LMFDB - Embedded newform 1200.3.c.m.449.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,3,Mod(449,1200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1200, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("1200.449");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1200 = 2^{4} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1200.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$32.6976317232$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 138x^{12} + 3393x^{8} + 15208x^{4} + 1296 $ x^16 + 138x^12 + 3393x^8 + 15208*x^4 + 1296
Coefficient ring:	$\Z[a_1, \ldots, a_{37}]$
Coefficient ring index:	$ 2^{26}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.3
Root			$-1.09102 + 1.09102i$ of defining polynomial
Character	$\chi$	$=$	1200.449
Dual form			1200.3.c.m.449.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.79813 - 2.40140i) q^{3} +10.2132i q^{7} +(-2.53346 + 8.63606i) q^{9} +O(q^{10})$	$q+(-1.79813 - 2.40140i) q^{3} +10.2132i q^{7} +(-2.53346 + 8.63606i) q^{9} +8.19300i q^{11} +13.5822i q^{13} +15.4710 q^{17} -25.4934 q^{19} +(24.5261 - 18.3647i) q^{21} -17.9156 q^{23} +(25.2941 - 9.44491i) q^{27} -42.0022i q^{29} -38.4878 q^{31} +(19.6747 - 14.7321i) q^{33} +11.8387i q^{37} +(32.6163 - 24.4225i) q^{39} -46.3781i q^{41} -54.0181i q^{43} -43.0955 q^{47} -55.3102 q^{49} +(-27.8188 - 37.1521i) q^{51} +82.7421 q^{53} +(45.8404 + 61.2199i) q^{57} -45.8928i q^{59} -93.6873 q^{61} +(-88.2022 - 25.8748i) q^{63} -34.4995i q^{67} +(32.2147 + 43.0227i) q^{69} +68.0061i q^{71} +44.7191i q^{73} -83.6770 q^{77} -11.7499 q^{79} +(-68.1632 - 43.7582i) q^{81} +144.436 q^{83} +(-100.864 + 75.5255i) q^{87} -63.7094i q^{89} -138.718 q^{91} +(69.2061 + 92.4247i) q^{93} +63.9013i q^{97} +(-70.7552 - 20.7566i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 40 q^{9}+O(q^{10}) $ 16 * q - 40 * q^9	$ 16 q - 40 q^{9} - 16 q^{19} + 56 q^{21} - 240 q^{31} - 144 q^{39} - 128 q^{49} - 128 q^{51} + 16 q^{61} - 200 q^{69} + 176 q^{79} + 448 q^{81} - 1120 q^{91} + 64 q^{99}+O(q^{100}) $ 16 * q - 40 * q^9 - 16 * q^19 + 56 * q^21 - 240 * q^31 - 144 * q^39 - 128 * q^49 - 128 * q^51 + 16 * q^61 - 200 * q^69 + 176 * q^79 + 448 * q^81 - 1120 * q^91 + 64 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$.

$n$	$401$	$577$	$751$	$901$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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
$3$	−1.79813	−	2.40140i	−0.599377	−	0.800467i
$3$	−1.79813	−	2.40140i	−0.599377	−	0.800467i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$			10.2132i			1.45903i	0.683963	+	0.729517i	$0.260255\pi$
$7$			10.2132i			1.45903i	−0.683963	+	0.729517i	$0.739745\pi$
$8$		0			0
$9$	−2.53346	+	8.63606i	−0.281496	+	0.959563i
$10$		0			0
$11$			8.19300i			0.744818i	0.928069	+	0.372409i	$0.121468\pi$
$11$			8.19300i			0.744818i	−0.928069	+	0.372409i	$0.878532\pi$
$12$		0			0
$13$			13.5822i			1.04478i	0.852705	+	0.522392i	$0.174960\pi$
$13$			13.5822i			1.04478i	−0.852705	+	0.522392i	$0.825040\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	15.4710			0.910058			0.455029	−	0.890477i	$-0.349629\pi$
$17$	15.4710			0.910058			0.455029	+	0.890477i	$0.349629\pi$
$18$		0			0
$19$	−25.4934			−1.34176			−0.670879	−	0.741567i	$-0.734083\pi$
$19$	−25.4934			−1.34176			−0.670879	+	0.741567i	$0.734083\pi$
$20$		0			0
$21$	24.5261	−	18.3647i	1.16791	−	0.874511i
$22$		0			0
$23$	−17.9156			−0.778941			−0.389471	−	0.921039i	$-0.627342\pi$
$23$	−17.9156			−0.778941			−0.389471	+	0.921039i	$0.627342\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	25.2941	−	9.44491i	0.936820	−	0.349811i
$28$		0			0
$29$		−	42.0022i		−	1.44835i	−0.689615	−	0.724177i	$-0.742220\pi$
$29$		−	42.0022i		−	1.44835i	0.689615	−	0.724177i	$-0.257780\pi$
$30$		0			0
$31$	−38.4878			−1.24154			−0.620771	−	0.783992i	$-0.713180\pi$
$31$	−38.4878			−1.24154			−0.620771	+	0.783992i	$0.713180\pi$
$32$		0			0
$33$	19.6747	−	14.7321i	0.596202	−	0.446426i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			11.8387i			0.319965i	0.987120	+	0.159982i	$0.0511437\pi$
$37$			11.8387i			0.319965i	−0.987120	+	0.159982i	$0.948856\pi$
$38$		0			0
$39$	32.6163	−	24.4225i	0.836315	−	0.626219i
$40$		0			0
$41$		−	46.3781i		−	1.13117i	−0.824689	−	0.565587i	$-0.808650\pi$
$41$		−	46.3781i		−	1.13117i	0.824689	−	0.565587i	$-0.191350\pi$
$42$		0			0
$43$		−	54.0181i		−	1.25623i	−0.778119	−	0.628117i	$-0.783826\pi$
$43$		−	54.0181i		−	1.25623i	0.778119	−	0.628117i	$-0.216174\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−43.0955			−0.916925			−0.458462	−	0.888714i	$-0.651600\pi$
$47$	−43.0955			−0.916925			−0.458462	+	0.888714i	$0.651600\pi$
$48$		0			0
$49$	−55.3102			−1.12878
$50$		0			0
$51$	−27.8188	−	37.1521i	−0.545468	−	0.728472i
$52$		0			0
$53$	82.7421			1.56117			0.780586	−	0.625049i	$-0.214921\pi$
$53$	82.7421			1.56117			0.780586	+	0.625049i	$0.214921\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	45.8404	+	61.2199i	0.804218	+	1.07403i
$58$		0			0
$59$		−	45.8928i		−	0.777844i	−0.921271	−	0.388922i	$-0.872848\pi$
$59$		−	45.8928i		−	0.777844i	0.921271	−	0.388922i	$-0.127152\pi$
$60$		0			0
$61$	−93.6873			−1.53586			−0.767929	−	0.640535i	$-0.778713\pi$
$61$	−93.6873			−1.53586			−0.767929	+	0.640535i	$0.778713\pi$
$62$		0			0
$63$	−88.2022	−	25.8748i	−1.40003	−	0.410712i
$64$		0			0
$65$		0			0
$66$		0			0
$67$		−	34.4995i		−	0.514917i	−0.966289	−	0.257459i	$-0.917115\pi$
$67$		−	34.4995i		−	0.514917i	0.966289	−	0.257459i	$-0.0828851\pi$
$68$		0			0
$69$	32.2147	+	43.0227i	0.466879	+	0.623517i
$70$		0			0
$71$			68.0061i			0.957832i	0.877861	+	0.478916i	$0.158970\pi$
$71$			68.0061i			0.957832i	−0.877861	+	0.478916i	$0.841030\pi$
$72$		0			0
$73$			44.7191i			0.612591i	0.951937	+	0.306295i	$0.0990895\pi$
$73$			44.7191i			0.612591i	−0.951937	+	0.306295i	$0.900911\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−83.6770			−1.08671
$78$		0			0
$79$	−11.7499			−0.148733			−0.0743665	−	0.997231i	$-0.523693\pi$
$79$	−11.7499			−0.148733			−0.0743665	+	0.997231i	$0.523693\pi$
$80$		0			0
$81$	−68.1632	−	43.7582i	−0.841521	−	0.540225i
$82$		0			0
$83$	144.436			1.74020			0.870099	−	0.492877i	$-0.164055\pi$
$83$	144.436			1.74020			0.870099	+	0.492877i	$0.164055\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−100.864	+	75.5255i	−1.15936	+	0.868109i
$88$		0			0
$89$		−	63.7094i		−	0.715836i	−0.933753	−	0.357918i	$-0.883487\pi$
$89$		−	63.7094i		−	0.715836i	0.933753	−	0.357918i	$-0.116513\pi$
$90$		0			0
$91$	−138.718			−1.52438
$92$		0			0
$93$	69.2061	+	92.4247i	0.744151	+	0.993814i
$94$		0			0
$95$		0			0
$96$		0			0
$97$			63.9013i			0.658776i	0.944195	+	0.329388i	$0.106842\pi$
$97$			63.9013i			0.658776i	−0.944195	+	0.329388i	$0.893158\pi$
$98$		0			0
$99$	−70.7552	−	20.7566i	−0.714699	−	0.209663i
$100$		0			0
$101$		−	50.8769i		−	0.503731i	−0.967762	−	0.251866i	$-0.918956\pi$
$101$		−	50.8769i		−	0.503731i	0.967762	−	0.251866i	$-0.0810441\pi$
$102$		0			0
$103$			45.9491i			0.446108i	0.974806	+	0.223054i	$0.0716026\pi$
$103$			45.9491i			0.446108i	−0.974806	+	0.223054i	$0.928397\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−110.249			−1.03036			−0.515181	−	0.857082i	$-0.672275\pi$
$107$	−110.249			−1.03036			−0.515181	+	0.857082i	$0.672275\pi$
$108$		0			0
$109$	49.2797			0.452107			0.226054	−	0.974115i	$-0.427418\pi$
$109$	49.2797			0.452107			0.226054	+	0.974115i	$0.427418\pi$
$110$		0			0
$111$	28.4295	−	21.2875i	0.256121	−	0.191779i
$112$		0			0
$113$	−157.450			−1.39337			−0.696683	−	0.717379i	$-0.745342\pi$
$113$	−157.450			−1.39337			−0.696683	+	0.717379i	$0.745342\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−117.297	−	34.4099i	−1.00254	−	0.294102i
$118$		0			0
$119$			158.009i			1.32781i
$120$		0			0
$121$	53.8748			0.445246
$122$		0			0
$123$	−111.373	+	83.3939i	−0.905468	+	0.677999i
$124$		0			0
$125$		0			0
$126$		0			0
$127$			135.065i			1.06351i	0.846900	+	0.531753i	$0.178466\pi$
$127$			135.065i			1.06351i	−0.846900	+	0.531753i	$0.821534\pi$
$128$		0			0
$129$	−129.719	+	97.1315i	−1.00557	+	0.752957i
$130$		0			0
$131$		−	150.653i		−	1.15002i	−0.818146	−	0.575010i	$-0.804998\pi$
$131$		−	150.653i		−	1.15002i	0.818146	−	0.575010i	$-0.195002\pi$
$132$		0			0
$133$		−	260.370i		−	1.95767i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−139.086			−1.01523			−0.507613	−	0.861585i	$-0.669472\pi$
$137$	−139.086			−1.01523			−0.507613	+	0.861585i	$0.669472\pi$
$138$		0			0
$139$	110.296			0.793496			0.396748	−	0.917928i	$-0.370139\pi$
$139$	110.296			0.793496			0.396748	+	0.917928i	$0.370139\pi$
$140$		0			0
$141$	77.4912	+	103.490i	0.549583	+	0.733968i
$142$		0			0
$143$	−111.279			−0.778174
$144$		0			0
$145$		0			0
$146$		0			0
$147$	99.4549	+	132.822i	0.676564	+	0.903551i
$148$		0			0
$149$		−	90.8076i		−	0.609447i	−0.952441	−	0.304723i	$-0.901436\pi$
$149$		−	90.8076i		−	0.609447i	0.952441	−	0.304723i	$-0.0985640\pi$
$150$		0			0
$151$	−111.286			−0.736992			−0.368496	−	0.929629i	$-0.620127\pi$
$151$	−111.286			−0.736992			−0.368496	+	0.929629i	$0.620127\pi$
$152$		0			0
$153$	−39.1951	+	133.608i	−0.256177	+	0.873258i
$154$		0			0
$155$		0			0
$156$		0			0
$157$			142.755i			0.909269i	0.890678	+	0.454635i	$0.150230\pi$
$157$			142.755i			0.909269i	−0.890678	+	0.454635i	$0.849770\pi$
$158$		0			0
$159$	−148.781	−	198.697i	−0.935730	−	1.24967i
$160$		0			0
$161$		−	182.977i		−	1.13650i
$162$		0			0
$163$		−	170.179i		−	1.04405i	−0.852932	−	0.522023i	$-0.825178\pi$
$163$		−	170.179i		−	1.04405i	0.852932	−	0.522023i	$-0.174822\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−6.38780			−0.0382503			−0.0191251	−	0.999817i	$-0.506088\pi$
$167$	−6.38780			−0.0382503			−0.0191251	+	0.999817i	$0.506088\pi$
$168$		0			0
$169$	−15.4759			−0.0915734
$170$		0			0
$171$	64.5865	−	220.163i	0.377699	−	1.28750i
$172$		0			0
$173$	−238.403			−1.37805			−0.689027	−	0.724736i	$-0.741962\pi$
$173$	−238.403			−1.37805			−0.689027	+	0.724736i	$0.741962\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−110.207	+	82.5212i	−0.622638	+	0.466221i
$178$		0			0
$179$			243.541i			1.36057i	0.732950	+	0.680283i	$0.238143\pi$
$179$			243.541i			1.36057i	−0.732950	+	0.680283i	$0.761857\pi$
$180$		0			0
$181$	−325.449			−1.79806			−0.899030	−	0.437887i	$-0.855727\pi$
$181$	−325.449			−1.79806			−0.899030	+	0.437887i	$0.855727\pi$
$182$		0			0
$183$	168.462	+	224.981i	0.920557	+	1.22940i
$184$		0			0
$185$		0			0
$186$		0			0
$187$			126.754i			0.677828i
$188$		0			0
$189$	96.4631	+	258.335i	0.510387	+	1.36685i
$190$		0			0
$191$		−	166.001i		−	0.869113i	−0.900645	−	0.434556i	$-0.856905\pi$
$191$		−	166.001i		−	0.869113i	0.900645	−	0.434556i	$-0.143095\pi$
$192$		0			0
$193$		−	239.408i		−	1.24045i	−0.784422	−	0.620227i	$-0.787041\pi$
$193$		−	239.408i		−	1.24045i	0.784422	−	0.620227i	$-0.212959\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	257.518			1.30720			0.653600	−	0.756840i	$-0.273258\pi$
$197$	257.518			1.30720			0.653600	+	0.756840i	$0.273258\pi$
$198$		0			0
$199$	−78.4735			−0.394339			−0.197170	−	0.980369i	$-0.563175\pi$
$199$	−78.4735			−0.394339			−0.197170	+	0.980369i	$0.563175\pi$
$200$		0			0
$201$	−82.8471	+	62.0345i	−0.412174	+	0.308629i
$202$		0			0
$203$	428.979			2.11320
$204$		0			0
$205$		0			0
$206$		0			0
$207$	45.3886	−	154.721i	0.219268	−	0.747443i
$208$		0			0
$209$		−	208.867i		−	0.999365i
$210$		0			0
$211$	112.724			0.534237			0.267119	−	0.963664i	$-0.413928\pi$
$211$	112.724			0.534237			0.267119	+	0.963664i	$0.413928\pi$
$212$		0			0
$213$	163.310	−	122.284i	0.766713	−	0.574102i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		−	393.085i		−	1.81145i
$218$		0			0
$219$	107.389	−	80.4108i	0.490359	−	0.367172i
$220$		0			0
$221$			210.130i			0.950814i
$222$		0			0
$223$		−	204.686i		−	0.917872i	−0.888469	−	0.458936i	$-0.848231\pi$
$223$		−	204.686i		−	0.917872i	0.888469	−	0.458936i	$-0.151769\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	115.071			0.506920			0.253460	−	0.967346i	$-0.418431\pi$
$227$	115.071			0.506920			0.253460	+	0.967346i	$0.418431\pi$
$228$		0			0
$229$	129.458			0.565317			0.282659	−	0.959221i	$-0.408784\pi$
$229$	129.458			0.565317			0.282659	+	0.959221i	$0.408784\pi$
$230$		0			0
$231$	150.462	+	200.942i	0.651351	+	0.869879i
$232$		0			0
$233$	−250.150			−1.07361			−0.536803	−	0.843708i	$-0.680368\pi$
$233$	−250.150			−1.07361			−0.536803	+	0.843708i	$0.680368\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	21.1278	+	28.2162i	0.0891470	+	0.119056i
$238$		0			0
$239$			49.2556i			0.206090i	0.994677	+	0.103045i	$0.0328586\pi$
$239$			49.2556i			0.206090i	−0.994677	+	0.103045i	$0.967141\pi$
$240$		0			0
$241$	−457.672			−1.89905			−0.949526	−	0.313688i	$-0.898435\pi$
$241$	−457.672			−1.89905			−0.949526	+	0.313688i	$0.898435\pi$
$242$		0			0
$243$	17.4851	+	242.370i	0.0719551	+	0.997408i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		−	346.256i		−	1.40185i
$248$		0			0
$249$	−259.715	−	346.850i	−1.04303	−	1.39297i
$250$		0			0
$251$			119.717i			0.476959i	0.971148	+	0.238479i	$0.0766489\pi$
$251$			119.717i			0.476959i	−0.971148	+	0.238479i	$0.923351\pi$
$252$		0			0
$253$		−	146.783i		−	0.580169i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	111.805			0.435041			0.217520	−	0.976056i	$-0.430203\pi$
$257$	111.805			0.435041			0.217520	+	0.976056i	$0.430203\pi$
$258$		0			0
$259$	−120.911			−0.466839
$260$		0			0
$261$	362.734	+	106.411i	1.38979	+	0.407705i
$262$		0			0
$263$	55.3652			0.210514			0.105257	−	0.994445i	$-0.466433\pi$
$263$	55.3652			0.210514			0.105257	+	0.994445i	$0.466433\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−152.992	+	114.558i	−0.573003	+	0.429055i
$268$		0			0
$269$			309.427i			1.15028i	0.818053	+	0.575142i	$0.195053\pi$
$269$			309.427i			1.15028i	−0.818053	+	0.575142i	$0.804947\pi$
$270$		0			0
$271$	194.062			0.716096			0.358048	−	0.933703i	$-0.383442\pi$
$271$	194.062			0.716096			0.358048	+	0.933703i	$0.383442\pi$
$272$		0			0
$273$	249.433	+	333.118i	0.913675	+	1.22021i
$274$		0			0
$275$		0			0
$276$		0			0
$277$			5.20416i			0.0187876i	0.999956	+	0.00939379i	$0.00299018\pi$
$277$			5.20416i			0.0187876i	−0.999956	+	0.00939379i	$0.997010\pi$
$278$		0			0
$279$	97.5073	−	332.383i	0.349489	−	1.19134i
$280$		0			0
$281$			240.004i			0.854105i	0.904227	+	0.427053i	$0.140448\pi$
$281$			240.004i			0.854105i	−0.904227	+	0.427053i	$0.859552\pi$
$282$		0			0
$283$			153.109i			0.541022i	0.962717	+	0.270511i	$0.0871927\pi$
$283$			153.109i			0.541022i	−0.962717	+	0.270511i	$0.912807\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	473.671			1.65042
$288$		0			0
$289$	−49.6485			−0.171794
$290$		0			0
$291$	153.453	−	114.903i	0.527329	−	0.394855i
$292$		0			0
$293$	169.252			0.577653			0.288827	−	0.957381i	$-0.406735\pi$
$293$	169.252			0.577653			0.288827	+	0.957381i	$0.406735\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	77.3821	+	207.235i	0.260546	+	0.697760i
$298$		0			0
$299$		−	243.334i		−	0.813825i
$300$		0			0
$301$	551.699			1.83289
$302$		0			0
$303$	−122.176	+	91.4832i	−0.403220	+	0.301925i
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	330.401i		−	1.07623i	−0.842873	−	0.538113i	$-0.819137\pi$
$307$		−	330.401i		−	1.07623i	0.842873	−	0.538113i	$-0.180863\pi$
$308$		0			0
$309$	110.342	−	82.6224i	0.357095	−	0.267387i
$310$		0			0
$311$			133.596i			0.429570i	0.976661	+	0.214785i	$0.0689050\pi$
$311$			133.596i			0.429570i	−0.976661	+	0.214785i	$0.931095\pi$
$312$		0			0
$313$			459.981i			1.46959i	0.678290	+	0.734794i	$0.262721\pi$
$313$			459.981i			1.46959i	−0.678290	+	0.734794i	$0.737279\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−298.150			−0.940536			−0.470268	−	0.882524i	$-0.655843\pi$
$317$	−298.150			−0.940536			−0.470268	+	0.882524i	$0.655843\pi$
$318$		0			0
$319$	344.124			1.07876
$320$		0			0
$321$	198.241	+	264.751i	0.617574	+	0.824771i
$322$		0			0
$323$	−394.408			−1.22108
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−88.6112	−	118.340i	−0.270982	−	0.361897i
$328$		0			0
$329$		−	440.144i		−	1.33782i
$330$		0			0
$331$	−397.599			−1.20120			−0.600602	−	0.799548i	$-0.705073\pi$
$331$	−397.599			−1.20120			−0.600602	+	0.799548i	$0.705073\pi$
$332$		0			0
$333$	−102.240	−	29.9929i	−0.307026	−	0.0900687i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	427.942i		−	1.26986i	−0.772571	−	0.634929i	$-0.781030\pi$
$337$		−	427.942i		−	1.26986i	0.772571	−	0.634929i	$-0.218970\pi$
$338$		0			0
$339$	283.116	+	378.101i	0.835151	+	1.11534i
$340$		0			0
$341$		−	315.331i		−	0.924723i
$342$		0			0
$343$		−	64.4477i		−	0.187894i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−138.137			−0.398090			−0.199045	−	0.979990i	$-0.563784\pi$
$347$	−138.137			−0.398090			−0.199045	+	0.979990i	$0.563784\pi$
$348$		0			0
$349$	301.421			0.863669			0.431835	−	0.901953i	$-0.357866\pi$
$349$	301.421			0.863669			0.431835	+	0.901953i	$0.357866\pi$
$350$		0			0
$351$	128.283	+	343.550i	0.365477	+	0.978775i
$352$		0			0
$353$	142.622			0.404028			0.202014	−	0.979383i	$-0.435251\pi$
$353$	142.622			0.404028			0.202014	+	0.979383i	$0.435251\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	379.443	−	284.120i	1.06286	−	0.795856i
$358$		0			0
$359$			3.05663i			0.00851429i	0.999991	+	0.00425715i	$0.00135510\pi$
$359$			3.05663i			0.00851429i	−0.999991	+	0.00425715i	$0.998645\pi$
$360$		0			0
$361$	288.913			0.800313
$362$		0			0
$363$	−96.8739	−	129.375i	−0.266870	−	0.356405i
$364$		0			0
$365$		0			0
$366$		0			0
$367$			370.072i			1.00837i	0.863595	+	0.504186i	$0.168207\pi$
$367$			370.072i			1.00837i	−0.863595	+	0.504186i	$0.831793\pi$
$368$		0			0
$369$	400.525	+	117.497i	1.08543	+	0.318420i
$370$		0			0
$371$			845.065i			2.27780i
$372$		0			0
$373$		−	455.556i		−	1.22133i	−0.791889	−	0.610665i	$-0.790902\pi$
$373$		−	455.556i		−	1.22133i	0.791889	−	0.610665i	$-0.209098\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	570.482			1.51322
$378$		0			0
$379$	−380.051			−1.00277			−0.501387	−	0.865223i	$-0.667177\pi$
$379$	−380.051			−1.00277			−0.501387	+	0.865223i	$0.667177\pi$
$380$		0			0
$381$	324.346	−	242.865i	0.851301	−	0.637440i
$382$		0			0
$383$	−164.078			−0.428402			−0.214201	−	0.976790i	$-0.568715\pi$
$383$	−164.078			−0.428402			−0.214201	+	0.976790i	$0.568715\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	466.503	+	136.853i	1.20544	+	0.353624i
$388$		0			0
$389$			562.622i			1.44633i	0.690676	+	0.723164i	$0.257313\pi$
$389$			562.622i			1.44633i	−0.690676	+	0.723164i	$0.742687\pi$
$390$		0			0
$391$	−277.173			−0.708882
$392$		0			0
$393$	−361.778	+	270.893i	−0.920553	+	0.689295i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	93.4668i		−	0.235433i	−0.993047	−	0.117716i	$-0.962443\pi$
$397$		−	93.4668i		−	0.235433i	0.993047	−	0.117716i	$-0.0375574\pi$
$398$		0			0
$399$	−625.253	+	468.179i	−1.56705	+	1.17338i
$400$		0			0
$401$		−	23.4831i		−	0.0585613i	−0.999571	−	0.0292807i	$-0.990678\pi$
$401$		−	23.4831i		−	0.0585613i	0.999571	−	0.0292807i	$-0.00932166\pi$
$402$		0			0
$403$		−	522.749i		−	1.29714i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−96.9944			−0.238316
$408$		0			0
$409$	−697.472			−1.70531			−0.852655	−	0.522474i	$-0.825009\pi$
$409$	−697.472			−1.70531			−0.852655	+	0.522474i	$0.825009\pi$
$410$		0			0
$411$	250.095	+	334.001i	0.608503	+	0.812656i
$412$		0			0
$413$	468.714			1.13490
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−198.326	−	264.865i	−0.475603	−	0.635167i
$418$		0			0
$419$		−	115.537i		−	0.275744i	−0.990450	−	0.137872i	$-0.955974\pi$
$419$		−	115.537i		−	0.275744i	0.990450	−	0.137872i	$-0.0440262\pi$
$420$		0			0
$421$	117.396			0.278850			0.139425	−	0.990233i	$-0.455475\pi$
$421$	117.396			0.278850			0.139425	+	0.990233i	$0.455475\pi$
$422$		0			0
$423$	109.181	−	372.175i	0.258110	−	0.879847i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		−	956.851i		−	2.24087i
$428$		0			0
$429$	200.094	+	267.225i	0.466419	+	0.622903i
$430$		0			0
$431$		−	673.123i		−	1.56177i	−0.624675	−	0.780885i	$-0.714768\pi$
$431$		−	673.123i		−	1.56177i	0.624675	−	0.780885i	$-0.285232\pi$
$432$		0			0
$433$			392.655i			0.906824i	0.891301	+	0.453412i	$0.149793\pi$
$433$			392.655i			0.906824i	−0.891301	+	0.453412i	$0.850207\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	456.731			1.04515
$438$		0			0
$439$	409.353			0.932468			0.466234	−	0.884661i	$-0.345611\pi$
$439$	409.353			0.932468			0.466234	+	0.884661i	$0.345611\pi$
$440$		0			0
$441$	140.126	−	477.662i	0.317746	−	1.08313i
$442$		0			0
$443$	−355.935			−0.803464			−0.401732	−	0.915757i	$-0.631592\pi$
$443$	−355.935			−0.803464			−0.401732	+	0.915757i	$0.631592\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−218.065	+	163.284i	−0.487842	+	0.365288i
$448$		0			0
$449$			294.636i			0.656205i	0.944642	+	0.328103i	$0.106409\pi$
$449$			294.636i			0.656205i	−0.944642	+	0.328103i	$0.893591\pi$
$450$		0			0
$451$	379.976			0.842519
$452$		0			0
$453$	200.106	+	267.242i	0.441736	+	0.589938i
$454$		0			0
$455$		0			0
$456$		0			0
$457$			491.814i			1.07618i	0.842887	+	0.538090i	$0.180854\pi$
$457$			491.814i			1.07618i	−0.842887	+	0.538090i	$0.819146\pi$
$458$		0			0
$459$	391.325	−	146.122i	0.852561	−	0.318349i
$460$		0			0
$461$			42.6832i			0.0925883i	0.998928	+	0.0462942i	$0.0147412\pi$
$461$			42.6832i			0.0925883i	−0.998928	+	0.0462942i	$0.985259\pi$
$462$		0			0
$463$		−	422.582i		−	0.912705i	−0.889799	−	0.456352i	$-0.849156\pi$
$463$		−	422.582i		−	0.912705i	0.889799	−	0.456352i	$-0.150844\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−203.497			−0.435754			−0.217877	−	0.975976i	$-0.569913\pi$
$467$	−203.497			−0.435754			−0.217877	+	0.975976i	$0.569913\pi$
$468$		0			0
$469$	352.351			0.751282
$470$		0			0
$471$	342.813	−	256.693i	0.727840	−	0.544995i
$472$		0			0
$473$	442.570			0.935666
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−209.624	+	714.566i	−0.439463	+	1.49804i
$478$		0			0
$479$			485.253i			1.01305i	0.862224	+	0.506527i	$0.169071\pi$
$479$			485.253i			1.01305i	−0.862224	+	0.506527i	$0.830929\pi$
$480$		0			0
$481$	−160.795			−0.334294
$482$		0			0
$483$	−439.401	+	329.016i	−0.909732	+	0.681192i
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	140.013i		−	0.287500i	−0.989614	−	0.143750i	$-0.954084\pi$
$487$		−	140.013i		−	0.287500i	0.989614	−	0.143750i	$-0.0459162\pi$
$488$		0			0
$489$	−408.669	+	306.005i	−0.835724	+	0.625776i
$490$		0			0
$491$			20.4450i			0.0416396i	0.999783	+	0.0208198i	$0.00662763\pi$
$491$			20.4450i			0.0416396i	−0.999783	+	0.0208198i	$0.993372\pi$
$492$		0			0
$493$		−	649.816i		−	1.31809i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−694.562			−1.39751
$498$		0			0
$499$	49.2051			0.0986074			0.0493037	−	0.998784i	$-0.484300\pi$
$499$	49.2051			0.0986074			0.0493037	+	0.998784i	$0.484300\pi$
$500$		0			0
$501$	11.4861	+	15.3397i	0.0229263	+	0.0306181i
$502$		0			0
$503$	−149.060			−0.296343			−0.148171	−	0.988962i	$-0.547339\pi$
$503$	−149.060			−0.296343			−0.148171	+	0.988962i	$0.547339\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	27.8277	+	37.1639i	0.0548870	+	0.0733015i
$508$		0			0
$509$			655.375i			1.28757i	0.765205	+	0.643787i	$0.222638\pi$
$509$			655.375i			1.28757i	−0.765205	+	0.643787i	$0.777362\pi$
$510$		0			0
$511$	−456.727			−0.893790
$512$		0			0
$513$	−644.834	+	240.783i	−1.25699	+	0.469362i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	353.081i		−	0.682942i
$518$		0			0
$519$	428.680	+	572.502i	0.825973	+	1.10309i
$520$		0			0
$521$			419.868i			0.805890i	0.915224	+	0.402945i	$0.132013\pi$
$521$			419.868i			0.805890i	−0.915224	+	0.402945i	$0.867987\pi$
$522$		0			0
$523$			146.678i			0.280454i	0.990119	+	0.140227i	$0.0447833\pi$
$523$			146.678i			0.280454i	−0.990119	+	0.140227i	$0.955217\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−595.444			−1.12988
$528$		0			0
$529$	−208.030			−0.393251
$530$		0			0
$531$	396.333	+	116.268i	0.746390	+	0.218960i
$532$		0			0
$533$	629.917			1.18183
$534$		0			0
$535$		0			0
$536$		0			0
$537$	584.840	−	437.919i	1.08909	−	0.815491i
$538$		0			0
$539$		−	453.156i		−	0.840735i
$540$		0			0
$541$	7.07300			0.0130739			0.00653697	−	0.999979i	$-0.497919\pi$
$541$	7.07300			0.0130739			0.00653697	+	0.999979i	$0.497919\pi$
$542$		0			0
$543$	585.199	+	781.534i	1.07772	+	1.43929i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	3.46995i		−	0.00634360i	−0.999995	−	0.00317180i	$-0.998990\pi$
$547$		−	3.46995i		−	0.00634360i	0.999995	−	0.00317180i	$-0.00100962\pi$
$548$		0			0
$549$	237.353	−	809.090i	0.432337	−	1.47375i
$550$		0			0
$551$			1070.78i			1.94334i
$552$		0			0
$553$		−	120.005i		−	0.217006i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−654.467			−1.17499			−0.587493	−	0.809230i	$-0.699885\pi$
$557$	−654.467			−1.17499			−0.587493	+	0.809230i	$0.699885\pi$
$558$		0			0
$559$	733.684			1.31249
$560$		0			0
$561$	304.387	−	227.920i	0.542579	−	0.406274i
$562$		0			0
$563$	−546.529			−0.970745			−0.485373	−	0.874307i	$-0.661316\pi$
$563$	−546.529			−0.970745			−0.485373	+	0.874307i	$0.661316\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	446.913	−	696.167i	0.788207	−	1.22781i
$568$		0			0
$569$		−	746.774i		−	1.31243i	−0.754573	−	0.656216i	$-0.772156\pi$
$569$		−	746.774i		−	1.31243i	0.754573	−	0.656216i	$-0.227844\pi$
$570$		0			0
$571$	230.358			0.403430			0.201715	−	0.979444i	$-0.435349\pi$
$571$	230.358			0.403430			0.201715	+	0.979444i	$0.435349\pi$
$572$		0			0
$573$	−398.634	+	298.491i	−0.695696	+	0.520926i
$574$		0			0
$575$		0			0
$576$		0			0
$577$			525.585i			0.910893i	0.890263	+	0.455446i	$0.150520\pi$
$577$			525.585i			0.910893i	−0.890263	+	0.455446i	$0.849480\pi$
$578$		0			0
$579$	−574.914	+	430.486i	−0.992943	+	0.743499i
$580$		0			0
$581$			1475.16i			2.53901i
$582$		0			0
$583$			677.906i			1.16279i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	75.3619			0.128385			0.0641924	−	0.997938i	$-0.479553\pi$
$587$	75.3619			0.128385			0.0641924	+	0.997938i	$0.479553\pi$
$588$		0			0
$589$	981.185			1.66585
$590$		0			0
$591$	−463.051	−	618.405i	−0.783505	−	1.04637i
$592$		0			0
$593$	−1142.87			−1.92727			−0.963633	−	0.267230i	$-0.913892\pi$
$593$	−1142.87			−1.92727			−0.963633	+	0.267230i	$0.913892\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	141.106	+	188.446i	0.236358	+	0.315656i
$598$		0			0
$599$		−	1060.02i		−	1.76965i	−0.465927	−	0.884823i	$-0.654279\pi$
$599$		−	1060.02i		−	1.76965i	0.465927	−	0.884823i	$-0.345721\pi$
$600$		0			0
$601$	−530.900			−0.883361			−0.441681	−	0.897172i	$-0.645618\pi$
$601$	−530.900			−0.883361			−0.441681	+	0.897172i	$0.645618\pi$
$602$		0			0
$603$	297.940	+	87.4030i	0.494095	+	0.144947i
$604$		0			0
$605$		0			0
$606$		0			0
$607$			719.995i			1.18615i	0.805146	+	0.593077i	$0.202087\pi$
$607$			719.995i			1.18615i	−0.805146	+	0.593077i	$0.797913\pi$
$608$		0			0
$609$	−771.360	−	1030.15i	−1.26660	−	1.69154i
$610$		0			0
$611$		−	585.331i		−	0.957988i
$612$		0			0
$613$			216.348i			0.352932i	0.984307	+	0.176466i	$0.0564666\pi$
$613$			216.348i			0.352932i	−0.984307	+	0.176466i	$0.943533\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	231.190			0.374701			0.187350	−	0.982293i	$-0.440010\pi$
$617$	231.190			0.374701			0.187350	+	0.982293i	$0.440010\pi$
$618$		0			0
$619$	−689.633			−1.11411			−0.557054	−	0.830476i	$-0.688068\pi$
$619$	−689.633			−1.11411			−0.557054	+	0.830476i	$0.688068\pi$
$620$		0			0
$621$	−453.161	+	169.212i	−0.729728	+	0.272482i
$622$		0			0
$623$	650.679			1.04443
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−501.574	+	375.570i	−0.799959	+	0.598996i
$628$		0			0
$629$			183.156i			0.291187i
$630$		0			0
$631$	616.973			0.977770			0.488885	−	0.872348i	$-0.337404\pi$
$631$	616.973			0.977770			0.488885	+	0.872348i	$0.337404\pi$
$632$		0			0
$633$	−202.692	−	270.696i	−0.320209	−	0.427639i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		−	751.234i		−	1.17933i
$638$		0			0
$639$	−587.305	−	172.291i	−0.919100	−	0.269626i
$640$		0			0
$641$		−	1131.85i		−	1.76576i	−0.469598	−	0.882880i	$-0.655601\pi$
$641$		−	1131.85i		−	1.76576i	0.469598	−	0.882880i	$-0.344399\pi$
$642$		0			0
$643$		−	637.481i		−	0.991418i	−0.868489	−	0.495709i	$-0.834908\pi$
$643$		−	637.481i		−	0.991418i	0.868489	−	0.495709i	$-0.165092\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	800.782			1.23769			0.618843	−	0.785515i	$-0.287602\pi$
$647$	800.782			1.23769			0.618843	+	0.785515i	$0.287602\pi$
$648$		0			0
$649$	375.999			0.579352
$650$		0			0
$651$	−943.955	+	706.818i	−1.45001	+	1.08574i
$652$		0			0
$653$	313.185			0.479610			0.239805	−	0.970821i	$-0.422916\pi$
$653$	313.185			0.479610			0.239805	+	0.970821i	$0.422916\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−386.197	−	113.294i	−0.587819	−	0.172442i
$658$		0			0
$659$		−	878.311i		−	1.33279i	−0.745597	−	0.666397i	$-0.767836\pi$
$659$		−	878.311i		−	1.33279i	0.745597	−	0.666397i	$-0.232164\pi$
$660$		0			0
$661$	200.421			0.303208			0.151604	−	0.988441i	$-0.451556\pi$
$661$	200.421			0.303208			0.151604	+	0.988441i	$0.451556\pi$
$662$		0			0
$663$	504.606	−	377.841i	0.761096	−	0.569896i
$664$		0			0
$665$		0			0
$666$		0			0
$667$			752.497i			1.12818i
$668$		0			0
$669$	−491.532	+	368.051i	−0.734727	+	0.550151i
$670$		0			0
$671$		−	767.580i		−	1.14393i
$672$		0			0
$673$			60.8622i			0.0904341i	0.998977	+	0.0452171i	$0.0143980\pi$
$673$			60.8622i			0.0904341i	−0.998977	+	0.0452171i	$0.985602\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−415.595			−0.613877			−0.306939	−	0.951729i	$-0.599305\pi$
$677$	−415.595			−0.613877			−0.306939	+	0.951729i	$0.599305\pi$
$678$		0			0
$679$	−652.639			−0.961177
$680$		0			0
$681$	−206.912	−	276.331i	−0.303836	−	0.405772i
$682$		0			0
$683$	434.504			0.636170			0.318085	−	0.948062i	$-0.396960\pi$
$683$	434.504			0.636170			0.318085	+	0.948062i	$0.396960\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−232.782	−	310.880i	−0.338838	−	0.452518i
$688$		0			0
$689$			1123.82i			1.63109i
$690$		0			0
$691$	166.630			0.241144			0.120572	−	0.992705i	$-0.461527\pi$
$691$	166.630			0.241144			0.120572	+	0.992705i	$0.461527\pi$
$692$		0			0
$693$	211.992	−	722.640i	0.305905	−	1.04277i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	717.516i		−	1.02943i
$698$		0			0
$699$	449.802	+	600.711i	0.643494	+	0.859386i
$700$		0			0
$701$			248.345i			0.354272i	0.984186	+	0.177136i	$0.0566833\pi$
$701$			248.345i			0.354272i	−0.984186	+	0.177136i	$0.943317\pi$
$702$		0			0
$703$		−	301.809i		−	0.429315i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	519.617			0.734961
$708$		0			0
$709$	24.1911			0.0341200			0.0170600	−	0.999854i	$-0.494569\pi$
$709$	24.1911			0.0341200			0.0170600	+	0.999854i	$0.494569\pi$
$710$		0			0
$711$	29.7679	−	101.473i	0.0418677	−	0.142719i
$712$		0			0
$713$	689.534			0.967088
$714$		0			0
$715$		0			0
$716$		0			0
$717$	118.282	−	88.5679i	0.164969	−	0.123526i
$718$		0			0
$719$		−	323.584i		−	0.450048i	−0.974353	−	0.225024i	$-0.927754\pi$
$719$		−	323.584i		−	0.450048i	0.974353	−	0.225024i	$-0.0722460\pi$
$720$		0			0
$721$	−469.289			−0.650886
$722$		0			0
$723$	822.953	+	1099.05i	1.13825	+	1.52013i
$724$		0			0
$725$		0			0
$726$		0			0
$727$			454.949i			0.625789i	0.949788	+	0.312895i	$0.101299\pi$
$727$			454.949i			0.625789i	−0.949788	+	0.312895i	$0.898701\pi$
$728$		0			0
$729$	550.588	−	477.802i	0.755264	−	0.655421i
$730$		0			0
$731$		−	835.713i		−	1.14325i
$732$		0			0
$733$		−	131.588i		−	0.179520i	−0.995963	−	0.0897600i	$-0.971390\pi$
$733$		−	131.588i		−	0.179520i	0.995963	−	0.0897600i	$-0.0286100\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	282.654			0.383520
$738$		0			0
$739$	1033.59			1.39864			0.699319	−	0.714810i	$-0.253487\pi$
$739$	1033.59			1.39864			0.699319	+	0.714810i	$0.253487\pi$
$740$		0			0
$741$	−831.500	+	622.613i	−1.12213	+	0.840234i
$742$		0			0
$743$	439.535			0.591569			0.295784	−	0.955255i	$-0.404419\pi$
$743$	439.535			0.591569			0.295784	+	0.955255i	$0.404419\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−365.924	+	1247.36i	−0.489858	+	1.66983i
$748$		0			0
$749$		−	1126.00i		−	1.50333i
$750$		0			0
$751$	−496.343			−0.660909			−0.330455	−	0.943822i	$-0.607202\pi$
$751$	−496.343			−0.660909			−0.330455	+	0.943822i	$0.607202\pi$
$752$		0			0
$753$	287.488	−	215.266i	0.381790	−	0.285878i
$754$		0			0
$755$		0			0
$756$		0			0
$757$			468.694i			0.619147i	0.950875	+	0.309574i	$0.100186\pi$
$757$			468.694i			0.619147i	−0.950875	+	0.309574i	$0.899814\pi$
$758$		0			0
$759$	−352.485	+	263.935i	−0.464406	+	0.347740i
$760$		0			0
$761$		−	156.940i		−	0.206229i	−0.994669	−	0.103115i	$-0.967119\pi$
$761$		−	156.940i		−	0.206229i	0.994669	−	0.103115i	$-0.0328808\pi$
$762$		0			0
$763$			503.305i			0.659640i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	623.325			0.812679
$768$		0			0
$769$	−1188.63			−1.54569			−0.772844	−	0.634596i	$-0.781166\pi$
$769$	−1188.63			−1.54569			−0.772844	+	0.634596i	$0.781166\pi$
$770$		0			0
$771$	−201.041	−	268.490i	−0.260753	−	0.348236i
$772$		0			0
$773$	−950.291			−1.22935			−0.614677	−	0.788779i	$-0.710714\pi$
$773$	−950.291			−1.22935			−0.614677	+	0.788779i	$0.710714\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	217.414	+	290.357i	0.279813	+	0.373690i
$778$		0			0
$779$			1182.34i			1.51776i
$780$		0			0
$781$	−557.174			−0.713411
$782$		0			0
$783$	−396.707	−	1062.41i	−0.506650	−	1.35685i
$784$		0			0
$785$		0			0
$786$		0			0
$787$		−	1056.60i		−	1.34257i	−0.741200	−	0.671284i	$-0.765743\pi$
$787$		−	1056.60i		−	1.34257i	0.741200	−	0.671284i	$-0.234257\pi$
$788$		0			0
$789$	−99.5538	−	132.954i	−0.126177	−	0.168510i
$790$		0			0
$791$		−	1608.08i		−	2.03297i
$792$		0			0
$793$		−	1272.48i		−	1.60464i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−78.7126			−0.0987612			−0.0493806	−	0.998780i	$-0.515725\pi$
$797$	−78.7126			−0.0987612			−0.0493806	+	0.998780i	$0.515725\pi$
$798$		0			0
$799$	−666.730			−0.834455
$800$		0			0
$801$	550.198	+	161.405i	0.686889	+	0.201505i
$802$		0			0
$803$	−366.384			−0.456268
$804$		0			0
$805$		0			0
$806$		0			0
$807$	743.058	−	556.389i	0.920765	−	0.689454i
$808$		0			0
$809$			176.199i			0.217799i	0.994053	+	0.108899i	$0.0347327\pi$
$809$			176.199i			0.217799i	−0.994053	+	0.108899i	$0.965267\pi$
$810$		0			0
$811$	−1103.89			−1.36115			−0.680574	−	0.732679i	$-0.738270\pi$
$811$	−1103.89			−1.36115			−0.680574	+	0.732679i	$0.738270\pi$
$812$		0			0
$813$	−348.949	−	466.021i	−0.429211	−	0.573211i
$814$		0			0
$815$		0			0
$816$		0			0
$817$			1377.10i			1.68556i
$818$		0			0
$819$	351.437	−	1197.98i	0.429105	−	1.46273i
$820$		0			0
$821$		−	1321.32i		−	1.60940i	−0.593679	−	0.804702i	$-0.702325\pi$
$821$		−	1321.32i		−	1.60940i	0.593679	−	0.804702i	$-0.297675\pi$
$822$		0			0
$823$			760.814i			0.924439i	0.886765	+	0.462220i	$0.152947\pi$
$823$			760.814i			0.924439i	−0.886765	+	0.462220i	$0.847053\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−610.599			−0.738331			−0.369165	−	0.929364i	$-0.620356\pi$
$827$	−610.599			−0.738331			−0.369165	+	0.929364i	$0.620356\pi$
$828$		0			0
$829$	−265.766			−0.320586			−0.160293	−	0.987069i	$-0.551244\pi$
$829$	−265.766			−0.320586			−0.160293	+	0.987069i	$0.551244\pi$
$830$		0			0
$831$	12.4973	−	9.35776i	0.0150388	−	0.0112608i
$832$		0			0
$833$	−855.704			−1.02726
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−973.516	+	363.514i	−1.16310	+	0.434306i
$838$		0			0
$839$		−	48.5299i		−	0.0578425i	−0.999582	−	0.0289213i	$-0.990793\pi$
$839$		−	48.5299i		−	0.0578425i	0.999582	−	0.0289213i	$-0.00920721\pi$
$840$		0			0
$841$	−923.188			−1.09773
$842$		0			0
$843$	576.345	−	431.558i	0.683683	−	0.511931i
$844$		0			0
$845$		0			0
$846$		0			0
$847$			550.236i			0.649629i
$848$		0			0
$849$	367.677	−	275.310i	0.433071	−	0.324276i
$850$		0			0
$851$		−	212.098i		−	0.249234i
$852$		0			0
$853$			331.180i			0.388253i	0.980976	+	0.194127i	$0.0621873\pi$
$853$			331.180i			0.388253i	−0.980976	+	0.194127i	$0.937813\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	159.395			0.185992			0.0929961	−	0.995666i	$-0.470356\pi$
$857$	159.395			0.185992			0.0929961	+	0.995666i	$0.470356\pi$
$858$		0			0
$859$	−1635.01			−1.90339			−0.951694	−	0.307049i	$-0.900658\pi$
$859$	−1635.01			−1.90339			−0.951694	+	0.307049i	$0.900658\pi$
$860$		0			0
$861$	−851.722	−	1137.47i	−0.989224	−	1.32111i
$862$		0			0
$863$	−152.237			−0.176405			−0.0882025	−	0.996103i	$-0.528112\pi$
$863$	−152.237			−0.176405			−0.0882025	+	0.996103i	$0.528112\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	89.2744	+	119.226i	0.102969	+	0.137516i
$868$		0			0
$869$		−	96.2669i		−	0.110779i
$870$		0			0
$871$	468.578			0.537977
$872$		0			0
$873$	−551.856	−	161.891i	−0.632137	−	0.185443i
$874$		0			0
$875$		0			0
$876$		0			0
$877$			521.588i			0.594742i	0.954762	+	0.297371i	$0.0961097\pi$
$877$			521.588i			0.594742i	−0.954762	+	0.297371i	$0.903890\pi$
$878$		0			0
$879$	−304.338	−	406.443i	−0.346232	−	0.462392i
$880$		0			0
$881$			821.701i			0.932692i	0.884603	+	0.466346i	$0.154430\pi$
$881$			821.701i			0.932692i	−0.884603	+	0.466346i	$0.845570\pi$
$882$		0			0
$883$			747.791i			0.846876i	0.905925	+	0.423438i	$0.139177\pi$
$883$			747.791i			0.846876i	−0.905925	+	0.423438i	$0.860823\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1160.21			−1.30801			−0.654006	−	0.756490i	$-0.726913\pi$
$887$	−1160.21			−1.30801			−0.654006	+	0.756490i	$0.726913\pi$
$888$		0			0
$889$	−1379.45			−1.55169
$890$		0			0
$891$	358.511	−	558.461i	0.402369	−	0.626780i
$892$		0			0
$893$	1098.65			1.23029
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−584.342	+	437.546i	−0.651440	+	0.487788i
$898$		0			0
$899$			1616.57i			1.79819i
$900$		0			0
$901$	1280.10			1.42076
$902$		0			0
$903$	−992.027	−	1324.85i	−1.09859	−	1.46717i
$904$		0			0
$905$		0			0
$906$		0			0
$907$			354.207i			0.390526i	0.980751	+	0.195263i	$0.0625560\pi$
$907$			354.207i			0.390526i	−0.980751	+	0.195263i	$0.937444\pi$
$908$		0			0
$909$	439.376	+	128.894i	0.483362	+	0.141798i
$910$		0			0
$911$			1461.82i			1.60463i	0.596900	+	0.802316i	$0.296399\pi$
$911$			1461.82i			1.60463i	−0.596900	+	0.802316i	$0.703601\pi$
$912$		0			0
$913$			1183.37i			1.29613i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1538.65			1.67792
$918$		0			0
$919$	600.104			0.652997			0.326499	−	0.945198i	$-0.394131\pi$
$919$	600.104			0.652997			0.326499	+	0.945198i	$0.394131\pi$
$920$		0			0
$921$	−793.427	+	594.105i	−0.861484	+	0.645065i
$922$		0			0
$923$	−923.672			−1.00073
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−396.819	−	116.410i	−0.428068	−	0.125577i
$928$		0			0
$929$			243.828i			0.262462i	0.991352	+	0.131231i	$0.0418930\pi$
$929$			243.828i			0.262462i	−0.991352	+	0.131231i	$0.958107\pi$
$930$		0			0
$931$	1410.05			1.51455
$932$		0			0
$933$	320.818	−	240.223i	0.343856	−	0.257474i
$934$		0			0
$935$		0			0
$936$		0			0
$937$			894.637i			0.954789i	0.878689	+	0.477395i	$0.158419\pi$
$937$			894.637i			0.954789i	−0.878689	+	0.477395i	$0.841581\pi$
$938$		0			0
$939$	1104.60	−	827.105i	1.17636	−	0.880837i
$940$		0			0
$941$		−	592.336i		−	0.629475i	−0.949179	−	0.314737i	$-0.898084\pi$
$941$		−	592.336i		−	0.629475i	0.949179	−	0.314737i	$-0.101916\pi$
$942$		0			0
$943$			830.894i			0.881118i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	526.526			0.555994			0.277997	−	0.960582i	$-0.410330\pi$
$947$	526.526			0.555994			0.277997	+	0.960582i	$0.410330\pi$
$948$		0			0
$949$	−607.383			−0.640025
$950$		0			0
$951$	536.112	+	715.977i	0.563735	+	0.752868i
$952$		0			0
$953$	117.158			0.122936			0.0614678	−	0.998109i	$-0.480422\pi$
$953$	117.158			0.122936			0.0614678	+	0.998109i	$0.480422\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−618.780	−	826.381i	−0.646583	−	0.863512i
$958$		0			0
$959$		−	1420.52i		−	1.48125i
$960$		0			0
$961$	520.311			0.541427
$962$		0			0
$963$	279.311	−	952.114i	0.290042	−	0.988696i
$964$		0			0
$965$		0			0
$966$		0			0
$967$			811.294i			0.838981i	0.907760	+	0.419490i	$0.137791\pi$
$967$			811.294i			0.838981i	−0.907760	+	0.419490i	$0.862209\pi$
$968$		0			0
$969$	709.197	+	947.132i	0.731885	+	0.977432i
$970$		0			0
$971$		−	1052.66i		−	1.08410i	−0.840346	−	0.542050i	$-0.817648\pi$
$971$		−	1052.66i		−	1.08410i	0.840346	−	0.542050i	$-0.182352\pi$
$972$		0			0
$973$			1126.48i			1.15774i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−830.832			−0.850391			−0.425195	−	0.905102i	$-0.639795\pi$
$977$	−830.832			−0.850391			−0.425195	+	0.905102i	$0.639795\pi$
$978$		0			0
$979$	521.971			0.533167
$980$		0			0
$981$	−124.848	+	425.582i	−0.127266	+	0.433825i
$982$		0			0
$983$	−1056.30			−1.07457			−0.537283	−	0.843402i	$-0.680549\pi$
$983$	−1056.30			−1.07457			−0.537283	+	0.843402i	$0.680549\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−1056.96	+	791.436i	−1.07088	+	0.801861i
$988$		0			0
$989$			967.768i			0.978532i
$990$		0			0
$991$	−1592.65			−1.60712			−0.803559	−	0.595225i	$-0.797063\pi$
$991$	−1592.65			−1.60712			−0.803559	+	0.595225i	$0.797063\pi$
$992$		0			0
$993$	714.934	+	954.795i	0.719974	+	0.961525i
$994$		0			0
$995$		0			0
$996$		0			0
$997$			978.023i			0.980966i	0.871451	+	0.490483i	$0.163180\pi$
$997$			978.023i			0.980966i	−0.871451	+	0.490483i	$0.836820\pi$
$998$		0			0
$999$	111.815	+	299.450i	0.111927	+	0.299749i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.3.c.m.449.3		16
3.2	odd	2	inner	1200.3.c.m.449.13		16
4.3	odd	2		600.3.c.d.449.14		16
5.2	odd	4		1200.3.l.x.401.4		8
5.3	odd	4		240.3.l.d.161.5		8
5.4	even	2	inner	1200.3.c.m.449.14		16
12.11	even	2		600.3.c.d.449.4		16
15.2	even	4		1200.3.l.x.401.3		8
15.8	even	4		240.3.l.d.161.6		8
15.14	odd	2	inner	1200.3.c.m.449.4		16
20.3	even	4		120.3.l.a.41.4	yes	8
20.7	even	4		600.3.l.f.401.5		8
20.19	odd	2		600.3.c.d.449.3		16
40.3	even	4		960.3.l.h.641.5		8
40.13	odd	4		960.3.l.g.641.4		8
60.23	odd	4		120.3.l.a.41.3	✓	8
60.47	odd	4		600.3.l.f.401.6		8
60.59	even	2		600.3.c.d.449.13		16
120.53	even	4		960.3.l.g.641.3		8
120.83	odd	4		960.3.l.h.641.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.3.l.a.41.3	✓	8	60.23	odd	4
120.3.l.a.41.4	yes	8	20.3	even	4
240.3.l.d.161.5		8	5.3	odd	4
240.3.l.d.161.6		8	15.8	even	4
600.3.c.d.449.3		16	20.19	odd	2
600.3.c.d.449.4		16	12.11	even	2
600.3.c.d.449.13		16	60.59	even	2
600.3.c.d.449.14		16	4.3	odd	2
600.3.l.f.401.5		8	20.7	even	4
600.3.l.f.401.6		8	60.47	odd	4
960.3.l.g.641.3		8	120.53	even	4
960.3.l.g.641.4		8	40.13	odd	4
960.3.l.h.641.5		8	40.3	even	4
960.3.l.h.641.6		8	120.83	odd	4
1200.3.c.m.449.3		16	1.1	even	1	trivial
1200.3.c.m.449.4		16	15.14	odd	2	inner
1200.3.c.m.449.13		16	3.2	odd	2	inner
1200.3.c.m.449.14		16	5.4	even	2	inner
1200.3.l.x.401.3		8	15.2	even	4
1200.3.l.x.401.4		8	5.2	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:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1200.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.79813 - 2.40140i) q^{3} +10.2132i q^{7} +(-2.53346 + 8.63606i) q^{9} +O(q^{10})\)	\(q+(-1.79813 - 2.40140i) q^{3} +10.2132i q^{7} +(-2.53346 + 8.63606i) q^{9} +8.19300i q^{11} +13.5822i q^{13} +15.4710 q^{17} -25.4934 q^{19} +(24.5261 - 18.3647i) q^{21} -17.9156 q^{23} +(25.2941 - 9.44491i) q^{27} -42.0022i q^{29} -38.4878 q^{31} +(19.6747 - 14.7321i) q^{33} +11.8387i q^{37} +(32.6163 - 24.4225i) q^{39} -46.3781i q^{41} -54.0181i q^{43} -43.0955 q^{47} -55.3102 q^{49} +(-27.8188 - 37.1521i) q^{51} +82.7421 q^{53} +(45.8404 + 61.2199i) q^{57} -45.8928i q^{59} -93.6873 q^{61} +(-88.2022 - 25.8748i) q^{63} -34.4995i q^{67} +(32.2147 + 43.0227i) q^{69} +68.0061i q^{71} +44.7191i q^{73} -83.6770 q^{77} -11.7499 q^{79} +(-68.1632 - 43.7582i) q^{81} +144.436 q^{83} +(-100.864 + 75.5255i) q^{87} -63.7094i q^{89} -138.718 q^{91} +(69.2061 + 92.4247i) q^{93} +63.9013i q^{97} +(-70.7552 - 20.7566i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 40 q^{9}+O(q^{10}) \) 16 * q - 40 * q^9	\( 16 q - 40 q^{9} - 16 q^{19} + 56 q^{21} - 240 q^{31} - 144 q^{39} - 128 q^{49} - 128 q^{51} + 16 q^{61} - 200 q^{69} + 176 q^{79} + 448 q^{81} - 1120 q^{91} + 64 q^{99}+O(q^{100}) \) 16 * q - 40 * q^9 - 16 * q^19 + 56 * q^21 - 240 * q^31 - 144 * q^39 - 128 * q^49 - 128 * q^51 + 16 * q^61 - 200 * q^69 + 176 * q^79 + 448 * q^81 - 1120 * q^91 + 64 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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