LMFDB - Embedded newform 825.4.c.p.199.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,4,Mod(199,825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("825.199");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	825.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:	$48.6765757547$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 54x^{6} + 913x^{4} + 5292x^{2} + 8464 $ x^8 + 54x^6 + 913x^4 + 5292*x^2 + 8464
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 165)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.6
Root			$2.63835i$ of defining polynomial
Character	$\chi$	$=$	825.199
Dual form			825.4.c.p.199.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.63835i q^{2} -3.00000i q^{3} -5.23763 q^{4} +10.9151 q^{6} +20.8444i q^{7} +10.0505i q^{8} -9.00000 q^{9} +O(q^{10})$	\(q+3.63835i q^{2} -3.00000i q^{3} -5.23763 q^{4} +10.9151 q^{6} +20.8444i q^{7} +10.0505i q^{8} -9.00000 q^{9} -11.0000 q^{11} +15.7129i q^{12} -67.4988i q^{13} -75.8394 q^{14} -78.4683 q^{16} -57.8120i q^{17} -32.7452i q^{18} +7.98646 q^{19} +62.5333 q^{21} -40.0219i q^{22} -67.5185i q^{23} +30.1515 q^{24} +245.585 q^{26} +27.0000i q^{27} -109.175i q^{28} +56.1558 q^{29} -127.085 q^{31} -205.091i q^{32} +33.0000i q^{33} +210.341 q^{34} +47.1386 q^{36} -95.4222i q^{37} +29.0576i q^{38} -202.497 q^{39} -485.903 q^{41} +227.518i q^{42} +146.216i q^{43} +57.6139 q^{44} +245.656 q^{46} +164.296i q^{47} +235.405i q^{48} -91.4901 q^{49} -173.436 q^{51} +353.534i q^{52} -431.492i q^{53} -98.2356 q^{54} -209.497 q^{56} -23.9594i q^{57} +204.315i q^{58} +804.178 q^{59} -120.847 q^{61} -462.381i q^{62} -187.600i q^{63} +118.449 q^{64} -120.066 q^{66} -371.469i q^{67} +302.798i q^{68} -202.555 q^{69} +529.835 q^{71} -90.4545i q^{72} -1059.19i q^{73} +347.180 q^{74} -41.8301 q^{76} -229.289i q^{77} -736.754i q^{78} +168.663 q^{79} +81.0000 q^{81} -1767.89i q^{82} +144.130i q^{83} -327.526 q^{84} -531.986 q^{86} -168.468i q^{87} -110.555i q^{88} +1400.20 q^{89} +1406.97 q^{91} +353.637i q^{92} +381.256i q^{93} -597.767 q^{94} -615.274 q^{96} +29.6912i q^{97} -332.873i q^{98} +99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 52 q^{4} + 24 q^{6} - 72 q^{9}+O(q^{10}) $ 8 * q - 52 * q^4 + 24 * q^6 - 72 * q^9	$ 8 q - 52 q^{4} + 24 q^{6} - 72 q^{9} - 88 q^{11} + 104 q^{14} + 132 q^{16} - 272 q^{19} + 204 q^{21} - 288 q^{24} - 640 q^{26} - 104 q^{29} + 984 q^{31} - 488 q^{34} + 468 q^{36} - 12 q^{39} + 536 q^{41} + 572 q^{44} + 736 q^{46} + 992 q^{49} + 444 q^{51} - 216 q^{54} - 1704 q^{56} + 2064 q^{59} + 232 q^{61} + 1836 q^{64} - 264 q^{66} + 384 q^{69} - 1840 q^{71} + 5712 q^{74} + 3144 q^{76} - 2304 q^{79} + 648 q^{81} + 120 q^{84} + 472 q^{86} + 2128 q^{89} + 5560 q^{91} + 2864 q^{94} + 1248 q^{96} + 792 q^{99}+O(q^{100}) $ 8 * q - 52 * q^4 + 24 * q^6 - 72 * q^9 - 88 * q^11 + 104 * q^14 + 132 * q^16 - 272 * q^19 + 204 * q^21 - 288 * q^24 - 640 * q^26 - 104 * q^29 + 984 * q^31 - 488 * q^34 + 468 * q^36 - 12 * q^39 + 536 * q^41 + 572 * q^44 + 736 * q^46 + 992 * q^49 + 444 * q^51 - 216 * q^54 - 1704 * q^56 + 2064 * q^59 + 232 * q^61 + 1836 * q^64 - 264 * q^66 + 384 * q^69 - 1840 * q^71 + 5712 * q^74 + 3144 * q^76 - 2304 * q^79 + 648 * q^81 + 120 * q^84 + 472 * q^86 + 2128 * q^89 + 5560 * q^91 + 2864 * q^94 + 1248 * q^96 + 792 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/825\mathbb{Z}\right)^\times$.

$n$	$376$	$551$	$727$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	3.63835i	1.28635i	0.765718	+	0.643176i	$0.222384\pi$
$2$	3.63835i	1.28635i	−0.765718	+	0.643176i	$0.777616\pi$
$3$	− 3.00000i	− 0.577350i
$3$	− 3.00000i	− 0.577350i
$4$	−5.23763	−0.654703
$5$	0	0
$5$	0	0
$6$	10.9151	0.742676
$7$	20.8444i	1.12549i	0.826629	+	0.562747i	$0.190255\pi$
$7$	20.8444i	1.12549i	−0.826629	+	0.562747i	$0.809745\pi$
$8$	10.0505i	0.444173i
$9$	−9.00000	−0.333333
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	15.7129i	0.377993i
$13$	− 67.4988i	− 1.44006i	−0.693942	−	0.720031i	$-0.744128\pi$
$13$	− 67.4988i	− 1.44006i	0.693942	−	0.720031i	$-0.255872\pi$
$14$	−75.8394	−1.44778
$15$	0	0
$16$	−78.4683	−1.22607
$17$	− 57.8120i	− 0.824792i	−0.911005	−	0.412396i	$-0.864692\pi$
$17$	− 57.8120i	− 0.824792i	0.911005	−	0.412396i	$-0.135308\pi$
$18$	− 32.7452i	− 0.428784i
$19$	7.98646	0.0964326	0.0482163	−	0.998837i	$-0.484646\pi$
$19$	7.98646	0.0964326	0.0482163	+	0.998837i	$0.484646\pi$
$20$	0	0
$21$	62.5333	0.649804
$22$	− 40.0219i	− 0.387850i
$23$	− 67.5185i	− 0.612112i	−0.952013	−	0.306056i	$-0.900990\pi$
$23$	− 67.5185i	− 0.612112i	0.952013	−	0.306056i	$-0.0990095\pi$
$24$	30.1515	0.256444
$25$	0	0
$26$	245.585	1.85243
$27$	27.0000i	0.192450i
$28$	− 109.175i	− 0.736864i
$29$	56.1558	0.359582	0.179791	−	0.983705i	$-0.442458\pi$
$29$	56.1558	0.359582	0.179791	+	0.983705i	$0.442458\pi$
$30$	0	0
$31$	−127.085	−0.736296	−0.368148	−	0.929767i	$-0.620008\pi$
$31$	−127.085	−0.736296	−0.368148	+	0.929767i	$0.620008\pi$
$32$	− 205.091i	− 1.13298i
$33$	33.0000i	0.174078i
$34$	210.341	1.06097
$35$	0	0
$36$	47.1386	0.218234
$37$	− 95.4222i	− 0.423981i	−0.977272	−	0.211991i	$-0.932005\pi$
$37$	− 95.4222i	− 0.423981i	0.977272	−	0.211991i	$-0.0679946\pi$
$38$	29.0576i	0.124046i
$39$	−202.497	−0.831420
$40$	0	0
$41$	−485.903	−1.85086	−0.925431	−	0.378917i	$-0.876297\pi$
$41$	−485.903	−1.85086	−0.925431	+	0.378917i	$0.876297\pi$
$42$	227.518i	0.835877i
$43$	146.216i	0.518552i	0.965803	+	0.259276i	$0.0834840\pi$
$43$	146.216i	0.518552i	−0.965803	+	0.259276i	$0.916516\pi$
$44$	57.6139	0.197400
$45$	0	0
$46$	245.656	0.787392
$47$	164.296i	0.509894i	0.966955	+	0.254947i	$0.0820580\pi$
$47$	164.296i	0.509894i	−0.966955	+	0.254947i	$0.917942\pi$
$48$	235.405i	0.707870i
$49$	−91.4901	−0.266735
$50$	0	0
$51$	−173.436	−0.476194
$52$	353.534i	0.942813i
$53$	− 431.492i	− 1.11830i	−0.829066	−	0.559151i	$-0.811127\pi$
$53$	− 431.492i	− 1.11830i	0.829066	−	0.559151i	$-0.188873\pi$
$54$	−98.2356	−0.247559
$55$	0	0
$56$	−209.497	−0.499914
$57$	− 23.9594i	− 0.0556754i
$58$	204.315i	0.462549i
$59$	804.178	1.77449	0.887246	−	0.461296i	$-0.152615\pi$
$59$	804.178	1.77449	0.887246	+	0.461296i	$0.152615\pi$
$60$	0	0
$61$	−120.847	−0.253653	−0.126826	−	0.991925i	$-0.540479\pi$
$61$	−120.847	−0.253653	−0.126826	+	0.991925i	$0.540479\pi$
$62$	− 462.381i	− 0.947137i
$63$	− 187.600i	− 0.375164i
$64$	118.449	0.231346
$65$	0	0
$66$	−120.066	−0.223925
$67$	− 371.469i	− 0.677346i	−0.940904	−	0.338673i	$-0.890022\pi$
$67$	− 371.469i	− 0.677346i	0.940904	−	0.338673i	$-0.109978\pi$
$68$	302.798i	0.539994i
$69$	−202.555	−0.353403
$70$	0	0
$71$	529.835	0.885631	0.442816	−	0.896613i	$-0.353980\pi$
$71$	529.835	0.885631	0.442816	+	0.896613i	$0.353980\pi$
$72$	− 90.4545i	− 0.148058i
$73$	− 1059.19i	− 1.69820i	−0.528232	−	0.849100i	$-0.677145\pi$
$73$	− 1059.19i	− 1.69820i	0.528232	−	0.849100i	$-0.322855\pi$
$74$	347.180	0.545389
$75$	0	0
$76$	−41.8301	−0.0631348
$77$	− 229.289i	− 0.339349i
$78$	− 736.754i	− 1.06950i
$79$	168.663	0.240204	0.120102	−	0.992762i	$-0.461678\pi$
$79$	168.663	0.240204	0.120102	+	0.992762i	$0.461678\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	− 1767.89i	− 2.38086i
$83$	144.130i	0.190606i	0.995448	+	0.0953032i	$0.0303820\pi$
$83$	144.130i	0.190606i	−0.995448	+	0.0953032i	$0.969618\pi$
$84$	−327.526	−0.425429
$85$	0	0
$86$	−531.986	−0.667041
$87$	− 168.468i	− 0.207605i
$88$	− 110.555i	− 0.133923i
$89$	1400.20	1.66765	0.833823	−	0.552032i	$-0.186147\pi$
$89$	1400.20	1.66765	0.833823	+	0.552032i	$0.186147\pi$
$90$	0	0
$91$	1406.97	1.62078
$92$	353.637i	0.400752i
$93$	381.256i	0.425101i
$94$	−597.767	−0.655904
$95$	0	0
$96$	−615.274	−0.654127
$97$	29.6912i	0.0310792i	0.999879	+	0.0155396i	$0.00494661\pi$
$97$	29.6912i	0.0310792i	−0.999879	+	0.0155396i	$0.995053\pi$
$98$	− 332.873i	− 0.343115i
$99$	99.0000	0.100504
$100$	0	0
$101$	−17.5982	−0.0173375	−0.00866874	−	0.999962i	$-0.502759\pi$
$101$	−17.5982	−0.0173375	−0.00866874	+	0.999962i	$0.502759\pi$
$102$	− 631.022i	− 0.612553i
$103$	− 1423.03i	− 1.36131i	−0.732604	−	0.680655i	$-0.761695\pi$
$103$	− 1423.03i	− 1.36131i	0.732604	−	0.680655i	$-0.238305\pi$
$104$	678.397	0.639637
$105$	0	0
$106$	1569.92	1.43853
$107$	− 1335.15i	− 1.20629i	−0.797630	−	0.603147i	$-0.793913\pi$
$107$	− 1335.15i	− 1.20629i	0.797630	−	0.603147i	$-0.206087\pi$
$108$	− 141.416i	− 0.125998i
$109$	−1565.39	−1.37557	−0.687784	−	0.725916i	$-0.741416\pi$
$109$	−1565.39	−1.37557	−0.687784	+	0.725916i	$0.741416\pi$
$110$	0	0
$111$	−286.266	−0.244786
$112$	− 1635.63i	− 1.37993i
$113$	− 1176.83i	− 0.979709i	−0.871804	−	0.489854i	$-0.837050\pi$
$113$	− 1176.83i	− 0.979709i	0.871804	−	0.489854i	$-0.162950\pi$
$114$	87.1728	0.0716182
$115$	0	0
$116$	−294.123	−0.235420
$117$	607.490i	0.480021i
$118$	2925.89i	2.28262i
$119$	1205.06	0.928298
$120$	0	0
$121$	121.000	0.0909091
$122$	− 439.683i	− 0.326287i
$123$	1457.71i	1.06860i
$124$	665.625	0.482056
$125$	0	0
$126$	682.555	0.482594
$127$	311.155i	0.217406i	0.994074	+	0.108703i	$0.0346697\pi$
$127$	311.155i	0.217406i	−0.994074	+	0.108703i	$0.965330\pi$
$128$	− 1209.77i	− 0.835388i
$129$	438.648	0.299386
$130$	0	0
$131$	582.818	0.388711	0.194355	−	0.980931i	$-0.437739\pi$
$131$	582.818	0.388711	0.194355	+	0.980931i	$0.437739\pi$
$132$	− 172.842i	− 0.113969i
$133$	166.473i	0.108534i
$134$	1351.54	0.871306
$135$	0	0
$136$	581.039	0.366351
$137$	− 2367.98i	− 1.47672i	−0.674410	−	0.738358i	$-0.735602\pi$
$137$	− 2367.98i	− 1.47672i	0.674410	−	0.738358i	$-0.264398\pi$
$138$	− 736.969i	− 0.454601i
$139$	−2573.48	−1.57036	−0.785179	−	0.619269i	$-0.787429\pi$
$139$	−2573.48	−1.57036	−0.785179	+	0.619269i	$0.787429\pi$
$140$	0	0
$141$	492.888	0.294388
$142$	1927.73i	1.13923i
$143$	742.487i	0.434195i
$144$	706.215	0.408689
$145$	0	0
$146$	3853.70	2.18448
$147$	274.470i	0.153999i
$148$	499.786i	0.277582i
$149$	−1992.38	−1.09545	−0.547724	−	0.836659i	$-0.684506\pi$
$149$	−1992.38	−1.09545	−0.547724	+	0.836659i	$0.684506\pi$
$150$	0	0
$151$	2961.20	1.59589	0.797944	−	0.602731i	$-0.205921\pi$
$151$	2961.20	1.59589	0.797944	+	0.602731i	$0.205921\pi$
$152$	80.2679i	0.0428328i
$153$	520.308i	0.274931i
$154$	834.234	0.436522
$155$	0	0
$156$	1060.60	0.544334
$157$	1302.56i	0.662139i	0.943606	+	0.331070i	$0.107409\pi$
$157$	1302.56i	0.662139i	−0.943606	+	0.331070i	$0.892591\pi$
$158$	613.657i	0.308987i
$159$	−1294.48	−0.645652
$160$	0	0
$161$	1407.38	0.688928
$162$	294.707i	0.142928i
$163$	563.163i	0.270615i	0.990804	+	0.135308i	$0.0432023\pi$
$163$	563.163i	0.270615i	−0.990804	+	0.135308i	$0.956798\pi$
$164$	2544.98	1.21176
$165$	0	0
$166$	−524.396	−0.245187
$167$	1128.96i	0.523124i	0.965187	+	0.261562i	$0.0842376\pi$
$167$	1128.96i	0.523124i	−0.965187	+	0.261562i	$0.915762\pi$
$168$	628.490i	0.288626i
$169$	−2359.09	−1.07378
$170$	0	0
$171$	−71.8782	−0.0321442
$172$	− 765.825i	− 0.339498i
$173$	− 3091.97i	− 1.35883i	−0.733754	−	0.679416i	$-0.762233\pi$
$173$	− 3091.97i	− 1.35883i	0.733754	−	0.679416i	$-0.237767\pi$
$174$	612.945	0.267053
$175$	0	0
$176$	863.151	0.369673
$177$	− 2412.53i	− 1.02450i
$178$	5094.41i	2.14518i
$179$	−2879.27	−1.20227	−0.601136	−	0.799147i	$-0.705285\pi$
$179$	−2879.27	−1.20227	−0.601136	+	0.799147i	$0.705285\pi$
$180$	0	0
$181$	−702.137	−0.288339	−0.144170	−	0.989553i	$-0.546051\pi$
$181$	−702.137	−0.288339	−0.144170	+	0.989553i	$0.546051\pi$
$182$	5119.07i	2.08489i
$183$	362.540i	0.146447i
$184$	678.594	0.271884
$185$	0	0
$186$	−1387.14	−0.546830
$187$	635.932i	0.248684i
$188$	− 860.521i	− 0.333829i
$189$	−562.799	−0.216601
$190$	0	0
$191$	4294.28	1.62682	0.813412	−	0.581688i	$-0.197608\pi$
$191$	4294.28	1.62682	0.813412	+	0.581688i	$0.197608\pi$
$192$	− 355.348i	− 0.133568i
$193$	3888.11i	1.45012i	0.688687	+	0.725059i	$0.258187\pi$
$193$	3888.11i	1.45012i	−0.688687	+	0.725059i	$0.741813\pi$
$194$	−108.027	−0.0399788
$195$	0	0
$196$	479.191	0.174632
$197$	− 3062.26i	− 1.10750i	−0.832684	−	0.553748i	$-0.813197\pi$
$197$	− 3062.26i	− 1.10750i	0.832684	−	0.553748i	$-0.186803\pi$
$198$	360.197i	0.129283i
$199$	−4874.64	−1.73645	−0.868226	−	0.496168i	$-0.834740\pi$
$199$	−4874.64	−1.73645	−0.868226	+	0.496168i	$0.834740\pi$
$200$	0	0
$201$	−1114.41	−0.391066
$202$	− 64.0285i	− 0.0223021i
$203$	1170.54i	0.404707i
$204$	908.393	0.311766
$205$	0	0
$206$	5177.47	1.75112
$207$	607.666i	0.204037i
$208$	5296.52i	1.76561i
$209$	−87.8511	−0.0290755
$210$	0	0
$211$	−4324.02	−1.41080	−0.705398	−	0.708811i	$-0.749232\pi$
$211$	−4324.02	−1.41080	−0.705398	+	0.708811i	$0.749232\pi$
$212$	2259.99i	0.732156i
$213$	− 1589.50i	− 0.511319i
$214$	4857.74	1.55172
$215$	0	0
$216$	−271.363	−0.0854812
$217$	− 2649.02i	− 0.828697i
$218$	− 5695.43i	− 1.76947i
$219$	−3177.56	−0.980456
$220$	0	0
$221$	−3902.24	−1.18775
$222$	− 1041.54i	− 0.314881i
$223$	6205.69i	1.86351i	0.363084	+	0.931757i	$0.381724\pi$
$223$	6205.69i	1.86351i	−0.363084	+	0.931757i	$0.618276\pi$
$224$	4275.01	1.27516
$225$	0	0
$226$	4281.73	1.26025
$227$	1535.81i	0.449055i	0.974468	+	0.224528i	$0.0720839\pi$
$227$	1535.81i	0.449055i	−0.974468	+	0.224528i	$0.927916\pi$
$228$	125.490i	0.0364509i
$229$	−94.7362	−0.0273378	−0.0136689	−	0.999907i	$-0.504351\pi$
$229$	−94.7362	−0.0273378	−0.0136689	+	0.999907i	$0.504351\pi$
$230$	0	0
$231$	−687.866	−0.195923
$232$	564.394i	0.159717i
$233$	− 654.983i	− 0.184160i	−0.995752	−	0.0920801i	$-0.970648\pi$
$233$	− 654.983i	− 0.184160i	0.995752	−	0.0920801i	$-0.0293516\pi$
$234$	−2210.26	−0.617476
$235$	0	0
$236$	−4211.98	−1.16177
$237$	− 505.990i	− 0.138682i
$238$	4384.43i	1.19412i
$239$	5660.64	1.53204	0.766018	−	0.642819i	$-0.222235\pi$
$239$	5660.64	1.53204	0.766018	+	0.642819i	$0.222235\pi$
$240$	0	0
$241$	−4156.88	−1.11107	−0.555536	−	0.831493i	$-0.687487\pi$
$241$	−4156.88	−1.11107	−0.555536	+	0.831493i	$0.687487\pi$
$242$	440.241i	0.116941i
$243$	− 243.000i	− 0.0641500i
$244$	632.950	0.166067
$245$	0	0
$246$	−5303.66	−1.37459
$247$	− 539.077i	− 0.138869i
$248$	− 1277.27i	− 0.327043i
$249$	432.390	0.110047
$250$	0	0
$251$	−7156.50	−1.79966	−0.899829	−	0.436243i	$-0.856309\pi$
$251$	−7156.50	−1.79966	−0.899829	+	0.436243i	$0.856309\pi$
$252$	982.578i	0.245621i
$253$	742.703i	0.184559i
$254$	−1132.09	−0.279660
$255$	0	0
$256$	5349.17	1.30595
$257$	− 2188.36i	− 0.531152i	−0.964090	−	0.265576i	$-0.914438\pi$
$257$	− 2188.36i	− 0.531152i	0.964090	−	0.265576i	$-0.0855621\pi$
$258$	1595.96i	0.385116i
$259$	1989.02	0.477188
$260$	0	0
$261$	−505.403	−0.119861
$262$	2120.50i	0.500019i
$263$	− 757.388i	− 0.177576i	−0.996051	−	0.0887881i	$-0.971701\pi$
$263$	− 757.388i	− 0.177576i	0.996051	−	0.0887881i	$-0.0282994\pi$
$264$	−331.666	−0.0773207
$265$	0	0
$266$	−605.689	−0.139613
$267$	− 4200.59i	− 0.962816i
$268$	1945.62i	0.443461i
$269$	6782.64	1.53734	0.768671	−	0.639645i	$-0.220919\pi$
$269$	6782.64	1.53734	0.768671	+	0.639645i	$0.220919\pi$
$270$	0	0
$271$	7040.87	1.57824	0.789119	−	0.614241i	$-0.210538\pi$
$271$	7040.87	1.57824	0.789119	+	0.614241i	$0.210538\pi$
$272$	4536.41i	1.01125i
$273$	− 4220.92i	− 0.935758i
$274$	8615.54	1.89958
$275$	0	0
$276$	1060.91	0.231374
$277$	− 3211.46i	− 0.696599i	−0.937383	−	0.348300i	$-0.886759\pi$
$277$	− 3211.46i	− 0.696599i	0.937383	−	0.348300i	$-0.113241\pi$
$278$	− 9363.24i	− 2.02003i
$279$	1143.77	0.245432
$280$	0	0
$281$	3986.15	0.846241	0.423120	−	0.906073i	$-0.360935\pi$
$281$	3986.15	0.846241	0.423120	+	0.906073i	$0.360935\pi$
$282$	1793.30i	0.378686i
$283$	− 7838.75i	− 1.64652i	−0.567664	−	0.823260i	$-0.692153\pi$
$283$	− 7838.75i	− 1.64652i	0.567664	−	0.823260i	$-0.307847\pi$
$284$	−2775.08	−0.579826
$285$	0	0
$286$	−2701.43	−0.558528
$287$	− 10128.4i	− 2.08313i
$288$	1845.82i	0.377660i
$289$	1570.77	0.319718
$290$	0	0
$291$	89.0735	0.0179436
$292$	5547.63i	1.11182i
$293$	5430.65i	1.08280i	0.840764	+	0.541402i	$0.182106\pi$
$293$	5430.65i	1.08280i	−0.840764	+	0.541402i	$0.817894\pi$
$294$	−998.620	−0.198098
$295$	0	0
$296$	959.040	0.188321
$297$	− 297.000i	− 0.0580259i
$298$	− 7248.97i	− 1.40913i
$299$	−4557.42	−0.881480
$300$	0	0
$301$	−3047.79	−0.583627
$302$	10773.9i	2.05288i
$303$	52.7946i	0.0100098i
$304$	−626.684	−0.118233
$305$	0	0
$306$	−1893.06	−0.353658
$307$	8045.25i	1.49566i	0.663892	+	0.747828i	$0.268903\pi$
$307$	8045.25i	1.49566i	−0.663892	+	0.747828i	$0.731097\pi$
$308$	1200.93i	0.222173i
$309$	−4269.08	−0.785952
$310$	0	0
$311$	−4712.06	−0.859152	−0.429576	−	0.903031i	$-0.641337\pi$
$311$	−4712.06	−0.859152	−0.429576	+	0.903031i	$0.641337\pi$
$312$	− 2035.19i	− 0.369295i
$313$	1425.19i	0.257369i	0.991686	+	0.128684i	$0.0410754\pi$
$313$	1425.19i	0.257369i	−0.991686	+	0.128684i	$0.958925\pi$
$314$	−4739.19	−0.851744
$315$	0	0
$316$	−883.395	−0.157262
$317$	− 2031.54i	− 0.359944i	−0.983672	−	0.179972i	$-0.942399\pi$
$317$	− 2031.54i	− 0.359944i	0.983672	−	0.179972i	$-0.0576008\pi$
$318$	− 4709.76i	− 0.830536i
$319$	−617.714	−0.108418
$320$	0	0
$321$	−4005.44	−0.696454
$322$	5120.56i	0.886204i
$323$	− 461.713i	− 0.0795369i
$324$	−424.248	−0.0727448
$325$	0	0
$326$	−2048.99	−0.348107
$327$	4696.16i	0.794184i
$328$	− 4883.57i	− 0.822103i
$329$	−3424.65	−0.573882
$330$	0	0
$331$	316.790	0.0526054	0.0263027	−	0.999654i	$-0.491627\pi$
$331$	316.790	0.0526054	0.0263027	+	0.999654i	$0.491627\pi$
$332$	− 754.899i	− 0.124791i
$333$	858.799i	0.141327i
$334$	−4107.57	−0.672922
$335$	0	0
$336$	−4906.88	−0.796703
$337$	4441.65i	0.717958i	0.933346	+	0.358979i	$0.116875\pi$
$337$	4441.65i	0.717958i	−0.933346	+	0.358979i	$0.883125\pi$
$338$	− 8583.22i	− 1.38126i
$339$	−3530.50	−0.565635
$340$	0	0
$341$	1397.94	0.222002
$342$	− 261.518i	− 0.0413488i
$343$	5242.58i	0.825285i
$344$	−1469.54	−0.230327
$345$	0	0
$346$	11249.7	1.74794
$347$	258.695i	0.0400215i	0.999800	+	0.0200108i	$0.00637005\pi$
$347$	258.695i	0.0400215i	−0.999800	+	0.0200108i	$0.993630\pi$
$348$	882.370i	0.135920i
$349$	−9929.62	−1.52298	−0.761491	−	0.648176i	$-0.775532\pi$
$349$	−9929.62	−1.52298	−0.761491	+	0.648176i	$0.775532\pi$
$350$	0	0
$351$	1822.47	0.277140
$352$	2256.01i	0.341607i
$353$	1008.73i	0.152095i	0.997104	+	0.0760474i	$0.0242300\pi$
$353$	1008.73i	0.152095i	−0.997104	+	0.0760474i	$0.975770\pi$
$354$	8777.66	1.31787
$355$	0	0
$356$	−7333.70	−1.09181
$357$	− 3615.17i	− 0.535953i
$358$	− 10475.8i	− 1.54655i
$359$	−5813.26	−0.854630	−0.427315	−	0.904103i	$-0.640540\pi$
$359$	−5813.26	−0.854630	−0.427315	+	0.904103i	$0.640540\pi$
$360$	0	0
$361$	−6795.22	−0.990701
$362$	− 2554.62i	− 0.370906i
$363$	− 363.000i	− 0.0524864i
$364$	−7369.21	−1.06113
$365$	0	0
$366$	−1319.05	−0.188382
$367$	− 10247.0i	− 1.45747i	−0.684798	−	0.728733i	$-0.740110\pi$
$367$	− 10247.0i	− 1.45747i	0.684798	−	0.728733i	$-0.259890\pi$
$368$	5298.06i	0.750490i
$369$	4373.13	0.616954
$370$	0	0
$371$	8994.20	1.25864
$372$	− 1996.88i	− 0.278315i
$373$	2202.63i	0.305759i	0.988245	+	0.152879i	$0.0488546\pi$
$373$	2202.63i	0.305759i	−0.988245	+	0.152879i	$0.951145\pi$
$374$	−2313.75	−0.319896
$375$	0	0
$376$	−1651.26	−0.226481
$377$	− 3790.45i	− 0.517821i
$378$	− 2047.66i	− 0.278626i
$379$	1851.13	0.250887	0.125444	−	0.992101i	$-0.459965\pi$
$379$	1851.13	0.250887	0.125444	+	0.992101i	$0.459965\pi$
$380$	0	0
$381$	933.464	0.125519
$382$	15624.1i	2.09267i
$383$	− 5880.65i	− 0.784562i	−0.919846	−	0.392281i	$-0.871686\pi$
$383$	− 5880.65i	− 0.784562i	0.919846	−	0.392281i	$-0.128314\pi$
$384$	−3629.31	−0.482311
$385$	0	0
$386$	−14146.3	−1.86536
$387$	− 1315.95i	− 0.172851i
$388$	− 155.511i	− 0.0203476i
$389$	6963.66	0.907639	0.453820	−	0.891094i	$-0.350061\pi$
$389$	6963.66	0.907639	0.453820	+	0.891094i	$0.350061\pi$
$390$	0	0
$391$	−3903.38	−0.504865
$392$	− 919.520i	− 0.118477i
$393$	− 1748.46i	− 0.224422i
$394$	11141.6	1.42463
$395$	0	0
$396$	−518.525	−0.0658002
$397$	3024.31i	0.382332i	0.981558	+	0.191166i	$0.0612269\pi$
$397$	3024.31i	0.382332i	−0.981558	+	0.191166i	$0.938773\pi$
$398$	− 17735.7i	− 2.23369i
$399$	499.420	0.0626623
$400$	0	0
$401$	−13392.7	−1.66783	−0.833914	−	0.551895i	$-0.813905\pi$
$401$	−13392.7	−1.66783	−0.833914	+	0.551895i	$0.813905\pi$
$402$	− 4054.61i	− 0.503049i
$403$	8578.11i	1.06031i
$404$	92.1728	0.0113509
$405$	0	0
$406$	−4258.83	−0.520596
$407$	1049.64i	0.127835i
$408$	− 1743.12i	− 0.211513i
$409$	−7637.16	−0.923308	−0.461654	−	0.887060i	$-0.652744\pi$
$409$	−7637.16	−0.923308	−0.461654	+	0.887060i	$0.652744\pi$
$410$	0	0
$411$	−7103.93	−0.852582
$412$	7453.28i	0.891254i
$413$	16762.6i	1.99718i
$414$	−2210.91	−0.262464
$415$	0	0
$416$	−13843.4	−1.63156
$417$	7720.44i	0.906647i
$418$	− 319.633i	− 0.0374014i
$419$	−12523.9	−1.46022	−0.730112	−	0.683328i	$-0.760532\pi$
$419$	−12523.9	−1.46022	−0.730112	+	0.683328i	$0.760532\pi$
$420$	0	0
$421$	−11150.0	−1.29078	−0.645391	−	0.763852i	$-0.723306\pi$
$421$	−11150.0	−1.29078	−0.645391	+	0.763852i	$0.723306\pi$
$422$	− 15732.3i	− 1.81478i
$423$	− 1478.66i	− 0.169965i
$424$	4336.71	0.496720
$425$	0	0
$426$	5783.18	0.657737
$427$	− 2518.98i	− 0.285485i
$428$	6993.00i	0.789765i
$429$	2227.46	0.250683
$430$	0	0
$431$	−2093.77	−0.233998	−0.116999	−	0.993132i	$-0.537327\pi$
$431$	−2093.77	−0.233998	−0.116999	+	0.993132i	$0.537327\pi$
$432$	− 2118.64i	− 0.235957i
$433$	4391.47i	0.487392i	0.969852	+	0.243696i	$0.0783599\pi$
$433$	4391.47i	0.487392i	−0.969852	+	0.243696i	$0.921640\pi$
$434$	9638.07	1.06600
$435$	0	0
$436$	8198.91	0.900589
$437$	− 539.234i	− 0.0590276i
$438$	− 11561.1i	− 1.26121i
$439$	−15614.6	−1.69759	−0.848797	−	0.528719i	$-0.822672\pi$
$439$	−15614.6	−1.69759	−0.848797	+	0.528719i	$0.822672\pi$
$440$	0	0
$441$	823.411	0.0889116
$442$	− 14197.7i	− 1.52787i
$443$	− 7909.57i	− 0.848296i	−0.905593	−	0.424148i	$-0.860574\pi$
$443$	− 7909.57i	− 0.848296i	0.905593	−	0.424148i	$-0.139426\pi$
$444$	1499.36	0.160262
$445$	0	0
$446$	−22578.5	−2.39714
$447$	5977.13i	0.632458i
$448$	2469.01i	0.260379i
$449$	14958.7	1.57226	0.786128	−	0.618064i	$-0.212083\pi$
$449$	14958.7	1.57226	0.786128	+	0.618064i	$0.212083\pi$
$450$	0	0
$451$	5344.93	0.558056
$452$	6163.81i	0.641419i
$453$	− 8883.61i	− 0.921387i
$454$	−5587.84	−0.577644
$455$	0	0
$456$	240.804	0.0247295
$457$	14584.6i	1.49286i	0.665464	+	0.746430i	$0.268234\pi$
$457$	14584.6i	1.49286i	−0.665464	+	0.746430i	$0.731766\pi$
$458$	− 344.684i	− 0.0351660i
$459$	1560.92	0.158731
$460$	0	0
$461$	−1802.35	−0.182091	−0.0910454	−	0.995847i	$-0.529021\pi$
$461$	−1802.35	−0.182091	−0.0910454	+	0.995847i	$0.529021\pi$
$462$	− 2502.70i	− 0.252026i
$463$	− 6753.79i	− 0.677916i	−0.940801	−	0.338958i	$-0.889926\pi$
$463$	− 6753.79i	− 0.677916i	0.940801	−	0.338958i	$-0.110074\pi$
$464$	−4406.45	−0.440872
$465$	0	0
$466$	2383.06	0.236895
$467$	− 1744.55i	− 0.172866i	−0.996258	−	0.0864329i	$-0.972453\pi$
$467$	− 1744.55i	− 0.172866i	0.996258	−	0.0864329i	$-0.0275468\pi$
$468$	− 3181.80i	− 0.314271i
$469$	7743.06	0.762349
$470$	0	0
$471$	3907.69	0.382286
$472$	8082.39i	0.788182i
$473$	− 1608.38i	− 0.156349i
$474$	1840.97	0.178394
$475$	0	0
$476$	−6311.64	−0.607760
$477$	3883.43i	0.372767i
$478$	20595.4i	1.97074i
$479$	−16930.6	−1.61499	−0.807495	−	0.589874i	$-0.799177\pi$
$479$	−16930.6	−1.61499	−0.807495	+	0.589874i	$0.799177\pi$
$480$	0	0
$481$	−6440.88	−0.610559
$482$	− 15124.2i	− 1.42923i
$483$	− 4222.15i	− 0.397753i
$484$	−633.753	−0.0595185
$485$	0	0
$486$	884.120	0.0825196
$487$	− 3932.10i	− 0.365874i	−0.983125	−	0.182937i	$-0.941440\pi$
$487$	− 3932.10i	− 0.365874i	0.983125	−	0.182937i	$-0.0585604\pi$
$488$	− 1214.57i	− 0.112666i
$489$	1689.49	0.156240
$490$	0	0
$491$	11477.9	1.05497	0.527484	−	0.849565i	$-0.323136\pi$
$491$	11477.9	1.05497	0.527484	+	0.849565i	$0.323136\pi$
$492$	− 7634.94i	− 0.699613i
$493$	− 3246.48i	− 0.296580i
$494$	1961.35	0.178635
$495$	0	0
$496$	9972.16	0.902749
$497$	11044.1i	0.996772i
$498$	1573.19i	0.141559i
$499$	−12806.6	−1.14890	−0.574451	−	0.818539i	$-0.694785\pi$
$499$	−12806.6	−1.14890	−0.574451	+	0.818539i	$0.694785\pi$
$500$	0	0
$501$	3386.89	0.302026
$502$	− 26037.9i	− 2.31500i
$503$	− 1319.41i	− 0.116958i	−0.998289	−	0.0584789i	$-0.981375\pi$
$503$	− 1319.41i	− 0.116958i	0.998289	−	0.0584789i	$-0.0186250\pi$
$504$	1885.47	0.166638
$505$	0	0
$506$	−2702.22	−0.237408
$507$	7077.28i	0.619947i
$508$	− 1629.71i	− 0.142336i
$509$	−1658.45	−0.144419	−0.0722097	−	0.997389i	$-0.523005\pi$
$509$	−1658.45	−0.144419	−0.0722097	+	0.997389i	$0.523005\pi$
$510$	0	0
$511$	22078.2	1.91131
$512$	9784.02i	0.844524i
$513$	215.635i	0.0185585i
$514$	7962.02	0.683248
$515$	0	0
$516$	−2297.48	−0.196009
$517$	− 1807.26i	− 0.153739i
$518$	7236.76i	0.613832i
$519$	−9275.90	−0.784522
$520$	0	0
$521$	2790.40	0.234644	0.117322	−	0.993094i	$-0.462569\pi$
$521$	2790.40	0.234644	0.117322	+	0.993094i	$0.462569\pi$
$522$	− 1838.83i	− 0.154183i
$523$	2440.70i	0.204062i	0.994781	+	0.102031i	$0.0325341\pi$
$523$	2440.70i	0.204062i	−0.994781	+	0.102031i	$0.967466\pi$
$524$	−3052.59	−0.254490
$525$	0	0
$526$	2755.64	0.228426
$527$	7347.05i	0.607292i
$528$	− 2589.45i	− 0.213431i
$529$	7608.25	0.625319
$530$	0	0
$531$	−7237.60	−0.591498
$532$	− 871.925i	− 0.0710578i
$533$	32797.9i	2.66536i
$534$	15283.2	1.23852
$535$	0	0
$536$	3733.45	0.300859
$537$	8637.81i	0.694132i
$538$	24677.6i	1.97756i
$539$	1006.39	0.0804236
$540$	0	0
$541$	−7756.41	−0.616403	−0.308202	−	0.951321i	$-0.599727\pi$
$541$	−7756.41	−0.616403	−0.308202	+	0.951321i	$0.599727\pi$
$542$	25617.2i	2.03017i
$543$	2106.41i	0.166473i
$544$	−11856.7	−0.934474
$545$	0	0
$546$	15357.2	1.20371
$547$	− 13056.6i	− 1.02058i	−0.860002	−	0.510290i	$-0.829538\pi$
$547$	− 13056.6i	− 1.02058i	0.860002	−	0.510290i	$-0.170462\pi$
$548$	12402.6i	0.966810i
$549$	1087.62	0.0845510
$550$	0	0
$551$	448.487	0.0346754
$552$	− 2035.78i	− 0.156972i
$553$	3515.69i	0.270348i
$554$	11684.4	0.896072
$555$	0	0
$556$	13478.9	1.02812
$557$	− 1837.48i	− 0.139778i	−0.997555	−	0.0698892i	$-0.977735\pi$
$557$	− 1837.48i	− 0.139778i	0.997555	−	0.0698892i	$-0.0222646\pi$
$558$	4161.43i	0.315712i
$559$	9869.42	0.746748
$560$	0	0
$561$	1907.80	0.143578
$562$	14503.0i	1.08856i
$563$	− 11473.6i	− 0.858890i	−0.903093	−	0.429445i	$-0.858709\pi$
$563$	− 11473.6i	− 0.858890i	0.903093	−	0.429445i	$-0.141291\pi$
$564$	−2581.56	−0.192736
$565$	0	0
$566$	28520.1	2.11801
$567$	1688.40i	0.125055i
$568$	5325.10i	0.393374i
$569$	−9698.61	−0.714564	−0.357282	−	0.933997i	$-0.616297\pi$
$569$	−9698.61	−0.714564	−0.357282	+	0.933997i	$0.616297\pi$
$570$	0	0
$571$	14019.8	1.02752	0.513758	−	0.857935i	$-0.328253\pi$
$571$	14019.8	1.02752	0.513758	+	0.857935i	$0.328253\pi$
$572$	− 3888.87i	− 0.284269i
$573$	− 12882.8i	− 0.939247i
$574$	36850.6	2.67964
$575$	0	0
$576$	−1066.04	−0.0771155
$577$	3912.25i	0.282269i	0.989990	+	0.141134i	$0.0450750\pi$
$577$	3912.25i	0.282269i	−0.989990	+	0.141134i	$0.954925\pi$
$578$	5715.03i	0.411270i
$579$	11664.3	0.837226
$580$	0	0
$581$	−3004.31	−0.214526
$582$	324.081i	0.0230818i
$583$	4746.41i	0.337181i
$584$	10645.4	0.754295
$585$	0	0
$586$	−19758.6	−1.39287
$587$	11148.5i	0.783899i	0.919987	+	0.391949i	$0.128199\pi$
$587$	11148.5i	0.783899i	−0.919987	+	0.391949i	$0.871801\pi$
$588$	− 1437.57i	− 0.100824i
$589$	−1014.96	−0.0710030
$590$	0	0
$591$	−9186.77	−0.639413
$592$	7487.61i	0.519829i
$593$	17155.0i	1.18798i	0.804473	+	0.593989i	$0.202448\pi$
$593$	17155.0i	1.18798i	−0.804473	+	0.593989i	$0.797552\pi$
$594$	1080.59	0.0746418
$595$	0	0
$596$	10435.3	0.717194
$597$	14623.9i	1.00254i
$598$	− 16581.5i	− 1.13389i
$599$	2891.03	0.197203	0.0986014	−	0.995127i	$-0.468563\pi$
$599$	2891.03	0.197203	0.0986014	+	0.995127i	$0.468563\pi$
$600$	0	0
$601$	11439.4	0.776412	0.388206	−	0.921573i	$-0.373095\pi$
$601$	11439.4	0.776412	0.388206	+	0.921573i	$0.373095\pi$
$602$	− 11088.9i	− 0.750750i
$603$	3343.22i	0.225782i
$604$	−15509.7	−1.04483
$605$	0	0
$606$	−192.085	−0.0128761
$607$	− 7768.78i	− 0.519481i	−0.965678	−	0.259741i	$-0.916363\pi$
$607$	− 7768.78i	− 0.519481i	0.965678	−	0.259741i	$-0.0836370\pi$
$608$	− 1637.96i	− 0.109256i
$609$	3511.61	0.233658
$610$	0	0
$611$	11089.8	0.734279
$612$	− 2725.18i	− 0.179998i
$613$	6469.74i	0.426281i	0.977022	+	0.213140i	$0.0683692\pi$
$613$	6469.74i	0.426281i	−0.977022	+	0.213140i	$0.931631\pi$
$614$	−29271.5	−1.92394
$615$	0	0
$616$	2304.46	0.150730
$617$	− 18323.7i	− 1.19560i	−0.801646	−	0.597799i	$-0.796042\pi$
$617$	− 18323.7i	− 1.19560i	0.801646	−	0.597799i	$-0.203958\pi$
$618$	− 15532.4i	− 1.01101i
$619$	16697.1	1.08419	0.542096	−	0.840317i	$-0.317631\pi$
$619$	16697.1	1.08419	0.542096	+	0.840317i	$0.317631\pi$
$620$	0	0
$621$	1823.00	0.117801
$622$	− 17144.1i	− 1.10517i
$623$	29186.3i	1.87692i
$624$	15889.6	1.01938
$625$	0	0
$626$	−5185.35	−0.331067
$627$	263.553i	0.0167868i
$628$	− 6822.34i	− 0.433505i
$629$	−5516.54	−0.349696
$630$	0	0
$631$	20458.8	1.29073	0.645367	−	0.763873i	$-0.276705\pi$
$631$	20458.8	1.29073	0.645367	+	0.763873i	$0.276705\pi$
$632$	1695.15i	0.106692i
$633$	12972.1i	0.814524i
$634$	7391.45	0.463016
$635$	0	0
$636$	6779.98	0.422710
$637$	6175.47i	0.384115i
$638$	− 2247.46i	− 0.139464i
$639$	−4768.51	−0.295210
$640$	0	0
$641$	4676.71	0.288173	0.144087	−	0.989565i	$-0.453976\pi$
$641$	4676.71	0.288173	0.144087	+	0.989565i	$0.453976\pi$
$642$	− 14573.2i	− 0.895886i
$643$	2321.97i	0.142410i	0.997462	+	0.0712049i	$0.0226844\pi$
$643$	2321.97i	0.142410i	−0.997462	+	0.0712049i	$0.977316\pi$
$644$	−7371.35	−0.451043
$645$	0	0
$646$	1679.88	0.102312
$647$	18149.3i	1.10282i	0.834236	+	0.551408i	$0.185909\pi$
$647$	18149.3i	1.10282i	−0.834236	+	0.551408i	$0.814091\pi$
$648$	814.090i	0.0493526i
$649$	−8845.96	−0.535030
$650$	0	0
$651$	−7947.06	−0.478448
$652$	− 2949.64i	− 0.177173i
$653$	− 23089.9i	− 1.38374i	−0.722024	−	0.691868i	$-0.756788\pi$
$653$	− 23089.9i	− 1.38374i	0.722024	−	0.691868i	$-0.243212\pi$
$654$	−17086.3	−1.02160
$655$	0	0
$656$	38128.0	2.26928
$657$	9532.69i	0.566067i
$658$	− 12460.1i	− 0.738215i
$659$	−415.639	−0.0245690	−0.0122845	−	0.999925i	$-0.503910\pi$
$659$	−415.639	−0.0245690	−0.0122845	+	0.999925i	$0.503910\pi$
$660$	0	0
$661$	12044.0	0.708710	0.354355	−	0.935111i	$-0.384700\pi$
$661$	12044.0	0.708710	0.354355	+	0.935111i	$0.384700\pi$
$662$	1152.60i	0.0676691i
$663$	11706.7i	0.685749i
$664$	−1448.58	−0.0846622
$665$	0	0
$666$	−3124.62	−0.181796
$667$	− 3791.56i	− 0.220105i
$668$	− 5913.09i	− 0.342491i
$669$	18617.1	1.07590
$670$	0	0
$671$	1329.31	0.0764792
$672$	− 12825.0i	− 0.736215i
$673$	− 8699.88i	− 0.498300i	−0.968465	−	0.249150i	$-0.919849\pi$
$673$	− 8699.88i	− 0.498300i	0.968465	−	0.249150i	$-0.0801512\pi$
$674$	−16160.3	−0.923547
$675$	0	0
$676$	12356.0	0.703007
$677$	− 29389.8i	− 1.66845i	−0.551424	−	0.834225i	$-0.685915\pi$
$677$	− 29389.8i	− 1.66845i	0.551424	−	0.834225i	$-0.314085\pi$
$678$	− 12845.2i	− 0.727606i
$679$	−618.895	−0.0349794
$680$	0	0
$681$	4607.44	0.259262
$682$	5086.20i	0.285573i
$683$	− 12261.0i	− 0.686905i	−0.939170	−	0.343452i	$-0.888404\pi$
$683$	− 12261.0i	− 0.686905i	0.939170	−	0.343452i	$-0.111596\pi$
$684$	376.471	0.0210449
$685$	0	0
$686$	−19074.4	−1.06161
$687$	284.209i	0.0157835i
$688$	− 11473.3i	− 0.635780i
$689$	−29125.2	−1.61042
$690$	0	0
$691$	−22711.4	−1.25034	−0.625168	−	0.780490i	$-0.714970\pi$
$691$	−22711.4	−1.25034	−0.625168	+	0.780490i	$0.714970\pi$
$692$	16194.6i	0.889631i
$693$	2063.60i	0.113116i
$694$	−941.224	−0.0514818
$695$	0	0
$696$	1693.18	0.0922125
$697$	28091.0i	1.52658i
$698$	− 36127.5i	− 1.95909i
$699$	−1964.95	−0.106325
$700$	0	0
$701$	−21070.9	−1.13529	−0.567643	−	0.823275i	$-0.692145\pi$
$701$	−21070.9	−1.13529	−0.567643	+	0.823275i	$0.692145\pi$
$702$	6630.79i	0.356500i
$703$	− 762.086i	− 0.0408856i
$704$	−1302.94	−0.0697536
$705$	0	0
$706$	−3670.13	−0.195648
$707$	− 366.824i	− 0.0195132i
$708$	12636.0i	0.670746i
$709$	−8521.30	−0.451374	−0.225687	−	0.974200i	$-0.572463\pi$
$709$	−8521.30	−0.451374	−0.225687	+	0.974200i	$0.572463\pi$
$710$	0	0
$711$	−1517.97	−0.0800679
$712$	14072.7i	0.740724i
$713$	8580.61i	0.450696i
$714$	13153.3	0.689425
$715$	0	0
$716$	15080.5	0.787132
$717$	− 16981.9i	− 0.884521i
$718$	− 21150.7i	− 1.09936i
$719$	−8217.19	−0.426216	−0.213108	−	0.977029i	$-0.568359\pi$
$719$	−8217.19	−0.426216	−0.213108	+	0.977029i	$0.568359\pi$
$720$	0	0
$721$	29662.1	1.53214
$722$	− 24723.4i	− 1.27439i
$723$	12470.6i	0.641478i
$724$	3677.53	0.188777
$725$	0	0
$726$	1320.72	0.0675160
$727$	− 20870.8i	− 1.06473i	−0.846516	−	0.532363i	$-0.821304\pi$
$727$	− 20870.8i	− 1.06473i	0.846516	−	0.532363i	$-0.178696\pi$
$728$	14140.8i	0.719907i
$729$	−729.000	−0.0370370
$730$	0	0
$731$	8453.04	0.427698
$732$	− 1898.85i	− 0.0958790i
$733$	33776.4i	1.70199i	0.525171	+	0.850997i	$0.324002\pi$
$733$	33776.4i	1.70199i	−0.525171	+	0.850997i	$0.675998\pi$
$734$	37282.3	1.87482
$735$	0	0
$736$	−13847.5	−0.693512
$737$	4086.16i	0.204228i
$738$	15911.0i	0.793620i
$739$	14342.3	0.713922	0.356961	−	0.934119i	$-0.383813\pi$
$739$	14342.3	0.713922	0.356961	+	0.934119i	$0.383813\pi$
$740$	0	0
$741$	−1617.23	−0.0801761
$742$	32724.1i	1.61906i
$743$	19225.5i	0.949279i	0.880180	+	0.474640i	$0.157422\pi$
$743$	19225.5i	0.949279i	−0.880180	+	0.474640i	$0.842578\pi$
$744$	−3831.81	−0.188819
$745$	0	0
$746$	−8013.96	−0.393313
$747$	− 1297.17i	− 0.0635354i
$748$	− 3330.77i	− 0.162814i
$749$	27830.4	1.35768
$750$	0	0
$751$	−19629.4	−0.953776	−0.476888	−	0.878964i	$-0.658235\pi$
$751$	−19629.4	−0.953776	−0.476888	+	0.878964i	$0.658235\pi$
$752$	− 12892.0i	− 0.625164i
$753$	21469.5i	1.03903i
$754$	13791.0	0.666100
$755$	0	0
$756$	2947.73	0.141810
$757$	− 26801.5i	− 1.28681i	−0.765525	−	0.643406i	$-0.777521\pi$
$757$	− 26801.5i	− 1.28681i	0.765525	−	0.643406i	$-0.222479\pi$
$758$	6735.07i	0.322729i
$759$	2228.11	0.106555
$760$	0	0
$761$	6490.31	0.309164	0.154582	−	0.987980i	$-0.450597\pi$
$761$	6490.31	0.309164	0.154582	+	0.987980i	$0.450597\pi$
$762$	3396.27i	0.161462i
$763$	− 32629.6i	− 1.54819i
$764$	−22491.8	−1.06509
$765$	0	0
$766$	21395.9	1.00922
$767$	− 54281.1i	− 2.55538i
$768$	− 16047.5i	− 0.753991i
$769$	−14356.1	−0.673204	−0.336602	−	0.941647i	$-0.609278\pi$
$769$	−14356.1	−0.673204	−0.336602	+	0.941647i	$0.609278\pi$
$770$	0	0
$771$	−6565.07	−0.306661
$772$	− 20364.5i	− 0.949397i
$773$	− 19127.6i	− 0.890004i	−0.895529	−	0.445002i	$-0.853203\pi$
$773$	− 19127.6i	− 0.890004i	0.895529	−	0.445002i	$-0.146797\pi$
$774$	4787.88	0.222347
$775$	0	0
$776$	−298.411	−0.0138045
$777$	− 5967.06i	− 0.275505i
$778$	25336.3i	1.16754i
$779$	−3880.65	−0.178483
$780$	0	0
$781$	−5828.18	−0.267028
$782$	− 14201.9i	− 0.649435i
$783$	1516.21i	0.0692016i
$784$	7179.07	0.327035
$785$	0	0
$786$	6361.50	0.288686
$787$	23509.7i	1.06484i	0.846480	+	0.532420i	$0.178717\pi$
$787$	23509.7i	1.06484i	−0.846480	+	0.532420i	$0.821283\pi$
$788$	16039.0i	0.725082i
$789$	−2272.16	−0.102524
$790$	0	0
$791$	24530.4	1.10266
$792$	994.999i	0.0446411i
$793$	8157.01i	0.365276i
$794$	−11003.5	−0.491814
$795$	0	0
$796$	25531.5	1.13686
$797$	43145.1i	1.91754i	0.284187	+	0.958769i	$0.408276\pi$
$797$	43145.1i	1.91754i	−0.284187	+	0.958769i	$0.591724\pi$
$798$	1817.07i	0.0806058i
$799$	9498.27	0.420557
$800$	0	0
$801$	−12601.8	−0.555882
$802$	− 48727.3i	− 2.14541i
$803$	11651.1i	0.512027i
$804$	5836.85	0.256032
$805$	0	0
$806$	−31210.2	−1.36394
$807$	− 20347.9i	− 0.887584i
$808$	− 176.871i	− 0.00770085i
$809$	23470.4	1.02000	0.509998	−	0.860176i	$-0.329646\pi$
$809$	23470.4	1.02000	0.509998	+	0.860176i	$0.329646\pi$
$810$	0	0
$811$	−4906.52	−0.212443	−0.106221	−	0.994343i	$-0.533875\pi$
$811$	−4906.52	−0.212443	−0.106221	+	0.994343i	$0.533875\pi$
$812$	− 6130.83i	− 0.264963i
$813$	− 21122.6i	− 0.911196i
$814$	−3818.98	−0.164441
$815$	0	0
$816$	13609.2	0.583846
$817$	1167.75i	0.0500054i
$818$	− 27786.7i	− 1.18770i
$819$	−12662.8	−0.540260
$820$	0	0
$821$	45229.6	1.92268	0.961342	−	0.275356i	$-0.0887958\pi$
$821$	45229.6	1.92268	0.961342	+	0.275356i	$0.0887958\pi$
$822$	− 25846.6i	− 1.09672i
$823$	− 24192.1i	− 1.02465i	−0.858793	−	0.512323i	$-0.828785\pi$
$823$	− 24192.1i	− 1.02465i	0.858793	−	0.512323i	$-0.171215\pi$
$824$	14302.1	0.604657
$825$	0	0
$826$	−60988.4	−2.56908
$827$	2031.70i	0.0854282i	0.999087	+	0.0427141i	$0.0136005\pi$
$827$	2031.70i	0.0854282i	−0.999087	+	0.0427141i	$0.986400\pi$
$828$	− 3182.73i	− 0.133584i
$829$	−17010.6	−0.712671	−0.356335	−	0.934358i	$-0.615974\pi$
$829$	−17010.6	−0.712671	−0.356335	+	0.934358i	$0.615974\pi$
$830$	0	0
$831$	−9634.38	−0.402182
$832$	− 7995.20i	− 0.333153i
$833$	5289.22i	0.220001i
$834$	−28089.7	−1.16627
$835$	0	0
$836$	460.131	0.0190358
$837$	− 3431.30i	− 0.141700i
$838$	− 45566.5i	− 1.87836i
$839$	−3184.10	−0.131022	−0.0655109	−	0.997852i	$-0.520868\pi$
$839$	−3184.10	−0.131022	−0.0655109	+	0.997852i	$0.520868\pi$
$840$	0	0
$841$	−21235.5	−0.870701
$842$	− 40567.8i	− 1.66040i
$843$	− 11958.4i	− 0.488577i
$844$	22647.6	0.923653
$845$	0	0
$846$	5379.90	0.218635
$847$	2522.18i	0.102318i
$848$	33858.4i	1.37111i
$849$	−23516.2	−0.950619
$850$	0	0
$851$	−6442.76	−0.259524
$852$	8325.23i	0.334762i
$853$	− 20566.8i	− 0.825548i	−0.910833	−	0.412774i	$-0.864560\pi$
$853$	− 20566.8i	− 0.825548i	0.910833	−	0.412774i	$-0.135440\pi$
$854$	9164.94	0.367234
$855$	0	0
$856$	13418.9	0.535804
$857$	48125.8i	1.91826i	0.282971	+	0.959129i	$0.408680\pi$
$857$	48125.8i	1.91826i	−0.282971	+	0.959129i	$0.591320\pi$
$858$	8104.30i	0.322466i
$859$	22013.7	0.874387	0.437194	−	0.899367i	$-0.355972\pi$
$859$	22013.7	0.874387	0.437194	+	0.899367i	$0.355972\pi$
$860$	0	0
$861$	−30385.1	−1.20270
$862$	− 7617.86i	− 0.301004i
$863$	23417.0i	0.923667i	0.886967	+	0.461834i	$0.152808\pi$
$863$	23417.0i	0.923667i	−0.886967	+	0.461834i	$0.847192\pi$
$864$	5537.47	0.218042
$865$	0	0
$866$	−15977.7	−0.626958
$867$	− 4712.32i	− 0.184589i
$868$	13874.6i	0.542550i
$869$	−1855.30	−0.0724241
$870$	0	0
$871$	−25073.7	−0.975421
$872$	− 15732.9i	− 0.610991i
$873$	− 267.221i	− 0.0103597i
$874$	1961.92	0.0759303
$875$	0	0
$876$	16642.9	0.641908
$877$	− 29017.9i	− 1.11729i	−0.829407	−	0.558645i	$-0.811321\pi$
$877$	− 29017.9i	− 1.11729i	0.829407	−	0.558645i	$-0.188679\pi$
$878$	− 56811.4i	− 2.18370i
$879$	16291.9	0.625158
$880$	0	0
$881$	−33241.1	−1.27119	−0.635597	−	0.772021i	$-0.719246\pi$
$881$	−33241.1	−1.27119	−0.635597	+	0.772021i	$0.719246\pi$
$882$	2995.86i	0.114372i
$883$	5618.39i	0.214127i	0.994252	+	0.107063i	$0.0341448\pi$
$883$	5618.39i	0.214127i	−0.994252	+	0.107063i	$0.965855\pi$
$884$	20438.5	0.777625
$885$	0	0
$886$	28777.8	1.09121
$887$	− 14334.2i	− 0.542612i	−0.962493	−	0.271306i	$-0.912545\pi$
$887$	− 14334.2i	− 0.542612i	0.962493	−	0.271306i	$-0.0874555\pi$
$888$	− 2877.12i	− 0.108727i
$889$	−6485.84	−0.244688
$890$	0	0
$891$	−891.000	−0.0335013
$892$	− 32503.1i	− 1.22005i
$893$	1312.14i	0.0491704i
$894$	−21746.9	−0.813564
$895$	0	0
$896$	25217.0	0.940223
$897$	13672.3i	0.508922i
$898$	54424.9i	2.02247i
$899$	−7136.58	−0.264759
$900$	0	0
$901$	−24945.4	−0.922367
$902$	19446.8i	0.717856i
$903$	9143.37i	0.336957i
$904$	11827.7	0.435161
$905$	0	0
$906$	32321.7	1.18523
$907$	− 36659.3i	− 1.34207i	−0.741428	−	0.671033i	$-0.765851\pi$
$907$	− 36659.3i	− 1.34207i	0.741428	−	0.671033i	$-0.234149\pi$
$908$	− 8044.02i	− 0.293998i
$909$	158.384	0.00577916
$910$	0	0
$911$	−9050.97	−0.329168	−0.164584	−	0.986363i	$-0.552628\pi$
$911$	−9050.97	−0.329168	−0.164584	+	0.986363i	$0.552628\pi$
$912$	1880.05i	0.0682618i
$913$	− 1585.43i	− 0.0574700i
$914$	−53063.8	−1.92035
$915$	0	0
$916$	496.193	0.0178981
$917$	12148.5i	0.437491i
$918$	5679.19i	0.204184i
$919$	46444.5	1.66710	0.833549	−	0.552445i	$-0.186305\pi$
$919$	46444.5	1.66710	0.833549	+	0.552445i	$0.186305\pi$
$920$	0	0
$921$	24135.7	0.863518
$922$	− 6557.59i	− 0.234233i
$923$	− 35763.2i	− 1.27536i
$924$	3602.79	0.128272
$925$	0	0
$926$	24572.7	0.872039
$927$	12807.2i	0.453770i
$928$	− 11517.1i	− 0.407400i
$929$	18695.8	0.660267	0.330133	−	0.943934i	$-0.392906\pi$
$929$	18695.8	0.660267	0.330133	+	0.943934i	$0.392906\pi$
$930$	0	0
$931$	−730.682	−0.0257219
$932$	3430.55i	0.120570i
$933$	14136.2i	0.496032i
$934$	6347.30	0.222366
$935$	0	0
$936$	−6105.57	−0.213212
$937$	− 13323.6i	− 0.464530i	−0.972653	−	0.232265i	$-0.925386\pi$
$937$	− 13323.6i	− 0.464530i	0.972653	−	0.232265i	$-0.0746136\pi$
$938$	28172.0i	0.980649i
$939$	4275.57	0.148592
$940$	0	0
$941$	−9500.50	−0.329126	−0.164563	−	0.986367i	$-0.552621\pi$
$941$	−9500.50	−0.329126	−0.164563	+	0.986367i	$0.552621\pi$
$942$	14217.6i	0.491755i
$943$	32807.4i	1.13293i
$944$	−63102.5	−2.17565
$945$	0	0
$946$	5851.85	0.201120
$947$	15179.4i	0.520869i	0.965491	+	0.260435i	$0.0838658\pi$
$947$	15179.4i	0.520869i	−0.965491	+	0.260435i	$0.916134\pi$
$948$	2650.18i	0.0907953i
$949$	−71494.0	−2.44551
$950$	0	0
$951$	−6094.61	−0.207814
$952$	12111.4i	0.412325i
$953$	− 5051.55i	− 0.171706i	−0.996308	−	0.0858530i	$-0.972638\pi$
$953$	− 5051.55i	− 0.171706i	0.996308	−	0.0858530i	$-0.0273615\pi$
$954$	−14129.3	−0.479510
$955$	0	0
$956$	−29648.3	−1.00303
$957$	1853.14i	0.0625952i
$958$	− 61599.6i	− 2.07745i
$959$	49359.1	1.66203
$960$	0	0
$961$	−13640.3	−0.457867
$962$	− 23434.2i	− 0.785395i
$963$	12016.3i	0.402098i
$964$	21772.2	0.727422
$965$	0	0
$966$	15361.7	0.511650
$967$	15341.1i	0.510172i	0.966918	+	0.255086i	$0.0821037\pi$
$967$	15341.1i	0.510172i	−0.966918	+	0.255086i	$0.917896\pi$
$968$	1216.11i	0.0403794i
$969$	−1385.14	−0.0459206
$970$	0	0
$971$	−5397.65	−0.178392	−0.0891961	−	0.996014i	$-0.528430\pi$
$971$	−5397.65	−0.178392	−0.0891961	+	0.996014i	$0.528430\pi$
$972$	1272.74i	0.0419992i
$973$	− 53642.7i	− 1.76743i
$974$	14306.4	0.470643
$975$	0	0
$976$	9482.63	0.310995
$977$	24180.4i	0.791812i	0.918291	+	0.395906i	$0.129569\pi$
$977$	24180.4i	0.791812i	−0.918291	+	0.395906i	$0.870431\pi$
$978$	6146.96i	0.200980i
$979$	−15402.2	−0.502814
$980$	0	0
$981$	14088.5	0.458523
$982$	41760.5i	1.35706i
$983$	− 15159.6i	− 0.491877i	−0.969285	−	0.245938i	$-0.920904\pi$
$983$	− 15159.6i	− 0.491877i	0.969285	−	0.245938i	$-0.0790961\pi$
$984$	−14650.7	−0.474642
$985$	0	0
$986$	11811.9	0.381507
$987$	10274.0i	0.331331i
$988$	2823.48i	0.0909180i
$989$	9872.29	0.317412
$990$	0	0
$991$	−3654.97	−0.117158	−0.0585792	−	0.998283i	$-0.518657\pi$
$991$	−3654.97	−0.117158	−0.0585792	+	0.998283i	$0.518657\pi$
$992$	26064.1i	0.834210i
$993$	− 950.371i	− 0.0303717i
$994$	−40182.4	−1.28220
$995$	0	0
$996$	−2264.70	−0.0720479
$997$	46219.9i	1.46820i	0.679040	+	0.734102i	$0.262396\pi$
$997$	46219.9i	1.46820i	−0.679040	+	0.734102i	$0.737604\pi$
$998$	− 46595.0i	− 1.47789i
$999$	2576.40	0.0815952

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.4.c.p.199.6		8
5.2	odd	4		825.4.a.t.1.2		4
5.3	odd	4		165.4.a.h.1.3	✓	4
5.4	even	2	inner	825.4.c.p.199.3		8
15.2	even	4		2475.4.a.be.1.3		4
15.8	even	4		495.4.a.m.1.2		4
55.43	even	4		1815.4.a.t.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.4.a.h.1.3	✓	4	5.3	odd	4
495.4.a.m.1.2		4	15.8	even	4
825.4.a.t.1.2		4	5.2	odd	4
825.4.c.p.199.3		8	5.4	even	2	inner
825.4.c.p.199.6		8	1.1	even	1	trivial
1815.4.a.t.1.2		4	55.43	even	4
2475.4.a.be.1.3		4	15.2	even	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:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	825.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.63835i q^{2} -3.00000i q^{3} -5.23763 q^{4} +10.9151 q^{6} +20.8444i q^{7} +10.0505i q^{8} -9.00000 q^{9} +O(q^{10})\)	\(q+3.63835i q^{2} -3.00000i q^{3} -5.23763 q^{4} +10.9151 q^{6} +20.8444i q^{7} +10.0505i q^{8} -9.00000 q^{9} -11.0000 q^{11} +15.7129i q^{12} -67.4988i q^{13} -75.8394 q^{14} -78.4683 q^{16} -57.8120i q^{17} -32.7452i q^{18} +7.98646 q^{19} +62.5333 q^{21} -40.0219i q^{22} -67.5185i q^{23} +30.1515 q^{24} +245.585 q^{26} +27.0000i q^{27} -109.175i q^{28} +56.1558 q^{29} -127.085 q^{31} -205.091i q^{32} +33.0000i q^{33} +210.341 q^{34} +47.1386 q^{36} -95.4222i q^{37} +29.0576i q^{38} -202.497 q^{39} -485.903 q^{41} +227.518i q^{42} +146.216i q^{43} +57.6139 q^{44} +245.656 q^{46} +164.296i q^{47} +235.405i q^{48} -91.4901 q^{49} -173.436 q^{51} +353.534i q^{52} -431.492i q^{53} -98.2356 q^{54} -209.497 q^{56} -23.9594i q^{57} +204.315i q^{58} +804.178 q^{59} -120.847 q^{61} -462.381i q^{62} -187.600i q^{63} +118.449 q^{64} -120.066 q^{66} -371.469i q^{67} +302.798i q^{68} -202.555 q^{69} +529.835 q^{71} -90.4545i q^{72} -1059.19i q^{73} +347.180 q^{74} -41.8301 q^{76} -229.289i q^{77} -736.754i q^{78} +168.663 q^{79} +81.0000 q^{81} -1767.89i q^{82} +144.130i q^{83} -327.526 q^{84} -531.986 q^{86} -168.468i q^{87} -110.555i q^{88} +1400.20 q^{89} +1406.97 q^{91} +353.637i q^{92} +381.256i q^{93} -597.767 q^{94} -615.274 q^{96} +29.6912i q^{97} -332.873i q^{98} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 52 q^{4} + 24 q^{6} - 72 q^{9}+O(q^{10}) \) 8 * q - 52 * q^4 + 24 * q^6 - 72 * q^9	\( 8 q - 52 q^{4} + 24 q^{6} - 72 q^{9} - 88 q^{11} + 104 q^{14} + 132 q^{16} - 272 q^{19} + 204 q^{21} - 288 q^{24} - 640 q^{26} - 104 q^{29} + 984 q^{31} - 488 q^{34} + 468 q^{36} - 12 q^{39} + 536 q^{41} + 572 q^{44} + 736 q^{46} + 992 q^{49} + 444 q^{51} - 216 q^{54} - 1704 q^{56} + 2064 q^{59} + 232 q^{61} + 1836 q^{64} - 264 q^{66} + 384 q^{69} - 1840 q^{71} + 5712 q^{74} + 3144 q^{76} - 2304 q^{79} + 648 q^{81} + 120 q^{84} + 472 q^{86} + 2128 q^{89} + 5560 q^{91} + 2864 q^{94} + 1248 q^{96} + 792 q^{99}+O(q^{100}) \) 8 * q - 52 * q^4 + 24 * q^6 - 72 * q^9 - 88 * q^11 + 104 * q^14 + 132 * q^16 - 272 * q^19 + 204 * q^21 - 288 * q^24 - 640 * q^26 - 104 * q^29 + 984 * q^31 - 488 * q^34 + 468 * q^36 - 12 * q^39 + 536 * q^41 + 572 * q^44 + 736 * q^46 + 992 * q^49 + 444 * q^51 - 216 * q^54 - 1704 * q^56 + 2064 * q^59 + 232 * q^61 + 1836 * q^64 - 264 * q^66 + 384 * q^69 - 1840 * q^71 + 5712 * q^74 + 3144 * q^76 - 2304 * q^79 + 648 * q^81 + 120 * q^84 + 472 * q^86 + 2128 * q^89 + 5560 * q^91 + 2864 * q^94 + 1248 * q^96 + 792 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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