LMFDB - Embedded newform 3100.3.d.e.1301.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3100,3,Mod(1301,3100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3100, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3100.1301");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 3100 = 2^{2} \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	3100.d (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:	$84.4688819517$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 114 x^{18} + 5280 x^{16} + 128422 x^{14} + 1776819 x^{12} + 14249420 x^{10} + 65297060 x^{8} + \cdots + 20793600 $ x^20 + 114x^18 + 5280x^16 + 128422x^14 + 1776819x^12 + 14249420x^10 + 65297060x^8 + 162416640x^6 + 199076000x^4 + 107961600*x^2 + 20793600
Coefficient ring:	$\Z[a_1, \ldots, a_{31}]$
Coefficient ring index:	$ 2^{23}\cdot 3 $
Twist minimal:	no (minimal twist has level 620)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1301.3
Root			$-4.37942i$ of defining polynomial
Character	$\chi$	$=$	3100.1301
Dual form			3100.3.d.e.1301.18

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.37942i q^{3} +3.17309 q^{7} -10.1793 q^{9} +O(q^{10})$	$q-4.37942i q^{3} +3.17309 q^{7} -10.1793 q^{9} -21.9048i q^{11} +10.7062i q^{13} +29.8794i q^{17} +26.6507 q^{19} -13.8963i q^{21} +9.73695i q^{23} +5.16467i q^{27} -12.5940i q^{29} +(-27.6074 - 14.1007i) q^{31} -95.9302 q^{33} -48.4755i q^{37} +46.8868 q^{39} -68.8299 q^{41} -36.0647i q^{43} +5.79288 q^{47} -38.9315 q^{49} +130.855 q^{51} +11.5038i q^{53} -116.714i q^{57} +18.1346 q^{59} -97.3036i q^{61} -32.2999 q^{63} -88.4233 q^{67} +42.6422 q^{69} +28.8155 q^{71} +27.2353i q^{73} -69.5059i q^{77} +78.8596i q^{79} -68.9955 q^{81} -37.4532i q^{83} -55.1543 q^{87} -123.760i q^{89} +33.9717i q^{91} +(-61.7530 + 120.904i) q^{93} -102.117 q^{97} +222.976i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 12 q^{7} - 48 q^{9}+O(q^{10}) $ 20 * q - 12 * q^7 - 48 * q^9	$ 20 q - 12 q^{7} - 48 q^{9} + 60 q^{19} + 8 q^{31} - 68 q^{33} + 28 q^{39} - 80 q^{41} + 48 q^{47} + 84 q^{49} + 344 q^{51} + 160 q^{59} + 232 q^{63} + 180 q^{67} - 140 q^{69} - 108 q^{71} + 336 q^{81} + 236 q^{87} + 332 q^{93} + 112 q^{97}+O(q^{100}) $ 20 * q - 12 * q^7 - 48 * q^9 + 60 * q^19 + 8 * q^31 - 68 * q^33 + 28 * q^39 - 80 * q^41 + 48 * q^47 + 84 * q^49 + 344 * q^51 + 160 * q^59 + 232 * q^63 + 180 * q^67 - 140 * q^69 - 108 * q^71 + 336 * q^81 + 236 * q^87 + 332 * q^93 + 112 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3100\mathbb{Z}\right)^\times$.

$n$	$1551$	$1801$	$2977$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$		−	4.37942i		−	1.45981i	−0.683551	−	0.729903i	$-0.739565\pi$
$3$		−	4.37942i		−	1.45981i	0.683551	−	0.729903i	$-0.260435\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	3.17309			0.453299			0.226649	−	0.973976i	$-0.427223\pi$
$7$	3.17309			0.453299			0.226649	+	0.973976i	$0.427223\pi$
$8$		0			0
$9$	−10.1793			−1.13103
$10$		0			0
$11$		−	21.9048i		−	1.99134i	−0.0929327	−	0.995672i	$-0.529624\pi$
$11$		−	21.9048i		−	1.99134i	0.0929327	−	0.995672i	$-0.470376\pi$
$12$		0			0
$13$			10.7062i			0.823552i	0.911285	+	0.411776i	$0.135091\pi$
$13$			10.7062i			0.823552i	−0.911285	+	0.411776i	$0.864909\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			29.8794i			1.75761i	0.477177	+	0.878807i	$0.341660\pi$
$17$			29.8794i			1.75761i	−0.477177	+	0.878807i	$0.658340\pi$
$18$		0			0
$19$	26.6507			1.40267			0.701333	−	0.712834i	$-0.252589\pi$
$19$	26.6507			1.40267			0.701333	+	0.712834i	$0.252589\pi$
$20$		0			0
$21$		−	13.8963i		−	0.661728i
$22$		0			0
$23$			9.73695i			0.423346i	0.977341	+	0.211673i	$0.0678911\pi$
$23$			9.73695i			0.423346i	−0.977341	+	0.211673i	$0.932109\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$			5.16467i			0.191284i
$28$		0			0
$29$		−	12.5940i		−	0.434275i	−0.976141	−	0.217138i	$-0.930328\pi$
$29$		−	12.5940i		−	0.434275i	0.976141	−	0.217138i	$-0.0696720\pi$
$30$		0			0
$31$	−27.6074	−	14.1007i	−0.890562	−	0.454862i
$31$	−27.6074	−	14.1007i	−0.890562	−	0.454862i
$32$		0			0
$33$	−95.9302			−2.90698
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	48.4755i		−	1.31015i	−0.755564	−	0.655074i	$-0.772637\pi$
$37$		−	48.4755i		−	1.31015i	0.755564	−	0.655074i	$-0.227363\pi$
$38$		0			0
$39$	46.8868			1.20223
$40$		0			0
$41$	−68.8299			−1.67878			−0.839390	−	0.543530i	$-0.817087\pi$
$41$	−68.8299			−1.67878			−0.839390	+	0.543530i	$0.817087\pi$
$42$		0			0
$43$		−	36.0647i		−	0.838713i	−0.907822	−	0.419357i	$-0.862256\pi$
$43$		−	36.0647i		−	0.838713i	0.907822	−	0.419357i	$-0.137744\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	5.79288			0.123253			0.0616264	−	0.998099i	$-0.480371\pi$
$47$	5.79288			0.123253			0.0616264	+	0.998099i	$0.480371\pi$
$48$		0			0
$49$	−38.9315			−0.794520
$50$		0			0
$51$	130.855			2.56578
$52$		0			0
$53$			11.5038i			0.217054i	0.994094	+	0.108527i	$0.0346133\pi$
$53$			11.5038i			0.217054i	−0.994094	+	0.108527i	$0.965387\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		−	116.714i		−	2.04762i
$58$		0			0
$59$	18.1346			0.307367			0.153683	−	0.988120i	$-0.450886\pi$
$59$	18.1346			0.307367			0.153683	+	0.988120i	$0.450886\pi$
$60$		0			0
$61$		−	97.3036i		−	1.59514i	−0.603226	−	0.797570i	$-0.706118\pi$
$61$		−	97.3036i		−	1.59514i	0.603226	−	0.797570i	$-0.293882\pi$
$62$		0			0
$63$	−32.2999			−0.512696
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−88.4233			−1.31975			−0.659875	−	0.751375i	$-0.729391\pi$
$67$	−88.4233			−1.31975			−0.659875	+	0.751375i	$0.729391\pi$
$68$		0			0
$69$	42.6422			0.618003
$70$		0			0
$71$	28.8155			0.405852			0.202926	−	0.979194i	$-0.434955\pi$
$71$	28.8155			0.405852			0.202926	+	0.979194i	$0.434955\pi$
$72$		0			0
$73$			27.2353i			0.373087i	0.982447	+	0.186543i	$0.0597285\pi$
$73$			27.2353i			0.373087i	−0.982447	+	0.186543i	$0.940272\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	69.5059i		−	0.902674i
$78$		0			0
$79$			78.8596i			0.998223i	0.866538	+	0.499112i	$0.166340\pi$
$79$			78.8596i			0.998223i	−0.866538	+	0.499112i	$0.833660\pi$
$80$		0			0
$81$	−68.9955			−0.851796
$82$		0			0
$83$		−	37.4532i		−	0.451243i	−0.974215	−	0.225621i	$-0.927559\pi$
$83$		−	37.4532i		−	0.451243i	0.974215	−	0.225621i	$-0.0724412\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−55.1543			−0.633958
$88$		0			0
$89$		−	123.760i		−	1.39056i	−0.718739	−	0.695280i	$-0.755280\pi$
$89$		−	123.760i		−	1.39056i	0.718739	−	0.695280i	$-0.244720\pi$
$90$		0			0
$91$			33.9717i			0.373315i
$92$		0			0
$93$	−61.7530	+	120.904i	−0.664010	+	1.30005i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−102.117			−1.05275			−0.526377	−	0.850251i	$-0.676450\pi$
$97$	−102.117			−1.05275			−0.526377	+	0.850251i	$0.676450\pi$
$98$		0			0
$99$			222.976i			2.25228i
$100$		0			0
$101$	12.1447			0.120244			0.0601222	−	0.998191i	$-0.480851\pi$
$101$	12.1447			0.120244			0.0601222	+	0.998191i	$0.480851\pi$
$102$		0			0
$103$	−137.839			−1.33825			−0.669124	−	0.743151i	$-0.733330\pi$
$103$	−137.839			−1.33825			−0.669124	+	0.743151i	$0.733330\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−46.3858			−0.433512			−0.216756	−	0.976226i	$-0.569548\pi$
$107$	−46.3858			−0.433512			−0.216756	+	0.976226i	$0.569548\pi$
$108$		0			0
$109$	89.8462			0.824277			0.412138	−	0.911121i	$-0.364782\pi$
$109$	89.8462			0.824277			0.412138	+	0.911121i	$0.364782\pi$
$110$		0			0
$111$	−212.295			−1.91256
$112$		0			0
$113$	81.6265			0.722358			0.361179	−	0.932496i	$-0.382374\pi$
$113$	81.6265			0.722358			0.361179	+	0.932496i	$0.382374\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		−	108.981i		−	0.931465i
$118$		0			0
$119$			94.8102i			0.796724i
$120$		0			0
$121$	−358.820			−2.96545
$122$		0			0
$123$			301.435i			2.45069i
$124$		0			0
$125$		0			0
$126$		0			0
$127$			35.2926i			0.277894i	0.990300	+	0.138947i	$0.0443718\pi$
$127$			35.2926i			0.277894i	−0.990300	+	0.138947i	$0.955628\pi$
$128$		0			0
$129$	−157.942			−1.22436
$130$		0			0
$131$	143.068			1.09212			0.546060	−	0.837746i	$-0.316127\pi$
$131$	143.068			1.09212			0.546060	+	0.837746i	$0.316127\pi$
$132$		0			0
$133$	84.5649			0.635827
$134$		0			0
$135$		0			0
$136$		0			0
$137$			76.9814i			0.561908i	0.959721	+	0.280954i	$0.0906508\pi$
$137$			76.9814i			0.561908i	−0.959721	+	0.280954i	$0.909349\pi$
$138$		0			0
$139$		−	41.6873i		−	0.299908i	−0.988693	−	0.149954i	$-0.952087\pi$
$139$		−	41.6873i		−	0.299908i	0.988693	−	0.149954i	$-0.0479126\pi$
$140$		0			0
$141$		−	25.3694i		−	0.179925i
$142$		0			0
$143$	234.517			1.63998
$144$		0			0
$145$		0			0
$146$		0			0
$147$			170.497i			1.15985i
$148$		0			0
$149$	−220.223			−1.47801			−0.739003	−	0.673702i	$-0.764703\pi$
$149$	−220.223			−1.47801			−0.739003	+	0.673702i	$0.764703\pi$
$150$		0			0
$151$		−	292.506i		−	1.93712i	−0.248771	−	0.968562i	$-0.580027\pi$
$151$		−	292.506i		−	1.93712i	0.248771	−	0.968562i	$-0.419973\pi$
$152$		0			0
$153$		−	304.152i		−	1.98792i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	230.414			1.46760			0.733802	−	0.679363i	$-0.237744\pi$
$157$	230.414			1.46760			0.733802	+	0.679363i	$0.237744\pi$
$158$		0			0
$159$	50.3801			0.316856
$160$		0			0
$161$			30.8962i			0.191902i
$162$		0			0
$163$	57.3892			0.352081			0.176040	−	0.984383i	$-0.443671\pi$
$163$	57.3892			0.352081			0.176040	+	0.984383i	$0.443671\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	220.623i		−	1.32110i	−0.750783	−	0.660549i	$-0.770323\pi$
$167$		−	220.623i		−	1.32110i	0.750783	−	0.660549i	$-0.229677\pi$
$168$		0			0
$169$	54.3778			0.321762
$170$		0			0
$171$	−271.285			−1.58646
$172$		0			0
$173$	−189.101			−1.09307			−0.546534	−	0.837437i	$-0.684053\pi$
$173$	−189.101			−1.09307			−0.546534	+	0.837437i	$0.684053\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		−	79.4192i		−	0.448696i
$178$		0			0
$179$		−	189.267i		−	1.05736i	−0.848823	−	0.528678i	$-0.822688\pi$
$179$		−	189.267i		−	1.05736i	0.848823	−	0.528678i	$-0.177312\pi$
$180$		0			0
$181$			135.389i			0.748003i	0.927428	+	0.374001i	$0.122014\pi$
$181$			135.389i			0.748003i	−0.927428	+	0.374001i	$0.877986\pi$
$182$		0			0
$183$	−426.133			−2.32860
$184$		0			0
$185$		0			0
$186$		0			0
$187$	654.503			3.50002
$188$		0			0
$189$			16.3880i			0.0867088i
$190$		0			0
$191$	170.416			0.892233			0.446116	−	0.894975i	$-0.352807\pi$
$191$	170.416			0.892233			0.446116	+	0.894975i	$0.352807\pi$
$192$		0			0
$193$	−231.933			−1.20172			−0.600862	−	0.799353i	$-0.705176\pi$
$193$	−231.933			−1.20172			−0.600862	+	0.799353i	$0.705176\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	83.3134i		−	0.422911i	−0.977388	−	0.211455i	$-0.932180\pi$
$197$		−	83.3134i		−	0.422911i	0.977388	−	0.211455i	$-0.0678203\pi$
$198$		0			0
$199$			213.014i			1.07042i	0.844718	+	0.535212i	$0.179768\pi$
$199$			213.014i			1.07042i	−0.844718	+	0.535212i	$0.820232\pi$
$200$		0			0
$201$			387.243i			1.92658i
$202$		0			0
$203$		−	39.9618i		−	0.196856i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		−	99.1154i		−	0.478818i
$208$		0			0
$209$		−	583.777i		−	2.79319i
$210$		0			0
$211$	−190.163			−0.901247			−0.450623	−	0.892714i	$-0.648798\pi$
$211$	−190.163			−0.901247			−0.450623	+	0.892714i	$0.648798\pi$
$212$		0			0
$213$		−	126.195i		−	0.592465i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−87.6008	−	44.7429i	−0.403691	−	0.206188i
$218$		0			0
$219$	119.275			0.544634
$220$		0			0
$221$	−319.895			−1.44749
$222$		0			0
$223$		−	41.5786i		−	0.186451i	−0.995645	−	0.0932256i	$-0.970282\pi$
$223$		−	41.5786i		−	0.186451i	0.995645	−	0.0932256i	$-0.0297178\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−33.4102			−0.147182			−0.0735908	−	0.997289i	$-0.523446\pi$
$227$	−33.4102			−0.147182			−0.0735908	+	0.997289i	$0.523446\pi$
$228$		0			0
$229$			371.589i			1.62266i	0.584589	+	0.811330i	$0.301256\pi$
$229$			371.589i			1.62266i	−0.584589	+	0.811330i	$0.698744\pi$
$230$		0			0
$231$	−304.395			−1.31773
$232$		0			0
$233$	−459.007			−1.96999			−0.984993	−	0.172595i	$-0.944785\pi$
$233$	−459.007			−1.96999			−0.984993	+	0.172595i	$0.944785\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	345.359			1.45721
$238$		0			0
$239$			202.508i			0.847315i	0.905822	+	0.423658i	$0.139254\pi$
$239$			202.508i			0.847315i	−0.905822	+	0.423658i	$0.860746\pi$
$240$		0			0
$241$			376.370i			1.56170i	0.624718	+	0.780851i	$0.285214\pi$
$241$			376.370i			1.56170i	−0.624718	+	0.780851i	$0.714786\pi$
$242$		0			0
$243$			348.642i			1.43474i
$244$		0			0
$245$		0			0
$246$		0			0
$247$			285.327i			1.15517i
$248$		0			0
$249$	−164.023			−0.658727
$250$		0			0
$251$		−	270.334i		−	1.07703i	−0.842616	−	0.538515i	$-0.818986\pi$
$251$		−	270.334i		−	1.07703i	0.842616	−	0.538515i	$-0.181014\pi$
$252$		0			0
$253$	213.286			0.843027
$254$		0			0
$255$		0			0
$256$		0			0
$257$	289.963			1.12826			0.564131	−	0.825686i	$-0.309211\pi$
$257$	289.963			1.12826			0.564131	+	0.825686i	$0.309211\pi$
$258$		0			0
$259$		−	153.817i		−	0.593889i
$260$		0			0
$261$			128.198i			0.491180i
$262$		0			0
$263$		−	264.996i		−	1.00759i	−0.863824	−	0.503794i	$-0.831937\pi$
$263$		−	264.996i		−	1.00759i	0.863824	−	0.503794i	$-0.168063\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−541.996			−2.02995
$268$		0			0
$269$		−	258.522i		−	0.961047i	−0.876982	−	0.480523i	$-0.840447\pi$
$269$		−	258.522i		−	0.961047i	0.876982	−	0.480523i	$-0.159553\pi$
$270$		0			0
$271$			357.865i			1.32053i	0.751031	+	0.660267i	$0.229557\pi$
$271$			357.865i			1.32053i	−0.751031	+	0.660267i	$0.770443\pi$
$272$		0			0
$273$	148.776			0.544968
$274$		0			0
$275$		0			0
$276$		0			0
$277$		−	95.7194i		−	0.345558i	−0.984961	−	0.172779i	$-0.944725\pi$
$277$		−	95.7194i		−	0.345558i	0.984961	−	0.172779i	$-0.0552746\pi$
$278$		0			0
$279$	281.024	+	143.536i	1.00726	+	0.514464i
$280$		0			0
$281$	447.854			1.59379			0.796894	−	0.604119i	$-0.206475\pi$
$281$	447.854			1.59379			0.796894	+	0.604119i	$0.206475\pi$
$282$		0			0
$283$	−172.581			−0.609825			−0.304913	−	0.952380i	$-0.598627\pi$
$283$	−172.581			−0.609825			−0.304913	+	0.952380i	$0.598627\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−218.404			−0.760988
$288$		0			0
$289$	−603.781			−2.08921
$290$		0			0
$291$			447.214i			1.53682i
$292$		0			0
$293$	−338.816			−1.15637			−0.578185	−	0.815906i	$-0.696239\pi$
$293$	−338.816			−1.15637			−0.578185	+	0.815906i	$0.696239\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	113.131			0.380912
$298$		0			0
$299$	−104.246			−0.348647
$300$		0			0
$301$		−	114.436i		−	0.380188i
$302$		0			0
$303$		−	53.1866i		−	0.175533i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	193.683			0.630890			0.315445	−	0.948944i	$-0.397846\pi$
$307$	193.683			0.630890			0.315445	+	0.948944i	$0.397846\pi$
$308$		0			0
$309$			603.657i			1.95358i
$310$		0			0
$311$	76.4200			0.245724			0.122862	−	0.992424i	$-0.460793\pi$
$311$	76.4200			0.245724			0.122862	+	0.992424i	$0.460793\pi$
$312$		0			0
$313$			70.1981i			0.224275i	0.993693	+	0.112138i	$0.0357697\pi$
$313$			70.1981i			0.224275i	−0.993693	+	0.112138i	$0.964230\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	414.973			1.30906			0.654532	−	0.756034i	$-0.272866\pi$
$317$	414.973			1.30906			0.654532	+	0.756034i	$0.272866\pi$
$318$		0			0
$319$	−275.869			−0.864792
$320$		0			0
$321$			203.143i			0.632844i
$322$		0			0
$323$			796.307i			2.46535i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		−	393.474i		−	1.20328i
$328$		0			0
$329$	18.3813			0.0558703
$330$		0			0
$331$		−	191.784i		−	0.579409i	−0.957116	−	0.289705i	$-0.906443\pi$
$331$		−	191.784i		−	0.579409i	0.957116	−	0.289705i	$-0.0935570\pi$
$332$		0			0
$333$			493.447i			1.48182i
$334$		0			0
$335$		0			0
$336$		0			0
$337$			612.613i			1.81784i	0.416968	+	0.908921i	$0.363093\pi$
$337$			612.613i			1.81784i	−0.416968	+	0.908921i	$0.636907\pi$
$338$		0			0
$339$		−	357.477i		−	1.05450i
$340$		0			0
$341$	−308.873	+	604.735i	−0.905787	+	1.77342i
$342$		0			0
$343$	−279.015			−0.813454
$344$		0			0
$345$		0			0
$346$		0			0
$347$			145.960i			0.420634i	0.977633	+	0.210317i	$0.0674497\pi$
$347$			145.960i			0.420634i	−0.977633	+	0.210317i	$0.932550\pi$
$348$		0			0
$349$	480.073			1.37557			0.687784	−	0.725915i	$-0.258584\pi$
$349$	480.073			1.37557			0.687784	+	0.725915i	$0.258584\pi$
$350$		0			0
$351$	−55.2939			−0.157532
$352$		0			0
$353$			150.889i			0.427448i	0.976894	+	0.213724i	$0.0685594\pi$
$353$			150.889i			0.427448i	−0.976894	+	0.213724i	$0.931441\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	415.214			1.16306
$358$		0			0
$359$	−271.826			−0.757174			−0.378587	−	0.925566i	$-0.623590\pi$
$359$	−271.826			−0.757174			−0.378587	+	0.925566i	$0.623590\pi$
$360$		0			0
$361$	349.257			0.967472
$362$		0			0
$363$			1571.42i			4.32899i
$364$		0			0
$365$		0			0
$366$		0			0
$367$			390.547i			1.06416i	0.846694	+	0.532080i	$0.178589\pi$
$367$			390.547i			1.06416i	−0.846694	+	0.532080i	$0.821411\pi$
$368$		0			0
$369$	700.641			1.89876
$370$		0			0
$371$			36.5027i			0.0983901i
$372$		0			0
$373$	246.577			0.661064			0.330532	−	0.943795i	$-0.392772\pi$
$373$	246.577			0.661064			0.330532	+	0.943795i	$0.392772\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	134.833			0.357648
$378$		0			0
$379$	−533.513			−1.40769			−0.703843	−	0.710356i	$-0.748534\pi$
$379$	−533.513			−1.40769			−0.703843	+	0.710356i	$0.748534\pi$
$380$		0			0
$381$	154.561			0.405672
$382$		0			0
$383$			52.7243i			0.137661i	0.997628	+	0.0688307i	$0.0219268\pi$
$383$			52.7243i			0.137661i	−0.997628	+	0.0688307i	$0.978073\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$			367.113i			0.948613i
$388$		0			0
$389$		−	143.777i		−	0.369607i	−0.982775	−	0.184803i	$-0.940835\pi$
$389$		−	143.777i		−	0.369607i	0.982775	−	0.184803i	$-0.0591649\pi$
$390$		0			0
$391$	−290.935			−0.744079
$392$		0			0
$393$		−	626.554i		−	1.59428i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−332.656			−0.837924			−0.418962	−	0.908004i	$-0.637606\pi$
$397$	−332.656			−0.837924			−0.418962	+	0.908004i	$0.637606\pi$
$398$		0			0
$399$		−	370.345i		−	0.928184i
$400$		0			0
$401$			33.1046i			0.0825551i	0.999148	+	0.0412775i	$0.0131428\pi$
$401$			33.1046i			0.0825551i	−0.999148	+	0.0412775i	$0.986857\pi$
$402$		0			0
$403$	150.965	−	295.570i	0.374603	−	0.733424i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−1061.85			−2.60896
$408$		0			0
$409$			168.197i			0.411240i	0.978632	+	0.205620i	$0.0659210\pi$
$409$			168.197i			0.411240i	−0.978632	+	0.205620i	$0.934079\pi$
$410$		0			0
$411$	337.134			0.820277
$412$		0			0
$413$	57.5429			0.139329
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−182.566			−0.437808
$418$		0			0
$419$	269.033			0.642084			0.321042	−	0.947065i	$-0.395967\pi$
$419$	269.033			0.642084			0.321042	+	0.947065i	$0.395967\pi$
$420$		0			0
$421$	−514.900			−1.22304			−0.611521	−	0.791228i	$-0.709442\pi$
$421$	−514.900			−1.22304			−0.611521	+	0.791228i	$0.709442\pi$
$422$		0			0
$423$	−58.9675			−0.139403
$424$		0			0
$425$		0			0
$426$		0			0
$427$		−	308.753i		−	0.723075i
$428$		0			0
$429$		−	1027.05i		−	2.39405i
$430$		0			0
$431$	179.927			0.417465			0.208733	−	0.977973i	$-0.433066\pi$
$431$	179.927			0.417465			0.208733	+	0.977973i	$0.433066\pi$
$432$		0			0
$433$		−	503.365i		−	1.16251i	−0.813723	−	0.581253i	$-0.802563\pi$
$433$		−	503.365i		−	1.16251i	0.813723	−	0.581253i	$-0.197437\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			259.496i			0.593813i
$438$		0			0
$439$	−233.370			−0.531595			−0.265798	−	0.964029i	$-0.585635\pi$
$439$	−233.370			−0.531595			−0.265798	+	0.964029i	$0.585635\pi$
$440$		0			0
$441$	396.296			0.898629
$442$		0			0
$443$	−583.212			−1.31651			−0.658253	−	0.752797i	$-0.728704\pi$
$443$	−583.212			−1.31651			−0.658253	+	0.752797i	$0.728704\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$			964.448i			2.15760i
$448$		0			0
$449$			152.533i			0.339717i	0.985468	+	0.169859i	$0.0543311\pi$
$449$			152.533i			0.339717i	−0.985468	+	0.169859i	$0.945669\pi$
$450$		0			0
$451$			1507.71i			3.34303i
$452$		0			0
$453$	−1281.01			−2.82783
$454$		0			0
$455$		0			0
$456$		0			0
$457$			417.048i			0.912578i	0.889832	+	0.456289i	$0.150822\pi$
$457$			417.048i			0.912578i	−0.889832	+	0.456289i	$0.849178\pi$
$458$		0			0
$459$	−154.317			−0.336204
$460$		0			0
$461$			85.1724i			0.184756i	0.995724	+	0.0923779i	$0.0294468\pi$
$461$			85.1724i			0.184756i	−0.995724	+	0.0923779i	$0.970553\pi$
$462$		0			0
$463$		−	661.319i		−	1.42834i	−0.699975	−	0.714168i	$-0.746805\pi$
$463$		−	661.319i		−	1.42834i	0.699975	−	0.714168i	$-0.253195\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	866.167			1.85475			0.927374	−	0.374136i	$-0.122061\pi$
$467$	866.167			1.85475			0.927374	+	0.374136i	$0.122061\pi$
$468$		0			0
$469$	−280.575			−0.598241
$470$		0			0
$471$		−	1009.08i		−	2.14242i
$472$		0			0
$473$	−789.989			−1.67017
$474$		0			0
$475$		0			0
$476$		0			0
$477$		−	117.101i		−	0.245495i
$478$		0			0
$479$	396.326			0.827403			0.413702	−	0.910413i	$-0.364236\pi$
$479$	396.326			0.827403			0.413702	+	0.910413i	$0.364236\pi$
$480$		0			0
$481$	518.987			1.07898
$482$		0			0
$483$	135.308			0.280140
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	400.739i		−	0.822873i	−0.911438	−	0.411436i	$-0.865027\pi$
$487$		−	400.739i		−	0.822873i	0.911438	−	0.411436i	$-0.134973\pi$
$488$		0			0
$489$		−	251.331i		−	0.513970i
$490$		0			0
$491$			22.2174i			0.0452493i	0.999744	+	0.0226247i	$0.00720227\pi$
$491$			22.2174i			0.0452493i	−0.999744	+	0.0226247i	$0.992798\pi$
$492$		0			0
$493$	376.301			0.763288
$494$		0			0
$495$		0			0
$496$		0			0
$497$	91.4342			0.183972
$498$		0			0
$499$		−	224.526i		−	0.449953i	−0.974364	−	0.224976i	$-0.927770\pi$
$499$		−	224.526i		−	0.449953i	0.974364	−	0.224976i	$-0.0722305\pi$
$500$		0			0
$501$	−966.202			−1.92855
$502$		0			0
$503$	−120.740			−0.240039			−0.120020	−	0.992772i	$-0.538296\pi$
$503$	−120.740			−0.240039			−0.120020	+	0.992772i	$0.538296\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		−	238.143i		−	0.469710i
$508$		0			0
$509$		−	763.572i		−	1.50014i	−0.661357	−	0.750071i	$-0.730019\pi$
$509$		−	763.572i		−	1.50014i	0.661357	−	0.750071i	$-0.269981\pi$
$510$		0			0
$511$			86.4202i			0.169120i
$512$		0			0
$513$			137.642i			0.268308i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	126.892i		−	0.245439i
$518$		0			0
$519$			828.151i			1.59567i
$520$		0			0
$521$	444.974			0.854077			0.427038	−	0.904234i	$-0.359557\pi$
$521$	444.974			0.854077			0.427038	+	0.904234i	$0.359557\pi$
$522$		0			0
$523$		−	499.676i		−	0.955404i	−0.878522	−	0.477702i	$-0.841470\pi$
$523$		−	499.676i		−	0.955404i	0.878522	−	0.477702i	$-0.158530\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	421.322	−	824.894i	0.799472	−	1.56526i
$528$		0			0
$529$	434.192			0.820778
$530$		0			0
$531$	−184.598			−0.347642
$532$		0			0
$533$		−	736.906i		−	1.38256i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−828.877			−1.54353
$538$		0			0
$539$			852.786i			1.58216i
$540$		0			0
$541$	−347.974			−0.643206			−0.321603	−	0.946875i	$-0.604222\pi$
$541$	−347.974			−0.643206			−0.321603	+	0.946875i	$0.604222\pi$
$542$		0			0
$543$	592.923			1.09194
$544$		0			0
$545$		0			0
$546$		0			0
$547$	266.288			0.486815			0.243407	−	0.969924i	$-0.421735\pi$
$547$	266.288			0.486815			0.243407	+	0.969924i	$0.421735\pi$
$548$		0			0
$549$			990.483i			1.80416i
$550$		0			0
$551$		−	335.638i		−	0.609143i
$552$		0			0
$553$			250.229i			0.452493i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	818.119i		−	1.46880i	−0.678719	−	0.734398i	$-0.737465\pi$
$557$		−	818.119i		−	1.46880i	0.678719	−	0.734398i	$-0.262535\pi$
$558$		0			0
$559$	386.115			0.690724
$560$		0			0
$561$		−	2866.34i		−	5.10935i
$562$		0			0
$563$	−577.339			−1.02547			−0.512734	−	0.858548i	$-0.671367\pi$
$563$	−577.339			−1.02547			−0.512734	+	0.858548i	$0.671367\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−218.929			−0.386118
$568$		0			0
$569$		−	552.701i		−	0.971355i	−0.874138	−	0.485677i	$-0.838573\pi$
$569$		−	552.701i		−	0.971355i	0.874138	−	0.485677i	$-0.161427\pi$
$570$		0			0
$571$			221.341i			0.387637i	0.981037	+	0.193818i	$0.0620872\pi$
$571$			221.341i			0.387637i	−0.981037	+	0.193818i	$0.937913\pi$
$572$		0			0
$573$		−	746.325i		−	1.30249i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	320.225			0.554982			0.277491	−	0.960728i	$-0.410497\pi$
$577$	320.225			0.554982			0.277491	+	0.960728i	$0.410497\pi$
$578$		0			0
$579$			1015.73i			1.75428i
$580$		0			0
$581$		−	118.842i		−	0.204548i
$582$		0			0
$583$	251.989			0.432228
$584$		0			0
$585$		0			0
$586$		0			0
$587$			506.883i			0.863515i	0.901990	+	0.431757i	$0.142106\pi$
$587$			506.883i			0.863515i	−0.901990	+	0.431757i	$0.857894\pi$
$588$		0			0
$589$	−735.756	−	375.793i	−1.24916	−	0.638020i
$590$		0			0
$591$	−364.864			−0.617368
$592$		0			0
$593$	986.747			1.66399			0.831996	−	0.554782i	$-0.187198\pi$
$593$	986.747			1.66399			0.831996	+	0.554782i	$0.187198\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	932.880			1.56261
$598$		0			0
$599$	746.964			1.24702			0.623509	−	0.781816i	$-0.285706\pi$
$599$	746.964			1.24702			0.623509	+	0.781816i	$0.285706\pi$
$600$		0			0
$601$			368.950i			0.613893i	0.951727	+	0.306946i	$0.0993073\pi$
$601$			368.950i			0.613893i	−0.951727	+	0.306946i	$0.900693\pi$
$602$		0			0
$603$	900.088			1.49268
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−856.570			−1.41115			−0.705577	−	0.708634i	$-0.749312\pi$
$607$	−856.570			−1.41115			−0.705577	+	0.708634i	$0.749312\pi$
$608$		0			0
$609$	−175.010			−0.287372
$610$		0			0
$611$			62.0196i			0.101505i
$612$		0			0
$613$			357.438i			0.583097i	0.956556	+	0.291548i	$0.0941704\pi$
$613$			357.438i			0.583097i	−0.956556	+	0.291548i	$0.905830\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	227.390			0.368541			0.184270	−	0.982876i	$-0.441008\pi$
$617$	227.390			0.368541			0.184270	+	0.982876i	$0.441008\pi$
$618$		0			0
$619$			13.8294i			0.0223416i	0.999938	+	0.0111708i	$0.00355585\pi$
$619$			13.8294i			0.0223416i	−0.999938	+	0.0111708i	$0.996444\pi$
$620$		0			0
$621$	−50.2881			−0.0809793
$622$		0			0
$623$		−	392.701i		−	0.630339i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−2556.60			−4.07752
$628$		0			0
$629$	1448.42			2.30274
$630$		0			0
$631$		−	229.859i		−	0.364277i	−0.983273	−	0.182139i	$-0.941698\pi$
$631$		−	229.859i		−	0.364277i	0.983273	−	0.182139i	$-0.0583019\pi$
$632$		0			0
$633$			832.804i			1.31565i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		−	416.808i		−	0.654329i
$638$		0			0
$639$	−293.322			−0.459032
$640$		0			0
$641$			186.314i			0.290662i	0.989383	+	0.145331i	$0.0464246\pi$
$641$			186.314i			0.290662i	−0.989383	+	0.145331i	$0.953575\pi$
$642$		0			0
$643$			187.737i			0.291971i	0.989287	+	0.145985i	$0.0466352\pi$
$643$			187.737i			0.291971i	−0.989287	+	0.145985i	$0.953365\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	185.216i		−	0.286268i	−0.989703	−	0.143134i	$-0.954282\pi$
$647$		−	185.216i		−	0.286268i	0.989703	−	0.143134i	$-0.0457180\pi$
$648$		0			0
$649$		−	397.236i		−	0.612073i
$650$		0			0
$651$	−195.948	+	383.641i	−0.300995	+	0.589310i
$652$		0			0
$653$	−527.549			−0.807885			−0.403942	−	0.914784i	$-0.632360\pi$
$653$	−527.549			−0.807885			−0.403942	+	0.914784i	$0.632360\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		−	277.237i		−	0.421974i
$658$		0			0
$659$	−10.3222			−0.0156634			−0.00783171	−	0.999969i	$-0.502493\pi$
$659$	−10.3222			−0.0156634			−0.00783171	+	0.999969i	$0.502493\pi$
$660$		0			0
$661$	480.618			0.727108			0.363554	−	0.931573i	$-0.381563\pi$
$661$	480.618			0.727108			0.363554	+	0.931573i	$0.381563\pi$
$662$		0			0
$663$			1400.95i			2.11305i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	122.627			0.183849
$668$		0			0
$669$	−182.090			−0.272183
$670$		0			0
$671$	−2131.41			−3.17647
$672$		0			0
$673$			1149.20i			1.70758i	0.520620	+	0.853788i	$0.325701\pi$
$673$			1149.20i			1.70758i	−0.520620	+	0.853788i	$0.674299\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	905.301i		−	1.33722i	−0.743611	−	0.668612i	$-0.766889\pi$
$677$		−	905.301i		−	1.33722i	0.743611	−	0.668612i	$-0.233111\pi$
$678$		0			0
$679$	−324.027			−0.477212
$680$		0			0
$681$			146.317i			0.214856i
$682$		0			0
$683$	360.313			0.527544			0.263772	−	0.964585i	$-0.415033\pi$
$683$	360.313			0.527544			0.263772	+	0.964585i	$0.415033\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	1627.34			2.36877
$688$		0			0
$689$	−123.162			−0.178755
$690$		0			0
$691$	−258.880			−0.374645			−0.187323	−	0.982298i	$-0.559981\pi$
$691$	−258.880			−0.374645			−0.187323	+	0.982298i	$0.559981\pi$
$692$		0			0
$693$			707.522i			1.02095i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	2056.60i		−	2.95065i
$698$		0			0
$699$			2010.18i			2.87580i
$700$		0			0
$701$	628.260			0.896234			0.448117	−	0.893975i	$-0.352095\pi$
$701$	628.260			0.896234			0.448117	+	0.893975i	$0.352095\pi$
$702$		0			0
$703$		−	1291.90i		−	1.83770i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	38.5362			0.0545066
$708$		0			0
$709$		−	1407.10i		−	1.98463i	−0.123751	−	0.992313i	$-0.539493\pi$
$709$		−	1407.10i		−	1.98463i	0.123751	−	0.992313i	$-0.460507\pi$
$710$		0			0
$711$		−	802.736i		−	1.12902i
$712$		0			0
$713$	137.298	−	268.812i	0.192564	−	0.377016i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	886.869			1.23692
$718$		0			0
$719$			743.741i			1.03441i	0.855862	+	0.517205i	$0.173028\pi$
$719$			743.741i			1.03441i	−0.855862	+	0.517205i	$0.826972\pi$
$720$		0			0
$721$	−437.377			−0.606626
$722$		0			0
$723$	1648.28			2.27978
$724$		0			0
$725$		0			0
$726$		0			0
$727$	1319.89			1.81553			0.907766	−	0.419477i	$-0.137787\pi$
$727$	1319.89			1.81553			0.907766	+	0.419477i	$0.137787\pi$
$728$		0			0
$729$	905.890			1.24265
$730$		0			0
$731$	1077.59			1.47413
$732$		0			0
$733$	266.946			0.364183			0.182091	−	0.983282i	$-0.441713\pi$
$733$	266.946			0.364183			0.182091	+	0.983282i	$0.441713\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			1936.89i			2.62808i
$738$		0			0
$739$		−	714.076i		−	0.966273i	−0.875545	−	0.483136i	$-0.839498\pi$
$739$		−	714.076i		−	0.966273i	0.875545	−	0.483136i	$-0.160502\pi$
$740$		0			0
$741$	1249.56			1.68632
$742$		0			0
$743$			74.8223i			0.100703i	0.998732	+	0.0503515i	$0.0160342\pi$
$743$			74.8223i			0.100703i	−0.998732	+	0.0503515i	$0.983966\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$			381.247i			0.510371i
$748$		0			0
$749$	−147.186			−0.196511
$750$		0			0
$751$	−131.577			−0.175202			−0.0876009	−	0.996156i	$-0.527920\pi$
$751$	−131.577			−0.175202			−0.0876009	+	0.996156i	$0.527920\pi$
$752$		0			0
$753$	−1183.91			−1.57225
$754$		0			0
$755$		0			0
$756$		0			0
$757$			368.843i			0.487244i	0.969870	+	0.243622i	$0.0783356\pi$
$757$			368.843i			0.487244i	−0.969870	+	0.243622i	$0.921664\pi$
$758$		0			0
$759$		−	934.068i		−	1.23066i
$760$		0			0
$761$		−	514.706i		−	0.676355i	−0.941082	−	0.338178i	$-0.890190\pi$
$761$		−	514.706i		−	0.676355i	0.941082	−	0.338178i	$-0.109810\pi$
$762$		0			0
$763$	285.090			0.373644
$764$		0			0
$765$		0			0
$766$		0			0
$767$			194.153i			0.253133i
$768$		0			0
$769$	−782.294			−1.01729			−0.508644	−	0.860977i	$-0.669853\pi$
$769$	−782.294			−1.01729			−0.508644	+	0.860977i	$0.669853\pi$
$770$		0			0
$771$		−	1269.87i		−	1.64704i
$772$		0			0
$773$		−	1068.37i		−	1.38211i	−0.722800	−	0.691057i	$-0.757145\pi$
$773$		−	1068.37i		−	1.38211i	0.722800	−	0.691057i	$-0.242855\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−673.630			−0.866962
$778$		0			0
$779$	−1834.36			−2.35477
$780$		0			0
$781$		−	631.197i		−	0.808191i
$782$		0			0
$783$	65.0437			0.0830699
$784$		0			0
$785$		0			0
$786$		0			0
$787$		−	742.554i		−	0.943525i	−0.881726	−	0.471763i	$-0.843618\pi$
$787$		−	742.554i		−	0.943525i	0.881726	−	0.471763i	$-0.156382\pi$
$788$		0			0
$789$	−1160.53			−1.47088
$790$		0			0
$791$	259.008			0.327444
$792$		0			0
$793$	1041.75			1.31368
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	1232.92i		−	1.54695i	−0.633830	−	0.773473i	$-0.718518\pi$
$797$		−	1232.92i		−	1.54695i	0.633830	−	0.773473i	$-0.281482\pi$
$798$		0			0
$799$			173.088i			0.216631i
$800$		0			0
$801$			1259.79i			1.57277i
$802$		0			0
$803$	596.584			0.742944
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−1132.17			−1.40294
$808$		0			0
$809$			707.180i			0.874141i	0.899427	+	0.437071i	$0.143984\pi$
$809$			707.180i			0.874141i	−0.899427	+	0.437071i	$0.856016\pi$
$810$		0			0
$811$	198.481			0.244736			0.122368	−	0.992485i	$-0.460951\pi$
$811$	198.481			0.244736			0.122368	+	0.992485i	$0.460951\pi$
$812$		0			0
$813$	1567.24			1.92772
$814$		0			0
$815$		0			0
$816$		0			0
$817$		−	961.147i		−	1.17643i
$818$		0			0
$819$		−	345.808i		−	0.422232i
$820$		0			0
$821$		−	372.466i		−	0.453673i	−0.973933	−	0.226837i	$-0.927162\pi$
$821$		−	372.466i		−	0.453673i	0.973933	−	0.226837i	$-0.0728384\pi$
$822$		0			0
$823$		−	795.950i		−	0.967132i	−0.875308	−	0.483566i	$-0.839341\pi$
$823$		−	795.950i		−	0.967132i	0.875308	−	0.483566i	$-0.160659\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	359.898i		−	0.435185i	−0.976040	−	0.217592i	$-0.930180\pi$
$827$		−	359.898i		−	0.435185i	0.976040	−	0.217592i	$-0.0698203\pi$
$828$		0			0
$829$		−	249.911i		−	0.301461i	−0.988575	−	0.150730i	$-0.951837\pi$
$829$		−	249.911i		−	0.301461i	0.988575	−	0.150730i	$-0.0481626\pi$
$830$		0			0
$831$	−419.195			−0.504447
$832$		0			0
$833$		−	1163.25i		−	1.39646i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	72.8256	−	142.583i	0.0870078	−	0.170350i
$838$		0			0
$839$	1.73361			0.00206628			0.00103314	−	0.999999i	$-0.499671\pi$
$839$	1.73361			0.00206628			0.00103314	+	0.999999i	$0.499671\pi$
$840$		0			0
$841$	682.392			0.811405
$842$		0			0
$843$		−	1961.34i		−	2.32662i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−1138.57			−1.34424
$848$		0			0
$849$			755.802i			0.890227i
$850$		0			0
$851$	472.004			0.554646
$852$		0			0
$853$	−898.359			−1.05318			−0.526588	−	0.850121i	$-0.676529\pi$
$853$	−898.359			−1.05318			−0.526588	+	0.850121i	$0.676529\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1426.13			−1.66410			−0.832048	−	0.554704i	$-0.812832\pi$
$857$	−1426.13			−1.66410			−0.832048	+	0.554704i	$0.812832\pi$
$858$		0			0
$859$			166.412i			0.193728i	0.995298	+	0.0968638i	$0.0308811\pi$
$859$			166.412i			0.193728i	−0.995298	+	0.0968638i	$0.969119\pi$
$860$		0			0
$861$			956.481i			1.11090i
$862$		0			0
$863$			679.129i			0.786940i	0.919337	+	0.393470i	$0.128726\pi$
$863$			679.129i			0.786940i	−0.919337	+	0.393470i	$0.871274\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$			2644.21i			3.04984i
$868$		0			0
$869$	1727.40			1.98781
$870$		0			0
$871$		−	946.676i		−	1.08688i
$872$		0			0
$873$	1039.48			1.19070
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−1330.13			−1.51668			−0.758342	−	0.651857i	$-0.773990\pi$
$877$	−1330.13			−1.51668			−0.758342	+	0.651857i	$0.773990\pi$
$878$		0			0
$879$			1483.82i			1.68808i
$880$		0			0
$881$		−	372.975i		−	0.423354i	−0.977340	−	0.211677i	$-0.932107\pi$
$881$		−	372.975i		−	0.423354i	0.977340	−	0.211677i	$-0.0678926\pi$
$882$		0			0
$883$		−	1286.13i		−	1.45655i	−0.685285	−	0.728275i	$-0.740322\pi$
$883$		−	1286.13i		−	1.45655i	0.685285	−	0.728275i	$-0.259678\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1114.04			−1.25596			−0.627979	−	0.778230i	$-0.716118\pi$
$887$	−1114.04			−1.25596			−0.627979	+	0.778230i	$0.716118\pi$
$888$		0			0
$889$			111.986i			0.125969i
$890$		0			0
$891$			1511.33i			1.69622i
$892$		0			0
$893$	154.384			0.172882
$894$		0			0
$895$		0			0
$896$		0			0
$897$			456.535i			0.508957i
$898$		0			0
$899$	−177.584	+	347.687i	−0.197535	+	0.386749i
$900$		0			0
$901$	−343.728			−0.381496
$902$		0			0
$903$	−501.165			−0.555000
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−964.204			−1.06307			−0.531535	−	0.847036i	$-0.678385\pi$
$907$	−964.204			−1.06307			−0.531535	+	0.847036i	$0.678385\pi$
$908$		0			0
$909$	−123.624			−0.136000
$910$		0			0
$911$			326.462i			0.358355i	0.983817	+	0.179178i	$0.0573437\pi$
$911$			326.462i			0.358355i	−0.983817	+	0.179178i	$0.942656\pi$
$912$		0			0
$913$	−820.404			−0.898580
$914$		0			0
$915$		0			0
$916$		0			0
$917$	453.967			0.495057
$918$		0			0
$919$	−1233.96			−1.34272			−0.671361	−	0.741130i	$-0.734290\pi$
$919$	−1233.96			−1.34272			−0.671361	+	0.741130i	$0.734290\pi$
$920$		0			0
$921$		−	848.220i		−	0.920978i
$922$		0			0
$923$			308.504i			0.334240i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	1403.11			1.51360
$928$		0			0
$929$		−	997.308i		−	1.07353i	−0.843732	−	0.536764i	$-0.819647\pi$
$929$		−	997.308i		−	1.07353i	0.843732	−	0.536764i	$-0.180353\pi$
$930$		0			0
$931$	−1037.55			−1.11445
$932$		0			0
$933$		−	334.675i		−	0.358709i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−788.023			−0.841006			−0.420503	−	0.907291i	$-0.638146\pi$
$937$	−788.023			−0.841006			−0.420503	+	0.907291i	$0.638146\pi$
$938$		0			0
$939$	307.427			0.327398
$940$		0			0
$941$		−	1138.15i		−	1.20951i	−0.796413	−	0.604754i	$-0.793272\pi$
$941$		−	1138.15i		−	1.20951i	0.796413	−	0.604754i	$-0.206728\pi$
$942$		0			0
$943$		−	670.194i		−	0.710704i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			339.643i			0.358651i	0.983790	+	0.179326i	$0.0573916\pi$
$947$			339.643i			0.358651i	−0.983790	+	0.179326i	$0.942608\pi$
$948$		0			0
$949$	−291.586			−0.307256
$950$		0			0
$951$		−	1817.34i		−	1.91098i
$952$		0			0
$953$			1516.62i			1.59141i	0.605682	+	0.795706i	$0.292900\pi$
$953$			1516.62i			1.59141i	−0.605682	+	0.795706i	$0.707100\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$			1208.14i			1.26243i
$958$		0			0
$959$			244.269i			0.254712i
$960$		0			0
$961$	563.339	+	778.569i	0.586201	+	0.810166i
$962$		0			0
$963$	472.176			0.490317
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	232.357i		−	0.240286i	−0.992757	−	0.120143i	$-0.961665\pi$
$967$		−	232.357i		−	0.240286i	0.992757	−	0.120143i	$-0.0383354\pi$
$968$		0			0
$969$	3487.36			3.59893
$970$		0			0
$971$	−98.8188			−0.101770			−0.0508850	−	0.998705i	$-0.516204\pi$
$971$	−98.8188			−0.101770			−0.0508850	+	0.998705i	$0.516204\pi$
$972$		0			0
$973$		−	132.277i		−	0.135948i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	866.051			0.886439			0.443220	−	0.896413i	$-0.353836\pi$
$977$	866.051			0.886439			0.443220	+	0.896413i	$0.353836\pi$
$978$		0			0
$979$	−2710.94			−2.76909
$980$		0			0
$981$	−914.572			−0.932285
$982$		0			0
$983$		−	1190.98i		−	1.21157i	−0.795627	−	0.605787i	$-0.792858\pi$
$983$		−	1190.98i		−	1.21157i	0.795627	−	0.605787i	$-0.207142\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		−	80.4995i		−	0.0815598i
$988$		0			0
$989$	351.160			0.355066
$990$		0			0
$991$			109.095i			0.110086i	0.998484	+	0.0550429i	$0.0175296\pi$
$991$			109.095i			0.110086i	−0.998484	+	0.0550429i	$0.982470\pi$
$992$		0			0
$993$	−839.904			−0.845825
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−102.381			−0.102689			−0.0513447	−	0.998681i	$-0.516351\pi$
$997$	−102.381			−0.102689			−0.0513447	+	0.998681i	$0.516351\pi$
$998$		0			0
$999$	250.360			0.250611

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3100.3.d.e.1301.3		20
5.2	odd	4		3100.3.f.c.1549.5		40
5.3	odd	4		3100.3.f.c.1549.36		40
5.4	even	2		620.3.d.a.61.18	yes	20
31.30	odd	2	inner	3100.3.d.e.1301.18		20
155.92	even	4		3100.3.f.c.1549.35		40
155.123	even	4		3100.3.f.c.1549.6		40
155.154	odd	2		620.3.d.a.61.3	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
620.3.d.a.61.3	✓	20	155.154	odd	2
620.3.d.a.61.18	yes	20	5.4	even	2
3100.3.d.e.1301.3		20	1.1	even	1	trivial
3100.3.d.e.1301.18		20	31.30	odd	2	inner
3100.3.f.c.1549.5		40	5.2	odd	4
3100.3.f.c.1549.6		40	155.123	even	4
3100.3.f.c.1549.35		40	155.92	even	4
3100.3.f.c.1549.36		40	5.3	odd	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:	\( N \)	\(=\)	\( 3100 = 2^{2} \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	3100.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-4.37942i q^{3} +3.17309 q^{7} -10.1793 q^{9} +O(q^{10})\)	\(q-4.37942i q^{3} +3.17309 q^{7} -10.1793 q^{9} -21.9048i q^{11} +10.7062i q^{13} +29.8794i q^{17} +26.6507 q^{19} -13.8963i q^{21} +9.73695i q^{23} +5.16467i q^{27} -12.5940i q^{29} +(-27.6074 - 14.1007i) q^{31} -95.9302 q^{33} -48.4755i q^{37} +46.8868 q^{39} -68.8299 q^{41} -36.0647i q^{43} +5.79288 q^{47} -38.9315 q^{49} +130.855 q^{51} +11.5038i q^{53} -116.714i q^{57} +18.1346 q^{59} -97.3036i q^{61} -32.2999 q^{63} -88.4233 q^{67} +42.6422 q^{69} +28.8155 q^{71} +27.2353i q^{73} -69.5059i q^{77} +78.8596i q^{79} -68.9955 q^{81} -37.4532i q^{83} -55.1543 q^{87} -123.760i q^{89} +33.9717i q^{91} +(-61.7530 + 120.904i) q^{93} -102.117 q^{97} +222.976i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 12 q^{7} - 48 q^{9}+O(q^{10}) \) 20 * q - 12 * q^7 - 48 * q^9	\( 20 q - 12 q^{7} - 48 q^{9} + 60 q^{19} + 8 q^{31} - 68 q^{33} + 28 q^{39} - 80 q^{41} + 48 q^{47} + 84 q^{49} + 344 q^{51} + 160 q^{59} + 232 q^{63} + 180 q^{67} - 140 q^{69} - 108 q^{71} + 336 q^{81} + 236 q^{87} + 332 q^{93} + 112 q^{97}+O(q^{100}) \) 20 * q - 12 * q^7 - 48 * q^9 + 60 * q^19 + 8 * q^31 - 68 * q^33 + 28 * q^39 - 80 * q^41 + 48 * q^47 + 84 * q^49 + 344 * q^51 + 160 * q^59 + 232 * q^63 + 180 * q^67 - 140 * q^69 - 108 * q^71 + 336 * q^81 + 236 * q^87 + 332 * q^93 + 112 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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