LMFDB - Embedded newform 2475.4.a.bp.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,4,Mod(1,2475)]

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, 0, 0]))

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

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

chi := DirichletCharacter("2475.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2475.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:	$146.029727264$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 41x^{5} + 40x^{4} + 424x^{3} - 168x^{2} - 1042x - 388 $ x^7 - 2x^6 - 41x^5 + 40x^4 + 424x^3 - 168x^2 - 1042x - 388
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 495)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.30247$ of defining polynomial
Character	$\chi$	$=$	2475.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.30247 q^{2} -2.69863 q^{4} -33.7164 q^{7} +24.6333 q^{8} +O(q^{10})$	$q-2.30247 q^{2} -2.69863 q^{4} -33.7164 q^{7} +24.6333 q^{8} +11.0000 q^{11} -37.9156 q^{13} +77.6309 q^{14} -35.1283 q^{16} +44.2817 q^{17} -42.3986 q^{19} -25.3272 q^{22} -30.6793 q^{23} +87.2995 q^{26} +90.9881 q^{28} +194.548 q^{29} -82.9601 q^{31} -116.184 q^{32} -101.957 q^{34} -445.236 q^{37} +97.6216 q^{38} +206.184 q^{41} -197.228 q^{43} -29.6850 q^{44} +70.6382 q^{46} +275.301 q^{47} +793.792 q^{49} +102.320 q^{52} +382.718 q^{53} -830.544 q^{56} -447.941 q^{58} +771.622 q^{59} -210.576 q^{61} +191.013 q^{62} +548.537 q^{64} -452.672 q^{67} -119.500 q^{68} -310.139 q^{71} +174.350 q^{73} +1025.14 q^{74} +114.418 q^{76} -370.880 q^{77} +1191.11 q^{79} -474.732 q^{82} +183.669 q^{83} +454.111 q^{86} +270.966 q^{88} -613.710 q^{89} +1278.38 q^{91} +82.7923 q^{92} -633.873 q^{94} +1204.77 q^{97} -1827.68 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 5 q^{2} + 33 q^{4} - 30 q^{7} - 45 q^{8}+O(q^{10}) $ 7 * q - 5 * q^2 + 33 * q^4 - 30 * q^7 - 45 * q^8	$ 7 q - 5 q^{2} + 33 q^{4} - 30 q^{7} - 45 q^{8} + 77 q^{11} - 38 q^{13} + 20 q^{14} + 309 q^{16} - 12 q^{17} + 226 q^{19} - 55 q^{22} - 334 q^{23} - 372 q^{26} - 812 q^{28} - 258 q^{29} + 336 q^{31} - 485 q^{32} + 78 q^{34} - 466 q^{37} + 494 q^{38} - 258 q^{41} - 308 q^{43} + 363 q^{44} + 98 q^{46} - 546 q^{47} + 735 q^{49} - 512 q^{52} - 110 q^{53} + 20 q^{56} - 1362 q^{58} - 68 q^{59} + 1096 q^{61} - 356 q^{62} + 2761 q^{64} - 2268 q^{67} + 1186 q^{68} - 166 q^{71} - 200 q^{73} - 1710 q^{74} + 3310 q^{76} - 330 q^{77} + 2152 q^{79} + 1006 q^{82} - 370 q^{83} + 106 q^{86} - 495 q^{88} - 252 q^{89} + 2768 q^{91} - 3774 q^{92} + 2218 q^{94} - 3698 q^{97} - 697 q^{98}+O(q^{100}) $ 7 * q - 5 * q^2 + 33 * q^4 - 30 * q^7 - 45 * q^8 + 77 * q^11 - 38 * q^13 + 20 * q^14 + 309 * q^16 - 12 * q^17 + 226 * q^19 - 55 * q^22 - 334 * q^23 - 372 * q^26 - 812 * q^28 - 258 * q^29 + 336 * q^31 - 485 * q^32 + 78 * q^34 - 466 * q^37 + 494 * q^38 - 258 * q^41 - 308 * q^43 + 363 * q^44 + 98 * q^46 - 546 * q^47 + 735 * q^49 - 512 * q^52 - 110 * q^53 + 20 * q^56 - 1362 * q^58 - 68 * q^59 + 1096 * q^61 - 356 * q^62 + 2761 * q^64 - 2268 * q^67 + 1186 * q^68 - 166 * q^71 - 200 * q^73 - 1710 * q^74 + 3310 * q^76 - 330 * q^77 + 2152 * q^79 + 1006 * q^82 - 370 * q^83 + 106 * q^86 - 495 * q^88 - 252 * q^89 + 2768 * q^91 - 3774 * q^92 + 2218 * q^94 - 3698 * q^97 - 697 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.30247	−0.814046	−0.407023	−	0.913418i	$-0.633433\pi$
$2$	−2.30247	−0.814046	−0.407023	+	0.913418i	$0.633433\pi$
$3$	0	0
$3$	0	0
$4$	−2.69863	−0.337329
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−33.7164	−1.82051	−0.910256	−	0.414046i	$-0.864115\pi$
$7$	−33.7164	−1.82051	−0.910256	+	0.414046i	$0.864115\pi$
$8$	24.6333	1.08865
$9$	0	0
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	−37.9156	−0.808915	−0.404457	−	0.914557i	$-0.632540\pi$
$13$	−37.9156	−0.808915	−0.404457	+	0.914557i	$0.632540\pi$
$14$	77.6309	1.48198
$15$	0	0
$16$	−35.1283	−0.548880
$17$	44.2817	0.631758	0.315879	−	0.948799i	$-0.397701\pi$
$17$	44.2817	0.631758	0.315879	+	0.948799i	$0.397701\pi$
$18$	0	0
$19$	−42.3986	−0.511943	−0.255971	−	0.966684i	$-0.582395\pi$
$19$	−42.3986	−0.511943	−0.255971	+	0.966684i	$0.582395\pi$
$20$	0	0
$21$	0	0
$22$	−25.3272	−0.245444
$23$	−30.6793	−0.278134	−0.139067	−	0.990283i	$-0.544410\pi$
$23$	−30.6793	−0.278134	−0.139067	+	0.990283i	$0.544410\pi$
$24$	0	0
$25$	0	0
$26$	87.2995	0.658494
$27$	0	0
$28$	90.9881	0.614112
$29$	194.548	1.24575	0.622873	−	0.782323i	$-0.285965\pi$
$29$	194.548	1.24575	0.622873	+	0.782323i	$0.285965\pi$
$30$	0	0
$31$	−82.9601	−0.480648	−0.240324	−	0.970693i	$-0.577254\pi$
$31$	−82.9601	−0.480648	−0.240324	+	0.970693i	$0.577254\pi$
$32$	−116.184	−0.641834
$33$	0	0
$34$	−101.957	−0.514280
$35$	0	0
$36$	0	0
$37$	−445.236	−1.97828	−0.989140	−	0.146980i	$-0.953045\pi$
$37$	−445.236	−1.97828	−0.989140	+	0.146980i	$0.953045\pi$
$38$	97.6216	0.416745
$39$	0	0
$40$	0	0
$41$	206.184	0.785378	0.392689	−	0.919671i	$-0.371545\pi$
$41$	206.184	0.785378	0.392689	+	0.919671i	$0.371545\pi$
$42$	0	0
$43$	−197.228	−0.699464	−0.349732	−	0.936850i	$-0.613727\pi$
$43$	−197.228	−0.699464	−0.349732	+	0.936850i	$0.613727\pi$
$44$	−29.6850	−0.101709
$45$	0	0
$46$	70.6382	0.226414
$47$	275.301	0.854401	0.427201	−	0.904157i	$-0.359500\pi$
$47$	275.301	0.854401	0.427201	+	0.904157i	$0.359500\pi$
$48$	0	0
$49$	793.792	2.31426
$50$	0	0
$51$	0	0
$52$	102.320	0.272871
$53$	382.718	0.991894	0.495947	−	0.868353i	$-0.334821\pi$
$53$	382.718	0.991894	0.495947	+	0.868353i	$0.334821\pi$
$54$	0	0
$55$	0	0
$56$	−830.544	−1.98190
$57$	0	0
$58$	−447.941	−1.01409
$59$	771.622	1.70265	0.851327	−	0.524635i	$-0.175798\pi$
$59$	771.622	1.70265	0.851327	+	0.524635i	$0.175798\pi$
$60$	0	0
$61$	−210.576	−0.441992	−0.220996	−	0.975275i	$-0.570931\pi$
$61$	−210.576	−0.441992	−0.220996	+	0.975275i	$0.570931\pi$
$62$	191.013	0.391269
$63$	0	0
$64$	548.537	1.07136
$65$	0	0
$66$	0	0
$67$	−452.672	−0.825413	−0.412707	−	0.910864i	$-0.635416\pi$
$67$	−452.672	−0.825413	−0.412707	+	0.910864i	$0.635416\pi$
$68$	−119.500	−0.213111
$69$	0	0
$70$	0	0
$71$	−310.139	−0.518405	−0.259203	−	0.965823i	$-0.583460\pi$
$71$	−310.139	−0.518405	−0.259203	+	0.965823i	$0.583460\pi$
$72$	0	0
$73$	174.350	0.279536	0.139768	−	0.990184i	$-0.455364\pi$
$73$	174.350	0.279536	0.139768	+	0.990184i	$0.455364\pi$
$74$	1025.14	1.61041
$75$	0	0
$76$	114.418	0.172693
$77$	−370.880	−0.548905
$78$	0	0
$79$	1191.11	1.69634	0.848168	−	0.529727i	$-0.177706\pi$
$79$	1191.11	1.69634	0.848168	+	0.529727i	$0.177706\pi$
$80$	0	0
$81$	0	0
$82$	−474.732	−0.639334
$83$	183.669	0.242896	0.121448	−	0.992598i	$-0.461246\pi$
$83$	183.669	0.242896	0.121448	+	0.992598i	$0.461246\pi$
$84$	0	0
$85$	0	0
$86$	454.111	0.569396
$87$	0	0
$88$	270.966	0.328240
$89$	−613.710	−0.730934	−0.365467	−	0.930824i	$-0.619091\pi$
$89$	−613.710	−0.730934	−0.365467	+	0.930824i	$0.619091\pi$
$90$	0	0
$91$	1278.38	1.47264
$92$	82.7923	0.0938227
$93$	0	0
$94$	−633.873	−0.695522
$95$	0	0
$96$	0	0
$97$	1204.77	1.26109	0.630545	−	0.776152i	$-0.282831\pi$
$97$	1204.77	1.26109	0.630545	+	0.776152i	$0.282831\pi$
$98$	−1827.68	−1.88392
$99$	0	0
$100$	0	0
$101$	−1389.76	−1.36917	−0.684586	−	0.728932i	$-0.740017\pi$
$101$	−1389.76	−1.36917	−0.684586	+	0.728932i	$0.740017\pi$
$102$	0	0
$103$	981.947	0.939361	0.469680	−	0.882837i	$-0.344369\pi$
$103$	981.947	0.939361	0.469680	+	0.882837i	$0.344369\pi$
$104$	−933.985	−0.880623
$105$	0	0
$106$	−881.197	−0.807448
$107$	738.649	0.667364	0.333682	−	0.942686i	$-0.391709\pi$
$107$	738.649	0.667364	0.333682	+	0.942686i	$0.391709\pi$
$108$	0	0
$109$	599.887	0.527144	0.263572	−	0.964640i	$-0.415099\pi$
$109$	599.887	0.527144	0.263572	+	0.964640i	$0.415099\pi$
$110$	0	0
$111$	0	0
$112$	1184.40	0.999242
$113$	817.177	0.680297	0.340149	−	0.940372i	$-0.389523\pi$
$113$	817.177	0.680297	0.340149	+	0.940372i	$0.389523\pi$
$114$	0	0
$115$	0	0
$116$	−525.014	−0.420227
$117$	0	0
$118$	−1776.64	−1.38604
$119$	−1493.02	−1.15012
$120$	0	0
$121$	121.000	0.0909091
$122$	484.845	0.359802
$123$	0	0
$124$	223.879	0.162137
$125$	0	0
$126$	0	0
$127$	−916.072	−0.640065	−0.320033	−	0.947407i	$-0.603694\pi$
$127$	−916.072	−0.640065	−0.320033	+	0.947407i	$0.603694\pi$
$128$	−333.515	−0.230304
$129$	0	0
$130$	0	0
$131$	903.159	0.602361	0.301181	−	0.953567i	$-0.402619\pi$
$131$	903.159	0.602361	0.301181	+	0.953567i	$0.402619\pi$
$132$	0	0
$133$	1429.53	0.931998
$134$	1042.26	0.671924
$135$	0	0
$136$	1090.80	0.687762
$137$	2373.28	1.48002	0.740009	−	0.672597i	$-0.234821\pi$
$137$	2373.28	1.48002	0.740009	+	0.672597i	$0.234821\pi$
$138$	0	0
$139$	126.777	0.0773601	0.0386801	−	0.999252i	$-0.487685\pi$
$139$	126.777	0.0773601	0.0386801	+	0.999252i	$0.487685\pi$
$140$	0	0
$141$	0	0
$142$	714.086	0.422006
$143$	−417.071	−0.243897
$144$	0	0
$145$	0	0
$146$	−401.436	−0.227555
$147$	0	0
$148$	1201.53	0.667332
$149$	−1448.83	−0.796596	−0.398298	−	0.917256i	$-0.630399\pi$
$149$	−1448.83	−0.796596	−0.398298	+	0.917256i	$0.630399\pi$
$150$	0	0
$151$	1536.43	0.828033	0.414016	−	0.910269i	$-0.364126\pi$
$151$	1536.43	0.828033	0.414016	+	0.910269i	$0.364126\pi$
$152$	−1044.42	−0.557325
$153$	0	0
$154$	853.940	0.446834
$155$	0	0
$156$	0	0
$157$	−2928.34	−1.48858	−0.744289	−	0.667857i	$-0.767212\pi$
$157$	−2928.34	−1.48858	−0.744289	+	0.667857i	$0.767212\pi$
$158$	−2742.50	−1.38090
$159$	0	0
$160$	0	0
$161$	1034.39	0.506346
$162$	0	0
$163$	2064.31	0.991957	0.495979	−	0.868335i	$-0.334809\pi$
$163$	2064.31	0.991957	0.495979	+	0.868335i	$0.334809\pi$
$164$	−556.415	−0.264931
$165$	0	0
$166$	−422.893	−0.197728
$167$	2896.40	1.34210	0.671049	−	0.741413i	$-0.265844\pi$
$167$	2896.40	1.34210	0.671049	+	0.741413i	$0.265844\pi$
$168$	0	0
$169$	−759.408	−0.345657
$170$	0	0
$171$	0	0
$172$	532.245	0.235950
$173$	−2938.08	−1.29120	−0.645601	−	0.763675i	$-0.723393\pi$
$173$	−2938.08	−1.29120	−0.645601	+	0.763675i	$0.723393\pi$
$174$	0	0
$175$	0	0
$176$	−386.411	−0.165493
$177$	0	0
$178$	1413.05	0.595013
$179$	−3352.50	−1.39987	−0.699937	−	0.714204i	$-0.746789\pi$
$179$	−3352.50	−1.39987	−0.699937	+	0.714204i	$0.746789\pi$
$180$	0	0
$181$	−4167.19	−1.71130	−0.855649	−	0.517556i	$-0.826842\pi$
$181$	−4167.19	−1.71130	−0.855649	+	0.517556i	$0.826842\pi$
$182$	−2943.42	−1.19880
$183$	0	0
$184$	−755.732	−0.302790
$185$	0	0
$186$	0	0
$187$	487.099	0.190482
$188$	−742.938	−0.288215
$189$	0	0
$190$	0	0
$191$	1663.88	0.630337	0.315169	−	0.949036i	$-0.397939\pi$
$191$	1663.88	0.630337	0.315169	+	0.949036i	$0.397939\pi$
$192$	0	0
$193$	3148.25	1.17418	0.587088	−	0.809523i	$-0.300274\pi$
$193$	3148.25	1.17418	0.587088	+	0.809523i	$0.300274\pi$
$194$	−2773.94	−1.02659
$195$	0	0
$196$	−2142.16	−0.780669
$197$	−5034.38	−1.82074	−0.910368	−	0.413800i	$-0.864201\pi$
$197$	−5034.38	−1.82074	−0.910368	+	0.413800i	$0.864201\pi$
$198$	0	0
$199$	1918.95	0.683571	0.341785	−	0.939778i	$-0.388968\pi$
$199$	1918.95	0.683571	0.341785	+	0.939778i	$0.388968\pi$
$200$	0	0
$201$	0	0
$202$	3199.88	1.11457
$203$	−6559.44	−2.26790
$204$	0	0
$205$	0	0
$206$	−2260.90	−0.764683
$207$	0	0
$208$	1331.91	0.443997
$209$	−466.385	−0.154357
$210$	0	0
$211$	−1200.62	−0.391725	−0.195862	−	0.980631i	$-0.562751\pi$
$211$	−1200.62	−0.391725	−0.195862	+	0.980631i	$0.562751\pi$
$212$	−1032.82	−0.334595
$213$	0	0
$214$	−1700.72	−0.543265
$215$	0	0
$216$	0	0
$217$	2797.11	0.875025
$218$	−1381.22	−0.429120
$219$	0	0
$220$	0	0
$221$	−1678.97	−0.511039
$222$	0	0
$223$	−3956.12	−1.18799	−0.593994	−	0.804469i	$-0.702450\pi$
$223$	−3956.12	−1.18799	−0.593994	+	0.804469i	$0.702450\pi$
$224$	3917.31	1.16847
$225$	0	0
$226$	−1881.53	−0.553793
$227$	5849.85	1.71043	0.855216	−	0.518272i	$-0.173424\pi$
$227$	5849.85	1.71043	0.855216	+	0.518272i	$0.173424\pi$
$228$	0	0
$229$	483.283	0.139459	0.0697297	−	0.997566i	$-0.477786\pi$
$229$	483.283	0.139459	0.0697297	+	0.997566i	$0.477786\pi$
$230$	0	0
$231$	0	0
$232$	4792.35	1.35618
$233$	−656.670	−0.184635	−0.0923173	−	0.995730i	$-0.529427\pi$
$233$	−656.670	−0.184635	−0.0923173	+	0.995730i	$0.529427\pi$
$234$	0	0
$235$	0	0
$236$	−2082.33	−0.574355
$237$	0	0
$238$	3437.63	0.936253
$239$	2394.73	0.648125	0.324063	−	0.946036i	$-0.394951\pi$
$239$	2394.73	0.648125	0.324063	+	0.946036i	$0.394951\pi$
$240$	0	0
$241$	−1308.22	−0.349668	−0.174834	−	0.984598i	$-0.555939\pi$
$241$	−1308.22	−0.349668	−0.174834	+	0.984598i	$0.555939\pi$
$242$	−278.599	−0.0740042
$243$	0	0
$244$	568.268	0.149097
$245$	0	0
$246$	0	0
$247$	1607.57	0.414118
$248$	−2043.58	−0.523256
$249$	0	0
$250$	0	0
$251$	−5173.00	−1.30086	−0.650432	−	0.759565i	$-0.725412\pi$
$251$	−5173.00	−1.30086	−0.650432	+	0.759565i	$0.725412\pi$
$252$	0	0
$253$	−337.472	−0.0838605
$254$	2109.23	0.521042
$255$	0	0
$256$	−3620.39	−0.883884
$257$	7589.68	1.84215	0.921073	−	0.389390i	$-0.127314\pi$
$257$	7589.68	1.84215	0.921073	+	0.389390i	$0.127314\pi$
$258$	0	0
$259$	15011.7	3.60148
$260$	0	0
$261$	0	0
$262$	−2079.50	−0.490350
$263$	6237.14	1.46235	0.731176	−	0.682189i	$-0.238972\pi$
$263$	6237.14	1.46235	0.731176	+	0.682189i	$0.238972\pi$
$264$	0	0
$265$	0	0
$266$	−3291.44	−0.758689
$267$	0	0
$268$	1221.60	0.278436
$269$	2250.12	0.510009	0.255004	−	0.966940i	$-0.417923\pi$
$269$	2250.12	0.510009	0.255004	+	0.966940i	$0.417923\pi$
$270$	0	0
$271$	−2157.04	−0.483508	−0.241754	−	0.970338i	$-0.577723\pi$
$271$	−2157.04	−0.483508	−0.241754	+	0.970338i	$0.577723\pi$
$272$	−1555.54	−0.346759
$273$	0	0
$274$	−5464.39	−1.20480
$275$	0	0
$276$	0	0
$277$	−6886.28	−1.49371	−0.746853	−	0.664990i	$-0.768436\pi$
$277$	−6886.28	−1.49371	−0.746853	+	0.664990i	$0.768436\pi$
$278$	−291.899	−0.0629747
$279$	0	0
$280$	0	0
$281$	2279.62	0.483953	0.241977	−	0.970282i	$-0.422204\pi$
$281$	2279.62	0.483953	0.241977	+	0.970282i	$0.422204\pi$
$282$	0	0
$283$	5289.42	1.11104	0.555519	−	0.831504i	$-0.312520\pi$
$283$	5289.42	1.11104	0.555519	+	0.831504i	$0.312520\pi$
$284$	836.953	0.174873
$285$	0	0
$286$	960.294	0.198543
$287$	−6951.77	−1.42979
$288$	0	0
$289$	−2952.13	−0.600881
$290$	0	0
$291$	0	0
$292$	−470.508	−0.0942958
$293$	−6130.70	−1.22239	−0.611193	−	0.791482i	$-0.709310\pi$
$293$	−6130.70	−1.22239	−0.611193	+	0.791482i	$0.709310\pi$
$294$	0	0
$295$	0	0
$296$	−10967.6	−2.15365
$297$	0	0
$298$	3335.89	0.648465
$299$	1163.22	0.224987
$300$	0	0
$301$	6649.80	1.27338
$302$	−3537.58	−0.674057
$303$	0	0
$304$	1489.39	0.280995
$305$	0	0
$306$	0	0
$307$	3627.26	0.674328	0.337164	−	0.941446i	$-0.390532\pi$
$307$	3627.26	0.674328	0.337164	+	0.941446i	$0.390532\pi$
$308$	1000.87	0.185162
$309$	0	0
$310$	0	0
$311$	−701.327	−0.127873	−0.0639367	−	0.997954i	$-0.520366\pi$
$311$	−701.327	−0.127873	−0.0639367	+	0.997954i	$0.520366\pi$
$312$	0	0
$313$	−7338.97	−1.32531	−0.662657	−	0.748923i	$-0.730571\pi$
$313$	−7338.97	−1.32531	−0.662657	+	0.748923i	$0.730571\pi$
$314$	6742.41	1.21177
$315$	0	0
$316$	−3214.38	−0.572224
$317$	7376.07	1.30688	0.653440	−	0.756978i	$-0.273325\pi$
$317$	7376.07	1.30688	0.653440	+	0.756978i	$0.273325\pi$
$318$	0	0
$319$	2140.03	0.375607
$320$	0	0
$321$	0	0
$322$	−2381.66	−0.412189
$323$	−1877.48	−0.323424
$324$	0	0
$325$	0	0
$326$	−4753.01	−0.807499
$327$	0	0
$328$	5078.98	0.855000
$329$	−9282.16	−1.55545
$330$	0	0
$331$	−7801.26	−1.29546	−0.647729	−	0.761871i	$-0.724281\pi$
$331$	−7801.26	−1.29546	−0.647729	+	0.761871i	$0.724281\pi$
$332$	−495.657	−0.0819358
$333$	0	0
$334$	−6668.88	−1.09253
$335$	0	0
$336$	0	0
$337$	−2829.95	−0.457439	−0.228720	−	0.973492i	$-0.573454\pi$
$337$	−2829.95	−0.457439	−0.228720	+	0.973492i	$0.573454\pi$
$338$	1748.51	0.281381
$339$	0	0
$340$	0	0
$341$	−912.561	−0.144921
$342$	0	0
$343$	−15199.1	−2.39263
$344$	−4858.36	−0.761469
$345$	0	0
$346$	6764.83	1.05110
$347$	1629.68	0.252120	0.126060	−	0.992023i	$-0.459767\pi$
$347$	1629.68	0.252120	0.126060	+	0.992023i	$0.459767\pi$
$348$	0	0
$349$	−2409.67	−0.369590	−0.184795	−	0.982777i	$-0.559162\pi$
$349$	−2409.67	−0.369590	−0.184795	+	0.982777i	$0.559162\pi$
$350$	0	0
$351$	0	0
$352$	−1278.03	−0.193520
$353$	−720.287	−0.108603	−0.0543017	−	0.998525i	$-0.517293\pi$
$353$	−720.287	−0.108603	−0.0543017	+	0.998525i	$0.517293\pi$
$354$	0	0
$355$	0	0
$356$	1656.18	0.246565
$357$	0	0
$358$	7719.03	1.13956
$359$	3839.51	0.564461	0.282230	−	0.959347i	$-0.408926\pi$
$359$	3839.51	0.564461	0.282230	+	0.959347i	$0.408926\pi$
$360$	0	0
$361$	−5061.36	−0.737914
$362$	9594.84	1.39308
$363$	0	0
$364$	−3449.87	−0.496764
$365$	0	0
$366$	0	0
$367$	−3599.11	−0.511913	−0.255957	−	0.966688i	$-0.582390\pi$
$367$	−3599.11	−0.511913	−0.255957	+	0.966688i	$0.582390\pi$
$368$	1077.71	0.152662
$369$	0	0
$370$	0	0
$371$	−12903.9	−1.80576
$372$	0	0
$373$	2214.60	0.307419	0.153710	−	0.988116i	$-0.450878\pi$
$373$	2214.60	0.307419	0.153710	+	0.988116i	$0.450878\pi$
$374$	−1121.53	−0.155061
$375$	0	0
$376$	6781.58	0.930142
$377$	−7376.40	−1.00770
$378$	0	0
$379$	−13631.2	−1.84747	−0.923733	−	0.383037i	$-0.874878\pi$
$379$	−13631.2	−1.84747	−0.923733	+	0.383037i	$0.874878\pi$
$380$	0	0
$381$	0	0
$382$	−3831.04	−0.513124
$383$	7154.42	0.954501	0.477250	−	0.878767i	$-0.341634\pi$
$383$	7154.42	0.954501	0.477250	+	0.878767i	$0.341634\pi$
$384$	0	0
$385$	0	0
$386$	−7248.75	−0.955833
$387$	0	0
$388$	−3251.23	−0.425403
$389$	−3300.42	−0.430175	−0.215088	−	0.976595i	$-0.569004\pi$
$389$	−3300.42	−0.430175	−0.215088	+	0.976595i	$0.569004\pi$
$390$	0	0
$391$	−1358.53	−0.175713
$392$	19553.7	2.51942
$393$	0	0
$394$	11591.5	1.48216
$395$	0	0
$396$	0	0
$397$	−3855.87	−0.487457	−0.243729	−	0.969843i	$-0.578371\pi$
$397$	−3855.87	−0.487457	−0.243729	+	0.969843i	$0.578371\pi$
$398$	−4418.32	−0.556458
$399$	0	0
$400$	0	0
$401$	9412.25	1.17213	0.586066	−	0.810263i	$-0.300676\pi$
$401$	9412.25	1.17213	0.586066	+	0.810263i	$0.300676\pi$
$402$	0	0
$403$	3145.48	0.388803
$404$	3750.46	0.461862
$405$	0	0
$406$	15102.9	1.84617
$407$	−4897.59	−0.596474
$408$	0	0
$409$	10135.1	1.22530	0.612649	−	0.790355i	$-0.290104\pi$
$409$	10135.1	1.22530	0.612649	+	0.790355i	$0.290104\pi$
$410$	0	0
$411$	0	0
$412$	−2649.92	−0.316874
$413$	−26016.3	−3.09970
$414$	0	0
$415$	0	0
$416$	4405.20	0.519189
$417$	0	0
$418$	1073.84	0.125653
$419$	6764.61	0.788718	0.394359	−	0.918956i	$-0.370967\pi$
$419$	6764.61	0.788718	0.394359	+	0.918956i	$0.370967\pi$
$420$	0	0
$421$	12417.9	1.43756	0.718778	−	0.695240i	$-0.244702\pi$
$421$	12417.9	1.43756	0.718778	+	0.695240i	$0.244702\pi$
$422$	2764.38	0.318882
$423$	0	0
$424$	9427.61	1.07982
$425$	0	0
$426$	0	0
$427$	7099.86	0.804652
$428$	−1993.34	−0.225121
$429$	0	0
$430$	0	0
$431$	−15819.9	−1.76802	−0.884010	−	0.467467i	$-0.845167\pi$
$431$	−15819.9	−1.76802	−0.884010	+	0.467467i	$0.845167\pi$
$432$	0	0
$433$	−15554.9	−1.72637	−0.863185	−	0.504887i	$-0.831534\pi$
$433$	−15554.9	−1.72637	−0.863185	+	0.504887i	$0.831534\pi$
$434$	−6440.26	−0.712310
$435$	0	0
$436$	−1618.87	−0.177821
$437$	1300.76	0.142389
$438$	0	0
$439$	−3128.47	−0.340123	−0.170061	−	0.985433i	$-0.554397\pi$
$439$	−3128.47	−0.340123	−0.170061	+	0.985433i	$0.554397\pi$
$440$	0	0
$441$	0	0
$442$	3865.77	0.416009
$443$	−10581.2	−1.13482	−0.567411	−	0.823435i	$-0.692055\pi$
$443$	−10581.2	−1.13482	−0.567411	+	0.823435i	$0.692055\pi$
$444$	0	0
$445$	0	0
$446$	9108.84	0.967077
$447$	0	0
$448$	−18494.7	−1.95043
$449$	−8549.76	−0.898637	−0.449318	−	0.893372i	$-0.648333\pi$
$449$	−8549.76	−0.898637	−0.449318	+	0.893372i	$0.648333\pi$
$450$	0	0
$451$	2268.02	0.236800
$452$	−2205.26	−0.229484
$453$	0	0
$454$	−13469.1	−1.39237
$455$	0	0
$456$	0	0
$457$	−10920.2	−1.11778	−0.558888	−	0.829243i	$-0.688772\pi$
$457$	−10920.2	−1.11778	−0.558888	+	0.829243i	$0.688772\pi$
$458$	−1112.74	−0.113526
$459$	0	0
$460$	0	0
$461$	−10700.4	−1.08106	−0.540528	−	0.841326i	$-0.681775\pi$
$461$	−10700.4	−1.08106	−0.540528	+	0.841326i	$0.681775\pi$
$462$	0	0
$463$	−713.866	−0.0716548	−0.0358274	−	0.999358i	$-0.511407\pi$
$463$	−713.866	−0.0716548	−0.0358274	+	0.999358i	$0.511407\pi$
$464$	−6834.14	−0.683765
$465$	0	0
$466$	1511.96	0.150301
$467$	−11686.7	−1.15802	−0.579010	−	0.815320i	$-0.696561\pi$
$467$	−11686.7	−1.15802	−0.579010	+	0.815320i	$0.696561\pi$
$468$	0	0
$469$	15262.4	1.50267
$470$	0	0
$471$	0	0
$472$	19007.6	1.85359
$473$	−2169.50	−0.210896
$474$	0	0
$475$	0	0
$476$	4029.11	0.387970
$477$	0	0
$478$	−5513.78	−0.527604
$479$	1443.85	0.137727	0.0688633	−	0.997626i	$-0.478063\pi$
$479$	1443.85	0.137727	0.0688633	+	0.997626i	$0.478063\pi$
$480$	0	0
$481$	16881.4	1.60026
$482$	3012.15	0.284646
$483$	0	0
$484$	−326.535	−0.0306663
$485$	0	0
$486$	0	0
$487$	−7658.37	−0.712595	−0.356297	−	0.934373i	$-0.615961\pi$
$487$	−7658.37	−0.712595	−0.356297	+	0.934373i	$0.615961\pi$
$488$	−5187.18	−0.481174
$489$	0	0
$490$	0	0
$491$	3394.27	0.311978	0.155989	−	0.987759i	$-0.450144\pi$
$491$	3394.27	0.311978	0.155989	+	0.987759i	$0.450144\pi$
$492$	0	0
$493$	8614.91	0.787010
$494$	−3701.38	−0.337111
$495$	0	0
$496$	2914.25	0.263818
$497$	10456.8	0.943763
$498$	0	0
$499$	−11997.3	−1.07630	−0.538149	−	0.842850i	$-0.680876\pi$
$499$	−11997.3	−1.07630	−0.538149	+	0.842850i	$0.680876\pi$
$500$	0	0
$501$	0	0
$502$	11910.7	1.05896
$503$	−11969.4	−1.06101	−0.530506	−	0.847681i	$-0.677998\pi$
$503$	−11969.4	−1.06101	−0.530506	+	0.847681i	$0.677998\pi$
$504$	0	0
$505$	0	0
$506$	777.020	0.0682663
$507$	0	0
$508$	2472.14	0.215913
$509$	−18724.8	−1.63057	−0.815286	−	0.579059i	$-0.803420\pi$
$509$	−18724.8	−1.63057	−0.815286	+	0.579059i	$0.803420\pi$
$510$	0	0
$511$	−5878.45	−0.508899
$512$	11004.0	0.949826
$513$	0	0
$514$	−17475.0	−1.49959
$515$	0	0
$516$	0	0
$517$	3028.32	0.257612
$518$	−34564.0	−2.93177
$519$	0	0
$520$	0	0
$521$	18285.5	1.53762	0.768812	−	0.639475i	$-0.220848\pi$
$521$	18285.5	1.53762	0.768812	+	0.639475i	$0.220848\pi$
$522$	0	0
$523$	19025.4	1.59068	0.795338	−	0.606167i	$-0.207294\pi$
$523$	19025.4	1.59068	0.795338	+	0.606167i	$0.207294\pi$
$524$	−2437.30	−0.203194
$525$	0	0
$526$	−14360.8	−1.19042
$527$	−3673.61	−0.303653
$528$	0	0
$529$	−11225.8	−0.922642
$530$	0	0
$531$	0	0
$532$	−3857.77	−0.314390
$533$	−7817.58	−0.635304
$534$	0	0
$535$	0	0
$536$	−11150.8	−0.898584
$537$	0	0
$538$	−5180.84	−0.415171
$539$	8731.72	0.697777
$540$	0	0
$541$	4886.50	0.388331	0.194166	−	0.980969i	$-0.437800\pi$
$541$	4886.50	0.388331	0.194166	+	0.980969i	$0.437800\pi$
$542$	4966.51	0.393598
$543$	0	0
$544$	−5144.84	−0.405484
$545$	0	0
$546$	0	0
$547$	−12672.6	−0.990570	−0.495285	−	0.868731i	$-0.664936\pi$
$547$	−12672.6	−0.990570	−0.495285	+	0.868731i	$0.664936\pi$
$548$	−6404.60	−0.499254
$549$	0	0
$550$	0	0
$551$	−8248.57	−0.637751
$552$	0	0
$553$	−40159.9	−3.08820
$554$	15855.4	1.21594
$555$	0	0
$556$	−342.124	−0.0260958
$557$	−8274.65	−0.629458	−0.314729	−	0.949182i	$-0.601914\pi$
$557$	−8274.65	−0.629458	−0.314729	+	0.949182i	$0.601914\pi$
$558$	0	0
$559$	7478.00	0.565807
$560$	0	0
$561$	0	0
$562$	−5248.76	−0.393960
$563$	9174.35	0.686772	0.343386	−	0.939194i	$-0.388426\pi$
$563$	9174.35	0.686772	0.343386	+	0.939194i	$0.388426\pi$
$564$	0	0
$565$	0	0
$566$	−12178.7	−0.904435
$567$	0	0
$568$	−7639.75	−0.564360
$569$	4686.05	0.345254	0.172627	−	0.984987i	$-0.444774\pi$
$569$	4686.05	0.345254	0.172627	+	0.984987i	$0.444774\pi$
$570$	0	0
$571$	−19892.5	−1.45792	−0.728962	−	0.684554i	$-0.759997\pi$
$571$	−19892.5	−1.45792	−0.728962	+	0.684554i	$0.759997\pi$
$572$	1125.52	0.0822736
$573$	0	0
$574$	16006.2	1.16391
$575$	0	0
$576$	0	0
$577$	2566.17	0.185149	0.0925746	−	0.995706i	$-0.470490\pi$
$577$	2566.17	0.185149	0.0925746	+	0.995706i	$0.470490\pi$
$578$	6797.19	0.489145
$579$	0	0
$580$	0	0
$581$	−6192.66	−0.442194
$582$	0	0
$583$	4209.90	0.299067
$584$	4294.82	0.304316
$585$	0	0
$586$	14115.7	0.995078
$587$	−19097.4	−1.34282	−0.671409	−	0.741087i	$-0.734310\pi$
$587$	−19097.4	−1.34282	−0.671409	+	0.741087i	$0.734310\pi$
$588$	0	0
$589$	3517.40	0.246064
$590$	0	0
$591$	0	0
$592$	15640.4	1.08584
$593$	598.368	0.0414368	0.0207184	−	0.999785i	$-0.493405\pi$
$593$	598.368	0.0414368	0.0207184	+	0.999785i	$0.493405\pi$
$594$	0	0
$595$	0	0
$596$	3909.86	0.268715
$597$	0	0
$598$	−2678.29	−0.183149
$599$	1927.27	0.131463	0.0657315	−	0.997837i	$-0.479062\pi$
$599$	1927.27	0.131463	0.0657315	+	0.997837i	$0.479062\pi$
$600$	0	0
$601$	11578.8	0.785869	0.392935	−	0.919566i	$-0.371460\pi$
$601$	11578.8	0.785869	0.392935	+	0.919566i	$0.371460\pi$
$602$	−15311.0	−1.03659
$603$	0	0
$604$	−4146.27	−0.279320
$605$	0	0
$606$	0	0
$607$	−1284.90	−0.0859182	−0.0429591	−	0.999077i	$-0.513679\pi$
$607$	−1284.90	−0.0859182	−0.0429591	+	0.999077i	$0.513679\pi$
$608$	4926.06	0.328582
$609$	0	0
$610$	0	0
$611$	−10438.2	−0.691138
$612$	0	0
$613$	−24325.7	−1.60278	−0.801391	−	0.598141i	$-0.795906\pi$
$613$	−24325.7	−1.60278	−0.801391	+	0.598141i	$0.795906\pi$
$614$	−8351.65	−0.548934
$615$	0	0
$616$	−9135.99	−0.597564
$617$	−18016.9	−1.17558	−0.587789	−	0.809014i	$-0.700002\pi$
$617$	−18016.9	−1.17558	−0.587789	+	0.809014i	$0.700002\pi$
$618$	0	0
$619$	−20542.5	−1.33388	−0.666941	−	0.745110i	$-0.732397\pi$
$619$	−20542.5	−1.33388	−0.666941	+	0.745110i	$0.732397\pi$
$620$	0	0
$621$	0	0
$622$	1614.79	0.104095
$623$	20692.0	1.33067
$624$	0	0
$625$	0	0
$626$	16897.8	1.07887
$627$	0	0
$628$	7902.52	0.502141
$629$	−19715.8	−1.24979
$630$	0	0
$631$	5441.98	0.343331	0.171665	−	0.985155i	$-0.445085\pi$
$631$	5441.98	0.343331	0.171665	+	0.985155i	$0.445085\pi$
$632$	29341.0	1.84671
$633$	0	0
$634$	−16983.2	−1.06386
$635$	0	0
$636$	0	0
$637$	−30097.1	−1.87204
$638$	−4927.35	−0.305761
$639$	0	0
$640$	0	0
$641$	4499.67	0.277264	0.138632	−	0.990344i	$-0.455729\pi$
$641$	4499.67	0.277264	0.138632	+	0.990344i	$0.455729\pi$
$642$	0	0
$643$	−2929.30	−0.179659	−0.0898293	−	0.995957i	$-0.528632\pi$
$643$	−2929.30	−0.179659	−0.0898293	+	0.995957i	$0.528632\pi$
$644$	−2791.45	−0.170805
$645$	0	0
$646$	4322.85	0.263282
$647$	20128.9	1.22311	0.611554	−	0.791203i	$-0.290545\pi$
$647$	20128.9	1.22311	0.611554	+	0.791203i	$0.290545\pi$
$648$	0	0
$649$	8487.84	0.513370
$650$	0	0
$651$	0	0
$652$	−5570.81	−0.334616
$653$	−13802.2	−0.827141	−0.413571	−	0.910472i	$-0.635719\pi$
$653$	−13802.2	−0.827141	−0.413571	+	0.910472i	$0.635719\pi$
$654$	0	0
$655$	0	0
$656$	−7242.88	−0.431078
$657$	0	0
$658$	21371.9	1.26621
$659$	29377.3	1.73653	0.868267	−	0.496097i	$-0.165234\pi$
$659$	29377.3	1.73653	0.868267	+	0.496097i	$0.165234\pi$
$660$	0	0
$661$	−22438.5	−1.32036	−0.660179	−	0.751108i	$-0.729520\pi$
$661$	−22438.5	−1.32036	−0.660179	+	0.751108i	$0.729520\pi$
$662$	17962.2	1.05456
$663$	0	0
$664$	4524.38	0.264428
$665$	0	0
$666$	0	0
$667$	−5968.60	−0.346484
$668$	−7816.33	−0.452729
$669$	0	0
$670$	0	0
$671$	−2316.34	−0.133266
$672$	0	0
$673$	31739.0	1.81790	0.908951	−	0.416903i	$-0.136884\pi$
$673$	31739.0	1.81790	0.908951	+	0.416903i	$0.136884\pi$
$674$	6515.87	0.372377
$675$	0	0
$676$	2049.36	0.116600
$677$	−28959.9	−1.64404	−0.822022	−	0.569456i	$-0.807154\pi$
$677$	−28959.9	−1.64404	−0.822022	+	0.569456i	$0.807154\pi$
$678$	0	0
$679$	−40620.4	−2.29583
$680$	0	0
$681$	0	0
$682$	2101.14	0.117972
$683$	20978.4	1.17528	0.587640	−	0.809122i	$-0.300057\pi$
$683$	20978.4	1.17528	0.587640	+	0.809122i	$0.300057\pi$
$684$	0	0
$685$	0	0
$686$	34995.4	1.94771
$687$	0	0
$688$	6928.27	0.383921
$689$	−14511.0	−0.802358
$690$	0	0
$691$	−14713.7	−0.810037	−0.405018	−	0.914309i	$-0.632735\pi$
$691$	−14713.7	−0.810037	−0.405018	+	0.914309i	$0.632735\pi$
$692$	7928.80	0.435560
$693$	0	0
$694$	−3752.29	−0.205238
$695$	0	0
$696$	0	0
$697$	9130.17	0.496169
$698$	5548.20	0.300863
$699$	0	0
$700$	0	0
$701$	−30033.8	−1.61821	−0.809103	−	0.587667i	$-0.800046\pi$
$701$	−30033.8	−1.61821	−0.809103	+	0.587667i	$0.800046\pi$
$702$	0	0
$703$	18877.4	1.01277
$704$	6033.91	0.323028
$705$	0	0
$706$	1658.44	0.0884082
$707$	46857.7	2.49260
$708$	0	0
$709$	6202.27	0.328535	0.164267	−	0.986416i	$-0.447474\pi$
$709$	6202.27	0.328535	0.164267	+	0.986416i	$0.447474\pi$
$710$	0	0
$711$	0	0
$712$	−15117.7	−0.795729
$713$	2545.16	0.133684
$714$	0	0
$715$	0	0
$716$	9047.17	0.472219
$717$	0	0
$718$	−8840.35	−0.459497
$719$	28650.6	1.48608	0.743038	−	0.669250i	$-0.233384\pi$
$719$	28650.6	1.48608	0.743038	+	0.669250i	$0.233384\pi$
$720$	0	0
$721$	−33107.7	−1.71012
$722$	11653.6	0.600696
$723$	0	0
$724$	11245.7	0.577271
$725$	0	0
$726$	0	0
$727$	11144.9	0.568558	0.284279	−	0.958742i	$-0.408246\pi$
$727$	11144.9	0.568558	0.284279	+	0.958742i	$0.408246\pi$
$728$	31490.6	1.60318
$729$	0	0
$730$	0	0
$731$	−8733.58	−0.441892
$732$	0	0
$733$	28546.2	1.43844	0.719221	−	0.694781i	$-0.244499\pi$
$733$	28546.2	1.43844	0.719221	+	0.694781i	$0.244499\pi$
$734$	8286.84	0.416721
$735$	0	0
$736$	3564.46	0.178516
$737$	−4979.39	−0.248871
$738$	0	0
$739$	−19992.4	−0.995175	−0.497587	−	0.867414i	$-0.665781\pi$
$739$	−19992.4	−0.995175	−0.497587	+	0.867414i	$0.665781\pi$
$740$	0	0
$741$	0	0
$742$	29710.8	1.46997
$743$	10763.2	0.531446	0.265723	−	0.964049i	$-0.414389\pi$
$743$	10763.2	0.531446	0.265723	+	0.964049i	$0.414389\pi$
$744$	0	0
$745$	0	0
$746$	−5099.04	−0.250253
$747$	0	0
$748$	−1314.50	−0.0642553
$749$	−24904.6	−1.21494
$750$	0	0
$751$	3243.24	0.157586	0.0787932	−	0.996891i	$-0.474893\pi$
$751$	3243.24	0.157586	0.0787932	+	0.996891i	$0.474893\pi$
$752$	−9670.87	−0.468963
$753$	0	0
$754$	16983.9	0.820316
$755$	0	0
$756$	0	0
$757$	−11503.0	−0.552288	−0.276144	−	0.961116i	$-0.589057\pi$
$757$	−11503.0	−0.552288	−0.276144	+	0.961116i	$0.589057\pi$
$758$	31385.5	1.50392
$759$	0	0
$760$	0	0
$761$	−36993.2	−1.76216	−0.881080	−	0.472968i	$-0.843183\pi$
$761$	−36993.2	−1.76216	−0.881080	+	0.472968i	$0.843183\pi$
$762$	0	0
$763$	−20226.0	−0.959672
$764$	−4490.22	−0.212631
$765$	0	0
$766$	−16472.8	−0.777007
$767$	−29256.5	−1.37730
$768$	0	0
$769$	−21702.1	−1.01768	−0.508841	−	0.860861i	$-0.669926\pi$
$769$	−21702.1	−1.01768	−0.508841	+	0.860861i	$0.669926\pi$
$770$	0	0
$771$	0	0
$772$	−8495.98	−0.396084
$773$	14968.0	0.696459	0.348230	−	0.937409i	$-0.386783\pi$
$773$	14968.0	0.696459	0.348230	+	0.937409i	$0.386783\pi$
$774$	0	0
$775$	0	0
$776$	29677.4	1.37288
$777$	0	0
$778$	7599.13	0.350182
$779$	−8741.91	−0.402069
$780$	0	0
$781$	−3411.53	−0.156305
$782$	3127.98	0.143039
$783$	0	0
$784$	−27884.6	−1.27025
$785$	0	0
$786$	0	0
$787$	43780.7	1.98299	0.991494	−	0.130149i	$-0.0415455\pi$
$787$	43780.7	1.98299	0.991494	+	0.130149i	$0.0415455\pi$
$788$	13586.0	0.614187
$789$	0	0
$790$	0	0
$791$	−27552.2	−1.23849
$792$	0	0
$793$	7984.12	0.357534
$794$	8878.02	0.396813
$795$	0	0
$796$	−5178.54	−0.230588
$797$	−4398.73	−0.195497	−0.0977484	−	0.995211i	$-0.531164\pi$
$797$	−4398.73	−0.195497	−0.0977484	+	0.995211i	$0.531164\pi$
$798$	0	0
$799$	12190.8	0.539775
$800$	0	0
$801$	0	0
$802$	−21671.4	−0.954170
$803$	1917.85	0.0842834
$804$	0	0
$805$	0	0
$806$	−7242.37	−0.316503
$807$	0	0
$808$	−34234.4	−1.49055
$809$	164.162	0.00713427	0.00356714	−	0.999994i	$-0.498865\pi$
$809$	164.162	0.00713427	0.00356714	+	0.999994i	$0.498865\pi$
$810$	0	0
$811$	−35517.8	−1.53785	−0.768926	−	0.639338i	$-0.779209\pi$
$811$	−35517.8	−1.53785	−0.768926	+	0.639338i	$0.779209\pi$
$812$	17701.5	0.765028
$813$	0	0
$814$	11276.6	0.485557
$815$	0	0
$816$	0	0
$817$	8362.18	0.358085
$818$	−23335.7	−0.997449
$819$	0	0
$820$	0	0
$821$	8770.33	0.372822	0.186411	−	0.982472i	$-0.440314\pi$
$821$	8770.33	0.372822	0.186411	+	0.982472i	$0.440314\pi$
$822$	0	0
$823$	−9344.99	−0.395803	−0.197902	−	0.980222i	$-0.563413\pi$
$823$	−9344.99	−0.395803	−0.197902	+	0.980222i	$0.563413\pi$
$824$	24188.6	1.02263
$825$	0	0
$826$	59901.7	2.52330
$827$	−37820.4	−1.59026	−0.795130	−	0.606439i	$-0.792597\pi$
$827$	−37820.4	−1.59026	−0.795130	+	0.606439i	$0.792597\pi$
$828$	0	0
$829$	8605.31	0.360525	0.180262	−	0.983619i	$-0.442305\pi$
$829$	8605.31	0.360525	0.180262	+	0.983619i	$0.442305\pi$
$830$	0	0
$831$	0	0
$832$	−20798.1	−0.866641
$833$	35150.5	1.46206
$834$	0	0
$835$	0	0
$836$	1258.60	0.0520690
$837$	0	0
$838$	−15575.3	−0.642053
$839$	46477.5	1.91249	0.956247	−	0.292560i	$-0.0945071\pi$
$839$	46477.5	1.91249	0.956247	+	0.292560i	$0.0945071\pi$
$840$	0	0
$841$	13459.9	0.551883
$842$	−28591.8	−1.17024
$843$	0	0
$844$	3240.03	0.132140
$845$	0	0
$846$	0	0
$847$	−4079.68	−0.165501
$848$	−13444.2	−0.544431
$849$	0	0
$850$	0	0
$851$	13659.5	0.550226
$852$	0	0
$853$	−3867.32	−0.155234	−0.0776169	−	0.996983i	$-0.524731\pi$
$853$	−3867.32	−0.155234	−0.0776169	+	0.996983i	$0.524731\pi$
$854$	−16347.2	−0.655024
$855$	0	0
$856$	18195.4	0.726524
$857$	37944.2	1.51243	0.756213	−	0.654325i	$-0.227047\pi$
$857$	37944.2	1.51243	0.756213	+	0.654325i	$0.227047\pi$
$858$	0	0
$859$	9438.58	0.374901	0.187451	−	0.982274i	$-0.439978\pi$
$859$	9438.58	0.374901	0.187451	+	0.982274i	$0.439978\pi$
$860$	0	0
$861$	0	0
$862$	36424.8	1.43925
$863$	−2626.80	−0.103612	−0.0518061	−	0.998657i	$-0.516498\pi$
$863$	−2626.80	−0.103612	−0.0518061	+	0.998657i	$0.516498\pi$
$864$	0	0
$865$	0	0
$866$	35814.6	1.40534
$867$	0	0
$868$	−7548.38	−0.295171
$869$	13102.2	0.511465
$870$	0	0
$871$	17163.3	0.667689
$872$	14777.2	0.573874
$873$	0	0
$874$	−2994.96	−0.115911
$875$	0	0
$876$	0	0
$877$	39926.8	1.53732	0.768662	−	0.639655i	$-0.220923\pi$
$877$	39926.8	1.53732	0.768662	+	0.639655i	$0.220923\pi$
$878$	7203.22	0.276876
$879$	0	0
$880$	0	0
$881$	48387.5	1.85041	0.925207	−	0.379462i	$-0.123891\pi$
$881$	48387.5	1.85041	0.925207	+	0.379462i	$0.123891\pi$
$882$	0	0
$883$	24690.2	0.940986	0.470493	−	0.882404i	$-0.344076\pi$
$883$	24690.2	0.940986	0.470493	+	0.882404i	$0.344076\pi$
$884$	4530.92	0.172388
$885$	0	0
$886$	24362.8	0.923797
$887$	−25504.5	−0.965453	−0.482726	−	0.875771i	$-0.660353\pi$
$887$	−25504.5	−0.965453	−0.482726	+	0.875771i	$0.660353\pi$
$888$	0	0
$889$	30886.6	1.16525
$890$	0	0
$891$	0	0
$892$	10676.1	0.400743
$893$	−11672.4	−0.437405
$894$	0	0
$895$	0	0
$896$	11244.9	0.419271
$897$	0	0
$898$	19685.6	0.731532
$899$	−16139.7	−0.598765
$900$	0	0
$901$	16947.4	0.626638
$902$	−5222.05	−0.192766
$903$	0	0
$904$	20129.8	0.740604
$905$	0	0
$906$	0	0
$907$	−4050.02	−0.148268	−0.0741338	−	0.997248i	$-0.523619\pi$
$907$	−4050.02	−0.148268	−0.0741338	+	0.997248i	$0.523619\pi$
$908$	−15786.6	−0.576979
$909$	0	0
$910$	0	0
$911$	10405.9	0.378445	0.189223	−	0.981934i	$-0.439403\pi$
$911$	10405.9	0.378445	0.189223	+	0.981934i	$0.439403\pi$
$912$	0	0
$913$	2020.36	0.0732358
$914$	25143.4	0.909922
$915$	0	0
$916$	−1304.20	−0.0470438
$917$	−30451.2	−1.09661
$918$	0	0
$919$	31719.1	1.13854	0.569268	−	0.822152i	$-0.307227\pi$
$919$	31719.1	1.13854	0.569268	+	0.822152i	$0.307227\pi$
$920$	0	0
$921$	0	0
$922$	24637.3	0.880029
$923$	11759.1	0.419346
$924$	0	0
$925$	0	0
$926$	1643.65	0.0583303
$927$	0	0
$928$	−22603.4	−0.799562
$929$	−36958.3	−1.30523	−0.652617	−	0.757688i	$-0.726329\pi$
$929$	−36958.3	−1.30523	−0.652617	+	0.757688i	$0.726329\pi$
$930$	0	0
$931$	−33655.7	−1.18477
$932$	1772.11	0.0622827
$933$	0	0
$934$	26908.2	0.942681
$935$	0	0
$936$	0	0
$937$	10365.2	0.361383	0.180691	−	0.983540i	$-0.442167\pi$
$937$	10365.2	0.361383	0.180691	+	0.983540i	$0.442167\pi$
$938$	−35141.3	−1.22325
$939$	0	0
$940$	0	0
$941$	10350.4	0.358570	0.179285	−	0.983797i	$-0.442622\pi$
$941$	10350.4	0.358570	0.179285	+	0.983797i	$0.442622\pi$
$942$	0	0
$943$	−6325.58	−0.218440
$944$	−27105.8	−0.934552
$945$	0	0
$946$	4995.22	0.171679
$947$	−24145.1	−0.828521	−0.414260	−	0.910158i	$-0.635960\pi$
$947$	−24145.1	−0.828521	−0.414260	+	0.910158i	$0.635960\pi$
$948$	0	0
$949$	−6610.59	−0.226121
$950$	0	0
$951$	0	0
$952$	−36777.9	−1.25208
$953$	53268.6	1.81064	0.905319	−	0.424732i	$-0.139632\pi$
$953$	53268.6	1.81064	0.905319	+	0.424732i	$0.139632\pi$
$954$	0	0
$955$	0	0
$956$	−6462.49	−0.218632
$957$	0	0
$958$	−3324.41	−0.112116
$959$	−80018.2	−2.69439
$960$	0	0
$961$	−22908.6	−0.768978
$962$	−38868.9	−1.30268
$963$	0	0
$964$	3530.42	0.117953
$965$	0	0
$966$	0	0
$967$	10029.7	0.333541	0.166771	−	0.985996i	$-0.446666\pi$
$967$	10029.7	0.333541	0.166771	+	0.985996i	$0.446666\pi$
$968$	2980.63	0.0989679
$969$	0	0
$970$	0	0
$971$	42152.5	1.39314	0.696569	−	0.717489i	$-0.254709\pi$
$971$	42152.5	1.39314	0.696569	+	0.717489i	$0.254709\pi$
$972$	0	0
$973$	−4274.45	−0.140835
$974$	17633.2	0.580085
$975$	0	0
$976$	7397.18	0.242601
$977$	−40437.3	−1.32416	−0.662079	−	0.749434i	$-0.730326\pi$
$977$	−40437.3	−1.32416	−0.662079	+	0.749434i	$0.730326\pi$
$978$	0	0
$979$	−6750.81	−0.220385
$980$	0	0
$981$	0	0
$982$	−7815.21	−0.253965
$983$	40951.5	1.32874	0.664369	−	0.747405i	$-0.268701\pi$
$983$	40951.5	1.32874	0.664369	+	0.747405i	$0.268701\pi$
$984$	0	0
$985$	0	0
$986$	−19835.6	−0.640663
$987$	0	0
$988$	−4338.24	−0.139694
$989$	6050.81	0.194545
$990$	0	0
$991$	255.910	0.00820306	0.00410153	−	0.999992i	$-0.498694\pi$
$991$	255.910	0.00820306	0.00410153	+	0.999992i	$0.498694\pi$
$992$	9638.67	0.308496
$993$	0	0
$994$	−24076.4	−0.768266
$995$	0	0
$996$	0	0
$997$	−14729.2	−0.467883	−0.233941	−	0.972251i	$-0.575162\pi$
$997$	−14729.2	−0.467883	−0.233941	+	0.972251i	$0.575162\pi$
$998$	27623.4	0.876156
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.4.a.bp.1.3		7
3.2	odd	2		2475.4.a.bt.1.5		7
5.4	even	2		495.4.a.p.1.5	yes	7
15.14	odd	2		495.4.a.o.1.3	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
495.4.a.o.1.3	✓	7	15.14	odd	2
495.4.a.p.1.5	yes	7	5.4	even	2
2475.4.a.bp.1.3		7	1.1	even	1	trivial
2475.4.a.bt.1.5		7	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2475.a (trivial)

\(f(q)\)	\(=\)	\(q-2.30247 q^{2} -2.69863 q^{4} -33.7164 q^{7} +24.6333 q^{8} +O(q^{10})\)	\(q-2.30247 q^{2} -2.69863 q^{4} -33.7164 q^{7} +24.6333 q^{8} +11.0000 q^{11} -37.9156 q^{13} +77.6309 q^{14} -35.1283 q^{16} +44.2817 q^{17} -42.3986 q^{19} -25.3272 q^{22} -30.6793 q^{23} +87.2995 q^{26} +90.9881 q^{28} +194.548 q^{29} -82.9601 q^{31} -116.184 q^{32} -101.957 q^{34} -445.236 q^{37} +97.6216 q^{38} +206.184 q^{41} -197.228 q^{43} -29.6850 q^{44} +70.6382 q^{46} +275.301 q^{47} +793.792 q^{49} +102.320 q^{52} +382.718 q^{53} -830.544 q^{56} -447.941 q^{58} +771.622 q^{59} -210.576 q^{61} +191.013 q^{62} +548.537 q^{64} -452.672 q^{67} -119.500 q^{68} -310.139 q^{71} +174.350 q^{73} +1025.14 q^{74} +114.418 q^{76} -370.880 q^{77} +1191.11 q^{79} -474.732 q^{82} +183.669 q^{83} +454.111 q^{86} +270.966 q^{88} -613.710 q^{89} +1278.38 q^{91} +82.7923 q^{92} -633.873 q^{94} +1204.77 q^{97} -1827.68 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 5 q^{2} + 33 q^{4} - 30 q^{7} - 45 q^{8}+O(q^{10}) \) 7 * q - 5 * q^2 + 33 * q^4 - 30 * q^7 - 45 * q^8	\( 7 q - 5 q^{2} + 33 q^{4} - 30 q^{7} - 45 q^{8} + 77 q^{11} - 38 q^{13} + 20 q^{14} + 309 q^{16} - 12 q^{17} + 226 q^{19} - 55 q^{22} - 334 q^{23} - 372 q^{26} - 812 q^{28} - 258 q^{29} + 336 q^{31} - 485 q^{32} + 78 q^{34} - 466 q^{37} + 494 q^{38} - 258 q^{41} - 308 q^{43} + 363 q^{44} + 98 q^{46} - 546 q^{47} + 735 q^{49} - 512 q^{52} - 110 q^{53} + 20 q^{56} - 1362 q^{58} - 68 q^{59} + 1096 q^{61} - 356 q^{62} + 2761 q^{64} - 2268 q^{67} + 1186 q^{68} - 166 q^{71} - 200 q^{73} - 1710 q^{74} + 3310 q^{76} - 330 q^{77} + 2152 q^{79} + 1006 q^{82} - 370 q^{83} + 106 q^{86} - 495 q^{88} - 252 q^{89} + 2768 q^{91} - 3774 q^{92} + 2218 q^{94} - 3698 q^{97} - 697 q^{98}+O(q^{100}) \) 7 * q - 5 * q^2 + 33 * q^4 - 30 * q^7 - 45 * q^8 + 77 * q^11 - 38 * q^13 + 20 * q^14 + 309 * q^16 - 12 * q^17 + 226 * q^19 - 55 * q^22 - 334 * q^23 - 372 * q^26 - 812 * q^28 - 258 * q^29 + 336 * q^31 - 485 * q^32 + 78 * q^34 - 466 * q^37 + 494 * q^38 - 258 * q^41 - 308 * q^43 + 363 * q^44 + 98 * q^46 - 546 * q^47 + 735 * q^49 - 512 * q^52 - 110 * q^53 + 20 * q^56 - 1362 * q^58 - 68 * q^59 + 1096 * q^61 - 356 * q^62 + 2761 * q^64 - 2268 * q^67 + 1186 * q^68 - 166 * q^71 - 200 * q^73 - 1710 * q^74 + 3310 * q^76 - 330 * q^77 + 2152 * q^79 + 1006 * q^82 - 370 * q^83 + 106 * q^86 - 495 * q^88 - 252 * q^89 + 2768 * q^91 - 3774 * q^92 + 2218 * q^94 - 3698 * q^97 - 697 * q^98

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