LMFDB - Embedded newform 7650.2.a.cx.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7650,2,Mod(1,7650)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7650, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("7650.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7650 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7650.a (trivial)

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:	yes
Analytic conductor:	$61.0855575463$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 510)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	7650.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.23607 q^{7} -1.00000 q^{8} -2.00000 q^{11} +5.23607 q^{13} +1.23607 q^{14} +1.00000 q^{16} -1.00000 q^{17} -6.47214 q^{19} +2.00000 q^{22} -4.00000 q^{23} -5.23607 q^{26} -1.23607 q^{28} +4.00000 q^{29} +2.47214 q^{31} -1.00000 q^{32} +1.00000 q^{34} +2.00000 q^{37} +6.47214 q^{38} +7.70820 q^{41} +12.1803 q^{43} -2.00000 q^{44} +4.00000 q^{46} -10.4721 q^{47} -5.47214 q^{49} +5.23607 q^{52} +10.9443 q^{53} +1.23607 q^{56} -4.00000 q^{58} -11.7082 q^{59} +4.47214 q^{61} -2.47214 q^{62} +1.00000 q^{64} -1.70820 q^{67} -1.00000 q^{68} -11.2361 q^{71} +8.76393 q^{73} -2.00000 q^{74} -6.47214 q^{76} +2.47214 q^{77} +0.944272 q^{79} -7.70820 q^{82} -10.4721 q^{83} -12.1803 q^{86} +2.00000 q^{88} -12.4721 q^{89} -6.47214 q^{91} -4.00000 q^{92} +10.4721 q^{94} -1.70820 q^{97} +5.47214 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} + 2 q^{7} - 2 q^{8} - 4 q^{11} + 6 q^{13} - 2 q^{14} + 2 q^{16} - 2 q^{17} - 4 q^{19} + 4 q^{22} - 8 q^{23} - 6 q^{26} + 2 q^{28} + 8 q^{29} - 4 q^{31} - 2 q^{32} + 2 q^{34} + 4 q^{37}+ \cdots + 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^7 - 2 * q^8 - 4 * q^11 + 6 * q^13 - 2 * q^14 + 2 * q^16 - 2 * q^17 - 4 * q^19 + 4 * q^22 - 8 * q^23 - 6 * q^26 + 2 * q^28 + 8 * q^29 - 4 * q^31 - 2 * q^32 + 2 * q^34 + 4 * q^37 + 4 * q^38 + 2 * q^41 + 2 * q^43 - 4 * q^44 + 8 * q^46 - 12 * q^47 - 2 * q^49 + 6 * q^52 + 4 * q^53 - 2 * q^56 - 8 * q^58 - 10 * q^59 + 4 * q^62 + 2 * q^64 + 10 * q^67 - 2 * q^68 - 18 * q^71 + 22 * q^73 - 4 * q^74 - 4 * q^76 - 4 * q^77 - 16 * q^79 - 2 * q^82 - 12 * q^83 - 2 * q^86 + 4 * q^88 - 16 * q^89 - 4 * q^91 - 8 * q^92 + 12 * q^94 + 10 * q^97 + 2 * q^98

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
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.23607	−0.467190	−0.233595	−	0.972334i	$-0.575049\pi$
$7$	−1.23607	−0.467190	−0.233595	+	0.972334i	$0.575049\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	5.23607	1.45222	0.726112	−	0.687576i	$-0.241325\pi$
$13$	5.23607	1.45222	0.726112	+	0.687576i	$0.241325\pi$
$14$	1.23607	0.330353
$15$	0	0
$16$	1.00000	0.250000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−6.47214	−1.48481	−0.742405	−	0.669951i	$-0.766315\pi$
$19$	−6.47214	−1.48481	−0.742405	+	0.669951i	$0.766315\pi$
$20$	0	0
$21$	0	0
$22$	2.00000	0.426401
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	0	0
$26$	−5.23607	−1.02688
$27$	0	0
$28$	−1.23607	−0.233595
$29$	4.00000	0.742781	0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000	0.742781	0.371391	+	0.928477i	$0.378881\pi$
$30$	0	0
$31$	2.47214	0.444009	0.222004	−	0.975046i	$-0.428740\pi$
$31$	2.47214	0.444009	0.222004	+	0.975046i	$0.428740\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	1.00000	0.171499
$35$	0	0
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	6.47214	1.04992
$39$	0	0
$40$	0	0
$41$	7.70820	1.20382	0.601910	−	0.798564i	$-0.294407\pi$
$41$	7.70820	1.20382	0.601910	+	0.798564i	$0.294407\pi$
$42$	0	0
$43$	12.1803	1.85748	0.928742	−	0.370726i	$-0.120891\pi$
$43$	12.1803	1.85748	0.928742	+	0.370726i	$0.120891\pi$
$44$	−2.00000	−0.301511
$45$	0	0
$46$	4.00000	0.589768
$47$	−10.4721	−1.52752	−0.763759	−	0.645501i	$-0.776648\pi$
$47$	−10.4721	−1.52752	−0.763759	+	0.645501i	$0.776648\pi$
$48$	0	0
$49$	−5.47214	−0.781734
$50$	0	0
$51$	0	0
$52$	5.23607	0.726112
$53$	10.9443	1.50331	0.751656	−	0.659556i	$-0.229256\pi$
$53$	10.9443	1.50331	0.751656	+	0.659556i	$0.229256\pi$
$54$	0	0
$55$	0	0
$56$	1.23607	0.165177
$57$	0	0
$58$	−4.00000	−0.525226
$59$	−11.7082	−1.52428	−0.762139	−	0.647413i	$-0.775851\pi$
$59$	−11.7082	−1.52428	−0.762139	+	0.647413i	$0.775851\pi$
$60$	0	0
$61$	4.47214	0.572598	0.286299	−	0.958140i	$-0.407575\pi$
$61$	4.47214	0.572598	0.286299	+	0.958140i	$0.407575\pi$
$62$	−2.47214	−0.313962
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−1.70820	−0.208690	−0.104345	−	0.994541i	$-0.533275\pi$
$67$	−1.70820	−0.208690	−0.104345	+	0.994541i	$0.533275\pi$
$68$	−1.00000	−0.121268
$69$	0	0
$70$	0	0
$71$	−11.2361	−1.33348	−0.666738	−	0.745292i	$-0.732310\pi$
$71$	−11.2361	−1.33348	−0.666738	+	0.745292i	$0.732310\pi$
$72$	0	0
$73$	8.76393	1.02574	0.512870	−	0.858466i	$-0.328582\pi$
$73$	8.76393	1.02574	0.512870	+	0.858466i	$0.328582\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	−6.47214	−0.742405
$77$	2.47214	0.281726
$78$	0	0
$79$	0.944272	0.106239	0.0531194	−	0.998588i	$-0.483084\pi$
$79$	0.944272	0.106239	0.0531194	+	0.998588i	$0.483084\pi$
$80$	0	0
$81$	0	0
$82$	−7.70820	−0.851229
$83$	−10.4721	−1.14947	−0.574733	−	0.818341i	$-0.694894\pi$
$83$	−10.4721	−1.14947	−0.574733	+	0.818341i	$0.694894\pi$
$84$	0	0
$85$	0	0
$86$	−12.1803	−1.31344
$87$	0	0
$88$	2.00000	0.213201
$89$	−12.4721	−1.32204	−0.661022	−	0.750367i	$-0.729877\pi$
$89$	−12.4721	−1.32204	−0.661022	+	0.750367i	$0.729877\pi$
$90$	0	0
$91$	−6.47214	−0.678464
$92$	−4.00000	−0.417029
$93$	0	0
$94$	10.4721	1.08012
$95$	0	0
$96$	0	0
$97$	−1.70820	−0.173442	−0.0867209	−	0.996233i	$-0.527639\pi$
$97$	−1.70820	−0.173442	−0.0867209	+	0.996233i	$0.527639\pi$
$98$	5.47214	0.552769
$99$	0	0
$100$	0	0
$101$	1.70820	0.169973	0.0849863	−	0.996382i	$-0.472915\pi$
$101$	1.70820	0.169973	0.0849863	+	0.996382i	$0.472915\pi$
$102$	0	0
$103$	10.9443	1.07837	0.539186	−	0.842187i	$-0.318732\pi$
$103$	10.9443	1.07837	0.539186	+	0.842187i	$0.318732\pi$
$104$	−5.23607	−0.513439
$105$	0	0
$106$	−10.9443	−1.06300
$107$	−8.00000	−0.773389	−0.386695	−	0.922208i	$-0.626383\pi$
$107$	−8.00000	−0.773389	−0.386695	+	0.922208i	$0.626383\pi$
$108$	0	0
$109$	−7.52786	−0.721039	−0.360519	−	0.932752i	$-0.617401\pi$
$109$	−7.52786	−0.721039	−0.360519	+	0.932752i	$0.617401\pi$
$110$	0	0
$111$	0	0
$112$	−1.23607	−0.116797
$113$	10.0000	0.940721	0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000	0.940721	0.470360	+	0.882474i	$0.344124\pi$
$114$	0	0
$115$	0	0
$116$	4.00000	0.371391
$117$	0	0
$118$	11.7082	1.07783
$119$	1.23607	0.113310
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−4.47214	−0.404888
$123$	0	0
$124$	2.47214	0.222004
$125$	0	0
$126$	0	0
$127$	14.0000	1.24230	0.621150	−	0.783692i	$-0.286666\pi$
$127$	14.0000	1.24230	0.621150	+	0.783692i	$0.286666\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0	0
$131$	−12.4721	−1.08970	−0.544848	−	0.838535i	$-0.683413\pi$
$131$	−12.4721	−1.08970	−0.544848	+	0.838535i	$0.683413\pi$
$132$	0	0
$133$	8.00000	0.693688
$134$	1.70820	0.147566
$135$	0	0
$136$	1.00000	0.0857493
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	0	0
$139$	1.52786	0.129592	0.0647959	−	0.997899i	$-0.479360\pi$
$139$	1.52786	0.129592	0.0647959	+	0.997899i	$0.479360\pi$
$140$	0	0
$141$	0	0
$142$	11.2361	0.942910
$143$	−10.4721	−0.875724
$144$	0	0
$145$	0	0
$146$	−8.76393	−0.725308
$147$	0	0
$148$	2.00000	0.164399
$149$	−12.1803	−0.997852	−0.498926	−	0.866644i	$-0.666272\pi$
$149$	−12.1803	−0.997852	−0.498926	+	0.866644i	$0.666272\pi$
$150$	0	0
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	6.47214	0.524960
$153$	0	0
$154$	−2.47214	−0.199210
$155$	0	0
$156$	0	0
$157$	10.1803	0.812480	0.406240	−	0.913767i	$-0.366840\pi$
$157$	10.1803	0.812480	0.406240	+	0.913767i	$0.366840\pi$
$158$	−0.944272	−0.0751222
$159$	0	0
$160$	0	0
$161$	4.94427	0.389663
$162$	0	0
$163$	−22.4721	−1.76015	−0.880077	−	0.474831i	$-0.842509\pi$
$163$	−22.4721	−1.76015	−0.880077	+	0.474831i	$0.842509\pi$
$164$	7.70820	0.601910
$165$	0	0
$166$	10.4721	0.812795
$167$	13.8885	1.07473	0.537364	−	0.843350i	$-0.319420\pi$
$167$	13.8885	1.07473	0.537364	+	0.843350i	$0.319420\pi$
$168$	0	0
$169$	14.4164	1.10895
$170$	0	0
$171$	0	0
$172$	12.1803	0.928742
$173$	15.8885	1.20798	0.603992	−	0.796991i	$-0.293576\pi$
$173$	15.8885	1.20798	0.603992	+	0.796991i	$0.293576\pi$
$174$	0	0
$175$	0	0
$176$	−2.00000	−0.150756
$177$	0	0
$178$	12.4721	0.934826
$179$	7.70820	0.576138	0.288069	−	0.957610i	$-0.406987\pi$
$179$	7.70820	0.576138	0.288069	+	0.957610i	$0.406987\pi$
$180$	0	0
$181$	0.472136	0.0350936	0.0175468	−	0.999846i	$-0.494414\pi$
$181$	0.472136	0.0350936	0.0175468	+	0.999846i	$0.494414\pi$
$182$	6.47214	0.479747
$183$	0	0
$184$	4.00000	0.294884
$185$	0	0
$186$	0	0
$187$	2.00000	0.146254
$188$	−10.4721	−0.763759
$189$	0	0
$190$	0	0
$191$	−5.52786	−0.399982	−0.199991	−	0.979798i	$-0.564091\pi$
$191$	−5.52786	−0.399982	−0.199991	+	0.979798i	$0.564091\pi$
$192$	0	0
$193$	−23.5967	−1.69853	−0.849266	−	0.527966i	$-0.822955\pi$
$193$	−23.5967	−1.69853	−0.849266	+	0.527966i	$0.822955\pi$
$194$	1.70820	0.122642
$195$	0	0
$196$	−5.47214	−0.390867
$197$	−15.8885	−1.13201	−0.566006	−	0.824401i	$-0.691512\pi$
$197$	−15.8885	−1.13201	−0.566006	+	0.824401i	$0.691512\pi$
$198$	0	0
$199$	−8.94427	−0.634043	−0.317021	−	0.948418i	$-0.602683\pi$
$199$	−8.94427	−0.634043	−0.317021	+	0.948418i	$0.602683\pi$
$200$	0	0
$201$	0	0
$202$	−1.70820	−0.120189
$203$	−4.94427	−0.347020
$204$	0	0
$205$	0	0
$206$	−10.9443	−0.762524
$207$	0	0
$208$	5.23607	0.363056
$209$	12.9443	0.895374
$210$	0	0
$211$	−24.9443	−1.71723	−0.858617	−	0.512617i	$-0.828676\pi$
$211$	−24.9443	−1.71723	−0.858617	+	0.512617i	$0.828676\pi$
$212$	10.9443	0.751656
$213$	0	0
$214$	8.00000	0.546869
$215$	0	0
$216$	0	0
$217$	−3.05573	−0.207436
$218$	7.52786	0.509851
$219$	0	0
$220$	0	0
$221$	−5.23607	−0.352216
$222$	0	0
$223$	9.05573	0.606416	0.303208	−	0.952924i	$-0.401942\pi$
$223$	9.05573	0.606416	0.303208	+	0.952924i	$0.401942\pi$
$224$	1.23607	0.0825883
$225$	0	0
$226$	−10.0000	−0.665190
$227$	−9.88854	−0.656326	−0.328163	−	0.944621i	$-0.606429\pi$
$227$	−9.88854	−0.656326	−0.328163	+	0.944621i	$0.606429\pi$
$228$	0	0
$229$	−21.4164	−1.41524	−0.707618	−	0.706595i	$-0.750230\pi$
$229$	−21.4164	−1.41524	−0.707618	+	0.706595i	$0.750230\pi$
$230$	0	0
$231$	0	0
$232$	−4.00000	−0.262613
$233$	−5.41641	−0.354841	−0.177420	−	0.984135i	$-0.556775\pi$
$233$	−5.41641	−0.354841	−0.177420	+	0.984135i	$0.556775\pi$
$234$	0	0
$235$	0	0
$236$	−11.7082	−0.762139
$237$	0	0
$238$	−1.23607	−0.0801224
$239$	21.8885	1.41585	0.707926	−	0.706287i	$-0.249631\pi$
$239$	21.8885	1.41585	0.707926	+	0.706287i	$0.249631\pi$
$240$	0	0
$241$	9.41641	0.606564	0.303282	−	0.952901i	$-0.401918\pi$
$241$	9.41641	0.606564	0.303282	+	0.952901i	$0.401918\pi$
$242$	7.00000	0.449977
$243$	0	0
$244$	4.47214	0.286299
$245$	0	0
$246$	0	0
$247$	−33.8885	−2.15628
$248$	−2.47214	−0.156981
$249$	0	0
$250$	0	0
$251$	10.1803	0.642577	0.321289	−	0.946981i	$-0.395884\pi$
$251$	10.1803	0.642577	0.321289	+	0.946981i	$0.395884\pi$
$252$	0	0
$253$	8.00000	0.502956
$254$	−14.0000	−0.878438
$255$	0	0
$256$	1.00000	0.0625000
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	0	0
$259$	−2.47214	−0.153611
$260$	0	0
$261$	0	0
$262$	12.4721	0.770531
$263$	4.00000	0.246651	0.123325	−	0.992366i	$-0.460644\pi$
$263$	4.00000	0.246651	0.123325	+	0.992366i	$0.460644\pi$
$264$	0	0
$265$	0	0
$266$	−8.00000	−0.490511
$267$	0	0
$268$	−1.70820	−0.104345
$269$	7.05573	0.430195	0.215098	−	0.976593i	$-0.430993\pi$
$269$	7.05573	0.430195	0.215098	+	0.976593i	$0.430993\pi$
$270$	0	0
$271$	4.00000	0.242983	0.121491	−	0.992592i	$-0.461232\pi$
$271$	4.00000	0.242983	0.121491	+	0.992592i	$0.461232\pi$
$272$	−1.00000	−0.0606339
$273$	0	0
$274$	−2.00000	−0.120824
$275$	0	0
$276$	0	0
$277$	26.3607	1.58386	0.791930	−	0.610612i	$-0.209077\pi$
$277$	26.3607	1.58386	0.791930	+	0.610612i	$0.209077\pi$
$278$	−1.52786	−0.0916352
$279$	0	0
$280$	0	0
$281$	−1.41641	−0.0844958	−0.0422479	−	0.999107i	$-0.513452\pi$
$281$	−1.41641	−0.0844958	−0.0422479	+	0.999107i	$0.513452\pi$
$282$	0	0
$283$	−25.8885	−1.53891	−0.769457	−	0.638699i	$-0.779473\pi$
$283$	−25.8885	−1.53891	−0.769457	+	0.638699i	$0.779473\pi$
$284$	−11.2361	−0.666738
$285$	0	0
$286$	10.4721	0.619230
$287$	−9.52786	−0.562412
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	0	0
$292$	8.76393	0.512870
$293$	−29.4164	−1.71852	−0.859262	−	0.511535i	$-0.829077\pi$
$293$	−29.4164	−1.71852	−0.859262	+	0.511535i	$0.829077\pi$
$294$	0	0
$295$	0	0
$296$	−2.00000	−0.116248
$297$	0	0
$298$	12.1803	0.705588
$299$	−20.9443	−1.21124
$300$	0	0
$301$	−15.0557	−0.867798
$302$	4.00000	0.230174
$303$	0	0
$304$	−6.47214	−0.371202
$305$	0	0
$306$	0	0
$307$	4.18034	0.238585	0.119292	−	0.992859i	$-0.461937\pi$
$307$	4.18034	0.238585	0.119292	+	0.992859i	$0.461937\pi$
$308$	2.47214	0.140863
$309$	0	0
$310$	0	0
$311$	−19.5967	−1.11123	−0.555615	−	0.831440i	$-0.687517\pi$
$311$	−19.5967	−1.11123	−0.555615	+	0.831440i	$0.687517\pi$
$312$	0	0
$313$	−21.7082	−1.22702	−0.613510	−	0.789687i	$-0.710243\pi$
$313$	−21.7082	−1.22702	−0.613510	+	0.789687i	$0.710243\pi$
$314$	−10.1803	−0.574510
$315$	0	0
$316$	0.944272	0.0531194
$317$	−11.8885	−0.667727	−0.333864	−	0.942621i	$-0.608352\pi$
$317$	−11.8885	−0.667727	−0.333864	+	0.942621i	$0.608352\pi$
$318$	0	0
$319$	−8.00000	−0.447914
$320$	0	0
$321$	0	0
$322$	−4.94427	−0.275534
$323$	6.47214	0.360119
$324$	0	0
$325$	0	0
$326$	22.4721	1.24462
$327$	0	0
$328$	−7.70820	−0.425614
$329$	12.9443	0.713641
$330$	0	0
$331$	−3.41641	−0.187783	−0.0938914	−	0.995582i	$-0.529931\pi$
$331$	−3.41641	−0.187783	−0.0938914	+	0.995582i	$0.529931\pi$
$332$	−10.4721	−0.574733
$333$	0	0
$334$	−13.8885	−0.759947
$335$	0	0
$336$	0	0
$337$	−20.1803	−1.09929	−0.549647	−	0.835397i	$-0.685238\pi$
$337$	−20.1803	−1.09929	−0.549647	+	0.835397i	$0.685238\pi$
$338$	−14.4164	−0.784149
$339$	0	0
$340$	0	0
$341$	−4.94427	−0.267747
$342$	0	0
$343$	15.4164	0.832408
$344$	−12.1803	−0.656720
$345$	0	0
$346$	−15.8885	−0.854173
$347$	22.8328	1.22573	0.612865	−	0.790188i	$-0.290017\pi$
$347$	22.8328	1.22573	0.612865	+	0.790188i	$0.290017\pi$
$348$	0	0
$349$	−28.4721	−1.52408	−0.762039	−	0.647531i	$-0.775802\pi$
$349$	−28.4721	−1.52408	−0.762039	+	0.647531i	$0.775802\pi$
$350$	0	0
$351$	0	0
$352$	2.00000	0.106600
$353$	14.9443	0.795403	0.397702	−	0.917515i	$-0.369808\pi$
$353$	14.9443	0.795403	0.397702	+	0.917515i	$0.369808\pi$
$354$	0	0
$355$	0	0
$356$	−12.4721	−0.661022
$357$	0	0
$358$	−7.70820	−0.407391
$359$	9.52786	0.502861	0.251431	−	0.967875i	$-0.419099\pi$
$359$	9.52786	0.502861	0.251431	+	0.967875i	$0.419099\pi$
$360$	0	0
$361$	22.8885	1.20466
$362$	−0.472136	−0.0248149
$363$	0	0
$364$	−6.47214	−0.339232
$365$	0	0
$366$	0	0
$367$	−5.81966	−0.303784	−0.151892	−	0.988397i	$-0.548537\pi$
$367$	−5.81966	−0.303784	−0.151892	+	0.988397i	$0.548537\pi$
$368$	−4.00000	−0.208514
$369$	0	0
$370$	0	0
$371$	−13.5279	−0.702332
$372$	0	0
$373$	−1.23607	−0.0640012	−0.0320006	−	0.999488i	$-0.510188\pi$
$373$	−1.23607	−0.0640012	−0.0320006	+	0.999488i	$0.510188\pi$
$374$	−2.00000	−0.103418
$375$	0	0
$376$	10.4721	0.540059
$377$	20.9443	1.07868
$378$	0	0
$379$	−1.52786	−0.0784811	−0.0392406	−	0.999230i	$-0.512494\pi$
$379$	−1.52786	−0.0784811	−0.0392406	+	0.999230i	$0.512494\pi$
$380$	0	0
$381$	0	0
$382$	5.52786	0.282830
$383$	−17.8885	−0.914062	−0.457031	−	0.889451i	$-0.651087\pi$
$383$	−17.8885	−0.914062	−0.457031	+	0.889451i	$0.651087\pi$
$384$	0	0
$385$	0	0
$386$	23.5967	1.20104
$387$	0	0
$388$	−1.70820	−0.0867209
$389$	−20.1803	−1.02318	−0.511592	−	0.859229i	$-0.670944\pi$
$389$	−20.1803	−1.02318	−0.511592	+	0.859229i	$0.670944\pi$
$390$	0	0
$391$	4.00000	0.202289
$392$	5.47214	0.276385
$393$	0	0
$394$	15.8885	0.800453
$395$	0	0
$396$	0	0
$397$	3.88854	0.195160	0.0975802	−	0.995228i	$-0.468890\pi$
$397$	3.88854	0.195160	0.0975802	+	0.995228i	$0.468890\pi$
$398$	8.94427	0.448336
$399$	0	0
$400$	0	0
$401$	36.0689	1.80119	0.900597	−	0.434655i	$-0.143130\pi$
$401$	36.0689	1.80119	0.900597	+	0.434655i	$0.143130\pi$
$402$	0	0
$403$	12.9443	0.644800
$404$	1.70820	0.0849863
$405$	0	0
$406$	4.94427	0.245380
$407$	−4.00000	−0.198273
$408$	0	0
$409$	26.3607	1.30345	0.651726	−	0.758455i	$-0.274045\pi$
$409$	26.3607	1.30345	0.651726	+	0.758455i	$0.274045\pi$
$410$	0	0
$411$	0	0
$412$	10.9443	0.539186
$413$	14.4721	0.712127
$414$	0	0
$415$	0	0
$416$	−5.23607	−0.256719
$417$	0	0
$418$	−12.9443	−0.633125
$419$	−6.36068	−0.310740	−0.155370	−	0.987856i	$-0.549657\pi$
$419$	−6.36068	−0.310740	−0.155370	+	0.987856i	$0.549657\pi$
$420$	0	0
$421$	−37.4164	−1.82356	−0.911782	−	0.410674i	$-0.865293\pi$
$421$	−37.4164	−1.82356	−0.911782	+	0.410674i	$0.865293\pi$
$422$	24.9443	1.21427
$423$	0	0
$424$	−10.9443	−0.531501
$425$	0	0
$426$	0	0
$427$	−5.52786	−0.267512
$428$	−8.00000	−0.386695
$429$	0	0
$430$	0	0
$431$	−31.5967	−1.52196	−0.760981	−	0.648774i	$-0.775282\pi$
$431$	−31.5967	−1.52196	−0.760981	+	0.648774i	$0.775282\pi$
$432$	0	0
$433$	28.3607	1.36293	0.681464	−	0.731852i	$-0.261344\pi$
$433$	28.3607	1.36293	0.681464	+	0.731852i	$0.261344\pi$
$434$	3.05573	0.146680
$435$	0	0
$436$	−7.52786	−0.360519
$437$	25.8885	1.23842
$438$	0	0
$439$	13.5279	0.645650	0.322825	−	0.946459i	$-0.395368\pi$
$439$	13.5279	0.645650	0.322825	+	0.946459i	$0.395368\pi$
$440$	0	0
$441$	0	0
$442$	5.23607	0.249054
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	0	0
$445$	0	0
$446$	−9.05573	−0.428801
$447$	0	0
$448$	−1.23607	−0.0583987
$449$	−17.5967	−0.830442	−0.415221	−	0.909721i	$-0.636296\pi$
$449$	−17.5967	−0.830442	−0.415221	+	0.909721i	$0.636296\pi$
$450$	0	0
$451$	−15.4164	−0.725930
$452$	10.0000	0.470360
$453$	0	0
$454$	9.88854	0.464092
$455$	0	0
$456$	0	0
$457$	−16.3607	−0.765320	−0.382660	−	0.923889i	$-0.624992\pi$
$457$	−16.3607	−0.765320	−0.382660	+	0.923889i	$0.624992\pi$
$458$	21.4164	1.00072
$459$	0	0
$460$	0	0
$461$	30.0689	1.40045	0.700224	−	0.713923i	$-0.253084\pi$
$461$	30.0689	1.40045	0.700224	+	0.713923i	$0.253084\pi$
$462$	0	0
$463$	−18.9443	−0.880415	−0.440207	−	0.897896i	$-0.645095\pi$
$463$	−18.9443	−0.880415	−0.440207	+	0.897896i	$0.645095\pi$
$464$	4.00000	0.185695
$465$	0	0
$466$	5.41641	0.250910
$467$	−0.944272	−0.0436957	−0.0218478	−	0.999761i	$-0.506955\pi$
$467$	−0.944272	−0.0436957	−0.0218478	+	0.999761i	$0.506955\pi$
$468$	0	0
$469$	2.11146	0.0974980
$470$	0	0
$471$	0	0
$472$	11.7082	0.538914
$473$	−24.3607	−1.12011
$474$	0	0
$475$	0	0
$476$	1.23607	0.0566551
$477$	0	0
$478$	−21.8885	−1.00116
$479$	35.5967	1.62646	0.813228	−	0.581945i	$-0.197708\pi$
$479$	35.5967	1.62646	0.813228	+	0.581945i	$0.197708\pi$
$480$	0	0
$481$	10.4721	0.477488
$482$	−9.41641	−0.428906
$483$	0	0
$484$	−7.00000	−0.318182
$485$	0	0
$486$	0	0
$487$	−16.2918	−0.738252	−0.369126	−	0.929379i	$-0.620343\pi$
$487$	−16.2918	−0.738252	−0.369126	+	0.929379i	$0.620343\pi$
$488$	−4.47214	−0.202444
$489$	0	0
$490$	0	0
$491$	−12.2918	−0.554721	−0.277360	−	0.960766i	$-0.589460\pi$
$491$	−12.2918	−0.554721	−0.277360	+	0.960766i	$0.589460\pi$
$492$	0	0
$493$	−4.00000	−0.180151
$494$	33.8885	1.52472
$495$	0	0
$496$	2.47214	0.111002
$497$	13.8885	0.622986
$498$	0	0
$499$	−16.0000	−0.716258	−0.358129	−	0.933672i	$-0.616585\pi$
$499$	−16.0000	−0.716258	−0.358129	+	0.933672i	$0.616585\pi$
$500$	0	0
$501$	0	0
$502$	−10.1803	−0.454371
$503$	27.4164	1.22244	0.611219	−	0.791462i	$-0.290680\pi$
$503$	27.4164	1.22244	0.611219	+	0.791462i	$0.290680\pi$
$504$	0	0
$505$	0	0
$506$	−8.00000	−0.355643
$507$	0	0
$508$	14.0000	0.621150
$509$	−1.34752	−0.0597280	−0.0298640	−	0.999554i	$-0.509507\pi$
$509$	−1.34752	−0.0597280	−0.0298640	+	0.999554i	$0.509507\pi$
$510$	0	0
$511$	−10.8328	−0.479216
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	18.0000	0.793946
$515$	0	0
$516$	0	0
$517$	20.9443	0.921128
$518$	2.47214	0.108619
$519$	0	0
$520$	0	0
$521$	−5.81966	−0.254964	−0.127482	−	0.991841i	$-0.540689\pi$
$521$	−5.81966	−0.254964	−0.127482	+	0.991841i	$0.540689\pi$
$522$	0	0
$523$	−14.2918	−0.624937	−0.312468	−	0.949928i	$-0.601156\pi$
$523$	−14.2918	−0.624937	−0.312468	+	0.949928i	$0.601156\pi$
$524$	−12.4721	−0.544848
$525$	0	0
$526$	−4.00000	−0.174408
$527$	−2.47214	−0.107688
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	0	0
$532$	8.00000	0.346844
$533$	40.3607	1.74822
$534$	0	0
$535$	0	0
$536$	1.70820	0.0737832
$537$	0	0
$538$	−7.05573	−0.304194
$539$	10.9443	0.471403
$540$	0	0
$541$	−31.3050	−1.34590	−0.672952	−	0.739686i	$-0.734974\pi$
$541$	−31.3050	−1.34590	−0.672952	+	0.739686i	$0.734974\pi$
$542$	−4.00000	−0.171815
$543$	0	0
$544$	1.00000	0.0428746
$545$	0	0
$546$	0	0
$547$	0.944272	0.0403742	0.0201871	−	0.999796i	$-0.493574\pi$
$547$	0.944272	0.0403742	0.0201871	+	0.999796i	$0.493574\pi$
$548$	2.00000	0.0854358
$549$	0	0
$550$	0	0
$551$	−25.8885	−1.10289
$552$	0	0
$553$	−1.16718	−0.0496337
$554$	−26.3607	−1.11996
$555$	0	0
$556$	1.52786	0.0647959
$557$	25.0557	1.06165	0.530823	−	0.847483i	$-0.321883\pi$
$557$	25.0557	1.06165	0.530823	+	0.847483i	$0.321883\pi$
$558$	0	0
$559$	63.7771	2.69748
$560$	0	0
$561$	0	0
$562$	1.41641	0.0597476
$563$	−2.11146	−0.0889873	−0.0444936	−	0.999010i	$-0.514167\pi$
$563$	−2.11146	−0.0889873	−0.0444936	+	0.999010i	$0.514167\pi$
$564$	0	0
$565$	0	0
$566$	25.8885	1.08818
$567$	0	0
$568$	11.2361	0.471455
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	36.3607	1.52165	0.760824	−	0.648959i	$-0.224795\pi$
$571$	36.3607	1.52165	0.760824	+	0.648959i	$0.224795\pi$
$572$	−10.4721	−0.437862
$573$	0	0
$574$	9.52786	0.397685
$575$	0	0
$576$	0	0
$577$	−35.4164	−1.47440	−0.737202	−	0.675672i	$-0.763853\pi$
$577$	−35.4164	−1.47440	−0.737202	+	0.675672i	$0.763853\pi$
$578$	−1.00000	−0.0415945
$579$	0	0
$580$	0	0
$581$	12.9443	0.537019
$582$	0	0
$583$	−21.8885	−0.906531
$584$	−8.76393	−0.362654
$585$	0	0
$586$	29.4164	1.21518
$587$	5.52786	0.228159	0.114080	−	0.993472i	$-0.463608\pi$
$587$	5.52786	0.228159	0.114080	+	0.993472i	$0.463608\pi$
$588$	0	0
$589$	−16.0000	−0.659269
$590$	0	0
$591$	0	0
$592$	2.00000	0.0821995
$593$	−30.9443	−1.27073	−0.635364	−	0.772212i	$-0.719150\pi$
$593$	−30.9443	−1.27073	−0.635364	+	0.772212i	$0.719150\pi$
$594$	0	0
$595$	0	0
$596$	−12.1803	−0.498926
$597$	0	0
$598$	20.9443	0.856475
$599$	−25.5279	−1.04304	−0.521520	−	0.853239i	$-0.674635\pi$
$599$	−25.5279	−1.04304	−0.521520	+	0.853239i	$0.674635\pi$
$600$	0	0
$601$	−28.4721	−1.16140	−0.580701	−	0.814117i	$-0.697222\pi$
$601$	−28.4721	−1.16140	−0.580701	+	0.814117i	$0.697222\pi$
$602$	15.0557	0.613626
$603$	0	0
$604$	−4.00000	−0.162758
$605$	0	0
$606$	0	0
$607$	−31.1246	−1.26331	−0.631655	−	0.775250i	$-0.717624\pi$
$607$	−31.1246	−1.26331	−0.631655	+	0.775250i	$0.717624\pi$
$608$	6.47214	0.262480
$609$	0	0
$610$	0	0
$611$	−54.8328	−2.21830
$612$	0	0
$613$	−42.1803	−1.70365	−0.851824	−	0.523828i	$-0.824503\pi$
$613$	−42.1803	−1.70365	−0.851824	+	0.523828i	$0.824503\pi$
$614$	−4.18034	−0.168705
$615$	0	0
$616$	−2.47214	−0.0996052
$617$	20.4721	0.824177	0.412089	−	0.911144i	$-0.364799\pi$
$617$	20.4721	0.824177	0.412089	+	0.911144i	$0.364799\pi$
$618$	0	0
$619$	−48.9443	−1.96724	−0.983618	−	0.180264i	$-0.942305\pi$
$619$	−48.9443	−1.96724	−0.983618	+	0.180264i	$0.942305\pi$
$620$	0	0
$621$	0	0
$622$	19.5967	0.785758
$623$	15.4164	0.617645
$624$	0	0
$625$	0	0
$626$	21.7082	0.867634
$627$	0	0
$628$	10.1803	0.406240
$629$	−2.00000	−0.0797452
$630$	0	0
$631$	−9.88854	−0.393657	−0.196828	−	0.980438i	$-0.563064\pi$
$631$	−9.88854	−0.393657	−0.196828	+	0.980438i	$0.563064\pi$
$632$	−0.944272	−0.0375611
$633$	0	0
$634$	11.8885	0.472154
$635$	0	0
$636$	0	0
$637$	−28.6525	−1.13525
$638$	8.00000	0.316723
$639$	0	0
$640$	0	0
$641$	−11.7082	−0.462446	−0.231223	−	0.972901i	$-0.574273\pi$
$641$	−11.7082	−0.462446	−0.231223	+	0.972901i	$0.574273\pi$
$642$	0	0
$643$	−27.4164	−1.08120	−0.540599	−	0.841281i	$-0.681802\pi$
$643$	−27.4164	−1.08120	−0.540599	+	0.841281i	$0.681802\pi$
$644$	4.94427	0.194832
$645$	0	0
$646$	−6.47214	−0.254643
$647$	20.9443	0.823404	0.411702	−	0.911318i	$-0.364934\pi$
$647$	20.9443	0.823404	0.411702	+	0.911318i	$0.364934\pi$
$648$	0	0
$649$	23.4164	0.919174
$650$	0	0
$651$	0	0
$652$	−22.4721	−0.880077
$653$	3.52786	0.138056	0.0690280	−	0.997615i	$-0.478010\pi$
$653$	3.52786	0.138056	0.0690280	+	0.997615i	$0.478010\pi$
$654$	0	0
$655$	0	0
$656$	7.70820	0.300955
$657$	0	0
$658$	−12.9443	−0.504620
$659$	−28.2918	−1.10209	−0.551046	−	0.834475i	$-0.685771\pi$
$659$	−28.2918	−1.10209	−0.551046	+	0.834475i	$0.685771\pi$
$660$	0	0
$661$	13.4164	0.521838	0.260919	−	0.965361i	$-0.415974\pi$
$661$	13.4164	0.521838	0.260919	+	0.965361i	$0.415974\pi$
$662$	3.41641	0.132782
$663$	0	0
$664$	10.4721	0.406398
$665$	0	0
$666$	0	0
$667$	−16.0000	−0.619522
$668$	13.8885	0.537364
$669$	0	0
$670$	0	0
$671$	−8.94427	−0.345290
$672$	0	0
$673$	4.18034	0.161140	0.0805701	−	0.996749i	$-0.474326\pi$
$673$	4.18034	0.161140	0.0805701	+	0.996749i	$0.474326\pi$
$674$	20.1803	0.777318
$675$	0	0
$676$	14.4164	0.554477
$677$	9.05573	0.348040	0.174020	−	0.984742i	$-0.444324\pi$
$677$	9.05573	0.348040	0.174020	+	0.984742i	$0.444324\pi$
$678$	0	0
$679$	2.11146	0.0810303
$680$	0	0
$681$	0	0
$682$	4.94427	0.189326
$683$	32.9443	1.26058	0.630289	−	0.776361i	$-0.282936\pi$
$683$	32.9443	1.26058	0.630289	+	0.776361i	$0.282936\pi$
$684$	0	0
$685$	0	0
$686$	−15.4164	−0.588601
$687$	0	0
$688$	12.1803	0.464371
$689$	57.3050	2.18314
$690$	0	0
$691$	−4.94427	−0.188089	−0.0940445	−	0.995568i	$-0.529980\pi$
$691$	−4.94427	−0.188089	−0.0940445	+	0.995568i	$0.529980\pi$
$692$	15.8885	0.603992
$693$	0	0
$694$	−22.8328	−0.866722
$695$	0	0
$696$	0	0
$697$	−7.70820	−0.291969
$698$	28.4721	1.07769
$699$	0	0
$700$	0	0
$701$	−11.5967	−0.438003	−0.219002	−	0.975725i	$-0.570280\pi$
$701$	−11.5967	−0.438003	−0.219002	+	0.975725i	$0.570280\pi$
$702$	0	0
$703$	−12.9443	−0.488202
$704$	−2.00000	−0.0753778
$705$	0	0
$706$	−14.9443	−0.562435
$707$	−2.11146	−0.0794095
$708$	0	0
$709$	−39.5279	−1.48450	−0.742250	−	0.670123i	$-0.766241\pi$
$709$	−39.5279	−1.48450	−0.742250	+	0.670123i	$0.766241\pi$
$710$	0	0
$711$	0	0
$712$	12.4721	0.467413
$713$	−9.88854	−0.370329
$714$	0	0
$715$	0	0
$716$	7.70820	0.288069
$717$	0	0
$718$	−9.52786	−0.355577
$719$	36.5410	1.36275	0.681375	−	0.731934i	$-0.261382\pi$
$719$	36.5410	1.36275	0.681375	+	0.731934i	$0.261382\pi$
$720$	0	0
$721$	−13.5279	−0.503804
$722$	−22.8885	−0.851823
$723$	0	0
$724$	0.472136	0.0175468
$725$	0	0
$726$	0	0
$727$	−23.8885	−0.885977	−0.442989	−	0.896527i	$-0.646082\pi$
$727$	−23.8885	−0.885977	−0.442989	+	0.896527i	$0.646082\pi$
$728$	6.47214	0.239873
$729$	0	0
$730$	0	0
$731$	−12.1803	−0.450506
$732$	0	0
$733$	38.5410	1.42355	0.711773	−	0.702410i	$-0.247893\pi$
$733$	38.5410	1.42355	0.711773	+	0.702410i	$0.247893\pi$
$734$	5.81966	0.214808
$735$	0	0
$736$	4.00000	0.147442
$737$	3.41641	0.125845
$738$	0	0
$739$	33.5279	1.23334	0.616671	−	0.787221i	$-0.288481\pi$
$739$	33.5279	1.23334	0.616671	+	0.787221i	$0.288481\pi$
$740$	0	0
$741$	0	0
$742$	13.5279	0.496624
$743$	−33.5279	−1.23002	−0.615009	−	0.788520i	$-0.710848\pi$
$743$	−33.5279	−1.23002	−0.615009	+	0.788520i	$0.710848\pi$
$744$	0	0
$745$	0	0
$746$	1.23607	0.0452557
$747$	0	0
$748$	2.00000	0.0731272
$749$	9.88854	0.361320
$750$	0	0
$751$	29.8885	1.09065	0.545324	−	0.838225i	$-0.316407\pi$
$751$	29.8885	1.09065	0.545324	+	0.838225i	$0.316407\pi$
$752$	−10.4721	−0.381880
$753$	0	0
$754$	−20.9443	−0.762745
$755$	0	0
$756$	0	0
$757$	−37.5967	−1.36648	−0.683239	−	0.730195i	$-0.739429\pi$
$757$	−37.5967	−1.36648	−0.683239	+	0.730195i	$0.739429\pi$
$758$	1.52786	0.0554945
$759$	0	0
$760$	0	0
$761$	−52.8328	−1.91519	−0.957594	−	0.288121i	$-0.906969\pi$
$761$	−52.8328	−1.91519	−0.957594	+	0.288121i	$0.906969\pi$
$762$	0	0
$763$	9.30495	0.336862
$764$	−5.52786	−0.199991
$765$	0	0
$766$	17.8885	0.646339
$767$	−61.3050	−2.21359
$768$	0	0
$769$	−34.3607	−1.23908	−0.619539	−	0.784966i	$-0.712680\pi$
$769$	−34.3607	−1.23908	−0.619539	+	0.784966i	$0.712680\pi$
$770$	0	0
$771$	0	0
$772$	−23.5967	−0.849266
$773$	−33.4164	−1.20190	−0.600952	−	0.799285i	$-0.705212\pi$
$773$	−33.4164	−1.20190	−0.600952	+	0.799285i	$0.705212\pi$
$774$	0	0
$775$	0	0
$776$	1.70820	0.0613209
$777$	0	0
$778$	20.1803	0.723500
$779$	−49.8885	−1.78744
$780$	0	0
$781$	22.4721	0.804116
$782$	−4.00000	−0.143040
$783$	0	0
$784$	−5.47214	−0.195433
$785$	0	0
$786$	0	0
$787$	46.4721	1.65655	0.828276	−	0.560320i	$-0.189322\pi$
$787$	46.4721	1.65655	0.828276	+	0.560320i	$0.189322\pi$
$788$	−15.8885	−0.566006
$789$	0	0
$790$	0	0
$791$	−12.3607	−0.439495
$792$	0	0
$793$	23.4164	0.831541
$794$	−3.88854	−0.137999
$795$	0	0
$796$	−8.94427	−0.317021
$797$	−46.0000	−1.62940	−0.814702	−	0.579880i	$-0.803099\pi$
$797$	−46.0000	−1.62940	−0.814702	+	0.579880i	$0.803099\pi$
$798$	0	0
$799$	10.4721	0.370478
$800$	0	0
$801$	0	0
$802$	−36.0689	−1.27364
$803$	−17.5279	−0.618545
$804$	0	0
$805$	0	0
$806$	−12.9443	−0.455943
$807$	0	0
$808$	−1.70820	−0.0600944
$809$	−47.1246	−1.65681	−0.828407	−	0.560127i	$-0.810752\pi$
$809$	−47.1246	−1.65681	−0.828407	+	0.560127i	$0.810752\pi$
$810$	0	0
$811$	42.2492	1.48357	0.741785	−	0.670637i	$-0.233979\pi$
$811$	42.2492	1.48357	0.741785	+	0.670637i	$0.233979\pi$
$812$	−4.94427	−0.173510
$813$	0	0
$814$	4.00000	0.140200
$815$	0	0
$816$	0	0
$817$	−78.8328	−2.75801
$818$	−26.3607	−0.921680
$819$	0	0
$820$	0	0
$821$	−15.4164	−0.538036	−0.269018	−	0.963135i	$-0.586699\pi$
$821$	−15.4164	−0.538036	−0.269018	+	0.963135i	$0.586699\pi$
$822$	0	0
$823$	38.1803	1.33088	0.665441	−	0.746450i	$-0.268243\pi$
$823$	38.1803	1.33088	0.665441	+	0.746450i	$0.268243\pi$
$824$	−10.9443	−0.381262
$825$	0	0
$826$	−14.4721	−0.503550
$827$	23.0557	0.801726	0.400863	−	0.916138i	$-0.368710\pi$
$827$	23.0557	0.801726	0.400863	+	0.916138i	$0.368710\pi$
$828$	0	0
$829$	34.0000	1.18087	0.590434	−	0.807086i	$-0.298956\pi$
$829$	34.0000	1.18087	0.590434	+	0.807086i	$0.298956\pi$
$830$	0	0
$831$	0	0
$832$	5.23607	0.181528
$833$	5.47214	0.189598
$834$	0	0
$835$	0	0
$836$	12.9443	0.447687
$837$	0	0
$838$	6.36068	0.219726
$839$	22.0689	0.761902	0.380951	−	0.924595i	$-0.375597\pi$
$839$	22.0689	0.761902	0.380951	+	0.924595i	$0.375597\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	37.4164	1.28945
$843$	0	0
$844$	−24.9443	−0.858617
$845$	0	0
$846$	0	0
$847$	8.65248	0.297303
$848$	10.9443	0.375828
$849$	0	0
$850$	0	0
$851$	−8.00000	−0.274236
$852$	0	0
$853$	−6.36068	−0.217786	−0.108893	−	0.994054i	$-0.534731\pi$
$853$	−6.36068	−0.217786	−0.108893	+	0.994054i	$0.534731\pi$
$854$	5.52786	0.189160
$855$	0	0
$856$	8.00000	0.273434
$857$	−50.3607	−1.72029	−0.860144	−	0.510051i	$-0.829626\pi$
$857$	−50.3607	−1.72029	−0.860144	+	0.510051i	$0.829626\pi$
$858$	0	0
$859$	7.05573	0.240738	0.120369	−	0.992729i	$-0.461592\pi$
$859$	7.05573	0.240738	0.120369	+	0.992729i	$0.461592\pi$
$860$	0	0
$861$	0	0
$862$	31.5967	1.07619
$863$	21.3050	0.725229	0.362614	−	0.931939i	$-0.381884\pi$
$863$	21.3050	0.725229	0.362614	+	0.931939i	$0.381884\pi$
$864$	0	0
$865$	0	0
$866$	−28.3607	−0.963735
$867$	0	0
$868$	−3.05573	−0.103718
$869$	−1.88854	−0.0640645
$870$	0	0
$871$	−8.94427	−0.303065
$872$	7.52786	0.254926
$873$	0	0
$874$	−25.8885	−0.875693
$875$	0	0
$876$	0	0
$877$	−40.4721	−1.36665	−0.683323	−	0.730116i	$-0.739466\pi$
$877$	−40.4721	−1.36665	−0.683323	+	0.730116i	$0.739466\pi$
$878$	−13.5279	−0.456543
$879$	0	0
$880$	0	0
$881$	7.12461	0.240034	0.120017	−	0.992772i	$-0.461705\pi$
$881$	7.12461	0.240034	0.120017	+	0.992772i	$0.461705\pi$
$882$	0	0
$883$	−27.2361	−0.916567	−0.458283	−	0.888806i	$-0.651535\pi$
$883$	−27.2361	−0.916567	−0.458283	+	0.888806i	$0.651535\pi$
$884$	−5.23607	−0.176108
$885$	0	0
$886$	4.00000	0.134383
$887$	−8.94427	−0.300319	−0.150160	−	0.988662i	$-0.547979\pi$
$887$	−8.94427	−0.300319	−0.150160	+	0.988662i	$0.547979\pi$
$888$	0	0
$889$	−17.3050	−0.580389
$890$	0	0
$891$	0	0
$892$	9.05573	0.303208
$893$	67.7771	2.26807
$894$	0	0
$895$	0	0
$896$	1.23607	0.0412941
$897$	0	0
$898$	17.5967	0.587211
$899$	9.88854	0.329801
$900$	0	0
$901$	−10.9443	−0.364607
$902$	15.4164	0.513310
$903$	0	0
$904$	−10.0000	−0.332595
$905$	0	0
$906$	0	0
$907$	4.36068	0.144794	0.0723970	−	0.997376i	$-0.476935\pi$
$907$	4.36068	0.144794	0.0723970	+	0.997376i	$0.476935\pi$
$908$	−9.88854	−0.328163
$909$	0	0
$910$	0	0
$911$	12.5410	0.415503	0.207751	−	0.978182i	$-0.433386\pi$
$911$	12.5410	0.415503	0.207751	+	0.978182i	$0.433386\pi$
$912$	0	0
$913$	20.9443	0.693154
$914$	16.3607	0.541163
$915$	0	0
$916$	−21.4164	−0.707618
$917$	15.4164	0.509095
$918$	0	0
$919$	52.7214	1.73912	0.869559	−	0.493830i	$-0.164403\pi$
$919$	52.7214	1.73912	0.869559	+	0.493830i	$0.164403\pi$
$920$	0	0
$921$	0	0
$922$	−30.0689	−0.990266
$923$	−58.8328	−1.93651
$924$	0	0
$925$	0	0
$926$	18.9443	0.622547
$927$	0	0
$928$	−4.00000	−0.131306
$929$	−24.0689	−0.789674	−0.394837	−	0.918751i	$-0.629199\pi$
$929$	−24.0689	−0.789674	−0.394837	+	0.918751i	$0.629199\pi$
$930$	0	0
$931$	35.4164	1.16073
$932$	−5.41641	−0.177420
$933$	0	0
$934$	0.944272	0.0308975
$935$	0	0
$936$	0	0
$937$	17.5279	0.572610	0.286305	−	0.958138i	$-0.407573\pi$
$937$	17.5279	0.572610	0.286305	+	0.958138i	$0.407573\pi$
$938$	−2.11146	−0.0689415
$939$	0	0
$940$	0	0
$941$	−13.3050	−0.433729	−0.216865	−	0.976202i	$-0.569583\pi$
$941$	−13.3050	−0.433729	−0.216865	+	0.976202i	$0.569583\pi$
$942$	0	0
$943$	−30.8328	−1.00405
$944$	−11.7082	−0.381070
$945$	0	0
$946$	24.3607	0.792034
$947$	−48.7214	−1.58323	−0.791616	−	0.611019i	$-0.790760\pi$
$947$	−48.7214	−1.58323	−0.791616	+	0.611019i	$0.790760\pi$
$948$	0	0
$949$	45.8885	1.48961
$950$	0	0
$951$	0	0
$952$	−1.23607	−0.0400612
$953$	−26.9443	−0.872811	−0.436405	−	0.899750i	$-0.643749\pi$
$953$	−26.9443	−0.872811	−0.436405	+	0.899750i	$0.643749\pi$
$954$	0	0
$955$	0	0
$956$	21.8885	0.707926
$957$	0	0
$958$	−35.5967	−1.15008
$959$	−2.47214	−0.0798294
$960$	0	0
$961$	−24.8885	−0.802856
$962$	−10.4721	−0.337635
$963$	0	0
$964$	9.41641	0.303282
$965$	0	0
$966$	0	0
$967$	−0.472136	−0.0151829	−0.00759143	−	0.999971i	$-0.502416\pi$
$967$	−0.472136	−0.0151829	−0.00759143	+	0.999971i	$0.502416\pi$
$968$	7.00000	0.224989
$969$	0	0
$970$	0	0
$971$	22.7639	0.730529	0.365265	−	0.930904i	$-0.380978\pi$
$971$	22.7639	0.730529	0.365265	+	0.930904i	$0.380978\pi$
$972$	0	0
$973$	−1.88854	−0.0605439
$974$	16.2918	0.522023
$975$	0	0
$976$	4.47214	0.143150
$977$	−26.9443	−0.862024	−0.431012	−	0.902346i	$-0.641843\pi$
$977$	−26.9443	−0.862024	−0.431012	+	0.902346i	$0.641843\pi$
$978$	0	0
$979$	24.9443	0.797222
$980$	0	0
$981$	0	0
$982$	12.2918	0.392247
$983$	−11.4164	−0.364127	−0.182063	−	0.983287i	$-0.558278\pi$
$983$	−11.4164	−0.364127	−0.182063	+	0.983287i	$0.558278\pi$
$984$	0	0
$985$	0	0
$986$	4.00000	0.127386
$987$	0	0
$988$	−33.8885	−1.07814
$989$	−48.7214	−1.54925
$990$	0	0
$991$	−10.8328	−0.344116	−0.172058	−	0.985087i	$-0.555042\pi$
$991$	−10.8328	−0.344116	−0.172058	+	0.985087i	$0.555042\pi$
$992$	−2.47214	−0.0784904
$993$	0	0
$994$	−13.8885	−0.440518
$995$	0	0
$996$	0	0
$997$	41.0557	1.30025	0.650124	−	0.759828i	$-0.274717\pi$
$997$	41.0557	1.30025	0.650124	+	0.759828i	$0.274717\pi$
$998$	16.0000	0.506471
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7650.2.a.cx.1.1		2
3.2	odd	2		2550.2.a.bk.1.1		2
5.2	odd	4		1530.2.d.f.919.2		4
5.3	odd	4		1530.2.d.f.919.3		4
5.4	even	2		7650.2.a.da.1.2		2
15.2	even	4		510.2.d.b.409.3	yes	4
15.8	even	4		510.2.d.b.409.2	✓	4
15.14	odd	2		2550.2.a.bh.1.2		2
60.23	odd	4		4080.2.m.m.2449.4		4
60.47	odd	4		4080.2.m.m.2449.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
510.2.d.b.409.2	✓	4	15.8	even	4
510.2.d.b.409.3	yes	4	15.2	even	4
1530.2.d.f.919.2		4	5.2	odd	4
1530.2.d.f.919.3		4	5.3	odd	4
2550.2.a.bh.1.2		2	15.14	odd	2
2550.2.a.bk.1.1		2	3.2	odd	2
4080.2.m.m.2449.1		4	60.47	odd	4
4080.2.m.m.2449.4		4	60.23	odd	4
7650.2.a.cx.1.1		2	1.1	even	1	trivial
7650.2.a.da.1.2		2	5.4	even	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:	\( N \)	\(=\)	\( 7650 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7650.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.23607 q^{7} -1.00000 q^{8} -2.00000 q^{11} +5.23607 q^{13} +1.23607 q^{14} +1.00000 q^{16} -1.00000 q^{17} -6.47214 q^{19} +2.00000 q^{22} -4.00000 q^{23} -5.23607 q^{26} -1.23607 q^{28} +4.00000 q^{29} +2.47214 q^{31} -1.00000 q^{32} +1.00000 q^{34} +2.00000 q^{37} +6.47214 q^{38} +7.70820 q^{41} +12.1803 q^{43} -2.00000 q^{44} +4.00000 q^{46} -10.4721 q^{47} -5.47214 q^{49} +5.23607 q^{52} +10.9443 q^{53} +1.23607 q^{56} -4.00000 q^{58} -11.7082 q^{59} +4.47214 q^{61} -2.47214 q^{62} +1.00000 q^{64} -1.70820 q^{67} -1.00000 q^{68} -11.2361 q^{71} +8.76393 q^{73} -2.00000 q^{74} -6.47214 q^{76} +2.47214 q^{77} +0.944272 q^{79} -7.70820 q^{82} -10.4721 q^{83} -12.1803 q^{86} +2.00000 q^{88} -12.4721 q^{89} -6.47214 q^{91} -4.00000 q^{92} +10.4721 q^{94} -1.70820 q^{97} +5.47214 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{7} - 2 q^{8} - 4 q^{11} + 6 q^{13} - 2 q^{14} + 2 q^{16} - 2 q^{17} - 4 q^{19} + 4 q^{22} - 8 q^{23} - 6 q^{26} + 2 q^{28} + 8 q^{29} - 4 q^{31} - 2 q^{32} + 2 q^{34} + 4 q^{37}+ \cdots + 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^7 - 2 * q^8 - 4 * q^11 + 6 * q^13 - 2 * q^14 + 2 * q^16 - 2 * q^17 - 4 * q^19 + 4 * q^22 - 8 * q^23 - 6 * q^26 + 2 * q^28 + 8 * q^29 - 4 * q^31 - 2 * q^32 + 2 * q^34 + 4 * q^37 + 4 * q^38 + 2 * q^41 + 2 * q^43 - 4 * q^44 + 8 * q^46 - 12 * q^47 - 2 * q^49 + 6 * q^52 + 4 * q^53 - 2 * q^56 - 8 * q^58 - 10 * q^59 + 4 * q^62 + 2 * q^64 + 10 * q^67 - 2 * q^68 - 18 * q^71 + 22 * q^73 - 4 * q^74 - 4 * q^76 - 4 * q^77 - 16 * q^79 - 2 * q^82 - 12 * q^83 - 2 * q^86 + 4 * q^88 - 16 * q^89 - 4 * q^91 - 8 * q^92 + 12 * q^94 + 10 * q^97 + 2 * q^98

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