LMFDB - Embedded newform 1800.3.v.n.1657.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1800,3,Mod(793,1800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1800, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 0, 3]))

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

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

chi := DirichletCharacter("1800.793");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1800.v (of order $4$, degree $2$, 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:	$49.0464475849$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1657.1
Root			$1.22474 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	1800.1657
Dual form			1800.3.v.n.793.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.550510 + 0.550510i) q^{7} +O(q^{10})$	$q+(0.550510 + 0.550510i) q^{7} +1.55051 q^{11} +(-9.55051 + 9.55051i) q^{13} +(11.1464 + 11.1464i) q^{17} -12.6969i q^{19} +(-21.3485 + 21.3485i) q^{23} -44.0454i q^{29} -44.4949 q^{31} +(-20.6515 - 20.6515i) q^{37} +48.2929 q^{41} +(36.2929 - 36.2929i) q^{43} +(42.5403 + 42.5403i) q^{47} -48.3939i q^{49} +(-54.4949 + 54.4949i) q^{53} +47.4393i q^{59} -59.8888 q^{61} +(-81.2827 - 81.2827i) q^{67} -87.5959 q^{71} +(-75.9898 + 75.9898i) q^{73} +(0.853572 + 0.853572i) q^{77} -97.3031i q^{79} +(41.0556 - 41.0556i) q^{83} +52.2020i q^{89} -10.5153 q^{91} +(37.0000 + 37.0000i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{7}+O(q^{10}) $ 4 * q + 12 * q^7	$ 4 q + 12 q^{7} + 16 q^{11} - 48 q^{13} - 24 q^{17} - 56 q^{23} - 80 q^{31} - 112 q^{37} + 56 q^{41} + 8 q^{43} - 16 q^{47} - 120 q^{53} - 24 q^{61} + 8 q^{67} - 272 q^{71} - 108 q^{73} + 72 q^{77} + 272 q^{83} - 336 q^{91} + 148 q^{97}+O(q^{100}) $ 4 * q + 12 * q^7 + 16 * q^11 - 48 * q^13 - 24 * q^17 - 56 * q^23 - 80 * q^31 - 112 * q^37 + 56 * q^41 + 8 * q^43 - 16 * q^47 - 120 * q^53 - 24 * q^61 + 8 * q^67 - 272 * q^71 - 108 * q^73 + 72 * q^77 + 272 * q^83 - 336 * q^91 + 148 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1001$	$1351$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$	$1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	0.550510	+	0.550510i	0.0786443	+	0.0786443i	0.745335	−	0.666690i	$-0.232290\pi$
$7$	0.550510	+	0.550510i	0.0786443	+	0.0786443i	−0.666690	+	0.745335i	$0.732290\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.55051			0.140955			0.0704777	−	0.997513i	$-0.477548\pi$
$11$	1.55051			0.140955			0.0704777	+	0.997513i	$0.477548\pi$
$12$		0			0
$13$	−9.55051	+	9.55051i	−0.734655	+	0.734655i	−0.971538	−	0.236883i	$-0.923874\pi$
$13$	−9.55051	+	9.55051i	−0.734655	+	0.734655i	0.236883	+	0.971538i	$0.423874\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	11.1464	+	11.1464i	0.655672	+	0.655672i	0.954353	−	0.298681i	$-0.0965466\pi$
$17$	11.1464	+	11.1464i	0.655672	+	0.655672i	−0.298681	+	0.954353i	$0.596547\pi$
$18$		0			0
$19$		−	12.6969i		−	0.668260i	−0.942527	−	0.334130i	$-0.891558\pi$
$19$		−	12.6969i		−	0.668260i	0.942527	−	0.334130i	$-0.108442\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−21.3485	+	21.3485i	−0.928194	+	0.928194i	−0.997589	−	0.0693950i	$-0.977893\pi$
$23$	−21.3485	+	21.3485i	−0.928194	+	0.928194i	0.0693950	+	0.997589i	$0.477893\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	44.0454i		−	1.51881i	−0.650620	−	0.759404i	$-0.725491\pi$
$29$		−	44.0454i		−	1.51881i	0.650620	−	0.759404i	$-0.274509\pi$
$30$		0			0
$31$	−44.4949			−1.43532			−0.717660	−	0.696394i	$-0.754787\pi$
$31$	−44.4949			−1.43532			−0.717660	+	0.696394i	$0.754787\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−20.6515	−	20.6515i	−0.558149	−	0.558149i	0.370631	−	0.928780i	$-0.379142\pi$
$37$	−20.6515	−	20.6515i	−0.558149	−	0.558149i	−0.928780	+	0.370631i	$0.879142\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	48.2929			1.17787			0.588937	−	0.808179i	$-0.299546\pi$
$41$	48.2929			1.17787			0.588937	+	0.808179i	$0.299546\pi$
$42$		0			0
$43$	36.2929	−	36.2929i	0.844020	−	0.844020i	−0.145359	−	0.989379i	$-0.546434\pi$
$43$	36.2929	−	36.2929i	0.844020	−	0.844020i	0.989379	+	0.145359i	$0.0464337\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	42.5403	+	42.5403i	0.905113	+	0.905113i	0.995873	−	0.0907599i	$-0.0289296\pi$
$47$	42.5403	+	42.5403i	0.905113	+	0.905113i	−0.0907599	+	0.995873i	$0.528930\pi$
$48$		0			0
$49$		−	48.3939i		−	0.987630i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−54.4949	+	54.4949i	−1.02821	+	1.02821i	−0.0286151	+	0.999591i	$0.509110\pi$
$53$	−54.4949	+	54.4949i	−1.02821	+	1.02821i	−0.999591	+	0.0286151i	$0.990890\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$			47.4393i			0.804056i	0.915627	+	0.402028i	$0.131694\pi$
$59$			47.4393i			0.804056i	−0.915627	+	0.402028i	$0.868306\pi$
$60$		0			0
$61$	−59.8888			−0.981783			−0.490892	−	0.871221i	$-0.663329\pi$
$61$	−59.8888			−0.981783			−0.490892	+	0.871221i	$0.663329\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−81.2827	−	81.2827i	−1.21317	−	1.21317i	−0.969977	−	0.243197i	$-0.921804\pi$
$67$	−81.2827	−	81.2827i	−1.21317	−	1.21317i	−0.243197	−	0.969977i	$-0.578196\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−87.5959			−1.23375			−0.616873	−	0.787063i	$-0.711601\pi$
$71$	−87.5959			−1.23375			−0.616873	+	0.787063i	$0.711601\pi$
$72$		0			0
$73$	−75.9898	+	75.9898i	−1.04096	+	1.04096i	−0.0418314	+	0.999125i	$0.513319\pi$
$73$	−75.9898	+	75.9898i	−1.04096	+	1.04096i	−0.999125	+	0.0418314i	$0.986681\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.853572	+	0.853572i	0.0110853	+	0.0110853i
$78$		0			0
$79$		−	97.3031i		−	1.23168i	−0.787870	−	0.615842i	$-0.788816\pi$
$79$		−	97.3031i		−	1.23168i	0.787870	−	0.615842i	$-0.211184\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	41.0556	−	41.0556i	0.494646	−	0.494646i	−0.415120	−	0.909766i	$-0.636261\pi$
$83$	41.0556	−	41.0556i	0.494646	−	0.494646i	0.909766	+	0.415120i	$0.136261\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$			52.2020i			0.586540i	0.956030	+	0.293270i	$0.0947435\pi$
$89$			52.2020i			0.586540i	−0.956030	+	0.293270i	$0.905257\pi$
$90$		0			0
$91$	−10.5153			−0.115553
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	37.0000	+	37.0000i	0.381443	+	0.381443i	0.871622	−	0.490179i	$-0.163068\pi$
$97$	37.0000	+	37.0000i	0.381443	+	0.381443i	−0.490179	+	0.871622i	$0.663068\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	18.3383			0.181567			0.0907835	−	0.995871i	$-0.471063\pi$
$101$	18.3383			0.181567			0.0907835	+	0.995871i	$0.471063\pi$
$102$		0			0
$103$	10.6413	−	10.6413i	0.103314	−	0.103314i	−0.653560	−	0.756874i	$-0.726725\pi$
$103$	10.6413	−	10.6413i	0.103314	−	0.103314i	0.756874	+	0.653560i	$0.226725\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−66.9444	−	66.9444i	−0.625648	−	0.625648i	0.321322	−	0.946970i	$-0.395873\pi$
$107$	−66.9444	−	66.9444i	−0.625648	−	0.625648i	−0.946970	+	0.321322i	$0.895873\pi$
$108$		0			0
$109$			97.2827i			0.892501i	0.894908	+	0.446251i	$0.147241\pi$
$109$			97.2827i			0.892501i	−0.894908	+	0.446251i	$0.852759\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	30.1362	−	30.1362i	0.266692	−	0.266692i	−0.561074	−	0.827766i	$-0.689612\pi$
$113$	30.1362	−	30.1362i	0.266692	−	0.266692i	0.827766	+	0.561074i	$0.189612\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$			12.2724i			0.103130i
$120$		0			0
$121$	−118.596			−0.980132
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−66.0556	−	66.0556i	−0.520123	−	0.520123i	0.397486	−	0.917608i	$-0.369883\pi$
$127$	−66.0556	−	66.0556i	−0.520123	−	0.520123i	−0.917608	+	0.397486i	$0.869883\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−86.8536			−0.663004			−0.331502	−	0.943454i	$-0.607555\pi$
$131$	−86.8536			−0.663004			−0.331502	+	0.943454i	$0.607555\pi$
$132$		0			0
$133$	6.98979	−	6.98979i	0.0525548	−	0.0525548i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−71.5301	−	71.5301i	−0.522118	−	0.522118i	0.396093	−	0.918210i	$-0.370366\pi$
$137$	−71.5301	−	71.5301i	−0.522118	−	0.522118i	−0.918210	+	0.396093i	$0.870366\pi$
$138$		0			0
$139$			23.2122i			0.166995i	0.996508	+	0.0834973i	$0.0266090\pi$
$139$			23.2122i			0.166995i	−0.996508	+	0.0834973i	$0.973391\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−14.8082	+	14.8082i	−0.103554	+	0.103554i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	137.530i		−	0.923021i	−0.887135	−	0.461510i	$-0.847308\pi$
$149$		−	137.530i		−	0.923021i	0.887135	−	0.461510i	$-0.152692\pi$
$150$		0			0
$151$	−291.394			−1.92976			−0.964880	−	0.262690i	$-0.915390\pi$
$151$	−291.394			−1.92976			−0.964880	+	0.262690i	$0.915390\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	52.6311	+	52.6311i	0.335230	+	0.335230i	0.854569	−	0.519339i	$-0.173822\pi$
$157$	52.6311	+	52.6311i	0.335230	+	0.335230i	−0.519339	+	0.854569i	$0.673822\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−23.5051			−0.145994
$162$		0			0
$163$	−117.576	+	117.576i	−0.721322	+	0.721322i	−0.968875	−	0.247552i	$-0.920374\pi$
$163$	−117.576	+	117.576i	−0.721322	+	0.721322i	0.247552	+	0.968875i	$0.420374\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−146.338	−	146.338i	−0.876277	−	0.876277i	0.116870	−	0.993147i	$-0.462714\pi$
$167$	−146.338	−	146.338i	−0.876277	−	0.876277i	−0.993147	+	0.116870i	$0.962714\pi$
$168$		0			0
$169$		−	13.4245i		−	0.0794349i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	3.75255	−	3.75255i	0.0216910	−	0.0216910i	−0.696178	−	0.717869i	$-0.745118\pi$
$173$	3.75255	−	3.75255i	0.0216910	−	0.0216910i	0.717869	+	0.696178i	$0.245118\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	77.2372i		−	0.431493i	−0.976449	−	0.215746i	$-0.930782\pi$
$179$		−	77.2372i		−	0.431493i	0.976449	−	0.215746i	$-0.0692185\pi$
$180$		0			0
$181$	160.656			0.887603			0.443801	−	0.896125i	$-0.353630\pi$
$181$	160.656			0.887603			0.443801	+	0.896125i	$0.353630\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	17.2827	+	17.2827i	0.0924206	+	0.0924206i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−251.868			−1.31868			−0.659341	−	0.751844i	$-0.729165\pi$
$191$	−251.868			−1.31868			−0.659341	+	0.751844i	$0.729165\pi$
$192$		0			0
$193$	−173.384	+	173.384i	−0.898361	+	0.898361i	−0.995291	−	0.0969302i	$-0.969098\pi$
$193$	−173.384	+	173.384i	−0.898361	+	0.898361i	0.0969302	+	0.995291i	$0.469098\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−2.09082	−	2.09082i	−0.0106133	−	0.0106133i	0.701780	−	0.712394i	$-0.252389\pi$
$197$	−2.09082	−	2.09082i	−0.0106133	−	0.0106133i	−0.712394	+	0.701780i	$0.752389\pi$
$198$		0			0
$199$		−	304.565i		−	1.53048i	−0.643746	−	0.765239i	$-0.722621\pi$
$199$		−	304.565i		−	1.53048i	0.643746	−	0.765239i	$-0.277379\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	24.2474	−	24.2474i	0.119446	−	0.119446i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	19.6867i		−	0.0941949i
$210$		0			0
$211$	273.151			1.29455			0.647277	−	0.762255i	$-0.275908\pi$
$211$	273.151			1.29455			0.647277	+	0.762255i	$0.275908\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−24.4949	−	24.4949i	−0.112880	−	0.112880i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−212.908			−0.963385
$222$		0			0
$223$	−174.662	+	174.662i	−0.783236	+	0.783236i	−0.980376	−	0.197139i	$-0.936835\pi$
$223$	−174.662	+	174.662i	−0.783236	+	0.783236i	0.197139	+	0.980376i	$0.436835\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−77.6163	−	77.6163i	−0.341922	−	0.341922i	0.515167	−	0.857090i	$-0.327730\pi$
$227$	−77.6163	−	77.6163i	−0.341922	−	0.341922i	−0.857090	+	0.515167i	$0.827730\pi$
$228$		0			0
$229$			8.96938i			0.0391676i	0.999808	+	0.0195838i	$0.00623412\pi$
$229$			8.96938i			0.0391676i	−0.999808	+	0.0195838i	$0.993766\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−50.4699	+	50.4699i	−0.216609	+	0.216609i	−0.807068	−	0.590459i	$-0.798947\pi$
$233$	−50.4699	+	50.4699i	−0.216609	+	0.216609i	0.590459	+	0.807068i	$0.298947\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	279.737i		−	1.17045i	−0.810872	−	0.585223i	$-0.801007\pi$
$239$		−	279.737i		−	1.17045i	0.810872	−	0.585223i	$-0.198993\pi$
$240$		0			0
$241$	−397.131			−1.64784			−0.823922	−	0.566703i	$-0.808219\pi$
$241$	−397.131			−1.64784			−0.823922	+	0.566703i	$0.808219\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	121.262	+	121.262i	0.490940	+	0.490940i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−53.8638			−0.214597			−0.107298	−	0.994227i	$-0.534220\pi$
$251$	−53.8638			−0.214597			−0.107298	+	0.994227i	$0.534220\pi$
$252$		0			0
$253$	−33.1010	+	33.1010i	−0.130834	+	0.130834i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	309.732	+	309.732i	1.20518	+	1.20518i	0.972569	+	0.232614i	$0.0747280\pi$
$257$	309.732	+	309.732i	1.20518	+	1.20518i	0.232614	+	0.972569i	$0.425272\pi$
$258$		0			0
$259$		−	22.7378i		−	0.0877906i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−128.854	+	128.854i	−0.489938	+	0.489938i	−0.908286	−	0.418349i	$-0.862609\pi$
$263$	−128.854	+	128.854i	−0.489938	+	0.489938i	0.418349	+	0.908286i	$0.362609\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$			112.227i			0.417201i	0.978001	+	0.208600i	$0.0668908\pi$
$269$			112.227i			0.417201i	−0.978001	+	0.208600i	$0.933109\pi$
$270$		0			0
$271$	167.616			0.618510			0.309255	−	0.950979i	$-0.399920\pi$
$271$	167.616			0.618510			0.309255	+	0.950979i	$0.399920\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	298.136	+	298.136i	1.07630	+	1.07630i	0.996838	+	0.0794665i	$0.0253217\pi$
$277$	298.136	+	298.136i	1.07630	+	1.07630i	0.0794665	+	0.996838i	$0.474678\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	336.434			1.19727			0.598636	−	0.801021i	$-0.295709\pi$
$281$	336.434			1.19727			0.598636	+	0.801021i	$0.295709\pi$
$282$		0			0
$283$	−20.1612	+	20.1612i	−0.0712411	+	0.0712411i	−0.741830	−	0.670588i	$-0.766042\pi$
$283$	−20.1612	+	20.1612i	−0.0712411	+	0.0712411i	0.670588	+	0.741830i	$0.266042\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	26.5857	+	26.5857i	0.0926331	+	0.0926331i
$288$		0			0
$289$		−	40.5143i		−	0.140188i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−246.586	+	246.586i	−0.841589	+	0.841589i	−0.989066	−	0.147476i	$-0.952885\pi$
$293$	−246.586	+	246.586i	−0.841589	+	0.841589i	0.147476	+	0.989066i	$0.452885\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	407.778i		−	1.36380i
$300$		0			0
$301$	39.9592			0.132755
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−365.373	−	365.373i	−1.19014	−	1.19014i	−0.977027	−	0.213114i	$-0.931639\pi$
$307$	−365.373	−	365.373i	−1.19014	−	1.19014i	−0.213114	−	0.977027i	$-0.568361\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	154.384			0.496411			0.248205	−	0.968707i	$-0.420159\pi$
$311$	154.384			0.496411			0.248205	+	0.968707i	$0.420159\pi$
$312$		0			0
$313$	258.959	−	258.959i	0.827346	−	0.827346i	−0.159803	−	0.987149i	$-0.551086\pi$
$313$	258.959	−	258.959i	0.827346	−	0.827346i	0.987149	+	0.159803i	$0.0510860\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−2.85357	−	2.85357i	−0.00900180	−	0.00900180i	0.702592	−	0.711593i	$-0.252026\pi$
$317$	−2.85357	−	2.85357i	−0.00900180	−	0.00900180i	−0.711593	+	0.702592i	$0.752026\pi$
$318$		0			0
$319$		−	68.2929i		−	0.214084i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	141.526	−	141.526i	0.438159	−	0.438159i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$			46.8377i			0.142364i
$330$		0			0
$331$	−497.646			−1.50346			−0.751731	−	0.659470i	$-0.770781\pi$
$331$	−497.646			−1.50346			−0.751731	+	0.659470i	$0.770781\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−274.757	−	274.757i	−0.815303	−	0.815303i	0.170120	−	0.985423i	$-0.445584\pi$
$337$	−274.757	−	274.757i	−0.815303	−	0.815303i	−0.985423	+	0.170120i	$0.945584\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−68.9898			−0.202316
$342$		0			0
$343$	53.6163	−	53.6163i	0.156316	−	0.156316i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−465.848	−	465.848i	−1.34250	−	1.34250i	−0.893563	−	0.448939i	$-0.851802\pi$
$347$	−465.848	−	465.848i	−1.34250	−	1.34250i	−0.448939	−	0.893563i	$-0.648198\pi$
$348$		0			0
$349$		−	74.7173i		−	0.214090i	−0.994254	−	0.107045i	$-0.965861\pi$
$349$		−	74.7173i		−	0.214090i	0.994254	−	0.107045i	$-0.0341389\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−34.8332	+	34.8332i	−0.0986775	+	0.0986775i	−0.754722	−	0.656045i	$-0.772228\pi$
$353$	−34.8332	+	34.8332i	−0.0986775	+	0.0986775i	0.656045	+	0.754722i	$0.272228\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$			129.889i			0.361807i	0.983501	+	0.180904i	$0.0579022\pi$
$359$			129.889i			0.361807i	−0.983501	+	0.180904i	$0.942098\pi$
$360$		0			0
$361$	199.788			0.553429
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	160.056	+	160.056i	0.436119	+	0.436119i	0.890704	−	0.454585i	$-0.150212\pi$
$367$	160.056	+	160.056i	0.436119	+	0.436119i	−0.454585	+	0.890704i	$0.650212\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−60.0000			−0.161725
$372$		0			0
$373$	30.0250	−	30.0250i	0.0804960	−	0.0804960i	−0.665712	−	0.746208i	$-0.731872\pi$
$373$	30.0250	−	30.0250i	0.0804960	−	0.0804960i	0.746208	+	0.665712i	$0.231872\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	420.656	+	420.656i	1.11580	+	1.11580i
$378$		0			0
$379$			424.343i			1.11964i	0.828615	+	0.559819i	$0.189129\pi$
$379$			424.343i			1.11964i	−0.828615	+	0.559819i	$0.810871\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	325.460	−	325.460i	0.849764	−	0.849764i	−0.140339	−	0.990103i	$-0.544819\pi$
$383$	325.460	−	325.460i	0.849764	−	0.849764i	0.990103	+	0.140339i	$0.0448193\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$			698.116i			1.79464i	0.441378	+	0.897321i	$0.354490\pi$
$389$			698.116i			1.79464i	−0.441378	+	0.897321i	$0.645510\pi$
$390$		0			0
$391$	−475.918			−1.21718
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−257.984	−	257.984i	−0.649834	−	0.649834i	0.303119	−	0.952953i	$-0.401972\pi$
$397$	−257.984	−	257.984i	−0.649834	−	0.649834i	−0.952953	+	0.303119i	$0.901972\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	357.151			0.890651			0.445325	−	0.895369i	$-0.353088\pi$
$401$	357.151			0.890651			0.445325	+	0.895369i	$0.353088\pi$
$402$		0			0
$403$	424.949	−	424.949i	1.05446	−	1.05446i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−32.0204	−	32.0204i	−0.0786742	−	0.0786742i
$408$		0			0
$409$		−	66.3837i		−	0.162307i	−0.996702	−	0.0811536i	$-0.974140\pi$
$409$		−	66.3837i		−	0.162307i	0.996702	−	0.0811536i	$-0.0258604\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−26.1158	+	26.1158i	−0.0632344	+	0.0632344i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$			414.772i			0.989909i	0.868919	+	0.494955i	$0.164815\pi$
$419$			414.772i			0.989909i	−0.868919	+	0.494955i	$0.835185\pi$
$420$		0			0
$421$	762.727			1.81170			0.905851	−	0.423597i	$-0.139233\pi$
$421$	762.727			1.81170			0.905851	+	0.423597i	$0.139233\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−32.9694	−	32.9694i	−0.0772117	−	0.0772117i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−21.3439			−0.0495218			−0.0247609	−	0.999693i	$-0.507882\pi$
$431$	−21.3439			−0.0495218			−0.0247609	+	0.999693i	$0.507882\pi$
$432$		0			0
$433$	−288.868	+	288.868i	−0.667132	+	0.667132i	−0.957051	−	0.289919i	$-0.906372\pi$
$433$	−288.868	+	288.868i	−0.667132	+	0.667132i	0.289919	+	0.957051i	$0.406372\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	271.060	+	271.060i	0.620275	+	0.620275i
$438$		0			0
$439$			410.182i			0.934355i	0.884164	+	0.467177i	$0.154729\pi$
$439$			410.182i			0.934355i	−0.884164	+	0.467177i	$0.845271\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	262.747	−	262.747i	0.593108	−	0.593108i	−0.345362	−	0.938470i	$-0.612244\pi$
$443$	262.747	−	262.747i	0.593108	−	0.593108i	0.938470	+	0.345362i	$0.112244\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$			841.242i			1.87359i	0.349879	+	0.936795i	$0.386223\pi$
$449$			841.242i			1.87359i	−0.349879	+	0.936795i	$0.613777\pi$
$450$		0			0
$451$	74.8786			0.166028
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	23.8286	+	23.8286i	0.0521413	+	0.0521413i	0.732697	−	0.680555i	$-0.238261\pi$
$457$	23.8286	+	23.8286i	0.0521413	+	0.0521413i	−0.680555	+	0.732697i	$0.738261\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−44.3179			−0.0961342			−0.0480671	−	0.998844i	$-0.515306\pi$
$461$	−44.3179			−0.0961342			−0.0480671	+	0.998844i	$0.515306\pi$
$462$		0			0
$463$	193.116	−	193.116i	0.417097	−	0.417097i	−0.467105	−	0.884202i	$-0.654703\pi$
$463$	193.116	−	193.116i	0.417097	−	0.417097i	0.884202	+	0.467105i	$0.154703\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	377.369	+	377.369i	0.808070	+	0.808070i	0.984342	−	0.176271i	$-0.0564036\pi$
$467$	377.369	+	377.369i	0.808070	+	0.808070i	−0.176271	+	0.984342i	$0.556404\pi$
$468$		0			0
$469$		−	89.4939i		−	0.190818i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	56.2724	−	56.2724i	0.118969	−	0.118969i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	112.141i		−	0.234114i	−0.993125	−	0.117057i	$-0.962654\pi$
$479$		−	112.141i		−	0.234114i	0.993125	−	0.117057i	$-0.0373461\pi$
$480$		0			0
$481$	394.465			0.820094
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	41.9444	+	41.9444i	0.0861281	+	0.0861281i	0.748858	−	0.662730i	$-0.230602\pi$
$487$	41.9444	+	41.9444i	0.0861281	+	0.0861281i	−0.662730	+	0.748858i	$0.730602\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	926.468			1.88690			0.943450	−	0.331515i	$-0.107560\pi$
$491$	926.468			1.88690			0.943450	+	0.331515i	$0.107560\pi$
$492$		0			0
$493$	490.949	−	490.949i	0.995840	−	0.995840i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−48.2225	−	48.2225i	−0.0970271	−	0.0970271i
$498$		0			0
$499$			67.5255i			0.135322i	0.997708	+	0.0676608i	$0.0215536\pi$
$499$			67.5255i			0.135322i	−0.997708	+	0.0676608i	$0.978446\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−180.470	+	180.470i	−0.358787	+	0.358787i	−0.863366	−	0.504579i	$-0.831648\pi$
$503$	−180.470	+	180.470i	−0.358787	+	0.358787i	0.504579	+	0.863366i	$0.331648\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	920.772i		−	1.80898i	−0.426493	−	0.904491i	$-0.640251\pi$
$509$		−	920.772i		−	1.80898i	0.426493	−	0.904491i	$-0.359749\pi$
$510$		0			0
$511$	−83.6663			−0.163731
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	65.9592	+	65.9592i	0.127581	+	0.127581i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	333.687			0.640474			0.320237	−	0.947338i	$-0.396238\pi$
$521$	333.687			0.640474			0.320237	+	0.947338i	$0.396238\pi$
$522$		0			0
$523$	380.474	−	380.474i	0.727485	−	0.727485i	−0.242633	−	0.970118i	$-0.578011\pi$
$523$	380.474	−	380.474i	0.727485	−	0.727485i	0.970118	+	0.242633i	$0.0780112\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−495.959	−	495.959i	−0.941099	−	0.941099i
$528$		0			0
$529$		−	382.514i		−	0.723089i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−461.221	+	461.221i	−0.865331	+	0.865331i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	75.0352i		−	0.139212i
$540$		0			0
$541$	156.515			0.289307			0.144654	−	0.989482i	$-0.453793\pi$
$541$	156.515			0.289307			0.144654	+	0.989482i	$0.453793\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	12.6061	+	12.6061i	0.0230459	+	0.0230459i	0.718536	−	0.695490i	$-0.244813\pi$
$547$	12.6061	+	12.6061i	0.0230459	+	0.0230459i	−0.695490	+	0.718536i	$0.744813\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−559.242			−1.01496
$552$		0			0
$553$	53.5663	−	53.5663i	0.0968650	−	0.0968650i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−21.0194	−	21.0194i	−0.0377368	−	0.0377368i	0.687987	−	0.725723i	$-0.258495\pi$
$557$	−21.0194	−	21.0194i	−0.0377368	−	0.0377368i	−0.725723	+	0.687987i	$0.758495\pi$
$558$		0			0
$559$			693.231i			1.24013i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	360.536	−	360.536i	0.640383	−	0.640383i	−0.310266	−	0.950650i	$-0.600418\pi$
$563$	360.536	−	360.536i	0.640383	−	0.640383i	0.950650	+	0.310266i	$0.100418\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$			486.504i			0.855016i	0.904012	+	0.427508i	$0.140608\pi$
$569$			486.504i			0.855016i	−0.904012	+	0.427508i	$0.859392\pi$
$570$		0			0
$571$	447.040			0.782907			0.391453	−	0.920198i	$-0.371972\pi$
$571$	447.040			0.782907			0.391453	+	0.920198i	$0.371972\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	248.444	+	248.444i	0.430579	+	0.430579i	0.888825	−	0.458247i	$-0.151522\pi$
$577$	248.444	+	248.444i	0.430579	+	0.430579i	−0.458247	+	0.888825i	$0.651522\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	45.2031			0.0778022
$582$		0			0
$583$	−84.4949	+	84.4949i	−0.144931	+	0.144931i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	547.712	+	547.712i	0.933069	+	0.933069i	0.997897	−	0.0648271i	$-0.0206496\pi$
$587$	547.712	+	547.712i	0.933069	+	0.933069i	−0.0648271	+	0.997897i	$0.520650\pi$
$588$		0			0
$589$			564.949i			0.959166i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−78.9444	+	78.9444i	−0.133127	+	0.133127i	−0.770530	−	0.637403i	$-0.780009\pi$
$593$	−78.9444	+	78.9444i	−0.133127	+	0.133127i	0.637403	+	0.770530i	$0.280009\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$			381.807i			0.637408i	0.947854	+	0.318704i	$0.103248\pi$
$599$			381.807i			0.637408i	−0.947854	+	0.318704i	$0.896752\pi$
$600$		0			0
$601$	−231.757			−0.385619			−0.192810	−	0.981236i	$-0.561760\pi$
$601$	−231.757			−0.385619			−0.192810	+	0.981236i	$0.561760\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−96.8434	−	96.8434i	−0.159544	−	0.159544i	0.622821	−	0.782365i	$-0.285987\pi$
$607$	−96.8434	−	96.8434i	−0.159544	−	0.159544i	−0.782365	+	0.622821i	$0.785987\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−812.563			−1.32989
$612$		0			0
$613$	−105.712	+	105.712i	−0.172450	+	0.172450i	−0.788055	−	0.615605i	$-0.788912\pi$
$613$	−105.712	+	105.712i	−0.172450	+	0.172450i	0.615605	+	0.788055i	$0.288912\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−41.9250	−	41.9250i	−0.0679498	−	0.0679498i	0.672315	−	0.740265i	$-0.265300\pi$
$617$	−41.9250	−	41.9250i	−0.0679498	−	0.0679498i	−0.740265	+	0.672315i	$0.765300\pi$
$618$		0			0
$619$		−	434.363i		−	0.701718i	−0.936428	−	0.350859i	$-0.885890\pi$
$619$		−	434.363i		−	0.701718i	0.936428	−	0.350859i	$-0.114110\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−28.7378	+	28.7378i	−0.0461280	+	0.0461280i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	460.382i		−	0.731926i
$630$		0			0
$631$	816.413			1.29384			0.646920	−	0.762558i	$-0.276057\pi$
$631$	816.413			1.29384			0.646920	+	0.762558i	$0.276057\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	462.186	+	462.186i	0.725567	+	0.725567i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	937.959			1.46327			0.731637	−	0.681694i	$-0.238756\pi$
$641$	937.959			1.46327			0.731637	+	0.681694i	$0.238756\pi$
$642$		0			0
$643$	−62.8786	+	62.8786i	−0.0977894	+	0.0977894i	−0.754309	−	0.656520i	$-0.772028\pi$
$643$	−62.8786	+	62.8786i	−0.0977894	+	0.0977894i	0.656520	+	0.754309i	$0.272028\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−86.3087	−	86.3087i	−0.133398	−	0.133398i	0.637255	−	0.770653i	$-0.280070\pi$
$647$	−86.3087	−	86.3087i	−0.133398	−	0.133398i	−0.770653	+	0.637255i	$0.780070\pi$
$648$		0			0
$649$			73.5551i			0.113336i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	294.904	−	294.904i	0.451613	−	0.451613i	−0.444276	−	0.895890i	$-0.646539\pi$
$653$	294.904	−	294.904i	0.451613	−	0.451613i	0.895890	+	0.444276i	$0.146539\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	1156.80i		−	1.75539i	−0.479220	−	0.877695i	$-0.659081\pi$
$659$		−	1156.80i		−	1.75539i	0.479220	−	0.877695i	$-0.340919\pi$
$660$		0			0
$661$	908.838			1.37494			0.687472	−	0.726211i	$-0.258720\pi$
$661$	908.838			1.37494			0.687472	+	0.726211i	$0.258720\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	940.302	+	940.302i	1.40975	+	1.40975i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−92.8582			−0.138388
$672$		0			0
$673$	−833.756	+	833.756i	−1.23886	+	1.23886i	−0.278400	+	0.960465i	$0.589804\pi$
$673$	−833.756	+	833.756i	−1.23886	+	1.23886i	−0.960465	+	0.278400i	$0.910196\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−200.257	−	200.257i	−0.295800	−	0.295800i	0.543566	−	0.839366i	$-0.317074\pi$
$677$	−200.257	−	200.257i	−0.295800	−	0.295800i	−0.839366	+	0.543566i	$0.817074\pi$
$678$		0			0
$679$			40.7378i			0.0599967i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	156.025	−	156.025i	0.228441	−	0.228441i	−0.583600	−	0.812041i	$-0.698357\pi$
$683$	156.025	−	156.025i	0.228441	−	0.228441i	0.812041	+	0.583600i	$0.198357\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	1040.91i		−	1.51075i
$690$		0			0
$691$	774.940			1.12148			0.560738	−	0.827993i	$-0.310518\pi$
$691$	774.940			1.12148			0.560738	+	0.827993i	$0.310518\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	538.293	+	538.293i	0.772300	+	0.772300i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−280.309			−0.399870			−0.199935	−	0.979809i	$-0.564073\pi$
$701$	−280.309			−0.399870			−0.199935	+	0.979809i	$0.564073\pi$
$702$		0			0
$703$	−262.211	+	262.211i	−0.372989	+	0.372989i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	10.0954	+	10.0954i	0.0142792	+	0.0142792i
$708$		0			0
$709$			926.686i			1.30703i	0.756913	+	0.653516i	$0.226707\pi$
$709$			926.686i			1.30703i	−0.756913	+	0.653516i	$0.773293\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	949.898	−	949.898i	1.33226	−	1.33226i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$			938.565i			1.30538i	0.757627	+	0.652688i	$0.226359\pi$
$719$			938.565i			1.30538i	−0.757627	+	0.652688i	$0.773641\pi$
$720$		0			0
$721$	11.7163			0.0162501
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−404.803	−	404.803i	−0.556812	−	0.556812i	0.371586	−	0.928398i	$-0.378814\pi$
$727$	−404.803	−	404.803i	−0.556812	−	0.556812i	−0.928398	+	0.371586i	$0.878814\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	809.071			1.10680
$732$		0			0
$733$	644.529	−	644.529i	0.879303	−	0.879303i	−0.114159	−	0.993462i	$-0.536417\pi$
$733$	644.529	−	644.529i	0.879303	−	0.879303i	0.993462	+	0.114159i	$0.0364175\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−126.030	−	126.030i	−0.171004	−	0.171004i
$738$		0			0
$739$		−	1182.11i		−	1.59961i	−0.600262	−	0.799803i	$-0.704937\pi$
$739$		−	1182.11i		−	1.59961i	0.600262	−	0.799803i	$-0.295063\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−570.681	+	570.681i	−0.768077	+	0.768077i	−0.977768	−	0.209691i	$-0.932754\pi$
$743$	−570.681	+	570.681i	−0.768077	+	0.768077i	0.209691	+	0.977768i	$0.432754\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	73.7071i		−	0.0984074i
$750$		0			0
$751$	180.050			0.239747			0.119873	−	0.992789i	$-0.461751\pi$
$751$	180.050			0.239747			0.119873	+	0.992789i	$0.461751\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	187.176	+	187.176i	0.247260	+	0.247260i	0.819845	−	0.572585i	$-0.194059\pi$
$757$	187.176	+	187.176i	0.247260	+	0.247260i	−0.572585	+	0.819845i	$0.694059\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−912.130			−1.19859			−0.599297	−	0.800527i	$-0.704553\pi$
$761$	−912.130			−1.19859			−0.599297	+	0.800527i	$0.704553\pi$
$762$		0			0
$763$	−53.5551	+	53.5551i	−0.0701902	+	0.0701902i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−453.069	−	453.069i	−0.590703	−	0.590703i
$768$		0			0
$769$			201.778i			0.262390i	0.991357	+	0.131195i	$0.0418813\pi$
$769$			201.778i			0.262390i	−0.991357	+	0.131195i	$0.958119\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−147.793	+	147.793i	−0.191195	+	0.191195i	−0.796212	−	0.605018i	$-0.793166\pi$
$773$	−147.793	+	147.793i	−0.191195	+	0.191195i	0.605018	+	0.796212i	$0.293166\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	613.171i		−	0.787126i
$780$		0			0
$781$	−135.818			−0.173903
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	447.576	+	447.576i	0.568711	+	0.568711i	0.931767	−	0.363056i	$-0.118267\pi$
$787$	447.576	+	447.576i	0.568711	+	0.568711i	−0.363056	+	0.931767i	$0.618267\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	33.1806			0.0419477
$792$		0			0
$793$	571.968	−	571.968i	0.721272	−	0.721272i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−128.338	−	128.338i	−0.161027	−	0.161027i	0.621995	−	0.783021i	$-0.286323\pi$
$797$	−128.338	−	128.338i	−0.161027	−	0.161027i	−0.783021	+	0.621995i	$0.786323\pi$
$798$		0			0
$799$			948.345i			1.18691i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−117.823	+	117.823i	−0.146728	+	0.146728i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$			699.212i			0.864292i	0.901804	+	0.432146i	$0.142243\pi$
$809$			699.212i			0.864292i	−0.901804	+	0.432146i	$0.857757\pi$
$810$		0			0
$811$	−90.5041			−0.111596			−0.0557978	−	0.998442i	$-0.517770\pi$
$811$	−90.5041			−0.111596			−0.0557978	+	0.998442i	$0.517770\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−460.808	−	460.808i	−0.564025	−	0.564025i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−250.783			−0.305461			−0.152730	−	0.988268i	$-0.548807\pi$
$821$	−250.783			−0.305461			−0.152730	+	0.988268i	$0.548807\pi$
$822$		0			0
$823$	27.0954	−	27.0954i	0.0329227	−	0.0329227i	−0.690454	−	0.723377i	$-0.742589\pi$
$823$	27.0954	−	27.0954i	0.0329227	−	0.0329227i	0.723377	+	0.690454i	$0.242589\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	589.930	+	589.930i	0.713337	+	0.713337i	0.967232	−	0.253895i	$-0.0817117\pi$
$827$	589.930	+	589.930i	0.713337	+	0.713337i	−0.253895	+	0.967232i	$0.581712\pi$
$828$		0			0
$829$			1576.77i			1.90202i	0.309160	+	0.951010i	$0.399952\pi$
$829$			1576.77i			1.90202i	−0.309160	+	0.951010i	$0.600048\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	539.419	−	539.419i	0.647562	−	0.647562i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$			963.523i			1.14842i	0.818708	+	0.574209i	$0.194690\pi$
$839$			963.523i			1.14842i	−0.818708	+	0.574209i	$0.805310\pi$
$840$		0			0
$841$	−1099.00			−1.30678
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−65.2883	−	65.2883i	−0.0770818	−	0.0770818i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	881.757			1.03614
$852$		0			0
$853$	−254.166	+	254.166i	−0.297967	+	0.297967i	−0.840217	−	0.542250i	$-0.817573\pi$
$853$	−254.166	+	254.166i	−0.297967	+	0.297967i	0.542250	+	0.840217i	$0.317573\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−699.206	−	699.206i	−0.815876	−	0.815876i	0.169632	−	0.985508i	$-0.445742\pi$
$857$	−699.206	−	699.206i	−0.815876	−	0.815876i	−0.985508	+	0.169632i	$0.945742\pi$
$858$		0			0
$859$		−	246.708i		−	0.287204i	−0.989636	−	0.143602i	$-0.954131\pi$
$859$		−	246.708i		−	0.287204i	0.989636	−	0.143602i	$-0.0458685\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−1051.01	+	1051.01i	−1.21786	+	1.21786i	−0.249483	+	0.968379i	$0.580261\pi$
$863$	−1051.01	+	1051.01i	−1.21786	+	1.21786i	−0.968379	+	0.249483i	$0.919739\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	150.869i		−	0.173613i
$870$		0			0
$871$	1552.58			1.78253
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−30.2974	−	30.2974i	−0.0345467	−	0.0345467i	0.689622	−	0.724169i	$-0.257777\pi$
$877$	−30.2974	−	30.2974i	−0.0345467	−	0.0345467i	−0.724169	+	0.689622i	$0.757777\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−751.294			−0.852774			−0.426387	−	0.904541i	$-0.640214\pi$
$881$	−751.294			−0.852774			−0.426387	+	0.904541i	$0.640214\pi$
$882$		0			0
$883$	666.929	−	666.929i	0.755298	−	0.755298i	−0.220164	−	0.975463i	$-0.570659\pi$
$883$	666.929	−	666.929i	0.755298	−	0.755298i	0.975463	+	0.220164i	$0.0706594\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−269.662	−	269.662i	−0.304015	−	0.304015i	0.538567	−	0.842583i	$-0.318966\pi$
$887$	−269.662	−	269.662i	−0.304015	−	0.304015i	−0.842583	+	0.538567i	$0.818966\pi$
$888$		0			0
$889$		−	72.7286i		−	0.0818094i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	540.132	−	540.132i	0.604851	−	0.604851i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$			1959.80i			2.17997i
$900$		0			0
$901$	−1214.85			−1.34833
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−578.515	−	578.515i	−0.637834	−	0.637834i	0.312187	−	0.950021i	$-0.398938\pi$
$907$	−578.515	−	578.515i	−0.637834	−	0.637834i	−0.950021	+	0.312187i	$0.898938\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−90.4745			−0.0993134			−0.0496567	−	0.998766i	$-0.515813\pi$
$911$	−90.4745			−0.0993134			−0.0496567	+	0.998766i	$0.515813\pi$
$912$		0			0
$913$	63.6571	−	63.6571i	0.0697231	−	0.0697231i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−47.8138	−	47.8138i	−0.0521415	−	0.0521415i
$918$		0			0
$919$		−	1467.08i		−	1.59639i	−0.602402	−	0.798193i	$-0.705790\pi$
$919$		−	1467.08i		−	1.59639i	0.602402	−	0.798193i	$-0.294210\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	836.586	−	836.586i	0.906377	−	0.906377i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	907.444i		−	0.976796i	−0.872621	−	0.488398i	$-0.837581\pi$
$929$		−	907.444i		−	0.976796i	0.872621	−	0.488398i	$-0.162419\pi$
$930$		0			0
$931$	−614.454			−0.659994
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−158.898	−	158.898i	−0.169582	−	0.169582i	0.617214	−	0.786795i	$-0.288261\pi$
$937$	−158.898	−	158.898i	−0.169582	−	0.169582i	−0.786795	+	0.617214i	$0.788261\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	666.497			0.708286			0.354143	−	0.935191i	$-0.384773\pi$
$941$	666.497			0.708286			0.354143	+	0.935191i	$0.384773\pi$
$942$		0			0
$943$	−1030.98	+	1030.98i	−1.09330	+	1.09330i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1038.08	+	1038.08i	1.09618	+	1.09618i	0.994854	+	0.101323i	$0.0323077\pi$
$947$	1038.08	+	1038.08i	1.09618	+	1.09618i	0.101323	+	0.994854i	$0.467692\pi$
$948$		0			0
$949$		−	1451.48i		−	1.52949i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−262.045	+	262.045i	−0.274969	+	0.274969i	−0.831097	−	0.556128i	$-0.812286\pi$
$953$	−262.045	+	262.045i	−0.274969	+	0.274969i	0.556128	+	0.831097i	$0.312286\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	78.7561i		−	0.0821232i
$960$		0			0
$961$	1018.80			1.06014
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−1277.98	−	1277.98i	−1.32160	−	1.32160i	−0.912484	−	0.409111i	$-0.865839\pi$
$967$	−1277.98	−	1277.98i	−1.32160	−	1.32160i	−0.409111	−	0.912484i	$-0.634161\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−1640.26			−1.68924			−0.844622	−	0.535363i	$-0.820175\pi$
$971$	−1640.26			−1.68924			−0.844622	+	0.535363i	$0.820175\pi$
$972$		0			0
$973$	−12.7786	+	12.7786i	−0.0131332	+	0.0131332i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−761.480	−	761.480i	−0.779406	−	0.779406i	0.200323	−	0.979730i	$-0.435801\pi$
$977$	−761.480	−	761.480i	−0.779406	−	0.779406i	−0.979730	+	0.200323i	$0.935801\pi$
$978$		0			0
$979$			80.9398i			0.0826760i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	215.035	−	215.035i	0.218754	−	0.218754i	−0.589219	−	0.807973i	$-0.700565\pi$
$983$	215.035	−	215.035i	0.218754	−	0.218754i	0.807973	+	0.589219i	$0.200565\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$			1549.59i			1.56683i
$990$		0			0
$991$	−1240.62			−1.25189			−0.625946	−	0.779867i	$-0.715287\pi$
$991$	−1240.62			−1.25189			−0.625946	+	0.779867i	$0.715287\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−899.358	−	899.358i	−0.902064	−	0.902064i	0.0935507	−	0.995615i	$-0.470178\pi$
$997$	−899.358	−	899.358i	−0.902064	−	0.902064i	−0.995615	+	0.0935507i	$0.970178\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1800.3.v.n.1657.1		4
3.2	odd	2		600.3.u.e.457.1		4
5.2	odd	4		360.3.v.b.73.1		4
5.3	odd	4	inner	1800.3.v.n.793.1		4
5.4	even	2		360.3.v.b.217.1		4
12.11	even	2		1200.3.bg.e.1057.2		4
15.2	even	4		120.3.u.a.73.2	✓	4
15.8	even	4		600.3.u.e.193.1		4
15.14	odd	2		120.3.u.a.97.2	yes	4
20.7	even	4		720.3.bh.g.433.1		4
20.19	odd	2		720.3.bh.g.577.1		4
60.23	odd	4		1200.3.bg.e.193.2		4
60.47	odd	4		240.3.bg.c.193.1		4
60.59	even	2		240.3.bg.c.97.1		4
120.29	odd	2		960.3.bg.c.577.1		4
120.59	even	2		960.3.bg.d.577.2		4
120.77	even	4		960.3.bg.c.193.1		4
120.107	odd	4		960.3.bg.d.193.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.3.u.a.73.2	✓	4	15.2	even	4
120.3.u.a.97.2	yes	4	15.14	odd	2
240.3.bg.c.97.1		4	60.59	even	2
240.3.bg.c.193.1		4	60.47	odd	4
360.3.v.b.73.1		4	5.2	odd	4
360.3.v.b.217.1		4	5.4	even	2
600.3.u.e.193.1		4	15.8	even	4
600.3.u.e.457.1		4	3.2	odd	2
720.3.bh.g.433.1		4	20.7	even	4
720.3.bh.g.577.1		4	20.19	odd	2
960.3.bg.c.193.1		4	120.77	even	4
960.3.bg.c.577.1		4	120.29	odd	2
960.3.bg.d.193.2		4	120.107	odd	4
960.3.bg.d.577.2		4	120.59	even	2
1200.3.bg.e.193.2		4	60.23	odd	4
1200.3.bg.e.1057.2		4	12.11	even	2
1800.3.v.n.793.1		4	5.3	odd	4	inner
1800.3.v.n.1657.1		4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1800.v (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.550510 + 0.550510i) q^{7} +O(q^{10})\)	\(q+(0.550510 + 0.550510i) q^{7} +1.55051 q^{11} +(-9.55051 + 9.55051i) q^{13} +(11.1464 + 11.1464i) q^{17} -12.6969i q^{19} +(-21.3485 + 21.3485i) q^{23} -44.0454i q^{29} -44.4949 q^{31} +(-20.6515 - 20.6515i) q^{37} +48.2929 q^{41} +(36.2929 - 36.2929i) q^{43} +(42.5403 + 42.5403i) q^{47} -48.3939i q^{49} +(-54.4949 + 54.4949i) q^{53} +47.4393i q^{59} -59.8888 q^{61} +(-81.2827 - 81.2827i) q^{67} -87.5959 q^{71} +(-75.9898 + 75.9898i) q^{73} +(0.853572 + 0.853572i) q^{77} -97.3031i q^{79} +(41.0556 - 41.0556i) q^{83} +52.2020i q^{89} -10.5153 q^{91} +(37.0000 + 37.0000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{7}+O(q^{10}) \) 4 * q + 12 * q^7	\( 4 q + 12 q^{7} + 16 q^{11} - 48 q^{13} - 24 q^{17} - 56 q^{23} - 80 q^{31} - 112 q^{37} + 56 q^{41} + 8 q^{43} - 16 q^{47} - 120 q^{53} - 24 q^{61} + 8 q^{67} - 272 q^{71} - 108 q^{73} + 72 q^{77} + 272 q^{83} - 336 q^{91} + 148 q^{97}+O(q^{100}) \) 4 * q + 12 * q^7 + 16 * q^11 - 48 * q^13 - 24 * q^17 - 56 * q^23 - 80 * q^31 - 112 * q^37 + 56 * q^41 + 8 * q^43 - 16 * q^47 - 120 * q^53 - 24 * q^61 + 8 * q^67 - 272 * q^71 - 108 * q^73 + 72 * q^77 + 272 * q^83 - 336 * q^91 + 148 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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