LMFDB - Embedded newform 2325.2.c.k.1024.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2325,2,Mod(1024,2325)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2325, 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("2325.1024");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2325 = 3 \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2325.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:	$18.5652184699$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.5089536.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 16x^{2} - 24x + 18 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 16x^2 - 24*x + 18
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 465)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1024.6
Root			$1.66044 - 1.66044i$ of defining polynomial
Character	$\chi$	$=$	2325.1024
Dual form			2325.2.c.k.1024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.51414i q^{2} +1.00000i q^{3} -4.32088 q^{4} -2.51414 q^{6} +0.485863i q^{7} -5.83502i q^{8} -1.00000 q^{9} +5.02827 q^{11} -4.32088i q^{12} -3.51414i q^{13} -1.22153 q^{14} +6.02827 q^{16} -1.32088i q^{17} -2.51414i q^{18} +6.64177 q^{19} -0.485863 q^{21} +12.6418i q^{22} -0.292611i q^{23} +5.83502 q^{24} +8.83502 q^{26} -1.00000i q^{27} -2.09936i q^{28} +9.86330 q^{29} +1.00000 q^{31} +3.48586i q^{32} +5.02827i q^{33} +3.32088 q^{34} +4.32088 q^{36} +5.51414i q^{37} +16.6983i q^{38} +3.51414 q^{39} -7.02827 q^{41} -1.22153i q^{42} +1.02827i q^{43} -21.7266 q^{44} +0.735663 q^{46} -6.93438i q^{47} +6.02827i q^{48} +6.76394 q^{49} +1.32088 q^{51} +15.1842i q^{52} +1.70739i q^{53} +2.51414 q^{54} +2.83502 q^{56} +6.64177i q^{57} +24.7977i q^{58} +2.19325 q^{59} -2.00000 q^{61} +2.51414i q^{62} -0.485863i q^{63} +3.29261 q^{64} -12.6418 q^{66} +9.12763i q^{67} +5.70739i q^{68} +0.292611 q^{69} +13.4768 q^{71} +5.83502i q^{72} -12.5424i q^{73} -13.8633 q^{74} -28.6983 q^{76} +2.44305i q^{77} +8.83502i q^{78} +0.349158 q^{79} +1.00000 q^{81} -17.6700i q^{82} -10.9344i q^{83} +2.09936 q^{84} -2.58522 q^{86} +9.86330i q^{87} -29.3401i q^{88} -5.03374 q^{89} +1.70739 q^{91} +1.26434i q^{92} +1.00000i q^{93} +17.4340 q^{94} -3.48586 q^{96} +10.4431i q^{97} +17.0055i q^{98} -5.02827 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 10 q^{4} - 2 q^{6} - 6 q^{9} + 4 q^{11} + 16 q^{14} + 10 q^{16} + 8 q^{19} - 16 q^{21} + 6 q^{24} + 24 q^{26} + 4 q^{29} + 6 q^{31} + 4 q^{34} + 10 q^{36} + 8 q^{39} - 16 q^{41} - 20 q^{44} - 32 q^{46}+ \cdots - 4 q^{99}+O(q^{100}) $ 6 * q - 10 * q^4 - 2 * q^6 - 6 * q^9 + 4 * q^11 + 16 * q^14 + 10 * q^16 + 8 * q^19 - 16 * q^21 + 6 * q^24 + 24 * q^26 + 4 * q^29 + 6 * q^31 + 4 * q^34 + 10 * q^36 + 8 * q^39 - 16 * q^41 - 20 * q^44 - 32 * q^46 - 22 * q^49 - 8 * q^51 + 2 * q^54 - 12 * q^56 + 16 * q^59 - 12 * q^61 + 30 * q^64 - 44 * q^66 + 12 * q^69 + 20 * q^71 - 28 * q^74 - 88 * q^76 - 40 * q^79 + 6 * q^81 + 20 * q^84 - 36 * q^86 - 16 * q^89 - 16 * q^94 - 34 * q^96 - 4 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2325\mathbb{Z}\right)^\times$.

$n$	$652$	$776$	$1801$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$	2.51414i	1.77776i	0.458137	+	0.888882i	$0.348517\pi$
$2$	2.51414i	1.77776i	−0.458137	+	0.888882i	$0.651483\pi$
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	−4.32088	−2.16044
$5$	0	0
$5$	0	0
$6$	−2.51414	−1.02639
$7$	0.485863i	0.183639i	0.995776	+	0.0918195i	$0.0292683\pi$
$7$	0.485863i	0.183639i	−0.995776	+	0.0918195i	$0.970732\pi$
$8$	− 5.83502i	− 2.06299i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	5.02827	1.51608	0.758041	−	0.652207i	$-0.226157\pi$
$11$	5.02827	1.51608	0.758041	+	0.652207i	$0.226157\pi$
$12$	− 4.32088i	− 1.24733i
$13$	− 3.51414i	− 0.974646i	−0.873222	−	0.487323i	$-0.837973\pi$
$13$	− 3.51414i	− 0.974646i	0.873222	−	0.487323i	$-0.162027\pi$
$14$	−1.22153	−0.326467
$15$	0	0
$16$	6.02827	1.50707
$17$	− 1.32088i	− 0.320362i	−0.987088	−	0.160181i	$-0.948792\pi$
$17$	− 1.32088i	− 0.320362i	0.987088	−	0.160181i	$-0.0512077\pi$
$18$	− 2.51414i	− 0.592588i
$19$	6.64177	1.52373	0.761863	−	0.647738i	$-0.224285\pi$
$19$	6.64177	1.52373	0.761863	+	0.647738i	$0.224285\pi$
$20$	0	0
$21$	−0.485863	−0.106024
$22$	12.6418i	2.69523i
$23$	− 0.292611i	− 0.0610135i	−0.999535	−	0.0305068i	$-0.990288\pi$
$23$	− 0.292611i	− 0.0610135i	0.999535	−	0.0305068i	$-0.00971211\pi$
$24$	5.83502	1.19107
$25$	0	0
$26$	8.83502	1.73269
$27$	− 1.00000i	− 0.192450i
$28$	− 2.09936i	− 0.396741i
$29$	9.86330	1.83157	0.915784	−	0.401671i	$-0.131571\pi$
$29$	9.86330	1.83157	0.915784	+	0.401671i	$0.131571\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	3.48586i	0.616219i
$33$	5.02827i	0.875310i
$34$	3.32088	0.569527
$35$	0	0
$36$	4.32088	0.720147
$37$	5.51414i	0.906519i	0.891379	+	0.453259i	$0.149739\pi$
$37$	5.51414i	0.906519i	−0.891379	+	0.453259i	$0.850261\pi$
$38$	16.6983i	2.70882i
$39$	3.51414	0.562712
$40$	0	0
$41$	−7.02827	−1.09763	−0.548816	−	0.835943i	$-0.684921\pi$
$41$	−7.02827	−1.09763	−0.548816	+	0.835943i	$0.684921\pi$
$42$	− 1.22153i	− 0.188486i
$43$	1.02827i	0.156810i	0.996922	+	0.0784051i	$0.0249828\pi$
$43$	1.02827i	0.156810i	−0.996922	+	0.0784051i	$0.975017\pi$
$44$	−21.7266	−3.27541
$45$	0	0
$46$	0.735663	0.108468
$47$	− 6.93438i	− 1.01148i	−0.862685	−	0.505742i	$-0.831219\pi$
$47$	− 6.93438i	− 1.01148i	0.862685	−	0.505742i	$-0.168781\pi$
$48$	6.02827i	0.870106i
$49$	6.76394	0.966277
$50$	0	0
$51$	1.32088	0.184961
$52$	15.1842i	2.10567i
$53$	1.70739i	0.234528i	0.993101	+	0.117264i	$0.0374124\pi$
$53$	1.70739i	0.234528i	−0.993101	+	0.117264i	$0.962588\pi$
$54$	2.51414	0.342131
$55$	0	0
$56$	2.83502	0.378846
$57$	6.64177i	0.879724i
$58$	24.7977i	3.25609i
$59$	2.19325	0.285537	0.142769	−	0.989756i	$-0.454400\pi$
$59$	2.19325	0.285537	0.142769	+	0.989756i	$0.454400\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	2.51414i	0.319296i
$63$	− 0.485863i	− 0.0612130i
$64$	3.29261	0.411576
$65$	0	0
$66$	−12.6418	−1.55609
$67$	9.12763i	1.11512i	0.830137	+	0.557559i	$0.188262\pi$
$67$	9.12763i	1.11512i	−0.830137	+	0.557559i	$0.811738\pi$
$68$	5.70739i	0.692123i
$69$	0.292611	0.0352262
$70$	0	0
$71$	13.4768	1.59940	0.799700	−	0.600399i	$-0.204992\pi$
$71$	13.4768	1.59940	0.799700	+	0.600399i	$0.204992\pi$
$72$	5.83502i	0.687664i
$73$	− 12.5424i	− 1.46798i	−0.679161	−	0.733989i	$-0.737656\pi$
$73$	− 12.5424i	− 1.46798i	0.679161	−	0.733989i	$-0.262344\pi$
$74$	−13.8633	−1.61158
$75$	0	0
$76$	−28.6983	−3.29192
$77$	2.44305i	0.278412i
$78$	8.83502i	1.00037i
$79$	0.349158	0.0392834	0.0196417	−	0.999807i	$-0.493747\pi$
$79$	0.349158	0.0392834	0.0196417	+	0.999807i	$0.493747\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 17.6700i	− 1.95133i
$83$	− 10.9344i	− 1.20020i	−0.799923	−	0.600102i	$-0.795127\pi$
$83$	− 10.9344i	− 1.20020i	0.799923	−	0.600102i	$-0.204873\pi$
$84$	2.09936	0.229059
$85$	0	0
$86$	−2.58522	−0.278772
$87$	9.86330i	1.05746i
$88$	− 29.3401i	− 3.12766i
$89$	−5.03374	−0.533575	−0.266788	−	0.963755i	$-0.585962\pi$
$89$	−5.03374	−0.533575	−0.266788	+	0.963755i	$0.585962\pi$
$90$	0	0
$91$	1.70739	0.178983
$92$	1.26434i	0.131816i
$93$	1.00000i	0.103695i
$94$	17.4340	1.79818
$95$	0	0
$96$	−3.48586	−0.355774
$97$	10.4431i	1.06033i	0.847894	+	0.530166i	$0.177870\pi$
$97$	10.4431i	1.06033i	−0.847894	+	0.530166i	$0.822130\pi$
$98$	17.0055i	1.71781i
$99$	−5.02827	−0.505361
$100$	0	0
$101$	−11.6135	−1.15559	−0.577793	−	0.816183i	$-0.696086\pi$
$101$	−11.6135	−1.15559	−0.577793	+	0.816183i	$0.696086\pi$
$102$	3.32088i	0.328817i
$103$	3.76940i	0.371410i	0.982606	+	0.185705i	$0.0594569\pi$
$103$	3.76940i	0.371410i	−0.982606	+	0.185705i	$0.940543\pi$
$104$	−20.5051	−2.01069
$105$	0	0
$106$	−4.29261	−0.416935
$107$	− 6.73566i	− 0.651161i	−0.945514	−	0.325581i	$-0.894440\pi$
$107$	− 6.73566i	− 0.651161i	0.945514	−	0.325581i	$-0.105560\pi$
$108$	4.32088i	0.415777i
$109$	3.90611	0.374137	0.187069	−	0.982347i	$-0.440101\pi$
$109$	3.90611	0.374137	0.187069	+	0.982347i	$0.440101\pi$
$110$	0	0
$111$	−5.51414	−0.523379
$112$	2.92892i	0.276757i
$113$	15.0848i	1.41906i	0.704675	+	0.709530i	$0.251093\pi$
$113$	15.0848i	1.41906i	−0.704675	+	0.709530i	$0.748907\pi$
$114$	−16.6983	−1.56394
$115$	0	0
$116$	−42.6182	−3.95700
$117$	3.51414i	0.324882i
$118$	5.51414i	0.507617i
$119$	0.641769	0.0588309
$120$	0	0
$121$	14.2835	1.29850
$122$	− 5.02827i	− 0.455239i
$123$	− 7.02827i	− 0.633718i
$124$	−4.32088	−0.388027
$125$	0	0
$126$	1.22153	0.108822
$127$	− 6.25526i	− 0.555065i	−0.960716	−	0.277532i	$-0.910483\pi$
$127$	− 6.25526i	− 0.555065i	0.960716	−	0.277532i	$-0.0895166\pi$
$128$	15.2498i	1.34790i
$129$	−1.02827	−0.0905345
$130$	0	0
$131$	−22.1186	−1.93251	−0.966254	−	0.257592i	$-0.917071\pi$
$131$	−22.1186	−1.93251	−0.966254	+	0.257592i	$0.917071\pi$
$132$	− 21.7266i	− 1.89106i
$133$	3.22699i	0.279816i
$134$	−22.9481	−1.98242
$135$	0	0
$136$	−7.70739	−0.660903
$137$	− 14.6044i	− 1.24774i	−0.781528	−	0.623870i	$-0.785559\pi$
$137$	− 14.6044i	− 1.24774i	0.781528	−	0.623870i	$-0.214441\pi$
$138$	0.735663i	0.0626238i
$139$	−12.8970	−1.09391	−0.546956	−	0.837161i	$-0.684214\pi$
$139$	−12.8970	−1.09391	−0.546956	+	0.837161i	$0.684214\pi$
$140$	0	0
$141$	6.93438	0.583980
$142$	33.8825i	2.84336i
$143$	− 17.6700i	− 1.47764i
$144$	−6.02827	−0.502356
$145$	0	0
$146$	31.5333	2.60972
$147$	6.76394i	0.557880i
$148$	− 23.8259i	− 1.95848i
$149$	−16.3118	−1.33632	−0.668158	−	0.744020i	$-0.732917\pi$
$149$	−16.3118	−1.33632	−0.668158	+	0.744020i	$0.732917\pi$
$150$	0	0
$151$	19.3774	1.57691	0.788457	−	0.615090i	$-0.210881\pi$
$151$	19.3774	1.57691	0.788457	+	0.615090i	$0.210881\pi$
$152$	− 38.7549i	− 3.14343i
$153$	1.32088i	0.106787i
$154$	−6.14217	−0.494950
$155$	0	0
$156$	−15.1842	−1.21571
$157$	− 3.80128i	− 0.303375i	−0.988428	−	0.151688i	$-0.951529\pi$
$157$	− 3.80128i	− 0.303375i	0.988428	−	0.151688i	$-0.0484708\pi$
$158$	0.877832i	0.0698366i
$159$	−1.70739	−0.135405
$160$	0	0
$161$	0.142169	0.0112045
$162$	2.51414i	0.197529i
$163$	− 12.5424i	− 0.982397i	−0.871048	−	0.491199i	$-0.836559\pi$
$163$	− 12.5424i	− 0.982397i	0.871048	−	0.491199i	$-0.163441\pi$
$164$	30.3684	2.37137
$165$	0	0
$166$	27.4905	2.13368
$167$	22.2553i	1.72216i	0.508466	+	0.861082i	$0.330213\pi$
$167$	22.2553i	1.72216i	−0.508466	+	0.861082i	$0.669787\pi$
$168$	2.83502i	0.218727i
$169$	0.650842	0.0500647
$170$	0	0
$171$	−6.64177	−0.507909
$172$	− 4.44305i	− 0.338780i
$173$	10.9717i	0.834165i	0.908869	+	0.417082i	$0.136947\pi$
$173$	10.9717i	0.834165i	−0.908869	+	0.417082i	$0.863053\pi$
$174$	−24.7977	−1.87991
$175$	0	0
$176$	30.3118	2.28484
$177$	2.19325i	0.164855i
$178$	− 12.6555i	− 0.948570i
$179$	15.9253	1.19031	0.595157	−	0.803610i	$-0.297090\pi$
$179$	15.9253	1.19031	0.595157	+	0.803610i	$0.297090\pi$
$180$	0	0
$181$	−8.05655	−0.598838	−0.299419	−	0.954122i	$-0.596793\pi$
$181$	−8.05655	−0.598838	−0.299419	+	0.954122i	$0.596793\pi$
$182$	4.29261i	0.318189i
$183$	− 2.00000i	− 0.147844i
$184$	−1.70739	−0.125870
$185$	0	0
$186$	−2.51414	−0.184345
$187$	− 6.64177i	− 0.485694i
$188$	29.9627i	2.18525i
$189$	0.485863	0.0353413
$190$	0	0
$191$	6.19325	0.448128	0.224064	−	0.974574i	$-0.428068\pi$
$191$	6.19325	0.448128	0.224064	+	0.974574i	$0.428068\pi$
$192$	3.29261i	0.237624i
$193$	2.25526i	0.162337i	0.996700	+	0.0811687i	$0.0258653\pi$
$193$	2.25526i	0.162337i	−0.996700	+	0.0811687i	$0.974135\pi$
$194$	−26.2553	−1.88502
$195$	0	0
$196$	−29.2262	−2.08759
$197$	9.32088i	0.664086i	0.943264	+	0.332043i	$0.107738\pi$
$197$	9.32088i	0.664086i	−0.943264	+	0.332043i	$0.892262\pi$
$198$	− 12.6418i	− 0.898411i
$199$	−16.5479	−1.17305	−0.586524	−	0.809932i	$-0.699504\pi$
$199$	−16.5479	−1.17305	−0.586524	+	0.809932i	$0.699504\pi$
$200$	0	0
$201$	−9.12763	−0.643814
$202$	− 29.1979i	− 2.05436i
$203$	4.79221i	0.336347i
$204$	−5.70739	−0.399597
$205$	0	0
$206$	−9.47679	−0.660279
$207$	0.292611i	0.0203378i
$208$	− 21.1842i	− 1.46886i
$209$	33.3966	2.31009
$210$	0	0
$211$	25.0101	1.72177	0.860884	−	0.508801i	$-0.169911\pi$
$211$	25.0101	1.72177	0.860884	+	0.508801i	$0.169911\pi$
$212$	− 7.37743i	− 0.506684i
$213$	13.4768i	0.923414i
$214$	16.9344	1.15761
$215$	0	0
$216$	−5.83502	−0.397023
$217$	0.485863i	0.0329825i
$218$	9.82048i	0.665127i
$219$	12.5424	0.847538
$220$	0	0
$221$	−4.64177	−0.312239
$222$	− 13.8633i	− 0.930443i
$223$	− 19.2835i	− 1.29132i	−0.763625	−	0.645661i	$-0.776582\pi$
$223$	− 19.2835i	− 1.29132i	0.763625	−	0.645661i	$-0.223418\pi$
$224$	−1.69365	−0.113162
$225$	0	0
$226$	−37.9253	−2.52275
$227$	23.4340i	1.55537i	0.628655	+	0.777684i	$0.283606\pi$
$227$	23.4340i	1.55537i	−0.628655	+	0.777684i	$0.716394\pi$
$228$	− 28.6983i	− 1.90059i
$229$	25.0283	1.65391	0.826957	−	0.562265i	$-0.190070\pi$
$229$	25.0283	1.65391	0.826957	+	0.562265i	$0.190070\pi$
$230$	0	0
$231$	−2.44305	−0.160741
$232$	− 57.5525i	− 3.77851i
$233$	− 18.7175i	− 1.22623i	−0.789995	−	0.613113i	$-0.789917\pi$
$233$	− 18.7175i	− 1.22623i	0.789995	−	0.613113i	$-0.210083\pi$
$234$	−8.83502	−0.577563
$235$	0	0
$236$	−9.47679	−0.616887
$237$	0.349158i	0.0226803i
$238$	1.61350i	0.104587i
$239$	3.86876	0.250249	0.125125	−	0.992141i	$-0.460067\pi$
$239$	3.86876	0.250249	0.125125	+	0.992141i	$0.460067\pi$
$240$	0	0
$241$	−1.61350	−0.103934	−0.0519672	−	0.998649i	$-0.516549\pi$
$241$	−1.61350	−0.103934	−0.0519672	+	0.998649i	$0.516549\pi$
$242$	35.9108i	2.30843i
$243$	1.00000i	0.0641500i
$244$	8.64177	0.553233
$245$	0	0
$246$	17.6700	1.12660
$247$	− 23.3401i	− 1.48509i
$248$	− 5.83502i	− 0.370524i
$249$	10.9344	0.692938
$250$	0	0
$251$	−25.4713	−1.60774	−0.803868	−	0.594808i	$-0.797228\pi$
$251$	−25.4713	−1.60774	−0.803868	+	0.594808i	$0.797228\pi$
$252$	2.09936i	0.132247i
$253$	− 1.47133i	− 0.0925015i
$254$	15.7266	0.986774
$255$	0	0
$256$	−31.7549	−1.98468
$257$	18.0192i	1.12401i	0.827135	+	0.562003i	$0.189969\pi$
$257$	18.0192i	1.12401i	−0.827135	+	0.562003i	$0.810031\pi$
$258$	− 2.58522i	− 0.160949i
$259$	−2.67912	−0.166472
$260$	0	0
$261$	−9.86330	−0.610523
$262$	− 55.6091i	− 3.43554i
$263$	− 1.15951i	− 0.0714987i	−0.999361	−	0.0357494i	$-0.988618\pi$
$263$	− 1.15951i	− 0.0714987i	0.999361	−	0.0357494i	$-0.0113818\pi$
$264$	29.3401	1.80576
$265$	0	0
$266$	−8.11310	−0.497446
$267$	− 5.03374i	− 0.308060i
$268$	− 39.4394i	− 2.40915i
$269$	7.80675	0.475986	0.237993	−	0.971267i	$-0.423511\pi$
$269$	7.80675	0.475986	0.237993	+	0.971267i	$0.423511\pi$
$270$	0	0
$271$	17.7831	1.08025	0.540124	−	0.841585i	$-0.318377\pi$
$271$	17.7831	1.08025	0.540124	+	0.841585i	$0.318377\pi$
$272$	− 7.96265i	− 0.482807i
$273$	1.70739i	0.103336i
$274$	36.7175	2.21819
$275$	0	0
$276$	−1.26434	−0.0761041
$277$	25.9390i	1.55853i	0.626697	+	0.779263i	$0.284406\pi$
$277$	25.9390i	1.55853i	−0.626697	+	0.779263i	$0.715594\pi$
$278$	− 32.4249i	− 1.94472i
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	−13.0283	−0.777202	−0.388601	−	0.921406i	$-0.627041\pi$
$281$	−13.0283	−0.777202	−0.388601	+	0.921406i	$0.627041\pi$
$282$	17.4340i	1.03818i
$283$	− 15.9006i	− 0.945195i	−0.881278	−	0.472598i	$-0.843316\pi$
$283$	− 15.9006i	− 0.945195i	0.881278	−	0.472598i	$-0.156684\pi$
$284$	−58.2317	−3.45541
$285$	0	0
$286$	44.4249	2.62690
$287$	− 3.41478i	− 0.201568i
$288$	− 3.48586i	− 0.205406i
$289$	15.2553	0.897368
$290$	0	0
$291$	−10.4431	−0.612183
$292$	54.1943i	3.17148i
$293$	29.4340i	1.71955i	0.510672	+	0.859776i	$0.329397\pi$
$293$	29.4340i	1.71955i	−0.510672	+	0.859776i	$0.670603\pi$
$294$	−17.0055	−0.991779
$295$	0	0
$296$	32.1751	1.87014
$297$	− 5.02827i	− 0.291770i
$298$	− 41.0101i	− 2.37565i
$299$	−1.02827	−0.0594666
$300$	0	0
$301$	−0.499600	−0.0287965
$302$	48.7175i	2.80338i
$303$	− 11.6135i	− 0.667178i
$304$	40.0384	2.29636
$305$	0	0
$306$	−3.32088	−0.189842
$307$	9.96812i	0.568911i	0.958689	+	0.284455i	$0.0918127\pi$
$307$	9.96812i	0.568911i	−0.958689	+	0.284455i	$0.908187\pi$
$308$	− 10.5561i	− 0.601492i
$309$	−3.76940	−0.214434
$310$	0	0
$311$	15.9945	0.906967	0.453483	−	0.891265i	$-0.350181\pi$
$311$	15.9945	0.906967	0.453483	+	0.891265i	$0.350181\pi$
$312$	− 20.5051i	− 1.16087i
$313$	22.8542i	1.29180i	0.763423	+	0.645899i	$0.223517\pi$
$313$	22.8542i	1.29180i	−0.763423	+	0.645899i	$0.776483\pi$
$314$	9.55695	0.539330
$315$	0	0
$316$	−1.50867	−0.0848695
$317$	14.6044i	0.820266i	0.912026	+	0.410133i	$0.134518\pi$
$317$	14.6044i	0.820266i	−0.912026	+	0.410133i	$0.865482\pi$
$318$	− 4.29261i	− 0.240718i
$319$	49.5953	2.77681
$320$	0	0
$321$	6.73566	0.375948
$322$	0.357432i	0.0199189i
$323$	− 8.77301i	− 0.488143i
$324$	−4.32088	−0.240049
$325$	0	0
$326$	31.5333	1.74647
$327$	3.90611i	0.216008i
$328$	41.0101i	2.26441i
$329$	3.36916	0.185748
$330$	0	0
$331$	−8.16137	−0.448589	−0.224295	−	0.974521i	$-0.572008\pi$
$331$	−8.16137	−0.448589	−0.224295	+	0.974521i	$0.572008\pi$
$332$	47.2462i	2.59297i
$333$	− 5.51414i	− 0.302173i
$334$	−55.9528	−3.06160
$335$	0	0
$336$	−2.92892	−0.159785
$337$	− 11.8825i	− 0.647281i	−0.946180	−	0.323640i	$-0.895093\pi$
$337$	− 11.8825i	− 0.647281i	0.946180	−	0.323640i	$-0.104907\pi$
$338$	1.63631i	0.0890033i
$339$	−15.0848	−0.819295
$340$	0	0
$341$	5.02827	0.272296
$342$	− 16.6983i	− 0.902942i
$343$	6.68739i	0.361085i
$344$	6.00000	0.323498
$345$	0	0
$346$	−27.5844	−1.48295
$347$	− 18.3684i	− 0.986065i	−0.870011	−	0.493033i	$-0.835888\pi$
$347$	− 18.3684i	− 0.986065i	0.870011	−	0.493033i	$-0.164112\pi$
$348$	− 42.6182i	− 2.28457i
$349$	−11.2462	−0.601995	−0.300997	−	0.953625i	$-0.597320\pi$
$349$	−11.2462	−0.601995	−0.300997	+	0.953625i	$0.597320\pi$
$350$	0	0
$351$	−3.51414	−0.187571
$352$	17.5279i	0.934239i
$353$	− 25.3593i	− 1.34974i	−0.737937	−	0.674869i	$-0.764200\pi$
$353$	− 25.3593i	− 1.34974i	0.737937	−	0.674869i	$-0.235800\pi$
$354$	−5.51414	−0.293073
$355$	0	0
$356$	21.7502	1.15276
$357$	0.641769i	0.0339660i
$358$	40.0384i	2.11610i
$359$	−15.2890	−0.806923	−0.403461	−	0.914997i	$-0.632193\pi$
$359$	−15.2890	−0.806923	−0.403461	+	0.914997i	$0.632193\pi$
$360$	0	0
$361$	25.1131	1.32174
$362$	− 20.2553i	− 1.06459i
$363$	14.2835i	0.749691i
$364$	−7.37743	−0.386683
$365$	0	0
$366$	5.02827	0.262832
$367$	6.05655i	0.316149i	0.987427	+	0.158075i	$0.0505287\pi$
$367$	6.05655i	0.316149i	−0.987427	+	0.158075i	$0.949471\pi$
$368$	− 1.76394i	− 0.0919516i
$369$	7.02827	0.365877
$370$	0	0
$371$	−0.829557	−0.0430685
$372$	− 4.32088i	− 0.224027i
$373$	− 8.06748i	− 0.417718i	−0.977946	−	0.208859i	$-0.933025\pi$
$373$	− 8.06748i	− 0.417718i	0.977946	−	0.208859i	$-0.0669750\pi$
$374$	16.6983	0.863449
$375$	0	0
$376$	−40.4623	−2.08668
$377$	− 34.6610i	− 1.78513i
$378$	1.22153i	0.0628285i
$379$	36.8114	1.89088	0.945438	−	0.325803i	$-0.105635\pi$
$379$	36.8114	1.89088	0.945438	+	0.325803i	$0.105635\pi$
$380$	0	0
$381$	6.25526	0.320467
$382$	15.5707i	0.796666i
$383$	3.63270i	0.185622i	0.995684	+	0.0928111i	$0.0295853\pi$
$383$	3.63270i	0.185622i	−0.995684	+	0.0928111i	$0.970415\pi$
$384$	−15.2498	−0.778213
$385$	0	0
$386$	−5.67004	−0.288598
$387$	− 1.02827i	− 0.0522701i
$388$	− 45.1232i	− 2.29078i
$389$	−15.2781	−0.774629	−0.387315	−	0.921948i	$-0.626597\pi$
$389$	−15.2781	−0.774629	−0.387315	+	0.921948i	$0.626597\pi$
$390$	0	0
$391$	−0.386505	−0.0195464
$392$	− 39.4677i	− 1.99342i
$393$	− 22.1186i	− 1.11573i
$394$	−23.4340	−1.18059
$395$	0	0
$396$	21.7266	1.09180
$397$	− 18.5852i	− 0.932766i	−0.884583	−	0.466383i	$-0.845557\pi$
$397$	− 18.5852i	− 0.932766i	0.884583	−	0.466383i	$-0.154443\pi$
$398$	− 41.6036i	− 2.08540i
$399$	−3.22699	−0.161552
$400$	0	0
$401$	6.19325	0.309276	0.154638	−	0.987971i	$-0.450579\pi$
$401$	6.19325	0.309276	0.154638	+	0.987971i	$0.450579\pi$
$402$	− 22.9481i	− 1.14455i
$403$	− 3.51414i	− 0.175052i
$404$	50.1806	2.49658
$405$	0	0
$406$	−12.0483	−0.597946
$407$	27.7266i	1.37436i
$408$	− 7.70739i	− 0.381573i
$409$	32.0565	1.58509	0.792547	−	0.609811i	$-0.208755\pi$
$409$	32.0565	1.58509	0.792547	+	0.609811i	$0.208755\pi$
$410$	0	0
$411$	14.6044	0.720383
$412$	− 16.2871i	− 0.802410i
$413$	1.06562i	0.0524357i
$414$	−0.735663	−0.0361559
$415$	0	0
$416$	12.2498	0.600596
$417$	− 12.8970i	− 0.631570i
$418$	83.9637i	4.10680i
$419$	−9.78860	−0.478205	−0.239102	−	0.970994i	$-0.576853\pi$
$419$	−9.78860	−0.478205	−0.239102	+	0.970994i	$0.576853\pi$
$420$	0	0
$421$	−14.2745	−0.695695	−0.347847	−	0.937551i	$-0.613087\pi$
$421$	−14.2745	−0.695695	−0.347847	+	0.937551i	$0.613087\pi$
$422$	62.8789i	3.06090i
$423$	6.93438i	0.337161i
$424$	9.96265	0.483829
$425$	0	0
$426$	−33.8825	−1.64161
$427$	− 0.971726i	− 0.0470251i
$428$	29.1040i	1.40680i
$429$	17.6700	0.853118
$430$	0	0
$431$	−22.8350	−1.09992	−0.549962	−	0.835190i	$-0.685358\pi$
$431$	−22.8350	−1.09992	−0.549962	+	0.835190i	$0.685358\pi$
$432$	− 6.02827i	− 0.290035i
$433$	− 13.8259i	− 0.664433i	−0.943203	−	0.332216i	$-0.892204\pi$
$433$	− 13.8259i	− 0.664433i	0.943203	−	0.332216i	$-0.107796\pi$
$434$	−1.22153	−0.0586351
$435$	0	0
$436$	−16.8778	−0.808302
$437$	− 1.94345i	− 0.0929679i
$438$	31.5333i	1.50672i
$439$	−39.8506	−1.90197	−0.950983	−	0.309243i	$-0.899924\pi$
$439$	−39.8506	−1.90197	−0.950983	+	0.309243i	$0.899924\pi$
$440$	0	0
$441$	−6.76394	−0.322092
$442$	− 11.6700i	− 0.555087i
$443$	10.1504i	0.482262i	0.970493	+	0.241131i	$0.0775184\pi$
$443$	10.1504i	0.482262i	−0.970493	+	0.241131i	$0.922482\pi$
$444$	23.8259	1.13073
$445$	0	0
$446$	48.4815	2.29566
$447$	− 16.3118i	− 0.771522i
$448$	1.59976i	0.0755815i
$449$	3.75020	0.176983	0.0884914	−	0.996077i	$-0.471795\pi$
$449$	3.75020	0.176983	0.0884914	+	0.996077i	$0.471795\pi$
$450$	0	0
$451$	−35.3401	−1.66410
$452$	− 65.1798i	− 3.06580i
$453$	19.3774i	0.910431i
$454$	−58.9162	−2.76508
$455$	0	0
$456$	38.7549	1.81486
$457$	− 1.70193i	− 0.0796127i	−0.999207	−	0.0398064i	$-0.987326\pi$
$457$	− 1.70193i	− 0.0796127i	0.999207	−	0.0398064i	$-0.0126741\pi$
$458$	62.9245i	2.94027i
$459$	−1.32088	−0.0616536
$460$	0	0
$461$	7.40931	0.345086	0.172543	−	0.985002i	$-0.444802\pi$
$461$	7.40931	0.345086	0.172543	+	0.985002i	$0.444802\pi$
$462$	− 6.14217i	− 0.285760i
$463$	30.3865i	1.41218i	0.708122	+	0.706090i	$0.249543\pi$
$463$	30.3865i	1.41218i	−0.708122	+	0.706090i	$0.750457\pi$
$464$	59.4586	2.76030
$465$	0	0
$466$	47.0584	2.17994
$467$	17.1040i	0.791480i	0.918363	+	0.395740i	$0.129512\pi$
$467$	17.1040i	0.791480i	−0.918363	+	0.395740i	$0.870488\pi$
$468$	− 15.1842i	− 0.701889i
$469$	−4.43478	−0.204779
$470$	0	0
$471$	3.80128	0.175154
$472$	− 12.7977i	− 0.589061i
$473$	5.17044i	0.237737i
$474$	−0.877832	−0.0403202
$475$	0	0
$476$	−2.77301	−0.127101
$477$	− 1.70739i	− 0.0781760i
$478$	9.72659i	0.444884i
$479$	−21.2890	−0.972719	−0.486360	−	0.873759i	$-0.661676\pi$
$479$	−21.2890	−0.972719	−0.486360	+	0.873759i	$0.661676\pi$
$480$	0	0
$481$	19.3774	0.883535
$482$	− 4.05655i	− 0.184771i
$483$	0.142169i	0.00646890i
$484$	−61.7175	−2.80534
$485$	0	0
$486$	−2.51414	−0.114044
$487$	20.4996i	0.928926i	0.885593	+	0.464463i	$0.153753\pi$
$487$	20.4996i	0.928926i	−0.885593	+	0.464463i	$0.846247\pi$
$488$	11.6700i	0.528278i
$489$	12.5424	0.567187
$490$	0	0
$491$	−31.0667	−1.40202	−0.701010	−	0.713152i	$-0.747267\pi$
$491$	−31.0667	−1.40202	−0.701010	+	0.713152i	$0.747267\pi$
$492$	30.3684i	1.36911i
$493$	− 13.0283i	− 0.586764i
$494$	58.6802	2.64015
$495$	0	0
$496$	6.02827	0.270677
$497$	6.54787i	0.293712i
$498$	27.4905i	1.23188i
$499$	−15.3401	−0.686717	−0.343358	−	0.939205i	$-0.611565\pi$
$499$	−15.3401	−0.686717	−0.343358	+	0.939205i	$0.611565\pi$
$500$	0	0
$501$	−22.2553	−0.994292
$502$	− 64.0384i	− 2.85817i
$503$	14.9344i	0.665891i	0.942946	+	0.332946i	$0.108043\pi$
$503$	14.9344i	0.665891i	−0.942946	+	0.332946i	$0.891957\pi$
$504$	−2.83502	−0.126282
$505$	0	0
$506$	3.69912	0.164446
$507$	0.650842i	0.0289049i
$508$	27.0283i	1.19919i
$509$	1.22153	0.0541432	0.0270716	−	0.999633i	$-0.491382\pi$
$509$	1.22153	0.0541432	0.0270716	+	0.999633i	$0.491382\pi$
$510$	0	0
$511$	6.09389	0.269578
$512$	− 49.3365i	− 2.18038i
$513$	− 6.64177i	− 0.293241i
$514$	−45.3027	−1.99822
$515$	0	0
$516$	4.44305	0.195594
$517$	− 34.8680i	− 1.53349i
$518$	− 6.73566i	− 0.295948i
$519$	−10.9717	−0.481605
$520$	0	0
$521$	4.57429	0.200403	0.100202	−	0.994967i	$-0.468051\pi$
$521$	4.57429	0.200403	0.100202	+	0.994967i	$0.468051\pi$
$522$	− 24.7977i	− 1.08536i
$523$	29.7831i	1.30233i	0.758938	+	0.651163i	$0.225719\pi$
$523$	29.7831i	1.30233i	−0.758938	+	0.651163i	$0.774281\pi$
$524$	95.5717	4.17507
$525$	0	0
$526$	2.91518	0.127108
$527$	− 1.32088i	− 0.0575386i
$528$	30.3118i	1.31915i
$529$	22.9144	0.996277
$530$	0	0
$531$	−2.19325	−0.0951790
$532$	− 13.9435i	− 0.604525i
$533$	24.6983i	1.06980i
$534$	12.6555	0.547657
$535$	0	0
$536$	53.2599	2.30048
$537$	15.9253i	0.687228i
$538$	19.6272i	0.846190i
$539$	34.0109	1.46495
$540$	0	0
$541$	−3.04748	−0.131021	−0.0655106	−	0.997852i	$-0.520868\pi$
$541$	−3.04748	−0.131021	−0.0655106	+	0.997852i	$0.520868\pi$
$542$	44.7092i	1.92043i
$543$	− 8.05655i	− 0.345740i
$544$	4.60442	0.197413
$545$	0	0
$546$	−4.29261	−0.183707
$547$	9.82595i	0.420127i	0.977688	+	0.210064i	$0.0673671\pi$
$547$	9.82595i	0.420127i	−0.977688	+	0.210064i	$0.932633\pi$
$548$	63.1040i	2.69567i
$549$	2.00000	0.0853579
$550$	0	0
$551$	65.5097	2.79081
$552$	− 1.70739i	− 0.0726713i
$553$	0.169643i	0.00721396i
$554$	−65.2143	−2.77069
$555$	0	0
$556$	55.7266	2.36333
$557$	− 38.7922i	− 1.64368i	−0.569719	−	0.821839i	$-0.692948\pi$
$557$	− 38.7922i	− 1.64368i	0.569719	−	0.821839i	$-0.307052\pi$
$558$	− 2.51414i	− 0.106432i
$559$	3.61350	0.152835
$560$	0	0
$561$	6.64177	0.280416
$562$	− 32.7549i	− 1.38168i
$563$	− 2.73566i	− 0.115294i	−0.998337	−	0.0576472i	$-0.981640\pi$
$563$	− 2.73566i	− 0.115294i	0.998337	−	0.0576472i	$-0.0183599\pi$
$564$	−29.9627	−1.26166
$565$	0	0
$566$	39.9764	1.68033
$567$	0.485863i	0.0204043i
$568$	− 78.6374i	− 3.29955i
$569$	−4.39197	−0.184121	−0.0920605	−	0.995753i	$-0.529345\pi$
$569$	−4.39197	−0.184121	−0.0920605	+	0.995753i	$0.529345\pi$
$570$	0	0
$571$	9.67004	0.404679	0.202339	−	0.979315i	$-0.435146\pi$
$571$	9.67004	0.404679	0.202339	+	0.979315i	$0.435146\pi$
$572$	76.3502i	3.19236i
$573$	6.19325i	0.258727i
$574$	8.58522	0.358340
$575$	0	0
$576$	−3.29261	−0.137192
$577$	31.4148i	1.30781i	0.756575	+	0.653907i	$0.226871\pi$
$577$	31.4148i	1.30781i	−0.756575	+	0.653907i	$0.773129\pi$
$578$	38.3538i	1.59531i
$579$	−2.25526	−0.0937256
$580$	0	0
$581$	5.31261	0.220404
$582$	− 26.2553i	− 1.08832i
$583$	8.58522i	0.355564i
$584$	−73.1852	−3.02843
$585$	0	0
$586$	−74.0011	−3.05696
$587$	− 27.2161i	− 1.12333i	−0.827366	−	0.561664i	$-0.810162\pi$
$587$	− 27.2161i	− 1.12333i	0.827366	−	0.561664i	$-0.189838\pi$
$588$	− 29.2262i	− 1.20527i
$589$	6.64177	0.273669
$590$	0	0
$591$	−9.32088	−0.383410
$592$	33.2407i	1.36619i
$593$	− 32.5561i	− 1.33692i	−0.743748	−	0.668460i	$-0.766954\pi$
$593$	− 32.5561i	− 1.33692i	0.743748	−	0.668460i	$-0.233046\pi$
$594$	12.6418	0.518698
$595$	0	0
$596$	70.4815	2.88703
$597$	− 16.5479i	− 0.677259i
$598$	− 2.58522i	− 0.105718i
$599$	−27.8067	−1.13615	−0.568076	−	0.822976i	$-0.692312\pi$
$599$	−27.8067	−1.13615	−0.568076	+	0.822976i	$0.692312\pi$
$600$	0	0
$601$	17.6519	0.720036	0.360018	−	0.932945i	$-0.382771\pi$
$601$	17.6519	0.720036	0.360018	+	0.932945i	$0.382771\pi$
$602$	− 1.25606i	− 0.0511933i
$603$	− 9.12763i	− 0.371706i
$604$	−83.7276	−3.40683
$605$	0	0
$606$	29.1979	1.18608
$607$	44.9673i	1.82517i	0.408890	+	0.912584i	$0.365916\pi$
$607$	44.9673i	1.82517i	−0.408890	+	0.912584i	$0.634084\pi$
$608$	23.1523i	0.938950i
$609$	−4.79221	−0.194190
$610$	0	0
$611$	−24.3684	−0.985838
$612$	− 5.70739i	− 0.230708i
$613$	− 33.7694i	− 1.36393i	−0.731383	−	0.681967i	$-0.761125\pi$
$613$	− 33.7694i	− 1.36393i	0.731383	−	0.681967i	$-0.238875\pi$
$614$	−25.0612	−1.01139
$615$	0	0
$616$	14.2553	0.574361
$617$	− 26.1131i	− 1.05127i	−0.850709	−	0.525637i	$-0.823827\pi$
$617$	− 26.1131i	− 1.05127i	0.850709	−	0.525637i	$-0.176173\pi$
$618$	− 9.47679i	− 0.381212i
$619$	20.8597	0.838422	0.419211	−	0.907889i	$-0.362307\pi$
$619$	20.8597	0.838422	0.419211	+	0.907889i	$0.362307\pi$
$620$	0	0
$621$	−0.292611	−0.0117421
$622$	40.2125i	1.61237i
$623$	− 2.44571i	− 0.0979852i
$624$	21.1842	0.848046
$625$	0	0
$626$	−57.4586	−2.29651
$627$	33.3966i	1.33373i
$628$	16.4249i	0.655425i
$629$	7.28354	0.290414
$630$	0	0
$631$	−20.5105	−0.816511	−0.408256	−	0.912868i	$-0.633863\pi$
$631$	−20.5105	−0.816511	−0.408256	+	0.912868i	$0.633863\pi$
$632$	− 2.03735i	− 0.0810413i
$633$	25.0101i	0.994063i
$634$	−36.7175	−1.45824
$635$	0	0
$636$	7.37743	0.292534
$637$	− 23.7694i	− 0.941778i
$638$	124.690i	4.93650i
$639$	−13.4768	−0.533134
$640$	0	0
$641$	23.9945	0.947727	0.473864	−	0.880598i	$-0.342859\pi$
$641$	23.9945	0.947727	0.473864	+	0.880598i	$0.342859\pi$
$642$	16.9344i	0.668347i
$643$	13.6700i	0.539094i	0.962987	+	0.269547i	$0.0868739\pi$
$643$	13.6700i	0.539094i	−0.962987	+	0.269547i	$0.913126\pi$
$644$	−0.614295	−0.0242066
$645$	0	0
$646$	22.0565	0.867803
$647$	− 11.1523i	− 0.438442i	−0.975675	−	0.219221i	$-0.929648\pi$
$647$	− 11.1523i	− 0.438442i	0.975675	−	0.219221i	$-0.0703516\pi$
$648$	− 5.83502i	− 0.229221i
$649$	11.0283	0.432898
$650$	0	0
$651$	−0.485863	−0.0190425
$652$	54.1943i	2.12241i
$653$	39.1896i	1.53361i	0.641881	+	0.766805i	$0.278154\pi$
$653$	39.1896i	1.53361i	−0.641881	+	0.766805i	$0.721846\pi$
$654$	−9.82048	−0.384011
$655$	0	0
$656$	−42.3684	−1.65421
$657$	12.5424i	0.489326i
$658$	8.47053i	0.330216i
$659$	31.6464	1.23277	0.616385	−	0.787445i	$-0.288597\pi$
$659$	31.6464	1.23277	0.616385	+	0.787445i	$0.288597\pi$
$660$	0	0
$661$	−3.59535	−0.139843	−0.0699215	−	0.997553i	$-0.522275\pi$
$661$	−3.59535	−0.139843	−0.0699215	+	0.997553i	$0.522275\pi$
$662$	− 20.5188i	− 0.797486i
$663$	− 4.64177i	− 0.180271i
$664$	−63.8023	−2.47601
$665$	0	0
$666$	13.8633	0.537192
$667$	− 2.88611i	− 0.111750i
$668$	− 96.1624i	− 3.72064i
$669$	19.2835	0.745545
$670$	0	0
$671$	−10.0565	−0.388229
$672$	− 1.69365i	− 0.0653340i
$673$	7.24073i	0.279110i	0.990214	+	0.139555i	$0.0445671\pi$
$673$	7.24073i	0.279110i	−0.990214	+	0.139555i	$0.955433\pi$
$674$	29.8742	1.15071
$675$	0	0
$676$	−2.81221	−0.108162
$677$	38.5188i	1.48040i	0.672388	+	0.740199i	$0.265269\pi$
$677$	38.5188i	1.48040i	−0.672388	+	0.740199i	$0.734731\pi$
$678$	− 37.9253i	− 1.45651i
$679$	−5.07389	−0.194718
$680$	0	0
$681$	−23.4340	−0.897992
$682$	12.6418i	0.484078i
$683$	− 40.0192i	− 1.53129i	−0.643262	−	0.765646i	$-0.722419\pi$
$683$	− 40.0192i	− 1.53129i	0.643262	−	0.765646i	$-0.277581\pi$
$684$	28.6983	1.09731
$685$	0	0
$686$	−16.8130	−0.641924
$687$	25.0283i	0.954888i
$688$	6.19872i	0.236324i
$689$	6.00000	0.228582
$690$	0	0
$691$	−11.8013	−0.448942	−0.224471	−	0.974481i	$-0.572065\pi$
$691$	−11.8013	−0.448942	−0.224471	+	0.974481i	$0.572065\pi$
$692$	− 47.4076i	− 1.80217i
$693$	− 2.44305i	− 0.0928039i
$694$	46.1806	1.75299
$695$	0	0
$696$	57.5525	2.18152
$697$	9.28354i	0.351639i
$698$	− 28.2745i	− 1.07020i
$699$	18.7175	0.707962
$700$	0	0
$701$	5.17044	0.195285	0.0976425	−	0.995222i	$-0.468870\pi$
$701$	5.17044	0.195285	0.0976425	+	0.995222i	$0.468870\pi$
$702$	− 8.83502i	− 0.333456i
$703$	36.6236i	1.38129i
$704$	16.5561	0.623983
$705$	0	0
$706$	63.7567	2.39952
$707$	− 5.64257i	− 0.212211i
$708$	− 9.47679i	− 0.356160i
$709$	−47.7722	−1.79412	−0.897062	−	0.441906i	$-0.854303\pi$
$709$	−47.7722	−1.79412	−0.897062	+	0.441906i	$0.854303\pi$
$710$	0	0
$711$	−0.349158	−0.0130945
$712$	29.3720i	1.10076i
$713$	− 0.292611i	− 0.0109584i
$714$	−1.61350	−0.0603835
$715$	0	0
$716$	−68.8114	−2.57160
$717$	3.86876i	0.144481i
$718$	− 38.4386i	− 1.43452i
$719$	−23.0848	−0.860919	−0.430459	−	0.902610i	$-0.641648\pi$
$719$	−23.0848	−0.860919	−0.430459	+	0.902610i	$0.641648\pi$
$720$	0	0
$721$	−1.83141	−0.0682054
$722$	63.1378i	2.34974i
$723$	− 1.61350i	− 0.0600065i
$724$	34.8114	1.29376
$725$	0	0
$726$	−35.9108	−1.33277
$727$	− 22.2125i	− 0.823814i	−0.911226	−	0.411907i	$-0.864863\pi$
$727$	− 22.2125i	− 0.823814i	0.911226	−	0.411907i	$-0.135137\pi$
$728$	− 9.96265i	− 0.369241i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	1.35823	0.0502360
$732$	8.64177i	0.319409i
$733$	43.9072i	1.62175i	0.585221	+	0.810874i	$0.301008\pi$
$733$	43.9072i	1.62175i	−0.585221	+	0.810874i	$0.698992\pi$
$734$	−15.2270	−0.562038
$735$	0	0
$736$	1.02000	0.0375977
$737$	45.8962i	1.69061i
$738$	17.6700i	0.650443i
$739$	−6.86690	−0.252603	−0.126302	−	0.991992i	$-0.540311\pi$
$739$	−6.86690	−0.252603	−0.126302	+	0.991992i	$0.540311\pi$
$740$	0	0
$741$	23.3401	0.857419
$742$	− 2.08562i	− 0.0765656i
$743$	− 29.2726i	− 1.07391i	−0.843612	−	0.536954i	$-0.819575\pi$
$743$	− 29.2726i	− 1.07391i	0.843612	−	0.536954i	$-0.180425\pi$
$744$	5.83502	0.213922
$745$	0	0
$746$	20.2827	0.742604
$747$	10.9344i	0.400068i
$748$	28.6983i	1.04931i
$749$	3.27261	0.119579
$750$	0	0
$751$	9.67004	0.352865	0.176432	−	0.984313i	$-0.443544\pi$
$751$	9.67004	0.352865	0.176432	+	0.984313i	$0.443544\pi$
$752$	− 41.8023i	− 1.52437i
$753$	− 25.4713i	− 0.928227i
$754$	87.1424	3.17354
$755$	0	0
$756$	−2.09936	−0.0763529
$757$	− 49.4104i	− 1.79585i	−0.440148	−	0.897925i	$-0.645074\pi$
$757$	− 49.4104i	− 1.79585i	0.440148	−	0.897925i	$-0.354926\pi$
$758$	92.5489i	3.36153i
$759$	1.47133	0.0534058
$760$	0	0
$761$	47.2599	1.71317	0.856586	−	0.516005i	$-0.172581\pi$
$761$	47.2599	1.71317	0.856586	+	0.516005i	$0.172581\pi$
$762$	15.7266i	0.569714i
$763$	1.89783i	0.0687062i
$764$	−26.7603	−0.968155
$765$	0	0
$766$	−9.13310	−0.329992
$767$	− 7.70739i	− 0.278298i
$768$	− 31.7549i	− 1.14585i
$769$	−9.49053	−0.342237	−0.171119	−	0.985250i	$-0.554738\pi$
$769$	−9.49053	−0.342237	−0.171119	+	0.985250i	$0.554738\pi$
$770$	0	0
$771$	−18.0192	−0.648946
$772$	− 9.74474i	− 0.350721i
$773$	− 41.7567i	− 1.50188i	−0.660368	−	0.750942i	$-0.729600\pi$
$773$	− 41.7567i	− 1.50188i	0.660368	−	0.750942i	$-0.270400\pi$
$774$	2.58522	0.0929239
$775$	0	0
$776$	60.9354	2.18745
$777$	− 2.67912i	− 0.0961127i
$778$	− 38.4112i	− 1.37711i
$779$	−46.6802	−1.67249
$780$	0	0
$781$	67.7650	2.42482
$782$	− 0.971726i	− 0.0347489i
$783$	− 9.86330i	− 0.352485i
$784$	40.7749	1.45625
$785$	0	0
$786$	55.6091	1.98351
$787$	− 40.0950i	− 1.42923i	−0.699518	−	0.714615i	$-0.746602\pi$
$787$	− 40.0950i	− 1.42923i	0.699518	−	0.714615i	$-0.253398\pi$
$788$	− 40.2745i	− 1.43472i
$789$	1.15951	0.0412798
$790$	0	0
$791$	−7.32916	−0.260595
$792$	29.3401i	1.04255i
$793$	7.02827i	0.249581i
$794$	46.7258	1.65824
$795$	0	0
$796$	71.5015	2.53430
$797$	− 31.1150i	− 1.10215i	−0.834456	−	0.551074i	$-0.814218\pi$
$797$	− 31.1150i	− 1.10215i	0.834456	−	0.551074i	$-0.185782\pi$
$798$	− 8.11310i	− 0.287200i
$799$	−9.15951	−0.324040
$800$	0	0
$801$	5.03374	0.177858
$802$	15.5707i	0.549820i
$803$	− 63.0667i	− 2.22557i
$804$	39.4394	1.39092
$805$	0	0
$806$	8.83502	0.311200
$807$	7.80675i	0.274811i
$808$	67.7650i	2.38396i
$809$	−46.6291	−1.63939	−0.819696	−	0.572799i	$-0.805857\pi$
$809$	−46.6291	−1.63939	−0.819696	+	0.572799i	$0.805857\pi$
$810$	0	0
$811$	−47.0283	−1.65139	−0.825693	−	0.564120i	$-0.809216\pi$
$811$	−47.0283	−1.65139	−0.825693	+	0.564120i	$0.809216\pi$
$812$	− 20.7066i	− 0.726659i
$813$	17.7831i	0.623682i
$814$	−69.7084	−2.44328
$815$	0	0
$816$	7.96265	0.278749
$817$	6.82956i	0.238936i
$818$	80.5946i	2.81792i
$819$	−1.70739	−0.0596610
$820$	0	0
$821$	35.3912	1.23516	0.617580	−	0.786508i	$-0.288113\pi$
$821$	35.3912	1.23516	0.617580	+	0.786508i	$0.288113\pi$
$822$	36.7175i	1.28067i
$823$	17.2161i	0.600114i	0.953921	+	0.300057i	$0.0970057\pi$
$823$	17.2161i	0.600114i	−0.953921	+	0.300057i	$0.902994\pi$
$824$	21.9945	0.766216
$825$	0	0
$826$	−2.67912	−0.0932184
$827$	16.3310i	0.567885i	0.958841	+	0.283942i	$0.0916425\pi$
$827$	16.3310i	0.567885i	−0.958841	+	0.283942i	$0.908358\pi$
$828$	− 1.26434i	− 0.0439387i
$829$	29.0667	1.00953	0.504764	−	0.863258i	$-0.331580\pi$
$829$	29.0667	1.00953	0.504764	+	0.863258i	$0.331580\pi$
$830$	0	0
$831$	−25.9390	−0.899815
$832$	− 11.5707i	− 0.401141i
$833$	− 8.93438i	− 0.309558i
$834$	32.4249	1.12278
$835$	0	0
$836$	−144.303	−4.99082
$837$	− 1.00000i	− 0.0345651i
$838$	− 24.6099i	− 0.850134i
$839$	−27.6646	−0.955087	−0.477544	−	0.878608i	$-0.658473\pi$
$839$	−27.6646	−0.955087	−0.477544	+	0.878608i	$0.658473\pi$
$840$	0	0
$841$	68.2846	2.35464
$842$	− 35.8880i	− 1.23678i
$843$	− 13.0283i	− 0.448718i
$844$	−108.066	−3.71978
$845$	0	0
$846$	−17.4340	−0.599393
$847$	6.93984i	0.238456i
$848$	10.2926i	0.353450i
$849$	15.9006	0.545709
$850$	0	0
$851$	1.61350	0.0553099
$852$	− 58.2317i	− 1.99498i
$853$	− 32.5105i	− 1.11314i	−0.830801	−	0.556570i	$-0.812117\pi$
$853$	− 32.5105i	− 1.11314i	0.830801	−	0.556570i	$-0.187883\pi$
$854$	2.44305	0.0835995
$855$	0	0
$856$	−39.3027	−1.34334
$857$	− 45.9144i	− 1.56841i	−0.620505	−	0.784203i	$-0.713072\pi$
$857$	− 45.9144i	− 1.56841i	0.620505	−	0.784203i	$-0.286928\pi$
$858$	44.4249i	1.51664i
$859$	19.6026	0.668831	0.334415	−	0.942426i	$-0.391461\pi$
$859$	19.6026	0.668831	0.334415	+	0.942426i	$0.391461\pi$
$860$	0	0
$861$	3.41478	0.116375
$862$	− 57.4104i	− 1.95540i
$863$	− 46.1696i	− 1.57163i	−0.618460	−	0.785816i	$-0.712243\pi$
$863$	− 46.1696i	− 1.57163i	0.618460	−	0.785816i	$-0.287757\pi$
$864$	3.48586	0.118591
$865$	0	0
$866$	34.7603	1.18120
$867$	15.2553i	0.518096i
$868$	− 2.09936i	− 0.0712569i
$869$	1.75566	0.0595568
$870$	0	0
$871$	32.0757	1.08685
$872$	− 22.7922i	− 0.771842i
$873$	− 10.4431i	− 0.353444i
$874$	4.88611	0.165275
$875$	0	0
$876$	−54.1943	−1.83106
$877$	11.3017i	0.381631i	0.981626	+	0.190815i	$0.0611132\pi$
$877$	11.3017i	0.381631i	−0.981626	+	0.190815i	$0.938887\pi$
$878$	− 100.190i	− 3.38125i
$879$	−29.4340	−0.992784
$880$	0	0
$881$	−9.60803	−0.323703	−0.161851	−	0.986815i	$-0.551747\pi$
$881$	−9.60803	−0.323703	−0.161851	+	0.986815i	$0.551747\pi$
$882$	− 17.0055i	− 0.572604i
$883$	33.5279i	1.12830i	0.825671	+	0.564151i	$0.190797\pi$
$883$	33.5279i	1.12830i	−0.825671	+	0.564151i	$0.809203\pi$
$884$	20.0565	0.674575
$885$	0	0
$886$	−25.5196	−0.857348
$887$	− 9.76394i	− 0.327841i	−0.986474	−	0.163920i	$-0.947586\pi$
$887$	− 9.76394i	− 0.327841i	0.986474	−	0.163920i	$-0.0524140\pi$
$888$	32.1751i	1.07973i
$889$	3.03920	0.101932
$890$	0	0
$891$	5.02827	0.168454
$892$	83.3219i	2.78982i
$893$	− 46.0565i	− 1.54122i
$894$	41.0101	1.37158
$895$	0	0
$896$	−7.40931	−0.247528
$897$	− 1.02827i	− 0.0343331i
$898$	9.42852i	0.314634i
$899$	9.86330	0.328959
$900$	0	0
$901$	2.25526	0.0751337
$902$	− 88.8498i	− 2.95838i
$903$	− 0.499600i	− 0.0166257i
$904$	88.0203	2.92751
$905$	0	0
$906$	−48.7175	−1.61853
$907$	− 9.18418i	− 0.304956i	−0.988307	−	0.152478i	$-0.951275\pi$
$907$	− 9.18418i	− 0.304956i	0.988307	−	0.152478i	$-0.0487253\pi$
$908$	− 101.256i	− 3.36028i
$909$	11.6135	0.385195
$910$	0	0
$911$	−12.2553	−0.406035	−0.203018	−	0.979175i	$-0.565075\pi$
$911$	−12.2553	−0.406035	−0.203018	+	0.979175i	$0.565075\pi$
$912$	40.0384i	1.32580i
$913$	− 54.9811i	− 1.81961i
$914$	4.27887	0.141533
$915$	0	0
$916$	−108.144	−3.57319
$917$	− 10.7466i	− 0.354884i
$918$	− 3.32088i	− 0.109606i
$919$	47.5663	1.56907	0.784533	−	0.620087i	$-0.212903\pi$
$919$	47.5663	1.56907	0.784533	+	0.620087i	$0.212903\pi$
$920$	0	0
$921$	−9.96812	−0.328461
$922$	18.6280i	0.613482i
$923$	− 47.3593i	− 1.55885i
$924$	10.5561	0.347272
$925$	0	0
$926$	−76.3958	−2.51052
$927$	− 3.76940i	− 0.123803i
$928$	34.3821i	1.12865i
$929$	−23.5333	−0.772104	−0.386052	−	0.922477i	$-0.626161\pi$
$929$	−23.5333	−0.772104	−0.386052	+	0.922477i	$0.626161\pi$
$930$	0	0
$931$	44.9245	1.47234
$932$	80.8762i	2.64919i
$933$	15.9945i	0.523638i
$934$	−43.0019	−1.40706
$935$	0	0
$936$	20.5051	0.670229
$937$	− 56.7258i	− 1.85315i	−0.376109	−	0.926575i	$-0.622738\pi$
$937$	− 56.7258i	− 1.85315i	0.376109	−	0.926575i	$-0.377262\pi$
$938$	− 11.1496i	− 0.364049i
$939$	−22.8542	−0.745819
$940$	0	0
$941$	−33.5333	−1.09316	−0.546578	−	0.837408i	$-0.684070\pi$
$941$	−33.5333	−1.09316	−0.546578	+	0.837408i	$0.684070\pi$
$942$	9.55695i	0.311382i
$943$	2.05655i	0.0669704i
$944$	13.2215	0.430324
$945$	0	0
$946$	−12.9992	−0.422640
$947$	− 52.0685i	− 1.69200i	−0.533183	−	0.846000i	$-0.679004\pi$
$947$	− 52.0685i	− 1.69200i	0.533183	−	0.846000i	$-0.320996\pi$
$948$	− 1.50867i	− 0.0489994i
$949$	−44.0757	−1.43076
$950$	0	0
$951$	−14.6044	−0.473581
$952$	− 3.74474i	− 0.121368i
$953$	11.1896i	0.362468i	0.983440	+	0.181234i	$0.0580092\pi$
$953$	11.1896i	0.362468i	−0.983440	+	0.181234i	$0.941991\pi$
$954$	4.29261	0.138978
$955$	0	0
$956$	−16.7165	−0.540649
$957$	49.5953i	1.60319i
$958$	− 53.5235i	− 1.72926i
$959$	7.09575	0.229134
$960$	0	0
$961$	1.00000	0.0322581
$962$	48.7175i	1.57072i
$963$	6.73566i	0.217054i
$964$	6.97173	0.224544
$965$	0	0
$966$	−0.357432	−0.0115002
$967$	36.8296i	1.18436i	0.805806	+	0.592179i	$0.201732\pi$
$967$	36.8296i	1.18436i	−0.805806	+	0.592179i	$0.798268\pi$
$968$	− 83.3448i	− 2.67880i
$969$	8.77301	0.281830
$970$	0	0
$971$	18.7494	0.601697	0.300848	−	0.953672i	$-0.402730\pi$
$971$	18.7494	0.601697	0.300848	+	0.953672i	$0.402730\pi$
$972$	− 4.32088i	− 0.138592i
$973$	− 6.26619i	− 0.200885i
$974$	−51.5388	−1.65141
$975$	0	0
$976$	−12.0565	−0.385921
$977$	− 50.2262i	− 1.60688i	−0.595387	−	0.803439i	$-0.703001\pi$
$977$	− 50.2262i	− 1.60688i	0.595387	−	0.803439i	$-0.296999\pi$
$978$	31.5333i	1.00832i
$979$	−25.3110	−0.808943
$980$	0	0
$981$	−3.90611	−0.124712
$982$	− 78.1059i	− 2.49246i
$983$	− 15.5279i	− 0.495262i	−0.968854	−	0.247631i	$-0.920348\pi$
$983$	− 15.5279i	− 0.495262i	0.968854	−	0.247631i	$-0.0796521\pi$
$984$	−41.0101	−1.30736
$985$	0	0
$986$	32.7549	1.04313
$987$	3.36916i	0.107242i
$988$	100.850i	3.20846i
$989$	0.300884	0.00956755
$990$	0	0
$991$	−20.1022	−0.638566	−0.319283	−	0.947659i	$-0.603442\pi$
$991$	−20.1022	−0.638566	−0.319283	+	0.947659i	$0.603442\pi$
$992$	3.48586i	0.110676i
$993$	− 8.16137i	− 0.258993i
$994$	−16.4623	−0.522151
$995$	0	0
$996$	−47.2462	−1.49705
$997$	− 20.8186i	− 0.659333i	−0.944098	−	0.329666i	$-0.893064\pi$
$997$	− 20.8186i	− 0.659333i	0.944098	−	0.329666i	$-0.106936\pi$
$998$	− 38.5671i	− 1.22082i
$999$	5.51414	0.174460

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2325.2.c.k.1024.6		6
5.2	odd	4		2325.2.a.r.1.1		3
5.3	odd	4		465.2.a.e.1.3	✓	3
5.4	even	2	inner	2325.2.c.k.1024.1		6
15.2	even	4		6975.2.a.bf.1.3		3
15.8	even	4		1395.2.a.j.1.1		3
20.3	even	4		7440.2.a.bs.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
465.2.a.e.1.3	✓	3	5.3	odd	4
1395.2.a.j.1.1		3	15.8	even	4
2325.2.a.r.1.1		3	5.2	odd	4
2325.2.c.k.1024.1		6	5.4	even	2	inner
2325.2.c.k.1024.6		6	1.1	even	1	trivial
6975.2.a.bf.1.3		3	15.2	even	4
7440.2.a.bs.1.3		3	20.3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2325 = 3 \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2325.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.51414i q^{2} +1.00000i q^{3} -4.32088 q^{4} -2.51414 q^{6} +0.485863i q^{7} -5.83502i q^{8} -1.00000 q^{9} +5.02827 q^{11} -4.32088i q^{12} -3.51414i q^{13} -1.22153 q^{14} +6.02827 q^{16} -1.32088i q^{17} -2.51414i q^{18} +6.64177 q^{19} -0.485863 q^{21} +12.6418i q^{22} -0.292611i q^{23} +5.83502 q^{24} +8.83502 q^{26} -1.00000i q^{27} -2.09936i q^{28} +9.86330 q^{29} +1.00000 q^{31} +3.48586i q^{32} +5.02827i q^{33} +3.32088 q^{34} +4.32088 q^{36} +5.51414i q^{37} +16.6983i q^{38} +3.51414 q^{39} -7.02827 q^{41} -1.22153i q^{42} +1.02827i q^{43} -21.7266 q^{44} +0.735663 q^{46} -6.93438i q^{47} +6.02827i q^{48} +6.76394 q^{49} +1.32088 q^{51} +15.1842i q^{52} +1.70739i q^{53} +2.51414 q^{54} +2.83502 q^{56} +6.64177i q^{57} +24.7977i q^{58} +2.19325 q^{59} -2.00000 q^{61} +2.51414i q^{62} -0.485863i q^{63} +3.29261 q^{64} -12.6418 q^{66} +9.12763i q^{67} +5.70739i q^{68} +0.292611 q^{69} +13.4768 q^{71} +5.83502i q^{72} -12.5424i q^{73} -13.8633 q^{74} -28.6983 q^{76} +2.44305i q^{77} +8.83502i q^{78} +0.349158 q^{79} +1.00000 q^{81} -17.6700i q^{82} -10.9344i q^{83} +2.09936 q^{84} -2.58522 q^{86} +9.86330i q^{87} -29.3401i q^{88} -5.03374 q^{89} +1.70739 q^{91} +1.26434i q^{92} +1.00000i q^{93} +17.4340 q^{94} -3.48586 q^{96} +10.4431i q^{97} +17.0055i q^{98} -5.02827 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 10 q^{4} - 2 q^{6} - 6 q^{9} + 4 q^{11} + 16 q^{14} + 10 q^{16} + 8 q^{19} - 16 q^{21} + 6 q^{24} + 24 q^{26} + 4 q^{29} + 6 q^{31} + 4 q^{34} + 10 q^{36} + 8 q^{39} - 16 q^{41} - 20 q^{44} - 32 q^{46}+ \cdots - 4 q^{99}+O(q^{100}) \) 6 * q - 10 * q^4 - 2 * q^6 - 6 * q^9 + 4 * q^11 + 16 * q^14 + 10 * q^16 + 8 * q^19 - 16 * q^21 + 6 * q^24 + 24 * q^26 + 4 * q^29 + 6 * q^31 + 4 * q^34 + 10 * q^36 + 8 * q^39 - 16 * q^41 - 20 * q^44 - 32 * q^46 - 22 * q^49 - 8 * q^51 + 2 * q^54 - 12 * q^56 + 16 * q^59 - 12 * q^61 + 30 * q^64 - 44 * q^66 + 12 * q^69 + 20 * q^71 - 28 * q^74 - 88 * q^76 - 40 * q^79 + 6 * q^81 + 20 * q^84 - 36 * q^86 - 16 * q^89 - 16 * q^94 - 34 * q^96 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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