
$67$	491.105	0.895492	0.447746	−	0.894161i	$-0.352227\pi$
$67$	491.105	0.895492	0.447746	+	0.894161i	$0.352227\pi$
$68$	0	0
$69$	−224.066	−0.390932
$70$	0	0
$71$	402.503	0.672792	0.336396	−	0.941721i	$-0.390792\pi$
$71$	402.503	0.672792	0.336396	+	0.941721i	$0.390792\pi$
$72$	0	0
$73$	−388.314	−0.622585	−0.311293	−	0.950314i	$-0.600762\pi$
$73$	−388.314	−0.622585	−0.311293	+	0.950314i	$0.600762\pi$
$74$	0	0
$75$	−393.009	−0.605077
$76$	0	0
$77$	77.0000	0.113961
$78$	0	0
$79$	1035.36	1.47452	0.737262	−	0.675607i	$-0.236118\pi$
$79$	1035.36	1.47452	0.737262	+	0.675607i	$0.236118\pi$
$80$	0	0
$81$	−1.46447	−0.00200888
$82$	0	0
$83$	1039.64	1.37489	0.687443	−	0.726238i	$-0.258733\pi$
$83$	1039.64	1.37489	0.687443	+	0.726238i	$0.258733\pi$
$84$	0	0
$85$	53.7379	0.0685729
$86$	0	0
$87$	814.554	1.00379
$88$	0	0
$89$	−639.555	−0.761716	−0.380858	−	0.924634i	$-0.624371\pi$
$89$	−639.555	−0.761716	−0.380858	+	0.924634i	$0.624371\pi$
$90$	0	0
$91$	−140.434	−0.161775
$92$	0	0
$93$	−323.254	−0.360429
$94$	0	0
$95$	235.948	0.254818
$96$	0	0
$97$	−225.195	−0.235723	−0.117861	−	0.993030i	$-0.537604\pi$
$97$	−225.195	−0.235723	−0.117861	+	0.993030i	$0.537604\pi$
$98$	0	0
$99$	183.289	0.186073
$100$	0	0
$101$	−1534.79	−1.51205	−0.756027	−	0.654540i	$-0.772862\pi$
$101$	−1534.79	−1.51205	−0.756027	+	0.654540i	$0.772862\pi$
$102$	0	0
$103$	−369.183	−0.353171	−0.176586	−	0.984285i	$-0.556505\pi$
$103$	−369.183	−0.353171	−0.176586	+	0.984285i	$0.556505\pi$
$104$	0	0
$105$	−37.4200	−0.0347792
$106$	0	0
$107$	1205.11	1.08881	0.544403	−	0.838824i	$-0.316756\pi$
$107$	1205.11	1.08881	0.544403	+	0.838824i	$0.316756\pi$
$108$	0	0
$109$	1237.83	1.08773	0.543867	−	0.839172i	$-0.316960\pi$
$109$	1237.83	1.08773	0.543867	+	0.839172i	$0.316960\pi$
$110$	0	0
$111$	−700.675	−0.599145
$112$	0	0
$113$	−1192.43	−0.992690	−0.496345	−	0.868125i	$-0.665325\pi$
$113$	−1192.43	−0.992690	−0.496345	+	0.868125i	$0.665325\pi$
$114$	0	0
$115$	−115.870	−0.0939561
$116$	0	0
$117$	−334.287	−0.264144
$118$	0	0
$119$	−226.244	−0.174284
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	−479.508	−0.351511
$124$	0	0
$125$	−411.067	−0.294135
$126$	0	0
$127$	1896.03	1.32477	0.662383	−	0.749166i	$-0.269545\pi$
$127$	1896.03	1.32477	0.662383	+	0.749166i	$0.269545\pi$
$128$	0	0
$129$	1277.46	0.871892
$130$	0	0
$131$	1211.98	0.808330	0.404165	−	0.914686i	$-0.367562\pi$
$131$	1211.98	0.808330	0.404165	+	0.914686i	$0.367562\pi$
$132$	0	0
$133$	−993.375	−0.647643
$134$	0	0
$135$	−233.408	−0.148804
$136$	0	0
$137$	1605.18	1.00102	0.500509	−	0.865731i	$-0.333146\pi$
$137$	1605.18	1.00102	0.500509	+	0.865731i	$0.333146\pi$
$138$	0	0
$139$	11.8609	0.00723762	0.00361881	−	0.999993i	$-0.498848\pi$
$139$	11.8609	0.00723762	0.00361881	+	0.999993i	$0.498848\pi$
$140$	0	0
$141$	813.305	0.485763
$142$	0	0
$143$	−220.682	−0.129052
$144$	0	0
$145$	421.227	0.241248
$146$	0	0
$147$	157.544	0.0883944
$148$	0	0
$149$	−259.818	−0.142853	−0.0714266	−	0.997446i	$-0.522755\pi$
$149$	−259.818	−0.142853	−0.0714266	+	0.997446i	$0.522755\pi$
$150$	0	0
$151$	−922.967	−0.497417	−0.248709	−	0.968578i	$-0.580006\pi$
$151$	−922.967	−0.497417	−0.248709	+	0.968578i	$0.580006\pi$
$152$	0	0
$153$	−538.547	−0.284568
$154$	0	0
$155$	−167.163	−0.0866250
$156$	0	0
$157$	3152.24	1.60240	0.801198	−	0.598399i	$-0.204196\pi$
$157$	3152.24	1.60240	0.801198	+	0.598399i	$0.204196\pi$
$158$	0	0
$159$	718.071	0.358156
$160$	0	0
$161$	487.830	0.238798
$162$	0	0
$163$	3666.82	1.76201	0.881006	−	0.473106i	$-0.156867\pi$
$163$	3666.82	1.76201	0.881006	+	0.473106i	$0.156867\pi$
$164$	0	0
$165$	−58.8029	−0.0277442
$166$	0	0
$167$	−794.515	−0.368152	−0.184076	−	0.982912i	$-0.558929\pi$
$167$	−794.515	−0.368152	−0.184076	+	0.982912i	$0.558929\pi$
$168$	0	0
$169$	−1794.51	−0.816802
$170$	0	0
$171$	−2364.61	−1.05746
$172$	0	0
$173$	682.260	0.299834	0.149917	−	0.988699i	$-0.452099\pi$
$173$	682.260	0.299834	0.149917	+	0.988699i	$0.452099\pi$
$174$	0	0
$175$	855.649	0.369606
$176$	0	0
$177$	953.771	0.405027
$178$	0	0
$179$	869.316	0.362993	0.181496	−	0.983392i	$-0.441906\pi$
$179$	869.316	0.362993	0.181496	+	0.983392i	$0.441906\pi$
$180$	0	0
$181$	3350.62	1.37597	0.687983	−	0.725727i	$-0.258496\pi$
$181$	3350.62	1.37597	0.687983	+	0.725727i	$0.258496\pi$
$182$	0	0
$183$	1536.99	0.620859
$184$	0	0
$185$	−362.337	−0.143998
$186$	0	0
$187$	−355.527	−0.139030
$188$	0	0
$189$	982.681	0.378199
$190$	0	0
$191$	−1262.94	−0.478448	−0.239224	−	0.970964i	$-0.576893\pi$
$191$	−1262.94	−0.478448	−0.239224	+	0.970964i	$0.576893\pi$
$192$	0	0
$193$	−3915.80	−1.46044	−0.730221	−	0.683211i	$-0.760583\pi$
$193$	−3915.80	−1.46044	−0.730221	+	0.683211i	$0.760583\pi$
$194$	0	0
$195$	107.246	0.0393848
$196$	0	0
$197$	−951.755	−0.344212	−0.172106	−	0.985078i	$-0.555057\pi$
$197$	−951.755	−0.344212	−0.172106	+	0.985078i	$0.555057\pi$
$198$	0	0
$199$	−2116.50	−0.753945	−0.376972	−	0.926224i	$-0.623035\pi$
$199$	−2116.50	−0.753945	−0.376972	+	0.926224i	$0.623035\pi$
$200$	0	0
$201$	1578.99	0.554095
$202$	0	0
$203$	−1773.43	−0.613153
$204$	0	0
$205$	−247.966	−0.0844815
$206$	0	0
$207$	1161.22	0.389905
$208$	0	0
$209$	−1561.02	−0.516640
$210$	0	0
$211$	−5593.44	−1.82497	−0.912483	−	0.409114i	$-0.865838\pi$
$211$	−5593.44	−1.82497	−0.912483	+	0.409114i	$0.865838\pi$
$212$	0	0
$213$	1294.12	0.416298
$214$	0	0
$215$	660.608	0.209549
$216$	0	0
$217$	703.781	0.220165
$218$	0	0
$219$	−1248.50	−0.385231
$220$	0	0
$221$	648.417	0.197363
$222$	0	0
$223$	3499.61	1.05090	0.525451	−	0.850824i	$-0.323897\pi$
$223$	3499.61	1.05090	0.525451	+	0.850824i	$0.323897\pi$
$224$	0	0
$225$	2036.77	0.603487
$226$	0	0
$227$	592.105	0.173125	0.0865625	−	0.996246i	$-0.472412\pi$
$227$	592.105	0.173125	0.0865625	+	0.996246i	$0.472412\pi$
$228$	0	0
$229$	5116.16	1.47635	0.738177	−	0.674607i	$-0.235687\pi$
$229$	5116.16	1.47635	0.738177	+	0.674607i	$0.235687\pi$
$230$	0	0
$231$	247.568	0.0705143
$232$	0	0
$233$	3092.28	0.869452	0.434726	−	0.900563i	$-0.356845\pi$
$233$	3092.28	0.869452	0.434726	+	0.900563i	$0.356845\pi$
$234$	0	0
$235$	420.581	0.116748
$236$	0	0
$237$	3328.87	0.912378
$238$	0	0
$239$	6257.16	1.69348	0.846740	−	0.532006i	$-0.178562\pi$
$239$	6257.16	1.69348	0.846740	+	0.532006i	$0.178562\pi$
$240$	0	0
$241$	−3733.06	−0.997791	−0.498896	−	0.866662i	$-0.666261\pi$
$241$	−3733.06	−0.997791	−0.498896	+	0.866662i	$0.666261\pi$
$242$	0	0
$243$	3785.63	0.999377
$244$	0	0
$245$	81.4699	0.0212446
$246$	0	0
$247$	2847.02	0.733406
$248$	0	0
$249$	3342.63	0.850725
$250$	0	0
$251$	−5644.56	−1.41945	−0.709723	−	0.704480i	$-0.751180\pi$
$251$	−5644.56	−1.41945	−0.709723	+	0.704480i	$0.751180\pi$
$252$	0	0
$253$	766.590	0.190495
$254$	0	0
$255$	172.777	0.0424302
$256$	0	0
$257$	2732.25	0.663163	0.331581	−	0.943427i	$-0.392418\pi$
$257$	2732.25	0.663163	0.331581	+	0.943427i	$0.392418\pi$
$258$	0	0
$259$	1525.49	0.365983
$260$	0	0
$261$	−4221.43	−1.00115
$262$	0	0
$263$	−5054.53	−1.18508	−0.592539	−	0.805542i	$-0.701875\pi$
$263$	−5054.53	−1.18508	−0.592539	+	0.805542i	$0.701875\pi$
$264$	0	0
$265$	371.333	0.0860786
$266$	0	0
$267$	−2056.28	−0.471320
$268$	0	0
$269$	6554.20	1.48556	0.742782	−	0.669533i	$-0.233506\pi$
$269$	6554.20	1.48556	0.742782	+	0.669533i	$0.233506\pi$
$270$	0	0
$271$	−47.8353	−0.0107225	−0.00536123	−	0.999986i	$-0.501707\pi$
$271$	−47.8353	−0.0107225	−0.00536123	+	0.999986i	$0.501707\pi$
$272$	0	0
$273$	−451.521	−0.100100
$274$	0	0
$275$	1344.59	0.294843
$276$	0	0
$277$	5009.62	1.08664	0.543319	−	0.839526i	$-0.317167\pi$
$277$	5009.62	1.08664	0.543319	+	0.839526i	$0.317167\pi$
$278$	0	0
$279$	1675.27	0.359482
$280$	0	0
$281$	−1335.46	−0.283512	−0.141756	−	0.989902i	$-0.545275\pi$
$281$	−1335.46	−0.283512	−0.141756	+	0.989902i	$0.545275\pi$
$282$	0	0
$283$	−6909.27	−1.45128	−0.725642	−	0.688073i	$-0.758457\pi$
$283$	−6909.27	−1.45128	−0.725642	+	0.688073i	$0.758457\pi$
$284$	0	0
$285$	758.614	0.157672
$286$	0	0
$287$	1043.97	0.214717
$288$	0	0
$289$	−3868.38	−0.787376
$290$	0	0
$291$	−724.042	−0.145856
$292$	0	0
$293$	4225.63	0.842538	0.421269	−	0.906936i	$-0.361585\pi$
$293$	4225.63	0.842538	0.421269	+	0.906936i	$0.361585\pi$
$294$	0	0
$295$	493.220	0.0973437
$296$	0	0
$297$	1544.21	0.301698
$298$	0	0
$299$	−1398.12	−0.270420
$300$	0	0
$301$	−2781.25	−0.532587
$302$	0	0
$303$	−4934.63	−0.935600
$304$	0	0
$305$	794.816	0.149216
$306$	0	0
$307$	−9806.66	−1.82311	−0.911557	−	0.411174i	$-0.865119\pi$
$307$	−9806.66	−1.82311	−0.911557	+	0.411174i	$0.865119\pi$
$308$	0	0
$309$	−1186.99	−0.218529
$310$	0	0
$311$	9073.33	1.65435	0.827173	−	0.561947i	$-0.189948\pi$
$311$	9073.33	1.65435	0.827173	+	0.561947i	$0.189948\pi$
$312$	0	0
$313$	−4069.65	−0.734921	−0.367460	−	0.930039i	$-0.619773\pi$
$313$	−4069.65	−0.734921	−0.367460	+	0.930039i	$0.619773\pi$
$314$	0	0
$315$	193.929	0.0346879
$316$	0	0
$317$	−9513.29	−1.68555	−0.842775	−	0.538265i	$-0.819080\pi$
$317$	−9513.29	−1.68555	−0.842775	+	0.538265i	$0.819080\pi$
$318$	0	0
$319$	−2786.81	−0.489127
$320$	0	0
$321$	3874.64	0.673710
$322$	0	0
$323$	4586.64	0.790116
$324$	0	0
$325$	−2452.29	−0.418550
$326$	0	0
$327$	3979.85	0.673047
$328$	0	0
$329$	−1770.71	−0.296724
$330$	0	0
$331$	5635.24	0.935772	0.467886	−	0.883789i	$-0.345016\pi$
$331$	5635.24	0.935772	0.467886	+	0.883789i	$0.345016\pi$
$332$	0	0
$333$	3631.25	0.597571
$334$	0	0
$335$	816.536	0.133170
$336$	0	0
$337$	3496.75	0.565222	0.282611	−	0.959235i	$-0.408799\pi$
$337$	3496.75	0.565222	0.282611	+	0.959235i	$0.408799\pi$
$338$	0	0
$339$	−3833.86	−0.614238
$340$	0	0
$341$	1105.94	0.175631
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	−372.543	−0.0581364
$346$	0	0
$347$	5767.79	0.892309	0.446154	−	0.894956i	$-0.352793\pi$
$347$	5767.79	0.892309	0.446154	+	0.894956i	$0.352793\pi$
$348$	0	0
$349$	−10993.9	−1.68622	−0.843112	−	0.537737i	$-0.819279\pi$
$349$	−10993.9	−1.68622	−0.843112	+	0.537737i	$0.819279\pi$
$350$	0	0
$351$	−2816.37	−0.428281
$352$	0	0
$353$	1064.90	0.160564	0.0802818	−	0.996772i	$-0.474418\pi$
$353$	1064.90	0.160564	0.0802818	+	0.996772i	$0.474418\pi$
$354$	0	0
$355$	669.221	0.100052
$356$	0	0
$357$	−727.415	−0.107840
$358$	0	0
$359$	4558.21	0.670120	0.335060	−	0.942197i	$-0.391243\pi$
$359$	4558.21	0.670120	0.335060	+	0.942197i	$0.391243\pi$
$360$	0	0
$361$	13279.6	1.93609
$362$	0	0
$363$	389.036	0.0562510
$364$	0	0
$365$	−645.631	−0.0925859
$366$	0	0
$367$	−7916.99	−1.12606	−0.563029	−	0.826437i	$-0.690364\pi$
$367$	−7916.99	−1.12606	−0.563029	+	0.826437i	$0.690364\pi$
$368$	0	0
$369$	2485.05	0.350587
$370$	0	0
$371$	−1563.37	−0.218776
$372$	0	0
$373$	8660.16	1.20216	0.601081	−	0.799188i	$-0.294737\pi$
$373$	8660.16	1.20216	0.601081	+	0.799188i	$0.294737\pi$
$374$	0	0
$375$	−1321.65	−0.181999
$376$	0	0
$377$	5082.65	0.694349
$378$	0	0
$379$	6904.78	0.935817	0.467908	−	0.883777i	$-0.345008\pi$
$379$	6904.78	0.935817	0.467908	+	0.883777i	$0.345008\pi$
$380$	0	0
$381$	6096.06	0.819713
$382$	0	0
$383$	−4010.64	−0.535076	−0.267538	−	0.963547i	$-0.586210\pi$
$383$	−4010.64	−0.535076	−0.267538	+	0.963547i	$0.586210\pi$
$384$	0	0
$385$	128.024	0.0169473
$386$	0	0
$387$	−6620.44	−0.869602
$388$	0	0
$389$	−2267.42	−0.295534	−0.147767	−	0.989022i	$-0.547209\pi$
$389$	−2267.42	−0.295534	−0.147767	+	0.989022i	$0.547209\pi$
$390$	0	0
$391$	−2252.43	−0.291330
$392$	0	0
$393$	3896.73	0.500163
$394$	0	0
$395$	1721.45	0.219280
$396$	0	0
$397$	85.1528	0.0107650	0.00538249	−	0.999986i	$-0.498287\pi$
$397$	85.1528	0.0107650	0.00538249	+	0.999986i	$0.498287\pi$
$398$	0	0
$399$	−3193.87	−0.400736
$400$	0	0
$401$	−4274.68	−0.532338	−0.266169	−	0.963926i	$-0.585758\pi$
$401$	−4274.68	−0.532338	−0.266169	+	0.963926i	$0.585758\pi$
$402$	0	0
$403$	−2017.04	−0.249320
$404$	0	0
$405$	−2.43491	−0.000298744 0
$406$	0	0
$407$	2397.20	0.291953
$408$	0	0
$409$	−2364.68	−0.285883	−0.142941	−	0.989731i	$-0.545656\pi$
$409$	−2364.68	−0.285883	−0.142941	+	0.989731i	$0.545656\pi$
$410$	0	0
$411$	5160.93	0.619391
$412$	0	0
$413$	−2076.53	−0.247407
$414$	0	0
$415$	1728.56	0.204462
$416$	0	0
$417$	38.1349	0.00447836
$418$	0	0
$419$	−6947.76	−0.810072	−0.405036	−	0.914301i	$-0.632741\pi$
$419$	−6947.76	−0.810072	−0.405036	+	0.914301i	$0.632741\pi$
$420$	0	0
$421$	−2597.93	−0.300749	−0.150375	−	0.988629i	$-0.548048\pi$
$421$	−2597.93	−0.300749	−0.150375	+	0.988629i	$0.548048\pi$
$422$	0	0
$423$	−4214.96	−0.484487
$424$	0	0
$425$	−3950.73	−0.450914
$426$	0	0
$427$	−3346.29	−0.379246
$428$	0	0
$429$	−709.532	−0.0798521
$430$	0	0
$431$	7801.64	0.871906	0.435953	−	0.899969i	$-0.356411\pi$
$431$	7801.64	0.871906	0.435953	+	0.899969i	$0.356411\pi$
$432$	0	0
$433$	6979.45	0.774622	0.387311	−	0.921949i	$-0.373404\pi$
$433$	6979.45	0.774622	0.387311	+	0.921949i	$0.373404\pi$
$434$	0	0
$435$	1354.32	0.149275
$436$	0	0
$437$	−9889.76	−1.08259
$438$	0	0
$439$	3608.84	0.392347	0.196174	−	0.980569i	$-0.437148\pi$
$439$	3608.84	0.392347	0.196174	+	0.980569i	$0.437148\pi$
$440$	0	0
$441$	−816.470	−0.0881622
$442$	0	0
$443$	−1122.75	−0.120414	−0.0602070	−	0.998186i	$-0.519176\pi$
$443$	−1122.75	−0.120414	−0.0602070	+	0.998186i	$0.519176\pi$
$444$	0	0
$445$	−1063.36	−0.113276
$446$	0	0
$447$	−835.361	−0.0883920
$448$	0	0
$449$	−12321.6	−1.29508	−0.647539	−	0.762032i	$-0.724202\pi$
$449$	−12321.6	−1.29508	−0.647539	+	0.762032i	$0.724202\pi$
$450$	0	0
$451$	1640.53	0.171285
$452$	0	0
$453$	−2967.50	−0.307782
$454$	0	0
$455$	−233.493	−0.0240579
$456$	0	0
$457$	−12104.8	−1.23903	−0.619515	−	0.784985i	$-0.712671\pi$
$457$	−12104.8	−1.23903	−0.619515	+	0.784985i	$0.712671\pi$
$458$	0	0
$459$	−4537.27	−0.461398
$460$	0	0
$461$	5659.54	0.571781	0.285891	−	0.958262i	$-0.407711\pi$
$461$	5659.54	0.571781	0.285891	+	0.958262i	$0.407711\pi$
$462$	0	0
$463$	−2843.60	−0.285429	−0.142714	−	0.989764i	$-0.545583\pi$
$463$	−2843.60	−0.285429	−0.142714	+	0.989764i	$0.545583\pi$
$464$	0	0
$465$	−537.459	−0.0536001
$466$	0	0
$467$	−8784.79	−0.870475	−0.435237	−	0.900316i	$-0.643336\pi$
$467$	−8784.79	−0.870475	−0.435237	+	0.900316i	$0.643336\pi$
$468$	0	0
$469$	−3437.73	−0.338464
$470$	0	0
$471$	10135.0	0.991500
$472$	0	0
$473$	−4370.54	−0.424858
$474$	0	0
$475$	−17346.5	−1.67561
$476$	0	0
$477$	−3721.41	−0.357215
$478$	0	0
$479$	16512.2	1.57508	0.787540	−	0.616263i	$-0.211354\pi$
$479$	16512.2	1.57508	0.787540	+	0.616263i	$0.211354\pi$
$480$	0	0
$481$	−4372.07	−0.414447
$482$	0	0
$483$	1568.46	0.147759
$484$	0	0
$485$	−374.421	−0.0350548
$486$	0	0
$487$	−957.337	−0.0890782	−0.0445391	−	0.999008i	$-0.514182\pi$
$487$	−957.337	−0.0890782	−0.0445391	+	0.999008i	$0.514182\pi$
$488$	0	0
$489$	11789.5	1.09026
$490$	0	0
$491$	−9880.85	−0.908180	−0.454090	−	0.890956i	$-0.650036\pi$
$491$	−9880.85	−0.908180	−0.454090	+	0.890956i	$0.650036\pi$
$492$	0	0
$493$	8188.32	0.748039
$494$	0	0
$495$	304.746	0.0276713
$496$	0	0
$497$	−2817.52	−0.254292
$498$	0	0
$499$	3832.03	0.343778	0.171889	−	0.985116i	$-0.445013\pi$
$499$	3832.03	0.343778	0.171889	+	0.985116i	$0.445013\pi$
$500$	0	0
$501$	−2554.51	−0.227798
$502$	0	0
$503$	−15930.7	−1.41215	−0.706077	−	0.708135i	$-0.749537\pi$
$503$	−15930.7	−1.41215	−0.706077	+	0.708135i	$0.749537\pi$
$504$	0	0
$505$	−2551.82	−0.224861
$506$	0	0
$507$	−5769.68	−0.505405
$508$	0	0
$509$	−7336.50	−0.638870	−0.319435	−	0.947608i	$-0.603493\pi$
$509$	−7336.50	−0.638870	−0.319435	+	0.947608i	$0.603493\pi$
$510$	0	0
$511$	2718.20	0.235315
$512$	0	0
$513$	−19921.9	−1.71456
$514$	0	0
$515$	−613.822	−0.0525208
$516$	0	0
$517$	−2782.54	−0.236704
$518$	0	0
$519$	2193.58	0.185525
$520$	0	0
$521$	−17236.3	−1.44939	−0.724697	−	0.689068i	$-0.758020\pi$
$521$	−17236.3	−1.44939	−0.724697	+	0.689068i	$0.758020\pi$
$522$	0	0
$523$	18167.9	1.51898	0.759490	−	0.650519i	$-0.225449\pi$
$523$	18167.9	1.51898	0.759490	+	0.650519i	$0.225449\pi$
$524$	0	0
$525$	2751.06	0.228697
$526$	0	0
$527$	−3249.52	−0.268598
$528$	0	0
$529$	−7310.30	−0.600830
$530$	0	0
$531$	−4942.92	−0.403963
$532$	0	0
$533$	−2992.03	−0.243151
$534$	0	0
$535$	2003.68	0.161919
$536$	0	0
$537$	2795.00	0.224606
$538$	0	0
$539$	−539.000	−0.0430730
$540$	0	0
$541$	19988.1	1.58845	0.794227	−	0.607621i	$-0.207876\pi$
$541$	19988.1	1.58845	0.794227	+	0.607621i	$0.207876\pi$
$542$	0	0
$543$	10772.8	0.851394
$544$	0	0
$545$	2058.09	0.161759
$546$	0	0
$547$	14834.8	1.15958	0.579788	−	0.814767i	$-0.303135\pi$
$547$	14834.8	1.15958	0.579788	+	0.814767i	$0.303135\pi$
$548$	0	0
$549$	−7965.43	−0.619228
$550$	0	0
$551$	35952.6	2.77973
$552$	0	0
$553$	−7247.54	−0.557318
$554$	0	0
$555$	−1164.98	−0.0891001
$556$	0	0
$557$	−19884.1	−1.51260	−0.756298	−	0.654227i	$-0.772994\pi$
$557$	−19884.1	−1.51260	−0.756298	+	0.654227i	$0.772994\pi$
$558$	0	0
$559$	7971.09	0.603115
$560$	0	0
$561$	−1143.08	−0.0860266
$562$	0	0
$563$	−23139.3	−1.73216	−0.866080	−	0.499905i	$-0.833368\pi$
$563$	−23139.3	−1.73216	−0.866080	+	0.499905i	$0.833368\pi$
$564$	0	0
$565$	−1982.59	−0.147625
$566$	0	0
$567$	10.2513	0.000759284 0
$568$	0	0
$569$	−10922.2	−0.804716	−0.402358	−	0.915482i	$-0.631809\pi$
$569$	−10922.2	−0.804716	−0.402358	+	0.915482i	$0.631809\pi$
$570$	0	0
$571$	−16602.7	−1.21682	−0.608409	−	0.793623i	$-0.708192\pi$
$571$	−16602.7	−1.21682	−0.608409	+	0.793623i	$0.708192\pi$
$572$	0	0
$573$	−4060.59	−0.296045
$574$	0	0
$575$	8518.60	0.617827
$576$	0	0
$577$	−2235.24	−0.161273	−0.0806363	−	0.996744i	$-0.525695\pi$
$577$	−2235.24	−0.161273	−0.0806363	+	0.996744i	$0.525695\pi$
$578$	0	0
$579$	−12590.0	−0.903664
$580$	0	0
$581$	−7277.49	−0.519658
$582$	0	0
$583$	−2456.72	−0.174523
$584$	0	0
$585$	−555.802	−0.0392813
$586$	0	0
$587$	−2889.23	−0.203154	−0.101577	−	0.994828i	$-0.532389\pi$
$587$	−2889.23	−0.203154	−0.101577	+	0.994828i	$0.532389\pi$
$588$	0	0
$589$	−14267.7	−0.998118
$590$	0	0
$591$	−3060.06	−0.212985
$592$	0	0
$593$	−10163.4	−0.703810	−0.351905	−	0.936036i	$-0.614466\pi$
$593$	−10163.4	−0.703810	−0.351905	+	0.936036i	$0.614466\pi$
$594$	0	0
$595$	−376.165	−0.0259181
$596$	0	0
$597$	−6804.93	−0.466511
$598$	0	0
$599$	−11109.2	−0.757779	−0.378889	−	0.925442i	$-0.623694\pi$
$599$	−11109.2	−0.757779	−0.378889	+	0.925442i	$0.623694\pi$
$600$	0	0
$601$	23157.4	1.57173	0.785864	−	0.618399i	$-0.212219\pi$
$601$	23157.4	1.57173	0.785864	+	0.618399i	$0.212219\pi$
$602$	0	0
$603$	−8183.10	−0.552640
$604$	0	0
$605$	201.181	0.0135193
$606$	0	0
$607$	11721.8	0.783810	0.391905	−	0.920006i	$-0.371816\pi$
$607$	11721.8	0.783810	0.391905	+	0.920006i	$0.371816\pi$
$608$	0	0
$609$	−5701.88	−0.379395
$610$	0	0
$611$	5074.86	0.336018
$612$	0	0
$613$	15061.8	0.992400	0.496200	−	0.868208i	$-0.334728\pi$
$613$	15061.8	0.992400	0.496200	+	0.868208i	$0.334728\pi$
$614$	0	0
$615$	−797.255	−0.0522739
$616$	0	0
$617$	−14396.5	−0.939351	−0.469675	−	0.882839i	$-0.655629\pi$
$617$	−14396.5	−0.939351	−0.469675	+	0.882839i	$0.655629\pi$
$618$	0	0
$619$	−6049.69	−0.392823	−0.196412	−	0.980522i	$-0.562929\pi$
$619$	−6049.69	−0.392823	−0.196412	+	0.980522i	$0.562929\pi$
$620$	0	0
$621$	9783.30	0.632190
$622$	0	0
$623$	4476.89	0.287902
$624$	0	0
$625$	14596.0	0.934143
$626$	0	0
$627$	−5018.94	−0.319677
$628$	0	0
$629$	−7043.55	−0.446494
$630$	0	0
$631$	−3751.60	−0.236686	−0.118343	−	0.992973i	$-0.537758\pi$
$631$	−3751.60	−0.236686	−0.118343	+	0.992973i	$0.537758\pi$
$632$	0	0
$633$	−17983.9	−1.12922
$634$	0	0
$635$	3152.43	0.197009
$636$	0	0
$637$	983.040	0.0611451
$638$	0	0
$639$	−6706.76	−0.415204
$640$	0	0
$641$	17626.2	1.08610	0.543052	−	0.839699i	$-0.317269\pi$
$641$	17626.2	1.08610	0.543052	+	0.839699i	$0.317269\pi$
$642$	0	0
$643$	13582.5	0.833036	0.416518	−	0.909127i	$-0.363250\pi$
$643$	13582.5	0.833036	0.416518	+	0.909127i	$0.363250\pi$
$644$	0	0
$645$	2123.97	0.129661
$646$	0	0
$647$	−16975.7	−1.03151	−0.515753	−	0.856737i	$-0.672488\pi$
$647$	−16975.7	−1.03151	−0.515753	+	0.856737i	$0.672488\pi$
$648$	0	0
$649$	−3263.12	−0.197363
$650$	0	0
$651$	2262.78	0.136229
$652$	0	0
$653$	−31204.5	−1.87002	−0.935012	−	0.354617i	$-0.884611\pi$
$653$	−31204.5	−1.87002	−0.935012	+	0.354617i	$0.884611\pi$
$654$	0	0
$655$	2015.10	0.120208
$656$	0	0
$657$	6470.34	0.384219
$658$	0	0
$659$	−3579.67	−0.211599	−0.105800	−	0.994387i	$-0.533740\pi$
$659$	−3579.67	−0.211599	−0.105800	+	0.994387i	$0.533740\pi$
$660$	0	0
$661$	−22084.1	−1.29950	−0.649752	−	0.760146i	$-0.725127\pi$
$661$	−22084.1	−1.29950	−0.649752	+	0.760146i	$0.725127\pi$
$662$	0	0
$663$	2084.78	0.122121
$664$	0	0
$665$	−1651.64	−0.0963123
$666$	0	0
$667$	−17655.7	−1.02494
$668$	0	0
$669$	11251.9	0.650257
$670$	0	0
$671$	−5258.45	−0.302534
$672$	0	0
$673$	−17629.5	−1.00976	−0.504879	−	0.863190i	$-0.668463\pi$
$673$	−17629.5	−1.00976	−0.504879	+	0.863190i	$0.668463\pi$
$674$	0	0
$675$	17159.8	0.978491
$676$	0	0
$677$	8098.30	0.459738	0.229869	−	0.973222i	$-0.426170\pi$
$677$	8098.30	0.459738	0.229869	+	0.973222i	$0.426170\pi$
$678$	0	0
$679$	1576.37	0.0890948
$680$	0	0
$681$	1903.72	0.107123
$682$	0	0
$683$	−23594.4	−1.32184	−0.660920	−	0.750457i	$-0.729834\pi$
$683$	−23594.4	−1.32184	−0.660920	+	0.750457i	$0.729834\pi$
$684$	0	0
$685$	2668.85	0.148864
$686$	0	0
$687$	16449.3	0.913510
$688$	0	0
$689$	4480.62	0.247747
$690$	0	0
$691$	−5069.15	−0.279073	−0.139537	−	0.990217i	$-0.544561\pi$
$691$	−5069.15	−0.279073	−0.139537	+	0.990217i	$0.544561\pi$
$692$	0	0
$693$	−1283.02	−0.0703291
$694$	0	0
$695$	19.7206	0.00107632
$696$	0	0
$697$	−4820.27	−0.261952
$698$	0	0
$699$	9942.23	0.537983
$700$	0	0
$701$	−9534.12	−0.513693	−0.256847	−	0.966452i	$-0.582683\pi$
$701$	−9534.12	−0.513693	−0.256847	+	0.966452i	$0.582683\pi$
$702$	0	0
$703$	−30926.2	−1.65918
$704$	0	0
$705$	1352.24	0.0722389
$706$	0	0
$707$	10743.5	0.571503
$708$	0	0
$709$	−13154.8	−0.696813	−0.348407	−	0.937343i	$-0.613277\pi$
$709$	−13154.8	−0.696813	−0.348407	+	0.937343i	$0.613277\pi$
$710$	0	0
$711$	−17251.9	−0.909981
$712$	0	0
$713$	7006.65	0.368024
$714$	0	0
$715$	−366.918	−0.0191915
$716$	0	0
$717$	20117.9	1.04786
$718$	0	0
$719$	37015.2	1.91993	0.959967	−	0.280113i	$-0.0903719\pi$
$719$	37015.2	1.91993	0.959967	+	0.280113i	$0.0903719\pi$
$720$	0	0
$721$	2584.28	0.133486
$722$	0	0
$723$	−12002.4	−0.617394
$724$	0	0
$725$	−30968.0	−1.58637
$726$	0	0
$727$	−298.213	−0.0152134	−0.00760668	−	0.999971i	$-0.502421\pi$
$727$	−298.213	−0.0152134	−0.00760668	+	0.999971i	$0.502421\pi$
$728$	0	0
$729$	12211.0	0.620384
$730$	0	0
$731$	12841.7	0.649750
$732$	0	0
$733$	3933.89	0.198229	0.0991143	−	0.995076i	$-0.468399\pi$
$733$	3933.89	0.198229	0.0991143	+	0.995076i	$0.468399\pi$
$734$	0	0
$735$	261.940	0.0131453
$736$	0	0
$737$	−5402.15	−0.270001
$738$	0	0
$739$	−14401.9	−0.716892	−0.358446	−	0.933551i	$-0.616693\pi$
$739$	−14401.9	−0.716892	−0.358446	+	0.933551i	$0.616693\pi$
$740$	0	0
$741$	9153.66	0.453803
$742$	0	0
$743$	7974.58	0.393754	0.196877	−	0.980428i	$-0.436920\pi$
$743$	7974.58	0.393754	0.196877	+	0.980428i	$0.436920\pi$
$744$	0	0
$745$	−431.987	−0.0212440
$746$	0	0
$747$	−17323.2	−0.848491
$748$	0	0
$749$	−8435.76	−0.411530
$750$	0	0
$751$	5185.64	0.251966	0.125983	−	0.992032i	$-0.459791\pi$
$751$	5185.64	0.251966	0.125983	+	0.992032i	$0.459791\pi$
$752$	0	0
$753$	−18148.2	−0.878298
$754$	0	0
$755$	−1534.57	−0.0739719
$756$	0	0
$757$	26128.7	1.25451	0.627255	−	0.778814i	$-0.284178\pi$
$757$	26128.7	1.25451	0.627255	+	0.778814i	$0.284178\pi$
$758$	0	0
$759$	2464.72	0.117871
$760$	0	0
$761$	22228.3	1.05884	0.529418	−	0.848361i	$-0.322410\pi$
$761$	22228.3	1.05884	0.529418	+	0.848361i	$0.322410\pi$
$762$	0	0
$763$	−8664.83	−0.411125
$764$	0	0
$765$	−895.416	−0.0423187
$766$	0	0
$767$	5951.34	0.280170
$768$	0	0
$769$	15293.2	0.717150	0.358575	−	0.933501i	$-0.383263\pi$
$769$	15293.2	0.717150	0.358575	+	0.933501i	$0.383263\pi$
$770$	0	0
$771$	8784.65	0.410339
$772$	0	0
$773$	2289.71	0.106540	0.0532699	−	0.998580i	$-0.483036\pi$
$773$	2289.71	0.106540	0.0532699	+	0.998580i	$0.483036\pi$
$774$	0	0
$775$	12289.6	0.569619
$776$	0	0
$777$	4904.73	0.226456
$778$	0	0
$779$	−21164.4	−0.973420
$780$	0	0
$781$	−4427.53	−0.202855
$782$	0	0
$783$	−35565.6	−1.62326
$784$	0	0
$785$	5241.08	0.238296
$786$	0	0
$787$	−30359.4	−1.37509	−0.687545	−	0.726141i	$-0.741312\pi$
$787$	−30359.4	−1.37509	−0.687545	+	0.726141i	$0.741312\pi$
$788$	0	0
$789$	−16251.2	−0.733280
$790$	0	0
$791$	8346.98	0.375202
$792$	0	0
$793$	9590.48	0.429468
$794$	0	0
$795$	1193.90	0.0532621
$796$	0	0
$797$	−5819.31	−0.258633	−0.129316	−	0.991603i	$-0.541278\pi$
$797$	−5819.31	−0.258633	−0.129316	+	0.991603i	$0.541278\pi$
$798$	0	0
$799$	8175.77	0.362000
$800$	0	0
$801$	10656.7	0.470082
$802$	0	0
$803$	4271.45	0.187717
$804$	0	0
$805$	811.092	0.0355121
$806$	0	0
$807$	21072.9	0.919209
$808$	0	0
$809$	19148.1	0.832151	0.416075	−	0.909330i	$-0.363405\pi$
$809$	19148.1	0.832151	0.416075	+	0.909330i	$0.363405\pi$
$810$	0	0
$811$	−36755.4	−1.59144	−0.795718	−	0.605667i	$-0.792906\pi$
$811$	−36755.4	−1.59144	−0.795718	+	0.605667i	$0.792906\pi$
$812$	0	0
$813$	−153.799	−0.00663463
$814$	0	0
$815$	6096.65	0.262032
$816$	0	0
$817$	56384.2	2.41449
$818$	0	0
$819$	2340.01	0.0998369
$820$	0	0
$821$	25490.3	1.08358	0.541789	−	0.840515i	$-0.317747\pi$
$821$	25490.3	1.08358	0.541789	+	0.840515i	$0.317747\pi$
$822$	0	0
$823$	11737.5	0.497138	0.248569	−	0.968614i	$-0.420040\pi$
$823$	11737.5	0.497138	0.248569	+	0.968614i	$0.420040\pi$
$824$	0	0
$825$	4323.10	0.182437
$826$	0	0
$827$	5174.57	0.217578	0.108789	−	0.994065i	$-0.465303\pi$
$827$	5174.57	0.217578	0.108789	+	0.994065i	$0.465303\pi$
$828$	0	0
$829$	−16641.5	−0.697205	−0.348602	−	0.937271i	$-0.613344\pi$
$829$	−16641.5	−0.697205	−0.348602	+	0.937271i	$0.613344\pi$
$830$	0	0
$831$	16106.8	0.672369
$832$	0	0
$833$	1583.71	0.0658731
$834$	0	0
$835$	−1321.00	−0.0547487
$836$	0	0
$837$	14114.1	0.582863
$838$	0	0
$839$	−30566.0	−1.25775	−0.628877	−	0.777505i	$-0.716485\pi$
$839$	−30566.0	−1.25775	−0.628877	+	0.777505i	$0.716485\pi$
$840$	0	0
$841$	39795.5	1.63170
$842$	0	0
$843$	−4293.73	−0.175426
$844$	0	0
$845$	−2983.65	−0.121468
$846$	0	0
$847$	−847.000	−0.0343604
$848$	0	0
$849$	−22214.5	−0.897997
$850$	0	0
$851$	15187.4	0.611770
$852$	0	0
$853$	35198.5	1.41286	0.706431	−	0.707781i	$-0.250304\pi$
$853$	35198.5	1.41286	0.706431	+	0.707781i	$0.250304\pi$
$854$	0	0
$855$	−3931.52	−0.157257
$856$	0	0
$857$	26714.4	1.06482	0.532409	−	0.846488i	$-0.321287\pi$
$857$	26714.4	1.06482	0.532409	+	0.846488i	$0.321287\pi$
$858$	0	0
$859$	−4232.34	−0.168109	−0.0840544	−	0.996461i	$-0.526787\pi$
$859$	−4232.34	−0.168109	−0.0840544	+	0.996461i	$0.526787\pi$
$860$	0	0
$861$	3356.56	0.132859
$862$	0	0
$863$	13399.9	0.528549	0.264274	−	0.964448i	$-0.414868\pi$
$863$	13399.9	0.528549	0.264274	+	0.964448i	$0.414868\pi$
$864$	0	0
$865$	1134.36	0.0445889
$866$	0	0
$867$	−12437.5	−0.487197
$868$	0	0
$869$	−11389.0	−0.444586
$870$	0	0
$871$	9852.56	0.383285
$872$	0	0
$873$	3752.35	0.145473
$874$	0	0
$875$	2877.47	0.111173
$876$	0	0
$877$	−3285.49	−0.126503	−0.0632515	−	0.997998i	$-0.520147\pi$
$877$	−3285.49	−0.126503	−0.0632515	+	0.997998i	$0.520147\pi$
$878$	0	0
$879$	13586.1	0.521329
$880$	0	0
$881$	−25909.5	−0.990821	−0.495411	−	0.868659i	$-0.664982\pi$
$881$	−25909.5	−0.990821	−0.495411	+	0.868659i	$0.664982\pi$
$882$	0	0
$883$	32019.0	1.22030	0.610149	−	0.792286i	$-0.291109\pi$
$883$	32019.0	1.22030	0.610149	+	0.792286i	$0.291109\pi$
$884$	0	0
$885$	1585.79	0.0602325
$886$	0	0
$887$	3013.04	0.114056	0.0570282	−	0.998373i	$-0.481838\pi$
$887$	3013.04	0.114056	0.0570282	+	0.998373i	$0.481838\pi$
$888$	0	0
$889$	−13272.2	−0.500714
$890$	0	0
$891$	16.1092	0.000605699 0
$892$	0	0
$893$	35897.5	1.34520
$894$	0	0
$895$	1445.37	0.0539814
$896$	0	0
$897$	−4495.21	−0.167325
$898$	0	0
$899$	−25471.5	−0.944964
$900$	0	0
$901$	7218.43	0.266904
$902$	0	0
$903$	−8942.22	−0.329544
$904$	0	0
$905$	5570.92	0.204623
$906$	0	0
$907$	−48589.1	−1.77880	−0.889401	−	0.457128i	$-0.848878\pi$
$907$	−48589.1	−1.77880	−0.889401	+	0.457128i	$0.848878\pi$
$908$	0	0
$909$	25573.7	0.933142
$910$	0	0
$911$	7878.62	0.286532	0.143266	−	0.989684i	$-0.454240\pi$
$911$	7878.62	0.286532	0.143266	+	0.989684i	$0.454240\pi$
$912$	0	0
$913$	−11436.1	−0.414544
$914$	0	0
$915$	2555.47	0.0923293
$916$	0	0
$917$	−8483.87	−0.305520
$918$	0	0
$919$	18781.1	0.674135	0.337068	−	0.941480i	$-0.390565\pi$
$919$	18781.1	0.674135	0.337068	+	0.941480i	$0.390565\pi$
$920$	0	0
$921$	−31530.1	−1.12807
$922$	0	0
$923$	8075.02	0.287966
$924$	0	0
$925$	26638.5	0.946885
$926$	0	0
$927$	6151.56	0.217955
$928$	0	0
$929$	−21425.0	−0.756655	−0.378327	−	0.925672i	$-0.623501\pi$
$929$	−21425.0	−0.756655	−0.378327	+	0.925672i	$0.623501\pi$
$930$	0	0
$931$	6953.62	0.244786
$932$	0	0
$933$	29172.4	1.02364
$934$	0	0
$935$	−591.117	−0.0206755
$936$	0	0
$937$	35959.5	1.25373	0.626866	−	0.779127i	$-0.284337\pi$
$937$	35959.5	1.25373	0.626866	+	0.779127i	$0.284337\pi$
$938$	0	0
$939$	−13084.6	−0.454740
$940$	0	0
$941$	−21072.2	−0.730004	−0.365002	−	0.931007i	$-0.618932\pi$
$941$	−21072.2	−0.730004	−0.365002	+	0.931007i	$0.618932\pi$
$942$	0	0
$943$	10393.5	0.358917
$944$	0	0
$945$	1633.86	0.0562427
$946$	0	0
$947$	38281.3	1.31360	0.656798	−	0.754067i	$-0.271910\pi$
$947$	38281.3	1.31360	0.656798	+	0.754067i	$0.271910\pi$
$948$	0	0
$949$	−7790.37	−0.266476
$950$	0	0
$951$	−30586.9	−1.04295
$952$	0	0
$953$	21800.5	0.741015	0.370507	−	0.928830i	$-0.379184\pi$
$953$	21800.5	0.741015	0.370507	+	0.928830i	$0.379184\pi$
$954$	0	0
$955$	−2099.84	−0.0711509
$956$	0	0
$957$	−8960.09	−0.302653
$958$	0	0
$959$	−11236.2	−0.378350
$960$	0	0
$961$	−19682.7	−0.660692
$962$	0	0
$963$	−20080.3	−0.671941
$964$	0	0
$965$	−6510.60	−0.217185
$966$	0	0
$967$	−820.914	−0.0272997	−0.0136499	−	0.999907i	$-0.504345\pi$
$967$	−820.914	−0.0272997	−0.0136499	+	0.999907i	$0.504345\pi$
$968$	0	0
$969$	14746.9	0.488893
$970$	0	0
$971$	−10758.7	−0.355575	−0.177787	−	0.984069i	$-0.556894\pi$
$971$	−10758.7	−0.355575	−0.177787	+	0.984069i	$0.556894\pi$
$972$	0	0
$973$	−83.0264	−0.00273556
$974$	0	0
$975$	−7884.55	−0.258982
$976$	0	0
$977$	−11492.8	−0.376344	−0.188172	−	0.982136i	$-0.560256\pi$
$977$	−11492.8	−0.376344	−0.188172	+	0.982136i	$0.560256\pi$
$978$	0	0
$979$	7035.11	0.229666
$980$	0	0
$981$	−20625.6	−0.671279
$982$	0	0
$983$	39183.0	1.27136	0.635678	−	0.771955i	$-0.280721\pi$
$983$	39183.0	1.27136	0.635678	+	0.771955i	$0.280721\pi$
$984$	0	0
$985$	−1582.44	−0.0511885
$986$	0	0
$987$	−5693.14	−0.183601
$988$	0	0
$989$	−27689.4	−0.890264
$990$	0	0
$991$	8273.94	0.265217	0.132609	−	0.991168i	$-0.457665\pi$
$991$	8273.94	0.265217	0.132609	+	0.991168i	$0.457665\pi$
$992$	0	0
$993$	18118.3	0.579019
$994$	0	0
$995$	−3519.01	−0.112121
$996$	0	0
$997$	53608.3	1.70290	0.851450	−	0.524436i	$-0.175724\pi$
$997$	53608.3	1.70290	0.851450	+	0.524436i	$0.175724\pi$
$998$	0	0
$999$	30593.3	0.968899

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1232.4.a.u.1.3		4
4.3	odd	2		616.4.a.f.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
616.4.a.f.1.2	✓	4	4.3	odd	2
1232.4.a.u.1.3		4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1232.a (trivial)

\(f(q)\)	\(=\)	\(q+3.21517 q^{3} +1.66265 q^{5} -7.00000 q^{7} -16.6627 q^{9} +O(q^{10})\)	\(q+3.21517 q^{3} +1.66265 q^{5} -7.00000 q^{7} -16.6627 q^{9} -11.0000 q^{11} +20.0620 q^{13} +5.34571 q^{15} +32.3206 q^{17} +141.911 q^{19} -22.5062 q^{21} -69.6900 q^{23} -122.236 q^{25} -140.383 q^{27} +253.347 q^{29} -100.540 q^{31} -35.3669 q^{33} -11.6386 q^{35} -217.928 q^{37} +64.5029 q^{39} -149.139 q^{41} +397.322 q^{43} -27.7042 q^{45} +252.958 q^{47} +49.0000 q^{49} +103.916 q^{51} +223.338 q^{53} -18.2892 q^{55} +456.268 q^{57} +296.647 q^{59} +478.041 q^{61} +116.639 q^{63} +33.3562 q^{65} +491.105 q^{67} -224.066 q^{69} +402.503 q^{71} -388.314 q^{73} -393.009 q^{75} +77.0000 q^{77} +1035.36 q^{79} -1.46447 q^{81} +1039.64 q^{83} +53.7379 q^{85} +814.554 q^{87} -639.555 q^{89} -140.434 q^{91} -323.254 q^{93} +235.948 q^{95} -225.195 q^{97} +183.289 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 3 q^{3} - 13 q^{5} - 28 q^{7} - 47 q^{9}+O(q^{10}) \) 4 * q + 3 * q^3 - 13 * q^5 - 28 * q^7 - 47 * q^9	\( 4 q + 3 q^{3} - 13 q^{5} - 28 q^{7} - 47 q^{9} - 44 q^{11} - 34 q^{13} - 63 q^{15} - 58 q^{17} - 60 q^{19} - 21 q^{21} + 93 q^{23} - 19 q^{25} - 63 q^{27} - 144 q^{29} + 129 q^{31} - 33 q^{33} + 91 q^{35} - 187 q^{37} + 244 q^{39} + 110 q^{41} + 360 q^{43} - 286 q^{45} + 438 q^{47} + 196 q^{49} + 902 q^{51} + 56 q^{53} + 143 q^{55} - 94 q^{57} + 1209 q^{59} + 104 q^{61} + 329 q^{63} - 1256 q^{65} + 1075 q^{67} + 451 q^{69} + 963 q^{71} - 646 q^{73} + 804 q^{75} + 308 q^{77} + 1838 q^{79} - 656 q^{81} + 1238 q^{83} - 278 q^{85} + 914 q^{87} - 1453 q^{89} + 238 q^{91} + 521 q^{93} + 2730 q^{95} + 573 q^{97} + 517 q^{99}+O(q^{100}) \) 4 * q + 3 * q^3 - 13 * q^5 - 28 * q^7 - 47 * q^9 - 44 * q^11 - 34 * q^13 - 63 * q^15 - 58 * q^17 - 60 * q^19 - 21 * q^21 + 93 * q^23 - 19 * q^25 - 63 * q^27 - 144 * q^29 + 129 * q^31 - 33 * q^33 + 91 * q^35 - 187 * q^37 + 244 * q^39 + 110 * q^41 + 360 * q^43 - 286 * q^45 + 438 * q^47 + 196 * q^49 + 902 * q^51 + 56 * q^53 + 143 * q^55 - 94 * q^57 + 1209 * q^59 + 104 * q^61 + 329 * q^63 - 1256 * q^65 + 1075 * q^67 + 451 * q^69 + 963 * q^71 - 646 * q^73 + 804 * q^75 + 308 * q^77 + 1838 * q^79 - 656 * q^81 + 1238 * q^83 - 278 * q^85 + 914 * q^87 - 1453 * q^89 + 238 * q^91 + 521 * q^93 + 2730 * q^95 + 573 * q^97 + 517 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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