LMFDB - Embedded newform 3150.3.c.b.449.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3150,3,Mod(449,3150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3150, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("3150.449");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	3150.c (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:	$85.8312832735$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.157351936.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + x^{4} + 16 $ x^8 + x^4 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 126)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.3
Root			$-0.581861 + 1.28897i$ of defining polynomial
Character	$\chi$	$=$	3150.449
Dual form			3150.3.c.b.449.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} +2.00000 q^{4} +2.64575i q^{7} -2.82843 q^{8} +O(q^{10})$	$q-1.41421 q^{2} +2.00000 q^{4} +2.64575i q^{7} -2.82843 q^{8} -17.7951i q^{11} -2.58301i q^{13} -3.74166i q^{14} +4.00000 q^{16} -25.8681 q^{17} -20.0000 q^{19} +25.1660i q^{22} -17.7951 q^{23} +3.65292i q^{26} +5.29150i q^{28} +11.9034i q^{29} -17.1660 q^{31} -5.65685 q^{32} +36.5830 q^{34} +38.0000i q^{37} +28.2843 q^{38} +15.7338i q^{41} +43.4980i q^{43} -35.5901i q^{44} +25.1660 q^{46} -16.9706 q^{47} -7.00000 q^{49} -5.16601i q^{52} +85.5571 q^{53} -7.48331i q^{56} -16.8340i q^{58} +1.64899i q^{59} +100.332 q^{61} +24.2764 q^{62} +8.00000 q^{64} +36.6640i q^{67} -51.7362 q^{68} -17.7951i q^{71} -28.9150i q^{73} -53.7401i q^{74} -40.0000 q^{76} +47.0813 q^{77} -118.332 q^{79} -22.2510i q^{82} +120.443 q^{83} -61.5155i q^{86} +50.3320i q^{88} -139.475i q^{89} +6.83399 q^{91} -35.5901 q^{92} +24.0000 q^{94} +44.4131i q^{97} +9.89949 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{4}+O(q^{10}) $ 8 * q + 16 * q^4	$ 8 q + 16 q^{4} + 32 q^{16} - 160 q^{19} + 32 q^{31} + 208 q^{34} + 32 q^{46} - 56 q^{49} + 464 q^{61} + 64 q^{64} - 320 q^{76} - 608 q^{79} + 224 q^{91} + 192 q^{94}+O(q^{100}) $ 8 * q + 16 * q^4 + 32 * q^16 - 160 * q^19 + 32 * q^31 + 208 * q^34 + 32 * q^46 - 56 * q^49 + 464 * q^61 + 64 * q^64 - 320 * q^76 - 608 * q^79 + 224 * q^91 + 192 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$2801$
$\chi(n)$	$-1$	$1$	$-1$

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 $

$2$	−1.41421	−0.707107
$2$	−1.41421	−0.707107
$3$	0	0
$3$	0	0
$4$	2.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.64575i	0.377964i
$7$	2.64575i	0.377964i
$8$	−2.82843	−0.353553
$9$	0	0
$10$	0	0
$11$	− 17.7951i	− 1.61773i	−0.587993	−	0.808866i	$-0.700082\pi$
$11$	− 17.7951i	− 1.61773i	0.587993	−	0.808866i	$-0.299918\pi$
$12$	0	0
$13$	− 2.58301i	− 0.198693i	−0.995053	−	0.0993464i	$-0.968325\pi$
$13$	− 2.58301i	− 0.198693i	0.995053	−	0.0993464i	$-0.0316752\pi$
$14$	− 3.74166i	− 0.267261i
$15$	0	0
$16$	4.00000	0.250000
$17$	−25.8681	−1.52165	−0.760826	−	0.648956i	$-0.775206\pi$
$17$	−25.8681	−1.52165	−0.760826	+	0.648956i	$0.775206\pi$
$18$	0	0
$19$	−20.0000	−1.05263	−0.526316	−	0.850289i	$-0.676427\pi$
$19$	−20.0000	−1.05263	−0.526316	+	0.850289i	$0.676427\pi$
$20$	0	0
$21$	0	0
$22$	25.1660i	1.14391i
$23$	−17.7951	−0.773698	−0.386849	−	0.922143i	$-0.626437\pi$
$23$	−17.7951	−0.773698	−0.386849	+	0.922143i	$0.626437\pi$
$24$	0	0
$25$	0	0
$26$	3.65292i	0.140497i
$27$	0	0
$28$	5.29150i	0.188982i
$29$	11.9034i	0.410463i	0.978713	+	0.205232i	$0.0657947\pi$
$29$	11.9034i	0.410463i	−0.978713	+	0.205232i	$0.934205\pi$
$30$	0	0
$31$	−17.1660	−0.553742	−0.276871	−	0.960907i	$-0.589298\pi$
$31$	−17.1660	−0.553742	−0.276871	+	0.960907i	$0.589298\pi$
$32$	−5.65685	−0.176777
$33$	0	0
$34$	36.5830	1.07597
$35$	0	0
$36$	0	0
$37$	38.0000i	1.02703i	0.858082	+	0.513514i	$0.171656\pi$
$37$	38.0000i	1.02703i	−0.858082	+	0.513514i	$0.828344\pi$
$38$	28.2843	0.744323
$39$	0	0
$40$	0	0
$41$	15.7338i	0.383752i	0.981419	+	0.191876i	$0.0614571\pi$
$41$	15.7338i	0.383752i	−0.981419	+	0.191876i	$0.938543\pi$
$42$	0	0
$43$	43.4980i	1.01158i	0.862656	+	0.505791i	$0.168799\pi$
$43$	43.4980i	1.01158i	−0.862656	+	0.505791i	$0.831201\pi$
$44$	− 35.5901i	− 0.808866i
$45$	0	0
$46$	25.1660	0.547087
$47$	−16.9706	−0.361076	−0.180538	−	0.983568i	$-0.557784\pi$
$47$	−16.9706	−0.361076	−0.180538	+	0.983568i	$0.557784\pi$
$48$	0	0
$49$	−7.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	− 5.16601i	− 0.0993464i
$53$	85.5571	1.61429	0.807143	−	0.590356i	$-0.201013\pi$
$53$	85.5571	1.61429	0.807143	+	0.590356i	$0.201013\pi$
$54$	0	0
$55$	0	0
$56$	− 7.48331i	− 0.133631i
$57$	0	0
$58$	− 16.8340i	− 0.290241i
$59$	1.64899i	0.0279489i	0.999902	+	0.0139745i	$0.00444836\pi$
$59$	1.64899i	0.0279489i	−0.999902	+	0.0139745i	$0.995552\pi$
$60$	0	0
$61$	100.332	1.64479	0.822394	−	0.568919i	$-0.192638\pi$
$61$	100.332	1.64479	0.822394	+	0.568919i	$0.192638\pi$
$62$	24.2764	0.391555
$63$	0	0
$64$	8.00000	0.125000
$65$	0	0
$66$	0	0
$67$	36.6640i	0.547225i	0.961840	+	0.273612i	$0.0882185\pi$
$67$	36.6640i	0.547225i	−0.961840	+	0.273612i	$0.911781\pi$
$68$	−51.7362	−0.760826
$69$	0	0
$70$	0	0
$71$	− 17.7951i	− 0.250635i	−0.992117	−	0.125317i	$-0.960005\pi$
$71$	− 17.7951i	− 0.250635i	0.992117	−	0.125317i	$-0.0399949\pi$
$72$	0	0
$73$	− 28.9150i	− 0.396096i	−0.980192	−	0.198048i	$-0.936540\pi$
$73$	− 28.9150i	− 0.396096i	0.980192	−	0.198048i	$-0.0634602\pi$
$74$	− 53.7401i	− 0.726218i
$75$	0	0
$76$	−40.0000	−0.526316
$77$	47.0813	0.611445
$78$	0	0
$79$	−118.332	−1.49787	−0.748937	−	0.662641i	$-0.769435\pi$
$79$	−118.332	−1.49787	−0.748937	+	0.662641i	$0.769435\pi$
$80$	0	0
$81$	0	0
$82$	− 22.2510i	− 0.271353i
$83$	120.443	1.45112	0.725560	−	0.688159i	$-0.241581\pi$
$83$	120.443	1.45112	0.725560	+	0.688159i	$0.241581\pi$
$84$	0	0
$85$	0	0
$86$	− 61.5155i	− 0.715297i
$87$	0	0
$88$	50.3320i	0.571955i
$89$	− 139.475i	− 1.56713i	−0.621309	−	0.783566i	$-0.713399\pi$
$89$	− 139.475i	− 1.56713i	0.621309	−	0.783566i	$-0.286601\pi$
$90$	0	0
$91$	6.83399	0.0750988
$92$	−35.5901	−0.386849
$93$	0	0
$94$	24.0000	0.255319
$95$	0	0
$96$	0	0
$97$	44.4131i	0.457867i	0.973442	+	0.228933i	$0.0735238\pi$
$97$	44.4131i	0.457867i	−0.973442	+	0.228933i	$0.926476\pi$
$98$	9.89949	0.101015
$99$	0	0
$100$	0	0
$101$	− 31.8799i	− 0.315642i	−0.987468	−	0.157821i	$-0.949553\pi$
$101$	− 31.8799i	− 0.315642i	0.987468	−	0.157821i	$-0.0504470\pi$
$102$	0	0
$103$	− 4.50197i	− 0.0437084i	−0.999761	−	0.0218542i	$-0.993043\pi$
$103$	− 4.50197i	− 0.0437084i	0.999761	−	0.0218542i	$-0.00695697\pi$
$104$	7.30584i	0.0702485i
$105$	0	0
$106$	−120.996	−1.14147
$107$	172.179	1.60915	0.804575	−	0.593851i	$-0.202393\pi$
$107$	172.179	1.60915	0.804575	+	0.593851i	$0.202393\pi$
$108$	0	0
$109$	177.830	1.63147	0.815734	−	0.578427i	$-0.196333\pi$
$109$	177.830	1.63147	0.815734	+	0.578427i	$0.196333\pi$
$110$	0	0
$111$	0	0
$112$	10.5830i	0.0944911i
$113$	−31.3475	−0.277411	−0.138706	−	0.990334i	$-0.544294\pi$
$113$	−31.3475	−0.277411	−0.138706	+	0.990334i	$0.544294\pi$
$114$	0	0
$115$	0	0
$116$	23.8069i	0.205232i
$117$	0	0
$118$	− 2.33202i	− 0.0197629i
$119$	− 68.4405i	− 0.575131i
$120$	0	0
$121$	−195.664	−1.61706
$122$	−141.891	−1.16304
$123$	0	0
$124$	−34.3320	−0.276871
$125$	0	0
$126$	0	0
$127$	214.332i	1.68765i	0.536616	+	0.843827i	$0.319703\pi$
$127$	214.332i	1.68765i	−0.536616	+	0.843827i	$0.680297\pi$
$128$	−11.3137	−0.0883883
$129$	0	0
$130$	0	0
$131$	− 91.4488i	− 0.698082i	−0.937107	−	0.349041i	$-0.886507\pi$
$131$	− 91.4488i	− 0.698082i	0.937107	−	0.349041i	$-0.113493\pi$
$132$	0	0
$133$	− 52.9150i	− 0.397857i
$134$	− 51.8508i	− 0.386946i
$135$	0	0
$136$	73.1660	0.537985
$137$	−106.891	−0.780223	−0.390111	−	0.920768i	$-0.627563\pi$
$137$	−106.891	−0.780223	−0.390111	+	0.920768i	$0.627563\pi$
$138$	0	0
$139$	121.328	0.872864	0.436432	−	0.899737i	$-0.356242\pi$
$139$	121.328	0.872864	0.436432	+	0.899737i	$0.356242\pi$
$140$	0	0
$141$	0	0
$142$	25.1660i	0.177225i
$143$	−45.9647	−0.321432
$144$	0	0
$145$	0	0
$146$	40.8920i	0.280082i
$147$	0	0
$148$	76.0000i	0.513514i
$149$	− 17.6749i	− 0.118623i	−0.998240	−	0.0593117i	$-0.981109\pi$
$149$	− 17.6749i	− 0.118623i	0.998240	−	0.0593117i	$-0.0188906\pi$
$150$	0	0
$151$	−50.8340	−0.336649	−0.168324	−	0.985732i	$-0.553836\pi$
$151$	−50.8340	−0.336649	−0.168324	+	0.985732i	$0.553836\pi$
$152$	56.5685	0.372161
$153$	0	0
$154$	−66.5830	−0.432357
$155$	0	0
$156$	0	0
$157$	68.9961i	0.439465i	0.975560	+	0.219733i	$0.0705185\pi$
$157$	68.9961i	0.439465i	−0.975560	+	0.219733i	$0.929481\pi$
$158$	167.347	1.05916
$159$	0	0
$160$	0	0
$161$	− 47.0813i	− 0.292430i
$162$	0	0
$163$	166.996i	1.02452i	0.858832	+	0.512258i	$0.171191\pi$
$163$	166.996i	1.02452i	−0.858832	+	0.512258i	$0.828809\pi$
$164$	31.4676i	0.191876i
$165$	0	0
$166$	−170.332	−1.02610
$167$	120.443	0.721215	0.360608	−	0.932718i	$-0.382569\pi$
$167$	120.443	0.721215	0.360608	+	0.932718i	$0.382569\pi$
$168$	0	0
$169$	162.328	0.960521
$170$	0	0
$171$	0	0
$172$	86.9961i	0.505791i
$173$	91.8610	0.530989	0.265494	−	0.964112i	$-0.414465\pi$
$173$	91.8610	0.530989	0.265494	+	0.964112i	$0.414465\pi$
$174$	0	0
$175$	0	0
$176$	− 71.1802i	− 0.404433i
$177$	0	0
$178$	197.247i	1.10813i
$179$	133.291i	0.744643i	0.928104	+	0.372321i	$0.121438\pi$
$179$	133.291i	0.744643i	−0.928104	+	0.372321i	$0.878562\pi$
$180$	0	0
$181$	83.0850	0.459033	0.229517	−	0.973305i	$-0.426286\pi$
$181$	83.0850	0.459033	0.229517	+	0.973305i	$0.426286\pi$
$182$	−9.66472	−0.0531029
$183$	0	0
$184$	50.3320	0.273544
$185$	0	0
$186$	0	0
$187$	460.324i	2.46163i
$188$	−33.9411	−0.180538
$189$	0	0
$190$	0	0
$191$	− 41.3616i	− 0.216553i	−0.994121	−	0.108276i	$-0.965467\pi$
$191$	− 41.3616i	− 0.216553i	0.994121	−	0.108276i	$-0.0345331\pi$
$192$	0	0
$193$	− 134.000i	− 0.694301i	−0.937810	−	0.347150i	$-0.887149\pi$
$193$	− 134.000i	− 0.694301i	0.937810	−	0.347150i	$-0.112851\pi$
$194$	− 62.8095i	− 0.323761i
$195$	0	0
$196$	−14.0000	−0.0714286
$197$	−68.8269	−0.349375	−0.174688	−	0.984624i	$-0.555892\pi$
$197$	−68.8269	−0.349375	−0.174688	+	0.984624i	$0.555892\pi$
$198$	0	0
$199$	278.494	1.39947	0.699734	−	0.714404i	$-0.253302\pi$
$199$	278.494	1.39947	0.699734	+	0.714404i	$0.253302\pi$
$200$	0	0
$201$	0	0
$202$	45.0850i	0.223193i
$203$	−31.4935	−0.155140
$204$	0	0
$205$	0	0
$206$	6.36674i	0.0309065i
$207$	0	0
$208$	− 10.3320i	− 0.0496732i
$209$	355.901i	1.70288i
$210$	0	0
$211$	−211.498	−1.00236	−0.501180	−	0.865343i	$-0.667101\pi$
$211$	−211.498	−1.00236	−0.501180	+	0.865343i	$0.667101\pi$
$212$	171.114	0.807143
$213$	0	0
$214$	−243.498	−1.13784
$215$	0	0
$216$	0	0
$217$	− 45.4170i	− 0.209295i
$218$	−251.490	−1.15362
$219$	0	0
$220$	0	0
$221$	66.8174i	0.302341i
$222$	0	0
$223$	− 222.494i	− 0.997731i	−0.866679	−	0.498866i	$-0.833750\pi$
$223$	− 222.494i	− 0.997731i	0.866679	−	0.498866i	$-0.166250\pi$
$224$	− 14.9666i	− 0.0668153i
$225$	0	0
$226$	44.3320	0.196159
$227$	101.823	0.448561	0.224281	−	0.974525i	$-0.427997\pi$
$227$	101.823	0.448561	0.224281	+	0.974525i	$0.427997\pi$
$228$	0	0
$229$	163.085	0.712161	0.356081	−	0.934455i	$-0.384113\pi$
$229$	163.085	0.712161	0.356081	+	0.934455i	$0.384113\pi$
$230$	0	0
$231$	0	0
$232$	− 33.6680i	− 0.145121i
$233$	−362.858	−1.55733	−0.778664	−	0.627441i	$-0.784102\pi$
$233$	−362.858	−1.55733	−0.778664	+	0.627441i	$0.784102\pi$
$234$	0	0
$235$	0	0
$236$	3.29798i	0.0139745i
$237$	0	0
$238$	96.7895i	0.406679i
$239$	− 177.126i	− 0.741113i	−0.928810	−	0.370557i	$-0.879167\pi$
$239$	− 177.126i	− 0.741113i	0.928810	−	0.370557i	$-0.120833\pi$
$240$	0	0
$241$	152.753	0.633830	0.316915	−	0.948454i	$-0.397353\pi$
$241$	152.753	0.633830	0.316915	+	0.948454i	$0.397353\pi$
$242$	276.711	1.14343
$243$	0	0
$244$	200.664	0.822394
$245$	0	0
$246$	0	0
$247$	51.6601i	0.209150i
$248$	48.5528	0.195777
$249$	0	0
$250$	0	0
$251$	− 356.382i	− 1.41985i	−0.704278	−	0.709924i	$-0.748729\pi$
$251$	− 356.382i	− 1.41985i	0.704278	−	0.709924i	$-0.251271\pi$
$252$	0	0
$253$	316.664i	1.25164i
$254$	− 303.111i	− 1.19335i
$255$	0	0
$256$	16.0000	0.0625000
$257$	59.5689	0.231786	0.115893	−	0.993262i	$-0.463027\pi$
$257$	59.5689	0.231786	0.115893	+	0.993262i	$0.463027\pi$
$258$	0	0
$259$	−100.539	−0.388180
$260$	0	0
$261$	0	0
$262$	129.328i	0.493619i
$263$	7.42045	0.0282146	0.0141073	−	0.999900i	$-0.495509\pi$
$263$	7.42045	0.0282146	0.0141073	+	0.999900i	$0.495509\pi$
$264$	0	0
$265$	0	0
$266$	74.8331i	0.281328i
$267$	0	0
$268$	73.3281i	0.273612i
$269$	− 430.207i	− 1.59928i	−0.600477	−	0.799642i	$-0.705023\pi$
$269$	− 430.207i	− 1.59928i	0.600477	−	0.799642i	$-0.294977\pi$
$270$	0	0
$271$	−41.1660	−0.151904	−0.0759520	−	0.997111i	$-0.524200\pi$
$271$	−41.1660	−0.151904	−0.0759520	+	0.997111i	$0.524200\pi$
$272$	−103.472	−0.380413
$273$	0	0
$274$	151.166	0.551701
$275$	0	0
$276$	0	0
$277$	32.0000i	0.115523i	0.998330	+	0.0577617i	$0.0183964\pi$
$277$	32.0000i	0.115523i	−0.998330	+	0.0577617i	$0.981604\pi$
$278$	−171.584	−0.617208
$279$	0	0
$280$	0	0
$281$	− 17.0907i	− 0.0608211i	−0.999537	−	0.0304106i	$-0.990319\pi$
$281$	− 17.0907i	− 0.0608211i	0.999537	−	0.0304106i	$-0.00968147\pi$
$282$	0	0
$283$	− 439.660i	− 1.55357i	−0.629766	−	0.776785i	$-0.716849\pi$
$283$	− 439.660i	− 1.55357i	0.629766	−	0.776785i	$-0.283151\pi$
$284$	− 35.5901i	− 0.125317i
$285$	0	0
$286$	65.0039	0.227286
$287$	−41.6278	−0.145045
$288$	0	0
$289$	380.158	1.31543
$290$	0	0
$291$	0	0
$292$	− 57.8301i	− 0.198048i
$293$	394.377	1.34600	0.672998	−	0.739644i	$-0.265006\pi$
$293$	394.377	1.34600	0.672998	+	0.739644i	$0.265006\pi$
$294$	0	0
$295$	0	0
$296$	− 107.480i	− 0.363109i
$297$	0	0
$298$	24.9961i	0.0838794i
$299$	45.9647i	0.153728i
$300$	0	0
$301$	−115.085	−0.382342
$302$	71.8901	0.238047
$303$	0	0
$304$	−80.0000	−0.263158
$305$	0	0
$306$	0	0
$307$	− 23.3360i	− 0.0760129i	−0.999277	−	0.0380064i	$-0.987899\pi$
$307$	− 23.3360i	− 0.0760129i	0.999277	−	0.0380064i	$-0.0121007\pi$
$308$	94.1626	0.305723
$309$	0	0
$310$	0	0
$311$	527.256i	1.69536i	0.530511	+	0.847678i	$0.322000\pi$
$311$	527.256i	1.69536i	−0.530511	+	0.847678i	$0.678000\pi$
$312$	0	0
$313$	295.328i	0.943540i	0.881722	+	0.471770i	$0.156385\pi$
$313$	295.328i	0.943540i	−0.881722	+	0.471770i	$0.843615\pi$
$314$	− 97.5752i	− 0.310749i
$315$	0	0
$316$	−236.664	−0.748937
$317$	107.475	0.339037	0.169518	−	0.985527i	$-0.445779\pi$
$317$	107.475	0.339037	0.169518	+	0.985527i	$0.445779\pi$
$318$	0	0
$319$	211.822	0.664019
$320$	0	0
$321$	0	0
$322$	66.5830i	0.206780i
$323$	517.362	1.60174
$324$	0	0
$325$	0	0
$326$	− 236.168i	− 0.724442i
$327$	0	0
$328$	− 44.5020i	− 0.135677i
$329$	− 44.8999i	− 0.136474i
$330$	0	0
$331$	−273.490	−0.826254	−0.413127	−	0.910673i	$-0.635563\pi$
$331$	−273.490	−0.826254	−0.413127	+	0.910673i	$0.635563\pi$
$332$	240.886	0.725560
$333$	0	0
$334$	−170.332	−0.509976
$335$	0	0
$336$	0	0
$337$	− 341.166i	− 1.01236i	−0.862427	−	0.506181i	$-0.831057\pi$
$337$	− 341.166i	− 1.01236i	0.862427	−	0.506181i	$-0.168943\pi$
$338$	−229.567	−0.679191
$339$	0	0
$340$	0	0
$341$	305.470i	0.895807i
$342$	0	0
$343$	− 18.5203i	− 0.0539949i
$344$	− 123.031i	− 0.357648i
$345$	0	0
$346$	−129.911	−0.375466
$347$	116.320	0.335217	0.167609	−	0.985854i	$-0.446395\pi$
$347$	116.320	0.335217	0.167609	+	0.985854i	$0.446395\pi$
$348$	0	0
$349$	158.324	0.453651	0.226825	−	0.973935i	$-0.427165\pi$
$349$	158.324	0.453651	0.226825	+	0.973935i	$0.427165\pi$
$350$	0	0
$351$	0	0
$352$	100.664i	0.285977i
$353$	230.339	0.652519	0.326260	−	0.945280i	$-0.394212\pi$
$353$	230.339	0.652519	0.326260	+	0.945280i	$0.394212\pi$
$354$	0	0
$355$	0	0
$356$	− 278.949i	− 0.783566i
$357$	0	0
$358$	− 188.502i	− 0.526542i
$359$	171.698i	0.478269i	0.970987	+	0.239134i	$0.0768636\pi$
$359$	171.698i	0.478269i	−0.970987	+	0.239134i	$0.923136\pi$
$360$	0	0
$361$	39.0000	0.108033
$362$	−117.500	−0.324585
$363$	0	0
$364$	13.6680	0.0375494
$365$	0	0
$366$	0	0
$367$	517.490i	1.41005i	0.709180	+	0.705027i	$0.249065\pi$
$367$	517.490i	1.41005i	−0.709180	+	0.705027i	$0.750935\pi$
$368$	−71.1802	−0.193425
$369$	0	0
$370$	0	0
$371$	226.363i	0.610143i
$372$	0	0
$373$	233.336i	0.625566i	0.949825	+	0.312783i	$0.101261\pi$
$373$	233.336i	0.625566i	−0.949825	+	0.312783i	$0.898739\pi$
$374$	− 650.997i	− 1.74063i
$375$	0	0
$376$	48.0000	0.127660
$377$	30.7466	0.0815560
$378$	0	0
$379$	−441.166	−1.16403	−0.582013	−	0.813179i	$-0.697735\pi$
$379$	−441.166	−1.16403	−0.582013	+	0.813179i	$0.697735\pi$
$380$	0	0
$381$	0	0
$382$	58.4941i	0.153126i
$383$	213.060	0.556292	0.278146	−	0.960539i	$-0.410280\pi$
$383$	213.060	0.556292	0.278146	+	0.960539i	$0.410280\pi$
$384$	0	0
$385$	0	0
$386$	189.505i	0.490945i
$387$	0	0
$388$	88.8261i	0.228933i
$389$	− 565.096i	− 1.45269i	−0.687331	−	0.726344i	$-0.741218\pi$
$389$	− 565.096i	− 1.45269i	0.687331	−	0.726344i	$-0.258782\pi$
$390$	0	0
$391$	460.324	1.17730
$392$	19.7990	0.0505076
$393$	0	0
$394$	97.3360	0.247046
$395$	0	0
$396$	0	0
$397$	498.324i	1.25522i	0.778526	+	0.627612i	$0.215968\pi$
$397$	498.324i	1.25522i	−0.778526	+	0.627612i	$0.784032\pi$
$398$	−393.850	−0.989573
$399$	0	0
$400$	0	0
$401$	193.392i	0.482275i	0.970491	+	0.241138i	$0.0775205\pi$
$401$	193.392i	0.482275i	−0.970491	+	0.241138i	$0.922480\pi$
$402$	0	0
$403$	44.3399i	0.110025i
$404$	− 63.7598i	− 0.157821i
$405$	0	0
$406$	44.5385	0.109701
$407$	676.212	1.66145
$408$	0	0
$409$	454.243	1.11062	0.555309	−	0.831644i	$-0.312600\pi$
$409$	454.243	1.11062	0.555309	+	0.831644i	$0.312600\pi$
$410$	0	0
$411$	0	0
$412$	− 9.00394i	− 0.0218542i
$413$	−4.36281	−0.0105637
$414$	0	0
$415$	0	0
$416$	14.6117i	0.0351242i
$417$	0	0
$418$	− 503.320i	− 1.20412i
$419$	339.411i	0.810051i	0.914305	+	0.405025i	$0.132737\pi$
$419$	339.411i	0.810051i	−0.914305	+	0.405025i	$0.867263\pi$
$420$	0	0
$421$	247.320	0.587459	0.293729	−	0.955889i	$-0.405104\pi$
$421$	247.320	0.587459	0.293729	+	0.955889i	$0.405104\pi$
$422$	299.103	0.708776
$423$	0	0
$424$	−241.992	−0.570736
$425$	0	0
$426$	0	0
$427$	265.454i	0.621671i
$428$	344.358	0.804575
$429$	0	0
$430$	0	0
$431$	456.419i	1.05898i	0.848317	+	0.529489i	$0.177616\pi$
$431$	456.419i	1.05898i	−0.848317	+	0.529489i	$0.822384\pi$
$432$	0	0
$433$	637.984i	1.47340i	0.676217	+	0.736702i	$0.263618\pi$
$433$	637.984i	1.47340i	−0.676217	+	0.736702i	$0.736382\pi$
$434$	64.2293i	0.147994i
$435$	0	0
$436$	355.660	0.815734
$437$	355.901	0.814419
$438$	0	0
$439$	784.146	1.78621	0.893105	−	0.449848i	$-0.148522\pi$
$439$	784.146	1.78621	0.893105	+	0.449848i	$0.148522\pi$
$440$	0	0
$441$	0	0
$442$	− 94.4941i	− 0.213788i
$443$	472.222	1.06596	0.532981	−	0.846127i	$-0.321072\pi$
$443$	472.222	1.06596	0.532981	+	0.846127i	$0.321072\pi$
$444$	0	0
$445$	0	0
$446$	314.654i	0.705503i
$447$	0	0
$448$	21.1660i	0.0472456i
$449$	739.852i	1.64778i	0.566752	+	0.823888i	$0.308200\pi$
$449$	739.852i	1.64778i	−0.566752	+	0.823888i	$0.691800\pi$
$450$	0	0
$451$	279.984	0.620808
$452$	−62.6949	−0.138706
$453$	0	0
$454$	−144.000	−0.317181
$455$	0	0
$456$	0	0
$457$	− 248.324i	− 0.543379i	−0.962385	−	0.271689i	$-0.912418\pi$
$457$	− 248.324i	− 0.543379i	0.962385	−	0.271689i	$-0.0875824\pi$
$458$	−230.637	−0.503574
$459$	0	0
$460$	0	0
$461$	355.970i	0.772168i	0.922464	+	0.386084i	$0.126173\pi$
$461$	355.970i	0.772168i	−0.922464	+	0.386084i	$0.873827\pi$
$462$	0	0
$463$	6.33202i	0.0136761i	0.999977	+	0.00683804i	$0.00217663\pi$
$463$	6.33202i	0.0136761i	−0.999977	+	0.00683804i	$0.997823\pi$
$464$	47.6137i	0.102616i
$465$	0	0
$466$	513.158	1.10120
$467$	−878.691	−1.88156	−0.940782	−	0.339011i	$-0.889907\pi$
$467$	−878.691	−1.88156	−0.940782	+	0.339011i	$0.889907\pi$
$468$	0	0
$469$	−97.0039	−0.206831
$470$	0	0
$471$	0	0
$472$	− 4.66404i	− 0.00988144i
$473$	774.050	1.63647
$474$	0	0
$475$	0	0
$476$	− 136.881i	− 0.287565i
$477$	0	0
$478$	250.494i	0.524046i
$479$	− 224.396i	− 0.468468i	−0.972180	−	0.234234i	$-0.924742\pi$
$479$	− 224.396i	− 0.468468i	0.972180	−	0.234234i	$-0.0752581\pi$
$480$	0	0
$481$	98.1542	0.204063
$482$	−216.025	−0.448185
$483$	0	0
$484$	−391.328	−0.808529
$485$	0	0
$486$	0	0
$487$	− 717.490i	− 1.47329i	−0.676282	−	0.736643i	$-0.736410\pi$
$487$	− 717.490i	− 1.47329i	0.676282	−	0.736643i	$-0.263590\pi$
$488$	−283.782	−0.581520
$489$	0	0
$490$	0	0
$491$	274.002i	0.558050i	0.960284	+	0.279025i	$0.0900112\pi$
$491$	274.002i	0.558050i	−0.960284	+	0.279025i	$0.909989\pi$
$492$	0	0
$493$	− 307.919i	− 0.624582i
$494$	− 73.0584i	− 0.147892i
$495$	0	0
$496$	−68.6640	−0.138436
$497$	47.0813	0.0947310
$498$	0	0
$499$	728.810	1.46054	0.730271	−	0.683158i	$-0.239394\pi$
$499$	728.810	1.46054	0.730271	+	0.683158i	$0.239394\pi$
$500$	0	0
$501$	0	0
$502$	504.000i	1.00398i
$503$	−594.657	−1.18222	−0.591111	−	0.806590i	$-0.701310\pi$
$503$	−594.657	−1.18222	−0.591111	+	0.806590i	$0.701310\pi$
$504$	0	0
$505$	0	0
$506$	− 447.831i	− 0.885041i
$507$	0	0
$508$	428.664i	0.843827i
$509$	994.015i	1.95288i	0.215795	+	0.976439i	$0.430766\pi$
$509$	994.015i	1.95288i	−0.215795	+	0.976439i	$0.569234\pi$
$510$	0	0
$511$	76.5020	0.149710
$512$	−22.6274	−0.0441942
$513$	0	0
$514$	−84.2431	−0.163897
$515$	0	0
$516$	0	0
$517$	301.992i	0.584124i
$518$	142.183	0.274485
$519$	0	0
$520$	0	0
$521$	− 40.8459i	− 0.0783990i	−0.999231	−	0.0391995i	$-0.987519\pi$
$521$	− 40.8459i	− 0.0783990i	0.999231	−	0.0391995i	$-0.0124808\pi$
$522$	0	0
$523$	232.000i	0.443595i	0.975093	+	0.221797i	$0.0711923\pi$
$523$	232.000i	0.443595i	−0.975093	+	0.221797i	$0.928808\pi$
$524$	− 182.898i	− 0.349041i
$525$	0	0
$526$	−10.4941	−0.0199507
$527$	444.052	0.842603
$528$	0	0
$529$	−212.336	−0.401391
$530$	0	0
$531$	0	0
$532$	− 105.830i	− 0.198929i
$533$	40.6405	0.0762487
$534$	0	0
$535$	0	0
$536$	− 103.702i	− 0.193473i
$537$	0	0
$538$	608.405i	1.13086i
$539$	124.565i	0.231105i
$540$	0	0
$541$	250.332	0.462721	0.231360	−	0.972868i	$-0.425682\pi$
$541$	250.332	0.462721	0.231360	+	0.972868i	$0.425682\pi$
$542$	58.2175	0.107412
$543$	0	0
$544$	146.332	0.268993
$545$	0	0
$546$	0	0
$547$	888.324i	1.62399i	0.583662	+	0.811996i	$0.301619\pi$
$547$	888.324i	1.62399i	−0.583662	+	0.811996i	$0.698381\pi$
$548$	−213.781	−0.390111
$549$	0	0
$550$	0	0
$551$	− 238.069i	− 0.432066i
$552$	0	0
$553$	− 313.077i	− 0.566143i
$554$	− 45.2548i	− 0.0816874i
$555$	0	0
$556$	242.656	0.436432
$557$	316.309	0.567879	0.283940	−	0.958842i	$-0.408358\pi$
$557$	316.309	0.567879	0.283940	+	0.958842i	$0.408358\pi$
$558$	0	0
$559$	112.356	0.200994
$560$	0	0
$561$	0	0
$562$	24.1699i	0.0430070i
$563$	−58.4690	−0.103853	−0.0519263	−	0.998651i	$-0.516536\pi$
$563$	−58.4690	−0.103853	−0.0519263	+	0.998651i	$0.516536\pi$
$564$	0	0
$565$	0	0
$566$	621.773i	1.09854i
$567$	0	0
$568$	50.3320i	0.0886127i
$569$	221.665i	0.389570i	0.980846	+	0.194785i	$0.0624009\pi$
$569$	221.665i	0.389570i	−0.980846	+	0.194785i	$0.937599\pi$
$570$	0	0
$571$	−487.644	−0.854018	−0.427009	−	0.904247i	$-0.640433\pi$
$571$	−487.644	−0.854018	−0.427009	+	0.904247i	$0.640433\pi$
$572$	−91.9294	−0.160716
$573$	0	0
$574$	58.8706	0.102562
$575$	0	0
$576$	0	0
$577$	− 487.328i	− 0.844589i	−0.906459	−	0.422295i	$-0.861225\pi$
$577$	− 487.328i	− 0.844589i	0.906459	−	0.422295i	$-0.138775\pi$
$578$	−537.625	−0.930147
$579$	0	0
$580$	0	0
$581$	318.662i	0.548472i
$582$	0	0
$583$	− 1522.49i	− 2.61148i
$584$	81.7840i	0.140041i
$585$	0	0
$586$	−557.733	−0.951763
$587$	−445.701	−0.759286	−0.379643	−	0.925133i	$-0.623953\pi$
$587$	−445.701	−0.759286	−0.379643	+	0.925133i	$0.623953\pi$
$588$	0	0
$589$	343.320	0.582887
$590$	0	0
$591$	0	0
$592$	152.000i	0.256757i
$593$	276.648	0.466523	0.233261	−	0.972414i	$-0.425060\pi$
$593$	276.648	0.466523	0.233261	+	0.972414i	$0.425060\pi$
$594$	0	0
$595$	0	0
$596$	− 35.3498i	− 0.0593117i
$597$	0	0
$598$	− 65.0039i	− 0.108702i
$599$	− 82.3793i	− 0.137528i	−0.997633	−	0.0687641i	$-0.978094\pi$
$599$	− 82.3793i	− 0.137528i	0.997633	−	0.0687641i	$-0.0219056\pi$
$600$	0	0
$601$	−418.000	−0.695507	−0.347754	−	0.937586i	$-0.613055\pi$
$601$	−418.000	−0.695507	−0.347754	+	0.937586i	$0.613055\pi$
$602$	162.755	0.270357
$603$	0	0
$604$	−101.668	−0.168324
$605$	0	0
$606$	0	0
$607$	76.8419i	0.126593i	0.997995	+	0.0632964i	$0.0201614\pi$
$607$	76.8419i	0.126593i	−0.997995	+	0.0632964i	$0.979839\pi$
$608$	113.137	0.186081
$609$	0	0
$610$	0	0
$611$	43.8351i	0.0717431i
$612$	0	0
$613$	− 59.3281i	− 0.0967832i	−0.998828	−	0.0483916i	$-0.984590\pi$
$613$	− 59.3281i	− 0.0967832i	0.998828	−	0.0483916i	$-0.0154095\pi$
$614$	33.0020i	0.0537492i
$615$	0	0
$616$	−133.166	−0.216179
$617$	29.1143	0.0471869	0.0235935	−	0.999722i	$-0.492489\pi$
$617$	29.1143	0.0471869	0.0235935	+	0.999722i	$0.492489\pi$
$618$	0	0
$619$	−455.644	−0.736098	−0.368049	−	0.929806i	$-0.619974\pi$
$619$	−455.644	−0.736098	−0.368049	+	0.929806i	$0.619974\pi$
$620$	0	0
$621$	0	0
$622$	− 745.652i	− 1.19880i
$623$	369.015	0.592320
$624$	0	0
$625$	0	0
$626$	− 417.657i	− 0.667184i
$627$	0	0
$628$	137.992i	0.219733i
$629$	− 982.987i	− 1.56278i
$630$	0	0
$631$	−45.0039	−0.0713216	−0.0356608	−	0.999364i	$-0.511354\pi$
$631$	−45.0039	−0.0713216	−0.0356608	+	0.999364i	$0.511354\pi$
$632$	334.693	0.529578
$633$	0	0
$634$	−151.992	−0.239735
$635$	0	0
$636$	0	0
$637$	18.0810i	0.0283847i
$638$	−299.562	−0.469533
$639$	0	0
$640$	0	0
$641$	641.223i	1.00035i	0.865925	+	0.500174i	$0.166731\pi$
$641$	641.223i	1.00035i	−0.865925	+	0.500174i	$0.833269\pi$
$642$	0	0
$643$	604.000i	0.939347i	0.882840	+	0.469673i	$0.155628\pi$
$643$	604.000i	0.939347i	−0.882840	+	0.469673i	$0.844372\pi$
$644$	− 94.1626i	− 0.146215i
$645$	0	0
$646$	−731.660	−1.13260
$647$	−179.600	−0.277588	−0.138794	−	0.990321i	$-0.544323\pi$
$647$	−179.600	−0.277588	−0.138794	+	0.990321i	$0.544323\pi$
$648$	0	0
$649$	29.3438	0.0452139
$650$	0	0
$651$	0	0
$652$	333.992i	0.512258i
$653$	−392.092	−0.600447	−0.300224	−	0.953869i	$-0.597061\pi$
$653$	−392.092	−0.600447	−0.300224	+	0.953869i	$0.597061\pi$
$654$	0	0
$655$	0	0
$656$	62.9353i	0.0959379i
$657$	0	0
$658$	63.4980i	0.0965016i
$659$	1266.54i	1.92191i	0.276701	+	0.960956i	$0.410759\pi$
$659$	1266.54i	1.92191i	−0.276701	+	0.960956i	$0.589241\pi$
$660$	0	0
$661$	917.644	1.38827	0.694133	−	0.719846i	$-0.255788\pi$
$661$	917.644	1.38827	0.694133	+	0.719846i	$0.255788\pi$
$662$	386.773	0.584250
$663$	0	0
$664$	−340.664	−0.513048
$665$	0	0
$666$	0	0
$667$	− 211.822i	− 0.317574i
$668$	240.886	0.360608
$669$	0	0
$670$	0	0
$671$	− 1785.41i	− 2.66083i
$672$	0	0
$673$	152.008i	0.225866i	0.993603	+	0.112933i	$0.0360246\pi$
$673$	152.008i	0.225866i	−0.993603	+	0.112933i	$0.963975\pi$
$674$	482.482i	0.715848i
$675$	0	0
$676$	324.656	0.480261
$677$	163.178	0.241031	0.120516	−	0.992711i	$-0.461545\pi$
$677$	163.178	0.241031	0.120516	+	0.992711i	$0.461545\pi$
$678$	0	0
$679$	−117.506	−0.173057
$680$	0	0
$681$	0	0
$682$	− 432.000i	− 0.633431i
$683$	324.914	0.475716	0.237858	−	0.971300i	$-0.423555\pi$
$683$	324.914	0.475716	0.237858	+	0.971300i	$0.423555\pi$
$684$	0	0
$685$	0	0
$686$	26.1916i	0.0381802i
$687$	0	0
$688$	173.992i	0.252896i
$689$	− 220.995i	− 0.320747i
$690$	0	0
$691$	1218.98	1.76408	0.882041	−	0.471173i	$-0.156169\pi$
$691$	1218.98	1.76408	0.882041	+	0.471173i	$0.156169\pi$
$692$	183.722	0.265494
$693$	0	0
$694$	−164.502	−0.237035
$695$	0	0
$696$	0	0
$697$	− 407.004i	− 0.583937i
$698$	−223.904	−0.320780
$699$	0	0
$700$	0	0
$701$	427.202i	0.609417i	0.952446	+	0.304709i	$0.0985591\pi$
$701$	427.202i	0.609417i	−0.952446	+	0.304709i	$0.901441\pi$
$702$	0	0
$703$	− 760.000i	− 1.08108i
$704$	− 142.360i	− 0.202217i
$705$	0	0
$706$	−325.749	−0.461401
$707$	84.3463	0.119302
$708$	0	0
$709$	−71.4980	−0.100843	−0.0504217	−	0.998728i	$-0.516057\pi$
$709$	−71.4980	−0.100843	−0.0504217	+	0.998728i	$0.516057\pi$
$710$	0	0
$711$	0	0
$712$	394.494i	0.554065i
$713$	305.470	0.428429
$714$	0	0
$715$	0	0
$716$	266.582i	0.372321i
$717$	0	0
$718$	− 242.818i	− 0.338187i
$719$	− 111.030i	− 0.154422i	−0.997015	−	0.0772112i	$-0.975398\pi$
$719$	− 111.030i	− 0.154422i	0.997015	−	0.0772112i	$-0.0246016\pi$
$720$	0	0
$721$	11.9111	0.0165202
$722$	−55.1543	−0.0763910
$723$	0	0
$724$	166.170	0.229517
$725$	0	0
$726$	0	0
$727$	− 1338.82i	− 1.84157i	−0.390076	−	0.920783i	$-0.627551\pi$
$727$	− 1338.82i	− 1.84157i	0.390076	−	0.920783i	$-0.372449\pi$
$728$	−19.3294	−0.0265514
$729$	0	0
$730$	0	0
$731$	− 1125.21i	− 1.53928i
$732$	0	0
$733$	− 49.0771i	− 0.0669538i	−0.999439	−	0.0334769i	$-0.989342\pi$
$733$	− 49.0771i	− 0.0669538i	0.999439	−	0.0334769i	$-0.0106580\pi$
$734$	− 731.842i	− 0.997059i
$735$	0	0
$736$	100.664	0.136772
$737$	652.439	0.885263
$738$	0	0
$739$	−1430.32	−1.93548	−0.967738	−	0.251960i	$-0.918925\pi$
$739$	−1430.32	−1.93548	−0.967738	+	0.251960i	$0.918925\pi$
$740$	0	0
$741$	0	0
$742$	− 320.125i	− 0.431436i
$743$	−875.736	−1.17865	−0.589325	−	0.807896i	$-0.700606\pi$
$743$	−875.736	−1.17865	−0.589325	+	0.807896i	$0.700606\pi$
$744$	0	0
$745$	0	0
$746$	− 329.987i	− 0.442342i
$747$	0	0
$748$	920.648i	1.23081i
$749$	455.543i	0.608202i
$750$	0	0
$751$	320.826	0.427199	0.213599	−	0.976921i	$-0.431481\pi$
$751$	320.826	0.427199	0.213599	+	0.976921i	$0.431481\pi$
$752$	−67.8823	−0.0902690
$753$	0	0
$754$	−43.4823	−0.0576688
$755$	0	0
$756$	0	0
$757$	289.830i	0.382867i	0.981506	+	0.191433i	$0.0613136\pi$
$757$	289.830i	0.382867i	−0.981506	+	0.191433i	$0.938686\pi$
$758$	623.903	0.823091
$759$	0	0
$760$	0	0
$761$	704.657i	0.925962i	0.886368	+	0.462981i	$0.153220\pi$
$761$	704.657i	0.925962i	−0.886368	+	0.462981i	$0.846780\pi$
$762$	0	0
$763$	470.494i	0.616637i
$764$	− 82.7231i	− 0.108276i
$765$	0	0
$766$	−301.312	−0.393358
$767$	4.25934	0.00555325
$768$	0	0
$769$	117.320	0.152562	0.0762810	−	0.997086i	$-0.475695\pi$
$769$	117.320	0.152562	0.0762810	+	0.997086i	$0.475695\pi$
$770$	0	0
$771$	0	0
$772$	− 268.000i	− 0.347150i
$773$	−658.005	−0.851235	−0.425618	−	0.904903i	$-0.639943\pi$
$773$	−658.005	−0.851235	−0.425618	+	0.904903i	$0.639943\pi$
$774$	0	0
$775$	0	0
$776$	− 125.619i	− 0.161880i
$777$	0	0
$778$	799.166i	1.02721i
$779$	− 314.676i	− 0.403949i
$780$	0	0
$781$	−316.664	−0.405460
$782$	−650.997	−0.832477
$783$	0	0
$784$	−28.0000	−0.0357143
$785$	0	0
$786$	0	0
$787$	− 1200.65i	− 1.52560i	−0.646634	−	0.762801i	$-0.723824\pi$
$787$	− 1200.65i	− 1.52560i	0.646634	−	0.762801i	$-0.276176\pi$
$788$	−137.654	−0.174688
$789$	0	0
$790$	0	0
$791$	− 82.9376i	− 0.104852i
$792$	0	0
$793$	− 259.158i	− 0.326807i
$794$	− 704.737i	− 0.887578i
$795$	0	0
$796$	556.988	0.699734
$797$	797.411	1.00052	0.500258	−	0.865876i	$-0.333239\pi$
$797$	797.411	1.00052	0.500258	+	0.865876i	$0.333239\pi$
$798$	0	0
$799$	438.996	0.549432
$800$	0	0
$801$	0	0
$802$	− 273.498i	− 0.341020i
$803$	−514.545	−0.640778
$804$	0	0
$805$	0	0
$806$	− 62.7061i	− 0.0777991i
$807$	0	0
$808$	90.1699i	0.111596i
$809$	− 156.016i	− 0.192851i	−0.995340	−	0.0964254i	$-0.969259\pi$
$809$	− 156.016i	− 0.192851i	0.995340	−	0.0964254i	$-0.0307409\pi$
$810$	0	0
$811$	−598.316	−0.737751	−0.368876	−	0.929479i	$-0.620257\pi$
$811$	−598.316	−0.737751	−0.368876	+	0.929479i	$0.620257\pi$
$812$	−62.9870	−0.0775702
$813$	0	0
$814$	−956.308	−1.17483
$815$	0	0
$816$	0	0
$817$	− 869.961i	− 1.06482i
$818$	−642.397	−0.785326
$819$	0	0
$820$	0	0
$821$	980.978i	1.19486i	0.801922	+	0.597429i	$0.203811\pi$
$821$	980.978i	1.19486i	−0.801922	+	0.597429i	$0.796189\pi$
$822$	0	0
$823$	431.336i	0.524102i	0.965054	+	0.262051i	$0.0843989\pi$
$823$	431.336i	0.524102i	−0.965054	+	0.262051i	$0.915601\pi$
$824$	12.7335i	0.0154533i
$825$	0	0
$826$	6.16995	0.00746967
$827$	−1219.41	−1.47449	−0.737247	−	0.675623i	$-0.763875\pi$
$827$	−1219.41	−1.47449	−0.737247	+	0.675623i	$0.763875\pi$
$828$	0	0
$829$	770.081	0.928928	0.464464	−	0.885592i	$-0.346247\pi$
$829$	770.081	0.928928	0.464464	+	0.885592i	$0.346247\pi$
$830$	0	0
$831$	0	0
$832$	− 20.6640i	− 0.0248366i
$833$	181.077	0.217379
$834$	0	0
$835$	0	0
$836$	711.802i	0.851438i
$837$	0	0
$838$	− 480.000i	− 0.572792i
$839$	1310.03i	1.56142i	0.624894	+	0.780710i	$0.285142\pi$
$839$	1310.03i	1.56142i	−0.624894	+	0.780710i	$0.714858\pi$
$840$	0	0
$841$	699.308	0.831520
$842$	−349.764	−0.415396
$843$	0	0
$844$	−422.996	−0.501180
$845$	0	0
$846$	0	0
$847$	− 517.678i	− 0.611191i
$848$	342.229	0.403571
$849$	0	0
$850$	0	0
$851$	− 676.212i	− 0.794609i
$852$	0	0
$853$	− 898.988i	− 1.05391i	−0.849892	−	0.526957i	$-0.823333\pi$
$853$	− 898.988i	− 1.05391i	0.849892	−	0.526957i	$-0.176667\pi$
$854$	− 375.408i	− 0.439588i
$855$	0	0
$856$	−486.996	−0.568921
$857$	−746.156	−0.870660	−0.435330	−	0.900271i	$-0.643368\pi$
$857$	−746.156	−0.870660	−0.435330	+	0.900271i	$0.643368\pi$
$858$	0	0
$859$	991.984	1.15481	0.577406	−	0.816457i	$-0.304065\pi$
$859$	991.984	1.15481	0.577406	+	0.816457i	$0.304065\pi$
$860$	0	0
$861$	0	0
$862$	− 645.474i	− 0.748810i
$863$	209.418	0.242663	0.121332	−	0.992612i	$-0.461284\pi$
$863$	209.418	0.242663	0.121332	+	0.992612i	$0.461284\pi$
$864$	0	0
$865$	0	0
$866$	− 902.246i	− 1.04185i
$867$	0	0
$868$	− 90.8340i	− 0.104647i
$869$	2105.73i	2.42316i
$870$	0	0
$871$	94.7034	0.108730
$872$	−502.979	−0.576811
$873$	0	0
$874$	−503.320	−0.575881
$875$	0	0
$876$	0	0
$877$	− 865.304i	− 0.986664i	−0.869841	−	0.493332i	$-0.835779\pi$
$877$	− 865.304i	− 0.986664i	0.869841	−	0.493332i	$-0.164221\pi$
$878$	−1108.95	−1.26304
$879$	0	0
$880$	0	0
$881$	995.046i	1.12945i	0.825279	+	0.564725i	$0.191018\pi$
$881$	995.046i	1.12945i	−0.825279	+	0.564725i	$0.808982\pi$
$882$	0	0
$883$	− 101.474i	− 0.114920i	−0.998348	−	0.0574600i	$-0.981700\pi$
$883$	− 101.474i	− 0.114920i	0.998348	−	0.0574600i	$-0.0183002\pi$
$884$	133.635i	0.151171i
$885$	0	0
$886$	−667.822	−0.753750
$887$	−1074.09	−1.21093	−0.605464	−	0.795873i	$-0.707012\pi$
$887$	−1074.09	−1.21093	−0.605464	+	0.795873i	$0.707012\pi$
$888$	0	0
$889$	−567.069	−0.637873
$890$	0	0
$891$	0	0
$892$	− 444.988i	− 0.498866i
$893$	339.411	0.380080
$894$	0	0
$895$	0	0
$896$	− 29.9333i	− 0.0334077i
$897$	0	0
$898$	− 1046.31i	− 1.16515i
$899$	− 204.334i	− 0.227291i
$900$	0	0
$901$	−2213.20	−2.45638
$902$	−395.958	−0.438977
$903$	0	0
$904$	88.6640	0.0980797
$905$	0	0
$906$	0	0
$907$	− 432.162i	− 0.476474i	−0.971207	−	0.238237i	$-0.923430\pi$
$907$	− 432.162i	− 0.476474i	0.971207	−	0.238237i	$-0.0765695\pi$
$908$	203.647	0.224281
$909$	0	0
$910$	0	0
$911$	− 104.984i	− 0.115241i	−0.998339	−	0.0576205i	$-0.981649\pi$
$911$	− 104.984i	− 0.115241i	0.998339	−	0.0576205i	$-0.0183513\pi$
$912$	0	0
$913$	− 2143.29i	− 2.34752i
$914$	351.183i	0.384227i
$915$	0	0
$916$	326.170	0.356081
$917$	241.951	0.263850
$918$	0	0
$919$	91.8379	0.0999325	0.0499662	−	0.998751i	$-0.484089\pi$
$919$	91.8379	0.0999325	0.0499662	+	0.998751i	$0.484089\pi$
$920$	0	0
$921$	0	0
$922$	− 503.417i	− 0.546005i
$923$	−45.9647	−0.0497993
$924$	0	0
$925$	0	0
$926$	− 8.95483i	− 0.00967044i
$927$	0	0
$928$	− 67.3360i	− 0.0725603i
$929$	1309.00i	1.40904i	0.709682	+	0.704522i	$0.248838\pi$
$929$	1309.00i	1.40904i	−0.709682	+	0.704522i	$0.751162\pi$
$930$	0	0
$931$	140.000	0.150376
$932$	−725.715	−0.778664
$933$	0	0
$934$	1242.66	1.33047
$935$	0	0
$936$	0	0
$937$	1262.00i	1.34685i	0.739255	+	0.673426i	$0.235178\pi$
$937$	1262.00i	1.34685i	−0.739255	+	0.673426i	$0.764822\pi$
$938$	137.184	0.146252
$939$	0	0
$940$	0	0
$941$	− 1315.97i	− 1.39849i	−0.714884	−	0.699243i	$-0.753521\pi$
$941$	− 1315.97i	− 1.39849i	0.714884	−	0.699243i	$-0.246479\pi$
$942$	0	0
$943$	− 279.984i	− 0.296908i
$944$	6.59595i	0.00698724i
$945$	0	0
$946$	−1094.67	−1.15716
$947$	−486.582	−0.513814	−0.256907	−	0.966436i	$-0.582703\pi$
$947$	−486.582	−0.513814	−0.256907	+	0.966436i	$0.582703\pi$
$948$	0	0
$949$	−74.6877	−0.0787014
$950$	0	0
$951$	0	0
$952$	193.579i	0.203339i
$953$	43.3711	0.0455100	0.0227550	−	0.999741i	$-0.492756\pi$
$953$	43.3711	0.0455100	0.0227550	+	0.999741i	$0.492756\pi$
$954$	0	0
$955$	0	0
$956$	− 354.252i	− 0.370557i
$957$	0	0
$958$	317.344i	0.331257i
$959$	− 282.806i	− 0.294896i
$960$	0	0
$961$	−666.328	−0.693369
$962$	−138.811	−0.144294
$963$	0	0
$964$	305.506	0.316915
$965$	0	0
$966$	0	0
$967$	1648.99i	1.70526i	0.522514	+	0.852631i	$0.324994\pi$
$967$	1648.99i	1.70526i	−0.522514	+	0.852631i	$0.675006\pi$
$968$	553.421	0.571716
$969$	0	0
$970$	0	0
$971$	518.323i	0.533803i	0.963724	+	0.266902i	$0.0859999\pi$
$971$	518.323i	0.533803i	−0.963724	+	0.266902i	$0.914000\pi$
$972$	0	0
$973$	321.004i	0.329912i
$974$	1014.68i	1.04177i
$975$	0	0
$976$	401.328	0.411197
$977$	−109.948	−0.112536	−0.0562682	−	0.998416i	$-0.517920\pi$
$977$	−109.948	−0.112536	−0.0562682	+	0.998416i	$0.517920\pi$
$978$	0	0
$979$	−2481.96	−2.53520
$980$	0	0
$981$	0	0
$982$	− 387.498i	− 0.394601i
$983$	−589.710	−0.599909	−0.299954	−	0.953954i	$-0.596971\pi$
$983$	−589.710	−0.599909	−0.299954	+	0.953954i	$0.596971\pi$
$984$	0	0
$985$	0	0
$986$	435.463i	0.441646i
$987$	0	0
$988$	103.320i	0.104575i
$989$	− 774.050i	− 0.782659i
$990$	0	0
$991$	713.474	0.719954	0.359977	−	0.932961i	$-0.382785\pi$
$991$	713.474	0.719954	0.359977	+	0.932961i	$0.382785\pi$
$992$	97.1056	0.0978887
$993$	0	0
$994$	−66.5830	−0.0669849
$995$	0	0
$996$	0	0
$997$	1222.99i	1.22667i	0.789824	+	0.613334i	$0.210172\pi$
$997$	1222.99i	1.22667i	−0.789824	+	0.613334i	$0.789828\pi$
$998$	−1030.69	−1.03276
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3150.3.c.b.449.3		8
3.2	odd	2	inner	3150.3.c.b.449.8		8
5.2	odd	4		3150.3.e.e.701.1		4
5.3	odd	4		126.3.b.a.71.3	yes	4
5.4	even	2	inner	3150.3.c.b.449.5		8
15.2	even	4		3150.3.e.e.701.3		4
15.8	even	4		126.3.b.a.71.2	✓	4
15.14	odd	2	inner	3150.3.c.b.449.2		8
20.3	even	4		1008.3.d.a.449.1		4
35.3	even	12		882.3.s.i.863.1		8
35.13	even	4		882.3.b.f.197.4		4
35.18	odd	12		882.3.s.e.863.2		8
35.23	odd	12		882.3.s.e.557.3		8
35.33	even	12		882.3.s.i.557.4		8
40.3	even	4		4032.3.d.j.449.4		4
40.13	odd	4		4032.3.d.i.449.4		4
45.13	odd	12		1134.3.q.c.1079.1		8
45.23	even	12		1134.3.q.c.1079.4		8
45.38	even	12		1134.3.q.c.701.1		8
45.43	odd	12		1134.3.q.c.701.4		8
60.23	odd	4		1008.3.d.a.449.4		4
105.23	even	12		882.3.s.e.557.2		8
105.38	odd	12		882.3.s.i.863.4		8
105.53	even	12		882.3.s.e.863.3		8
105.68	odd	12		882.3.s.i.557.1		8
105.83	odd	4		882.3.b.f.197.1		4
120.53	even	4		4032.3.d.i.449.1		4
120.83	odd	4		4032.3.d.j.449.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.3.b.a.71.2	✓	4	15.8	even	4
126.3.b.a.71.3	yes	4	5.3	odd	4
882.3.b.f.197.1		4	105.83	odd	4
882.3.b.f.197.4		4	35.13	even	4
882.3.s.e.557.2		8	105.23	even	12
882.3.s.e.557.3		8	35.23	odd	12
882.3.s.e.863.2		8	35.18	odd	12
882.3.s.e.863.3		8	105.53	even	12
882.3.s.i.557.1		8	105.68	odd	12
882.3.s.i.557.4		8	35.33	even	12
882.3.s.i.863.1		8	35.3	even	12
882.3.s.i.863.4		8	105.38	odd	12
1008.3.d.a.449.1		4	20.3	even	4
1008.3.d.a.449.4		4	60.23	odd	4
1134.3.q.c.701.1		8	45.38	even	12
1134.3.q.c.701.4		8	45.43	odd	12
1134.3.q.c.1079.1		8	45.13	odd	12
1134.3.q.c.1079.4		8	45.23	even	12
3150.3.c.b.449.2		8	15.14	odd	2	inner
3150.3.c.b.449.3		8	1.1	even	1	trivial
3150.3.c.b.449.5		8	5.4	even	2	inner
3150.3.c.b.449.8		8	3.2	odd	2	inner
3150.3.e.e.701.1		4	5.2	odd	4
3150.3.e.e.701.3		4	15.2	even	4
4032.3.d.i.449.1		4	120.53	even	4
4032.3.d.i.449.4		4	40.13	odd	4
4032.3.d.j.449.1		4	120.83	odd	4
4032.3.d.j.449.4		4	40.3	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:	\( N \)	\(=\)	\( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	3150.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} +2.00000 q^{4} +2.64575i q^{7} -2.82843 q^{8} +O(q^{10})\)	\(q-1.41421 q^{2} +2.00000 q^{4} +2.64575i q^{7} -2.82843 q^{8} -17.7951i q^{11} -2.58301i q^{13} -3.74166i q^{14} +4.00000 q^{16} -25.8681 q^{17} -20.0000 q^{19} +25.1660i q^{22} -17.7951 q^{23} +3.65292i q^{26} +5.29150i q^{28} +11.9034i q^{29} -17.1660 q^{31} -5.65685 q^{32} +36.5830 q^{34} +38.0000i q^{37} +28.2843 q^{38} +15.7338i q^{41} +43.4980i q^{43} -35.5901i q^{44} +25.1660 q^{46} -16.9706 q^{47} -7.00000 q^{49} -5.16601i q^{52} +85.5571 q^{53} -7.48331i q^{56} -16.8340i q^{58} +1.64899i q^{59} +100.332 q^{61} +24.2764 q^{62} +8.00000 q^{64} +36.6640i q^{67} -51.7362 q^{68} -17.7951i q^{71} -28.9150i q^{73} -53.7401i q^{74} -40.0000 q^{76} +47.0813 q^{77} -118.332 q^{79} -22.2510i q^{82} +120.443 q^{83} -61.5155i q^{86} +50.3320i q^{88} -139.475i q^{89} +6.83399 q^{91} -35.5901 q^{92} +24.0000 q^{94} +44.4131i q^{97} +9.89949 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{4}+O(q^{10}) \) 8 * q + 16 * q^4	\( 8 q + 16 q^{4} + 32 q^{16} - 160 q^{19} + 32 q^{31} + 208 q^{34} + 32 q^{46} - 56 q^{49} + 464 q^{61} + 64 q^{64} - 320 q^{76} - 608 q^{79} + 224 q^{91} + 192 q^{94}+O(q^{100}) \) 8 * q + 16 * q^4 + 32 * q^16 - 160 * q^19 + 32 * q^31 + 208 * q^34 + 32 * q^46 - 56 * q^49 + 464 * q^61 + 64 * q^64 - 320 * q^76 - 608 * q^79 + 224 * q^91 + 192 * q^94

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