LMFDB - Embedded newform 1200.3.c.f.449.4

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:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 15)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.4
Root			$0.618034i$ of defining polynomial
Character	$\chi$	$=$	1200.449
Dual form			1200.3.c.f.449.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.23607 + 2.00000i) q^{3} -6.00000i q^{7} +(1.00000 + 8.94427i) q^{9} +O(q^{10})$	$q+(2.23607 + 2.00000i) q^{3} -6.00000i q^{7} +(1.00000 + 8.94427i) q^{9} -4.47214i q^{11} +16.0000i q^{13} +4.47214 q^{17} -2.00000 q^{19} +(12.0000 - 13.4164i) q^{21} +13.4164 q^{23} +(-15.6525 + 22.0000i) q^{27} +31.3050i q^{29} +18.0000 q^{31} +(8.94427 - 10.0000i) q^{33} +16.0000i q^{37} +(-32.0000 + 35.7771i) q^{39} +62.6099i q^{41} -16.0000i q^{43} +49.1935 q^{47} +13.0000 q^{49} +(10.0000 + 8.94427i) q^{51} -4.47214 q^{53} +(-4.47214 - 4.00000i) q^{57} +4.47214i q^{59} +82.0000 q^{61} +(53.6656 - 6.00000i) q^{63} +24.0000i q^{67} +(30.0000 + 26.8328i) q^{69} +125.220i q^{71} -74.0000i q^{73} -26.8328 q^{77} +138.000 q^{79} +(-79.0000 + 17.8885i) q^{81} -93.9149 q^{83} +(-62.6099 + 70.0000i) q^{87} -107.331i q^{89} +96.0000 q^{91} +(40.2492 + 36.0000i) q^{93} +166.000i q^{97} +(40.0000 - 4.47214i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^9	$ 4 q + 4 q^{9} - 8 q^{19} + 48 q^{21} + 72 q^{31} - 128 q^{39} + 52 q^{49} + 40 q^{51} + 328 q^{61} + 120 q^{69} + 552 q^{79} - 316 q^{81} + 384 q^{91} + 160 q^{99}+O(q^{100}) $ 4 * q + 4 * q^9 - 8 * q^19 + 48 * q^21 + 72 * q^31 - 128 * q^39 + 52 * q^49 + 40 * q^51 + 328 * q^61 + 120 * q^69 + 552 * q^79 - 316 * q^81 + 384 * q^91 + 160 * 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
$3$	2.23607	+	2.00000i	0.745356	+	0.666667i
$3$	2.23607	+	2.00000i	0.745356	+	0.666667i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$		−	6.00000i		−	0.857143i	−0.903508	−	0.428571i	$-0.859017\pi$
$7$		−	6.00000i		−	0.857143i	0.903508	−	0.428571i	$-0.140983\pi$
$8$		0			0
$9$	1.00000	+	8.94427i	0.111111	+	0.993808i
$10$		0			0
$11$		−	4.47214i		−	0.406558i	−0.979121	−	0.203279i	$-0.934840\pi$
$11$		−	4.47214i		−	0.406558i	0.979121	−	0.203279i	$-0.0651598\pi$
$12$		0			0
$13$			16.0000i			1.23077i	0.788227	+	0.615385i	$0.210999\pi$
$13$			16.0000i			1.23077i	−0.788227	+	0.615385i	$0.789001\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	4.47214			0.263067			0.131533	−	0.991312i	$-0.458010\pi$
$17$	4.47214			0.263067			0.131533	+	0.991312i	$0.458010\pi$
$18$		0			0
$19$	−2.00000			−0.105263			−0.0526316	−	0.998614i	$-0.516761\pi$
$19$	−2.00000			−0.105263			−0.0526316	+	0.998614i	$0.516761\pi$
$20$		0			0
$21$	12.0000	−	13.4164i	0.571429	−	0.638877i
$22$		0			0
$23$	13.4164			0.583322			0.291661	−	0.956522i	$-0.405792\pi$
$23$	13.4164			0.583322			0.291661	+	0.956522i	$0.405792\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−15.6525	+	22.0000i	−0.579721	+	0.814815i
$28$		0			0
$29$			31.3050i			1.07948i	0.841831	+	0.539741i	$0.181478\pi$
$29$			31.3050i			1.07948i	−0.841831	+	0.539741i	$0.818522\pi$
$30$		0			0
$31$	18.0000			0.580645			0.290323	−	0.956929i	$-0.406237\pi$
$31$	18.0000			0.580645			0.290323	+	0.956929i	$0.406237\pi$
$32$		0			0
$33$	8.94427	−	10.0000i	0.271039	−	0.303030i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			16.0000i			0.432432i	0.976346	+	0.216216i	$0.0693716\pi$
$37$			16.0000i			0.432432i	−0.976346	+	0.216216i	$0.930628\pi$
$38$		0			0
$39$	−32.0000	+	35.7771i	−0.820513	+	0.917361i
$40$		0			0
$41$			62.6099i			1.52707i	0.645766	+	0.763535i	$0.276538\pi$
$41$			62.6099i			1.52707i	−0.645766	+	0.763535i	$0.723462\pi$
$42$		0			0
$43$		−	16.0000i		−	0.372093i	−0.982541	−	0.186047i	$-0.940432\pi$
$43$		−	16.0000i		−	0.372093i	0.982541	−	0.186047i	$-0.0595675\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	49.1935			1.04667			0.523335	−	0.852127i	$-0.324688\pi$
$47$	49.1935			1.04667			0.523335	+	0.852127i	$0.324688\pi$
$48$		0			0
$49$	13.0000			0.265306
$50$		0			0
$51$	10.0000	+	8.94427i	0.196078	+	0.175378i
$52$		0			0
$53$	−4.47214			−0.0843799			−0.0421900	−	0.999110i	$-0.513433\pi$
$53$	−4.47214			−0.0843799			−0.0421900	+	0.999110i	$0.513433\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−4.47214	−	4.00000i	−0.0784585	−	0.0701754i
$58$		0			0
$59$			4.47214i			0.0757989i	0.999282	+	0.0378995i	$0.0120667\pi$
$59$			4.47214i			0.0757989i	−0.999282	+	0.0378995i	$0.987933\pi$
$60$		0			0
$61$	82.0000			1.34426			0.672131	−	0.740432i	$-0.265379\pi$
$61$	82.0000			1.34426			0.672131	+	0.740432i	$0.265379\pi$
$62$		0			0
$63$	53.6656	−	6.00000i	0.851835	−	0.0952381i
$64$		0			0
$65$		0			0
$66$		0			0
$67$			24.0000i			0.358209i	0.983830	+	0.179104i	$0.0573200\pi$
$67$			24.0000i			0.358209i	−0.983830	+	0.179104i	$0.942680\pi$
$68$		0			0
$69$	30.0000	+	26.8328i	0.434783	+	0.388881i
$70$		0			0
$71$			125.220i			1.76366i	0.471568	+	0.881830i	$0.343688\pi$
$71$			125.220i			1.76366i	−0.471568	+	0.881830i	$0.656312\pi$
$72$		0			0
$73$		−	74.0000i		−	1.01370i	−0.862035	−	0.506849i	$-0.830810\pi$
$73$		−	74.0000i		−	1.01370i	0.862035	−	0.506849i	$-0.169190\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−26.8328			−0.348478
$78$		0			0
$79$	138.000			1.74684			0.873418	−	0.486972i	$-0.161899\pi$
$79$	138.000			1.74684			0.873418	+	0.486972i	$0.161899\pi$
$80$		0			0
$81$	−79.0000	+	17.8885i	−0.975309	+	0.220846i
$82$		0			0
$83$	−93.9149			−1.13150			−0.565752	−	0.824575i	$-0.691414\pi$
$83$	−93.9149			−1.13150			−0.565752	+	0.824575i	$0.691414\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−62.6099	+	70.0000i	−0.719654	+	0.804598i
$88$		0			0
$89$		−	107.331i		−	1.20597i	−0.797753	−	0.602985i	$-0.793978\pi$
$89$		−	107.331i		−	1.20597i	0.797753	−	0.602985i	$-0.206022\pi$
$90$		0			0
$91$	96.0000			1.05495
$92$		0			0
$93$	40.2492	+	36.0000i	0.432787	+	0.387097i
$94$		0			0
$95$		0			0
$96$		0			0
$97$			166.000i			1.71134i	0.517522	+	0.855670i	$0.326855\pi$
$97$			166.000i			1.71134i	−0.517522	+	0.855670i	$0.673145\pi$
$98$		0			0
$99$	40.0000	−	4.47214i	0.404040	−	0.0451731i
$100$		0			0
$101$			67.0820i			0.664179i	0.943248	+	0.332089i	$0.107754\pi$
$101$			67.0820i			0.664179i	−0.943248	+	0.332089i	$0.892246\pi$
$102$		0			0
$103$		−	26.0000i		−	0.252427i	−0.992003	−	0.126214i	$-0.959718\pi$
$103$		−	26.0000i		−	0.252427i	0.992003	−	0.126214i	$-0.0402825\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−201.246			−1.88080			−0.940402	−	0.340064i	$-0.889551\pi$
$107$	−201.246			−1.88080			−0.940402	+	0.340064i	$0.889551\pi$
$108$		0			0
$109$	−38.0000			−0.348624			−0.174312	−	0.984690i	$-0.555770\pi$
$109$	−38.0000			−0.348624			−0.174312	+	0.984690i	$0.555770\pi$
$110$		0			0
$111$	−32.0000	+	35.7771i	−0.288288	+	0.322316i
$112$		0			0
$113$	−31.3050			−0.277035			−0.138517	−	0.990360i	$-0.544234\pi$
$113$	−31.3050			−0.277035			−0.138517	+	0.990360i	$0.544234\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−143.108	+	16.0000i	−1.22315	+	0.136752i
$118$		0			0
$119$		−	26.8328i		−	0.225486i
$120$		0			0
$121$	101.000			0.834711
$122$		0			0
$123$	−125.220	+	140.000i	−1.01805	+	1.13821i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	26.0000i		−	0.204724i	−0.994747	−	0.102362i	$-0.967360\pi$
$127$		−	26.0000i		−	0.204724i	0.994747	−	0.102362i	$-0.0326401\pi$
$128$		0			0
$129$	32.0000	−	35.7771i	0.248062	−	0.277342i
$130$		0			0
$131$		−	13.4164i		−	0.102415i	−0.998688	−	0.0512077i	$-0.983693\pi$
$131$		−	13.4164i		−	0.102415i	0.998688	−	0.0512077i	$-0.0163070\pi$
$132$		0			0
$133$			12.0000i			0.0902256i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−120.748			−0.881370			−0.440685	−	0.897662i	$-0.645264\pi$
$137$	−120.748			−0.881370			−0.440685	+	0.897662i	$0.645264\pi$
$138$		0			0
$139$	−82.0000			−0.589928			−0.294964	−	0.955508i	$-0.595308\pi$
$139$	−82.0000			−0.589928			−0.294964	+	0.955508i	$0.595308\pi$
$140$		0			0
$141$	110.000	+	98.3870i	0.780142	+	0.697780i
$142$		0			0
$143$	71.5542			0.500379
$144$		0			0
$145$		0			0
$146$		0			0
$147$	29.0689	+	26.0000i	0.197748	+	0.176871i
$148$		0			0
$149$			111.803i			0.750358i	0.926952	+	0.375179i	$0.122419\pi$
$149$			111.803i			0.750358i	−0.926952	+	0.375179i	$0.877581\pi$
$150$		0			0
$151$	158.000			1.04636			0.523179	−	0.852223i	$-0.324746\pi$
$151$	158.000			1.04636			0.523179	+	0.852223i	$0.324746\pi$
$152$		0			0
$153$	4.47214	+	40.0000i	0.0292296	+	0.261438i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	164.000i		−	1.04459i	−0.852766	−	0.522293i	$-0.825077\pi$
$157$		−	164.000i		−	1.04459i	0.852766	−	0.522293i	$-0.174923\pi$
$158$		0			0
$159$	−10.0000	−	8.94427i	−0.0628931	−	0.0562533i
$160$		0			0
$161$		−	80.4984i		−	0.499990i
$162$		0			0
$163$		−	236.000i		−	1.44785i	−0.689877	−	0.723926i	$-0.742336\pi$
$163$		−	236.000i		−	1.44785i	0.689877	−	0.723926i	$-0.257664\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−93.9149			−0.562364			−0.281182	−	0.959654i	$-0.590727\pi$
$167$	−93.9149			−0.562364			−0.281182	+	0.959654i	$0.590727\pi$
$168$		0			0
$169$	−87.0000			−0.514793
$170$		0			0
$171$	−2.00000	−	17.8885i	−0.0116959	−	0.104611i
$172$		0			0
$173$	−13.4164			−0.0775515			−0.0387757	−	0.999248i	$-0.512346\pi$
$173$	−13.4164			−0.0775515			−0.0387757	+	0.999248i	$0.512346\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−8.94427	+	10.0000i	−0.0505326	+	0.0564972i
$178$		0			0
$179$			192.302i			1.07431i	0.843483	+	0.537156i	$0.180501\pi$
$179$			192.302i			1.07431i	−0.843483	+	0.537156i	$0.819499\pi$
$180$		0			0
$181$	2.00000			0.0110497			0.00552486	−	0.999985i	$-0.498241\pi$
$181$	2.00000			0.0110497			0.00552486	+	0.999985i	$0.498241\pi$
$182$		0			0
$183$	183.358	+	164.000i	1.00195	+	0.896175i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		−	20.0000i		−	0.106952i
$188$		0			0
$189$	132.000	+	93.9149i	0.698413	+	0.496904i
$190$		0			0
$191$			205.718i			1.07706i	0.842607	+	0.538529i	$0.181020\pi$
$191$			205.718i			1.07706i	−0.842607	+	0.538529i	$0.818980\pi$
$192$		0			0
$193$		−	214.000i		−	1.10881i	−0.832248	−	0.554404i	$-0.812946\pi$
$193$		−	214.000i		−	1.10881i	0.832248	−	0.554404i	$-0.187054\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−93.9149			−0.476725			−0.238363	−	0.971176i	$-0.576611\pi$
$197$	−93.9149			−0.476725			−0.238363	+	0.971176i	$0.576611\pi$
$198$		0			0
$199$	−242.000			−1.21608			−0.608040	−	0.793906i	$-0.708044\pi$
$199$	−242.000			−1.21608			−0.608040	+	0.793906i	$0.708044\pi$
$200$		0			0
$201$	−48.0000	+	53.6656i	−0.238806	+	0.266993i
$202$		0			0
$203$	187.830			0.925270
$204$		0			0
$205$		0			0
$206$		0			0
$207$	13.4164	+	120.000i	0.0648136	+	0.579710i
$208$		0			0
$209$			8.94427i			0.0427956i
$210$		0			0
$211$	−2.00000			−0.00947867			−0.00473934	−	0.999989i	$-0.501509\pi$
$211$	−2.00000			−0.00947867			−0.00473934	+	0.999989i	$0.501509\pi$
$212$		0			0
$213$	−250.440	+	280.000i	−1.17577	+	1.31455i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		−	108.000i		−	0.497696i
$218$		0			0
$219$	148.000	−	165.469i	0.675799	−	0.755566i
$220$		0			0
$221$			71.5542i			0.323775i
$222$		0			0
$223$		−	86.0000i		−	0.385650i	−0.981233	−	0.192825i	$-0.938235\pi$
$223$		−	86.0000i		−	0.385650i	0.981233	−	0.192825i	$-0.0617650\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	58.1378			0.256114			0.128057	−	0.991767i	$-0.459126\pi$
$227$	58.1378			0.256114			0.128057	+	0.991767i	$0.459126\pi$
$228$		0			0
$229$	282.000			1.23144			0.615721	−	0.787965i	$-0.288865\pi$
$229$	282.000			1.23144			0.615721	+	0.787965i	$0.288865\pi$
$230$		0			0
$231$	−60.0000	−	53.6656i	−0.259740	−	0.232319i
$232$		0			0
$233$	362.243			1.55469			0.777346	−	0.629074i	$-0.216566\pi$
$233$	362.243			1.55469			0.777346	+	0.629074i	$0.216566\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	308.577	+	276.000i	1.30201	+	1.16456i
$238$		0			0
$239$		−	250.440i		−	1.04786i	−0.851760	−	0.523932i	$-0.824464\pi$
$239$		−	250.440i		−	1.04786i	0.851760	−	0.523932i	$-0.175536\pi$
$240$		0			0
$241$	262.000			1.08714			0.543568	−	0.839365i	$-0.317073\pi$
$241$	262.000			1.08714			0.543568	+	0.839365i	$0.317073\pi$
$242$		0			0
$243$	−212.426	−	118.000i	−0.874183	−	0.485597i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		−	32.0000i		−	0.129555i
$248$		0			0
$249$	−210.000	−	187.830i	−0.843373	−	0.754336i
$250$		0			0
$251$		−	469.574i		−	1.87081i	−0.353573	−	0.935407i	$-0.615033\pi$
$251$		−	469.574i		−	1.87081i	0.353573	−	0.935407i	$-0.384967\pi$
$252$		0			0
$253$		−	60.0000i		−	0.237154i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	201.246			0.783059			0.391529	−	0.920166i	$-0.371946\pi$
$257$	201.246			0.783059			0.391529	+	0.920166i	$0.371946\pi$
$258$		0			0
$259$	96.0000			0.370656
$260$		0			0
$261$	−280.000	+	31.3050i	−1.07280	+	0.119942i
$262$		0			0
$263$	−58.1378			−0.221056			−0.110528	−	0.993873i	$-0.535254\pi$
$263$	−58.1378			−0.221056			−0.110528	+	0.993873i	$0.535254\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	214.663	−	240.000i	0.803979	−	0.898876i
$268$		0			0
$269$			371.187i			1.37988i	0.723867	+	0.689939i	$0.242363\pi$
$269$			371.187i			1.37988i	−0.723867	+	0.689939i	$0.757637\pi$
$270$		0			0
$271$	−82.0000			−0.302583			−0.151292	−	0.988489i	$-0.548343\pi$
$271$	−82.0000			−0.302583			−0.151292	+	0.988489i	$0.548343\pi$
$272$		0			0
$273$	214.663	+	192.000i	0.786310	+	0.703297i
$274$		0			0
$275$		0			0
$276$		0			0
$277$		−	24.0000i		−	0.0866426i	−0.999061	−	0.0433213i	$-0.986206\pi$
$277$		−	24.0000i		−	0.0866426i	0.999061	−	0.0433213i	$-0.0137939\pi$
$278$		0			0
$279$	18.0000	+	160.997i	0.0645161	+	0.577050i
$280$		0			0
$281$		−	187.830i		−	0.668433i	−0.942496	−	0.334217i	$-0.891528\pi$
$281$		−	187.830i		−	0.668433i	0.942496	−	0.334217i	$-0.108472\pi$
$282$		0			0
$283$			144.000i			0.508834i	0.967095	+	0.254417i	$0.0818836\pi$
$283$			144.000i			0.508834i	−0.967095	+	0.254417i	$0.918116\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	375.659			1.30892
$288$		0			0
$289$	−269.000			−0.930796
$290$		0			0
$291$	−332.000	+	371.187i	−1.14089	+	1.27556i
$292$		0			0
$293$	−469.574			−1.60264			−0.801321	−	0.598234i	$-0.795869\pi$
$293$	−469.574			−1.60264			−0.801321	+	0.598234i	$0.795869\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	98.3870	+	70.0000i	0.331269	+	0.235690i
$298$		0			0
$299$			214.663i			0.717935i
$300$		0			0
$301$	−96.0000			−0.318937
$302$		0			0
$303$	−134.164	+	150.000i	−0.442786	+	0.495050i
$304$		0			0
$305$		0			0
$306$		0			0
$307$			184.000i			0.599349i	0.954042	+	0.299674i	$0.0968780\pi$
$307$			184.000i			0.599349i	−0.954042	+	0.299674i	$0.903122\pi$
$308$		0			0
$309$	52.0000	−	58.1378i	0.168285	−	0.188148i
$310$		0			0
$311$		−	160.997i		−	0.517675i	−0.965921	−	0.258837i	$-0.916661\pi$
$311$		−	160.997i		−	0.517675i	0.965921	−	0.258837i	$-0.0833394\pi$
$312$		0			0
$313$		−	394.000i		−	1.25879i	−0.777087	−	0.629393i	$-0.783304\pi$
$313$		−	394.000i		−	1.25879i	0.777087	−	0.629393i	$-0.216696\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	451.686			1.42488			0.712438	−	0.701735i	$-0.247591\pi$
$317$	451.686			1.42488			0.712438	+	0.701735i	$0.247591\pi$
$318$		0			0
$319$	140.000			0.438871
$320$		0			0
$321$	−450.000	−	402.492i	−1.40187	−	1.25387i
$322$		0			0
$323$	−8.94427			−0.0276912
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−84.9706	−	76.0000i	−0.259849	−	0.232416i
$328$		0			0
$329$		−	295.161i		−	0.897146i
$330$		0			0
$331$	198.000			0.598187			0.299094	−	0.954224i	$-0.403316\pi$
$331$	198.000			0.598187			0.299094	+	0.954224i	$0.403316\pi$
$332$		0			0
$333$	−143.108	+	16.0000i	−0.429755	+	0.0480480i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	394.000i		−	1.16914i	−0.811343	−	0.584570i	$-0.801263\pi$
$337$		−	394.000i		−	1.16914i	0.811343	−	0.584570i	$-0.198737\pi$
$338$		0			0
$339$	−70.0000	−	62.6099i	−0.206490	−	0.184690i
$340$		0			0
$341$		−	80.4984i		−	0.236066i
$342$		0			0
$343$		−	372.000i		−	1.08455i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	183.358			0.528408			0.264204	−	0.964467i	$-0.414891\pi$
$347$	183.358			0.528408			0.264204	+	0.964467i	$0.414891\pi$
$348$		0			0
$349$	362.000			1.03725			0.518625	−	0.855002i	$-0.326444\pi$
$349$	362.000			1.03725			0.518625	+	0.855002i	$0.326444\pi$
$350$		0			0
$351$	−352.000	−	250.440i	−1.00285	−	0.713503i
$352$		0			0
$353$	308.577			0.874157			0.437078	−	0.899423i	$-0.356013\pi$
$353$	308.577			0.874157			0.437078	+	0.899423i	$0.356013\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	53.6656	−	60.0000i	0.150324	−	0.168067i
$358$		0			0
$359$			295.161i			0.822175i	0.911596	+	0.411088i	$0.134851\pi$
$359$			295.161i			0.822175i	−0.911596	+	0.411088i	$0.865149\pi$
$360$		0			0
$361$	−357.000			−0.988920
$362$		0			0
$363$	225.843	+	202.000i	0.622157	+	0.556474i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		−	186.000i		−	0.506812i	−0.967360	−	0.253406i	$-0.918449\pi$
$367$		−	186.000i		−	0.506812i	0.967360	−	0.253406i	$-0.0815509\pi$
$368$		0			0
$369$	−560.000	+	62.6099i	−1.51762	+	0.169675i
$370$		0			0
$371$			26.8328i			0.0723256i
$372$		0			0
$373$		−	44.0000i		−	0.117962i	−0.998259	−	0.0589812i	$-0.981215\pi$
$373$		−	44.0000i		−	0.117962i	0.998259	−	0.0589812i	$-0.0187852\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−500.879			−1.32859
$378$		0			0
$379$	−362.000			−0.955145			−0.477573	−	0.878592i	$-0.658483\pi$
$379$	−362.000			−0.955145			−0.477573	+	0.878592i	$0.658483\pi$
$380$		0			0
$381$	52.0000	−	58.1378i	0.136483	−	0.152593i
$382$		0			0
$383$	−362.243			−0.945804			−0.472902	−	0.881115i	$-0.656794\pi$
$383$	−362.243			−0.945804			−0.472902	+	0.881115i	$0.656794\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	143.108	−	16.0000i	0.369789	−	0.0413437i
$388$		0			0
$389$		−	442.741i		−	1.13815i	−0.822285	−	0.569076i	$-0.807301\pi$
$389$		−	442.741i		−	1.13815i	0.822285	−	0.569076i	$-0.192699\pi$
$390$		0			0
$391$	60.0000			0.153453
$392$		0			0
$393$	26.8328	−	30.0000i	0.0682769	−	0.0763359i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	124.000i		−	0.312343i	−0.987730	−	0.156171i	$-0.950085\pi$
$397$		−	124.000i		−	0.312343i	0.987730	−	0.156171i	$-0.0499152\pi$
$398$		0			0
$399$	−24.0000	+	26.8328i	−0.0601504	+	0.0672502i
$400$		0			0
$401$		−	268.328i		−	0.669148i	−0.942370	−	0.334574i	$-0.891408\pi$
$401$		−	268.328i		−	0.669148i	0.942370	−	0.334574i	$-0.108592\pi$
$402$		0			0
$403$			288.000i			0.714640i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	71.5542			0.175809
$408$		0			0
$409$	−458.000			−1.11980			−0.559902	−	0.828559i	$-0.689161\pi$
$409$	−458.000			−1.11980			−0.559902	+	0.828559i	$0.689161\pi$
$410$		0			0
$411$	−270.000	−	241.495i	−0.656934	−	0.587580i
$412$		0			0
$413$	26.8328			0.0649705
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−183.358	−	164.000i	−0.439706	−	0.393285i
$418$		0			0
$419$		−	594.794i		−	1.41956i	−0.704425	−	0.709778i	$-0.748795\pi$
$419$		−	594.794i		−	1.41956i	0.704425	−	0.709778i	$-0.251205\pi$
$420$		0			0
$421$	562.000			1.33492			0.667458	−	0.744647i	$-0.267382\pi$
$421$	562.000			1.33492			0.667458	+	0.744647i	$0.267382\pi$
$422$		0			0
$423$	49.1935	+	440.000i	0.116297	+	1.04019i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		−	492.000i		−	1.15222i
$428$		0			0
$429$	160.000	+	143.108i	0.372960	+	0.333586i
$430$		0			0
$431$		−	348.827i		−	0.809342i	−0.914462	−	0.404671i	$-0.867386\pi$
$431$		−	348.827i		−	0.809342i	0.914462	−	0.404671i	$-0.132614\pi$
$432$		0			0
$433$			226.000i			0.521940i	0.965347	+	0.260970i	$0.0840424\pi$
$433$			226.000i			0.521940i	−0.965347	+	0.260970i	$0.915958\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−26.8328			−0.0614023
$438$		0			0
$439$	−2.00000			−0.00455581			−0.00227790	−	0.999997i	$-0.500725\pi$
$439$	−2.00000			−0.00455581			−0.00227790	+	0.999997i	$0.500725\pi$
$440$		0			0
$441$	13.0000	+	116.276i	0.0294785	+	0.263663i
$442$		0			0
$443$	201.246			0.454280			0.227140	−	0.973862i	$-0.427062\pi$
$443$	201.246			0.454280			0.227140	+	0.973862i	$0.427062\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−223.607	+	250.000i	−0.500239	+	0.559284i
$448$		0			0
$449$		−	313.050i		−	0.697215i	−0.937269	−	0.348607i	$-0.886655\pi$
$449$		−	313.050i		−	0.697215i	0.937269	−	0.348607i	$-0.113345\pi$
$450$		0			0
$451$	280.000			0.620843
$452$		0			0
$453$	353.299	+	316.000i	0.779909	+	0.697572i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	334.000i		−	0.730853i	−0.930840	−	0.365427i	$-0.880923\pi$
$457$		−	334.000i		−	0.730853i	0.930840	−	0.365427i	$-0.119077\pi$
$458$		0			0
$459$	−70.0000	+	98.3870i	−0.152505	+	0.214351i
$460$		0			0
$461$			93.9149i			0.203720i	0.994799	+	0.101860i	$0.0324794\pi$
$461$			93.9149i			0.203720i	−0.994799	+	0.101860i	$0.967521\pi$
$462$		0			0
$463$		−	366.000i		−	0.790497i	−0.918574	−	0.395248i	$-0.870659\pi$
$463$		−	366.000i		−	0.790497i	0.918574	−	0.395248i	$-0.129341\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−451.686			−0.967207			−0.483604	−	0.875287i	$-0.660672\pi$
$467$	−451.686			−0.967207			−0.483604	+	0.875287i	$0.660672\pi$
$468$		0			0
$469$	144.000			0.307036
$470$		0			0
$471$	328.000	−	366.715i	0.696391	−	0.778588i
$472$		0			0
$473$	−71.5542			−0.151277
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−4.47214	−	40.0000i	−0.00937555	−	0.0838574i
$478$		0			0
$479$		−	590.322i		−	1.23240i	−0.787588	−	0.616202i	$-0.788670\pi$
$479$		−	590.322i		−	1.23240i	0.787588	−	0.616202i	$-0.211330\pi$
$480$		0			0
$481$	−256.000			−0.532225
$482$		0			0
$483$	160.997	−	180.000i	0.333327	−	0.372671i
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	886.000i		−	1.81930i	−0.415374	−	0.909651i	$-0.636349\pi$
$487$		−	886.000i		−	1.81930i	0.415374	−	0.909651i	$-0.363651\pi$
$488$		0			0
$489$	472.000	−	527.712i	0.965235	−	1.07917i
$490$		0			0
$491$			406.964i			0.828848i	0.910084	+	0.414424i	$0.136017\pi$
$491$			406.964i			0.828848i	−0.910084	+	0.414424i	$0.863983\pi$
$492$		0			0
$493$			140.000i			0.283976i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	751.319			1.51171
$498$		0			0
$499$	−2.00000			−0.00400802			−0.00200401	−	0.999998i	$-0.500638\pi$
$499$	−2.00000			−0.00400802			−0.00200401	+	0.999998i	$0.500638\pi$
$500$		0			0
$501$	−210.000	−	187.830i	−0.419162	−	0.374910i
$502$		0			0
$503$	−219.135			−0.435655			−0.217828	−	0.975987i	$-0.569897\pi$
$503$	−219.135			−0.435655			−0.217828	+	0.975987i	$0.569897\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−194.538	−	174.000i	−0.383704	−	0.343195i
$508$		0			0
$509$			800.512i			1.57272i	0.617771	+	0.786358i	$0.288036\pi$
$509$			800.512i			1.57272i	−0.617771	+	0.786358i	$0.711964\pi$
$510$		0			0
$511$	−444.000			−0.868885
$512$		0			0
$513$	31.3050	−	44.0000i	0.0610233	−	0.0857700i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	220.000i		−	0.425532i
$518$		0			0
$519$	−30.0000	−	26.8328i	−0.0578035	−	0.0517010i
$520$		0			0
$521$		−	527.712i		−	1.01288i	−0.862274	−	0.506441i	$-0.830961\pi$
$521$		−	527.712i		−	1.01288i	0.862274	−	0.506441i	$-0.169039\pi$
$522$		0			0
$523$		−	376.000i		−	0.718929i	−0.933159	−	0.359465i	$-0.882959\pi$
$523$		−	376.000i		−	0.718929i	0.933159	−	0.359465i	$-0.117041\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	80.4984			0.152748
$528$		0			0
$529$	−349.000			−0.659735
$530$		0			0
$531$	−40.0000	+	4.47214i	−0.0753296	+	0.00842210i
$532$		0			0
$533$	−1001.76			−1.87947
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−384.604	+	430.000i	−0.716208	+	0.800745i
$538$		0			0
$539$		−	58.1378i		−	0.107862i
$540$		0			0
$541$	−198.000			−0.365989			−0.182994	−	0.983114i	$-0.558579\pi$
$541$	−198.000			−0.365989			−0.182994	+	0.983114i	$0.558579\pi$
$542$		0			0
$543$	4.47214	+	4.00000i	0.00823598	+	0.00736648i
$544$		0			0
$545$		0			0
$546$		0			0
$547$			1024.00i			1.87203i	0.351961	+	0.936015i	$0.385515\pi$
$547$			1024.00i			1.87203i	−0.351961	+	0.936015i	$0.614485\pi$
$548$		0			0
$549$	82.0000	+	733.430i	0.149362	+	1.33594i
$550$		0			0
$551$		−	62.6099i		−	0.113630i
$552$		0			0
$553$		−	828.000i		−	1.49729i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	67.0820			0.120435			0.0602173	−	0.998185i	$-0.480821\pi$
$557$	67.0820			0.120435			0.0602173	+	0.998185i	$0.480821\pi$
$558$		0			0
$559$	256.000			0.457961
$560$		0			0
$561$	40.0000	−	44.7214i	0.0713012	−	0.0797172i
$562$		0			0
$563$	254.912			0.452774			0.226387	−	0.974037i	$-0.427309\pi$
$563$	254.912			0.452774			0.226387	+	0.974037i	$0.427309\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	107.331	+	474.000i	0.189297	+	0.835979i
$568$		0			0
$569$		−	858.650i		−	1.50905i	−0.656271	−	0.754526i	$-0.727867\pi$
$569$		−	858.650i		−	1.50905i	0.656271	−	0.754526i	$-0.272133\pi$
$570$		0			0
$571$	−962.000			−1.68476			−0.842382	−	0.538881i	$-0.818847\pi$
$571$	−962.000			−1.68476			−0.842382	+	0.538881i	$0.818847\pi$
$572$		0			0
$573$	−411.437	+	460.000i	−0.718039	+	0.802792i
$574$		0			0
$575$		0			0
$576$		0			0
$577$			886.000i			1.53553i	0.640732	+	0.767764i	$0.278631\pi$
$577$			886.000i			1.53553i	−0.640732	+	0.767764i	$0.721369\pi$
$578$		0			0
$579$	428.000	−	478.519i	0.739206	−	0.826457i
$580$		0			0
$581$			563.489i			0.969861i
$582$		0			0
$583$			20.0000i			0.0343053i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	657.404			1.11994			0.559969	−	0.828513i	$-0.310813\pi$
$587$	657.404			1.11994			0.559969	+	0.828513i	$0.310813\pi$
$588$		0			0
$589$	−36.0000			−0.0611205
$590$		0			0
$591$	−210.000	−	187.830i	−0.355330	−	0.317817i
$592$		0			0
$593$	111.803			0.188539			0.0942693	−	0.995547i	$-0.469949\pi$
$593$	111.803			0.188539			0.0942693	+	0.995547i	$0.469949\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−541.128	−	484.000i	−0.906413	−	0.810720i
$598$		0			0
$599$			223.607i			0.373300i	0.982426	+	0.186650i	$0.0597631\pi$
$599$			223.607i			0.373300i	−0.982426	+	0.186650i	$0.940237\pi$
$600$		0			0
$601$	2.00000			0.00332779			0.00166389	−	0.999999i	$-0.499470\pi$
$601$	2.00000			0.00332779			0.00166389	+	0.999999i	$0.499470\pi$
$602$		0			0
$603$	−214.663	+	24.0000i	−0.355991	+	0.0398010i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		−	506.000i		−	0.833608i	−0.908996	−	0.416804i	$-0.863150\pi$
$607$		−	506.000i		−	0.833608i	0.908996	−	0.416804i	$-0.136850\pi$
$608$		0			0
$609$	420.000	+	375.659i	0.689655	+	0.616846i
$610$		0			0
$611$			787.096i			1.28821i
$612$		0			0
$613$			556.000i			0.907015i	0.891253	+	0.453507i	$0.149827\pi$
$613$			556.000i			0.907015i	−0.891253	+	0.453507i	$0.850173\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	93.9149			0.152212			0.0761060	−	0.997100i	$-0.475751\pi$
$617$	93.9149			0.152212			0.0761060	+	0.997100i	$0.475751\pi$
$618$		0			0
$619$	−802.000			−1.29564			−0.647819	−	0.761794i	$-0.724319\pi$
$619$	−802.000			−1.29564			−0.647819	+	0.761794i	$0.724319\pi$
$620$		0			0
$621$	−210.000	+	295.161i	−0.338164	+	0.475299i
$622$		0			0
$623$	−643.988			−1.03369
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−17.8885	+	20.0000i	−0.0285304	+	0.0318979i
$628$		0			0
$629$			71.5542i			0.113759i
$630$		0			0
$631$	698.000			1.10618			0.553090	−	0.833121i	$-0.313448\pi$
$631$	698.000			1.10618			0.553090	+	0.833121i	$0.313448\pi$
$632$		0			0
$633$	−4.47214	−	4.00000i	−0.00706499	−	0.00631912i
$634$		0			0
$635$		0			0
$636$		0			0
$637$			208.000i			0.326531i
$638$		0			0
$639$	−1120.00	+	125.220i	−1.75274	+	0.195962i
$640$		0			0
$641$			912.316i			1.42327i	0.702550	+	0.711635i	$0.252045\pi$
$641$			912.316i			1.42327i	−0.702550	+	0.711635i	$0.747955\pi$
$642$		0			0
$643$		−	156.000i		−	0.242613i	−0.992615	−	0.121306i	$-0.961292\pi$
$643$		−	156.000i		−	0.242613i	0.992615	−	0.121306i	$-0.0387084\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−755.791			−1.16815			−0.584073	−	0.811701i	$-0.698542\pi$
$647$	−755.791			−1.16815			−0.584073	+	0.811701i	$0.698542\pi$
$648$		0			0
$649$	20.0000			0.0308166
$650$		0			0
$651$	216.000	−	241.495i	0.331797	−	0.370961i
$652$		0			0
$653$	487.463			0.746497			0.373249	−	0.927731i	$-0.378244\pi$
$653$	487.463			0.746497			0.373249	+	0.927731i	$0.378244\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	661.876	−	74.0000i	1.00742	−	0.112633i
$658$		0			0
$659$			406.964i			0.617548i	0.951135	+	0.308774i	$0.0999187\pi$
$659$			406.964i			0.617548i	−0.951135	+	0.308774i	$0.900081\pi$
$660$		0			0
$661$	682.000			1.03177			0.515885	−	0.856658i	$-0.327463\pi$
$661$	682.000			1.03177			0.515885	+	0.856658i	$0.327463\pi$
$662$		0			0
$663$	−143.108	+	160.000i	−0.215850	+	0.241327i
$664$		0			0
$665$		0			0
$666$		0			0
$667$			420.000i			0.629685i
$668$		0			0
$669$	172.000	−	192.302i	0.257100	−	0.287447i
$670$		0			0
$671$		−	366.715i		−	0.546520i
$672$		0			0
$673$		−	894.000i		−	1.32838i	−0.747564	−	0.664190i	$-0.768777\pi$
$673$		−	894.000i		−	1.32838i	0.747564	−	0.664190i	$-0.231223\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−550.073			−0.812515			−0.406258	−	0.913759i	$-0.633166\pi$
$677$	−550.073			−0.812515			−0.406258	+	0.913759i	$0.633166\pi$
$678$		0			0
$679$	996.000			1.46686
$680$		0			0
$681$	130.000	+	116.276i	0.190896	+	0.170742i
$682$		0			0
$683$	442.741			0.648231			0.324115	−	0.946018i	$-0.394933\pi$
$683$	442.741			0.648231			0.324115	+	0.946018i	$0.394933\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	630.571	+	564.000i	0.917862	+	0.820961i
$688$		0			0
$689$		−	71.5542i		−	0.103852i
$690$		0			0
$691$	758.000			1.09696			0.548480	−	0.836163i	$-0.315207\pi$
$691$	758.000			1.09696			0.548480	+	0.836163i	$0.315207\pi$
$692$		0			0
$693$	−26.8328	−	240.000i	−0.0387198	−	0.346320i
$694$		0			0
$695$		0			0
$696$		0			0
$697$			280.000i			0.401722i
$698$		0			0
$699$	810.000	+	724.486i	1.15880	+	1.03646i
$700$		0			0
$701$		−	782.624i		−	1.11644i	−0.829693	−	0.558220i	$-0.811485\pi$
$701$		−	782.624i		−	1.11644i	0.829693	−	0.558220i	$-0.188515\pi$
$702$		0			0
$703$		−	32.0000i		−	0.0455192i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	402.492			0.569296
$708$		0			0
$709$	2.00000			0.00282087			0.00141044	−	0.999999i	$-0.499551\pi$
$709$	2.00000			0.00282087			0.00141044	+	0.999999i	$0.499551\pi$
$710$		0			0
$711$	138.000	+	1234.31i	0.194093	+	1.73602i
$712$		0			0
$713$	241.495			0.338703
$714$		0			0
$715$		0			0
$716$		0			0
$717$	500.879	−	560.000i	0.698576	−	0.781032i
$718$		0			0
$719$			858.650i			1.19423i	0.802156	+	0.597114i	$0.203686\pi$
$719$			858.650i			1.19423i	−0.802156	+	0.597114i	$0.796314\pi$
$720$		0			0
$721$	−156.000			−0.216366
$722$		0			0
$723$	585.850	+	524.000i	0.810304	+	0.724758i
$724$		0			0
$725$		0			0
$726$		0			0
$727$			674.000i			0.927098i	0.886071	+	0.463549i	$0.153424\pi$
$727$			674.000i			0.927098i	−0.886071	+	0.463549i	$0.846576\pi$
$728$		0			0
$729$	−239.000	−	688.709i	−0.327846	−	0.944731i
$730$		0			0
$731$		−	71.5542i		−	0.0978853i
$732$		0			0
$733$			656.000i			0.894952i	0.894296	+	0.447476i	$0.147677\pi$
$733$			656.000i			0.894952i	−0.894296	+	0.447476i	$0.852323\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	107.331			0.145633
$738$		0			0
$739$	598.000			0.809202			0.404601	−	0.914493i	$-0.367410\pi$
$739$	598.000			0.809202			0.404601	+	0.914493i	$0.367410\pi$
$740$		0			0
$741$	64.0000	−	71.5542i	0.0863698	−	0.0965643i
$742$		0			0
$743$	782.624			1.05333			0.526665	−	0.850073i	$-0.323442\pi$
$743$	782.624			1.05333			0.526665	+	0.850073i	$0.323442\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−93.9149	−	840.000i	−0.125723	−	1.12450i
$748$		0			0
$749$			1207.48i			1.61212i
$750$		0			0
$751$	338.000			0.450067			0.225033	−	0.974351i	$-0.427751\pi$
$751$	338.000			0.450067			0.225033	+	0.974351i	$0.427751\pi$
$752$		0			0
$753$	939.149	−	1050.00i	1.24721	−	1.39442i
$754$		0			0
$755$		0			0
$756$		0			0
$757$			656.000i			0.866579i	0.901255	+	0.433289i	$0.142647\pi$
$757$			656.000i			0.866579i	−0.901255	+	0.433289i	$0.857353\pi$
$758$		0			0
$759$	120.000	−	134.164i	0.158103	−	0.176764i
$760$		0			0
$761$			295.161i			0.387859i	0.981015	+	0.193930i	$0.0621234\pi$
$761$			295.161i			0.387859i	−0.981015	+	0.193930i	$0.937877\pi$
$762$		0			0
$763$			228.000i			0.298820i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−71.5542			−0.0932910
$768$		0			0
$769$	82.0000			0.106632			0.0533160	−	0.998578i	$-0.483021\pi$
$769$	82.0000			0.106632			0.0533160	+	0.998578i	$0.483021\pi$
$770$		0			0
$771$	450.000	+	402.492i	0.583658	+	0.522039i
$772$		0			0
$773$	1059.90			1.37115			0.685573	−	0.728004i	$-0.259552\pi$
$773$	1059.90			1.37115			0.685573	+	0.728004i	$0.259552\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	214.663	+	192.000i	0.276271	+	0.247104i
$778$		0			0
$779$		−	125.220i		−	0.160744i
$780$		0			0
$781$	560.000			0.717029
$782$		0			0
$783$	−688.709	−	490.000i	−0.879577	−	0.625798i
$784$		0			0
$785$		0			0
$786$		0			0
$787$		−	536.000i		−	0.681067i	−0.940232	−	0.340534i	$-0.889392\pi$
$787$		−	536.000i		−	0.681067i	0.940232	−	0.340534i	$-0.110608\pi$
$788$		0			0
$789$	−130.000	−	116.276i	−0.164766	−	0.147371i
$790$		0			0
$791$			187.830i			0.237459i
$792$		0			0
$793$			1312.00i			1.65448i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−406.964			−0.510620			−0.255310	−	0.966859i	$-0.582178\pi$
$797$	−406.964			−0.510620			−0.255310	+	0.966859i	$0.582178\pi$
$798$		0			0
$799$	220.000			0.275344
$800$		0			0
$801$	960.000	−	107.331i	1.19850	−	0.133997i
$802$		0			0
$803$	−330.938			−0.412127
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−742.375	+	830.000i	−0.919919	+	1.02850i
$808$		0			0
$809$			1091.20i			1.34883i	0.738354	+	0.674414i	$0.235603\pi$
$809$			1091.20i			1.34883i	−0.738354	+	0.674414i	$0.764397\pi$
$810$		0			0
$811$	558.000			0.688039			0.344020	−	0.938962i	$-0.388211\pi$
$811$	558.000			0.688039			0.344020	+	0.938962i	$0.388211\pi$
$812$		0			0
$813$	−183.358	−	164.000i	−0.225532	−	0.201722i
$814$		0			0
$815$		0			0
$816$		0			0
$817$			32.0000i			0.0391677i
$818$		0			0
$819$	96.0000	+	858.650i	0.117216	+	1.04841i
$820$		0			0
$821$			389.076i			0.473905i	0.971521	+	0.236952i	$0.0761485\pi$
$821$			389.076i			0.473905i	−0.971521	+	0.236952i	$0.923851\pi$
$822$		0			0
$823$			214.000i			0.260024i	0.991512	+	0.130012i	$0.0415016\pi$
$823$			214.000i			0.260024i	−0.991512	+	0.130012i	$0.958498\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−31.3050			−0.0378536			−0.0189268	−	0.999821i	$-0.506025\pi$
$827$	−31.3050			−0.0378536			−0.0189268	+	0.999821i	$0.506025\pi$
$828$		0			0
$829$	−318.000			−0.383595			−0.191797	−	0.981435i	$-0.561432\pi$
$829$	−318.000			−0.383595			−0.191797	+	0.981435i	$0.561432\pi$
$830$		0			0
$831$	48.0000	−	53.6656i	0.0577617	−	0.0645796i
$832$		0			0
$833$	58.1378			0.0697932
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−281.745	+	396.000i	−0.336612	+	0.473118i
$838$		0			0
$839$			62.6099i			0.0746244i	0.999304	+	0.0373122i	$0.0118796\pi$
$839$			62.6099i			0.0746244i	−0.999304	+	0.0373122i	$0.988120\pi$
$840$		0			0
$841$	−139.000			−0.165279
$842$		0			0
$843$	375.659	−	420.000i	0.445622	−	0.498221i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		−	606.000i		−	0.715466i
$848$		0			0
$849$	−288.000	+	321.994i	−0.339223	+	0.379262i
$850$		0			0
$851$			214.663i			0.252247i
$852$		0			0
$853$		−	684.000i		−	0.801876i	−0.916105	−	0.400938i	$-0.868684\pi$
$853$		−	684.000i		−	0.801876i	0.916105	−	0.400938i	$-0.131316\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1498.17			−1.74815			−0.874076	−	0.485790i	$-0.838532\pi$
$857$	−1498.17			−1.74815			−0.874076	+	0.485790i	$0.838532\pi$
$858$		0			0
$859$	−842.000			−0.980210			−0.490105	−	0.871664i	$-0.663041\pi$
$859$	−842.000			−0.980210			−0.490105	+	0.871664i	$0.663041\pi$
$860$		0			0
$861$	840.000	+	751.319i	0.975610	+	0.872612i
$862$		0			0
$863$	1015.17			1.17633			0.588166	−	0.808740i	$-0.299850\pi$
$863$	1015.17			1.17633			0.588166	+	0.808740i	$0.299850\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−601.502	−	538.000i	−0.693774	−	0.620531i
$868$		0			0
$869$		−	617.155i		−	0.710190i
$870$		0			0
$871$	−384.000			−0.440873
$872$		0			0
$873$	−1484.75	+	166.000i	−1.70074	+	0.190149i
$874$		0			0
$875$		0			0
$876$		0			0
$877$			156.000i			0.177879i	0.996037	+	0.0889396i	$0.0283478\pi$
$877$			156.000i			0.177879i	−0.996037	+	0.0889396i	$0.971652\pi$
$878$		0			0
$879$	−1050.00	−	939.149i	−1.19454	−	1.06843i
$880$		0			0
$881$			125.220i			0.142134i	0.997472	+	0.0710669i	$0.0226404\pi$
$881$			125.220i			0.142134i	−0.997472	+	0.0710669i	$0.977360\pi$
$882$		0			0
$883$			964.000i			1.09173i	0.837872	+	0.545866i	$0.183799\pi$
$883$			964.000i			1.09173i	−0.837872	+	0.545866i	$0.816201\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−952.565			−1.07392			−0.536959	−	0.843608i	$-0.680427\pi$
$887$	−952.565			−1.07392			−0.536959	+	0.843608i	$0.680427\pi$
$888$		0			0
$889$	−156.000			−0.175478
$890$		0			0
$891$	80.0000	+	353.299i	0.0897868	+	0.396519i
$892$		0			0
$893$	−98.3870			−0.110176
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−429.325	+	480.000i	−0.478623	+	0.535117i
$898$		0			0
$899$			563.489i			0.626795i
$900$		0			0
$901$	−20.0000			−0.0221976
$902$		0			0
$903$	−214.663	−	192.000i	−0.237722	−	0.212625i
$904$		0			0
$905$		0			0
$906$		0			0
$907$			1284.00i			1.41566i	0.706385	+	0.707828i	$0.250325\pi$
$907$			1284.00i			1.41566i	−0.706385	+	0.707828i	$0.749675\pi$
$908$		0			0
$909$	−600.000	+	67.0820i	−0.660066	+	0.0737976i
$910$		0			0
$911$			62.6099i			0.0687266i	0.999409	+	0.0343633i	$0.0109403\pi$
$911$			62.6099i			0.0687266i	−0.999409	+	0.0343633i	$0.989060\pi$
$912$		0			0
$913$			420.000i			0.460022i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−80.4984			−0.0877846
$918$		0			0
$919$	418.000			0.454842			0.227421	−	0.973797i	$-0.426971\pi$
$919$	418.000			0.454842			0.227421	+	0.973797i	$0.426971\pi$
$920$		0			0
$921$	−368.000	+	411.437i	−0.399566	+	0.446728i
$922$		0			0
$923$	−2003.52			−2.17066
$924$		0			0
$925$		0			0
$926$		0			0
$927$	232.551	−	26.0000i	0.250864	−	0.0280475i
$928$		0			0
$929$		−	169.941i		−	0.182929i	−0.995808	−	0.0914646i	$-0.970845\pi$
$929$		−	169.941i		−	0.182929i	0.995808	−	0.0914646i	$-0.0291548\pi$
$930$		0			0
$931$	−26.0000			−0.0279270
$932$		0			0
$933$	321.994	−	360.000i	0.345117	−	0.385852i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	534.000i		−	0.569904i	−0.958542	−	0.284952i	$-0.908022\pi$
$937$		−	534.000i		−	0.569904i	0.958542	−	0.284952i	$-0.0919777\pi$
$938$		0			0
$939$	788.000	−	881.011i	0.839191	−	0.938244i
$940$		0			0
$941$			129.692i			0.137824i	0.997623	+	0.0689118i	$0.0219527\pi$
$941$			129.692i			0.137824i	−0.997623	+	0.0689118i	$0.978047\pi$
$942$		0			0
$943$			840.000i			0.890774i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1516.05			−1.60090			−0.800451	−	0.599398i	$-0.795407\pi$
$947$	−1516.05			−1.60090			−0.800451	+	0.599398i	$0.795407\pi$
$948$		0			0
$949$	1184.00			1.24763
$950$		0			0
$951$	1010.00	+	903.371i	1.06204	+	0.949917i
$952$		0			0
$953$	−406.964			−0.427035			−0.213518	−	0.976939i	$-0.568492\pi$
$953$	−406.964			−0.427035			−0.213518	+	0.976939i	$0.568492\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	313.050	+	280.000i	0.327115	+	0.292581i
$958$		0			0
$959$			724.486i			0.755460i
$960$		0			0
$961$	−637.000			−0.662851
$962$		0			0
$963$	−201.246	−	1800.00i	−0.208978	−	1.86916i
$964$		0			0
$965$		0			0
$966$		0			0
$967$			674.000i			0.697001i	0.937309	+	0.348501i	$0.113309\pi$
$967$			674.000i			0.697001i	−0.937309	+	0.348501i	$0.886691\pi$
$968$		0			0
$969$	−20.0000	−	17.8885i	−0.0206398	−	0.0184608i
$970$		0			0
$971$		−	1328.22i		−	1.36789i	−0.729532	−	0.683947i	$-0.760262\pi$
$971$		−	1328.22i		−	1.36789i	0.729532	−	0.683947i	$-0.239738\pi$
$972$		0			0
$973$			492.000i			0.505653i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−371.187			−0.379926			−0.189963	−	0.981791i	$-0.560837\pi$
$977$	−371.187			−0.379926			−0.189963	+	0.981791i	$0.560837\pi$
$978$		0			0
$979$	−480.000			−0.490296
$980$		0			0
$981$	−38.0000	−	339.882i	−0.0387360	−	0.346465i
$982$		0			0
$983$	442.741			0.450398			0.225199	−	0.974313i	$-0.427697\pi$
$983$	442.741			0.450398			0.225199	+	0.974313i	$0.427697\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	590.322	−	660.000i	0.598097	−	0.668693i
$988$		0			0
$989$		−	214.663i		−	0.217050i
$990$		0			0
$991$	−962.000			−0.970737			−0.485368	−	0.874310i	$-0.661314\pi$
$991$	−962.000			−0.970737			−0.485368	+	0.874310i	$0.661314\pi$
$992$		0			0
$993$	442.741	+	396.000i	0.445862	+	0.398792i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		−	24.0000i		−	0.0240722i	−0.999928	−	0.0120361i	$-0.996169\pi$
$997$		−	24.0000i		−	0.0240722i	0.999928	−	0.0120361i	$-0.00383130\pi$
$998$		0			0
$999$	−352.000	−	250.440i	−0.352352	−	0.250690i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.3.c.f.449.4		4
3.2	odd	2	inner	1200.3.c.f.449.2		4
4.3	odd	2		75.3.d.b.74.1		4
5.2	odd	4		240.3.l.b.161.1		2
5.3	odd	4		1200.3.l.g.401.2		2
5.4	even	2	inner	1200.3.c.f.449.1		4
12.11	even	2		75.3.d.b.74.3		4
15.2	even	4		240.3.l.b.161.2		2
15.8	even	4		1200.3.l.g.401.1		2
15.14	odd	2	inner	1200.3.c.f.449.3		4
20.3	even	4		75.3.c.e.26.2		2
20.7	even	4		15.3.c.a.11.1	✓	2
20.19	odd	2		75.3.d.b.74.4		4
40.27	even	4		960.3.l.c.641.1		2
40.37	odd	4		960.3.l.b.641.2		2
60.23	odd	4		75.3.c.e.26.1		2
60.47	odd	4		15.3.c.a.11.2	yes	2
60.59	even	2		75.3.d.b.74.2		4
120.77	even	4		960.3.l.b.641.1		2
120.107	odd	4		960.3.l.c.641.2		2
180.7	even	12		405.3.i.b.296.1		4
180.47	odd	12		405.3.i.b.296.2		4
180.67	even	12		405.3.i.b.26.2		4
180.167	odd	12		405.3.i.b.26.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.3.c.a.11.1	✓	2	20.7	even	4
15.3.c.a.11.2	yes	2	60.47	odd	4
75.3.c.e.26.1		2	60.23	odd	4
75.3.c.e.26.2		2	20.3	even	4
75.3.d.b.74.1		4	4.3	odd	2
75.3.d.b.74.2		4	60.59	even	2
75.3.d.b.74.3		4	12.11	even	2
75.3.d.b.74.4		4	20.19	odd	2
240.3.l.b.161.1		2	5.2	odd	4
240.3.l.b.161.2		2	15.2	even	4
405.3.i.b.26.1		4	180.167	odd	12
405.3.i.b.26.2		4	180.67	even	12
405.3.i.b.296.1		4	180.7	even	12
405.3.i.b.296.2		4	180.47	odd	12
960.3.l.b.641.1		2	120.77	even	4
960.3.l.b.641.2		2	40.37	odd	4
960.3.l.c.641.1		2	40.27	even	4
960.3.l.c.641.2		2	120.107	odd	4
1200.3.c.f.449.1		4	5.4	even	2	inner
1200.3.c.f.449.2		4	3.2	odd	2	inner
1200.3.c.f.449.3		4	15.14	odd	2	inner
1200.3.c.f.449.4		4	1.1	even	1	trivial
1200.3.l.g.401.1		2	15.8	even	4
1200.3.l.g.401.2		2	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:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1200.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.23607 + 2.00000i) q^{3} -6.00000i q^{7} +(1.00000 + 8.94427i) q^{9} +O(q^{10})\)	\(q+(2.23607 + 2.00000i) q^{3} -6.00000i q^{7} +(1.00000 + 8.94427i) q^{9} -4.47214i q^{11} +16.0000i q^{13} +4.47214 q^{17} -2.00000 q^{19} +(12.0000 - 13.4164i) q^{21} +13.4164 q^{23} +(-15.6525 + 22.0000i) q^{27} +31.3050i q^{29} +18.0000 q^{31} +(8.94427 - 10.0000i) q^{33} +16.0000i q^{37} +(-32.0000 + 35.7771i) q^{39} +62.6099i q^{41} -16.0000i q^{43} +49.1935 q^{47} +13.0000 q^{49} +(10.0000 + 8.94427i) q^{51} -4.47214 q^{53} +(-4.47214 - 4.00000i) q^{57} +4.47214i q^{59} +82.0000 q^{61} +(53.6656 - 6.00000i) q^{63} +24.0000i q^{67} +(30.0000 + 26.8328i) q^{69} +125.220i q^{71} -74.0000i q^{73} -26.8328 q^{77} +138.000 q^{79} +(-79.0000 + 17.8885i) q^{81} -93.9149 q^{83} +(-62.6099 + 70.0000i) q^{87} -107.331i q^{89} +96.0000 q^{91} +(40.2492 + 36.0000i) q^{93} +166.000i q^{97} +(40.0000 - 4.47214i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^9	\( 4 q + 4 q^{9} - 8 q^{19} + 48 q^{21} + 72 q^{31} - 128 q^{39} + 52 q^{49} + 40 q^{51} + 328 q^{61} + 120 q^{69} + 552 q^{79} - 316 q^{81} + 384 q^{91} + 160 q^{99}+O(q^{100}) \) 4 * q + 4 * q^9 - 8 * q^19 + 48 * q^21 + 72 * q^31 - 128 * q^39 + 52 * q^49 + 40 * q^51 + 328 * q^61 + 120 * q^69 + 552 * q^79 - 316 * q^81 + 384 * q^91 + 160 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more