LMFDB - Embedded newform 1150.4.b.o.599.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1150,4,Mod(599,1150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1150, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1150.599");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1150 = 2 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1150.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$67.8521965066$
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} + 124x^{6} + 4272x^{4} + 28129x^{2} + 4 $ x^8 + 124x^6 + 4272x^4 + 28129*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 230)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			599.8
Root			$8.04090i$ of defining polynomial
Character	$\chi$	$=$	1150.599
Dual form			1150.4.b.o.599.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.00000i q^{2} +5.04090i q^{3} -4.00000 q^{4} -10.0818 q^{6} +5.03071i q^{7} -8.00000i q^{8} +1.58932 q^{9} -5.58219 q^{11} -20.1636i q^{12} -62.7277i q^{13} -10.0614 q^{14} +16.0000 q^{16} -19.7435i q^{17} +3.17864i q^{18} -158.545 q^{19} -25.3593 q^{21} -11.1644i q^{22} +23.0000i q^{23} +40.3272 q^{24} +125.455 q^{26} +144.116i q^{27} -20.1228i q^{28} +35.5033 q^{29} +282.041 q^{31} +32.0000i q^{32} -28.1393i q^{33} +39.4870 q^{34} -6.35728 q^{36} -139.981i q^{37} -317.090i q^{38} +316.204 q^{39} +227.680 q^{41} -50.7186i q^{42} -436.962i q^{43} +22.3288 q^{44} -46.0000 q^{46} +90.2701i q^{47} +80.6544i q^{48} +317.692 q^{49} +99.5250 q^{51} +250.911i q^{52} -330.183i q^{53} -288.232 q^{54} +40.2457 q^{56} -799.209i q^{57} +71.0066i q^{58} +796.203 q^{59} -568.580 q^{61} +564.081i q^{62} +7.99541i q^{63} -64.0000 q^{64} +56.2785 q^{66} +85.1419i q^{67} +78.9740i q^{68} -115.941 q^{69} -369.578 q^{71} -12.7146i q^{72} +310.188i q^{73} +279.963 q^{74} +634.180 q^{76} -28.0824i q^{77} +632.408i q^{78} +1325.46 q^{79} -683.562 q^{81} +455.360i q^{82} +158.806i q^{83} +101.437 q^{84} +873.924 q^{86} +178.969i q^{87} +44.6575i q^{88} +1233.89 q^{89} +315.565 q^{91} -92.0000i q^{92} +1421.74i q^{93} -180.540 q^{94} -161.309 q^{96} -106.389i q^{97} +635.384i q^{98} -8.87189 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 32 q^{4} + 56 q^{6} - 128 q^{9} + 42 q^{11} - 32 q^{14} + 128 q^{16} - 346 q^{19} - 240 q^{21} - 224 q^{24} + 280 q^{26} + 236 q^{29} + 34 q^{31} - 224 q^{34} + 512 q^{36} + 442 q^{39} + 278 q^{41}+ \cdots + 5490 q^{99}+O(q^{100}) $ 8 * q - 32 * q^4 + 56 * q^6 - 128 * q^9 + 42 * q^11 - 32 * q^14 + 128 * q^16 - 346 * q^19 - 240 * q^21 - 224 * q^24 + 280 * q^26 + 236 * q^29 + 34 * q^31 - 224 * q^34 + 512 * q^36 + 442 * q^39 + 278 * q^41 - 168 * q^44 - 368 * q^46 + 248 * q^49 + 878 * q^51 - 1556 * q^54 + 128 * q^56 + 906 * q^59 - 654 * q^61 - 512 * q^64 - 356 * q^66 + 644 * q^69 + 390 * q^71 + 1372 * q^74 + 1384 * q^76 + 2280 * q^79 + 2912 * q^81 + 960 * q^84 - 200 * q^86 + 4340 * q^89 + 1114 * q^91 - 1468 * q^94 + 896 * q^96 + 5490 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1150\mathbb{Z}\right)^\times$.

$n$	$51$	$277$
$\chi(n)$	$1$	$-1$

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 $

$2$	2.00000i	0.707107i
$2$	2.00000i	0.707107i
$3$	5.04090i	0.970122i	0.874480	+	0.485061i	$0.161203\pi$
$3$	5.04090i	0.970122i	−0.874480	+	0.485061i	$0.838797\pi$
$4$	−4.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−10.0818	−0.685980
$7$	5.03071i	0.271633i	0.990734	+	0.135816i	$0.0433657\pi$
$7$	5.03071i	0.271633i	−0.990734	+	0.135816i	$0.956634\pi$
$8$	− 8.00000i	− 0.353553i
$9$	1.58932	0.0588637
$10$	0	0
$11$	−5.58219	−0.153008	−0.0765042	−	0.997069i	$-0.524376\pi$
$11$	−5.58219	−0.153008	−0.0765042	+	0.997069i	$0.524376\pi$
$12$	− 20.1636i	− 0.485061i
$13$	− 62.7277i	− 1.33827i	−0.743140	−	0.669136i	$-0.766664\pi$
$13$	− 62.7277i	− 1.33827i	0.743140	−	0.669136i	$-0.233336\pi$
$14$	−10.0614	−0.192073
$15$	0	0
$16$	16.0000	0.250000
$17$	− 19.7435i	− 0.281677i	−0.990033	−	0.140838i	$-0.955020\pi$
$17$	− 19.7435i	− 0.281677i	0.990033	−	0.140838i	$-0.0449798\pi$
$18$	3.17864i	0.0416229i
$19$	−158.545	−1.91435	−0.957177	−	0.289505i	$-0.906509\pi$
$19$	−158.545	−1.91435	−0.957177	+	0.289505i	$0.906509\pi$
$20$	0	0
$21$	−25.3593	−0.263517
$22$	− 11.1644i	− 0.108193i
$23$	23.0000i	0.208514i
$23$	23.0000i	0.208514i
$24$	40.3272	0.342990
$25$	0	0
$26$	125.455	0.946301
$27$	144.116i	1.02723i
$28$	− 20.1228i	− 0.135816i
$29$	35.5033	0.227338	0.113669	−	0.993519i	$-0.463740\pi$
$29$	35.5033	0.227338	0.113669	+	0.993519i	$0.463740\pi$
$30$	0	0
$31$	282.041	1.63406	0.817032	−	0.576592i	$-0.195618\pi$
$31$	282.041	1.63406	0.817032	+	0.576592i	$0.195618\pi$
$32$	32.0000i	0.176777i
$33$	− 28.1393i	− 0.148437i
$34$	39.4870	0.199175
$35$	0	0
$36$	−6.35728	−0.0294319
$37$	− 139.981i	− 0.621967i	−0.950415	−	0.310984i	$-0.899342\pi$
$37$	− 139.981i	− 0.621967i	0.950415	−	0.310984i	$-0.100658\pi$
$38$	− 317.090i	− 1.35365i
$39$	316.204	1.29829
$40$	0	0
$41$	227.680	0.867260	0.433630	−	0.901091i	$-0.357232\pi$
$41$	227.680	0.867260	0.433630	+	0.901091i	$0.357232\pi$
$42$	− 50.7186i	− 0.186335i
$43$	− 436.962i	− 1.54968i	−0.632159	−	0.774838i	$-0.717831\pi$
$43$	− 436.962i	− 1.54968i	0.632159	−	0.774838i	$-0.282169\pi$
$44$	22.3288	0.0765042
$45$	0	0
$46$	−46.0000	−0.147442
$47$	90.2701i	0.280154i	0.990141	+	0.140077i	$0.0447350\pi$
$47$	90.2701i	0.280154i	−0.990141	+	0.140077i	$0.955265\pi$
$48$	80.6544i	0.242530i
$49$	317.692	0.926216
$50$	0	0
$51$	99.5250	0.273261
$52$	250.911i	0.669136i
$53$	− 330.183i	− 0.855737i	−0.903841	−	0.427869i	$-0.859265\pi$
$53$	− 330.183i	− 0.855737i	0.903841	−	0.427869i	$-0.140735\pi$
$54$	−288.232	−0.726359
$55$	0	0
$56$	40.2457	0.0960367
$57$	− 799.209i	− 1.85716i
$58$	71.0066i	0.160752i
$59$	796.203	1.75689	0.878447	−	0.477839i	$-0.158580\pi$
$59$	796.203	1.75689	0.878447	+	0.477839i	$0.158580\pi$
$60$	0	0
$61$	−568.580	−1.19343	−0.596715	−	0.802453i	$-0.703528\pi$
$61$	−568.580	−1.19343	−0.596715	+	0.802453i	$0.703528\pi$
$62$	564.081i	1.15546i
$63$	7.99541i	0.0159893i
$64$	−64.0000	−0.125000
$65$	0	0
$66$	56.2785	0.104961
$67$	85.1419i	0.155250i	0.996983	+	0.0776250i	$0.0247337\pi$
$67$	85.1419i	0.155250i	−0.996983	+	0.0776250i	$0.975266\pi$
$68$	78.9740i	0.140838i
$69$	−115.941	−0.202284
$70$	0	0
$71$	−369.578	−0.617758	−0.308879	−	0.951101i	$-0.599954\pi$
$71$	−369.578	−0.617758	−0.308879	+	0.951101i	$0.599954\pi$
$72$	− 12.7146i	− 0.0208115i
$73$	310.188i	0.497325i	0.968590	+	0.248662i	$0.0799910\pi$
$73$	310.188i	0.497325i	−0.968590	+	0.248662i	$0.920009\pi$
$74$	279.963	0.439797
$75$	0	0
$76$	634.180	0.957177
$77$	− 28.0824i	− 0.0415621i
$78$	632.408i	0.918027i
$79$	1325.46	1.88766	0.943832	−	0.330427i	$-0.107193\pi$
$79$	1325.46	1.88766	0.943832	+	0.330427i	$0.107193\pi$
$80$	0	0
$81$	−683.562	−0.937671
$82$	455.360i	0.613246i
$83$	158.806i	0.210014i	0.994471	+	0.105007i	$0.0334865\pi$
$83$	158.806i	0.210014i	−0.994471	+	0.105007i	$0.966513\pi$
$84$	101.437	0.131758
$85$	0	0
$86$	873.924	1.09579
$87$	178.969i	0.220545i
$88$	44.6575i	0.0540967i
$89$	1233.89	1.46957	0.734787	−	0.678298i	$-0.237282\pi$
$89$	1233.89	1.46957	0.734787	+	0.678298i	$0.237282\pi$
$90$	0	0
$91$	315.565	0.363519
$92$	− 92.0000i	− 0.104257i
$93$	1421.74i	1.58524i
$94$	−180.540	−0.198099
$95$	0	0
$96$	−161.309	−0.171495
$97$	− 106.389i	− 0.111363i	−0.998449	−	0.0556814i	$-0.982267\pi$
$97$	− 106.389i	− 0.111363i	0.998449	−	0.0556814i	$-0.0177331\pi$
$98$	635.384i	0.654933i
$99$	−8.87189	−0.00900665
$100$	0	0
$101$	642.676	0.633155	0.316578	−	0.948567i	$-0.397466\pi$
$101$	642.676	0.633155	0.316578	+	0.948567i	$0.397466\pi$
$102$	199.050i	0.193224i
$103$	− 1621.99i	− 1.55165i	−0.630951	−	0.775823i	$-0.717335\pi$
$103$	− 1621.99i	− 1.55165i	0.630951	−	0.775823i	$-0.282665\pi$
$104$	−501.822	−0.473150
$105$	0	0
$106$	660.365	0.605098
$107$	1490.86i	1.34698i	0.739196	+	0.673490i	$0.235205\pi$
$107$	1490.86i	1.34698i	−0.739196	+	0.673490i	$0.764795\pi$
$108$	− 576.464i	− 0.513613i
$109$	−1204.00	−1.05800	−0.529001	−	0.848621i	$-0.677433\pi$
$109$	−1204.00	−1.05800	−0.529001	+	0.848621i	$0.677433\pi$
$110$	0	0
$111$	705.632	0.603384
$112$	80.4914i	0.0679082i
$113$	− 276.771i	− 0.230411i	−0.993342	−	0.115206i	$-0.963247\pi$
$113$	− 276.771i	− 0.230411i	0.993342	−	0.115206i	$-0.0367527\pi$
$114$	1598.42	1.31321
$115$	0	0
$116$	−142.013	−0.113669
$117$	− 99.6944i	− 0.0787757i
$118$	1592.41i	1.24231i
$119$	99.3239	0.0765126
$120$	0	0
$121$	−1299.84	−0.976588
$122$	− 1137.16i	− 0.843883i
$123$	1147.71i	0.841348i
$124$	−1128.16	−0.817032
$125$	0	0
$126$	−15.9908	−0.0113062
$127$	1552.79i	1.08494i	0.840075	+	0.542471i	$0.182511\pi$
$127$	1552.79i	1.08494i	−0.840075	+	0.542471i	$0.817489\pi$
$128$	− 128.000i	− 0.0883883i
$129$	2202.68	1.50338
$130$	0	0
$131$	−313.873	−0.209337	−0.104669	−	0.994507i	$-0.533378\pi$
$131$	−313.873	−0.209337	−0.104669	+	0.994507i	$0.533378\pi$
$132$	112.557i	0.0742184i
$133$	− 797.594i	− 0.520001i
$134$	−170.284	−0.109778
$135$	0	0
$136$	−157.948	−0.0995877
$137$	− 1066.09i	− 0.664835i	−0.943132	−	0.332417i	$-0.892136\pi$
$137$	− 1066.09i	− 0.664835i	0.943132	−	0.332417i	$-0.107864\pi$
$138$	− 231.881i	− 0.143037i
$139$	1594.79	0.973152	0.486576	−	0.873638i	$-0.338246\pi$
$139$	1594.79	0.973152	0.486576	+	0.873638i	$0.338246\pi$
$140$	0	0
$141$	−455.043	−0.271784
$142$	− 739.155i	− 0.436821i
$143$	350.158i	0.204767i
$144$	25.4291	0.0147159
$145$	0	0
$146$	−620.375	−0.351662
$147$	1601.45i	0.898542i
$148$	559.925i	0.310984i
$149$	1096.91	0.603102	0.301551	−	0.953450i	$-0.402496\pi$
$149$	1096.91	0.603102	0.301551	+	0.953450i	$0.402496\pi$
$150$	0	0
$151$	2734.41	1.47366	0.736832	−	0.676076i	$-0.236321\pi$
$151$	2734.41	1.47366	0.736832	+	0.676076i	$0.236321\pi$
$152$	1268.36i	0.676826i
$153$	− 31.3788i	− 0.0165805i
$154$	56.1647	0.0293889
$155$	0	0
$156$	−1264.82	−0.649143
$157$	− 1439.63i	− 0.731813i	−0.930652	−	0.365907i	$-0.880759\pi$
$157$	− 1439.63i	− 0.731813i	0.930652	−	0.365907i	$-0.119241\pi$
$158$	2650.91i	1.33478i
$159$	1664.42	0.830169
$160$	0	0
$161$	−115.706	−0.0566394
$162$	− 1367.12i	− 0.663034i
$163$	2995.82i	1.43957i	0.694195	+	0.719787i	$0.255760\pi$
$163$	2995.82i	1.43957i	−0.694195	+	0.719787i	$0.744240\pi$
$164$	−910.721	−0.433630
$165$	0	0
$166$	−317.611	−0.148503
$167$	− 351.532i	− 0.162888i	−0.996678	−	0.0814442i	$-0.974047\pi$
$167$	− 351.532i	− 0.162888i	0.996678	−	0.0814442i	$-0.0259532\pi$
$168$	202.874i	0.0931673i
$169$	−1737.76	−0.790971
$170$	0	0
$171$	−251.979	−0.112686
$172$	1747.85i	0.774838i
$173$	3701.77i	1.62682i	0.581690	+	0.813411i	$0.302392\pi$
$173$	3701.77i	1.62682i	−0.581690	+	0.813411i	$0.697608\pi$
$174$	−357.937	−0.155949
$175$	0	0
$176$	−89.3150	−0.0382521
$177$	4013.58i	1.70440i
$178$	2467.78i	1.03915i
$179$	−3584.09	−1.49658	−0.748290	−	0.663372i	$-0.769125\pi$
$179$	−3584.09	−1.49658	−0.748290	+	0.663372i	$0.769125\pi$
$180$	0	0
$181$	783.672	0.321822	0.160911	−	0.986969i	$-0.448557\pi$
$181$	783.672	0.321822	0.160911	+	0.986969i	$0.448557\pi$
$182$	631.130i	0.257046i
$183$	− 2866.16i	− 1.15777i
$184$	184.000	0.0737210
$185$	0	0
$186$	−2843.48	−1.12093
$187$	110.212i	0.0430989i
$188$	− 361.080i	− 0.140077i
$189$	−725.005	−0.279029
$190$	0	0
$191$	1491.95	0.565201	0.282601	−	0.959238i	$-0.408803\pi$
$191$	1491.95	0.565201	0.282601	+	0.959238i	$0.408803\pi$
$192$	− 322.618i	− 0.121265i
$193$	− 3091.09i	− 1.15286i	−0.817147	−	0.576430i	$-0.804446\pi$
$193$	− 3091.09i	− 1.15286i	0.817147	−	0.576430i	$-0.195554\pi$
$194$	212.778	0.0787454
$195$	0	0
$196$	−1270.77	−0.463108
$197$	2820.30i	1.01999i	0.860178	+	0.509995i	$0.170353\pi$
$197$	2820.30i	1.01999i	−0.860178	+	0.509995i	$0.829647\pi$
$198$	− 17.7438i	− 0.00636866i
$199$	−1316.11	−0.468825	−0.234413	−	0.972137i	$-0.575317\pi$
$199$	−1316.11	−0.468825	−0.234413	+	0.972137i	$0.575317\pi$
$200$	0	0
$201$	−429.192	−0.150611
$202$	1285.35i	0.447708i
$203$	178.607i	0.0617524i
$204$	−398.100	−0.136630
$205$	0	0
$206$	3243.98	1.09718
$207$	36.5544i	0.0122739i
$208$	− 1003.64i	− 0.334568i
$209$	885.028	0.292912
$210$	0	0
$211$	−3399.19	−1.10905	−0.554525	−	0.832167i	$-0.687100\pi$
$211$	−3399.19	−1.10905	−0.554525	+	0.832167i	$0.687100\pi$
$212$	1320.73i	0.427869i
$213$	− 1863.00i	− 0.599300i
$214$	−2981.72	−0.952458
$215$	0	0
$216$	1152.93	0.363180
$217$	1418.86i	0.443865i
$218$	− 2408.00i	− 0.748121i
$219$	−1563.62	−0.482466
$220$	0	0
$221$	−1238.46	−0.376960
$222$	1411.26i	0.426657i
$223$	− 864.660i	− 0.259650i	−0.991537	−	0.129825i	$-0.958558\pi$
$223$	− 864.660i	− 0.259650i	0.991537	−	0.129825i	$-0.0414416\pi$
$224$	−160.983	−0.0480184
$225$	0	0
$226$	553.543	0.162925
$227$	− 1979.40i	− 0.578754i	−0.957215	−	0.289377i	$-0.906552\pi$
$227$	− 1979.40i	− 0.578754i	0.957215	−	0.289377i	$-0.0934481\pi$
$228$	3196.84i	0.928578i
$229$	3113.53	0.898462	0.449231	−	0.893416i	$-0.351698\pi$
$229$	3113.53	0.898462	0.449231	+	0.893416i	$0.351698\pi$
$230$	0	0
$231$	141.560	0.0403203
$232$	− 284.026i	− 0.0803761i
$233$	− 6018.65i	− 1.69225i	−0.532983	−	0.846126i	$-0.678929\pi$
$233$	− 6018.65i	− 1.69225i	0.532983	−	0.846126i	$-0.321071\pi$
$234$	199.389	0.0557028
$235$	0	0
$236$	−3184.81	−0.878447
$237$	6681.49i	1.83126i
$238$	198.648i	0.0541026i
$239$	4224.88	1.14345	0.571725	−	0.820445i	$-0.306274\pi$
$239$	4224.88	1.14345	0.571725	+	0.820445i	$0.306274\pi$
$240$	0	0
$241$	1394.24	0.372658	0.186329	−	0.982487i	$-0.440341\pi$
$241$	1394.24	0.372658	0.186329	+	0.982487i	$0.440341\pi$
$242$	− 2599.68i	− 0.690552i
$243$	445.360i	0.117571i
$244$	2274.32	0.596715
$245$	0	0
$246$	−2295.43	−0.594923
$247$	9945.16i	2.56192i
$248$	− 2256.32i	− 0.577729i
$249$	−800.523	−0.203739
$250$	0	0
$251$	−5968.83	−1.50099	−0.750497	−	0.660874i	$-0.770186\pi$
$251$	−5968.83	−1.50099	−0.750497	+	0.660874i	$0.770186\pi$
$252$	− 31.9816i	− 0.00799466i
$253$	− 128.390i	− 0.0319045i
$254$	−3105.57	−0.767169
$255$	0	0
$256$	256.000	0.0625000
$257$	− 1636.06i	− 0.397100i	−0.980091	−	0.198550i	$-0.936377\pi$
$257$	− 1636.06i	− 0.397100i	0.980091	−	0.198550i	$-0.0636232\pi$
$258$	4405.37i	1.06305i
$259$	704.206	0.168947
$260$	0	0
$261$	56.4261	0.0133820
$262$	− 627.745i	− 0.148024i
$263$	− 2995.55i	− 0.702333i	−0.936313	−	0.351166i	$-0.885785\pi$
$263$	− 2995.55i	− 0.702333i	0.936313	−	0.351166i	$-0.114215\pi$
$264$	−225.114	−0.0524803
$265$	0	0
$266$	1595.19	0.367696
$267$	6219.92i	1.42567i
$268$	− 340.568i	− 0.0776250i
$269$	5190.17	1.17640	0.588198	−	0.808717i	$-0.299838\pi$
$269$	5190.17	1.17640	0.588198	+	0.808717i	$0.299838\pi$
$270$	0	0
$271$	6720.43	1.50641	0.753205	−	0.657786i	$-0.228507\pi$
$271$	6720.43	1.50641	0.753205	+	0.657786i	$0.228507\pi$
$272$	− 315.896i	− 0.0704192i
$273$	1590.73i	0.352657i
$274$	2132.18	0.470109
$275$	0	0
$276$	463.763	0.101142
$277$	− 6114.74i	− 1.32635i	−0.748464	−	0.663176i	$-0.769208\pi$
$277$	− 6114.74i	− 1.32635i	0.748464	−	0.663176i	$-0.230792\pi$
$278$	3189.58i	0.688123i
$279$	448.253	0.0961871
$280$	0	0
$281$	−3549.07	−0.753450	−0.376725	−	0.926325i	$-0.622950\pi$
$281$	−3549.07	−0.753450	−0.376725	+	0.926325i	$0.622950\pi$
$282$	− 910.085i	− 0.192180i
$283$	− 3103.11i	− 0.651806i	−0.945403	−	0.325903i	$-0.894332\pi$
$283$	− 3103.11i	− 0.651806i	0.945403	−	0.325903i	$-0.105668\pi$
$284$	1478.31	0.308879
$285$	0	0
$286$	−700.316	−0.144792
$287$	1145.39i	0.235576i
$288$	50.8583i	0.0104057i
$289$	4523.19	0.920658
$290$	0	0
$291$	536.297	0.108035
$292$	− 1240.75i	− 0.248662i
$293$	− 1218.51i	− 0.242957i	−0.992594	−	0.121478i	$-0.961237\pi$
$293$	− 1218.51i	− 0.242957i	0.992594	−	0.121478i	$-0.0387635\pi$
$294$	−3202.91	−0.635365
$295$	0	0
$296$	−1119.85	−0.219899
$297$	− 804.482i	− 0.157174i
$298$	2193.82i	0.426458i
$299$	1442.74	0.279049
$300$	0	0
$301$	2198.23	0.420943
$302$	5468.82i	1.04204i
$303$	3239.67i	0.614238i
$304$	−2536.72	−0.478588
$305$	0	0
$306$	62.7575	0.0117242
$307$	2564.45i	0.476745i	0.971174	+	0.238373i	$0.0766140\pi$
$307$	2564.45i	0.476745i	−0.971174	+	0.238373i	$0.923386\pi$
$308$	112.329i	0.0207811i
$309$	8176.30	1.50529
$310$	0	0
$311$	−92.6253	−0.0168884	−0.00844421	−	0.999964i	$-0.502688\pi$
$311$	−92.6253	−0.0168884	−0.00844421	+	0.999964i	$0.502688\pi$
$312$	− 2529.63i	− 0.459014i
$313$	4276.44i	0.772263i	0.922444	+	0.386132i	$0.126189\pi$
$313$	4276.44i	0.772263i	−0.922444	+	0.386132i	$0.873811\pi$
$314$	2879.25	0.517470
$315$	0	0
$316$	−5301.82	−0.943832
$317$	− 2964.24i	− 0.525199i	−0.964905	−	0.262600i	$-0.915420\pi$
$317$	− 2964.24i	− 0.525199i	0.964905	−	0.262600i	$-0.0845798\pi$
$318$	3328.84i	0.587018i
$319$	−198.186	−0.0347846
$320$	0	0
$321$	−7515.27	−1.30673
$322$	− 231.413i	− 0.0400501i
$323$	3130.23i	0.539229i
$324$	2734.25	0.468836
$325$	0	0
$326$	−5991.63	−1.01793
$327$	− 6069.24i	− 1.02639i
$328$	− 1821.44i	− 0.306623i
$329$	−454.123	−0.0760991
$330$	0	0
$331$	−1348.81	−0.223979	−0.111989	−	0.993709i	$-0.535722\pi$
$331$	−1348.81	−0.223979	−0.111989	+	0.993709i	$0.535722\pi$
$332$	− 635.223i	− 0.105007i
$333$	− 222.475i	− 0.0366113i
$334$	703.064	0.115180
$335$	0	0
$336$	−405.749	−0.0658792
$337$	− 5764.30i	− 0.931755i	−0.884849	−	0.465878i	$-0.845739\pi$
$337$	− 5764.30i	− 0.931755i	0.884849	−	0.465878i	$-0.154261\pi$
$338$	− 3475.53i	− 0.559301i
$339$	1395.18	0.223527
$340$	0	0
$341$	−1574.40	−0.250026
$342$	− 503.958i	− 0.0796810i
$343$	3323.75i	0.523223i
$344$	−3495.70	−0.547894
$345$	0	0
$346$	−7403.53	−1.15034
$347$	4066.16i	0.629057i	0.949248	+	0.314528i	$0.101846\pi$
$347$	4066.16i	0.629057i	−0.949248	+	0.314528i	$0.898154\pi$
$348$	− 715.875i	− 0.110273i
$349$	−8407.82	−1.28957	−0.644786	−	0.764363i	$-0.723053\pi$
$349$	−8407.82	−1.28957	−0.644786	+	0.764363i	$0.723053\pi$
$350$	0	0
$351$	9040.06	1.37471
$352$	− 178.630i	− 0.0270483i
$353$	− 256.068i	− 0.0386095i	−0.999814	−	0.0193047i	$-0.993855\pi$
$353$	− 256.068i	− 0.0386095i	0.999814	−	0.0193047i	$-0.00614527\pi$
$354$	−8027.16	−1.20519
$355$	0	0
$356$	−4935.56	−0.734787
$357$	500.682i	0.0742266i
$358$	− 7168.19i	− 1.05824i
$359$	12052.5	1.77189	0.885943	−	0.463794i	$-0.153512\pi$
$359$	12052.5	1.77189	0.885943	+	0.463794i	$0.153512\pi$
$360$	0	0
$361$	18277.5	2.66475
$362$	1567.34i	0.227563i
$363$	− 6552.36i	− 0.947410i
$364$	−1262.26	−0.181759
$365$	0	0
$366$	5732.31	0.818669
$367$	− 9245.40i	− 1.31500i	−0.753453	−	0.657502i	$-0.771613\pi$
$367$	− 9245.40i	− 1.31500i	0.753453	−	0.657502i	$-0.228387\pi$
$368$	368.000i	0.0521286i
$369$	361.857	0.0510502
$370$	0	0
$371$	1661.05	0.232446
$372$	− 5686.96i	− 0.792621i
$373$	869.265i	0.120667i	0.998178	+	0.0603335i	$0.0192164\pi$
$373$	869.265i	0.120667i	−0.998178	+	0.0603335i	$0.980784\pi$
$374$	−220.424	−0.0304755
$375$	0	0
$376$	722.161	0.0990495
$377$	− 2227.04i	− 0.304240i
$378$	− 1450.01i	− 0.197303i
$379$	−1278.76	−0.173313	−0.0866566	−	0.996238i	$-0.527618\pi$
$379$	−1278.76	−0.173313	−0.0866566	+	0.996238i	$0.527618\pi$
$380$	0	0
$381$	−7827.44	−1.05252
$382$	2983.89i	0.399658i
$383$	5343.75i	0.712932i	0.934308	+	0.356466i	$0.116018\pi$
$383$	5343.75i	0.712932i	−0.934308	+	0.356466i	$0.883982\pi$
$384$	645.235	0.0857475
$385$	0	0
$386$	6182.19	0.815194
$387$	− 694.473i	− 0.0912198i
$388$	425.557i	0.0556814i
$389$	708.346	0.0923254	0.0461627	−	0.998934i	$-0.485301\pi$
$389$	708.346	0.0923254	0.0461627	+	0.998934i	$0.485301\pi$
$390$	0	0
$391$	454.101	0.0587336
$392$	− 2541.54i	− 0.327467i
$393$	− 1582.20i	− 0.203083i
$394$	−5640.60	−0.721241
$395$	0	0
$396$	35.4875	0.00450332
$397$	− 5924.01i	− 0.748911i	−0.927245	−	0.374455i	$-0.877830\pi$
$397$	− 5924.01i	− 0.748911i	0.927245	−	0.374455i	$-0.122170\pi$
$398$	− 2632.21i	− 0.331510i
$399$	4020.59	0.504464
$400$	0	0
$401$	−11393.6	−1.41888	−0.709441	−	0.704765i	$-0.751052\pi$
$401$	−11393.6	−1.41888	−0.709441	+	0.704765i	$0.751052\pi$
$402$	− 858.384i	− 0.106498i
$403$	− 17691.8i	− 2.18682i
$404$	−2570.71	−0.316578
$405$	0	0
$406$	−357.214	−0.0436656
$407$	781.402i	0.0951663i
$408$	− 796.200i	− 0.0966122i
$409$	−13804.4	−1.66891	−0.834457	−	0.551073i	$-0.814218\pi$
$409$	−13804.4	−1.66891	−0.834457	+	0.551073i	$0.814218\pi$
$410$	0	0
$411$	5374.06	0.644971
$412$	6487.96i	0.775823i
$413$	4005.47i	0.477230i
$414$	−73.1087	−0.00867898
$415$	0	0
$416$	2007.29	0.236575
$417$	8039.17i	0.944076i
$418$	1770.06i	0.207120i
$419$	14795.8	1.72512	0.862558	−	0.505958i	$-0.168861\pi$
$419$	14795.8	1.72512	0.862558	+	0.505958i	$0.168861\pi$
$420$	0	0
$421$	5804.66	0.671976	0.335988	−	0.941866i	$-0.390930\pi$
$421$	5804.66	0.671976	0.335988	+	0.941866i	$0.390930\pi$
$422$	− 6798.37i	− 0.784217i
$423$	143.468i	0.0164909i
$424$	−2641.46	−0.302549
$425$	0	0
$426$	3726.01	0.423769
$427$	− 2860.36i	− 0.324175i
$428$	− 5963.44i	− 0.673490i
$429$	−1765.11	−0.198649
$430$	0	0
$431$	15268.2	1.70636	0.853181	−	0.521615i	$-0.174670\pi$
$431$	15268.2	1.70636	0.853181	+	0.521615i	$0.174670\pi$
$432$	2305.85i	0.256807i
$433$	− 2250.62i	− 0.249788i	−0.992170	−	0.124894i	$-0.960141\pi$
$433$	− 2250.62i	− 0.249788i	0.992170	−	0.124894i	$-0.0398591\pi$
$434$	−2837.73	−0.313860
$435$	0	0
$436$	4816.00	0.529001
$437$	− 3646.53i	− 0.399170i
$438$	− 3127.25i	− 0.341155i
$439$	−10366.2	−1.12700	−0.563499	−	0.826117i	$-0.690545\pi$
$439$	−10366.2	−1.12700	−0.563499	+	0.826117i	$0.690545\pi$
$440$	0	0
$441$	504.914	0.0545205
$442$	− 2476.93i	− 0.266551i
$443$	3881.11i	0.416246i	0.978103	+	0.208123i	$0.0667355\pi$
$443$	3881.11i	0.416246i	−0.978103	+	0.208123i	$0.933264\pi$
$444$	−2822.53	−0.301692
$445$	0	0
$446$	1729.32	0.183600
$447$	5529.41i	0.585083i
$448$	− 321.965i	− 0.0339541i
$449$	10146.5	1.06647	0.533233	−	0.845968i	$-0.320977\pi$
$449$	10146.5	1.06647	0.533233	+	0.845968i	$0.320977\pi$
$450$	0	0
$451$	−1270.95	−0.132698
$452$	1107.09i	0.115206i
$453$	13783.9i	1.42963i
$454$	3958.79	0.409241
$455$	0	0
$456$	−6393.68	−0.656604
$457$	− 4728.87i	− 0.484042i	−0.970271	−	0.242021i	$-0.922190\pi$
$457$	− 4728.87i	− 0.484042i	0.970271	−	0.242021i	$-0.0778103\pi$
$458$	6227.06i	0.635308i
$459$	2845.35	0.289346
$460$	0	0
$461$	8380.69	0.846698	0.423349	−	0.905967i	$-0.360854\pi$
$461$	8380.69	0.846698	0.423349	+	0.905967i	$0.360854\pi$
$462$	283.121i	0.0285108i
$463$	− 6845.17i	− 0.687089i	−0.939136	−	0.343545i	$-0.888372\pi$
$463$	− 6845.17i	− 0.687089i	0.939136	−	0.343545i	$-0.111628\pi$
$464$	568.053	0.0568345
$465$	0	0
$466$	12037.3	1.19660
$467$	− 14884.0i	− 1.47484i	−0.675436	−	0.737418i	$-0.736045\pi$
$467$	− 14884.0i	− 1.47484i	0.675436	−	0.737418i	$-0.263955\pi$
$468$	398.778i	0.0393878i
$469$	−428.324	−0.0421710
$470$	0	0
$471$	7257.01	0.709948
$472$	− 6369.62i	− 0.621156i
$473$	2439.20i	0.237114i
$474$	−13363.0	−1.29490
$475$	0	0
$476$	−397.295	−0.0382563
$477$	− 524.766i	− 0.0503719i
$478$	8449.76i	0.808542i
$479$	−6048.69	−0.576976	−0.288488	−	0.957483i	$-0.593153\pi$
$479$	−6048.69	−0.576976	−0.288488	+	0.957483i	$0.593153\pi$
$480$	0	0
$481$	−8780.71	−0.832361
$482$	2788.47i	0.263509i
$483$	− 583.264i	− 0.0549471i
$484$	5199.36	0.488294
$485$	0	0
$486$	−890.720	−0.0831355
$487$	3199.35i	0.297693i	0.988860	+	0.148846i	$0.0475560\pi$
$487$	3199.35i	0.297693i	−0.988860	+	0.148846i	$0.952444\pi$
$488$	4548.64i	0.421941i
$489$	−15101.6	−1.39656
$490$	0	0
$491$	−1614.41	−0.148385	−0.0741926	−	0.997244i	$-0.523638\pi$
$491$	−1614.41	−0.148385	−0.0741926	+	0.997244i	$0.523638\pi$
$492$	− 4590.85i	− 0.420674i
$493$	− 700.960i	− 0.0640358i
$494$	−19890.3	−1.81155
$495$	0	0
$496$	4512.65	0.408516
$497$	− 1859.24i	− 0.167803i
$498$	− 1601.05i	− 0.144066i
$499$	5392.04	0.483729	0.241864	−	0.970310i	$-0.422241\pi$
$499$	5392.04	0.483729	0.241864	+	0.970310i	$0.422241\pi$
$500$	0	0
$501$	1772.04	0.158022
$502$	− 11937.7i	− 1.06136i
$503$	− 12625.4i	− 1.11917i	−0.828775	−	0.559583i	$-0.810961\pi$
$503$	− 12625.4i	− 1.11917i	0.828775	−	0.559583i	$-0.189039\pi$
$504$	63.9633	0.00565308
$505$	0	0
$506$	256.781	0.0225599
$507$	− 8759.89i	− 0.767338i
$508$	− 6211.15i	− 0.542471i
$509$	−7065.76	−0.615294	−0.307647	−	0.951501i	$-0.599541\pi$
$509$	−7065.76	−0.615294	−0.307647	+	0.951501i	$0.599541\pi$
$510$	0	0
$511$	−1560.46	−0.135090
$512$	512.000i	0.0441942i
$513$	− 22848.9i	− 1.96647i
$514$	3272.12	0.280792
$515$	0	0
$516$	−8810.73	−0.751688
$517$	− 503.905i	− 0.0428660i
$518$	1408.41i	0.119463i
$519$	−18660.2	−1.57822
$520$	0	0
$521$	−8399.20	−0.706287	−0.353144	−	0.935569i	$-0.614887\pi$
$521$	−8399.20	−0.706287	−0.353144	+	0.935569i	$0.614887\pi$
$522$	112.852i	0.00946247i
$523$	7002.06i	0.585428i	0.956200	+	0.292714i	$0.0945584\pi$
$523$	7002.06i	0.585428i	−0.956200	+	0.292714i	$0.905442\pi$
$524$	1255.49	0.104669
$525$	0	0
$526$	5991.10	0.496624
$527$	− 5568.47i	− 0.460278i
$528$	− 450.228i	− 0.0371092i
$529$	−529.000	−0.0434783
$530$	0	0
$531$	1265.42	0.103417
$532$	3190.37i	0.260001i
$533$	− 14281.9i	− 1.16063i
$534$	−12439.8	−1.00810
$535$	0	0
$536$	681.136	0.0548891
$537$	− 18067.1i	− 1.45186i
$538$	10380.3i	0.831838i
$539$	−1773.42	−0.141719
$540$	0	0
$541$	3338.13	0.265282	0.132641	−	0.991164i	$-0.457654\pi$
$541$	3338.13	0.265282	0.132641	+	0.991164i	$0.457654\pi$
$542$	13440.9i	1.06519i
$543$	3950.41i	0.312207i
$544$	631.792	0.0497939
$545$	0	0
$546$	−3181.46	−0.249366
$547$	9609.09i	0.751106i	0.926801	+	0.375553i	$0.122547\pi$
$547$	9609.09i	0.751106i	−0.926801	+	0.375553i	$0.877453\pi$
$548$	4264.37i	0.332417i
$549$	−903.656	−0.0702497
$550$	0	0
$551$	−5628.87	−0.435205
$552$	927.526i	0.0715183i
$553$	6667.98i	0.512751i
$554$	12229.5	0.937872
$555$	0	0
$556$	−6379.15	−0.486576
$557$	− 11931.1i	− 0.907608i	−0.891102	−	0.453804i	$-0.850067\pi$
$557$	− 11931.1i	− 0.907608i	0.891102	−	0.453804i	$-0.149933\pi$
$558$	896.506i	0.0680146i
$559$	−27409.6	−2.07389
$560$	0	0
$561$	−555.567	−0.0418112
$562$	− 7098.13i	− 0.532770i
$563$	10082.4i	0.754746i	0.926061	+	0.377373i	$0.123173\pi$
$563$	10082.4i	0.754746i	−0.926061	+	0.377373i	$0.876827\pi$
$564$	1820.17	0.135892
$565$	0	0
$566$	6206.23	0.460896
$567$	− 3438.80i	− 0.254702i
$568$	2956.62i	0.218410i
$569$	18710.5	1.37853	0.689267	−	0.724507i	$-0.257933\pi$
$569$	18710.5	1.37853	0.689267	+	0.724507i	$0.257933\pi$
$570$	0	0
$571$	13510.1	0.990159	0.495079	−	0.868848i	$-0.335139\pi$
$571$	13510.1	0.990159	0.495079	+	0.868848i	$0.335139\pi$
$572$	− 1400.63i	− 0.102383i
$573$	7520.75i	0.548314i
$574$	−2290.79	−0.166578
$575$	0	0
$576$	−101.717	−0.00735797
$577$	11582.3i	0.835662i	0.908525	+	0.417831i	$0.137210\pi$
$577$	11582.3i	0.835662i	−0.908525	+	0.417831i	$0.862790\pi$
$578$	9046.39i	0.651004i
$579$	15581.9	1.11841
$580$	0	0
$581$	−798.905	−0.0570468
$582$	1072.59i	0.0763926i
$583$	1843.14i	0.130935i
$584$	2481.50	0.175831
$585$	0	0
$586$	2437.03	0.171796
$587$	− 6019.34i	− 0.423245i	−0.977351	−	0.211623i	$-0.932125\pi$
$587$	− 6019.34i	− 0.423245i	0.977351	−	0.211623i	$-0.0678747\pi$
$588$	− 6405.81i	− 0.449271i
$589$	−44716.1	−3.12818
$590$	0	0
$591$	−14216.8	−0.989514
$592$	− 2239.70i	− 0.155492i
$593$	− 26299.3i	− 1.82122i	−0.413270	−	0.910608i	$-0.635613\pi$
$593$	− 26299.3i	− 1.82122i	0.413270	−	0.910608i	$-0.364387\pi$
$594$	1608.96	0.111139
$595$	0	0
$596$	−4387.63	−0.301551
$597$	− 6634.36i	− 0.454818i
$598$	2885.47i	0.197317i
$599$	27368.0	1.86682	0.933409	−	0.358813i	$-0.116819\pi$
$599$	27368.0	1.86682	0.933409	+	0.358813i	$0.116819\pi$
$600$	0	0
$601$	−3472.33	−0.235673	−0.117836	−	0.993033i	$-0.537596\pi$
$601$	−3472.33	−0.235673	−0.117836	+	0.993033i	$0.537596\pi$
$602$	4396.46i	0.297652i
$603$	135.318i	0.00913859i
$604$	−10937.6	−0.736832
$605$	0	0
$606$	−6479.33	−0.434332
$607$	5061.10i	0.338425i	0.985580	+	0.169212i	$0.0541224\pi$
$607$	5061.10i	0.338425i	−0.985580	+	0.169212i	$0.945878\pi$
$608$	− 5073.44i	− 0.338413i
$609$	−900.339	−0.0599074
$610$	0	0
$611$	5662.44	0.374922
$612$	125.515i	0.00829027i
$613$	28272.6i	1.86283i	0.363953	+	0.931417i	$0.381427\pi$
$613$	28272.6i	1.86283i	−0.363953	+	0.931417i	$0.618573\pi$
$614$	−5128.90	−0.337110
$615$	0	0
$616$	−224.659	−0.0146944
$617$	− 2222.47i	− 0.145013i	−0.997368	−	0.0725066i	$-0.976900\pi$
$617$	− 2222.47i	− 0.145013i	0.997368	−	0.0725066i	$-0.0230999\pi$
$618$	16352.6i	1.06440i
$619$	10498.2	0.681679	0.340840	−	0.940121i	$-0.389289\pi$
$619$	10498.2	0.681679	0.340840	+	0.940121i	$0.389289\pi$
$620$	0	0
$621$	−3314.67	−0.214192
$622$	− 185.251i	− 0.0119419i
$623$	6207.34i	0.399185i
$624$	5059.27	0.324572
$625$	0	0
$626$	−8552.87	−0.546073
$627$	4461.34i	0.284160i
$628$	5758.51i	0.365907i
$629$	−2763.72	−0.175194
$630$	0	0
$631$	−11402.2	−0.719355	−0.359678	−	0.933077i	$-0.617113\pi$
$631$	−11402.2	−0.719355	−0.359678	+	0.933077i	$0.617113\pi$
$632$	− 10603.6i	− 0.667390i
$633$	− 17135.0i	− 1.07591i
$634$	5928.47	0.371372
$635$	0	0
$636$	−6657.67	−0.415085
$637$	− 19928.1i	− 1.23953i
$638$	− 396.372i	− 0.0245964i
$639$	−587.377	−0.0363635
$640$	0	0
$641$	26128.3	1.60999	0.804996	−	0.593280i	$-0.202167\pi$
$641$	26128.3	1.60999	0.804996	+	0.593280i	$0.202167\pi$
$642$	− 15030.5i	− 0.924001i
$643$	− 863.502i	− 0.0529599i	−0.999649	−	0.0264799i	$-0.991570\pi$
$643$	− 863.502i	− 0.0529599i	0.999649	−	0.0264799i	$-0.00842981\pi$
$644$	462.825	0.0283197
$645$	0	0
$646$	−6260.47	−0.381292
$647$	− 5477.43i	− 0.332828i	−0.986056	−	0.166414i	$-0.946781\pi$
$647$	− 5477.43i	− 0.332828i	0.986056	−	0.166414i	$-0.0532188\pi$
$648$	5468.50i	0.331517i
$649$	−4444.55	−0.268820
$650$	0	0
$651$	−7152.36	−0.430604
$652$	− 11983.3i	− 0.719787i
$653$	− 19233.2i	− 1.15261i	−0.817235	−	0.576305i	$-0.804494\pi$
$653$	− 19233.2i	− 1.15261i	0.817235	−	0.576305i	$-0.195506\pi$
$654$	12138.5	0.725768
$655$	0	0
$656$	3642.88	0.216815
$657$	492.987i	0.0292744i
$658$	− 908.245i	− 0.0538102i
$659$	−9952.51	−0.588307	−0.294154	−	0.955758i	$-0.595038\pi$
$659$	−9952.51	−0.588307	−0.294154	+	0.955758i	$0.595038\pi$
$660$	0	0
$661$	3047.96	0.179352	0.0896762	−	0.995971i	$-0.471417\pi$
$661$	3047.96	0.179352	0.0896762	+	0.995971i	$0.471417\pi$
$662$	− 2697.61i	− 0.158377i
$663$	− 6242.98i	− 0.365697i
$664$	1270.45	0.0742513
$665$	0	0
$666$	444.950	0.0258881
$667$	816.576i	0.0474032i
$668$	1406.13i	0.0814442i
$669$	4358.67	0.251892
$670$	0	0
$671$	3173.92	0.182605
$672$	− 811.498i	− 0.0465836i
$673$	− 16245.9i	− 0.930508i	−0.885177	−	0.465254i	$-0.845963\pi$
$673$	− 16245.9i	− 0.930508i	0.885177	−	0.465254i	$-0.154037\pi$
$674$	11528.6	0.658850
$675$	0	0
$676$	6951.05	0.395486
$677$	− 30056.6i	− 1.70631i	−0.521659	−	0.853154i	$-0.674687\pi$
$677$	− 30056.6i	− 1.70631i	0.521659	−	0.853154i	$-0.325313\pi$
$678$	2790.35i	0.158057i
$679$	535.213	0.0302498
$680$	0	0
$681$	9977.94	0.561462
$682$	− 3148.81i	− 0.176795i
$683$	− 10064.8i	− 0.563866i	−0.959434	−	0.281933i	$-0.909024\pi$
$683$	− 10064.8i	− 0.563866i	0.959434	−	0.281933i	$-0.0909755\pi$
$684$	1007.92	0.0563430
$685$	0	0
$686$	−6647.50	−0.369975
$687$	15695.0i	0.871617i
$688$	− 6991.39i	− 0.387419i
$689$	−20711.6	−1.14521
$690$	0	0
$691$	10260.7	0.564886	0.282443	−	0.959284i	$-0.408855\pi$
$691$	10260.7	0.564886	0.282443	+	0.959284i	$0.408855\pi$
$692$	− 14807.1i	− 0.813411i
$693$	− 44.6319i	− 0.00244650i
$694$	−8132.31	−0.444810
$695$	0	0
$696$	1431.75	0.0779746
$697$	− 4495.21i	− 0.244287i
$698$	− 16815.6i	− 0.911865i
$699$	30339.4	1.64169
$700$	0	0
$701$	−19815.6	−1.06766	−0.533828	−	0.845593i	$-0.679247\pi$
$701$	−19815.6	−1.06766	−0.533828	+	0.845593i	$0.679247\pi$
$702$	18080.1i	0.972066i
$703$	22193.3i	1.19067i
$704$	357.260	0.0191261
$705$	0	0
$706$	512.136	0.0273010
$707$	3233.12i	0.171986i
$708$	− 16054.3i	− 0.852201i
$709$	30533.5	1.61736	0.808681	−	0.588247i	$-0.200182\pi$
$709$	30533.5	1.61736	0.808681	+	0.588247i	$0.200182\pi$
$710$	0	0
$711$	2106.57	0.111115
$712$	− 9871.12i	− 0.519573i
$713$	6486.93i	0.340726i
$714$	−1001.36	−0.0524861
$715$	0	0
$716$	14336.4	0.748290
$717$	21297.2i	1.10929i
$718$	24105.0i	1.25291i
$719$	−9125.45	−0.473327	−0.236663	−	0.971592i	$-0.576054\pi$
$719$	−9125.45	−0.473327	−0.236663	+	0.971592i	$0.576054\pi$
$720$	0	0
$721$	8159.77	0.421478
$722$	36555.0i	1.88426i
$723$	7028.21i	0.361524i
$724$	−3134.69	−0.160911
$725$	0	0
$726$	13104.7	0.669920
$727$	2635.53i	0.134452i	0.997738	+	0.0672259i	$0.0214148\pi$
$727$	2635.53i	0.134452i	−0.997738	+	0.0672259i	$0.978585\pi$
$728$	− 2524.52i	− 0.128523i
$729$	−20701.2	−1.05173
$730$	0	0
$731$	−8627.17	−0.436508
$732$	11464.6i	0.578886i
$733$	− 10778.0i	− 0.543102i	−0.962424	−	0.271551i	$-0.912463\pi$
$733$	− 10778.0i	− 0.543102i	0.962424	−	0.271551i	$-0.0875366\pi$
$734$	18490.8	0.929848
$735$	0	0
$736$	−736.000	−0.0368605
$737$	− 475.278i	− 0.0237545i
$738$	723.714i	0.0360979i
$739$	1758.72	0.0875449	0.0437725	−	0.999042i	$-0.486062\pi$
$739$	1758.72	0.0875449	0.0437725	+	0.999042i	$0.486062\pi$
$740$	0	0
$741$	−50132.6	−2.48538
$742$	3322.11i	0.164364i
$743$	− 9737.97i	− 0.480823i	−0.970671	−	0.240412i	$-0.922718\pi$
$743$	− 9737.97i	− 0.480823i	0.970671	−	0.240412i	$-0.0772824\pi$
$744$	11373.9	0.560467
$745$	0	0
$746$	−1738.53	−0.0853245
$747$	252.393i	0.0123622i
$748$	− 440.848i	− 0.0215495i
$749$	−7500.08	−0.365884
$750$	0	0
$751$	−23795.7	−1.15621	−0.578107	−	0.815961i	$-0.696208\pi$
$751$	−23795.7	−1.15621	−0.578107	+	0.815961i	$0.696208\pi$
$752$	1444.32i	0.0700385i
$753$	− 30088.3i	− 1.45615i
$754$	4454.08	0.215130
$755$	0	0
$756$	2900.02	0.139514
$757$	23647.6i	1.13538i	0.823241	+	0.567692i	$0.192163\pi$
$757$	23647.6i	1.13538i	−0.823241	+	0.567692i	$0.807837\pi$
$758$	− 2557.53i	− 0.122551i
$759$	647.203	0.0309512
$760$	0	0
$761$	−4606.09	−0.219410	−0.109705	−	0.993964i	$-0.534991\pi$
$761$	−4606.09	−0.219410	−0.109705	+	0.993964i	$0.534991\pi$
$762$	− 15654.9i	− 0.744248i
$763$	− 6056.98i	− 0.287388i
$764$	−5967.79	−0.282601
$765$	0	0
$766$	−10687.5	−0.504119
$767$	− 49944.0i	− 2.35120i
$768$	1290.47i	0.0606326i
$769$	−17760.9	−0.832867	−0.416433	−	0.909166i	$-0.636720\pi$
$769$	−17760.9	−0.832867	−0.416433	+	0.909166i	$0.636720\pi$
$770$	0	0
$771$	8247.22	0.385235
$772$	12364.4i	0.576430i
$773$	− 7147.47i	− 0.332570i	−0.986078	−	0.166285i	$-0.946823\pi$
$773$	− 7147.47i	− 0.332570i	0.986078	−	0.166285i	$-0.0531772\pi$
$774$	1388.95	0.0645021
$775$	0	0
$776$	−851.114	−0.0393727
$777$	3549.83i	0.163899i
$778$	1416.69i	0.0652839i
$779$	−36097.6	−1.66024
$780$	0	0
$781$	2063.05	0.0945222
$782$	908.201i	0.0415310i
$783$	5116.59i	0.233528i
$784$	5083.07	0.231554
$785$	0	0
$786$	3164.40	0.143601
$787$	− 31719.1i	− 1.43668i	−0.695694	−	0.718338i	$-0.744903\pi$
$787$	− 31719.1i	− 1.43668i	0.695694	−	0.718338i	$-0.255097\pi$
$788$	− 11281.2i	− 0.509995i
$789$	15100.3	0.681348
$790$	0	0
$791$	1392.36	0.0625872
$792$	70.9751i	0.00318433i
$793$	35665.7i	1.59713i
$794$	11848.0	0.529560
$795$	0	0
$796$	5264.42	0.234413
$797$	30828.3i	1.37013i	0.728481	+	0.685066i	$0.240227\pi$
$797$	30828.3i	1.37013i	−0.728481	+	0.685066i	$0.759773\pi$
$798$	8041.18i	0.356710i
$799$	1782.25	0.0789129
$800$	0	0
$801$	1961.05	0.0865046
$802$	− 22787.3i	− 1.00330i
$803$	− 1731.53i	− 0.0760949i
$804$	1716.77	0.0753057
$805$	0	0
$806$	35383.5	1.54632
$807$	26163.2i	1.14125i
$808$	− 5141.41i	− 0.223854i
$809$	−36679.2	−1.59403	−0.797015	−	0.603959i	$-0.793589\pi$
$809$	−36679.2	−1.59403	−0.797015	+	0.603959i	$0.793589\pi$
$810$	0	0
$811$	−12210.0	−0.528672	−0.264336	−	0.964431i	$-0.585153\pi$
$811$	−12210.0	−0.528672	−0.264336	+	0.964431i	$0.585153\pi$
$812$	− 714.427i	− 0.0308762i
$813$	33877.0i	1.46140i
$814$	−1562.80	−0.0672927
$815$	0	0
$816$	1592.40	0.0683152
$817$	69278.2i	2.96663i
$818$	− 27608.9i	− 1.18010i
$819$	501.534	0.0213981
$820$	0	0
$821$	−9996.47	−0.424944	−0.212472	−	0.977167i	$-0.568151\pi$
$821$	−9996.47	−0.424944	−0.212472	+	0.977167i	$0.568151\pi$
$822$	10748.1i	0.456063i
$823$	42868.8i	1.81569i	0.419306	+	0.907845i	$0.362273\pi$
$823$	42868.8i	1.81569i	−0.419306	+	0.907845i	$0.637727\pi$
$824$	−12975.9	−0.548590
$825$	0	0
$826$	−8010.93	−0.337453
$827$	− 15833.4i	− 0.665759i	−0.942969	−	0.332880i	$-0.891980\pi$
$827$	− 15833.4i	− 0.665759i	0.942969	−	0.332880i	$-0.108020\pi$
$828$	− 146.217i	− 0.00613697i
$829$	17722.3	0.742484	0.371242	−	0.928536i	$-0.378932\pi$
$829$	17722.3	0.742484	0.371242	+	0.928536i	$0.378932\pi$
$830$	0	0
$831$	30823.8	1.28672
$832$	4014.57i	0.167284i
$833$	− 6272.35i	− 0.260893i
$834$	−16078.3	−0.667563
$835$	0	0
$836$	−3540.11	−0.146456
$837$	40646.5i	1.67855i
$838$	29591.7i	1.21984i
$839$	40528.4	1.66769	0.833847	−	0.551996i	$-0.186134\pi$
$839$	40528.4	1.66769	0.833847	+	0.551996i	$0.186134\pi$
$840$	0	0
$841$	−23128.5	−0.948317
$842$	11609.3i	0.475159i
$843$	− 17890.5i	− 0.730938i
$844$	13596.7	0.554525
$845$	0	0
$846$	−286.936	−0.0116608
$847$	− 6539.11i	− 0.265273i
$848$	− 5282.92i	− 0.213934i
$849$	15642.5	0.632331
$850$	0	0
$851$	3219.57	0.129689
$852$	7452.02i	0.299650i
$853$	− 8116.97i	− 0.325814i	−0.986641	−	0.162907i	$-0.947913\pi$
$853$	− 8116.97i	− 0.325814i	0.986641	−	0.162907i	$-0.0520871\pi$
$854$	5720.73	0.229226
$855$	0	0
$856$	11926.9	0.476229
$857$	16285.5i	0.649126i	0.945864	+	0.324563i	$0.105217\pi$
$857$	16285.5i	0.649126i	−0.945864	+	0.324563i	$0.894783\pi$
$858$	− 3530.22i	− 0.140466i
$859$	26487.5	1.05209	0.526043	−	0.850458i	$-0.323675\pi$
$859$	26487.5	1.05209	0.526043	+	0.850458i	$0.323675\pi$
$860$	0	0
$861$	−5773.81	−0.228538
$862$	30536.3i	1.20658i
$863$	− 40396.2i	− 1.59340i	−0.604376	−	0.796699i	$-0.706578\pi$
$863$	− 40396.2i	− 1.59340i	0.604376	−	0.796699i	$-0.293422\pi$
$864$	−4611.71	−0.181590
$865$	0	0
$866$	4501.25	0.176627
$867$	22801.0i	0.893151i
$868$	− 5675.46i	− 0.221933i
$869$	−7398.94	−0.288828
$870$	0	0
$871$	5340.76	0.207767
$872$	9632.00i	0.374060i
$873$	− 169.087i	− 0.00655523i
$874$	7293.07	0.282256
$875$	0	0
$876$	6254.50	0.241233
$877$	10962.9i	0.422109i	0.977474	+	0.211054i	$0.0676897\pi$
$877$	10962.9i	0.422109i	−0.977474	+	0.211054i	$0.932310\pi$
$878$	− 20732.4i	− 0.796908i
$879$	6142.40	0.235698
$880$	0	0
$881$	23906.5	0.914222	0.457111	−	0.889410i	$-0.348884\pi$
$881$	23906.5	0.914222	0.457111	+	0.889410i	$0.348884\pi$
$882$	1009.83i	0.0385518i
$883$	30509.1i	1.16275i	0.813634	+	0.581377i	$0.197486\pi$
$883$	30509.1i	1.16275i	−0.813634	+	0.581377i	$0.802514\pi$
$884$	4953.86	0.188480
$885$	0	0
$886$	−7762.22	−0.294331
$887$	− 40034.3i	− 1.51547i	−0.652565	−	0.757733i	$-0.726307\pi$
$887$	− 40034.3i	− 1.51547i	0.652565	−	0.757733i	$-0.273693\pi$
$888$	− 5645.06i	− 0.213328i
$889$	−7811.62	−0.294706
$890$	0	0
$891$	3815.77	0.143472
$892$	3458.64i	0.129825i
$893$	− 14311.9i	− 0.536314i
$894$	−11058.8	−0.413716
$895$	0	0
$896$	643.931	0.0240092
$897$	7272.69i	0.270711i
$898$	20293.0i	0.754106i
$899$	10013.4	0.371485
$900$	0	0
$901$	−6518.96	−0.241041
$902$	− 2541.91i	− 0.0938318i
$903$	11081.1i	0.408366i
$904$	−2214.17	−0.0814626
$905$	0	0
$906$	−27567.8	−1.01090
$907$	− 2843.48i	− 0.104097i	−0.998645	−	0.0520486i	$-0.983425\pi$
$907$	− 2843.48i	− 0.104097i	0.998645	−	0.0520486i	$-0.0165751\pi$
$908$	7917.58i	0.289377i
$909$	1021.42	0.0372699
$910$	0	0
$911$	42268.6	1.53723	0.768617	−	0.639710i	$-0.220945\pi$
$911$	42268.6	1.53723	0.768617	+	0.639710i	$0.220945\pi$
$912$	− 12787.4i	− 0.464289i
$913$	− 886.483i	− 0.0321340i
$914$	9457.74	0.342269
$915$	0	0
$916$	−12454.1	−0.449231
$917$	− 1579.00i	− 0.0568629i
$918$	5690.71i	0.204598i
$919$	12699.6	0.455844	0.227922	−	0.973679i	$-0.426807\pi$
$919$	12699.6	0.455844	0.227922	+	0.973679i	$0.426807\pi$
$920$	0	0
$921$	−12927.1	−0.462501
$922$	16761.4i	0.598706i
$923$	23182.8i	0.826728i
$924$	−566.242	−0.0201602
$925$	0	0
$926$	13690.3	0.485845
$927$	− 2577.86i	− 0.0913357i
$928$	1136.11i	0.0401880i
$929$	24889.6	0.879011	0.439505	−	0.898240i	$-0.355154\pi$
$929$	24889.6	0.879011	0.439505	+	0.898240i	$0.355154\pi$
$930$	0	0
$931$	−50368.5	−1.77310
$932$	24074.6i	0.846126i
$933$	− 466.915i	− 0.0163838i
$934$	29768.0	1.04287
$935$	0	0
$936$	−797.555	−0.0278514
$937$	9585.48i	0.334199i	0.985940	+	0.167099i	$0.0534401\pi$
$937$	9585.48i	0.334199i	−0.985940	+	0.167099i	$0.946560\pi$
$938$	− 856.649i	− 0.0298194i
$939$	−21557.1	−0.749189
$940$	0	0
$941$	−21146.9	−0.732592	−0.366296	−	0.930498i	$-0.619374\pi$
$941$	−21146.9	−0.732592	−0.366296	+	0.930498i	$0.619374\pi$
$942$	14514.0i	0.502009i
$943$	5236.65i	0.180836i
$944$	12739.2	0.439224
$945$	0	0
$946$	−4878.41	−0.167665
$947$	48747.2i	1.67272i	0.548177	+	0.836362i	$0.315322\pi$
$947$	48747.2i	1.67272i	−0.548177	+	0.836362i	$0.684678\pi$
$948$	− 26726.0i	− 0.915632i
$949$	19457.3	0.665556
$950$	0	0
$951$	14942.4	0.509507
$952$	− 794.591i	− 0.0270513i
$953$	− 24925.9i	− 0.847249i	−0.905838	−	0.423625i	$-0.860758\pi$
$953$	− 24925.9i	− 0.847249i	0.905838	−	0.423625i	$-0.139242\pi$
$954$	1049.53	0.0356183
$955$	0	0
$956$	−16899.5	−0.571725
$957$	− 999.036i	− 0.0337453i
$958$	− 12097.4i	− 0.407984i
$959$	5363.20	0.180591
$960$	0	0
$961$	49755.9	1.67017
$962$	− 17561.4i	− 0.588568i
$963$	2369.45i	0.0792882i
$964$	−5576.95	−0.186329
$965$	0	0
$966$	1166.53	0.0388534
$967$	− 1881.92i	− 0.0625838i	−0.999510	−	0.0312919i	$-0.990038\pi$
$967$	− 1881.92i	− 0.0625838i	0.999510	−	0.0312919i	$-0.00996215\pi$
$968$	10398.7i	0.345276i
$969$	−15779.2	−0.523117
$970$	0	0
$971$	−14637.3	−0.483763	−0.241881	−	0.970306i	$-0.577764\pi$
$971$	−14637.3	−0.483763	−0.241881	+	0.970306i	$0.577764\pi$
$972$	− 1781.44i	− 0.0587857i
$973$	8022.92i	0.264340i
$974$	−6398.70	−0.210501
$975$	0	0
$976$	−9097.28	−0.298358
$977$	42591.4i	1.39470i	0.716733	+	0.697348i	$0.245637\pi$
$977$	42591.4i	1.39470i	−0.716733	+	0.697348i	$0.754363\pi$
$978$	− 30203.2i	− 0.987518i
$979$	−6887.81	−0.224857
$980$	0	0
$981$	−1913.54	−0.0622780
$982$	− 3228.81i	− 0.104924i
$983$	48858.3i	1.58529i	0.609684	+	0.792644i	$0.291296\pi$
$983$	48858.3i	1.58529i	−0.609684	+	0.792644i	$0.708704\pi$
$984$	9181.71	0.297462
$985$	0	0
$986$	1401.92	0.0452801
$987$	− 2289.19i	− 0.0738254i
$988$	− 39780.6i	− 1.28096i
$989$	10050.1	0.323130
$990$	0	0
$991$	−56522.1	−1.81179	−0.905895	−	0.423502i	$-0.860801\pi$
$991$	−56522.1	−1.81179	−0.905895	+	0.423502i	$0.860801\pi$
$992$	9025.30i	0.288864i
$993$	− 6799.19i	− 0.217287i
$994$	3718.48	0.118655
$995$	0	0
$996$	3202.09	0.101870
$997$	− 20767.9i	− 0.659706i	−0.944032	−	0.329853i	$-0.893001\pi$
$997$	− 20767.9i	− 0.659706i	0.944032	−	0.329853i	$-0.106999\pi$
$998$	10784.1i	0.342048i
$999$	20173.5	0.638902

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1150.4.b.o.599.8		8
5.2	odd	4		1150.4.a.n.1.4		4
5.3	odd	4		230.4.a.j.1.1	✓	4
5.4	even	2	inner	1150.4.b.o.599.1		8
15.8	even	4		2070.4.a.bg.1.3		4
20.3	even	4		1840.4.a.k.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.4.a.j.1.1	✓	4	5.3	odd	4
1150.4.a.n.1.4		4	5.2	odd	4
1150.4.b.o.599.1		8	5.4	even	2	inner
1150.4.b.o.599.8		8	1.1	even	1	trivial
1840.4.a.k.1.4		4	20.3	even	4
2070.4.a.bg.1.3		4	15.8	even	4

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:	\( N \)	\(=\)	\( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1150.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{2} +5.04090i q^{3} -4.00000 q^{4} -10.0818 q^{6} +5.03071i q^{7} -8.00000i q^{8} +1.58932 q^{9} -5.58219 q^{11} -20.1636i q^{12} -62.7277i q^{13} -10.0614 q^{14} +16.0000 q^{16} -19.7435i q^{17} +3.17864i q^{18} -158.545 q^{19} -25.3593 q^{21} -11.1644i q^{22} +23.0000i q^{23} +40.3272 q^{24} +125.455 q^{26} +144.116i q^{27} -20.1228i q^{28} +35.5033 q^{29} +282.041 q^{31} +32.0000i q^{32} -28.1393i q^{33} +39.4870 q^{34} -6.35728 q^{36} -139.981i q^{37} -317.090i q^{38} +316.204 q^{39} +227.680 q^{41} -50.7186i q^{42} -436.962i q^{43} +22.3288 q^{44} -46.0000 q^{46} +90.2701i q^{47} +80.6544i q^{48} +317.692 q^{49} +99.5250 q^{51} +250.911i q^{52} -330.183i q^{53} -288.232 q^{54} +40.2457 q^{56} -799.209i q^{57} +71.0066i q^{58} +796.203 q^{59} -568.580 q^{61} +564.081i q^{62} +7.99541i q^{63} -64.0000 q^{64} +56.2785 q^{66} +85.1419i q^{67} +78.9740i q^{68} -115.941 q^{69} -369.578 q^{71} -12.7146i q^{72} +310.188i q^{73} +279.963 q^{74} +634.180 q^{76} -28.0824i q^{77} +632.408i q^{78} +1325.46 q^{79} -683.562 q^{81} +455.360i q^{82} +158.806i q^{83} +101.437 q^{84} +873.924 q^{86} +178.969i q^{87} +44.6575i q^{88} +1233.89 q^{89} +315.565 q^{91} -92.0000i q^{92} +1421.74i q^{93} -180.540 q^{94} -161.309 q^{96} -106.389i q^{97} +635.384i q^{98} -8.87189 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 32 q^{4} + 56 q^{6} - 128 q^{9} + 42 q^{11} - 32 q^{14} + 128 q^{16} - 346 q^{19} - 240 q^{21} - 224 q^{24} + 280 q^{26} + 236 q^{29} + 34 q^{31} - 224 q^{34} + 512 q^{36} + 442 q^{39} + 278 q^{41}+ \cdots + 5490 q^{99}+O(q^{100}) \) 8 * q - 32 * q^4 + 56 * q^6 - 128 * q^9 + 42 * q^11 - 32 * q^14 + 128 * q^16 - 346 * q^19 - 240 * q^21 - 224 * q^24 + 280 * q^26 + 236 * q^29 + 34 * q^31 - 224 * q^34 + 512 * q^36 + 442 * q^39 + 278 * q^41 - 168 * q^44 - 368 * q^46 + 248 * q^49 + 878 * q^51 - 1556 * q^54 + 128 * q^56 + 906 * q^59 - 654 * q^61 - 512 * q^64 - 356 * q^66 + 644 * q^69 + 390 * q^71 + 1372 * q^74 + 1384 * q^76 + 2280 * q^79 + 2912 * q^81 + 960 * q^84 - 200 * q^86 + 4340 * q^89 + 1114 * q^91 - 1468 * q^94 + 896 * q^96 + 5490 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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