LMFDB - Embedded newform 3150.3.e.e.701.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3150,3,Mod(701,3150)]

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, 0, 0]))

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

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

chi := DirichletCharacter("3150.701");

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.e (of order $2$, degree $1$, 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:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 8x^{2} + 9 $ x^4 + 8*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 126)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			701.3
Root			$-2.57794i$ of defining polynomial
Character	$\chi$	$=$	3150.701
Dual form			3150.3.e.e.701.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{2} -2.00000 q^{4} -2.64575 q^{7} -2.82843i q^{8} +O(q^{10})$	$q+1.41421i q^{2} -2.00000 q^{4} -2.64575 q^{7} -2.82843i q^{8} +17.7951i q^{11} -2.58301 q^{13} -3.74166i q^{14} +4.00000 q^{16} +25.8681i q^{17} +20.0000 q^{19} -25.1660 q^{22} -17.7951i q^{23} -3.65292i q^{26} +5.29150 q^{28} +11.9034i q^{29} -17.1660 q^{31} +5.65685i q^{32} -36.5830 q^{34} -38.0000 q^{37} +28.2843i q^{38} -15.7338i q^{41} +43.4980 q^{43} -35.5901i q^{44} +25.1660 q^{46} +16.9706i q^{47} +7.00000 q^{49} +5.16601 q^{52} +85.5571i q^{53} +7.48331i q^{56} -16.8340 q^{58} +1.64899i q^{59} +100.332 q^{61} -24.2764i q^{62} -8.00000 q^{64} -36.6640 q^{67} -51.7362i q^{68} +17.7951i q^{71} -28.9150 q^{73} -53.7401i q^{74} -40.0000 q^{76} -47.0813i q^{77} +118.332 q^{79} +22.2510 q^{82} +120.443i q^{83} +61.5155i q^{86} +50.3320 q^{88} -139.475i q^{89} +6.83399 q^{91} +35.5901i q^{92} -24.0000 q^{94} -44.4131 q^{97} +9.89949i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{4}+O(q^{10}) $ 4 * q - 8 * q^4	$ 4 q - 8 q^{4} + 32 q^{13} + 16 q^{16} + 80 q^{19} - 16 q^{22} + 16 q^{31} - 104 q^{34} - 152 q^{37} - 80 q^{43} + 16 q^{46} + 28 q^{49} - 64 q^{52} - 152 q^{58} + 232 q^{61} - 32 q^{64} + 192 q^{67} + 96 q^{73} - 160 q^{76} + 304 q^{79} + 216 q^{82} + 32 q^{88} + 112 q^{91} - 96 q^{94} + 288 q^{97}+O(q^{100}) $ 4 * q - 8 * q^4 + 32 * q^13 + 16 * q^16 + 80 * q^19 - 16 * q^22 + 16 * q^31 - 104 * q^34 - 152 * q^37 - 80 * q^43 + 16 * q^46 + 28 * q^49 - 64 * q^52 - 152 * q^58 + 232 * q^61 - 32 * q^64 + 192 * q^67 + 96 * q^73 - 160 * q^76 + 304 * q^79 + 216 * q^82 + 32 * q^88 + 112 * q^91 - 96 * q^94 + 288 * q^97

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 $

$2$	1.41421i	0.707107i
$2$	1.41421i	0.707107i
$3$	0	0
$3$	0	0
$4$	−2.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.64575	−0.377964
$7$	−2.64575	−0.377964
$8$	− 2.82843i	− 0.353553i
$9$	0	0
$10$	0	0
$11$	17.7951i	1.61773i	0.587993	+	0.808866i	$0.299918\pi$
$11$	17.7951i	1.61773i	−0.587993	+	0.808866i	$0.700082\pi$
$12$	0	0
$13$	−2.58301	−0.198693	−0.0993464	−	0.995053i	$-0.531675\pi$
$13$	−2.58301	−0.198693	−0.0993464	+	0.995053i	$0.531675\pi$
$14$	− 3.74166i	− 0.267261i
$15$	0	0
$16$	4.00000	0.250000
$17$	25.8681i	1.52165i	0.648956	+	0.760826i	$0.275206\pi$
$17$	25.8681i	1.52165i	−0.648956	+	0.760826i	$0.724794\pi$
$18$	0	0
$19$	20.0000	1.05263	0.526316	−	0.850289i	$-0.323573\pi$
$19$	20.0000	1.05263	0.526316	+	0.850289i	$0.323573\pi$
$20$	0	0
$21$	0	0
$22$	−25.1660	−1.14391
$23$	− 17.7951i	− 0.773698i	−0.922143	−	0.386849i	$-0.873563\pi$
$23$	− 17.7951i	− 0.773698i	0.922143	−	0.386849i	$-0.126437\pi$
$24$	0	0
$25$	0	0
$26$	− 3.65292i	− 0.140497i
$27$	0	0
$28$	5.29150	0.188982
$29$	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	$0.589298\pi$
$32$	5.65685i	0.176777i
$33$	0	0
$34$	−36.5830	−1.07597
$35$	0	0
$36$	0	0
$37$	−38.0000	−1.02703	−0.513514	−	0.858082i	$-0.671656\pi$
$37$	−38.0000	−1.02703	−0.513514	+	0.858082i	$0.671656\pi$
$38$	28.2843i	0.744323i
$39$	0	0
$40$	0	0
$41$	− 15.7338i	− 0.383752i	−0.981419	−	0.191876i	$-0.938543\pi$
$41$	− 15.7338i	− 0.383752i	0.981419	−	0.191876i	$-0.0614571\pi$
$42$	0	0
$43$	43.4980	1.01158	0.505791	−	0.862656i	$-0.331201\pi$
$43$	43.4980	1.01158	0.505791	+	0.862656i	$0.331201\pi$
$44$	− 35.5901i	− 0.808866i
$45$	0	0
$46$	25.1660	0.547087
$47$	16.9706i	0.361076i	0.983568	+	0.180538i	$0.0577838\pi$
$47$	16.9706i	0.361076i	−0.983568	+	0.180538i	$0.942216\pi$
$48$	0	0
$49$	7.00000	0.142857
$50$	0	0
$51$	0	0
$52$	5.16601	0.0993464
$53$	85.5571i	1.61429i	0.590356	+	0.807143i	$0.298987\pi$
$53$	85.5571i	1.61429i	−0.590356	+	0.807143i	$0.701013\pi$
$54$	0	0
$55$	0	0
$56$	7.48331i	0.133631i
$57$	0	0
$58$	−16.8340	−0.290241
$59$	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	$0.192638\pi$
$62$	− 24.2764i	− 0.391555i
$63$	0	0
$64$	−8.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	−36.6640	−0.547225	−0.273612	−	0.961840i	$-0.588219\pi$
$67$	−36.6640	−0.547225	−0.273612	+	0.961840i	$0.588219\pi$
$68$	− 51.7362i	− 0.760826i
$69$	0	0
$70$	0	0
$71$	17.7951i	0.250635i	0.992117	+	0.125317i	$0.0399949\pi$
$71$	17.7951i	0.250635i	−0.992117	+	0.125317i	$0.960005\pi$
$72$	0	0
$73$	−28.9150	−0.396096	−0.198048	−	0.980192i	$-0.563460\pi$
$73$	−28.9150	−0.396096	−0.198048	+	0.980192i	$0.563460\pi$
$74$	− 53.7401i	− 0.726218i
$75$	0	0
$76$	−40.0000	−0.526316
$77$	− 47.0813i	− 0.611445i
$78$	0	0
$79$	118.332	1.49787	0.748937	−	0.662641i	$-0.230565\pi$
$79$	118.332	1.49787	0.748937	+	0.662641i	$0.230565\pi$
$80$	0	0
$81$	0	0
$82$	22.2510	0.271353
$83$	120.443i	1.45112i	0.688159	+	0.725560i	$0.258419\pi$
$83$	120.443i	1.45112i	−0.688159	+	0.725560i	$0.741581\pi$
$84$	0	0
$85$	0	0
$86$	61.5155i	0.715297i
$87$	0	0
$88$	50.3320	0.571955
$89$	− 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
$92$	35.5901i	0.386849i
$93$	0	0
$94$	−24.0000	−0.255319
$95$	0	0
$96$	0	0
$97$	−44.4131	−0.457867	−0.228933	−	0.973442i	$-0.573524\pi$
$97$	−44.4131	−0.457867	−0.228933	+	0.973442i	$0.573524\pi$
$98$	9.89949i	0.101015i
$99$	0	0
$100$	0	0
$101$	31.8799i	0.315642i	0.987468	+	0.157821i	$0.0504470\pi$
$101$	31.8799i	0.315642i	−0.987468	+	0.157821i	$0.949553\pi$
$102$	0	0
$103$	−4.50197	−0.0437084	−0.0218542	−	0.999761i	$-0.506957\pi$
$103$	−4.50197	−0.0437084	−0.0218542	+	0.999761i	$0.506957\pi$
$104$	7.30584i	0.0702485i
$105$	0	0
$106$	−120.996	−1.14147
$107$	− 172.179i	− 1.60915i	−0.593851	−	0.804575i	$-0.702393\pi$
$107$	− 172.179i	− 1.60915i	0.593851	−	0.804575i	$-0.297607\pi$
$108$	0	0
$109$	−177.830	−1.63147	−0.815734	−	0.578427i	$-0.803667\pi$
$109$	−177.830	−1.63147	−0.815734	+	0.578427i	$0.803667\pi$
$110$	0	0
$111$	0	0
$112$	−10.5830	−0.0944911
$113$	− 31.3475i	− 0.277411i	−0.990334	−	0.138706i	$-0.955706\pi$
$113$	− 31.3475i	− 0.277411i	0.990334	−	0.138706i	$-0.0442942\pi$
$114$	0	0
$115$	0	0
$116$	− 23.8069i	− 0.205232i
$117$	0	0
$118$	−2.33202	−0.0197629
$119$	− 68.4405i	− 0.575131i
$120$	0	0
$121$	−195.664	−1.61706
$122$	141.891i	1.16304i
$123$	0	0
$124$	34.3320	0.276871
$125$	0	0
$126$	0	0
$127$	−214.332	−1.68765	−0.843827	−	0.536616i	$-0.819703\pi$
$127$	−214.332	−1.68765	−0.843827	+	0.536616i	$0.819703\pi$
$128$	− 11.3137i	− 0.0883883i
$129$	0	0
$130$	0	0
$131$	91.4488i	0.698082i	0.937107	+	0.349041i	$0.113493\pi$
$131$	91.4488i	0.698082i	−0.937107	+	0.349041i	$0.886507\pi$
$132$	0	0
$133$	−52.9150	−0.397857
$134$	− 51.8508i	− 0.386946i
$135$	0	0
$136$	73.1660	0.537985
$137$	106.891i	0.780223i	0.920768	+	0.390111i	$0.127563\pi$
$137$	106.891i	0.780223i	−0.920768	+	0.390111i	$0.872437\pi$
$138$	0	0
$139$	−121.328	−0.872864	−0.436432	−	0.899737i	$-0.643758\pi$
$139$	−121.328	−0.872864	−0.436432	+	0.899737i	$0.643758\pi$
$140$	0	0
$141$	0	0
$142$	−25.1660	−0.177225
$143$	− 45.9647i	− 0.321432i
$144$	0	0
$145$	0	0
$146$	− 40.8920i	− 0.280082i
$147$	0	0
$148$	76.0000	0.513514
$149$	− 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	$0.553836\pi$
$152$	− 56.5685i	− 0.372161i
$153$	0	0
$154$	66.5830	0.432357
$155$	0	0
$156$	0	0
$157$	−68.9961	−0.439465	−0.219733	−	0.975560i	$-0.570519\pi$
$157$	−68.9961	−0.439465	−0.219733	+	0.975560i	$0.570519\pi$
$158$	167.347i	1.05916i
$159$	0	0
$160$	0	0
$161$	47.0813i	0.292430i
$162$	0	0
$163$	166.996	1.02452	0.512258	−	0.858832i	$-0.328809\pi$
$163$	166.996	1.02452	0.512258	+	0.858832i	$0.328809\pi$
$164$	31.4676i	0.191876i
$165$	0	0
$166$	−170.332	−1.02610
$167$	− 120.443i	− 0.721215i	−0.932718	−	0.360608i	$-0.882569\pi$
$167$	− 120.443i	− 0.721215i	0.932718	−	0.360608i	$-0.117431\pi$
$168$	0	0
$169$	−162.328	−0.960521
$170$	0	0
$171$	0	0
$172$	−86.9961	−0.505791
$173$	91.8610i	0.530989i	0.964112	+	0.265494i	$0.0855351\pi$
$173$	91.8610i	0.530989i	−0.964112	+	0.265494i	$0.914465\pi$
$174$	0	0
$175$	0	0
$176$	71.1802i	0.404433i
$177$	0	0
$178$	197.247	1.10813
$179$	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	$0.426286\pi$
$182$	9.66472i	0.0531029i
$183$	0	0
$184$	−50.3320	−0.273544
$185$	0	0
$186$	0	0
$187$	−460.324	−2.46163
$188$	− 33.9411i	− 0.180538i
$189$	0	0
$190$	0	0
$191$	41.3616i	0.216553i	0.994121	+	0.108276i	$0.0345331\pi$
$191$	41.3616i	0.216553i	−0.994121	+	0.108276i	$0.965467\pi$
$192$	0	0
$193$	−134.000	−0.694301	−0.347150	−	0.937810i	$-0.612851\pi$
$193$	−134.000	−0.694301	−0.347150	+	0.937810i	$0.612851\pi$
$194$	− 62.8095i	− 0.323761i
$195$	0	0
$196$	−14.0000	−0.0714286
$197$	68.8269i	0.349375i	0.984624	+	0.174688i	$0.0558916\pi$
$197$	68.8269i	0.349375i	−0.984624	+	0.174688i	$0.944108\pi$
$198$	0	0
$199$	−278.494	−1.39947	−0.699734	−	0.714404i	$-0.746698\pi$
$199$	−278.494	−1.39947	−0.699734	+	0.714404i	$0.746698\pi$
$200$	0	0
$201$	0	0
$202$	−45.0850	−0.223193
$203$	− 31.4935i	− 0.155140i
$204$	0	0
$205$	0	0
$206$	− 6.36674i	− 0.0309065i
$207$	0	0
$208$	−10.3320	−0.0496732
$209$	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	$0.667101\pi$
$212$	− 171.114i	− 0.807143i
$213$	0	0
$214$	243.498	1.13784
$215$	0	0
$216$	0	0
$217$	45.4170	0.209295
$218$	− 251.490i	− 1.15362i
$219$	0	0
$220$	0	0
$221$	− 66.8174i	− 0.302341i
$222$	0	0
$223$	−222.494	−0.997731	−0.498866	−	0.866679i	$-0.666250\pi$
$223$	−222.494	−0.997731	−0.498866	+	0.866679i	$0.666250\pi$
$224$	− 14.9666i	− 0.0668153i
$225$	0	0
$226$	44.3320	0.196159
$227$	− 101.823i	− 0.448561i	−0.974525	−	0.224281i	$-0.927997\pi$
$227$	− 101.823i	− 0.448561i	0.974525	−	0.224281i	$-0.0720032\pi$
$228$	0	0
$229$	−163.085	−0.712161	−0.356081	−	0.934455i	$-0.615887\pi$
$229$	−163.085	−0.712161	−0.356081	+	0.934455i	$0.615887\pi$
$230$	0	0
$231$	0	0
$232$	33.6680	0.145121
$233$	− 362.858i	− 1.55733i	−0.627441	−	0.778664i	$-0.715898\pi$
$233$	− 362.858i	− 1.55733i	0.627441	−	0.778664i	$-0.284102\pi$
$234$	0	0
$235$	0	0
$236$	− 3.29798i	− 0.0139745i
$237$	0	0
$238$	96.7895	0.406679
$239$	− 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	$0.397353\pi$
$242$	− 276.711i	− 1.14343i
$243$	0	0
$244$	−200.664	−0.822394
$245$	0	0
$246$	0	0
$247$	−51.6601	−0.209150
$248$	48.5528i	0.195777i
$249$	0	0
$250$	0	0
$251$	356.382i	1.41985i	0.704278	+	0.709924i	$0.251271\pi$
$251$	356.382i	1.41985i	−0.704278	+	0.709924i	$0.748729\pi$
$252$	0	0
$253$	316.664	1.25164
$254$	− 303.111i	− 1.19335i
$255$	0	0
$256$	16.0000	0.0625000
$257$	− 59.5689i	− 0.231786i	−0.993262	−	0.115893i	$-0.963027\pi$
$257$	− 59.5689i	− 0.231786i	0.993262	−	0.115893i	$-0.0369729\pi$
$258$	0	0
$259$	100.539	0.388180
$260$	0	0
$261$	0	0
$262$	−129.328	−0.493619
$263$	7.42045i	0.0282146i	0.999900	+	0.0141073i	$0.00449065\pi$
$263$	7.42045i	0.0282146i	−0.999900	+	0.0141073i	$0.995509\pi$
$264$	0	0
$265$	0	0
$266$	− 74.8331i	− 0.281328i
$267$	0	0
$268$	73.3281	0.273612
$269$	− 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	$0.524200\pi$
$272$	103.472i	0.380413i
$273$	0	0
$274$	−151.166	−0.551701
$275$	0	0
$276$	0	0
$277$	−32.0000	−0.115523	−0.0577617	−	0.998330i	$-0.518396\pi$
$277$	−32.0000	−0.115523	−0.0577617	+	0.998330i	$0.518396\pi$
$278$	− 171.584i	− 0.617208i
$279$	0	0
$280$	0	0
$281$	17.0907i	0.0608211i	0.999537	+	0.0304106i	$0.00968147\pi$
$281$	17.0907i	0.0608211i	−0.999537	+	0.0304106i	$0.990319\pi$
$282$	0	0
$283$	−439.660	−1.55357	−0.776785	−	0.629766i	$-0.783151\pi$
$283$	−439.660	−1.55357	−0.776785	+	0.629766i	$0.783151\pi$
$284$	− 35.5901i	− 0.125317i
$285$	0	0
$286$	65.0039	0.227286
$287$	41.6278i	0.145045i
$288$	0	0
$289$	−380.158	−1.31543
$290$	0	0
$291$	0	0
$292$	57.8301	0.198048
$293$	394.377i	1.34600i	0.739644	+	0.672998i	$0.234994\pi$
$293$	394.377i	1.34600i	−0.739644	+	0.672998i	$0.765006\pi$
$294$	0	0
$295$	0	0
$296$	107.480i	0.363109i
$297$	0	0
$298$	24.9961	0.0838794
$299$	45.9647i	0.153728i
$300$	0	0
$301$	−115.085	−0.382342
$302$	− 71.8901i	− 0.238047i
$303$	0	0
$304$	80.0000	0.263158
$305$	0	0
$306$	0	0
$307$	23.3360	0.0760129	0.0380064	−	0.999277i	$-0.487899\pi$
$307$	23.3360	0.0760129	0.0380064	+	0.999277i	$0.487899\pi$
$308$	94.1626i	0.305723i
$309$	0	0
$310$	0	0
$311$	− 527.256i	− 1.69536i	−0.530511	−	0.847678i	$-0.678000\pi$
$311$	− 527.256i	− 1.69536i	0.530511	−	0.847678i	$-0.322000\pi$
$312$	0	0
$313$	295.328	0.943540	0.471770	−	0.881722i	$-0.343615\pi$
$313$	295.328	0.943540	0.471770	+	0.881722i	$0.343615\pi$
$314$	− 97.5752i	− 0.310749i
$315$	0	0
$316$	−236.664	−0.748937
$317$	− 107.475i	− 0.339037i	−0.985527	−	0.169518i	$-0.945779\pi$
$317$	− 107.475i	− 0.339037i	0.985527	−	0.169518i	$-0.0542212\pi$
$318$	0	0
$319$	−211.822	−0.664019
$320$	0	0
$321$	0	0
$322$	−66.5830	−0.206780
$323$	517.362i	1.60174i
$324$	0	0
$325$	0	0
$326$	236.168i	0.724442i
$327$	0	0
$328$	−44.5020	−0.135677
$329$	− 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	$0.635563\pi$
$332$	− 240.886i	− 0.725560i
$333$	0	0
$334$	170.332	0.509976
$335$	0	0
$336$	0	0
$337$	341.166	1.01236	0.506181	−	0.862427i	$-0.331057\pi$
$337$	341.166	1.01236	0.506181	+	0.862427i	$0.331057\pi$
$338$	− 229.567i	− 0.679191i
$339$	0	0
$340$	0	0
$341$	− 305.470i	− 0.895807i
$342$	0	0
$343$	−18.5203	−0.0539949
$344$	− 123.031i	− 0.357648i
$345$	0	0
$346$	−129.911	−0.375466
$347$	− 116.320i	− 0.335217i	−0.985854	−	0.167609i	$-0.946395\pi$
$347$	− 116.320i	− 0.335217i	0.985854	−	0.167609i	$-0.0536045\pi$
$348$	0	0
$349$	−158.324	−0.453651	−0.226825	−	0.973935i	$-0.572835\pi$
$349$	−158.324	−0.453651	−0.226825	+	0.973935i	$0.572835\pi$
$350$	0	0
$351$	0	0
$352$	−100.664	−0.285977
$353$	230.339i	0.652519i	0.945280	+	0.326260i	$0.105788\pi$
$353$	230.339i	0.652519i	−0.945280	+	0.326260i	$0.894212\pi$
$354$	0	0
$355$	0	0
$356$	278.949i	0.783566i
$357$	0	0
$358$	−188.502	−0.526542
$359$	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
$362$	117.500i	0.324585i
$363$	0	0
$364$	−13.6680	−0.0375494
$365$	0	0
$366$	0	0
$367$	−517.490	−1.41005	−0.705027	−	0.709180i	$-0.749065\pi$
$367$	−517.490	−1.41005	−0.705027	+	0.709180i	$0.749065\pi$
$368$	− 71.1802i	− 0.193425i
$369$	0	0
$370$	0	0
$371$	− 226.363i	− 0.610143i
$372$	0	0
$373$	233.336	0.625566	0.312783	−	0.949825i	$-0.398739\pi$
$373$	233.336	0.625566	0.312783	+	0.949825i	$0.398739\pi$
$374$	− 650.997i	− 1.74063i
$375$	0	0
$376$	48.0000	0.127660
$377$	− 30.7466i	− 0.0815560i
$378$	0	0
$379$	441.166	1.16403	0.582013	−	0.813179i	$-0.302265\pi$
$379$	441.166	1.16403	0.582013	+	0.813179i	$0.302265\pi$
$380$	0	0
$381$	0	0
$382$	−58.4941	−0.153126
$383$	213.060i	0.556292i	0.960539	+	0.278146i	$0.0897200\pi$
$383$	213.060i	0.556292i	−0.960539	+	0.278146i	$0.910280\pi$
$384$	0	0
$385$	0	0
$386$	− 189.505i	− 0.490945i
$387$	0	0
$388$	88.8261	0.228933
$389$	− 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
$392$	− 19.7990i	− 0.0505076i
$393$	0	0
$394$	−97.3360	−0.247046
$395$	0	0
$396$	0	0
$397$	−498.324	−1.25522	−0.627612	−	0.778526i	$-0.715968\pi$
$397$	−498.324	−1.25522	−0.627612	+	0.778526i	$0.715968\pi$
$398$	− 393.850i	− 0.989573i
$399$	0	0
$400$	0	0
$401$	− 193.392i	− 0.482275i	−0.970491	−	0.241138i	$-0.922480\pi$
$401$	− 193.392i	− 0.482275i	0.970491	−	0.241138i	$-0.0775205\pi$
$402$	0	0
$403$	44.3399	0.110025
$404$	− 63.7598i	− 0.157821i
$405$	0	0
$406$	44.5385	0.109701
$407$	− 676.212i	− 1.66145i
$408$	0	0
$409$	−454.243	−1.11062	−0.555309	−	0.831644i	$-0.687400\pi$
$409$	−454.243	−1.11062	−0.555309	+	0.831644i	$0.687400\pi$
$410$	0	0
$411$	0	0
$412$	9.00394	0.0218542
$413$	− 4.36281i	− 0.0105637i
$414$	0	0
$415$	0	0
$416$	− 14.6117i	− 0.0351242i
$417$	0	0
$418$	−503.320	−1.20412
$419$	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	$0.405104\pi$
$422$	− 299.103i	− 0.708776i
$423$	0	0
$424$	241.992	0.570736
$425$	0	0
$426$	0	0
$427$	−265.454	−0.621671
$428$	344.358i	0.804575i
$429$	0	0
$430$	0	0
$431$	− 456.419i	− 1.05898i	−0.848317	−	0.529489i	$-0.822384\pi$
$431$	− 456.419i	− 1.05898i	0.848317	−	0.529489i	$-0.177616\pi$
$432$	0	0
$433$	637.984	1.47340	0.736702	−	0.676217i	$-0.236382\pi$
$433$	637.984	1.47340	0.736702	+	0.676217i	$0.236382\pi$
$434$	64.2293i	0.147994i
$435$	0	0
$436$	355.660	0.815734
$437$	− 355.901i	− 0.814419i
$438$	0	0
$439$	−784.146	−1.78621	−0.893105	−	0.449848i	$-0.851478\pi$
$439$	−784.146	−1.78621	−0.893105	+	0.449848i	$0.851478\pi$
$440$	0	0
$441$	0	0
$442$	94.4941	0.213788
$443$	472.222i	1.06596i	0.846127	+	0.532981i	$0.178928\pi$
$443$	472.222i	1.06596i	−0.846127	+	0.532981i	$0.821072\pi$
$444$	0	0
$445$	0	0
$446$	− 314.654i	− 0.705503i
$447$	0	0
$448$	21.1660	0.0472456
$449$	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
$452$	62.6949i	0.138706i
$453$	0	0
$454$	144.000	0.317181
$455$	0	0
$456$	0	0
$457$	248.324	0.543379	0.271689	−	0.962385i	$-0.412418\pi$
$457$	248.324	0.543379	0.271689	+	0.962385i	$0.412418\pi$
$458$	− 230.637i	− 0.503574i
$459$	0	0
$460$	0	0
$461$	− 355.970i	− 0.772168i	−0.922464	−	0.386084i	$-0.873827\pi$
$461$	− 355.970i	− 0.772168i	0.922464	−	0.386084i	$-0.126173\pi$
$462$	0	0
$463$	6.33202	0.0136761	0.00683804	−	0.999977i	$-0.497823\pi$
$463$	6.33202	0.0136761	0.00683804	+	0.999977i	$0.497823\pi$
$464$	47.6137i	0.102616i
$465$	0	0
$466$	513.158	1.10120
$467$	878.691i	1.88156i	0.339011	+	0.940782i	$0.389907\pi$
$467$	878.691i	1.88156i	−0.339011	+	0.940782i	$0.610093\pi$
$468$	0	0
$469$	97.0039	0.206831
$470$	0	0
$471$	0	0
$472$	4.66404	0.00988144
$473$	774.050i	1.63647i
$474$	0	0
$475$	0	0
$476$	136.881i	0.287565i
$477$	0	0
$478$	250.494	0.524046
$479$	− 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
$482$	216.025i	0.448185i
$483$	0	0
$484$	391.328	0.808529
$485$	0	0
$486$	0	0
$487$	717.490	1.47329	0.736643	−	0.676282i	$-0.236410\pi$
$487$	717.490	1.47329	0.736643	+	0.676282i	$0.236410\pi$
$488$	− 283.782i	− 0.581520i
$489$	0	0
$490$	0	0
$491$	− 274.002i	− 0.558050i	−0.960284	−	0.279025i	$-0.909989\pi$
$491$	− 274.002i	− 0.558050i	0.960284	−	0.279025i	$-0.0900112\pi$
$492$	0	0
$493$	−307.919	−0.624582
$494$	− 73.0584i	− 0.147892i
$495$	0	0
$496$	−68.6640	−0.138436
$497$	− 47.0813i	− 0.0947310i
$498$	0	0
$499$	−728.810	−1.46054	−0.730271	−	0.683158i	$-0.760606\pi$
$499$	−728.810	−1.46054	−0.730271	+	0.683158i	$0.760606\pi$
$500$	0	0
$501$	0	0
$502$	−504.000	−1.00398
$503$	− 594.657i	− 1.18222i	−0.806590	−	0.591111i	$-0.798690\pi$
$503$	− 594.657i	− 1.18222i	0.806590	−	0.591111i	$-0.201310\pi$
$504$	0	0
$505$	0	0
$506$	447.831i	0.885041i
$507$	0	0
$508$	428.664	0.843827
$509$	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
$512$	22.6274i	0.0441942i
$513$	0	0
$514$	84.2431	0.163897
$515$	0	0
$516$	0	0
$517$	−301.992	−0.584124
$518$	142.183i	0.274485i
$519$	0	0
$520$	0	0
$521$	40.8459i	0.0783990i	0.999231	+	0.0391995i	$0.0124808\pi$
$521$	40.8459i	0.0783990i	−0.999231	+	0.0391995i	$0.987519\pi$
$522$	0	0
$523$	232.000	0.443595	0.221797	−	0.975093i	$-0.428808\pi$
$523$	232.000	0.443595	0.221797	+	0.975093i	$0.428808\pi$
$524$	− 182.898i	− 0.349041i
$525$	0	0
$526$	−10.4941	−0.0199507
$527$	− 444.052i	− 0.842603i
$528$	0	0
$529$	212.336	0.401391
$530$	0	0
$531$	0	0
$532$	105.830	0.198929
$533$	40.6405i	0.0762487i
$534$	0	0
$535$	0	0
$536$	103.702i	0.193473i
$537$	0	0
$538$	608.405	1.13086
$539$	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	$0.425682\pi$
$542$	− 58.2175i	− 0.107412i
$543$	0	0
$544$	−146.332	−0.268993
$545$	0	0
$546$	0	0
$547$	−888.324	−1.62399	−0.811996	−	0.583662i	$-0.801619\pi$
$547$	−888.324	−1.62399	−0.811996	+	0.583662i	$0.801619\pi$
$548$	− 213.781i	− 0.390111i
$549$	0	0
$550$	0	0
$551$	238.069i	0.432066i
$552$	0	0
$553$	−313.077	−0.566143
$554$	− 45.2548i	− 0.0816874i
$555$	0	0
$556$	242.656	0.436432
$557$	− 316.309i	− 0.567879i	−0.958842	−	0.283940i	$-0.908358\pi$
$557$	− 316.309i	− 0.567879i	0.958842	−	0.283940i	$-0.0916415\pi$
$558$	0	0
$559$	−112.356	−0.200994
$560$	0	0
$561$	0	0
$562$	−24.1699	−0.0430070
$563$	− 58.4690i	− 0.103853i	−0.998651	−	0.0519263i	$-0.983464\pi$
$563$	− 58.4690i	− 0.103853i	0.998651	−	0.0519263i	$-0.0165361\pi$
$564$	0	0
$565$	0	0
$566$	− 621.773i	− 1.09854i
$567$	0	0
$568$	50.3320	0.0886127
$569$	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	$0.640433\pi$
$572$	91.9294i	0.160716i
$573$	0	0
$574$	−58.8706	−0.102562
$575$	0	0
$576$	0	0
$577$	487.328	0.844589	0.422295	−	0.906459i	$-0.361225\pi$
$577$	487.328	0.844589	0.422295	+	0.906459i	$0.361225\pi$
$578$	− 537.625i	− 0.930147i
$579$	0	0
$580$	0	0
$581$	− 318.662i	− 0.548472i
$582$	0	0
$583$	−1522.49	−2.61148
$584$	81.7840i	0.140041i
$585$	0	0
$586$	−557.733	−0.951763
$587$	445.701i	0.759286i	0.925133	+	0.379643i	$0.123953\pi$
$587$	445.701i	0.759286i	−0.925133	+	0.379643i	$0.876047\pi$
$588$	0	0
$589$	−343.320	−0.582887
$590$	0	0
$591$	0	0
$592$	−152.000	−0.256757
$593$	276.648i	0.466523i	0.972414	+	0.233261i	$0.0749397\pi$
$593$	276.648i	0.466523i	−0.972414	+	0.233261i	$0.925060\pi$
$594$	0	0
$595$	0	0
$596$	35.3498i	0.0593117i
$597$	0	0
$598$	−65.0039	−0.108702
$599$	− 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	$0.613055\pi$
$602$	− 162.755i	− 0.270357i
$603$	0	0
$604$	101.668	0.168324
$605$	0	0
$606$	0	0
$607$	−76.8419	−0.126593	−0.0632964	−	0.997995i	$-0.520161\pi$
$607$	−76.8419	−0.126593	−0.0632964	+	0.997995i	$0.520161\pi$
$608$	113.137i	0.186081i
$609$	0	0
$610$	0	0
$611$	− 43.8351i	− 0.0717431i
$612$	0	0
$613$	−59.3281	−0.0967832	−0.0483916	−	0.998828i	$-0.515410\pi$
$613$	−59.3281	−0.0967832	−0.0483916	+	0.998828i	$0.515410\pi$
$614$	33.0020i	0.0537492i
$615$	0	0
$616$	−133.166	−0.216179
$617$	− 29.1143i	− 0.0471869i	−0.999722	−	0.0235935i	$-0.992489\pi$
$617$	− 29.1143i	− 0.0471869i	0.999722	−	0.0235935i	$-0.00751073\pi$
$618$	0	0
$619$	455.644	0.736098	0.368049	−	0.929806i	$-0.380026\pi$
$619$	455.644	0.736098	0.368049	+	0.929806i	$0.380026\pi$
$620$	0	0
$621$	0	0
$622$	745.652	1.19880
$623$	369.015i	0.592320i
$624$	0	0
$625$	0	0
$626$	417.657i	0.667184i
$627$	0	0
$628$	137.992	0.219733
$629$	− 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	$0.511354\pi$
$632$	− 334.693i	− 0.529578i
$633$	0	0
$634$	151.992	0.239735
$635$	0	0
$636$	0	0
$637$	−18.0810	−0.0283847
$638$	− 299.562i	− 0.469533i
$639$	0	0
$640$	0	0
$641$	− 641.223i	− 1.00035i	−0.865925	−	0.500174i	$-0.833269\pi$
$641$	− 641.223i	− 1.00035i	0.865925	−	0.500174i	$-0.166731\pi$
$642$	0	0
$643$	604.000	0.939347	0.469673	−	0.882840i	$-0.344372\pi$
$643$	604.000	0.939347	0.469673	+	0.882840i	$0.344372\pi$
$644$	− 94.1626i	− 0.146215i
$645$	0	0
$646$	−731.660	−1.13260
$647$	179.600i	0.277588i	0.990321	+	0.138794i	$0.0443226\pi$
$647$	179.600i	0.277588i	−0.990321	+	0.138794i	$0.955677\pi$
$648$	0	0
$649$	−29.3438	−0.0452139
$650$	0	0
$651$	0	0
$652$	−333.992	−0.512258
$653$	− 392.092i	− 0.600447i	−0.953869	−	0.300224i	$-0.902939\pi$
$653$	− 392.092i	− 0.600447i	0.953869	−	0.300224i	$-0.0970613\pi$
$654$	0	0
$655$	0	0
$656$	− 62.9353i	− 0.0959379i
$657$	0	0
$658$	63.4980	0.0965016
$659$	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	$0.255788\pi$
$662$	− 386.773i	− 0.584250i
$663$	0	0
$664$	340.664	0.513048
$665$	0	0
$666$	0	0
$667$	211.822	0.317574
$668$	240.886i	0.360608i
$669$	0	0
$670$	0	0
$671$	1785.41i	2.66083i
$672$	0	0
$673$	152.008	0.225866	0.112933	−	0.993603i	$-0.463975\pi$
$673$	152.008	0.225866	0.112933	+	0.993603i	$0.463975\pi$
$674$	482.482i	0.715848i
$675$	0	0
$676$	324.656	0.480261
$677$	− 163.178i	− 0.241031i	−0.992711	−	0.120516i	$-0.961545\pi$
$677$	− 163.178i	− 0.241031i	0.992711	−	0.120516i	$-0.0384548\pi$
$678$	0	0
$679$	117.506	0.173057
$680$	0	0
$681$	0	0
$682$	432.000	0.633431
$683$	324.914i	0.475716i	0.971300	+	0.237858i	$0.0764453\pi$
$683$	324.914i	0.475716i	−0.971300	+	0.237858i	$0.923555\pi$
$684$	0	0
$685$	0	0
$686$	− 26.1916i	− 0.0381802i
$687$	0	0
$688$	173.992	0.252896
$689$	− 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	$0.156169\pi$
$692$	− 183.722i	− 0.265494i
$693$	0	0
$694$	164.502	0.237035
$695$	0	0
$696$	0	0
$697$	407.004	0.583937
$698$	− 223.904i	− 0.320780i
$699$	0	0
$700$	0	0
$701$	− 427.202i	− 0.609417i	−0.952446	−	0.304709i	$-0.901441\pi$
$701$	− 427.202i	− 0.609417i	0.952446	−	0.304709i	$-0.0985591\pi$
$702$	0	0
$703$	−760.000	−1.08108
$704$	− 142.360i	− 0.202217i
$705$	0	0
$706$	−325.749	−0.461401
$707$	− 84.3463i	− 0.119302i
$708$	0	0
$709$	71.4980	0.100843	0.0504217	−	0.998728i	$-0.483943\pi$
$709$	71.4980	0.100843	0.0504217	+	0.998728i	$0.483943\pi$
$710$	0	0
$711$	0	0
$712$	−394.494	−0.554065
$713$	305.470i	0.428429i
$714$	0	0
$715$	0	0
$716$	− 266.582i	− 0.372321i
$717$	0	0
$718$	−242.818	−0.338187
$719$	− 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
$722$	55.1543i	0.0763910i
$723$	0	0
$724$	−166.170	−0.229517
$725$	0	0
$726$	0	0
$727$	1338.82	1.84157	0.920783	−	0.390076i	$-0.127551\pi$
$727$	1338.82	1.84157	0.920783	+	0.390076i	$0.127551\pi$
$728$	− 19.3294i	− 0.0265514i
$729$	0	0
$730$	0	0
$731$	1125.21i	1.53928i
$732$	0	0
$733$	−49.0771	−0.0669538	−0.0334769	−	0.999439i	$-0.510658\pi$
$733$	−49.0771	−0.0669538	−0.0334769	+	0.999439i	$0.510658\pi$
$734$	− 731.842i	− 0.997059i
$735$	0	0
$736$	100.664	0.136772
$737$	− 652.439i	− 0.885263i
$738$	0	0
$739$	1430.32	1.93548	0.967738	−	0.251960i	$-0.0810751\pi$
$739$	1430.32	1.93548	0.967738	+	0.251960i	$0.0810751\pi$
$740$	0	0
$741$	0	0
$742$	320.125	0.431436
$743$	− 875.736i	− 1.17865i	−0.807896	−	0.589325i	$-0.799394\pi$
$743$	− 875.736i	− 1.17865i	0.807896	−	0.589325i	$-0.200606\pi$
$744$	0	0
$745$	0	0
$746$	329.987i	0.442342i
$747$	0	0
$748$	920.648	1.23081
$749$	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	$0.431481\pi$
$752$	67.8823i	0.0902690i
$753$	0	0
$754$	43.4823	0.0576688
$755$	0	0
$756$	0	0
$757$	−289.830	−0.382867	−0.191433	−	0.981506i	$-0.561314\pi$
$757$	−289.830	−0.382867	−0.191433	+	0.981506i	$0.561314\pi$
$758$	623.903i	0.823091i
$759$	0	0
$760$	0	0
$761$	− 704.657i	− 0.925962i	−0.886368	−	0.462981i	$-0.846780\pi$
$761$	− 704.657i	− 0.925962i	0.886368	−	0.462981i	$-0.153220\pi$
$762$	0	0
$763$	470.494	0.616637
$764$	− 82.7231i	− 0.108276i
$765$	0	0
$766$	−301.312	−0.393358
$767$	− 4.25934i	− 0.00555325i
$768$	0	0
$769$	−117.320	−0.152562	−0.0762810	−	0.997086i	$-0.524305\pi$
$769$	−117.320	−0.152562	−0.0762810	+	0.997086i	$0.524305\pi$
$770$	0	0
$771$	0	0
$772$	268.000	0.347150
$773$	− 658.005i	− 0.851235i	−0.904903	−	0.425618i	$-0.860057\pi$
$773$	− 658.005i	− 0.851235i	0.904903	−	0.425618i	$-0.139943\pi$
$774$	0	0
$775$	0	0
$776$	125.619i	0.161880i
$777$	0	0
$778$	799.166	1.02721
$779$	− 314.676i	− 0.403949i
$780$	0	0
$781$	−316.664	−0.405460
$782$	650.997i	0.832477i
$783$	0	0
$784$	28.0000	0.0357143
$785$	0	0
$786$	0	0
$787$	1200.65	1.52560	0.762801	−	0.646634i	$-0.223824\pi$
$787$	1200.65	1.52560	0.762801	+	0.646634i	$0.223824\pi$
$788$	− 137.654i	− 0.174688i
$789$	0	0
$790$	0	0
$791$	82.9376i	0.104852i
$792$	0	0
$793$	−259.158	−0.326807
$794$	− 704.737i	− 0.887578i
$795$	0	0
$796$	556.988	0.699734
$797$	− 797.411i	− 1.00052i	−0.865876	−	0.500258i	$-0.833239\pi$
$797$	− 797.411i	− 1.00052i	0.865876	−	0.500258i	$-0.166761\pi$
$798$	0	0
$799$	−438.996	−0.549432
$800$	0	0
$801$	0	0
$802$	273.498	0.341020
$803$	− 514.545i	− 0.640778i
$804$	0	0
$805$	0	0
$806$	62.7061i	0.0777991i
$807$	0	0
$808$	90.1699	0.111596
$809$	− 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	$0.620257\pi$
$812$	62.9870i	0.0775702i
$813$	0	0
$814$	956.308	1.17483
$815$	0	0
$816$	0	0
$817$	869.961	1.06482
$818$	− 642.397i	− 0.785326i
$819$	0	0
$820$	0	0
$821$	− 980.978i	− 1.19486i	−0.801922	−	0.597429i	$-0.796189\pi$
$821$	− 980.978i	− 1.19486i	0.801922	−	0.597429i	$-0.203811\pi$
$822$	0	0
$823$	431.336	0.524102	0.262051	−	0.965054i	$-0.415601\pi$
$823$	431.336	0.524102	0.262051	+	0.965054i	$0.415601\pi$
$824$	12.7335i	0.0154533i
$825$	0	0
$826$	6.16995	0.00746967
$827$	1219.41i	1.47449i	0.675623	+	0.737247i	$0.263875\pi$
$827$	1219.41i	1.47449i	−0.675623	+	0.737247i	$0.736125\pi$
$828$	0	0
$829$	−770.081	−0.928928	−0.464464	−	0.885592i	$-0.653753\pi$
$829$	−770.081	−0.928928	−0.464464	+	0.885592i	$0.653753\pi$
$830$	0	0
$831$	0	0
$832$	20.6640	0.0248366
$833$	181.077i	0.217379i
$834$	0	0
$835$	0	0
$836$	− 711.802i	− 0.851438i
$837$	0	0
$838$	−480.000	−0.572792
$839$	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
$842$	349.764i	0.415396i
$843$	0	0
$844$	422.996	0.501180
$845$	0	0
$846$	0	0
$847$	517.678	0.611191
$848$	342.229i	0.403571i
$849$	0	0
$850$	0	0
$851$	676.212i	0.794609i
$852$	0	0
$853$	−898.988	−1.05391	−0.526957	−	0.849892i	$-0.676667\pi$
$853$	−898.988	−1.05391	−0.526957	+	0.849892i	$0.676667\pi$
$854$	− 375.408i	− 0.439588i
$855$	0	0
$856$	−486.996	−0.568921
$857$	746.156i	0.870660i	0.900271	+	0.435330i	$0.143368\pi$
$857$	746.156i	0.870660i	−0.900271	+	0.435330i	$0.856632\pi$
$858$	0	0
$859$	−991.984	−1.15481	−0.577406	−	0.816457i	$-0.695935\pi$
$859$	−991.984	−1.15481	−0.577406	+	0.816457i	$0.695935\pi$
$860$	0	0
$861$	0	0
$862$	645.474	0.748810
$863$	209.418i	0.242663i	0.992612	+	0.121332i	$0.0387164\pi$
$863$	209.418i	0.242663i	−0.992612	+	0.121332i	$0.961284\pi$
$864$	0	0
$865$	0	0
$866$	902.246i	1.04185i
$867$	0	0
$868$	−90.8340	−0.104647
$869$	2105.73i	2.42316i
$870$	0	0
$871$	94.7034	0.108730
$872$	502.979i	0.576811i
$873$	0	0
$874$	503.320	0.575881
$875$	0	0
$876$	0	0
$877$	865.304	0.986664	0.493332	−	0.869841i	$-0.335779\pi$
$877$	865.304	0.986664	0.493332	+	0.869841i	$0.335779\pi$
$878$	− 1108.95i	− 1.26304i
$879$	0	0
$880$	0	0
$881$	− 995.046i	− 1.12945i	−0.825279	−	0.564725i	$-0.808982\pi$
$881$	− 995.046i	− 1.12945i	0.825279	−	0.564725i	$-0.191018\pi$
$882$	0	0
$883$	−101.474	−0.114920	−0.0574600	−	0.998348i	$-0.518300\pi$
$883$	−101.474	−0.114920	−0.0574600	+	0.998348i	$0.518300\pi$
$884$	133.635i	0.151171i
$885$	0	0
$886$	−667.822	−0.753750
$887$	1074.09i	1.21093i	0.795873	+	0.605464i	$0.207012\pi$
$887$	1074.09i	1.21093i	−0.795873	+	0.605464i	$0.792988\pi$
$888$	0	0
$889$	567.069	0.637873
$890$	0	0
$891$	0	0
$892$	444.988	0.498866
$893$	339.411i	0.380080i
$894$	0	0
$895$	0	0
$896$	29.9333i	0.0334077i
$897$	0	0
$898$	−1046.31	−1.16515
$899$	− 204.334i	− 0.227291i
$900$	0	0
$901$	−2213.20	−2.45638
$902$	395.958i	0.438977i
$903$	0	0
$904$	−88.6640	−0.0980797
$905$	0	0
$906$	0	0
$907$	432.162	0.476474	0.238237	−	0.971207i	$-0.423430\pi$
$907$	432.162	0.476474	0.238237	+	0.971207i	$0.423430\pi$
$908$	203.647i	0.224281i
$909$	0	0
$910$	0	0
$911$	104.984i	0.115241i	0.998339	+	0.0576205i	$0.0183513\pi$
$911$	104.984i	0.115241i	−0.998339	+	0.0576205i	$0.981649\pi$
$912$	0	0
$913$	−2143.29	−2.34752
$914$	351.183i	0.384227i
$915$	0	0
$916$	326.170	0.356081
$917$	− 241.951i	− 0.263850i
$918$	0	0
$919$	−91.8379	−0.0999325	−0.0499662	−	0.998751i	$-0.515911\pi$
$919$	−91.8379	−0.0999325	−0.0499662	+	0.998751i	$0.515911\pi$
$920$	0	0
$921$	0	0
$922$	503.417	0.546005
$923$	− 45.9647i	− 0.0497993i
$924$	0	0
$925$	0	0
$926$	8.95483i	0.00967044i
$927$	0	0
$928$	−67.3360	−0.0725603
$929$	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
$932$	725.715i	0.778664i
$933$	0	0
$934$	−1242.66	−1.33047
$935$	0	0
$936$	0	0
$937$	−1262.00	−1.34685	−0.673426	−	0.739255i	$-0.735178\pi$
$937$	−1262.00	−1.34685	−0.673426	+	0.739255i	$0.735178\pi$
$938$	137.184i	0.146252i
$939$	0	0
$940$	0	0
$941$	1315.97i	1.39849i	0.714884	+	0.699243i	$0.246479\pi$
$941$	1315.97i	1.39849i	−0.714884	+	0.699243i	$0.753521\pi$
$942$	0	0
$943$	−279.984	−0.296908
$944$	6.59595i	0.00698724i
$945$	0	0
$946$	−1094.67	−1.15716
$947$	486.582i	0.513814i	0.966436	+	0.256907i	$0.0827034\pi$
$947$	486.582i	0.513814i	−0.966436	+	0.256907i	$0.917297\pi$
$948$	0	0
$949$	74.6877	0.0787014
$950$	0	0
$951$	0	0
$952$	−193.579	−0.203339
$953$	43.3711i	0.0455100i	0.999741	+	0.0227550i	$0.00724377\pi$
$953$	43.3711i	0.0455100i	−0.999741	+	0.0227550i	$0.992756\pi$
$954$	0	0
$955$	0	0
$956$	354.252i	0.370557i
$957$	0	0
$958$	317.344	0.331257
$959$	− 282.806i	− 0.294896i
$960$	0	0
$961$	−666.328	−0.693369
$962$	138.811i	0.144294i
$963$	0	0
$964$	−305.506	−0.316915
$965$	0	0
$966$	0	0
$967$	−1648.99	−1.70526	−0.852631	−	0.522514i	$-0.824994\pi$
$967$	−1648.99	−1.70526	−0.852631	+	0.522514i	$0.824994\pi$
$968$	553.421i	0.571716i
$969$	0	0
$970$	0	0
$971$	− 518.323i	− 0.533803i	−0.963724	−	0.266902i	$-0.914000\pi$
$971$	− 518.323i	− 0.533803i	0.963724	−	0.266902i	$-0.0859999\pi$
$972$	0	0
$973$	321.004	0.329912
$974$	1014.68i	1.04177i
$975$	0	0
$976$	401.328	0.411197
$977$	109.948i	0.112536i	0.998416	+	0.0562682i	$0.0179202\pi$
$977$	109.948i	0.112536i	−0.998416	+	0.0562682i	$0.982080\pi$
$978$	0	0
$979$	2481.96	2.53520
$980$	0	0
$981$	0	0
$982$	387.498	0.394601
$983$	− 589.710i	− 0.599909i	−0.953954	−	0.299954i	$-0.903029\pi$
$983$	− 589.710i	− 0.599909i	0.953954	−	0.299954i	$-0.0969715\pi$
$984$	0	0
$985$	0	0
$986$	− 435.463i	− 0.441646i
$987$	0	0
$988$	103.320	0.104575
$989$	− 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	$0.382785\pi$
$992$	− 97.1056i	− 0.0978887i
$993$	0	0
$994$	66.5830	0.0669849
$995$	0	0
$996$	0	0
$997$	−1222.99	−1.22667	−0.613334	−	0.789824i	$-0.710172\pi$
$997$	−1222.99	−1.22667	−0.613334	+	0.789824i	$0.710172\pi$
$998$	− 1030.69i	− 1.03276i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3150.3.e.e.701.3		4
3.2	odd	2	inner	3150.3.e.e.701.1		4
5.2	odd	4		3150.3.c.b.449.2		8
5.3	odd	4		3150.3.c.b.449.8		8
5.4	even	2		126.3.b.a.71.2	✓	4
15.2	even	4		3150.3.c.b.449.5		8
15.8	even	4		3150.3.c.b.449.3		8
15.14	odd	2		126.3.b.a.71.3	yes	4
20.19	odd	2		1008.3.d.a.449.4		4
35.4	even	6		882.3.s.e.863.3		8
35.9	even	6		882.3.s.e.557.2		8
35.19	odd	6		882.3.s.i.557.1		8
35.24	odd	6		882.3.s.i.863.4		8
35.34	odd	2		882.3.b.f.197.1		4
40.19	odd	2		4032.3.d.j.449.1		4
40.29	even	2		4032.3.d.i.449.1		4
45.4	even	6		1134.3.q.c.1079.4		8
45.14	odd	6		1134.3.q.c.1079.1		8
45.29	odd	6		1134.3.q.c.701.4		8
45.34	even	6		1134.3.q.c.701.1		8
60.59	even	2		1008.3.d.a.449.1		4
105.44	odd	6		882.3.s.e.557.3		8
105.59	even	6		882.3.s.i.863.1		8
105.74	odd	6		882.3.s.e.863.2		8
105.89	even	6		882.3.s.i.557.4		8
105.104	even	2		882.3.b.f.197.4		4
120.29	odd	2		4032.3.d.i.449.4		4
120.59	even	2		4032.3.d.j.449.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.3.b.a.71.2	✓	4	5.4	even	2
126.3.b.a.71.3	yes	4	15.14	odd	2
882.3.b.f.197.1		4	35.34	odd	2
882.3.b.f.197.4		4	105.104	even	2
882.3.s.e.557.2		8	35.9	even	6
882.3.s.e.557.3		8	105.44	odd	6
882.3.s.e.863.2		8	105.74	odd	6
882.3.s.e.863.3		8	35.4	even	6
882.3.s.i.557.1		8	35.19	odd	6
882.3.s.i.557.4		8	105.89	even	6
882.3.s.i.863.1		8	105.59	even	6
882.3.s.i.863.4		8	35.24	odd	6
1008.3.d.a.449.1		4	60.59	even	2
1008.3.d.a.449.4		4	20.19	odd	2
1134.3.q.c.701.1		8	45.34	even	6
1134.3.q.c.701.4		8	45.29	odd	6
1134.3.q.c.1079.1		8	45.14	odd	6
1134.3.q.c.1079.4		8	45.4	even	6
3150.3.c.b.449.2		8	5.2	odd	4
3150.3.c.b.449.3		8	15.8	even	4
3150.3.c.b.449.5		8	15.2	even	4
3150.3.c.b.449.8		8	5.3	odd	4
3150.3.e.e.701.1		4	3.2	odd	2	inner
3150.3.e.e.701.3		4	1.1	even	1	trivial
4032.3.d.i.449.1		4	40.29	even	2
4032.3.d.i.449.4		4	120.29	odd	2
4032.3.d.j.449.1		4	40.19	odd	2
4032.3.d.j.449.4		4	120.59	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	3150.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} -2.00000 q^{4} -2.64575 q^{7} -2.82843i q^{8} +O(q^{10})\)	\(q+1.41421i q^{2} -2.00000 q^{4} -2.64575 q^{7} -2.82843i q^{8} +17.7951i q^{11} -2.58301 q^{13} -3.74166i q^{14} +4.00000 q^{16} +25.8681i q^{17} +20.0000 q^{19} -25.1660 q^{22} -17.7951i q^{23} -3.65292i q^{26} +5.29150 q^{28} +11.9034i q^{29} -17.1660 q^{31} +5.65685i q^{32} -36.5830 q^{34} -38.0000 q^{37} +28.2843i q^{38} -15.7338i q^{41} +43.4980 q^{43} -35.5901i q^{44} +25.1660 q^{46} +16.9706i q^{47} +7.00000 q^{49} +5.16601 q^{52} +85.5571i q^{53} +7.48331i q^{56} -16.8340 q^{58} +1.64899i q^{59} +100.332 q^{61} -24.2764i q^{62} -8.00000 q^{64} -36.6640 q^{67} -51.7362i q^{68} +17.7951i q^{71} -28.9150 q^{73} -53.7401i q^{74} -40.0000 q^{76} -47.0813i q^{77} +118.332 q^{79} +22.2510 q^{82} +120.443i q^{83} +61.5155i q^{86} +50.3320 q^{88} -139.475i q^{89} +6.83399 q^{91} +35.5901i q^{92} -24.0000 q^{94} -44.4131 q^{97} +9.89949i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4}+O(q^{10}) \) 4 * q - 8 * q^4	\( 4 q - 8 q^{4} + 32 q^{13} + 16 q^{16} + 80 q^{19} - 16 q^{22} + 16 q^{31} - 104 q^{34} - 152 q^{37} - 80 q^{43} + 16 q^{46} + 28 q^{49} - 64 q^{52} - 152 q^{58} + 232 q^{61} - 32 q^{64} + 192 q^{67} + 96 q^{73} - 160 q^{76} + 304 q^{79} + 216 q^{82} + 32 q^{88} + 112 q^{91} - 96 q^{94} + 288 q^{97}+O(q^{100}) \) 4 * q - 8 * q^4 + 32 * q^13 + 16 * q^16 + 80 * q^19 - 16 * q^22 + 16 * q^31 - 104 * q^34 - 152 * q^37 - 80 * q^43 + 16 * q^46 + 28 * q^49 - 64 * q^52 - 152 * q^58 + 232 * q^61 - 32 * q^64 + 192 * q^67 + 96 * q^73 - 160 * q^76 + 304 * q^79 + 216 * q^82 + 32 * q^88 + 112 * q^91 - 96 * q^94 + 288 * q^97

Introduction

L-functions

Modular forms

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Database

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