LMFDB - Embedded newform 1350.4.c.x.649.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1350,4,Mod(649,1350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1350, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1350.649");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1350 = 2 \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1350.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:	$79.6525785077$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{209})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 105x^{2} + 2704 $ x^4 + 105*x^2 + 2704
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.2
Root			$-6.72842i$ of defining polynomial
Character	$\chi$	$=$	1350.649
Dual form			1350.4.c.x.649.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{2} -4.00000 q^{4} +28.1852i q^{7} +8.00000i q^{8} +O(q^{10})$	$q-2.00000i q^{2} -4.00000 q^{4} +28.1852i q^{7} +8.00000i q^{8} -52.3705 q^{11} -25.1852i q^{13} +56.3705 q^{14} +16.0000 q^{16} +40.3705i q^{17} +0.185248 q^{19} +104.741i q^{22} +152.741i q^{23} -50.3705 q^{26} -112.741i q^{28} -28.3705 q^{29} +117.370 q^{31} -32.0000i q^{32} +80.7410 q^{34} -375.223i q^{37} -0.370497i q^{38} -145.111 q^{41} +289.593i q^{43} +209.482 q^{44} +305.482 q^{46} +57.8525i q^{47} -451.408 q^{49} +100.741i q^{52} +726.593i q^{53} -225.482 q^{56} +56.7410i q^{58} -185.482 q^{59} -279.002 q^{61} -234.741i q^{62} -64.0000 q^{64} -939.779i q^{67} -161.482i q^{68} +858.669 q^{71} -955.223i q^{73} -750.446 q^{74} -0.740994 q^{76} -1476.08i q^{77} +259.000 q^{79} +290.223i q^{82} -1311.49i q^{83} +579.187 q^{86} -418.964i q^{88} -247.554 q^{89} +709.852 q^{91} -610.964i q^{92} +115.705 q^{94} -37.7393i q^{97} +902.816i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 16 q^{4}+O(q^{10}) $ 4 * q - 16 * q^4	$ 4 q - 16 q^{4} - 36 q^{11} + 52 q^{14} + 64 q^{16} - 86 q^{19} - 28 q^{26} + 60 q^{29} + 296 q^{31} - 24 q^{34} - 60 q^{41} + 144 q^{44} + 528 q^{46} - 678 q^{49} - 208 q^{56} - 48 q^{59} + 1226 q^{61} - 256 q^{64} + 312 q^{71} - 920 q^{74} + 344 q^{76} + 1036 q^{79} - 112 q^{86} - 3072 q^{89} + 1972 q^{91} - 1272 q^{94}+O(q^{100}) $ 4 * q - 16 * q^4 - 36 * q^11 + 52 * q^14 + 64 * q^16 - 86 * q^19 - 28 * q^26 + 60 * q^29 + 296 * q^31 - 24 * q^34 - 60 * q^41 + 144 * q^44 + 528 * q^46 - 678 * q^49 - 208 * q^56 - 48 * q^59 + 1226 * q^61 - 256 * q^64 + 312 * q^71 - 920 * q^74 + 344 * q^76 + 1036 * q^79 - 112 * q^86 - 3072 * q^89 + 1972 * q^91 - 1272 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times$.

$n$	$1001$	$1027$
$\chi(n)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	− 2.00000i	− 0.707107i
$2$	− 2.00000i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−4.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	28.1852i	1.52186i	0.648834	+	0.760930i	$0.275257\pi$
$7$	28.1852i	1.52186i	−0.648834	+	0.760930i	$0.724743\pi$
$8$	8.00000i	0.353553i
$9$	0	0
$10$	0	0
$11$	−52.3705	−1.43548	−0.717741	−	0.696310i	$-0.754824\pi$
$11$	−52.3705	−1.43548	−0.717741	+	0.696310i	$0.754824\pi$
$12$	0	0
$13$	− 25.1852i	− 0.537318i	−0.963235	−	0.268659i	$-0.913420\pi$
$13$	− 25.1852i	− 0.537318i	0.963235	−	0.268659i	$-0.0865804\pi$
$14$	56.3705	1.07612
$15$	0	0
$16$	16.0000	0.250000
$17$	40.3705i	0.575958i	0.957637	+	0.287979i	$0.0929833\pi$
$17$	40.3705i	0.575958i	−0.957637	+	0.287979i	$0.907017\pi$
$18$	0	0
$19$	0.185248	0.00223678	0.00111839	−	0.999999i	$-0.499644\pi$
$19$	0.185248	0.00223678	0.00111839	+	0.999999i	$0.499644\pi$
$20$	0	0
$21$	0	0
$22$	104.741i	1.01504i
$23$	152.741i	1.38473i	0.721549	+	0.692363i	$0.243430\pi$
$23$	152.741i	1.38473i	−0.721549	+	0.692363i	$0.756570\pi$
$24$	0	0
$25$	0	0
$26$	−50.3705	−0.379941
$27$	0	0
$28$	− 112.741i	− 0.760930i
$29$	−28.3705	−0.181664	−0.0908322	−	0.995866i	$-0.528953\pi$
$29$	−28.3705	−0.181664	−0.0908322	+	0.995866i	$0.528953\pi$
$30$	0	0
$31$	117.370	0.680012	0.340006	−	0.940423i	$-0.389571\pi$
$31$	117.370	0.680012	0.340006	+	0.940423i	$0.389571\pi$
$32$	− 32.0000i	− 0.176777i
$33$	0	0
$34$	80.7410	0.407264
$35$	0	0
$36$	0	0
$37$	− 375.223i	− 1.66720i	−0.552371	−	0.833598i	$-0.686277\pi$
$37$	− 375.223i	− 1.66720i	0.552371	−	0.833598i	$-0.313723\pi$
$38$	− 0.370497i	− 0.00158165i
$39$	0	0
$40$	0	0
$41$	−145.111	−0.552747	−0.276373	−	0.961050i	$-0.589133\pi$
$41$	−145.111	−0.552747	−0.276373	+	0.961050i	$0.589133\pi$
$42$	0	0
$43$	289.593i	1.02704i	0.858079	+	0.513519i	$0.171658\pi$
$43$	289.593i	1.02704i	−0.858079	+	0.513519i	$0.828342\pi$
$44$	209.482	0.717741
$45$	0	0
$46$	305.482	0.979149
$47$	57.8525i	0.179546i	0.995962	+	0.0897729i	$0.0286141\pi$
$47$	57.8525i	0.179546i	−0.995962	+	0.0897729i	$0.971386\pi$
$48$	0	0
$49$	−451.408	−1.31606
$50$	0	0
$51$	0	0
$52$	100.741i	0.268659i
$53$	726.593i	1.88312i	0.336847	+	0.941559i	$0.390639\pi$
$53$	726.593i	1.88312i	−0.336847	+	0.941559i	$0.609361\pi$
$54$	0	0
$55$	0	0
$56$	−225.482	−0.538059
$57$	0	0
$58$	56.7410i	0.128456i
$59$	−185.482	−0.409283	−0.204642	−	0.978837i	$-0.565603\pi$
$59$	−185.482	−0.409283	−0.204642	+	0.978837i	$0.565603\pi$
$60$	0	0
$61$	−279.002	−0.585615	−0.292807	−	0.956171i	$-0.594589\pi$
$61$	−279.002	−0.585615	−0.292807	+	0.956171i	$0.594589\pi$
$62$	− 234.741i	− 0.480841i
$63$	0	0
$64$	−64.0000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 939.779i	− 1.71362i	−0.515636	−	0.856808i	$-0.672444\pi$
$67$	− 939.779i	− 1.71362i	0.515636	−	0.856808i	$-0.327556\pi$
$68$	− 161.482i	− 0.287979i
$69$	0	0
$70$	0	0
$71$	858.669	1.43529	0.717643	−	0.696412i	$-0.245221\pi$
$71$	858.669	1.43529	0.717643	+	0.696412i	$0.245221\pi$
$72$	0	0
$73$	− 955.223i	− 1.53151i	−0.643131	−	0.765756i	$-0.722365\pi$
$73$	− 955.223i	− 1.53151i	0.643131	−	0.765756i	$-0.277635\pi$
$74$	−750.446	−1.17889
$75$	0	0
$76$	−0.740994	−0.00111839
$77$	− 1476.08i	− 2.18460i
$78$	0	0
$79$	259.000	0.368858	0.184429	−	0.982846i	$-0.440956\pi$
$79$	259.000	0.368858	0.184429	+	0.982846i	$0.440956\pi$
$80$	0	0
$81$	0	0
$82$	290.223i	0.390851i
$83$	− 1311.49i	− 1.73439i	−0.497970	−	0.867194i	$-0.665921\pi$
$83$	− 1311.49i	− 1.73439i	0.497970	−	0.867194i	$-0.334079\pi$
$84$	0	0
$85$	0	0
$86$	579.187	0.726225
$87$	0	0
$88$	− 418.964i	− 0.507519i
$89$	−247.554	−0.294839	−0.147420	−	0.989074i	$-0.547097\pi$
$89$	−247.554	−0.294839	−0.147420	+	0.989074i	$0.547097\pi$
$90$	0	0
$91$	709.852	0.817723
$92$	− 610.964i	− 0.692363i
$93$	0	0
$94$	115.705	0.126958
$95$	0	0
$96$	0	0
$97$	− 37.7393i	− 0.0395035i	−0.999805	−	0.0197518i	$-0.993712\pi$
$97$	− 37.7393i	− 0.0395035i	0.999805	−	0.0197518i	$-0.00628759\pi$
$98$	902.816i	0.930594i
$99$	0	0
$100$	0	0
$101$	−1458.59	−1.43698	−0.718492	−	0.695535i	$-0.755168\pi$
$101$	−1458.59	−1.43698	−0.718492	+	0.695535i	$0.755168\pi$
$102$	0	0
$103$	− 1009.22i	− 0.965453i	−0.875771	−	0.482727i	$-0.839646\pi$
$103$	− 1009.22i	− 0.965453i	0.875771	−	0.482727i	$-0.160354\pi$
$104$	201.482	0.189971
$105$	0	0
$106$	1453.19	1.33157
$107$	− 1779.41i	− 1.60768i	−0.594844	−	0.803841i	$-0.702786\pi$
$107$	− 1779.41i	− 1.60768i	0.594844	−	0.803841i	$-0.297214\pi$
$108$	0	0
$109$	−397.741	−0.349511	−0.174755	−	0.984612i	$-0.555913\pi$
$109$	−397.741	−0.349511	−0.174755	+	0.984612i	$0.555913\pi$
$110$	0	0
$111$	0	0
$112$	450.964i	0.380465i
$113$	− 25.0360i	− 0.0208424i	−0.999946	−	0.0104212i	$-0.996683\pi$
$113$	− 25.0360i	− 0.0208424i	0.999946	−	0.0104212i	$-0.00331723\pi$
$114$	0	0
$115$	0	0
$116$	113.482	0.0908322
$117$	0	0
$118$	370.964i	0.289407i
$119$	−1137.85	−0.876527
$120$	0	0
$121$	1411.67	1.06061
$122$	558.003i	0.414092i
$123$	0	0
$124$	−469.482	−0.340006
$125$	0	0
$126$	0	0
$127$	674.669i	0.471395i	0.971826	+	0.235698i	$0.0757375\pi$
$127$	674.669i	0.471395i	−0.971826	+	0.235698i	$0.924263\pi$
$128$	128.000i	0.0883883i
$129$	0	0
$130$	0	0
$131$	−1233.85	−0.822917	−0.411459	−	0.911428i	$-0.634981\pi$
$131$	−1233.85	−0.822917	−0.411459	+	0.911428i	$0.634981\pi$
$132$	0	0
$133$	5.22127i	0.00340407i
$134$	−1879.56	−1.21171
$135$	0	0
$136$	−322.964	−0.203632
$137$	130.003i	0.0810726i	0.999178	+	0.0405363i	$0.0129066\pi$
$137$	130.003i	0.0810726i	−0.999178	+	0.0405363i	$0.987093\pi$
$138$	0	0
$139$	1173.89	0.716318	0.358159	−	0.933661i	$-0.383405\pi$
$139$	1173.89	0.716318	0.358159	+	0.933661i	$0.383405\pi$
$140$	0	0
$141$	0	0
$142$	− 1717.34i	− 1.01490i
$143$	1318.96i	0.771310i
$144$	0	0
$145$	0	0
$146$	−1910.45	−1.08294
$147$	0	0
$148$	1500.89i	0.833598i
$149$	−860.666	−0.473211	−0.236606	−	0.971606i	$-0.576035\pi$
$149$	−860.666	−0.473211	−0.236606	+	0.971606i	$0.576035\pi$
$150$	0	0
$151$	2259.22	1.21757	0.608785	−	0.793335i	$-0.291657\pi$
$151$	2259.22	1.21757	0.608785	+	0.793335i	$0.291657\pi$
$152$	1.48199i	0 0.000790823i
$153$	0	0
$154$	−2952.15	−1.54475
$155$	0	0
$156$	0	0
$157$	− 932.705i	− 0.474127i	−0.971494	−	0.237064i	$-0.923815\pi$
$157$	− 932.705i	− 0.474127i	0.971494	−	0.237064i	$-0.0761850\pi$
$158$	− 518.000i	− 0.260822i
$159$	0	0
$160$	0	0
$161$	−4305.04	−2.10736
$162$	0	0
$163$	258.746i	0.124335i	0.998066	+	0.0621674i	$0.0198013\pi$
$163$	258.746i	0.124335i	−0.998066	+	0.0621674i	$0.980199\pi$
$164$	580.446	0.276373
$165$	0	0
$166$	−2622.97	−1.22640
$167$	− 4005.85i	− 1.85618i	−0.372355	−	0.928090i	$-0.621450\pi$
$167$	− 4005.85i	− 1.85618i	0.372355	−	0.928090i	$-0.378550\pi$
$168$	0	0
$169$	1562.70	0.711290
$170$	0	0
$171$	0	0
$172$	− 1158.37i	− 0.513519i
$173$	3935.27i	1.72944i	0.502256	+	0.864719i	$0.332503\pi$
$173$	3935.27i	1.72944i	−0.502256	+	0.864719i	$0.667497\pi$
$174$	0	0
$175$	0	0
$176$	−837.928	−0.358870
$177$	0	0
$178$	495.108i	0.208483i
$179$	−3269.48	−1.36521	−0.682605	−	0.730788i	$-0.739153\pi$
$179$	−3269.48	−1.36521	−0.682605	+	0.730788i	$0.739153\pi$
$180$	0	0
$181$	−4608.30	−1.89244	−0.946222	−	0.323517i	$-0.895135\pi$
$181$	−4608.30	−1.89244	−0.946222	+	0.323517i	$0.895135\pi$
$182$	− 1419.70i	− 0.578217i
$183$	0	0
$184$	−1221.93	−0.489575
$185$	0	0
$186$	0	0
$187$	− 2114.22i	− 0.826777i
$188$	− 231.410i	− 0.0897729i
$189$	0	0
$190$	0	0
$191$	−2808.96	−1.06413	−0.532066	−	0.846703i	$-0.678584\pi$
$191$	−2808.96	−1.06413	−0.532066	+	0.846703i	$0.678584\pi$
$192$	0	0
$193$	− 4077.45i	− 1.52073i	−0.649495	−	0.760366i	$-0.725019\pi$
$193$	− 4077.45i	− 1.52073i	0.649495	−	0.760366i	$-0.274981\pi$
$194$	−75.4786	−0.0279332
$195$	0	0
$196$	1805.63	0.658029
$197$	1133.86i	0.410072i	0.978754	+	0.205036i	$0.0657311\pi$
$197$	1133.86i	0.410072i	−0.978754	+	0.205036i	$0.934269\pi$
$198$	0	0
$199$	2180.82	0.776855	0.388427	−	0.921479i	$-0.373018\pi$
$199$	2180.82	0.776855	0.388427	+	0.921479i	$0.373018\pi$
$200$	0	0
$201$	0	0
$202$	2917.19i	1.01610i
$203$	− 799.630i	− 0.276468i
$204$	0	0
$205$	0	0
$206$	−2018.45	−0.682679
$207$	0	0
$208$	− 402.964i	− 0.134329i
$209$	−9.70155	−0.00321086
$210$	0	0
$211$	1606.30	0.524088	0.262044	−	0.965056i	$-0.415603\pi$
$211$	1606.30	0.524088	0.262044	+	0.965056i	$0.415603\pi$
$212$	− 2906.37i	− 0.941559i
$213$	0	0
$214$	−3558.82	−1.13680
$215$	0	0
$216$	0	0
$217$	3308.12i	1.03488i
$218$	795.482i	0.247141i
$219$	0	0
$220$	0	0
$221$	1016.74	0.309472
$222$	0	0
$223$	4369.74i	1.31220i	0.754676	+	0.656098i	$0.227794\pi$
$223$	4369.74i	1.31220i	−0.754676	+	0.656098i	$0.772206\pi$
$224$	901.928	0.269029
$225$	0	0
$226$	−50.0720	−0.0147378
$227$	− 4021.34i	− 1.17580i	−0.808935	−	0.587898i	$-0.799956\pi$
$227$	− 4021.34i	− 1.17580i	0.808935	−	0.587898i	$-0.200044\pi$
$228$	0	0
$229$	−1273.16	−0.367391	−0.183696	−	0.982983i	$-0.558806\pi$
$229$	−1273.16	−0.367391	−0.183696	+	0.982983i	$0.558806\pi$
$230$	0	0
$231$	0	0
$232$	− 226.964i	− 0.0642281i
$233$	− 513.118i	− 0.144273i	−0.997395	−	0.0721363i	$-0.977018\pi$
$233$	− 513.118i	− 0.144273i	0.997395	−	0.0721363i	$-0.0229816\pi$
$234$	0	0
$235$	0	0
$236$	741.928	0.204642
$237$	0	0
$238$	2275.70i	0.619799i
$239$	−4717.41	−1.27675	−0.638377	−	0.769724i	$-0.720394\pi$
$239$	−4717.41	−1.27675	−0.638377	+	0.769724i	$0.720394\pi$
$240$	0	0
$241$	−3118.97	−0.833652	−0.416826	−	0.908986i	$-0.636858\pi$
$241$	−3118.97	−0.833652	−0.416826	+	0.908986i	$0.636858\pi$
$242$	− 2823.34i	− 0.749963i
$243$	0	0
$244$	1116.01	0.292807
$245$	0	0
$246$	0	0
$247$	− 4.66553i	− 0.00120186i
$248$	938.964i	0.240420i
$249$	0	0
$250$	0	0
$251$	92.9674	0.0233787	0.0116893	−	0.999932i	$-0.496279\pi$
$251$	92.9674	0.0233787	0.0116893	+	0.999932i	$0.496279\pi$
$252$	0	0
$253$	− 7999.12i	− 1.98775i
$254$	1349.34	0.333327
$255$	0	0
$256$	256.000	0.0625000
$257$	− 5256.96i	− 1.27595i	−0.770055	−	0.637977i	$-0.779771\pi$
$257$	− 5256.96i	− 1.27595i	0.770055	−	0.637977i	$-0.220229\pi$
$258$	0	0
$259$	10575.8	2.53724
$260$	0	0
$261$	0	0
$262$	2467.70i	0.581891i
$263$	6339.49i	1.48635i	0.669098	+	0.743174i	$0.266680\pi$
$263$	6339.49i	1.48635i	−0.669098	+	0.743174i	$0.733320\pi$
$264$	0	0
$265$	0	0
$266$	10.4425	0.00240704
$267$	0	0
$268$	3759.11i	0.856808i
$269$	−5113.64	−1.15905	−0.579524	−	0.814955i	$-0.696762\pi$
$269$	−5113.64	−1.15905	−0.579524	+	0.814955i	$0.696762\pi$
$270$	0	0
$271$	628.568	0.140896	0.0704479	−	0.997515i	$-0.477557\pi$
$271$	628.568	0.140896	0.0704479	+	0.997515i	$0.477557\pi$
$272$	645.928i	0.143989i
$273$	0	0
$274$	260.007	0.0573270
$275$	0	0
$276$	0	0
$277$	1148.71i	0.249168i	0.992209	+	0.124584i	$0.0397596\pi$
$277$	1148.71i	0.249168i	−0.992209	+	0.124584i	$0.960240\pi$
$278$	− 2347.78i	− 0.506513i
$279$	0	0
$280$	0	0
$281$	957.420	0.203256	0.101628	−	0.994822i	$-0.467595\pi$
$281$	957.420	0.203256	0.101628	+	0.994822i	$0.467595\pi$
$282$	0	0
$283$	3965.97i	0.833048i	0.909125	+	0.416524i	$0.136752\pi$
$283$	3965.97i	0.833048i	−0.909125	+	0.416524i	$0.863248\pi$
$284$	−3434.68	−0.717643
$285$	0	0
$286$	2637.93	0.545398
$287$	− 4090.00i	− 0.841203i
$288$	0	0
$289$	3283.22	0.668273
$290$	0	0
$291$	0	0
$292$	3820.89i	0.765756i
$293$	− 6738.24i	− 1.34352i	−0.740768	−	0.671761i	$-0.765538\pi$
$293$	− 6738.24i	− 1.34352i	0.740768	−	0.671761i	$-0.234462\pi$
$294$	0	0
$295$	0	0
$296$	3001.78	0.589443
$297$	0	0
$298$	1721.33i	0.334611i
$299$	3846.82	0.744038
$300$	0	0
$301$	−8162.26	−1.56301
$302$	− 4518.45i	− 0.860952i
$303$	0	0
$304$	2.96398	0.000559196 0
$305$	0	0
$306$	0	0
$307$	− 5215.46i	− 0.969584i	−0.874630	−	0.484792i	$-0.838895\pi$
$307$	− 5215.46i	− 0.969584i	0.874630	−	0.484792i	$-0.161105\pi$
$308$	5904.30i	1.09230i
$309$	0	0
$310$	0	0
$311$	9552.38	1.74169	0.870845	−	0.491558i	$-0.163572\pi$
$311$	9552.38	1.74169	0.870845	+	0.491558i	$0.163572\pi$
$312$	0	0
$313$	6337.63i	1.14449i	0.820084	+	0.572243i	$0.193927\pi$
$313$	6337.63i	1.14449i	−0.820084	+	0.572243i	$0.806073\pi$
$314$	−1865.41	−0.335259
$315$	0	0
$316$	−1036.00	−0.184429
$317$	− 7892.84i	− 1.39844i	−0.714906	−	0.699221i	$-0.753531\pi$
$317$	− 7892.84i	− 1.39844i	0.714906	−	0.699221i	$-0.246469\pi$
$318$	0	0
$319$	1485.78	0.260776
$320$	0	0
$321$	0	0
$322$	8610.09i	1.49013i
$323$	7.47857i	0.00128829i
$324$	0	0
$325$	0	0
$326$	517.492	0.0879179
$327$	0	0
$328$	− 1160.89i	− 0.195425i
$329$	−1630.59	−0.273244
$330$	0	0
$331$	−10482.8	−1.74075	−0.870373	−	0.492393i	$-0.836122\pi$
$331$	−10482.8	−1.74075	−0.870373	+	0.492393i	$0.836122\pi$
$332$	5245.94i	0.867194i
$333$	0	0
$334$	−8011.70	−1.31252
$335$	0	0
$336$	0	0
$337$	− 4156.75i	− 0.671907i	−0.941879	−	0.335953i	$-0.890942\pi$
$337$	− 4156.75i	− 0.671907i	0.941879	−	0.335953i	$-0.109058\pi$
$338$	− 3125.41i	− 0.502958i
$339$	0	0
$340$	0	0
$341$	−6146.75	−0.976144
$342$	0	0
$343$	− 3055.51i	− 0.480998i
$344$	−2316.75	−0.363112
$345$	0	0
$346$	7870.53	1.22290
$347$	631.122i	0.0976380i	0.998808	+	0.0488190i	$0.0155458\pi$
$347$	631.122i	0.0976380i	−0.998808	+	0.0488190i	$0.984454\pi$
$348$	0	0
$349$	−3245.69	−0.497816	−0.248908	−	0.968527i	$-0.580072\pi$
$349$	−3245.69	−0.497816	−0.248908	+	0.968527i	$0.580072\pi$
$350$	0	0
$351$	0	0
$352$	1675.86i	0.253760i
$353$	10729.8i	1.61782i	0.587935	+	0.808908i	$0.299941\pi$
$353$	10729.8i	1.61782i	−0.587935	+	0.808908i	$0.700059\pi$
$354$	0	0
$355$	0	0
$356$	990.216	0.147420
$357$	0	0
$358$	6538.96i	0.965349i
$359$	7941.27	1.16748	0.583738	−	0.811942i	$-0.301589\pi$
$359$	7941.27	1.16748	0.583738	+	0.811942i	$0.301589\pi$
$360$	0	0
$361$	−6858.97	−0.999995
$362$	9216.61i	1.33816i
$363$	0	0
$364$	−2839.41	−0.408861
$365$	0	0
$366$	0	0
$367$	− 9929.13i	− 1.41225i	−0.708087	−	0.706126i	$-0.750441\pi$
$367$	− 9929.13i	− 1.41225i	0.708087	−	0.706126i	$-0.249559\pi$
$368$	2443.86i	0.346182i
$369$	0	0
$370$	0	0
$371$	−20479.2	−2.86584
$372$	0	0
$373$	− 2381.16i	− 0.330540i	−0.986248	−	0.165270i	$-0.947150\pi$
$373$	− 2381.16i	− 0.330540i	0.986248	−	0.165270i	$-0.0528496\pi$
$374$	−4228.45	−0.584620
$375$	0	0
$376$	−462.820	−0.0634790
$377$	714.518i	0.0976115i
$378$	0	0
$379$	−12135.5	−1.64475	−0.822374	−	0.568947i	$-0.807351\pi$
$379$	−12135.5	−1.64475	−0.822374	+	0.568947i	$0.807351\pi$
$380$	0	0
$381$	0	0
$382$	5617.92i	0.752455i
$383$	3033.87i	0.404762i	0.979307	+	0.202381i	$0.0648679\pi$
$383$	3033.87i	0.404762i	−0.979307	+	0.202381i	$0.935132\pi$
$384$	0	0
$385$	0	0
$386$	−8154.90	−1.07532
$387$	0	0
$388$	150.957i	0.0197518i
$389$	5478.74	0.714096	0.357048	−	0.934086i	$-0.383783\pi$
$389$	5478.74	0.714096	0.357048	+	0.934086i	$0.383783\pi$
$390$	0	0
$391$	−6166.23	−0.797544
$392$	− 3611.27i	− 0.465297i
$393$	0	0
$394$	2267.72	0.289965
$395$	0	0
$396$	0	0
$397$	3120.04i	0.394434i	0.980360	+	0.197217i	$0.0631903\pi$
$397$	3120.04i	0.394434i	−0.980360	+	0.197217i	$0.936810\pi$
$398$	− 4361.64i	− 0.549319i
$399$	0	0
$400$	0	0
$401$	10870.7	1.35375	0.676877	−	0.736097i	$-0.263333\pi$
$401$	10870.7	1.35375	0.676877	+	0.736097i	$0.263333\pi$
$402$	0	0
$403$	− 2956.01i	− 0.365382i
$404$	5834.37	0.718492
$405$	0	0
$406$	−1599.26	−0.195492
$407$	19650.6i	2.39323i
$408$	0	0
$409$	12519.4	1.51356	0.756780	−	0.653670i	$-0.226771\pi$
$409$	12519.4	1.51356	0.756780	+	0.653670i	$0.226771\pi$
$410$	0	0
$411$	0	0
$412$	4036.89i	0.482727i
$413$	− 5227.86i	− 0.622872i
$414$	0	0
$415$	0	0
$416$	−805.928	−0.0949853
$417$	0	0
$418$	19.4031i	0.00227042i
$419$	3922.10	0.457296	0.228648	−	0.973509i	$-0.426569\pi$
$419$	3922.10	0.457296	0.228648	+	0.973509i	$0.426569\pi$
$420$	0	0
$421$	11969.5	1.38564	0.692822	−	0.721108i	$-0.256367\pi$
$421$	11969.5	1.38564	0.692822	+	0.721108i	$0.256367\pi$
$422$	− 3212.61i	− 0.370586i
$423$	0	0
$424$	−5812.75	−0.665783
$425$	0	0
$426$	0	0
$427$	− 7863.73i	− 0.891224i
$428$	7117.64i	0.803841i
$429$	0	0
$430$	0	0
$431$	−1940.67	−0.216888	−0.108444	−	0.994103i	$-0.534587\pi$
$431$	−1940.67	−0.216888	−0.108444	+	0.994103i	$0.534587\pi$
$432$	0	0
$433$	− 1393.50i	− 0.154659i	−0.997006	−	0.0773297i	$-0.975361\pi$
$433$	− 1393.50i	− 0.154659i	0.997006	−	0.0773297i	$-0.0246394\pi$
$434$	6616.23	0.731773
$435$	0	0
$436$	1590.96	0.174755
$437$	28.2950i	0.00309733i
$438$	0	0
$439$	17781.5	1.93317	0.966587	−	0.256339i	$-0.0825165\pi$
$439$	17781.5	1.93317	0.966587	+	0.256339i	$0.0825165\pi$
$440$	0	0
$441$	0	0
$442$	− 2033.48i	− 0.218830i
$443$	− 14243.7i	− 1.52763i	−0.645435	−	0.763815i	$-0.723324\pi$
$443$	− 14243.7i	− 1.52763i	0.645435	−	0.763815i	$-0.276676\pi$
$444$	0	0
$445$	0	0
$446$	8739.49	0.927863
$447$	0	0
$448$	− 1803.86i	− 0.190233i
$449$	3047.47	0.320310	0.160155	−	0.987092i	$-0.448801\pi$
$449$	3047.47	0.320310	0.160155	+	0.987092i	$0.448801\pi$
$450$	0	0
$451$	7599.56	0.793458
$452$	100.144i	0.0104212i
$453$	0	0
$454$	−8042.68	−0.831413
$455$	0	0
$456$	0	0
$457$	974.453i	0.0997439i	0.998756	+	0.0498720i	$0.0158813\pi$
$457$	974.453i	0.0997439i	−0.998756	+	0.0498720i	$0.984119\pi$
$458$	2546.32i	0.259785i
$459$	0	0
$460$	0	0
$461$	−6079.10	−0.614169	−0.307085	−	0.951682i	$-0.599353\pi$
$461$	−6079.10	−0.614169	−0.307085	+	0.951682i	$0.599353\pi$
$462$	0	0
$463$	− 4226.11i	− 0.424198i	−0.977248	−	0.212099i	$-0.931970\pi$
$463$	− 4226.11i	− 0.424198i	0.977248	−	0.212099i	$-0.0680300\pi$
$464$	−453.928	−0.0454161
$465$	0	0
$466$	−1026.24	−0.102016
$467$	5482.66i	0.543271i	0.962400	+	0.271635i	$0.0875644\pi$
$467$	5482.66i	0.543271i	−0.962400	+	0.271635i	$0.912436\pi$
$468$	0	0
$469$	26487.9	2.60788
$470$	0	0
$471$	0	0
$472$	− 1483.86i	− 0.144703i
$473$	− 15166.2i	− 1.47429i
$474$	0	0
$475$	0	0
$476$	4551.41	0.438264
$477$	0	0
$478$	9434.83i	0.902801i
$479$	−12919.7	−1.23240	−0.616199	−	0.787591i	$-0.711328\pi$
$479$	−12919.7	−1.23240	−0.616199	+	0.787591i	$0.711328\pi$
$480$	0	0
$481$	−9450.08	−0.895814
$482$	6237.93i	0.589481i
$483$	0	0
$484$	−5646.68	−0.530304
$485$	0	0
$486$	0	0
$487$	19124.1i	1.77946i	0.456488	+	0.889730i	$0.349107\pi$
$487$	19124.1i	1.77946i	−0.456488	+	0.889730i	$0.650893\pi$
$488$	− 2232.01i	− 0.207046i
$489$	0	0
$490$	0	0
$491$	15409.8	1.41637	0.708184	−	0.706028i	$-0.249515\pi$
$491$	15409.8	1.41637	0.708184	+	0.706028i	$0.249515\pi$
$492$	0	0
$493$	− 1145.33i	− 0.104631i
$494$	−9.33106	−0.000849846 0
$495$	0	0
$496$	1877.93	0.170003
$497$	24201.8i	2.18430i
$498$	0	0
$499$	−8302.79	−0.744858	−0.372429	−	0.928061i	$-0.621475\pi$
$499$	−8302.79	−0.744858	−0.372429	+	0.928061i	$0.621475\pi$
$500$	0	0
$501$	0	0
$502$	− 185.935i	− 0.0165312i
$503$	18380.6i	1.62932i	0.579937	+	0.814661i	$0.303077\pi$
$503$	18380.6i	1.62932i	−0.579937	+	0.814661i	$0.696923\pi$
$504$	0	0
$505$	0	0
$506$	−15998.2	−1.40555
$507$	0	0
$508$	− 2698.68i	− 0.235698i
$509$	−14596.5	−1.27107	−0.635537	−	0.772070i	$-0.719221\pi$
$509$	−14596.5	−1.27107	−0.635537	+	0.772070i	$0.719221\pi$
$510$	0	0
$511$	26923.2	2.33075
$512$	− 512.000i	− 0.0441942i
$513$	0	0
$514$	−10513.9	−0.902236
$515$	0	0
$516$	0	0
$517$	− 3029.76i	− 0.257735i
$518$	− 21151.5i	− 1.79410i
$519$	0	0
$520$	0	0
$521$	18819.6	1.58254	0.791268	−	0.611469i	$-0.209421\pi$
$521$	18819.6	1.58254	0.791268	+	0.611469i	$0.209421\pi$
$522$	0	0
$523$	− 17834.8i	− 1.49113i	−0.666433	−	0.745565i	$-0.732180\pi$
$523$	− 17834.8i	− 1.49113i	0.666433	−	0.745565i	$-0.267820\pi$
$524$	4935.41	0.411459
$525$	0	0
$526$	12679.0	1.05101
$527$	4738.31i	0.391658i
$528$	0	0
$529$	−11162.8	−0.917466
$530$	0	0
$531$	0	0
$532$	− 20.8851i	− 0.00170204i
$533$	3654.67i	0.297001i
$534$	0	0
$535$	0	0
$536$	7518.23	0.605855
$537$	0	0
$538$	10227.3i	0.819571i
$539$	23640.5	1.88918
$540$	0	0
$541$	−5983.39	−0.475501	−0.237751	−	0.971326i	$-0.576410\pi$
$541$	−5983.39	−0.475501	−0.237751	+	0.971326i	$0.576410\pi$
$542$	− 1257.14i	− 0.0996284i
$543$	0	0
$544$	1291.86	0.101816
$545$	0	0
$546$	0	0
$547$	− 23979.7i	− 1.87440i	−0.348790	−	0.937201i	$-0.613407\pi$
$547$	− 23979.7i	− 1.87440i	0.348790	−	0.937201i	$-0.386593\pi$
$548$	− 520.014i	− 0.0405363i
$549$	0	0
$550$	0	0
$551$	−5.25559	−0.000406344 0
$552$	0	0
$553$	7299.98i	0.561350i
$554$	2297.42	0.176188
$555$	0	0
$556$	−4695.57	−0.358159
$557$	− 5738.04i	− 0.436496i	−0.975893	−	0.218248i	$-0.929966\pi$
$557$	− 5738.04i	− 0.436496i	0.975893	−	0.218248i	$-0.0700342\pi$
$558$	0	0
$559$	7293.48	0.551845
$560$	0	0
$561$	0	0
$562$	− 1914.84i	− 0.143724i
$563$	− 18955.4i	− 1.41896i	−0.704724	−	0.709481i	$-0.748929\pi$
$563$	− 18955.4i	− 1.41896i	0.704724	−	0.709481i	$-0.251071\pi$
$564$	0	0
$565$	0	0
$566$	7931.95	0.589054
$567$	0	0
$568$	6869.35i	0.507450i
$569$	8104.14	0.597089	0.298544	−	0.954396i	$-0.403499\pi$
$569$	8104.14	0.597089	0.298544	+	0.954396i	$0.403499\pi$
$570$	0	0
$571$	−8698.74	−0.637532	−0.318766	−	0.947833i	$-0.603268\pi$
$571$	−8698.74	−0.637532	−0.318766	+	0.947833i	$0.603268\pi$
$572$	− 5275.86i	− 0.385655i
$573$	0	0
$574$	−8180.01	−0.594820
$575$	0	0
$576$	0	0
$577$	− 14736.8i	− 1.06326i	−0.846976	−	0.531631i	$-0.821579\pi$
$577$	− 14736.8i	− 1.06326i	0.846976	−	0.531631i	$-0.178421\pi$
$578$	− 6566.45i	− 0.472540i
$579$	0	0
$580$	0	0
$581$	36964.5	2.63950
$582$	0	0
$583$	− 38052.1i	− 2.70318i
$584$	7641.78	0.541471
$585$	0	0
$586$	−13476.5	−0.950014
$587$	15591.8i	1.09632i	0.836373	+	0.548161i	$0.184672\pi$
$587$	15591.8i	1.09632i	−0.836373	+	0.548161i	$0.815328\pi$
$588$	0	0
$589$	21.7427	0.00152104
$590$	0	0
$591$	0	0
$592$	− 6003.57i	− 0.416799i
$593$	− 16988.3i	− 1.17644i	−0.808702	−	0.588218i	$-0.799830\pi$
$593$	− 16988.3i	− 1.17644i	0.808702	−	0.588218i	$-0.200170\pi$
$594$	0	0
$595$	0	0
$596$	3442.66	0.236606
$597$	0	0
$598$	− 7693.64i	− 0.526114i
$599$	−12007.6	−0.819060	−0.409530	−	0.912297i	$-0.634307\pi$
$599$	−12007.6	−0.819060	−0.409530	+	0.912297i	$0.634307\pi$
$600$	0	0
$601$	4717.09	0.320157	0.160078	−	0.987104i	$-0.448825\pi$
$601$	4717.09	0.320157	0.160078	+	0.987104i	$0.448825\pi$
$602$	16324.5i	1.10521i
$603$	0	0
$604$	−9036.90	−0.608785
$605$	0	0
$606$	0	0
$607$	6237.70i	0.417101i	0.978012	+	0.208551i	$0.0668746\pi$
$607$	6237.70i	0.417101i	−0.978012	+	0.208551i	$0.933125\pi$
$608$	− 5.92795i	0 0.000395411i
$609$	0	0
$610$	0	0
$611$	1457.03	0.0964731
$612$	0	0
$613$	− 3203.02i	− 0.211042i	−0.994417	−	0.105521i	$-0.966349\pi$
$613$	− 3203.02i	− 0.211042i	0.994417	−	0.105521i	$-0.0336510\pi$
$614$	−10430.9	−0.685599
$615$	0	0
$616$	11808.6	0.772374
$617$	− 14971.6i	− 0.976878i	−0.872598	−	0.488439i	$-0.837566\pi$
$617$	− 14971.6i	− 0.976878i	0.872598	−	0.488439i	$-0.162434\pi$
$618$	0	0
$619$	−14437.5	−0.937464	−0.468732	−	0.883340i	$-0.655289\pi$
$619$	−14437.5	−0.937464	−0.468732	+	0.883340i	$0.655289\pi$
$620$	0	0
$621$	0	0
$622$	− 19104.8i	− 1.23156i
$623$	− 6977.37i	− 0.448704i
$624$	0	0
$625$	0	0
$626$	12675.3	0.809274
$627$	0	0
$628$	3730.82i	0.237064i
$629$	15147.9	0.960235
$630$	0	0
$631$	17955.5	1.13280	0.566399	−	0.824131i	$-0.308336\pi$
$631$	17955.5	1.13280	0.566399	+	0.824131i	$0.308336\pi$
$632$	2072.00i	0.130411i
$633$	0	0
$634$	−15785.7	−0.988847
$635$	0	0
$636$	0	0
$637$	11368.8i	0.707142i
$638$	− 2971.55i	− 0.184396i
$639$	0	0
$640$	0	0
$641$	−8385.78	−0.516721	−0.258361	−	0.966049i	$-0.583182\pi$
$641$	−8385.78	−0.516721	−0.258361	+	0.966049i	$0.583182\pi$
$642$	0	0
$643$	1823.57i	0.111843i	0.998435	+	0.0559213i	$0.0178096\pi$
$643$	1823.57i	0.111843i	−0.998435	+	0.0559213i	$0.982190\pi$
$644$	17220.2	1.05368
$645$	0	0
$646$	14.9571	0.000910961 0
$647$	27666.4i	1.68111i	0.541726	+	0.840555i	$0.317771\pi$
$647$	27666.4i	1.68111i	−0.541726	+	0.840555i	$0.682229\pi$
$648$	0	0
$649$	9713.78	0.587518
$650$	0	0
$651$	0	0
$652$	− 1034.98i	− 0.0621674i
$653$	9649.68i	0.578287i	0.957286	+	0.289143i	$0.0933704\pi$
$653$	9649.68i	0.578287i	−0.957286	+	0.289143i	$0.906630\pi$
$654$	0	0
$655$	0	0
$656$	−2321.78	−0.138187
$657$	0	0
$658$	3261.17i	0.193212i
$659$	30673.0	1.81313	0.906563	−	0.422070i	$-0.138696\pi$
$659$	30673.0	1.81313	0.906563	+	0.422070i	$0.138696\pi$
$660$	0	0
$661$	9087.92	0.534764	0.267382	−	0.963591i	$-0.413841\pi$
$661$	9087.92	0.534764	0.267382	+	0.963591i	$0.413841\pi$
$662$	20965.6i	1.23089i
$663$	0	0
$664$	10491.9	0.613199
$665$	0	0
$666$	0	0
$667$	− 4333.34i	− 0.251555i
$668$	16023.4i	0.928090i
$669$	0	0
$670$	0	0
$671$	14611.5	0.840639
$672$	0	0
$673$	20309.5i	1.16326i	0.813453	+	0.581630i	$0.197585\pi$
$673$	20309.5i	1.16326i	−0.813453	+	0.581630i	$0.802415\pi$
$674$	−8313.50	−0.475110
$675$	0	0
$676$	−6250.81	−0.355645
$677$	29276.8i	1.66204i	0.556243	+	0.831019i	$0.312242\pi$
$677$	29276.8i	1.66204i	−0.556243	+	0.831019i	$0.687758\pi$
$678$	0	0
$679$	1063.69	0.0601189
$680$	0	0
$681$	0	0
$682$	12293.5i	0.690238i
$683$	3919.18i	0.219565i	0.993956	+	0.109783i	$0.0350155\pi$
$683$	3919.18i	0.219565i	−0.993956	+	0.109783i	$0.964985\pi$
$684$	0	0
$685$	0	0
$686$	−6111.03	−0.340117
$687$	0	0
$688$	4633.50i	0.256759i
$689$	18299.4	1.01183
$690$	0	0
$691$	−20951.6	−1.15345	−0.576727	−	0.816937i	$-0.695670\pi$
$691$	−20951.6	−1.15345	−0.576727	+	0.816937i	$0.695670\pi$
$692$	− 15741.1i	− 0.864719i
$693$	0	0
$694$	1262.24	0.0690405
$695$	0	0
$696$	0	0
$697$	− 5858.22i	− 0.318359i
$698$	6491.38i	0.352009i
$699$	0	0
$700$	0	0
$701$	−12614.4	−0.679659	−0.339829	−	0.940487i	$-0.610369\pi$
$701$	−12614.4	−0.679659	−0.339829	+	0.940487i	$0.610369\pi$
$702$	0	0
$703$	− 69.5095i	− 0.00372916i
$704$	3351.71	0.179435
$705$	0	0
$706$	21459.6	1.14397
$707$	− 41110.8i	− 2.18689i
$708$	0	0
$709$	55.0908	0.00291817	0.00145908	−	0.999999i	$-0.499536\pi$
$709$	55.0908	0.00291817	0.00145908	+	0.999999i	$0.499536\pi$
$710$	0	0
$711$	0	0
$712$	− 1980.43i	− 0.104241i
$713$	17927.3i	0.941630i
$714$	0	0
$715$	0	0
$716$	13077.9	0.682605
$717$	0	0
$718$	− 15882.5i	− 0.825531i
$719$	−16400.3	−0.850662	−0.425331	−	0.905038i	$-0.639842\pi$
$719$	−16400.3	−0.850662	−0.425331	+	0.905038i	$0.639842\pi$
$720$	0	0
$721$	28445.2	1.46929
$722$	13717.9i	0.707103i
$723$	0	0
$724$	18433.2	0.946222
$725$	0	0
$726$	0	0
$727$	2987.48i	0.152407i	0.997092	+	0.0762033i	$0.0242798\pi$
$727$	2987.48i	0.152407i	−0.997092	+	0.0762033i	$0.975720\pi$
$728$	5678.82i	0.289109i
$729$	0	0
$730$	0	0
$731$	−11691.0	−0.591530
$732$	0	0
$733$	18077.6i	0.910928i	0.890254	+	0.455464i	$0.150527\pi$
$733$	18077.6i	0.910928i	−0.890254	+	0.455464i	$0.849473\pi$
$734$	−19858.3	−0.998612
$735$	0	0
$736$	4887.71	0.244787
$737$	49216.7i	2.45986i
$738$	0	0
$739$	−12841.8	−0.639234	−0.319617	−	0.947547i	$-0.603554\pi$
$739$	−12841.8	−0.639234	−0.319617	+	0.947547i	$0.603554\pi$
$740$	0	0
$741$	0	0
$742$	40958.4i	2.02646i
$743$	− 27072.5i	− 1.33673i	−0.743832	−	0.668367i	$-0.766994\pi$
$743$	− 27072.5i	− 1.33673i	0.743832	−	0.668367i	$-0.233006\pi$
$744$	0	0
$745$	0	0
$746$	−4762.31	−0.233727
$747$	0	0
$748$	8456.89i	0.413388i
$749$	50153.1	2.44667
$750$	0	0
$751$	6646.78	0.322962	0.161481	−	0.986876i	$-0.448373\pi$
$751$	6646.78	0.322962	0.161481	+	0.986876i	$0.448373\pi$
$752$	925.640i	0.0448864i
$753$	0	0
$754$	1429.04	0.0690218
$755$	0	0
$756$	0	0
$757$	4798.97i	0.230412i	0.993342	+	0.115206i	$0.0367528\pi$
$757$	4798.97i	0.230412i	−0.993342	+	0.115206i	$0.963247\pi$
$758$	24271.0i	1.16301i
$759$	0	0
$760$	0	0
$761$	−16722.8	−0.796585	−0.398293	−	0.917258i	$-0.630397\pi$
$761$	−16722.8	−0.796585	−0.398293	+	0.917258i	$0.630397\pi$
$762$	0	0
$763$	− 11210.4i	− 0.531907i
$764$	11235.8	0.532066
$765$	0	0
$766$	6067.75	0.286210
$767$	4671.41i	0.219915i
$768$	0	0
$769$	11706.8	0.548972	0.274486	−	0.961591i	$-0.411492\pi$
$769$	11706.8	0.548972	0.274486	+	0.961591i	$0.411492\pi$
$770$	0	0
$771$	0	0
$772$	16309.8i	0.760366i
$773$	− 360.830i	− 0.0167893i	−0.999965	−	0.00839467i	$-0.997328\pi$
$773$	− 360.830i	− 0.0167893i	0.999965	−	0.00839467i	$-0.00267214\pi$
$774$	0	0
$775$	0	0
$776$	301.914	0.0139666
$777$	0	0
$778$	− 10957.5i	− 0.504942i
$779$	−26.8817	−0.00123637
$780$	0	0
$781$	−44968.9	−2.06033
$782$	12332.5i	0.563949i
$783$	0	0
$784$	−7222.53	−0.329015
$785$	0	0
$786$	0	0
$787$	− 34121.3i	− 1.54548i	−0.634723	−	0.772740i	$-0.718886\pi$
$787$	− 34121.3i	− 1.54548i	0.634723	−	0.772740i	$-0.281114\pi$
$788$	− 4535.44i	− 0.205036i
$789$	0	0
$790$	0	0
$791$	705.647	0.0317192
$792$	0	0
$793$	7026.73i	0.314661i
$794$	6240.07	0.278907
$795$	0	0
$796$	−8723.27	−0.388427
$797$	34883.5i	1.55036i	0.631740	+	0.775180i	$0.282341\pi$
$797$	34883.5i	1.55036i	−0.631740	+	0.775180i	$0.717659\pi$
$798$	0	0
$799$	−2335.53	−0.103411
$800$	0	0
$801$	0	0
$802$	− 21741.3i	− 0.957248i
$803$	50025.5i	2.19846i
$804$	0	0
$805$	0	0
$806$	−5912.01	−0.258364
$807$	0	0
$808$	− 11668.7i	− 0.508051i
$809$	−21141.8	−0.918796	−0.459398	−	0.888231i	$-0.651935\pi$
$809$	−21141.8	−0.918796	−0.459398	+	0.888231i	$0.651935\pi$
$810$	0	0
$811$	22430.1	0.971178	0.485589	−	0.874187i	$-0.338605\pi$
$811$	22430.1	0.971178	0.485589	+	0.874187i	$0.338605\pi$
$812$	3198.52i	0.138234i
$813$	0	0
$814$	39301.2	1.69227
$815$	0	0
$816$	0	0
$817$	53.6467i	0.00229726i
$818$	− 25038.9i	− 1.07025i
$819$	0	0
$820$	0	0
$821$	−166.134	−0.00706226	−0.00353113	−	0.999994i	$-0.501124\pi$
$821$	−166.134	−0.00706226	−0.00353113	+	0.999994i	$0.501124\pi$
$822$	0	0
$823$	− 42816.6i	− 1.81348i	−0.421691	−	0.906740i	$-0.638563\pi$
$823$	− 42816.6i	− 1.81348i	0.421691	−	0.906740i	$-0.361437\pi$
$824$	8073.78	0.341339
$825$	0	0
$826$	−10455.7	−0.440437
$827$	− 17074.6i	− 0.717949i	−0.933347	−	0.358974i	$-0.883127\pi$
$827$	− 17074.6i	− 0.717949i	0.933347	−	0.358974i	$-0.116873\pi$
$828$	0	0
$829$	32106.1	1.34510	0.672552	−	0.740050i	$-0.265198\pi$
$829$	32106.1	1.34510	0.672552	+	0.740050i	$0.265198\pi$
$830$	0	0
$831$	0	0
$832$	1611.86i	0.0671647i
$833$	− 18223.6i	− 0.757995i
$834$	0	0
$835$	0	0
$836$	38.8062	0.00160543
$837$	0	0
$838$	− 7844.20i	− 0.323357i
$839$	−1673.80	−0.0688750	−0.0344375	−	0.999407i	$-0.510964\pi$
$839$	−1673.80	−0.0688750	−0.0344375	+	0.999407i	$0.510964\pi$
$840$	0	0
$841$	−23584.1	−0.966998
$842$	− 23938.9i	− 0.979799i
$843$	0	0
$844$	−6425.21	−0.262044
$845$	0	0
$846$	0	0
$847$	39788.2i	1.61410i
$848$	11625.5i	0.470780i
$849$	0	0
$850$	0	0
$851$	57311.9	2.30861
$852$	0	0
$853$	24897.4i	0.999380i	0.866204	+	0.499690i	$0.166553\pi$
$853$	24897.4i	0.999380i	−0.866204	+	0.499690i	$0.833447\pi$
$854$	−15727.5	−0.630191
$855$	0	0
$856$	14235.3	0.568402
$857$	− 35144.2i	− 1.40082i	−0.713740	−	0.700411i	$-0.753000\pi$
$857$	− 35144.2i	− 1.40082i	0.713740	−	0.700411i	$-0.247000\pi$
$858$	0	0
$859$	−48782.2	−1.93763	−0.968817	−	0.247779i	$-0.920299\pi$
$859$	−48782.2	−1.93763	−0.968817	+	0.247779i	$0.920299\pi$
$860$	0	0
$861$	0	0
$862$	3881.33i	0.153363i
$863$	− 17753.7i	− 0.700283i	−0.936697	−	0.350142i	$-0.886133\pi$
$863$	− 17753.7i	− 0.700283i	0.936697	−	0.350142i	$-0.113867\pi$
$864$	0	0
$865$	0	0
$866$	−2787.01	−0.109361
$867$	0	0
$868$	− 13232.5i	− 0.517441i
$869$	−13564.0	−0.529489
$870$	0	0
$871$	−23668.6	−0.920756
$872$	− 3181.93i	− 0.123571i
$873$	0	0
$874$	56.5901	0.00219015
$875$	0	0
$876$	0	0
$877$	− 37927.7i	− 1.46035i	−0.683261	−	0.730175i	$-0.739439\pi$
$877$	− 37927.7i	− 1.46035i	0.683261	−	0.730175i	$-0.260561\pi$
$878$	− 35562.9i	− 1.36696i
$879$	0	0
$880$	0	0
$881$	−11233.1	−0.429573	−0.214787	−	0.976661i	$-0.568906\pi$
$881$	−11233.1	−0.429573	−0.214787	+	0.976661i	$0.568906\pi$
$882$	0	0
$883$	34945.8i	1.33185i	0.746021	+	0.665923i	$0.231962\pi$
$883$	34945.8i	1.33185i	−0.746021	+	0.665923i	$0.768038\pi$
$884$	−4066.96	−0.154736
$885$	0	0
$886$	−28487.5	−1.08020
$887$	− 3461.39i	− 0.131028i	−0.997852	−	0.0655141i	$-0.979131\pi$
$887$	− 3461.39i	− 0.131028i	0.997852	−	0.0655141i	$-0.0208687\pi$
$888$	0	0
$889$	−19015.7	−0.717398
$890$	0	0
$891$	0	0
$892$	− 17479.0i	− 0.656098i
$893$	10.7171i	0 0.000401605i
$894$	0	0
$895$	0	0
$896$	−3607.71	−0.134515
$897$	0	0
$898$	− 6094.94i	− 0.226493i
$899$	−3329.86	−0.123534
$900$	0	0
$901$	−29332.9	−1.08460
$902$	− 15199.1i	− 0.561059i
$903$	0	0
$904$	200.288	0.00736890
$905$	0	0
$906$	0	0
$907$	− 389.271i	− 0.0142509i	−0.999975	−	0.00712543i	$-0.997732\pi$
$907$	− 389.271i	− 0.0142509i	0.999975	−	0.00712543i	$-0.00226811\pi$
$908$	16085.4i	0.587898i
$909$	0	0
$910$	0	0
$911$	10917.2	0.397039	0.198520	−	0.980097i	$-0.436387\pi$
$911$	10917.2	0.397039	0.198520	+	0.980097i	$0.436387\pi$
$912$	0	0
$913$	68683.1i	2.48968i
$914$	1948.91	0.0705296
$915$	0	0
$916$	5092.63	0.183696
$917$	− 34776.4i	− 1.25237i
$918$	0	0
$919$	−26616.2	−0.955372	−0.477686	−	0.878531i	$-0.658524\pi$
$919$	−26616.2	−0.955372	−0.477686	+	0.878531i	$0.658524\pi$
$920$	0	0
$921$	0	0
$922$	12158.2i	0.434283i
$923$	− 21625.8i	− 0.771204i
$924$	0	0
$925$	0	0
$926$	−8452.22	−0.299954
$927$	0	0
$928$	907.856i	0.0321140i
$929$	−34287.3	−1.21090	−0.605452	−	0.795882i	$-0.707008\pi$
$929$	−34287.3	−1.21090	−0.605452	+	0.795882i	$0.707008\pi$
$930$	0	0
$931$	−83.6227	−0.00294374
$932$	2052.47i	0.0721363i
$933$	0	0
$934$	10965.3	0.384150
$935$	0	0
$936$	0	0
$937$	36879.5i	1.28581i	0.765947	+	0.642903i	$0.222270\pi$
$937$	36879.5i	1.28581i	−0.765947	+	0.642903i	$0.777730\pi$
$938$	− 52975.8i	− 1.84405i
$939$	0	0
$940$	0	0
$941$	−7912.95	−0.274129	−0.137064	−	0.990562i	$-0.543767\pi$
$941$	−7912.95	−0.274129	−0.137064	+	0.990562i	$0.543767\pi$
$942$	0	0
$943$	− 22164.5i	− 0.765402i
$944$	−2967.71	−0.102321
$945$	0	0
$946$	−30332.3	−1.04248
$947$	15131.7i	0.519232i	0.965712	+	0.259616i	$0.0835960\pi$
$947$	15131.7i	0.519232i	−0.965712	+	0.259616i	$0.916404\pi$
$948$	0	0
$949$	−24057.5	−0.822909
$950$	0	0
$951$	0	0
$952$	− 9102.82i	− 0.309899i
$953$	22870.8i	0.777395i	0.921365	+	0.388697i	$0.127075\pi$
$953$	22870.8i	0.777395i	−0.921365	+	0.388697i	$0.872925\pi$
$954$	0	0
$955$	0	0
$956$	18869.7	0.638377
$957$	0	0
$958$	25839.5i	0.871436i
$959$	−3664.18	−0.123381
$960$	0	0
$961$	−16015.2	−0.537584
$962$	18900.2i	0.633436i
$963$	0	0
$964$	12475.9	0.416826
$965$	0	0
$966$	0	0
$967$	− 2121.16i	− 0.0705398i	−0.999378	−	0.0352699i	$-0.988771\pi$
$967$	− 2121.16i	− 0.0705398i	0.999378	−	0.0352699i	$-0.0112291\pi$
$968$	11293.4i	0.374981i
$969$	0	0
$970$	0	0
$971$	−1094.05	−0.0361584	−0.0180792	−	0.999837i	$-0.505755\pi$
$971$	−1094.05	−0.0361584	−0.0180792	+	0.999837i	$0.505755\pi$
$972$	0	0
$973$	33086.4i	1.09014i
$974$	38248.2	1.25827
$975$	0	0
$976$	−4464.03	−0.146404
$977$	− 45694.7i	− 1.49632i	−0.663519	−	0.748159i	$-0.730938\pi$
$977$	− 45694.7i	− 1.49632i	0.663519	−	0.748159i	$-0.269062\pi$
$978$	0	0
$979$	12964.5	0.423236
$980$	0	0
$981$	0	0
$982$	− 30819.7i	− 1.00152i
$983$	− 7658.58i	− 0.248495i	−0.992251	−	0.124248i	$-0.960348\pi$
$983$	− 7658.58i	− 0.248495i	0.992251	−	0.124248i	$-0.0396517\pi$
$984$	0	0
$985$	0	0
$986$	−2290.66	−0.0739853
$987$	0	0
$988$	18.6621i	0 0.000600932i
$989$	−44232.8	−1.42216
$990$	0	0
$991$	39774.1	1.27494	0.637470	−	0.770475i	$-0.279981\pi$
$991$	39774.1	1.27494	0.637470	+	0.770475i	$0.279981\pi$
$992$	− 3755.86i	− 0.120210i
$993$	0	0
$994$	48403.6	1.54454
$995$	0	0
$996$	0	0
$997$	47834.0i	1.51948i	0.650229	+	0.759739i	$0.274673\pi$
$997$	47834.0i	1.51948i	−0.650229	+	0.759739i	$0.725327\pi$
$998$	16605.6i	0.526694i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1350.4.c.x.649.2		4
3.2	odd	2		1350.4.c.y.649.4		4
5.2	odd	4		1350.4.a.bk.1.1	yes	2
5.3	odd	4		1350.4.a.bh.1.2	yes	2
5.4	even	2	inner	1350.4.c.x.649.3		4
15.2	even	4		1350.4.a.bd.1.1	✓	2
15.8	even	4		1350.4.a.bo.1.2	yes	2
15.14	odd	2		1350.4.c.y.649.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1350.4.a.bd.1.1	✓	2	15.2	even	4
1350.4.a.bh.1.2	yes	2	5.3	odd	4
1350.4.a.bk.1.1	yes	2	5.2	odd	4
1350.4.a.bo.1.2	yes	2	15.8	even	4
1350.4.c.x.649.2		4	1.1	even	1	trivial
1350.4.c.x.649.3		4	5.4	even	2	inner
1350.4.c.y.649.1		4	15.14	odd	2
1350.4.c.y.649.4		4	3.2	odd	2

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:	\( N \)	\(=\)	\( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1350.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{2} -4.00000 q^{4} +28.1852i q^{7} +8.00000i q^{8} +O(q^{10})\)	\(q-2.00000i q^{2} -4.00000 q^{4} +28.1852i q^{7} +8.00000i q^{8} -52.3705 q^{11} -25.1852i q^{13} +56.3705 q^{14} +16.0000 q^{16} +40.3705i q^{17} +0.185248 q^{19} +104.741i q^{22} +152.741i q^{23} -50.3705 q^{26} -112.741i q^{28} -28.3705 q^{29} +117.370 q^{31} -32.0000i q^{32} +80.7410 q^{34} -375.223i q^{37} -0.370497i q^{38} -145.111 q^{41} +289.593i q^{43} +209.482 q^{44} +305.482 q^{46} +57.8525i q^{47} -451.408 q^{49} +100.741i q^{52} +726.593i q^{53} -225.482 q^{56} +56.7410i q^{58} -185.482 q^{59} -279.002 q^{61} -234.741i q^{62} -64.0000 q^{64} -939.779i q^{67} -161.482i q^{68} +858.669 q^{71} -955.223i q^{73} -750.446 q^{74} -0.740994 q^{76} -1476.08i q^{77} +259.000 q^{79} +290.223i q^{82} -1311.49i q^{83} +579.187 q^{86} -418.964i q^{88} -247.554 q^{89} +709.852 q^{91} -610.964i q^{92} +115.705 q^{94} -37.7393i q^{97} +902.816i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 16 q^{4}+O(q^{10}) \) 4 * q - 16 * q^4	\( 4 q - 16 q^{4} - 36 q^{11} + 52 q^{14} + 64 q^{16} - 86 q^{19} - 28 q^{26} + 60 q^{29} + 296 q^{31} - 24 q^{34} - 60 q^{41} + 144 q^{44} + 528 q^{46} - 678 q^{49} - 208 q^{56} - 48 q^{59} + 1226 q^{61} - 256 q^{64} + 312 q^{71} - 920 q^{74} + 344 q^{76} + 1036 q^{79} - 112 q^{86} - 3072 q^{89} + 1972 q^{91} - 1272 q^{94}+O(q^{100}) \) 4 * q - 16 * q^4 - 36 * q^11 + 52 * q^14 + 64 * q^16 - 86 * q^19 - 28 * q^26 + 60 * q^29 + 296 * q^31 - 24 * q^34 - 60 * q^41 + 144 * q^44 + 528 * q^46 - 678 * q^49 - 208 * q^56 - 48 * q^59 + 1226 * q^61 - 256 * q^64 + 312 * q^71 - 920 * q^74 + 344 * q^76 + 1036 * q^79 - 112 * q^86 - 3072 * q^89 + 1972 * q^91 - 1272 * q^94

Introduction

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