LMFDB - Embedded newform 675.4.b.o.649.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [675,4,Mod(649,675)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("675.649"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(675, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	675.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,-52,0,0,0,0,0,0,0,0,0,0,0,148,0,0,-692] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$39.8262892539$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 2x^{6} - 38x^{5} + 650x^{4} - 2138x^{3} + 3698x^{2} - 3182x + 1369 $ x^8 - 2x^7 + 2x^6 - 38x^5 + 650x^4 - 2138x^3 + 3698x^2 - 3182*x + 1369
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.2
Root			$-3.80561 - 3.80561i$ of defining polynomial
Character	$\chi$	$=$	675.649
Dual form			675.4.b.o.649.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-4.90965i q^{2} -16.1047 q^{4} +2.10469i q^{7} +39.7912i q^{8} +40.3052 q^{11} -61.7328i q^{13} +10.3333 q^{14} +66.5234 q^{16} -99.2210i q^{17} -134.523 q^{19} -197.884i q^{22} +76.4985i q^{23} -303.087 q^{26} -33.8953i q^{28} -236.691 q^{29} +243.628 q^{31} -8.27738i q^{32} -487.141 q^{34} +57.4187i q^{37} +660.463i q^{38} -411.383 q^{41} -102.884i q^{43} -649.102 q^{44} +375.581 q^{46} +260.442i q^{47} +338.570 q^{49} +994.187i q^{52} +75.4706i q^{53} -83.7480 q^{56} +1162.07i q^{58} +39.2772 q^{59} -675.851 q^{61} -1196.13i q^{62} +491.548 q^{64} -601.176i q^{67} +1597.92i q^{68} +222.192 q^{71} +297.419i q^{73} +281.906 q^{74} +2166.46 q^{76} +84.8297i q^{77} +95.6999 q^{79} +2019.75i q^{82} -3.08384i q^{83} -505.126 q^{86} +1603.79i q^{88} -1342.73 q^{89} +129.928 q^{91} -1231.99i q^{92} +1278.68 q^{94} +1482.57i q^{97} -1662.26i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 52 q^{4} + 148 q^{16} - 692 q^{19} + 1488 q^{31} - 1592 q^{34} + 3312 q^{46} + 1556 q^{49} + 356 q^{61} - 140 q^{64} + 8188 q^{76} - 4152 q^{79} + 5496 q^{91} + 2392 q^{94}+O(q^{100}) $ 8 * q - 52 * q^4 + 148 * q^16 - 692 * q^19 + 1488 * q^31 - 1592 * q^34 + 3312 * q^46 + 1556 * q^49 + 356 * q^61 - 140 * q^64 + 8188 * q^76 - 4152 * q^79 + 5496 * q^91 + 2392 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/675\mathbb{Z}\right)^\times$.

$n$	$326$	$352$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 4.90965i	− 1.73582i	−0.496718	−	0.867912i	$-0.665462\pi$
$2$	− 4.90965i	− 1.73582i	0.496718	−	0.867912i	$-0.334538\pi$
$3$	0	0
$3$	0	0
$4$	−16.1047	−2.01309
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.10469i	0.113642i	0.998384	+	0.0568212i	$0.0180965\pi$
$7$	2.10469i	0.113642i	−0.998384	+	0.0568212i	$0.981904\pi$
$8$	39.7912i	1.75854i
$9$	0	0
$10$	0	0
$11$	40.3052	1.10477	0.552385	−	0.833589i	$-0.313718\pi$
$11$	40.3052	1.10477	0.552385	+	0.833589i	$0.313718\pi$
$12$	0	0
$13$	− 61.7328i	− 1.31705i	−0.752561	−	0.658523i	$-0.771182\pi$
$13$	− 61.7328i	− 1.31705i	0.752561	−	0.658523i	$-0.228818\pi$
$14$	10.3333	0.197263
$15$	0	0
$16$	66.5234	1.03943
$17$	− 99.2210i	− 1.41557i	−0.706430	−	0.707783i	$-0.749695\pi$
$17$	− 99.2210i	− 1.41557i	0.706430	−	0.707783i	$-0.250305\pi$
$18$	0	0
$19$	−134.523	−1.62430	−0.812152	−	0.583445i	$-0.801704\pi$
$19$	−134.523	−1.62430	−0.812152	+	0.583445i	$0.801704\pi$
$20$	0	0
$21$	0	0
$22$	− 197.884i	− 1.91769i
$23$	76.4985i	0.693524i	0.937953	+	0.346762i	$0.112719\pi$
$23$	76.4985i	0.693524i	−0.937953	+	0.346762i	$0.887281\pi$
$24$	0	0
$25$	0	0
$26$	−303.087	−2.28616
$27$	0	0
$28$	− 33.8953i	− 0.228772i
$29$	−236.691	−1.51560	−0.757801	−	0.652486i	$-0.773726\pi$
$29$	−236.691	−1.51560	−0.757801	+	0.652486i	$0.773726\pi$
$30$	0	0
$31$	243.628	1.41151	0.705756	−	0.708455i	$-0.250607\pi$
$31$	243.628	1.41151	0.705756	+	0.708455i	$0.250607\pi$
$32$	− 8.27738i	− 0.0457265i
$33$	0	0
$34$	−487.141	−2.45717
$35$	0	0
$36$	0	0
$37$	57.4187i	0.255124i	0.991831	+	0.127562i	$0.0407152\pi$
$37$	57.4187i	0.255124i	−0.991831	+	0.127562i	$0.959285\pi$
$38$	660.463i	2.81951i
$39$	0	0
$40$	0	0
$41$	−411.383	−1.56701	−0.783503	−	0.621389i	$-0.786569\pi$
$41$	−411.383	−1.56701	−0.783503	+	0.621389i	$0.786569\pi$
$42$	0	0
$43$	− 102.884i	− 0.364877i	−0.983217	−	0.182439i	$-0.941601\pi$
$43$	− 102.884i	− 0.364877i	0.983217	−	0.182439i	$-0.0583991\pi$
$44$	−649.102	−2.22400
$45$	0	0
$46$	375.581	1.20384
$47$	260.442i	0.808283i	0.914696	+	0.404142i	$0.132430\pi$
$47$	260.442i	0.808283i	−0.914696	+	0.404142i	$0.867570\pi$
$48$	0	0
$49$	338.570	0.987085
$50$	0	0
$51$	0	0
$52$	994.187i	2.65133i
$53$	75.4706i	0.195598i	0.995206	+	0.0977989i	$0.0311802\pi$
$53$	75.4706i	0.195598i	−0.995206	+	0.0977989i	$0.968820\pi$
$54$	0	0
$55$	0	0
$56$	−83.7480	−0.199845
$57$	0	0
$58$	1162.07i	2.63082i
$59$	39.2772	0.0866688	0.0433344	−	0.999061i	$-0.486202\pi$
$59$	39.2772	0.0866688	0.0433344	+	0.999061i	$0.486202\pi$
$60$	0	0
$61$	−675.851	−1.41859	−0.709294	−	0.704912i	$-0.750986\pi$
$61$	−675.851	−1.41859	−0.709294	+	0.704912i	$0.750986\pi$
$62$	− 1196.13i	− 2.45014i
$63$	0	0
$64$	491.548	0.960055
$65$	0	0
$66$	0	0
$67$	− 601.176i	− 1.09620i	−0.836413	−	0.548100i	$-0.815351\pi$
$67$	− 601.176i	− 1.09620i	0.836413	−	0.548100i	$-0.184649\pi$
$68$	1597.92i	2.84966i
$69$	0	0
$70$	0	0
$71$	222.192	0.371400	0.185700	−	0.982607i	$-0.440545\pi$
$71$	222.192	0.371400	0.185700	+	0.982607i	$0.440545\pi$
$72$	0	0
$73$	297.419i	0.476852i	0.971161	+	0.238426i	$0.0766315\pi$
$73$	297.419i	0.476852i	−0.971161	+	0.238426i	$0.923368\pi$
$74$	281.906	0.442850
$75$	0	0
$76$	2166.46	3.26986
$77$	84.8297i	0.125549i
$78$	0	0
$79$	95.6999	0.136292	0.0681461	−	0.997675i	$-0.478292\pi$
$79$	95.6999	0.136292	0.0681461	+	0.997675i	$0.478292\pi$
$80$	0	0
$81$	0	0
$82$	2019.75i	2.72005i
$83$	− 3.08384i	− 0.00407826i	−0.999998	−	0.00203913i	$-0.999351\pi$
$83$	− 3.08384i	− 0.00407826i	0.999998	−	0.00203913i	$-0.000649076\pi$
$84$	0	0
$85$	0	0
$86$	−505.126	−0.633363
$87$	0	0
$88$	1603.79i	1.94278i
$89$	−1342.73	−1.59920	−0.799601	−	0.600532i	$-0.794956\pi$
$89$	−1342.73	−1.59920	−0.799601	+	0.600532i	$0.794956\pi$
$90$	0	0
$91$	129.928	0.149672
$92$	− 1231.99i	− 1.39612i
$93$	0	0
$94$	1278.68	1.40304
$95$	0	0
$96$	0	0
$97$	1482.57i	1.55188i	0.630806	+	0.775941i	$0.282724\pi$
$97$	1482.57i	1.55188i	−0.630806	+	0.775941i	$0.717276\pi$
$98$	− 1662.26i	− 1.71341i
$99$	0	0
$100$	0	0
$101$	−1535.82	−1.51306	−0.756532	−	0.653957i	$-0.773108\pi$
$101$	−1535.82	−1.51306	−0.756532	+	0.653957i	$0.773108\pi$
$102$	0	0
$103$	356.581i	0.341116i	0.985348	+	0.170558i	$0.0545571\pi$
$103$	356.581i	0.341116i	−0.985348	+	0.170558i	$0.945443\pi$
$104$	2456.42	2.31608
$105$	0	0
$106$	370.534	0.339523
$107$	1580.23i	1.42773i	0.700285	+	0.713864i	$0.253056\pi$
$107$	1580.23i	1.42773i	−0.700285	+	0.713864i	$0.746944\pi$
$108$	0	0
$109$	−1189.98	−1.04568	−0.522842	−	0.852430i	$-0.675128\pi$
$109$	−1189.98	−1.04568	−0.522842	+	0.852430i	$0.675128\pi$
$110$	0	0
$111$	0	0
$112$	140.011i	0.118123i
$113$	− 369.022i	− 0.307209i	−0.988132	−	0.153605i	$-0.950912\pi$
$113$	− 369.022i	− 0.307209i	0.988132	−	0.153605i	$-0.0490882\pi$
$114$	0	0
$115$	0	0
$116$	3811.84	3.05104
$117$	0	0
$118$	− 192.837i	− 0.150442i
$119$	208.829	0.160868
$120$	0	0
$121$	293.506	0.220516
$122$	3318.20i	2.46242i
$123$	0	0
$124$	−3923.55	−2.84150
$125$	0	0
$126$	0	0
$127$	− 1414.84i	− 0.988560i	−0.869303	−	0.494280i	$-0.835432\pi$
$127$	− 1414.84i	− 0.988560i	0.869303	−	0.494280i	$-0.164568\pi$
$128$	− 2479.55i	− 1.71221i
$129$	0	0
$130$	0	0
$131$	−155.161	−0.103484	−0.0517422	−	0.998660i	$-0.516477\pi$
$131$	−155.161	−0.103484	−0.0517422	+	0.998660i	$0.516477\pi$
$132$	0	0
$133$	− 283.130i	− 0.184590i
$134$	−2951.57	−1.90281
$135$	0	0
$136$	3948.12	2.48933
$137$	2447.63i	1.52639i	0.646169	+	0.763194i	$0.276370\pi$
$137$	2447.63i	1.52639i	−0.646169	+	0.763194i	$0.723630\pi$
$138$	0	0
$139$	−1463.98	−0.893329	−0.446665	−	0.894702i	$-0.647388\pi$
$139$	−1463.98	−0.893329	−0.446665	+	0.894702i	$0.647388\pi$
$140$	0	0
$141$	0	0
$142$	− 1090.89i	− 0.644685i
$143$	− 2488.15i	− 1.45503i
$144$	0	0
$145$	0	0
$146$	1460.22	0.827732
$147$	0	0
$148$	− 924.711i	− 0.513586i
$149$	3363.34	1.84923	0.924616	−	0.380899i	$-0.124386\pi$
$149$	3363.34	1.84923	0.924616	+	0.380899i	$0.124386\pi$
$150$	0	0
$151$	2004.07	1.08006	0.540029	−	0.841647i	$-0.318413\pi$
$151$	2004.07	1.08006	0.540029	+	0.841647i	$0.318413\pi$
$152$	− 5352.85i	− 2.85640i
$153$	0	0
$154$	416.485	0.217930
$155$	0	0
$156$	0	0
$157$	− 951.144i	− 0.483500i	−0.970339	−	0.241750i	$-0.922279\pi$
$157$	− 951.144i	− 0.483500i	0.970339	−	0.241750i	$-0.0777214\pi$
$158$	− 469.853i	− 0.236579i
$159$	0	0
$160$	0	0
$161$	−161.005	−0.0788137
$162$	0	0
$163$	− 3152.95i	− 1.51508i	−0.652790	−	0.757539i	$-0.726402\pi$
$163$	− 3152.95i	− 1.51508i	0.652790	−	0.757539i	$-0.273598\pi$
$164$	6625.19	3.15452
$165$	0	0
$166$	−15.1406	−0.00707914
$167$	1886.55i	0.874165i	0.899421	+	0.437083i	$0.143988\pi$
$167$	1886.55i	0.874165i	−0.899421	+	0.437083i	$0.856012\pi$
$168$	0	0
$169$	−1613.94	−0.734610
$170$	0	0
$171$	0	0
$172$	1656.92i	0.734529i
$173$	− 1036.73i	− 0.455615i	−0.973706	−	0.227808i	$-0.926844\pi$
$173$	− 1036.73i	− 0.455615i	0.973706	−	0.227808i	$-0.0731558\pi$
$174$	0	0
$175$	0	0
$176$	2681.24	1.14833
$177$	0	0
$178$	6592.33i	2.77593i
$179$	−3790.25	−1.58266	−0.791332	−	0.611387i	$-0.790612\pi$
$179$	−3790.25	−1.58266	−0.791332	+	0.611387i	$0.790612\pi$
$180$	0	0
$181$	946.498	0.388689	0.194344	−	0.980933i	$-0.437742\pi$
$181$	946.498	0.388689	0.194344	+	0.980933i	$0.437742\pi$
$182$	− 637.902i	− 0.259805i
$183$	0	0
$184$	−3043.97	−1.21959
$185$	0	0
$186$	0	0
$187$	− 3999.12i	− 1.56387i
$188$	− 4194.33i	− 1.62714i
$189$	0	0
$190$	0	0
$191$	2618.92	0.992137	0.496068	−	0.868284i	$-0.334777\pi$
$191$	2618.92	0.992137	0.496068	+	0.868284i	$0.334777\pi$
$192$	0	0
$193$	− 2587.20i	− 0.964927i	−0.875916	−	0.482463i	$-0.839742\pi$
$193$	− 2587.20i	− 0.964927i	0.875916	−	0.482463i	$-0.160258\pi$
$194$	7278.92	2.69379
$195$	0	0
$196$	−5452.57	−1.98709
$197$	− 628.328i	− 0.227241i	−0.993524	−	0.113621i	$-0.963755\pi$
$197$	− 628.328i	− 0.227241i	0.993524	−	0.113621i	$-0.0362448\pi$
$198$	0	0
$199$	2892.44	1.03035	0.515175	−	0.857085i	$-0.327727\pi$
$199$	2892.44	1.03035	0.515175	+	0.857085i	$0.327727\pi$
$200$	0	0
$201$	0	0
$202$	7540.32i	2.62641i
$203$	− 498.161i	− 0.172237i
$204$	0	0
$205$	0	0
$206$	1750.69	0.592118
$207$	0	0
$208$	− 4106.68i	− 1.36898i
$209$	−5421.99	−1.79448
$210$	0	0
$211$	−2864.38	−0.934560	−0.467280	−	0.884109i	$-0.654766\pi$
$211$	−2864.38	−0.934560	−0.467280	+	0.884109i	$0.654766\pi$
$212$	− 1215.43i	− 0.393755i
$213$	0	0
$214$	7758.39	2.47828
$215$	0	0
$216$	0	0
$217$	512.761i	0.160408i
$218$	5842.39i	1.81512i
$219$	0	0
$220$	0	0
$221$	−6125.19	−1.86437
$222$	0	0
$223$	2244.32i	0.673950i	0.941514	+	0.336975i	$0.109404\pi$
$223$	2244.32i	0.673950i	−0.941514	+	0.336975i	$0.890596\pi$
$224$	17.4213	0.00519647
$225$	0	0
$226$	−1811.77	−0.533261
$227$	5959.64i	1.74253i	0.490809	+	0.871267i	$0.336701\pi$
$227$	5959.64i	1.74253i	−0.490809	+	0.871267i	$0.663299\pi$
$228$	0	0
$229$	−2181.56	−0.629527	−0.314764	−	0.949170i	$-0.601925\pi$
$229$	−2181.56	−0.629527	−0.314764	+	0.949170i	$0.601925\pi$
$230$	0	0
$231$	0	0
$232$	− 9418.23i	− 2.66525i
$233$	− 4013.04i	− 1.12834i	−0.825659	−	0.564170i	$-0.809196\pi$
$233$	− 4013.04i	− 1.12834i	0.825659	−	0.564170i	$-0.190804\pi$
$234$	0	0
$235$	0	0
$236$	−632.547	−0.174472
$237$	0	0
$238$	− 1025.28i	− 0.279239i
$239$	−6489.56	−1.75638	−0.878190	−	0.478312i	$-0.841249\pi$
$239$	−6489.56	−1.75638	−0.878190	+	0.478312i	$0.841249\pi$
$240$	0	0
$241$	1886.47	0.504224	0.252112	−	0.967698i	$-0.418875\pi$
$241$	1886.47	0.504224	0.252112	+	0.967698i	$0.418875\pi$
$242$	− 1441.01i	− 0.382777i
$243$	0	0
$244$	10884.4	2.85574
$245$	0	0
$246$	0	0
$247$	8304.51i	2.13928i
$248$	9694.25i	2.48220i
$249$	0	0
$250$	0	0
$251$	−6307.08	−1.58605	−0.793026	−	0.609187i	$-0.791496\pi$
$251$	−6307.08	−1.58605	−0.793026	+	0.609187i	$0.791496\pi$
$252$	0	0
$253$	3083.29i	0.766184i
$254$	−6946.39	−1.71597
$255$	0	0
$256$	−8241.35	−2.01205
$257$	4384.12i	1.06410i	0.846713	+	0.532050i	$0.178578\pi$
$257$	4384.12i	1.06410i	−0.846713	+	0.532050i	$0.821422\pi$
$258$	0	0
$259$	−120.848	−0.0289929
$260$	0	0
$261$	0	0
$262$	761.785i	0.179631i
$263$	2927.40i	0.686353i	0.939271	+	0.343177i	$0.111503\pi$
$263$	2927.40i	0.686353i	−0.939271	+	0.343177i	$0.888497\pi$
$264$	0	0
$265$	0	0
$266$	−1390.07	−0.320416
$267$	0	0
$268$	9681.76i	2.20674i
$269$	1869.35	0.423703	0.211852	−	0.977302i	$-0.432051\pi$
$269$	1869.35	0.423703	0.211852	+	0.977302i	$0.432051\pi$
$270$	0	0
$271$	−111.800	−0.0250603	−0.0125302	−	0.999921i	$-0.503989\pi$
$271$	−111.800	−0.0250603	−0.0125302	+	0.999921i	$0.503989\pi$
$272$	− 6600.52i	− 1.47138i
$273$	0	0
$274$	12017.0	2.64954
$275$	0	0
$276$	0	0
$277$	− 4454.02i	− 0.966124i	−0.875586	−	0.483062i	$-0.839525\pi$
$277$	− 4454.02i	− 0.966124i	0.875586	−	0.483062i	$-0.160475\pi$
$278$	7187.61i	1.55066i
$279$	0	0
$280$	0	0
$281$	2236.64	0.474828	0.237414	−	0.971409i	$-0.423700\pi$
$281$	2236.64	0.474828	0.237414	+	0.971409i	$0.423700\pi$
$282$	0	0
$283$	− 1629.67i	− 0.342309i	−0.985244	−	0.171155i	$-0.945250\pi$
$283$	− 1629.67i	− 0.342309i	0.985244	−	0.171155i	$-0.0547498\pi$
$284$	−3578.34	−0.747660
$285$	0	0
$286$	−12216.0	−2.52568
$287$	− 865.832i	− 0.178078i
$288$	0	0
$289$	−4931.81	−1.00383
$290$	0	0
$291$	0	0
$292$	− 4789.84i	− 0.959945i
$293$	− 2540.47i	− 0.506538i	−0.967396	−	0.253269i	$-0.918494\pi$
$293$	− 2540.47i	− 0.506538i	0.967396	−	0.253269i	$-0.0815058\pi$
$294$	0	0
$295$	0	0
$296$	−2284.76	−0.448645
$297$	0	0
$298$	− 16512.8i	− 3.20994i
$299$	4722.47	0.913403
$300$	0	0
$301$	216.539	0.0414655
$302$	− 9839.27i	− 1.87479i
$303$	0	0
$304$	−8948.96	−1.68835
$305$	0	0
$306$	0	0
$307$	− 4752.32i	− 0.883482i	−0.897143	−	0.441741i	$-0.854361\pi$
$307$	− 4752.32i	− 0.883482i	0.897143	−	0.441741i	$-0.145639\pi$
$308$	− 1366.16i	− 0.252740i
$309$	0	0
$310$	0	0
$311$	−7665.82	−1.39771	−0.698857	−	0.715262i	$-0.746308\pi$
$311$	−7665.82	−1.39771	−0.698857	+	0.715262i	$0.746308\pi$
$312$	0	0
$313$	− 7864.29i	− 1.42018i	−0.704112	−	0.710089i	$-0.748655\pi$
$313$	− 7864.29i	− 1.42018i	0.704112	−	0.710089i	$-0.251345\pi$
$314$	−4669.78	−0.839271
$315$	0	0
$316$	−1541.22	−0.274368
$317$	− 3845.65i	− 0.681367i	−0.940178	−	0.340684i	$-0.889342\pi$
$317$	− 3845.65i	− 0.681367i	0.940178	−	0.340684i	$-0.110658\pi$
$318$	0	0
$319$	−9539.88	−1.67439
$320$	0	0
$321$	0	0
$322$	790.481i	0.136807i
$323$	13347.5i	2.29931i
$324$	0	0
$325$	0	0
$326$	−15479.9	−2.62991
$327$	0	0
$328$	− 16369.4i	− 2.75564i
$329$	−548.148	−0.0918553
$330$	0	0
$331$	−700.894	−0.116389	−0.0581943	−	0.998305i	$-0.518534\pi$
$331$	−700.894	−0.116389	−0.0581943	+	0.998305i	$0.518534\pi$
$332$	49.6643i	0.00820989i
$333$	0	0
$334$	9262.30	1.51740
$335$	0	0
$336$	0	0
$337$	− 12070.9i	− 1.95118i	−0.219610	−	0.975588i	$-0.570479\pi$
$337$	− 12070.9i	− 1.95118i	0.219610	−	0.975588i	$-0.429521\pi$
$338$	7923.88i	1.27515i
$339$	0	0
$340$	0	0
$341$	9819.47	1.55940
$342$	0	0
$343$	1434.49i	0.225817i
$344$	4093.89	0.641651
$345$	0	0
$346$	−5090.01	−0.790868
$347$	− 8064.60i	− 1.24764i	−0.781569	−	0.623819i	$-0.785580\pi$
$347$	− 8064.60i	− 1.24764i	0.781569	−	0.623819i	$-0.214420\pi$
$348$	0	0
$349$	−8726.16	−1.33840	−0.669199	−	0.743083i	$-0.733363\pi$
$349$	−8726.16	−1.33840	−0.669199	+	0.743083i	$0.733363\pi$
$350$	0	0
$351$	0	0
$352$	− 333.621i	− 0.0505173i
$353$	6125.68i	0.923618i	0.886979	+	0.461809i	$0.152799\pi$
$353$	6125.68i	0.923618i	−0.886979	+	0.461809i	$0.847201\pi$
$354$	0	0
$355$	0	0
$356$	21624.2	3.21933
$357$	0	0
$358$	18608.8i	2.74723i
$359$	567.297	0.0834005	0.0417002	−	0.999130i	$-0.486723\pi$
$359$	567.297	0.0834005	0.0417002	+	0.999130i	$0.486723\pi$
$360$	0	0
$361$	11237.6	1.63837
$362$	− 4646.98i	− 0.674695i
$363$	0	0
$364$	−2092.45	−0.301303
$365$	0	0
$366$	0	0
$367$	380.770i	0.0541581i	0.999633	+	0.0270791i	$0.00862058\pi$
$367$	380.770i	0.0541581i	−0.999633	+	0.0270791i	$0.991379\pi$
$368$	5088.95i	0.720869i
$369$	0	0
$370$	0	0
$371$	−158.842	−0.0222282
$372$	0	0
$373$	1049.75i	0.145721i	0.997342	+	0.0728604i	$0.0232128\pi$
$373$	1049.75i	0.145721i	−0.997342	+	0.0728604i	$0.976787\pi$
$374$	−19634.3	−2.71461
$375$	0	0
$376$	−10363.3	−1.42140
$377$	14611.6i	1.99612i
$378$	0	0
$379$	−3574.95	−0.484520	−0.242260	−	0.970211i	$-0.577889\pi$
$379$	−3574.95	−0.484520	−0.242260	+	0.970211i	$0.577889\pi$
$380$	0	0
$381$	0	0
$382$	− 12858.0i	− 1.72217i
$383$	− 14221.2i	− 1.89731i	−0.316310	−	0.948656i	$-0.602444\pi$
$383$	− 14221.2i	− 1.89731i	0.316310	−	0.948656i	$-0.397556\pi$
$384$	0	0
$385$	0	0
$386$	−12702.3	−1.67494
$387$	0	0
$388$	− 23876.4i	− 3.12407i
$389$	−7982.31	−1.04041	−0.520205	−	0.854042i	$-0.674144\pi$
$389$	−7982.31	−1.04041	−0.520205	+	0.854042i	$0.674144\pi$
$390$	0	0
$391$	7590.26	0.981729
$392$	13472.1i	1.73583i
$393$	0	0
$394$	−3084.87	−0.394451
$395$	0	0
$396$	0	0
$397$	− 2355.10i	− 0.297731i	−0.988857	−	0.148865i	$-0.952438\pi$
$397$	− 2355.10i	− 0.297731i	0.988857	−	0.148865i	$-0.0475621\pi$
$398$	− 14200.9i	− 1.78851i
$399$	0	0
$400$	0	0
$401$	−6419.98	−0.799498	−0.399749	−	0.916625i	$-0.630903\pi$
$401$	−6419.98	−0.799498	−0.399749	+	0.916625i	$0.630903\pi$
$402$	0	0
$403$	− 15039.8i	− 1.85903i
$404$	24733.8	3.04593
$405$	0	0
$406$	−2445.80	−0.298973
$407$	2314.27i	0.281853i
$408$	0	0
$409$	8435.21	1.01979	0.509895	−	0.860236i	$-0.329684\pi$
$409$	8435.21	1.01979	0.509895	+	0.860236i	$0.329684\pi$
$410$	0	0
$411$	0	0
$412$	− 5742.63i	− 0.686697i
$413$	82.6662i	0.00984925i
$414$	0	0
$415$	0	0
$416$	−510.986	−0.0602239
$417$	0	0
$418$	26620.1i	3.11491i
$419$	9663.61	1.12673	0.563363	−	0.826210i	$-0.309507\pi$
$419$	9663.61	1.12673	0.563363	+	0.826210i	$0.309507\pi$
$420$	0	0
$421$	−3723.76	−0.431081	−0.215540	−	0.976495i	$-0.569151\pi$
$421$	−3723.76	−0.431081	−0.215540	+	0.976495i	$0.569151\pi$
$422$	14063.1i	1.62223i
$423$	0	0
$424$	−3003.07	−0.343966
$425$	0	0
$426$	0	0
$427$	− 1422.46i	− 0.161212i
$428$	− 25449.2i	− 2.87414i
$429$	0	0
$430$	0	0
$431$	2671.60	0.298577	0.149288	−	0.988794i	$-0.452302\pi$
$431$	2671.60	0.298577	0.149288	+	0.988794i	$0.452302\pi$
$432$	0	0
$433$	− 2723.52i	− 0.302273i	−0.988513	−	0.151137i	$-0.951707\pi$
$433$	− 2723.52i	− 0.302273i	0.988513	−	0.151137i	$-0.0482933\pi$
$434$	2517.48	0.278440
$435$	0	0
$436$	19164.3	2.10505
$437$	− 10290.8i	− 1.12649i
$438$	0	0
$439$	−8281.91	−0.900396	−0.450198	−	0.892929i	$-0.648647\pi$
$439$	−8281.91	−0.900396	−0.450198	+	0.892929i	$0.648647\pi$
$440$	0	0
$441$	0	0
$442$	30072.6i	3.23621i
$443$	− 6286.69i	− 0.674243i	−0.941461	−	0.337121i	$-0.890547\pi$
$443$	− 6286.69i	− 0.674243i	0.941461	−	0.337121i	$-0.109453\pi$
$444$	0	0
$445$	0	0
$446$	11018.8	1.16986
$447$	0	0
$448$	1034.56i	0.109103i
$449$	4737.77	0.497971	0.248986	−	0.968507i	$-0.419903\pi$
$449$	4737.77	0.497971	0.248986	+	0.968507i	$0.419903\pi$
$450$	0	0
$451$	−16580.9	−1.73118
$452$	5942.98i	0.618439i
$453$	0	0
$454$	29259.8	3.02473
$455$	0	0
$456$	0	0
$457$	13151.5i	1.34617i	0.739565	+	0.673085i	$0.235031\pi$
$457$	13151.5i	1.34617i	−0.739565	+	0.673085i	$0.764969\pi$
$458$	10710.7i	1.09275i
$459$	0	0
$460$	0	0
$461$	18163.7	1.83507	0.917533	−	0.397658i	$-0.130177\pi$
$461$	18163.7	1.83507	0.917533	+	0.397658i	$0.130177\pi$
$462$	0	0
$463$	− 12788.0i	− 1.28361i	−0.766869	−	0.641803i	$-0.778187\pi$
$463$	− 12788.0i	− 1.28361i	0.766869	−	0.641803i	$-0.221813\pi$
$464$	−15745.5	−1.57536
$465$	0	0
$466$	−19702.6	−1.95860
$467$	− 3978.90i	− 0.394265i	−0.980377	−	0.197132i	$-0.936837\pi$
$467$	− 3978.90i	− 0.394265i	0.980377	−	0.197132i	$-0.0631629\pi$
$468$	0	0
$469$	1265.29	0.124575
$470$	0	0
$471$	0	0
$472$	1562.89i	0.152410i
$473$	− 4146.77i	− 0.403105i
$474$	0	0
$475$	0	0
$476$	−3363.13	−0.323842
$477$	0	0
$478$	31861.5i	3.04877i
$479$	−1095.93	−0.104539	−0.0522694	−	0.998633i	$-0.516645\pi$
$479$	−1095.93	−0.104539	−0.0522694	+	0.998633i	$0.516645\pi$
$480$	0	0
$481$	3544.62	0.336010
$482$	− 9261.90i	− 0.875244i
$483$	0	0
$484$	−4726.83	−0.443917
$485$	0	0
$486$	0	0
$487$	− 15345.9i	− 1.42790i	−0.700196	−	0.713951i	$-0.746904\pi$
$487$	− 15345.9i	− 1.42790i	0.700196	−	0.713951i	$-0.253096\pi$
$488$	− 26892.9i	− 2.49464i
$489$	0	0
$490$	0	0
$491$	−2148.29	−0.197457	−0.0987283	−	0.995114i	$-0.531477\pi$
$491$	−2148.29	−0.197457	−0.0987283	+	0.995114i	$0.531477\pi$
$492$	0	0
$493$	23484.7i	2.14544i
$494$	40772.3	3.71342
$495$	0	0
$496$	16207.0	1.46717
$497$	467.645i	0.0422068i
$498$	0	0
$499$	−771.203	−0.0691860	−0.0345930	−	0.999401i	$-0.511013\pi$
$499$	−771.203	−0.0691860	−0.0345930	+	0.999401i	$0.511013\pi$
$500$	0	0
$501$	0	0
$502$	30965.6i	2.75311i
$503$	− 4031.11i	− 0.357333i	−0.983910	−	0.178666i	$-0.942822\pi$
$503$	− 4031.11i	− 0.357333i	0.983910	−	0.178666i	$-0.0571783\pi$
$504$	0	0
$505$	0	0
$506$	15137.9	1.32996
$507$	0	0
$508$	22785.6i	1.99006i
$509$	7337.75	0.638979	0.319489	−	0.947590i	$-0.396489\pi$
$509$	7337.75	0.638979	0.319489	+	0.947590i	$0.396489\pi$
$510$	0	0
$511$	−625.973	−0.0541907
$512$	20625.7i	1.78035i
$513$	0	0
$514$	21524.5	1.84709
$515$	0	0
$516$	0	0
$517$	10497.1i	0.892967i
$518$	593.324i	0.0503266i
$519$	0	0
$520$	0	0
$521$	−6972.35	−0.586304	−0.293152	−	0.956066i	$-0.594704\pi$
$521$	−6972.35	−0.586304	−0.293152	+	0.956066i	$0.594704\pi$
$522$	0	0
$523$	− 9040.75i	− 0.755878i	−0.925830	−	0.377939i	$-0.876633\pi$
$523$	− 9040.75i	− 0.755878i	0.925830	−	0.377939i	$-0.123367\pi$
$524$	2498.81	0.208323
$525$	0	0
$526$	14372.5	1.19139
$527$	− 24173.0i	− 1.99809i
$528$	0	0
$529$	6314.97	0.519025
$530$	0	0
$531$	0	0
$532$	4559.71i	0.371595i
$533$	25395.8i	2.06382i
$534$	0	0
$535$	0	0
$536$	23921.5	1.92771
$537$	0	0
$538$	− 9177.85i	− 0.735475i
$539$	13646.1	1.09050
$540$	0	0
$541$	22956.9	1.82439	0.912196	−	0.409755i	$-0.134386\pi$
$541$	22956.9	1.82439	0.912196	+	0.409755i	$0.134386\pi$
$542$	548.898i	0.0435003i
$543$	0	0
$544$	−821.290	−0.0647289
$545$	0	0
$546$	0	0
$547$	734.019i	0.0573755i	0.999588	+	0.0286877i	$0.00913284\pi$
$547$	734.019i	0.0573755i	−0.999588	+	0.0286877i	$0.990867\pi$
$548$	− 39418.3i	− 3.07275i
$549$	0	0
$550$	0	0
$551$	31840.5	2.46180
$552$	0	0
$553$	201.418i	0.0154886i
$554$	−21867.7	−1.67702
$555$	0	0
$556$	23576.9	1.79835
$557$	− 8428.97i	− 0.641197i	−0.947215	−	0.320599i	$-0.896116\pi$
$557$	− 8428.97i	− 0.641197i	0.947215	−	0.320599i	$-0.103884\pi$
$558$	0	0
$559$	−6351.34	−0.480560
$560$	0	0
$561$	0	0
$562$	− 10981.1i	− 0.824218i
$563$	− 9091.27i	− 0.680553i	−0.940325	−	0.340276i	$-0.889479\pi$
$563$	− 9091.27i	− 0.680553i	0.940325	−	0.340276i	$-0.110521\pi$
$564$	0	0
$565$	0	0
$566$	−8001.09	−0.594189
$567$	0	0
$568$	8841.30i	0.653121i
$569$	6766.12	0.498507	0.249253	−	0.968438i	$-0.419815\pi$
$569$	6766.12	0.498507	0.249253	+	0.968438i	$0.419815\pi$
$570$	0	0
$571$	−3864.13	−0.283203	−0.141602	−	0.989924i	$-0.545225\pi$
$571$	−3864.13	−0.283203	−0.141602	+	0.989924i	$0.545225\pi$
$572$	40070.9i	2.92911i
$573$	0	0
$574$	−4250.93	−0.309112
$575$	0	0
$576$	0	0
$577$	− 3572.62i	− 0.257765i	−0.991660	−	0.128882i	$-0.958861\pi$
$577$	− 3572.62i	− 0.257765i	0.991660	−	0.128882i	$-0.0411390\pi$
$578$	24213.5i	1.74247i
$579$	0	0
$580$	0	0
$581$	6.49052	0.000463463 0
$582$	0	0
$583$	3041.85i	0.216090i
$584$	−11834.6	−0.838564
$585$	0	0
$586$	−12472.8	−0.879262
$587$	10186.2i	0.716236i	0.933676	+	0.358118i	$0.116581\pi$
$587$	10186.2i	0.716236i	−0.933676	+	0.358118i	$0.883419\pi$
$588$	0	0
$589$	−32773.7	−2.29273
$590$	0	0
$591$	0	0
$592$	3819.69i	0.265183i
$593$	2691.78i	0.186405i	0.995647	+	0.0932025i	$0.0297104\pi$
$593$	2691.78i	0.186405i	−0.995647	+	0.0932025i	$0.970290\pi$
$594$	0	0
$595$	0	0
$596$	−54165.6	−3.72266
$597$	0	0
$598$	− 23185.7i	− 1.58551i
$599$	−4236.48	−0.288978	−0.144489	−	0.989506i	$-0.546154\pi$
$599$	−4236.48	−0.288978	−0.144489	+	0.989506i	$0.546154\pi$
$600$	0	0
$601$	10749.6	0.729595	0.364798	−	0.931087i	$-0.381138\pi$
$601$	10749.6	0.729595	0.364798	+	0.931087i	$0.381138\pi$
$602$	− 1063.13i	− 0.0719768i
$603$	0	0
$604$	−32274.9	−2.17425
$605$	0	0
$606$	0	0
$607$	− 19499.1i	− 1.30386i	−0.758278	−	0.651932i	$-0.773959\pi$
$607$	− 19499.1i	− 1.30386i	0.758278	−	0.651932i	$-0.226041\pi$
$608$	1113.50i	0.0742738i
$609$	0	0
$610$	0	0
$611$	16077.8	1.06455
$612$	0	0
$613$	− 1509.82i	− 0.0994796i	−0.998762	−	0.0497398i	$-0.984161\pi$
$613$	− 1509.82i	− 0.0994796i	0.998762	−	0.0497398i	$-0.0158392\pi$
$614$	−23332.2	−1.53357
$615$	0	0
$616$	−3375.48	−0.220782
$617$	− 27905.1i	− 1.82077i	−0.413758	−	0.910387i	$-0.635784\pi$
$617$	− 27905.1i	− 1.82077i	0.413758	−	0.910387i	$-0.364216\pi$
$618$	0	0
$619$	−1448.88	−0.0940801	−0.0470401	−	0.998893i	$-0.514979\pi$
$619$	−1448.88	−0.0940801	−0.0470401	+	0.998893i	$0.514979\pi$
$620$	0	0
$621$	0	0
$622$	37636.5i	2.42619i
$623$	− 2826.02i	− 0.181737i
$624$	0	0
$625$	0	0
$626$	−38610.9	−2.46518
$627$	0	0
$628$	15317.9i	0.973327i
$629$	5697.15	0.361145
$630$	0	0
$631$	7476.02	0.471657	0.235829	−	0.971795i	$-0.424220\pi$
$631$	7476.02	0.471657	0.235829	+	0.971795i	$0.424220\pi$
$632$	3808.01i	0.239675i
$633$	0	0
$634$	−18880.8	−1.18273
$635$	0	0
$636$	0	0
$637$	− 20900.9i	− 1.30004i
$638$	46837.5i	2.90645i
$639$	0	0
$640$	0	0
$641$	50.3581	0.00310301	0.00155150	−	0.999999i	$-0.499506\pi$
$641$	50.3581	0.00310301	0.00155150	+	0.999999i	$0.499506\pi$
$642$	0	0
$643$	16728.8i	1.02600i	0.858387	+	0.513002i	$0.171467\pi$
$643$	16728.8i	1.02600i	−0.858387	+	0.513002i	$0.828533\pi$
$644$	2592.94	0.158659
$645$	0	0
$646$	65531.8	3.99120
$647$	− 13149.5i	− 0.799010i	−0.916731	−	0.399505i	$-0.869182\pi$
$647$	− 13149.5i	− 0.799010i	0.916731	−	0.399505i	$-0.130818\pi$
$648$	0	0
$649$	1583.07	0.0957490
$650$	0	0
$651$	0	0
$652$	50777.2i	3.04998i
$653$	− 4131.96i	− 0.247620i	−0.992306	−	0.123810i	$-0.960489\pi$
$653$	− 4131.96i	− 0.247620i	0.992306	−	0.123810i	$-0.0395114\pi$
$654$	0	0
$655$	0	0
$656$	−27366.6	−1.62879
$657$	0	0
$658$	2691.22i	0.159445i
$659$	−4141.95	−0.244837	−0.122419	−	0.992479i	$-0.539065\pi$
$659$	−4141.95	−0.244837	−0.122419	+	0.992479i	$0.539065\pi$
$660$	0	0
$661$	−25593.3	−1.50600	−0.753000	−	0.658021i	$-0.771394\pi$
$661$	−25593.3	−1.50600	−0.753000	+	0.658021i	$0.771394\pi$
$662$	3441.15i	0.202030i
$663$	0	0
$664$	122.710	0.00717178
$665$	0	0
$666$	0	0
$667$	− 18106.5i	− 1.05111i
$668$	− 30382.3i	− 1.75977i
$669$	0	0
$670$	0	0
$671$	−27240.3	−1.56721
$672$	0	0
$673$	− 27185.3i	− 1.55708i	−0.627593	−	0.778541i	$-0.715960\pi$
$673$	− 27185.3i	− 1.55708i	0.627593	−	0.778541i	$-0.284040\pi$
$674$	−59264.1	−3.38690
$675$	0	0
$676$	25992.0	1.47883
$677$	− 3359.82i	− 0.190736i	−0.995442	−	0.0953680i	$-0.969597\pi$
$677$	− 3359.82i	− 0.190736i	0.995442	−	0.0953680i	$-0.0304028\pi$
$678$	0	0
$679$	−3120.35	−0.176360
$680$	0	0
$681$	0	0
$682$	− 48210.2i	− 2.70684i
$683$	25176.1i	1.41045i	0.708985	+	0.705223i	$0.249153\pi$
$683$	25176.1i	1.41045i	−0.708985	+	0.705223i	$0.750847\pi$
$684$	0	0
$685$	0	0
$686$	7042.86	0.391979
$687$	0	0
$688$	− 6844.22i	− 0.379264i
$689$	4659.01	0.257611
$690$	0	0
$691$	−5980.72	−0.329258	−0.164629	−	0.986356i	$-0.552643\pi$
$691$	−5980.72	−0.329258	−0.164629	+	0.986356i	$0.552643\pi$
$692$	16696.3i	0.917193i
$693$	0	0
$694$	−39594.4	−2.16568
$695$	0	0
$696$	0	0
$697$	40817.8i	2.21820i
$698$	42842.4i	2.32322i
$699$	0	0
$700$	0	0
$701$	8195.53	0.441570	0.220785	−	0.975322i	$-0.429138\pi$
$701$	8195.53	0.441570	0.220785	+	0.975322i	$0.429138\pi$
$702$	0	0
$703$	− 7724.17i	− 0.414399i
$704$	19811.9	1.06064
$705$	0	0
$706$	30075.0	1.60324
$707$	− 3232.41i	− 0.171948i
$708$	0	0
$709$	−6700.56	−0.354929	−0.177465	−	0.984127i	$-0.556790\pi$
$709$	−6700.56	−0.354929	−0.177465	+	0.984127i	$0.556790\pi$
$710$	0	0
$711$	0	0
$712$	− 53428.8i	− 2.81226i
$713$	18637.2i	0.978918i
$714$	0	0
$715$	0	0
$716$	61040.8	3.18604
$717$	0	0
$718$	− 2785.23i	− 0.144769i
$719$	−319.573	−0.0165759	−0.00828794	−	0.999966i	$-0.502638\pi$
$719$	−319.573	−0.0165759	−0.00828794	+	0.999966i	$0.502638\pi$
$720$	0	0
$721$	−750.492	−0.0387653
$722$	− 55172.5i	− 2.84392i
$723$	0	0
$724$	−15243.1	−0.782464
$725$	0	0
$726$	0	0
$727$	− 4869.19i	− 0.248402i	−0.992257	−	0.124201i	$-0.960363\pi$
$727$	− 4869.19i	− 0.248402i	0.992257	−	0.124201i	$-0.0396367\pi$
$728$	5170.00i	0.263204i
$729$	0	0
$730$	0	0
$731$	−10208.3	−0.516508
$732$	0	0
$733$	− 26969.4i	− 1.35899i	−0.733681	−	0.679495i	$-0.762199\pi$
$733$	− 26969.4i	− 1.35899i	0.733681	−	0.679495i	$-0.237801\pi$
$734$	1869.45	0.0940090
$735$	0	0
$736$	633.208	0.0317124
$737$	− 24230.5i	− 1.21105i
$738$	0	0
$739$	29544.7	1.47066	0.735331	−	0.677708i	$-0.237027\pi$
$739$	29544.7	1.47066	0.735331	+	0.677708i	$0.237027\pi$
$740$	0	0
$741$	0	0
$742$	779.859i	0.0385842i
$743$	37991.6i	1.87588i	0.346800	+	0.937939i	$0.387268\pi$
$743$	37991.6i	1.87588i	−0.346800	+	0.937939i	$0.612732\pi$
$744$	0	0
$745$	0	0
$746$	5153.90	0.252946
$747$	0	0
$748$	64404.6i	3.14821i
$749$	−3325.89	−0.162250
$750$	0	0
$751$	−18817.9	−0.914349	−0.457174	−	0.889377i	$-0.651138\pi$
$751$	−18817.9	−0.914349	−0.457174	+	0.889377i	$0.651138\pi$
$752$	17325.5i	0.840153i
$753$	0	0
$754$	71738.0	3.46491
$755$	0	0
$756$	0	0
$757$	− 6255.42i	− 0.300339i	−0.988660	−	0.150170i	$-0.952018\pi$
$757$	− 6255.42i	− 0.300339i	0.988660	−	0.150170i	$-0.0479820\pi$
$758$	17551.8i	0.841041i
$759$	0	0
$760$	0	0
$761$	4900.19	0.233419	0.116709	−	0.993166i	$-0.462765\pi$
$761$	4900.19	0.233419	0.116709	+	0.993166i	$0.462765\pi$
$762$	0	0
$763$	− 2504.54i	− 0.118834i
$764$	−42176.8	−1.99726
$765$	0	0
$766$	−69821.3	−3.29340
$767$	− 2424.69i	− 0.114147i
$768$	0	0
$769$	−21378.4	−1.00250	−0.501252	−	0.865301i	$-0.667127\pi$
$769$	−21378.4	−1.00250	−0.501252	+	0.865301i	$0.667127\pi$
$770$	0	0
$771$	0	0
$772$	41666.1i	1.94248i
$773$	7694.82i	0.358038i	0.983846	+	0.179019i	$0.0572924\pi$
$773$	7694.82i	0.358038i	−0.983846	+	0.179019i	$0.942708\pi$
$774$	0	0
$775$	0	0
$776$	−58993.4	−2.72904
$777$	0	0
$778$	39190.4i	1.80597i
$779$	55340.6	2.54529
$780$	0	0
$781$	8955.50	0.410311
$782$	− 37265.5i	− 1.70411i
$783$	0	0
$784$	22522.9	1.02600
$785$	0	0
$786$	0	0
$787$	26712.1i	1.20989i	0.796267	+	0.604945i	$0.206805\pi$
$787$	26712.1i	1.20989i	−0.796267	+	0.604945i	$0.793195\pi$
$788$	10119.0i	0.457456i
$789$	0	0
$790$	0	0
$791$	776.675	0.0349120
$792$	0	0
$793$	41722.2i	1.86835i
$794$	−11562.7	−0.516808
$795$	0	0
$796$	−46581.8	−2.07418
$797$	12894.2i	0.573070i	0.958070	+	0.286535i	$0.0925035\pi$
$797$	12894.2i	0.573070i	−0.958070	+	0.286535i	$0.907497\pi$
$798$	0	0
$799$	25841.3	1.14418
$800$	0	0
$801$	0	0
$802$	31519.9i	1.38779i
$803$	11987.5i	0.526812i
$804$	0	0
$805$	0	0
$806$	−73840.4	−3.22695
$807$	0	0
$808$	− 61111.9i	− 2.66078i
$809$	−13581.6	−0.590241	−0.295120	−	0.955460i	$-0.595360\pi$
$809$	−13581.6	−0.590241	−0.295120	+	0.955460i	$0.595360\pi$
$810$	0	0
$811$	29019.1	1.25647	0.628235	−	0.778023i	$-0.283777\pi$
$811$	29019.1	1.25647	0.628235	+	0.778023i	$0.283777\pi$
$812$	8022.72i	0.346727i
$813$	0	0
$814$	11362.3	0.489247
$815$	0	0
$816$	0	0
$817$	13840.4i	0.592672i
$818$	− 41414.0i	− 1.77018i
$819$	0	0
$820$	0	0
$821$	7950.00	0.337950	0.168975	−	0.985620i	$-0.445954\pi$
$821$	7950.00	0.337950	0.168975	+	0.985620i	$0.445954\pi$
$822$	0	0
$823$	− 10499.7i	− 0.444708i	−0.974966	−	0.222354i	$-0.928626\pi$
$823$	− 10499.7i	− 0.444708i	0.974966	−	0.222354i	$-0.0713741\pi$
$824$	−14188.8	−0.599867
$825$	0	0
$826$	405.862	0.0170966
$827$	27097.0i	1.13936i	0.821865	+	0.569682i	$0.192934\pi$
$827$	27097.0i	1.13936i	−0.821865	+	0.569682i	$0.807066\pi$
$828$	0	0
$829$	−325.331	−0.0136299	−0.00681497	−	0.999977i	$-0.502169\pi$
$829$	−325.331	−0.0136299	−0.00681497	+	0.999977i	$0.502169\pi$
$830$	0	0
$831$	0	0
$832$	− 30344.7i	− 1.26444i
$833$	− 33593.3i	− 1.39728i
$834$	0	0
$835$	0	0
$836$	87319.4	3.61245
$837$	0	0
$838$	− 47444.9i	− 1.95580i
$839$	−30523.3	−1.25600	−0.627999	−	0.778214i	$-0.716126\pi$
$839$	−30523.3	−1.25600	−0.627999	+	0.778214i	$0.716126\pi$
$840$	0	0
$841$	31633.8	1.29705
$842$	18282.4i	0.748281i
$843$	0	0
$844$	46130.0	1.88135
$845$	0	0
$846$	0	0
$847$	617.739i	0.0250599i
$848$	5020.56i	0.203310i
$849$	0	0
$850$	0	0
$851$	−4392.45	−0.176935
$852$	0	0
$853$	32915.4i	1.32122i	0.750728	+	0.660612i	$0.229703\pi$
$853$	32915.4i	1.32122i	−0.750728	+	0.660612i	$0.770297\pi$
$854$	−6983.76	−0.279835
$855$	0	0
$856$	−62879.3	−2.51071
$857$	18915.5i	0.753959i	0.926222	+	0.376979i	$0.123037\pi$
$857$	18915.5i	0.753959i	−0.926222	+	0.376979i	$0.876963\pi$
$858$	0	0
$859$	31407.3	1.24750	0.623750	−	0.781624i	$-0.285608\pi$
$859$	31407.3	1.24750	0.623750	+	0.781624i	$0.285608\pi$
$860$	0	0
$861$	0	0
$862$	− 13116.6i	− 0.518277i
$863$	− 39038.2i	− 1.53983i	−0.638146	−	0.769916i	$-0.720298\pi$
$863$	− 39038.2i	− 1.53983i	0.638146	−	0.769916i	$-0.279702\pi$
$864$	0	0
$865$	0	0
$866$	−13371.6	−0.524693
$867$	0	0
$868$	− 8257.85i	− 0.322914i
$869$	3857.20	0.150571
$870$	0	0
$871$	−37112.3	−1.44375
$872$	− 47350.8i	− 1.83888i
$873$	0	0
$874$	−50524.5	−1.95540
$875$	0	0
$876$	0	0
$877$	47360.9i	1.82356i	0.410677	+	0.911781i	$0.365293\pi$
$877$	47360.9i	1.82356i	−0.410677	+	0.911781i	$0.634707\pi$
$878$	40661.3i	1.56293i
$879$	0	0
$880$	0	0
$881$	−5167.57	−0.197616	−0.0988081	−	0.995107i	$-0.531503\pi$
$881$	−5167.57	−0.197616	−0.0988081	+	0.995107i	$0.531503\pi$
$882$	0	0
$883$	31450.1i	1.19862i	0.800518	+	0.599308i	$0.204558\pi$
$883$	31450.1i	1.19862i	−0.800518	+	0.599308i	$0.795442\pi$
$884$	98644.3	3.75313
$885$	0	0
$886$	−30865.4	−1.17037
$887$	7264.18i	0.274980i	0.990503	+	0.137490i	$0.0439035\pi$
$887$	7264.18i	0.274980i	−0.990503	+	0.137490i	$0.956096\pi$
$888$	0	0
$889$	2977.80	0.112342
$890$	0	0
$891$	0	0
$892$	− 36144.1i	− 1.35672i
$893$	− 35035.5i	− 1.31290i
$894$	0	0
$895$	0	0
$896$	5218.68	0.194580
$897$	0	0
$898$	− 23260.8i	− 0.864391i
$899$	−57664.6	−2.13929
$900$	0	0
$901$	7488.27	0.276882
$902$	81406.2i	3.00502i
$903$	0	0
$904$	14683.8	0.540239
$905$	0	0
$906$	0	0
$907$	− 9535.80i	− 0.349097i	−0.984649	−	0.174549i	$-0.944153\pi$
$907$	− 9535.80i	− 0.349097i	0.984649	−	0.174549i	$-0.0558466\pi$
$908$	− 95978.2i	− 3.50787i
$909$	0	0
$910$	0	0
$911$	50779.1	1.84675	0.923373	−	0.383904i	$-0.125421\pi$
$911$	50779.1	1.84675	0.923373	+	0.383904i	$0.125421\pi$
$912$	0	0
$913$	− 124.295i	− 0.00450554i
$914$	64569.1	2.33671
$915$	0	0
$916$	35133.4	1.26729
$917$	− 326.564i	− 0.0117602i
$918$	0	0
$919$	44669.7	1.60339	0.801696	−	0.597732i	$-0.203931\pi$
$919$	44669.7	1.60339	0.801696	+	0.597732i	$0.203931\pi$
$920$	0	0
$921$	0	0
$922$	− 89177.2i	− 3.18535i
$923$	− 13716.6i	− 0.489151i
$924$	0	0
$925$	0	0
$926$	−62784.8	−2.22812
$927$	0	0
$928$	1959.18i	0.0693032i
$929$	23431.4	0.827513	0.413756	−	0.910388i	$-0.364217\pi$
$929$	23431.4	0.827513	0.413756	+	0.910388i	$0.364217\pi$
$930$	0	0
$931$	−45545.6	−1.60333
$932$	64628.8i	2.27144i
$933$	0	0
$934$	−19535.0	−0.684375
$935$	0	0
$936$	0	0
$937$	− 25234.2i	− 0.879792i	−0.898049	−	0.439896i	$-0.855015\pi$
$937$	− 25234.2i	− 0.879792i	0.898049	−	0.439896i	$-0.144985\pi$
$938$	− 6212.12i	− 0.216240i
$939$	0	0
$940$	0	0
$941$	−47812.7	−1.65637	−0.828187	−	0.560452i	$-0.810628\pi$
$941$	−47812.7	−1.65637	−0.828187	+	0.560452i	$0.810628\pi$
$942$	0	0
$943$	− 31470.2i	− 1.08676i
$944$	2612.86	0.0900860
$945$	0	0
$946$	−20359.2	−0.699720
$947$	− 26024.4i	− 0.893007i	−0.894782	−	0.446504i	$-0.852669\pi$
$947$	− 26024.4i	− 0.893007i	0.894782	−	0.446504i	$-0.147331\pi$
$948$	0	0
$949$	18360.5	0.628037
$950$	0	0
$951$	0	0
$952$	8309.56i	0.282893i
$953$	− 32403.9i	− 1.10143i	−0.834692	−	0.550716i	$-0.814355\pi$
$953$	− 32403.9i	− 1.10143i	0.834692	−	0.550716i	$-0.185645\pi$
$954$	0	0
$955$	0	0
$956$	104512.	3.53574
$957$	0	0
$958$	5380.61i	0.181461i
$959$	−5151.49	−0.173462
$960$	0	0
$961$	29563.7	0.992369
$962$	− 17402.9i	− 0.583254i
$963$	0	0
$964$	−30381.0	−1.01505
$965$	0	0
$966$	0	0
$967$	− 50840.1i	− 1.69070i	−0.534213	−	0.845350i	$-0.679392\pi$
$967$	− 50840.1i	− 1.69070i	0.534213	−	0.845350i	$-0.320608\pi$
$968$	11679.0i	0.387785i
$969$	0	0
$970$	0	0
$971$	17336.6	0.572975	0.286488	−	0.958084i	$-0.407512\pi$
$971$	17336.6	0.572975	0.286488	+	0.958084i	$0.407512\pi$
$972$	0	0
$973$	− 3081.21i	− 0.101520i
$974$	−75342.9	−2.47859
$975$	0	0
$976$	−44960.0	−1.47452
$977$	− 48360.1i	− 1.58360i	−0.610781	−	0.791799i	$-0.709145\pi$
$977$	− 48360.1i	− 1.58360i	0.610781	−	0.791799i	$-0.290855\pi$
$978$	0	0
$979$	−54118.9	−1.76675
$980$	0	0
$981$	0	0
$982$	10547.4i	0.342750i
$983$	56102.8i	1.82035i	0.414226	+	0.910174i	$0.364052\pi$
$983$	56102.8i	1.82035i	−0.414226	+	0.910174i	$0.635948\pi$
$984$	0	0
$985$	0	0
$986$	115302.	3.72410
$987$	0	0
$988$	− 133742.i	− 4.30656i
$989$	7870.50	0.253051
$990$	0	0
$991$	−46186.7	−1.48049	−0.740247	−	0.672335i	$-0.765291\pi$
$991$	−46186.7	−1.48049	−0.740247	+	0.672335i	$0.765291\pi$
$992$	− 2016.60i	− 0.0645436i
$993$	0	0
$994$	2295.98	0.0732635
$995$	0	0
$996$	0	0
$997$	16073.7i	0.510592i	0.966863	+	0.255296i	$0.0821728\pi$
$997$	16073.7i	0.510592i	−0.966863	+	0.255296i	$0.917827\pi$
$998$	3786.34i	0.120095i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	675.4.b.o.649.2		8
3.2	odd	2	inner	675.4.b.o.649.8		8
5.2	odd	4		675.4.a.x.1.4	yes	4
5.3	odd	4		675.4.a.w.1.1	✓	4
5.4	even	2	inner	675.4.b.o.649.7		8
15.2	even	4		675.4.a.x.1.1	yes	4
15.8	even	4		675.4.a.w.1.4	yes	4
15.14	odd	2	inner	675.4.b.o.649.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
675.4.a.w.1.1	✓	4	5.3	odd	4
675.4.a.w.1.4	yes	4	15.8	even	4
675.4.a.x.1.1	yes	4	15.2	even	4
675.4.a.x.1.4	yes	4	5.2	odd	4
675.4.b.o.649.1		8	15.14	odd	2	inner
675.4.b.o.649.2		8	1.1	even	1	trivial
675.4.b.o.649.7		8	5.4	even	2	inner
675.4.b.o.649.8		8	3.2	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	675.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.90965i q^{2} -16.1047 q^{4} +2.10469i q^{7} +39.7912i q^{8} +40.3052 q^{11} -61.7328i q^{13} +10.3333 q^{14} +66.5234 q^{16} -99.2210i q^{17} -134.523 q^{19} -197.884i q^{22} +76.4985i q^{23} -303.087 q^{26} -33.8953i q^{28} -236.691 q^{29} +243.628 q^{31} -8.27738i q^{32} -487.141 q^{34} +57.4187i q^{37} +660.463i q^{38} -411.383 q^{41} -102.884i q^{43} -649.102 q^{44} +375.581 q^{46} +260.442i q^{47} +338.570 q^{49} +994.187i q^{52} +75.4706i q^{53} -83.7480 q^{56} +1162.07i q^{58} +39.2772 q^{59} -675.851 q^{61} -1196.13i q^{62} +491.548 q^{64} -601.176i q^{67} +1597.92i q^{68} +222.192 q^{71} +297.419i q^{73} +281.906 q^{74} +2166.46 q^{76} +84.8297i q^{77} +95.6999 q^{79} +2019.75i q^{82} -3.08384i q^{83} -505.126 q^{86} +1603.79i q^{88} -1342.73 q^{89} +129.928 q^{91} -1231.99i q^{92} +1278.68 q^{94} +1482.57i q^{97} -1662.26i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 52 q^{4} + 148 q^{16} - 692 q^{19} + 1488 q^{31} - 1592 q^{34} + 3312 q^{46} + 1556 q^{49} + 356 q^{61} - 140 q^{64} + 8188 q^{76} - 4152 q^{79} + 5496 q^{91} + 2392 q^{94}+O(q^{100}) \) 8 * q - 52 * q^4 + 148 * q^16 - 692 * q^19 + 1488 * q^31 - 1592 * q^34 + 3312 * q^46 + 1556 * q^49 + 356 * q^61 - 140 * q^64 + 8188 * q^76 - 4152 * q^79 + 5496 * q^91 + 2392 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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