LMFDB - Embedded newform 950.3.d.a.949.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [950,3,Mod(949,950)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(950, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("950.949");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 950 = 2 \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	950.d (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:	$25.8856251142$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 38)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			949.2
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	950.949
Dual form			950.3.d.a.949.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} +2.82843 q^{3} +2.00000 q^{4} -4.00000 q^{6} +5.00000i q^{7} -2.82843 q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.41421 q^{2} +2.82843 q^{3} +2.00000 q^{4} -4.00000 q^{6} +5.00000i q^{7} -2.82843 q^{8} -1.00000 q^{9} +5.00000 q^{11} +5.65685 q^{12} -16.9706 q^{13} -7.07107i q^{14} +4.00000 q^{16} -25.0000i q^{17} +1.41421 q^{18} -19.0000 q^{19} +14.1421i q^{21} -7.07107 q^{22} +10.0000i q^{23} -8.00000 q^{24} +24.0000 q^{26} -28.2843 q^{27} +10.0000i q^{28} -42.4264i q^{29} -42.4264i q^{31} -5.65685 q^{32} +14.1421 q^{33} +35.3553i q^{34} -2.00000 q^{36} +25.4558 q^{37} +26.8701 q^{38} -48.0000 q^{39} -42.4264i q^{41} -20.0000i q^{42} -5.00000i q^{43} +10.0000 q^{44} -14.1421i q^{46} +5.00000i q^{47} +11.3137 q^{48} +24.0000 q^{49} -70.7107i q^{51} -33.9411 q^{52} -25.4558 q^{53} +40.0000 q^{54} -14.1421i q^{56} -53.7401 q^{57} +60.0000i q^{58} -84.8528i q^{59} +95.0000 q^{61} +60.0000i q^{62} -5.00000i q^{63} +8.00000 q^{64} -20.0000 q^{66} -110.309 q^{67} -50.0000i q^{68} +28.2843i q^{69} +2.82843 q^{72} +25.0000i q^{73} -36.0000 q^{74} -38.0000 q^{76} +25.0000i q^{77} +67.8823 q^{78} -42.4264i q^{79} -71.0000 q^{81} +60.0000i q^{82} +130.000i q^{83} +28.2843i q^{84} +7.07107i q^{86} -120.000i q^{87} -14.1421 q^{88} +127.279i q^{89} -84.8528i q^{91} +20.0000i q^{92} -120.000i q^{93} -7.07107i q^{94} -16.0000 q^{96} -16.9706 q^{97} -33.9411 q^{98} -5.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{4} - 16 q^{6} - 4 q^{9}+O(q^{10}) $ 4 * q + 8 * q^4 - 16 * q^6 - 4 * q^9	$ 4 q + 8 q^{4} - 16 q^{6} - 4 q^{9} + 20 q^{11} + 16 q^{16} - 76 q^{19} - 32 q^{24} + 96 q^{26} - 8 q^{36} - 192 q^{39} + 40 q^{44} + 96 q^{49} + 160 q^{54} + 380 q^{61} + 32 q^{64} - 80 q^{66} - 144 q^{74} - 152 q^{76} - 284 q^{81} - 64 q^{96} - 20 q^{99}+O(q^{100}) $ 4 * q + 8 * q^4 - 16 * q^6 - 4 * q^9 + 20 * q^11 + 16 * q^16 - 76 * q^19 - 32 * q^24 + 96 * q^26 - 8 * q^36 - 192 * q^39 + 40 * q^44 + 96 * q^49 + 160 * q^54 + 380 * q^61 + 32 * q^64 - 80 * q^66 - 144 * q^74 - 152 * q^76 - 284 * q^81 - 64 * q^96 - 20 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/950\mathbb{Z}\right)^\times$.

$n$	$77$	$401$
$\chi(n)$	$-1$	$-1$

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 $

$2$	−1.41421	−0.707107
$2$	−1.41421	−0.707107
$3$	2.82843	0.942809	0.471405	−	0.881917i	$-0.343747\pi$
$3$	2.82843	0.942809	0.471405	+	0.881917i	$0.343747\pi$
$4$	2.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−4.00000	−0.666667
$7$	5.00000i	0.714286i	0.934050	+	0.357143i	$0.116249\pi$
$7$	5.00000i	0.714286i	−0.934050	+	0.357143i	$0.883751\pi$
$8$	−2.82843	−0.353553
$9$	−1.00000	−0.111111
$10$	0	0
$11$	5.00000	0.454545	0.227273	−	0.973831i	$-0.427019\pi$
$11$	5.00000	0.454545	0.227273	+	0.973831i	$0.427019\pi$
$12$	5.65685	0.471405
$13$	−16.9706	−1.30543	−0.652714	−	0.757604i	$-0.726370\pi$
$13$	−16.9706	−1.30543	−0.652714	+	0.757604i	$0.726370\pi$
$14$	− 7.07107i	− 0.505076i
$15$	0	0
$16$	4.00000	0.250000
$17$	− 25.0000i	− 1.47059i	−0.677748	−	0.735294i	$-0.737044\pi$
$17$	− 25.0000i	− 1.47059i	0.677748	−	0.735294i	$-0.262956\pi$
$18$	1.41421	0.0785674
$19$	−19.0000	−1.00000
$19$	−19.0000	−1.00000
$20$	0	0
$21$	14.1421i	0.673435i
$22$	−7.07107	−0.321412
$23$	10.0000i	0.434783i	0.976085	+	0.217391i	$0.0697548\pi$
$23$	10.0000i	0.434783i	−0.976085	+	0.217391i	$0.930245\pi$
$24$	−8.00000	−0.333333
$25$	0	0
$26$	24.0000	0.923077
$27$	−28.2843	−1.04757
$28$	10.0000i	0.357143i
$29$	− 42.4264i	− 1.46298i	−0.681852	−	0.731490i	$-0.738825\pi$
$29$	− 42.4264i	− 1.46298i	0.681852	−	0.731490i	$-0.261175\pi$
$30$	0	0
$31$	− 42.4264i	− 1.36859i	−0.729204	−	0.684297i	$-0.760109\pi$
$31$	− 42.4264i	− 1.36859i	0.729204	−	0.684297i	$-0.239891\pi$
$32$	−5.65685	−0.176777
$33$	14.1421	0.428550
$34$	35.3553i	1.03986i
$35$	0	0
$36$	−2.00000	−0.0555556
$37$	25.4558	0.687996	0.343998	−	0.938970i	$-0.388219\pi$
$37$	25.4558	0.687996	0.343998	+	0.938970i	$0.388219\pi$
$38$	26.8701	0.707107
$39$	−48.0000	−1.23077
$40$	0	0
$41$	− 42.4264i	− 1.03479i	−0.855747	−	0.517395i	$-0.826902\pi$
$41$	− 42.4264i	− 1.03479i	0.855747	−	0.517395i	$-0.173098\pi$
$42$	− 20.0000i	− 0.476190i
$43$	− 5.00000i	− 0.116279i	−0.998308	−	0.0581395i	$-0.981483\pi$
$43$	− 5.00000i	− 0.116279i	0.998308	−	0.0581395i	$-0.0185168\pi$
$44$	10.0000	0.227273
$45$	0	0
$46$	− 14.1421i	− 0.307438i
$47$	5.00000i	0.106383i	0.998584	+	0.0531915i	$0.0169394\pi$
$47$	5.00000i	0.106383i	−0.998584	+	0.0531915i	$0.983061\pi$
$48$	11.3137	0.235702
$49$	24.0000	0.489796
$50$	0	0
$51$	− 70.7107i	− 1.38648i
$52$	−33.9411	−0.652714
$53$	−25.4558	−0.480299	−0.240149	−	0.970736i	$-0.577196\pi$
$53$	−25.4558	−0.480299	−0.240149	+	0.970736i	$0.577196\pi$
$54$	40.0000	0.740741
$55$	0	0
$56$	− 14.1421i	− 0.252538i
$57$	−53.7401	−0.942809
$58$	60.0000i	1.03448i
$59$	− 84.8528i	− 1.43818i	−0.694915	−	0.719092i	$-0.744558\pi$
$59$	− 84.8528i	− 1.43818i	0.694915	−	0.719092i	$-0.255442\pi$
$60$	0	0
$61$	95.0000	1.55738	0.778689	−	0.627411i	$-0.215885\pi$
$61$	95.0000	1.55738	0.778689	+	0.627411i	$0.215885\pi$
$62$	60.0000i	0.967742i
$63$	− 5.00000i	− 0.0793651i
$64$	8.00000	0.125000
$65$	0	0
$66$	−20.0000	−0.303030
$67$	−110.309	−1.64640	−0.823199	−	0.567753i	$-0.807813\pi$
$67$	−110.309	−1.64640	−0.823199	+	0.567753i	$0.807813\pi$
$68$	− 50.0000i	− 0.735294i
$69$	28.2843i	0.409917i
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	2.82843	0.0392837
$73$	25.0000i	0.342466i	0.985231	+	0.171233i	$0.0547750\pi$
$73$	25.0000i	0.342466i	−0.985231	+	0.171233i	$0.945225\pi$
$74$	−36.0000	−0.486486
$75$	0	0
$76$	−38.0000	−0.500000
$77$	25.0000i	0.324675i
$78$	67.8823	0.870285
$79$	− 42.4264i	− 0.537043i	−0.963274	−	0.268522i	$-0.913465\pi$
$79$	− 42.4264i	− 0.537043i	0.963274	−	0.268522i	$-0.0865351\pi$
$80$	0	0
$81$	−71.0000	−0.876543
$82$	60.0000i	0.731707i
$83$	130.000i	1.56627i	0.621855	+	0.783133i	$0.286379\pi$
$83$	130.000i	1.56627i	−0.621855	+	0.783133i	$0.713621\pi$
$84$	28.2843i	0.336718i
$85$	0	0
$86$	7.07107i	0.0822217i
$87$	− 120.000i	− 1.37931i
$88$	−14.1421	−0.160706
$89$	127.279i	1.43010i	0.699071	+	0.715052i	$0.253597\pi$
$89$	127.279i	1.43010i	−0.699071	+	0.715052i	$0.746403\pi$
$90$	0	0
$91$	− 84.8528i	− 0.932449i
$92$	20.0000i	0.217391i
$93$	− 120.000i	− 1.29032i
$94$	− 7.07107i	− 0.0752241i
$95$	0	0
$96$	−16.0000	−0.166667
$97$	−16.9706	−0.174954	−0.0874771	−	0.996167i	$-0.527880\pi$
$97$	−16.9706	−0.174954	−0.0874771	+	0.996167i	$0.527880\pi$
$98$	−33.9411	−0.346338
$99$	−5.00000	−0.0505051
$100$	0	0
$101$	50.0000	0.495050	0.247525	−	0.968882i	$-0.420383\pi$
$101$	50.0000	0.495050	0.247525	+	0.968882i	$0.420383\pi$
$102$	100.000i	0.980392i
$103$	−16.9706	−0.164763	−0.0823814	−	0.996601i	$-0.526253\pi$
$103$	−16.9706	−0.164763	−0.0823814	+	0.996601i	$0.526253\pi$
$104$	48.0000	0.461538
$105$	0	0
$106$	36.0000	0.339623
$107$	101.823	0.951620	0.475810	−	0.879548i	$-0.342155\pi$
$107$	101.823	0.951620	0.475810	+	0.879548i	$0.342155\pi$
$108$	−56.5685	−0.523783
$109$	− 127.279i	− 1.16770i	−0.811862	−	0.583850i	$-0.801546\pi$
$109$	− 127.279i	− 1.16770i	0.811862	−	0.583850i	$-0.198454\pi$
$110$	0	0
$111$	72.0000	0.648649
$112$	20.0000i	0.178571i
$113$	110.309	0.976183	0.488091	−	0.872793i	$-0.337693\pi$
$113$	110.309	0.976183	0.488091	+	0.872793i	$0.337693\pi$
$114$	76.0000	0.666667
$115$	0	0
$116$	− 84.8528i	− 0.731490i
$117$	16.9706	0.145048
$118$	120.000i	1.01695i
$119$	125.000	1.05042
$120$	0	0
$121$	−96.0000	−0.793388
$122$	−134.350	−1.10123
$123$	− 120.000i	− 0.975610i
$124$	− 84.8528i	− 0.684297i
$125$	0	0
$126$	7.07107i	0.0561196i
$127$	−229.103	−1.80396	−0.901979	−	0.431780i	$-0.857886\pi$
$127$	−229.103	−1.80396	−0.901979	+	0.431780i	$0.857886\pi$
$128$	−11.3137	−0.0883883
$129$	− 14.1421i	− 0.109629i
$130$	0	0
$131$	−163.000	−1.24427	−0.622137	−	0.782908i	$-0.713735\pi$
$131$	−163.000	−1.24427	−0.622137	+	0.782908i	$0.713735\pi$
$132$	28.2843	0.214275
$133$	− 95.0000i	− 0.714286i
$134$	156.000	1.16418
$135$	0	0
$136$	70.7107i	0.519931i
$137$	95.0000i	0.693431i	0.937970	+	0.346715i	$0.112703\pi$
$137$	95.0000i	0.693431i	−0.937970	+	0.346715i	$0.887297\pi$
$138$	− 40.0000i	− 0.289855i
$139$	−125.000	−0.899281	−0.449640	−	0.893210i	$-0.648448\pi$
$139$	−125.000	−0.899281	−0.449640	+	0.893210i	$0.648448\pi$
$140$	0	0
$141$	14.1421i	0.100299i
$142$	0	0
$143$	−84.8528	−0.593376
$144$	−4.00000	−0.0277778
$145$	0	0
$146$	− 35.3553i	− 0.242160i
$147$	67.8823	0.461784
$148$	50.9117	0.343998
$149$	−215.000	−1.44295	−0.721477	−	0.692439i	$-0.756536\pi$
$149$	−215.000	−1.44295	−0.721477	+	0.692439i	$0.756536\pi$
$150$	0	0
$151$	− 84.8528i	− 0.561939i	−0.959717	−	0.280970i	$-0.909344\pi$
$151$	− 84.8528i	− 0.561939i	0.959717	−	0.280970i	$-0.0906560\pi$
$152$	53.7401	0.353553
$153$	25.0000i	0.163399i
$154$	− 35.3553i	− 0.229580i
$155$	0	0
$156$	−96.0000	−0.615385
$157$	− 190.000i	− 1.21019i	−0.796153	−	0.605096i	$-0.793135\pi$
$157$	− 190.000i	− 1.21019i	0.796153	−	0.605096i	$-0.206865\pi$
$158$	60.0000i	0.379747i
$159$	−72.0000	−0.452830
$160$	0	0
$161$	−50.0000	−0.310559
$162$	100.409	0.619810
$163$	− 110.000i	− 0.674847i	−0.941353	−	0.337423i	$-0.890445\pi$
$163$	− 110.000i	− 0.674847i	0.941353	−	0.337423i	$-0.109555\pi$
$164$	− 84.8528i	− 0.517395i
$165$	0	0
$166$	− 183.848i	− 1.10752i
$167$	−59.3970	−0.355670	−0.177835	−	0.984060i	$-0.556909\pi$
$167$	−59.3970	−0.355670	−0.177835	+	0.984060i	$0.556909\pi$
$168$	− 40.0000i	− 0.238095i
$169$	119.000	0.704142
$170$	0	0
$171$	19.0000	0.111111
$172$	− 10.0000i	− 0.0581395i
$173$	−186.676	−1.07905	−0.539527	−	0.841969i	$-0.681397\pi$
$173$	−186.676	−1.07905	−0.539527	+	0.841969i	$0.681397\pi$
$174$	169.706i	0.975320i
$175$	0	0
$176$	20.0000	0.113636
$177$	− 240.000i	− 1.35593i
$178$	− 180.000i	− 1.01124i
$179$	127.279i	0.711057i	0.934665	+	0.355529i	$0.115699\pi$
$179$	127.279i	0.711057i	−0.934665	+	0.355529i	$0.884301\pi$
$180$	0	0
$181$	− 254.558i	− 1.40640i	−0.710992	−	0.703200i	$-0.751754\pi$
$181$	− 254.558i	− 1.40640i	0.710992	−	0.703200i	$-0.248246\pi$
$182$	120.000i	0.659341i
$183$	268.701	1.46831
$184$	− 28.2843i	− 0.153719i
$185$	0	0
$186$	169.706i	0.912396i
$187$	− 125.000i	− 0.668449i
$188$	10.0000i	0.0531915i
$189$	− 141.421i	− 0.748261i
$190$	0	0
$191$	293.000	1.53403	0.767016	−	0.641628i	$-0.221741\pi$
$191$	293.000	1.53403	0.767016	+	0.641628i	$0.221741\pi$
$192$	22.6274	0.117851
$193$	59.3970	0.307756	0.153878	−	0.988090i	$-0.450824\pi$
$193$	59.3970	0.307756	0.153878	+	0.988090i	$0.450824\pi$
$194$	24.0000	0.123711
$195$	0	0
$196$	48.0000	0.244898
$197$	− 70.0000i	− 0.355330i	−0.984091	−	0.177665i	$-0.943146\pi$
$197$	− 70.0000i	− 0.355330i	0.984091	−	0.177665i	$-0.0568543\pi$
$198$	7.07107	0.0357125
$199$	−173.000	−0.869347	−0.434673	−	0.900588i	$-0.643136\pi$
$199$	−173.000	−0.869347	−0.434673	+	0.900588i	$0.643136\pi$
$200$	0	0
$201$	−312.000	−1.55224
$202$	−70.7107	−0.350053
$203$	212.132	1.04499
$204$	− 141.421i	− 0.693242i
$205$	0	0
$206$	24.0000	0.116505
$207$	− 10.0000i	− 0.0483092i
$208$	−67.8823	−0.326357
$209$	−95.0000	−0.454545
$210$	0	0
$211$	84.8528i	0.402146i	0.979576	+	0.201073i	$0.0644429\pi$
$211$	84.8528i	0.402146i	−0.979576	+	0.201073i	$0.935557\pi$
$212$	−50.9117	−0.240149
$213$	0	0
$214$	−144.000	−0.672897
$215$	0	0
$216$	80.0000	0.370370
$217$	212.132	0.977567
$218$	180.000i	0.825688i
$219$	70.7107i	0.322880i
$220$	0	0
$221$	424.264i	1.91975i
$222$	−101.823	−0.458664
$223$	−364.867	−1.63618	−0.818088	−	0.575094i	$-0.804966\pi$
$223$	−364.867	−1.63618	−0.818088	+	0.575094i	$0.804966\pi$
$224$	− 28.2843i	− 0.126269i
$225$	0	0
$226$	−156.000	−0.690265
$227$	−67.8823	−0.299041	−0.149520	−	0.988759i	$-0.547773\pi$
$227$	−67.8823	−0.299041	−0.149520	+	0.988759i	$0.547773\pi$
$228$	−107.480	−0.471405
$229$	145.000	0.633188	0.316594	−	0.948561i	$-0.397461\pi$
$229$	145.000	0.633188	0.316594	+	0.948561i	$0.397461\pi$
$230$	0	0
$231$	70.7107i	0.306107i
$232$	120.000i	0.517241i
$233$	− 335.000i	− 1.43777i	−0.695130	−	0.718884i	$-0.744653\pi$
$233$	− 335.000i	− 1.43777i	0.695130	−	0.718884i	$-0.255347\pi$
$234$	−24.0000	−0.102564
$235$	0	0
$236$	− 169.706i	− 0.719092i
$237$	− 120.000i	− 0.506329i
$238$	−176.777	−0.742759
$239$	−197.000	−0.824268	−0.412134	−	0.911123i	$-0.635216\pi$
$239$	−197.000	−0.824268	−0.412134	+	0.911123i	$0.635216\pi$
$240$	0	0
$241$	296.985i	1.23230i	0.787628	+	0.616151i	$0.211309\pi$
$241$	296.985i	1.23230i	−0.787628	+	0.616151i	$0.788691\pi$
$242$	135.765	0.561010
$243$	53.7401	0.221153
$244$	190.000	0.778689
$245$	0	0
$246$	169.706i	0.689860i
$247$	322.441	1.30543
$248$	120.000i	0.483871i
$249$	367.696i	1.47669i
$250$	0	0
$251$	173.000	0.689243	0.344622	−	0.938742i	$-0.388007\pi$
$251$	173.000	0.689243	0.344622	+	0.938742i	$0.388007\pi$
$252$	− 10.0000i	− 0.0396825i
$253$	50.0000i	0.197628i
$254$	324.000	1.27559
$255$	0	0
$256$	16.0000	0.0625000
$257$	67.8823	0.264133	0.132067	−	0.991241i	$-0.457839\pi$
$257$	67.8823	0.264133	0.132067	+	0.991241i	$0.457839\pi$
$258$	20.0000i	0.0775194i
$259$	127.279i	0.491426i
$260$	0	0
$261$	42.4264i	0.162553i
$262$	230.517	0.879835
$263$	355.000i	1.34981i	0.737905	+	0.674905i	$0.235815\pi$
$263$	355.000i	1.34981i	−0.737905	+	0.674905i	$0.764185\pi$
$264$	−40.0000	−0.151515
$265$	0	0
$266$	134.350i	0.505076i
$267$	360.000i	1.34831i
$268$	−220.617	−0.823199
$269$	381.838i	1.41947i	0.704468	+	0.709735i	$0.251186\pi$
$269$	381.838i	1.41947i	−0.704468	+	0.709735i	$0.748814\pi$
$270$	0	0
$271$	110.000	0.405904	0.202952	−	0.979189i	$-0.434946\pi$
$271$	110.000	0.405904	0.202952	+	0.979189i	$0.434946\pi$
$272$	− 100.000i	− 0.367647i
$273$	− 240.000i	− 0.879121i
$274$	− 134.350i	− 0.490330i
$275$	0	0
$276$	56.5685i	0.204958i
$277$	− 265.000i	− 0.956679i	−0.878175	−	0.478339i	$-0.841239\pi$
$277$	− 265.000i	− 0.956679i	0.878175	−	0.478339i	$-0.158761\pi$
$278$	176.777	0.635887
$279$	42.4264i	0.152066i
$280$	0	0
$281$	424.264i	1.50984i	0.655819	+	0.754918i	$0.272323\pi$
$281$	424.264i	1.50984i	−0.655819	+	0.754918i	$0.727677\pi$
$282$	− 20.0000i	− 0.0709220i
$283$	− 125.000i	− 0.441696i	−0.975308	−	0.220848i	$-0.929118\pi$
$283$	− 125.000i	− 0.441696i	0.975308	−	0.220848i	$-0.0708825\pi$
$284$	0	0
$285$	0	0
$286$	120.000	0.419580
$287$	212.132	0.739136
$288$	5.65685	0.0196419
$289$	−336.000	−1.16263
$290$	0	0
$291$	−48.0000	−0.164948
$292$	50.0000i	0.171233i
$293$	186.676	0.637120	0.318560	−	0.947903i	$-0.396801\pi$
$293$	186.676	0.637120	0.318560	+	0.947903i	$0.396801\pi$
$294$	−96.0000	−0.326531
$295$	0	0
$296$	−72.0000	−0.243243
$297$	−141.421	−0.476166
$298$	304.056	1.02032
$299$	− 169.706i	− 0.567577i
$300$	0	0
$301$	25.0000	0.0830565
$302$	120.000i	0.397351i
$303$	141.421	0.466737
$304$	−76.0000	−0.250000
$305$	0	0
$306$	− 35.3553i	− 0.115540i
$307$	−280.014	−0.912099	−0.456049	−	0.889955i	$-0.650736\pi$
$307$	−280.014	−0.912099	−0.456049	+	0.889955i	$0.650736\pi$
$308$	50.0000i	0.162338i
$309$	−48.0000	−0.155340
$310$	0	0
$311$	−235.000	−0.755627	−0.377814	−	0.925882i	$-0.623324\pi$
$311$	−235.000	−0.755627	−0.377814	+	0.925882i	$0.623324\pi$
$312$	135.765	0.435143
$313$	310.000i	0.990415i	0.868775	+	0.495208i	$0.164908\pi$
$313$	310.000i	0.990415i	−0.868775	+	0.495208i	$0.835092\pi$
$314$	268.701i	0.855734i
$315$	0	0
$316$	− 84.8528i	− 0.268522i
$317$	186.676	0.588884	0.294442	−	0.955669i	$-0.404866\pi$
$317$	186.676	0.588884	0.294442	+	0.955669i	$0.404866\pi$
$318$	101.823	0.320199
$319$	− 212.132i	− 0.664991i
$320$	0	0
$321$	288.000	0.897196
$322$	70.7107	0.219598
$323$	475.000i	1.47059i
$324$	−142.000	−0.438272
$325$	0	0
$326$	155.563i	0.477189i
$327$	− 360.000i	− 1.10092i
$328$	120.000i	0.365854i
$329$	−25.0000	−0.0759878
$330$	0	0
$331$	296.985i	0.897235i	0.893724	+	0.448618i	$0.148083\pi$
$331$	296.985i	0.897235i	−0.893724	+	0.448618i	$0.851917\pi$
$332$	260.000i	0.783133i
$333$	−25.4558	−0.0764440
$334$	84.0000	0.251497
$335$	0	0
$336$	56.5685i	0.168359i
$337$	526.087	1.56109	0.780545	−	0.625099i	$-0.214942\pi$
$337$	526.087	1.56109	0.780545	+	0.625099i	$0.214942\pi$
$338$	−168.291	−0.497904
$339$	312.000	0.920354
$340$	0	0
$341$	− 212.132i	− 0.622088i
$342$	−26.8701	−0.0785674
$343$	365.000i	1.06414i
$344$	14.1421i	0.0411109i
$345$	0	0
$346$	264.000	0.763006
$347$	125.000i	0.360231i	0.983646	+	0.180115i	$0.0576471\pi$
$347$	125.000i	0.360231i	−0.983646	+	0.180115i	$0.942353\pi$
$348$	− 240.000i	− 0.689655i
$349$	−23.0000	−0.0659026	−0.0329513	−	0.999457i	$-0.510491\pi$
$349$	−23.0000	−0.0659026	−0.0329513	+	0.999457i	$0.510491\pi$
$350$	0	0
$351$	480.000	1.36752
$352$	−28.2843	−0.0803530
$353$	− 410.000i	− 1.16147i	−0.814092	−	0.580737i	$-0.802765\pi$
$353$	− 410.000i	− 1.16147i	0.814092	−	0.580737i	$-0.197235\pi$
$354$	339.411i	0.958789i
$355$	0	0
$356$	254.558i	0.715052i
$357$	353.553	0.990346
$358$	− 180.000i	− 0.502793i
$359$	475.000	1.32312	0.661560	−	0.749892i	$-0.269895\pi$
$359$	475.000	1.32312	0.661560	+	0.749892i	$0.269895\pi$
$360$	0	0
$361$	361.000	1.00000
$362$	360.000i	0.994475i
$363$	−271.529	−0.748014
$364$	− 169.706i	− 0.466224i
$365$	0	0
$366$	−380.000	−1.03825
$367$	230.000i	0.626703i	0.949637	+	0.313351i	$0.101452\pi$
$367$	230.000i	0.626703i	−0.949637	+	0.313351i	$0.898548\pi$
$368$	40.0000i	0.108696i
$369$	42.4264i	0.114977i
$370$	0	0
$371$	− 127.279i	− 0.343071i
$372$	− 240.000i	− 0.645161i
$373$	−67.8823	−0.181990	−0.0909950	−	0.995851i	$-0.529005\pi$
$373$	−67.8823	−0.181990	−0.0909950	+	0.995851i	$0.529005\pi$
$374$	176.777i	0.472665i
$375$	0	0
$376$	− 14.1421i	− 0.0376121i
$377$	720.000i	1.90981i
$378$	200.000i	0.529101i
$379$	254.558i	0.671658i	0.941923	+	0.335829i	$0.109016\pi$
$379$	254.558i	0.671658i	−0.941923	+	0.335829i	$0.890984\pi$
$380$	0	0
$381$	−648.000	−1.70079
$382$	−414.365	−1.08472
$383$	−144.250	−0.376631	−0.188316	−	0.982109i	$-0.560303\pi$
$383$	−144.250	−0.376631	−0.188316	+	0.982109i	$0.560303\pi$
$384$	−32.0000	−0.0833333
$385$	0	0
$386$	−84.0000	−0.217617
$387$	5.00000i	0.0129199i
$388$	−33.9411	−0.0874771
$389$	553.000	1.42159	0.710797	−	0.703397i	$-0.248334\pi$
$389$	553.000	1.42159	0.710797	+	0.703397i	$0.248334\pi$
$390$	0	0
$391$	250.000	0.639386
$392$	−67.8823	−0.173169
$393$	−461.034	−1.17311
$394$	98.9949i	0.251256i
$395$	0	0
$396$	−10.0000	−0.0252525
$397$	335.000i	0.843829i	0.906636	+	0.421914i	$0.138642\pi$
$397$	335.000i	0.843829i	−0.906636	+	0.421914i	$0.861358\pi$
$398$	244.659	0.614721
$399$	− 268.701i	− 0.673435i
$400$	0	0
$401$	212.132i	0.529008i	0.964385	+	0.264504i	$0.0852082\pi$
$401$	212.132i	0.529008i	−0.964385	+	0.264504i	$0.914792\pi$
$402$	441.235	1.09760
$403$	720.000i	1.78660i
$404$	100.000	0.247525
$405$	0	0
$406$	−300.000	−0.738916
$407$	127.279	0.312725
$408$	200.000i	0.490196i
$409$	− 721.249i	− 1.76344i	−0.471769	−	0.881722i	$-0.656384\pi$
$409$	− 721.249i	− 1.76344i	0.471769	−	0.881722i	$-0.343616\pi$
$410$	0	0
$411$	268.701i	0.653773i
$412$	−33.9411	−0.0823814
$413$	424.264	1.02727
$414$	14.1421i	0.0341597i
$415$	0	0
$416$	96.0000	0.230769
$417$	−353.553	−0.847850
$418$	134.350	0.321412
$419$	−62.0000	−0.147971	−0.0739857	−	0.997259i	$-0.523572\pi$
$419$	−62.0000	−0.147971	−0.0739857	+	0.997259i	$0.523572\pi$
$420$	0	0
$421$	296.985i	0.705427i	0.935731	+	0.352714i	$0.114741\pi$
$421$	296.985i	0.705427i	−0.935731	+	0.352714i	$0.885259\pi$
$422$	− 120.000i	− 0.284360i
$423$	− 5.00000i	− 0.0118203i
$424$	72.0000	0.169811
$425$	0	0
$426$	0	0
$427$	475.000i	1.11241i
$428$	203.647	0.475810
$429$	−240.000	−0.559441
$430$	0	0
$431$	− 509.117i	− 1.18125i	−0.806948	−	0.590623i	$-0.798882\pi$
$431$	− 509.117i	− 1.18125i	0.806948	−	0.590623i	$-0.201118\pi$
$432$	−113.137	−0.261891
$433$	−229.103	−0.529105	−0.264553	−	0.964371i	$-0.585224\pi$
$433$	−229.103	−0.529105	−0.264553	+	0.964371i	$0.585224\pi$
$434$	−300.000	−0.691244
$435$	0	0
$436$	− 254.558i	− 0.583850i
$437$	− 190.000i	− 0.434783i
$438$	− 100.000i	− 0.228311i
$439$	− 806.102i	− 1.83622i	−0.396323	−	0.918111i	$-0.629714\pi$
$439$	− 806.102i	− 1.83622i	0.396323	−	0.918111i	$-0.370286\pi$
$440$	0	0
$441$	−24.0000	−0.0544218
$442$	− 600.000i	− 1.35747i
$443$	− 365.000i	− 0.823928i	−0.911200	−	0.411964i	$-0.864843\pi$
$443$	− 365.000i	− 0.823928i	0.911200	−	0.411964i	$-0.135157\pi$
$444$	144.000	0.324324
$445$	0	0
$446$	516.000	1.15695
$447$	−608.112	−1.36043
$448$	40.0000i	0.0892857i
$449$	763.675i	1.70084i	0.526108	+	0.850418i	$0.323651\pi$
$449$	763.675i	1.70084i	−0.526108	+	0.850418i	$0.676349\pi$
$450$	0	0
$451$	− 212.132i	− 0.470359i
$452$	220.617	0.488091
$453$	− 240.000i	− 0.529801i
$454$	96.0000	0.211454
$455$	0	0
$456$	152.000	0.333333
$457$	− 265.000i	− 0.579869i	−0.957047	−	0.289934i	$-0.906367\pi$
$457$	− 265.000i	− 0.579869i	0.957047	−	0.289934i	$-0.0936335\pi$
$458$	−205.061	−0.447731
$459$	707.107i	1.54054i
$460$	0	0
$461$	−553.000	−1.19957	−0.599783	−	0.800163i	$-0.704746\pi$
$461$	−553.000	−1.19957	−0.599783	+	0.800163i	$0.704746\pi$
$462$	− 100.000i	− 0.216450i
$463$	− 485.000i	− 1.04752i	−0.851867	−	0.523758i	$-0.824530\pi$
$463$	− 485.000i	− 1.04752i	0.851867	−	0.523758i	$-0.175470\pi$
$464$	− 169.706i	− 0.365745i
$465$	0	0
$466$	473.762i	1.01666i
$467$	− 115.000i	− 0.246253i	−0.992391	−	0.123126i	$-0.960708\pi$
$467$	− 115.000i	− 0.246253i	0.992391	−	0.123126i	$-0.0392920\pi$
$468$	33.9411	0.0725238
$469$	− 551.543i	− 1.17600i
$470$	0	0
$471$	− 537.401i	− 1.14098i
$472$	240.000i	0.508475i
$473$	− 25.0000i	− 0.0528541i
$474$	169.706i	0.358029i
$475$	0	0
$476$	250.000	0.525210
$477$	25.4558	0.0533665
$478$	278.600	0.582845
$479$	490.000	1.02296	0.511482	−	0.859294i	$-0.329097\pi$
$479$	490.000	1.02296	0.511482	+	0.859294i	$0.329097\pi$
$480$	0	0
$481$	−432.000	−0.898129
$482$	− 420.000i	− 0.871369i
$483$	−141.421	−0.292798
$484$	−192.000	−0.396694
$485$	0	0
$486$	−76.0000	−0.156379
$487$	610.940	1.25450	0.627249	−	0.778819i	$-0.284181\pi$
$487$	610.940	1.25450	0.627249	+	0.778819i	$0.284181\pi$
$488$	−268.701	−0.550616
$489$	− 311.127i	− 0.636252i
$490$	0	0
$491$	−82.0000	−0.167006	−0.0835031	−	0.996508i	$-0.526611\pi$
$491$	−82.0000	−0.167006	−0.0835031	+	0.996508i	$0.526611\pi$
$492$	− 240.000i	− 0.487805i
$493$	−1060.66	−2.15144
$494$	−456.000	−0.923077
$495$	0	0
$496$	− 169.706i	− 0.342148i
$497$	0	0
$498$	− 520.000i	− 1.04418i
$499$	−485.000	−0.971944	−0.485972	−	0.873974i	$-0.661534\pi$
$499$	−485.000	−0.971944	−0.485972	+	0.873974i	$0.661534\pi$
$500$	0	0
$501$	−168.000	−0.335329
$502$	−244.659	−0.487368
$503$	250.000i	0.497018i	0.968630	+	0.248509i	$0.0799405\pi$
$503$	250.000i	0.497018i	−0.968630	+	0.248509i	$0.920059\pi$
$504$	14.1421i	0.0280598i
$505$	0	0
$506$	− 70.7107i	− 0.139744i
$507$	336.583	0.663871
$508$	−458.205	−0.901979
$509$	169.706i	0.333410i	0.986007	+	0.166705i	$0.0533127\pi$
$509$	169.706i	0.333410i	−0.986007	+	0.166705i	$0.946687\pi$
$510$	0	0
$511$	−125.000	−0.244618
$512$	−22.6274	−0.0441942
$513$	537.401	1.04757
$514$	−96.0000	−0.186770
$515$	0	0
$516$	− 28.2843i	− 0.0548145i
$517$	25.0000i	0.0483559i
$518$	− 180.000i	− 0.347490i
$519$	−528.000	−1.01734
$520$	0	0
$521$	− 127.279i	− 0.244298i	−0.992512	−	0.122149i	$-0.961021\pi$
$521$	− 127.279i	− 0.244298i	0.992512	−	0.122149i	$-0.0389786\pi$
$522$	− 60.0000i	− 0.114943i
$523$	356.382	0.681418	0.340709	−	0.940169i	$-0.389333\pi$
$523$	356.382	0.681418	0.340709	+	0.940169i	$0.389333\pi$
$524$	−326.000	−0.622137
$525$	0	0
$526$	− 502.046i	− 0.954460i
$527$	−1060.66	−2.01264
$528$	56.5685	0.107137
$529$	429.000	0.810964
$530$	0	0
$531$	84.8528i	0.159798i
$532$	− 190.000i	− 0.357143i
$533$	720.000i	1.35084i
$534$	− 509.117i	− 0.953402i
$535$	0	0
$536$	312.000	0.582090
$537$	360.000i	0.670391i
$538$	− 540.000i	− 1.00372i
$539$	120.000	0.222635
$540$	0	0
$541$	−25.0000	−0.0462107	−0.0231054	−	0.999733i	$-0.507355\pi$
$541$	−25.0000	−0.0462107	−0.0231054	+	0.999733i	$0.507355\pi$
$542$	−155.563	−0.287018
$543$	− 720.000i	− 1.32597i
$544$	141.421i	0.259966i
$545$	0	0
$546$	339.411i	0.621632i
$547$	−16.9706	−0.0310248	−0.0155124	−	0.999880i	$-0.504938\pi$
$547$	−16.9706	−0.0310248	−0.0155124	+	0.999880i	$0.504938\pi$
$548$	190.000i	0.346715i
$549$	−95.0000	−0.173042
$550$	0	0
$551$	806.102i	1.46298i
$552$	− 80.0000i	− 0.144928i
$553$	212.132	0.383602
$554$	374.767i	0.676474i
$555$	0	0
$556$	−250.000	−0.449640
$557$	− 745.000i	− 1.33752i	−0.743477	−	0.668761i	$-0.766825\pi$
$557$	− 745.000i	− 1.33752i	0.743477	−	0.668761i	$-0.233175\pi$
$558$	− 60.0000i	− 0.107527i
$559$	84.8528i	0.151794i
$560$	0	0
$561$	− 353.553i	− 0.630220i
$562$	− 600.000i	− 1.06762i
$563$	313.955	0.557647	0.278824	−	0.960342i	$-0.410056\pi$
$563$	313.955	0.557647	0.278824	+	0.960342i	$0.410056\pi$
$564$	28.2843i	0.0501494i
$565$	0	0
$566$	176.777i	0.312326i
$567$	− 355.000i	− 0.626102i
$568$	0	0
$569$	− 424.264i	− 0.745631i	−0.927905	−	0.372816i	$-0.878392\pi$
$569$	− 424.264i	− 0.745631i	0.927905	−	0.372816i	$-0.121608\pi$
$570$	0	0
$571$	1070.00	1.87391	0.936953	−	0.349456i	$-0.113634\pi$
$571$	1070.00	1.87391	0.936953	+	0.349456i	$0.113634\pi$
$572$	−169.706	−0.296688
$573$	828.729	1.44630
$574$	−300.000	−0.522648
$575$	0	0
$576$	−8.00000	−0.0138889
$577$	− 25.0000i	− 0.0433276i	−0.999765	−	0.0216638i	$-0.993104\pi$
$577$	− 25.0000i	− 0.0433276i	0.999765	−	0.0216638i	$-0.00689633\pi$
$578$	475.176	0.822103
$579$	168.000	0.290155
$580$	0	0
$581$	−650.000	−1.11876
$582$	67.8823	0.116636
$583$	−127.279	−0.218318
$584$	− 70.7107i	− 0.121080i
$585$	0	0
$586$	−264.000	−0.450512
$587$	725.000i	1.23509i	0.786534	+	0.617547i	$0.211873\pi$
$587$	725.000i	1.23509i	−0.786534	+	0.617547i	$0.788127\pi$
$588$	135.765	0.230892
$589$	806.102i	1.36859i
$590$	0	0
$591$	− 197.990i	− 0.335008i
$592$	101.823	0.171999
$593$	− 650.000i	− 1.09612i	−0.836439	−	0.548061i	$-0.815366\pi$
$593$	− 650.000i	− 1.09612i	0.836439	−	0.548061i	$-0.184634\pi$
$594$	200.000	0.336700
$595$	0	0
$596$	−430.000	−0.721477
$597$	−489.318	−0.819628
$598$	240.000i	0.401338i
$599$	296.985i	0.495801i	0.968785	+	0.247901i	$0.0797406\pi$
$599$	296.985i	0.495801i	−0.968785	+	0.247901i	$0.920259\pi$
$600$	0	0
$601$	− 848.528i	− 1.41186i	−0.708281	−	0.705930i	$-0.750529\pi$
$601$	− 848.528i	− 1.41186i	0.708281	−	0.705930i	$-0.249471\pi$
$602$	−35.3553	−0.0587298
$603$	110.309	0.182933
$604$	− 169.706i	− 0.280970i
$605$	0	0
$606$	−200.000	−0.330033
$607$	271.529	0.447329	0.223665	−	0.974666i	$-0.428198\pi$
$607$	271.529	0.447329	0.223665	+	0.974666i	$0.428198\pi$
$608$	107.480	0.176777
$609$	600.000	0.985222
$610$	0	0
$611$	− 84.8528i	− 0.138875i
$612$	50.0000i	0.0816993i
$613$	− 1055.00i	− 1.72104i	−0.509413	−	0.860522i	$-0.670138\pi$
$613$	− 1055.00i	− 1.72104i	0.509413	−	0.860522i	$-0.329862\pi$
$614$	396.000	0.644951
$615$	0	0
$616$	− 70.7107i	− 0.114790i
$617$	− 505.000i	− 0.818476i	−0.912428	−	0.409238i	$-0.865794\pi$
$617$	− 505.000i	− 0.818476i	0.912428	−	0.409238i	$-0.134206\pi$
$618$	67.8823	0.109842
$619$	130.000	0.210016	0.105008	−	0.994471i	$-0.466513\pi$
$619$	130.000	0.210016	0.105008	+	0.994471i	$0.466513\pi$
$620$	0	0
$621$	− 282.843i	− 0.455463i
$622$	332.340	0.534309
$623$	−636.396	−1.02150
$624$	−192.000	−0.307692
$625$	0	0
$626$	− 438.406i	− 0.700329i
$627$	−268.701	−0.428550
$628$	− 380.000i	− 0.605096i
$629$	− 636.396i	− 1.01176i
$630$	0	0
$631$	−475.000	−0.752773	−0.376387	−	0.926463i	$-0.622834\pi$
$631$	−475.000	−0.752773	−0.376387	+	0.926463i	$0.622834\pi$
$632$	120.000i	0.189873i
$633$	240.000i	0.379147i
$634$	−264.000	−0.416404
$635$	0	0
$636$	−144.000	−0.226415
$637$	−407.294	−0.639393
$638$	300.000i	0.470219i
$639$	0	0
$640$	0	0
$641$	− 848.528i	− 1.32376i	−0.749611	−	0.661878i	$-0.769760\pi$
$641$	− 848.528i	− 1.32376i	0.749611	−	0.661878i	$-0.230240\pi$
$642$	−407.294	−0.634414
$643$	955.000i	1.48523i	0.669721	+	0.742613i	$0.266414\pi$
$643$	955.000i	1.48523i	−0.669721	+	0.742613i	$0.733586\pi$
$644$	−100.000	−0.155280
$645$	0	0
$646$	− 671.751i	− 1.03986i
$647$	965.000i	1.49150i	0.666226	+	0.745750i	$0.267908\pi$
$647$	965.000i	1.49150i	−0.666226	+	0.745750i	$0.732092\pi$
$648$	200.818	0.309905
$649$	− 424.264i	− 0.653720i
$650$	0	0
$651$	600.000	0.921659
$652$	− 220.000i	− 0.337423i
$653$	− 935.000i	− 1.43185i	−0.698176	−	0.715926i	$-0.746005\pi$
$653$	− 935.000i	− 1.43185i	0.698176	−	0.715926i	$-0.253995\pi$
$654$	509.117i	0.778466i
$655$	0	0
$656$	− 169.706i	− 0.258698i
$657$	− 25.0000i	− 0.0380518i
$658$	35.3553	0.0537315
$659$	84.8528i	0.128760i	0.997925	+	0.0643800i	$0.0205070\pi$
$659$	84.8528i	0.128760i	−0.997925	+	0.0643800i	$0.979493\pi$
$660$	0	0
$661$	− 678.823i	− 1.02696i	−0.858101	−	0.513481i	$-0.828356\pi$
$661$	− 678.823i	− 1.02696i	0.858101	−	0.513481i	$-0.171644\pi$
$662$	− 420.000i	− 0.634441i
$663$	1200.00i	1.80995i
$664$	− 367.696i	− 0.553758i
$665$	0	0
$666$	36.0000	0.0540541
$667$	424.264	0.636078
$668$	−118.794	−0.177835
$669$	−1032.00	−1.54260
$670$	0	0
$671$	475.000	0.707899
$672$	− 80.0000i	− 0.119048i
$673$	−186.676	−0.277379	−0.138690	−	0.990336i	$-0.544289\pi$
$673$	−186.676	−0.277379	−0.138690	+	0.990336i	$0.544289\pi$
$674$	−744.000	−1.10386
$675$	0	0
$676$	238.000	0.352071
$677$	−907.925	−1.34110	−0.670550	−	0.741864i	$-0.733942\pi$
$677$	−907.925	−1.34110	−0.670550	+	0.741864i	$0.733942\pi$
$678$	−441.235	−0.650789
$679$	− 84.8528i	− 0.124967i
$680$	0	0
$681$	−192.000	−0.281938
$682$	300.000i	0.439883i
$683$	1120.06	1.63991	0.819954	−	0.572429i	$-0.193999\pi$
$683$	1120.06	1.63991	0.819954	+	0.572429i	$0.193999\pi$
$684$	38.0000	0.0555556
$685$	0	0
$686$	− 516.188i	− 0.752461i
$687$	410.122	0.596975
$688$	− 20.0000i	− 0.0290698i
$689$	432.000	0.626996
$690$	0	0
$691$	−715.000	−1.03473	−0.517366	−	0.855764i	$-0.673087\pi$
$691$	−715.000	−1.03473	−0.517366	+	0.855764i	$0.673087\pi$
$692$	−373.352	−0.539527
$693$	− 25.0000i	− 0.0360750i
$694$	− 176.777i	− 0.254721i
$695$	0	0
$696$	339.411i	0.487660i
$697$	−1060.66	−1.52175
$698$	32.5269	0.0466002
$699$	− 947.523i	− 1.35554i
$700$	0	0
$701$	−430.000	−0.613409	−0.306705	−	0.951805i	$-0.599226\pi$
$701$	−430.000	−0.613409	−0.306705	+	0.951805i	$0.599226\pi$
$702$	−678.823	−0.966984
$703$	−483.661	−0.687996
$704$	40.0000	0.0568182
$705$	0	0
$706$	579.828i	0.821285i
$707$	250.000i	0.353607i
$708$	− 480.000i	− 0.677966i
$709$	382.000	0.538787	0.269394	−	0.963030i	$-0.413177\pi$
$709$	382.000	0.538787	0.269394	+	0.963030i	$0.413177\pi$
$710$	0	0
$711$	42.4264i	0.0596715i
$712$	− 360.000i	− 0.505618i
$713$	424.264	0.595041
$714$	−500.000	−0.700280
$715$	0	0
$716$	254.558i	0.355529i
$717$	−557.200	−0.777127
$718$	−671.751	−0.935587
$719$	115.000	0.159944	0.0799722	−	0.996797i	$-0.474517\pi$
$719$	115.000	0.159944	0.0799722	+	0.996797i	$0.474517\pi$
$720$	0	0
$721$	− 84.8528i	− 0.117688i
$722$	−510.531	−0.707107
$723$	840.000i	1.16183i
$724$	− 509.117i	− 0.703200i
$725$	0	0
$726$	384.000	0.528926
$727$	− 1075.00i	− 1.47868i	−0.673333	−	0.739340i	$-0.735138\pi$
$727$	− 1075.00i	− 1.47868i	0.673333	−	0.739340i	$-0.264862\pi$
$728$	240.000i	0.329670i
$729$	791.000	1.08505
$730$	0	0
$731$	−125.000	−0.170999
$732$	537.401	0.734155
$733$	− 530.000i	− 0.723056i	−0.932361	−	0.361528i	$-0.882255\pi$
$733$	− 530.000i	− 0.723056i	0.932361	−	0.361528i	$-0.117745\pi$
$734$	− 325.269i	− 0.443146i
$735$	0	0
$736$	− 56.5685i	− 0.0768594i
$737$	−551.543	−0.748363
$738$	− 60.0000i	− 0.0813008i
$739$	547.000	0.740189	0.370095	−	0.928994i	$-0.379325\pi$
$739$	547.000	0.740189	0.370095	+	0.928994i	$0.379325\pi$
$740$	0	0
$741$	912.000	1.23077
$742$	180.000i	0.242588i
$743$	−958.837	−1.29049	−0.645247	−	0.763974i	$-0.723245\pi$
$743$	−958.837	−1.29049	−0.645247	+	0.763974i	$0.723245\pi$
$744$	339.411i	0.456198i
$745$	0	0
$746$	96.0000	0.128686
$747$	− 130.000i	− 0.174029i
$748$	− 250.000i	− 0.334225i
$749$	509.117i	0.679729i
$750$	0	0
$751$	− 169.706i	− 0.225973i	−0.993597	−	0.112986i	$-0.963958\pi$
$751$	− 169.706i	− 0.225973i	0.993597	−	0.112986i	$-0.0360417\pi$
$752$	20.0000i	0.0265957i
$753$	489.318	0.649825
$754$	− 1018.23i	− 1.35044i
$755$	0	0
$756$	− 282.843i	− 0.374131i
$757$	1055.00i	1.39366i	0.717237	+	0.696830i	$0.245407\pi$
$757$	1055.00i	1.39366i	−0.717237	+	0.696830i	$0.754593\pi$
$758$	− 360.000i	− 0.474934i
$759$	141.421i	0.186326i
$760$	0	0
$761$	215.000	0.282523	0.141261	−	0.989972i	$-0.454884\pi$
$761$	215.000	0.282523	0.141261	+	0.989972i	$0.454884\pi$
$762$	916.410	1.20264
$763$	636.396	0.834071
$764$	586.000	0.767016
$765$	0	0
$766$	204.000	0.266319
$767$	1440.00i	1.87744i
$768$	45.2548	0.0589256
$769$	145.000	0.188557	0.0942783	−	0.995546i	$-0.469946\pi$
$769$	145.000	0.188557	0.0942783	+	0.995546i	$0.469946\pi$
$770$	0	0
$771$	192.000	0.249027
$772$	118.794	0.153878
$773$	407.294	0.526900	0.263450	−	0.964673i	$-0.415140\pi$
$773$	407.294	0.526900	0.263450	+	0.964673i	$0.415140\pi$
$774$	− 7.07107i	− 0.00913575i
$775$	0	0
$776$	48.0000	0.0618557
$777$	360.000i	0.463320i
$778$	−782.060	−1.00522
$779$	806.102i	1.03479i
$780$	0	0
$781$	0	0
$782$	−353.553	−0.452114
$783$	1200.00i	1.53257i
$784$	96.0000	0.122449
$785$	0	0
$786$	652.000	0.829517
$787$	−186.676	−0.237200	−0.118600	−	0.992942i	$-0.537841\pi$
$787$	−186.676	−0.237200	−0.118600	+	0.992942i	$0.537841\pi$
$788$	− 140.000i	− 0.177665i
$789$	1004.09i	1.27261i
$790$	0	0
$791$	551.543i	0.697273i
$792$	14.1421	0.0178562
$793$	−1612.20	−2.03304
$794$	− 473.762i	− 0.596677i
$795$	0	0
$796$	−346.000	−0.434673
$797$	704.278	0.883662	0.441831	−	0.897098i	$-0.354329\pi$
$797$	704.278	0.883662	0.441831	+	0.897098i	$0.354329\pi$
$798$	380.000i	0.476190i
$799$	125.000	0.156446
$800$	0	0
$801$	− 127.279i	− 0.158900i
$802$	− 300.000i	− 0.374065i
$803$	125.000i	0.155666i
$804$	−624.000	−0.776119
$805$	0	0
$806$	− 1018.23i	− 1.26332i
$807$	1080.00i	1.33829i
$808$	−141.421	−0.175026
$809$	457.000	0.564895	0.282447	−	0.959283i	$-0.408854\pi$
$809$	457.000	0.564895	0.282447	+	0.959283i	$0.408854\pi$
$810$	0	0
$811$	509.117i	0.627764i	0.949462	+	0.313882i	$0.101630\pi$
$811$	509.117i	0.627764i	−0.949462	+	0.313882i	$0.898370\pi$
$812$	424.264	0.522493
$813$	311.127	0.382690
$814$	−180.000	−0.221130
$815$	0	0
$816$	− 282.843i	− 0.346621i
$817$	95.0000i	0.116279i
$818$	1020.00i	1.24694i
$819$	84.8528i	0.103605i
$820$	0	0
$821$	167.000	0.203410	0.101705	−	0.994815i	$-0.467570\pi$
$821$	167.000	0.203410	0.101705	+	0.994815i	$0.467570\pi$
$822$	− 380.000i	− 0.462287i
$823$	1315.00i	1.59781i	0.601455	+	0.798906i	$0.294588\pi$
$823$	1315.00i	1.59781i	−0.601455	+	0.798906i	$0.705412\pi$
$824$	48.0000	0.0582524
$825$	0	0
$826$	−600.000	−0.726392
$827$	534.573	0.646400	0.323200	−	0.946331i	$-0.395241\pi$
$827$	534.573	0.646400	0.323200	+	0.946331i	$0.395241\pi$
$828$	− 20.0000i	− 0.0241546i
$829$	− 763.675i	− 0.921201i	−0.887608	−	0.460600i	$-0.847634\pi$
$829$	− 763.675i	− 0.921201i	0.887608	−	0.460600i	$-0.152366\pi$
$830$	0	0
$831$	− 749.533i	− 0.901965i
$832$	−135.765	−0.163178
$833$	− 600.000i	− 0.720288i
$834$	500.000	0.599520
$835$	0	0
$836$	−190.000	−0.227273
$837$	1200.00i	1.43369i
$838$	87.6812	0.104632
$839$	339.411i	0.404543i	0.979330	+	0.202271i	$0.0648323\pi$
$839$	339.411i	0.404543i	−0.979330	+	0.202271i	$0.935168\pi$
$840$	0	0
$841$	−959.000	−1.14031
$842$	− 420.000i	− 0.498812i
$843$	1200.00i	1.42349i
$844$	169.706i	0.201073i
$845$	0	0
$846$	7.07107i	0.00835824i
$847$	− 480.000i	− 0.566706i
$848$	−101.823	−0.120075
$849$	− 353.553i	− 0.416435i
$850$	0	0
$851$	254.558i	0.299129i
$852$	0	0
$853$	− 770.000i	− 0.902696i	−0.892348	−	0.451348i	$-0.850943\pi$
$853$	− 770.000i	− 0.902696i	0.892348	−	0.451348i	$-0.149057\pi$
$854$	− 671.751i	− 0.786594i
$855$	0	0
$856$	−288.000	−0.336449
$857$	1255.82	1.46537	0.732685	−	0.680568i	$-0.238267\pi$
$857$	1255.82	1.46537	0.732685	+	0.680568i	$0.238267\pi$
$858$	339.411	0.395584
$859$	−557.000	−0.648428	−0.324214	−	0.945984i	$-0.605100\pi$
$859$	−557.000	−0.648428	−0.324214	+	0.945984i	$0.605100\pi$
$860$	0	0
$861$	600.000	0.696864
$862$	720.000i	0.835267i
$863$	992.778	1.15038	0.575190	−	0.818020i	$-0.304928\pi$
$863$	992.778	1.15038	0.575190	+	0.818020i	$0.304928\pi$
$864$	160.000	0.185185
$865$	0	0
$866$	324.000	0.374134
$867$	−950.352	−1.09614
$868$	424.264	0.488783
$869$	− 212.132i	− 0.244111i
$870$	0	0
$871$	1872.00	2.14925
$872$	360.000i	0.412844i
$873$	16.9706	0.0194394
$874$	268.701i	0.307438i
$875$	0	0
$876$	141.421i	0.161440i
$877$	−186.676	−0.212858	−0.106429	−	0.994320i	$-0.533942\pi$
$877$	−186.676	−0.212858	−0.106429	+	0.994320i	$0.533942\pi$
$878$	1140.00i	1.29841i
$879$	528.000	0.600683
$880$	0	0
$881$	−25.0000	−0.0283768	−0.0141884	−	0.999899i	$-0.504516\pi$
$881$	−25.0000	−0.0283768	−0.0141884	+	0.999899i	$0.504516\pi$
$882$	33.9411	0.0384820
$883$	− 965.000i	− 1.09287i	−0.837503	−	0.546433i	$-0.815985\pi$
$883$	− 965.000i	− 1.09287i	0.837503	−	0.546433i	$-0.184015\pi$
$884$	848.528i	0.959873i
$885$	0	0
$886$	516.188i	0.582605i
$887$	780.646	0.880097	0.440048	−	0.897974i	$-0.354961\pi$
$887$	780.646	0.880097	0.440048	+	0.897974i	$0.354961\pi$
$888$	−203.647	−0.229332
$889$	− 1145.51i	− 1.28854i
$890$	0	0
$891$	−355.000	−0.398429
$892$	−729.734	−0.818088
$893$	− 95.0000i	− 0.106383i
$894$	860.000	0.961969
$895$	0	0
$896$	− 56.5685i	− 0.0631345i
$897$	− 480.000i	− 0.535117i
$898$	− 1080.00i	− 1.20267i
$899$	−1800.00	−2.00222
$900$	0	0
$901$	636.396i	0.706322i
$902$	300.000i	0.332594i
$903$	70.7107	0.0783064
$904$	−312.000	−0.345133
$905$	0	0
$906$	339.411i	0.374626i
$907$	313.955	0.346147	0.173074	−	0.984909i	$-0.444630\pi$
$907$	313.955	0.346147	0.173074	+	0.984909i	$0.444630\pi$
$908$	−135.765	−0.149520
$909$	−50.0000	−0.0550055
$910$	0	0
$911$	933.381i	1.02457i	0.858816	+	0.512284i	$0.171200\pi$
$911$	933.381i	1.02457i	−0.858816	+	0.512284i	$0.828800\pi$
$912$	−214.960	−0.235702
$913$	650.000i	0.711939i
$914$	374.767i	0.410029i
$915$	0	0
$916$	290.000	0.316594
$917$	− 815.000i	− 0.888768i
$918$	− 1000.00i	− 1.08932i
$919$	538.000	0.585419	0.292709	−	0.956201i	$-0.405443\pi$
$919$	538.000	0.585419	0.292709	+	0.956201i	$0.405443\pi$
$920$	0	0
$921$	−792.000	−0.859935
$922$	782.060	0.848221
$923$	0	0
$924$	141.421i	0.153053i
$925$	0	0
$926$	685.894i	0.740706i
$927$	16.9706	0.0183070
$928$	240.000i	0.258621i
$929$	742.000	0.798708	0.399354	−	0.916797i	$-0.369234\pi$
$929$	742.000	0.798708	0.399354	+	0.916797i	$0.369234\pi$
$930$	0	0
$931$	−456.000	−0.489796
$932$	− 670.000i	− 0.718884i
$933$	−664.680	−0.712412
$934$	162.635i	0.174127i
$935$	0	0
$936$	−48.0000	−0.0512821
$937$	335.000i	0.357524i	0.983892	+	0.178762i	$0.0572092\pi$
$937$	335.000i	0.357524i	−0.983892	+	0.178762i	$0.942791\pi$
$938$	780.000i	0.831557i
$939$	876.812i	0.933773i
$940$	0	0
$941$	− 424.264i	− 0.450865i	−0.974259	−	0.225433i	$-0.927620\pi$
$941$	− 424.264i	− 0.450865i	0.974259	−	0.225433i	$-0.0723795\pi$
$942$	760.000i	0.806794i
$943$	424.264	0.449909
$944$	− 339.411i	− 0.359546i
$945$	0	0
$946$	35.3553i	0.0373735i
$947$	− 1210.00i	− 1.27772i	−0.769323	−	0.638860i	$-0.779406\pi$
$947$	− 1210.00i	− 1.27772i	0.769323	−	0.638860i	$-0.220594\pi$
$948$	− 240.000i	− 0.253165i
$949$	− 424.264i	− 0.447064i
$950$	0	0
$951$	528.000	0.555205
$952$	−353.553	−0.371380
$953$	992.778	1.04174	0.520870	−	0.853636i	$-0.325608\pi$
$953$	992.778	1.04174	0.520870	+	0.853636i	$0.325608\pi$
$954$	−36.0000	−0.0377358
$955$	0	0
$956$	−394.000	−0.412134
$957$	− 600.000i	− 0.626959i
$958$	−692.965	−0.723345
$959$	−475.000	−0.495308
$960$	0	0
$961$	−839.000	−0.873049
$962$	610.940	0.635073
$963$	−101.823	−0.105736
$964$	593.970i	0.616151i
$965$	0	0
$966$	200.000	0.207039
$967$	350.000i	0.361944i	0.983488	+	0.180972i	$0.0579244\pi$
$967$	350.000i	0.361944i	−0.983488	+	0.180972i	$0.942076\pi$
$968$	271.529	0.280505
$969$	1343.50i	1.38648i
$970$	0	0
$971$	254.558i	0.262161i	0.991372	+	0.131081i	$0.0418447\pi$
$971$	254.558i	0.262161i	−0.991372	+	0.131081i	$0.958155\pi$
$972$	107.480	0.110576
$973$	− 625.000i	− 0.642343i
$974$	−864.000	−0.887064
$975$	0	0
$976$	380.000	0.389344
$977$	398.808	0.408197	0.204098	−	0.978950i	$-0.434574\pi$
$977$	398.808	0.408197	0.204098	+	0.978950i	$0.434574\pi$
$978$	440.000i	0.449898i
$979$	636.396i	0.650047i
$980$	0	0
$981$	127.279i	0.129744i
$982$	115.966	0.118091
$983$	−695.793	−0.707826	−0.353913	−	0.935278i	$-0.615149\pi$
$983$	−695.793	−0.707826	−0.353913	+	0.935278i	$0.615149\pi$
$984$	339.411i	0.344930i
$985$	0	0
$986$	1500.00	1.52130
$987$	−70.7107	−0.0716420
$988$	644.881	0.652714
$989$	50.0000	0.0505561
$990$	0	0
$991$	381.838i	0.385305i	0.981267	+	0.192653i	$0.0617091\pi$
$991$	381.838i	0.385305i	−0.981267	+	0.192653i	$0.938291\pi$
$992$	240.000i	0.241935i
$993$	840.000i	0.845921i
$994$	0	0
$995$	0	0
$996$	735.391i	0.738344i
$997$	− 265.000i	− 0.265797i	−0.991130	−	0.132899i	$-0.957572\pi$
$997$	− 265.000i	− 0.265797i	0.991130	−	0.132899i	$-0.0424285\pi$
$998$	685.894	0.687268
$999$	−720.000	−0.720721

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.3.d.a.949.2		4
5.2	odd	4		950.3.c.a.151.1		2
5.3	odd	4		38.3.b.a.37.2	yes	2
5.4	even	2	inner	950.3.d.a.949.3		4
15.8	even	4		342.3.d.a.37.1		2
19.18	odd	2	inner	950.3.d.a.949.4		4
20.3	even	4		304.3.e.c.113.1		2
40.3	even	4		1216.3.e.i.1025.2		2
40.13	odd	4		1216.3.e.j.1025.1		2
60.23	odd	4		2736.3.o.h.721.1		2
95.18	even	4		38.3.b.a.37.1	✓	2
95.37	even	4		950.3.c.a.151.2		2
95.94	odd	2	inner	950.3.d.a.949.1		4
285.113	odd	4		342.3.d.a.37.2		2
380.303	odd	4		304.3.e.c.113.2		2
760.493	even	4		1216.3.e.j.1025.2		2
760.683	odd	4		1216.3.e.i.1025.1		2
1140.683	even	4		2736.3.o.h.721.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.3.b.a.37.1	✓	2	95.18	even	4
38.3.b.a.37.2	yes	2	5.3	odd	4
304.3.e.c.113.1		2	20.3	even	4
304.3.e.c.113.2		2	380.303	odd	4
342.3.d.a.37.1		2	15.8	even	4
342.3.d.a.37.2		2	285.113	odd	4
950.3.c.a.151.1		2	5.2	odd	4
950.3.c.a.151.2		2	95.37	even	4
950.3.d.a.949.1		4	95.94	odd	2	inner
950.3.d.a.949.2		4	1.1	even	1	trivial
950.3.d.a.949.3		4	5.4	even	2	inner
950.3.d.a.949.4		4	19.18	odd	2	inner
1216.3.e.i.1025.1		2	760.683	odd	4
1216.3.e.i.1025.2		2	40.3	even	4
1216.3.e.j.1025.1		2	40.13	odd	4
1216.3.e.j.1025.2		2	760.493	even	4
2736.3.o.h.721.1		2	60.23	odd	4
2736.3.o.h.721.2		2	1140.683	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:	\( N \)	\(=\)	\( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	950.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} +2.82843 q^{3} +2.00000 q^{4} -4.00000 q^{6} +5.00000i q^{7} -2.82843 q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.41421 q^{2} +2.82843 q^{3} +2.00000 q^{4} -4.00000 q^{6} +5.00000i q^{7} -2.82843 q^{8} -1.00000 q^{9} +5.00000 q^{11} +5.65685 q^{12} -16.9706 q^{13} -7.07107i q^{14} +4.00000 q^{16} -25.0000i q^{17} +1.41421 q^{18} -19.0000 q^{19} +14.1421i q^{21} -7.07107 q^{22} +10.0000i q^{23} -8.00000 q^{24} +24.0000 q^{26} -28.2843 q^{27} +10.0000i q^{28} -42.4264i q^{29} -42.4264i q^{31} -5.65685 q^{32} +14.1421 q^{33} +35.3553i q^{34} -2.00000 q^{36} +25.4558 q^{37} +26.8701 q^{38} -48.0000 q^{39} -42.4264i q^{41} -20.0000i q^{42} -5.00000i q^{43} +10.0000 q^{44} -14.1421i q^{46} +5.00000i q^{47} +11.3137 q^{48} +24.0000 q^{49} -70.7107i q^{51} -33.9411 q^{52} -25.4558 q^{53} +40.0000 q^{54} -14.1421i q^{56} -53.7401 q^{57} +60.0000i q^{58} -84.8528i q^{59} +95.0000 q^{61} +60.0000i q^{62} -5.00000i q^{63} +8.00000 q^{64} -20.0000 q^{66} -110.309 q^{67} -50.0000i q^{68} +28.2843i q^{69} +2.82843 q^{72} +25.0000i q^{73} -36.0000 q^{74} -38.0000 q^{76} +25.0000i q^{77} +67.8823 q^{78} -42.4264i q^{79} -71.0000 q^{81} +60.0000i q^{82} +130.000i q^{83} +28.2843i q^{84} +7.07107i q^{86} -120.000i q^{87} -14.1421 q^{88} +127.279i q^{89} -84.8528i q^{91} +20.0000i q^{92} -120.000i q^{93} -7.07107i q^{94} -16.0000 q^{96} -16.9706 q^{97} -33.9411 q^{98} -5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{4} - 16 q^{6} - 4 q^{9}+O(q^{10}) \) 4 * q + 8 * q^4 - 16 * q^6 - 4 * q^9	\( 4 q + 8 q^{4} - 16 q^{6} - 4 q^{9} + 20 q^{11} + 16 q^{16} - 76 q^{19} - 32 q^{24} + 96 q^{26} - 8 q^{36} - 192 q^{39} + 40 q^{44} + 96 q^{49} + 160 q^{54} + 380 q^{61} + 32 q^{64} - 80 q^{66} - 144 q^{74} - 152 q^{76} - 284 q^{81} - 64 q^{96} - 20 q^{99}+O(q^{100}) \) 4 * q + 8 * q^4 - 16 * q^6 - 4 * q^9 + 20 * q^11 + 16 * q^16 - 76 * q^19 - 32 * q^24 + 96 * q^26 - 8 * q^36 - 192 * q^39 + 40 * q^44 + 96 * q^49 + 160 * q^54 + 380 * q^61 + 32 * q^64 - 80 * q^66 - 144 * q^74 - 152 * q^76 - 284 * q^81 - 64 * q^96 - 20 * q^99

Introduction

L-functions

Modular forms

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Database

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