LMFDB - Embedded newform 2475.2.c.r.199.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,2,Mod(199,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2475.199");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2475.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:	$19.7629745003$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.350464.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 4x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 165)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.3
Root			$0.403032 - 0.403032i$ of defining polynomial
Character	$\chi$	$=$	2475.199
Dual form			2475.2.c.r.199.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.193937i q^{2} +1.96239 q^{4} -3.35026i q^{7} -0.768452i q^{8} +O(q^{10})$	$q-0.193937i q^{2} +1.96239 q^{4} -3.35026i q^{7} -0.768452i q^{8} -1.00000 q^{11} +2.96239i q^{13} -0.649738 q^{14} +3.77575 q^{16} -4.57452i q^{17} +4.31265 q^{19} +0.193937i q^{22} +6.70052i q^{23} +0.574515 q^{26} -6.57452i q^{28} -3.61213 q^{29} +9.92478 q^{31} -2.26916i q^{32} -0.887166 q^{34} +2.00000i q^{37} -0.836381i q^{38} +4.38787 q^{41} -9.27504i q^{43} -1.96239 q^{44} +1.29948 q^{46} -9.92478i q^{47} -4.22425 q^{49} +5.81336i q^{52} -4.70052i q^{53} -2.57452 q^{56} +0.700523i q^{58} +10.7005 q^{59} -8.70052 q^{61} -1.92478i q^{62} +7.11142 q^{64} -5.92478i q^{67} -8.97698i q^{68} -9.92478 q^{71} -7.73813i q^{73} +0.387873 q^{74} +8.46310 q^{76} +3.35026i q^{77} -11.5369 q^{79} -0.850969i q^{82} -10.8872i q^{83} -1.79877 q^{86} +0.768452i q^{88} -2.77575 q^{89} +9.92478 q^{91} +13.1490i q^{92} -1.92478 q^{94} -0.0752228i q^{97} +0.819237i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 10 q^{4}+O(q^{10}) $ 6 * q - 10 * q^4	$ 6 q - 10 q^{4} - 6 q^{11} - 24 q^{14} + 26 q^{16} - 16 q^{19} - 20 q^{26} - 20 q^{29} + 16 q^{31} + 60 q^{34} + 28 q^{41} + 10 q^{44} + 48 q^{46} - 22 q^{49} + 8 q^{56} + 24 q^{59} - 12 q^{61} - 26 q^{64} - 16 q^{71} + 4 q^{74} + 96 q^{76} - 24 q^{79} + 16 q^{86} - 20 q^{89} + 16 q^{91} + 32 q^{94}+O(q^{100}) $ 6 * q - 10 * q^4 - 6 * q^11 - 24 * q^14 + 26 * q^16 - 16 * q^19 - 20 * q^26 - 20 * q^29 + 16 * q^31 + 60 * q^34 + 28 * q^41 + 10 * q^44 + 48 * q^46 - 22 * q^49 + 8 * q^56 + 24 * q^59 - 12 * q^61 - 26 * q^64 - 16 * q^71 + 4 * q^74 + 96 * q^76 - 24 * q^79 + 16 * q^86 - 20 * q^89 + 16 * q^91 + 32 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2475\mathbb{Z}\right)^\times$.

$n$	$551$	$2026$	$2377$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	− 0.193937i	− 0.137134i	−0.997647	−	0.0685669i	$-0.978157\pi$
$2$	− 0.193937i	− 0.137134i	0.997647	−	0.0685669i	$-0.0218427\pi$
$3$	0	0
$3$	0	0
$4$	1.96239	0.981194
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 3.35026i	− 1.26628i	−0.774037	−	0.633140i	$-0.781766\pi$
$7$	− 3.35026i	− 1.26628i	0.774037	−	0.633140i	$-0.218234\pi$
$8$	− 0.768452i	− 0.271689i
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	2.96239i	0.821619i	0.911721	+	0.410809i	$0.134754\pi$
$13$	2.96239i	0.821619i	−0.911721	+	0.410809i	$0.865246\pi$
$14$	−0.649738	−0.173650
$15$	0	0
$16$	3.77575	0.943937
$17$	− 4.57452i	− 1.10948i	−0.832023	−	0.554741i	$-0.812817\pi$
$17$	− 4.57452i	− 1.10948i	0.832023	−	0.554741i	$-0.187183\pi$
$18$	0	0
$19$	4.31265	0.989390	0.494695	−	0.869067i	$-0.335280\pi$
$19$	4.31265	0.989390	0.494695	+	0.869067i	$0.335280\pi$
$20$	0	0
$21$	0	0
$22$	0.193937i	0.0413474i
$23$	6.70052i	1.39716i	0.715534	+	0.698578i	$0.246183\pi$
$23$	6.70052i	1.39716i	−0.715534	+	0.698578i	$0.753817\pi$
$24$	0	0
$25$	0	0
$26$	0.574515	0.112672
$27$	0	0
$28$	− 6.57452i	− 1.24247i
$29$	−3.61213	−0.670755	−0.335378	−	0.942084i	$-0.608864\pi$
$29$	−3.61213	−0.670755	−0.335378	+	0.942084i	$0.608864\pi$
$30$	0	0
$31$	9.92478	1.78254	0.891271	−	0.453470i	$-0.149814\pi$
$31$	9.92478	1.78254	0.891271	+	0.453470i	$0.149814\pi$
$32$	− 2.26916i	− 0.401134i
$33$	0	0
$34$	−0.887166	−0.152148
$35$	0	0
$36$	0	0
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	− 0.836381i	− 0.135679i
$39$	0	0
$40$	0	0
$41$	4.38787	0.685271	0.342635	−	0.939468i	$-0.388680\pi$
$41$	4.38787	0.685271	0.342635	+	0.939468i	$0.388680\pi$
$42$	0	0
$43$	− 9.27504i	− 1.41443i	−0.706998	−	0.707215i	$-0.749951\pi$
$43$	− 9.27504i	− 1.41443i	0.706998	−	0.707215i	$-0.250049\pi$
$44$	−1.96239	−0.295841
$45$	0	0
$46$	1.29948	0.191597
$47$	− 9.92478i	− 1.44768i	−0.689969	−	0.723839i	$-0.742376\pi$
$47$	− 9.92478i	− 1.44768i	0.689969	−	0.723839i	$-0.257624\pi$
$48$	0	0
$49$	−4.22425	−0.603465
$50$	0	0
$51$	0	0
$52$	5.81336i	0.806168i
$53$	− 4.70052i	− 0.645667i	−0.946456	−	0.322833i	$-0.895365\pi$
$53$	− 4.70052i	− 0.645667i	0.946456	−	0.322833i	$-0.104635\pi$
$54$	0	0
$55$	0	0
$56$	−2.57452	−0.344034
$57$	0	0
$58$	0.700523i	0.0919832i
$59$	10.7005	1.39309	0.696545	−	0.717513i	$-0.254720\pi$
$59$	10.7005	1.39309	0.696545	+	0.717513i	$0.254720\pi$
$60$	0	0
$61$	−8.70052	−1.11399	−0.556994	−	0.830517i	$-0.688045\pi$
$61$	−8.70052	−1.11399	−0.556994	+	0.830517i	$0.688045\pi$
$62$	− 1.92478i	− 0.244447i
$63$	0	0
$64$	7.11142	0.888927
$65$	0	0
$66$	0	0
$67$	− 5.92478i	− 0.723827i	−0.932212	−	0.361913i	$-0.882124\pi$
$67$	− 5.92478i	− 0.723827i	0.932212	−	0.361913i	$-0.117876\pi$
$68$	− 8.97698i	− 1.08862i
$69$	0	0
$70$	0	0
$71$	−9.92478	−1.17785	−0.588927	−	0.808186i	$-0.700450\pi$
$71$	−9.92478	−1.17785	−0.588927	+	0.808186i	$0.700450\pi$
$72$	0	0
$73$	− 7.73813i	− 0.905680i	−0.891592	−	0.452840i	$-0.850411\pi$
$73$	− 7.73813i	− 0.905680i	0.891592	−	0.452840i	$-0.149589\pi$
$74$	0.387873	0.0450893
$75$	0	0
$76$	8.46310	0.970784
$77$	3.35026i	0.381798i
$78$	0	0
$79$	−11.5369	−1.29800	−0.649002	−	0.760787i	$-0.724813\pi$
$79$	−11.5369	−1.29800	−0.649002	+	0.760787i	$0.724813\pi$
$80$	0	0
$81$	0	0
$82$	− 0.850969i	− 0.0939738i
$83$	− 10.8872i	− 1.19502i	−0.801861	−	0.597511i	$-0.796156\pi$
$83$	− 10.8872i	− 1.19502i	0.801861	−	0.597511i	$-0.203844\pi$
$84$	0	0
$85$	0	0
$86$	−1.79877	−0.193966
$87$	0	0
$88$	0.768452i	0.0819173i
$89$	−2.77575	−0.294229	−0.147114	−	0.989120i	$-0.546999\pi$
$89$	−2.77575	−0.294229	−0.147114	+	0.989120i	$0.546999\pi$
$90$	0	0
$91$	9.92478	1.04040
$92$	13.1490i	1.37088i
$93$	0	0
$94$	−1.92478	−0.198526
$95$	0	0
$96$	0	0
$97$	− 0.0752228i	− 0.00763772i	−0.999993	−	0.00381886i	$-0.998784\pi$
$97$	− 0.0752228i	− 0.00763772i	0.999993	−	0.00381886i	$-0.00121558\pi$
$98$	0.819237i	0.0827555i
$99$	0	0
$100$	0	0
$101$	15.0884	1.50135	0.750676	−	0.660671i	$-0.229728\pi$
$101$	15.0884	1.50135	0.750676	+	0.660671i	$0.229728\pi$
$102$	0	0
$103$	− 3.22425i	− 0.317695i	−0.987303	−	0.158848i	$-0.949222\pi$
$103$	− 3.22425i	− 0.317695i	0.987303	−	0.158848i	$-0.0507778\pi$
$104$	2.27645	0.223225
$105$	0	0
$106$	−0.911603	−0.0885427
$107$	− 0.962389i	− 0.0930376i	−0.998917	−	0.0465188i	$-0.985187\pi$
$107$	− 0.962389i	− 0.0930376i	0.998917	−	0.0465188i	$-0.0148127\pi$
$108$	0	0
$109$	−11.4010	−1.09202	−0.546011	−	0.837778i	$-0.683854\pi$
$109$	−11.4010	−1.09202	−0.546011	+	0.837778i	$0.683854\pi$
$110$	0	0
$111$	0	0
$112$	− 12.6497i	− 1.19529i
$113$	6.00000i	0.564433i	0.959351	+	0.282216i	$0.0910696\pi$
$113$	6.00000i	0.564433i	−0.959351	+	0.282216i	$0.908930\pi$
$114$	0	0
$115$	0	0
$116$	−7.08840	−0.658141
$117$	0	0
$118$	− 2.07522i	− 0.191040i
$119$	−15.3258	−1.40492
$120$	0	0
$121$	1.00000	0.0909091
$122$	1.68735i	0.152765i
$123$	0	0
$124$	19.4763	1.74902
$125$	0	0
$126$	0	0
$127$	14.5745i	1.29328i	0.762796	+	0.646640i	$0.223826\pi$
$127$	14.5745i	1.29328i	−0.762796	+	0.646640i	$0.776174\pi$
$128$	− 5.91748i	− 0.523037i
$129$	0	0
$130$	0	0
$131$	5.92478	0.517650	0.258825	−	0.965924i	$-0.416665\pi$
$131$	5.92478	0.517650	0.258825	+	0.965924i	$0.416665\pi$
$132$	0	0
$133$	− 14.4485i	− 1.25284i
$134$	−1.14903	−0.0992612
$135$	0	0
$136$	−3.51530	−0.301434
$137$	13.8496i	1.18325i	0.806214	+	0.591624i	$0.201513\pi$
$137$	13.8496i	1.18325i	−0.806214	+	0.591624i	$0.798487\pi$
$138$	0	0
$139$	−13.6121	−1.15457	−0.577283	−	0.816544i	$-0.695887\pi$
$139$	−13.6121	−1.15457	−0.577283	+	0.816544i	$0.695887\pi$
$140$	0	0
$141$	0	0
$142$	1.92478i	0.161524i
$143$	− 2.96239i	− 0.247727i
$144$	0	0
$145$	0	0
$146$	−1.50071	−0.124199
$147$	0	0
$148$	3.92478i	0.322615i
$149$	1.53690	0.125908	0.0629540	−	0.998016i	$-0.479948\pi$
$149$	1.53690	0.125908	0.0629540	+	0.998016i	$0.479948\pi$
$150$	0	0
$151$	−6.76116	−0.550215	−0.275108	−	0.961413i	$-0.588713\pi$
$151$	−6.76116	−0.550215	−0.275108	+	0.961413i	$0.588713\pi$
$152$	− 3.31406i	− 0.268806i
$153$	0	0
$154$	0.649738	0.0523574
$155$	0	0
$156$	0	0
$157$	5.47627i	0.437054i	0.975831	+	0.218527i	$0.0701252\pi$
$157$	5.47627i	0.437054i	−0.975831	+	0.218527i	$0.929875\pi$
$158$	2.23743i	0.178000i
$159$	0	0
$160$	0	0
$161$	22.4485	1.76919
$162$	0	0
$163$	12.6253i	0.988890i	0.869209	+	0.494445i	$0.164629\pi$
$163$	12.6253i	0.988890i	−0.869209	+	0.494445i	$0.835371\pi$
$164$	8.61071	0.672384
$165$	0	0
$166$	−2.11142	−0.163878
$167$	18.3634i	1.42101i	0.703695	+	0.710503i	$0.251532\pi$
$167$	18.3634i	1.42101i	−0.703695	+	0.710503i	$0.748468\pi$
$168$	0	0
$169$	4.22425	0.324943
$170$	0	0
$171$	0	0
$172$	− 18.2012i	− 1.38783i
$173$	8.57452i	0.651908i	0.945386	+	0.325954i	$0.105686\pi$
$173$	8.57452i	0.651908i	−0.945386	+	0.325954i	$0.894314\pi$
$174$	0	0
$175$	0	0
$176$	−3.77575	−0.284608
$177$	0	0
$178$	0.538319i	0.0403487i
$179$	14.1768	1.05962	0.529812	−	0.848115i	$-0.322263\pi$
$179$	14.1768	1.05962	0.529812	+	0.848115i	$0.322263\pi$
$180$	0	0
$181$	−5.22425	−0.388316	−0.194158	−	0.980970i	$-0.562197\pi$
$181$	−5.22425	−0.388316	−0.194158	+	0.980970i	$0.562197\pi$
$182$	− 1.92478i	− 0.142674i
$183$	0	0
$184$	5.14903	0.379592
$185$	0	0
$186$	0	0
$187$	4.57452i	0.334522i
$188$	− 19.4763i	− 1.42045i
$189$	0	0
$190$	0	0
$191$	16.6253	1.20296	0.601482	−	0.798886i	$-0.294577\pi$
$191$	16.6253	1.20296	0.601482	+	0.798886i	$0.294577\pi$
$192$	0	0
$193$	− 16.3634i	− 1.17787i	−0.808182	−	0.588933i	$-0.799548\pi$
$193$	− 16.3634i	− 1.17787i	0.808182	−	0.588933i	$-0.200452\pi$
$194$	−0.0145884	−0.00104739
$195$	0	0
$196$	−8.28963	−0.592116
$197$	− 20.4241i	− 1.45515i	−0.686026	−	0.727577i	$-0.740646\pi$
$197$	− 20.4241i	− 1.45515i	0.686026	−	0.727577i	$-0.259354\pi$
$198$	0	0
$199$	8.62530	0.611431	0.305716	−	0.952123i	$-0.401104\pi$
$199$	8.62530	0.611431	0.305716	+	0.952123i	$0.401104\pi$
$200$	0	0
$201$	0	0
$202$	− 2.92619i	− 0.205886i
$203$	12.1016i	0.849364i
$204$	0	0
$205$	0	0
$206$	−0.625301	−0.0435668
$207$	0	0
$208$	11.1852i	0.775556i
$209$	−4.31265	−0.298312
$210$	0	0
$211$	9.08840	0.625671	0.312836	−	0.949807i	$-0.398721\pi$
$211$	9.08840	0.625671	0.312836	+	0.949807i	$0.398721\pi$
$212$	− 9.22425i	− 0.633524i
$213$	0	0
$214$	−0.186642	−0.0127586
$215$	0	0
$216$	0	0
$217$	− 33.2506i	− 2.25720i
$218$	2.21108i	0.149753i
$219$	0	0
$220$	0	0
$221$	13.5515	0.911572
$222$	0	0
$223$	− 6.70052i	− 0.448700i	−0.974509	−	0.224350i	$-0.927974\pi$
$223$	− 6.70052i	− 0.448700i	0.974509	−	0.224350i	$-0.0720259\pi$
$224$	−7.60228	−0.507949
$225$	0	0
$226$	1.16362	0.0774028
$227$	16.9624i	1.12583i	0.826514	+	0.562917i	$0.190321\pi$
$227$	16.9624i	1.12583i	−0.826514	+	0.562917i	$0.809679\pi$
$228$	0	0
$229$	−25.8496	−1.70819	−0.854093	−	0.520120i	$-0.825887\pi$
$229$	−25.8496	−1.70819	−0.854093	+	0.520120i	$0.825887\pi$
$230$	0	0
$231$	0	0
$232$	2.77575i	0.182237i
$233$	19.2750i	1.26275i	0.775478	+	0.631375i	$0.217509\pi$
$233$	19.2750i	1.26275i	−0.775478	+	0.631375i	$0.782491\pi$
$234$	0	0
$235$	0	0
$236$	20.9986	1.36689
$237$	0	0
$238$	2.97224i	0.192662i
$239$	26.5501	1.71738	0.858691	−	0.512494i	$-0.171278\pi$
$239$	26.5501	1.71738	0.858691	+	0.512494i	$0.171278\pi$
$240$	0	0
$241$	28.5501	1.83907	0.919536	−	0.393006i	$-0.128565\pi$
$241$	28.5501	1.83907	0.919536	+	0.393006i	$0.128565\pi$
$242$	− 0.193937i	− 0.0124667i
$243$	0	0
$244$	−17.0738	−1.09304
$245$	0	0
$246$	0	0
$247$	12.7757i	0.812901i
$248$	− 7.62672i	− 0.484297i
$249$	0	0
$250$	0	0
$251$	−29.9248	−1.88884	−0.944418	−	0.328748i	$-0.893373\pi$
$251$	−29.9248	−1.88884	−0.944418	+	0.328748i	$0.893373\pi$
$252$	0	0
$253$	− 6.70052i	− 0.421258i
$254$	2.82653	0.177352
$255$	0	0
$256$	13.0752	0.817201
$257$	8.70052i	0.542724i	0.962477	+	0.271362i	$0.0874740\pi$
$257$	8.70052i	0.542724i	−0.962477	+	0.271362i	$0.912526\pi$
$258$	0	0
$259$	6.70052	0.416350
$260$	0	0
$261$	0	0
$262$	− 1.14903i	− 0.0709874i
$263$	− 12.2882i	− 0.757724i	−0.925453	−	0.378862i	$-0.876316\pi$
$263$	− 12.2882i	− 0.757724i	0.925453	−	0.378862i	$-0.123684\pi$
$264$	0	0
$265$	0	0
$266$	−2.80209	−0.171807
$267$	0	0
$268$	− 11.6267i	− 0.710215i
$269$	−5.84955	−0.356654	−0.178327	−	0.983971i	$-0.557068\pi$
$269$	−5.84955	−0.356654	−0.178327	+	0.983971i	$0.557068\pi$
$270$	0	0
$271$	−5.08840	−0.309098	−0.154549	−	0.987985i	$-0.549392\pi$
$271$	−5.08840	−0.309098	−0.154549	+	0.987985i	$0.549392\pi$
$272$	− 17.2722i	− 1.04728i
$273$	0	0
$274$	2.68594	0.162263
$275$	0	0
$276$	0	0
$277$	− 1.41090i	− 0.0847725i	−0.999101	−	0.0423863i	$-0.986504\pi$
$277$	− 1.41090i	− 0.0847725i	0.999101	−	0.0423863i	$-0.0134960\pi$
$278$	2.63989i	0.158330i
$279$	0	0
$280$	0	0
$281$	4.38787	0.261759	0.130879	−	0.991398i	$-0.458220\pi$
$281$	4.38787	0.261759	0.130879	+	0.991398i	$0.458220\pi$
$282$	0	0
$283$	26.5745i	1.57969i	0.613306	+	0.789845i	$0.289839\pi$
$283$	26.5745i	1.57969i	−0.613306	+	0.789845i	$0.710161\pi$
$284$	−19.4763	−1.15570
$285$	0	0
$286$	−0.574515	−0.0339718
$287$	− 14.7005i	− 0.867744i
$288$	0	0
$289$	−3.92619	−0.230952
$290$	0	0
$291$	0	0
$292$	− 15.1852i	− 0.888648i
$293$	3.42548i	0.200119i	0.994981	+	0.100059i	$0.0319033\pi$
$293$	3.42548i	0.200119i	−0.994981	+	0.100059i	$0.968097\pi$
$294$	0	0
$295$	0	0
$296$	1.53690	0.0893307
$297$	0	0
$298$	− 0.298062i	− 0.0172663i
$299$	−19.8496	−1.14793
$300$	0	0
$301$	−31.0738	−1.79106
$302$	1.31124i	0.0754531i
$303$	0	0
$304$	16.2835	0.933921
$305$	0	0
$306$	0	0
$307$	16.6497i	0.950251i	0.879918	+	0.475125i	$0.157597\pi$
$307$	16.6497i	0.950251i	−0.879918	+	0.475125i	$0.842403\pi$
$308$	6.57452i	0.374618i
$309$	0	0
$310$	0	0
$311$	−32.9986	−1.87118	−0.935589	−	0.353091i	$-0.885131\pi$
$311$	−32.9986	−1.87118	−0.935589	+	0.353091i	$0.885131\pi$
$312$	0	0
$313$	15.4010i	0.870519i	0.900305	+	0.435259i	$0.143343\pi$
$313$	15.4010i	0.870519i	−0.900305	+	0.435259i	$0.856657\pi$
$314$	1.06205	0.0599349
$315$	0	0
$316$	−22.6399	−1.27359
$317$	2.15045i	0.120781i	0.998175	+	0.0603905i	$0.0192346\pi$
$317$	2.15045i	0.120781i	−0.998175	+	0.0603905i	$0.980765\pi$
$318$	0	0
$319$	3.61213	0.202240
$320$	0	0
$321$	0	0
$322$	− 4.35359i	− 0.242616i
$323$	− 19.7283i	− 1.09771i
$324$	0	0
$325$	0	0
$326$	2.44851	0.135610
$327$	0	0
$328$	− 3.37187i	− 0.186180i
$329$	−33.2506	−1.83316
$330$	0	0
$331$	−14.5501	−0.799745	−0.399872	−	0.916571i	$-0.630946\pi$
$331$	−14.5501	−0.799745	−0.399872	+	0.916571i	$0.630946\pi$
$332$	− 21.3649i	− 1.17255i
$333$	0	0
$334$	3.56134	0.194868
$335$	0	0
$336$	0	0
$337$	− 16.2619i	− 0.885840i	−0.896561	−	0.442920i	$-0.853943\pi$
$337$	− 16.2619i	− 0.885840i	0.896561	−	0.442920i	$-0.146057\pi$
$338$	− 0.819237i	− 0.0445606i
$339$	0	0
$340$	0	0
$341$	−9.92478	−0.537457
$342$	0	0
$343$	− 9.29948i	− 0.502125i
$344$	−7.12742	−0.384285
$345$	0	0
$346$	1.66291	0.0893987
$347$	− 0.962389i	− 0.0516637i	−0.999666	−	0.0258319i	$-0.991777\pi$
$347$	− 0.962389i	− 0.0516637i	0.999666	−	0.0258319i	$-0.00822345\pi$
$348$	0	0
$349$	−20.7005	−1.10807	−0.554037	−	0.832492i	$-0.686913\pi$
$349$	−20.7005	−1.10807	−0.554037	+	0.832492i	$0.686913\pi$
$350$	0	0
$351$	0	0
$352$	2.26916i	0.120947i
$353$	− 20.5501i	− 1.09377i	−0.837208	−	0.546885i	$-0.815813\pi$
$353$	− 20.5501i	− 1.09377i	0.837208	−	0.546885i	$-0.184187\pi$
$354$	0	0
$355$	0	0
$356$	−5.44709	−0.288695
$357$	0	0
$358$	− 2.74940i	− 0.145310i
$359$	17.9248	0.946034	0.473017	−	0.881053i	$-0.343165\pi$
$359$	17.9248	0.946034	0.473017	+	0.881053i	$0.343165\pi$
$360$	0	0
$361$	−0.401047	−0.0211077
$362$	1.01317i	0.0532512i
$363$	0	0
$364$	19.4763	1.02083
$365$	0	0
$366$	0	0
$367$	29.6531i	1.54788i	0.633261	+	0.773939i	$0.281716\pi$
$367$	29.6531i	1.54788i	−0.633261	+	0.773939i	$0.718284\pi$
$368$	25.2995i	1.31883i
$369$	0	0
$370$	0	0
$371$	−15.7480	−0.817595
$372$	0	0
$373$	− 9.13918i	− 0.473209i	−0.971606	−	0.236604i	$-0.923965\pi$
$373$	− 9.13918i	− 0.473209i	0.971606	−	0.236604i	$-0.0760345\pi$
$374$	0.887166	0.0458743
$375$	0	0
$376$	−7.62672	−0.393318
$377$	− 10.7005i	− 0.551105i
$378$	0	0
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	0	0
$381$	0	0
$382$	− 3.22425i	− 0.164967i
$383$	34.9234i	1.78450i	0.451541	+	0.892250i	$0.350874\pi$
$383$	34.9234i	1.78450i	−0.451541	+	0.892250i	$0.649126\pi$
$384$	0	0
$385$	0	0
$386$	−3.17347	−0.161525
$387$	0	0
$388$	− 0.147616i	− 0.00749408i
$389$	2.77575	0.140736	0.0703680	−	0.997521i	$-0.477583\pi$
$389$	2.77575	0.140736	0.0703680	+	0.997521i	$0.477583\pi$
$390$	0	0
$391$	30.6516	1.55012
$392$	3.24614i	0.163955i
$393$	0	0
$394$	−3.96097	−0.199551
$395$	0	0
$396$	0	0
$397$	19.9248i	0.999996i	0.866027	+	0.499998i	$0.166666\pi$
$397$	19.9248i	0.999996i	−0.866027	+	0.499998i	$0.833334\pi$
$398$	− 1.67276i	− 0.0838479i
$399$	0	0
$400$	0	0
$401$	−2.00000	−0.0998752	−0.0499376	−	0.998752i	$-0.515902\pi$
$401$	−2.00000	−0.0998752	−0.0499376	+	0.998752i	$0.515902\pi$
$402$	0	0
$403$	29.4010i	1.46457i
$404$	29.6093	1.47312
$405$	0	0
$406$	2.34694	0.116477
$407$	− 2.00000i	− 0.0991363i
$408$	0	0
$409$	13.0738	0.646458	0.323229	−	0.946321i	$-0.395232\pi$
$409$	13.0738	0.646458	0.323229	+	0.946321i	$0.395232\pi$
$410$	0	0
$411$	0	0
$412$	− 6.32724i	− 0.311721i
$413$	− 35.8496i	− 1.76404i
$414$	0	0
$415$	0	0
$416$	6.72213	0.329580
$417$	0	0
$418$	0.836381i	0.0409087i
$419$	7.22425	0.352928	0.176464	−	0.984307i	$-0.443534\pi$
$419$	7.22425	0.352928	0.176464	+	0.984307i	$0.443534\pi$
$420$	0	0
$421$	30.6253	1.49259	0.746293	−	0.665618i	$-0.231832\pi$
$421$	30.6253	1.49259	0.746293	+	0.665618i	$0.231832\pi$
$422$	− 1.76257i	− 0.0858007i
$423$	0	0
$424$	−3.61213	−0.175420
$425$	0	0
$426$	0	0
$427$	29.1490i	1.41062i
$428$	− 1.88858i	− 0.0912880i
$429$	0	0
$430$	0	0
$431$	33.8759	1.63174	0.815872	−	0.578232i	$-0.196257\pi$
$431$	33.8759	1.63174	0.815872	+	0.578232i	$0.196257\pi$
$432$	0	0
$433$	− 9.47627i	− 0.455400i	−0.973731	−	0.227700i	$-0.926879\pi$
$433$	− 9.47627i	− 0.455400i	0.973731	−	0.227700i	$-0.0731206\pi$
$434$	−6.44851	−0.309538
$435$	0	0
$436$	−22.3733	−1.07149
$437$	28.8970i	1.38233i
$438$	0	0
$439$	29.4617	1.40613	0.703065	−	0.711126i	$-0.251814\pi$
$439$	29.4617	1.40613	0.703065	+	0.711126i	$0.251814\pi$
$440$	0	0
$441$	0	0
$442$	− 2.62813i	− 0.125007i
$443$	19.0738i	0.906224i	0.891454	+	0.453112i	$0.149686\pi$
$443$	19.0738i	0.906224i	−0.891454	+	0.453112i	$0.850314\pi$
$444$	0	0
$445$	0	0
$446$	−1.29948	−0.0615320
$447$	0	0
$448$	− 23.8251i	− 1.12563i
$449$	35.8759	1.69309	0.846544	−	0.532318i	$-0.178679\pi$
$449$	35.8759	1.69309	0.846544	+	0.532318i	$0.178679\pi$
$450$	0	0
$451$	−4.38787	−0.206617
$452$	11.7743i	0.553818i
$453$	0	0
$454$	3.28963	0.154390
$455$	0	0
$456$	0	0
$457$	− 5.28963i	− 0.247438i	−0.992317	−	0.123719i	$-0.960518\pi$
$457$	− 5.28963i	− 0.247438i	0.992317	−	0.123719i	$-0.0394822\pi$
$458$	5.01317i	0.234250i
$459$	0	0
$460$	0	0
$461$	−36.3390	−1.69248	−0.846238	−	0.532805i	$-0.821138\pi$
$461$	−36.3390	−1.69248	−0.846238	+	0.532805i	$0.821138\pi$
$462$	0	0
$463$	10.5501i	0.490304i	0.969485	+	0.245152i	$0.0788378\pi$
$463$	10.5501i	0.490304i	−0.969485	+	0.245152i	$0.921162\pi$
$464$	−13.6385	−0.633150
$465$	0	0
$466$	3.73813	0.173166
$467$	18.7005i	0.865357i	0.901548	+	0.432679i	$0.142431\pi$
$467$	18.7005i	0.865357i	−0.901548	+	0.432679i	$0.857569\pi$
$468$	0	0
$469$	−19.8496	−0.916567
$470$	0	0
$471$	0	0
$472$	− 8.22284i	− 0.378487i
$473$	9.27504i	0.426467i
$474$	0	0
$475$	0	0
$476$	−30.0752	−1.37850
$477$	0	0
$478$	− 5.14903i	− 0.235511i
$479$	−9.29948	−0.424904	−0.212452	−	0.977172i	$-0.568145\pi$
$479$	−9.29948	−0.424904	−0.212452	+	0.977172i	$0.568145\pi$
$480$	0	0
$481$	−5.92478	−0.270147
$482$	− 5.53690i	− 0.252199i
$483$	0	0
$484$	1.96239	0.0891995
$485$	0	0
$486$	0	0
$487$	35.4763i	1.60758i	0.594911	+	0.803792i	$0.297187\pi$
$487$	35.4763i	1.60758i	−0.594911	+	0.803792i	$0.702813\pi$
$488$	6.68594i	0.302658i
$489$	0	0
$490$	0	0
$491$	−24.7757	−1.11811	−0.559057	−	0.829129i	$-0.688837\pi$
$491$	−24.7757	−1.11811	−0.559057	+	0.829129i	$0.688837\pi$
$492$	0	0
$493$	16.5237i	0.744191i
$494$	2.47768	0.111476
$495$	0	0
$496$	37.4734	1.68261
$497$	33.2506i	1.49149i
$498$	0	0
$499$	−14.1768	−0.634640	−0.317320	−	0.948318i	$-0.602783\pi$
$499$	−14.1768	−0.634640	−0.317320	+	0.948318i	$0.602783\pi$
$500$	0	0
$501$	0	0
$502$	5.80351i	0.259023i
$503$	8.43866i	0.376261i	0.982144	+	0.188131i	$0.0602428\pi$
$503$	8.43866i	0.376261i	−0.982144	+	0.188131i	$0.939757\pi$
$504$	0	0
$505$	0	0
$506$	−1.29948	−0.0577688
$507$	0	0
$508$	28.6009i	1.26896i
$509$	1.10299	0.0488890	0.0244445	−	0.999701i	$-0.492218\pi$
$509$	1.10299	0.0488890	0.0244445	+	0.999701i	$0.492218\pi$
$510$	0	0
$511$	−25.9248	−1.14684
$512$	− 14.3707i	− 0.635103i
$513$	0	0
$514$	1.68735	0.0744258
$515$	0	0
$516$	0	0
$517$	9.92478i	0.436491i
$518$	− 1.29948i	− 0.0570957i
$519$	0	0
$520$	0	0
$521$	12.4485	0.545379	0.272690	−	0.962102i	$-0.412087\pi$
$521$	12.4485	0.545379	0.272690	+	0.962102i	$0.412087\pi$
$522$	0	0
$523$	30.0508i	1.31403i	0.753878	+	0.657015i	$0.228181\pi$
$523$	30.0508i	1.31403i	−0.753878	+	0.657015i	$0.771819\pi$
$524$	11.6267	0.507915
$525$	0	0
$526$	−2.38313	−0.103910
$527$	− 45.4010i	− 1.97770i
$528$	0	0
$529$	−21.8970	−0.952044
$530$	0	0
$531$	0	0
$532$	− 28.3536i	− 1.22928i
$533$	12.9986i	0.563031i
$534$	0	0
$535$	0	0
$536$	−4.55291	−0.196656
$537$	0	0
$538$	1.13444i	0.0489093i
$539$	4.22425	0.181951
$540$	0	0
$541$	−18.0000	−0.773880	−0.386940	−	0.922105i	$-0.626468\pi$
$541$	−18.0000	−0.773880	−0.386940	+	0.922105i	$0.626468\pi$
$542$	0.986826i	0.0423878i
$543$	0	0
$544$	−10.3803	−0.445052
$545$	0	0
$546$	0	0
$547$	14.3028i	0.611544i	0.952105	+	0.305772i	$0.0989145\pi$
$547$	14.3028i	0.611544i	−0.952105	+	0.305772i	$0.901086\pi$
$548$	27.1782i	1.16100i
$549$	0	0
$550$	0	0
$551$	−15.5778	−0.663638
$552$	0	0
$553$	38.6516i	1.64364i
$554$	−0.273624	−0.0116252
$555$	0	0
$556$	−26.7123	−1.13285
$557$	− 11.7988i	− 0.499930i	−0.968255	−	0.249965i	$-0.919581\pi$
$557$	− 11.7988i	− 0.499930i	0.968255	−	0.249965i	$-0.0804191\pi$
$558$	0	0
$559$	27.4763	1.16212
$560$	0	0
$561$	0	0
$562$	− 0.850969i	− 0.0358960i
$563$	− 30.4847i	− 1.28478i	−0.766379	−	0.642389i	$-0.777944\pi$
$563$	− 30.4847i	− 1.28478i	0.766379	−	0.642389i	$-0.222056\pi$
$564$	0	0
$565$	0	0
$566$	5.15377	0.216629
$567$	0	0
$568$	7.62672i	0.320010i
$569$	−27.0884	−1.13560	−0.567802	−	0.823165i	$-0.692206\pi$
$569$	−27.0884	−1.13560	−0.567802	+	0.823165i	$0.692206\pi$
$570$	0	0
$571$	7.28489	0.304863	0.152432	−	0.988314i	$-0.451290\pi$
$571$	7.28489	0.304863	0.152432	+	0.988314i	$0.451290\pi$
$572$	− 5.81336i	− 0.243069i
$573$	0	0
$574$	−2.85097	−0.118997
$575$	0	0
$576$	0	0
$577$	31.6239i	1.31652i	0.752791	+	0.658260i	$0.228707\pi$
$577$	31.6239i	1.31652i	−0.752791	+	0.658260i	$0.771293\pi$
$578$	0.761432i	0.0316714i
$579$	0	0
$580$	0	0
$581$	−36.4749	−1.51323
$582$	0	0
$583$	4.70052i	0.194676i
$584$	−5.94639	−0.246063
$585$	0	0
$586$	0.664327	0.0274431
$587$	33.1490i	1.36821i	0.729385	+	0.684103i	$0.239806\pi$
$587$	33.1490i	1.36821i	−0.729385	+	0.684103i	$0.760194\pi$
$588$	0	0
$589$	42.8021	1.76363
$590$	0	0
$591$	0	0
$592$	7.55149i	0.310364i
$593$	− 34.4993i	− 1.41672i	−0.705853	−	0.708358i	$-0.749436\pi$
$593$	− 34.4993i	− 1.41672i	0.705853	−	0.708358i	$-0.250564\pi$
$594$	0	0
$595$	0	0
$596$	3.01600	0.123540
$597$	0	0
$598$	3.84955i	0.157420i
$599$	−14.4485	−0.590350	−0.295175	−	0.955443i	$-0.595378\pi$
$599$	−14.4485	−0.590350	−0.295175	+	0.955443i	$0.595378\pi$
$600$	0	0
$601$	−15.9248	−0.649585	−0.324793	−	0.945785i	$-0.605295\pi$
$601$	−15.9248	−0.649585	−0.324793	+	0.945785i	$0.605295\pi$
$602$	6.02635i	0.245616i
$603$	0	0
$604$	−13.2680	−0.539868
$605$	0	0
$606$	0	0
$607$	14.5745i	0.591561i	0.955256	+	0.295781i	$0.0955798\pi$
$607$	14.5745i	0.591561i	−0.955256	+	0.295781i	$0.904420\pi$
$608$	− 9.78609i	− 0.396878i
$609$	0	0
$610$	0	0
$611$	29.4010	1.18944
$612$	0	0
$613$	16.4123i	0.662887i	0.943475	+	0.331443i	$0.107536\pi$
$613$	16.4123i	0.662887i	−0.943475	+	0.331443i	$0.892464\pi$
$614$	3.22899	0.130312
$615$	0	0
$616$	2.57452	0.103730
$617$	− 17.8496i	− 0.718596i	−0.933223	−	0.359298i	$-0.883016\pi$
$617$	− 17.8496i	− 0.718596i	0.933223	−	0.359298i	$-0.116984\pi$
$618$	0	0
$619$	0.402462	0.0161763	0.00808815	−	0.999967i	$-0.497425\pi$
$619$	0.402462	0.0161763	0.00808815	+	0.999967i	$0.497425\pi$
$620$	0	0
$621$	0	0
$622$	6.39963i	0.256602i
$623$	9.29948i	0.372576i
$624$	0	0
$625$	0	0
$626$	2.98683	0.119378
$627$	0	0
$628$	10.7466i	0.428835i
$629$	9.14903	0.364796
$630$	0	0
$631$	−38.0263	−1.51380	−0.756902	−	0.653528i	$-0.773288\pi$
$631$	−38.0263	−1.51380	−0.756902	+	0.653528i	$0.773288\pi$
$632$	8.86556i	0.352653i
$633$	0	0
$634$	0.417050	0.0165632
$635$	0	0
$636$	0	0
$637$	− 12.5139i	− 0.495818i
$638$	− 0.700523i	− 0.0277340i
$639$	0	0
$640$	0	0
$641$	28.0263	1.10697	0.553487	−	0.832858i	$-0.313297\pi$
$641$	28.0263	1.10697	0.553487	+	0.832858i	$0.313297\pi$
$642$	0	0
$643$	− 4.62530i	− 0.182404i	−0.995832	−	0.0912020i	$-0.970929\pi$
$643$	− 4.62530i	− 0.182404i	0.995832	−	0.0912020i	$-0.0290709\pi$
$644$	44.0527	1.73592
$645$	0	0
$646$	−3.82604	−0.150533
$647$	23.5778i	0.926941i	0.886112	+	0.463470i	$0.153396\pi$
$647$	23.5778i	0.926941i	−0.886112	+	0.463470i	$0.846604\pi$
$648$	0	0
$649$	−10.7005	−0.420032
$650$	0	0
$651$	0	0
$652$	24.7757i	0.970293i
$653$	2.25202i	0.0881282i	0.999029	+	0.0440641i	$0.0140306\pi$
$653$	2.25202i	0.0881282i	−0.999029	+	0.0440641i	$0.985969\pi$
$654$	0	0
$655$	0	0
$656$	16.5675	0.646852
$657$	0	0
$658$	6.44851i	0.251389i
$659$	−41.4010	−1.61276	−0.806378	−	0.591401i	$-0.798575\pi$
$659$	−41.4010	−1.61276	−0.806378	+	0.591401i	$0.798575\pi$
$660$	0	0
$661$	3.40105	0.132285	0.0661427	−	0.997810i	$-0.478931\pi$
$661$	3.40105	0.132285	0.0661427	+	0.997810i	$0.478931\pi$
$662$	2.82179i	0.109672i
$663$	0	0
$664$	−8.36626	−0.324674
$665$	0	0
$666$	0	0
$667$	− 24.2031i	− 0.937149i
$668$	36.0362i	1.39428i
$669$	0	0
$670$	0	0
$671$	8.70052	0.335880
$672$	0	0
$673$	0.887166i	0.0341977i	0.999854	+	0.0170989i	$0.00544300\pi$
$673$	0.887166i	0.0341977i	−0.999854	+	0.0170989i	$0.994557\pi$
$674$	−3.15377	−0.121479
$675$	0	0
$676$	8.28963	0.318832
$677$	18.9018i	0.726453i	0.931701	+	0.363227i	$0.118325\pi$
$677$	18.9018i	0.726453i	−0.931701	+	0.363227i	$0.881675\pi$
$678$	0	0
$679$	−0.252016	−0.00967149
$680$	0	0
$681$	0	0
$682$	1.92478i	0.0737035i
$683$	20.8773i	0.798848i	0.916766	+	0.399424i	$0.130790\pi$
$683$	20.8773i	0.798848i	−0.916766	+	0.399424i	$0.869210\pi$
$684$	0	0
$685$	0	0
$686$	−1.80351	−0.0688583
$687$	0	0
$688$	− 35.0202i	− 1.33513i
$689$	13.9248	0.530492
$690$	0	0
$691$	−2.44851	−0.0931456	−0.0465728	−	0.998915i	$-0.514830\pi$
$691$	−2.44851	−0.0931456	−0.0465728	+	0.998915i	$0.514830\pi$
$692$	16.8265i	0.639649i
$693$	0	0
$694$	−0.186642	−0.00708485
$695$	0	0
$696$	0	0
$697$	− 20.0724i	− 0.760296i
$698$	4.01459i	0.151954i
$699$	0	0
$700$	0	0
$701$	2.98683	0.112811	0.0564054	−	0.998408i	$-0.482036\pi$
$701$	2.98683	0.112811	0.0564054	+	0.998408i	$0.482036\pi$
$702$	0	0
$703$	8.62530i	0.325309i
$704$	−7.11142	−0.268022
$705$	0	0
$706$	−3.98541	−0.149993
$707$	− 50.5501i	− 1.90113i
$708$	0	0
$709$	−24.1768	−0.907979	−0.453989	−	0.891007i	$-0.650000\pi$
$709$	−24.1768	−0.907979	−0.453989	+	0.891007i	$0.650000\pi$
$710$	0	0
$711$	0	0
$712$	2.13303i	0.0799386i
$713$	66.5012i	2.49049i
$714$	0	0
$715$	0	0
$716$	27.8204	1.03970
$717$	0	0
$718$	− 3.47627i	− 0.129733i
$719$	−30.0263	−1.11979	−0.559897	−	0.828562i	$-0.689159\pi$
$719$	−30.0263	−1.11979	−0.559897	+	0.828562i	$0.689159\pi$
$720$	0	0
$721$	−10.8021	−0.402291
$722$	0.0777777i	0.00289459i
$723$	0	0
$724$	−10.2520	−0.381013
$725$	0	0
$726$	0	0
$727$	− 14.9525i	− 0.554559i	−0.960789	−	0.277279i	$-0.910567\pi$
$727$	− 14.9525i	− 0.554559i	0.960789	−	0.277279i	$-0.0894328\pi$
$728$	− 7.62672i	− 0.282665i
$729$	0	0
$730$	0	0
$731$	−42.4288	−1.56929
$732$	0	0
$733$	− 19.1128i	− 0.705949i	−0.935633	−	0.352974i	$-0.885170\pi$
$733$	− 19.1128i	− 0.705949i	0.935633	−	0.352974i	$-0.114830\pi$
$734$	5.75081	0.212266
$735$	0	0
$736$	15.2046	0.560447
$737$	5.92478i	0.218242i
$738$	0	0
$739$	3.31406	0.121910	0.0609549	−	0.998141i	$-0.480585\pi$
$739$	3.31406	0.121910	0.0609549	+	0.998141i	$0.480585\pi$
$740$	0	0
$741$	0	0
$742$	3.05411i	0.112120i
$743$	34.9887i	1.28361i	0.766867	+	0.641806i	$0.221815\pi$
$743$	34.9887i	1.28361i	−0.766867	+	0.641806i	$0.778185\pi$
$744$	0	0
$745$	0	0
$746$	−1.77242	−0.0648930
$747$	0	0
$748$	8.97698i	0.328231i
$749$	−3.22425	−0.117812
$750$	0	0
$751$	−26.9234	−0.982447	−0.491224	−	0.871033i	$-0.663450\pi$
$751$	−26.9234	−0.982447	−0.491224	+	0.871033i	$0.663450\pi$
$752$	− 37.4734i	− 1.36652i
$753$	0	0
$754$	−2.07522	−0.0755752
$755$	0	0
$756$	0	0
$757$	− 15.9248i	− 0.578796i	−0.957209	−	0.289398i	$-0.906545\pi$
$757$	− 15.9248i	− 0.578796i	0.957209	−	0.289398i	$-0.0934551\pi$
$758$	− 3.87873i	− 0.140882i
$759$	0	0
$760$	0	0
$761$	30.9380	1.12150	0.560750	−	0.827985i	$-0.310513\pi$
$761$	30.9380	1.12150	0.560750	+	0.827985i	$0.310513\pi$
$762$	0	0
$763$	38.1965i	1.38281i
$764$	32.6253	1.18034
$765$	0	0
$766$	6.77292	0.244715
$767$	31.6991i	1.14459i
$768$	0	0
$769$	−9.32582	−0.336298	−0.168149	−	0.985762i	$-0.553779\pi$
$769$	−9.32582	−0.336298	−0.168149	+	0.985762i	$0.553779\pi$
$770$	0	0
$771$	0	0
$772$	− 32.1114i	− 1.15572i
$773$	− 44.7005i	− 1.60777i	−0.594787	−	0.803883i	$-0.702764\pi$
$773$	− 44.7005i	− 1.60777i	0.594787	−	0.803883i	$-0.297236\pi$
$774$	0	0
$775$	0	0
$776$	−0.0578051	−0.00207508
$777$	0	0
$778$	− 0.538319i	− 0.0192997i
$779$	18.9234	0.678000
$780$	0	0
$781$	9.92478	0.355136
$782$	− 5.94448i	− 0.212574i
$783$	0	0
$784$	−15.9497	−0.569633
$785$	0	0
$786$	0	0
$787$	− 21.6775i	− 0.772719i	−0.922348	−	0.386360i	$-0.873732\pi$
$787$	− 21.6775i	− 0.772719i	0.922348	−	0.386360i	$-0.126268\pi$
$788$	− 40.0800i	− 1.42779i
$789$	0	0
$790$	0	0
$791$	20.1016	0.714730
$792$	0	0
$793$	− 25.7743i	− 0.915273i
$794$	3.86414	0.137133
$795$	0	0
$796$	16.9262	0.599933
$797$	22.7466i	0.805725i	0.915261	+	0.402862i	$0.131985\pi$
$797$	22.7466i	0.805725i	−0.915261	+	0.402862i	$0.868015\pi$
$798$	0	0
$799$	−45.4010	−1.60617
$800$	0	0
$801$	0	0
$802$	0.387873i	0.0136963i
$803$	7.73813i	0.273073i
$804$	0	0
$805$	0	0
$806$	5.70194	0.200842
$807$	0	0
$808$	− 11.5947i	− 0.407900i
$809$	−23.6121	−0.830158	−0.415079	−	0.909785i	$-0.636246\pi$
$809$	−23.6121	−0.830158	−0.415079	+	0.909785i	$0.636246\pi$
$810$	0	0
$811$	−26.0870	−0.916038	−0.458019	−	0.888942i	$-0.651441\pi$
$811$	−26.0870	−0.916038	−0.458019	+	0.888942i	$0.651441\pi$
$812$	23.7480i	0.833391i
$813$	0	0
$814$	−0.387873	−0.0135949
$815$	0	0
$816$	0	0
$817$	− 40.0000i	− 1.39942i
$818$	− 2.53549i	− 0.0886513i
$819$	0	0
$820$	0	0
$821$	54.4142	1.89907	0.949535	−	0.313662i	$-0.101556\pi$
$821$	54.4142	1.89907	0.949535	+	0.313662i	$0.101556\pi$
$822$	0	0
$823$	0.121269i	0.00422716i	0.999998	+	0.00211358i	$0.000672774\pi$
$823$	0.121269i	0.00422716i	−0.999998	+	0.00211358i	$0.999327\pi$
$824$	−2.47768	−0.0863142
$825$	0	0
$826$	−6.95254	−0.241910
$827$	− 18.2130i	− 0.633328i	−0.948538	−	0.316664i	$-0.897437\pi$
$827$	− 18.2130i	− 0.633328i	0.948538	−	0.316664i	$-0.102563\pi$
$828$	0	0
$829$	−13.0738	−0.454072	−0.227036	−	0.973886i	$-0.572904\pi$
$829$	−13.0738	−0.454072	−0.227036	+	0.973886i	$0.572904\pi$
$830$	0	0
$831$	0	0
$832$	21.0668i	0.730359i
$833$	19.3239i	0.669534i
$834$	0	0
$835$	0	0
$836$	−8.46310	−0.292702
$837$	0	0
$838$	− 1.40105i	− 0.0483984i
$839$	−26.5501	−0.916610	−0.458305	−	0.888795i	$-0.651543\pi$
$839$	−26.5501	−0.916610	−0.458305	+	0.888795i	$0.651543\pi$
$840$	0	0
$841$	−15.9525	−0.550088
$842$	− 5.93937i	− 0.204684i
$843$	0	0
$844$	17.8350	0.613905
$845$	0	0
$846$	0	0
$847$	− 3.35026i	− 0.115116i
$848$	− 17.7480i	− 0.609468i
$849$	0	0
$850$	0	0
$851$	−13.4010	−0.459382
$852$	0	0
$853$	40.6155i	1.39065i	0.718697	+	0.695323i	$0.244739\pi$
$853$	40.6155i	1.39065i	−0.718697	+	0.695323i	$0.755261\pi$
$854$	5.65306	0.193444
$855$	0	0
$856$	−0.739549	−0.0252773
$857$	20.1721i	0.689064i	0.938775	+	0.344532i	$0.111962\pi$
$857$	20.1721i	0.689064i	−0.938775	+	0.344532i	$0.888038\pi$
$858$	0	0
$859$	−21.8035	−0.743926	−0.371963	−	0.928248i	$-0.621315\pi$
$859$	−21.8035	−0.743926	−0.371963	+	0.928248i	$0.621315\pi$
$860$	0	0
$861$	0	0
$862$	− 6.56978i	− 0.223767i
$863$	− 35.4274i	− 1.20596i	−0.797755	−	0.602981i	$-0.793979\pi$
$863$	− 35.4274i	− 1.20596i	0.797755	−	0.602981i	$-0.206021\pi$
$864$	0	0
$865$	0	0
$866$	−1.83780	−0.0624508
$867$	0	0
$868$	− 65.2506i	− 2.21475i
$869$	11.5369	0.391363
$870$	0	0
$871$	17.5515	0.594710
$872$	8.76116i	0.296690i
$873$	0	0
$874$	5.60419	0.189564
$875$	0	0
$876$	0	0
$877$	14.0362i	0.473969i	0.971513	+	0.236984i	$0.0761590\pi$
$877$	14.0362i	0.473969i	−0.971513	+	0.236984i	$0.923841\pi$
$878$	− 5.71370i	− 0.192828i
$879$	0	0
$880$	0	0
$881$	21.0738	0.709995	0.354997	−	0.934867i	$-0.384482\pi$
$881$	21.0738	0.709995	0.354997	+	0.934867i	$0.384482\pi$
$882$	0	0
$883$	− 42.1476i	− 1.41838i	−0.705017	−	0.709190i	$-0.749061\pi$
$883$	− 42.1476i	− 1.41838i	0.705017	−	0.709190i	$-0.250939\pi$
$884$	26.5933	0.894429
$885$	0	0
$886$	3.69911	0.124274
$887$	6.93604i	0.232889i	0.993197	+	0.116445i	$0.0371498\pi$
$887$	6.93604i	0.232889i	−0.993197	+	0.116445i	$0.962850\pi$
$888$	0	0
$889$	48.8284	1.63765
$890$	0	0
$891$	0	0
$892$	− 13.1490i	− 0.440262i
$893$	− 42.8021i	− 1.43232i
$894$	0	0
$895$	0	0
$896$	−19.8251	−0.662311
$897$	0	0
$898$	− 6.95765i	− 0.232180i
$899$	−35.8496	−1.19565
$900$	0	0
$901$	−21.5026	−0.716356
$902$	0.850969i	0.0283342i
$903$	0	0
$904$	4.61071	0.153350
$905$	0	0
$906$	0	0
$907$	− 53.2017i	− 1.76653i	−0.468870	−	0.883267i	$-0.655339\pi$
$907$	− 53.2017i	− 1.76653i	0.468870	−	0.883267i	$-0.344661\pi$
$908$	33.2868i	1.10466i
$909$	0	0
$910$	0	0
$911$	−36.4749	−1.20847	−0.604233	−	0.796808i	$-0.706520\pi$
$911$	−36.4749	−1.20847	−0.604233	+	0.796808i	$0.706520\pi$
$912$	0	0
$913$	10.8872i	0.360313i
$914$	−1.02585	−0.0339322
$915$	0	0
$916$	−50.7269	−1.67606
$917$	− 19.8496i	− 0.655490i
$918$	0	0
$919$	−9.73340	−0.321075	−0.160538	−	0.987030i	$-0.551323\pi$
$919$	−9.73340	−0.321075	−0.160538	+	0.987030i	$0.551323\pi$
$920$	0	0
$921$	0	0
$922$	7.04746i	0.232096i
$923$	− 29.4010i	− 0.967747i
$924$	0	0
$925$	0	0
$926$	2.04605	0.0672372
$927$	0	0
$928$	8.19649i	0.269063i
$929$	−24.1768	−0.793215	−0.396607	−	0.917988i	$-0.629813\pi$
$929$	−24.1768	−0.793215	−0.396607	+	0.917988i	$0.629813\pi$
$930$	0	0
$931$	−18.2177	−0.597062
$932$	37.8251i	1.23900i
$933$	0	0
$934$	3.62672	0.118670
$935$	0	0
$936$	0	0
$937$	7.48612i	0.244561i	0.992496	+	0.122280i	$0.0390207\pi$
$937$	7.48612i	0.244561i	−0.992496	+	0.122280i	$0.960979\pi$
$938$	3.84955i	0.125692i
$939$	0	0
$940$	0	0
$941$	21.2360	0.692274	0.346137	−	0.938184i	$-0.387493\pi$
$941$	21.2360	0.692274	0.346137	+	0.938184i	$0.387493\pi$
$942$	0	0
$943$	29.4010i	0.957430i
$944$	40.4025	1.31499
$945$	0	0
$946$	1.79877	0.0584830
$947$	− 15.4763i	− 0.502911i	−0.967869	−	0.251456i	$-0.919091\pi$
$947$	− 15.4763i	− 0.502911i	0.967869	−	0.251456i	$-0.0809092\pi$
$948$	0	0
$949$	22.9234	0.744124
$950$	0	0
$951$	0	0
$952$	11.7772i	0.381700i
$953$	32.0508i	1.03823i	0.854705	+	0.519113i	$0.173738\pi$
$953$	32.0508i	1.03823i	−0.854705	+	0.519113i	$0.826262\pi$
$954$	0	0
$955$	0	0
$956$	52.1016	1.68509
$957$	0	0
$958$	1.80351i	0.0582687i
$959$	46.3996	1.49832
$960$	0	0
$961$	67.5012	2.17746
$962$	1.14903i	0.0370462i
$963$	0	0
$964$	56.0263	1.80449
$965$	0	0
$966$	0	0
$967$	17.3766i	0.558794i	0.960176	+	0.279397i	$0.0901346\pi$
$967$	17.3766i	0.558794i	−0.960176	+	0.279397i	$0.909865\pi$
$968$	− 0.768452i	− 0.0246990i
$969$	0	0
$970$	0	0
$971$	−36.2031	−1.16181	−0.580907	−	0.813970i	$-0.697302\pi$
$971$	−36.2031	−1.16181	−0.580907	+	0.813970i	$0.697302\pi$
$972$	0	0
$973$	45.6042i	1.46200i
$974$	6.88015	0.220454
$975$	0	0
$976$	−32.8510	−1.05153
$977$	− 28.1476i	− 0.900522i	−0.892897	−	0.450261i	$-0.851331\pi$
$977$	− 28.1476i	− 0.900522i	0.892897	−	0.450261i	$-0.148669\pi$
$978$	0	0
$979$	2.77575	0.0887132
$980$	0	0
$981$	0	0
$982$	4.80492i	0.153331i
$983$	− 7.07381i	− 0.225619i	−0.993617	−	0.112810i	$-0.964015\pi$
$983$	− 7.07381i	− 0.225619i	0.993617	−	0.112810i	$-0.0359851\pi$
$984$	0	0
$985$	0	0
$986$	3.20456	0.102054
$987$	0	0
$988$	25.0710i	0.797614i
$989$	62.1476	1.97618
$990$	0	0
$991$	44.4260	1.41124	0.705619	−	0.708592i	$-0.250669\pi$
$991$	44.4260	1.41124	0.705619	+	0.708592i	$0.250669\pi$
$992$	− 22.5209i	− 0.715039i
$993$	0	0
$994$	6.44851	0.204534
$995$	0	0
$996$	0	0
$997$	28.4847i	0.902120i	0.892494	+	0.451060i	$0.148954\pi$
$997$	28.4847i	0.902120i	−0.892494	+	0.451060i	$0.851046\pi$
$998$	2.74940i	0.0870307i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.2.c.r.199.3		6
3.2	odd	2		825.2.c.g.199.4		6
5.2	odd	4		495.2.a.e.1.2		3
5.3	odd	4		2475.2.a.bb.1.2		3
5.4	even	2	inner	2475.2.c.r.199.4		6
15.2	even	4		165.2.a.c.1.2	✓	3
15.8	even	4		825.2.a.k.1.2		3
15.14	odd	2		825.2.c.g.199.3		6
20.7	even	4		7920.2.a.cj.1.1		3
55.32	even	4		5445.2.a.z.1.2		3
60.47	odd	4		2640.2.a.be.1.1		3
105.62	odd	4		8085.2.a.bk.1.2		3
165.32	odd	4		1815.2.a.m.1.2		3
165.98	odd	4		9075.2.a.cf.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.a.c.1.2	✓	3	15.2	even	4
495.2.a.e.1.2		3	5.2	odd	4
825.2.a.k.1.2		3	15.8	even	4
825.2.c.g.199.3		6	15.14	odd	2
825.2.c.g.199.4		6	3.2	odd	2
1815.2.a.m.1.2		3	165.32	odd	4
2475.2.a.bb.1.2		3	5.3	odd	4
2475.2.c.r.199.3		6	1.1	even	1	trivial
2475.2.c.r.199.4		6	5.4	even	2	inner
2640.2.a.be.1.1		3	60.47	odd	4
5445.2.a.z.1.2		3	55.32	even	4
7920.2.a.cj.1.1		3	20.7	even	4
8085.2.a.bk.1.2		3	105.62	odd	4
9075.2.a.cf.1.2		3	165.98	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2475.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.193937i q^{2} +1.96239 q^{4} -3.35026i q^{7} -0.768452i q^{8} +O(q^{10})\)	\(q-0.193937i q^{2} +1.96239 q^{4} -3.35026i q^{7} -0.768452i q^{8} -1.00000 q^{11} +2.96239i q^{13} -0.649738 q^{14} +3.77575 q^{16} -4.57452i q^{17} +4.31265 q^{19} +0.193937i q^{22} +6.70052i q^{23} +0.574515 q^{26} -6.57452i q^{28} -3.61213 q^{29} +9.92478 q^{31} -2.26916i q^{32} -0.887166 q^{34} +2.00000i q^{37} -0.836381i q^{38} +4.38787 q^{41} -9.27504i q^{43} -1.96239 q^{44} +1.29948 q^{46} -9.92478i q^{47} -4.22425 q^{49} +5.81336i q^{52} -4.70052i q^{53} -2.57452 q^{56} +0.700523i q^{58} +10.7005 q^{59} -8.70052 q^{61} -1.92478i q^{62} +7.11142 q^{64} -5.92478i q^{67} -8.97698i q^{68} -9.92478 q^{71} -7.73813i q^{73} +0.387873 q^{74} +8.46310 q^{76} +3.35026i q^{77} -11.5369 q^{79} -0.850969i q^{82} -10.8872i q^{83} -1.79877 q^{86} +0.768452i q^{88} -2.77575 q^{89} +9.92478 q^{91} +13.1490i q^{92} -1.92478 q^{94} -0.0752228i q^{97} +0.819237i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 10 q^{4}+O(q^{10}) \) 6 * q - 10 * q^4	\( 6 q - 10 q^{4} - 6 q^{11} - 24 q^{14} + 26 q^{16} - 16 q^{19} - 20 q^{26} - 20 q^{29} + 16 q^{31} + 60 q^{34} + 28 q^{41} + 10 q^{44} + 48 q^{46} - 22 q^{49} + 8 q^{56} + 24 q^{59} - 12 q^{61} - 26 q^{64} - 16 q^{71} + 4 q^{74} + 96 q^{76} - 24 q^{79} + 16 q^{86} - 20 q^{89} + 16 q^{91} + 32 q^{94}+O(q^{100}) \) 6 * q - 10 * q^4 - 6 * q^11 - 24 * q^14 + 26 * q^16 - 16 * q^19 - 20 * q^26 - 20 * q^29 + 16 * q^31 + 60 * q^34 + 28 * q^41 + 10 * q^44 + 48 * q^46 - 22 * q^49 + 8 * q^56 + 24 * q^59 - 12 * q^61 - 26 * q^64 - 16 * q^71 + 4 * q^74 + 96 * q^76 - 24 * q^79 + 16 * q^86 - 20 * q^89 + 16 * q^91 + 32 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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