LMFDB - Embedded newform 2475.2.c.o.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:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 825)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.3
Root			$-0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	2475.199
Dual form			2475.2.c.o.199.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.414214i q^{2} +1.82843 q^{4} -2.41421i q^{7} +1.58579i q^{8} +O(q^{10})$	$q+0.414214i q^{2} +1.82843 q^{4} -2.41421i q^{7} +1.58579i q^{8} +1.00000 q^{11} -2.82843i q^{13} +1.00000 q^{14} +3.00000 q^{16} +0.414214i q^{17} -3.58579 q^{19} +0.414214i q^{22} +1.00000i q^{23} +1.17157 q^{26} -4.41421i q^{28} +6.82843 q^{29} +8.48528 q^{31} +4.41421i q^{32} -0.171573 q^{34} -5.82843i q^{37} -1.48528i q^{38} -8.89949 q^{41} +0.343146i q^{43} +1.82843 q^{44} -0.414214 q^{46} -9.48528i q^{47} +1.17157 q^{49} -5.17157i q^{52} -3.65685i q^{53} +3.82843 q^{56} +2.82843i q^{58} +11.0000 q^{59} +3.17157 q^{61} +3.51472i q^{62} +4.17157 q^{64} -11.6569i q^{67} +0.757359i q^{68} -2.17157 q^{71} +3.17157i q^{73} +2.41421 q^{74} -6.55635 q^{76} -2.41421i q^{77} -4.75736 q^{79} -3.68629i q^{82} +12.4853i q^{83} -0.142136 q^{86} +1.58579i q^{88} -7.65685 q^{89} -6.82843 q^{91} +1.82843i q^{92} +3.92893 q^{94} +0.171573i q^{97} +0.485281i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4}+O(q^{10}) $ 4 * q - 4 * q^4	$ 4 q - 4 q^{4} + 4 q^{11} + 4 q^{14} + 12 q^{16} - 20 q^{19} + 16 q^{26} + 16 q^{29} - 12 q^{34} + 4 q^{41} - 4 q^{44} + 4 q^{46} + 16 q^{49} + 4 q^{56} + 44 q^{59} + 24 q^{61} + 28 q^{64} - 20 q^{71} + 4 q^{74} + 36 q^{76} - 36 q^{79} + 56 q^{86} - 8 q^{89} - 16 q^{91} + 44 q^{94}+O(q^{100}) $ 4 * q - 4 * q^4 + 4 * q^11 + 4 * q^14 + 12 * q^16 - 20 * q^19 + 16 * q^26 + 16 * q^29 - 12 * q^34 + 4 * q^41 - 4 * q^44 + 4 * q^46 + 16 * q^49 + 4 * q^56 + 44 * q^59 + 24 * q^61 + 28 * q^64 - 20 * q^71 + 4 * q^74 + 36 * q^76 - 36 * q^79 + 56 * q^86 - 8 * q^89 - 16 * q^91 + 44 * 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 $

$2$	0.414214i	0.292893i	0.989219	+	0.146447i	$0.0467837\pi$
$2$	0.414214i	0.292893i	−0.989219	+	0.146447i	$0.953216\pi$
$3$	0	0
$3$	0	0
$4$	1.82843	0.914214
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.41421i	− 0.912487i	−0.889855	−	0.456243i	$-0.849195\pi$
$7$	− 2.41421i	− 0.912487i	0.889855	−	0.456243i	$-0.150805\pi$
$8$	1.58579i	0.560660i
$9$	0	0
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	− 2.82843i	− 0.784465i	−0.919866	−	0.392232i	$-0.871703\pi$
$13$	− 2.82843i	− 0.784465i	0.919866	−	0.392232i	$-0.128297\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	3.00000	0.750000
$17$	0.414214i	0.100462i	0.998738	+	0.0502308i	$0.0159957\pi$
$17$	0.414214i	0.100462i	−0.998738	+	0.0502308i	$0.984004\pi$
$18$	0	0
$19$	−3.58579	−0.822636	−0.411318	−	0.911492i	$-0.634931\pi$
$19$	−3.58579	−0.822636	−0.411318	+	0.911492i	$0.634931\pi$
$20$	0	0
$21$	0	0
$22$	0.414214i	0.0883106i
$23$	1.00000i	0.208514i	0.994550	+	0.104257i	$0.0332465\pi$
$23$	1.00000i	0.208514i	−0.994550	+	0.104257i	$0.966753\pi$
$24$	0	0
$25$	0	0
$26$	1.17157	0.229764
$27$	0	0
$28$	− 4.41421i	− 0.834208i
$29$	6.82843	1.26801	0.634004	−	0.773330i	$-0.281410\pi$
$29$	6.82843	1.26801	0.634004	+	0.773330i	$0.281410\pi$
$30$	0	0
$31$	8.48528	1.52400	0.762001	−	0.647576i	$-0.224217\pi$
$31$	8.48528	1.52400	0.762001	+	0.647576i	$0.224217\pi$
$32$	4.41421i	0.780330i
$33$	0	0
$34$	−0.171573	−0.0294245
$35$	0	0
$36$	0	0
$37$	− 5.82843i	− 0.958188i	−0.877764	−	0.479094i	$-0.840965\pi$
$37$	− 5.82843i	− 0.958188i	0.877764	−	0.479094i	$-0.159035\pi$
$38$	− 1.48528i	− 0.240944i
$39$	0	0
$40$	0	0
$41$	−8.89949	−1.38987	−0.694934	−	0.719074i	$-0.744566\pi$
$41$	−8.89949	−1.38987	−0.694934	+	0.719074i	$0.744566\pi$
$42$	0	0
$43$	0.343146i	0.0523292i	0.999658	+	0.0261646i	$0.00832941\pi$
$43$	0.343146i	0.0523292i	−0.999658	+	0.0261646i	$0.991671\pi$
$44$	1.82843	0.275646
$45$	0	0
$46$	−0.414214	−0.0610725
$47$	− 9.48528i	− 1.38357i	−0.722103	−	0.691785i	$-0.756824\pi$
$47$	− 9.48528i	− 1.38357i	0.722103	−	0.691785i	$-0.243176\pi$
$48$	0	0
$49$	1.17157	0.167368
$50$	0	0
$51$	0	0
$52$	− 5.17157i	− 0.717168i
$53$	− 3.65685i	− 0.502308i	−0.967947	−	0.251154i	$-0.919190\pi$
$53$	− 3.65685i	− 0.502308i	0.967947	−	0.251154i	$-0.0808100\pi$
$54$	0	0
$55$	0	0
$56$	3.82843	0.511595
$57$	0	0
$58$	2.82843i	0.371391i
$59$	11.0000	1.43208	0.716039	−	0.698060i	$-0.245953\pi$
$59$	11.0000	1.43208	0.716039	+	0.698060i	$0.245953\pi$
$60$	0	0
$61$	3.17157	0.406078	0.203039	−	0.979171i	$-0.434918\pi$
$61$	3.17157	0.406078	0.203039	+	0.979171i	$0.434918\pi$
$62$	3.51472i	0.446370i
$63$	0	0
$64$	4.17157	0.521447
$65$	0	0
$66$	0	0
$67$	− 11.6569i	− 1.42411i	−0.702123	−	0.712056i	$-0.747764\pi$
$67$	− 11.6569i	− 1.42411i	0.702123	−	0.712056i	$-0.252236\pi$
$68$	0.757359i	0.0918433i
$69$	0	0
$70$	0	0
$71$	−2.17157	−0.257718	−0.128859	−	0.991663i	$-0.541132\pi$
$71$	−2.17157	−0.257718	−0.128859	+	0.991663i	$0.541132\pi$
$72$	0	0
$73$	3.17157i	0.371205i	0.982625	+	0.185602i	$0.0594236\pi$
$73$	3.17157i	0.371205i	−0.982625	+	0.185602i	$0.940576\pi$
$74$	2.41421	0.280647
$75$	0	0
$76$	−6.55635	−0.752065
$77$	− 2.41421i	− 0.275125i
$78$	0	0
$79$	−4.75736	−0.535245	−0.267622	−	0.963524i	$-0.586238\pi$
$79$	−4.75736	−0.535245	−0.267622	+	0.963524i	$0.586238\pi$
$80$	0	0
$81$	0	0
$82$	− 3.68629i	− 0.407083i
$83$	12.4853i	1.37044i	0.728337	+	0.685219i	$0.240293\pi$
$83$	12.4853i	1.37044i	−0.728337	+	0.685219i	$0.759707\pi$
$84$	0	0
$85$	0	0
$86$	−0.142136	−0.0153269
$87$	0	0
$88$	1.58579i	0.169045i
$89$	−7.65685	−0.811625	−0.405812	−	0.913956i	$-0.633011\pi$
$89$	−7.65685	−0.811625	−0.405812	+	0.913956i	$0.633011\pi$
$90$	0	0
$91$	−6.82843	−0.715814
$92$	1.82843i	0.190627i
$93$	0	0
$94$	3.92893	0.405238
$95$	0	0
$96$	0	0
$97$	0.171573i	0.0174206i	0.999962	+	0.00871029i	$0.00277261\pi$
$97$	0.171573i	0.0174206i	−0.999962	+	0.00871029i	$0.997227\pi$
$98$	0.485281i	0.0490208i
$99$	0	0
$100$	0	0
$101$	4.89949	0.487518	0.243759	−	0.969836i	$-0.421619\pi$
$101$	4.89949	0.487518	0.243759	+	0.969836i	$0.421619\pi$
$102$	0	0
$103$	2.34315i	0.230877i	0.993315	+	0.115439i	$0.0368273\pi$
$103$	2.34315i	0.230877i	−0.993315	+	0.115439i	$0.963173\pi$
$104$	4.48528	0.439818
$105$	0	0
$106$	1.51472	0.147122
$107$	− 17.3137i	− 1.67378i	−0.547372	−	0.836890i	$-0.684372\pi$
$107$	− 17.3137i	− 1.67378i	0.547372	−	0.836890i	$-0.315628\pi$
$108$	0	0
$109$	17.3137	1.65835	0.829176	−	0.558987i	$-0.188810\pi$
$109$	17.3137	1.65835	0.829176	+	0.558987i	$0.188810\pi$
$110$	0	0
$111$	0	0
$112$	− 7.24264i	− 0.684365i
$113$	− 10.0000i	− 0.940721i	−0.882474	−	0.470360i	$-0.844124\pi$
$113$	− 10.0000i	− 0.940721i	0.882474	−	0.470360i	$-0.155876\pi$
$114$	0	0
$115$	0	0
$116$	12.4853	1.15923
$117$	0	0
$118$	4.55635i	0.419446i
$119$	1.00000	0.0916698
$120$	0	0
$121$	1.00000	0.0909091
$122$	1.31371i	0.118938i
$123$	0	0
$124$	15.5147	1.39326
$125$	0	0
$126$	0	0
$127$	− 1.24264i	− 0.110267i	−0.998479	−	0.0551333i	$-0.982442\pi$
$127$	− 1.24264i	− 0.110267i	0.998479	−	0.0551333i	$-0.0175584\pi$
$128$	10.5563i	0.933058i
$129$	0	0
$130$	0	0
$131$	−1.17157	−0.102361	−0.0511804	−	0.998689i	$-0.516298\pi$
$131$	−1.17157	−0.102361	−0.0511804	+	0.998689i	$0.516298\pi$
$132$	0	0
$133$	8.65685i	0.750644i
$134$	4.82843	0.417113
$135$	0	0
$136$	−0.656854	−0.0563248
$137$	− 16.1421i	− 1.37912i	−0.724231	−	0.689558i	$-0.757805\pi$
$137$	− 16.1421i	− 1.37912i	0.724231	−	0.689558i	$-0.242195\pi$
$138$	0	0
$139$	−14.9706	−1.26979	−0.634893	−	0.772600i	$-0.718956\pi$
$139$	−14.9706	−1.26979	−0.634893	+	0.772600i	$0.718956\pi$
$140$	0	0
$141$	0	0
$142$	− 0.899495i	− 0.0754839i
$143$	− 2.82843i	− 0.236525i
$144$	0	0
$145$	0	0
$146$	−1.31371	−0.108723
$147$	0	0
$148$	− 10.6569i	− 0.875988i
$149$	−17.7279	−1.45233	−0.726164	−	0.687522i	$-0.758699\pi$
$149$	−17.7279	−1.45233	−0.726164	+	0.687522i	$0.758699\pi$
$150$	0	0
$151$	14.0000	1.13930	0.569652	−	0.821886i	$-0.307078\pi$
$151$	14.0000	1.13930	0.569652	+	0.821886i	$0.307078\pi$
$152$	− 5.68629i	− 0.461219i
$153$	0	0
$154$	1.00000	0.0805823
$155$	0	0
$156$	0	0
$157$	6.00000i	0.478852i	0.970915	+	0.239426i	$0.0769593\pi$
$157$	6.00000i	0.478852i	−0.970915	+	0.239426i	$0.923041\pi$
$158$	− 1.97056i	− 0.156770i
$159$	0	0
$160$	0	0
$161$	2.41421	0.190267
$162$	0	0
$163$	23.7990i	1.86408i	0.362354	+	0.932040i	$0.381973\pi$
$163$	23.7990i	1.86408i	−0.362354	+	0.932040i	$0.618027\pi$
$164$	−16.2721	−1.27064
$165$	0	0
$166$	−5.17157	−0.401392
$167$	17.7990i	1.37733i	0.725081	+	0.688664i	$0.241802\pi$
$167$	17.7990i	1.37733i	−0.725081	+	0.688664i	$0.758198\pi$
$168$	0	0
$169$	5.00000	0.384615
$170$	0	0
$171$	0	0
$172$	0.627417i	0.0478401i
$173$	18.5563i	1.41081i	0.708803	+	0.705407i	$0.249236\pi$
$173$	18.5563i	1.41081i	−0.708803	+	0.705407i	$0.750764\pi$
$174$	0	0
$175$	0	0
$176$	3.00000	0.226134
$177$	0	0
$178$	− 3.17157i	− 0.237719i
$179$	−22.7990	−1.70408	−0.852038	−	0.523480i	$-0.824634\pi$
$179$	−22.7990	−1.70408	−0.852038	+	0.523480i	$0.824634\pi$
$180$	0	0
$181$	11.9706	0.889765	0.444882	−	0.895589i	$-0.353245\pi$
$181$	11.9706	0.889765	0.444882	+	0.895589i	$0.353245\pi$
$182$	− 2.82843i	− 0.209657i
$183$	0	0
$184$	−1.58579	−0.116906
$185$	0	0
$186$	0	0
$187$	0.414214i	0.0302903i
$188$	− 17.3431i	− 1.26488i
$189$	0	0
$190$	0	0
$191$	−11.8284	−0.855875	−0.427937	−	0.903808i	$-0.640760\pi$
$191$	−11.8284	−0.855875	−0.427937	+	0.903808i	$0.640760\pi$
$192$	0	0
$193$	19.3137i	1.39023i	0.718898	+	0.695116i	$0.244647\pi$
$193$	19.3137i	1.39023i	−0.718898	+	0.695116i	$0.755353\pi$
$194$	−0.0710678	−0.00510237
$195$	0	0
$196$	2.14214	0.153010
$197$	− 13.2426i	− 0.943499i	−0.881733	−	0.471750i	$-0.843623\pi$
$197$	− 13.2426i	− 0.943499i	0.881733	−	0.471750i	$-0.156377\pi$
$198$	0	0
$199$	−5.17157	−0.366603	−0.183302	−	0.983057i	$-0.558678\pi$
$199$	−5.17157	−0.366603	−0.183302	+	0.983057i	$0.558678\pi$
$200$	0	0
$201$	0	0
$202$	2.02944i	0.142791i
$203$	− 16.4853i	− 1.15704i
$204$	0	0
$205$	0	0
$206$	−0.970563	−0.0676223
$207$	0	0
$208$	− 8.48528i	− 0.588348i
$209$	−3.58579	−0.248034
$210$	0	0
$211$	9.31371	0.641182	0.320591	−	0.947218i	$-0.396118\pi$
$211$	9.31371	0.641182	0.320591	+	0.947218i	$0.396118\pi$
$212$	− 6.68629i	− 0.459216i
$213$	0	0
$214$	7.17157	0.490239
$215$	0	0
$216$	0	0
$217$	− 20.4853i	− 1.39063i
$218$	7.17157i	0.485720i
$219$	0	0
$220$	0	0
$221$	1.17157	0.0788085
$222$	0	0
$223$	26.8284i	1.79656i	0.439419	+	0.898282i	$0.355184\pi$
$223$	26.8284i	1.79656i	−0.439419	+	0.898282i	$0.644816\pi$
$224$	10.6569	0.712041
$225$	0	0
$226$	4.14214	0.275531
$227$	1.51472i	0.100535i	0.998736	+	0.0502677i	$0.0160075\pi$
$227$	1.51472i	0.100535i	−0.998736	+	0.0502677i	$0.983993\pi$
$228$	0	0
$229$	19.4853	1.28762	0.643812	−	0.765184i	$-0.277352\pi$
$229$	19.4853	1.28762	0.643812	+	0.765184i	$0.277352\pi$
$230$	0	0
$231$	0	0
$232$	10.8284i	0.710921i
$233$	− 14.5563i	− 0.953618i	−0.879007	−	0.476809i	$-0.841793\pi$
$233$	− 14.5563i	− 0.953618i	0.879007	−	0.476809i	$-0.158207\pi$
$234$	0	0
$235$	0	0
$236$	20.1127	1.30923
$237$	0	0
$238$	0.414214i	0.0268495i
$239$	12.3431	0.798412	0.399206	−	0.916861i	$-0.369286\pi$
$239$	12.3431	0.798412	0.399206	+	0.916861i	$0.369286\pi$
$240$	0	0
$241$	−14.1421	−0.910975	−0.455488	−	0.890242i	$-0.650535\pi$
$241$	−14.1421	−0.910975	−0.455488	+	0.890242i	$0.650535\pi$
$242$	0.414214i	0.0266267i
$243$	0	0
$244$	5.79899	0.371242
$245$	0	0
$246$	0	0
$247$	10.1421i	0.645329i
$248$	13.4558i	0.854447i
$249$	0	0
$250$	0	0
$251$	24.9706	1.57613	0.788064	−	0.615593i	$-0.211084\pi$
$251$	24.9706	1.57613	0.788064	+	0.615593i	$0.211084\pi$
$252$	0	0
$253$	1.00000i	0.0628695i
$254$	0.514719	0.0322963
$255$	0	0
$256$	3.97056	0.248160
$257$	13.3137i	0.830486i	0.909710	+	0.415243i	$0.136304\pi$
$257$	13.3137i	0.830486i	−0.909710	+	0.415243i	$0.863696\pi$
$258$	0	0
$259$	−14.0711	−0.874334
$260$	0	0
$261$	0	0
$262$	− 0.485281i	− 0.0299808i
$263$	− 10.9706i	− 0.676474i	−0.941061	−	0.338237i	$-0.890169\pi$
$263$	− 10.9706i	− 0.676474i	0.941061	−	0.338237i	$-0.109831\pi$
$264$	0	0
$265$	0	0
$266$	−3.58579	−0.219859
$267$	0	0
$268$	− 21.3137i	− 1.30194i
$269$	23.7990	1.45105	0.725525	−	0.688196i	$-0.241597\pi$
$269$	23.7990	1.45105	0.725525	+	0.688196i	$0.241597\pi$
$270$	0	0
$271$	−10.8995	−0.662097	−0.331049	−	0.943614i	$-0.607402\pi$
$271$	−10.8995	−0.662097	−0.331049	+	0.943614i	$0.607402\pi$
$272$	1.24264i	0.0753462i
$273$	0	0
$274$	6.68629	0.403934
$275$	0	0
$276$	0	0
$277$	− 0.828427i	− 0.0497754i	−0.999690	−	0.0248877i	$-0.992077\pi$
$277$	− 0.828427i	− 0.0497754i	0.999690	−	0.0248877i	$-0.00792281\pi$
$278$	− 6.20101i	− 0.371912i
$279$	0	0
$280$	0	0
$281$	−17.9289	−1.06955	−0.534775	−	0.844994i	$-0.679604\pi$
$281$	−17.9289	−1.06955	−0.534775	+	0.844994i	$0.679604\pi$
$282$	0	0
$283$	− 18.8995i	− 1.12346i	−0.827321	−	0.561729i	$-0.810136\pi$
$283$	− 18.8995i	− 1.12346i	0.827321	−	0.561729i	$-0.189864\pi$
$284$	−3.97056	−0.235610
$285$	0	0
$286$	1.17157	0.0692766
$287$	21.4853i	1.26824i
$288$	0	0
$289$	16.8284	0.989907
$290$	0	0
$291$	0	0
$292$	5.79899i	0.339360i
$293$	20.4142i	1.19261i	0.802758	+	0.596306i	$0.203365\pi$
$293$	20.4142i	1.19261i	−0.802758	+	0.596306i	$0.796635\pi$
$294$	0	0
$295$	0	0
$296$	9.24264	0.537218
$297$	0	0
$298$	− 7.34315i	− 0.425377i
$299$	2.82843	0.163572
$300$	0	0
$301$	0.828427	0.0477497
$302$	5.79899i	0.333694i
$303$	0	0
$304$	−10.7574	−0.616977
$305$	0	0
$306$	0	0
$307$	29.3137i	1.67302i	0.547950	+	0.836511i	$0.315408\pi$
$307$	29.3137i	1.67302i	−0.547950	+	0.836511i	$0.684592\pi$
$308$	− 4.41421i	− 0.251523i
$309$	0	0
$310$	0	0
$311$	−2.34315	−0.132868	−0.0664338	−	0.997791i	$-0.521162\pi$
$311$	−2.34315	−0.132868	−0.0664338	+	0.997791i	$0.521162\pi$
$312$	0	0
$313$	− 1.14214i	− 0.0645573i	−0.999479	−	0.0322787i	$-0.989724\pi$
$313$	− 1.14214i	− 0.0645573i	0.999479	−	0.0322787i	$-0.0102764\pi$
$314$	−2.48528	−0.140253
$315$	0	0
$316$	−8.69848	−0.489328
$317$	− 25.1716i	− 1.41378i	−0.707325	−	0.706888i	$-0.750098\pi$
$317$	− 25.1716i	− 1.41378i	0.707325	−	0.706888i	$-0.249902\pi$
$318$	0	0
$319$	6.82843	0.382319
$320$	0	0
$321$	0	0
$322$	1.00000i	0.0557278i
$323$	− 1.48528i	− 0.0826433i
$324$	0	0
$325$	0	0
$326$	−9.85786	−0.545977
$327$	0	0
$328$	− 14.1127i	− 0.779243i
$329$	−22.8995	−1.26249
$330$	0	0
$331$	−3.85786	−0.212047	−0.106024	−	0.994364i	$-0.533812\pi$
$331$	−3.85786	−0.212047	−0.106024	+	0.994364i	$0.533812\pi$
$332$	22.8284i	1.25287i
$333$	0	0
$334$	−7.37258	−0.403410
$335$	0	0
$336$	0	0
$337$	− 24.1421i	− 1.31511i	−0.753408	−	0.657553i	$-0.771592\pi$
$337$	− 24.1421i	− 1.31511i	0.753408	−	0.657553i	$-0.228408\pi$
$338$	2.07107i	0.112651i
$339$	0	0
$340$	0	0
$341$	8.48528	0.459504
$342$	0	0
$343$	− 19.7279i	− 1.06521i
$344$	−0.544156	−0.0293389
$345$	0	0
$346$	−7.68629	−0.413218
$347$	26.8284i	1.44023i	0.693857	+	0.720113i	$0.255910\pi$
$347$	26.8284i	1.44023i	−0.693857	+	0.720113i	$0.744090\pi$
$348$	0	0
$349$	−14.4853	−0.775379	−0.387690	−	0.921790i	$-0.626727\pi$
$349$	−14.4853	−0.775379	−0.387690	+	0.921790i	$0.626727\pi$
$350$	0	0
$351$	0	0
$352$	4.41421i	0.235278i
$353$	12.4853i	0.664524i	0.943187	+	0.332262i	$0.107812\pi$
$353$	12.4853i	0.664524i	−0.943187	+	0.332262i	$0.892188\pi$
$354$	0	0
$355$	0	0
$356$	−14.0000	−0.741999
$357$	0	0
$358$	− 9.44365i	− 0.499112i
$359$	32.4853	1.71451	0.857254	−	0.514894i	$-0.172169\pi$
$359$	32.4853	1.71451	0.857254	+	0.514894i	$0.172169\pi$
$360$	0	0
$361$	−6.14214	−0.323270
$362$	4.95837i	0.260606i
$363$	0	0
$364$	−12.4853	−0.654407
$365$	0	0
$366$	0	0
$367$	− 21.3137i	− 1.11257i	−0.830993	−	0.556283i	$-0.812227\pi$
$367$	− 21.3137i	− 1.11257i	0.830993	−	0.556283i	$-0.187773\pi$
$368$	3.00000i	0.156386i
$369$	0	0
$370$	0	0
$371$	−8.82843	−0.458349
$372$	0	0
$373$	12.3431i	0.639104i	0.947569	+	0.319552i	$0.103532\pi$
$373$	12.3431i	0.639104i	−0.947569	+	0.319552i	$0.896468\pi$
$374$	−0.171573	−0.00887182
$375$	0	0
$376$	15.0416	0.775713
$377$	− 19.3137i	− 0.994707i
$378$	0	0
$379$	−14.8284	−0.761685	−0.380843	−	0.924640i	$-0.624366\pi$
$379$	−14.8284	−0.761685	−0.380843	+	0.924640i	$0.624366\pi$
$380$	0	0
$381$	0	0
$382$	− 4.89949i	− 0.250680i
$383$	20.0000i	1.02195i	0.859595	+	0.510976i	$0.170716\pi$
$383$	20.0000i	1.02195i	−0.859595	+	0.510976i	$0.829284\pi$
$384$	0	0
$385$	0	0
$386$	−8.00000	−0.407189
$387$	0	0
$388$	0.313708i	0.0159261i
$389$	−6.34315	−0.321610	−0.160805	−	0.986986i	$-0.551409\pi$
$389$	−6.34315	−0.321610	−0.160805	+	0.986986i	$0.551409\pi$
$390$	0	0
$391$	−0.414214	−0.0209477
$392$	1.85786i	0.0938363i
$393$	0	0
$394$	5.48528	0.276344
$395$	0	0
$396$	0	0
$397$	31.9411i	1.60308i	0.597942	+	0.801540i	$0.295985\pi$
$397$	31.9411i	1.60308i	−0.597942	+	0.801540i	$0.704015\pi$
$398$	− 2.14214i	− 0.107376i
$399$	0	0
$400$	0	0
$401$	7.79899	0.389463	0.194731	−	0.980857i	$-0.437616\pi$
$401$	7.79899	0.389463	0.194731	+	0.980857i	$0.437616\pi$
$402$	0	0
$403$	− 24.0000i	− 1.19553i
$404$	8.95837	0.445696
$405$	0	0
$406$	6.82843	0.338889
$407$	− 5.82843i	− 0.288904i
$408$	0	0
$409$	−24.1421	−1.19375	−0.596876	−	0.802334i	$-0.703592\pi$
$409$	−24.1421	−1.19375	−0.596876	+	0.802334i	$0.703592\pi$
$410$	0	0
$411$	0	0
$412$	4.28427i	0.211071i
$413$	− 26.5563i	− 1.30675i
$414$	0	0
$415$	0	0
$416$	12.4853	0.612141
$417$	0	0
$418$	− 1.48528i	− 0.0726475i
$419$	−8.51472	−0.415971	−0.207986	−	0.978132i	$-0.566691\pi$
$419$	−8.51472	−0.415971	−0.207986	+	0.978132i	$0.566691\pi$
$420$	0	0
$421$	−27.0000	−1.31590	−0.657950	−	0.753062i	$-0.728576\pi$
$421$	−27.0000	−1.31590	−0.657950	+	0.753062i	$0.728576\pi$
$422$	3.85786i	0.187798i
$423$	0	0
$424$	5.79899	0.281624
$425$	0	0
$426$	0	0
$427$	− 7.65685i	− 0.370541i
$428$	− 31.6569i	− 1.53019i
$429$	0	0
$430$	0	0
$431$	−6.82843	−0.328914	−0.164457	−	0.986384i	$-0.552587\pi$
$431$	−6.82843	−0.328914	−0.164457	+	0.986384i	$0.552587\pi$
$432$	0	0
$433$	− 9.31371i	− 0.447588i	−0.974636	−	0.223794i	$-0.928156\pi$
$433$	− 9.31371i	− 0.447588i	0.974636	−	0.223794i	$-0.0718443\pi$
$434$	8.48528	0.407307
$435$	0	0
$436$	31.6569	1.51609
$437$	− 3.58579i	− 0.171531i
$438$	0	0
$439$	−27.7279	−1.32338	−0.661691	−	0.749777i	$-0.730161\pi$
$439$	−27.7279	−1.32338	−0.661691	+	0.749777i	$0.730161\pi$
$440$	0	0
$441$	0	0
$442$	0.485281i	0.0230825i
$443$	25.9706i	1.23390i	0.787003	+	0.616949i	$0.211632\pi$
$443$	25.9706i	1.23390i	−0.787003	+	0.616949i	$0.788368\pi$
$444$	0	0
$445$	0	0
$446$	−11.1127	−0.526202
$447$	0	0
$448$	− 10.0711i	− 0.475813i
$449$	10.4853	0.494831	0.247416	−	0.968909i	$-0.420419\pi$
$449$	10.4853	0.494831	0.247416	+	0.968909i	$0.420419\pi$
$450$	0	0
$451$	−8.89949	−0.419061
$452$	− 18.2843i	− 0.860020i
$453$	0	0
$454$	−0.627417	−0.0294461
$455$	0	0
$456$	0	0
$457$	32.1421i	1.50355i	0.659422	+	0.751773i	$0.270801\pi$
$457$	32.1421i	1.50355i	−0.659422	+	0.751773i	$0.729199\pi$
$458$	8.07107i	0.377136i
$459$	0	0
$460$	0	0
$461$	−40.7696	−1.89883	−0.949414	−	0.314028i	$-0.898321\pi$
$461$	−40.7696	−1.89883	−0.949414	+	0.314028i	$0.898321\pi$
$462$	0	0
$463$	− 1.02944i	− 0.0478420i	−0.999714	−	0.0239210i	$-0.992385\pi$
$463$	− 1.02944i	− 0.0478420i	0.999714	−	0.0239210i	$-0.00761502\pi$
$464$	20.4853	0.951005
$465$	0	0
$466$	6.02944	0.279308
$467$	34.6274i	1.60237i	0.598420	+	0.801183i	$0.295796\pi$
$467$	34.6274i	1.60237i	−0.598420	+	0.801183i	$0.704204\pi$
$468$	0	0
$469$	−28.1421	−1.29948
$470$	0	0
$471$	0	0
$472$	17.4437i	0.802909i
$473$	0.343146i	0.0157779i
$474$	0	0
$475$	0	0
$476$	1.82843	0.0838058
$477$	0	0
$478$	5.11270i	0.233849i
$479$	−7.51472	−0.343356	−0.171678	−	0.985153i	$-0.554919\pi$
$479$	−7.51472	−0.343356	−0.171678	+	0.985153i	$0.554919\pi$
$480$	0	0
$481$	−16.4853	−0.751664
$482$	− 5.85786i	− 0.266818i
$483$	0	0
$484$	1.82843	0.0831103
$485$	0	0
$486$	0	0
$487$	10.4853i	0.475133i	0.971371	+	0.237567i	$0.0763498\pi$
$487$	10.4853i	0.475133i	−0.971371	+	0.237567i	$0.923650\pi$
$488$	5.02944i	0.227672i
$489$	0	0
$490$	0	0
$491$	−4.14214	−0.186932	−0.0934660	−	0.995622i	$-0.529795\pi$
$491$	−4.14214	−0.186932	−0.0934660	+	0.995622i	$0.529795\pi$
$492$	0	0
$493$	2.82843i	0.127386i
$494$	−4.20101	−0.189012
$495$	0	0
$496$	25.4558	1.14300
$497$	5.24264i	0.235165i
$498$	0	0
$499$	−40.8284	−1.82773	−0.913866	−	0.406017i	$-0.866918\pi$
$499$	−40.8284	−1.82773	−0.913866	+	0.406017i	$0.866918\pi$
$500$	0	0
$501$	0	0
$502$	10.3431i	0.461637i
$503$	22.2843i	0.993607i	0.867863	+	0.496803i	$0.165493\pi$
$503$	22.2843i	0.993607i	−0.867863	+	0.496803i	$0.834507\pi$
$504$	0	0
$505$	0	0
$506$	−0.414214	−0.0184140
$507$	0	0
$508$	− 2.27208i	− 0.100807i
$509$	−40.6274	−1.80078	−0.900389	−	0.435085i	$-0.856718\pi$
$509$	−40.6274	−1.80078	−0.900389	+	0.435085i	$0.856718\pi$
$510$	0	0
$511$	7.65685	0.338719
$512$	22.7574i	1.00574i
$513$	0	0
$514$	−5.51472	−0.243244
$515$	0	0
$516$	0	0
$517$	− 9.48528i	− 0.417162i
$518$	− 5.82843i	− 0.256086i
$519$	0	0
$520$	0	0
$521$	7.85786	0.344259	0.172130	−	0.985074i	$-0.444935\pi$
$521$	7.85786	0.344259	0.172130	+	0.985074i	$0.444935\pi$
$522$	0	0
$523$	0.213203i	0.00932274i	0.999989	+	0.00466137i	$0.00148376\pi$
$523$	0.213203i	0.00932274i	−0.999989	+	0.00466137i	$0.998516\pi$
$524$	−2.14214	−0.0935796
$525$	0	0
$526$	4.54416	0.198135
$527$	3.51472i	0.153104i
$528$	0	0
$529$	22.0000	0.956522
$530$	0	0
$531$	0	0
$532$	15.8284i	0.686249i
$533$	25.1716i	1.09030i
$534$	0	0
$535$	0	0
$536$	18.4853	0.798443
$537$	0	0
$538$	9.85786i	0.425003i
$539$	1.17157	0.0504632
$540$	0	0
$541$	−11.3137	−0.486414	−0.243207	−	0.969974i	$-0.578199\pi$
$541$	−11.3137	−0.486414	−0.243207	+	0.969974i	$0.578199\pi$
$542$	− 4.51472i	− 0.193924i
$543$	0	0
$544$	−1.82843	−0.0783932
$545$	0	0
$546$	0	0
$547$	− 17.8701i	− 0.764068i	−0.924148	−	0.382034i	$-0.875224\pi$
$547$	− 17.8701i	− 0.764068i	0.924148	−	0.382034i	$-0.124776\pi$
$548$	− 29.5147i	− 1.26081i
$549$	0	0
$550$	0	0
$551$	−24.4853	−1.04311
$552$	0	0
$553$	11.4853i	0.488404i
$554$	0.343146	0.0145789
$555$	0	0
$556$	−27.3726	−1.16086
$557$	10.8284i	0.458815i	0.973330	+	0.229408i	$0.0736789\pi$
$557$	10.8284i	0.458815i	−0.973330	+	0.229408i	$0.926321\pi$
$558$	0	0
$559$	0.970563	0.0410504
$560$	0	0
$561$	0	0
$562$	− 7.42641i	− 0.313264i
$563$	− 7.31371i	− 0.308236i	−0.988052	−	0.154118i	$-0.950746\pi$
$563$	− 7.31371i	− 0.308236i	0.988052	−	0.154118i	$-0.0492536\pi$
$564$	0	0
$565$	0	0
$566$	7.82843	0.329053
$567$	0	0
$568$	− 3.44365i	− 0.144492i
$569$	6.75736	0.283283	0.141642	−	0.989918i	$-0.454762\pi$
$569$	6.75736	0.283283	0.141642	+	0.989918i	$0.454762\pi$
$570$	0	0
$571$	−42.9706	−1.79826	−0.899131	−	0.437680i	$-0.855800\pi$
$571$	−42.9706	−1.79826	−0.899131	+	0.437680i	$0.855800\pi$
$572$	− 5.17157i	− 0.216234i
$573$	0	0
$574$	−8.89949	−0.371458
$575$	0	0
$576$	0	0
$577$	− 9.97056i	− 0.415080i	−0.978227	−	0.207540i	$-0.933454\pi$
$577$	− 9.97056i	− 0.415080i	0.978227	−	0.207540i	$-0.0665457\pi$
$578$	6.97056i	0.289937i
$579$	0	0
$580$	0	0
$581$	30.1421	1.25051
$582$	0	0
$583$	− 3.65685i	− 0.151451i
$584$	−5.02944	−0.208120
$585$	0	0
$586$	−8.45584	−0.349308
$587$	25.3431i	1.04602i	0.852325	+	0.523012i	$0.175192\pi$
$587$	25.3431i	1.04602i	−0.852325	+	0.523012i	$0.824808\pi$
$588$	0	0
$589$	−30.4264	−1.25370
$590$	0	0
$591$	0	0
$592$	− 17.4853i	− 0.718641i
$593$	− 35.7990i	− 1.47009i	−0.678019	−	0.735044i	$-0.737161\pi$
$593$	− 35.7990i	− 1.47009i	0.678019	−	0.735044i	$-0.262839\pi$
$594$	0	0
$595$	0	0
$596$	−32.4142	−1.32774
$597$	0	0
$598$	1.17157i	0.0479092i
$599$	13.6863	0.559207	0.279603	−	0.960116i	$-0.409797\pi$
$599$	13.6863	0.559207	0.279603	+	0.960116i	$0.409797\pi$
$600$	0	0
$601$	−9.17157	−0.374116	−0.187058	−	0.982349i	$-0.559895\pi$
$601$	−9.17157	−0.374116	−0.187058	+	0.982349i	$0.559895\pi$
$602$	0.343146i	0.0139856i
$603$	0	0
$604$	25.5980	1.04157
$605$	0	0
$606$	0	0
$607$	30.9706i	1.25706i	0.777787	+	0.628528i	$0.216342\pi$
$607$	30.9706i	1.25706i	−0.777787	+	0.628528i	$0.783658\pi$
$608$	− 15.8284i	− 0.641927i
$609$	0	0
$610$	0	0
$611$	−26.8284	−1.08536
$612$	0	0
$613$	42.0000i	1.69636i	0.529705	+	0.848182i	$0.322303\pi$
$613$	42.0000i	1.69636i	−0.529705	+	0.848182i	$0.677697\pi$
$614$	−12.1421	−0.490017
$615$	0	0
$616$	3.82843	0.154252
$617$	38.1421i	1.53554i	0.640723	+	0.767772i	$0.278635\pi$
$617$	38.1421i	1.53554i	−0.640723	+	0.767772i	$0.721365\pi$
$618$	0	0
$619$	6.62742	0.266378	0.133189	−	0.991091i	$-0.457478\pi$
$619$	6.62742	0.266378	0.133189	+	0.991091i	$0.457478\pi$
$620$	0	0
$621$	0	0
$622$	− 0.970563i	− 0.0389160i
$623$	18.4853i	0.740597i
$624$	0	0
$625$	0	0
$626$	0.473088	0.0189084
$627$	0	0
$628$	10.9706i	0.437773i
$629$	2.41421	0.0962610
$630$	0	0
$631$	−18.6274	−0.741546	−0.370773	−	0.928724i	$-0.620907\pi$
$631$	−18.6274	−0.741546	−0.370773	+	0.928724i	$0.620907\pi$
$632$	− 7.54416i	− 0.300090i
$633$	0	0
$634$	10.4264	0.414086
$635$	0	0
$636$	0	0
$637$	− 3.31371i	− 0.131294i
$638$	2.82843i	0.111979i
$639$	0	0
$640$	0	0
$641$	25.5147	1.00777	0.503885	−	0.863771i	$-0.331903\pi$
$641$	25.5147	1.00777	0.503885	+	0.863771i	$0.331903\pi$
$642$	0	0
$643$	− 0.970563i	− 0.0382753i	−0.999817	−	0.0191376i	$-0.993908\pi$
$643$	− 0.970563i	− 0.0382753i	0.999817	−	0.0191376i	$-0.00609207\pi$
$644$	4.41421	0.173944
$645$	0	0
$646$	0.615224	0.0242057
$647$	− 28.6569i	− 1.12662i	−0.826247	−	0.563309i	$-0.809528\pi$
$647$	− 28.6569i	− 1.12662i	0.826247	−	0.563309i	$-0.190472\pi$
$648$	0	0
$649$	11.0000	0.431788
$650$	0	0
$651$	0	0
$652$	43.5147i	1.70417i
$653$	− 23.5147i	− 0.920202i	−0.887867	−	0.460101i	$-0.847813\pi$
$653$	− 23.5147i	− 0.920202i	0.887867	−	0.460101i	$-0.152187\pi$
$654$	0	0
$655$	0	0
$656$	−26.6985	−1.04240
$657$	0	0
$658$	− 9.48528i	− 0.369775i
$659$	−47.1127	−1.83525	−0.917625	−	0.397447i	$-0.869896\pi$
$659$	−47.1127	−1.83525	−0.917625	+	0.397447i	$0.869896\pi$
$660$	0	0
$661$	−23.3431	−0.907943	−0.453972	−	0.891016i	$-0.649993\pi$
$661$	−23.3431	−0.907943	−0.453972	+	0.891016i	$0.649993\pi$
$662$	− 1.59798i	− 0.0621072i
$663$	0	0
$664$	−19.7990	−0.768350
$665$	0	0
$666$	0	0
$667$	6.82843i	0.264398i
$668$	32.5442i	1.25917i
$669$	0	0
$670$	0	0
$671$	3.17157	0.122437
$672$	0	0
$673$	− 0.343146i	− 0.0132273i	−0.999978	−	0.00661365i	$-0.997895\pi$
$673$	− 0.343146i	− 0.0132273i	0.999978	−	0.00661365i	$-0.00210520\pi$
$674$	10.0000	0.385186
$675$	0	0
$676$	9.14214	0.351621
$677$	− 23.5147i	− 0.903744i	−0.892083	−	0.451872i	$-0.850756\pi$
$677$	− 23.5147i	− 0.903744i	0.892083	−	0.451872i	$-0.149244\pi$
$678$	0	0
$679$	0.414214	0.0158961
$680$	0	0
$681$	0	0
$682$	3.51472i	0.134586i
$683$	12.5147i	0.478862i	0.970913	+	0.239431i	$0.0769610\pi$
$683$	12.5147i	0.478862i	−0.970913	+	0.239431i	$0.923039\pi$
$684$	0	0
$685$	0	0
$686$	8.17157	0.311992
$687$	0	0
$688$	1.02944i	0.0392469i
$689$	−10.3431	−0.394042
$690$	0	0
$691$	−35.4558	−1.34880	−0.674402	−	0.738364i	$-0.735598\pi$
$691$	−35.4558	−1.34880	−0.674402	+	0.738364i	$0.735598\pi$
$692$	33.9289i	1.28978i
$693$	0	0
$694$	−11.1127	−0.421832
$695$	0	0
$696$	0	0
$697$	− 3.68629i	− 0.139628i
$698$	− 6.00000i	− 0.227103i
$699$	0	0
$700$	0	0
$701$	41.3848	1.56308	0.781541	−	0.623854i	$-0.214434\pi$
$701$	41.3848	1.56308	0.781541	+	0.623854i	$0.214434\pi$
$702$	0	0
$703$	20.8995i	0.788239i
$704$	4.17157	0.157222
$705$	0	0
$706$	−5.17157	−0.194635
$707$	− 11.8284i	− 0.444854i
$708$	0	0
$709$	29.1421	1.09446	0.547228	−	0.836984i	$-0.315683\pi$
$709$	29.1421	1.09446	0.547228	+	0.836984i	$0.315683\pi$
$710$	0	0
$711$	0	0
$712$	− 12.1421i	− 0.455046i
$713$	8.48528i	0.317776i
$714$	0	0
$715$	0	0
$716$	−41.6863	−1.55789
$717$	0	0
$718$	13.4558i	0.502168i
$719$	9.65685	0.360140	0.180070	−	0.983654i	$-0.442368\pi$
$719$	9.65685	0.360140	0.180070	+	0.983654i	$0.442368\pi$
$720$	0	0
$721$	5.65685	0.210672
$722$	− 2.54416i	− 0.0946837i
$723$	0	0
$724$	21.8873	0.813435
$725$	0	0
$726$	0	0
$727$	16.9706i	0.629403i	0.949191	+	0.314702i	$0.101904\pi$
$727$	16.9706i	0.629403i	−0.949191	+	0.314702i	$0.898096\pi$
$728$	− 10.8284i	− 0.401328i
$729$	0	0
$730$	0	0
$731$	−0.142136	−0.00525708
$732$	0	0
$733$	− 32.1421i	− 1.18720i	−0.804761	−	0.593598i	$-0.797707\pi$
$733$	− 32.1421i	− 1.18720i	0.804761	−	0.593598i	$-0.202293\pi$
$734$	8.82843	0.325863
$735$	0	0
$736$	−4.41421	−0.162710
$737$	− 11.6569i	− 0.429386i
$738$	0	0
$739$	−20.4142	−0.750949	−0.375474	−	0.926833i	$-0.622520\pi$
$739$	−20.4142	−0.750949	−0.375474	+	0.926833i	$0.622520\pi$
$740$	0	0
$741$	0	0
$742$	− 3.65685i	− 0.134247i
$743$	− 31.1127i	− 1.14141i	−0.821154	−	0.570707i	$-0.806669\pi$
$743$	− 31.1127i	− 1.14141i	0.821154	−	0.570707i	$-0.193331\pi$
$744$	0	0
$745$	0	0
$746$	−5.11270	−0.187189
$747$	0	0
$748$	0.757359i	0.0276918i
$749$	−41.7990	−1.52730
$750$	0	0
$751$	31.5980	1.15303	0.576513	−	0.817088i	$-0.304413\pi$
$751$	31.5980	1.15303	0.576513	+	0.817088i	$0.304413\pi$
$752$	− 28.4558i	− 1.03768i
$753$	0	0
$754$	8.00000	0.291343
$755$	0	0
$756$	0	0
$757$	− 10.6863i	− 0.388400i	−0.980962	−	0.194200i	$-0.937789\pi$
$757$	− 10.6863i	− 0.388400i	0.980962	−	0.194200i	$-0.0622111\pi$
$758$	− 6.14214i	− 0.223092i
$759$	0	0
$760$	0	0
$761$	5.17157	0.187469	0.0937347	−	0.995597i	$-0.470119\pi$
$761$	5.17157	0.187469	0.0937347	+	0.995597i	$0.470119\pi$
$762$	0	0
$763$	− 41.7990i	− 1.51323i
$764$	−21.6274	−0.782452
$765$	0	0
$766$	−8.28427	−0.299323
$767$	− 31.1127i	− 1.12341i
$768$	0	0
$769$	7.65685	0.276113	0.138057	−	0.990424i	$-0.455914\pi$
$769$	7.65685	0.276113	0.138057	+	0.990424i	$0.455914\pi$
$770$	0	0
$771$	0	0
$772$	35.3137i	1.27097i
$773$	− 0.828427i	− 0.0297965i	−0.999889	−	0.0148982i	$-0.995258\pi$
$773$	− 0.828427i	− 0.0297965i	0.999889	−	0.0148982i	$-0.00474243\pi$
$774$	0	0
$775$	0	0
$776$	−0.272078	−0.00976703
$777$	0	0
$778$	− 2.62742i	− 0.0941975i
$779$	31.9117	1.14335
$780$	0	0
$781$	−2.17157	−0.0777050
$782$	− 0.171573i	− 0.00613543i
$783$	0	0
$784$	3.51472	0.125526
$785$	0	0
$786$	0	0
$787$	− 7.92893i	− 0.282636i	−0.989964	−	0.141318i	$-0.954866\pi$
$787$	− 7.92893i	− 0.282636i	0.989964	−	0.141318i	$-0.0451340\pi$
$788$	− 24.2132i	− 0.862560i
$789$	0	0
$790$	0	0
$791$	−24.1421	−0.858396
$792$	0	0
$793$	− 8.97056i	− 0.318554i
$794$	−13.2304	−0.469531
$795$	0	0
$796$	−9.45584	−0.335154
$797$	− 40.9706i	− 1.45125i	−0.688089	−	0.725626i	$-0.741550\pi$
$797$	− 40.9706i	− 1.45125i	0.688089	−	0.725626i	$-0.258450\pi$
$798$	0	0
$799$	3.92893	0.138996
$800$	0	0
$801$	0	0
$802$	3.23045i	0.114071i
$803$	3.17157i	0.111922i
$804$	0	0
$805$	0	0
$806$	9.94113	0.350161
$807$	0	0
$808$	7.76955i	0.273332i
$809$	7.72792	0.271699	0.135850	−	0.990729i	$-0.456624\pi$
$809$	7.72792	0.271699	0.135850	+	0.990729i	$0.456624\pi$
$810$	0	0
$811$	10.2132	0.358634	0.179317	−	0.983791i	$-0.442611\pi$
$811$	10.2132	0.358634	0.179317	+	0.983791i	$0.442611\pi$
$812$	− 30.1421i	− 1.05778i
$813$	0	0
$814$	2.41421	0.0846181
$815$	0	0
$816$	0	0
$817$	− 1.23045i	− 0.0430479i
$818$	− 10.0000i	− 0.349642i
$819$	0	0
$820$	0	0
$821$	−15.7990	−0.551389	−0.275694	−	0.961245i	$-0.588908\pi$
$821$	−15.7990	−0.551389	−0.275694	+	0.961245i	$0.588908\pi$
$822$	0	0
$823$	14.9706i	0.521841i	0.965360	+	0.260921i	$0.0840260\pi$
$823$	14.9706i	0.521841i	−0.965360	+	0.260921i	$0.915974\pi$
$824$	−3.71573	−0.129444
$825$	0	0
$826$	11.0000	0.382739
$827$	12.6863i	0.441146i	0.975371	+	0.220573i	$0.0707927\pi$
$827$	12.6863i	0.441146i	−0.975371	+	0.220573i	$0.929207\pi$
$828$	0	0
$829$	47.9411	1.66506	0.832532	−	0.553977i	$-0.186890\pi$
$829$	47.9411	1.66506	0.832532	+	0.553977i	$0.186890\pi$
$830$	0	0
$831$	0	0
$832$	− 11.7990i	− 0.409056i
$833$	0.485281i	0.0168140i
$834$	0	0
$835$	0	0
$836$	−6.55635	−0.226756
$837$	0	0
$838$	− 3.52691i	− 0.121835i
$839$	−47.3137	−1.63345	−0.816725	−	0.577027i	$-0.804213\pi$
$839$	−47.3137	−1.63345	−0.816725	+	0.577027i	$0.804213\pi$
$840$	0	0
$841$	17.6274	0.607842
$842$	− 11.1838i	− 0.385418i
$843$	0	0
$844$	17.0294	0.586177
$845$	0	0
$846$	0	0
$847$	− 2.41421i	− 0.0829534i
$848$	− 10.9706i	− 0.376731i
$849$	0	0
$850$	0	0
$851$	5.82843	0.199796
$852$	0	0
$853$	19.1716i	0.656422i	0.944604	+	0.328211i	$0.106446\pi$
$853$	19.1716i	0.656422i	−0.944604	+	0.328211i	$0.893554\pi$
$854$	3.17157	0.108529
$855$	0	0
$856$	27.4558	0.938421
$857$	36.6985i	1.25360i	0.779182	+	0.626798i	$0.215635\pi$
$857$	36.6985i	1.25360i	−0.779182	+	0.626798i	$0.784365\pi$
$858$	0	0
$859$	20.4853	0.698949	0.349474	−	0.936946i	$-0.386360\pi$
$859$	20.4853	0.698949	0.349474	+	0.936946i	$0.386360\pi$
$860$	0	0
$861$	0	0
$862$	− 2.82843i	− 0.0963366i
$863$	− 28.6863i	− 0.976493i	−0.872706	−	0.488246i	$-0.837637\pi$
$863$	− 28.6863i	− 0.976493i	0.872706	−	0.488246i	$-0.162363\pi$
$864$	0	0
$865$	0	0
$866$	3.85786	0.131096
$867$	0	0
$868$	− 37.4558i	− 1.27133i
$869$	−4.75736	−0.161382
$870$	0	0
$871$	−32.9706	−1.11716
$872$	27.4558i	0.929772i
$873$	0	0
$874$	1.48528	0.0502404
$875$	0	0
$876$	0	0
$877$	15.1127i	0.510320i	0.966899	+	0.255160i	$0.0821281\pi$
$877$	15.1127i	0.510320i	−0.966899	+	0.255160i	$0.917872\pi$
$878$	− 11.4853i	− 0.387609i
$879$	0	0
$880$	0	0
$881$	24.0000	0.808581	0.404290	−	0.914631i	$-0.367519\pi$
$881$	24.0000	0.808581	0.404290	+	0.914631i	$0.367519\pi$
$882$	0	0
$883$	− 38.6274i	− 1.29992i	−0.759970	−	0.649958i	$-0.774786\pi$
$883$	− 38.6274i	− 1.29992i	0.759970	−	0.649958i	$-0.225214\pi$
$884$	2.14214	0.0720478
$885$	0	0
$886$	−10.7574	−0.361401
$887$	22.1421i	0.743460i	0.928341	+	0.371730i	$0.121235\pi$
$887$	22.1421i	0.743460i	−0.928341	+	0.371730i	$0.878765\pi$
$888$	0	0
$889$	−3.00000	−0.100617
$890$	0	0
$891$	0	0
$892$	49.0538i	1.64244i
$893$	34.0122i	1.13817i
$894$	0	0
$895$	0	0
$896$	25.4853	0.851403
$897$	0	0
$898$	4.34315i	0.144933i
$899$	57.9411	1.93244
$900$	0	0
$901$	1.51472	0.0504626
$902$	− 3.68629i	− 0.122740i
$903$	0	0
$904$	15.8579	0.527425
$905$	0	0
$906$	0	0
$907$	− 2.48528i	− 0.0825224i	−0.999148	−	0.0412612i	$-0.986862\pi$
$907$	− 2.48528i	− 0.0825224i	0.999148	−	0.0412612i	$-0.0131376\pi$
$908$	2.76955i	0.0919108i
$909$	0	0
$910$	0	0
$911$	−28.5147	−0.944735	−0.472367	−	0.881402i	$-0.656600\pi$
$911$	−28.5147	−0.944735	−0.472367	+	0.881402i	$0.656600\pi$
$912$	0	0
$913$	12.4853i	0.413203i
$914$	−13.3137	−0.440378
$915$	0	0
$916$	35.6274	1.17716
$917$	2.82843i	0.0934029i
$918$	0	0
$919$	−1.78680	−0.0589410	−0.0294705	−	0.999566i	$-0.509382\pi$
$919$	−1.78680	−0.0589410	−0.0294705	+	0.999566i	$0.509382\pi$
$920$	0	0
$921$	0	0
$922$	− 16.8873i	− 0.556154i
$923$	6.14214i	0.202171i
$924$	0	0
$925$	0	0
$926$	0.426407	0.0140126
$927$	0	0
$928$	30.1421i	0.989464i
$929$	−13.7990	−0.452730	−0.226365	−	0.974043i	$-0.572684\pi$
$929$	−13.7990	−0.452730	−0.226365	+	0.974043i	$0.572684\pi$
$930$	0	0
$931$	−4.20101	−0.137683
$932$	− 26.6152i	− 0.871811i
$933$	0	0
$934$	−14.3431	−0.469322
$935$	0	0
$936$	0	0
$937$	− 16.0000i	− 0.522697i	−0.965244	−	0.261349i	$-0.915833\pi$
$937$	− 16.0000i	− 0.522697i	0.965244	−	0.261349i	$-0.0841672\pi$
$938$	− 11.6569i	− 0.380610i
$939$	0	0
$940$	0	0
$941$	22.8995	0.746502	0.373251	−	0.927730i	$-0.378243\pi$
$941$	22.8995	0.746502	0.373251	+	0.927730i	$0.378243\pi$
$942$	0	0
$943$	− 8.89949i	− 0.289807i
$944$	33.0000	1.07406
$945$	0	0
$946$	−0.142136	−0.00462123
$947$	36.7990i	1.19581i	0.801568	+	0.597903i	$0.203999\pi$
$947$	36.7990i	1.19581i	−0.801568	+	0.597903i	$0.796001\pi$
$948$	0	0
$949$	8.97056	0.291197
$950$	0	0
$951$	0	0
$952$	1.58579i	0.0513956i
$953$	− 29.0416i	− 0.940751i	−0.882466	−	0.470375i	$-0.844119\pi$
$953$	− 29.0416i	− 0.940751i	0.882466	−	0.470375i	$-0.155881\pi$
$954$	0	0
$955$	0	0
$956$	22.5685	0.729919
$957$	0	0
$958$	− 3.11270i	− 0.100567i
$959$	−38.9706	−1.25843
$960$	0	0
$961$	41.0000	1.32258
$962$	− 6.82843i	− 0.220157i
$963$	0	0
$964$	−25.8579	−0.832826
$965$	0	0
$966$	0	0
$967$	38.0000i	1.22200i	0.791632	+	0.610999i	$0.209232\pi$
$967$	38.0000i	1.22200i	−0.791632	+	0.610999i	$0.790768\pi$
$968$	1.58579i	0.0509691i
$969$	0	0
$970$	0	0
$971$	50.3137	1.61464	0.807322	−	0.590111i	$-0.200916\pi$
$971$	50.3137	1.61464	0.807322	+	0.590111i	$0.200916\pi$
$972$	0	0
$973$	36.1421i	1.15866i
$974$	−4.34315	−0.139163
$975$	0	0
$976$	9.51472	0.304559
$977$	− 60.5685i	− 1.93776i	−0.247532	−	0.968880i	$-0.579620\pi$
$977$	− 60.5685i	− 1.93776i	0.247532	−	0.968880i	$-0.420380\pi$
$978$	0	0
$979$	−7.65685	−0.244714
$980$	0	0
$981$	0	0
$982$	− 1.71573i	− 0.0547511i
$983$	51.2843i	1.63571i	0.575421	+	0.817857i	$0.304838\pi$
$983$	51.2843i	1.63571i	−0.575421	+	0.817857i	$0.695162\pi$
$984$	0	0
$985$	0	0
$986$	−1.17157	−0.0373105
$987$	0	0
$988$	18.5442i	0.589968i
$989$	−0.343146	−0.0109114
$990$	0	0
$991$	−42.2843	−1.34320	−0.671602	−	0.740912i	$-0.734394\pi$
$991$	−42.2843	−1.34320	−0.671602	+	0.740912i	$0.734394\pi$
$992$	37.4558i	1.18922i
$993$	0	0
$994$	−2.17157	−0.0688781
$995$	0	0
$996$	0	0
$997$	− 47.2548i	− 1.49658i	−0.663374	−	0.748288i	$-0.730876\pi$
$997$	− 47.2548i	− 1.49658i	0.663374	−	0.748288i	$-0.269124\pi$
$998$	− 16.9117i	− 0.535330i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.2.c.o.199.3		4
3.2	odd	2		825.2.c.d.199.2		4
5.2	odd	4		2475.2.a.w.1.1		2
5.3	odd	4		2475.2.a.l.1.2		2
5.4	even	2	inner	2475.2.c.o.199.2		4
15.2	even	4		825.2.a.d.1.2	✓	2
15.8	even	4		825.2.a.f.1.1	yes	2
15.14	odd	2		825.2.c.d.199.3		4
165.32	odd	4		9075.2.a.ca.1.1		2
165.98	odd	4		9075.2.a.w.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.2.a.d.1.2	✓	2	15.2	even	4
825.2.a.f.1.1	yes	2	15.8	even	4
825.2.c.d.199.2		4	3.2	odd	2
825.2.c.d.199.3		4	15.14	odd	2
2475.2.a.l.1.2		2	5.3	odd	4
2475.2.a.w.1.1		2	5.2	odd	4
2475.2.c.o.199.2		4	5.4	even	2	inner
2475.2.c.o.199.3		4	1.1	even	1	trivial
9075.2.a.w.1.2		2	165.98	odd	4
9075.2.a.ca.1.1		2	165.32	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)

\(f(q)\)	\(=\)	\(q+0.414214i q^{2} +1.82843 q^{4} -2.41421i q^{7} +1.58579i q^{8} +O(q^{10})\)	\(q+0.414214i q^{2} +1.82843 q^{4} -2.41421i q^{7} +1.58579i q^{8} +1.00000 q^{11} -2.82843i q^{13} +1.00000 q^{14} +3.00000 q^{16} +0.414214i q^{17} -3.58579 q^{19} +0.414214i q^{22} +1.00000i q^{23} +1.17157 q^{26} -4.41421i q^{28} +6.82843 q^{29} +8.48528 q^{31} +4.41421i q^{32} -0.171573 q^{34} -5.82843i q^{37} -1.48528i q^{38} -8.89949 q^{41} +0.343146i q^{43} +1.82843 q^{44} -0.414214 q^{46} -9.48528i q^{47} +1.17157 q^{49} -5.17157i q^{52} -3.65685i q^{53} +3.82843 q^{56} +2.82843i q^{58} +11.0000 q^{59} +3.17157 q^{61} +3.51472i q^{62} +4.17157 q^{64} -11.6569i q^{67} +0.757359i q^{68} -2.17157 q^{71} +3.17157i q^{73} +2.41421 q^{74} -6.55635 q^{76} -2.41421i q^{77} -4.75736 q^{79} -3.68629i q^{82} +12.4853i q^{83} -0.142136 q^{86} +1.58579i q^{88} -7.65685 q^{89} -6.82843 q^{91} +1.82843i q^{92} +3.92893 q^{94} +0.171573i q^{97} +0.485281i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4}+O(q^{10}) \) 4 * q - 4 * q^4	\( 4 q - 4 q^{4} + 4 q^{11} + 4 q^{14} + 12 q^{16} - 20 q^{19} + 16 q^{26} + 16 q^{29} - 12 q^{34} + 4 q^{41} - 4 q^{44} + 4 q^{46} + 16 q^{49} + 4 q^{56} + 44 q^{59} + 24 q^{61} + 28 q^{64} - 20 q^{71} + 4 q^{74} + 36 q^{76} - 36 q^{79} + 56 q^{86} - 8 q^{89} - 16 q^{91} + 44 q^{94}+O(q^{100}) \) 4 * q - 4 * q^4 + 4 * q^11 + 4 * q^14 + 12 * q^16 - 20 * q^19 + 16 * q^26 + 16 * q^29 - 12 * q^34 + 4 * q^41 - 4 * q^44 + 4 * q^46 + 16 * q^49 + 4 * q^56 + 44 * q^59 + 24 * q^61 + 28 * q^64 - 20 * q^71 + 4 * q^74 + 36 * q^76 - 36 * q^79 + 56 * q^86 - 8 * q^89 - 16 * q^91 + 44 * q^94

Introduction

L-functions

Modular forms

Varieties

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Database

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