LMFDB - Embedded newform 6030.2.d.l.2411.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6030,2,Mod(2411,6030)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6030, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("6030.2411");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6030 = 2 \cdot 3^{2} \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6030.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$48.1497924188$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2411.9
Character	$\chi$	$=$	6030.2411
Dual form			6030.2.d.l.2411.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.58772i q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.58772i q^{7} +1.00000 q^{8} +1.00000 q^{10} +0.711582 q^{11} +6.95877i q^{13} -1.58772i q^{14} +1.00000 q^{16} -5.31770i q^{17} -1.17555 q^{19} +1.00000 q^{20} +0.711582 q^{22} -5.40166i q^{23} +1.00000 q^{25} +6.95877i q^{26} -1.58772i q^{28} -4.19576i q^{29} -5.83096i q^{31} +1.00000 q^{32} -5.31770i q^{34} -1.58772i q^{35} -5.35597 q^{37} -1.17555 q^{38} +1.00000 q^{40} +6.36412 q^{41} -7.11575i q^{43} +0.711582 q^{44} -5.40166i q^{46} -13.0943i q^{47} +4.47913 q^{49} +1.00000 q^{50} +6.95877i q^{52} +3.99176 q^{53} +0.711582 q^{55} -1.58772i q^{56} -4.19576i q^{58} +12.3817i q^{59} +11.4982i q^{61} -5.83096i q^{62} +1.00000 q^{64} +6.95877i q^{65} +(1.76399 - 7.99302i) q^{67} -5.31770i q^{68} -1.58772i q^{70} -4.17947i q^{71} +14.5463 q^{73} -5.35597 q^{74} -1.17555 q^{76} -1.12980i q^{77} -8.91625i q^{79} +1.00000 q^{80} +6.36412 q^{82} +16.7582i q^{83} -5.31770i q^{85} -7.11575i q^{86} +0.711582 q^{88} +12.9687i q^{89} +11.0486 q^{91} -5.40166i q^{92} -13.0943i q^{94} -1.17555 q^{95} -0.688811i q^{97} +4.47913 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 24 q^{2} + 24 q^{4} + 24 q^{5} + 24 q^{8}+O(q^{10}) $ 24 * q + 24 * q^2 + 24 * q^4 + 24 * q^5 + 24 * q^8	$ 24 q + 24 q^{2} + 24 q^{4} + 24 q^{5} + 24 q^{8} + 24 q^{10} + 12 q^{11} + 24 q^{16} + 4 q^{19} + 24 q^{20} + 12 q^{22} + 24 q^{25} + 24 q^{32} - 16 q^{37} + 4 q^{38} + 24 q^{40} + 8 q^{41} + 12 q^{44} - 20 q^{49} + 24 q^{50} + 24 q^{53} + 12 q^{55} + 24 q^{64} - 32 q^{67} - 4 q^{73} - 16 q^{74} + 4 q^{76} + 24 q^{80} + 8 q^{82} + 12 q^{88} + 4 q^{95} - 20 q^{98}+O(q^{100}) $ 24 * q + 24 * q^2 + 24 * q^4 + 24 * q^5 + 24 * q^8 + 24 * q^10 + 12 * q^11 + 24 * q^16 + 4 * q^19 + 24 * q^20 + 12 * q^22 + 24 * q^25 + 24 * q^32 - 16 * q^37 + 4 * q^38 + 24 * q^40 + 8 * q^41 + 12 * q^44 - 20 * q^49 + 24 * q^50 + 24 * q^53 + 12 * q^55 + 24 * q^64 - 32 * q^67 - 4 * q^73 - 16 * q^74 + 4 * q^76 + 24 * q^80 + 8 * q^82 + 12 * q^88 + 4 * q^95 - 20 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6030\mathbb{Z}\right)^\times$.

$n$	$1207$	$3151$	$4691$
$\chi(n)$	$1$	$-1$	$-1$

Coefficient data

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

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

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

$2$	1.00000			0.707107
$2$	1.00000			0.707107
$3$		0			0
$3$		0			0
$4$	1.00000			0.500000
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$		0			0
$7$		−	1.58772i		−	0.600103i	−0.953923	−	0.300052i	$-0.902996\pi$
$7$		−	1.58772i		−	0.600103i	0.953923	−	0.300052i	$-0.0970039\pi$
$8$	1.00000			0.353553
$9$		0			0
$10$	1.00000			0.316228
$11$	0.711582			0.214550			0.107275	−	0.994229i	$-0.465787\pi$
$11$	0.711582			0.214550			0.107275	+	0.994229i	$0.465787\pi$
$12$		0			0
$13$			6.95877i			1.93002i	0.262222	+	0.965008i	$0.415545\pi$
$13$			6.95877i			1.93002i	−0.262222	+	0.965008i	$0.584455\pi$
$14$		−	1.58772i		−	0.424337i
$15$		0			0
$16$	1.00000			0.250000
$17$		−	5.31770i		−	1.28973i	−0.764296	−	0.644865i	$-0.776913\pi$
$17$		−	5.31770i		−	1.28973i	0.764296	−	0.644865i	$-0.223087\pi$
$18$		0			0
$19$	−1.17555			−0.269689			−0.134845	−	0.990867i	$-0.543054\pi$
$19$	−1.17555			−0.269689			−0.134845	+	0.990867i	$0.543054\pi$
$20$	1.00000			0.223607
$21$		0			0
$22$	0.711582			0.151710
$23$		−	5.40166i		−	1.12632i	−0.826346	−	0.563162i	$-0.809585\pi$
$23$		−	5.40166i		−	1.12632i	0.826346	−	0.563162i	$-0.190415\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$			6.95877i			1.36473i
$27$		0			0
$28$		−	1.58772i		−	0.300052i
$29$		−	4.19576i		−	0.779134i	−0.920998	−	0.389567i	$-0.872625\pi$
$29$		−	4.19576i		−	0.779134i	0.920998	−	0.389567i	$-0.127375\pi$
$30$		0			0
$31$		−	5.83096i		−	1.04727i	−0.851942	−	0.523636i	$-0.824575\pi$
$31$		−	5.83096i		−	1.04727i	0.851942	−	0.523636i	$-0.175425\pi$
$32$	1.00000			0.176777
$33$		0			0
$34$		−	5.31770i		−	0.911977i
$35$		−	1.58772i		−	0.268374i
$36$		0			0
$37$	−5.35597			−0.880517			−0.440258	−	0.897871i	$-0.645113\pi$
$37$	−5.35597			−0.880517			−0.440258	+	0.897871i	$0.645113\pi$
$38$	−1.17555			−0.190699
$39$		0			0
$40$	1.00000			0.158114
$41$	6.36412			0.993908			0.496954	−	0.867777i	$-0.334452\pi$
$41$	6.36412			0.993908			0.496954	+	0.867777i	$0.334452\pi$
$42$		0			0
$43$		−	7.11575i		−	1.08514i	−0.840010	−	0.542571i	$-0.817451\pi$
$43$		−	7.11575i		−	1.08514i	0.840010	−	0.542571i	$-0.182549\pi$
$44$	0.711582			0.107275
$45$		0			0
$46$		−	5.40166i		−	0.796432i
$47$		−	13.0943i		−	1.91000i	−0.296599	−	0.955002i	$-0.595853\pi$
$47$		−	13.0943i		−	1.91000i	0.296599	−	0.955002i	$-0.404147\pi$
$48$		0			0
$49$	4.47913			0.639876
$50$	1.00000			0.141421
$51$		0			0
$52$			6.95877i			0.965008i
$53$	3.99176			0.548311			0.274155	−	0.961685i	$-0.411602\pi$
$53$	3.99176			0.548311			0.274155	+	0.961685i	$0.411602\pi$
$54$		0			0
$55$	0.711582			0.0959497
$56$		−	1.58772i		−	0.212169i
$57$		0			0
$58$		−	4.19576i		−	0.550931i
$59$			12.3817i			1.61196i	0.591944	+	0.805979i	$0.298361\pi$
$59$			12.3817i			1.61196i	−0.591944	+	0.805979i	$0.701639\pi$
$60$		0			0
$61$			11.4982i			1.47219i	0.676879	+	0.736094i	$0.263332\pi$
$61$			11.4982i			1.47219i	−0.676879	+	0.736094i	$0.736668\pi$
$62$		−	5.83096i		−	0.740533i
$63$		0			0
$64$	1.00000			0.125000
$65$			6.95877i			0.863129i
$66$		0			0
$67$	1.76399	−	7.99302i	0.215505	−	0.976503i
$67$	1.76399	−	7.99302i	0.215505	−	0.976503i
$68$		−	5.31770i		−	0.644865i
$69$		0			0
$70$		−	1.58772i		−	0.189769i
$71$		−	4.17947i		−	0.496012i	−0.968758	−	0.248006i	$-0.920225\pi$
$71$		−	4.17947i		−	0.496012i	0.968758	−	0.248006i	$-0.0797753\pi$
$72$		0			0
$73$	14.5463			1.70252			0.851260	−	0.524744i	$-0.175839\pi$
$73$	14.5463			1.70252			0.851260	+	0.524744i	$0.175839\pi$
$74$	−5.35597			−0.622619
$75$		0			0
$76$	−1.17555			−0.134845
$77$		−	1.12980i		−	0.128752i
$78$		0			0
$79$		−	8.91625i		−	1.00316i	−0.865112	−	0.501578i	$-0.832753\pi$
$79$		−	8.91625i		−	1.00316i	0.865112	−	0.501578i	$-0.167247\pi$
$80$	1.00000			0.111803
$81$		0			0
$82$	6.36412			0.702799
$83$			16.7582i			1.83945i	0.392560	+	0.919727i	$0.371590\pi$
$83$			16.7582i			1.83945i	−0.392560	+	0.919727i	$0.628410\pi$
$84$		0			0
$85$		−	5.31770i		−	0.576785i
$86$		−	7.11575i		−	0.767311i
$87$		0			0
$88$	0.711582			0.0758549
$89$			12.9687i			1.37468i	0.726336	+	0.687340i	$0.241222\pi$
$89$			12.9687i			1.37468i	−0.726336	+	0.687340i	$0.758778\pi$
$90$		0			0
$91$	11.0486			1.15821
$92$		−	5.40166i		−	0.563162i
$93$		0			0
$94$		−	13.0943i		−	1.35058i
$95$	−1.17555			−0.120609
$96$		0			0
$97$		−	0.688811i		−	0.0699381i	−0.999388	−	0.0349691i	$-0.988867\pi$
$97$		−	0.688811i		−	0.0699381i	0.999388	−	0.0349691i	$-0.0111333\pi$
$98$	4.47913			0.452461
$99$		0			0
$100$	1.00000			0.100000
$101$	11.8717			1.18128			0.590641	−	0.806935i	$-0.298875\pi$
$101$	11.8717			1.18128			0.590641	+	0.806935i	$0.298875\pi$
$102$		0			0
$103$	1.07806			0.106224			0.0531122	−	0.998589i	$-0.483086\pi$
$103$	1.07806			0.106224			0.0531122	+	0.998589i	$0.483086\pi$
$104$			6.95877i			0.682363i
$105$		0			0
$106$	3.99176			0.387714
$107$			6.76441i			0.653940i	0.945035	+	0.326970i	$0.106028\pi$
$107$			6.76441i			0.653940i	−0.945035	+	0.326970i	$0.893972\pi$
$108$		0			0
$109$		−	6.75749i		−	0.647250i	−0.946185	−	0.323625i	$-0.895098\pi$
$109$		−	6.75749i		−	0.647250i	0.946185	−	0.323625i	$-0.104902\pi$
$110$	0.711582			0.0678467
$111$		0			0
$112$		−	1.58772i		−	0.150026i
$113$	14.1799			1.33393			0.666965	−	0.745089i	$-0.267593\pi$
$113$	14.1799			1.33393			0.666965	+	0.745089i	$0.267593\pi$
$114$		0			0
$115$		−	5.40166i		−	0.503708i
$116$		−	4.19576i		−	0.389567i
$117$		0			0
$118$			12.3817i			1.13983i
$119$	−8.44304			−0.773972
$120$		0			0
$121$	−10.4937			−0.953968
$122$			11.4982i			1.04099i
$123$		0			0
$124$		−	5.83096i		−	0.523636i
$125$	1.00000			0.0894427
$126$		0			0
$127$	−3.31458			−0.294121			−0.147060	−	0.989127i	$-0.546981\pi$
$127$	−3.31458			−0.294121			−0.147060	+	0.989127i	$0.546981\pi$
$128$	1.00000			0.0883883
$129$		0			0
$130$			6.95877i			0.610324i
$131$		−	6.11885i		−	0.534606i	−0.963613	−	0.267303i	$-0.913868\pi$
$131$		−	6.11885i		−	0.534606i	0.963613	−	0.267303i	$-0.0861325\pi$
$132$		0			0
$133$			1.86645i			0.161841i
$134$	1.76399	−	7.99302i	0.152385	−	0.690492i
$135$		0			0
$136$		−	5.31770i		−	0.455989i
$137$	−2.12775			−0.181786			−0.0908930	−	0.995861i	$-0.528972\pi$
$137$	−2.12775			−0.181786			−0.0908930	+	0.995861i	$0.528972\pi$
$138$		0			0
$139$		−	1.55407i		−	0.131815i	−0.997826	−	0.0659073i	$-0.979006\pi$
$139$		−	1.55407i		−	0.131815i	0.997826	−	0.0659073i	$-0.0209942\pi$
$140$		−	1.58772i		−	0.134187i
$141$		0			0
$142$		−	4.17947i		−	0.350734i
$143$			4.95173i			0.414085i
$144$		0			0
$145$		−	4.19576i		−	0.348439i
$146$	14.5463			1.20386
$147$		0			0
$148$	−5.35597			−0.440258
$149$		−	21.2482i		−	1.74072i	−0.492419	−	0.870358i	$-0.663887\pi$
$149$		−	21.2482i		−	1.74072i	0.492419	−	0.870358i	$-0.336113\pi$
$150$		0			0
$151$	−18.4180			−1.49884			−0.749419	−	0.662096i	$-0.769667\pi$
$151$	−18.4180			−1.49884			−0.749419	+	0.662096i	$0.769667\pi$
$152$	−1.17555			−0.0953495
$153$		0			0
$154$		−	1.12980i		−	0.0910415i
$155$		−	5.83096i		−	0.468354i
$156$		0			0
$157$	12.8185			1.02302			0.511512	−	0.859276i	$-0.329086\pi$
$157$	12.8185			1.02302			0.511512	+	0.859276i	$0.329086\pi$
$158$		−	8.91625i		−	0.709339i
$159$		0			0
$160$	1.00000			0.0790569
$161$	−8.57635			−0.675911
$162$		0			0
$163$	16.2040			1.26920			0.634599	−	0.772842i	$-0.281165\pi$
$163$	16.2040			1.26920			0.634599	+	0.772842i	$0.281165\pi$
$164$	6.36412			0.496954
$165$		0			0
$166$			16.7582i			1.30069i
$167$		−	6.64308i		−	0.514057i	−0.966404	−	0.257028i	$-0.917257\pi$
$167$		−	6.64308i		−	0.514057i	0.966404	−	0.257028i	$-0.0827434\pi$
$168$		0			0
$169$	−35.4245			−2.72496
$170$		−	5.31770i		−	0.407849i
$171$		0			0
$172$		−	7.11575i		−	0.542571i
$173$		−	6.94551i		−	0.528057i	−0.964515	−	0.264029i	$-0.914949\pi$
$173$		−	6.94551i		−	0.528057i	0.964515	−	0.264029i	$-0.0850514\pi$
$174$		0			0
$175$		−	1.58772i		−	0.120021i
$176$	0.711582			0.0536375
$177$		0			0
$178$			12.9687i			0.972045i
$179$	14.6997			1.09871			0.549353	−	0.835591i	$-0.314874\pi$
$179$	14.6997			1.09871			0.549353	+	0.835591i	$0.314874\pi$
$180$		0			0
$181$	19.8624			1.47636			0.738180	−	0.674604i	$-0.235685\pi$
$181$	19.8624			1.47636			0.738180	+	0.674604i	$0.235685\pi$
$182$	11.0486			0.818977
$183$		0			0
$184$		−	5.40166i		−	0.398216i
$185$	−5.35597			−0.393779
$186$		0			0
$187$		−	3.78398i		−	0.276712i
$188$		−	13.0943i		−	0.955002i
$189$		0			0
$190$	−1.17555			−0.0852832
$191$	16.8296			1.21774			0.608872	−	0.793268i	$-0.291622\pi$
$191$	16.8296			1.21774			0.608872	+	0.793268i	$0.291622\pi$
$192$		0			0
$193$	−15.9265			−1.14642			−0.573209	−	0.819409i	$-0.694302\pi$
$193$	−15.9265			−1.14642			−0.573209	+	0.819409i	$0.694302\pi$
$194$		−	0.688811i		−	0.0494537i
$195$		0			0
$196$	4.47913			0.319938
$197$	−13.6290			−0.971023			−0.485511	−	0.874230i	$-0.661367\pi$
$197$	−13.6290			−0.971023			−0.485511	+	0.874230i	$0.661367\pi$
$198$		0			0
$199$	−0.654416			−0.0463903			−0.0231951	−	0.999731i	$-0.507384\pi$
$199$	−0.654416			−0.0463903			−0.0231951	+	0.999731i	$0.507384\pi$
$200$	1.00000			0.0707107
$201$		0			0
$202$	11.8717			0.835292
$203$	−6.66172			−0.467561
$204$		0			0
$205$	6.36412			0.444489
$206$	1.07806			0.0751119
$207$		0			0
$208$			6.95877i			0.482504i
$209$	−0.836498			−0.0578618
$210$		0			0
$211$	0.725900			0.0499730			0.0249865	−	0.999688i	$-0.492046\pi$
$211$	0.725900			0.0499730			0.0249865	+	0.999688i	$0.492046\pi$
$212$	3.99176			0.274155
$213$		0			0
$214$			6.76441i			0.462406i
$215$		−	7.11575i		−	0.485290i
$216$		0			0
$217$	−9.25796			−0.628472
$218$		−	6.75749i		−	0.457675i
$219$		0			0
$220$	0.711582			0.0479748
$221$	37.0046			2.48920
$222$		0			0
$223$	−13.0761			−0.875642			−0.437821	−	0.899062i	$-0.644250\pi$
$223$	−13.0761			−0.875642			−0.437821	+	0.899062i	$0.644250\pi$
$224$		−	1.58772i		−	0.106084i
$225$		0			0
$226$	14.1799			0.943230
$227$			11.9288i			0.791740i	0.918306	+	0.395870i	$0.129557\pi$
$227$			11.9288i			0.791740i	−0.918306	+	0.395870i	$0.870443\pi$
$228$		0			0
$229$			7.02413i			0.464167i	0.972696	+	0.232084i	$0.0745543\pi$
$229$			7.02413i			0.464167i	−0.972696	+	0.232084i	$0.925446\pi$
$230$		−	5.40166i		−	0.356175i
$231$		0			0
$232$		−	4.19576i		−	0.275465i
$233$	12.4117			0.813119			0.406559	−	0.913624i	$-0.366728\pi$
$233$	12.4117			0.813119			0.406559	+	0.913624i	$0.366728\pi$
$234$		0			0
$235$		−	13.0943i		−	0.854180i
$236$			12.3817i			0.805979i
$237$		0			0
$238$	−8.44304			−0.547281
$239$	−11.4278			−0.739205			−0.369603	−	0.929190i	$-0.620506\pi$
$239$	−11.4278			−0.739205			−0.369603	+	0.929190i	$0.620506\pi$
$240$		0			0
$241$	−24.5917			−1.58409			−0.792046	−	0.610462i	$-0.790984\pi$
$241$	−24.5917			−1.58409			−0.792046	+	0.610462i	$0.790984\pi$
$242$	−10.4937			−0.674557
$243$		0			0
$244$			11.4982i			0.736094i
$245$	4.47913			0.286161
$246$		0			0
$247$		−	8.18036i		−	0.520504i
$248$		−	5.83096i		−	0.370267i
$249$		0			0
$250$	1.00000			0.0632456
$251$	−6.82931			−0.431062			−0.215531	−	0.976497i	$-0.569148\pi$
$251$	−6.82931			−0.431062			−0.215531	+	0.976497i	$0.569148\pi$
$252$		0			0
$253$		−	3.84373i		−	0.241653i
$254$	−3.31458			−0.207975
$255$		0			0
$256$	1.00000			0.0625000
$257$			3.82314i			0.238481i	0.992865	+	0.119240i	$0.0380459\pi$
$257$			3.82314i			0.238481i	−0.992865	+	0.119240i	$0.961954\pi$
$258$		0			0
$259$			8.50381i			0.528401i
$260$			6.95877i			0.431565i
$261$		0			0
$262$		−	6.11885i		−	0.378024i
$263$		−	2.38656i		−	0.147161i	−0.997289	−	0.0735807i	$-0.976557\pi$
$263$		−	2.38656i		−	0.147161i	0.997289	−	0.0735807i	$-0.0234427\pi$
$264$		0			0
$265$	3.99176			0.245212
$266$			1.86645i			0.114439i
$267$		0			0
$268$	1.76399	−	7.99302i	0.107753	−	0.488251i
$269$		−	17.4694i		−	1.06513i	−0.846390	−	0.532564i	$-0.821229\pi$
$269$		−	17.4694i		−	1.06513i	0.846390	−	0.532564i	$-0.178771\pi$
$270$		0			0
$271$		−	12.4886i		−	0.758630i	−0.925268	−	0.379315i	$-0.876160\pi$
$271$		−	12.4886i		−	0.758630i	0.925268	−	0.379315i	$-0.123840\pi$
$272$		−	5.31770i		−	0.322433i
$273$		0			0
$274$	−2.12775			−0.128542
$275$	0.711582			0.0429100
$276$		0			0
$277$	2.26921			0.136344			0.0681718	−	0.997674i	$-0.478283\pi$
$277$	2.26921			0.136344			0.0681718	+	0.997674i	$0.478283\pi$
$278$		−	1.55407i		−	0.0932070i
$279$		0			0
$280$		−	1.58772i		−	0.0948847i
$281$	−22.8990			−1.36604			−0.683019	−	0.730401i	$-0.739333\pi$
$281$	−22.8990			−1.36604			−0.683019	+	0.730401i	$0.739333\pi$
$282$		0			0
$283$	2.00317			0.119076			0.0595379	−	0.998226i	$-0.481037\pi$
$283$	2.00317			0.119076			0.0595379	+	0.998226i	$0.481037\pi$
$284$		−	4.17947i		−	0.248006i
$285$		0			0
$286$			4.95173i			0.292802i
$287$		−	10.1045i		−	0.596447i
$288$		0			0
$289$	−11.2779			−0.663406
$290$		−	4.19576i		−	0.246384i
$291$		0			0
$292$	14.5463			0.851260
$293$			8.54078i			0.498957i	0.968380	+	0.249479i	$0.0802593\pi$
$293$			8.54078i			0.498957i	−0.968380	+	0.249479i	$0.919741\pi$
$294$		0			0
$295$			12.3817i			0.720890i
$296$	−5.35597			−0.311310
$297$		0			0
$298$		−	21.2482i		−	1.23087i
$299$	37.5889			2.17382
$300$		0			0
$301$	−11.2979			−0.651197
$302$	−18.4180			−1.05984
$303$		0			0
$304$	−1.17555			−0.0674223
$305$			11.4982i			0.658382i
$306$		0			0
$307$	−11.3122			−0.645620			−0.322810	−	0.946464i	$-0.604627\pi$
$307$	−11.3122			−0.645620			−0.322810	+	0.946464i	$0.604627\pi$
$308$		−	1.12980i		−	0.0643761i
$309$		0			0
$310$		−	5.83096i		−	0.331177i
$311$	14.0079			0.794315			0.397157	−	0.917751i	$-0.369997\pi$
$311$	14.0079			0.794315			0.397157	+	0.917751i	$0.369997\pi$
$312$		0			0
$313$		−	28.7123i		−	1.62291i	−0.584413	−	0.811456i	$-0.698675\pi$
$313$		−	28.7123i		−	1.62291i	0.584413	−	0.811456i	$-0.301325\pi$
$314$	12.8185			0.723387
$315$		0			0
$316$		−	8.91625i		−	0.501578i
$317$			0.296782i			0.0166690i	0.999965	+	0.00833448i	$0.00265298\pi$
$317$			0.296782i			0.0166690i	−0.999965	+	0.00833448i	$0.997347\pi$
$318$		0			0
$319$		−	2.98563i		−	0.167163i
$320$	1.00000			0.0559017
$321$		0			0
$322$	−8.57635			−0.477941
$323$			6.25120i			0.347826i
$324$		0			0
$325$			6.95877i			0.386003i
$326$	16.2040			0.897459
$327$		0			0
$328$	6.36412			0.351399
$329$	−20.7902			−1.14620
$330$		0			0
$331$		−	5.55383i		−	0.305266i	−0.988283	−	0.152633i	$-0.951225\pi$
$331$		−	5.55383i		−	0.305266i	0.988283	−	0.152633i	$-0.0487753\pi$
$332$			16.7582i			0.919727i
$333$		0			0
$334$		−	6.64308i		−	0.363493i
$335$	1.76399	−	7.99302i	0.0963768	−	0.436705i
$336$		0			0
$337$			27.1030i			1.47640i	0.674585	+	0.738198i	$0.264323\pi$
$337$			27.1030i			1.47640i	−0.674585	+	0.738198i	$0.735677\pi$
$338$	−35.4245			−1.92684
$339$		0			0
$340$		−	5.31770i		−	0.288393i
$341$		−	4.14921i		−	0.224692i
$342$		0			0
$343$		−	18.2257i		−	0.984095i
$344$		−	7.11575i		−	0.383656i
$345$		0			0
$346$		−	6.94551i		−	0.373393i
$347$	−6.96944			−0.374139			−0.187070	−	0.982347i	$-0.559899\pi$
$347$	−6.96944			−0.374139			−0.187070	+	0.982347i	$0.559899\pi$
$348$		0			0
$349$	2.65313			0.142019			0.0710093	−	0.997476i	$-0.477378\pi$
$349$	2.65313			0.142019			0.0710093	+	0.997476i	$0.477378\pi$
$350$		−	1.58772i		−	0.0848674i
$351$		0			0
$352$	0.711582			0.0379274
$353$	9.83072			0.523237			0.261618	−	0.965171i	$-0.415744\pi$
$353$	9.83072			0.523237			0.261618	+	0.965171i	$0.415744\pi$
$354$		0			0
$355$		−	4.17947i		−	0.221823i
$356$			12.9687i			0.687340i
$357$		0			0
$358$	14.6997			0.776902
$359$			11.3569i			0.599396i	0.954034	+	0.299698i	$0.0968859\pi$
$359$			11.3569i			0.599396i	−0.954034	+	0.299698i	$0.903114\pi$
$360$		0			0
$361$	−17.6181			−0.927268
$362$	19.8624			1.04394
$363$		0			0
$364$	11.0486			0.579104
$365$	14.5463			0.761390
$366$		0			0
$367$		−	22.3164i		−	1.16490i	−0.812865	−	0.582452i	$-0.802093\pi$
$367$		−	22.3164i		−	1.16490i	0.812865	−	0.582452i	$-0.197907\pi$
$368$		−	5.40166i		−	0.281581i
$369$		0			0
$370$	−5.35597			−0.278444
$371$		−	6.33782i		−	0.329043i
$372$		0			0
$373$			26.9999i			1.39800i	0.715120	+	0.699002i	$0.246372\pi$
$373$			26.9999i			1.39800i	−0.715120	+	0.699002i	$0.753628\pi$
$374$		−	3.78398i		−	0.195665i
$375$		0			0
$376$		−	13.0943i		−	0.675288i
$377$	29.1974			1.50374
$378$		0			0
$379$			3.06514i			0.157446i	0.996897	+	0.0787228i	$0.0250842\pi$
$379$			3.06514i			0.157446i	−0.996897	+	0.0787228i	$0.974916\pi$
$380$	−1.17555			−0.0603043
$381$		0			0
$382$	16.8296			0.861075
$383$	−23.7257			−1.21233			−0.606163	−	0.795340i	$-0.707292\pi$
$383$	−23.7257			−1.21233			−0.606163	+	0.795340i	$0.707292\pi$
$384$		0			0
$385$		−	1.12980i		−	0.0575797i
$386$	−15.9265			−0.810639
$387$		0			0
$388$		−	0.688811i		−	0.0349691i
$389$			29.6165i			1.50161i	0.660521	+	0.750807i	$0.270335\pi$
$389$			29.6165i			1.50161i	−0.660521	+	0.750807i	$0.729665\pi$
$390$		0			0
$391$	−28.7244			−1.45266
$392$	4.47913			0.226230
$393$		0			0
$394$	−13.6290			−0.686617
$395$		−	8.91625i		−	0.448625i
$396$		0			0
$397$	9.06475			0.454947			0.227473	−	0.973784i	$-0.426954\pi$
$397$	9.06475			0.454947			0.227473	+	0.973784i	$0.426954\pi$
$398$	−0.654416			−0.0328029
$399$		0			0
$400$	1.00000			0.0500000
$401$	37.6099			1.87815			0.939076	−	0.343711i	$-0.111684\pi$
$401$	37.6099			1.87815			0.939076	+	0.343711i	$0.111684\pi$
$402$		0			0
$403$	40.5763			2.02125
$404$	11.8717			0.590641
$405$		0			0
$406$	−6.66172			−0.330616
$407$	−3.81121			−0.188915
$408$		0			0
$409$			14.6139i			0.722609i	0.932448	+	0.361304i	$0.117668\pi$
$409$			14.6139i			0.722609i	−0.932448	+	0.361304i	$0.882332\pi$
$410$	6.36412			0.314301
$411$		0			0
$412$	1.07806			0.0531122
$413$	19.6587			0.967342
$414$		0			0
$415$			16.7582i			0.822628i
$416$			6.95877i			0.341182i
$417$		0			0
$418$	−0.836498			−0.0409145
$419$			23.8322i			1.16428i	0.813090	+	0.582138i	$0.197784\pi$
$419$			23.8322i			1.16428i	−0.813090	+	0.582138i	$0.802216\pi$
$420$		0			0
$421$	−10.6897			−0.520984			−0.260492	−	0.965476i	$-0.583885\pi$
$421$	−10.6897			−0.520984			−0.260492	+	0.965476i	$0.583885\pi$
$422$	0.725900			0.0353362
$423$		0			0
$424$	3.99176			0.193857
$425$		−	5.31770i		−	0.257946i
$426$		0			0
$427$	18.2559			0.883465
$428$			6.76441i			0.326970i
$429$		0			0
$430$		−	7.11575i		−	0.343152i
$431$			10.6293i			0.511995i	0.966677	+	0.255997i	$0.0824038\pi$
$431$			10.6293i			0.511995i	−0.966677	+	0.255997i	$0.917596\pi$
$432$		0			0
$433$		−	25.2986i		−	1.21577i	−0.794024	−	0.607886i	$-0.792018\pi$
$433$		−	25.2986i		−	1.21577i	0.794024	−	0.607886i	$-0.207982\pi$
$434$	−9.25796			−0.444396
$435$		0			0
$436$		−	6.75749i		−	0.323625i
$437$			6.34991i			0.303757i
$438$		0			0
$439$	−7.61607			−0.363495			−0.181748	−	0.983345i	$-0.558175\pi$
$439$	−7.61607			−0.363495			−0.181748	+	0.983345i	$0.558175\pi$
$440$	0.711582			0.0339233
$441$		0			0
$442$	37.0046			1.76013
$443$	−10.2755			−0.488203			−0.244102	−	0.969750i	$-0.578493\pi$
$443$	−10.2755			−0.488203			−0.244102	+	0.969750i	$0.578493\pi$
$444$		0			0
$445$			12.9687i			0.614775i
$446$	−13.0761			−0.619172
$447$		0			0
$448$		−	1.58772i		−	0.0750129i
$449$			8.13852i			0.384080i	0.981387	+	0.192040i	$0.0615104\pi$
$449$			8.13852i			0.384080i	−0.981387	+	0.192040i	$0.938490\pi$
$450$		0			0
$451$	4.52859			0.213243
$452$	14.1799			0.666965
$453$		0			0
$454$			11.9288i			0.559845i
$455$	11.0486			0.517967
$456$		0			0
$457$	20.7169			0.969097			0.484549	−	0.874764i	$-0.338984\pi$
$457$	20.7169			0.969097			0.484549	+	0.874764i	$0.338984\pi$
$458$			7.02413i			0.328216i
$459$		0			0
$460$		−	5.40166i		−	0.251854i
$461$			22.4215i			1.04427i	0.852861	+	0.522137i	$0.174865\pi$
$461$			22.4215i			1.04427i	−0.852861	+	0.522137i	$0.825135\pi$
$462$		0			0
$463$			14.5672i			0.676993i	0.940968	+	0.338497i	$0.109918\pi$
$463$			14.5672i			0.676993i	−0.940968	+	0.338497i	$0.890082\pi$
$464$		−	4.19576i		−	0.194783i
$465$		0			0
$466$	12.4117			0.574962
$467$		−	29.6127i		−	1.37031i	−0.728397	−	0.685155i	$-0.759734\pi$
$467$		−	29.6127i		−	1.37031i	0.728397	−	0.685155i	$-0.240266\pi$
$468$		0			0
$469$	−12.6907	−	2.80072i	−0.586003	−	0.129325i
$470$		−	13.0943i		−	0.603996i
$471$		0			0
$472$			12.3817i			0.569913i
$473$		−	5.06344i		−	0.232817i
$474$		0			0
$475$	−1.17555			−0.0539378
$476$	−8.44304			−0.386986
$477$		0			0
$478$	−11.4278			−0.522697
$479$			37.4532i			1.71128i	0.517570	+	0.855641i	$0.326837\pi$
$479$			37.4532i			1.71128i	−0.517570	+	0.855641i	$0.673163\pi$
$480$		0			0
$481$		−	37.2710i		−	1.69941i
$482$	−24.5917			−1.12012
$483$		0			0
$484$	−10.4937			−0.476984
$485$		−	0.688811i		−	0.0312773i
$486$		0			0
$487$			16.0328i			0.726517i	0.931688	+	0.363258i	$0.118336\pi$
$487$			16.0328i			0.726517i	−0.931688	+	0.363258i	$0.881664\pi$
$488$			11.4982i			0.520497i
$489$		0			0
$490$	4.47913			0.202347
$491$		−	23.3551i		−	1.05400i	−0.849865	−	0.527000i	$-0.823317\pi$
$491$		−	23.3551i		−	1.05400i	0.849865	−	0.527000i	$-0.176683\pi$
$492$		0			0
$493$	−22.3118			−1.00487
$494$		−	8.18036i		−	0.368052i
$495$		0			0
$496$		−	5.83096i		−	0.261818i
$497$	−6.63585			−0.297659
$498$		0			0
$499$			38.6758i			1.73137i	0.500593	+	0.865683i	$0.333115\pi$
$499$			38.6758i			1.73137i	−0.500593	+	0.865683i	$0.666885\pi$
$500$	1.00000			0.0447214
$501$		0			0
$502$	−6.82931			−0.304807
$503$	33.5035			1.49385			0.746923	−	0.664911i	$-0.231530\pi$
$503$	33.5035			1.49385			0.746923	+	0.664911i	$0.231530\pi$
$504$		0			0
$505$	11.8717			0.528285
$506$		−	3.84373i		−	0.170874i
$507$		0			0
$508$	−3.31458			−0.147060
$509$			15.6728i			0.694687i	0.937738	+	0.347343i	$0.112916\pi$
$509$			15.6728i			0.694687i	−0.937738	+	0.347343i	$0.887084\pi$
$510$		0			0
$511$		−	23.0956i		−	1.02169i
$512$	1.00000			0.0441942
$513$		0			0
$514$			3.82314i			0.168631i
$515$	1.07806			0.0475050
$516$		0			0
$517$		−	9.31768i		−	0.409791i
$518$			8.50381i			0.373636i
$519$		0			0
$520$			6.95877i			0.305162i
$521$	−21.6175			−0.947081			−0.473541	−	0.880772i	$-0.657024\pi$
$521$	−21.6175			−0.947081			−0.473541	+	0.880772i	$0.657024\pi$
$522$		0			0
$523$	−33.8893			−1.48187			−0.740937	−	0.671574i	$-0.765619\pi$
$523$	−33.8893			−1.48187			−0.740937	+	0.671574i	$0.765619\pi$
$524$		−	6.11885i		−	0.267303i
$525$		0			0
$526$		−	2.38656i		−	0.104059i
$527$	−31.0073			−1.35070
$528$		0			0
$529$	−6.17797			−0.268607
$530$	3.99176			0.173391
$531$		0			0
$532$			1.86645i			0.0809207i
$533$			44.2864i			1.91826i
$534$		0			0
$535$			6.76441i			0.292451i
$536$	1.76399	−	7.99302i	0.0761926	−	0.345246i
$537$		0			0
$538$		−	17.4694i		−	0.753159i
$539$	3.18727			0.137285
$540$		0			0
$541$			40.5143i			1.74184i	0.491421	+	0.870922i	$0.336478\pi$
$541$			40.5143i			1.74184i	−0.491421	+	0.870922i	$0.663522\pi$
$542$		−	12.4886i		−	0.536432i
$543$		0			0
$544$		−	5.31770i		−	0.227994i
$545$		−	6.75749i		−	0.289459i
$546$		0			0
$547$		−	22.6031i		−	0.966439i	−0.875499	−	0.483220i	$-0.839467\pi$
$547$		−	22.6031i		−	0.966439i	0.875499	−	0.483220i	$-0.160533\pi$
$548$	−2.12775			−0.0908930
$549$		0			0
$550$	0.711582			0.0303419
$551$			4.93232i			0.210124i
$552$		0			0
$553$	−14.1566			−0.601998
$554$	2.26921			0.0964095
$555$		0			0
$556$		−	1.55407i		−	0.0659073i
$557$			16.9769i			0.719333i	0.933081	+	0.359666i	$0.117109\pi$
$557$			16.9769i			0.719333i	−0.933081	+	0.359666i	$0.882891\pi$
$558$		0			0
$559$	49.5169			2.09434
$560$		−	1.58772i		−	0.0670936i
$561$		0			0
$562$	−22.8990			−0.965934
$563$	33.6900			1.41986			0.709932	−	0.704270i	$-0.248726\pi$
$563$	33.6900			1.41986			0.709932	+	0.704270i	$0.248726\pi$
$564$		0			0
$565$	14.1799			0.596551
$566$	2.00317			0.0841994
$567$		0			0
$568$		−	4.17947i		−	0.175367i
$569$		−	14.8402i		−	0.622134i	−0.950388	−	0.311067i	$-0.899314\pi$
$569$		−	14.8402i		−	0.622134i	0.950388	−	0.311067i	$-0.100686\pi$
$570$		0			0
$571$	−22.2947			−0.933005			−0.466503	−	0.884520i	$-0.654486\pi$
$571$	−22.2947			−0.933005			−0.466503	+	0.884520i	$0.654486\pi$
$572$			4.95173i			0.207042i
$573$		0			0
$574$		−	10.1045i		−	0.421752i
$575$		−	5.40166i		−	0.225265i
$576$		0			0
$577$			30.4024i			1.26567i	0.774287	+	0.632835i	$0.218109\pi$
$577$			30.4024i			1.26567i	−0.774287	+	0.632835i	$0.781891\pi$
$578$	−11.2779			−0.469099
$579$		0			0
$580$		−	4.19576i		−	0.174220i
$581$	26.6074			1.10386
$582$		0			0
$583$	2.84046			0.117640
$584$	14.5463			0.601932
$585$		0			0
$586$			8.54078i			0.352816i
$587$	−31.2439			−1.28958			−0.644788	−	0.764362i	$-0.723054\pi$
$587$	−31.2439			−1.28958			−0.644788	+	0.764362i	$0.723054\pi$
$588$		0			0
$589$			6.85457i			0.282438i
$590$			12.3817i			0.509746i
$591$		0			0
$592$	−5.35597			−0.220129
$593$	−17.7390			−0.728453			−0.364226	−	0.931310i	$-0.618667\pi$
$593$	−17.7390			−0.728453			−0.364226	+	0.931310i	$0.618667\pi$
$594$		0			0
$595$	−8.44304			−0.346131
$596$		−	21.2482i		−	0.870358i
$597$		0			0
$598$	37.5889			1.53713
$599$	32.3214			1.32062			0.660309	−	0.750994i	$-0.270425\pi$
$599$	32.3214			1.32062			0.660309	+	0.750994i	$0.270425\pi$
$600$		0			0
$601$	0.870034			0.0354894			0.0177447	−	0.999843i	$-0.494351\pi$
$601$	0.870034			0.0354894			0.0177447	+	0.999843i	$0.494351\pi$
$602$	−11.2979			−0.460466
$603$		0			0
$604$	−18.4180			−0.749419
$605$	−10.4937			−0.426628
$606$		0			0
$607$	−11.9640			−0.485605			−0.242802	−	0.970076i	$-0.578067\pi$
$607$	−11.9640			−0.485605			−0.242802	+	0.970076i	$0.578067\pi$
$608$	−1.17555			−0.0476747
$609$		0			0
$610$			11.4982i			0.465547i
$611$	91.1204			3.68634
$612$		0			0
$613$	3.07215			0.124083			0.0620415	−	0.998074i	$-0.480239\pi$
$613$	3.07215			0.124083			0.0620415	+	0.998074i	$0.480239\pi$
$614$	−11.3122			−0.456522
$615$		0			0
$616$		−	1.12980i		−	0.0455208i
$617$		−	0.848617i		−	0.0341640i	−0.999854	−	0.0170820i	$-0.994562\pi$
$617$		−	0.848617i		−	0.0341640i	0.999854	−	0.0170820i	$-0.00543764\pi$
$618$		0			0
$619$	−0.369648			−0.0148574			−0.00742871	−	0.999972i	$-0.502365\pi$
$619$	−0.369648			−0.0148574			−0.00742871	+	0.999972i	$0.502365\pi$
$620$		−	5.83096i		−	0.234177i
$621$		0			0
$622$	14.0079			0.561665
$623$	20.5907			0.824950
$624$		0			0
$625$	1.00000			0.0400000
$626$		−	28.7123i		−	1.14757i
$627$		0			0
$628$	12.8185			0.511512
$629$			28.4814i			1.13563i
$630$		0			0
$631$		−	37.4934i		−	1.49259i	−0.665616	−	0.746294i	$-0.731831\pi$
$631$		−	37.4934i		−	1.49259i	0.665616	−	0.746294i	$-0.268169\pi$
$632$		−	8.91625i		−	0.354669i
$633$		0			0
$634$			0.296782i			0.0117867i
$635$	−3.31458			−0.131535
$636$		0			0
$637$			31.1692i			1.23497i
$638$		−	2.98563i		−	0.118202i
$639$		0			0
$640$	1.00000			0.0395285
$641$	−27.3425			−1.07996			−0.539982	−	0.841676i	$-0.681569\pi$
$641$	−27.3425			−1.07996			−0.539982	+	0.841676i	$0.681569\pi$
$642$		0			0
$643$	−43.7866			−1.72678			−0.863388	−	0.504541i	$-0.831662\pi$
$643$	−43.7866			−1.72678			−0.863388	+	0.504541i	$0.831662\pi$
$644$	−8.57635			−0.337956
$645$		0			0
$646$			6.25120i			0.245950i
$647$	31.7689			1.24897			0.624483	−	0.781038i	$-0.285310\pi$
$647$	31.7689			1.24897			0.624483	+	0.781038i	$0.285310\pi$
$648$		0			0
$649$			8.81058i			0.345846i
$650$			6.95877i			0.272945i
$651$		0			0
$652$	16.2040			0.634599
$653$	−17.8148			−0.697148			−0.348574	−	0.937281i	$-0.613334\pi$
$653$	−17.8148			−0.697148			−0.348574	+	0.937281i	$0.613334\pi$
$654$		0			0
$655$		−	6.11885i		−	0.239083i
$656$	6.36412			0.248477
$657$		0			0
$658$	−20.7902			−0.810486
$659$		−	25.6761i		−	1.00020i	−0.865968	−	0.500099i	$-0.833297\pi$
$659$		−	25.6761i		−	1.00020i	0.865968	−	0.500099i	$-0.166703\pi$
$660$		0			0
$661$			5.66196i			0.220225i	0.993919	+	0.110112i	$0.0351211\pi$
$661$			5.66196i			0.220225i	−0.993919	+	0.110112i	$0.964879\pi$
$662$		−	5.55383i		−	0.215856i
$663$		0			0
$664$			16.7582i			0.650345i
$665$			1.86645i			0.0723776i
$666$		0			0
$667$	−22.6641			−0.877558
$668$		−	6.64308i		−	0.257028i
$669$		0			0
$670$	1.76399	−	7.99302i	0.0681487	−	0.308797i
$671$			8.18188i			0.315858i
$672$		0			0
$673$			13.7359i			0.529481i	0.964320	+	0.264740i	$0.0852862\pi$
$673$			13.7359i			0.529481i	−0.964320	+	0.264740i	$0.914714\pi$
$674$			27.1030i			1.04397i
$675$		0			0
$676$	−35.4245			−1.36248
$677$	28.3231			1.08854			0.544272	−	0.838909i	$-0.316806\pi$
$677$	28.3231			1.08854			0.544272	+	0.838909i	$0.316806\pi$
$678$		0			0
$679$	−1.09364			−0.0419701
$680$		−	5.31770i		−	0.203924i
$681$		0			0
$682$		−	4.14921i		−	0.158881i
$683$	−12.7350			−0.487290			−0.243645	−	0.969865i	$-0.578343\pi$
$683$	−12.7350			−0.487290			−0.243645	+	0.969865i	$0.578343\pi$
$684$		0			0
$685$	−2.12775			−0.0812972
$686$		−	18.2257i		−	0.695860i
$687$		0			0
$688$		−	7.11575i		−	0.271285i
$689$			27.7777i			1.05825i
$690$		0			0
$691$	−38.2371			−1.45461			−0.727303	−	0.686316i	$-0.759227\pi$
$691$	−38.2371			−1.45461			−0.727303	+	0.686316i	$0.759227\pi$
$692$		−	6.94551i		−	0.264029i
$693$		0			0
$694$	−6.96944			−0.264556
$695$		−	1.55407i		−	0.0589493i
$696$		0			0
$697$		−	33.8424i		−	1.28187i
$698$	2.65313			0.100422
$699$		0			0
$700$		−	1.58772i		−	0.0600103i
$701$	−31.6316			−1.19471			−0.597354	−	0.801978i	$-0.703781\pi$
$701$	−31.6316			−1.19471			−0.597354	+	0.801978i	$0.703781\pi$
$702$		0			0
$703$	6.29620			0.237466
$704$	0.711582			0.0268187
$705$		0			0
$706$	9.83072			0.369984
$707$		−	18.8490i		−	0.708891i
$708$		0			0
$709$	−48.2750			−1.81300			−0.906502	−	0.422200i	$-0.861258\pi$
$709$	−48.2750			−1.81300			−0.906502	+	0.422200i	$0.861258\pi$
$710$		−	4.17947i		−	0.156853i
$711$		0			0
$712$			12.9687i			0.486023i
$713$	−31.4969			−1.17957
$714$		0			0
$715$			4.95173i			0.185184i
$716$	14.6997			0.549353
$717$		0			0
$718$			11.3569i			0.423837i
$719$			20.1646i			0.752011i	0.926617	+	0.376005i	$0.122703\pi$
$719$			20.1646i			0.752011i	−0.926617	+	0.376005i	$0.877297\pi$
$720$		0			0
$721$		−	1.71166i		−	0.0637456i
$722$	−17.6181			−0.655677
$723$		0			0
$724$	19.8624			0.738180
$725$		−	4.19576i		−	0.155827i
$726$		0			0
$727$			0.827684i			0.0306971i	0.999882	+	0.0153486i	$0.00488579\pi$
$727$			0.827684i			0.0306971i	−0.999882	+	0.0153486i	$0.995114\pi$
$728$	11.0486			0.409489
$729$		0			0
$730$	14.5463			0.538384
$731$	−37.8394			−1.39954
$732$		0			0
$733$		−	45.9525i		−	1.69729i	−0.528960	−	0.848647i	$-0.677418\pi$
$733$		−	45.9525i		−	1.69729i	0.528960	−	0.848647i	$-0.322582\pi$
$734$		−	22.3164i		−	0.823712i
$735$		0			0
$736$		−	5.40166i		−	0.199108i
$737$	1.25522	−	5.68769i	0.0462366	−	0.209509i
$738$		0			0
$739$			38.1477i			1.40329i	0.712529	+	0.701643i	$0.247550\pi$
$739$			38.1477i			1.40329i	−0.712529	+	0.701643i	$0.752450\pi$
$740$	−5.35597			−0.196890
$741$		0			0
$742$		−	6.33782i		−	0.232669i
$743$		−	33.2686i		−	1.22051i	−0.792206	−	0.610253i	$-0.791068\pi$
$743$		−	33.2686i		−	1.22051i	0.792206	−	0.610253i	$-0.208932\pi$
$744$		0			0
$745$		−	21.2482i		−	0.778472i
$746$			26.9999i			0.988538i
$747$		0			0
$748$		−	3.78398i		−	0.138356i
$749$	10.7400			0.392432
$750$		0			0
$751$	41.3401			1.50852			0.754260	−	0.656576i	$-0.227996\pi$
$751$	41.3401			1.50852			0.754260	+	0.656576i	$0.227996\pi$
$752$		−	13.0943i		−	0.477501i
$753$		0			0
$754$	29.1974			1.06331
$755$	−18.4180			−0.670301
$756$		0			0
$757$		−	49.3729i		−	1.79449i	−0.441536	−	0.897244i	$-0.645566\pi$
$757$		−	49.3729i		−	1.79449i	0.441536	−	0.897244i	$-0.354434\pi$
$758$			3.06514i			0.111331i
$759$		0			0
$760$	−1.17555			−0.0426416
$761$		−	6.80637i		−	0.246731i	−0.992361	−	0.123365i	$-0.960631\pi$
$761$		−	6.80637i		−	0.246731i	0.992361	−	0.123365i	$-0.0393687\pi$
$762$		0			0
$763$	−10.7290			−0.388417
$764$	16.8296			0.608872
$765$		0			0
$766$	−23.7257			−0.857244
$767$	−86.1613			−3.11110
$768$		0			0
$769$		−	41.6839i		−	1.50316i	−0.659641	−	0.751581i	$-0.729292\pi$
$769$		−	41.6839i		−	1.50316i	0.659641	−	0.751581i	$-0.270708\pi$
$770$		−	1.12980i		−	0.0407150i
$771$		0			0
$772$	−15.9265			−0.573209
$773$		−	38.7281i		−	1.39295i	−0.717580	−	0.696476i	$-0.754750\pi$
$773$		−	38.7281i		−	1.39295i	0.717580	−	0.696476i	$-0.245250\pi$
$774$		0			0
$775$		−	5.83096i		−	0.209454i
$776$		−	0.688811i		−	0.0247269i
$777$		0			0
$778$			29.6165i			1.06180i
$779$	−7.48132			−0.268046
$780$		0			0
$781$		−	2.97404i		−	0.106419i
$782$	−28.7244			−1.02718
$783$		0			0
$784$	4.47913			0.159969
$785$	12.8185			0.457510
$786$		0			0
$787$			54.9221i			1.95776i	0.204429	+	0.978881i	$0.434466\pi$
$787$			54.9221i			1.95776i	−0.204429	+	0.978881i	$0.565534\pi$
$788$	−13.6290			−0.485511
$789$		0			0
$790$		−	8.91625i		−	0.317226i
$791$		−	22.5137i		−	0.800496i
$792$		0			0
$793$	−80.0130			−2.84134
$794$	9.06475			0.321696
$795$		0			0
$796$	−0.654416			−0.0231951
$797$			1.07204i			0.0379734i	0.999820	+	0.0189867i	$0.00604402\pi$
$797$			1.07204i			0.0379734i	−0.999820	+	0.0189867i	$0.993956\pi$
$798$		0			0
$799$	−69.6317			−2.46339
$800$	1.00000			0.0353553
$801$		0			0
$802$	37.6099			1.32805
$803$	10.3509			0.365276
$804$		0			0
$805$	−8.57635			−0.302277
$806$	40.5763			1.42924
$807$		0			0
$808$	11.8717			0.417646
$809$	−33.5064			−1.17802			−0.589011	−	0.808125i	$-0.700483\pi$
$809$	−33.5064			−1.17802			−0.589011	+	0.808125i	$0.700483\pi$
$810$		0			0
$811$		−	19.8154i		−	0.695813i	−0.937529	−	0.347907i	$-0.886893\pi$
$811$		−	19.8154i		−	0.695813i	0.937529	−	0.347907i	$-0.113107\pi$
$812$	−6.66172			−0.233780
$813$		0			0
$814$	−3.81121			−0.133583
$815$	16.2040			0.567603
$816$		0			0
$817$			8.36490i			0.292651i
$818$			14.6139i			0.510962i
$819$		0			0
$820$	6.36412			0.222245
$821$			33.5453i			1.17074i	0.810766	+	0.585370i	$0.199051\pi$
$821$			33.5453i			1.17074i	−0.810766	+	0.585370i	$0.800949\pi$
$822$		0			0
$823$	−29.3960			−1.02468			−0.512340	−	0.858782i	$-0.671221\pi$
$823$	−29.3960			−1.02468			−0.512340	+	0.858782i	$0.671221\pi$
$824$	1.07806			0.0375560
$825$		0			0
$826$	19.6587			0.684014
$827$		−	37.6353i		−	1.30871i	−0.756188	−	0.654354i	$-0.772941\pi$
$827$		−	37.6353i		−	1.30871i	0.756188	−	0.654354i	$-0.227059\pi$
$828$		0			0
$829$	−46.2311			−1.60567			−0.802836	−	0.596200i	$-0.796677\pi$
$829$	−46.2311			−1.60567			−0.802836	+	0.596200i	$0.796677\pi$
$830$			16.7582i			0.581686i
$831$		0			0
$832$			6.95877i			0.241252i
$833$		−	23.8187i		−	0.825268i
$834$		0			0
$835$		−	6.64308i		−	0.229893i
$836$	−0.836498			−0.0289309
$837$		0			0
$838$			23.8322i			0.823268i
$839$			54.8118i			1.89231i	0.323709	+	0.946157i	$0.395070\pi$
$839$			54.8118i			1.89231i	−0.323709	+	0.946157i	$0.604930\pi$
$840$		0			0
$841$	11.3956			0.392950
$842$	−10.6897			−0.368391
$843$		0			0
$844$	0.725900			0.0249865
$845$	−35.4245			−1.21864
$846$		0			0
$847$			16.6610i			0.572480i
$848$	3.99176			0.137078
$849$		0			0
$850$		−	5.31770i		−	0.182395i
$851$			28.9312i			0.991748i
$852$		0			0
$853$	6.75911			0.231427			0.115714	−	0.993283i	$-0.463084\pi$
$853$	6.75911			0.231427			0.115714	+	0.993283i	$0.463084\pi$
$854$	18.2559			0.624704
$855$		0			0
$856$			6.76441i			0.231203i
$857$	39.1263			1.33653			0.668265	−	0.743923i	$-0.267037\pi$
$857$	39.1263			1.33653			0.668265	+	0.743923i	$0.267037\pi$
$858$		0			0
$859$	8.06930			0.275321			0.137661	−	0.990479i	$-0.456042\pi$
$859$	8.06930			0.275321			0.137661	+	0.990479i	$0.456042\pi$
$860$		−	7.11575i		−	0.242645i
$861$		0			0
$862$			10.6293i			0.362035i
$863$		−	14.7703i		−	0.502788i	−0.967885	−	0.251394i	$-0.919111\pi$
$863$		−	14.7703i		−	0.502788i	0.967885	−	0.251394i	$-0.0808890\pi$
$864$		0			0
$865$		−	6.94551i		−	0.236154i
$866$		−	25.2986i		−	0.859681i
$867$		0			0
$868$	−9.25796			−0.314236
$869$		−	6.34464i		−	0.215227i
$870$		0			0
$871$	55.6216	+	12.2752i	1.88467	+	0.415928i
$872$		−	6.75749i		−	0.228837i
$873$		0			0
$874$			6.34991i			0.214789i
$875$		−	1.58772i		−	0.0536749i
$876$		0			0
$877$	32.7158			1.10473			0.552367	−	0.833601i	$-0.313725\pi$
$877$	32.7158			1.10473			0.552367	+	0.833601i	$0.313725\pi$
$878$	−7.61607			−0.257030
$879$		0			0
$880$	0.711582			0.0239874
$881$			4.31110i			0.145245i	0.997360	+	0.0726223i	$0.0231368\pi$
$881$			4.31110i			0.145245i	−0.997360	+	0.0726223i	$0.976863\pi$
$882$		0			0
$883$			31.0568i			1.04515i	0.852595	+	0.522573i	$0.175028\pi$
$883$			31.0568i			1.04515i	−0.852595	+	0.522573i	$0.824972\pi$
$884$	37.0046			1.24460
$885$		0			0
$886$	−10.2755			−0.345212
$887$		−	30.3583i		−	1.01933i	−0.860372	−	0.509667i	$-0.829769\pi$
$887$		−	30.3583i		−	1.01933i	0.860372	−	0.509667i	$-0.170231\pi$
$888$		0			0
$889$			5.26263i			0.176503i
$890$			12.9687i			0.434712i
$891$		0			0
$892$	−13.0761			−0.437821
$893$			15.3930i			0.515107i
$894$		0			0
$895$	14.6997			0.491356
$896$		−	1.58772i		−	0.0530422i
$897$		0			0
$898$			8.13852i			0.271586i
$899$	−24.4654			−0.815965
$900$		0			0
$901$		−	21.2270i		−	0.707173i
$902$	4.52859			0.150785
$903$		0			0
$904$	14.1799			0.471615
$905$	19.8624			0.660248
$906$		0			0
$907$	38.8734			1.29077			0.645385	−	0.763857i	$-0.276697\pi$
$907$	38.8734			1.29077			0.645385	+	0.763857i	$0.276697\pi$
$908$			11.9288i			0.395870i
$909$		0			0
$910$	11.0486			0.366258
$911$			53.5770i			1.77509i	0.460724	+	0.887543i	$0.347590\pi$
$911$			53.5770i			1.77509i	−0.460724	+	0.887543i	$0.652410\pi$
$912$		0			0
$913$			11.9248i			0.394655i
$914$	20.7169			0.685255
$915$		0			0
$916$			7.02413i			0.232084i
$917$	−9.71504			−0.320819
$918$		0			0
$919$			19.5848i			0.646044i	0.946392	+	0.323022i	$0.104699\pi$
$919$			19.5848i			0.646044i	−0.946392	+	0.323022i	$0.895301\pi$
$920$		−	5.40166i		−	0.178088i
$921$		0			0
$922$			22.4215i			0.738414i
$923$	29.0840			0.957311
$924$		0			0
$925$	−5.35597			−0.176103
$926$			14.5672i			0.478706i
$927$		0			0
$928$		−	4.19576i		−	0.137733i
$929$	−30.1263			−0.988411			−0.494205	−	0.869345i	$-0.664541\pi$
$929$	−30.1263			−0.988411			−0.494205	+	0.869345i	$0.664541\pi$
$930$		0			0
$931$	−5.26543			−0.172568
$932$	12.4117			0.406559
$933$		0			0
$934$		−	29.6127i		−	0.968956i
$935$		−	3.78398i		−	0.123749i
$936$		0			0
$937$		−	0.172972i		−	0.00565075i	−0.999996	−	0.00282538i	$-0.999101\pi$
$937$		−	0.172972i		−	0.00565075i	0.999996	−	0.00282538i	$-0.000899347\pi$
$938$	−12.6907	−	2.80072i	−0.414366	−	0.0914469i
$939$		0			0
$940$		−	13.0943i		−	0.427090i
$941$	24.4713			0.797741			0.398871	−	0.917007i	$-0.369402\pi$
$941$	24.4713			0.797741			0.398871	+	0.917007i	$0.369402\pi$
$942$		0			0
$943$		−	34.3768i		−	1.11946i
$944$			12.3817i			0.402990i
$945$		0			0
$946$		−	5.06344i		−	0.164627i
$947$		−	7.37457i		−	0.239641i	−0.992796	−	0.119821i	$-0.961768\pi$
$947$		−	7.37457i		−	0.239641i	0.992796	−	0.119821i	$-0.0382320\pi$
$948$		0			0
$949$			101.225i			3.28589i
$950$	−1.17555			−0.0381398
$951$		0			0
$952$	−8.44304			−0.273640
$953$			46.8633i			1.51805i	0.651061	+	0.759025i	$0.274324\pi$
$953$			46.8633i			1.51805i	−0.651061	+	0.759025i	$0.725676\pi$
$954$		0			0
$955$	16.8296			0.544592
$956$	−11.4278			−0.369603
$957$		0			0
$958$			37.4532i			1.21006i
$959$			3.37828i			0.109090i
$960$		0			0
$961$	−3.00014			−0.0967788
$962$		−	37.2710i		−	1.20166i
$963$		0			0
$964$	−24.5917			−0.792046
$965$	−15.9265			−0.512693
$966$		0			0
$967$	−0.146650			−0.00471596			−0.00235798	−	0.999997i	$-0.500751\pi$
$967$	−0.146650			−0.00471596			−0.00235798	+	0.999997i	$0.500751\pi$
$968$	−10.4937			−0.337279
$969$		0			0
$970$		−	0.688811i		−	0.0221164i
$971$			32.5582i			1.04484i	0.852687	+	0.522422i	$0.174971\pi$
$971$			32.5582i			1.04484i	−0.852687	+	0.522422i	$0.825029\pi$
$972$		0			0
$973$	−2.46744			−0.0791024
$974$			16.0328i			0.513725i
$975$		0			0
$976$			11.4982i			0.368047i
$977$			57.3365i			1.83436i	0.398476	+	0.917179i	$0.369539\pi$
$977$			57.3365i			1.83436i	−0.398476	+	0.917179i	$0.630461\pi$
$978$		0			0
$979$			9.22829i			0.294937i
$980$	4.47913			0.143081
$981$		0			0
$982$		−	23.3551i		−	0.745291i
$983$	51.7045			1.64912			0.824559	−	0.565776i	$-0.191423\pi$
$983$	51.7045			1.64912			0.824559	+	0.565776i	$0.191423\pi$
$984$		0			0
$985$	−13.6290			−0.434255
$986$	−22.3118			−0.710553
$987$		0			0
$988$		−	8.18036i		−	0.260252i
$989$	−38.4369			−1.22222
$990$		0			0
$991$		−	54.6212i		−	1.73510i	−0.497350	−	0.867550i	$-0.665693\pi$
$991$		−	54.6212i		−	1.73510i	0.497350	−	0.867550i	$-0.334307\pi$
$992$		−	5.83096i		−	0.185133i
$993$		0			0
$994$	−6.63585			−0.210476
$995$	−0.654416			−0.0207464
$996$		0			0
$997$	13.2691			0.420238			0.210119	−	0.977676i	$-0.432615\pi$
$997$	13.2691			0.420238			0.210119	+	0.977676i	$0.432615\pi$
$998$			38.6758i			1.22426i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6030.2.d.l.2411.9	yes	24
3.2	odd	2		6030.2.d.k.2411.9	✓	24
67.66	odd	2		6030.2.d.k.2411.16	yes	24
201.200	even	2	inner	6030.2.d.l.2411.16	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6030.2.d.k.2411.9	✓	24	3.2	odd	2
6030.2.d.k.2411.16	yes	24	67.66	odd	2
6030.2.d.l.2411.9	yes	24	1.1	even	1	trivial
6030.2.d.l.2411.16	yes	24	201.200	even	2	inner

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:	\( N \)	\(=\)	\( 6030 = 2 \cdot 3^{2} \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6030.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.58772i q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.58772i q^{7} +1.00000 q^{8} +1.00000 q^{10} +0.711582 q^{11} +6.95877i q^{13} -1.58772i q^{14} +1.00000 q^{16} -5.31770i q^{17} -1.17555 q^{19} +1.00000 q^{20} +0.711582 q^{22} -5.40166i q^{23} +1.00000 q^{25} +6.95877i q^{26} -1.58772i q^{28} -4.19576i q^{29} -5.83096i q^{31} +1.00000 q^{32} -5.31770i q^{34} -1.58772i q^{35} -5.35597 q^{37} -1.17555 q^{38} +1.00000 q^{40} +6.36412 q^{41} -7.11575i q^{43} +0.711582 q^{44} -5.40166i q^{46} -13.0943i q^{47} +4.47913 q^{49} +1.00000 q^{50} +6.95877i q^{52} +3.99176 q^{53} +0.711582 q^{55} -1.58772i q^{56} -4.19576i q^{58} +12.3817i q^{59} +11.4982i q^{61} -5.83096i q^{62} +1.00000 q^{64} +6.95877i q^{65} +(1.76399 - 7.99302i) q^{67} -5.31770i q^{68} -1.58772i q^{70} -4.17947i q^{71} +14.5463 q^{73} -5.35597 q^{74} -1.17555 q^{76} -1.12980i q^{77} -8.91625i q^{79} +1.00000 q^{80} +6.36412 q^{82} +16.7582i q^{83} -5.31770i q^{85} -7.11575i q^{86} +0.711582 q^{88} +12.9687i q^{89} +11.0486 q^{91} -5.40166i q^{92} -13.0943i q^{94} -1.17555 q^{95} -0.688811i q^{97} +4.47913 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 24 q^{2} + 24 q^{4} + 24 q^{5} + 24 q^{8}+O(q^{10}) \) 24 * q + 24 * q^2 + 24 * q^4 + 24 * q^5 + 24 * q^8	\( 24 q + 24 q^{2} + 24 q^{4} + 24 q^{5} + 24 q^{8} + 24 q^{10} + 12 q^{11} + 24 q^{16} + 4 q^{19} + 24 q^{20} + 12 q^{22} + 24 q^{25} + 24 q^{32} - 16 q^{37} + 4 q^{38} + 24 q^{40} + 8 q^{41} + 12 q^{44} - 20 q^{49} + 24 q^{50} + 24 q^{53} + 12 q^{55} + 24 q^{64} - 32 q^{67} - 4 q^{73} - 16 q^{74} + 4 q^{76} + 24 q^{80} + 8 q^{82} + 12 q^{88} + 4 q^{95} - 20 q^{98}+O(q^{100}) \) 24 * q + 24 * q^2 + 24 * q^4 + 24 * q^5 + 24 * q^8 + 24 * q^10 + 12 * q^11 + 24 * q^16 + 4 * q^19 + 24 * q^20 + 12 * q^22 + 24 * q^25 + 24 * q^32 - 16 * q^37 + 4 * q^38 + 24 * q^40 + 8 * q^41 + 12 * q^44 - 20 * q^49 + 24 * q^50 + 24 * q^53 + 12 * q^55 + 24 * q^64 - 32 * q^67 - 4 * q^73 - 16 * q^74 + 4 * q^76 + 24 * q^80 + 8 * q^82 + 12 * q^88 + 4 * q^95 - 20 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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